Nano Optics and Atomics_ Transport of Light and Matter Waves.pdf

SOCIETA ITALIANA DI FISICA

RENDICONTI

DELLA

SCUOLA INTERNAZIONALE DI FISICA

“ENRICO FERMI”

CLXXIII Corso

a cura di R. Kaiser e D. S. Wiersma

Direttori del Corso

e di

L. Fallani

VARENNA SUL LAGO DI COMO

VILLA MONASTERO

23 Giugno – 3 Luglio 2009

Nano-ottica e fisica atomica:

trasporto di luce e onde

di materia

2011


BOLOGNA-ITALY

ITALIAN PHYSICAL SOCIETY

PROCEEDINGS

OF THE

INTERNATIONAL SCHOOL OF PHYSICS

“ENRICO FERMI”

Course CLXXIII

edited by R. Kaiser and D. S. Wiersma

Directors of the Course

and

L. Fallani

VARENNA ON LAKE COMO

VILLA MONASTERO

23 June – 3 July 2009

Nano Optics and Atomics:

Transport of Light and Matter

Waves

2011

AMSTERDAM, OXFORD, TOKIO, WASHINGTON DC

Copyright c© 2011 by Societa Italiana di Fisica

All rights reserved. No part of this publication may be reproduced, stored in a retrieval

system, or transmitted, in any form or any means, electronic, mechanical, photocopying,

recording or otherwise, without the prior permission of the copyright owner.

ISSN 0074-784X (print)

ISSN 1879-8195 (online)

ISBN 978-1-60750-755-0 (print) (IOS Press)

ISBN 978-1-60750-756-7 (online) (IOS Press)

ISBN 978-88-7438-061-9 (SIF)

LCCN 2011927441

Production Manager Copy Editor

A. Oleandri M. Missiroli

jointly published and distributed by:

IOS PRESS

Nieuwe Hemweg 6B

1013 BG Amsterdam

The Netherlands

fax: +31 20 687 0019

[email protected]


Via Saragozza 12

40123 Bologna

Italy

fax: +39 051 581340

[email protected]

Distributor in the USA and Canada

IOS Press, Inc.

4502 Rachael Manor Drive

Fairfax, VA 22032

USA

fax: +1 703 323 3668

[email protected]

Proprieta Letteraria Riservata

Printed in Italy

Supported by

Istituto Nazionale di Fisica Nucleare (INFN)

Durham University, USA

This page intentionally left blank

INDICE

R. Kaiser, D. S. Wiersma and L. Fallani – Preface . . . . . . . . . . . . . . . . pag. XIII

Gruppo fotografico dei partecipanti al Corso . . . . . . . . . . . . . . . . . . . . . . . . . . � XVI

P. Wolfle – Anderson localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1

1. Basic notions of localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 11.1. Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1

1.2. Electrons and classical waves in disordered systems . . . . . . . . . . . . . . � 2

1.3. Strong localization and Anderson transition . . . . . . . . . . . . . . . . . . . . . � 3

1.4. Weak localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 5

1.5. One-dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 7

1.6. Quasi–one-dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 7

2. Theory of localization: fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . � 82.1. Thouless conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 9

2.2. Scaling theory of the conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 9

2.3. Renormalization group equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 10

2.4. Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 11

2.5. Dynamical scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 13

2.6. Symmetry classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 14

2.7. Non-linear σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 14

2.8. Fractal structure of critical wave functions . . . . . . . . . . . . . . . . . . . . . . � 15

2.9. Anderson transition in the kicked rotor model . . . . . . . . . . . . . . . . . . . � 16

3. Theory of localization: diagrammatic approaches . . . . . . . . . . . . . . . . . . . . . � 173.1. Self-consistent theory of localization . . . . . . . . . . . . . . . . . . . . . . . . . . . � 17

3.2. Results of the self-consistent theory of localization . . . . . . . . . . . . . . . � 20

3.3. Destruction of localization by decoherence processes . . . . . . . . . . . . . . � 21

4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 22

P. J. Steinhardt – Photonic properties of non-crystalline solids . . . . . . . . � 25

1. Some basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 262. Photonic band gaps in icosahedral quasicrystals . . . . . . . . . . . . . . . . . . . . . . � 313. Dependence of band gap width on symmetry in 2D . . . . . . . . . . . . . . . . . . . � 344. Finding optimal complete band gaps for 2D photonic quasicrystals . . . . . . � 375. Isotropic disordered photonic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 416. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 48

VII

VIII indice

R. C. Mesquita and A. G. Yodh – Diffuse optics: Fundamentals and tissue

applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 51

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 512. Light transport tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 53

2.1. Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 53

2.2. Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 53

2.3. Dynamic light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 55

2.4. Multiple light scattering in tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 55

2.5. Simple solutions of the photon diffusion equation . . . . . . . . . . . . . . . . � 58

3. Diffuse Optical Spectroscopy (DOS): monitoring . . . . . . . . . . . . . . . . . . . . . . � 594. Diffuse Optical Tomography (DOT): imaging . . . . . . . . . . . . . . . . . . . . . . . . � 615. Diffuse Correlation Spectroscopy (DCS): blood flow . . . . . . . . . . . . . . . . . . . � 626. Background on tissue hemodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 637. Validation and clinical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 658. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 71

J. H. Page – Ultrasonic wave transport in strongly scattering media . . . . . � 75

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 752. Acoustic wave transport in random media . . . . . . . . . . . . . . . . . . . . . . . . . . . � 76

2.1. Ballistic propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 77

2.2. Diffusive propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 81

3. Wave transport in ordered media: phononic crystals in 2D and 3D. . . . . . . � 844. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 91

J. H. Page – Anderson localization of ultrasound in three dimensions . . . . � 95

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 962. Mesoglasses: porous elastic solids with very strong scattering . . . . . . . . . . . � 973. Time-dependent transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 994. Transverse confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1025. Statistical approach to localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1076. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 112Appendix A. Schrodinger and Helmholtz equations in disordered media . . . . . . � 112

J. H. Page – Ultrasonic spectroscopy of complex media . . . . . . . . . . . . . . . . � 115

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1162. Diffusing Acoustic Wave Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1163. Probing food biomaterials with ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1244. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 129

M. Fink and M. Tanter – MultiWave imaging . . . . . . . . . . . . . . . . . . . . . . . � 133

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1332. Transcending classical diffraction limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1343. Wave-to-wave generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 136

indice IX

4. Wave-to-wave tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 1385. Wave-to-wave imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 139

5.1. Super-resolution in supersonic shear wave imaging . . . . . . . . . . . . . . . � 145

5.2. Clinical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 146

5.3. Shear wave spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 146

6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 149Appendix A. Waves propagation in tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 150

Appendix A.1. Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 150

Appendix A.1.1. Low frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 150

Appendix A.1.2. Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 150

Appendix A.1.3. Optical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 151

Appendix A.2. Mechanical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 151

M. Fink, J. de Rosny, G. Lerosey and A. Tourin – Time reversal

focusing and the diffraction limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 155

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1552. Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 157

2.1. An ideal time reversal experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 158

2.2. Time reversal in free space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 160

2.3. Time reversal through heterogeneous medium . . . . . . . . . . . . . . . . . . . � 161

2.4. An experimental point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 162

3. Time reversal in complex media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1623.1. One-channel time reversal in chaotic cavities . . . . . . . . . . . . . . . . . . . . � 164

3.2. Time reversal in open systems: random media . . . . . . . . . . . . . . . . . . . � 170

4. Focusing microwaves below the diffraction limit . . . . . . . . . . . . . . . . . . . . . . � 1735. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 176

L. Fallani and M. Inguscio – Ultracold atoms in bichromatic lattices . . . � 179

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1792. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 180

2.1. Ultracold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 180

2.2. Light forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 181

2.3. Crystals made of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 182

3. Monochromatic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1843.1. Energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 184

3.2. Tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 187

3.3. Adding interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 189

3.4. Mott insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 191

4. Bichromatic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 1954.1. Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 196

4.2. General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 197

4.2.1. Harper and Aubry-Andre model . . . . . . . . . . . . . . . . . . . . . . . . � 198

4.3. Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 200

4.4. Incommensurate lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 202

4.5. Localization in bichromatic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 203

X indice

4.5.1. Localized states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 203

4.5.2. Spectrum of the localized states . . . . . . . . . . . . . . . . . . . . . . . . � 205

4.6. Further considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 208

5. Anderson localization of matter waves in bichromatic lattices . . . . . . . . . . . � 209

5.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 210

5.2. Absence of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 211

5.3. Imaging the localized states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 214

5.4. Effect of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 215

6. Strongly interacting atoms in bichromatic lattices . . . . . . . . . . . . . . . . . . . . . � 218

6.1. Towards a Bose glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 220

6.2. Noise correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 223

6.3. Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 224

7. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 227

I. Bloch – Exploring strongly correlated ultracold bosonic and fermionic

quantum gases in optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 233

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 234

2. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 234

2.1. Optical dipole force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 234

2.2. Optical lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 237

2.2.1. 1D lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 237



3. Bose-Hubbard model of interacting bosons in optical lattices . . . . . . . . . . . � 240

3.1. Ground states of the Bose-Hubbard Hamiltonian . . . . . . . . . . . . . . . . � 241

3.2. Double-well case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 241

3.3. Multiple-well case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 243

3.4. Superfluid-to-Mott-insulator transition . . . . . . . . . . . . . . . . . . . . . . . . . � 244

4. Multi-orbital quantum phase diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 247

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 247

4.2. Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 248

4.3. Probing the energy scales via Fock state heterodyning . . . . . . . . . . . . � 249

4.4. Experimental setup and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 250

5. Compressible and incompressible quantum phases of fermionic spin mixturesin optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 250

5.1. Hubbard Hamiltonian in a trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 252

5.2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 255

5.3. Cloud compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 256

5.4. Entropy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 258

6. Controlled superexchange interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 259

6.1. Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 259

6.2. Time-resolved observation of superexchange interactions . . . . . . . . . . � 261

7. Quantum noise correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 262

7.1. Time-of-flight versus noise correlations . . . . . . . . . . . . . . . . . . . . . . . . . � 264

7.2. Noise correlations in bosonic Mott and fermionic band insulators . . . � 265

8. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 269

indice XI

Z. Hadzibabic and J. Dalibard – Two-dimensional Bose fluids: An atomic

physics perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pag. 273

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 2741.1. Absence of true long-range order in 2d . . . . . . . . . . . . . . . . . . . . . . . . . � 274

1.2. Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 275

2. The infinite uniform 2d Bose gas at low temperature . . . . . . . . . . . . . . . . . . � 2762.1. The ideal 2d Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 276

2.2. Interactions in a 2d Bose gas at low T . . . . . . . . . . . . . . . . . . . . . . . . . � 279

2.3. Suppression of density fluctuations and the low-energy Hamiltonian . � 280

2.4. Bogoliubov analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 282

2.5. Algebraic decay of correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 285

3. The Berezinskii-Kosterlitz-Thouless (BKT) transition in a 2d Bose gas . . . � 2873.1. The role of vortices and topological order . . . . . . . . . . . . . . . . . . . . . . . � 287

3.2. A simple physical picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 289

3.3. Results of the microscopic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 290

4. The 2d Bose gas in a finite box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 2924.1. The ideal Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 292

4.2. The interacting case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 293

4.3. Width of the critical region and crossover . . . . . . . . . . . . . . . . . . . . . . � 294

4.4. What comes first: BEC or BKT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 295

4.5. The case of anisotropic samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 296

5. The 2d Bose gas in a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 2985.1. The ideal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 298

5.2. LDA for an interacting gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 299

5.3. What comes first: BEC or BKT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 302

5.4. Width of the crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 303

6. Achieving a quasi-2d gas with cold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . � 3046.1. Experimental implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 304

6.2. Interactions in a 2d atomic gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 305

6.3. Residual excitation of the z-degree of freedom . . . . . . . . . . . . . . . . . . . � 307

7. Probing 2d atomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 3087.1. In situ density distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 308

7.2. Two-dimensional Time-of-Flight expansion . . . . . . . . . . . . . . . . . . . . . � 310

7.3. Three-dimensional Time of Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 311

7.4. Interference between independent planes . . . . . . . . . . . . . . . . . . . . . . . � 313

7.5. Interfering a single plane with itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 316

8. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 316

Elenco dei partecipanti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � 323


Preface

Many fundamental processes in physics involve transport. For a variety of physical

systems, e.g. electrons, light, cold atoms and sound, transport mechanisms eventually

reduce to different manifestations of wave transport. In the last decades, pushed by

the spectacular progresses in the control and engineering of matter at the nano-scale,

new regimes of wave transport became of strong interest. Indeed, fascinating effects

emerge when transport is studied at the “nano” level, when atoms behave like waves and

light propagation in nano-engineered structures acquires intriguing behaviors. This book

collects contributions from speakers and lecturers of the CLXXIII International School

of Physics “Enrico Fermi” which was held in Varenna (Italy) from June 23rd to July 3rd

2009. Different aspects of wave transport were covered during the school, from electrons

to light propagation, from sound to ultracold atoms. Considering the ubiquitous nature

of wave transport phenomena, the school was characterized by a strongly interdisciplinary

approach, with speakers, lecturers and students from different communities meeting and

sharing their knowledge and the often complementary points of view and approaches.

Among the different media in which waves can travel, periodic and disordered struc-

tures surely deserve particular attention. Interference of waves in periodic structures

results in the formation of energy bands, which are responsible for the conduction prop-

erties of electrons in solids. Periodic structures can be realized also for light and ultra-cold

atoms, in the form of photonic crystals or optical lattices, respectively, which allow the

observation of effects which have been originally predicted in the context of solid-state

physics. The most recent advances in the physics of ultra-cold atoms in optical lat-

tices are discussed in the contribution by I. Bloch concerning quantum simulation of

condensed-matter physics. Optical lattices also allow the production of low-dimensional

atomic systems, as discussed in the paper by Z. Hadzibabic and J. Dalibard devoted to

the investigation of transport and superfluidity in 2D bosonic quantum gases.

Disordered structures also show fascinating phenomena. Multiple scattering in ran-

dom media results in the localization of waves predicted by P. W. Anderson fifty years

ago for electrons moving in disordered crystals, and reviewed in this book in the opening

XIII

XIV Preface

article by P. Woefle. Also in this case, a phenomenon originally predicted for electrons in

crystals has been observed both for propagation of classical waves —light and sound— in

disordered media and very recently for ultracold atoms expanding in disordered optical

potentials. Transport of sound waves in different media, including Anderson localization

in disordered structures, is discussed in the contributions by J. Page.

At the border between periodic and disordered media, quasicrystals are topological

structures showing long-range order and absence of periodicity, which results in intriguing

properties that are described in the contribution by P. J. Steinhardt. Optical quasicrys-

tals can be realized for ultra-cold atoms and used to study Anderson localization of

matter waves, as discussed in the article by L. Fallani and M. Inguscio.

Knowledge of transport properties in complex systems is important not only for fun-

damental studies, but also for applications. Understanding the propagation of light is

extremely important for engineering new devices, as metamaterials and plasmonic mate-

rials, and for applications in the field of energy, biology and medicine. In this perspective,

the article by R. C. Mesquita and A. G. Yodh covers the application of diffuse optics

to medical imaging. Extending these concepts beyond the field of optics, in two differ-

ent contributions to this book, M. Fink and coworkers discuss multi-wave imaging for

medical applications and present theory and applications of time-reversal focusing.

The success of the Summer School was not only determined by the high quality of

the lectures, but also by the enthusiasm of the students and observers who attended the

Course. Their active participation resulted in the success of the two poster sessions (the

most interesting posters have been upgraded to invited presentations) and of the final

discussion session on future research perspectives. We would like to warmly thank all

the speakers, lecturers, participants and express our gratitude to the organizing team, in

particular Barbara Alzani of the Italian Physical Society for her passion and dedication

in the Course organization, as well as Ramona Brigatti and Marta Pigazzini for their

enthusiasm and assistance in Varenna. We also acknowledge financial support from the

Italian Physical Society through its president Luisa Cifarelli, and from the European

network Intercan.

Finally, the Summer School hosted a celebration in memoriam of Franco Bassani

(1929-2008), former president of the Italian Physical Society. On this occasion, friends

and colleagues Lucio Andreani, Luisa Cifarelli, Massimo Inguscio and Erio Tosatti pre-

sented several portraits of his scientific and personal life. Franco Bassani was an excellent

scientist and an outstanding man. We dedicate this book to his memory.

R. Kaiser, D. S. Wiersma and L. Fallani


Soci

età

Ital

iana

di F

isic

aSC

UO

LA IN

TER

NA

ZIO

NA

LE D

I FIS

ICA

«E.

FER

MI»

CLX

XII

I CO

RSO

- VA

REN

NA

SU

L LA

GO

DI C

OM

OV

ILLA

MO

NA

STER

O 2

3 G

iugn

o - 3

Lug

lio 2

009

1) T

. Pau

l 2

) A. U

llah

3) D

. Ela

m 4

) M. S

tratic

iuc

5) N

. Mer

cadi

er 6

) J. R

. Ott

7) B

. T. G

ebre

hiw

ot 8

) M. T

hore

son

9) W

. K. H

ildeb

rand

10) M

. Ber

ritta

11) B

. N. S

. Bha

ktha

12) E

. Kot

13) S

. Ris

t

14) J

. Har

twig

15) S

. T. S

eide

l16

) W. P

eete

rs17

) J. J

. Sae

nz18

) S. L

. Por

talu

pi19

) S. G

ualin

i20

) S. K

uryl

chyk

21) M

. Lar

cher

22) P

. L. K

osw

atta

23) L

. Rat

schb

ache

r24

) E. N

unes

-Per

eira

25) A

. Peñ

a26

) H. A

hler

s

27) P

. Bar

thel

emy

28) R

. Mup

para

pu29

) K. V

ynch

30) R

. Viv

ekan

anth

an31

) D. M

ilova

novi

c32

) J. A

rmijo

33

) F. J

. Val

divi

a Va

lero

34) M

. R. D

e Sai

nt

V

ince

nt35

) T. B

iena

imé

36) R

. Men

chòn

Enr

ich

37) Y

. Lah

ini

38) R

. Pug

atch

39) M

. Bau

mer

t40

) N. M

eyer

41) M

. Hol

ynsk

i42

) M. C

. Pig

azzi

ni43

) R. B

rigat

ti44

) T. W

asak

45) P

. Sza

nkow

ski

46) B

. Alz

ani

47) O

. Sliu

sare

nko

48) B

. Bar

nes

49) P

. Wöl

fle50

) R. H

ulet

51) A

. Asp

ect

52) N

. Fab

bri

53) L

. Cha

ntad

a

Sa

ntod

omin

go54

) M. I

ngus

cio

55) G

. Mar

et56

) D. S

. Wie

rsm

a57

) L. F

alla

ni58

) R. K

aise

r59

) J. P

age

60) P

. Ste

inha

rdt

61) C

. Hof

man

n

1

62) S

. Hoi

nka

63) A

. Rak

onja

c64

) T. K

itaga

wa

65) C

. Wut

tke

66) S

. Giu

dica

tti67

) T. H

artm

ann

68) K

. Put

tene

ers

69) I

. Prie

to70

) V. E

rem

eev

71) A

. Ben

seny

Cas

es72

) E. V

ogt

73) N

. Cio

banu

23

45

67

89

1011

1213

1415

16 1718

1920

2122

2324

2526

2728

2930

3132

3334

3536

3738

3940

41 4243

44

45

46

4748

4950

51

5253

5455

5657

5859

6061

62

63 6465

66 67

68

6970

71

7273


Proceedings of the International School of Physics “Enrico Fermi”Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser,D. S. Wiersma and L. Fallani(IOS, Amsterdam; SIF, Bologna)DOI 10.3254/978-1-60750-755-0-1

Anderson localization

P. Wolfle

Institut fur Theorie der Kondensierten Materie, and DFG-Center for Functional

Nanostructures, Universitat Karlsruhe - 76131 Karlsruhe, Germany

Summary. — The basic concepts of the theory of Anderson localization are re-viewed in the example of electrons in disordered solids. The lectures are organizedin three sections. In the first, the phenomenon of localization of quantum particlesor classical waves is introduced on a qualitative level. The regimes of strong andweak localization are discussed. Sample to sample fluctuations are considered forone-dimensional and quasi–one-dimensional systems. In sect. 2 the scaling theory ofthe Anderson localization transition is presented. The renormalization group theoryis introduced and results and consequences are presented. The classification of theAnderson transitions into universality classes is described. Basic concepts of thefractal structure of the wave functions at the critical point are reviewed. In sect. 3

the self-consistent theory of Anderson localization is presented. The effect of theelectron-electron interaction in destroying the phase coherence is briefly discussed.

1. – Basic notions of localization

1.1. Introductory remarks. – The localization of quantum particles or classical waves

by a static random potential or random fluctuations of the local parameters determining

wave propagation in a disordered medium has been studied for more than fifty years.

The very concept of localization was introduced in a seminal paper by P. W. Anderson

entitled “Absence of diffusion in certain random lattices” [1]. There it was shown that

c© Societa Italiana di Fisica 1

2 P. Wolfle

electrons may be localized by a random potential, so that diffusion would be suppressed.

That work had been triggered by the experimental observation of the apparent absence of

spin diffusion as detected in NMR signals of a disordered solid. The fundamental reason

for the localizing effect of a random potential is the multiple interference of wave compo-

nents scattered by randomly positioned scattering centers. The interference effect takes

place both with quantum particles and with classical waves, provided the propagation is

coherent.

It is interesting to note that the first application of the idea of localization concerned

the spin diffusion D of electrons and not the electrical conductivity σ. Anderson consid-

ered a tight-binding model of electrons on a crystal lattice, with energy levels at each site

chosen from a random distribution. The traditional view had been that scattering by the

random potential causes the Bloch waves to lose phase coherence on the length scale of

the mean-free path �. Nevertheless, the wave function was thought to remain extended

throughout the sample. Anderson pointed out that if the disorder is sufficiently strong,

the particles may become localized, in that the envelope of the wave function ψ(�r ) decays

exponentially from some point �r0 in space

(1) |ψ(�r )| ∼ exp(|�r − �r0|/ξ),

where ξ is the localization length.

There exist a number of review articles covering the earlier work on the Anderson

localization problem. The most complete account of the early work is given by Lee and

Ramakrishnan [2]. The seminal early work on interaction affects is presented in [3]. A

complete account of the early numerical work can be found in [39]. A path integral

formulation of weak localization can be found in [4]. Several more review articles and

books are cited along the way. In the following we will use units with Planck’s constant

h and Boltzmann’s constant kB equal to unity, unless stated otherwise.

1.2. Electrons and classical waves in disordered systems. – The wave function ψ(�r ) of

a single electron of mass m in a random potential V (�r ) obeys the stationary Schrodinger

equation

(2)

(−

h2

2m�∇2 + V (�r ) − E

)ψ(�r ) = 0.

In the simplest case V (�r ) may be taken to obey Gaussian statistics with 〈V (�r )V (�r ′)〉 =

〈V 2〉δ(�r − �r ′). Electrons propagating in the random potential V (�r ) will be scattered on

average after a time τ . For weak random potential the scattering rate is given by

(3)h

τ= πN(E) 〈V 2〉,

where N(E) is the density of states at the energy E of the electron. In a metal the

electrons carrying the charge current are those at the Fermi energy E = EF. Within the

time τ the electron travels a distance � = vτ , where v is its velocity.

Anderson localization 3

In close analogy the wave amplitude ψ(�r ) of a classical monochromatic wave of fre-

quency ω obeys the wave equation

(4)

(ω2

c2(�r )+ �∇2

)ψ(�r ) = 0.

Here c(�r ) is the wave velocity at position �r in an inhomogeneous medium, assumed to

be a randomly fluctuating quantity. A possible realization would be a system of spheres

of diameter r0 at random positions �ri, with wave velocity c0 < 1 inside the spheres and

c = 1 in the surrounding medium. The main difference between the Schrodinger equation

and the wave equation is that in the wave equation the “random potential” 1/c2(�r ) is

multiplied by ω2, so that disorder is suppressed in the limit ω → 0. By contrast, in the

quantum case disorder will be dominant in the limit of low energy E. A further difference

may arise if the wave amplitude is a vector quantity, as e.g. for electromagnetic waves

in d = 3 dimensions.

In real systems particles or wavepackets are not independent, but interact. Electrons

are coupled by the Coulomb interaction, leading to important effects that go much be-

yond the single-particle model. Some of these effects will be mentioned later. Similarly,

wavepackets interact via non-linear polarization of the medium.

Apart from these complications, the physics of electronic wavepackets and classical

wavepackets is quite similar. In the following we will present most of the discussion in

the language of electronic wavepackets.

1.3. Strong localization and Anderson transition. – The appearance of localized states

is easily understood in the limit of very strong disorder: localized orbitals will then

exist at positions where the random potential forms a deep well. The admixture of

adjacent orbitals by the hopping amplitudes will only cause a perturbation that does not

delocalize the particle. The reason is that nearby orbitals will have sufficiently different

energies so that the amount of admixture will be small. On the other hand, orbitals close

in energy will in general be spatially far apart, so that their overlap is exponentially

small. Thus, we can expect the wave functions in strongly disordered systems to be

exponentially localized. Whether the particles become delocalized, when the disorder

strength is reduced, is a much more complex question. In one dimension, it can be

shown rigorously that all states are localized, no matter how weak the disorder [5-7].

In three dimensions, the accepted view is that the particles are delocalized at weak

disorder. Localized and extended states of the same energy are not expected to coexist.

In the generic situation any small perturbation would lead to hybridization, delocalizing

any localized state. We can therefore assume that the localized and extended states of a

given energy are separated: as a function of increasing disorder strength η there will be

a sharp transition from delocalized to localized states at a critical disorder strength ηc.

A qualitative criterion as to when an Anderson transition is expected in 3d systems has

been proposed by Ioffe and Regel [59]. It states that as the mean free path � becomes

shorter with increasing disorder, the Anderson transition occurs when � is of the order

4 P. Wolfle

Fig. 1. – Phase diagram showing metallic (M) and insulating (I) regions of the tight-bindingmodel with site diagonal disorder (box distribution of width W ). Dots: numerical study [39];solid line: self-consistent theory [13]. The remaining lines are bounds on the energy spectrum.

of the wavelength λ of the particle (which amounts to the condition kF� ∼ 1 in metals,

where kF is the Fermi wave number). As we will see later, in 1d or 2d systems � may be

much longer than the wavelength and the particles are nonetheless localized. In fact, the

relevant mean free path here is the one with respect to momentum transfer. A similar sit-

uation exists when we fix the disorder strength, but vary the energy E. Electrons in states

near the bottom of the energy band are expected to be localized even by a weakly disor-

dered potential, whereas electrons in states near the band center (in dimension d = 3) will

be delocalized, provided the disorder is not too strong. Thus there exists a critical energy

Ec separating localized from delocalized states, the so-called mobility edge [8,9]. The elec-

tron mobility as a function of energy is identically zero on the localized side (at zero tem-

perature), and increases continuously with energy separation |E−Ec| in the delocalized,

or metallic, phase. The continuous character of this quantum phase transition, termed

Anderson transition, is a consequence of the scaling theory to be presented in sect. 2.

Historically the continuous nature of the metal-insulator transition in disordered solids

has been a point of controversy for many years. According to an earlier theory by [8,9], the

conductivity changes discontinuously at the transition, such that a “minimum metallic

conductivity” exists on the metallic side of the transition. Numerical simulations have

shown beyond doubt that the transition is in fact continuous at least in the absence of

interactions.

In the much more complex situation of interacting electrons the results obtained, e.g.

for electrons in the Hubbard model using the Dynamical Mean Field Theory (DMFT)

suggest that the metal-insulator transition at finite temperature is of first order, but

becomes continuous in the limit T → 0 [10,11].


Fig. 2. – Phase diagram showing regions of localized and extendes states for scalar waves propag-ing in a random system of point scatterers of average separation a and average dielectric constantε = 〈ε(�r )〉 = 〈1/c2(�r )〉.

The phase boundary separating localized and extended states in a disordered three-

dimensional system may be determined approximately by a variety of methods. For

electrons on a cubic lattice with nearest-neighbor hopping and one orbital per site of

random energy εi chosen from a box distribution in the interval [−W/2,W/2], the phase

diagram has been determined by numerical simulations [12] as shown in fig. 1. Also shown

is the result of an analytical approximation to be discussed in more detail in sect. 3 [13].

The agreement is quite satisfactory.

In fig. 2. the result of the analogous analytical approximation for the case of

scalar waves of fequency ω propagating in a medium with random dielectric constant

is shown [14]. Here the region of localized states is much smaller, the reason being that

waves of low frequency and correspondingly long wavelength average over the disorder.

The longer the wavelength, the smaller the effective disorder strength. Localization is

then most likely to appear in the regime of resonant scattering, i.e. when the wavelength

is comparable to the extension of the scattering centers.

1.4. Weak localization. – An electron or a wavepacket moving through a disordered

medium will be scattered by the random potential on the average after propagating

a distance �, the mean free path. On larger length scales the propagation is diffusive.

Weak localization is a consequence of destructive interference of two wave components

starting at some point and returning to the same point after traversing time-reversed

paths. Let the probability amplitudes for the wavepacket to move from point �r0 along

some path C back to �r0 be A, and the amplitude for the time-reversed path be Ar, then

the transition probability will be

w = |A + Ar|2 = wcl + wint,

6 P. Wolfle

where wcl = |A|2+ |Ar|2 and wint = 2Re(A∗Ar). For any two paths the interference term

wint may be positive or negative, and thus averages to zero. However, if the system is

time-reversal invariant, A = Ar, and the probability of return w is enhanced by a factor

of two compared to the probability wcl of a classical system

(5) w = 4|A|2 = 2wcl.

In that case the probability for transmission is reduced, which leads to a reduced dif-

fusion coefficient and a reduced conductivity. One may estimate the correction to the

conductivity in the following qualitative way. The relative change of the conductiv-

ity σ by the above interference effect is equal to the probability of interference of two

wavepackets of extension λ, the wavelength, after returning to the starting point. The

probability of return to the origin in time t of a particle diffusing in d dimension is given

by (4πDt)−d/2d3r, where D is the diffusion coefficient. Since the volume of interference

in the time interval [t, t + dt] is λd−1vdt, where λ is the wavelength and v is the velocity

of the wavepacket, one finds the quantum correction to the conductivity δσ, as [34, 15]

(6)δσ

σ0

≈ −

∫ τφ

τ

vλd−1dt

(4πDt)d/2=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

− 3

2π

√3

4πλ2

�2

(1 − τ

τφ

), d = 3,

− 1

2πλ� ln(τφ/τ), d = 2,

− 1√π

(√τφ

τ − 1)

, d = 1.

Here the expressions D = 1

dv2τ for the diffusion coefficient and � = vτ for the mean-

free path have been used. The mean-free time between successive elastic collisions is

τ . The Drude conductivity σ0 = e2nτ/m, with n the particle density. The upper limit

of the integral is the phase relaxation time τφ, i.e. the average time after which phase

coherence is lost due to inelastic or other phase-shifting processes. In subsect. 3.2 we will

present an estimate of 1/τφ. In order for weak-localization processes to exist at all, the

inequality τφ τ must hold. We note that the correction in three and two dimensions

depends on the ratio of wavelength λ to mean free path �, and gets smaller in the limit

of weak disorder, where λ/� 1. In two and one dimension, however, the correction

grows large in the limit τ/τφ → 0. Since one expects the phase relaxation rate 1/τφ for a

system in thermodynamic equilibrium to go to zero in the limit temperature T = 0 (see

below), the WL quantum correction will be large in any system in d = 1, 2 no matter

how weak the disorder is, in the limit of T → 0. As we will see, this behavior signals the

fact that there are no extended states in d = 1, 2 dimensions. The characteristic length

Lφ over which a wavepacket retains phase coherence is related to τφ by the diffusion

coefficient Lφ =√

Dτφ. In systems of restricted dimension like films of thickness a or

wires of diameter a the effective dimensionality of the system with respect to localization

is determined by the ratio Lφ/a: for Lφ a the system is three dimensional, while for

Lφ a diffusion over time τφ takes place in the restricted geometry of the film or wire

and the effective dimension is therefore 2 or 1.


1.5. One-dimensional systems. – One-dimensional systems in transport theory are

characterized by a single transmission channel along a wire or waveguide. Such systems

are not mathematical idealization of systems of point particles in strictly one dimen-

sion, but are actually realized. The conductance of a wire is given by the transmission

probability T for the quantum particle to propagate through the wire

(7) G =e2

2πhT

as first stated by Landauer [16, 17]. The dimensionless conductance of a given random

system depends exponentially on its length

(8) g = G/

(e2

2πh

)∼ e−αL,

where the average value of α is equal to the inverse localization length ξ, 〈α〉 = 1/ξ.

Here the angular brackets denote the appropriate weighted average over all realizations

of disorder of the system under consideration. The exponential dependence of g on the

randomly fluctuating (from sample to sample) quantity α causes g to have an extremely

wide probability distribution P (g). The average value of g is then very different from the

typical value of g (value of the maximum of P (g)). As first discussed in [18] the quantity

α has a normal (Gaussian) distribution, which in turn implies that the logarithm of g is

normally distributed, or

(9) P (g) ∼ exp

[−

ξ

4L

(ln

g

4+

L

ξ

)2]

.

(for details see, e.g., ref. [19]).

1.6. Quasi–one-dimensional systems. – Systems with many transport channels (wires),

but of one-dimensional character, such that the mean free path is much longer than the

diameter of the wire, are called quasi–one-dimensional. In this case the joint probability

distribution of the eigenvalues Ti of the transmission matrix as a function of the length

L of the wire may be calculated from a Fokker-Planck-type equation, called DMPK

equation [20,21]. The DMPK equation may be solved exactly in certain cases [22]. The

distribution of conductances P (g) may be calculated approximately from the distribution

of eigenvalues Ti [23], using the relation g = ΣiTi [17]. Depending on the value of L/ξ,

one may distinguish a localized regime (L/ξ 1), a metallic regime (L/ξ 1), and a

crossover regime (L/ξ ∼ 1).

In the localized regime the eigenvalues Ti are widely spaced and only the largest

eigenvalue contributes. Then P (g) is given by eq. (9), the result for the one-dimensional

case. In the metallic regime, the eigenvalues are densely packed around the limiting value

T = 1. Since the eigenvalues Ti are approximately statistically independent, one expects

8 P. Wolfle

Fig. 3. – Distribution of dimensionless conductance g of a quasi–one-dimensional wire of lengthL, with N channels and mean-free path L. Solid line: N�/L = 0.7; dashed line: N�/L = 0.2.

a Gaussian distribution P (g). The density of eigenvalues determines the width of the

Gaussian.

It turns out that the width is independent of the properties of the system [23]

(10) P (g) ∼ e−152

(g−γ)2

,

where γ = N�/L = ξ/L. This result holds for the simplest symmetry class, namely

unitary symmetry (see sect. 2). The sample-to-sample flucuations of the conductance

are hence universal [24, 25], and are one of the hallmarks of disordered systems.

In the crossover regime P (g) has a very asymmetric shape (see fig. 3). For ξ/L ∼ 1/2,

e.g., P (g) is given by a log-normal distribution centered at g ≈ 1/2 at g < 1 and a

sharply dropping exponential tail at g > 1, which has been termed “one-sided log-normal

distribution” [26].

2. – Theory of localization: fundamental concepts

The Anderson localization transition is a quantum phase transition, i.e. it is a tran-

sition at zero temperature tuned by a control parameter (strength of disorder, particle

energy, wave frequency). Unlike other quantum phase transitions, the Anderson transi-

tion does not appear to have an obvious order parameter. Nontheless, there appears to

be a dynamically generated length scale, the localization or correlation length ξ, which

tends to infinity as the transition is approached. Therefore, drawing an analogy with

magnetic phase transitions, Wegner early on proposed scaling properties [27]. Later, he

formulated a field-theoretic description of the Anderson transition in the form of a non-

linear sigma model (NLσM) of interacting matrices (rather than vectors, as for magnetic


systems) [28]. The NLσM was later formulated in the mathematically more tractable

supersymmetric form [29] (see below).

Many rigorous results especially on systems of restricted dimensions or in infinite

dimensions have been obtained within this framework (see, e.g., [29, 30]). However, a

systematic calculation of the properties of the Anderson transition in d = 3 dimensions

within the NLσM has not been possible so far. An approximate theory of the Anderson

transition, which takes into account the renormalization of the diffusion coefficient in a

self-consistent way, accounts well for the non-critical properties, but fails to describe the

critical behavior quantitatively [31, 32]. It will be discussed in sect. 3. In the following

we will mainly discuss a somewhat different approach, the scaling theory, which allows

to obtain the main features of Anderson localization without complex mathematical

formalism.

2.1. Thouless conductance. – The idea of scaling theory is to consider localization

behavior as a function of the system size L. The first studies along these lines were

performed by Thouless and collaborators [33]. They envisioned building a sample of

size (2L)d by putting building blocks of size Ld (cubes or squares) together. If the

building blocks are sufficiently large, i.e. are characterized by uniform disorder strength,

one should expect that the eigenstates of the sample of volume (2L)d should be entirely

determined by the properties of the building blocks. The eigenstates of the (2L)d sample

are linear combinations of those of the Ld sample and the amount of admixture of states

of neighboring blocks depends on the overlap integral and the energy denominator. The

energy denominator will be typically of the order of the mean level spacing in the Ld

sample, δε = (N0Ld)−1, where N0 is the density of the states at the energy of the particles

or wavepackets considered (in metals this energy is given by the Fermi energy). To

estimate the overlap, Thouless introduced the sensitivity of energy levels to the boundary

conditions. The energy shift ΔE obtained when the boundary conditions are changed

from periodic to antiperiodic is a measure of the extension of the eigenstates across the

volume Ld. Clearly, a state localized inside the Ld sample will have exponentially small

ΔE, whereas a delocalized state may have ΔE δε. The dimensionless parameter

ΔE/δε is the single parameter that characterizes the wave functions with respect to

their localization properties.

Thouless further noticed that the conductance G (and not the conductivity) is a

dimensionless quantity, when expressed in units of the quantum of conductance GQ =

e2/(2πh)

(11) g = G/(e2/2πh).

Then Thouless conjectured that g should be uniquely determined by the parameter

ΔE/δε.

2.2. Scaling theory of the conductance. – Along a different line Wegner [27] argued

that the Anderson localization transition should be described in the language of critical

phenomena of continuous (quantum) phase transitions. This requires the assumption of

10 P. Wolfle

a correlation length ξ diverging as a function of disorder strength η at the critical point

(12) ξ(η) ∼ |η − ηc|−ν .

The conductivity is then expected to obey the scaling law

(13) σ(η) ∼ ξ2−d ∼ (ηc − η)s; η < ηc, d > 2,

which follows from the fact that σ in units of e2/(2πh) has dimension (1/length)d−2,

and the only characteristic length near the transition is the correlation length ξ. By

comparing the conductivity exponent s with the exponent of ξ, one finds

(14) s = ν(d − 2).

On the other hand, the conductance g of a d-dimensional cube of length L, which for a

good metal of conductivity σ is given by g(L) = σLd−2, must obey the scaling property

(15) g(η;L) = Φ(L/ξ).

This means that g is a function of a single parameter Lξ , so that each value of L/ξ

corresponds to a value g.

2.3. Renormalization group equation. – It follows that g(L) obeys the renormalization

group (RG) equation

(16)d ln g

d lnL= β(g),

where β(g) is a function of g only, and does not depend on disorder. In a landmark paper,

Abrahams, Anderson, Licciardello and Ramakrishnan [34] proposed the above equation

and calculated the β-function in the limits of weak and strong disorder. A confirmation

of the assumption of scaling was obtained from a calculation of the next-order term [35].

At strong disorder we expect all states to be localized, with average localization length

ξ. It then follows that g(L) is an exponentially decreasing function of L

(17) g(L) ∼ exp(−L/ξ).

In comparison with the above (Ohmic) dependence g ∼ Ld−2, this is a very non-Ohmic

behavior. The β-function is then given by

(18) β(g) ∼ ln(g/gc) < 0.

At weak disorder one finds from g ∼ Ld−2

(19) β(g) = d − 2.


The important question of whether the system is delocalized (metal) or localized (insula-

tor) may be answered by integrating the RG equation from some starting point L0, where

g(L0) is known. Depending on whether β(g) is positive or negative along the integration

path, the conductance will scale to infinity or to zero, as L goes to infinity.

In d = 3 dimensions we have β(g) > 0 at large g, but β(g) < 0 at small g. Thus there

exists a critical point at g = gc, where β(gc) = 0, separating localized and delocalized

behavior.

In d = 1 dimension, on the other hand, β(g) < 0 at large and small g, and by

interpolation also for intermediate values of g, so that there is no transition in this case

and all states are localized.

The dimension d = 2 plays apparently a special role, as in this case β(g) → 0 for

g → ∞. In order to find out, whether β > 0 or < 0 for large g one has to calculate

the scale dependent (i.e. L-dependent) corrections to the Drude result at large g. This

is nothing but the weak-localization correction we already considered. For a system of

finite length L < Lφ we should replace 1

τφ= DL−2

φ in eq. (6) with DL−2, leading to

(20) g(L) = σ0 − a ln

(L

�

),

where a quantitative calculation [34] gives a = 2/π and σ0 = �/λ (in units of e2/h) has

been used. It follows that

(21) β(g) = −a

g, d = 2,

so that we can expect β(g) < 0 for all g, meaning that again all states are localized. This

result is valid for the “usual” type of disorder, i.e. in case all symmetries, in particular

time reversal symmetry (required for the weak-localization correction to be present)

are preserved. In case time reversal invariance is broken, e.g. by spin-flip scattering

at magnetic impurities, the weak-localization effect is somewhat suppressed, but not

completely removed. The first correction term in the β-function is then ∝ −1/g2 (see,

e.g., [29]) implying that still all states are localized. In the presence of a magnetic field

the situation is more complex, however, since the scaling of the Hall conductance is

coupled to the scaling of g. As a result, one finds exactly one extended state per Landau

energy level, which then gives rise to the quantum Hall effect [36]. On the other hand,

if spin-rotation invariance is broken, but time reversal invariance is preserved, as is the

case in the presence of spin-orbit scattering, the correction term is ∝ +1/g, i.e. it is

anti-localizing. In this case the β-function in d = 2 dimensions has a zero, implying the

existence of an Anderson transition [37].

2.4. Critical exponents. – In the neighborhood of the critical point at g = gc in d = 3

dimensions we may expand the β-function as

(22) β(g) =1

y

[g − gc

gc

], |g − gc| gc.

12 P. Wolfle

Fig. 4. – Renormalization group β-function in dimensions d = 1, 2, 3 for the orthogonal ensemble,calculated by the self-consistent theory [56].

Integrating the RG equation for g > gc from g(�) = g0 to β → 1 at large L we find

g(L) = σL, where

(23) σ ∼1

�(g(�) − gc)

y.

Since g(�) − gc ∝ ηc − η, we conclude that the inverse of the slope of the β-function, y,

is equal to the conductivity exponent s = y.

Similarly, one finds on the localized side (g < gc)

(24) g(L) ∼ gc exp[− c(gc − g(�))yL/�

]∼ gc exp(−L/ξ),

from which the localization length follows as

(25) ξ ∼ �|η − ηc|−y.

The critical exponent ν governing the localization length is therefore ν = y = s in d = 3

dimensions.

Since the critical conductance gc = O(1) in d = 3, analytical methods are not available

to calculate the β-function in the critical region in a quantitative way. A perturbative

expansion in 2 + ε dimensions, where gc 1 is available, but the expansion in ε is not

well behaved, so that it cannot be used to obtain quantitative results for s and ν in d = 3.


In fig. 4 the result of a calculation of the β-function using a self-consistent approxi-

mation is shown [38]. In that approximation, ν = 1 is found.

There exist, however, reliable results on ν from numerical studies, which give s = ν =

1.58 ± 0.02 [39,40].

2.5. Dynamical scaling. – The dynamical conductivity σ(ω) (the a.c. conductivity at

frequency ω) in the thermodynamic limit in d = 3 obeys the scaling law [41,42]

(26) σ(ω; η) =1

ξΦ(Lω/ξ),

where the scaling function Φ has been introduced in eq. (15). Here Lω is the typical

length a wavepacket travels in the time of one cycle, 1/ω. Since the motion is diffusive,

Lω =√

D(ω)/ω. It is important to note that the relevant diffusion coefficient is scale

dependent D = D(ω; η), which is related to the conductivity via the Einstein relation

(27) σ(ω) = hN(E)D(ω),

where N(E) is the density of states at the particle energy E.

At the Anderson transition, when ξ → ∞, we expect σ(ω) to be finite. It follows that

limξ→∞ Φ(Lω/ξ) ∼ ξ/Lω and consequently

(28) σ(ω; η) ∼1

Lω, η = ηc.

This is a self-consistent equation for σ(ω), with solution

(29) σ(ω) ∼ ω1/3, η = ηc.

To be more precise, ω in the above expressions should be replaced with the imaginary

frequency −iω, such that σ(ω) is a complex-valued quantity.

In more general notation, introducing the dynamical critical exponent z by σ(ω) ∼

ω1/z, we conclude that z = 3. Since the dynamical critical exponent determines how the

scaling of wave vector q and frequency ω of the critical fluctuations are related, ω ∼ qz,

we conclude that the initial scaling of the diffusion process, ω ∼ q2 is modified at the

transition as ω ∼ D(q)q2 ∼ q3, i.e. the wave vector-dependent diffusion coefficients scales

as D(q) ∼ q. The dynamical scaling is valid in a wide neighborhood of the critical point,

defined by ω > 1

τ (ξ/�)−z ∼ |η − ηc|νz, where νz ≈ 4.8. This scaling regime is accessible

in experiment, not only by measuring the dynamical conductivity directly, but also by

observing that at finite temperature the scaling in ω is cut off by the phase relaxation

rate 1/τφ [42, 43]. Therefore, assuming a single temperature power law 1/τφ ∼ T p, one

finds the following scaling law for the temperature-dependent d.c. conductivity:

(30) σ(T ; η) ∼ T p/3ΦT (ξT p/3).

14 P. Wolfle

Using this scaling law, one may in principle determine the critical exponent ν from

the temperature dependence of the conductivity in the vicinity of the critical point.

In the case of disordered metals or semiconductors, where studies of this type have

been performed, the effect of electron-electron interaction has to be taken into ac-

count. One major modification in the above is that the Einstein relation is changed:

the single-particle density of states, which is not critical is replaced with the compress-

ibility ∂n/∂μ (n = density, μ = chemical potential), which is expected to vanish at the

transition, i.e. the system becomes incompressible. Another change is that the frequency

cutoff is simply given by the temperature. The critical exponents determined from ex-

periment vary widely, from s = 0.5 [44], s = 1 [45] to s = 1.6 [46,47] and from z = 2 [46]

to z = 2.94 [47].

2.6. Symmetry classes. – So far we have mostly considered systems with all the avail-

able symmetries. As one may expect from a comparison to conventional thermodynamic

phase transitions, the critical behavior will depend on which of the symmetries are present

in a given system. The systems may be grouped into symmetry classes and within each

class the critical behavior is universal. Symmetry classes for disordered systems were

first introduced by [48] and [49] in the context of random Hamiltonian matrix models

(in short Random Matrix Theory (RMT)). The corresponding scheme considers time

reversal (T ) and spin rotation (S) symmetries. There are three possible combinations:

1) T and S preserved: Gaussian orthogonal ensemble (GOE) (the Hamiltonian ma-

trices are of orthogonal character).

2) T is violated: Gaussian unitary ensemble (GUE).

3) T preserved, but S violated: Gaussian symplectic ensemble (GSE).

In the presence of a strong magnetic field, GUE is realized, while in the presence of

spin-orbit interaction GSE has to be applied. Additional symmetries arise, when the

effective field theory has additional degrees of freedom [50]. For example electrons on

a bipartite lattice may be labeled by an isospin index specifying the sublattice. Under

certain conditions, e.g. disorder only in the hopping matrix elements, the Hamiltonian

matrix acquires a “chiral” symmetry. Three chiral classes may be identified. A further

class of symmetries may arise for the effective Hamiltonian of Bogolyubov quasiparti-

cles in superconductors. One finds four distinct classes of this socalled “Bogolyubov-de

Gennes” type (for a detailed discussion, see Evers and Mirlin [51]). The ten symmetry

classes may in fact be related to the known classical symmetric spaces [52].

2.7. Non-linear σ-model . – Any continuous phase transition is characterized by crit-

ical modes, which interact more and more strongly, as the transition is approached. In

the case of the Anderson localization transition the diffusion modes play the role of criti-

cal modes. An effective field theory should therefore be formulated in terms of (bosonic)

diffusion modes. A field-theoretic description of the Anderson model is, however, compli-

cated by the fact that there is only one particle in the system at any time (the contribution


of the many independent electrons in a metal may be added at the end of the calculation).

Technically speaking this implies that there are no internal closed loop diagrams allowed.

These unwanted contributions, appearing naturally in any quantum field theory, may be

projected out by one of three known methods. The first method, used by Wegner [28] in

his pioneering work, is the so-called replica trick. There one considers in addition to the

system under consideration N −1 identical replicas. Since any internal loop contribution

is proportional to the number of replicas, N , by taking the limit N → 0 at the end the

closed loop contributions are projected out. A mathematically better defined procedure

is obtained by adding to each Bose-type field a corresponding Fermi-type partner [61,29].

For any closed loop the contributions of Boson and Fermion fields cancel. In the latter

formulation the diffusion propagator may be represented as

(31) Φ(�r, �r ′;ω) =⟨GR

E+ω/2(�r, �r ′)GA

E−ω/2(�r ′, �r )

⟩=

∫d[Q]Qbb

12Qbb

21e−S[Q],

where the dynamics of the 4 × 4 matrix field Q is defined by the action of a non-linear

σ-model

(32) S[Q] =πN(E)

4

∫ddrStr

[− D(∇Q)2 − 2iωΛQ

],

where D is the diffusion coefficient, Λ = diag(1, 1,−1,−1), the matrix is normalized,

Q2 = 1, and Qbb12

denotes the boson-boson element of the retarded-advanced block (for

details, see [61, 29]). So far the above non-linear σ-model has been solved for simpler

limiting cases such as zero-dimensional and infinite-dimensional systems, weak disorder,

and to some extent systems close to the lower critical dimension d = 2, to name a few

important examples.

2.8. Fractal structure of critical wave functions. – The spatial structure of the wave

function at the critical point has been studied extensively (for a recent review, see [51]).

As early as 1980, Wegner introduced the idea that the critical wave functions have a

multifractal structure [53]. This means that the inverse participation ratios (IPR) Pq

defined by

(33) Pq =

∫ddr|ψ(�r )|2q, q real,

show anomalous scaling, when averaged over disorder:

(34) 〈Pq〉 = Ld〈|ψ(�r )|2q〉 ∼ L−τq ,

where a continous set of exponents τq has been introduced. One may define the fractal

dimensions Dq via Dq = (q − 1)/τq. In a metal one has Dq = d, while Dq = 0 in the

insulating phase. The spectrum of fractal dimensions may be characterized by a spectral

16 P. Wolfle

function f(α). The function f(α) characterizes the probability function of intensities

|ψ|2, which may be expressed as

(35) P (|ψ|2) ∼1

|ψ|2L−d+f(− ln |ψ2|/ ln L).

Calculating the moments 〈|ψ|2q〉 of the distribution P , one finds

(36) 〈Pq〉 ∼

∫dαL−qα+f(α).

In the limit of L → ∞ the α-integral may be done using a saddle-point approximation,

resulting in

(37) τq = qα0 − f(α0),

where α0 is obtained from q = f ′(α0). The spectral function f(α) is a convex function

defined for α ≥ 0, with a maximum at α0, where f(α0) = d. The function f(α) has been

determined in numerical studies for a number of cases of interest [51].

2.9. Anderson transition in the kicked rotor model . – A very useful analogue of local-

ization in a random potential in real space is localization of particles in momentum space.

This has been achieved in a system of atoms subject to a one-dimensional spatially peri-

odic potential varying periodically in a pulsed fashion in time (kicked rotor model). The

periodic kicking gives rise to exponential localization in momentum space (“dynamical

localization”) [62], provided the kicking is sufficiently strong driving the system into the

chaotic diffusion domain. Dynamical localization has been shown to be the analogue of

Anderson localization in one dimension [63]. The experimental observation of dynamical

localization in a system of cold atoms [64] may be considered as the first realization of

Anderson localization with atomic matter waves. In order to access the Anderson tran-

sition in the kicked rotor setup the “dimension” of the system has to be increased. This

may be done by introducing additional incommensurate periodicities, as expressed by

the following Hamiltonian:

(38) H =p2

2+ K cos x[1 + ε cos(ω2t) cos(ω3t)]

N−1∑n=0

δ(t − n).

Here x, p are the particle position and momentum, K is the pulse intensity, ω2, ω3 are

incommensurate frequencies, and appropriate units of mass, length and time have been

chosen. The above system has been shown to be essentially equivalent to the Anderson

model in three dimensions, with K and ε playing the role of inverse disorder strength

and effective dimensionality [65], the limit ε = 0 corresponding to one dimension. The

phase diagram in the ε-K plane shows regions of localized and extended states separated

by a phase boundary.


The above model has been realized with cesium atoms in a magneto-optical trap [66].

By monitoring the momentum distribution as a function of time, the two regions in the

phase diagram were clearly identified. It was possible to extract the critical behavior with

sufficient accuracy to allow for a determination of the critical exponent ν = 1.4 ± 0.3,

comparing rather well with the numerically determined value of ν = 1.6 ± 0.05.

3. – Theory of localization: diagrammatic approaches

The field-theoretic description in terms of the non-linear σ-model mentioned in the

beginning of sect. 2 is believed to be an exact framework within which the critical prop-

erties of the Anderson transition may be, in principle, exactly calculated. The mapping

of the initial microscopic model onto the NLσM requires a number of simplifications, so

that the non-critical properties like the critical disorder ηc, or the behavior in anisotropic

systems, or systems of finite extension are no longer well represented by the NLσM. In

addition, as mentioned already, it is not known how to solve the NLσM in cases of major

interest, such as in d = 3 dimensions.

It is therefore useful to consider approximation schemes, which on one hand keep the

information about the specific properties of the system and on the other hand account

approximately for the critical properties at the transition. Such a scheme is available at

least for the orthogonal ensemble (in which both, time reversal and spin rotation sym-

metry are conserved). This approach has been developed in [31] and is reviewed in [32].

It may be termed “self-consistent one-loop approximation” in the language of renormal-

ization group theory, but has in fact been derived following a somewhat different logic.

3.1. Self-consistent theory of localization. – The appropriate language to formulate

a microscopic theory of quantum transport or wave transport in disordered media is

renormalized perturbation theory in the disorder potential. The building blocks of this

theory for the model defined by eq. (2) are: i) the renormalized one-particle retarded

(advanced) Green’s functions averaged over disorder

(39) GR,Ak (E) =

[E − k2/2m − Σ

R,Ak (E)

]−1

,

where ΣR

k (E) = (ΣA

k (E))∗ is the self-energy, and ii) the random potential correlator

〈V 2〉. The self-energy Σ is a non-critical quantity and can be approximated by ΣR

k (E) �

−i/2τ , where 1/τ is the momentum relaxation rate entering the Drude formula of the

conductivity (assuming isotropic scattering).

The quantity of central interest here is the diffusion coefficient D. It follows from very

general considerations [54] that the density response function describing the change in

density caused by an external space- and time-dependent chemical potential χ = δn/δμ

is given by (in Fourier space)

(40) χ(�q, ω) =D(�q, ω)q2

−iω + D(�q, ω)q2χ0,

18 P. Wolfle

where D(�q, ω) is a generalized diffusion coefficient and NF is the density of states at

the Fermi level. The form of χ is dictated by particle number conservation. χ may be

expressed in terms of GR,A as

(41) χ(�q, ω) = −ω

2πi

∑ k, k′

Φ k k′(�q, ω) + χ0,

where χ0 = NF is the static compressibility (which is non-critical in the model of non-

interacting particles) and Φ may be expressed in terms of the irreducible vertex function

U as

(42) Φ k k′(�q, ω) = GR

k+GA

k−

⎡⎣δ k, k′

+∑ k′′

U k k′′(�q, ω)Φ k′′ k′

(�q, ω)

⎤⎦ ,

where k± = (�k±�q/2, E±ω/2). In terms of diagrams U is given by the sum of all particle-

hole irreducible diagrams of the four-point vertex function. By expressing GRGA as

(43) GR

k+GA

k−=

ΔGk

ω − �k · �q/m − ΔΣk

,

where ΔGk = GR

k+− GA

k−

and ΔΣk = ΣR

k+− ΣA

k−

, one may rewrite eq. (39) in the form

of a kinetic equation

(44)

(ω −

�k · �q

m− ΔΣk

)Φkk′ = −ΔGk

⎡⎣δ k k′

+∑ k′′

U k k′′Φ k′′ k′

⎤⎦ .

By summing eq. (42) over �k, �k′ one finds the continuity equation

(45) ωΦ(q, ω) − qΦj(q, ω) = 2πiNF,

with the density and the current density relaxation functions

(46) Φ(q, ω) =∑ k, k′

Φ k k′(q, ω); Φj(q, ω) =

∑ k, k′

�k · q

mΦ k k′

(q, ω),

where q = �q/|�q |.

Here the Ward identity ΔΣk =∑

k′U k k′

ΔG k′has been used [31]. In the hydrody-

namic limit, i.e. ωτ 1, q� 1, the current density is proportional to the gradient of

the density

(47) Φj + iqD(�q, ω)Φ = 0.


In fact, multiplying eq. (42) by �k · �q/m and summing over �k and �k′, one may derive

relation (47) and by comparison one finds

(48) D0/D(�q, ω) = 1 − η2E

mn

∑ k, k′

(�k · q)GR

k+GA

k−Ukk′(�q, ω)GR

k′

+GA

k′

−

(�k′ · q),

where the disorder parameter η = πNF〈V2〉 = 1

2πEτ and D0 = 1

dv2τ is the bare diffusion

coefficient.

As the Anderson transition is approached, the left-hand side of eq. (48) will diverge (at

q, ω → 0), and therefore the irreducible vertex U has to diverge. The leading divergent

contribution to U is given by the set of diagrams obtained by using a property of the

full vertex function Γ (the sum of all four-point vertex diagrams) in the presence of time

reversal symmetry:

(49) Γ k k′(�q, ω) = Γ

( k− k′+ q)/2,( k′− k+ q)/2(�k + �k′, ω).

This relation follows if one twists the particle-hole (p-h) diagrams of Γ so that the lower

line has its direction reversed, i.e. the diagram becomes a particle-particle (p-p) diagram.

Now, if time-reversal symmetry holds, one may reverse the arrow on the lower Green’s

function lines if one lets �k → −�k at the same time. This operation transforms p-p

diagrams back into p-h diagrams, so that an identity is established relating each diagram

Γ1 to its transformed diagram ΓT1, which is the above relation.

The leading singular diagrams of Γ are those leading to the diffusion pole

(50) ΓD =1

2πNFτ2

1

−iω + Dq2,

where D is the true diffusion coefficient. These diagrams are of the ladder type and

therefore reducible Their transformed counterparts ΓT

D are, however, irreducible and

contribute to U . We may therefore approximate the singular part of U by

(51) U sing

kk′ =1

2πNFτ2

1

−iω + D(�k + �k′)2.

In low-order perturbation theory U sing is given by the “maximally crossed diagrams”,

which when summed up give a result U sing,0 similar to eq. (51), with D replaced with

the bare diffusion constant D0. When U sing,0 is substituted as a vertex correction into

the conductivity diagram, the result is exactly the weak-localization correction discussed

in sect. 1.

When U sing from eq. (51) is substituted for U , eq. (48) for the diffusion coefficient

D(ω) (in the limit q → 0) leads to the following self-consistent equation for D(ω):

(52)D0

D(ω)= 1 +

k2−dF

πm

∫1/�

0

dQQd−1

−iω + D(ω)Q2.

20 P. Wolfle

Here we have assumed that a finite limit limq→0 D(q, ω) = D(ω) exists, and that Q is

limited to 1/� in the diffusive regime.

Equation (52) may be reexpressed as

(53)D(ω)

D0

= 1 − ηdk2−dF

∫1/�

0

dQQd−1

−iω/D(ω) + Q2.

It is useful to express (52) in position-energy space as [60]

(54)D0

D(ω)= 1 + 2π

k2−dF

mC(�r, �r ),

where C(�r, �r ′) is a solution of the diffusion equation (where a cutoff Q < 1/� has to be

applied to the spectrum of Q modes):

(55) [−iω + D(ω)∇2]C(�r, �r ′) = δ(�r − �r ′).

The above formulation allows to describe position-dependent diffusion processes, as ap-

pear near the sample surface in a confined geometry, e.g. transmission through a slab. In

that case the diffusion coefficient may be taken to be position-dependent, D = D(�r, ω),

and C(�r, �r ′) obeys the modified diffusion equation [67]

(56) [−iω + ∇D(�r, ω)∇]C(�r, �r ′) = δ(�r − �r ′).

The solution is subject to an appropriate boundary condition at the surface of the sample.

The theory accounts very well for the localization properties of acoustic waves transmitted

through a strongly scattering plate [67].

3.2. Results of the self-consistent theory of localization. – In d = 3 dimensions eq. (53)

has a solution in the limit ω → 0

(57) D = D0

(1 −

η

ηc

), η < ηc =

1√

3π,

indicating that the critical exponent of the conductivity is s = 1 in this approximation.

The ω-dependence of D(ω) at the critical point is obtained as [55]

(58) D(ω) = D0(ωτ)1/3, η = ηc,

implying a dynamical critical exponent z = 3, which is the exact result.

At stronger disorder, η > ηc, all states are localized and the localization length ξ

defined by ξ−2 = limω→0(−iω/D(ω)) is found as

(59) ξ =

√π

2�

∣∣∣∣1 −η

ηc

∣∣∣∣−1

,


so that the exponent ν = 1. For general d in the interval 2 < d < 4, one finds Wegner

scaling, s = ν(d − 2).

In dimensions d ≤ 2, there is no metallic-type solution. The localization length is

found as

ξ = �

[exp

1

η− 1

]1/2

, d = 2,(60)

ξ ∼= c1�, d = 1,

where the coefficient c1 ≈ 2.6, while the exact result is c1 = 4 [7].

The RG β-function has been derived from the self-consistent equation for the length-

dependent diffusion coefficient, where a lower cutoff 1/L has been applied to the Q-

integral in eq. (50). The result is [56] in d = 3 dimensions in the metallic regime

(61) β(g) =g − gc

g, g > gc =

1

π2

and in the localized regime

(62) β(g) = 1 −1

π2g

1 + x

1 + x2e−x −

x2

1 + x, g < gc,

where x = x(g) is the inverse function of

(63) g =1

π2(1 + x)e−x

(1 − x arctan

1

x

).

The β-functions in d = 1, 2, 3 obtained in this way are shown in fig. 4.

3.3. Destruction of localization by decoherence processes. – At any finite tempera-

ture inelastic processes, or more precisely, dephasing processes limit the phase coherence

of particles or wavepackets to a finite time interval τφ or equivalently, a characteristic

length Lφ. An important mechanism of dephasing for electrons in disordered metals is the

Coulomb interaction between electrons. Its contribution to 1/τφ may be estimated by the

following argument [57]: an electron moving through the system is subject to a fluctuating

electric potential δV (t) resulting from the Coulomb interaction with all the other elec-

trons. The corresponding energy shift eδV (t) leads to a phase shift in the wave function

(64) Δφ(t) =

∫dt1eδV (t1).

Assuming δV to be Gaussian distributed, the phase factor averages as

(65) 〈eiΔφ〉 = exp

[−

1

2

⟨(Δφ)2

⟩],

where

(66) 〈(Δφ)2 = e2

∫ t0

0

dt1

∫ t0

0

dt2⟨δV (t1)δV (t2)

⟩ != 2

t

τφ.

22 P. Wolfle

The thermal fluctuations of the voltage (Nyquist noise) are given by

(67)⟨δV (t1)δV (t2)

⟩= TRT δ(t1 − t2),

where T is the temperature and RT is the resistance of a volume of the size explored by

the particle in time h/T , which for a diffusing particle is LT = (D/T )1/2.

In d = 3 dimensions, RT = (σLT )−1, where σ may be approximated by the Drude

conductivity. It follows that

(68)1

τφ∼

1

τ

1√

EFτ

(T

EF

)3/2

, d = 3.

This result should be contrasted with the standard Fermi liquid result for the inelastic

scattering rate in the clean system, 1

τε∼ T 2. In d = 2 dimensions RT = (σa)−1, where

a ∼ k−1

Fis the thickness of the film. Then

(69)1

τφ∼

1

τ

T

EF

, d = 2.

A more quantitative calculation leads to an additional factor of ln(EFτ) [24].

Finally, in d = 1 dimensions it turns out that LT > Lφ. The relevant volume of the

resistor is therefore limited by Lφ, no longer by LT , and RT = k2

FLφ/σ. As a result, one

finds a self-consistent relation for 1/τφ, the solution of which is given by

(70)1

τφ∼

1

τ(Tτ)2/3, d = 1.

We note that in all dimensions 1/τφ tends to zero as T → 0. This appears to be in

agreement with observation. In some cases a plateau behavior of 1/τφ has been found in

experiment, which gave rise to the speculation that the zero-point fluctuations may cause

decoherence. Given a unique ground state, it is hardly possible for a particle in the system

to lose its phase coherence. Several physical mechanisms that may lead to a plateau of1

τφversus T have been identified. For a recent discussion of these issues, see [58].

4. – Conclusion

After fifty years of intense and wide-ranging studies Anderson localization is still a

lively field of research. Most of the current interest is concentrated on systems different

from the conventional disordered electron system, namely classical waves (light, electro-

magnetic microwaves, acoustic waves) or ultracold atoms. Even though the fundamental

concepts of Anderson transition are probably well understood by now, there still remain

a number of open questions. Some of those are related to the analytical theory of criti-

cal properties near the Anderson transition. Others concern the quantitative theory for

realistic materials, e.g. the question under which conditions precisely light or acoustic


waves become localized. As Anderson localization is a wave interference phenomenon,

the limitations of phase coherence are an important subject of study in this context. By

now Anderson localization has been observed in many different systems beyond doubt.

On the other hand, observation of the Anderson transition separating localized and ex-

tended states is a much more challenging task. Here the more recently studied classical

waves and atomic matter waves offer new perspectives, which will undoubtedly lead to

a deeper understanding of the localization phenomenon.

∗ ∗ ∗

I would like to thank my long term collaborators on the subject of Anderson localiza-

tion, H. Kroha, K. Muttalib, C. Soukoulis and D. Vollhardt, for many fruitful

discussions.

REFERENCES

[1] Anderson P. W., Phys. Rev., 109 (1958) 1492.[2] Lee P. A. and Ramakrishnan T. V., Rev. Mod. Phys., 57 (1985) 287.[3] Altshuler B. L. and Aronov A. G., Electron-Electron Interactions in Disordered

Systems, in Modern Problems in Condensed Matter Sciences, Vol. 10, edited by Efros

A. L. and Pollak M. (North-Holland, Amsterdam) 1985.[4] Chakravarty S. and Schmid A., Phys. Rep., 140 (1986) 193.[5] Mott N. F. and Twose W. D., Adv. Phys., 10 (1961) 107.[6] Borland R. E., Proc. R. Soc. London, Ser. A., 274 (1963) 529.[7] Berezinskii V. L., Sov. Phys. JETP, 38 (1974) 620.[8] Mott N. F., Metal-Insulator Transitions (Taylor and Francis, London) 1974.[9] Mott N. F. and Davis E. A., Electronic Processes in Non-crystalline Materials

(Clarendon Press, Oxford) 1979.[10] Kotliar G. and Vollhardt D., Physics Today, 57 (2004) 53.[11] Byczuk K., Hofstetter W. and Vollhardt D., Phys. Rev. Lett., 94 (2005) 056404.[12] Bulka B., Schreiber M. and Kramer B., Z. Phys. B, 66 (1987) 21.[13] Kroha J., Kopp T. and Wolfle P., Phys. Rev. B, 41 (1990) 888.[14] Kroha J., Soukoulis C. M. and Wolfle P., Phys. Rev. B, 47 (1993) 11093.[15] Gorkov L. P., Larkin A. I. and Khmelnitskii D. E., JETP, 30 (1979) 248.[16] Landauer R., IMB J. Res. Rev., 1 (1957) 223.[17] Landauer R., Philos. Mag., 21 (1970) 683.[18] Anderson P. W., Thouless E., Abrahams E. and Fisher D. S., Phys. Rev. B, 22

(198) 3519.[19] Altshuler B. L. and Prigodin V. N., JETP Lett., 45 (1987) 687.[20] Dorokhov U. N., JETP Lett., 36 (1982) 318.[21] Mello P. A., Pereyra P. and Kumar N., Ann. Phys. (N.Y.), 181 (1988) 290.[22] Beenakker C. W. J., Rev. Mod. Phys., 69 (1997) 731.[23] Muttalib K. A., Wolfle P. W. and Gopar V. A., Ann. Phys. (N.Y.), 308 (2003) 156.[24] Altshuler B. L., JETP Lett., 41 (1985) 648.[25] Lee P. A. and Stone A. D., Phys. Rev. Lett., 55 (1985) 1622.[26] Muttalib K. A. and Wolfle P. W., Phys. Rev. Lett., 83 (1999) 3013.[27] Wegner F., Z. Phys. B, 25 (1976) 327.[28] Wegner F., Z. Phys. B, 35 (1979) 207.

24 P. Wolfle

[29] Efetov K. B., Supersymmetry in Disorder and Chaos (Cambridge University Press,Cambridge) 1997.

[30] Mirlin A. D., Phys. Rep., 326 (2000) 259.[31] Vollhardt D. and Wolfle P. W., Phys. Rev. B, 22 (1980) 4666.[32] Vollhardt D. and Wolfle P. W., Electronic Phase Transitions (Modern Problems in

Condensed Matter Sciences), Vol. 32 (North-Holland, Amsterdam) 1992.[33] Thouless D., Phys. Rep., 13 (1974) 93.[34] Abrahams E., Anderson P. W., Licciardello D. and Ramakrishnan T. V., Phys.

Rev. Lett., 42 (1979) 673.[35] Gorkov L. P., Larkin A. I. and Khmelnitskii D. E., JETP Lett., 30 (1979) 248.[36] Pruisken A. M. M., in The Quantum Hall Effect, edited by Prange S. and Girvin S.

(Springer, Berlin) 1987.[37] Hikami S., Larkin A. I. and Nagaoka Y., Progr. Theor. Phys., 63 (19810) 707.[38] Vollhardt D. and Wolfle P., Phys. Rev. Lett., 48 (1982) 699.[39] Kramer B. and MacKinnon A., Rep. Progr. Phys., 56 (1993) 1469.[40] Slevin K. and Ohtsuki T., Phys. Rev. Lett., 82 (1999) 382.[41] Shapiro B. and Abrahams E., Phys. Rev. B, 24 (1981) 4889.[42] Imry Y., Gefen Y. and Bergman D., Phys. Rev. B, 26 (1982) 3436.[43] Thouless D., Phys. Rev. Lett., 39 (1977) 1167.[44] Paalanen M. A. et al., Phys. Rev. Lett., 48 (1982) 1284.[45] Field S. B. and Rosenbaum T. F., Phys. Rev. Lett., 55 (1985) 522.[46] Bogdanovich S., Sarachik M. P. and Bhatt R. N., Phys. Rev. Lett., 82 (1999) 137.[47] Waffenschmidt S., Pfleiderer C. and v. Lohneysen H., Phys. Rev. Lett., 83 (1999)

3005.[48] Wigner E., Ann. Math., 53 (1951) 36.[49] Dyson F. J., J. Math. Phys., 3 (1962) 140.[50] Gade R. and Wegner F., Nucl. Phys. B, 360 (1991) 213.[51] Evers F. and Mirlin A. D., Rev. Mod. Phys., 80 (2008) 1355.[52] Zirnbauer M. R., J. Math. Phys., 37 (1996) 4986.[53] Wegner F., Z. Phys. B, 36 (1980) 209.[54] Forster D., Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions

(Benjamin, Reading) 1975.[55] Shapiro B., Phys. Rev. Lett., 48 (1982) 823.[56] Vollhardt D. and Wolfle P., Phys. Rev. Lett., 48 (1982) 699.[57] Altshuler B., Aronov A. G. and Khmelnitskii D. E., J. Phys. C, 15 (1982) 7367.[58] von Delft J., Int. J. Mod. Phys. B, 22 (2008) 727.[59] Ioffe A. F. and Regel A. R., Prog. Semicond., 4 (1960) 237.[60] Skipetrov S. E. and van Tiggelen B. A., Phys. Rev. Lett., 96 (2006) 043602.[61] Efetov K. B., Adv. Phys., 32 (1983) 53.[62] Casati G., Chirikov B. V., Ford J. and Izrailev F. M., Stochastic Behavior of a

Quantum Pendulum Under Periodic Perturbation (Springer, Berlin) 1979, p. 334.[63] Grempel D. R., Prange R. E. and Fishman S., Phys. Rev. A, 29 (1984) 1639.[64] Moore F. L., Robinson J. C., Bharucha C. F., Sundaram B. and Raizen M. G.,

Phys. Rev. Lett., 75 (1995) 4598.[65] Casati G, Guarneri I. and Shepelyansky D. L., Phys. Rev. Lett., 62 (1989) 345.[66] Chabe J., Lemarie G., Gremaud B., Delande D., Szriftgiser P. and Garreau

J. C., Phys. Rev. Lett., 101 (2008) 255702.[67] Hefei Hu, Strybulevych A., Page J. H., Skipetrov S. E. and van Tigelen B. A.,

Nature Phys., 4 (2008) 945.


Photonic properties of non-crystalline solids

P. J. Steinhardt

Department of Physics & Princeton Center for Theoretical Science, Princeton University

Princeton, New Jersey, 08544 USA

Summary. — Photonic crystals, periodic arrangements of two or more dielectricmaterials, have been studied for more than two decades as a means of controllingand manipulating the flow of light. These lectures describe recent progress in de-signing non-periodic photonic solids. The aim is to find arrangements of dielectricmaterials that produce substantial complete photonic band gaps that block lightin all directions and for all polarizations over a range of frequencies. Methods aredescribed for constructing quasicrystalline examples with a wide range of rotationalsymmetries and a special class of isotropic, disordered photonic band gap materials.

Photonic band gap materials [1-3] are heterostructures composed of two or more ma-

terials with different dielectric constants arranged in a spatial configuration that forbids

the propagation of electromagnetic waves in a certain frequency range. They are the

photonic analogue of semiconductors, which are characterized by a finite electronic band

gap, and, hence, are potentially as important for photonic applications as semiconductors

are for electronic ones. Since their introduction in 1987, the understanding of photonic

band gap materials has evolved dramatically and their unusual properties have been con-

sidered for use in efficient radiation sources [4], sensors [5], and optical computer chips [6].


26 P. J. Steinhardt

To date, though, the only known dielectric heterostructures with sizable, complete band

gaps (Δω/ωC ≥ 10%, say, where Δω is the width of the band gap and ωC is the midpoint

frequency) have been crystalline (periodic) [1-3]. Crystalline structures have a limited

range of possible rotational symmetries and defect properties critical for controlling the

flow of light in applications.

For this reason, there has been increasing interest in non-crystalline photonic band

gap materials. In these lectures, we will discuss two classes that have been studied:

quasicrystals, with long-range quasiperiodic translational order and discrete rotational

symmetries that are forbidden for crystals (such as five-fold symmetry in two dimensions

and icosahedral symmetry in three dimensions); and a novel class of isotropic disordered

structures known as “hyperuniform.” As will be described below, examples of substantial

complete photonic band gaps (all direction and polarizations) have been found in both

classes. The disordered case is especially surprising since band gaps are normally associ-

ated with translational order. The resulting non-crystalline photonic band gap materials

have distinctive properties that are of interest both for basic physics and for applications.

For example, because the non-crystalline heterostructures are more isotropic (circular or

spherical), they can produce a wider band gap under some conditions. Also, compared to

photonic crystals, they generically have a more isotropic band gap, new types of photonic

modes and new kinds of defects. All of these distinctive properties may be desirable for

some applications.

This is a new direction in the study of photonic materials, one that has already offered

some surprises and may present more in the future. Also, many of the same principles can

be applied to phononics, electronic, plasmonic, shear wave and other physical properties.

For these reasons, the subject is opportune for bright students.

I caution that my personal expertise is in exploring novel patterns, packings and

structures and characterizing the mathematical properties relevant for physics. I am still

a novice in the subject of photonics, only slightly ahead of (most of) the students to

whom I am lecturing. The lectures represent work in which I have been directly involved

over the last few years and surely omit valuable work by other groups with which I am

not yet familiar; I apologize for these omissions and encourage students to investigate

the literature.

1. – Some basics

The physics underlying photonic band gaps is, in many ways, analogous to the physics

of electronic band gaps even though photonics involves purely classical wave phenomena

and electronic band gaps are quantum in nature. Electronic band gaps in the independent

electron approximation arise from solutions to the Schrodinger equation

(1)

[−

h2

2m∇2 + V (r)

]ψ(r) = Eψ(r),

Photonic properties of non-crystalline solids 27

where V (r) is the electron potential created by a configuration of atomic nuclei, ψ is the

electron wave function, and E is the energy of an electron eigenstate. An electronic stop

gap exists if there is a finite range of energies for which there is no eigenstate for any wave

vector k along some direction. An electronic band gap occurs if there is a finite range of

energies for which is there no eigenstate for any wave vector k, independent of direction.

Photonic band gaps arise from solutions of the classical Maxwell equations[�∇×

(1

ε(r)�∇

)×

]�H(r) =

(ω

c

)2

�H(r),(2)

[�∇× �∇×

]�E(r) = ε(r)

(ω

c

)2

�E(r),

where { �H, �E} are the magnetic and electric fields, respectively; ε(r) defines the distribu-

tion of dielectric material in the photonic material; and c is the speed of light. Typically,

ε(r) takes on two possible values correspond to the two dielectric constants of the dielec-

tric materials used to make the heterostructure. A photonic stop gap occurs of there is

a range of ω for which there is no solution for any wave vector k along some direction.

A photonic band gap occurs if there is a range of ω for which there is no solution for any

wave vector k, independent of direction.

Although the structures of the Schrodinger and Maxwell equations are similar, lead-

ing to many analogies between electronic and photonic band gaps, let us note here some

differences. First, one equation is quantum, causing the band gap to shrink to zero as

h → 0; the Maxwell equations are purely classical. The Schrodinger equation describes

a spin-1/2 particle, though ψ is treated as a scalar in this approximation; Maxwell’s

equations describe a massless vector quantity that can be expressed as a combination of

two polarizations, conventionally labeled TM (transverse magnetic) and TE (transverse

electric) models. For two-dimensional photonic materials (or three-dimensional materials

with azimuthal symmetry), where the wave vector lies in the plane, the labeling conven-

tion is that TM corresponds to modes in which the electric field oscillates in and out of the

plane; and TE corresponds to modes in which the electric field oscillates within the plane.

The Schrodinger equation for electron propagation is a single-electron approximation;

more precise treatments require inclusion of electron-electron interactions, which make

the equations inherently non-linear. Maxwell’s equations are inherently linear: there is

no photon-photon interaction. The electron problem involves massive quanta that satisfy

a parabolic dispersion relation in vacuo, ω ∝ k2. The photonics problem entails light,

which has a massless, linear dispersion relation in vacuo, ω ∝ k.

For the Schrodinger equation, the ratio h2/m, the electron Compton wavelength, sets

a fundamental length scale of atomic dimensions. Hence, an electronic band gap structure

is sensitive to the detailed arrangement of atoms and molecules and the interatomic

and intermolecular forces, over which there is limited control. For Maxwell’s equations,

the equations are scale-invariant (in the limit that the absorption length is negligible).

Photonic materials can, therefore, be constructed from macroscopic materials that can

be regarded as continuous (e.g., air and dielectric) and can be shaped at will. The

28 P. J. Steinhardt

challenge is to find the shapes and arrangements with optimal properties. Once the band

structure for a given photonic heterostructure is known, the band structure for a rescaled

heterostructure is the same, up to a rescaling of frequencies and wave vectors. This

is useful experimentally: a trial heterostructure can be constructed on any convenient

length scale where it is easy to measure the scattering radiation; and, then, the perfected

design can be rescaled for the desired frequency range needed for applications.

We will be interested in these lectures in examples of dielectric heterostructures that

produce the largest photonic band gaps for a given symmetry of the structure. The

largest band gaps occur when the wavelength of the radiation is comparable to the size

and/or separation of the scatterers. Two properties are most important in determining

the band structure. The first is long-range translational order. For practical purposes,

the degree of translational order in a structure can be determined by taking the Fourier

transform of dielectric density ε(r). For crystals and quasicrystals, the transform is a

discrete sum of Bragg peaks arranged in three-dimensional reciprocal space with the

same symmetry as the structure.

The periodicity of crystals insures that the wave vectors corresponding to the Bragg

peaks for a d-dimensional structure can be expressed as an integer linear combination of

d basis wave vectors, where basis vectors are defined to be integer linearly independent

(none can be expressed as a integer linear combination of the others). The Bragg wave

vectors form a periodic reciprocal lattice with a non-zero minimum spacing between wave

vectors along each direction. According to the well-known theorems of crystallography,

crystals can have only a limited set of rotational symmetries.

Quasicrystals are quasiperiodic structures with rotational symmetries that violate the

crystallographic theorems. The first examples of quasicrystalline solids were discovered

in 1984 by Shechtman et al., who observed icosahedral symmetry with five-fold rotation

axes in the electron diffraction pattern of an alloy of Al-Mn [7]. Independently, Levine and

Steinhardt [8] introduced the concept of quasicrystals and, then, showed that the Al-Mn

alloy fit well the predictions for an icosahedral quasicrystal. See ref. [9] for an introduction

and collection of foundational papers. Over the last twenty-five years, over a hundred

other quasicrystalline alloys have been synthesized in the laboratory [10], and, recently,

there is evidence of a natural quasicrystal [11]. Also, there have been notable studies of

impermanent non-linear photonic quasicrystals [12] produced by optical induction (only

existing when arrays of laser beams shine on non-linear optical material) that are beyond

the scope of these lectures.

For photonic materials, quasiperiodic means that ε(r) can be expressed as a finite

sum of periodic functions whose periods are incommensurate (the ratio of the periods is

an irrational number). Because they are not periodic, quasicrystals evade the crystallo-

graphic theorems and can exhibit any crystallographically forbidden symmetry, including

five-fold symmetry in two-dimensions and icosahedral symmetry in three dimensions, as

will be illustrated below. Yet, because ε(r) can be decomposed into a sum of periodic

functions, the Fourier transform is a discrete sum of Bragg (δ function) peaks, with

some significant differences compared to the crystal case. First, the wave vectors can

be expressed as a integer sum of basis wave vectors, but the number of (integer linearly


independent) basis vectors is greater than the number of dimensions. For example, four

basis vectors are required for the two-dimensional quasicrystal with five-fold symmetry

(e.g., any four vectors pointing from the center of a pentagon to the vertices), and six

basis wave vectors define the reciprocal lattice for three-dimensional quasicrystals with

icosahedral symmetry (e.g., along the six independent five-fold symmetry axes). As for

crystals, the reciprocal space pattern has the same rotational symmetry as the real space

structure. However, by constructing this reciprocal lattice, it is easy to see there is no

minimum distance between Bragg peaks; between any pair of Bragg peaks there are more

Bragg peaks. The heights (or intensities) of the Bragg peaks vary so that there is a clear

hierarchy of bright to arbitrarily dim peaks.

For crystals, the primitive unit cell of the reciprocal lattice is known as the Brillouin

zone. It corresponds to the polygon formed by joining the perpendicular bisectors of

the segments between the origin of the reciprocal lattice and the reciprocal lattice points

nearest to it. For the quasicrystal, there are no “nearest” points to the origin since

between any two points are more points. In practice, one joins the bisectors of the

segments between the origin and the brightest points about the origin with G ∼ 2π/a.

The irreducible Brillouin zone is the smallest section of the Brillouin zone that contains

all the symmetry axes of the complete zone.

Figure 1 illustrates the band structure for a photonic crystal composed of silicon

with an array of air holes arranged in a diamond (FCC) lattice. The inset shows the

Brillouin zone of the diamond structure with various symmetry directions indicated. The

band structure plot shows the frequency ω of all the photonic modes as the wave vector

k sweeps a path along the Brillouin zone that passes through the various symmetry

directions. Along any given direction, there is range of frequencies which contains no

modes; the range is the stop gap. There is also the range of frequency (indicated by the

shaded strip) which contains no modes along any direction; this is the complete photonic

band gap.

Long-range translational order and Bragg scattering are one contribution to band

gap formation. Another is resonant scattering by individual dielectric elements. The

Mie resonance effect, described in other lectures in this School, becomes stronger as the

dielectric contrast increases since this concentrates the electric field inside the high dielec-

tric components. For certain wavelengths and nearest-neighbor arrangements of dielectric

components, the Mie resonances by neighboring dielectric elements cause interference of

electromagnetic waves that block propagation.

The largest possible photonic band gaps result from the interplay between Bragg and

Mie scattering. However, the fact that Mie resonance relies only on the short-range geo-

metric order makes it conceivable that band gaps can occur in photonic materials with

no long-range order. Although completely random (Poisson) distribution of dielectric

components fails because there is insufficient local order for the Mie resonance to be ef-

fective, we will see that there exists a special class of isotropic, disordered “hyperuniform”

structures that can have substantial complete band gaps.

30 P. J. Steinhardt

Fig. 1. – (Color online) The photonic band structure (above) for photonic crystal comprisedof silicon with spherical air holes arranged in a diamond (FCC) lattice (below). The abscissaindicates the symmetry axes of the Brillouin, as shown in the inset of the upper figure. Thethin strip, which contains no bands, is an example of a complete photonic band gap.


Fig. 2. – (Color online) Experimental photonic structures and their Brillouin zones. (a) Stereo-lithographically produced icosahedral quasicrystal with 1 cm length rods. (b) Diamond structurewith 1 cm rods. (c) Triacontahedron, the effective Brillouin Zones of icosahedral symmetry. (d)Brillouin zone for the FCC/diamond structure. The dashed and dotted lines are trajectoriesreferenced in figs. 3 and 4.

2. – Photonic band gaps in icosahedral quasicrystals

To motivate the theoretical studies in subsequent sections, we begin by describing

an experimental investigation of photonic band gaps in three-dimensional icosahedral

quasicrystals [13]; this is work done in collaboration with W. Man, M. Megens, and P.

Chaikin. Although there had been some studies of 1D and 2D photonic quasicrystals [14-

17] prior to this investigation where exact (1D) or approximate (2D) band structures

can be calculated numerically, analogous calculations for the 3D case require enormous

computational resources and have not been performed to date.

An alternative is to do a series of physical experiments. For example, using stereo-

lithography, a photonic quasicrystal and a diamond (FCC) crystal with centimeter-scale

cells have been constructed and, then, used as the target for a series of microwave trans-

mission measurements. The quasicrystal consists of a quasiperiodic array of oblate and

acute rhombohedra [8] that can be constructed by projecting the points from a six-

dimensional hypercubic lattice into three dimensions, as described in ref. [9].

Figure 2 shows the effective Brillouin zone of the icosahedral structure with its irre-

ducible Brillouin zone highlighted in yellow. The effective Brillouin zone is constructed

32 P. J. Steinhardt

Fig. 3. – (Color online) Measured transmission for an icosahedral quasicrystal. (a) T (f, θ),transmission as a function of frequency and angle, for a rotation about a 2-fold rotation axis ofthe quasicrystal (corresponding to the dotted line in the triacontahedron Brillouin zone shown infig. 2) using two overlapping frequency bands. The dashed line is a 1/ cos (θ) curve characteristicof Bragg scattering from a Brillouin zone face. (b) T (f, θ), for a rotation about a 5-fold rotationaxis corresponding to the dashed line in the triacontahedron Brillouin zone. Inset: Schematicof the microwave horn and lens arrangement used for these measurements.

from the shell of brightest peaks nearest to k = 2π/a, where a is the edge length of the

rhombohedra (about 1 cm). The Brillouin zone is a rhombic triacontahedron composed

of 30 identical rhombi arranged with icosahedral symmetry. The triacontahedron, like

the icosahedron, has 5-, 3-, and 2-fold symmetry axes, as illustrated in the indicated

irreducible Brillouin zone. For comparison, the Brillouin zone of the diamond (FCC)

structure with its irreducible Brillouin zone is also shown.

Note that, as a measure of asphericity, along the edge of diamond structure’s irre-

ducible Brillouin zone the magnitude of k increases by 29.1% from L to W. Along the

edge of the effective irreducible triacontahedral Brillouin zone of the icosahedral struc-

ture, the magnitude of k increases by only 17.5% from the 2-fold to the 5-fold symmetry

points. Moreover the triacontahedron’s faces are identical and subtend smaller solid an-

gles. The question to be considered is whether this higher degree of sphericity for the

quasicrystal results in advantageous photonic properties.

Figure 3 shows the transmission T (f, θ) of microwave radiation through the icosa-

hedral sample as the frequency f and the angle of incidence θ are varied along the

trajectories indicated in fig. 2, which passes through the symmetry directions of the

Brillouin zone.

The rather complex spectra are consistent with Bragg scattering. As explained in the

previous section, to lie on the Brillouin zone, a wave vector must satisfy the condition

k·G = |G|2|/2, or equivalently, |k| = |G|/(2 cos θ). To lowest order, the center frequency


Fig. 4. – (Color online) Imaging of Brillouin zone for diamond and icosahedral quasicrystalstructures. (a) Brillouin zone for the diamond structure along 4-fold direction as seen in thecontour plot of the calculated frequency deviation (δf = f − (c/n)|k|/(2π) vs. k). (b)-(e):Brillouin zone in the plot of the measured T (r = f, θ) for the diamond lattice along the (b)4-fold and (c) 2-fold axis; and for the quasicrystal along the (d) 5-fold and 2-fold axes. Theinner decagon in (d) and the solid and dashed lines in (e) correspond to dashed and dotted linesindicated in the triacontahedral Brillouin zone. The dash-dotted line is a non-triacontahedralzone face.

of a stop gap is therefore fG = (c/n)|G|/(4π cos θ), where c is the speed of light and

n is the Bruggeman effective medium index of the dielectric heterostructure [18]. The

dashed curves in fig. 3 correspond to a 1/ cos θ angular dependence consistent with Bragg

scattering.

The results for the quasicrystal spectrum are simpler than one might expect given

that a quasicrystal has a dense set of Bragg spots (of zero measure), which was ignored

in constructing the effective Brillouin zone in fig. 2. One might have expected many gaps

and zone faces intersecting. On the contrary, the effective Brillouin zone approximation

appears to work well since there are only a few well-defined 1/ cos θ curves in fig. 3

and therefore few zone boundaries with sizable gap formation. To visualize the effective

Brillouin zone structure, the process is inverted by using the gaps to find the zone faces,

as shown in fig. 4 for both the diamond and quasicrystal structures. The zone plots are

made by making polar plots T (r = f, θ), as described in ref. [13].

The results show that there is a relatively well-defined effective Brillouin zone for the

icosahedral quasicrystal, all faces of which are consistent with the quasicrystal Bragg

pattern. Second, the Brillouin zone structure is surprisingly simple despite the fact that

a quasicrystal has a dense set of Bragg spots. Third, and a key result for photonics, the

34 P. J. Steinhardt

measured Brillouin zone is close to spherical, with the largest difference in gap center

corresponding to 17%, more spherical than for the best photonic crystal (diamond) for the

same dielectric constant ratio. Also, our experiments demonstrate that three-dimensional

quasicrystals exhibit sizable stop gaps on reasonably well-defined effective Brillouin zone

faces. Hence, despite the quasiperiodicity, much of the intuition built up for conventional

crystals may be applied.

These empirical results are based on materials (polymer and air) with a relatively low

dielectric constant ratio and with no attempt at finding structures with an optimally large

band gaps. The results encourage a more systematic investigation of how symmetry and

quasiperiodicity affect photonic properties. At present, such studies are only tractable

in 2D. The remainder of these lectures will focus on what has been learned in 2D.

3. – Dependence of band gap width on symmetry in 2D

Studies of photonic band gaps in two dimensions (or, equivalently, for three-

dimensional materials with azimuthal symmetry) are amenable to analytic and computa-

tional methods, as well as empirical methods. The studies are not purely academic: the

designs can be useful for some applications. However, the principal purpose of consider-

ing them in these lectures is to investigate how symmetry and the dielectric arrangement

can affect band gap properties. The wave vectors k propagate in the plane and the

polarization can be confined to purely TM (electric field oscillating in and out of the

plane), TE (electric field oscillating within the plane) or combined polarizations. Unlike

the case of three-dimensions, the two polarizations do not mix through scattering, which

simplifies the problem.

Figure 5 illustrates 2D photonic structures with TM, TE and full photonic band gaps

for photonic crystals. Rigorous computational searches show that the optimal TM band

gaps for large dielectric constant ratio are obtained for arrays of cylinders of high dielectric

embedded in a background (air) of low dielectric constant. In the frequency range of

interest, the TM modes confine the large electric-field oscillations to the cylinders. For TE

polarization, the optimum for photonic crystals is an in-plane network of high dielectric

material obtained by making an array of isolated holes of low dielectric. A full photonic

band gap (both polarizations) is a compromise between these two structures.

This section describes a study [19] designed to identify and compare crystalline and

quasicrystalline dielectric with the widest possible TM band gaps. The systematic ap-

proach is to treat the spatial configuration of the two dielectric components as a finite

sum of density waves, ρ(x), assigning regions where the sum exceeds a certain threshold a

high dielectric constant ε1 and all other regions to the low value ε0. As a practical matter,

the method is most efficient if the number of density waves for the optimal configuration

is small. This makes the method best suited for TM polarization for which the optimal

configurations tend to have smooth features of a single length scale, like the examples of


Fig. 5. – (Color online) Examples of optimal structures for 2D photonic crystals (or, equivalently,3D photonic crystals with azimuthal symmetry) with band gaps (all directions) for TM, TE andboth polarizations. Arrays of cylinders with high dielectric constant are optimal for TM only;arrays of holes in high dielectric constant material are optimal for TE only; a compromise isneeded to obtain a complete (all polarizations and directions) band gap.

cylinders in air shown in fig. 5. Thus, the dielectric constant at position x is given by

(3) ε(x) =

{ε1, if Re{ρ(x)} > 1,

ε0, otherwise,

where ρ is a sum of plane waves

(4) ρ(x) =

R∑r=1

n∑j=1

Ar exp [i(Gr,j · x + φr,j)] ,

and Ar is the (real) amplitude of the “ring” of reciprocal lattice vectors (defined below)

indexed by r; Gr,j is the j-th reciprocal lattice vector in ring r; φr,j is the phase of the j-th

plane wave in ring r; R is the total number of rings employed; and the system has n-fold

rotational symmetry. A ring is defined as a set of reciprocal lattice vectors that have equal

norms (for crystals) or approximately equal norms (for quasicrystal periodic approxi-

mants), and which have a predefined rotational symmetry. In particular, the r-th ring

with elements indexed by j and of wave number Gr is the set {Gr,j : |Gr,j | ≈ Gr,Gr,j ∈

{G}}, where {G} is the set of reciprocal lattice vectors. For example, the first ring of the

6-fold symmetric hexagonal lattice is composed of six non-zero reciprocal lattice vectors

closest to the origin in reciprocal space. The phases φr,j may be chosen to be the same for

all j within ring r without loss of generality, since structures are equivalent up to transla-

tions and phason shifts if the sum of their phases are equal (modulo 2π). For n even, set-

ting equal phases φr,j = φ is indistinguishable from rescaling the amplitudes Ar by cos φ;

36 P. J. Steinhardt

Γ E V Γk, Bloch wave vector

0

0.1

0.2

0.3

0.4

0.5

0.6

ωa/

(2πc)

Complete Bandgap

Γ E V Γk, Bloch wave vector

0

0.1

0.2

0.3

0.4

0.5

0.6

ωa/

(2πc) Complete Bandgap

(a) (b)

Fig. 6. – (Color online) Band structure of the 5-fold symmetric optimized structure, (a), and ofthe 6-fold symmetric optimized structure, (b), both with ε1/ε0 = 6. The path traversed throughthe BZ is from the center (Γ) to the center of an edge (E) to a close vertex (V ) and back to Γ.Here, ω is the frequency, a is the length of a quasiunit cell in (a) and a unit cell in (b), and c isthe speed of light. The quasicrystal lattice parameter a is smaller by a factor of (1/8) comparedto the periodic approximant unit cell. The band gap in (a) is highly isotropic as a result of thehigh symmetry of the structure. Here, the full gap in the 6-fold case (32.5%) is larger than thatin the 5-fold case (30.3%) in spite of the greater isotropy of the latter.

since we are allowing Ar to vary in our optimization scheme, the phases are redundant

variables and can be set to zero. In the optimization procedure, Ar and φr that maximize

the full photonic band gap between two chosen bands were found using the “steepest as-

cent” method (although “simulated annealing” [20] may also be used). For each lattice,

the full gap is optimized around a Bloch wave number on the order of k = π/a, where a

is the lattice parameter (or effective lattice parameter for quasicrystals), i.e., around the

first (effective) Brillouin zone. See ref. [19] for details of the optimization procedure.

The band structures of the optimized 5- and 6-fold configurations are shown in fig. 6

for dielectric contrast ratio 6.0. The 5-fold and other quasicrystal configurations are

treated as periodic approximants for the purposes of using conventional band structure

computational methods. That is, the band structure is computed as if the quasicrystal

configuration repeats over some large length scale (a supercell) and the limit is taken as

the supercell size gets large.

Optimized band gaps for n-fold symmetry are plotted in fig. 7 for each n over the

dielectric contrast range 1–20. The slopes of the band gap curves form a monotonically

decreasing function of the number of Bragg peaks per ring. This is because the gap at

the Brillouin zone edge is (to first order) proportional to the scattering amplitude across

the zone, and scattering power must be spread over a higher number of peaks per ring

for higher n. It is clear from fig. 7 that the higher-symmetry structures have greater gaps

at low contrast.

These TM-only studies confirm the hypothesis that high symmetry can be advantageous

for producing large band gaps under some conditions. In addition, the quasicrystalline

structures have gaps that are more isotropic than those of the crystals for all contrasts


0 5 10 15 20Dielectric contrast,

1 0

-20

0

20

40

60

Gap

(%

)

4 fold6 fold8 fold5 fold12 fold

Fig. 7. – (Color online) Normalized band gaps of optimized density wave structures for rotationalsymmetries n = 4, 5, 6, 8 and 12 as a function of the dielectric contrast of the two phases. Forreasons discussed in the text, structures of high symmetry tend to have larger band gaps forlow contrast, but smaller gaps for high contrast.

due to the fact that their effective Brillouin zones are more circular. A measure of the

isotropy is [19]

(5) I =mink,EBZ {ωh(k)} − maxk,EBZ {ωl(k)}

maxk,EBZ {ωh(k)} − mink,EBZ {ωl(k)},

where the minima and maxima are taken over all k on the boundary of the effective

Brillouin zone, and ωl and ωh refer to frequencies of bands just below and just above the

gap. It is clear from its definition that I ∈ [−1, 1]. When I is at its greatest, the gap is

perfectly isotropic, and it is at its lowest in a homogeneous material. Isotropy is plotted

against dielectric contrast for all rotational symmetries in fig. 8. The quasicrystals all

have higher isotropy than the crystals. I increases monotonically as the effective Brillouin

zone becomes more circular.

4. – Finding optimal complete band gaps for 2D photonic quasicrystals

The previous section explored how the width and isotropy of the optimal TM pho-

tonic band gap varies with symmetry. This section tackles the more challenging case of

optimizing the complete (TM+TE) band gap.

For the case of TM radiation only, the density wave results of the previous section

and other rigorous methods suggest that a nearly optimal photonic band gap is obtained

38 P. J. Steinhardt

0 5 10 15 20Dielectric contrast,

1 0

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Isot

ropy

,

4-fold6-fold8-fold5-fold12-fold

Fig. 8. – (Color online) The degree of isotropy, I (defined in the text), of the optimized dielectricconfigurations plotted against dielectric contrast. The quantity I is bounded from below by −1(for the case of a homogeneous medium) and from above by 1 (for the case where the upperand lower bands are perfectly flat). For all contrasts, quasicrystalline structures have higherisotropy. Note that the 5-fold case has greater isotropy than the 8-fold case because the formerhas an effective BZ that is closer to a circle than that of the latter. These zones are a decagonand an octagon, respectively.

by placing identical dielectric cylinders centered at each point and adjusting the radius.

To obtain the optimal photonic band gaps for TE radiation, where the electric field is

oriented in the plane of the scatters, necessitates a connected network of dielectric with air

pockets in between, as shown in fig. 5. For example, a commonly used configuration for

photonic crystals is inverted compared to the optimal structures for TM photonic band

gaps, i.e., placing an identical air cylinder at each point so as to produce a connected

network of dielectric material. However, in the case of quasicrystalline structures, this

method fails to produce sizable TE photonic band gaps. The main reason is that the

inverted structure has a very non-uniform distribution of dielectric scattering regions

that broadens the distribution of resonances.

As noted in the previous section, the density wave methods cannot be used for finding

an optimal TE (or complete) band gap structure because the optimal configurations in

these cases require many more waves. Exhaustive searches over all possible designs is well

known to be a daunting computational task, despite the recent development of optimiza-

tion methods, such as gradient-based approaches, and evolutionary methods [19, 21, 22].

The major difficulty in solving this inverse problem comes from the relatively large num-

ber of iterations required to achieve an optimal design and the high computational cost of

obtaining the band structure for complex distributions of dielectric materials, as needed

to simulate heterostructures without long-range order. For instance, the evolutionary

algorithms employed in [23] require over 1000 generations of designs to achieve fully

convergence.


Fig. 9. – (Color online) The protocol described in the text joins the vertices of a rhombus tiling(grey points) into a Delaunay triangular lattice (dashed lines) and, then, joins the centroids ofthe triangles into a network of cells whose vertices are trivalent (black segments and points). Toconstruct a photonic material with a complete band gap, the (black) edges are replaced with awall of dielectric of finite thickness and the vertices are replaced with cylinders of finite radius.

For these reasons, the recent development of a simpler design protocol by Florescu

et al. [24] that requires vastly less computational resources to produce a nearly optimal

design for a TE and complete band gap heterostructure is significant. The protocol can

be applied to a crystal or quasicrystal point pattern with any point symmetry.

The optimal pattern for TE modes obtained by the protocol is a planar, continuous

trivalent network (as in the case of the triangular lattice), which can be obtained from

the point pattern using the steps described in fig. 9. Namely, construct a Delaunay

tiling [25] from the original two-dimensional point pattern and follow the steps in fig. 9

to transform it into a tessellation of cells. Then decorate the cell edges with walls (along

the azimuthal direction) of dielectric material of uniform width w and vary the width of

the walls until the maximal TE band gap is obtained.

To obtain designs for complete photonic band gaps, the protocol is to optimally

overlap the TM and TE band gaps by decorating the vertices of the trivalent network

of cell walls with circular cylinders (black circles in fig. 9) of radius r. Then, for any

40 P. J. Steinhardt

Fig. 10. – Optimal photonic crystal and quasicrystal heterostructures and their diffraction pat-terns derived from: (a) a periodic 6-fold symmetric point pattern; (b) a 5-fold symmetric qua-sicrystal (Penrose) tiling; (c) an 8-fold symmetric quasicrystal tiling.

given set of dielectric materials, the maximal complete photonic band gap is achieved by

varying the only two free parameters, w and r.

Although a constrained optimization method like this is not guaranteed to produce the

absolute optimum over all possible designs, in examples where the absolute optimum is

known by rigorous optimization methods [21,22,19], the protocol produces a design whose

band gap is the same within the numerical error using exponentially less computational

resources.

For the optimization of the two degrees of freedom (w and r), the photonic band

structure must be computed as parameters are varied. Since the computational require-

ments are modest, one can use the conventional plane-wave expansion method [26]; we

generate the disordered pattern within a periodic box of size L much greater than the

average interparticle spacing and take the limit as L becomes large.

The photonic band gaps for heterostructures obtained by the protocol turn out to be

equivalent to the fundamental band gap in periodic systems in the sense that the spectral

location of the TM gap, for example, is determined by the resonant frequencies of the

scattering centers [27] and always occurs between band N and N + 1, with N precisely

the number of points per unit cell.

The examples illustrated here are based on the vertices of a Penrose tiling, [28] a

5-fold symmetric pattern composed of obtuse and acute rhombi and on the vertices of an

octagonal tiling, [29] an 8-fold symmetric pattern composed of squares and rhombi. The


protocol-optimized structures, along with a crystal 6-fold structure obtained by the same

protocol, is shown in fig. 10. Note that the quasicrystal diffraction patterns are all point-

like (Bragg peak) and have the symmetry of the real-space structure, with most of the

scattering intensity in a ring of peaks, conditions that are optimal for band gap formation.

TE band gaps: The optimal TE band gap designs correspond to the network of walls

connecting the centroids of the Delaunay tiling, as shown in fig. 10. In the case of the

5-fold symmetric quasicrystal, the optimum has wall width w/a = 0.103; the TE band

gap is Δω/ωC = 42.3%, the largest ever reported for a photonic quasicrystal. For the

octagonal quasicrystal, the optimal structure has a dielectric material width w/a = 0.106

along each edge, which produces a TE band gap of Δω/ωC = 39.2%.

The TE band gap formation is analogous to the TM case in that it involves an interplay

between scattering from individual cells and the Bragg scattering of the quasiperiodic

arrangement of scattering planes. As shown in fig. 11, the band gap size is related to the

degree of localization of the photonic states. In this case, the magnetic field (out of the

plane) of the lower band-edge states are concentrated in the air component and in the

dielectric for the upper band-edge states. Similar to the case of TM polarized radiation,

the band gap for the optimal structures always occurs between bands NP and NP + 1,

where NP is now the number of cells in the structure.

Complete band gap: For the complete band gap, the optimal structure consists of placing

a wall along each edge and a cylinder at each trihedral vertex of the network generated

by the protocol. The optimum for the 5-fold symmetric case has cylinder radius r/a =

0.157 and cell wall thickness w/a = 0.042. For the 8-fold symmetric point patterns, the

optimum is r/a = 0.167 and w/a = 0.014. The scattering properties of the individual

scattering centers and cells are again essential in the band gap opening and the complete

band gap occurs always between bands 3NP and 3NP + 1, where NP is the number of

points in the periodic approximant point pattern (there are 3NP total scattering units

in the system, 2NP dielectric disks and NP dielectric cells).

The resulting optimal 5-fold symmetric structure displays complete (TM and TE)

photonic band gap of 16.5% —the first complete band gap ever reported for a photonic

quasicrystal with 5-fold symmetry and comparable to the largest band gap (20%) found

for photonic crystals with the same dielectric contrast. The optimal 8-fold symmetric

structure has a full photonic band gap of 13.5%.

5. – Isotropic disordered photonic materials

The photonic structures considered above and in the literature to date exhibit long-

range translational order and Bragg peak scattering that contribute to the formation

of a band gap. But are these conditions essential to the formation of a band gap? In

this section, we show that the surprising answer is no: it is possible to design isotropic,

translationally disordered photonic materials with large complete photonic band gaps

using the same protocol described in the previous section beginning from a disordered

42 P. J. Steinhardt

Fig. 11. – (Color online) Panels (a)-(d) show the electric-field distribution for a 5-fold symmetricphotonic quasicrystal optimized for TM polarized radiation. The structure consists of dielectriccylinders of radius r/a = 0.177 placed at the vertices of a Penrose tiling and displays a TMphotonic band gap of 36.5%. Lower (a) and upper (c) band edge modes display a well-defineddegree of localization; modes just below the lower band edge (b) and just above the upper bandedge (d) display an extended character. Panels (e)-(h) show the magnetic-field distribution in5-fold symmetric quasicrystalline network optimized via the protocol for TE polarized radiation.The structure consists of trihedral network with wall thickness w/a = 0.102 and displays a TEphotonic band gap of 42.5%. The lower (e) and upper (g) band edge modes display a highdegree of spatial concentration and modes just below the lower band edge (f) and just abovethe upper band edge (h) display an extended character.


point pattern. The examples considered here are two-dimensional, though the same

approach can be used in three dimensions.

Obtaining complete photonic band gaps in dielectric materials without long-range

order is counterintuitive. In fact, the protocol cannot generate a complete photonic band

gap structure from an arbitrary point pattern; for example, it fails in the case of a random

Poisson pattern. Not all disordered points patterns are alike, though. There is a largely

unexplored zoology of disordered patterns distinguished according to how their density

fluctuations scale with volume [24,30].

Of special interest for photonic band gaps are point patterns that are hyperuni-

form [30], where the number variance σ2(R) ≡ 〈N2

R〉−〈NR〉2 within a spherical sampling

window of radius R (in d dimensions) grows more slowly than the window volume for

large R, i.e., more slowly than Rd. Among hyperuniform patterns, there is a further

subclassification. The 2D hyperuniform patterns considered in this section are restricted

to the most hyperuniform subclass in which the number variance grows like the window

surface area for large R, i.e., σ2(R) ∼ AR, up to small oscillations. (For a pattern in d

dimensions, σ2(R) ∼ Rβ must scale with a power β ≥ d − 1.) The coefficient A mea-

sures the degree of hyperuniformity within this subclass: smaller values of A are more

hyperuniform. In reciprocal space, hyperuniformity corresponds to having a structure

factor S(k) that tends to zero as the wave number k = |k| tends to zero (omitting for-

ward scattering), i.e., infinite wavelength density fluctuations vanish. As shown below,

patterns can be further subclassified according to the behavior of S(k) as k approaches

zero, which is important in optimizing their photonic properties.

All crystal and quasicrystal point patterns are hyperuniform, but it is considerably

more difficult to construct disordered hyperuniform point patterns. Recently, a collective

coordinate approach has been devised to explicitly produce point patterns with precisely

tuned wave scattering characteristics (that is to say, tuned S(k) for a fixed range of wave

numbers k), including a large class of disordered ones [30].

The motivation for considering hyperuniformity for photonics is the strong correlation

between the degree of hyperuniformity (smallness of A) for a variety two-dimensional

crystal structures as measured in ref. [31] and the resulting band gaps. For example, a

triangular lattice of parallel cylinders has the smallest value of A and the largest band

gap for light polarized with its electric field oscillating normal to the plane, whereas a

square lattice of cylinders has a larger value of A and a smaller photonic band gap.

Among disordered patterns, the most hyperuniform examples known are stealthy, so-

named because they are transparent to incident radiation (S(k) = 0) for a finite range of

wave numbers k < kC for some positive kC . Stealthy hyperuniform patterns are parame-

terized by kC or, equivalently, χ, the fraction of wave numbers k within the Brillouin zone

that are set to zero; as χ increases, kC and the degree of hyperuniformity increase, thus,

decreasing A in our definition of the number variance. Figure 12 presents four designs

of photonic structures with different values of χ and their structure factors, S(k).

When χ reaches a critical value χC (≈ 0.77 for two-dimensional systems) the pat-

tern develops long-range translational order, so the disordered patterns are restricted to

smaller χ.

44 P. J. Steinhardt

Fig. 12. – Four designs of isotropic dielectric heterostructures and their structure factors, S(k).Only panel (d) exhibits a complete photonic band gap. Panel (a) is a disordered network designderived from a Poisson (p = 2, non-hyperuniform) point pattern. Panel (b) shows a networkderived from a nearly hyperuniform equiluminous point pattern in which the structure factorS(k → 0) = S0 > 0 for k < kC . Panel (c) shows a network derived from a RSA point patternin which the structure factor S(k → 0) > 0 but there is more local geometric order than in(b). Panel (d) is derived from an isotropic, disordered, stealthy hyperuniform pattern withS(k) precisely zero within the inner disk. Note the two concentric shells of sharply increaseddensity just beyond the disk. These features sharpen as the ordering parameter χ increases;this trend coincides in real space with the exclusion zone increasing around each particle andthe emergence of complete photonic band gaps.

The largest photonic band gaps in hyperuniform patterns occur in the limit of large

dielectric contrast; For the purposes of illustration, the results below assume that the

photonic materials are composed of silicon (with dielectric constant ε = 11.56) and air.

For this dielectric constrast, a significant band gap begins to open for the stealthy hyper-

uniform designs for sufficiently large χ ≈ 0.35 (but well below χC), at a value where there

emerges a finite exclusion zone between neighboring points in the real space hyperuniform

pattern. That is, unlike a Poisson pattern which allows the spacing between neighboring

points to be arbitrarily small, there is a positive lower bound to the separation in hy-

peruniform patterns with χ > 0.35. The structures built around stealthy hyperuniform

patterns with χ = 0.5 are found to exhibit remarkably large TM (of 36.5%) and TE (of

29.6%) photonic band gaps, making them competitive with many of their periodic and

quasiperiodic counterparts. More importantly, there are complete photonic band gaps of

appreciable magnitude reaching values of about 10% of the central frequency for χ = 0.5.

A striking feature of the photonic band gaps is their isotropy. Using the isotropy

measure introduced earlier for quasicrystals, one finds that the most isotropic crystal

band gap has a variation of 20%, compared to less than 0.1% for the hyperuniform

disordered pattern in fig. 12d.


Even with hyperuniformity, the appearance of photonic band gaps in disordered struc-

tures presented is counterintuitive. There is not yet a rigorous explanation, but the

current conjecture, based on numerical evidence and physical arguments, is that com-

plete photonic band gaps can occur in disordered systems that exhibit a combination of

three mathematical properties: hyperuniformity, uniform local topology, and short-range

geometric order.

First, consider the evidence provided by numerical experiments to date. Photonic

crystals are hyperuniform (an automatic consequence of periodicity) and the known ex-

amples with the largest TM, TE and complete photonic band gaps satisfy the other two

conditions [22, 32, 33]. Numerical experiments indicate that hyperuniformity is a crucial

condition. For example, fig. 12b compares results for the hyperuniform pattern with net-

works generated from non-hyperuniform Poisson point patterns with p = 2, as in fig. 12a;

equiluminous point patterns with S(k → 0) > 0 for k < kC , where the non-zero constant

S(0) is made very small, as shown in fig. 12c; and with a random-sequential absorption

(RSA) point pattern [34] generated by randomly, irreversibly and sequentially placing

equal-sized circular disks in a large square box with periodic boundary conditions subject

to a non-overlap constraint until no more can be added. (It has been shown that such

two-dimensional RSA packings have S(k → 0) slightly positive at k = 0 and increas-

ing as a positive power of k for small k [35].) The latter two patterns are very nearly

hyperuniform presenting similar deviations from hyperuniformity, Se-lum(k → 0) = 0.05

and SRSA(k → 0) = 0.053; and the RSA network in fig. 12c exhibits uniform topological

order (trivalency) and well-defined short-range geometric order; furthermore, these two

patterns produce TM and TE band gaps separately. Yet, none of the three families of

non-hyperuniform patterns has been found to yield a complete photonic band gap.

Note also that hyperuniform stealthy patterns with χ < 0.35 (and keeping all other

parameters fixed) do not produce sizable complete photonic band gaps while those with

χ > 0.35 do. They are all hyperuniform, but they have different degrees of short-

range geometric order, the variance in the near-neighbor distribution of link lengths and

the distance between centers of neighboring links. This is evidence that short-range

geometrical order is essential, as well.

In principle, hyperuniformity and short-range geometrical order can be varied inde-

pendently, but it is notable that patterns with the highest degree of hyperuniformity

also possess the highest degree of short-range geometrical order and that, for the case of

stealthy patterns, both hyperuniformity and short-range geometric order increase as χ

increases.

To understand how hyperuniformity and short-range geometric order, when combined

with uniform local topology, can lead to a complete photonic band gap, consider that

the band gaps arise in the limit of large dielectric constant ratio. In this limit and for

the optimal link widths and cylinder radii, the interaction with electromagnetic waves is

in the Mie scattering limit. At frequencies near the Mie resonances (which coincide with

the photonic band gap lower band edge frequencies), the scattering of TM electromag-

netic waves in a heterostructure composed of parallel cylinders is similar to the scattering

of electrons by atomic orbitals in cases where the tight-binding approximation can be

46 P. J. Steinhardt

reliably applied [36]. The same applies for TE modes for any one direction k if, instead

of parallel azimuthal cylinders, there are parallel thick lines (or walls in the azimuthal

direction) in the plane and oriented perpendicular to k; however, to obtain a complete

band gap, some compromise must be found to enable band gap for all directions k. A

conjecture, based on comparison with rigorous optimization results, numerical experi-

ments, and arguments below, is that uniform local topology is advantageous for forming

optimal band gaps. In two dimensions, this is easiest to achieve in disordered structures

without disrupting the short-range geometric order if the networks are trivalent.

If the arrangement of dielectrics has local geometric order (the variance in link lengths

and inter-link distances is small), the propagation of light in the limit of high dielectric

constant ratio is described by a tight-binding model with nearly uniform coefficients. In

the analogous electronic problem, Weaire and Thorpe [37] proved that band gaps can ex-

ist in continuous random tetrahedrally coordinated networks, commonly used as models

for amorphous silicon and germanium. In addition to tight binding with nearly uni-

form coefficients, the derivation required uniform tetrahedral coordination. (Weaire and

Thorpe call networks satisfying these conditions topologically disordered.) The analogy

in two dimensions is a trivalent network. Although their proof discussed three dimen-

sions and tetrahedral-coordination specifically, we find that it can generalize to other

dimensions and networks with different uniform coordination. Note that our protocol

automatically imposes uniform topology (e.g., trivalency in two dimensions) and limits

variation of the tight-binding parameters by imposing local geometric order.

To complete their proof, Weaire and Thorpe added a mild stipulation that the density

has bounded variation, defined as the condition that the density remains between two

finite values as the volume is taken to infinity. This condition is satisfied by any homoge-

neous system, hyperuniform or not, and thus is much weaker than hyperuniformity, for

which σ2(R) = AR in the stealth two-dimensional examples. Bounded variation is not

sufficient for the photonics problem, though. A gap is needed simultaneously for both

TM and TE, and the gap centers must have values that allow an overlap. Also, the goal

is not simply to have a gap, but to have the widest gap possible. The evidence shows

that hyperuniformity is highly advantageous (perhaps even essential) for meeting these

added conditions.

The comparison to electronic band gaps is also useful in comparing states near the

band edges and continuum. For a perfectly ordered crystal (or photonic crystal), the

electronic (photonic) states at the band edge are propagating such that the electrons

(electromagnetic fields) sample many sites. If modest disorder is introduced, localized

states begin to fill in the gap so that the states just below and just above are localized.

Although formally the disordered heterostructures do not have equivalent propagating

states, an analogous phenomenon occurs. In the upper four panels of fig. 13, we compare

the azimuthal electric-field distribution for modes well below or well above the band gap

(upper two panels), which one might call extended since the field is distributed among

many sites; and then modes at the band edges, which are localized.

The formation of the TM band gap is closely related to the formation of electromag-

netic resonances localized within the dielectric cylinders (fig. 13(a) and (b)) and that


Fig. 13. – (Color online) (a), (b), (c) and (d): Electric-field distribution in hyperuniform dis-ordered structures for TM polarized radiation. The structure consists of dielectric cylinders(radius r/a = 0.189 and dielectric constant ε = 11.56) in air arranged according to a hyperuni-form distribution with χ = 0.5 and displays a TM photonic band gap of 36.5% of the centralfrequency. (a) Localized and (b) extended modes around the lower photonic band gap edge,and (c) localized and (d) extended modes around the upper photonic band gap edge. (e), (f),(g), and (h): Magnetic-field distribution in hyperuniform disordered structures for TE polarizedradiation. The structure consists of trihedral network architecture (wall thickness w/a = 0.101and dielectric constant ε = 11.56) obtained from a hyperuniform distribution with χ = 0.5and displays a TE photonic band gap of 31.5% of the central frequency. (e) Localized and (f)extended modes around the lower photonic band gap edge, and (g) localized and (h) extendedmodes around the upper photonic band gap edge.

48 P. J. Steinhardt

there is a strong correlation between the scattering properties of the individual scatter-

ers (dielectric cylinders) and the band gap location. In particular, the largest TM gap

occurs when the frequency of the first Mie resonance coincides with the lower edge of the

photonic band gap [27]. Analogous to the case of periodic systems, the electric field for

the lower band edge states is well localized in the cylinders (the high dielectric compo-

nent), thereby lowering their frequencies; and the electric field for the upper band edge

states is localized in the air fraction, increasing their frequencies (see fig. 13(c) and (d)).

As shown in fig. 13(e),(f),(g) and (h), an analogous behavior occurs for the azimuthal

magnetic-field distribution for TE modes: for the lower edge state, the azimuthal mag-

netic field is mostly localized inside the air fraction and presents nodal planes that pass

through the high index of refraction fraction of the structure, while the upper edge state

displays the opposite behavior.

6. – Discussion

The principal lesson from these lectures is that complete photonic band gaps are

possible for a much wider range of dielectric structural designs than previously consid-

ered. The possibilities have been extended here to quasicrystals with arbitrary rotational

symmetry and a class of isotropic disordered structures.

Although photonic crystals have the largest complete band gaps (consistent with

their being the most hyperuniform), quasicrystalline and disordered hyperuniform het-

erostructures with substantial complete photonic band gaps offer advantages for many

applications. Both are significantly more isotropic, which is advantageous for use as

highly efficient isotropic thermal radiation sources and waveguides with arbitrary bend-

ing angle. The properties of defects and channels [38] useful for controlling the flow of

light are different for crystal, quasicrystal and disordered structures. Quasicrystals, like

crystals, have a unique, reproducible band structure; by contrast, the band gaps for the

disordered structures have some modest random variation for different point distribu-

tions. Also, light with frequencies above or below the band edges are propagating modes

that are transmitted through photonic crystals and quasicrystals but are likely localized

modes in the case of hyperuniform disordered patterns, which give the former advantages

in some applications, such as light sources. On the other hand, due to their compatibil-

ity with general boundary constraints, photonic band gap structures based on disordered

hyperuniform patterns can provide a flexible optical insulator platform for planar optical

circuits. Moreover, eventual flaws that could seriously degrade the optical characteristics

of photonic crystals and perhaps quasicrystals are likely to have less effect on disordered

hyperuniform structures, therefore relaxing fabrication constraints.

Finally, we note that the lessons learned here have broader physical implications.

One is led to appreciate that all non-crystalline solids are not the same: as methods

of synthesizing solids and heterostructures advance, it will become possible to produce

different types and degrees of hyperuniformity, and, consequently, many distinct classes

of materials with novel photonic, electronic, phononic, plasmonic, shear wave and other

physical properties. The subject is in its infancy, with many possible directions to explore.


∗ ∗ ∗

These lectures are based on what I have learned about photonics from working with

a wonderful group of collaborators: S. Torquato, M. Florescu, P. Chaikin, M.

Rechtsman, W. Man, and M. Megens, who have been crucial in developing the ideas

and results described above. The figures and much of the text draw heavily from our

papers together. I would like to thank the Directors, D. Wiersma and R. Kaiser, for

inviting me to take part in this Summer School; I learned a great deal from the lectures

and from the outstanding students. I would also like to thank B. Alzani and her staff

for making my stay in Varenna so comfortable and memorable; and the Societa Italiana

di Fisica for funding this important and influential series of schools. This work was

supported in part by the National Science Foundation under Grant No. DMR-0606415

and by the National Science Foundation MRSEC program through New York University

under Grant No. DMR-0820341.

REFERENCES

[1] John S., Phys. Rev. Lett., 58 (1987) 2486.[2] Yablonovitch Y., Phys. Rev. Lett., 58 (1987) 2059.[3] Joannopoulos J. D., Villeneuve P. R. and Fan S., Solid State Commun., 102 (1997)

165.[4] Altug H., Englund D. and Vuckovic H., Nature Phys., 2 (2006) 484.[5] EI-Kady I., Taha M. M. R. and Su M. F., Appl. Phys. Lett., 88 (2006) 253109.[6] Chutinan A., John S. and Toader O., Phys. Rev. Lett., 90 (2003) 123901.[7] Schectman D., Blech I., Gratias D. and Cahn J. W., Phys. Rev. Lett., 53 (1984)

1951.[8] Levine D. and Steinhardt P. J., Phys. Rev. Lett., 53 (1984) 2477.[9] Steinhardt P. J. and Ostlund S., The Physics of Quasicrystals (World Scientific,

Singapore) 1987.[10] Janot C., Quasicrystals: A Primer (Oxford University Press, Oxford) 1994.[11] Bindi L., Steinhardt P. J., Yao N. and Lu P., Science, 324 (2009) 1306.[12] Freedman B., Bartal G., Segev M., Lifshitz R., Christodoulides D. N. and

Fleischer J. W., Nature, 440 (2006) 1166; Freedman B., Lifshitz R., Fleischer

J. W. and Segev M., Nat. Mater., 6 (2007) 776.[13] Man W., Megens M., Steinhardt P. J. and Chaikin P. M., Nature, 436 (2005) 993.[14] Chan Y. S., Chan C. T. and Liu Z. Y., Phys. Rev. Lett., 80 (1998) 956.[15] Dal Negro L., Oton C. J., Gaburro Z., Pavesi L., Johnson P., Lagendijk A.,

Righini R., Colocci M. and Wiersma D. S., Phys. Rev. Lett., 90 (2003) 055501.[16] Cheng S. S. M., Li L., Chan C. T. and Zhang Z. Q., Phys. Rev. B, 59 (1999) 4091.[17] Kaliteevski M. A., Brand S., Abram R. A., Krauss T. F., Millar P. and De La

Rue R. M., J. Phys.: Condens. Matter, 13 (2001) 10459.[18] Zeng X. C., Bergman D. J., Hui P. M. and Stroud D., Phys. Rev. B, 38 (1988) 10970.[19] Rechtsman M. C., Jeong H.-C., Chaikin P. M. and Torquato S., Phys. Rev. Lett.,

101 (2008) 073902.[20] Kirkpatrick S., Gelatt C. D. jr. and Vecchi M. P., Science, 220 (1983) 671.[21] Kao C., Osher S. and Yablonovitch E., Appl. Phys. B, 81 (2003) 235.[22] Sigmund O. and Hougaard K., Phys. Rev. Lett., 100 (2008) 153904.

50 P. J. Steinhardt

[23] Preble S., Lipson H. and Lipson M., Appl. Phys. Lett., 86 (2005) 061111.[24] Florescu M., Torquato S. and Steinhardt P. J., Proc. Natl. Acad. Sci. U.S.A., 106

(2009) 20658.[25] Preparata F. R. and Shamos M. I., Computational Geometry: An Introduction

(Springer-Verlag, New York) 1985.[26] Johnson S. G. and Joannopoulos J. D., Opt. Express, 8 (2001) 173.[27] Rockstuhl C., Peschel U. and Lederer F., Opt. Lett., 31 (2006) 1741.[28] Penrose R., Bull. Inst. Math. Appl., 10 (1974) 266.[29] Grunbaum B., Grunbaum Z. and Shephard G., Comput. Math. Appl., 12 (1986) 641.[30] Batten R. D., Stillinger F. H. and Torquato S., J. Appl. Phys., 104 (2008) 033504.[31] Torquato S. and Stillinger F. H., Phys. Rev. E, 68 (2003) 041113.[32] Fu H., Chen Y., Chen R. and Chang C., Opt. Express, 13 (2005) 7854.[33] Chan C. T., Datta S., Ho H.-M. and Soukoulis C., Phys. Rev. B, 50 (1994) 1988.[34] Torquato S., Random Heterogeneous Materials: Microstructure and Macroscopic

Properties (Springer-Verlag, New York) 2002.[35] Torquato S., Uche O. U. and Stillinger F. H., Phys. Rev. E, 74 (2006) 061308.[36] Lidorikis E., Sigalas M. M., Economou E. N. and Soukoulis C. M., Phys. Rev. Lett.,

81 (1998) 1405.[37] Weaire D., Phys. Rev. Lett., 26 (1971) 1541; Weaire D. and Thorpe M. F., Phys. Rev.

B, 4 (1971) 2508.[38] See, for example, Socolar J. E. S., Lubensky T. C. and Steinhardt P. J., Phys. Rev.

B, 34 (1986) 3345; see also ref. [9].


Diffuse optics: Fundamentals and tissue applications

R. C. Mesquita and A. G. Yodh

Department of Physics & Astronomy, University of Pennsylvania

209 South 33rd Street, Philadelphia, PA 19104-6396, USA

Summary. — The material in this paper is different from the mainstream topicsin this summer’s International School of Physics “Enrico Fermi”. It should becomeapparent, however, that the roots of these biomedical optics research problems sharecommon features with much of the light scattering and transport research taught inthe Varenna summer school. Here, our intention is to provide an informal review thatestablishes the roots of diffuse optics, and then demonstrates how diffuse optics isfinding application in medicine. This paper will have two main themes. After a briefmotivation of the problem, the first theme will provide a coherent discussion aboutlight transport in turbid media. The second theme is oriented towards problemsin biomedicine. As such, a short discussion of hemodynamics will be followed byrepresentative current work from our lab, particularly with breast and brain.

1. – Introduction

The dream of optics for in vivo biopsy has been with us in various contexts for many

years, and it continues to pop up in popular culture, particularly science fiction. Famous

examples come from the TV show Star Trek, wherein Dr. McCoy uses a “tricorder” device

to assess the condition of a patient, and from movies such as Minority Report, wherein

fiber optics pick up brain signals from special patients. In most cases, an instrument


52 R. C. Mesquita and A. G. Yodh

Fig. 1. – Illustration of a diffuse optical measurement on a subject’s arm. One source fiber andone detector fiber are shown, separated by approximately 2 cm.

shines light into the body and/or collects light from the body in transmission or reflection;

then a scientist employs the data to make a rapid diagnosis.

Without thinking too hard about the details, one might believe that such measure-

ments are possible. For example, if we use light in the near-infrared (700–950 nm) rather

than the uv, visible or mid- and far-infrared parts of the spectrum, then human tis-

sue has a window of low absorption. Within this window, the absorption of oxy- and

deoxy-hemoglobin is falling to zero, and the absorption of water has not started to in-

crease significantly. Thus, light from such near-infrared devices can penetrate deeply in

tissue. Furthermore, since each tissue chromophore has distinct spectral features, one

can readily envision using light transmission properties as a function of wavelength to

acquire sensitivity to tissue physiology, particularly to blood dynamics and edema. It

has thus turned out that, besides the obvious convenience of such a device for continuous

non-invasive measurement at the bedside, optical contrasts are complementary to other

kinds of medical diagnostics such as, for example, X-ray and ultrasound.

A typical diffuse optical measurement is shown in fig. 1. This image reveals what we

mean by “deep tissue”, i.e., big chunks of tissue located millimeters to centimeters below

the tissue surface. Typically an optical fiber, coupled to a light source such as a diode

laser, injects light into the tissue at the air-tissue interface, and a second optical fiber

collects remitted or transmitted light at another position on the air-tissue surface. In

practice, we vary source-detector position, light wavelength, light modulation, and even

the mode of light detection in order to derive physiological information about the tissue

in real time.

Currently, two limiting versions of this basic scheme are employed. One approach

is diffuse optical imaging. In this case, many source-detector pairs are placed on the

tissue surface, and the scientist attempts to reconstruct images of optical and physio-

logical properties in each of many volume elements (or voxels) within the tissue interior

based on measurements at the tissue surface. This imaging or tomographic scheme has

been particularly prevalent in breast imaging. The second approach is tissue monitoring.

Diffuse optics: Fundamentals and tissue applications 53

Here, a probe with few source-detector pairs is typically placed on (or near) tissues of

interest, and average properties of the underlying tissues are derived by fitting reflec-

tion/transmission data to simple light diffusion models. This monitoring approach has

been employed extensively in clinical studies of brain, muscle and tumor.

2. – Light transport tools

We now begin our discussion of light transport in tissues. In particular, we will intro-

duce key physical and physiological parameters, and we will discuss important underlying

concepts. We aim to facilitate understanding of the light diffusion problem in tissues,

with minimal use of mathematics. Readers interested in more formal discussions can

consult a variety of recent reviews [1-6] and references therein; a rather complete review

from our group, i.e., one that fleshes out the mathematics of many topics to be discussed

herein, should be available soon in the journal Reports on Progress in Physics.

The natural starting point for our discussion is the optically thin or single-scattering

sample. Typically such samples consist of molecules that can absorb light, and relatively

larger particle-like objects that scatter light significantly. In the typical “traditional

optics” experiment, the sample is illuminated with an incident light field (or incident

light intensity), and we are concerned with how much light remains in the beam after

the light traverses a distance L straight through the sample.

2.1. Absorption. – The best known attenuation effect from traditional optics is light

absorption (fig. 2a). Absorption is due to molecular chromophores in the sample and

is characterized by an absorption coefficient, μa. The classic result for a transmission

experiment of this kind is a law which states that the input light intensity is attenuated

exponentially with distance traveled through the sample. The absorption coefficient that

characterizes this attenuation depends on the concentration of chromophores in the sam-

ple and the chromophore cross-section or extinction coefficient —which in turn depends

on incident-light wavelength. Thus, by carrying out such an absorption measurement as

a function of input wavelength, one can learn which molecules are present, how many

molecules reside in the sample, and one can even learn very subtle details about the

molecule’s local environment via spectral shifts or spectral broadening. In the case of

tissue optics in the near infrared, the most important endogenous molecules are oxy- and

deoxy-hemoglobin, water and lipid.

2.2. Scattering . – A second important effect in traditional optics is scattering [7]

(fig. 2b). The scattering effect is characterized by several variables. The first param-

eter, by analogy with absorption, is the scattering coefficient, μs, which is essentially

the exponential decay rate of light intensity with distance traveled in the sample due to

scattering effects. The scattering coefficient depends on the concentration of scattering

particles and on the total scattering cross-section of these particles. This total scat-

tering cross-section is generally wavelength-dependent (e.g., depending on the particle

size, the particle index of refraction, and on light wavelength), although the scattering

cross-section usually has a much weaker wavelength dependence than typical molecular


Fig. 2. – Schematic representation of different effects that light encounters when traversing anoptically thin sample: (a) absorption, (b) scattering, and (c) dynamic scattering.

absorption spectra. Scattering experiments, however, can offer more information to the

experimenter. Though the incident light is not transmitted, it is also not lost. Rather,

it is sent off into different angular directions. This angular information can be measured

and is quantified by the so-called differential scattering cross-section (or scattering am-

plitude). Note that the integral of the differential scattering cross-section over all angles

equals the total scattering cross-section. The amount of light scattered into a particular

solid angle is thus proportional to the product of the incident light beam intensity and

the differential scattering cross-section at that angle.

Another factor that will eventually become of interest to us is the so-called anisotropy

factor, g. The anisotropy factor is the average value of the cosine of the scattering

angle for a typical scattering event. If the anisotropy factor, g, is near unity, then light

scattering is nearly forward. If the anisotropy factor is about one half, then scattering is

fairly isotropic and each scattering event is said to randomize the initial photon direction.

We introduce one more term in this context, because it turns out to be critical for light

diffusion: the reduced scattering coefficient, μ′s. The reciprocal of the reduced scattering

coefficient is called the photon random walk step or transport-mean-free pathlength; after

the incident light beam travels a photon random walk step in the scattering medium, only

e−1 of the input light remains in the incident beam. The reduced scattering coefficient

is simply related to the scattering coefficient and the anisotropy factor: μ′s = μs(1 − g).

The reduced scattering coefficient is the microscopic tissue scattering parameter that

survives the diffusion approximations to the linear transport equation, which will be

discussed shortly.


Scattering measurements such as this teach us about the microscopic objects that

cause substantial light scattering. Examples of such objects include particles (e.g.,

polystyrene, silica, PMMA, nanoparticles, etc.), cell organelles (e.g., the mitochondria,

nucleus, etc.), and cells (e.g., red blood cells, etc.). In principle one can learn about

scatterer concentration, about the fluids that surround the scatterers and whose indices

of refraction affect scattering strength, and more. This information is very useful, as was

the case of absorption, and it is complementary to the absorption information.

2.3. Dynamic light scattering. – The final traditional optics topic that we will describe

is called dynamic or quasi-elastic light scattering (i.e., DLS or QELS, respectively) [8-10].

In these experiments, the fluctuations of scattered light intensity (or electric field) are

measured as a function of time (fig. 2c). By analyzing these fluctuations, specifically

their temporal autocorrelation functions, we can learn about the motions of the scatter-

ers. In particular we can learn about how fast these particles are moving, how many

are moving, and in what manner they are moving (e.g., diffusive, ballistic, etc.). In the

typical dynamic light scattering (DLS) experiment, a point-like photon detector collects

light from the sample at an off-axis angle, i.e., an angle different from the input beam

propagation direction. The DLS effect is relatively easy to understand for particle-like

scatterers, and we shall adopt this approximation here. In the presence of the input

light field these particles acquire an oscillating induced-dipole moment. These oscillating

dipoles behave as antennas and re-radiate light into many off-axis angles. When the

particle-like objects move, the relative phases of these re-radiated dipole fields landing

at the detector will vary too, and the re-radiated fields will add both constructively

and destructively, depending on the particle configuration in the sample. Thus the to-

tal detected electric field (and intensity) will vary in time, and intensity fluctuations

are readily observed. These fluctuations of the electric field strength and light intensity

carry information about the dynamic properties of the medium. Specifically, they carry

information about the mean-square displacements of the particles. The normalized tem-

poral autocorrelation function of the scattered electric field, or the analogous temporal

intensity autocorrelation function, is measured in the DLS experiment. The decay rate

of the autocorrelation function depends on the particle motion; larger motions give faster

autocorrelation function decay rates. Quantitatively, the decay rate depends on the par-

ticle mean-square displacement (during the autocorrelation time interval), the scattering

angle, and the incident light wavelength. Often, the DLS autocorrelation function will

decay by e−1 when the particles move a distance of about one optical wavelength.

2.4. Multiple light scattering in tissues. – We have seen that the basic techniques of

traditional optics probe both scattering and absorption, and both effects lead to expo-

nential attenuation of the incident beam. Furthermore, the scattered beam can teach us

even more about the nature of the scatterer and its motions, e.g., if we measure other

quantities such as the temporal autocorrelation function. These traditional techniques

are rigorous and have been tested to a very substantial degree. The methods work.

Tissues are more complicated. Tissue samples are not optically thin. Tissues multiply

scatter light. One can envisage the transport of photons through tissue as a sequence


Fig. 3. – Pictorial representation of a light path due to multiple scattering in tissue.

of single scattering events (fig. 3). The light traverses the sample like a random walker.

Incident light travels some distance into the sample and then scatters, and then it travels

some distance along its new propagation direction and scatters again, and so on —many

times over before it eventually emerges from the sample. In this case, absorption, scat-

tering and correlation effects are seemingly scrambled together. Furthermore, the total

pathlength travelled by the incident light is many times the sample size. Thus, if we

want to learn something about these media, then we have to learn how to unscramble

this information from the emerging light fields.

The use of Maxwell’s full electromagnetic theory to address such a complex problem,

in general, is extraordinarily difficult [11]. Fortunately, there exists a useful approx-

imation called linear transport theory, which can deal with many aspects of the light

transport problem in highly scattering media [12, 13]. Linear transport theory is not a

perfect theory for the problem. For example, it ignores most (but not all) of the wave

aspects of the light fields. However, it turns out that the theory provides a very useful

starting point for many problems of light propagation in tissues. The key physical quan-

tity in the theory is called the radiance. The radiance is basically the light power per area

per solid angle at position r and time t in the sample, traveling in a particular direction,

Ω. The radiance scales as the absolute square of the light field at r, t and traveling in

direction Ω. Linear transport theory balances the radiance in each small volume of the

multiply scattering medium. This balance produces an equation for radiance, i.e., the

transport equation, that can, in principle, be used to solve for radiance throughout the

sample given some spatio-temporal distribution of absorption and scattering coefficients,

some distribution of differential scattering cross-section, boundary conditions, and initial

conditions. Of course, the equation is non-trivial to solve, and closed-form solutions only

exist for very simple geometries and conditions.

The physics behind the mathematics in linear transport theory becomes more evident

within a first-order approximation of the radiance balance equation. In this case the

diffusive nature of light transport becomes apparent. We will briefly outline some of the

steps involved in going from the fully general transport equation to the light diffusion

equation. In this regime, the two simplest and most important physical quantities associ-


ated with the radiance are the photon fluence rate and the photon flux. The fluence rate

at position r is a scalar function derived by integrating the radiance over a small sphere

centered on r. It is the isotropic part of the radiance. The photon flux is a vector. The

photon flux is also derived from an integration of the radiance over the solid angles, but

in this case it is derived from a vector integration that takes explicit account of the vector

nature of the radiance. The photon flux is the lowest-order anisotropic contribution to

the radiance.

To solve the transport equation, typically one carries out the so-called PN approxima-

tion. In the PN approximation the radiance, the source distribution, and the differential

scattering function are all expanded in terms of spherical harmonics or Legendre poly-

nomials. These expansions are then stuffed back into the transport equation and a set

of equations of different order are thus derived. Interestingly, it is straightforward to

show that the fluence rate depends only on the zeroth-order spherical harmonic, and

the flux depends only on the first-order spherical harmonics. Thus, in the lowest order,

i.e., in the so-called P1 approximation where the expansions are truncated at first order,

the radiance can be expressed precisely in terms of the fluence rate and flux. With a

little more work, the transport equation reduces to the photon diffusion equation for the

fluence rate. We write out the photon diffusion equation in its full glory below:

(1) ∇ · (D(r)∇Φ(r, t)) − vμa(r)Φ(r, t) + vS(r, t) =∂Φ(r, t)

∂t.

Here Φ(r, t) is the photon fluence rate (Joules/cm2 s), v is the velocity of light in the

medium (cm/s), and S(r, t) represents an isotropic source term proportional to the

number of photons emitted (at point r and time t) per unit volume per unit time

(Joules/cm3 s). The main new parameter in the problem is the photon diffusion coeffi-

cient D (cm2/s); D(r) = v/3(μ′s+μa), and (for small absorption) it depends primarily on

the reduced scattering coefficient, the reciprocal of which is the photon random walk step.

The analysis also shows explicitly that the photon flux is proportional to the gradient of

the fluence rate, as in standard diffusion problems.

We therefore arrive at the key mathematical result of this paper. It is worthwhile

at this point to consider some of the assumptions that went into this analysis, many of

which we have glossed over. First we assumed that the scattering length, (μs)−1, is much

less than the absorption length, (μa)−1, an assumption that is fine for the vast majority

of tissues. Second, we implicitly assumed that the fluence rate is significantly larger than

the flux; this approximation is generally fine, but it can break down near boundaries

and sources. We have assumed isotropic sources, an assumption that is reasonable as

long as we do not make measurements within a random walk step of the source. We

have assumed that the scattering angle of a typical scattering event does not depend

on incident angle, i.e., it depends only on the cosine of the angle between input and

output wave vectors. Finally, an assumption about the rate of change of the flux has

been made, which amounts to requiring that the time scale of source modulation is much

longer than the time between photon scattering events. Generally these assumptions


Fig. 4. – (a) Experimental setup for measurements in “ideal” turbid media, an aquarium filledwith milky Intralipid. A source fiber injects light into the medium near its center, and a movabledetector fiber measures amplitude and phase of the diffusive waves at different positions insample. (b) Constant phase contours of diffusive wave in the aquarium. Notice, the contoursare circular and emanate from the source. Inset: Measured phase and amplitude (in logarithm)as function of the source-detector separation. (c) Propagation through two different media; themedium containing source (S0) has larger scattering then the other medium. Refraction effectsare demonstrated. (d) Diffraction/scattering of diffusive wave due to an absorbing sphere ofdiameter a in the turbid medium [14,15].

are fine for tissues. The photon random walk step in most tissues is about 1mm and

the absorption lengths are on the order of 5 to 10 cm. It is important, however, that

our applications do not demand great precision from our measurements, otherwise the

affects of so many relatively innocuous approximations can wash out the effects of the

physiological perturbations we seek to probe.

2.5. Simple solutions of the photon diffusion equation. – In order to use the diffusion

equation for light, we must understand its solutions. We start with ideal case of infinite,

homogenous, turbid media. A good example of such a medium is an aquarium full

of milk (or Intralipid) and ink (see fig. 4a). This kind of tissue phantom sample can

be adjusted to have properties similar to tissues, without the clinical complications.

Working in the frequency domain, the source is amplitude-modulated at some frequency

ω (e.g., 100 MHz), and we look for solutions that oscillate at this same frequency. In this

case, the diffusion equation reduces to a very simple standard differential equation which

will give spherical wave solutions with a complex wave number that depends on sample

absorption and scattering, and on source modulation frequency. For a point oscillating

source, the solution to this differential equation is the Green’s function of the diffusion

equation for the infinite homogenous geometry. Notice (in fig. 4b insets) that the wave

attenuates exponentially with distance from the source, and that the phase of the wave

disturbance has an associated wavelength that depends on scattering, absorption and

frequency. The fundamental photon density disturbance is thus a kind of overdamped

wave. We call these disturbances diffuse photon density waves, or diffusive waves. It


turns out that these disturbances of light energy density behave in many ways like the

regular waves we know and love, except that properties such as disturbance wavelength

and medium effective index of refraction depend on factors such as the photon random

walk step.

Some examples of measurements we made to confirm these ideas using point sources

and detectors in an aquarium filled with Intralipid and ink are given in fig. 4b-d. In these

experiments, one source fiber is employed to launch a diffuse photon density wave (or

diffusive wave) modulated at 200 MHz into the medium. Another fiber coupled to a pho-

todetector measures the phase and amplitude of the diffusive wave at different positions

in the sample. In fig. 4b, one can see a nice spherical diffusive wave, whose phase and

amplitude variation with respect to source-detector separation can be used to obtain the

absorption and reduced scattering coefficients of the turbid medium. In the third figure

we divide the tank into two media with differing photon random walk steps. Refraction

type effects are apparent in this case. The last figure shows diffraction/scattering effects.

Thus far we have used a frequency domain picture to analyze the light propagation

problem in simple turbid media. We can also solve the problem in the time domain.

In this case, we seek the Green’s function solutions due to a point source emitting a

very short duration pulse in the infinite media. The result is a broadened light pulse

whose terminal slope is related to sample absorption and whose peak position is related

primarily to the light diffusion coefficient in the sample.

At this point it is worth reflecting on what has been gained. In fact, a lot has been

gained. We now have an experimental means to separate scattering from absorption in

turbid media by measuring the phase and amplitude of diffuse photon density waves or

by measuring the temporal broadening of a very short light pulse in the medium. Thus

we can derive the absorption coefficient and the scattering properties of the medium,

even though the medium is turbid. Next we explore the effects of heterogeneity.

3. – Diffuse Optical Spectroscopy (DOS): monitoring

We first consider piecewise continuous turbid media. A most important example in

practice is the semi-infinite medium, e.g., a tissue medium with an air-tissue interface.

To make progress on this problem, we need boundary conditions for the fluence rate at

the interface. The boundary condition is simple to state. We require that the radiance

coming into the medium at the boundary is equal to the Fresnel reflected radiance that

is traveling outward at the boundary. The result of carrying out this analysis is the

so-called partial flux boundary condition, which relates the fluence rate at the boundary

to its gradient at the same boundary. The proportionality constant in this relationship is

a length which can be explicitly derived and which depends on the indices of refraction of

the surrounding medium. One can solve the light diffusion problem exactly with partial

flux boundary conditions (for the semi-infinite medium), or one can invoke a fairly in-

nocuous approximation which leads to an even simpler boundary condition: the so-called

extrapolated zero-boundary condition. In this case, the true boundary condition is re-

placed by a zero-boundary condition at an extrapolated distance Ls above the boundary,


(b)(a)

Fig. 5. – (a) Schematic showing the probe on the baby and the probe with its sources (◦) anddetectors (×). (b) Measured amplitude and phase as a function of separation for different source-detector combinations (black dots). The line in both plots represents the best fit to the datapoints, and the slopes can be used to derive tissue absorption and scattering coefficients [16].

i.e., the fluence rate vanishes at the extrapolated distance. The zero-boundary condition

problem can then be solved by the classic method of images to derive a simple result for

the predicted diffuse light reflectance from a semi-infinite turbid medium.

Believe it or not, this light reflectance result for semi-infinite homogeneous turbid

media is THE workhorse result of the monitoring field. It performs fairly well in the

clinic, giving average optical properties quite accurately, even though the media are not

truly homogeneous or semi-infinite. As an example, fig. 5 shows a measurement of the

optical properties of a baby brain. We used a pad with many sources and detectors,

and placed it on the baby’s head. Then we measured the diffuse photon density wave

amplitude and phase as a function of source-detector separation. A fit to the data

using the extrapolated zero-boundary condition provides the average tissue scattering

and absorption coefficients at a single optical wavelength.

If the medium is slab-like instead of semi-infinite, it is straightforward to derive other

analytic results. One can also derive results for cylinders, spheres, spheres in slabs,

and more. We will not discuss these examples further. Suffice to say that for simple

monitoring applications, some sort of analytic model can always be found to fit for

measured phase and amplitude data, and, therefore, for deriving average scattering and

absorption coefficients.

Before turning to imaging and tomography, we briefly consider the critically impor-

tant problem of tissue diffuse optical spectroscopy (DOS). Thus far we have discussed

the reflectance problem at a single optical wavelength. Arguably the most important

feature of optics, however, is its potential for spectroscopy. For example, suppose we

assume that there are two chromophores in tissue, oxy- and deoxy-hemoglobin; both of

these chromophores will contribute to the measured absorption coefficient. The relative

amount that each chromophore contributes depends on its extinction coefficient and its

concentration. If we derive tissue absorption coefficients at multiple wavelengths —say

780 nm and 805 nm— then we generate two equations with two unknowns. The two un-

knowns are the concentrations of oxyhemoglobin ([HbO2]) and deoxyhemoglobin ([Hb]).


By solving these equations, using the data and the known chromophore extinction coef-

ficients, we can therefore derive [HbO2] and [Hb]. From this information we can further

derive total hemoglobin concentration ([THC] = [HbO2]+ [Hb]) and tissue blood oxygen

saturation (StO2 = [HbO2]/([HbO2] + [Hb])). [THC] and StO2 are important clinical

quantities that can thus be derived from relatively simple multi-wavelength, diffuse op-

tical reflectance measurements. These reflectance techniques go by the names Diffuse

Optical Spectroscopy (DOS) or Near-Infrared Spectroscopy (NIRS) in the biomedical

optics community.

4. – Diffuse Optical Tomography (DOT): imaging

Suppose the media we seek to understand are very heterogeneous. Furthermore, sup-

pose that these heterogeneities are the quantities which we seek to learn about. How does

one handle this situation? In this case, some kind of tomography or image reconstruction

is desirable [17-19]. Fortunately, it is conceptually straightforward to develop strategies

for this goal, because the heterogeneity problem is basically a problem that has already

been studied in the context of waves and scattering theory. The only difference for us is

that the waves are now diffuse photon density waves. In essence, in our experiments an

incident diffusive wave is launched from each source fiber into the medium and the wave

scatters from local optical property heterogeneities. From measurements of scattered

waves on the tissue surface, one can set up an inverse problem to work backwards from

the “perturbed” wave at the tissue surface to derive the heterogeneous optical properties

within the entire medium.

The standard approach for setting up the problem is to divide the absorption and light

diffusion coefficients into background and heterogeneous pieces, and then insert these

parameters back into the diffusion equation. Then we look for perturbative solutions.

Both the Born perturbation approach and the Rytov perturbation approach [20] will

work as long as scattering of diffusive waves by heterogeneities in the sample is weak.

As a concrete example, let us suppose the tissue optical property variation is only

due to absorption. In this case the scattered wave depends approximately on the value

of the incident wave at the heterogeneity times the strength of the heterogeneity times

a propagator that describes the diffusive wave transport from the heterogeneity to the

detector. The propagator is just the Green’s function for the diffusive wave.

Within this formulation, there will arise an integral equation for every source-detector

pair on the sample surface. The integral is over the entire sample volume. At this point

the sample volume is divided up into discrete-volume elements and the inverse problem

is formulated readily in the language of matrices. Here, a weight matrix, whose elements

are basically the product of Green’s functions, couples a vector of tissue optical properties

to a vector of measurements of the scattered diffusive waves. The absorption unknowns

in each volume element of the sample are represented as a vector; the measured scattered

signals for each source-detector pair are also represented by elements of a measurement

vector. The result is a set of linear equations that can be inverted, for example, by

the singular value decomposition technique [21], to derive an absorption tomogram. In


practice many other techniques can be used to solve this class of problems, though issues

of uniqueness and regularization always crop up. Most people nowadays solve the inverse

problem iteratively, a method of choice when the heterogeneities are not weak. DOT has

been demonstrated to work very well in phantoms. Its use in the clinic is promising and

is a subject of current research.

5. – Diffuse Correlation Spectroscopy (DCS): blood flow

We will discuss one last formal problem before moving to applications: Diffuse Cor-

relation Spectroscopy (DCS). DCS is a multiple-scattering correlation methodology for

measuring blood flow.

First, let us recall the single-scattering version of this problem: dynamic light scat-

tering. In the DLS experiment, a sample is illuminated. Illuminated particles in the

sample act like radiating dipoles. We collect their radiated fields with a photon detec-

tor located off at some scattering angle. The collected fields fluctuate and the intensity

fluctuates too, because the particles move and the relative phases of the radiated dipole

fields change as a result of this motion. Information about scatterer motions is contained

in the electric field and intensity temporal autocorrelation functions. The electric field

autocorrelation function decays exponentially at a rate that depends on how far the scat-

tering particles move in the correlation time interval, i.e., on the particles’ mean-square

displacement in the correlation time interval. Thus, we derive motional information by

measuring this decay rate.

Several schemes can be employed to analyze this problem in the multiple scatter-

ing limit. To keep things formally consistent, we will focus on the transport equation

methodology. Ackerson and coworkers [22, 23] suggested that it should be possible to

understand the propagation of temporal autocorrelation in turbid (and dynamic) media

primarily by replacing radiance in the transport equation with the normalized electric

field temporal autocorrelation function. This replacement is tantamount to exchanging

the function E∗(r, t,Ω)E(r, t,Ω) in the transport equation with E∗(r, t+τ,Ω)E(r, t,Ω);

here E(r, t,Ω) is the electric field of the diffusing light at r, t and traveling in the di-

rection Ω (hereafter we will supress the propagation direction in our notation), and τ is

the autocorrelation function time delay or interval. Then, following essentially the same

logic and mathematics as before, we arrive at a linear transport equation for correlation,

this time with the scattering source term that we have from DLS. We make the P1 ap-

proximation, as before, to convert the correlation transport equation to the Correlation

Diffusion Equation shown below [24,25]:

(2) ∇ · (D(r)∇G1(r, τ)) − v(μa(r) +

α

3μ′

sk2

0〈Δr2(τ)〉

)G1(r, τ) = vS(r, t),

where G1(r, τ) = 〈E∗(r, t + τ)E(r, t)〉 is the unnormalized temporal electric field auto-

correlation function; k0 is the wave vector of the fields in the medium, and 〈Δr2(τ)〉

is the mean-square displacement of the scattering particles in time τ . The brackets 〈〉


(a) (b)

Fig. 6. – (a) Schematic diagram of the cuff ischemia experiment. (b) Intensity temporal auto-correlation curves measured for different cuff pressures. Notice that the decay rate decreases ascuff pressure increases, indicating attenuation of blood flow (obtained from [29]).

denote ensemble averages, or averages, over time. The parameter α represents the frac-

tion of scattering events in the tissue that occur from moving particles, and S(r) is a

source term.

The correlation diffusion equation is a diffusion equation for electric field temporal

autocorrelation! Notice that in the limit that τ goes to zero, this equation reduces to the

conventional diffusion equation for fluence rate in the zero-frequency limit. Formally, this

result (eq. (2)) is the differential equation form of the diffusing wave spectroscopy (DWS)

technique [26-28] that Georg Maret invented and has also talked about in this summer

school. The differential equation form is particularly attractive, however, because almost

all the formal ideas we have discussed with respect to diffuse photon density waves will

apply to diffusing field temporal autocorrelation. That is, the solutions of the correlation

diffusion equation are formally the same as for fluence rate with only an additional

absorption term due to particle motions. Thus, it follows that analogous monitoring and

imaging measurements can be carried out with diffusing correlation.

Figure 6 shows an example of a blood flow measurement, based on cuff ischemia [29].

The measurement of correlation is carried out with source and detector on the forearm,

and it gives an autocorrelation function that decays in time. As the cuff pressure is

increased, the slope of the autocorrelation decay decreases. Thus the slope of this curve

can be used to define a blood flow index (BFI) for clinical experiments. The correlation

function decay rate is used to get the BFI. The BFI, in turn, can be related explicitly

to an effective diffusion constant (not the Einstein Brownian Diffusion constant), that

parameterizes the correlation function decay rate, times the factor alpha, which accounts

for the fraction of scattering events from moving scatterers. We have found that relative

changes in BFI, i.e., rBFI, provide a robust and quite accurate measure of relative blood

flow changes in a broad range of clinical scenarios [30-38].

6. – Background on tissue hemodynamics

Before we describe some physiological applications of diffuse optics, let us briefly re-

call the tissue parameters to which these tools are sensitive. Absorption information

provides access to the tissue concentrations of endogenous chromophores such as HbO2,


Hb, water and lipid, and also to the concentrations and local environments of exogenous

chromophores such as imaging contrast agents and drugs. Scattering information pro-

vides access to organelle concentrations, cell concentrations, and subtle changes in the

properties of background fluids. Correlation spectroscopy provides information about

the movement and flow of scattering “objects” in tissues such as red blood cells. Taken

together, the most prevalent application of these tools is for probing tissue hemodynam-

ics. In this case, absorption gives information about total hemoglobin concentration and

tissue blood oxygen saturation, and correlation spectroscopy gives information about

blood flow down to the tissue microvasculature level. Of course, all of these measure-

ments represent tissue-averaged quantities; the degree of tissue averaging depends on the

source-detector geometry.

The circulatory system is a network of channels of varying diameter with the purpose

to deliver nutrients to tissues and remove waste products of metabolism from tissues [39].

The arterial side of the vasculature is oxygen rich; it takes blood from heart and lungs

to tissues. The venial side of the vasculature is oxygen poor; it takes blood back to the

heart and lungs. The network is branched. The tubes start big, but break into smaller

and smaller tubes until finally (at the oxygen delivery points), the capillaries bring red

blood cells and plasma cells to within a hundred or so microns of every tissue cell in

the body. The arterioles, capillaries and venules therefore fill up a very sizable fraction

of the tissue space. Thus, it is their responses that we are typically measuring with

diffuse optics. Each red blood cell carries many molecules of hemoglobin (oxygenated or

deoxygenated), the primary oxygen carrier.

As we have noted, two forms of hemoglobin are important: oxygenated and deoxy-

genated. These hemoglobin molecules are typically in chemical equilibrium with dissolved

oxygen in the tissue. When dissolved oxygen is low in tissue, then oxygen is unloaded

off the HbO2 molecules and then the oxygen diffuses into the tissue. This equilibrium

between blood oxygen saturation and tissue pO2 is characterized by the classic “Hill

Curve” [39,40]. Remember that diffuse optics measures blood oxygen saturation, which

is closely related to tissue pO2 via the Hill Curve.

When tissue oxygen is low, then bad things can happen to tissues. Briefly, oxygen is

brought into tissues via the arterioles, and then some of this oxygen is used by the tissues

for metabolic processes, and the leftover oxygen is removed via the venules. Hypoxia

occurs when tissue oxygen is low. This condition can arise from a variety of effects.

For example, the amount of breathed oxygen could be low (as on a high mountain), or

the delivery of oxygen to the tissue could be impaired (e.g., by blockages in the feeding

arteries), or the tissue metabolism could be abnormally high, etc. Tissue hypoxia can

therefore reflect problematic clinical scenarios such as ischemic stroke, therapy resistant

hypoxic tumors, muscle disease, and more.

To recapitulate, tissue oxygen dynamics are important in clinical contexts. Diffuse

optics does not measure pO2 directly, but it does measure blood oxygen saturation, total

hemoglobin concentration, and blood flow. All of these hemodynamic ingredients are

useful to know, and they permit experimenters to develop a picture of the functioning

(or malfunctioning) tissue. In fact, measurement of changes in all three hemodynamic


quantities permits scientists to compute changes in another fundamental tissue prop-

erty: oxygen metabolism. Convincing recent work along these lines has been carried out

in all-optical functional experiments in brain, which probe changes in cerebral oxygen

metabolism [31,32].

7. – Validation and clinical examples

We will use the remainder of this paper to provide a set of representative illustrations

from our laboratory of the diffuse optical techniques in clinical and pre-clinical studies.

These examples are not intended to be comprehensive; nor are they intended to provide

a review of all activity in the field. Rather, they are intended to give a flavor for the

kinds of measurements that are becoming possible.

We will begin with some validation studies. One goal for our research group (as well

as for many researchers in biomedical optics) is to validate the diffuse optical methods.

For example, in fig. 7a we show a measurement of the Hill Curve that relates oxygen

partial pressure in tissue to blood oxygen saturation. The data in fig. 7a are derived from

a simple tissue phantom experiment. The tissue phantom had optical properties very

similar to those of human tissue and contained mouse erythrocytes (i.e., mouse blood).

Over the course of sample deoxygenation we measured both oxygen partial pressure

with needle electrodes (both Eppendorf histograph and Clark-style electrodes) and blood

oxygen saturation with a standard diffuse optical spectroscopy (DOS) set-up. It should

be apparent that the curve behaves as expected, thereby providing validation for diffuse

optical HbO2 and Hb quantification. In fig. 7b, in vivo mouse tumor experiments show

relative changes in DOS-measured StO2 (in figure, StO2 and SO2 both refer to blood

oxygen saturation) as a function of relative changes in Eppendorg histograph pO2 during

carbogen breathing. We see that the two techniques are again in quite good agreement.

Indeed there is a substantial literature that corroborates DOS (NIRS) measurements

of tissue blood oxygen saturation and blood volume concurrently with other medical

diagnostics and/or with the literature.

(a) (b)

Fig. 7. – (a) Oxyhemoglobin dissociation curve obtained from mouse erythrocytes in a tissuephantom. (b) Correlation between relative blood oxygen saturation (SO2), as measured bydiffuse optics, and relative pO2, as measured by an Eppendorf histograph, in fibrosarcomatumors in mice during carbogen breathing [41].


c( )

(b)

Heart Rate: Pulse-OxArterial O : Pusle-OxBlood Pressure: Arm-CuffE CO : Capnograph

2

2t3 6%CO2

RoomAir

Capnograph

CBF

E COt 2

Fig. 8. – (a) Schematic diagram of the hypercapnia experiment. (b) Measured end-tidal CO2

and CBF, from DCS, for a single subject. (c) Simultaneous skin/scalp flow measured by LaserDoppler for the same subject. Activation occurs between the thick solid lines in the figure.

Serious validation of diffuse correlation spectroscopy (DCS) has only begun relatively

recently [32, 36, 38, 42, 43]. In fig. 8 we show the functional blood flow response of an

adult to a hypercapnia perturbation [42]. In hypercapnia, the subject breathes excess

CO2 and blood flow is increased in his/her brain as a result. In practice we place the

DCS probe (with source-detector separation of 2.5 cm on the tissue surface) locally on

the head, even though hypercapnia is a whole-brain response. In DOS/DOT and DCS

measurements, light penetration depth through the scalp and skull and into the cortex

depends on source-detector separation; the penetration depth is typically one-third to

one-half of the source-detector separation on the surface of the head. Thus, the light

penetrates into approximately 0.5 cm of cortex in the present case. The increase in

blood flow with inspired pCO2 is evident from the data and is in quantitative agreement

with other measurements from the literature [44-46]. We also confirmed that the DCS

measurement is not simply measuring skin/scalp flow, by comparing the DCS response

to the much smaller (and noisier) Laser Doppler skin/scalp flow measurement.

As a second DCS validation example, we show data from experiments in brain-injured

patients in the neuro-intensive care unit [47]. In this case (fig. 9), we compared DCS

measurements of cerebral blood flow to flow measurements by xenon-CT. The latter is

a standard-of-care flow measurement in the clinic, but it cannot be used for continuous

bedside monitoring because of its complexity. Using the Xe-CT images, we can compare


Fig. 9. – (a) Top: single axial slice from non-contrast CT scan, with the region under DCSprobes outlined. Bottom: baseline (left) and after Xe infusion (right) CBF maps obtained fromXeCT scan. (b) Correlation between CBF as measured by DCS and xenon CT (color online;for more details, see [47]).

the same tissue volumes during various drug stimulations, such as increased dose of va-

sopressor drugs. A simple protocol was thus set up to induce cerebral blood flow changes

and to follow responses with both techniques. Again, we found that both techniques

were quite strongly correlated (fig. 9) [47].

Beyond validation, research in the field is oriented towards two classes of brain inves-

tigation. The first class of study is concerned with the function of normal brain, and the

second class of study is clinical, aimed to improve patient care. We will give one example

of each class.

First, we illustrate functional imaging through the skull. Many functional paradigms

have been developed to understand normal responses associated, for example, with vision,

with verbal fluency, with motor skills, and more. As a concrete example, consider the

classic motor stimuli functional activation problem: finger tapping. When you tap your

thumb against your forefingers, it is well known that a very small part of the motor

cortex is activated. In a recent experiment (fig. 10) we investigated whether the full

hemodynamic responses associated with finger tapping could be measured non-invasively

in vivo, through the skull. To this end, we built a small diffuse optical probe pad with

multiple source-detector separations (∼ 2.5 cm), and with the ability to carry out both

DOS and DCS measurements concurrently. We placed the probe on the heads of multiple

subjects, sometimes with finger tapping on and sometimes with finger tapping off, and

sometimes over the activated region of the motor cortex and sometimes a centimeter

away from this activation region.

Our observations are summarized concisely in fig. 10. With the probe in the “wrong

place”, activation was not observed. On the other hand, with the probe placed directly

over the motor activation site, we observed large perturbations in the concentrations of

oxy-, deoxy-hemoglobin, and in blood flow. Averaging over a small subject population

permits comparison with other techniques, and calculation of oxygen metabolism from

the all-optical probe! A most exciting aspect of this work is the quantification potential


Fig. 10. – (a) Schematic diagram showing the experimental setup. Hemodynamic responses aremeasured by diffuse optical methods as a function of time. In (b) and (c) data between dashedlines mark activation period. (b) Data with probe located over the somatomotor cortex, and(c) 1 cm off-center from the activation spot (color online) [32].

of diffuse optics. Indeed, many researchers are exploring all sorts of functional activation

paradigms with diffuse optics at the present time [43,48-52].

In considering clinical applications for brain, it is important to first reflect upon the

sorts of things that doctors measure at the present time, as well as upon the kinds of

physiological information that doctors need/want to know. Most of the current clinical

methods focus on metabolism-related problems and issues such as oxygen delivery, pres-

sure differences in the brain due to, for example, swelling, and flow autoregulation. In

fact, many treatment strategies for brain-injured patients basically aim to increase blood

flow to the injured parts of brain. For example, the normal brain has a broad range

of cerebral perfusion pressure conditions for which the brain autoregulatory apparatus

adjusts to keep oxygen delivery optimized. However, if a patient falls out of this normal

range due to stroke or head injury, then the vasculature does not respond as well, and

the patient might need drugs or other manipulations to ameliorate problems. The situa-

tion is exacerbated because the current diagnostic tools available to clinicians tend to be

very invasive (e.g., intracranial pressure and oxygen monitors, etc.) and/or very slow,

cumbersome and costly (e.g., MRI, Xe-CT, etc.). Thus a golden opportunity for diffuse

optics is evident: continuous cerebral blood flow and oxygenation monitoring.

A recent study, that illustrates the potential of diffuse optics for monitoring flow

treatment, was carried out in a critically ill population of ischemic stroke patients [37].

The idea of these experiments (fig. 11) was to longitudinally monitor cerebral hemody-

namics at the bedside induced by changes in head-of-bed positioning. Our hypothesis


Fig. 11. – (a) Schematic representation of the head-of-bed (HOB) manipulation. (b) CBFchanges in each hemisphere after each HOB manipulation in a healthy population. CBF changesin diseased population showed (c) impaired autoregulation in the injured hemisphere, but about25% of the population presented a paradoxical response (d), where CBF was decreased at −5degrees (color online) [37].

was that in response to this challenge, the impaired cerebral autoregulation would lead

to larger changes in cerebral hemodynamics in the infarcted hemisphere by comparison

to the “healthy”, contralateral hemisphere. To this end, diffuse optical measurements

were obtained from patients with acute hemispheric ischemic stroke (n = 17, mean age

65 years). The probes were placed on the forehead near the frontal poles. Cerebral blood

flow and the hemoglobin concentrations were measured at different head-of-bed (HOB)

positions of 30, 15, 0, −5 and 0 degrees, and normalized to their values at 30 degrees. A

clear differentiation was observed between two hemispheres that was statistically signif-

icant over the whole population. Interestingly, in roughly one-fourth of the patients we

observed that cerebral blood flow was not maximized at −5 degrees; rather it was very

small at this HOB angle. This paradoxical response was observed in traumatic brain

injury patients and was likely a result of a substantial increase in intracranial pressure,

a parameter that is not routinely monitored in ischemic stroke patients. This simple

example illustrates that diffuse optical instrumentation can be deployed at the bed-side

of critically ill patients, and that the methodology may be promising for use as a tool to

optimize patient care based on real-time cerebral hemodynamic measurements.

The last clinical example we will discuss concerns breast cancer detection, diagnosis

and monitoring based (primarily) on diffuse optical tomography. Even though diffuse

optics is a relative newcomer to the breast imaging field, the methodology could find uses

in this important field. Potential niches for diffuse optics include detection/screening in

high and intermediate risk populations, diagnosis between malignant and benign among

certain classes of call-back patients, and therapy monitoring. Collectively, our view is

that the diffuse optics field is just now at the point where we are starting to obtain

higher-fidelity images of breast cancer, and we are beginning to confirm and identify

application niches.

Here again, rather than provide a comprehensive review of the breast-DOT field, we

opt to show some illustrative results from our lab [33,53-55]. We have built a parallel-plate

soft compression device shown schematically in fig. 12. The instrument carries out mea-

surements in transmission and remission at six optical wavelengths, and it is capable of


Fig. 12. – Schematic representation of the parallel-plate diffuse optical tomography instrument.

both continuous-wave and frequency-modulated measurements. Approximately 105–106

measurements are effectively performed in ∼ 10 minutes. These measurements are then

used as input to the inverse problem, and a three-dimensional (3D) tomogram of breast

tissue physiological properties is thereby obtained [56,57].

A sample image, showing a slice out of the full 3D reconstruction, is given in fig. 13;

in this case an invasive ductal carcinoma is found. Notice that the tumor shows up in

some physiological variables (e.g. THC), but not in all (e.g. StO2). Furthermore, optical

indices based on multiple physiological/optical properties can be constructed to improve

tumor-to-normal contrast. Images such as these are based on endogenous tissue contrast,

which is typically relatively small (1.3× to 1.5×). Exogenous contrast agents, such as

Indocyanine Green (ICG), offer the potential for improved contrast. Figure 14 shows

recent images based on exogenous contrast via fluorescence-DOT and standard-DOT

endogenous contrast of the same tumor; notice that the ICG contrast is substantial and

is therefore likely to stimulate development of better molecular markers for tumors in

the future.

Particularly in breast cancer, we expect that innovations in instrumentation and re-

construction algorithms will continue to be developed and combined to improve image

fidelity and resolution. In addition, more in vivo breast cancer data will provide critical

insight and guidance for directed algorithm/instrumentation development. In searching

for enhanced differentiation between tumor and normal tissues, groups across the com-

munity are employing broader wavelength ranges to explore water, lipid and collagen

concentrations, bound water fraction, and refractive index [58-60], and they are even ex-

ploring blood flow and metabolic contrast in breast cancer [33,61]. Multi-modal imaging

and monitoring approaches can potentially overcome structural resolution limitations of

DOT, using the spatial information provided by other imaging modalities to constrain

the DOT inverse problem. These multi-modal approaches provide extra physiological in-

formation. DOT measurements have been made with concurrent MRI [62,63], 3D X-ray

mammography [64], and ultrasound [65]. Furthermore, advances in diffuse optical tomog-

raphy of breast are critical for exploitation of the advances of molecular imaging [66], an

emerging field of medicine with promise of new-generation optical-contrast agents.


Fig. 13. – Caudal-cranial view of diffuse optical images in a field of view (black arrows) coveringa tumoral region, with different parameters used as contrast [55].

Fig. 14. – Optical images from a breast with a tumoral region, using different parameters asconstrast. Notice the high contrast provided by ICG, compared to the others endogenous pa-rameters (see [53] for more details).

8. – Concluding remarks

It should be evident that the dream of using optics for in vivo biopsy is being realized.

In this informal review we have described these developments broadly, but we have also

left a substantial amount of important research out of our discussion. For example,

diffuse optics diagnostics have found uses in the study of muscle disease [36], in cancer

therapy monitoring of human subjects [55] and pre-clinical animal models [67, 68], in

studies of osteoarthritis [69] and skin disease [70]. We hope that our discourse will have

inspired the reader to explore the field further and perhaps even to join in the fun!


∗ ∗ ∗

The authors would like to acknowledge many fruitful discussions and interactions over

the years with colleagues from the biomedical community at the University of Pennsyl-

vania and throughout the world. At UPenn, much of this research was facilitated by

sustained collaborations with B. Chance, J. Greenberg, J. Detre, M. Schnall,

J. Culver, G. Yu, D. Boas, T. Durduran, R. Choe, and T. Busch. This work was

supported by the National Institutes of Health through NS-060653, HL-57835, RR-02305,

NS-45839, and CA-126187.

REFERENCES

[1] Yodh A. and Chance B., Phys. Today, 48 (1995) 34.[2] Hielscher A. H., Bluestone A. Y., Abdoulaev G. S., Klose A. D., Lasker J.,

Stewart M., Netz U. and Beuthan J., Disease Markers, 18 (2002) 313.[3] Yodh A. G. and Boas D. A., Functional Imaging with Diffusing Light, in Biomedical

Photonics, edited by Vo-Dinh T. (CRC Press, Boca Raton) 2003, pp. 21/1-45.[4] Obrig H. and Villringer A., J. Cereb. Blood Flow Metab., 23 (2003) 1.[5] Gibson A. and Dehghani H., Philos. Trans R. Soc. A, 367 (2009) 3055.[6] Wang L. V. and Wu H., Diffuse Optical Tomography, in Biomedical Optics: principles

and imaging (John Wiley & Sons, New Jersey) 2007, pp. 249-282.[7] van de Hulst H. C., Light Scattering by Small Particles (Dover, Mineola) 1981.[8] Clark N. A., Lunacek J. H. and Benedek G. B., Am. J. Phys., 38 (1970) 575.[9] Berne B. J. and Pecora R., Dynamic Light Scattering with Applications to Chemistry,

Biology, and Physics (Krieger, Malabar) 1990.[10] Chu B., Laser Light Scattering: Basic Principles and Practice (Academic, New York)

1991.[11] Arridge S. R. and Schotland J. C., Inverse Problems, 25 (2009) 123010.[12] Case K. M. and Zweifel P. F., Linear Transport Theory (Addison-Wesley, Reading)

1967.[13] Ishimaru A., Wave Propagation and Scattering in Random Media (Academic Press, San

Diego) 1978.[14] O’Leary M. A., Boas D. A., Chance B. and Yodh A. G., Phys. Rev. Lett., 69 (1992)

2658.[15] Boas D. A., O’Leary M. A., Chance B. and Yodh A. G., Phys. Rev. E, 47 (1993)

R2999.[16] Danen R. M., Wang Y., Li X. D., Thayer W. S. and Yodh A. G., Photochem.

Photobiol., 67 (1998) 33.[17] Arridge S. R., Inverse Problems, 15 (1999) R41.[18] Boas D. A., Brooks D. H., Miller E. L., DiMarzio C. A., Kilmer R. J. and Zhang

Q., IEEE Sign. Proc. Mag., November (2001) 57.[19] Gibson A. P., Hebden J. C. and Arridge S. R., Phys. Med. Biol., 50 (2005) R1.[20] Kak A. C. and Slaney M., Principles of Computerized Tomographic Imaging (IEEE

Press, New York) 1988.[21] Golub G. and Kahan W., J. SIAM Numer. Anal., 2 (1965) 205.[22] Ackerson B. J., Dougherty R. L., Reguigui N. M. and Nobbman U., J. Termophys.

Heat Trans., 6 (1992) 577.[23] Dougherty R. L., Ackerson B. J., Reguigui N. M., Dorri-Nowkoorani F. and

Nobbman U., J. Quantum Spectrosc. Radiat. Transfer, 52 (1994) 713.


[24] Boas D. A., Campbell L. E. and Yodh A. G., Phys. Rev. Lett., 75 (1995) 1855.[25] Boas D. A. and Yodh A. G., J. Opt. Soc. Am. A, 14 (1997) 192.[26] Maret G. and Wolf P. E., Z. Phys. B, 65 (1987) 409.[27] Stephen M. J., Phys. Rev. B, 37 (1988) 1.[28] Pine D. J., Weitz D. A., Chaikin P. M. and Herbolzheimer E., Phys. Rev. Lett., 60

(1988) 1134.[29] Boas D. A., Diffuse Photon Probes of Structural and Dynamical Properties of

Turbid Media: Theory and Biomedical Applications (Ph.D. dissertation, University ofPennsylvania) 1996.

[30] Cheung C., Culver J. P., Takahashi K., Greenberg J. H. and Yodh A. G., Phys.

Med. Biol., 46 (2001) 2053.[31] Culver J. P., Durduran T., Furuya D., Cheung C., Greenberg J. H. and Yodh

A. G., J. Cereb. Blood Flow Metab., 23 (2003) 911.[32] Durduran T., Yu G., Burnett M. G., Detre J. A., Greenberg J. H., Wang J.,

Zhou C. and Yodh A. G., Opt. Lett., 29 (2004) 1766.[33] Durduran T., Choe R., Yu G., Zhou C., Tchou J. C., Czerniecki B. J. and Yodh

A. G., Opt. Lett., 30 (2005) 2915.[34] Yu G., Durduran T., Zhou C., Wang H. W., Putt M. E., Saunders M., Seghal

C. M., Glatstein E., Yodh A. G. and Busch T. M., Clin. Cancer Res., 11 (2005) 3543.[35] Sunar U., Quon H., Durduran T., Zhang J., Du J., Zhou C., Yu G., Choe R.,

Kilger A., Lustig R., Loevner L., Nioka S., Chance B. and Yodh A. G., J. Biomed.

Opt., 11 (2006) 064021.[36] Yu G., Floyd T., Durduran T., Zhou C., Wang J. J., Detre J. A. and Yodh A. G.,

Opt. Express, 17 (2007) 1064.[37] Durduran T., Zhou C., Edlow B. L., Yu G., Choe R., Kim M. N., Cucchiara

B. L., Putt M. E., Shah Q., Kasner S. E., Greenberg J. H., Yodh A. G. andDetre J. A., Opt. Express, 17 (2009) 3884.

[38] Buckley E. M., Cook N. M., Durduran T., Kim M. N., Zhou C., Choe R., Yu G.,

Schultz S., Sehgal C. M., Licht D. J., Arger P. H., Putt M. E., Hurt H. H. andYodh A. G., Opt. Express, 17 (2009) 12571.

[39] Sherwood L., Fundamentals of Physiology: A Human Perspective (Thomson Brooks,Belmont) 2006.

[40] Widmaier E. P., Raff H. and Strang K. T., Human Physiology: The Mechanisms of

Body Function (McGraw-Hill, Los Angeles) 2007.[41] Wang H. W., Putt M. E., Emanuele M. J., Shin D. B., Glatstein E., Yodh A. G.

and Busch T. M., Cancer Res., 64 (2004) 7553.[42] Durduran T., Non-Invasive Measurements of Tissue Hemodynamics with Hybrid Diffuse

Optical Methods (Ph.D. dissertation, University of Pennsylvania) 2004.[43] Li J., Dietsche G., Iftime D., Skipetrov S. E., Maret G., Elbert T., Rockstroh

B. and Gisler T., J. Biomed. Opt., 10 (2005) 044002.[44] Sicard K. M. and Duong T. Q., Neuroimage, 25 (2005) 850.[45] Jones M., Berwick J., Hewson-Stoate N., Gias C. and Mayhew J., Neuroimage, 27

(2005) 609.[46] Huppert T. J., Jones P. B., Devor A., Dunn A. K., Teng I. C., Dale A. M. and

Boas D. A., J. Biomed. Opt., 14 (2009) 044038.[47] Kim M. N., Durduran T., Frangos S., Edlow B. L., Buckley E. M., Moss H. E.,

Zhou C., Yu G., Choe R., Maloney-Wilensky E., Wolf R. L., Grady M. S.,

Greenberg J. H., Levine J. M., Yodh A. G., Detre J. A. and Kofke W. A.,Neurocrit. Care, 12 (2010) 173.

[48] Villringer A. and Chance B., Trends Neurosc., 20 (1997) 435.


[49] Franceschini M. A., Thaker S., Themelis G., Krishnamoorthy K. K., Bortfeld

H., Diamond S. G., Boas D. A., Arvin K. and Grant P. E., Pediatr. Res., 61 (2007)546.

[50] Zeff B. W., White B. R., Dehghani H., Schlaggar B. L. and Culver J. P., Proc.

Natl. Acad. Sci. U.S.A., 104 (2007) 12169.[51] Jaillon F., Li J., Dietsche G., Elbert T. and Gisler T., Opt. Express, 15 (2007)

6643.[52] Becerra L., Harris W., Joseph D., Huppert T., Boas D. A. and Borsook D.,

Neuroimage, 41 (2008) 252.[53] Corlu A., Choe R., Durduran T., Rosen M. A., Schweiger M., Arridge S. R. and

Yodh A. G., Opt. Express, 15 (2007) 6696.[54] Konecky S. D., Choe R., Corlu A., Lee K., Wiener R., Srinivas S. M., Saffer

J. R., Freifelder R., Karp J. S., Hajjioui N., Azar F. and Yodh A. G., Med. Phys.,35 (2008) 446.

[55] Choe R., Konecky S. D., Corlu A., Lee K., Durduran T., Busch D. R., Pathak

S., Czerniecki B. J., Tchou J., Fraker D. L., DeMichele A., Chance B., Arridge

S. R., Schweiger M., Culver J. P., Schnall M. D., Putt M. E., Rosen M. A. andYodh A. G., J. Biomed. Opt., 14 (2009) 024020.

[56] Arridge S. R. and Schweiger M., Appl. Opt., 34 (1995) 8026.[57] Arridge S. R. and Schweiger M., Opt. Express, 2 (1998) 213.[58] Srinivasan S., Pogue B. W., Brooksby B., Jiang S., Dehghani H., Kogel C.,

Wells W. A., Poplack S. P. and Paulsen K. D., Technol. Cancer Res. Treat., 4

(2005) 513.[59] Cerussi A., Shah N., Hsiang D., Durkin A., Butler J. and Tromberg B. J., J.

Biomed. Opt., 11 (2006) 044005.[60] Srinivasan S., Pogue B. W., Carpenter C., Jiang S., Wells W. A., Poplack S. P.,

Kaufman P. A. and Paulsen K. D., Antioxidants Redox Signaling, 9 (2007) 1143.[61] Zhou C., Choe R., Shah N., Durduran T., Yu G., Durkin A., Hsiang D., Mehta

R., Butler J., Cerussi A., Tromberg B. J. and Yodh A. G., J. Biomed. Opt., 12

(2007) 051903.[62] Ntziachristos V., Yodh A. G., Schnall M. D. and Chance B., Neoplasia, 4 (2002)

347.[63] Carpenter C. M., Srinivasan S., Pogue B. W. and Paulsen K. D., Opt. Express, 16

(2008) 17903.[64] Fang Q., Carp S. A., Selb J., Boverman G., Zhang Q., Kopans D. B., Moore

R. H., Miller E. L., Brooks D. H. and Boas D. A., IEEE Trans. Med. Imaging, 28

(2009) 30.[65] Zhu Q., Tannenbaum S., Hegde P., Kane M., Xu C. and Kurtzman S. H., Neoplasia,

10 (2008) 1028.[66] Weissleder R. and Ntziachristos V., Nature Med., 9 (2003) 123.[67] Luckl J., Zhou C., Durduran T., Yodh A. G. and Greenberg J. H., J. Neurosci.

Res., 87 (2009) 1219.[68] Zhou C., Eucker S. A., Durduran T., Yu G., Ralston J., Friess S. H., Ichord

R. N., Margulies S. S. and Yodh A. G., J. Biomed. Opt., 14 (2009) 034015.[69] Hofmann G. O., Martick J., Grosstuck R., Hoffmann M., Lange M.,

Plettenberg H. K. W., Braunshweig R., Schilling O., Kaden I. and Spahn G.,Pathophysiol., 17 (2010) 1.

[70] Bednov A., Ulyanov S., Cheung C. and Yodh A. G., J. Biomed. Opt., 9 (2004) 347.


Ultrasonic wave transport in strongly scattering media

J. H. Page

Department of Physics and Astronomy, University of Manitoba

Winnipeg, MB Canada R3T 2N2

Summary. — Ultrasonic experiments are well suited to the investigation of classicalwave transport through strongly scattering media, and are playing a role that is oftencomplementary to investigations using light or microwaves. Advantages of ultrasonictechniques are their ability to readily detect the wave field (not just the intensity), toperform experiments resolved in both time and space, and to control the propertiesof the medium being investigated over a wide range of scattering contrasts. Thisfirst paper reviews what has been learned from ultrasonic experiments over the last15 years about the ballistic and diffusive propagation of classical waves throughstrongly scattering disordered media. These results are compared with studies ofordered media (phononic crystals), where band gaps and super-resolution focusinghave been observed.

1. – Introduction

For more than a decade, there has been growing interest in ultrasonic wave transport

in strongly scattering media. Just as for other classical waves, such as light and mi-

crowaves, much of this interest revolves around the many unusual wave phenomena have

been observed at intermediate frequencies, where the wavelengths are comparable to the

size of the scatterers [1]. Examples range from strikingly large variations in wave speeds

caused by strong resonant scattering (when a pulse can even appear to travel so quickly


76 J. H. Page

through a sample that its velocity is negative) to the inhibition of wave propagation that

can occur in very strongly scattering samples when the waves become localized. In seek-

ing to discover and understand such wave phenomena, ultrasonic experiments have an

important role to play, partly because of the relative ease with which the full wave field,

rather than just the intensity, can be measured. Thus, ultrasonic techniques give direct

experimental access to the wave function and/or Green’s function, allowing both phase

and amplitude information to be obtained. Ultrasound is also well adapted to pulsed

experiments, enabling the path length dependence of the transmission or reflection to

be resolved in time, usually in the near field. Furthermore, the fact that the scattering

contrast is governed by differences in both velocity and density enables the scattering

strength to be controlled over a very wide range. As a result, experiments with acous-

tic or elastic waves can make important contributions to both fundamental studies and

practical applications of wave scattering in complex media, and are often complementary

to optical and microwave methods for investigating these phenomena.

In this paper, I will review the progress that has been achieved over the last 15 years

in understanding how ultrasonic waves propagate through both random and ordered

media. The regime of interest here is one where multiple scattering dominates, but the

scattering is not so strong that the interference effects leading to Anderson localization

are present. (The latter is the subject of the second paper in this series, while the

third paper discusses applications such as Diffusing Acoustic Wave Spectroscopy.) To

illustrate the scope of information that is accessible to ultrasonic experiments in random

systems, sect. 2 summarizes results obtained for acoustic waves (longitudinal polarization

only) in a model system consisting of a suspension of glass beads in a liquid, where

a rather complete picture of wave transport has been achieved through transmission

measurements. Other types of acoustic scattering systems (plastic spheres and bubbles

surrounded by water), which lead to different wave behaviour, are also mentioned. By

contrast to the diffusive transport of energy that is seen in disordered systems, the

propagation of multiply scattered waves in ordered media is characterized by a coherent

multiply scattered wave field, leading to band gaps and unusual focusing phenomena.

These effects are described in the last major section of the paper on phononic crystals

(sect. 3).

2. – Acoustic wave transport in random media

Many features of ultrasound transport in strongly scattering media are demonstrated

by acoustic pulse propagation experiments that have been performed in a disordered

medium consisting of randomly packed 0.5-mm-radius glass beads immersed in wa-

ter [2-5]. Strong scattering in this model system is ensured by the large acoustic

impedance difference between glass and water (Zglass/Zwater ≈ 10, where Z = ρvp is

the acoustic impedance, ρ is the density and vp is the phase velocity). Although the

transmitted signals are dominated by multiply scattered waves at intermediate frequen-

cies, the phase sensitivity of piezoelectric ultrasonic transducers allows measurements to

be made of the weak signal that propagates ballistically through the medium without

Ultrasonic wave transport in strongly scattering media 77

Fig. 1. – Comparison of the total field (a), average field (b) and scattered field (c) transmit-ted through a thin sample (thickness, L = 1.7 mm) of glass spheres in water. The data arenormalized with respect to an input pulse of unit peak amplitude.

scattering out of the forward direction. This ballistic signal remains coherent both tem-

porarily and spatially with the input pulse, so that it can be extracted by averaging the

transmitted wave field over many speckles, a procedure that causes the multiply scattered

component to cancel because of the random phase fluctuations from speckle to speckle.

The separation of the ballistic pulse from the multiply scattered waves, whose energy

transport was found to be well described by the diffusion approximation, allows a very

complete picture of wave transport in strongly scattering media to be obtained.

2.1. Ballistic propagation. – Figure 1 shows an example of how the ballistic pulse

can be extracted from the total transmitted field when the sample is sufficiently thin.

The experiments were performed by enclosing the suspension of glass particles in a cell

with thin walls that are transparent to ultrasound, and then placing the sample in a

water tank between a plane-wave generating transducer and a subwavelength-diameter

hydrophone detector. The hydrophone position was scanned in a plane near the surface

of the sample to measure the transmitted signal in many independent coherence areas,

or speckles, using a grid separation of approximately a wavelength. The wave field

averaged over more than 100 speckles reveals the ballistic pulse, shown by the solid

curve in fig. 1(b). The multiply scattered waveforms (fig. 1(c)), often called the coda,

especially in the context of seismic waves, since they arrive after the ballistic pulse, can

then also be obtained by subtracting the ballistic pulse from the total transmitted field

in each speckle. This demonstration of coherent ballistic pulse propagation provides

convincing experimental evidence that a (uniform) effective medium can still be defined

78 J. H. Page

Fig. 2. – Frequency dependence of (a) the phase velocity, (b) the group and energy velocities,(c) the scattering and transport mean free paths, and (d) the diffusion coefficient. The dottedhorizontal line in (a) indicates the sound velocity in water. All data (symbols) and theory (solidand dashed curves) were measured/averaged over a small 5% variation in the bead size.

in the intermediate strongly scattering regime, a result that has been inferred less directly

in recent optical experiments [6].

The ballistic pulse contains information that is crucial for determining the frequency

dependence of the scattering properties of any sample, as it allows both the phase and

group velocities [vp = ω/k, vg = dω/dk] as well as the scattering mean free path ls [I(L) =

I(0) exp[−L/ls]], to be determined [5]. Here ω, k, L and I are the angular frequency,

wave vector, sample thickness and ultrasound intensity, respectively. Experimentally, vp

and ls are determined from the phase difference Δφ and amplitude ratio A(L)/A(0) of

the fast Fourier transforms (FFT) of the ballistic and input pulses (vp = ωL/Δφ, ls =

−L/ ln[A(L)/A(0)]2). The group velocity is accurately measured by digitally filtering the

ballistic and input pulses using a narrow Gaussian filter, whose bandwidth is chosen to

be sufficiently narrow that pulse distortion due to dispersion is negligible, and measuring

the delay tg between their peak arrival times (vg = L/tg).

Results for randomly close packed suspensions of glass beads in water are shown in

fig. 2 over a wide frequency range, corresponding to wavelengths from approximately

5a to 0.5a (dimensionless frequency range: 1 � kwa � 10, where kw is the wavelength

in water and a is the bead radius). For kwa > 2, strongly scattering is seen, with the

scattering mean free path approaching the bead radius, and the product kls ranging from

3 to 9. Both the phase and group velocities exhibit a considerable frequency dependence,

with the group velocity varying by more than a factor of 2. Note the very low values of

the group velocity near kwa = 2, where vg is substantially less than the sound velocities

in either water or glass (vw = 1.5 mm/μs, vglass = 5.6 or 3.4 mm/μs for longitudinal or

transverse polarizations).


The origin of these large velocity variations can be understood using an effective

medium model, based on a Spectral Function Approach (SpFA), which overcomes a

fundamental limitation of the traditional Coherent Potential Approximation (CPA) in the

intermediate frequency regime [3]. The scattering is calculated by modeling a typical glass

bead scatterer as an elastic sphere that is coated with a layer of water and embedded in a

homogeneous effective medium, which accounts for the presence of all the other scatterers.

The dispersion relation, ω versus k, for acoustic waves in the medium is then determined

by identifying the peaks of the spectral function, given by the negative imaginary part of

the Green’s function. The simple physical interpretation of the method is that these peaks

correspond to the locus of points in the frequency-wavevector plane where the scattering

is weakest, so that they delineate the modes that succeed in propagating through the

medium and identify the effective medium properties. The approach is accurate so long

as kls � 2 [5]. This dispersion relation enables vp and vg to be calculated, giving

the excellent agreement with the experimental data shown in fig. 2. In addition, the

scattering mean free path can be determined from the scattering cross section of the

coated elastic sphere [3, 5]. By calculating the energy density of a typical scatter as a

function of frequency, the sharp features in the group velocity near kwa ∼ 2 and above

kwa ∼ 5 were found to be associated with resonances of the fluid coating and solid spheres,

respectively, leading in the first case to a slowing down of the velocity by tortuosity of

the connected fluid pathways and in the second case to resonant trapping of energy in

the solid scatterers [5]. The overall mechanism underlying the frequency dependence

of the phase and group velocities can be understood as follows: because of the strong

coupling between the resonant scatterers, the uniform effective medium sensed by the

coherent ballistic propagation is very strongly renormalized, in much the same way as

quantum-mechanical resonances are shifted when there is strong coupling between them.

Thus, the ballistic pulse is still able to propagate coherently while being very strongly

affected by the scatterers. These experimental and theoretical results also show that the

group velocity remains well defined despite the strong scattering [3], thereby addressing

a question about the meaning of the group velocity in dispersive media that was raised

by Sommerfeld [7] and Brillouin [8] in the first part of the 20th century and discussed

more recently by Albada et al. [9].

Ultrasonic experiments on other types of suspensions with different acoustic properties

have also been performed to examine how ballistic pulse propagation is affected by the

strength and character of the scattering resonances. One interesting example is a slurry

of randomly close-packed plastic spheres in water, where gaps open up in the mode

spectrum due to scattering resonances having the character of interfacial or Stoneley

waves. These Stoneley-wave-like resonances involve both longitudinal and transverse

displacements inside the spheres, and compressional deformations of the surrounding

nearby liquid. As a result, a second longitudinal mode with slow velocities, due to the

coupling between these Stoneley wave resonances on adjacent spheres, is observed. This

slow mode was first discovered by Brillouin scattering experiments [10], which probe the

modes of the system by measuring the frequencies of the modes at fixed wave vector,

in contrast to ultrasonic pulse propagation experiments, which measure the velocities of

80 J. H. Page

(a)

2 4 6 8 101

2

3

4

5

Fast Slow

L 0.56 mm

L 2.46 mm

Theory

2ω

a /

vf

2 4 6 8

Fast longitudinal m

Stoneley branch

Fluid dispersion re

(b)

2 k a 2 k a

(c)

10

mode

elation

Fig. 3. – (a), (b) Dispersion curves for a suspension of PMMA spheres in water. In (a), experi-mental results are represented by symbols for two different sample thicknesses, and theoreticalpredictions from the peaks of the spectral function (shown in (b)) are shown by the solid curves.(c) Incident and transmitted pulses through a cloud of bubbles, which are shown in the greyscale background picture behind the graphs. The bubbles were generated by an electrolysismethod and had a radius of ∼ 15μm.

the modes at fixed frequency. Ultrasonic measurements of the dispersion relations are

shown in fig. 3(a), and compared with peaks in the spectral function (fig. 3(b)) predicted

by the SpFA model. Good overall agreement is found, confirming the basic character

of the unusual modes of this system. In contrast to the Brillouin scattering results,

the ultrasonic measurements reveal that because of absorption, longitudinal modes still

propagate inside the “gap” and interfere with the Stoneley wave branch, leading to rich

behaviour that provides a stringent test of the accuracy of the SpFA model.

A second example of the effects of very strong scattering is acoustic pulse propagation

through a suspension of bubbles. The acoustic properties of bubbly suspensions are dom-

inated by a low-frequency multipole resonance, leading to a wide range of unusual wave

phenomena such as anomalous dispersion and superradiance (e.g., see refs. [11-13]). One

remarkable consequence is shown in fig. 3(c), which provides compelling experimental

evidence that the group velocity is negative near the fundamental bubble resonance fre-

quency [14]. This unusual effect occurs because of pulse reshaping due to the anomalous

dispersion, which leads to constructive interference at the leading edge of the pulse and

destructive interference at the trailing edge; thus, the peak of the transmitted pulse

emerges from the sample before the peak of the input pulse has entered it, so that the

pulse transit time and hence group velocity is negative. It is noted that, at a given time,

the intensity of the incident wave is always greater than the transmitted one, so that

causality is not violated. In this case and in analogous examples for light [15], the group

velocity still accurately describes ballistic pulse propagation, providing the bandwidth

is sufficiently narrow, but can no longer correspond to the ballistic energy transport

velocity [16].


20

40

60

80

100

120 ρ 10.2 mm

ρ 15.2 mm

4Dt, with

D 0.45 mm2/μs

w 2 (m

m2)

1E-11

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

D = 0.435 ± 0.015 mm2/μs

τa = 12.0 ± 0.5 μs

Norm

aliz

ed I

nte

nsi

ty

L 20.5 mm

L 10.3 mm

L 5.31mm (a) (b)

0 10 20 30 40 50 600

Time (μs)

0 20 40 60 80 100

1E 11

Time (μs)

Fig. 4. – (a) Time dependence of the transmitted intensity, normalized so that the peak of thecorresponding input pulse is unity, for three randomly close-packed glass-bead-in-water sampleshaving different thicknesses L. The symbols represent experimental data taken with a pointsource, and the solid curves are fits to the predictions of the diffusion approximation, fromwhich the diffusion coefficient and absorption time are determined. (b) Time dependence ofthe mean square transverse width w

2 of the diffuse halo for a slab-shaped sample. The lineargrowth of w2 is characteristic of a diffusion process, enabling D to be measured from the slopeof the straight line fit (solid line) independent of absorption and boundary conditions.

2.2. Diffusive propagation. – Transport beyond the scale of the mean free path is

dominated by multiply scattered waves (fig. 1(c)). So long as the thickness of the sample

is greater than about three mean free paths and the scattering is not so strong that

kls ∼ 1, the transport of energy in ultrasonic experiments is well described using the

diffusion approximation [2, 17]. In this approximation, all phase information is ignored

and the quantity of interest is energy transport, which is treated as a random walk

process, characterized by the diffusion coefficient D = vEl∗/3. Here vE is the average

local velocity of energy transport, and l∗ is the transport mean free path, or distance over

which the direction of propagation is randomized. The transport and scattering mean free

paths are related by l∗ = ls/(1−〈cos θ〉), where θ is the scattering angle, and are therefore

equal only when the scattering is isotropic. Dynamic (pulsed) measurements, which probe

the distribution of multiply scattered path lengths in the time domain, are sensitive to

D, while steady state (continuous wave) experiments, such as the measurement of total

energy transmission, are sensitive to l∗/L.

To demonstrate the applicability of the diffusion approximation to acoustic wave

transport in strongly scattering media, and to measure D, l∗ and vE over a wide frequency

range, an extensive series of pulsed and quasi-continuous-wave experiments have been

performed on the same glass bead suspensions described above [2, 4, 17]. For examples

of other approaches to investigating diffusive transport of acoustic waves in 3D and

2D systems, see references [18, 19]. In refs. [2, 4, 17], slab-shaped samples were used

and the diffusion coefficient was measured from the temporal evolution of the average

82 J. H. Page

transmitted intensity, I(t), which was determined by averaging the square of the envelope

of the scattered sound field over a large number of independent speckles. Typical results,

which were obtained using a tightly focused incident pulse to create a point source, are

shown in fig. 4(a) for three different sample thicknesses, ranging from 7 to 30 scattering

mean free paths. Comparison of the experimental data with solutions of the diffusion

equation was facilitated by performing the measurements on slab-shaped samples, with

widths at least 10 times the thickness so that edge effects could be ignored (i.e., the

samples were excellent approximations to infinite slabs for the range of times over which

signals could be detected). Accounting for internal reflections at the front and back

faces of the slab and the possibility of dissipation inside the sample, the transmitted

intensity (flux) for a delta-function diffuse source of unit strength in time and position,

δ(t)δ(x−x′)δ(y−y′)δ(z−z′), is given by the solution of the diffusion equation with these

boundary conditions:

(1) I(t) = −D∂U

∂z

∣∣∣∣z=L

=e−ρ2/4Dte−t/τa

2πL2t

∞∑n=1

Ane−Dβ2nt/L2

,

where τa is the absorption time, βm are the positive roots of the transcendental equation

tan β = 2βK/(β2K2 − 1), K is equal to z0/L with z0 = (2l∗/3)(1 + R)/(1 − R) (z0

is known as the extrapolation length, since it is the distance outside the sample where

the diffuse energy extrapolates to zero), R is the angle-averaged reflectivity of diffuse

sound at the sample boundaries (calculated from the acoustic impedance mismatch),

and the coefficients An are given by an analytic function of βn, K and z′ [2]. Here

ρ =√

(x − x′)2 + (y − y′)2 is the transverse distance in the plane parallel to the slab

at which the intensity is detected relative to the point directly opposite the source.

The location of the diffuse source inside the sample, z′, has been shown by numerical

simulations to be equal to l∗ [20]. The solid curves in fig. 4(a) show the results of least-

squares fits of eq. (1) to this data set, with the initial increase of I(t) being sensitive

to D and the tail quite strongly influenced by τa. The good agreement between theory

and experiment demonstrates the validity of the diffusion approximation for multiply

scattered sound, enabling reliable measurements of both D and τa to be made.

One advantage of the point source geometry is that it enables the growth of the

diffuse halo to be measured in the transverse direction parallel to the surface of the slab.

This gives a method of measuring D directly, independent of boundary conditions and

absorption [2]. Experimentally, the average transmitted intensity at transverse distance

ρ (“off-axis”) and at ρ = 0 (“on-axis”) are measured by averaging over different sample

positions with source and detector positions fixed relative to each other. From eq. (1),

it can be seen that the ratio I(ρ, t)/I(0, t) is given simply by e−ρ2/4Dt = e−ρ2/w2(t), so

that the transverse width w(t) of the diffuse halo grows as the square root of time, as

expected for a diffuse process. Plotting w2(t) versus time enables D to be measured

from the slope of a straight line fit to the data, as shown in fig. 4(b). The excellent

agreement between the values of D measured directly from the transverse width, and

from the more cumbersome fits of eq. (1) to the time-of-flight profile, gives additional


confidence in the accuracy with which the diffusion coefficient can be measured in pulsed

transmission measurements.

Ultrasonic experiments can also be performed using a good approximation to a plane

wave source by placing the sample in the far field of a planar immersion transducer.

The solution of the diffusion equation for this experimental geometry can be obtained by

integrating eq. (1) over x′ and y′, giving Iplane(t) = 4DtIpoint(ρ = 0, t). Again, accurate

measurements of D can be obtained by fitting this expression to the measured time-of-

flight profiles for this geometry (e.g., see ref. [2]). This exact solution of the diffusion

equation is often approximated by the somewhat simpler expression

(2) U(t) ≈2e−t/τa

π(L + 2z0)

∞∑n=1

e−Dn2π2t/(L+2z0)2

sin

(nπ(z + z0)

L + 2z0

)sin

(nπ(z′ + z0)

L + 2z0

).

Equation (2) is a good approximation in many experimental situations, especially at long

times, but is not accurate for large values of the reflectivity R. At long times, in the

absence of absorption, eq. (2) is proportional to exp[−t/τD], with τD = (L+2z0)2/(π2D).

This gives a very simple result for the exponential decay of the time-of-flight profile in

terms of the diffusion time τD, which is determined by the effective thickness of the

sample L + 2z0 and the diffusion coefficient.

Results for the frequency dependence of the diffusion coefficient in the glass bead

suspensions are shown in fig. 2(d). A considerable variation, roughly a factor 3, is seen

over the range of frequencies investigated. To determine its origin, experiments were

also performed for very long pulses to attain quasi-continuous-wave conditions, so that

l∗ could be measured from the thickness dependence of the total transmitted intensity,

I(L) = fn(l∗/L, α =√

Dτa) (see ref. [2] for the complete expression). It was found

that l∗ has at most a very weak frequency dependence (fig. 2(c)), being approximately

equal to the diameter of the beads in the strong scattering regime. This weak frequency

dependence is also shown from calculations of l∗ using the SpFA model, where 〈cos θ〉

is determined from the average of cos θ weighted by the square of the angle-dependent

scattering amplitude (solid curve in fig. 2(c)). Hence the strong frequency dependence

of D must be due to the variation of vE , which was determined experimentally from

the ratio vE = 3D/l∗ using the measured values of D and l∗. Figure 2(b) compares the

measurements of the energy velocity with the group velocity, showing that both vE and

vg, which describe the transport of energy through the sample by diffusive and ballistic

waves, respectively, are remarkably similar in magnitude and frequency dependence. This

similarity between vE and vg, which was not anticipated from earlier theoretical work for

light [9], appears to hold quite generally except in cases of extreme dispersion, where the

group velocity loses its meaning as the ballistic energy transport velocity (even though

vg still describes narrow-band coherent pulse propagation accurately in such extreme

conditions). The comparison shown in fig. 2(b) suggests a simple physical picture for

vE and its relationship to vg. Even in the forward direction, the transport of energy is

strongly affected by scattering resonances, which lead to a large scattering delay near the

minima in vg. It is reasonable to expect that wave pulses scattered through a non-zero

84 J. H. Page

scattering angle will experience a similar, but not identical, scattering delay, so that in

this case vE should be similar to vg, with the relation between them taking into account

the additional angle-average scattering delay of the scattered waves [4].

These ideas can be formulated quantitatively by extending the SpFA model to cal-

culate the additional scattering delay experienced by a wave pulse. In this approach,

the angular dependence of the magnitude and phase shift in a typical scattering event

is calculated for each frequency component of the wave pulse from the complex scatter-

ing amplitude of the coated elastic sphere embedded in the effective medium. By in-

corporating these frequency-dependent phase and amplitude variations into the Fourier

components of the incident Gaussian pulse, and taking the inverse Fourier transform

to recover the scattered pulse, the corresponding time delay of the scattered pulse

envelope relative to the forward direction can be calculated for each scattering an-

gle. The intensity-weighted angular average of these additional scattering delays, Δtave,

can then be used to express the energy velocity in terms of the group velocity, giving

vE = l∗/(l ∗ /vg + Δtave) = vg/(1 − δm), where δm = Δtavevg/l∗. Note that, in this

approach, vE , l∗, vg and δm are all calculated in a renormalized effective medium, which

accounts for the effects of the multiple scattering that become especially pronounced

for high volume fractions of scatterers. Excellent quantitative agreement between the

predictions of this model and the experimental data was found not only vE and l∗ but

also for the diffusion coefficient that is calculated from them using D = vEl∗/3.

In summary, these ultrasonic experiments in a model system consisting of glass beads

in water have enabled a quantitative and comprehensive assessment of the applicability

of the diffusion approximation to the description of energy transport by multiply scat-

tered acoustic waves. By comparing the parameters that govern diffusive and ballistic

transport over a wide frequency range, a unified physical picture of energy transport in

strongly scattering media has emerged. In addition, the success of the SpFA model in

describing the experimental results for both ballistic and diffusive waves highlights the

relevance of an effective medium description even in the strongly scattering intermedi-

ate frequency regime. The methods developed in this work have facilitated both the

search for ultrasonic wave localization in more strongly scattering samples (see paper II,

this volume, p. 95) and the development of novel dynamic probes of multiply scattering

materials (see paper III, this volume, p. 115).

3. – Wave transport in ordered media: phononic crystals in 2D and 3D

The character of ultrasound transport in strongly scattering media is changed dramat-

ically when the scatterers are arranged in an ordered array to form a phononic crystal.

These materials are the acoustic and elastic counterparts of photonic crystals for light,

and have been the subject of increasing interest since the early 1990s [21]. Because it is

relatively easy to control the strength of the scattering contrast between the component

materials, phononic crystals may be viewed as ideal media in which to study the profound

effects of lattice structure on wave propagation. Much of the initial research concentrated

on phononic band gaps, which occur due to Bragg scattering when the wavelength is com-


0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

FR

EQ

UE

NC

Y (

MH

z)

1E-4 1E-3 0.01 0.1 10.0

0.2

TRANSMISSION COEFFICIENT

LU ΓX

WAVE VECT

4

8

12

16 MST (no absorption)

MST (with absorption)

Experiment

GR

OU

P V

EL

OC

ITY

(km

/s)

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 2 4 6 80

G

SAMPLE THICKNESS (mm)

0.0

0.2

KWX

TOR

Fig. 5. – Left panels: Amplitude transmission coefficient along the [111] direction for a 12-layerfcc phononic crystal made from tungsten carbide beads, and the corresponding band structure.Experimental data are shown by the symbols, and the results of MST calculations by the solidcurves. A photo of part of the surface of the crystal is shown in the insert on the left. Right panel:Thickness dependence of the group velocity at a frequency of 0.95 MHz in the first band gap.

parable to the lattice constants, leading to frequency bands where wave propagation is

forbidden. As a result, much is now known about the conditions under which phononic

crystals with compete band gaps can be fabricated, allowing wave transport in this fre-

quency range to be investigated and novel acoustic waveguides and noise blocking devices

to be constructed [22-30]. Methods for making compact phononic crystal sound insula-

tors have also been proposed and demonstrated [25]. More recently, attention has turned

to wave transport in the pass bands both below and above the band gaps, where unusual

negative refraction, diffraction and focusing effects have been observed [31-35].

To illustrate the main differences between ultrasonic wave transport in ordered and

disordered structures, consider the results that have been obtained for 3D phononic crys-

tals made from 0.8-mm-diameter tungsten-carbide beads surrounded by water [28,31]. In

this case, excellent crystal quality was assured by the availability of extremely monodis-

perse spheres due to the needs of the ballpoint pen industry, and meticulous hand-

assembly of the spheres in a custom-made mould. In transmission, multiple scattering

from the periodic array of scatterers leads to a transmitted pulse in the far field with

a well-defined, but coherent, coda, so that the entire transmitted pulse can be analysed

by the methods outlined in subsect. 2.1. Thus, ultrasonic pulsed techniques can readily

measure all the basic wave properties of the crystal, including the transmission coefficient

(from the ratio of the amplitudes of the FFTs of the transmitted and incident pulses)

and the band structure (from the phase shift at each frequency in the pulse, yielding the

variation of ω with k = Δφ/L).

Typical results for the 3D tungsten-carbide/water crystal can be found in fig. 5.

The left pair of panels shows the transmission coefficient and the band structure of

86 J. H. Page

this face-centred-cubic crystal, revealing a wide band gap due to Bragg scattering near

1 MHz (width ∼ 20% of the central frequency), where the spacing between layers of the

crystal is approximately equal to half the wavelength in water. In the [111] direction,

in which the experimental data were obtained, the gap is even wider, as shown by the

broad dip in the transmission coefficient, measured for a crystal consisting of 12 layers.

These results illustrate the relative ease with which wide band gaps can be obtained in

acoustics relative to optics, because of the large scattering contrast that can be achieved

in ultrasound (for this combination of solid spheres and liquid matrix, the longitudinal

impedance ratio is 60). Even wider gaps (∼ 100%) are found in solid-solid systems such

as steel beads in epoxy, where coupling with a resonance of the spheres enhances the band

gap considerably [29, 36]. In fig. 5 the experimental data are compared with predictions

of Multiple Scattering Theory (MST, indicated by the solid curves) [24], which is ideally

suited to calculating the properties of phononic crystals built from scattering elements

having simple geometric shapes such as spheres, where the scattering can be calculated

accurately with no adjustable parameters. Excellent agreement between theory and

experiment is found. Note that this agreement indicates that the band structure, which is

calculated for an infinite crystal, can be accurately measured by transmission experiments

in finite-thickness samples consisting of remarkably few layers.

The transmission coefficient in the middle of the gap (at f = 0.95 MHz) is found, both

experimentally and theoretically, to decrease exponentially with thickness as exp[−κL],

consistent with evanescent modes having imaginary wave vector κ. This suggests that

ultrasound is transmitted through the crystal by tunneling, whose dynamics can be

investigated through measurements of the group velocity [28]. The right panel of fig. 5

shows that the group velocity increases linearly with sample thickness, an unusual result

that is the classic signature of tunneling, implying that the tunneling time is indepen-

dent of thickness. For the thickest crystals, the magnitude of vg is remarkably fast —see

the horizontal arrows in the figure for the longitudinal velocities in the two constituent

materials. The solid and dashed curves in the figure are calculated using Multiple Scat-

tering Theory both without and with absorption, the latter being taken into account

by complex moduli of the constituents. It can be seen that the theory with absorption

gives a very satisfactory description of the experimental results, indicating how dissi-

pation, which has no counterpart in the quantum tunneling case, significantly affects

the measured tunneling time. This effect was interpreted using a so-called two modes

model, which allows the role of absorption to be understood in simple physical terms [28].

Absorption in the band gap of a phononic crystal cuts off the long multiple scattering

paths, making the destructive interference that gives rise to the band gap incomplete. As

a result, a small-amplitude propagating mode is produced in parallel with the dominant

tunneling mode, accounting for the reduction in the measured group velocity relative to

the predictions without absorption. This simple model was also found to give a good

quantitative explanation of the data [28].

Experiments on the same 3D tungsten-carbide/water phononic crystal were the first to

demonstrate ultrasound focusing by negative refraction [31] —another area of phononic

(and photonic) crystal research that is currently attracting considerable attention. At


Fig. 6. – (a) Band structure of a phononic crystal of steel rods in water (triangular lattice of1.02-mm-diameter rods with lattice constant a = 1.27 mm). The solid curves were calculatedusing Multiple Scattering Theory calculations, and the symbols represent experimental data.(b) Equifrequency contours at the three frequencies, 0.75, 0.85 and 0.95 MHz, in the secondpass band. (c) Snapshot the negatively-refracted pulse emerging from a phononic crystal prism(angles 30◦, 60◦ and 90◦, as shown) after a narrow-band pulse (central frequency of 0.85 MHz)was normally incident on the shortest face of the prism (in the direction of wide blue arrow).The data were measured by scanning a hydrophone in a rectangular grid, digitally filtering thepulses to narrow the bandwidth, and measuring the wave field at a particular moment in timeto construct the spatial variation of the field at that time.

frequencies in the pass band near 1.6 MHz in fig. 5, the initially diverging beam from

a quasi-point source was observed to be sharply focused in a plane that was quite far

from the crystal, where the focal spot could be easily measured. As is explained in

more detail below, focusing occurs because the group velocity inside the crystal is of

opposite sign to the wave vector, and as a result the direction of energy transport (which

is given by the group velocity) corresponds to a negative angle of refraction. In terms

of a simple ray picture, in which the rays are drawn parallel to the group velocity, the

wave vector components of the field from the source that are incident at angles ±θ

are refracted negatively as they enter the crystal, cross inside the crystal and are then

refracted negatively again as they leave the crystal, so that the emergent rays converge

to a focus on the far side of the crystal. The data in these initial experiments were

interpreted using a Fourier imaging model that accounted for this unusual wave transport

through the crystal, giving a quantitative explanation of the observed focusing effect [31].

88 J. H. Page

To explore the phenomena of negative refraction and focusing in phononic crystals in

more detail, a number of experiments and theoretical calculations have been performed

on 2D crystals [32-35]. The most direct observations of negative refraction were made

by Sukhovich et al. [34], who constructed a prism-shaped phononic crystal of steel rods,

arranged in a triangular lattice at a volume fraction of 58% and surrounded by water.

This crystal has the advantage of a relatively simple band structure, as shown by the

solid curves (MST) and symbols (experiment) in fig. 6(a). The second pass band, be-

tween the stop band along ΓM (the [1,1] direction) and the band gap near 1 MHz, has

a single branch, which appears quite isotropic. This isotropic behaviour is confirmed by

the equifrequency contours (fig. 6(b)), which characterize the variation with direction in

the magnitude of the wave vector at a given frequency. The contours are remarkably cir-

cular and shrink in radius as the frequency increases, indicating that the wave vector has

the same magnitude in all directions, and that the group velocity, �vg = ∇ kω(�k), points

towards the centre of each circular contour. Thus, for this band, the group velocity and

wave vector in the first Brillouin zone are antiparallel for all directions of propagation, a

situation resembling left-handed behaviour in negative index metamaterials [37]. A con-

sequence of vg and k being antiparallel is that waves arriving at the surface of the crystal

at non-normal incidence will be negatively refracted. This effect is demonstrated by the

experimental data shown in fig. 6(c), which was obtained by directing a narrowband

pulse with central frequency 0.85 MHz towards the shortest face of the prism at normal

incidence (see the wide blue arrow) and imaging the field that emerged from the longest

face using a miniature hydrophone. Since the wave pulse enters the crystal at normal

incidence, the pulse continues to travel inside the crystal in the original direction, which

is parallel to the group velocity. As the pulse leaves the crystal, the outgoing field pattern

is seen to bend backwards in the negative direction, showing according to Snell’s law that

the wave vector inside the crystal must also point in the negative direction, opposite to

the direction of the group velocity, as predicted from the equifrequency contour. To em-

phasize this point, the directions of the Bloch wave vector and group velocity inside the

crystal are also shown in fig. 6(c), as well as the direction of the refracted beam outside,

which is perpendicular to the wave fronts. (Note that to measure the direction of k, it is

crucial to measure the wave field and not just the intensity so that k can be determined

from the wave fronts, as the position of maximum intensity in the refracted beam in this

pulsed experiment is also influenced by the time the pulse reached the exit surface of the

crystal, with the earlier arrivals being closer to the top of the prism and corresponding

to the signals on the top left part of the measurement area.) Furthermore, the measured

refraction angle is given within experimental uncertainty by Snell’s law, using the value

of the wave vector inside the crystal predicted by MST, providing additional evidence

that the data can be quantitatively described in terms of negative refraction.

The direct observation of negative refraction in this 2D phononic crystal suggests that

it is a good system in which to investigate focusing by negative refraction in flat phononic

crystal lenses, and in particular to examine the ultimate image resolution that may be

possible. For this purpose, a rectangular-shaped six-layer crystal of steel rods with the

same crystal structure was constructed. Each layer contained 60 rods (to avoid edge


effects), and the layers were stacked in the ΓM direction, i.e. with the base of the trian-

gular unit cell parallel to the surface. To explore the resolution capabilities of the lens,

a narrow line source (width 0.55 mm, which is less than the wavelength in water at the

frequencies of interest) was built from piezoelectric polymer strips. When the crystal was

filled and surrounded by water, the best image of the source was measured at 0.70 MHz,

the lowest frequency at which the equifrequency contours are circular. However, the

image resolution, as determined by the Rayleigh criterion (resolution equals half the full

width of the peak Δ, i.e. the distance Δ/2 from the maximum to the adjacent minimum

(zero)), was only 1.15λ, where λ is the ultrasonic wavelength in water. This is not as

good as the diffraction limit of λ/2, which is obtained when all propagating components

of the field from a point source are brought to focus in the image plane, because the

equifrequency contours inside and outside the crystal were not matched, cutting off all

angles of incidence greater than 56.8◦ in this case. To overcome this limitation, a second

crystal was built with thin transparent walls to enable the liquid inside the crystal to

be replaced by methanol, which has a lower sound velocity than water, shrinking the

frequency axis of the dispersion curve by 74%. As a result, the size of the equifrequency

contours of both the crystal and the water outside were perfectly matched at a frequency

of 0.55 MHz in the second band. Thus, all angle negative refraction (AANR) is achieved

at this frequency, and all others down to the bottom of the band at 0.50 MHz. The image

obtained at 0.55 MHz, when the source was placed 1.6 mm from the opposite surface of

the crystal, is shown in fig. 7(a). A good focal pattern is clearly seen, with the focal

spot narrowly confined both perpendicular and parallel to the crystal surface. By fitting

a sinc function (fig. 7(b)), the transverse width of the image was measured to be 3.0 mm,

with a corresponding resolution of 0.55λ. This shows that a flat phononic crystal with

equifrequency contours matched to those of the medium outside is capable of producing

images with an excellent resolution approaching the diffraction limit [34].

To achieve super resolution (better than the diffraction limit), it is necessary to cap-

ture and amplify evanescent waves from the source —something that clearly did not

occur for the data shown in fig. 7(a). However, when the source was brought even closer

to the surface of the crystal, 0.1 mm or λ/25 away, it was found that significantly im-

proved resolution could be obtained [35]. The best resolution was found at a slightly

lower frequency, 0.53 MHz, as shown by the experimental results in figs. 7(c)(d) and

(f), which are compared with Finite Difference Time Domain (FDTD) simulations in

figs. 7(d)-(f). Both the experimental and theoretical resolutions, 0.37λ and 0.35λ, are

clearly better than the diffraction limit. The reason why super resolution can be attained

for this very small source-crystal separation is that some of the evanescent waves from

the source are now able to couple to a bound mode of the crystal, and hence become

amplified sufficiently to participate in image restoration. Evidence for the excitation of

this bound mode can be seen in the field patterns of figs. 7(d) and (e), which show several

subsidiary peaks that are largest at the crystal surface; these additional peaks (not seen

in (a)) decay rapidly with distance from the crystal, as expected for the evanescent decay

of bound crystal modes. Additional evidence for the existence of this bound mode was

obtained from FDTD calculations of the band structure of a finite crystal slab with the

90 J. H. Page

0.002

0.004 (b)

ude

(au

)

0.004ude

(au

)

-4 -2 0 2 4

2

4

6

x (mm)

z (m

m)

-4

2

4

6

z (m

m)

(a)

-4 -2 0 2 40.000

0.002

Am

pli

tu

x (mm)-4

0.000

0.002

Am

pli

tu

(f)(d)

-4 -2 0 2 4

x (mm)

4 -2 0 2 4

x (mm)

(c) (e)

0 2 4 6 8

z (mm)

-2 0 2 4

x (mm)

Fig. 7. – Contour maps of the ultrasonic amplitude (magnitude of the FFT of the wave fieldat the frequencies indicated) on the imaging side of a flat methanol-steel phononic crystal lensfor a 0.55-mm-wide line source, and corresponding plots of the amplitude through the focus.(a) Image measured at 0.55 MHz for a source-lens distance of 1.6 mm. (b) Amplitude parallel tothe lens surface (circles) through the focus in (a). The data are compared with a sinc function(red line), indicating a resolution Δ/2 = 0.55λ. (c) and (e) Images measured (c) and calculatedwith FDTD (e) for a frequency of 0.53 MHz when the source-lens distance is only 0.1 mm. Notethe appearance of a bound mode of the crystal, which decays evanescently as the distance fromthe surface (at z = 0) increases. (d) and (f) Comparison of experiment (circles) and theory(solid curves) for the transverse width of the focal spot (d) and its variation with distance fromthe surface of the crystal (f). Super resolution is evident from the half widths of the primarypeaks in (d), give a resolution of 0.37λ and 0.35λ for experiment and theory, respectively.

same number of layers as in the experiment. These calculations revealed a nearly flat

band that extends from 0.525 MHz at the water to 0.51 MHz at the zone boundary; as it

lies below the water line, this mode is bound to the crystal slab as it cannot propagate in

water. The best focusing is seen at 0.53 MHz, as this frequency lies between the frequency

for perfectly matched equifrequency contours (0.55 MHz) and the resonance frequencies

of the bound mode (0.51–0.525 MHz), but is still close enough to the bound mode that

it can be excited. Calculations of the field patterns inside the phononic crystal indicate

that this bound mode is a slab mode of the crystal, and not a surface mode.

This demonstration that super resolution can be achieved in practice with phononic

crystal lenses is enabling a detailed study of the many factors that can influence the op-

timum resolution. Perhaps the most interesting question concerns the mechanism that

sets the resolution limit for this crystal. This is determined by the largest transverse

wave vector kmax that the crystal will support, with the most logical choice for kmax

being the wave vector at the Brillouin zone boundary of the crystal along the ΓK di-

rection (parallel to the surface of the lens). (Note that since the bound mode that is

excited is a slab mode of the crystal, it is the bulk Brillouin zone boundary and not

the surface Brillouin zone boundary that sets the resolution limit, allowing better res-


olution to be achieved for this triangular lattice than would be found for the surface

modes that were considered by Luo et al. for photonic crystals.) This condition gives

kmax = 4π/3a. If we assume perfect transmission for all transverse wave vectors k⊥less than kmax, and zero transmission for k⊥ greater than kmax, then the image ampli-

tude will vary with distance x parallel to the crystal surface as |∫ kmax

−kmaxexp[ik⊥x]dk⊥| =

|2 sin(kmaxx)/(kmaxx)| so that the resolution limit Δmin/2 = π/kmax = 3a/4. This con-

dition gives Δmin/2 = 0.34λ at 0.53 MHz, which is very close to our experimental and

FDTD results.

In conclusion, these experimental and theoretical results demonstrate the conditions

needed to achieve optimal focusing: i) the equifrequency surfaces/contours should be

spherical/circular, ii) the equifrequency surfaces in the phononic crystal and in the

medium outside should be matched, and iii) the crystal should have a bound mode

at a frequency close to the operational frequency, in order to enable amplification of

evanescent waves from the source, for super resolution to be attained. The analysis

of the maximum possible resolution that can be obtained with the 2D methanol-steel

phononic crystal will be useful for designing new phononic crystal lenses in which the

super resolution may be enhanced.

4. – Conclusions

Experiments with ultrasonic waves are playing an increasing important role in probing

and understanding the rich diversity of wave phenomena that occur in strongly scatter-

ing media. In disordered media, the phase sensitivity of ultrasonic detectors enables

pulsed experiments to separate the coherent, forward-scattered signal, which propagates

ballistically through the medium, from the multiply scattered coda. Thus, transmission

experiments can be used to obtain a very complete set of measurements of wave transport

through the medium, allowing the parameters that describe both ballistic and diffusive

propagation to be compared over a wide frequency range. Such measurements have been

performed on a simple model system of glass beads in water, illustrating the potential of

ultrasound for gaining useful insights into the character of wave transport in the presence

of strong multiple scattering, and laying a useful foundation for future experiments on

more complex systems.

In the second part of this paper, the properties of ordered acoustic media, or phononic

crystals, have been reviewed. The main emphasis has been on focusing by negative re-

fraction, where super-resolution imaging has recently been demonstrated experimentally.

This area of research continues to grow, providing complementary information and ap-

plications to analogous optical experiments on photonic crystals.

∗ ∗ ∗

I would like to thank the many students and colleagues who have contributed to the

research that has been reviewed in this paper. Support from NSERC is also gratefully

acknowledged.

92 J. H. Page

REFERENCES

[1] Sheng P., Introduction to Wave Scattering, Localization and Mesoscopic Phenomena

(Academic Press, San Diego) 1995.

[2] Page J. H., Schreimer H. P., Bailey A. E. and Weitz D. A., Phys. Rev. E, 52 (1995)3106.

[3] Page J. H., Sheng P., Schreimer H. P., Jones I., Jing X. and Weitz D. A., Science,271 (1996) 634.

[4] Schreimer H. P., Cowan M. L., Page J. H., Sheng P., Liu Z. and Weitz D. A.,Phys. Rev. Lett., 79 (1997) 3166.

[5] Cowan M. L., Beaty K., Page J. H., Liu Z. and Sheng P, Phys. Rev. E, 58 (1998)6626.

[6] Faez S., Johnson P. M. and Lagendijk A., Phys. Rev. Lett., 103 (2009) 053903.

[7] Sommerfeld A., Ann. Phys., 44 (1914) 177.

[8] Brillouin L., Wave Propagation and Group Velocity (Academic Press, New York) 1960.

[9] van Albada M. P., van Tiggelen B. A., Lagendijk A. and Tip A. P., Phys. Rev.

Lett., 66 (1991) 3132.

[10] Liu J., Ye L., Weitz D. A. and Sheng P., Phys. Rev. Lett., 65 (1990) 2602.

[11] Leighton T. G., The Acoustic Bubble (Academic Press, San Diego) 1994.

[12] Leroy V., Strybulevych A., Page J. H. and Scanlon M. G., J. Acoust. Soc. Amer.,123 (2008) 1931.

[13] Leroy V., Strybulevych A., Scanlon M. G. and Page J. H., Euro. Phys. J., 29

(2009) 123.

[14] Leary D., de Bruyn J. R. and Page J. H., Bull. Am. Phys. Soc., 46(1) (2001) 947.

[15] Chu S. and Wong S., Phys. Rev. Lett., 48 (1982) 738.

[16] Oughstun K. E. and Sherman G. C., Electromagnetic Pulse Propagation in Causal

Dielectrics (Springer-Verlag, Berlin) 1994.

[17] Zhang Z. Q., Jones I. P., Schreimer H. P., Page J. H., Weitz D. A. and Sheng P.,Phys. Rev. E, 60 (1999) 4843.

[18] Weaver R. L. and Sachse W., J. Acoust. Soc. Am., 97 (1995) 2094.

[19] Tourin A., Derode A., Roux P., van Tiggelen B. A. and Fink M., Phys. Rev. Lett.,79 (1997) 3637.

[20] Durian D. J., Phys. Rev. E, 50 (1994) 857.

[21] Psorobas I. E. (Editor), Phononic Crystals: Sonic bandgap materials, a special issue ofZietschrift fur Kristallographie, 220 (2005).

[22] Economou E. N. and Sigalas M. M., Phys. Rev. B, 48 (1993) 13434; J. Acoust. Soc.

Am., 95 (1994) 1734.

[23] Kushwaha M. S., Halevi P., Dobrzynski L. and Djafari-Rouhani B., Phys. Rev.

Lett., 71 (1993) 2022; Kushwaha M. S., Djafari-Rouhani B., Dobrzynski L. andVasseur J. O., Eur. Phys. J. B, 3 (1998) 155.

[24] Kafesaki M. and Economou E. N., Phy. Rev. B, 60 (1999) 11993; Psarobas I. E.,

Stefanou N. and Modinos A., Phys. Rev. B, 62 (2000) 278; Liu Z., Chan C. T.,

Sheng P., Goertzen A. L. and Page J. H., Phys. Rev. B, 62 (2000) 2446.

[25] Liu Z., Zhang X., Mao Y., Zhu Y. Y., Yang Z., Chan C. T. and Sheng P., Science,289 (2000) 1734.

[26] Torres M., Montero de Espinosa F. R. and Aragn J. L., Phys. Rev. Lett., 86 (2001)4282.

[27] Vasseur J. O., Deymier P. A., Chenni B., Djafari-Rouhani B., Dobrzynski L. andPrevost D., Phys. Rev. Lett., 86 (2001) 3012.


[28] Yang S., Page J. H., Liu Z., Cowan M. L., Chan C. T. and Sheng P., Phys. Rev.

Lett., 88 (2002) 104301.[29] Page J. H., Yang S, Liu Z., Cowan M. L., Chan C. T. and Sheng P., Z. Kristallogr.,

220 (2005) 859.[30] Mei J., Liu Z., Shi J. and Tian D., Phys. Rev. B, 67 (2003) 245107.[31] Yang S., Page J. H., Liu Z., Cowan M. L., Chan C. T. and Sheng P., Phys. Rev.

Lett., 93 (2004) 024301.[32] Zhang X. and Liu Z., Appl. Phys. Lett., 85 (2004) 341.[33] Ke M., Liu Z., Cheng Z., Li J., Peng P. and Shi J., Solid State Comm., 142 (2007)

177.[34] Sukhovich A., Jing L. and Page J., Phys. Rev. B, 77 (2008) 014301.[35] Sukhovich A., Merheb B., Muralidharan K, Vasseur J. O., Pennec Y., Deymier

P. A. and Page J. H., Phys. Rev. Lett., 102 (2009) 154301.[36] Sainidou R., Stefanou N. and Modinos A., Phys. Rev. B, 66 (2002) 212301.[37] Pendry J. B., Phys. Rev. Lett., 85 (2000) 3966.[38] Luo C., Johnson S. G., Joannopoulos J. D. and Pendry J. B., Phys. Rev. B, 68

(2003) 045115.



Anderson localization of ultrasound

in three dimensions

J. H. Page



Summary. — Some fifty years after Anderson localization was first proposed,there is currently a resurgence of interest in this phenomenon, which has remainedone of the most challenging and fascinating aspects of wave transport in randommedia. This paper summarizes recent progress in demonstrating the localizationof ultrasound in a “mesoglass” made by assembling aluminum beads into a disor-dered three-dimensional elastic network. In this system, the disorder is sufficientlystrong that interference leads to trapping of the waves at intermediate frequencies,as demonstrated by studying three different fundamental aspects of Anderson lo-calization: time-dependent transmission, transverse confinement of the waves, andthe statistics of the non-Gaussian intensity fluctuations. Additional ultrasonic ex-periments have been performed to reveal the multifractal character of the wavefunctions near the Anderson transition. This is the first time that so many differentaspects of localization have been studied simultaneously, providing very convinc-ing evidence for the localization of ultrasonic waves in the presence of disorder inthree dimensions, and enabling new aspects of Anderson localization to be studiedexperimentally.


96 J. H. Page

1. – Introduction

During the 1980s, it was realized that Anderson localization [1,2] —the spatial trap-

ping of waves due to disorder— is not only a quantum effect, but is, more generally,

a phenomenon that may occur for any type of wave: quantum or classical. This phe-

nomenon results from the interference of waves that have been multiply scattered in a

disordered medium, and therefore should be observable so long as the disorder is suffi-

ciently strong and coherence is maintained, the latter condition being readily satisfied

for classical waves such as light, microwaves, sound, elastic waves, and even seismic

waves. To appreciate the analogy that exists between quantum and classical wave be-

haviour in disordered media, one need go no further than to compare the Schrodinger

and Helmholtz equations in the presence of disorder, as is outlined in appendix A. This

comparison shows that these equations for quantum and classical waves have the same

form, but with an important difference. This difference implies that it is only possible to

localize classical waves at intermediate frequencies where the wavelength is comparable

to the size of the scattering inhomogeneities, and not more or less trivially at very low

frequencies, as in the case for electrons. This absence of “bound states” for classical

waves, and the need to achieve sufficiently strong scattering, makes the localization of

classical waves challenging to observe in practice.

Not withstanding these challenges, there are several reasons why classical waves are

potentially better adapted to observing the phenomenon of Anderson localization di-

rectly. For electrons or other quantum particles (e.g., cold atoms), experiments must

be performed at low temperatures to minimize the effects of inelastic scattering, which

destroys phase coherence. There is no such restriction for classical waves. Classical waves

also have the advantage that the analogue of electron-electron interactions (nonlinear-

ities) can be avoided by suitable choices of materials and power levels. Perhaps most

significant is the versatility of experiments with classical waves, where measurements as

a function of both time and space are feasible, potentially yielding much more informa-

tion about localization than is possible by simply measuring the total transmittance at

a single frequency. The latter is equivalent to measuring the overall sample conductance

for electronic systems, the technique that has been used almost exclusively in studies of

electron localization.

Once these advantages of classical waves were appreciated, experimental work show-

ing strong localization of both acoustic and electromagnetic waves followed in one- and

two- dimensional systems (1D and 2D), as well as in quasi-1D waveguides [3-8]. These

were significant steps forward, as they permitted localized wave functions and their sta-

tistical properties to be studied directly, stimulating many new theoretical advances as

well. However, the central question in the field, whether or not classical waves could

be localized in three dimensions (3D), has been more difficult to answer, despite several

tour-de-force experiments in optics [9-11]. Three dimensions is especially important, as

it is only in 3D that scaling theory predicts the existence of a real transition from propa-

gating to localized modes [12]. In seeking experimental evidence, one of the problems, in

addition to the challenges mentioned above, has been absorption, which is always present

Anderson localization of ultrasound in three dimensions 97

to some extent for classical waves, and which leads to a reduction in total transmission

having the same dependence on sample thickness as localization. Thus, to demonstrate

3D localization convincingly, it is necessary to combine a number of experimental ap-

proaches that can probe key signatures of localization: e.g., anomalous dynamics (time

dependence), spatial confinement of the waves, and the statistics of the large intensity

fluctuations.

This paper describes the recent progress that has been achieved using ultrasonic

experiments, in combination with advances in the self-consistent theory of localization,

to unambiguously demonstrate Anderson localization in three dimensions [13]. One key

to this success has been the construction of sufficiently strongly scattering samples, which

are described in the next section (sect. 2). The following sections outline the three main

experimental approaches that have been exploited to obtain evidence of localization:

time-dependent transmission (sect. 3), transverse confinement (sect. 4) and statistics

(sect. 5). This last section ends with an example of an aspect of localization in 3D

that has not been accessible to experimental study previously, namely the structure of

localized wave functions as characterized by their multifractal properties [14].

2. – Mesoglasses: porous elastic solids with very strong scattering

In samples suitable for localization experiments, it is generally important to maxi-

mize the scattering strength and minimize the absorption (or dissipation). Despite the

very large scattering contrast between solids and liquids for ultrasonic waves, with differ-

ences in acoustic impedance as large as 10 to 60 being readily achievable, suspensions of

solid spherical particles in a fluid, as described in [15], were found to have insufficiently

strong scattering. This is true even in the intermediate frequency regime, where the

wavelength is comparable to the size of the scatterers and shape resonances can enhance

the scattering. The other problem with such samples is the relatively large dissipation,

one important contribution being viscous losses at the interface between the solid and

liquid phases. To avoid this difficulty, we decided to take a different approach and in-

vestigate porous, single-component solid systems instead. Our initial experiments were

performed on highly porous solid networks of well-sintered glass beads, revealing inter-

esting plateaus in both the diffusion coefficient and the density of states [16]. However,

in spite of very strong scattering, no evidence of localization was seen in these early

experiments. To make better samples for observing Anderson localization, three impor-

tant steps were taken: the glass particles were replaced by aluminum, thereby further

reducing the intrinsic absorption in the constituent particles; the particle radius a was

increased, allowing higher effective frequencies (ka) to be accessed using the same ultra-

sonic transducers; and a new way was developed for joining the particles together into

a solid network, allowing for better control of the interparticle contacts, and hence the

scattering strength. In this way, a disordered elastic network of aluminum beads was cre-

ated by brazing the beads together to form weak elastic bonds between the beads while

preserving their spherical shape, as shown in fig. 1. Such a structure may be viewed as a

“mesoglass” in which the beads are linked by narrow necks to form a disordered material

98 J. H. Page

Fig. 1. – One of the samples in which the localization of ultrasonic waves was observed in theintermediate frequency regime. The aluminum beads are brazed together with weak elastic linksto form a disordered solid network, in which the ultrasonic waves become trapped after strongmultiple scattering from the pores.

with mesoscopic particles as the building blocks rather than atoms, and with elastic bonds

between the particles rather than interatomic forces. The beads were monodisperse, with

a diameter of 4.11 ± 0.03mm, and the samples had an aluminum volume fraction of ap-

proximately 55%, corresponding to random loose packing of the beads before brazing.

The samples were slab-shaped, with circular cross sections of diameter much larger than

the thickness L, which ranged from 8 mm to 23mm.

To study the samples using ultrasonic immersion techniques [15,17], the samples were

first waterproofed with very thin plastic walls, so that the samples remained dry when

immersed in a water tank between the generating and detecting transducers. This pro-

cedure ensured that wave transport occurred through the aluminum network, where the

incident acoustic wave from the water (longitudinal polarization only) was converted into

an elastic wave in the solid (which supports both longitudinal and transverse polariza-

tions). Measurements were performed over a wide frequency range from 100kHz to sev-

eral MHz, since this corresponds to the intermediate frequency regime for this structure,

where the wavelength is comparable to the bead and pore sizes and very strong scatter-

ing is expected. Initial characterization of the samples was performed by measuring the

amplitude transmission coefficient, obtained from the ratio of the fast Fourier transforms

of the transmitted and input signals. These measurements confirmed the existence band

gaps in this type of system, as first reported by Turner and Weaver in 1998 [18]. The

gaps occur because the coupled resonances of the aluminum beads broaden to form pass

bands, with band gaps forming in between; wide band gaps are observed so long as the

coupling is not too strong, which is the case for our samples. For some of the samples, it

was possible to extract the coherent ballistic pulse and measure the longitudinal phase

and group velocities vp and vg, as well as the scattering mean free path l. Very strong

scattering was demonstrated by the observation that, outside the band gaps, the product


of wave vector and mean free path varied from nearly 1 to 2.5 over this frequency range,

thus approaching the Ioffe-Regel limit kl = 1. Although we were not able to measure kl

for transverse waves in these experiments, previous experiments on sintered glass bead

networks have shown similar values of kl for both transverse and longitudinal waves in

the strong scattering regime [19]. Localization is expected when kl � 1, but the exact

critical value klc at which the transition occurs is not known [20], and is likely to be

wave and sample dependent; thus, these initial measurements suggest that localization

of ultrasonic waves may indeed be possible in these samples.

3. – Time-dependent transmission

Our first experiment to investigate wave transport by multiply scattered waves in

these samples was performed using a short quasi-planar incident pulse and measuring

the time-dependent transmitted field with a miniature hydrophone. The hydrophone was

scanned over a square 55×55 grid parallel to, and within a few wavelengths of, the sample

surface. The grid separation was typically equal to the wavelength in water (for more

details on the method, see [15] and [17]). A schematic diagram of the setup is shown in

fig. 2(a). Several representative waveforms measured at different positions in the speckle

pattern are shown in fig. 2(b); this example was recorded for a two-cycle input pulse with

a central frequency of 0.25MHz, showing that data for a long range of propagation times,

corresponding to progressively longer and longer multiple scattering paths, is observable

in these samples. Before determining the time-dependent intensity I(t), the waveforms

were digitally filtered to limit the bandwidth to 5% of the central frequency of the pulse.

The average transmitted intensity I(t) was then determined by squaring the envelope

of the field at each position, averaging over each position in the speckle pattern, and

normalizing by the peak of the input pulse. Typical time-dependent intensity profiles

are shown in figs. 2(c) and (d), revealing the excellent signal-to-noise and large dynamic

range obtained at both low and high frequencies in the range of interest between 0.1 and

3 MHz.

Note that, even though both compressional (longitudinally polarized) and shear

(transversely polarized) waves are excited in these elastic materials, in which shear waves

typically propagate at roughly half the speed of longitudinal waves, a single intensity

profile I(t) is seen at all frequencies. This occurs because the polarizations mix at each

scattering event. Thus, after only a few scatterings, the total energy density becomes

equipartitioned between compressive and shear waves [21-23], according to their respec-

tive energy densities UL and UT . For weak disorder,

UL(ω) ∝ ρL(ω) =k2

2π2

dk

dω∝ ω2

v2

p,Lvg,L(1)

UT (ω) ∝ 2ω2

v2

p,T vg,T.(2)

Since the energy density depends on the inverse of the cube of the wave speeds, transverse

100 J. H. Page

0 100 200 300 400

-0.2

-0.1

0.0

0.1

0.2 SPECKLE 1

SPECKLE 12

SPECKLE 25

Wav

e F

ield

(a.

u.)

Time (μs)

1E

0.01f

y

(b)

0 100 200 300 400

1E-9

1E-8

1E-7

1E-6

1E-5 f = 2.4 MHz Experiment

Self consistent Theory

Diffusion Theory

Norm

aliz

ed I

nte

nsi

ty

Time (μs)

0 100 200 300 4001E-7

1E-6

1E-5

1E-4

1E-3

f = 0.2 MHz Experiment

Diffusion Theory

Norm

aliz

ed I

nte

nsi

ty

Time (μs)

(c)

(d)

Fig. 2. – (a) Schematic diagram indicating the experimental setup for measuring the time-dependent transmission. (b) The transmitted field measured by the hydrophone at three dif-ferent positions near the sample surface. The source was a short two-cycle pulse with a centralfrequency of 0.25 MHz. The peak of the input pulse occurs at t = 0. (c) Transmitted intensityI(t) at a representative frequency of 0.2 MHz in the diffuse regime for a sample with thick-ness L = 14.5 mm. The best fit to diffusion theory (solid red curve) with R = 0.85 yieldsD = 3.0 mm/μs and l∗ = 2.5 mm, with τa being to large to be measurable. (d) I(t) for the samesample at a frequency of 2.4 MHz in the localized regime. The data cannot be fitted by diffusiontheory (dashed blue curve), but is well fitted by the self-consistent theory of localization (solidred curve) with ξ = 15mm, l∗B = 2mm, DB = 16mm2/μs and τa = 160 μs.

waves dominate the energy transport. At sufficiently low frequencies, one might expect

that the transport is diffusive and that there are no renormalization effects due to inter-

ference, so that the Boltzmann diffusion coefficient DB = vEl∗B/3 and transport mean

free path l∗B are observed. In this regime, providing that the scattering is isotropic (l = l∗)and the energy velocity is equal to the group velocity, a simple relation can be derived

for the effective energy-density-weighted average diffusion coefficient:

(3) D =1

3

(l∗L

v2

p,L

+2l∗Tv2

p,T

)/(1

v2

p,Lvg,L+

2

v2

p,T vg,T

)=

1

3vE l∗.

Here, the effective transport mean free path is given by

(4) l∗ =

(l∗L

v2

p,L

+2l∗Tv2

p,T

)/(1

v2

p,L

+2

v2

p,T

)


and the effective energy velocity by

(5) vE =

(1

v2

p,L

+2

v2

p,T

)/(1

v2

p,Lvg,L+

2

v2

p,T vg,T

).

While the general case is more complex, these relations give some insight into the conse-

quences of energy equipartition, and provide a simple starting point for comparing data

with the predictions of the diffusion approximation whenever the ballistic parameters

can be measured.

For our brazed aluminum bead samples, the ultrasonic wave transport was indeed

found to be diffusive at low frequencies. The evidence for diffusive transport can be seen

in the exponential decay of the ensemble-averaged transmitted intensity at long times,

I(t) ∝ exp[−t/τD], which was observed for frequencies up to 0.4MHz. In this frequency

range, the entire time dependence of I(t) is well described by diffusion theory, as is il-

lustrated in fig. 2(c), which compares diffusion theory with data measured at 0.2MHz.

By fitting theory to experiment, the diffusion coefficient was determined, and the trans-

port mean free path was estimated from the dependence of the transmitted intensity on

boundary conditions. One consequence of the equipartition of elastic energy inside the

sample is that the internal reflection coefficient R is large, as the outside medium only

supports longitudinal waves; nonetheless R can still be reliably determined from the mea-

sured ballistic parameters by accounting for the angle-dependent reflection coefficients

for all polarizations [24], thereby reducing the number of fitting parameters. In this

frequency range, D was found to be roughly independent of frequency, consistent with

earlier experiments on sintered glass bead networks [16]. Significantly, we found that ab-

sorption, which attenuates I(t) by the factor exp[−t/τa], where τa is the absorption time,

was too small to measure in this frequency range, consistent with our expectations that

absorption would be much much lower for these samples than in most strongly scattering

acoustic systems, such as suspensions of particles in a fluid [17]. The success in inter-

preting these data using diffusion theory establishes that multiply scattered ultrasound

propagates diffusively in the lower part of the intermediate frequency range, which is the

diffusive regime for this system.

In the upper part of the intermediate frequency range (∼ 2 MHz), the time depen-

dence of I(t) shows qualitatively different behavior: at long times, I(t) decays more slowly

than in the diffusive regime, with a non-exponential tail that cannot be explained by the

diffusion approximation (fig. 2(d)). This behavior has been viewed as a slowing down of

the effective diffusion coefficient D(t) with propagation time, reflecting a time-dependent

renormalization of D due to interference effects associated with localization [7,10]. Phys-

ically, the waves become trapped by the disorder, but eventually manage to escape from

the sample, suggesting that Anderson localization may be occurring in these samples. To

interpret these data quantitatively, we exploit recent progress in the self-consistent the-

ory of localization, initially developed for electron localization by Vollhardt and Wolfle

in 1980 [25, 26]. The basic idea in this theory is to describe the renormalization of

the diffusion coefficient by accounting for constructive interferences between reciprocal

102 J. H. Page

paths, which lead to an increased probability that a quantum particle or classical wave

returns to the same spot. The recent progress [27,28] enables the dynamics of wave trans-

port to be predicted for experiments such as ours, which involve open three-dimensional

systems. The new aspect of this theory is the incorporation of boundary conditions self-

consistently, thereby accounting for the fact that the return probability is less reduced

near the boundaries, where the wave may escape, so that the renormalization of D by

interference is less there. This leads to a position-dependent dynamic diffusivity kernel

D(�r, t− t′). The solid curve in fig. 2(d) is a fit of the self-consistent theory to our data at

2.4MHz, and gives an excellent description of the experiment at all propagation times.

From this fit, we are able to determine the localization length ξ = 15mm for this sample.

This measurement of ξ is feasible since several key parameters (l and vp, and hence kl

and R) are known for our sample from independent ballistic measurements, leaving the

localization length, the bare transport mean free path l∗B, the diffusion time τD, and the

absorption time τa as fitting parameters. In the self-consistent theory, the localization

length is related the ratio of kl to its critical value at the mobility edge, χ = kl/(kl)c,

by ξ/l∗B = [6/(kl∗B)2c ]χ2/(1 − χ4). Thus ξ is positive in the localization regime where

χ < 1, and negative in the diffuse regime, where the absolute value of ξ plays the role

of a correlation length in the vicinity of a phase transition. The most important points

to emerge from the fitting are not only that the self-consistent theory describes the time

dependence of the measured I(t) very well, but also that it is only possible to fit to the

experimental measurements with the theory when ξ > 0. This gives strong evidence for

the dynamic localization of ultrasound in our sintered aluminum bead system.

4. – Transverse confinement

The measurements of the time-dependent transmission described in the previous sec-

tion give only indirect evidence of Anderson localization. Is it possible to observe local-

ization more directly? To answer this question, the quasi-plane wave source was replaced

by a point source (approximately a wavelength wide), and the transmitted intensity was

measured as a function of both position and time on the opposite face of the sample. The

point source was obtained by focusing the input pulse through a narrow aperture onto

the sample surface at the point ρ = 0, as shown schematically in fig. 3(a). The transmit-

ted wave field was measured with subwavelength resolution using a hydrophone, which

was moved over a range of transverse positions ρ for a given position of the source. The

transmitted intensity I(ρ, t) was calculated from the measured field as indicated in the

previous section. To average the intensity for each ρ and t over a large number (typically

552 = 3025) of speckle spots, the position of the sample was scanned in the x-y plane

parallel to the surface of the sample. Typical data for I(ρ, t) at 2.4MHz, measured on

the same sample for which I(t) is reported in fig. 2, are shown in fig. 3(b). As expected,

data for larger ρ start later because the distance from the source is greater; what was not

expected is that the curves for different ρ decay at essentially the same rate at long times,

i.e. they differ by a time-independent factor at long times. To understand this behaviour,

the crucial quantity is the ratio I(ρ, t)/I(0, t), which probes the dynamic spreading of the


11.0L = 14.5 mm; L / ξ = 1 ρ = 30 mm

25

0 20 40 60 80 100 120 140 160 180 2001E-11

1E-10

1E-9

1E-8

1E-7

1E-6 ρ 15 mm

ρ 20 mm

ρ 25 mm

ρ 30 mm

ρ 0 mm

ρ 10 mm

Norm

aliz

ed inte

nsi

ty

Time (μs)

(a)

d(c)

(b)

0.0 0.5 1.0 1.5 2.0

1E-4

1E-3

0.01

0.1

Expt Theory

t = 10 μs

t = 25 μs

t = 50 μs

t = 98 μs

t = 150 μs

t = 195 μs

exp[ (ρ /L) 2

]

I(ρ,t

) /

I(0,t

)

ρ / L

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

L 23.5 mm; L / ξ 3.6

ρ 25 mm

ρ 20 mm

ρ 15 mm

ρ 25 mm

ρ 20 mm

ρ 15 mm

w 2 /

L 2

Time (μs)

( )

Fig. 3. – (a) Schematic illustration (not to scale) of the setup for measuring the dynamic trans-verse confinement of the transmitted intensity emitted by a point source in the localizationregime. (b) Average time- and position-dependent intensity, I(ρ, t) at several positions ρ for thesetup shown in (a). The frequency (2.4 MHz) and sample are the same as in fig. 2(d). (c) Meansquare width w2

ρ(t) of the intensity ratio I(ρ, t)/I(0, t) as a function of time for the data shownin (b) and for a second sample with a thickness L = 23.5 mm. The frequency is 2.4MHz for bothsamples. The solid curves are the best fits of the self-consistent theory to the experimental data(symbols) with l∗B = 2mm, DB = 11 mm2/μs, ξ = 15mm for the L = 14.5 mm sample, and ξ =7mm for the L = 23.5 mm sample. Other parameters used in the calculations, kl = 1.8 and R =0.82, were determined from independent ballistic measurements. The dashed line shows the lin-ear time dependence of w2 that would occur for diffuse waves, using a value for D of 1.25 mm2/μs.(d) Dependence of the intensity ratio on distance ρ at selected times, showing the non-Gaussianprofile that is found both experimentally (symbols) and theoretically (solid curves).

intensity profile in a plane parallel to the surface of the sample. Most importantly, this

ratio is independent of absorption, since at each time, the absorption factor exp[−t/τa] is

the same for any ρ and therefore cancels in the ratio. We characterize this dynamic spatial

profile of the intensity by its width wρ(t), defined by I(ρ, t)/I(0, t) = exp[−ρ2/w2ρ(t)].

As shown in ref. [15], w2ρ(t) = 4Dt ∝ t in the diffuse regime, providing an accurate

method for measuring D that is independent of absorption and boundary conditions. By

contrast, in the localized regime, the transverse width wρ(t) exhibits completely different

behaviour, shown in fig. 3(c) for two samples of different thickness. Instead of increasing

104 J. H. Page

linearly with propagation time, w2ρ(t) saturates, approaching a constant value for each

ρ at long times. Furthermore, the transverse spatial profile of the intensity is no longer

Gaussian in the localized regime, since w2ρ(t) depends on ρ —another clear departure

from diffuse behaviour that is especially evident in the thinner sample. These results

indicate that the data cannot be explained by assuming a diffusion coefficient D(t) that

depends only on time, since in this case the magnitude of w(t) ∼ ∫D(t)dt would be

independent of ρ(1). The observed non-Gaussian shape of the spatial profiles is shown

explicitly in fig. 3(d) for a range of times separated by (almost) equal intervals at long

times, but by narrower intervals at early times. These data were measured on the thinner

sample. A Gaussian curve with width equal to the sample thickness (dashed curve) is

also included in this figure for comparison. Both figs. 3(c) and (d) show that the intensity

profile initially grows with time, but then converges to a constant profile at long times,

revealing how the initial propagation of the waves away from the source is brought to

a halt by localization. This is exactly what is meant by localization. These data are

therefore a very direct demonstration of 3D Anderson localization, and are, to the best

of our knowledge, the most direct observations of this phenomenon to date.

Additional information about Anderson localization in these samples can be obtained

by comparing the data with the predictions of the self-consistent theory. The solid curves

in fig. 3(c) demonstrate that the behaviour of the dynamic transverse width is accurately

predicted by the self-consistent theory, which gives an excellent fit to the data for all t

and ρ, with a single set of parameters for each sample. The fits give ξ = 15mm for the

thinner sample (L = 14.5mm —the same sample whose the time-dependent transmission

is plotted in fig. 2), and ξ = 7mm for the thicker sample (L = 23.5 mm). These results

suggest that the scattering is stronger in the thicker sample due to small differences in

microstructure, showing that ξ is very sensitive to the degree of disorder, quantified by

kl, near the localization threshold, as theory predicts. It is worth emphasizing explicitly

that the non-Gaussian character of the experimental intensity profiles is quantitatively

described by the self-consistent theory, showing the importance of accounting for the

position dependence of D in the theory. These measurements of the localization length

at 2.4MHz enable us to estimate the proximity to the mobility edge (kl)c, with kl being

only 1% below (kl)c at this frequency.

Compared with the plane wave case 2(d), these fits of the self-consistent theory for

w2ρ(t) provide a more accurate determination of the localization length. One reason is

the elimination of absorption, so there is one less parameter to fit. In addition, the self-

consistent theory predicts that the transverse width at long times depends predominantly

on both the localization length ξ and sample thickness L, and since it is straightforward

to measure L, the measurement of w2ρ(t) provides a more direct way of determining ξ.

(1) Indeed, if one were to use the solutions of the standard diffusion equation assuming onlythat the diffusion coefficient varies with time, serious inconsistencies are found when comparingD(t) extracted by this approach for planar and point source geometries, yielding results for D(t)differing by two orders of magnitude [29]. This comparison provides another indication of theinadequacy of this approach for our data.


0 50 100 150 200 250 300 3500.0

0.2

0.4

0.6

0.8

1.0

t 1/2

ρ = 25 mm

0.7 MHz

1.0 MHz

1.4 MHz

1.8 MHz

2.4 MHz

w 2 /

L 2

10 100 5000.1

0.2

0.3

0.4

0.5

0.6

0.70.80.9

1

t 2/3

t 1/2

1.0 MHz

1.4 MHz

1.8 MHz

2.4 MHz

w 2 /

L 2

(a)(b)

Time (μs) Time (μs)

Fig. 4. – (a) Time dependence of the transverse width for several frequencies in the strongscattering regime, showing that the width increases as the frequency is lowered. The meansquare width is normalized by the sample thickness squared (L = 23.5 mm for these data). Thedata are plotted for ρ = 25mm, but the dependence on ρ is not large for this thick sample(see fig. 3). A mobility edge can be inferred to lie at a frequency between 1 and 1.8 MHz forthis sample. (b) The same data plotted on doubly logarithmic scales to display the power-lawbehaviour of the time dependence near the mobility edge. At 1.0 MHz, the data are consistentwith a crossover from t2/3 to t1/2 behaviour, as predicted by the dynamic self-consistent theoryof localization.

For thick samples (L � ξ), the width is no longer influenced by ρ, i.e. the statistical

profile is again Gaussian, and the dependence of w2 at long times on L and ξ has been

shown to have a simple scaling form [30]: w2(t → ∞) ≈ 2Lξ(1 − ξ/L); thus, to leading

order, the long time limit of w2 is simply 2Lξ.

One of the interesting predictions of the theory is a strong and rapid renormalization

of the effective diffusion coefficient. As a result, DB cannot be measured directly even at

the earliest times at which transmission measurements can be made. The best fits give

surprisingly large values of DB, which imply vE > vp. Further theoretical work is needed

to understand these apparently very large values of the energy velocity in the localized

regime.

As the frequency is lowered, the transverse width increases, as is shown in fig. 4. One

would expect that in the localized regime, the width wρ(t) will always saturate at long

times, whereas in the diffuse regime, the width will continue to grow with time as the

energy density continues to expand in the transverse direction, albeit slowly near the

mobility edge. These expectations are confirmed by the self-consistent theory, where

recent calculations for thick samples show that the asymptotic value of w2(t) as t → ∞remains finite not only in the localized regime but even at the mobility edge, where the

saturation value is approximately equal to the thickness of the sample, w∞ ≈ L [30].

The experimental results plotted in fig. 4 appear consistent with these predictions. At

the lowest frequency shown (0.7MHz), the width clearly continues to increase without

limit, and its increase is almost linear, indicating that at this frequency, the transport is

106 J. H. Page

subdiffusive. At 1.0MHz, w2ρ(t)/L2 remains less than 1 throughout the range of times

where the signal could be measured, but is still increasing at the longest times, suggesting

that this frequency is very close to, but on the diffusive side of, the critical value fc at

the mobility edge. At higher frequencies, w2ρ(t)/L2 remains well below 1, and clearly

saturates at long times for frequencies above 1.8MHz, indicating that localization has

set in. These results illustrate the behaviour of the transverse width in the vicinity of the

Anderson transition, suggesting that it should be possible to use transverse confinement

to measure the variation of the localization length as the mobility edge is approached.

Work is currently in progress to examine this behaviour in detail.

Another interesting question concerns the time dependence of the transverse width,

especially at early times where it characterizes the initial growth of the spatial inten-

sity profile. In the localized regime, one might anticipate that the intensity would

spread out from the source at a slower rate than for diffuse waves; does this imply

that w2ρ(t) still increases as a power law, with a smaller exponent than in the diffuse

regime, or is the behaviour more complex? To examine possible power-law behaviour

more closely, the experimental results for w2ρ(t)/L2 are replotted in a doubly logarithmic

scale in fig. 4(b). This figure shows that such behaviour is exhibited at 1.0MHz near

the mobility edge, but that there appear to be two power-law regimes. The data are

consistent with an initial growth of wρ(t)2/L2 as t2/3, followed by a slower t1/2 regime,

which at this frequency extends up to the maximum time at which signals could be

recorded reliably. A t1/2 regime is also seen over a smaller range of times (∼ half a

decade) at 1.4MHz, before w2ρ(t)/L2 starts to level off towards a constant value (the

long-time data are not plotted here because the signal-to-noise ratio is poor at these

times, preventing an accurate measurement of how the width levels off, although it is

nonetheless clear that it drops below the t1/2 curve for times t > 220μs). These power

laws have been predicted by the self-consistent theory for the initial expansion of the

intensity from a point source located deep inside a thick sample (L � l) [30], where

a relatively simple estimate of the return probability may be obtained by neglecting

the position dependence of D. By solving for D(Ω) at the mobility edge, Cherroret etal. show that the mean-square radius of the 3D intensity profile grows as t2/3 at short

times (t � L2/DB), and as t1/2 at longer times (t � L2/DB) [30]. Assuming that the

mean-square transverse width for a slab sample scales with time in a similar way near

the mobility edge, this provides a qualitative explanation of the time dependence seen in

their numerical calculations [30] and in our experimental results (fig. 4). However, this

simple argument for the initial time dependence of w2(t) does not explain the saturation

at longer times.

To end this section, it is worth emphasizing again that these measurements of the

dynamic transverse confinement of the intensity due to localization are independent of

absorption, which has been a major obstacle to reaching definitive conclusions in previous

experiments [6, 9-11, 31]. The method provides a direct way of observing the trapping

of waves by disorder. The behaviour revealed in figs. 3(c) and (d) is both qualitatively

and quantitatively different to that seen for diffusive waves, and provides unambiguous

evidence for the localization of ultrasound in these 3D samples.


5. – Statistical approach to localization

The previous sections have examined the marked changes that occur in the temporal

and spatial profiles of the average intensity in the localized regime. Localization also leads

to very large fluctuations in the transmitted intensity, and the measurement and analysis

of their statistical properties can be used to reveal other signatures of localization [6]. To

investigate this statistical approach to localization, we have measured the large spatial

fluctuations of the intensity that occur in ultrasonic speckle patterns [13]. These were

measured by scanning the hydrophone in a plane near the surface of the samples when

illuminated on the opposite side with short pulse having a spatial profile that corresponds

quite closely to a broad Gaussian beam. By taking the Fourier transform of the measured

variations in the transmitted field, ψ(x, y, t), the variation of the intensity I(x, y) at each

frequency in the bandwidth of the incident pulse was determined, enabling the near-field

speckle patterns to be plotted, as illustrated in figs. 5(a) and (b) for frequencies in the

diffuse and localized regimes, respectively. Even by eye, a clear difference can be seen

between these two cases. In the diffuse regime, the speckles overlap and the overall

fluctuations are less. By contrast, localized speckle patterns are characterized by a few

very intense peaks, which are well separated from each other, so that the fluctuations

across the speckle pattern are very much larger.

These intensity fluctuations can be quantified by plotting their distribution functions,

shown in figs. 5(c) and (d), where we plot the probability P (I) of observing the different

values of the intensity normalized by the mean, I = I/〈I〉. In the diffuse regime, P (I)

is close to the well-known Rayleigh distribution, P (I) = exp[−I], for random wave fields

described by circular Gaussian statistics, such as can be observed for light from a laser

beam scattered off a rough, random surface. The small deviations seen for P (I) < 10−2

in fig. 5(c) can be explained by the leading-order corrections to Rayleigh statistics cal-

culated by Shnerb and Kaveh [32], and by Nieuwenhuizen and van Rossum [33]. Their

expression, P (I) = exp[I][1+ (I2 − 4I +2)/3g], contains only one parameter, the dimen-

sionless conductance g, and gives an excellent description of the experimental results,

with g = 11.4 ± 0.8 � 1. By contrast, near 2.4MHz in the localized regime (fig. 5(d)),

the intensity distribution function exhibits huge departures from Rayleigh statistics, with

greatly enhanced probability of observing large values of the normalized intensity, ex-

tending up to 50 times the average. To improve the accuracy of the measurements, P (I)

for the localized regime was determined from data for four equivalent samples over a

range of 101 frequencies between 2.35 and 2.45MHz. Figure 5(d) shows that the data

can be extremely well fitted over the entire range of intensities by Nieuwenhuizen and van

Rossum’s theory for P (I), yielding a value of g = 0.80±0.08. The theoretical expressions

used in the fits were determined for a broad Gaussian beam incident on a slab-shaped

sample, and account for interference processes dominated by the loopless connected dia-

grams [33]. For large intensities, the data can also be fitted by a stretched exponential,

exp[− 2

√gI

], with the same value of g, a simple analytic form of P (I) that can be

deduced from the complete intensity distribution derived in ref. [33]. These observations

108 J. H. Page

0 5 10 1510

6

105

104

103

102

101

100

Experiment, f = 0.20 MHz

NvR theory, g = 11.4

Rayleigh distribution

0 10 20 30 40 5010

6

105

104

103

102

101

100

Experiment, f = 2.4 MHz

NvR theory, g = 0.80

stretched exponential, g = 0.80

Rayleigh distribution

0

10

20

30

40

50

0

10

20

30

40

50

x (m

m)y (mm)0

0 6000

1 200

1 800

2 400

3 000

3 600

4 200

4 800

5 400

6 000

15

20

25

30

10

15

20

25

y (mm) x (m

m)

yyy

0

2 500

5 000

7 500

10 00

12 50

15 00

17 50

20 00

22 50

25 00

a

dc

bf = 0.20 MHz f = 2.40 MHz

I I

ˆ

P(I)

Fig. 5. – (a),(b) Comparison of the near field speckle patterns, showing the spatial variation ofthe intensity normalized by its average value, I(x, y)/〈I〉, at frequencies of 0.20 (a) and 2.4 MHz(b). In (a), the speckle pattern is typical of the diffuse regime, with broad overlapping specklespots, while in (b) the pattern is dominated by narrow intense peaks that are characteristic ofAnderson localization. In these two figures, the colour scale is different, but the z-axis scale(perpendicular to the plane) is the same so that the striking differences in these speckle patternscan be readily seen. (c) The measured probability distribution P (I) at 0.2MHz (open circles) isclose to the Rayleigh distribution (dashed blue line). The solid magenta curve is best fit of thetheory of ref. [33] to the data with g = 11.4. (d) At 2.4 MHz, the probability of observing largeintensities relative to the mean is very much greater than for diffuse waves. The solid curveshows the theory [33] for P (I) with g = 0.80, and is in excellent agreement with the experimentaldata (solid symbols). At large I � 25, the data can also be described by a stretched exponentialwith the same value of g (dotted curve). The large deviation from Rayleigh statistics withg < 1 provides additional evidence that the Anderson localization of ultrasound has occurredat frequencies near 2.4 MHz. (Nature Phys., 4 (2008) 945.)

of very large intensity fluctuations, for which the analysis of P (I) reveals g < 1, provide

additional evidence [6] that localization has occurred in our samples at high frequencies.

This interpretation is consistent with the Thouless criterion that g < 1 implies localiza-

tion. It is remarkable that such good agreement between theory and experiment for P (I)

has been found, as the theory was derived for the intensity in the far field and for g > 1;

this excellent agreement suggests a universality of the statistics of localized waves.


A simple way of characterizing the intensity fluctuations is measuring the variance,

〈I2〉. At 0.2MHz, we find 〈I2〉 = 1.12 ± 0.02, very close to the value of 1 for Rayleigh

statistics, while at 2.4MHz, a much larger value is found, 〈I2〉 = 2.74 ± 0.09. The

variance of the normalized speckle intensities can be directly related to the dimensionless

conductance, 〈I2〉 = 1 + 4/(3g) [6], providing an easier way of determining g. Using our

measured values of 〈I2〉, this relation gives g = 11.5± 2 at 0.2MHz, and g = 0.77± 0.04

at 2.4MHz, in good agreement with the values of g determined from fitting the intensity

distributions. Note that the localization condition g < 1 implies that localization will

be reached when the variance 〈I2〉 > 7/3 [6]. The measured variance at 2.4MHz is

larger than the threshold value 7/3, again supporting our conclusions that ultrasound is

localized at this frequency.

The ability of these ultrasonic experiments to measure the wave functions very near

the surface of a localized sample suggests that the spatial structure of localized wave

functions can now be investigated experimentally. There is a large body of theoretical

and numerical work that predicts that wave functions at the Anderson transition have

multifractal character —a striking relation between the spatial structure of wave func-

tions and their large fluctuations at criticality [34]. However, there have been virtually

no experimental studies until very recently. The following paragraphs outline recent

progress in using our ultrasonic data to examine this remarkable aspect of critical wave

functions close to the Anderson transition [14].

Multifractality implies that the moments of the wave function intensity, I(�r ) =

|ψ2(�r )|/ ∫ |ψ2(�r )|dd�r depend anomalously on length scale, with each moment scaling

as a power law with a different exponent. Note that I(�r ) is now normalized by the total

intensity, rather than the average intensity, and is therefore normalized in the same way

as |ψ2(�r )| for quantum systems. To characterize this length scale dependence experi-

mentally, one can either vary the size L of the samples, or, with a sample of a fixed size,

divide the sample into boxes of linear size b, and vary b. The latter is easier to implement

in practice, and is therefore used for our experimental data; it allows the size dependence

to be expressed in terms of the dimensionless scaling length Lg/b, where Lg is the size

of the speckle pattern over which the intensity is normalized. Note that in this analysis,

since we can only measure the wave function on or near the surface of the sample, the

dimension of the measurement space is d = 2, even though the sample is definitely three

dimensional. This procedure is illustrated in fig. 6, which shows the transmitted intensity

for three frequencies near 2.4MHz for a point source. (The point source geometry has

the advantage in this context of being more likely than an extended beam to excite a

single wave function.)

The length scale dependence of the moments of the intensity is quantified by the

generalized Inverse Participation Ratios (gIPR), which are defined as

(6) Pq =

n∑i=1

(IBi )q =

n∑i=1

[∫Bi

I(�r )dd�r

]q

.

Here IBi is the integrated probability inside box Bi of linear size b, with λ < b < Lg,

110 J. H. Page

15 10 5 0 5 10 15

15

10

5

0

5

10

15

x (mm)

y (m

m)

b

L = Lg

1510

50

510

1515

10

5

0

5

10

15

y (m

m)

x (mm)

1510

50

510

1515

10

5

0

5

10

15

y (m

m)

x (mm)

(a) (b) (c)

Fig. 6. – Speckle patterns at three representative frequencies near 2.4 MHz (2.35 MHz (a),2.375 MHz (b), and 2.425 MHz (c)) when a point source is incident on the opposite face ofthe sample. In (a) and (b), the speckle intensities very close to the sample surface are plotted asthree-dimensional intensity maps, so that the large fluctuations with position are easy to see. In(c), a contour plot is shown to illustrate the box-counting method used to determine the systemsize dependence in terms of the dimensionless scaling parameter Lg/b. In this example, the boxsize b is 2.

and the summation is performed over all of the n = (Lg/b)d boxes. By definition, P1 ≡ 1

and P0 ≡ n. The length scale dependence was studied experimentally by determining

the “typically averaged” gIPR for a single realization of disorder. In the critical regime,

the average gIPR are expected to scale anomalously with Lg/b as

(7) 〈Pq〉 ∼ (Lg/b)−τ(q) ≡ (Lg/b)−d(q−1)−Δq ,

where the exponent τ(q) is written in terms of the normal (Euclidean) dimension d(q−1)

and the anomalous dimension, Δq. Typical results for 〈Pq〉 from the ultrasonic data at

2.4MHz are shown in fig. 7(a) for integer values of q between −2 and 3. By plotting the

data on doubly logarithmic scales, power law behaviour is clearly seen over more than a

decade in Lg/b, as shown by the excellent fits of the data to straight lines. The slopes

of these linear fits yield τ(q), from which the anomalous exponents Δq were determined

by subtracting off the normal part d(q − 1). Average values of the anomalous exponents

were determined by averaging over many frequencies between 2.0 and 2.6MHz.

The q-dependence of the anomalous exponents is shown in fig. 7(b), where the results

for localized ultrasonic wave functions are compared with data from an optical speckle

pattern for diffuse waves. The behaviour seen for these two data sets is obviously very

different. For the diffuse optical data, the open circles in this figure show that Δq ≈ 0,

consistent with expectations for a normal (extended) wave function that Δq = 0 for every

q. By contrast, for multifractal wave functions, such as are expected in the critical regime

of the Anderson transition, τ(q) and hence Δq are expected to be continuous functions

of q, with substantial departures from Euclidean behaviour. This is precisely what fig. 7

shows for Δq determined from the ultrasonic data, clearly indicating that each intensity

moment has a different fractal exponent. In other words, the q-dependence of Δq in


-2 -1 0 1 2 3-2.0

-1.5

-1.0

-0.5

0.0

Δq ,

Δ1

-q

q1.4 1.2 1.0 0.8 0.6 0.4

2

0

2

4

6

8

10 q 2

q 1

q 0

q 1

q 2

q 3

log

Pq

log (b/L )

(a)

(b)

q log (b/Lg)

Fig. 7. – (a) An example of the dependence of the gIPR on box size at a frequency of 2.4 MHz,for integer values of q between −2 and 3. Box sizes of b = 2, 3, 4, 68, 12 and 24 were used.The grid size, Lg , was 55. (b) The anomalous exponents Δq measured for localized ultrasound(solid squares) and diffuse light (open circles). For the localized ultrasound exponents, the datamirrored relative to q = 1/2 are indicated by the dashed curve, showing the symmetry of themeasured values of Δq . The solid curve is a fit to the parabolic approximation.

fig. 7(b) reveals unambiguous evidence for multifractality of the localized ultrasound

wave functions.

The ultrasonic data in fig. 7(b) enables a recently predicted symmetry relation for

the anomalous exponents Δq to be tested experimentally. This relation, Δq = Δ1−q,

was predicted to hold exactly for multifractal wave functions at the Anderson transi-

tion [35]. The dashed blue curve in fig. 7(b) represents the experimental results reflected

about q = 1/2, showing that this symmetry relation is consistent with our data. As the

theoretical predictions of this symmetry relation were made in the context of particu-

lar symmetry classes of electronic systems, its observation in our ultrasonic experiments

provides evidence for the universality of critical properties at the Anderson transition.

The solid curve in fig. 7(b) is a fit of the predictions of the parabolic approximation

to our data. This analytic approximation, derived in first-order perturbation theory for

an Anderson transition in 2 + ε dimensions, gives the simple expression Δq = γq(1 − q).

This expression describes the ultrasonic data well, with γ = 0.21. The parabolic ap-

proximation also gives simple analytic expressions for two other important measures of

multifractality, the probability density function and the so-called singularity spectrum,

both of which were also measured for our ultrasonic data, as reported in ref. [14]. The

measurement of all three manifestations of multifractality, anomalous exponents, the (log-

normal) probability density function and the singularity spectrum, have demonstrated

an aspect of Anderson transitions that has not been studied experimentally before. This

opens up a number of interesting questions for future work, such as seeking an under-

standing of why the value of γ for the ultrasonic data is smaller than predicted for the 3D

Anderson tight-binding model [34], and exploring the relationship between multifractal

properties and the critical exponents governing the Anderson transition [36].

112 J. H. Page

6. – Conclusions

Ultrasonic experiments have several advantages for observing and studying Anderson

localization. As for all classical waves, they benefit from the convenience and versatility

that is associated with performing experiments at room temperature. More important is

the ability to readily investigate not only average wave transport at a single frequency,

but also the propagation of the multiply scattered wave fields resolved in time and space.

This has enabled the development of a new approach, transverse confinement, that has

permitted the most direct observation so far of Anderson localization in 3D, and provides

a valuable method for guiding future investigations of localization for any type of wave.

In particular, the measurement of the dynamic transverse confinement is a powerful way,

which is not affected by absorption, for assessing whether or not waves are localized; it

also enables the localization length to be measured.

By combining this approach with measurements of the time-dependent transmission

and the statistics of the large non-Gaussian intensity fluctuations, recent ultrasound ex-

periments, in conjunction with theoretical advances, have enabled the most unambiguous

demonstration of 3D Anderson localization of classical waves to date. These results sug-

gest that ultrasonic experiments on well-controlled samples may be able to investigate

previously unexplored aspects of 3D Anderson localization. One of these aspects, the

characterization of the multifractal spatial structure of wave functions near the Ander-

son transition, has already been realized. It is reasonable to anticipate that an even more

complete study of 3D Anderson localization using ultrasound is now within reach.

∗ ∗ ∗

I would like to acknowledge the theoretical contributions made by S. Skipetrov and

B. van Tiggelen to the work reviewed in this paper, and to Sanli Faez for suggesting

and carrying out the multifractal analysis. None of this research would have been possible

without the hard work of H. Hu and A. Strybulevych in my research group at the

University of Manitoba, who made the samples and performed the ultrasonic experiments.

Support from NSERC and a CNRS PICS grant is also gratefully acknowledged.

Appendix A.

Schrodinger and Helmholtz equations in disordered media

A quantum particle of energy E in a random potential V (�r ) is described by the wavefunction ψ(r) exp[−iEt/�], where ψ(�r ) obeys the time-independent Schrodinger equation

(A.1)

[− �

2

2m∇2 + V (�r )

]ψ(�r ) = Eψ(�r ).


A monochromatic scalar classical wave (e.g., sound, or light if one neglects polariza-tion) with angular frequency ω obeys the Helmholtz equation

(A.2)

[∇2 +

ω2

v2

]ψ(�r ) = 0,

where, for the example of acoustic waves, the wave function ψ(�r ) corresponds to thepressure. In a disordered medium, the wave velocity v(�r ) varies with position, and theHelmholtz equation can be rewritten in terms of the fluctuations of wave speed relativeto the average speed v0 as

(A.3)[−∇2 + σ(�r )

]ψ(�r ) =

ω2

v20

ψ(�r ).

Here σ(�r ) = ω2/v20− ω2/v2(�r ). This equation has the same form as the Schrodinger

equation if one considers σ(�r ) to play the role of an effective potential (the analogue of2mV (�r )/�

2), and ω2/v20 to play the role of an effective energy (the analogue of 2mE/�

2).This analogy indicates that similar wave phenomena can be expected for quantum parti-cles and classical waves. However, there is an important difference. Since σ(�r ) > ω2/v2

0

always, the classical wave case corresponds to the quantum case only when E > V (�r )for all �r. Thus classical waves can never be trapped by disorder in a trivial way anal-ogous to the trapping of a low-energy particle in the bottom of a potential well. Forclassical waves, lowering the frequency also lowers the effective disorder potential seenby the wave, since σ(�r ) is proportional to ω2, so that localization is most likely to oc-cur at intermediate frequencies where the wavelength is comparable with the size of theinhomogeneities and the scattering is greatest.

REFERENCES

[1] Anderson P. W., Phys. Rev., 109 (1958) 1492.

[2] For a recent non-specialist review of Anderson Localization, see Lagendijk A., van

Tiggelen B. A. and Wiersma D. S., Phys. Today, 62(8) (2009) 24.

[3] He H. and Maynard J. D., Phys. Rev. Lett., 57 (1986) 3171.

[4] Weaver R. L., Wave Motion, 12 (1990) 129.

[5] Dalichaouch R., Armstrong J. P., Schultz S., Platzman P. M. and McCall S. L.,Nature, 354 (1991) 53.

[6] Chabanov A. A., Stoytchev M. and Genack A. Z., Nature, 404 (2000) 850.

[7] Chabanov A. A., Zhang Z. Q. and Genack A. Z., Phys. Rev. Lett., 90 (2003) 203903.

[8] Schwartz T., Bartal G., Fishman S. and Segev M., Nature, 446 (2007) 52.

[9] Wiersma D. S., Bartolini P., Lagendijk A. and Righini R., Nature, 390 (1997) 671.

[10] Storzer M., Gross P., Aegerter C. M. and Maret G., Phys. Rev. Lett., 96 (2006)063904.

[11] Aegerter C. M., Storzer M. and Maret G., Europhys. Lett., 75 (2006) 562.

[12] Abrahams E., Anderson P. W., Licciardello D. C. and Ramakrishnan T. V., Phys.Rev. Lett., 42 (1979) 673.

[13] Hu H., Strybulevych A., Page J. H., Skipetrov S. E. and van Tiggelen B. A.,Nature Phys., 4 (2008) 945.

114 J. H. Page

[14] Faez S., Strybulevych A., Page J. H., Lagendijk A. and van Tiggelen B. A., Phys.Rev. Lett., 103 (2009) 155703.

[15] Page J. H., this volume, p. 75.[16] Page J. H., Hildebrand W. K., Beck J., Holmes R. and Bobowski J., Phys. Status

Solidi C, 1 (2004) 2925.[17] Page J. H., Schriemer H. P., Bailey A. E. and Weitz D. A., Phys. Rev. E, 52 (1995)

3106.[18] Turner J. A., Chambers M. E. and Weaver R. L., Acustica, 84 (1998) 628.[19] Schriemer H. P., Pachet N. G. and Page J. H., Waves in Random Media, 6 (1996)

361.[20] Van Tiggelen B. A., in Diffuse Waves in Complex Media, edited by Fouque J. P.

(Dordrecht, Kluwer) 1998, pp. 1-63.[21] Weaver R. L., J. Acoust. Soc. Am., 71 (1982) 1609.[22] Ryzhik L., Papanicolaou G. and Keller J. B., Wave Motion, 24 (1996) 327.[23] Hennino R., Tregoures N., N. M. Shapiro N. M., Margerin L., Campillo M., van

Tiggelen B. A. and Weaver R. L., Phys. Rev. Lett., 86 (2001) 3447.[24] Turner J. A. and Weaver R. L., J. Acoust. Soc. Am., 98 (1995) 2801.[25] Vollhardt D. and Wolfle P., Phys. Rev. B, 22 (1980) 4666; Phys. Rev. Lett., 48 (1982)

699.[26] Wolfle P., this volume, p. 1.[27] van Tiggelen B. A., Lagendijk A. and Wiersma D. S., Phys. Rev. Lett., 84 (2000)

4333.[28] Skipetrov S. E. and van Tiggelen B. A., Phys. Rev. Lett., 96 (2006) 043902.[29] Hu H., Ph.D. Thesis, University of Manitoba (2006).[30] Cherroret N., Skipetrov S. E. and van Tiggelen B. A., arXiv:0810.0767.[31] Scheffold F., Lenke R., Tweer R. and Maret G., Nature, 398 (1999) 206.[32] Shnerb N. and Kaveh M., Phys. Rev. B, 43 (1991) 1279.[33] Nieuwenhuizen Th. M. and van Rossum M. C. W., Phys. Rev. Lett., 74 (1995) 2674;

van Rossum M. C. W. and Nieuwenhuizen Th. M., Rev. Mod. Phys., 71 (1999) 313.[34] For a recent review, see Evers F. and Mirlin A. D., Rev. Mod. Phys., 80 (2008) 1355.[35] Mirlin A. D., Fyodorov Y. V., Mildenberger A. and Evers F., Phys. Rev. Lett., 97

(2006) 046803.[36] Grussbach H. and Schreiber M., Phys. Rev. B, 51 (1995) 663.


Ultrasonic spectroscopy of complex media

J. H. Page



Summary. — Mesoscopic wave physics underpins many of the new developmentsin ultrasonic spectroscopy for probing the physical properties of complex heteroge-neous materials. In this paper, two examples of recent progress are summarized.The first is Diffusing Acoustic Wave Spectroscopy (DAWS), which is a powerfulapproach for investigating the dynamics of strongly scattering media, one examplebeing velocity fluctuations in fluidized suspensions of particles. Recent advances inusing phase statistics to probe the particle dynamics are shown to give increasedsensitivity in some situations; this work has also led to new insights into the mean-ing of phase for multiply scattered waves. The second topic is the spectroscopy ofsoft food biomaterials, illustrated by experimental studies of ultrasonic velocity andattenuation in bread dough. Since wheat flour dough contains one of the strongestscatterers of ultrasonic waves (bubbles) dispersed in a viscoelastic matrix that is alsovery dissipative, appropriate ultrasonic techniques provide an excellent means forinvestigating its structure and dynamics. In addition to fundamental studies, unrav-eling the contributions of bubbles and matrix to dough properties is relevant to thebaking industry, because the bubbles ultimately grow into the voids that determinethe structural integrity of bread —an important quality attribute. The interpreta-tion of ultrasonic experiments on bread dough over three decades in frequency isgiving new insights into this complex material, as well as providing the basis fornew non-destructive methods of evaluating both dough processing behaviour andthe breadmaking potential of different flours.


116 J. H. Page

1. – Introduction

The fundamental studies of ultrasonic wave transport in strongly scattering random

media, described in refs. [1-5], have facilitated the development of ultrasonic techniques

for probing the physical properties of complex materials. Many such materials are meso-

scopic, with internal structures on length scales comparable with ultrasonic wavelengths,

and it is the structure and dynamics at this mesoscopic scale that determine their macro-

scopic physical properties. Familiar examples include foams, gels, slurries and a wide

range of food biomaterials, all of which are playing an increasing important role in in-

dustrial applications, and hence our prosperity. Mesoscopic structure, however, often

leads to multiple scattering of ultrasound, making traditional imaging methods impos-

sible and motivating the development and application of new approaches for extracting

useful information.

This paper reviews two examples of ultrasonic spectroscopy and their application to

novel materials characterization methods. The next section outlines Diffusing Acous-

tic Wave Spectroscopy (DAWS), a powerful technique in field fluctuation spectroscopy

for investigating the dynamics of strongly scattering media. Differences with the

complementary technique of Diffusing Optical Wave Spectroscopy are discussed, high-

lighting the advantages of DAWS in some contexts. DAWS is illustrated with ex-

periments on suspensions of particles and bubbles. Recent progress in probing dy-

namic properties using the phase of multiply scattered waves, which can readily be

measured for ultrasound but less easily for light, is summarized. Diffusing Acous-

tic Wave Spectroscopy, introduced in 2000 [6] and described in detail in ref. [7], has

also been reviewed in ref. [8], and more recently in a broader context in Physics To-

day [9]. The interested reader is encouraged to consult these references for additional

information.

Section 3 illustrates the characterization of food materials using ultrasound. Many

food materials are both strongly scattering and strongly absorbing for ultrasound, and

in cases where the multiple scattering coda is suppressed by dissipation, it is not feasible

to use techniques such as DAWS to probe their evolution during processing. Nonethe-

less information on the mechanical properties of foods is important in the preparation

and production of foods with appealing texture, which is crucial for making foods palat-

able to eat. This information can be obtained from spectroscopic techniques that rely

on ballistic propagation, and are especially valuable when data over a wide range of

ultrasonic frequencies are available. This approach is illustrated with experiments on

bread dough [10-19], where processing challenges encountered when incorporating nutri-

tional supplements may make ultrasonic monitoring techniques of particular value to the

functional foods industry.

2. – Diffusing Acoustic Wave Spectroscopy

Diffusing Acoustic Wave Spectroscopy (DAWS) determines the dynamics of a strongly

scattering medium from the temporal fluctuations of ultrasonic waves that are scattered

Ultrasonic spectroscopy of complex media 117

many times before leaving the medium [6,7]. Because multiply scattered waves are used,

the technique is extremely sensitive to the motion of the scatterers in the medium, or to

the evolution of the properties of the host material in which the scatterers are located;

this sensitivity results from the large number of scattering events that are involved, lead-

ing to long trajectories over which cumulative changes in the detected waves occur. As

the name suggests, there is much in common with the analogous technique of Diffusing

Optical Wave Spectroscopy (often simply abbreviated DWS) [20,21], although there are

differences in the way in which the measurements are made and in the range of appli-

cations for which the two techniques are well suited. One advantage of DAWS is that

the scattered wave field is measured, not the intensity, so that the field correlation func-

tion g1(τ) is determined directly. Thus, there is no need to invoke the Siegert relation

to interpret measurements of intensity correlation functions using models for the field

correlation functions. Another advantage of detecting the wave field in DAWS is that

the phase of the scattered fields can be exploited, offering the potential of better sensi-

tivity in some cases. The other major technical difference is the ease with which pulsed

measurements can be performed, enabling the detected changes to be monitored for a

fixed path length of the multiply scattered waves and therefore simplifying the analysis.

Finally, since ultrasonic wavelengths and wave periods are both larger (typically ∼ 1mm

and 1μs), DAWS is sensitive to dynamics on longer length scales than is possible with

light (or X-rays), enabling different types of materials and phenomena to be investigated.

By varying the frequency, this range of length scales can be extended significantly, and

can range up to kilometres for seismic applications.

Figure 1 shows two contrasting examples of evolving multiply scattered wave fields

that can be used to probe changes in the system under investigation. Figure 1(a) shows

a typical experimental setup in DAWS, where a pulsed incident wave from a generating

transducer propagates through a sample containing moving particles or bubbles (fig. 1(b))

and is detected by a hydrophone. A typical multiple scattering path is indicated by the

red arrows. Two segments of the transmitted field are shown in fig. 1(c), showing that at

early propagation times, almost no change in the transmitted field is seen, while at later

times, the wave field changes significantly as the scatterers move. Note that there are

two relevant times in this problem, the propagation time t of the waves in the medium,

which sets the sensitivity, and the evolution time T , which sets the time scale over which

the dynamics are recorded. In DAWS, the medium can be interrogated repeatedly on a

scale set by the pulse repetition time ΔT , which can be varied over a wide range to match

the rate at which the system is evolving; in this case, T = mΔT , where m is an integer.

Figure 1(c) shows that the waves are decorrelated in both amplitude and phase as the

evolution time increases, due to the motion of the bubbles in the suspension. Analysis

of the detected field fluctuations can be used to probe the velocities of the bubbles. By

contrast, the changes in the waveforms shown in fig. 1(d), which were detected on Mount

Merapi by a seismograph located 2 km away from the source, at evolution times separated

by two weeks, are shifted in propagation time but remain similar otherwise. In this case,

there is a uniform change in the medium, and the phase shift is related to a change in

the seismic wave speed.

118 J. H. Page

Ultrasonic

Transducer

Moving bubbles, or

fluidized particles

Hydrophone

Detector

2ry u

nit

s)

1 2 3 4 11 12 13 14

-2

-1

0

1

2

-2

-1

0

1

2

Wav

e fi

eld (

arbit

rary

unit

s)

Propagation time, t (μs)

(b)

(a)

(d)

(c)

1.0 1.5 12.0 12.5 13.0

-2

0

Wav

e fi

eld (

arbit

rar

Propagation time, t (s)

Fig. 1. – (a) Typical setup for measuring the dynamics of fluidized suspensions or bubbles us-ing Diffusing Acoustic Wave Spectroscopy. The red arrows indicate a multiple scattering paththrough the sample. (b) Photograph of ∼ 20-μm-diameter bubbles generated by an electrolysistechnique. These small bubbles move in complex swirling patterns through water. (c) Two seg-ments of the scattered wave fields transmitted through a fluidized suspension of glass particles,observed at three evolution times separated by 60 ms. The waveforms are very similar at earlypropagation times, but exhibit large fluctuations in both phase and amplitude a later times.(d) Waveforms detected by a seismograph on Mount Merapi, the site of an active volcano inIndonesia, after an air gun was used to generate short low frequency pulses 2 km away. Theevolution time interval between the two recorded signals (red and blue traces) is two weeks. Theshift in the phase of the waves can be most simply measured from a windowed cross correlationfunction of the fields at the two times; this phase shift is related to a small change in the seismicvelocity, as explained in ref. [9].

To determine the changes in the medium from the evolution of the scattered wave

fields, it is helpful to describe the multiply scattered waves detected at propagation time

t and evolution time T as the superposition of waves that have propagated along each

scattering path p. This can be shown explicitly by writing the measured field ψ(t, T ) as

the real part of a complex field (the complex analytic signal)

(1) Ψ(t, T ) = A(t, T )ei[ωt+Φ(t,T )] =∑

p

ap(t, T )ei[ωt+φp(t,T )].

Here ω is the central frequency of the pulse, Φ(t, T ) is the total phase of the scattered

waves at the detector and φp(t, T ) is the phase along a single multiple scattering path,


which may be conveniently called the “path phase”. As the medium evolves in time, the

waves still propagate along these scattering paths, but the path lengths change, so that ψ

is a function of both propagation time t and evolution time T . One direct way of relating

the changes in ψ(t, T ) to the dynamics of the medium is to take the autocorrelation

function of the field at a fixed propagation time ts, thereby selecting multiple scattering

paths with an average length s = ts/vE and a narrow path length range determined by

the source pulse width. The field autocorrelation function g1,ts(τ) is

(2) g1,ts(τ) =

∫ψ(T ) ψ(T + τ)dT∫ |ψ(T )|2dT

�⟨e−i Δφp(τ)

⟩,

where Δφp(τ) is the change in phase of a path containing n scattering events during the

evolution time interval τ . Here n = vEts/l∗, where vE and l∗ are the transport velocity

and mean free path of the multiply scattered waves. In general, the phase change for each

path can be written as the sum of the ensemble average phase shift 〈Δφpath(τ)〉 and the

deviation from the average value δφpath(τ), enabling the autocorrelation function (2) to

be written as a product of two factors, involving the average phase shift and its variance,

respectively:

(3) g1,ts(τ) � cos (〈Δφpath(τ)〉) exp

[−1

2

⟨δφ2

path(τ)⟩]

.

To obtain this result, the contribution to g1 from the ensemble average of 〈e−i δφpath(τ)〉is obtained to leading order using a cumulant expansion [6-8], and the real part of g1 is

taken, since this corresponds to the experimental situation.

A nonzero average phase shift arises when there is a uniform dilation of the medium

seen by the waves, such as can occur if there is a change in wave velocity, which shifts the

arrival time of all the scattered waves in the same way (e.g., see fig. 1(d)). In the case of

a small wave velocity change Δv, 〈Δφ〉 = −ωtΔv/v, where ω is the angular frequency.

When the scatterers are moving, such as for the example of moving bubbles in

fig. 1(b),(c), the dominant contribution to the decay of g1 comes from the path phase

variance 〈δφ2

path〉. In this case, the path phase variance can be related to the phase fluc-

tuations for each step j along a path, which are given by �kj ·Δ�rrel,j where �kj is the wave

vector of the wave scattered between the j-th to the (j + 1)-th scatterers, and Δ�rrel,j

is their relative displacement [6-8]. When the successive phase shifts along the paths,

as well as the directions of �kj and Δ�rrel,j , are uncorrelated, the field autocorrelation

function is given by

(4) g1,ts(τ) ≈ exp

[−nk2

6

⟨Δr2

rel(τ, l∗)

⟩].

(Here the average phase shift in eq. (3) has been set to zero, as is observed for fluidized

particles; then, 〈δφ2

path〉 = 〈Δφ2

path〉.) This equation shows that the decay of the corre-

lation function is determined by the relative mean square displacement of the scatterers

120 J. H. Page

Fig. 2. – (a) Mean square relative displacement of glass beads in a fluidized suspension measuredby Diffusing Acoustic Wave Spectroscopy. The data are plotted for several volume fractions ofbeads φvf , showing τ 2 behaviour indicative of ballistic particle motion at short times. (b) Rootmean square relative velocity, divided by the fluidization velocity Vf , as a function of the ul-trasound transport mean free path l∗ normalized by the bead radius a. The mean free pathl∗ determines the average particle separation at which the velocity fluctuations are measured.The solid curves are fits of eq. (5), with CV given by eq. (6), to the data for two volume frac-tions, enabling the particle velocity correlation length ξ to be measured at each volume fraction.(c) The particle velocity correlation function as a function of the average inter-particle sepa-rations R = l∗ at which the relative velocities are measured. The data show a good fit tothe exponential function exp[−R/ξ], confirming the form of the correlation function that wasassumed in (b). The different symbols represent data measured at different volume fractions ofscatterers, with all the data collapsing onto a common curve when CV is plotted as a functionof R/ξ.

that are separated, on average, by the average step length of the multiply scattered

waves, l∗. Measuring the field autocorrelation function thus enables the relative motion

of the scatterers to be determined on a length scale that can be tuned by the ultrasonic

frequency.

Typical DAWS results for the dynamics of fluidized particles, suspended by flowing

the liquid upwards to counteract sedimentation, are shown in fig. 2. In this example, the

scatterers are 1-mm-diameter glass spheres surrounded by a liquid mixture of water and

glycerol. At short evolution times, the motion of the particles is ballistic since the relative


mean square displacement grows quadratically with time, 〈Δr2

rel(τ, l∗)〉 = 〈ΔV 2

rel(l∗)〉τ2,

allowing the variance in the relative particle velocities to be measured directly from the

slope of 〈Δr2

rel(τ, l∗)〉 versus τ2. The root mean square relative velocity ΔVrel =

√〈ΔV 2

rel〉

determines the characteristic time scale of the motion, τDAWS ≡ 1/[√

nkΔVrel(l∗)], so

that, with this definition, the field autocorrelation function can be written in a very

simple way as g1(τ) = exp[− 1

6τ2/τDAWS].

By varying the frequency, the scattering strength and hence also l∗ was varied (see, for

example, ref. [1]), enabling the relative particle velocity to be measured over a wide range

of inter-particle distances R = l∗ inside the suspension. Figure 2(b) shows that at short

distances, the relative particle velocity increases as the square root of distance, but that at

longer distances it levels off to the value√

2Vrms, where Vrms is the absolute rms particle

velocity that can be measured directly in the single scattering regime using Dynamic

Sound Scattering [6]. The saturation value√

2Vrms is the relative velocity of particles

that move independently, indicating that all correlations in the motions become lost at

large inter-particle separations. These observations can be summarized mathematically

as follows:

(5)

⟨ΔV 2

rel(l∗)

⟩=

⟨(Δ�V (�r + l∗) − Δ�V (�r )

)2⟩

= 2〈ΔV 2〉 − 2⟨Δ�V (�r + l∗) · Δ�V (�r )

⟩= 2 V 2

rms(1 − CV (l∗)),

where

(6) CV (R) =

⟨Δ�V

(�r + �R

)· Δ�V (�r )

⟩〈ΔV 2〉 = exp[−R/ξ]

is the particle velocity correlation function, whose decay rate is determined by the ve-

locity correlation length ξ. Equations (5), (6) show how particle velocity correlation

function can be determined from the relative velocity fluctuations measured in DAWS

experiments, yielding the experimental results shown in fig. 2(c). The data in figs. 2(b)

and (c) show that the assumed exponential decay of the velocity correlations with dis-

tance is consistent with observations, and enable the correlation length to be measured

over a wide range of particle concentrations (with φvf varying from 0.08 to 0.50 in this

case). The correlation length ξ measures the range of distances over which the particles

move together, and is an important quantity for understanding the physics of fluidized

suspensions. Figure 2(c) also reveals a remarkable scaling of the velocity correlations

for different particle concentrations when CV is plotted as a function of R/ξ. Thus,

Diffusing Acoustic Wave Spectroscopy can provide information on the dynamics of sus-

pensions that is relevant both to fundamental studies of the motions in suspensions and

turbulent fluids, and to practical applications such as monitoring mixing processes, the

performance of chemical slurry bed reactors, and slurry flow in industrial processing.

122 J. H. Page

0.05

0.10

0.15

0.20(d)

τ 1 s

τ 2 s

103

102

101

100

101

(c)

τ 20 ms

0.1

1

(b)

P(Δ

Φ)

τ 100 ms

105

104

103

102

101

100

101

102

(a)

τ 2 ms

107

106

105

104

103

102

002 0 01 0 020 0110

7

106

105

<

Δr re

l2( τ

)> (

mm

2 )

104

103

102

101

100

101

from fits to P(ΔΦ)

from <ΔΦ2>

from g1

~ τ 2

< Δ

φp

ath

2 >

(rad

2)

(e)

-3 -2 -1 0 1 2 30.00

ΔΦ (rad)

-3 -2 -1 0 1 2 30.01

10

310

210

110

010

7

τ (s)

10

Fig. 3. – (a)-(d) The wrapped phase shift probability distribution P (ΔΦ) at several time intervalsτ . Experimental data are represented by the open symbols, and theoretical predictions bythe solid curves. The dashed curves show the approximate expression for P (ΔΦ) given byeq. (7). The only fitting parameter in the comparison of theory and experiment is the pathphase variance 〈Δφ2

path(τ )〉 at each time τ , yielding a value of τDAWS equal to 89 ms for thesedata. (e) Comparison of the relative mean square displacement of the particles determined fromthe phase data and from the field correlation function.

By capitalizing on the ability of ultrasonic techniques to measure the phase and

amplitude of the multiply scattered waves, DAWS has recently been extended to monitor

system dynamics by analysing the fluctuations of the phase of the waves in time-varying

systems [22]. Not only has this helped to advance our understanding of mesoscopic wave

physics, where the role of phase for multiply scattered waves has often been ignored,

but it has also provided a new approach for probing evolving media with increased

sensitivity in some situations. Experimentally, the wrapped phase Φ(t, T ) ∈ (−π : π]

was extracted from the measured field at any evolution time T by a numerical technique

that is equivalent to taking a Hilbert transform and obtaining the complex analytic signal

A(t, T ) exp[i[ωt + Φ(t, T )]]. Information on the dynamics of the medium is contained in

the phase difference over the time interval τ , ΔΦ(τ) = Φ(T + τ) − Φ(T ), which, since

the phase difference is still a phase, is best represented between −π and +π, and was

therefore re-wrapped ∈ (−π : π]. To relate this phase difference to the particle dynamics,

the relationship between the measured phase shift ΔΦ(τ) and the evolution of the path

phase Δφpath(τ) was established. The simplest way to do this is via the wrapped phase

difference probability distribution P (ΔΦ(τ)), which can be calculated for random wave

fields described by circular Gaussian statistics from the joint probability distribution of

the fields at two times Ti and Tj = Ti +τ [22]. Results for the same fluidized suspensions

of glass particles used for the data in fig. 2 are compared with theory in fig. 3(a)-(d),

showing how the statistics of the phase difference evolve as the scatterers move, with the

distribution becoming wider as the relative mean square displacement of the particles


increases. At long times when the fields are no longer correlated, P (ΔΦ(τ)) reaches a flat

distribution. At short time intervals τ and small ΔΦ, P (ΔΦ(τ)) has the simple form

(7) P (ΔΦ) =1

2

〈Δφ2

path〉[

〈Δφ2

path〉 + ΔΦ2

]3/2,

showing explicitly how the distribution depends on the path phase variance 〈Δφ2

path(τ)〉

and hence on the relative mean square displacement of the particles. While the general

expression for P (ΔΦ(τ)) is more complicated [22], it still only depends on one parameter,

the path phase variance, allowing the excellent fits of theory and experiment shown

in fig. 3 to accurately measure the mean square relative displacement of the particles.

Figure 3(e) shows that measurements of 〈Δr2

rel(τ)〉 from P (ΔΦ) and g1 are in superb

agreement over a wide range of evolution time intervals τ , validating this phase method.

The insert of this figure also shows an example where measurements of P (ΔΦ) yield more

accurate results. In this case, the presence of amplitude noise due to gain fluctuations

degrades the field correlation measurements of the particle dynamics at short times, but

has little effect on the phase statistics, which still give an accurate measurement of the

particle motions.

Another way of characterizing the dynamics is to measure the variance of the wrapped

phase difference 〈ΔΦ2(τ)〉. The variance of the measured phase shift is very different to

the path phase variance 〈Δφ2

path(τ)〉, since the phase of the measured field is determined

by the superposition of waves along all paths reaching the detector, while the path phase

variance is determined by the fluctuations in the phase along a typical path (see eq. (1)).

Remarkably, a universal relation has been found between the wrapped phase variance

and the path phase variance, as shown in fig. 4(a) by the solid curve. This universal

relation means that the particle dynamics can be determined directly from the measured

phase variance (open squares in fig. 3) —a simpler procedure than fitting the theoretical

expression for P (ΔΦ) to experimental data. Both methods work well for evolution times

that are short enough that 〈ΔΦ2(τ)〉 is less than its upper limit of π2/3, which occurs

when P (ΔΦ) has become flat.

Information on the dynamics can be followed to longer times by unwrapping the

phase, removing the jumps of 2π to determine the evolution of the cumulative phase

Φc(τ). Here, as an example, we consider the cumulative phase shift variance, which is

plotted as a function of τ/τDAWS in fig. 4(b). At early times, its increase with time is the

same as the wrapped phase variance, but at long times 〈ΔΦ2c(τ)〉 becomes proportional to

time, with 〈ΔΦ2c(τ)〉 = DΦτ , enabling the phase diffusion coefficient DΦ to be measured.

If the particles continue to move in ballistic trajectories at long times, DΦ = 1/τDAWS,

but if the relative motion slows down, due to deviations from ballistic particle trajectories

due to particle interactions, DΦ is reduced. The solid curve in fig. 4(b) shows a fit to a

simple empirical crossover model [6], indicating that the characteristic time τc for such

deviations to set in is about 7τDAWS. An interesting general point to emerge from this

analysis of the cumulative phase shift variance is that unwrapping the phase destroys the

124 J. H. Page

102

101

100

101

102

for <ΔΦ2

c>, no crossover

for <ΔΦ2

wrap>

< Δ

φp

ath

2 >

(r

ad

2)

0.1

1

10

(4/3)τ / τDAWS

<Δ

ΦC

2>

(a)

(b)

1E 3 0.01 0.1 1 1010

4

103

c

for <ΔΦ2

c>, τ

c = 7 τ

DAWS

for <ΔΦ2

c>, τ

c = 5 τ

DAWS

< ΔΦ 2

> (rad2 )

0.1 1 10

0.01

EXPERIMENT

THEORY (no crossover)

THEORY (τc 7τ

DAWS)

τ / τDAWS

Fig. 4. – (a) Relation between the measured phase shift variance 〈ΔΦ2(τ )〉 and the path phasevariance 〈Δφ2

path(τ )〉. For the wrapped phase, this relationship is universal (solid curve), whileunwrapping the phase destroys the universality, giving the cumulative phase shift variance〈ΔΦ2

c(τ )〉 greater sensitivity to the dynamics at long times. (b) The time dependence of the cu-mulative phase shift variance 〈ΔΦ2

c(τ )〉, showing a crossover to phase diffusion for times longerthat τDAWS, with a phase diffusion coefficient that is influenced by the long-time dynamics.

universal relationship between the measured phase variance and the path phase variance;

this actually has a positive benefit since it gives the cumulative phase shift variance

increased sensitivity to details of the particle motions at long times. The behaviour is

shown by the dashed and dotted curves in fig. 4(a).

Another advantage of examining the phase statistics has been demonstrated by theory

and experiment for the probability distributions of the phase derivatives with evolution

time. These distributions have been determined for Φ′, Φ′′, Φ′′′ and found to be remark-

ably sensitive to early time dynamics, allowing the relative particle motions to be deter-

mined up to the 6th power in time —something that simply could not be achieved from

measurements of the field correlation function. Another interesting quantity is the cumu-

lative phase correlation function, where current work is showing that, for evolving systems

such as the bubbly liquids, the phase correlation function can be used to investigate mo-

tions at remarkably long times, beyond those accessible to field correlation measurements.

Thus, progress in measuring and understanding the phase statistics and correlations of

multiply scattered fields is continuing to advance the capabilities of Diffusing Acoustic

Wave Spectroscopy for investigating the dynamics of strongly scattering materials.

3. – Probing food biomaterials with ultrasound

Many foods are heterogeneous on length scales that are comparable with the wave-

lengths of ultrasound in the 100 kHz to 10MHz range, making ultrasonic spectroscopy of


Fig. 5. – The ultrasonic attenuation and phase velocity as a function of frequency up to 10MHzfor a typical dough sample with a bubble concentration of 12%. The roman numerals and boxesindicate the three frequency regimes, as discussed in the text.

food materials a promising approach for investigating their mechanical properties, struc-

ture and dynamics. Because both scattering and dissipation of ultrasound are generally

strong in such materials, most information on their properties comes from ballistic veloc-

ity and attenuation measurements. In this section, I focus on one example, bread dough,

which contains one of the strongest scatterers of ultrasound, namely bubbles, with the

bubbles being dispersed in a viscoelastic matrix, which contributes to the ultrasonic ab-

sorption. Thus, the physics of how ultrasound propagates in dough is remarkably rich.

Understanding the effect of bubbles on the properties of dough is also critical to control-

ling the texture of bread, and hence its quality. As a result, ultrasonic characterization

of bubbles in bread dough is potentially important to the food industry.

Ultrasonic experiments on bread dough reveal different properties as the frequency is

varied [10-19]. Indeed, there are three important frequency regimes. These are identified

in fig. 5, which shows experiment and theory for the ultrasonic attenuation and phase

velocity in bread dough over almost three decades in frequency. At low frequencies,

f < 100 kHz, bubbles in dough drastically reduce the sound velocity, due to the large

compressibility of the bubbles. There is excellent sensitivity to the presence of bubbles

but no information on their sizes. The attenuation is relatively low in this frequency

regime, making experiments easier. For frequencies between 100kHz and 8MHz, there is

a strong resonant interaction between the ultrasonic waves and the bubbles, leading to a

very large variation in the velocity and attenuation. Their frequency dependence at these

intermediate frequencies depends on the bubble sizes, raising the interesting possibility

of extracting information on the bubble size distribution in this opaque medium from

126 J. H. Page

Fig. 6. – The ultrasonic velocity (a) and attenuation (b) in bread dough as a function of bubblevolume fraction at 50 kHz.

ultrasonic measurements. At high frequencies, f > 8MHz, the ultrasonic attenuation

and velocity depend on matrix properties only, enabling structural relaxations of the

molecular ingredients of the matrix to be probed.

The sensitivity of ultrasound to the concentration of bubbles in the low frequency

regime (at f ∼ 50 kHz) is shown in fig. 6. The dough samples were prepared by mixing

together a strong Canadian breadmaking flour (CWRS), salt and water, to produce a

lean-formula mechanically developed dough [23]. For these experiments, the bubble con-

centration was adjusted by varying the headspace pressure during mixing. As the bubble

concentration is increased, the ultrasonic velocity drops dramatically, reaching values less

than the velocity of sound in air for concentrations above 2%. This behaviour can be

understood qualitatively in Wood’s approximation for the low frequency compressibility

of a bubbly liquid. In this approximation, the average compressibility of the sample

κs is simply the volume-fraction-weighted average of the compressibilities of the bubble

inclusions i and surrounding matrix m

(8) κs = φvfκi + (1 − φvf)κm,

where φvf is the volume fraction of bubbles. Thus, since phase velocity and compress-

ibility are related by v =√

1/ρκ, where ρ is the density,

(9)1

ρsv2s

=φvf

ρiv2

i

+1 − φvf

ρmv2m

and the average sound velocity for concentrations of bubbles in this range reduces ap-

proximately to

(10) vs ≈ vair

√ρair

ρsφvf

.


The density ratio in the square root factor in this expression shows why the velocity

is so much less than the velocity in air (340m/s); the effective medium behaves as a

material with the low compressibility of air, but with a larger density. For dough, this

approximation underestimates the velocity at all volume fractions because it neglects the

shear modulus of the dough matrix, which can be included in a more complete (but also

more complicated) effective medium model [24]. For the higher volume fractions, where

the complex shear modulus μm = 0.39+ i0.14MPa can be reasonably extrapolated from

existing lower frequency shear rheology data on dough prepared at ambient pressure [14];

this model gives excellent agreement with experiment. At lower concentrations, however,

the measured velocities are larger than this prediction, suggesting that the shear modulus

of the dough matrix increases at low volume fractions. Thus the presence of bubbles in

the dough enables the shear properties of the dough to be investigated using longitudinal

waves —a considerable advantage as longitudinal ultrasonic measurements are easier to

perform in lossy materials such as dough.

As shown in fig. 6(b), the ultrasonic attenuation increases as the square root of the

volume fraction in this low frequency regime, a frequency dependence which is predicted

by effective medium theories (solid line) [24,16]. By treating the interaction of ultrasound

with bubbles in a viscoelastic medium, it can be shown, at frequencies well below the

resonance frequency ω0 of the bubbles, that the attenuation is predicted to have the form

(11) α =

√3φvf

a2

ω2Γ

ω30

.

Here a is the radius of the bubbles, and Γ is the damping rate, which at low frequen-

cies depends on viscous losses and thermal dissipation. If viscous losses dominate, the

dependence of α on bubble size in eq. (11) cancels out, since Γviscous = 4μ′′/ρωa2 and

the resonant frequency of the bubbles depends inversely on the bubble radius, ω0 ∝ a−1.

Thus, in this regime, the attenuation is sensitive only to the amount of gas entrained in

the bubbles, and not on how the gas is distributed, providing a good indicator of the

amount of gas entrained in the dough.

The sensitivity of these low-frequency measurements to bubble concentration is en-

abling the ultrasonic velocity and attenuation to be used to monitor dough mixing, where

reliable methods of determining optimum mixing conditions are of considerable value [18].

For doughs prepared with leavening agents, ultrasonic velocity and attenuation can be

used to monitor the growth of the bubbles due to incorporation of CO2 [12]. Low fre-

quency velocity measurements can also be used to assess dough quality [25], and since

these ultrasonic measurements can be performed on small samples, such measurements

are potentially very useful in wheat breeding programs.

At intermediate frequencies, the resonant coupling of ultrasound with the bubbles

causes the attenuation and phase velocity to exhibit a large frequency dependence, with

broad peaks that contain information on the bubble size distribution (fig. 7). To inter-

pret the experimental data, Leroy et al. [16] have used a model that extends the well-

established model for the resonant interactions of sound with bubbles in liquids [26-28]

128 J. H. Page

Fig. 7. – Frequency dependence of the ultrasonic attenuation (a1),(a2) and velocity (b1),(b2),showing the broad spectral features characteristic of resonant interactions with bubbles. Thegrey curves represent experimental data, and the solid black curves are theoretical predictionsof the model outlined in the text. (See ref. BIF for more information.) The numbers (1) and (2)identify the times after mixing at which the data were taken: 53 minutes for (1) and 90 minutesfor (2). The solid curves in (c1) and (c2) are the bubble size distributions inferred from theultrasonic data at these two times. The dotted curve in (c2) are the results of X-ray tomographymeasurements.

to viscoelastic materials, by incorporating a simple correction to the resonant frequency

proposed by Alekseev and Rybak [29]. Physically, the effects of finite shear rigidity are to

shift the resonant frequency to higher frequencies and also to weaken the resonance. This

model has been tested on transparent agar gels, where the bubble sizes can be measured

optically, and found to describe the data well [16]. Applying the same model to dough,

as described in ref. [16] and shown by the solid curves in figs. 7(a),(b), the bubble size

distribution can be estimated. The inferred bubble size distributions at two times after

mixing are shown in fig. 7(c). At the later time, the evolution of the size distribution had

slowed down sufficiently to enable bench-top X-ray tomography measurements to inde-

pendently measure the size distribution (dotted curve in fig. 7(c)) [15]. This comparison

indicates that the analysis of the ultrasonic data estimates smaller bubble sizes than the

X-ray measurement, although comparison between results is not straightforward because

the conditions of sample preparation were not the same in both cases. Work is continu-

ing to understand the origin of this discrepancy, so that the ultrasonic technique can be

developed to unambiguously determine bubble sizes. Since ultrasonic measurements can

be performed quickly, potentially even online, this information has practical relevance

for monitoring dough quality during breadmaking. Even though questions remain to

be resolved concerning the absolute sizes of the bubbles determined from the ultrasonic

velocity and attenuation, the shift of the resonance features in the ultrasonic data to

lower frequencies at the later observation time shows the effects of disproportionation in

the dough due to Ostwald ripening; this phenomenon leads to an increase in the aver-

age size of the bubbles with time as gas diffuses from the smaller bubbles to the larger


ones [30]. Such information on the dynamics of the bubbles in bread dough is valuable

for understanding the evolution of the bubble structure.

Measurements of ultrasonic velocity and attenuation in dough in the high frequency

range (above the bubble resonance regime, f > 8 MHz) reveal information on matrix

properties. The frequency dependence of the data show signatures of ultrasonic relax-

ation phenomena, which can be interpreted using a molecular relaxation model [31, 32].

Different fast relaxation times were observed for ambient-mixed dough (5 ns) and vacuum-

mixed dough (1 ns) [33]. These relaxation times may be associated with conformational

rearrangements in glutenin —the supermolecular structure of proteins in the gluten

matrix— perhaps due to the loop-to-train transition that is thought to play a role in

the elasticity of glutenin [34]. Thus data in this frequency range can probe ultrasonic

stress-induced changes in the secondary structure of gluten proteins that are important

for understanding the viscoelastic properties of this complex food material.

This example of ultrasonic spectroscopy shows that both ultrasonic velocity and at-

tenuation are sensitive probes of the gas cell structure of bread dough, enabling new

approaches to optimizing loaf quality to be developed. Ultrasound can be used to follow

the evolution of the gas cells (bubbles) throughout the entire breadmaking process, from

the initial entrainment of gas bubbles in dough during mixing, though the expansion of

the gas cells during proofing, all the way up to the final foam structure of bread. Ul-

trasound can also be used to probe changes in the viscoelastic properties of the dough

matrix. Remarkably, despite the complexity of dough and bread as mesoscopic materials,

their mechanical properties can be elucidated using relative simple physics models. This

combination of factors is leading to a new awareness of ultrasound’s potential to provide

novel information on technical issues of importance to the cereals processing industry.

4. – Conclusions

Ultrasonic spectroscopy is both contributing to and capitalizing on advances in the

wave physics of complex mesoscopic materials. As a result, new approaches that ex-

ploit the advantages of ultrasonic techniques are being developed to characterize the

structure and dynamics of this increasing important class of materials. This paper has

discussed two examples. The first was Diffusing Acoustic Wave Spectroscopy, which is

a sensitive technique for monitoring changes in materials in which conventional imaging

techniques are impossible due to multiple scattering, and which is complementary to Dif-

fusing Optical Wave Spectroscopy. DAWS is based on direct measurements of the field

autocorrelation function, and has been extended recently to probe dynamics using the

phase of multiply scattered waves. This approach has some advantages practically, as

well as being a way of advancing our understanding of phase in mesoscopic wave physics.

In this paper, the application of this technique to the investigation of particulate and

bubbly suspensions was demonstrated, but many more applications of this technique can

be envisaged (e.g., in process control).

The second example considered here was the characterization of biological materials

of importance in food science. Many such materials have internal length scales that

130 J. H. Page

are comparable with the wavelength of ultrasound, making ultrasonic spectroscopy a

particularly relevant approach. The studies of bread dough that were summarized in

this paper demonstrate how advances in physics underpin practical applications. The

latter are of considerable economic potential in the rapidly growing functional foods area,

where the interaction of functional ingredients with the bubble structure can damage the

taste and appearance of food products unless remedial action is taken. By monitoring

the properties at an early stage in production and helping to understand the dynamics

of these interactions, ultrasonic techniques can help overcome such problems.

∗ ∗ ∗I would like to thank the many students and colleagues who have contributed to the

research that has been reviewed in this paper. Support from NSERC is also gratefully

acknowledged.

REFERENCES

[1] Page J. H., this volume, p. 75.[2] Page J. H., Schreimer H. P., Bailey A. E. and Weitz D. A., Phys. Rev. E, 52 (1995)

3106.[3] Page J. H., Sheng P., Schreimer H. P., Jones I., Jing X. and Weitz D. A., Science,

271 (1996) 634.[4] Schreimer H. P., Cowan M. L., Page J. H., Sheng P., Liu Z. and Weitz D. A.,

Phys. Rev. Lett., 79 (1997) 3166.[5] Cowan M. L., Beaty K., Page J. H., Liu Z. and Sheng P., Phys. Rev. E, 58 (1998)

6626.[6] Cowan M. L., Page J. H. and Weitz D. A., Phys. Rev. Lett., 85 (2000) 453.[7] Cowan M. L., Jones I. P., Page J. H. and Weitz D. A., Phys. Rev. E, 65 (2002)

066605.[8] Cowan M. L., Page J. H., Weitz D. A. and van Tiggelen B. A., in Wave Scattering

in Complex Media: From Theory to Applications, edited by van Tiggelen B. A. andSkipetrov S. E. (NATO Science series, Kluwer Academic Publishers, Amsterdam) 2003,pp. 151-174.

[9] Snieder R. and Page J., Phys. Today, 60 (2007) 49.[10] Letang C., Piau M., Verdier C. and Lefebvre L., Ultrasonics, 39 (2001) 133.[11] Elmehdi H. M., Ph.D. Thesis, University of Manitoba (2001).[12] Elmehdi H. M., Page J. H. and Scanlon M. G., Trans. Inst. Chem. Eng. Part C: Food

Bioprod. Proc., 81 (2003) 217.[13] Elmehdi H. M., Page J. H. and Scanlon M. G., J. Cereal Sci., 38 (2003) 33.[14] Elmehdi H. M., Page J. H. and Scanlon M. G., Cereal Chem., 81 (2004) 504.[15] Bellido G. G., Scanlon M. G., Page J. H. and Hallgrimsson B., Food Res. Int., 39

(2006) 1058.[16] Leroy V., Fan Y., Strybulevych A. L., Bellido G. G., Page J. H. and Scanlon

M. G., in Bubbles in Food 2: Novelty, Health and Luxury, edited by Campbell G. M.,

Scanlon M. G. and Pyle D. L. (AACC Press, St Paul, MN) 2008, pp. 51-60.[17] Scanlon M. G., Page J. H. and Elmehdi H. M., in Bubbles in Food 2: Novelty, Health

and Luxury, edited by Campbell G. M., Scanlon M. G. and Pyle D. L. (AACC Press,St Paul, MN) 2008, pp. 217-230.


[18] Mehta K. L., Scanlon M. G., Sapirstein H. D. and Page J. H., J. Food Science, 74

(2009) E455.[19] Leroy V., Pitura K. M., Scanlon M. G. and Page J. H., J. Non-Newtonian Fluid

Mech., 165 (2010) 475.[20] Maret G. and Wolf P. E., Z. Phys. B, 65 (1987) 409.[21] Pine D. J., Weitz D. A., Chaikin P. M. and Herbolzheimer E., Phys. Rev. Lett., 60

(1988) 1134.[22] Cowan M. L., Anache-Menier D., Hildebrand W. K., Page J. H. and van Tiggelen

B. A., Phys. Rev. Lett., 99 (2007) 094301.[23] Preston K. R., Kilborn R. H. and Black H. C., Can. Inst. Food Sci. Technol. J., 15

(1982) 29.[24] Sheng P., in Homogenization and Effective Moduli of Materials and Media, edited by

Ericksen J. L., Kinderlehrer D., Kohn R. and Lions J.-L. (Springer) 1988, p. 196.[25] Scanlon M. G., Mehta K. L., Elmehdi H. M. and Page J. H., unpublished.[26] Foldy L. L., Phys. Rev., 67 (1945) 107.[27] Leighton T. G., The Acoustical Bubble (Academic Press, London) 1994.[28] Prosperetti A., J. Acoust. Soc. Am., 61 (1977) 17.[29] Alekseev V. N. and Rybak S. A., Acoust. Phys., 45 (1999) 535.[30] Van Vliet T., in Bubbles in Food, edited by Campbell G. M., Webb C., Pandiella

S. S. and Niranjan K. (Eagan Press, St. Paul, MN) 1999, pp. 121-127.[31] Litovitz T. A. and Davis C. M., in Physical Acoustics, Vol. IIA, edited by Mason

W. P. (Academic Press, New York) 1965, pp. 282-350.[32] Marvin R. S. and McKinney J. E., in Physical Acoustics, Vol. IIB, edited by Mason

W. P. (Academic Press, New York) 1965, pp. 160-230.[33] Scanlon M. G., Page J. H., Leroy V., Fan Y. and Mehta K. L., in Proceedings of

the 5th International Symposium on Food Rheology and Structure, edited by Fischer P.,

Pollard M. and Windhab E. J. (ETH, Zurich) 2009, pp. 378-381.[34] Belton P. S., J. Cereal Sci., 29 (1999) 103.



MultiWave imaging

M. Fink and M. Tanter

Institut Langevin, Ecole Superieure de Physique et de Chimie de la Ville de Paris

CNRS 7587, INSERM - 10 rue Vauquelin, Paris, 75005, France

Summary. — Interactions between waves can be turned into profit to break diffrac-tion limits and invent new kinds of medical images. It consists in productively com-bining two very different waves —one to provide contrast, another to provide spatialresolution— in order to build a new kind of image. Contrary to multimodality med-ical imaging that remains the superposition of two different images limited by theirrespective contrast/resolution couples, MultiWave imaging overcomes this limita-tion by providing a unique image of the most interesting contrast with the mostinteresting resolution. MultiWave imaging can benefit from three different potentialinteractions among waves that will be described in this paper.

1. – Introduction

The human body supports the propagation of many kinds of waves with very differ-

ent speeds ranging from some m/s for mechanical shear waves to thousands of m/s for

ultrasonic compressional waves and some hundreds of thousands km/s for optical and

electromagnetic waves. Each of these waves can provide a medical imaging modality

whose contrast is depending of the way the wave interacts with tissues (see appendix A).

As the image is built from the interaction of only one wave with the medium, spatial

resolution at one depth is linked to the order in which three spatial scales appear, i.e.

observation depth z, wavelength λ and mean free path l. The first situation, z < λ < l,


134 M. Fink and M. Tanter

corresponds to near-field imaging and is encountered in the field of near-field optics,

EMG or EEG imaging and electrical impedance tomography (EIT). The second situ-

ation, λ < z < l, corresponds to coherent wave propagation outside of the near field

of the object and is encountered in the field of ultrasonic imaging, optical computed

tomography, and CT or X-ray imaging. The third situation, λ < l < z, corresponds

to the diffusive regime where the wave loses its coherence through tissues interactions.

This multiple scattering regime is mainly encountered in deep optical imaging, known as

diffuse optical tomography [1]. For the first situation, spatial resolution is of the order

of the observation distance z. For the second one, spatial resolution depends mainly of

the wavelength and diffraction limits. For the third one, spatial resolution could at best

reach the mean free path l, but is generally of the order of the observation distance z.

In all these imaging situations, physicists try to reach the optimal limits of the res-

olution associated to each kind of wave. However, one single wave is sensitive only to

one given contrast associated to some physical properties of the medium. For example,

shear waves are carrying information on the visco-elasticity of tissues (shear modulus

and viscosity); ultrasonic waves are sensitive only to compression modulus and density.

For electromagnetic waves, low-frequency waves are sensitive to electrical conductivity

while optical waves are sensitive to optical absorption coefficient. Today, after having

pushed during decades the technological limits of these modalities, physicists are facing

the inherent physical limits of the contrast/resolution couple in each modality.

For medical imaging and diagnosis, physicians understood rapidly that the way to

overcome these limits was to combine different imaging modalities, such as PET/CT,

PET/MRI or Ultrasound/X-ray mammography. The basic idea of multimodality imag-

ing [2], such as, for example, the combination PET/CT, is to associate the high-resolution

morphological image of a first modality (CT) to an image of the second modality (PET)

that is poorly resolved, but provides a clinically interesting contrast (i.e. metabolic ac-

tivity). However, such multimodality imaging remains extremely costly and limited by

the inherent physical limits of each separate modality.

2. – Transcending classical diffraction limits

Is there any other solution than multimodality imaging to improve the diagnostic

capabilities? Two different scientific communities proposed new research directions. One

approach called molecular imaging was impulsed by chemists and biologists. It differs

from traditional imaging in that biomarkers are used to help image particular targets

or pathways. These biomarkers interact chemically with their surroundings and in turn

increase the image contrast.

The other approach was proposed independently by different groups in the physicist

community. It consists in productively combining two very different waves —one to

provide contrast, another to provide spatial resolution— in order to build a new kind

of image. We propose to unify these approaches under the general concept of Multi-

Wave imaging. Contrary to multimodality imaging that remains the superposition of

two images limited by their respective contrast/resolution couples, multiwave imaging

MultiWave imaging 135

Fig. 1. – (Colour on-line) Principle of MultiWave imaging. Three possible wave interactions:a) wave-to-wave generation, b) wave-to-wave tagging, c) wave-to-wave imaging. In a) and b) thered wave carries information about the desired contrast, the blue wave provides the resolution.

overcomes this limitation by providing a unique image of the most interesting contrast

with the most interesting resolution. The focus of this paper is on MultiWave imaging

and will be illustrated using several examples of wave interactions.

Multiwave imaging can benefit from three different potential interactions between

waves (fig. 1):

– In a first case, the interaction of the first wave with tissues during its propagation

can generate a second kind of wave. This is the case in thermoacoustic imaging or in

magnetoacoustic tomography where any kind of electromagnetic wave is absorbed

in some region causing either a transient change in temperature that radiates an

ultrasonic wave through thermal expansion or a tissue motion under some Lorentz

force.

– In a second case, a first wave that carries the information about the desired contrast

but either completely loses its coherence during propagation through tissues, or has

a large wavelength, can be tagged locally by a second kind of wave that remains

coherent and well focused. The tagging focal spot can then be steered at different

locations in order to build a complete image. This is the case of Acousto-optical


imaging (or acousto-optical tomography) where tissue displacements induced by

a focused ultrasound beam modulates the optical speckle pattern of photons trav-

elling through tissues. An image of the optical absorption is built with the sub-

millimetric resolution of the ultrasonic wave.

– In a third case, a first wave travelling much faster than the second one can be used

to produce a movie of the slow wave propagation. This is the case of Transient

Elastography, where ultrafast ultrasonic scanners can track the motion of tissues

scatterers induced by the propagation of low-speed shear waves. This last case is

relatively unique as it allows observing remotely the full movie of the near field of

the shear wave around each obstacle (even if these obstacles are located in the far

field of the two waves). A local inversion algorithm performed on this near-field

movie produces a shear elasticity image relying on a sub-millimetric resolution while

the shear wavelength is centimetric.

The concept of MultiWave imaging is particularly interesting for the estimation of

three physical parameters that remained difficult to map up to recently with a good

spatial resolution: optical absorption, that gives access to tissues color; electrical con-

ductivity that depends on ion concentration and mobility in tissue and on the amount

of intra- and extra-cellular fluids; and finally mechanical shear elasticity and viscosity.

3. – Wave-to-wave generation

All techniques based on wave/wave generation are related to some dissipative pro-

cesses that transform one part of a pulsed electromagnetic energy in some transient

tissue motion that radiates coherent ultrasonic waves. From the recording of the ul-

trasonic field on an array of piezoelectric transducers, one can deduce an image of the

ultrasonic sources. The fact that the ultrasonic speed is practically uniform in all tissue

and has a well-known value greatly simplifies the reconstruction process. An image of

the sources is then built with the sub-millimetric resolution of the ultrasonic wave.

In the thermo-acoustic approach both optical waves [3] and microwaves can be used.

An image of the optical absorption or of the tissue conductivity is built with the sub-

millimetric resolution of the radiated ultrasound. Microwave penetration allows deeper

exploration and first conductivity images of breast have been obtained with this modal-

ity, while vascular images on small animals have been obtained with the photo-acoustic

approach. Figure 2 illustrates a spectacular application using as a heating source a laser

with 532 nm wavelength and a wide-band ultrasonic transducer with a 2.25MHz central

frequency to receive the photo-acoustic wave. Figure 2 shows that blood vessels in the

cortical surface of small animals can be imaged with the skin and the skull intact. The

imaging depth is limited to 1 cm that is enough to image the entire brain of a small

animal.

Other modalities have been proposed to improve electrical impedance tomography,

known as magnetoacoustic tomography (MAT), where tissues are displaced by an electric

or a magnetic stimulation [5,6] to produce ultrasound. In the most interesting technique


Fig. 2. – a) Non-invasive photo-acoustic image of a superficial lesion (1 mm× 4 mm) in the rightcortex on rat’s cerebra acquired through skull. RH is the right cerebral hemisphere, LH theleft cerebral hemisphere and L the lesion. The blood vessels are clearly imaged. b) Open skullphotograph of the rat surface acquired after the photo-acoustic experiment. See ref. [4].

with magnetic induction (MAT-MI), tissues are put both in a strong static magnetic

field and in a time-varying magnetic field (MHz range). The time-varying magnetic field

induces eddy currents that interact with the static magnetic field to produce a Lorentz

force that induced ultrasonic waves, that can be also recorded by ultrasonic transducers.

In this approach the acoustic wave amplitude is proportional to the electrical conductivity

in the MHz range.

The wave equation governing the ultrasonic pressure field radiated in all these ap-

proaches can be written in a common way as

(1)

{Δ −

1

c2p

∂2

∂t2

}p(r, t) = s(r, t),

where cp is the ultrasound compressional speed (quasi-uniform in soft tissues) and where

the source term can correspond either to monopolar sources (dilation) s(r, t) = − Γ

c2p

∂H∂t

for thermo-acoustic imaging [3] or to dipolar source in magnetoacoustic imaging (MAT-

MI) s(r, t) = ∇ · (J × B0). In the last source term, B0 is the static magnetic field, and

J is the induced Eddy current that depends on the pulsed magnetic field B1(t) and on

the local conductivity σ(r) according to Faraday’s and Ohm’s Law. For B0 = 1 T and

for a typical value of the conductivity σ = 0.2 S/m, ultrasonic pressure field of some 10

millibars can be radiated, which is sufficient to be measurable for ultrasonic transducers

elements.

Different approaches can be conducted to map the source terms and the inverse

problem image reconstruction is typically based on the fact that in a medium with


constant speed the data recorded on the transducers array are spherical integrals of

pressure source. Back-propagation algorithm allows recovering function from integrals

over spheres (spherical radon transform). This step can also be accomplished by time-

reversing and back-propagating the acoustic data in a computer model of constant speed.

4. – Wave-to-wave tagging

Acousto-optic imaging combines, thanks to acousto-optic effects, ultrasound and light

in a different way from photoacoustic imaging that is directly related to a dissipative pro-

cess. A focused ultrasonic beam induces locally an ultrasonic modulation of a light beam

traversing a scattering medium. Light transmitted through an organ contains thus differ-

ent frequency components: the main component (the carrier) is centered at the incident

coherent optical beam frequency. It is related to the dffused photons that do not inter-

act with ultrasound. The sideband components are shifted by the ultrasound frequency.

The sideband photons which results from the interaction between light and ultrasound

are called “tagged photons”. The weight of these tagged photons components depends

on the optical absorption in the region of interest. Acousto-optic imaging serves in de-

tecting selectively the tagged photons. An image related both to the optical absorption

and to optical diffusion is then built up in scanning the focused ultrasonic beam over

the whole organ. Marks investigated this tagging technique for the first time in the

early 1990s’ [7]. Since then, many different groups have contributed to this field [8-12].

Two main mechanisms participate in the ultrasonic modulation of light in a scattering

media. One is based on the variation of the optical phase in response to ultrasound-

induced displacements scatterers. The displacement of scatterers modulates the physical

path lengths of light traversing the ultrasonic field. Multiply scattered light accumulates

modulated path lengths. Therefore, the intensity of the speckle associated to multiply

scattered light fluctuates with the ultrasonic frequency. A second mechanism is based

on the variation of the optical phase in response to ultrasonic modulation. As the result

of ultrasonic modulation of the index of refraction, the optical phase between scattering

events is modulated and the modulated phase causes also the speckle intensity to vary

with ultrasound.

Many coherent detection techniques have been proposed to detect the tagged pho-

tons. One of the most interesting techniques was the parallel detection scheme that uses

a source-synchronized lock-in technique in which a CCD camera works as a detector

array [10, 11]. An interesting improvement was proposed to increase the axial resolu-

tion of the acousto-optic images which was not as good as the lateral resolution with

monochromatic ultrasound. Wang and Ku replaced the ultrasound monochromatic exci-

tation with a frequency swept (chirped signal) that modulated also the gain of the optical

detectors [11].

However, the main difficulty of this technique in living tissues results from both the

motion of the scatterers due to the Brownian motion of the scatterers and to the tissue

inner motions (blood flow). This speckle decorrelation broadens the carrier and sideband

lines. Typically with 4 cm breast thickness, the speckle decorrelation time is in the ms


range and yields a 3 kHz broadening. Detector bandwidth in this range is needed. With a

mono detector (photodiode) there is of course no problem, but to achieve a good signal to

noise one needs a large optical etendue in detector plane (the product of the detector area

by the detector acceptance angle) to fit the etendue of the tagged photons source i.e. to

maximize the scattered light collection. This is why multidetectors like a CCD camera

have been investigated, but they suffer low image frequency rate (typically 100 Hz) which

is not enough to avoid broadening effect in living tissues. Faster cameras are not sensitive

enough. More recently different groups [12] proposed very promising tagged-photons

detection techniques based on the photorefractive crystal-based interferometry that can

give both a large etendue and a detection bandwidth in the kHz range (the response time

of GaAs photorefractive material is of the order of 1 ms).

It is interesting to note here that the tagging concept could be used not only in

acousto-optics but in many other fields of medical imaging such as, for example, electric

impedance tomography tagged by ultrasonic remote vibrations.

Here we discuss the concept of wave-to-wave tagging using two kinds of totally dif-

ferent waves. However, although it is not a multiwave technique, MR imaging can also

be interpreted as a tagging technique. It uses only one kind of wave combined with a

static field: a radio frequency electromagnetic wave that causes protons to absorb some

of its energy and to release it later at a resonance radio frequency. The spatial tagging is

achieved here through the addition of non-uniform magnetic fields to a static magnetic

field whose spatial gradient modifies the local Larmor frequency, allowing during the

reception mode to get a spatial resolution much better than the RF wavelength through

a frequency analysis of the received signal.

5. – Wave-to-wave imaging

The last example of multiwave imaging is perhaps the most fascinating. Indeed, the

wave interaction is here chiseled such that the near field of a wave around each obstacle

can be filmed in the far field of the imaging sensors. In this approach, the playground

consists in sonic shear waves and ultrasonic waves. These waves interact to produce a

quantitative and highly resolved image of deep organs stiffness.

Stiffness is characterized by Young’s Modulus E (in kPa) and is an important param-

eter in medicine. Stiffness changes are often linked to pathology [13] and the significant

dependence of E on the structural changes in the tissue is the basis for the palpatory di-

agnosis of various diseases, such as detection of cancer nodules in the breast or prostate.

Although it is strongly subjective, manual palpation is not only useful for screening

and diagnosis but also during interventions to effectively guide the surgeon toward the

pathological area.

In order to understand how to map tissue elasticity, we can in a first approximation

consider soft tissue as an isotropic elastic medium. The mechanical behaviour of such a

soft solid is characterized by two parameters, K (inverse of the compressibility κ = 1/K)

and μ, respectively, the bulk and shear modulus. The relationship between stiffness and


these parameters is described by the relationship

(2) E =9Kμ

3K + μ.

The property K � μ is a kind of definition of “soft solids” and human soft tissues belong

to this category. It implies straightforwardly a direct link between stiffness E and the

shear modulus, E = 3μ. Therefore to access stiffness properties, an elegant way consists

in using shear waves whose speed cs depends simply on the shear modulus cs =√μ/ρ.

Another important consequence of the big discrepancies between K and μ in soft solid

is that the compressionnal wave speed is much larger than the one of the shear waves

(from 1540 m/s for cp to some m/s for cs). This is a unique case where two mechanical

waves exhibit totally different wave speeds.

In conventional “single wave” imaging, only the compressional wave (and consequently

the contrast of bulk modulus) is used. This is the successful field of medical ultrasound

imaging. Now, could we use shear waves to image the shear modulus contrast of tissues?

As we have just seen, the MHz frequency range is forbidden due to shear tissue viscosity

and shear waves can only propagate on centimetric distances at low sonic frequencies.

The typical shear wave frequency ranges between 10 Hz and 1 kHz. For example, to prop-

agate on a 5 cm distance, we are limited to frequencies lower than 100 Hz, corresponding

to typical wavelengths of several centimeters. The use of shear waves in a “single wave”

imaging approach can only lead to poor results as it will rely on very bad spatial resolu-

tion. However, the contrast sensed by shear waves remains very relevant information for

the diagnosis.

How to solve this problem using a “multiwave” approach? We can take benefit from

the huge discrepancy between shear and compressional wave speeds. The idea is to use

the compressional waves at ultrasonic frequencies to produce ultrafast images of tissues

to image the propagation of low speed sonic shear waves. The goal is to obtain a movie

of the shear wave propagating inside organs with millimetric resolution. During their

propagation, shear waves induce local tissue displacements of the order of some tens

of microns around their equilibrium position. As the typical shear wave speed varies

between 1 m/s and 10 m/s, one needs at least to reach 10000 frames per second to follow

the shear wavefront millimeter by millimeter. Using such an ultrafast scanner, it could

be possible to estimate these local displacements between successive images.

Our group developed such an ultrafast scanner [14]. This is the first ultrasonic device

able to reach more than 10000 frames per second of deep-seated organs. In this device,

whose architecture emerged from the concept of time reversal mirrors [15,16], one trans-

mits several thousand times per second an ultrasonic beam widely spread in the whole

area of interest. This imaging sequence is very different from the one used in conventional

ultrasound scanners that insonicate the medium only using a very thin ultrasonic focused

beam that needs to be translated step by step to sequentially map the imaged area (see

fig. 3). Such a conventional echographic image results in more than 128 successive insoni-

cations. Taking into account the time of flight of backscattered ultrasound (a 20 cm back


Fig. 3. – From conventional to ultrafast ultrasonic imaging. In conventional ultrasound, thebackscattered echoes of hundreds of focused beams successively fired into the medium are neededto produce one ultrasonic image. In ultrafast imaging, a plane wave is fired and insonifies in asingle shot the whole medium and backscattered echoes are processed to produce the ultrasonicimage.

and forth propagation requires some 130μs), a typical frame rate of about 50 images

per second can be reached for sequential imaging. On the contrary, in an ultrafast scan-

ner, for each transmit beam, the backscattered echoes coming from a very large region

of interest are recorded by an array of some hundreds of piezoelectric transducers and

stored into large memories. Then, a fast algorithm transforms several thousand times

per second the backscattered echoes into an echographic image. Thanks to the fact that

ultrasonic wave speed is known and constant, this operation can be obtained through

a numerical time reversal refocusing. In order to track the local displacements induced

by shear wave propagation, successive ultrasonic images are compared. This is possi-

ble because the ultrasonic images are dominated by the so-called “speckle” noise that

originates from the random distribution of weak scatterers (Rayleigh scatterers much

smaller than the wavelength) that exist everywhere in tissues. Note that in soft tissue,

contrary to optics, ultrasonic backscattering is dominated by single scattering process

thus ensuring an unambiguous correspondence between the arrival time of the speckle

noise and the spatial location of the scatterers distribution. By cross-correlating in the

time domain the speckle noise observed from one frame to the other, a motion speckle

tracking algorithm enables to reconstruct a complete movie of the tissue displacements


Fig. 4. – Principle of elasticity imaging: In step 1, the ultrasonic probe generates a pushing forcein the focal area of the ultrasonic beam. In step 2, this radiation force generates a low-frequencyshear wave. The ultrasonic array switches into an ultrafast imaging mode. In step 3, the resultingultrasonic images are built and stored into memories. In step 4, successive ultrasonic imagesare compared using cross-correlation operations in order to image tissue displacement inducedby the propagation of the shear wave.

field along the ultrasonic beam direction (fig. 4). From this movie, one can locally deduce

the shear wave speed and thus the shear modulus μ (fig. 5).

How can we generate shear waves in the human body? Such shear waves already

exist naturally in our body. Each heart beat creates transient vibrations that propagate

near the cardiac muscles and along arteries. Our vocal cords are also producing shear

vibrations into nearby organs during speech.

These natural waves usually remain confined closed to their source and cannot be

used to assess all organs. External vibrators applied at the surface of our body can also

produce controlled vibrations that induce shear waves propagating from the surface to

deeper regions. Finally, instead of using heavy external vibrators, the most elegant way

to create a controlled shear wave source consists in using the radiation force induced

by ultrasonic focused beams into tissues. Indeed, by focusing an ultrasonic beam at a

given location in the organ, it is possible to create a volumic radiation force localized in

the focal spot and oriented along the beam axis. This force is due to the momentum


Fig. 5. – Generation of a supersonic shear source. a) Bursts of ultrasound are focused atsuccessive depths in the organ. Each burst creates a “pushing” radiation force at focus thatinduces a shear wave. As the “pushing” force is moved faster than the shear waves it generates,a supersonic regime is reached and shear waves accumulate on a Mach cone. b) Images (40 ×40mm2) of micrometric displacements induced in a tissue mimicking phantom obtained usingthe ultrasonic ultrafast imaging mode at different time steps.

transfer from the ultrasonic wave to the medium caused by nonlinearities, dissipation

and reflection and is proportional to the square of the ultrasonic pressure field. Thus,

the time profile of the applied force is linked to the beam intensity spatiotemporal profile.

The transmission of a 1 ms burst of focused ultrasound with a 5 MHz carrier frequency

will lead to an axial force in the kHz range. While remaining below the FDA limitations,

the radiation force of the ultrasonic beam permits to remotely create low-frequency shear

displacements of some tens of microns at several centimeters depths. Thus, it is possible

to generate the shear wave using the same array of piezoelectric transducers that is used

in ultrafast imaging. The use of the ultrasonic radiation force as a remote generator

of shear waves was proposed by Sarvazyan et al. in 1998 [17] and is used by several

research groups in medical imaging [18-21]. The transducers are used in a first step to

transmit a long burst of ultrasound focused at the desired location in the imaged area

(fig. 4, panel 1). The beam generates a low frequency pushing force at focus. When the

transmission (and consequently the pushing force) ends, tissues displaced in the focal

spot come back to their equilibrium position while generating a small localized source of

transient shear waves. It is the biomedical analog of a small earthquake created by the

shear force of a moving tectonic plate. The resulting shear wave begins to propagate in


the organ whereas the ultrasonic probe instantaneously switches into an ultrafast imaging

sequence in order to film this propagation (fig. 4, panel 2). The amplitude of the radiated

shear wave decreases quite rapidly due to the natural divergence associated to a small

shear source.

In order to extend the area sensed by the shear wave, we proposed an original solution

enabling the generation of weakly diffracting shear waves based on the remote creation of

a supersonic source radiating shear waves along a Mach cone [18]. This effect is the analog

of the Cherenkov electromagnetic radiation emitted by a beam of high-energy charged

particles passing through a transparent medium at a speed greater than the speed of light

in that medium [19]. In our configuration, Ultrasonic waves are successively focused at

different focal depths by changing the electronic delays between the signals transmitted

by the transducer elements. By moving the resulting shear source faster than the radiated

shear waves, one can induce constructive interference along a Mach cone (fig. 5). Such

a sonic boom is very efficient to create a shear wave of higher displacement that can

travel with minimized diffraction in a large area. Compared to the use of external

vibrators [22, 23], this technique enables to optimally polarize the shear displacement

field along the ultrasonic beam axis that corresponds to the most sensitive axis of the

speckle tracking algorithm.

From the 2D experimental movie of the shear displacements along the ultrasonic beam

axis (Oz) uz(x, z, t) induced by the supersonic push, one can have access to the shear

modulus map by solving locally the inverse problem of wave propagation. Indeed, the

elastodynamic wave equation characterizing the shear wave propagation {Δu = ρμ

∂2

∂t2 u}

can be inverted for each pixel (x, z) in the imaging plane and gives access to the local

shear modulus μ(x, z) [18]

(3) μ(x, z) =ρ∂2uz(x, z, t)

∂t2

∂2uz(x, z, t)

∂x2+∂2uz(x, z, t)

∂z2

,

where ρ is the medium density (almost constant in soft tissues). Surprisingly, this very

simple inversion approach (based on the calculation of second-order derivatives of the

displacement field both in time and space) reaches very good performances. It is mainly

due to the fact that the wave field uz(x, z, t) is experimentally measured everywhere

in the region of interest contrary to conventional imaging approaches where the field is

known only at boundaries and so requires complex inverse problem approaches.

An experiment conducted in a tissue mimicking phantom (made of Agar gelatin)

illustrates the interesting features of this technique. A conventional ultrasonic image of

this phantom reveals an almost perfectly homogeneous medium in terms of bulk modulus

(fig. 6A) although the phantom contains a centimetric stiffer inclusion. After generation

of the shear Mach cone, high frame rate images of tissue displacements induced in the

phantom are recorded during the resulting shear wave propagation. The shear wave is

clearly sensitive to the shear modulus contrast as it strongly accelerates while passing


Fig. 6. – Supersonic shear imaging in tissue mimicking phantoms. A) Images of local tissuedisplacements (gray scale ranging from −10μm to +10μm) at different time steps after thesupersonic shear source generation. One clearly sees that the shear wave is sensing the stiffnesscontrast as it is distorted while passing through a 10 mm stiff inclusion. B) On the contrary,the ultrasonic image of the medium does not reveal the inclusion, and C) quantitative image ofYoung’s modulus deduced from the shear wave movie (A).

through the stiffer region (fig. 6B). Then a local estimation of the shear wave speed

based on basic time-of-flight estimation enables to quantitatively map Young’s modulus

(fig. 6C). Young’s modulus image reveals a highly contrasted spherical inclusion twice

harder than the surrounding tissues.

5.1. Super-resolution in supersonic shear wave imaging. – Interestingly, one can notice

in fig. 6C that the resolution of the elasticity map is of the order of the millimeter,

despite the fact that the shear wavelength is of the order of a centimeter (as seen in


fig. 6A). This super-resolution capability of SSI can be explained by the fact that the local

displacements induced by the wave propagation are not only recorded on the boundaries

of the investigated medium (as, for example, in seismology) but also at every location

deep into the investigated medium. The estimation of a local displacement field in the

ultrasonic images provides as many virtual motion sensors (accelerometers) deep into

tissues and this even in the far field of the shear wave source. Such ultrasound-based

virtual motion sensors give a remote access to the local near field of the wave field around

each obstacle even if these obstacles are located in the far field of the two waves. The

evanescent waves created by the interaction of the incident shear wave with any shear

heterogeneities are recorded in the movie with a spatial sampling of some λ/20 in fig. 6.

The inverse algorithm based on the computation of the spatial and time derivative of

the field at each location (see eq. (3)) extracts this information. Therefore the spatial

resolution of the shear modulus image is no more limited by the classical diffraction limit

of shear wavelength but linked to the much smaller ultrasonic wavelength. We have

access to a complete movie of the near field.

5.2. Clinical applications. – A major advantage of this imaging technique, called “Su-

personic Shear Imaging” (SSI), is that it can be performed using conventional ultrasonic

probes and provides an additional imaging modality on a new generation of ultrasound

scanners. It gives spectacular in vivo results as, for example, breast cancer diagnosis [24].

Figure 7 shows two interesting cases obtained on breast cancer diagnosis. The 2D color

map of Young’s modulus of a patient breast is superimposed on the conventional ultra-

sonic image. In fig. 7b, two very small and stiff breast lesions (2 mm diameter invasive

ductal carcinomas) are visible on the highly contrasted elasticity image but undetected

on the conventional ultrasonic image. Figure 7a corresponds to a very stiff invasive

ductal carcinoma. This example is particularly interesting as it shows that the elasticity

information provides a new information clearly different from the conventional ultrasonic

image. The invasive ductal carcinoma presents very stiff boundaries (red color regions)

and a very soft core (blue region) corresponding to a necrotic region. These results were

confirmed by histology and emphasize the ability of SSI to provide new insights into the

radiological characterization of cancer lesions.

5.3. Shear wave spectroscopy . – Thanks to its ability to image very fast and transient

motion of shear waves, SSI can provide even much refined information about the me-

chanical properties of tissue than just a single estimation of Young’s modulus. Indeed, if

soft tissues are considered as a purely elastic medium, the shear wave speed cs is directly

linked to shear modulus μ via cs =√μ/ρ. As cs does not vary with frequency, the

time profile of the shear wave generated by the supersonic source is considered to be

undistorted during propagation. This approximation of a purely elastic medium leads

to the stiffness image provided in SSI by the estimation of the group speed of the shear

wave. However, in many organs tissues exhibit shear viscosity and signal processing of

the shear wave propagation movie can be refined to study this more complex biomechan-

ical behaviour. Viscosity affects the shear wave propagation speed. The time profile of


Fig. 7. – In vivo clinical application of the Supersonic Shear Imaging (SSI) technique for breastcancer diagnosis. Case (a) corresponds to an infiltrating ductal carcinoma (grade II) and showsthe ability of SSI to provide a biomechanical characterization of lesions. The superimposedYoung’s modulus image shows a lesion much stiffer than the surrounding tissue and a verysoft core corresponding to a necrosed area. This result was confirmed by histology. Case (b)corresponds to two very small infiltrating ductal carcinomas (Grade I &RH+). The millimetricresolution of Young’s modulus image enables to image these lesions which were not detected byX-ray mammography. (Courtesy 5of Supersonic Imagine, France.)

the plane shear wave is progressively distorted and attenuated during propagation. This

distortion is characterized by a frequency dependence of the shear wave speed and attenu-

ation that fully describes the rheological behaviour of tissue. The estimation of the shear

wave speed in the Fourier domain enables to provide the dispersion curve of the shear

wave phase speed cs(ω). As the radiation force of the ultrasonic burst (typically 100μs

burst) induces a transient excitation, the frequency bandwidth of the generated shear

wave is quite large, typically ranging from 100 to 1000 Hz. Such wide-band “shear wave

spectroscopy” [25] gives a much refined analysis of the complex mechanical behaviour of

tissue. Figure 8 shows an example of this analysis in transcostal liver imaging. Superim-

posed on the echographic image, a color map presents the quantitative Young’s modulus

image of liver (assuming a purely elastic medium). As it requires a better signal to noise


Fig. 8. – In vivo Young’s modulus image of a healthy volunteer liver using an ultrasonic probeplaced in the intercostal space. Superimposed on the ultrasonic image, Young’s modulus imageexhibits a stiff region corresponding to the intercostal muscle and a soft homogeneous regioncorresponding to the liver. Shear wave spectroscopy performed in the two boxes shows a strongdispersion of shear waves in liver (high shear viscosity) and a very low dispersion in the muscle(low shear viscosity).

ratio, shear wave spectroscopy is performed in a larger box (5×5 mm2) and provides the

full dispersion curve cs(ω). As a comparison, the dispersion curve is also estimated in

the intercostal muscle (almost purely elastic along the muscle fibers) and exhibits almost

no frequency dependence in that region. The slope of the dispersion curve is linked to

shear viscosity and many different rheological models can then be chosen to describe this

complex behaviour with a few determined parameters. Recent works in our lab focus on

demonstrating that such shear viscosity at the macroscopic level (some millimeters) is

related to the underlying architecture of tissues network at a much smaller scale. In par-

ticular, the combined estimation of stiffness and shear viscosity could provide a refined

estimation of liver fibrosis/steatosis degree in cyrrhotic or Hepatitis C patients.

In parallel to the advent of the SSI technology, the quest for stiffness imaging led to

extensive research efforts in the medical imaging community during the last twenty years.


The concept of stiffness imaging was introduced in the early 90s’ by J. Ophir et al. and

named Elastography [26]. Their technique is based on the ultrasonic imaging of tissue

deformations induced by a quasi-static compression of organs applied by the operator

at the surface of the body. Tissue deformations are obtained by acquiring two pre-

compression and post-compression ultrasonic images of the organ using a conventional

ultrasound scanner. So, static elastography is inherently a single-wave approach (based

on the single use of ultrasonic waves) that implies some important drawbacks. The

comparison of the two images enables only the mapping of the local tissue strain. This

strain image, called elastogram, is linked to stiffness as soft regions tend to exhibit a

higher strain than stiffer areas. However, even for a simple one-dimensional model,

the underlying link between local strain ζ and stiffness E (Young’s Modulus in kPa)

is strongly dependent on the local and unknown stress τ via the well-known relation

E = τ/ζ. Unfortunately, applying a quasi-static compression at the surface of the body

can create a very complex spatial distribution of stress that both prevents the assessment

of local stiffness and induces image artifacts.

Another elasticity imaging technique based on the imaging of shear wave propagation

using a MRI system (MR-Elastography) was introduced by Mayo Clinic in 1995 [27]. The

shear wave is generated by vibrators located at the surface of the body and a dedicated

MR sequence enables the local 3D displacements of tissues during shear wave propaga-

tion. Thanks to its MultiWave imaging nature, MRE succeeds in providing 3D quanti-

tative images of Young’s modulus with a millimetric resolution. This approach is never-

theless limited in clinical routine by huge acquisition times (typically tens of minutes).

6. – Conclusion

MultiWave imaging is a very general concept that can be extended to geophysics as

well as non-destructive testing and remote sensing. The future of MultiWave imaging

for clinical and biomedical applications is bright. Parallel to the advent of molecular

imaging that requires the injection of clinically approved biomarkers, this general wave

physics concept is today a fertile source of new ideas and technologies. Many MultiWave

imaging approaches can be performed as stand-alone techniques but could also be coupled

to molecular imaging strategies. Among all these smart combinations of waves, the

wave-to-wave imaging approach is quite exciting, as it is the only one to provide a

quantitative assessment of the desired contrast. The first MultiWave imaging application

that encounters today clinical success is elasticity imaging. It can be used for breast

cancer diagnosis, cardiovascular applications, abdominal or musculo-skeletal imaging and

will be soon applied to ophthalmology and dermatology. 2D ultrasonic arrays will also

facilitate new clinical applications such as image guidance for minimally invasive surgery

or the monitoring of thermal ablation treatments.

MultiWave imaging also introduces new scientific concepts transcending the conven-

tional limits of wave physics, such as (in the case of supersonic shear wave imaging)

providing a way to observe the near field of waves around heterogeneities from far-field

detectors. On the technology side, the first clinical ultrafast ultrasonic scanner provides


for the first time a way to image in real time many natural transient vibrations occurring

(permanently) in our body. This is perhaps the most fascinating kind of MultiWave

imaging where one of the waves naturally exists in living organs such as pulsatility vi-

brations induced in the heart, along arteries or mechanical vibrations induced in nerves,

neurons and muscle fibers by electric actions potentials.

Appendix A.

Waves propagation in tissues

Electromagnetic and mechanical waves can propagate through tissues with very dif-ferent regimes, depending mainly on their frequencies.

A.1. Electromagnetic waves

A.1.1. Low frequency. In Electrical Impedance Tomography (EIT), an image of the

conductivity or dielectric permittivity of tissue is inferred from surface electrical measure-ments. Conducting electrodes are attached to the skin and small alternating currents areapplied to some electrodes. The resulting electrical potentials are measured and an in-verse problem algorithm allows mapping conductivity and permittivity. In this technique,one generates electromagnetic waves within frequency range of kHz up to few MHz (theso called β dispersion region) but also sometimes as high as 10 MHz and as low as 100 Hz.At those frequencies, the wavelength is very large compared (> 100 m) to the organ di-mension. We are in the near-field regime and the electromagnetic field is dominated bythe frequency-dependent impedance of tissues. Biological impedance is characterized bya complex-valued conductivity σ∗ = σ − iωε that incorporates both the conduction cur-rent and the displacement current. The dielectric permittivity ε (farads/meter) resultsfrom capacitive energy storage due to the bound (dipolar) charge at cellular membranes,while the conductivity σ (siemens/meters) is dominated by ionic transfer in tissue and bythe amount of intra- and extra-cellular fluids. The injected current a low frequency willmainly flow through extracellular water; while at high frequencies the current will flowthrough both extra- and intra-cellular water. Note that at low frequency the imaginarypart of the complex conductivity is small and very precise measurements of the phaseare needed to map not only the conductivity but also the permittivity

A.1.2. Microwaves. Microwaves (500 MHz to 20 GHz) can also propagate deeply

through tissues. However, compared to LF electromagnetic waves, they are stronglyattenuated due to electrical conductivity that induces dissipative effects and heat deposi-tion, while the dielectric permittivity affects the wavelength of the microwave. Typically,around 1 GHz the average dielectric permittivity of normal breast tissue is 12.5, result-ing in a wavelength close to 10 cm, and the average ionic conductivity is 0.21 S m2/mol.These are values quite similar to the one of low-water content fat. In contrast, for somebreast cancer carcinoma the average dielectric permittivity was found to be close to 55,while the average conductivity increases to 1 S m2/mol. This has been attributed to anincrease in bound water and sodium in malignant cells. Various radar-based (backscat-ter response to ultra wide-band radar) and tomographic methods have been recentlyproposed for breast imaging but they still suffer of a lack of spatial resolution.


A.1.3. Optical waves. The use of light and near infrared (NIR) wavelength is motivated

by its relative low absorption in the so-called optical “therapeutic or diagnostic window”(650 to 1300 nm) and by the existence of optical contrast between healthy and malignanttissues in this spectrum. However in opaque tissue light loses very rapidly its coherencewhen propagating through tissues heterogeneities such as collagen fibers, cell membrane,organelles, extracellular matrix, etc. Therefore, both multiple scattering between indi-vidual heterogeneities and absorption are responsible for the light beam broadening anddecay as it travels through tissue. In this regime, light scattering is characterized by twolength parameters, i.e. the scattering length ls and the light transport mean free path ls

∗.The scattering length ls characterizes the memory of the optical phase and correspondsto the average distance that separates two scattering events. The light transport meanfree path ls

∗ characterises the memory of the light propagation direction. In tissues lsis typically 50 to 100μm, while ls

∗ is ten time larger (0.5 to 1 mm). Absorption of lightis also characterized by another length: the absorption length la which is in the 1 cm to10 cm range. Absorption strongly depends on the nature of tissue and is the main opticalcontrast that one tries to map.

X rays and Γ rays are also electromagnetic waves that propagate in tissue but theycan be interpreted as rays following geometrical optics approximation. They will not bediscussed here

A.2. Mechanical waves. – In a first approximation, soft tissues can be considered as

an isotropic elastic medium. Therefore two kinds of mechanical waves can propagate insuch medium: the well-known P and S waves of seismology. P waves are compressionalwaves traveling at a speed cp that depends mainly on the compressional bulk modulusK (inverse of the compressibility κ = 1/K) and shear waves (S waves) that propagateat a slower speed cs affected by the shear modulus μ. Soft human tissues are mainlycharacterized by the fact that K is huge in comparison with μ. K is almost uniform (lessthan 5% fluctuations) in all soft tissues with typical values of ∼ 1 GPa. Moreover, K doesnot change significantly with pathology as it is defined mainly by molecular compositionof tissue and short-range molecular interactions. Since most soft tissues are by about80% made of water, K is not much different from the water bulk modulus. On thecontrary, μ is very small with typical values of some kPa and it strongly varies from oneorgan to the other. μ is defined by cellular and higher levels of structural organizationof tissue and consequently is greatly affected by pathological or physiological changes intissue structure. For example, an infiltrating carcinoma can be characterized by a shearmodulus more than 100 times higher than surrounding healthy tissues.

As the P wave speed cp is given by cp =√

(K + (2/3)μ)/ρ ≈√K/ρ and the S

wave speed cs by√μ/ρ, we may notice that compressionnal waves propagate around

1500 m/s corresponding to the sound speed in water, while shear waves propagate ata much smaller speed with speed values ranging from 1 m/s to 10 m/s in soft humantissues.

Taking into account the fact that tissues are not only elastic media but also viscoelas-tic media, one has to introduce viscous effect (or rheological effect) in wave propagation.The main effect is that at ultrasonic frequency shear waves are so strongly attenuated byshear viscosity effects that they cannot be used to penetrate inside tissues. However com-pressionnal waves are not affected by shear viscosity but by bulk viscosity, which is muchweaker and depends on various mechanical relaxation processes. Typically compressionalwave attenuation enables to use frequencies ranging between 2 and 50MHz correspond-ing to imaging depth ranging, respectively, between 12 cm and 2 mm and associated towavelengths ranging from 0.8 mm to 30μm. In this range of frequencies soft tissues be-


have more like a fluid with a frequency linear attenuation (typically 0.1 dB/cm/MHz).This explains the success of ultrasound imaging in terms of spatial resolution for in-depthimaging. However, the contrast of ultrasound images remains poor as it is related onlyto the weak heterogeneities of the bulk modulus. Due to weak heterogeneities, ultrasoniccompressionnal waves can propagate on long distance without losing their coherence,resulting in a mean free path ls much larger than the observation distance. This meansthat in pulse echo-mode the backscattering echoes resulting from the distribution of het-erogeneities are obtained only from single scattering processes, which greatly simplifiedthe imaging modality, compared to diffuse optical tomography. Moreover the fact thatthe ultrasonic speed is practically constant and known in all tissues allows very simpleprocessing to recover an image.

To observe shear wave in tissues, we have to work at low sonic frequencies (less than1 kHz). Thus shear attenuation is low and shear waves may propagate deeply in tissues toreveal shear properties. For example, at 100 Hz, shear waves can propagate in glandularbreast tissue on nearly 10 cm, with a wavelength of 2 cm, corresponding to a shear speedof 2 m/s.

REFERENCES

[1] Yodh A. and Chance B., Phys. Today, 48 (1995) 34.[2] Cherry S. R., Annu. Rev. Biomed. Eng., 8 (2006) 35.[3] Emelianov S. Y., Li P. C. and O’Donnell M., Phys. Today, 62(8) (2009) 34.[4] Wang X., Pang Y., Ku G., Xie X., Stoica G. and Wang L. H. V., Nature Biotechnol.,

21 (2003) 803.[5] Towe B. C. and Islam M. R., IEEE Trans. Biomed. Eng., 35 (1988) 892.[6] Li X., Xu Y. and He B., J. Appl. Phys., 99 (2006) 066112.[7] Marks F. A., Tomlinson H. W. and Brooksby G. W., Photon Migration and Imaging

in Random Media and Tissues, Proc. SPIE, edited by Chance B. and Alfano R. R.,Vol. 1888, (1993) 500.

[8] Wang L.-H. V., Jacques S. L. and Zhao X.-M., Opt. Lett., 20 (1995) 629.[9] Leutz W. and Maret G., Physica B, 204 (1995) 14.

[10] Leveque S., Boccara A. C., Lebec M. and Saint-Jalmes H., Opt. Lett., 24 (1999)181.

[11] Wang L.-H. V and Ku G., Opt. Lett., 23 (1998) 975.[12] Gross M., Lesa M., Ramaz F., Delaye P., Roosen G. and Boccara A. C., Eur.

Phys. J. E, 28 (2009) 173.[13] Sarvazyan A. P., Acoustical Imaging, Vol. 21 (Plenum Press, New York) 1995, pp. 223–

240.[14] Sandrin L., Tanter M., Catheline S. and Fink M., IEEE Trans. Ultrason., Ferroelec.,

Freq. Contr., 49 (2002) 426.[15] Fink M., Phys. Today, 20 (1997) 34.[16] Fink M., Montaldo G. and Tanter M., Annu. Rev. Biomed. Eng., 5 (2003) 465.[17] Sarvazyan A. P., Rudenko O. V., Swanson S. D., Fowlkes J. B. and Emelianov

S. Y., Ultra. Med. Biol., 20 (1998) 1419.[18] Bercoff J., Tanter M. and Fink M., IEEE Trans. Ultrason., Ferroelec., Freq. Contr.,

51 (2004) 374.[19] Bercoff J., Tanter M. and Fink M., Appl. Phys. Lett., 84 (2004) 2202.[20] Nightingale K. R., Soo M. S., Nightingale R. W. and Trahey G. E., Ultra. Med.

Biol., 28 (2002) 227.


[21] Fatemi M. and Greenleaf J. F., Science, 280 (1998) 82.[22] Sandrin L., Tanter M., Gennisson J. L., Catheline S. and Fink M., IEEE Trans.

Ultrason., Ferroelec., Freq. Contr., 49 (2002) 436.[23] Catheline S., Thomas J.-L., Wu F. and Fink M., IEEE Trans. Ultrason., Ferroelec.,

Freq. Contr., 46 (1999) 1013.[24] Tanter M., Bercoff J., Athanasiou A., Deffieux T., Gennisson J.-L., Montaldo

G., Muller M., Tardivon A. and Fink M., Ultra. Med. Biol., 34 (2008) 1373.[25] Deffieux T., Montaldo G., Tanter M. and Fink M., IEEE Trans. Med. Imag., 28

(2009) 313.[26] Ophir J., Cespedes I., Ponnekanti H., Yasdi Y. and Li X., Ultrasonic Imaging, 13

(1991) 111.[27] Muthupillari R., Lomas D. J., Rossman P. J., Greenleaf J. F., Manduca A. and

Ehman R. L., Science, 269 (1995) 1854.



Time reversal focusing and the diffraction limit

M. Fink, J. de Rosny, G. Lerosey and A. Tourin

Institut Langevin, Ecole Superieure de Physique et de Chimie Industrielle de la Ville de Paris

UMR CNRS 7587 - 10 Rue Vauquelin, 75005 Paris, France

Summary. — Time reversal mirrors refocus an incident-wave field to the position ofthe original source, regardless of the complexity of the propagation medium. TRMshave now been implemented in a variety of physical scenarios from GHz Microwavesto MHz Ultrasonics and to hundreds of Hz in ocean acoustics. Common to this broadrange of scales is a remarkable robustness exemplified by observations at all scalesthat the more complex the medium (random or chaotic), the sharper the focus. ATRM acts as an antenna that uses complex environments to appear wider than itis, resulting, for a broad-band pulse, in a refocusing quality that does not dependon the TRM aperture. Moreover, when the complex environment is located in thenear field of the source, time reversal focusing opens completely new approachesto super-resolution. We will shown that, for a broad-band source located inside arandom metamaterial, a TRM located in the far field radiates a time-reversed wavethat interacts with the random medium to regenerate not only the propagating butalso the evanescent waves required to refocus below the diffraction limit.

1. – Introduction

Time reversal invariance of the wave equation in acoustics and electromagnetism al-

lows to build time reversal mirrors (TRMs) made of arrays of reversible antenna, allowing

an incident broad-band wave field to be sampled, recorded, time-reversed and re-emitted.


156 M. Fink, J. de Rosny, G. Lerosey and A. Tourin

Fig. 1. – (a) Recording step: A closed surface is filled with transducer elements. A point-likesource generates a wave front which is distorted by heterogeneities. The distorted pressure fieldis recorded on the cavity elements. (b) Time-reversed or reconstruction step: The recordedsignals are time-reversed and re-emitted by the cavity elements. The time-reversed pressurefield backpropagates and refocuses exactly on the initial source.

TRMs refocus the incident-wave field to the position of the original source regardless of

the complexity of the propagation medium. The first TRMs have been developed in the

field of Acoustics [1-3]. An acoustic source, located inside a lossless medium, radiates a

brief transient pulse that propagates and is potentially distorted by the medium. Time

reversal of the acoustic field would entail the reversal, at some instant, of every particle

velocity in the medium. This kind of instantaneous time reversal in the whole volume is

practically impossible to achieve. A more realistic alternative can be developed thanks

to the Helmoltz-Kirchoff integral theorem. The acoustic field radiated by a source could

be measured on every point of an enclosing surface (acoustic retina), and retransmitted

in time-reversed order, then the wave will travel back to its source, see fig. 1.

Both time reversal invariance and spatial reciprocity [4] are required to reconstruct

a time-reversed wave in the entire volume by means of this two-dimensional time re-

versal operation. From an experimental point of view a closed TRM consists of a two-

dimensional piezoelectric transducer array that samples the wave field over a closed

surface. An array pitch of the order of λ/2, where λ is the smallest wavelength of the

pressure field, is needed to ensure the recording of all the information on the wave field.

Each transducer is connected to its own electronic circuitry that consists of a receiving

amplifier, an A/D converter, a storage memory and a programmable transmitter able to

synthesize a time-reversed version of the stored signal.

Time reversal focusing and the diffraction limit 157

In practice, closed TRMs are difficult to realize and the TR operation is usually

performed on a limited angular area, thus apparently limiting focusing quality. A TRM

consists typically of a small number of elements or time-reversal channels. The major

interest of TRM, compared to classical focusing devices (lenses and beam forming) is

certainly the relation between the medium complexity and the size of the focal spot. A

TRM acts as an antenna that uses complex environments to appear wider than it is,

resulting in a refocusing quality that does not depend on the TRM aperture.

One spectacular result that is shown in this paper deals with complex environment

located in the near field of the source. Such environment can be made, for example, of

random or periodic distribution of resonating scattererers with a mean distance smaller

than the wavelength. It will be shown that, for a broad-band source located inside such

random metamaterials, a TRM located in the far field radiated a time-reversed wave that

interacts with the random medium to regenerate not only the propagating but also the

evanescent waves required to refocus below the diffraction limit. This focusing process is

very different from the one developed with superlenses made of negative index material

only valid for narrow-band signals. We will emphasize the role of the frequency diversity

in time reversal focusing.

2. – Basic principles

The basic theory employs a scalar wave formulation p(r, t) and, hence, is strictly

applicable to acoustic or ultrasound propagations in fluid. However, the basic ingredients

and conclusions apply equally well to elastic waves in solid and to electromagnetic fields.

Let us consider the propagation of an acoustic wave in a heterogeneous and non-

dissipative medium, whose compressibility κ(r) and density ρ(r) vary in space. By in-

troducing the sound speed c(r) = (ρ(r)κ(r))−1/2, one can obtain the wave propagation

equation for a given pressure field p(r, t)

(1) �∇ ·

(�∇p(r, t)

ρ(r)

)−

1

ρ(r)c(r)2∂2p(r, t)

∂t2= 0.

One can notice the particular behaviour of this wave equation regarding the time

variable t. Indeed, it only contains a second-order time derivative operator. This property

is the starting point of the time reversal principle. A straightforward consequence of this

property is that if p(r, t) is a solution of the wave equation, then p(r,−t) is also solution

of the problem. This property illustrates the invariance of the wave equation during a

time reversal operation, the so-called time reversal invariance. However, this property

is only valid in a non-dissipative medium. If wave propagation is affected by dissipation

effects, odd order time derivatives appear in the wave equation and the time reversal

invariance is lost. Nevertheless, one should here note that if the ultrasonic absorption

coefficient is sufficiently small in the frequency bandwidth of the ultrasonic waves used

for the experiments, the time reversal invariance remains valid.


In any propagation experiment, the acoustic sources and the boundary conditions

determine a unique solution p(r, t) in the fluid. The goal, in time reversal experiments, is

to modify the initial conditions in order to generate the dual solution p(r, T − t), where

T is a delay due to causality requirements. Cassereau and Fink [4] and Jackson and

Dowling [5] have studied theoretically the conditions necessary to ensure the generation

of p(r, T − t) in the entire volume of interest.

2.1. An ideal time reversal experiment. – Although reversible acoustic retinas usually

consist of discrete elements, it is convenient to examine the behavior of idealized contin-

uous retinas, defined by two-dimensional surfaces. In the case of a time reversal cavity,

we assume that the retina completely surrounds the source.

In a first step, let us consider a point-like source located at r0 inside a volume V

surrounded by a surface S, emitting a time modulation s(t). The inhomogeneous wave

equation is given by

(2) �∇ ·

(�∇p(r, t)

ρ

)−

1

ρc2∂2p(r, t)

∂t2= −δ(r − r0)s(t).

Note that contrary to eq. (1), the right part of this equation describes the source term

and this term may contain spatial and time singularities. Considering, for example, an

impulsive source s(t) = δ(t) at time 0, the causal solution to eq. (2) reduces to the

retarded Green’s function Gret(r, r0; t) that takes into account the heterogeneities and

the boundaries of the medium. Note that, to respect causality, only the causal Green’s

function (retarded) that satisfies the Sommerfeld radiation boundary condition at infinity

is selected, while the advanced Green’s function (the anti-causal) is neglected.

The initial goal of a perfect time-reversed experiment is to generate in the medium

this advanced Green’s function Gadv(r, r0; t) = Gret(r, r0;−t) by modifying the initial

conditions on the boundaries of the experiment. This would be an optimal way to

obtain super-resolution in a focusing experiment, because the advanced Green’s function

converges towards a spatial singularity. In a more realistic way, taking into account any

source modulation s(t) with a well-defined bandwidth, we are interested in generating

the volume of the experiment p(r,−t).

The so-called time reversal cavity approach was developed, by using the fact that a

wave field at any location inside a volume V (without source) can be predicted from

the knowledge of both the field and its normal derivative on the surrounding surface S.

Therefore a time reversal experiment can be conceived in the following way: During the

second step of the time reversal process, the initial source at r0 is removed and we create

on the surface of the cavity monopole and dipole sources that correspond to the time

reversal of those same components measured during the first step. The time reversal

operation is described by the transform t → −t and the secondary sources are

(3)

{ps(r, t) = G(r, r0;−t) ⊗ s(−t),

∂nps(r, t) = ∂nG(r, r0;−t) ⊗ s(−t),


where we use now and in the following the notation G(r, r0; t) instead of Gret(r, r0; t)

and neglect the causal delay T needed to record and re-emit the signals.

Due to these secondary sources on S, the time-reversed pressure field ptr(r, t) prop-

agates backward inside the cavity. It can be calculated using a modified version of the

Helmoltz-Kirchhoff integral, valid inside a zone without source

ptr(r, t)=

∫+∞

−∞

dt′∫∫

S

[G(r, r′; t− t′)∂nps(r′, t′)−ps(r

′, t′)∂nG(r, r′; t− t′)]d2r′

ρ(r′).(4)

Instead of directly computing this integral, there is a straightforward way to predict the

field ptr(r, t). Our initial goal was to radiate inside the volume surrounded by surface S,

the field p(r,−t) = G(r, r0;−t)⊗s(−t), with G(r, r0;−t) the advanced Green’s function.

However, the wave equation verified by p(r,−t) in the volume V can be obtained by

changing t into −t in eq. (2)

(5) �∇ ·

(�∇p(r,−t)

ρ

)−

1

ρc2∂2p(r,−t)

∂t2= −δ(r − r0)s(−t).

Therefore, to obtain a perfect time reversal field would require also that the original

active source that injected energy into the system in the initial step be replaced with a

sink (the time reversal of a source) that corresponds to the right term of eq. (4). This

means that to achieve a perfect time reversal, both the source has to be transformed into

a sink, and the field and its normal derivative on surface S have also to be time-reversed

(like in eq. (4)). The superposition of these two fields will give exactly p(r,−t). Therefore

p(r,−t) is given by the following sum:

(6) p(r,−t) = ptr(r, t) +G(r, r0; t) ⊗ s(−t).

For a source term with a Dirac excitation, we directly get for the time-reversed field

(7) ptr(r, t) = G(r, r0;−t) −G(r, r0; t).

This important result is, in some way, disappointing, because it means that reversing

an acoustic field using a closed TRM is not enough to radiate only the advanced wave

field. Complete time reversal requires not only to time-reverse the source but the original

source as well. Equation (7) can be interpreted as the difference of advanced and retarded

waves centered on the initial source position. The converging wave (advanced) collapses

at the origin and is followed by a diverging (retarded) wave. Thus the time-reversed

field observed as a function of time shows two wave fronts of opposite sign. The wave

re-emitted by the time reversal cavity looks like a convergent wave field during a given

period, but a wave field does not know how to stop. When the converging wave field

reaches the location of the initial source location, it collapses and then continues its

propagation as a diverging wave field.


To achieve a perfect time reversal, both the field on the surface of the cavity has to

be time-reversed, and the source has to be transformed into a sink [6,7]. In this manner

one may achieve time-reversed focusing below the diffraction limit. The role of the new

source term −δ(r−r0)s(−t) in eq. (5) is to transmit a diverging wave that exactly cancels

the outgoing spherical wave.

In a monochromatic approach, taking into account the evanescent waves concept, the

necessity of replacing a source with a sink in the complete time-reversed operation can

be interpreted as follows. In the first step a point-like source of size much smaller than

a wavelength radiates a field that can be described as a superposition of homogeneous

plane waves propagating in the various directions �k and of decaying, non-propagating,

evanescent plane waves [8]. The evanescent waves contain information on fine-scale fea-

tures of the source; they decay exponentially with distance and do not contribute in the

far field. If the TRM is located in the far field of the source, the time-reversed field

retransmitted by the mirror does not contain these evanescent components. The role of

the sink is to radiate exactly, with good timing, the evanescent waves that have been

lost during the first step. The resulting field contains the evanescent part that is needed

to focus below diffraction limits. Time reversal below the diffraction limit has been ex-

perimentally demonstrated in acoustics, using an acoustic sink placed at the focal point.

Focal spots of size λ/14 have been observed by de Rosny et al. [6]. One drawback is the

need to use an active source at the focusing point to exactly cancel the usual diverging

wave created during the focusing process.

2.2. Time reversal in free space. – For example, in the case of a homogeneous medium,

assuming that the retina does not perturb the field propagation (free-space assumption),

the free-space retarded Green’s function G0 reduces to a diverging spherical impulse wave

that depends only on r − r0 and propagates with sound speed c. Thus, neglecting the

causal time delay T , the time-reversed field can be written as

(8) ptr(r, t) ≺

{1

4π|r − r0|δ

(t+

|r − r0|

c

)−

1

4π|r − r0|δ

(t−

|r − r0|

c

)}⊗ s(−t),

that reduces to the time derivative of the source modulation at the origin

(9) ptr(r = r0, t) = −1

2πcs′(−t).

In the case of a narrow band excitation (monochromatic excitation of pulsation ω),

the interference between the converging and the diverging fields leads to the classical

diffraction limits. Indeed by calculating the Fourier transform of eq. (7) over the time

variable t, we obtain

Ptr(r, ω) =exp (−jk|r − r0|)

4π|r − r0|−

exp (jk|r − r0|)

4π|r − r0|(10)

= −2jsin (k|r − r0|)

4π|r − r0|= −2j Im G(r − r0, ω),


where j2 = −1. The time-reversed field at the initial source position is finite because it

is the difference between a converging and a diverging wave and not the sum (otherwise

it will have a discontinuity there).

As a consequence, the time-reversed field is focused on the initial source position, with

a focal spot size limited to one half-wavelength π/k that corresponds to the standard

formulation for the complex field modulus where k is the wave number and G(r− r0, ω)

is the monochromatic Green’s function. The point spread function is proportional to the

imaginary part of the monochromatic Green’s function.

2.3. Time reversal through heterogeneous medium. – In the case of a non-dissipative

heterogeneous medium surrounding the source, a similar interpretation can be given, but

the retarded Green’s function G(r, r0;ω) is no more dependent on r − r0 but is now a

function separately of both r and r0 taking into account, for example multiple scattering

process between heterogeneities

Ptr(r, ω) =

∫∫S

[∂nG

∗(r′, r0;ω)G(r, r′;ω) − G∗(r′, r0;ω)∂nG(r, r′;ω)] d2r′

ρ(r′)(11)

= −2j Im G(r, r0;ω).

Note that the field amplitude at the focal point is directly proportional to the LDOS,

the so-called local density of states that depends on the medium complexity around

the source point. In situation where the source is located inside a periodic or random

metamaterial with scatterers close to the source, it can happen that the LDOS be zero

for some source position and therefore the time-reversed wave will have a node at the

source point. However, for broad-band excitation, the resulting field takes advantage of

the frequency diversity.

For a broad-band excitation s(t) the time-reversed field is given by

(12) ptr(r, t) = −2j

∫Im G(r, r0;ω)S∗(ω) exp(jωt)dω,

where S(ω) is the Fourier transform of the source modulation. For an excitation with a

flat bandwidth Δω, the field at the collapse time (t = 0) reads

(13) ptr(r, t = 0) = −2j

∫Δω

Im G(r, r0;ω)dω.

Therefore the time-reversed field at the focus (source point) and at the collapse time is

given by

(14) ptr(r = r0, t = 0) = −2j

∫Δω

Im G(r0, r0;ω)dω.

Thus, the time-reversed field at the source point and at the focal time is directly propor-

tional to the number of modes excited by the source.


2.4. An experimental point of view . – From an experimental point of view a perfect

time reversal mirror verifying eq. (11) is an ideal description. Indeed, it is difficult to

measure both the field and its normal derivative at any point of the surface S. Ideal exper-

iments would be carried out with two transducer arrays that behave either as monopolar

or as dipolar transducers and that spatially sample the receiving and emitting surface.

If, however, the time reversal mirror is located in the far field of source and observation

points and heterogeneities, the expression can be simplified. In this case we may assume

that

(15) ∂nG(r, r′;ω) ≈ jkG(r, r′;ω).

Thus, the time-reversed field can be written as

(16) Ptr(r, ω) ≈ 2jω

ρc

∫∫S

G∗(r′, r0;ω)G(r, r′;ω)d2r′.

If we come back to the time domain, eq. (11) can be written as

(17) ptr(r, t) ≈2

ρc

∂

∂t

∫∫S

G(r′, r0;−t) ⊗G(r, r′; t)d2r′.

From an experimental point of view, it is not easy to measure and re-emit the field at any

point of a surface S: experiments are carried out with transducer arrays that spatially

sample the receiving and emitting surface. Assuming that the time reversal retina consists

of discrete elements located at position ri, this allows to replace the integration over S

in eq. (11) with a summation over N surface element positions

(18) ptr(r, t) = C∂

∂t

N∑i=1

G(ri, r0;−t) ⊗G(r, ri; t),

where C is a scaling factor. Taking into account spatial reciprocity, this expression can

be written a summation of cross-correlation functions

(19) ptr(r, t) = C∂

∂t

N∑i=1

G(r0, ri;−t) ⊗G(r, ri; t).

The spatial sampling of surface S by a set of elements may introduce grating lobes.

These lobes can be avoided by using an array pitch smaller than λmin/2, where λmin is

the smallest wavelength of interest.

3. – Time reversal in complex media

It is generally difficult to use acoustic arrays that completely surround the area of

interest, so the closed cavity is usually replaced with a TRM of finite angular aperture.


Fig. 2. – One part of the transducers is replaced with reflecting boundaries. In the first step(receive mode) the wave radiated by the source is recorded by a set of transducers through thereverberation inside the cavity. In the second step, the recorded signals are time-reversed andre-emitted by the transucers.

This yields an increase of the point spread function that is related to the limited angular

size of the mirror observed from the source. In the standard theory of diffraction in

homogeneous free space, the point spread function is related to the angular spectrum of

the aperture. For a closed time reversal mirror, the �k vectors of the radiated field span

the whole 4π solid angle and the focal spot dimension is minimal (λ/2). When a TRM

covers a limited solid angle, the spatial diversity of �k vectors that interact with the TRM

is reduced. Therefore the focal spot size is increased.

The main interest of focusing with TRM is that in media with complex structure the

spatial diversity of the �k vectors captured by a small TRM can be significantly increased.

Wave propagation in media with complex boundaries or random scattering medium can

increase the apparent aperture of the TRM, resulting in a focal spot size smaller than

that predicted by classical formulas.

The basic idea is to replace one part of the transducers needed to sample a closed

time reversal surface with reflecting boundaries that redirect one part of the incident

wave towards the TRM aperture (see fig. 2). When a source radiates a wave field inside

a closed cavity or in a waveguide, multiple reflections along the medium boundaries can

significantly increase the apparent aperture of the TRM. Thus spatial information on

the �k vectors that is usually lost with a finite aperture TRM is converted into the time

domain. The reversal quality then depends crucially on the duration of the time-reversal

window, i.e., the length of the recording that is reversed.


Such a concept is strongly related to a kaleidoscopic effect that appears thanks to

the multiple reverberations on the waveguide boundaries. Waves emitted by each trans-

ducer are multiply reflected, creating at each reflection “virtual” transducers that can

be observed from the desired focal point. Thus, we create a large virtual array from a

limited number of transducers and a small number of transducers is multiplied to create

a “kaleidoscopic” transducer array. Two different examples will be presented (a chaotic

cavity and a multiply scattering medium).

3.1. One-channel time reversal in chaotic cavities. – In this section, we are inter-

ested in multiply reflected waves: waves confined in closed reflecting cavities with non-

symmetrical geometry. With closed boundary conditions, no information can escape from

the system and a reverberant acoustic field is created. If, moreover, the geometry of the

cavity shows ergodic and mixing properties, one may hope to collect all information at

only one point. Ergodicity means that, due to the boundary geometry, any acoustic ray

radiated by a point source and multiply reflected would pass every location in the cavity.

Therefore, all the information about the source can be redirected towards a single time

reversal transducer. This is the regime of fully diffuse wave fields that can be also defined

as in room acoustics as an uncorrelated and isotropic mix of plane waves of all propaga-

tion directions [9,10]. Draeger and Fink [11-13] showed experimentally and theoretically

that in this particular case a time reversal focusing with λ/2 spot can be obtained using

only one TR channel operating in a closed cavity.

The first experiments were made with elastic waves propagating in a 2D cavity with

negligible absorption. They were carried out using guided elastic waves in a monocrys-

talline D-shaped silicon wafer known to have chaotic ray trajectories. This property

eliminates the effective regular gratings of the previous section. Silicon was selected also

for its weak absorption. Elastic waves in such a plate are akin to Lamb waves.

An aluminum cone coupled to a longitudinal transducer generated waves at one point

of the cavity. A second transducer was used as a receiver. The central frequency of the

transducers was 1 MHz and their relative bandwidth was 100% (Δω = 1 MHz). At this

frequency, only three propagating modes are possible (one flexural, one quasi-extensional,

one quasi-shear). The source was considered point-like and isotropic because the cone tip

is much smaller than the central wavelength. A heterodyne laser interferometer measures

the displacement field as a function of time at different points on the cavity. Assuming

that there is no mode conversion at the boundaries between the flexural mode and other

modes, we have only to deal with one field, the flexural-scalar field.

The experiment is a “two-step process” as described above: In the first step, one

of the transducers, located at point r0 (fig. 3), transmits a short omnidirectional sig-

nal of duration 0.5μs. Another transducer, located at rtrm, observes a long random-

looking signal that results from multiple reflections of along the boundaries of the cav-

ity. It continues for more than 50 milliseconds, corresponding to some hundred reflec-

tions at the boundaries. Then, a portion ΔT of the signal is selected, time-reversed

and re-emitted by point rtrm. As the time-reversed wave is a flexural wave that in-

duces vertical displacements of the silicon surface, it can be observed using the optical


Fig. 3. – Time reversal experiment conducted in a chaotic cavity with flexural waves. In a firststep, a point transducer located at point A transmits a 1μs long signal. The signal is recordedat point B by a second transducer. The signal spreads by more than 30 ms due to reverberation.In the second step of the experiment, a 1 ms portion of the recorded signal is time-reversed andretransmitted back in the cavity.

interferometer that scans the surface on different observation points r around the point

r0 (see fig. 3).

For time reversal windows of sufficiently long duration ΔT , one observes both an

impressive time recompression at point r0 and a refocusing of the time-reversed wave

around the origin (see figs. 4a and b for ΔT = 1 ms), with a focal spot whose radial

dimension is equal to half the wavelength of the flexural wave. Using reflections at the

boundaries, the time-reversed wave field converges towards the origin from all directions

and gives a circular spot, like the one that could be obtained with a closed time reversal

cavity covered with transducers. A complete study of the dependence of the spatio-

temporal side lobes around the origin shows a major result [11-13]: a time duration

ΔT of nearly 1 ms is enough to obtain a good focusing. For values of ΔT larger than

1 ms, the sidelobes shape and the signal-to-noise ratio (focal peak/sidelobes) does not

improve further. There is a saturation regime. Once the saturation regime is reached,

point rtrm will receive redundant information. The saturation regime is reached after a

time τHeisenberg called the Heisenberg time. It is the minimum time needed to resolve

the eigenmodes in the cavity. It can also be interpreted as the time it takes for all a

single ray to reach the vicinity of any point in the cavity within a distance λ/2. This

guarantees enough interference between all the multiply reflected waves to build each


Fig. 4. – (a) Time-reversed signal observed at point A. The observed signal is 210μs long;(b) time-reversed wave field observed at different times around point A on a square of 15 mm ×15 mm.

of the eigenmodes in the cavity. The mean distance δω between the eigenfrequencies is

related to the Heisenberg time, τHeisenberg = 1

δω .

The success of this time reversal experiment in closed chaotic cavity is particularly

interesting with respect to two aspects. Firstly, it proves the feasibility of acoustic time

reversal in cavities of complex geometry that give rise to chaotic ray dynamics. Para-

doxically, in the case of one-channel time-reversal, chaotic dynamics is not only harm-

less but even useful, as it guarantees ergodicity and mixing. Secondly, using a source

of vanishing aperture, there is an almost perfect focusing quality. The procedure ap-

proaches the performance of a closed TRM, which has an aperture of 360◦. Hence, a

one-point time reversal in a chaotic cavity produces better results than a limited aper-

ture TRM in an open system. Using reflections at the edge, focusing quality is not

aperture limited; the time-reversed collapsing wave front approaches the focal spot from

all directions.


Although one obtains excellent focusing, a one-channel time reversal is not perfect, as

a weak noise level throughout the system can be observed. There is a saturation regime

beyond the Heisenberg time. Residual temporal and spatial sidelobes persist even for

time reversal windows of duration larger than the Heisenberg time. They are due to

multiple reflections passing over the locations of the TR transducer and they have been

expressed in closed form by Draeger and Fink. Using an eigenmode analysis of the wave

field, they explain that, for long time reversal windows, there is a saturation regime that

limits the signal-to-noise ratio (SNR). To evaluate the time-reversed field for the elastic

wave in the one-channel experiment, we can use eq. (15) with a TRM located at unique

point rtrm and the vertical component of the displacement field ϕtr(r, t) is given by (note

that the time derivative of eq. (19) has disappeared because of the dimensionality of the

displacement field)

(20) ϕtr(r, t) ≺ G(rtrm, r0;−t) ⊗G(r, rtrm; t).

Taking into account the modal decomposition of the Green’s functions G(r, rtrm; t)

and G(rtrm, r0;−t) on each of the eigenmodes uj(r) of the cavity with eigenfrequency

ωj , we get

(21) G(r, rtrm; t) =∑

j

uj(r)uj(rtrm)sin(ωjt)

ωj(t > 0).

Under the assumption that the eigenmodes are not degenerated (valid for a chaotic

cavity), we calculate ϕtr(r, t) for a time window of duration longer than the Heizenberg

time of the cavity and we get

(22) ϕtr(r, t) ∝∑

i

1

ω2

i

ui(r)ui(r0)u2

i (rtrm) cos(ωit).

Note that at the focal time t = 0 (collapse) the directivity pattern of the time-reversed

wave field is

(23) ϕtr(r, t) ≺∑

i

1

ω2

i

ui(r)ui(r0)u2

i (rtrm).

Note that in a real experiment one has to take into account the limited bandwidth of

the source, so a spectral function S(ω) centered on center frequency ωc, with band width

Δω, must be introduced and we can write eq. (23) in the form

(24) ϕtr(r, 0) =∑

i

1

ω2

i

ui(r)ui(r0)u2

i (rtrm)S(ωi).

Thus the summation is limited to a finite number of modes, which is typical in our

experiment of the order of some hundreds. As we do not know the exact eigenmode


distribution for each chaotic cavity, we cannot evaluate this expression directly. However,

due to the ergodic properties of the cavity, one may use a statistical approach and consider

the average over different realizations, which consist in summing over different cavity

realizations. So we replace in eq. (24) the eigenmodes product with their expectation

values 〈. . .〉. We use also a qualitative argument proposed by Berry [14,15] to characterize

irregular modes in chaotic system. If chaotic rays support an irregular mode, it can be

considered as a superposition of a large number of plane waves with random direction

and phase. This implies that the amplitude of an eigenmode has a Gaussian distribution

with 〈u2

i 〉 = σ2 and a short-range isotropic correlation function given by a Bessel function

that reads

(25) 〈ui(r)ui(r0)〉 = J0(2π|r − r0|/λi),

with λi the wavelength corresponding to ωi. If r and r0 are sufficiently far apart from

rtrm not to be correlated, then

(26)⟨ui(r)ui(r0)u

2

i (rtrm)⟩

= 〈ui(r)ui(r0)〉⟨u2

i (rtrm)⟩.

One obtains finally

(27) 〈ϕtr(r, 0)〉 =∑

i

1

ω2

i

J0(2π|r − r0|/λi)σ2F (ωi).

The experimental results obtained in fig. 4b agree with this prediction and show that in a

chaotic cavity the spatial resolution is independent of the time reversal mirror aperture.

Indeed, with a one-channel time reversal mirror, the directivity pattern at t = 0 is close

to the Bessel function J0(2π|r − r0|/λc) corresponding to the central frequency of the

transducers. This means that the one-channel time-reversed field is a good estimate of

the imaginary part of the Green’s function (see eq. (14)) that was predicted for a closed

time reversal cavity made of a large number of antennas.

One can also observe, in fig. 4b, a very good estimate of the eigenmode correlation

function experimentally obtained with only one realization. A one-channel omnidirec-

tional transducer is able to refocus a wave in a chaotic cavity, and if the bandwidth is

very large, we do not have to use a TRM made of many transducers.

The focusing process described here is very different from the focusing techniques

used in monochromatic regime. Here, the frequency diversity is used to concentrate the

wave field at one time at one location. It is interesting to compare this focusing approach

for broad-band signals with phase conjugation of monochromatic signal. Time reversal

of p(r, t) is equivalent, for each spectral component P (r, ω), to complex conjugation. For

a single-frequency signal, time reversal is equivalent to complex conjugation of complex

amplitude. In a closed cavity, as above, if one works only at a single frequency (say that

of one of the eigenmodes ωi), one constructs only the eigenmode pattern correspond-

ing to the selected frequency. The refocusing process discussed above works only with


Fig. 5. – Schematic representation of a broad-band time reversal operation at the source (rightpart) and off the source (left part). Each arrow represents a different Fresnel vector correspond-ing to a frequency component. At the source position, all the phases are set back to 0, theamplitude of the resulting signal rises proportionally to the number of independent frequen-cies N . Outside the source, the different contributions are presumably decorrelated, and thestandard deviation of their sum rises as

√N .

broad-band pulses, over a bandwidth that includes a large number of eigenmodes. Here,

the averaging process that gives a good focusing is obtained by a sum over the different

modes in the cavity by assuming that in a chaotic cavity, we have a statistical decorre-

lation of the different eigenmodes, the time-reversed field can be computed by adding

the various frequency components (each individual mode) and it can be represented as

a sum of Fresnel vectors (fig. 5). At the source position, all these phase-conjugated

fields have a zero phase (this comes from the phase conjugation operation that exactly

compensates for the forward phase) and even if there is no amplitude focusing for each

spectral contributions, there is a constructive interference between all these fields at the

focusing time as∑

i |G(r0, rtrm;ωi)|2, where G(r0, rtrm;ω) is the Fourier transform of

G(r0, rtrm; t). Thus, the total field at the focusing time increases proportionally to the

number I of modes (or arrows). Outside the source position, at point r, we observe∑i G(r0, rtrm;ωi)G

∗(r, rtrm;ωi), the contributions of each individual mode are decor-

related because there is no more coherent phase compensation and therefore the total

length only increases as√I. On the whole, the focusing peak emerges at the focusing

time from the noise when the bandwidth is large enough to contain many different modes.

Ideally, if we could indefinitely expand the bandwidth, the background level on the direc-

tivity patterns should decrease as 1/√I. As the number of eigenfrequencies available in

the transducer bandwidth increases, the refocusing quality becomes better and the focal

spot pattern becomes closed to the ideal Bessel function.

As a conclusion, it must be emphasized that in a closed cavity a one-channel time

reversal mirror can focus with λ/2 resolution if the duration of time reversal window

is greater than or equal to the cavity Heisenberg time. Longer time windows do not

improve the focusing quality. However, larger bandwidth Δω reduces the side lobe levels

as 1/√

Δω.

Time reversal in reverberant cavities at audible frequencies has been shown to be an

efficient localizing technique in solid objects. The idea consists in detecting acoustic waves


in solid objects (for example a table or a glass plate) generated by a slight finger knock.

As in a reverberating object, a one channel TRM has the memory of many distinct source

locations, and the information location of an unknown source can then be extracted from

a simulated time reversal experiment in a computer. Any action, for example, turning

on the light or a compact disk player, can be associated with each source location. Thus,

the system transforms solid objects into interactive interfaces. Compared to the existing

acoustic techniques, it presents the great advantage of being simple and easily applicable

to inhomogeneous objects whatever their shapes. The number of possible touch locations

at the surface of objects is directly related to the number of independent time-reversed

focal spots that can be obtained. For example, a virtual keyboard can be drawn on the

surface of an object; the sound made by fingers when a text is captured, is used to localize

impacts. Then, the corresponding letters are displayed on a computer screen [16].

3.2. Time reversal in open systems: random media. – The ability to focus with a one-

channel time reversal mirror is not only limited to experiments conducted inside closed

cavity. Similar results have also been observed in time reversal experiments conducted

in open random media with multiple scattering [17-19]. Derode et al. carried out the

first experimental demonstration of the reversibility of an acoustic wave propagating

through a random collection of scatterers with strong multiple-scattering contributions.

A multiple-scattering sample is immersed between the source and an TRM array made

of 128 elements. The scattering medium consists of 2000 randomly distributed parallel

steel rods (diameter 0.8 mm) arrayed over a region of thickness L = 40 mm with average

distance between rods 2.3 mm. The elastic mean free path in this sample was found to

be 4 mm (see fig. 6). A source 30 cm away from the 128 elements TRM transmitted a

short (1μs) ultrasonic pulse (3 cycles of a 3.5 MHz, Δω = 1 MHz).

Figure 7a shows one part of the waveform received by one element of the TRM. It

spreads over more than 200μs, i.e. 200 times the initial pulse duration. After the arrival

of a first wave front corresponding to the ballistic wave, a long diffuse wave is observed due

to the multiple scattering. In the second step of the experiment, any number of signals

(between 1 and 128) is time-reversed and transmitted and a hydrophone measures the

time-reversed wave in the vicinity of the source. For a TRM of 128 elements, with a time

reversal window of 300μs, the time-reversed signal received on the source is represented

in fig. 7b: an impressive compression is observed, since the received signal lasts about

1μs, against over 300μs for the scattered signals. The directivity pattern of the TR field

is also plotted in fig. 8. It shows that the resolution (i.e. the beam width around the

source) is significantly finer than it is in the absence of scattering: the resolution is 30

times finer, and the background level is below −20 dB. Moreover fig. 9 shows that the

resolution is independent of the array aperture: even with only one transducer doing

the time reversal operation, the quality of focusing is quite good and the resolution

remains approximately the same as with an aperture 128 times larger. This is clearly

the same effect as observed with the closed cavity. High transverse spatial frequencies

of arbitrary k that would have been lost in a homogeneous medium are redirected by

the scatterers towards the array. Once again this result illustrates the difference between


Fig. 6. – Time reversal focusing through a random medium. In the first step the source (A)transmits a short pulse that propagates through the rods.The scattered waves are recorded on a128-element array (B). In the second step, N elements of the array (0 < N < 128) retransmit thetime-reversed signals through the rods.The piezoelectric element (A) is now used as a detector,and measures the signal reconstructed at the source position. It can also be translated alongthe x-axis while the same time-reversed signals are transmitted by B, in order to measure thedirectivity pattern.

phase conjugation and time reversal. If the experiment had been quasi-monochromatic

and the single array element had simply phase-conjugated one frequency component, the

conjugated wave field would never have focused on the source position. Indeed, whatever

its phase, there is no reason for a monochromatic wave emanating from a point source

to be focused in a particular place on the other side of a random sample. The phase-

conjugated field at one frequency in the source plane is perfectly random and verifies the

classical speckle distribution.

As for a broad-band signal in a closed cavity, an analysis similar to that of the last

paragraph can be conducted in order to predict the level of the side lobes around the

focal peak. A modal decomposition of the field is not directly applicable. However, if we

keep in mind that the focusing with one channel occurs only for a broad-band transducer,

we identify the number of uncorrelated spectral correlation length δω of the scattered

waves. Then there are Δω/δω uncorrelated bits of spectral information in the frequency

bandwidth, and the signal-to-noise is expected to vary like√

Δω/δω. To evaluate the

spectral correlation length, one can use the Wiener-Kinchin theorem [18] that gives the

spectral correlation function (averaged over the frequency bandwidth) as the Fourier


Fig. 7. – Experimental results. (a) Signal transmitted through the sample (L = 40 mm) andrecorded by the array element n◦ 64, and (b) signal recreated at the source after time reversal.

Fig. 8. – Directivity pattern of the time-reversed waves around the source position, in water(upper line) and through the rods (lower line), with a 16-element aperture. The sample thicknessis L = 40 mm. The −6 dB widths are 0.8 and 22 mm, respectively.


Fig. 9. – Directivity pattern of the time-reversed waves around the source position throughL = 40mm, with N = 128 transducers (bottom curve) and N = 1 transducer (top curve). The−6 dB resolutions are 0.84 and 0.9 mm, respectively.

transform of the “time of flight” distribution. In a multiply scattering medium, in the

diffusive approximation, it is well known that the typical spreading time (the so-called

Thouless time) is equal to τThouless = L2/D, where D is a diffusion coefficient related to

the mean free path and L is the thickness sample. Therefore δω = D/L2 so the number

of uncorrelated frequencies grows with L2, provided we can neglect dissipation effects

(τdissipat ≥ τThouless), where τdissipat is the dissipative time.

4. – Focusing microwaves below the diffraction limit

Super-resolution can be achieved with an acoustic sink but it has a severe drawback.

It needs to use an active source at the focusing point to exactly cancel the usual diverging

wave created during the focusing process. Since we know that the time-reversal focusing

spot at each frequency depends on the imaginary part of the Green function for any

heterogeneous medium, another approach consists to surround the focusing point by a

microstructured medium with length scales well below the wavelength; strong resonating

scatterers were placed in the near-field of the source. In this case, the microstructured

medium strongly modifies the spatial dependence of the imaginary part of the Green

function that now oscillates on scales much smaller than the wavelength. For a broad-

band pulse with enough frequency diversity, a time reversal will generate at the focal time

an interference between the imaginary part of the Green’s function at each frequency

(see eq. (13)). At the source point, the time-reversed field is directly proportional to

the number of modes excited inside the microstructure from the source. While at the

other points, the oscillations of the imaginary parts at different frequencies cancel their

effects. To predict the behaviour of time reversal in such medium, we have to know the

field correlations. A wave propagating in any medium can be characterized by a spatial


Fig. 10. – (a) A Time Reversal Mirror (TRM) made of eight commercial dipolar antennas oper-ating at 2.45 GHz (i.e., λ = 12 cm) is placed in a 1 m3 reverberating chamber. Ten wavelengthsaway from the TRM a subwavelength receiving array is placed, consisting of eight microstruc-tured antennas λ/30 apart from one another. (b) Details of one microstructured antenna.It consists of the core of a coaxial line which comes out 2 mm from an insultating layer andis surrounded by a microstructure consisting of a random distribution of thin copper wires.(c) Photograph of the 8-element subwavelength array surrounded by the random distribution ofcopper wires. Antennas #3 and #4 are indicated by the red and blue arrows.

correlation length and a spectral correlation length which have a pretty simple meaning.

The spatial correlation length of a medium represents, at a given frequency, the smallest

distance between two points which exhibits statistically different wave fields, while the

spectral correlation frequency δω measures the minimal frequency change that leads to

independent wave fields. If the correlation length of the medium is much smaller than

the wavelength, and if we use a bandwidth that contains several spectral correlation

lengths, one can achieve a focusing on a scale of the order of the correlation length of

the medium.


1

0

1

Am

plitu

de(a

u)

0 200 400 600 800

(a)

0 200 400 600 800

(b)

1

0

1

time(ns)

Am

plitu

de(a

u)

400 200 0 200 400

(c)

400 200 0 200 400

(d)

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

position in λ

Am

plitu

de(a

u)

(e)

λ/30

Fig. 11. – In (a) (respectively, (b)) is shown the signal received at one antenna of the TRMwhen a 10 ns pulse is sent from antenna #3 (respectively, #4) of the subwavelength array. Thesignals in (a) and (b) look significantly different although antenna #3 and antenna #4 aredistant from λ/30. In (c) (respectively, (d)) is shown the time compression obtained at antenna#3 (respectively, #4) obtained when the eight signals coming from antenna #3 (respectively,antenna #4) are time-reversed and sent back from the TRM. In (e) are shown the focusing spotsobtained around antenna #3 and #4. Their width is λ/30. Thus antenna #3 and #4 can beaddressed independently.

This is exactly the idea we exploit in the field of time reversal with microwaves [20]

to create focal spots much thinner than the wavelength. In a recent experiment [21] we

consider 8 possible focusing points placed in a strong reverberating chamber (fig. 10a).

Eight electromagnetic sources are placed at these 8 locations to be used in the learning

step of the TR process. These sources consist of wire antennas used at a central frequency

of 2.45 GHz (i.e., λ = 12 cm), with a bandwidth of 100 MHz. The pitch between them

is λ/30! These eight antennas form an array which will be referred to as the receiving

array. Each antenna in this array is surrounded by a microstructure consisting here of a

random distribution of thin copper wires (fig. 10b). The mean distance between the thin

copper wires was of the order of 1 mm (correlation length λ/100), while the frequency


correlation δω was of the order of 30 MHz, resulting in three independent speckle patterns

in the whole bandwidth. A TRM made of eight commercial dipolar antennas is placed

in the far field, ten wavelengths apart from the receiving array. The set “reverberant

chamber/TRM” acts as a virtual far-field time reversal cavity. When antenna marked

#3 in fig. 10 sends a short electromagnetic pulse (10 ns), the 8 signals received at the

TRM are much longer than the initial pulse due to strong reverberation in the chamber

(typically 500 ns). As an example, the signal received at one of the antennas of the

TRM is shown in fig. 11a. When antenna marked #4 is in its turn used as a source,

it is remarkable to point out that now the signal received at the same antenna in the

TRM (shown in fig. 11b) looks significantly different although sources #3 and #4 were

λ/30 apart from each other. When these signals are time-reversed and transmitted back,

the resulting waves converge, respectively, to antenna #3 and #4 where they recreate

pulses as short as the initial ones (fig. 11c and d). Measuring the signal received at

the other antennas of the receiving array gives access to the spatial focusing around

antennas #3 and #4 (fig. 11e). The remarkable result is that the two antennas can be

addressed independently since the focusing spots created around them have a size much

less than the wavelength (here typically λ/30): the diffraction limit is overcome although

the focusing points are in the far-field of the TRM!

Contrary to the acoustic sink experiment, in the microwave time reversal experiment,

the source remains passive and high spatial frequency components of the field are created

upon scattering at the disordered structure. Reciprocity ensures that the time-reversed

scattering process creates a sub wavelength focus around the source location [22]. The

initial evanescent waves created around the initial wire are converted into propagating

waves by the random distribution of wires. In the time-reversed step, these propagating

waves are playback, from the far field, with reverse �k. Spatial reciprocity ensures that

each propagating wave with a reverse �k interacts with the random distribution of wires

to recreate the initial evanescent waves around the focus.

Note that this approach can work not only when the experiment is conducted inside a

reverberant cavity [23]. Provided the microstructured medium generates enough multiple

scattering in the near field of the source resulting in a transmitted signal sufficiently

long, the time reversal signal from a small TRM located in free space leads to beat the

diffractions limit.

5. – Conclusion

We have shown that in the presence of multiple reflections or multiple scattering, a

small size time reversal mirror manages to focus a pulse back to the source with a spatial

resolution that beats the diffraction limit. The resolution is no more dependent on the

mirror aperture size but it is only limited by the spatial correlation of the wave field.

In these media, due to a sort of kaleidoscopic effect that creates virtual transducers, the

TRM appears to have an effective aperture that is much larger that its physical size.

Resolution can be improved in reverberating media using this concept. Time reversal

focusing opens also completely new approaches to super-resolution. We have show that


in a medium made of random distribution of sub-wavelength scatterers, a broad-band

time-reversed wave field interacts with the random medium to regenerate not only the

propagating but also the evanescent waves required to refocus below the diffraction limit.

Focal spots as small as λ/30 have been demonstrated with microwaves. This results in a

large increase of the information transfer rate by time reversal in such disordered media.

REFERENCES

[1] Fink M., IEEE Trans. Ultrason. Ferroelec. Freq. Contr., 39 (1992) 555.[2] Fink M., Phys Today, 50, 3 (1997) 34.[3] Fink M., Cassereau D., Derode A., Prada C., Roux P., Tanter M., Thomas J. L.

and Wu F., Rep. Progr. Phys., 63 (2000) 1933.[4] Cassereau D. and Fink M., IEEE Trans. Ultrason. Ferroelec. Freq. Contr., 39 (1992)

579.[5] Jackson D. R. and Dowling D. R., J. Acoust. Soc. Am., 89 (1991) 171.[6] de Rosny J. and Fink M., Phys. Rev. Lett., 89 (2002) 124301.[7] van Manen D-J., Robertsson J. and Curtis A., Phys. Rev. Lett., 94 (2005) 164301.[8] Nieto-Vesperinas M and Wolf E, J. Opt. Soc. Am. A, 2-9 (1985) 1429.[9] Weaver R., J. Acoust. Soc. Am., 71 (1982) 1608.

[10] Ebeling K. J., in Physical Acoustics, edited by Mason W. P., Vol. 17 (Academic, NewYork) 1984, pp. 233–309.

[11] Draeger C. and Fink M., Phys. Rev. Lett., 79 (1997) 407.[12] Draeger C. and Fink M., J. Acoust. Soc. Am., 105 (1999) 618.[13] Draeger C., Aime J-C. and Fink M., J. Acoust. Soc. Am., 105 (1999) 611.[14] Berry M. V., Chaotic Behaviour of Deterministic Systems (North Holland, Amsterdam)

1981, pp. 171-271.[15] McDonald S. W. and Kaufman A. N., Phys. Rev. A, 37 (1988) 3067.[16] Ing R. K., Quieffin N., Catheline S. and Fink M., Appl. Phys. Lett., 87 (2005) 204104.[17] Derode A., Roux P. and Fink M., Phys. Rev. Lett., 75, 23 (1995) 4206.[18] Derode A, Tourin A and Fink M, Phys. Rev. E, 64 (2001) 36606.[19] Blomgren P., Papanicolaou G. and Zhao H., J Acoust. Soc. Am., 111, 230.[20] Carminati R., Pierrat R., de Rosny J. and Fink M., Opt. Lett., 32 (2007) 3107.[21] Lerosey G., de Rosny J., Tourin A. and Fink M., Science, 315 (2007) 1120.[22] Carminati R., Saenz J., Greffet J. and Nieto-Vesperinas M., Phys. Rev. A, 62

(2000) 012712.[23] Li X. and Stockman M. I., Phys. Rev. B, 77 (2008) 195109.



Ultracold atoms in bichromatic lattices

L. Fallani and M. Inguscio

LENS, European Laboratory for Nonlinear Spectroscopy

Dipartimento di Fisica ed Astronomia, Universita di Firenze & INO-CNR

Via Nello Carrara 1, I-50019 Sesto Fiorentino (FI), Italy

Summary. — In this paper we illustrate the physics of ultracold atoms in bichro-matic optical lattices. The properties of biperiodic systems are presented in detail,with a particular focus on the localization transition for incommensurate lattices.We then present recent work on the experimental investigation of these systems.

1. – Introduction

The possibility of using far-detuned laser light to engineer trapping potentials for

cold atoms allows an unprecedented control of the forces exerted on a quantum system.

In this paper we discuss the physics of ultracold atoms in optical lattices, i.e. periodic

potentials created with laser standing waves, which allow the investigation of many fun-

damental problems in strong connection with the physics of condensed matter [1]. The

extremely long coherence times and the absence of defects in the lattice provide the ideal

environment for investigating the transport of quantum particles in periodic structures.

Furthermore, ultracold quantum gases in optical lattices represent promising resources

for studying many-body quantum-correlated systems with important implications in the

growing field of quantum simulation [2].


180 L. Fallani and M. Inguscio

In this paper we focus on the physics of bichromatic optical lattices produced by the

superposition of two standing waves with different spacing. We will especially consider

the case of incommensurate lattices, aperiodic potentials with discrete Fourier compo-

nents, which, similarly to quasicrystals, exhibit intriguing properties which interpolate

between the physics of ordered and disordered systems. While retaining peculiar prop-

erties of periodic lattices, as the existence of energy bands, they also support a quantum

phase transition from extended to Anderson-localized states, as in purely disordered

systems. This property has been investigated in a recent experiment, where Anderson

localization of matter waves has been observed [3].

This paper is structured according to the following scheme. In sect. 2 we will review

some basic notions related to the experimental realization of ultracold atomic gases in

optical lattices. In sect. 3 we will consider the case of monochromatic lattices, starting

from the quantum-mechanical theory of non-interacting particles in periodic potentials

and arriving at the physics of strongly interacting many-body systems. In sect. 4 we will

introduce the physics of bichromatic lattices, evidencing similarities and differences from

the uniform lattice case, and focusing on the localization transition for non-interacting

particles in incommensurate lattices. Section 5 will be devoted to the presentation of

recent LENS experiments in which bichromatic lattices have been used to study Ander-

son localization of non-interacting and weakly interacting Bose gases. Finally, in sect. 6

we will discuss the physics of strongly interacting disordered systems, showing the re-

sults of experimental work performed in this regime and discussing possible experimental

perspectives.

2. – Optical lattices

2.1. Ultracold atoms. – In the last decades a constantly growing interest has been de-

voted to the possibility of cooling and trapping atomic gases. After the first spectacular

demonstrations of this possibility, culminated in the 1997 and 2001 awards of the Nobel

Prize in Physics for laser cooling [4-6] and for the achievement of Bose-Einstein conden-

sation [7,8], it is now clear that ultracold gases can be used as versatile nano-laboratories

devoted to the investigation of many different aspects and open problems of quantum

mechanics.

The reason of this possibility lies in the remarkably low decoherence rates and in the

high degree of control that can be reached in the manipulation of neutral atoms. Properly

chosen configurations of laser beams permit to cool atomic gases to temperatures of few

microkelvin, which are unaccessible by the most advanced criogenic techniques. Optical

or magnetic traps can be easily operated, providing confinement of cold atoms in ultra-

high vacuum chambers, with long storage times and no thermal contact with the ambient.

With these techniques, it is possible to create quantum gases of either bosonic or fermionic

atoms, and eventually reach the degenerate regime, with the formation of Bose-Einstein

condensates or quantum-degenerate Fermi gases, respectively.

In this paper we will mostly concentrate on the physics of bosonic atoms below the

critical temperature for Bose-Einstein condensation [9]. Typical condensates of neutral

Ultracold atoms in bichromatic lattices 181

alkaline atoms are made of 105–106 particles at a temperature of few hundreds nanokelvin.

Condensates are created in either optical or magnetic traps, and generally have a spatial

extension of few tens of microns, which makes them easily observable via standard imag-

ing techniques. Owing to the small densities (typically 1014 atoms/cm3) and ultralow

temperatures, interactions between particles are limited to binary elastic s-wave colli-

sions. The strength of the interactions can be controlled by changing the density of the

gas (which is determined by the trapping potential) or by using a magnetic field tuned

in proximity of a Feshbach resonance (see sect. 5.1 for more details).

We will not enter here the description of the techniques by which it is possible to

create and probe Bose-Einstein condensates. An important resource on this topic is

provided by the Proceedings of the CXL International School of Physics “Enrico Fermi”

on “Bose-Einstein Condensation in Atomic Gases” [10].

2.2. Light forces. – Most laser cooling techniques rely on the near-resonant light-

matter interaction, in which the atoms are repeatedly excited to a higher-energy state

through cycles of absorption and spontaneous emission of photons. The mechanical

effects of this process originate the radiation pressure force, which, under suitable con-

ditions, provides the dissipative mechanism responsible for cooling. Optical trapping,

instead, relies on the off-resonant interaction, in which absorption is negligible and atoms

are not excited to higher-energy states. In this regime, the interaction is purely dispersive

and results in a conservative dipole force which can be used for trapping atoms for long

times (several seconds) without heating.

A complete theoretical analysis of these effects, founded on an elegant and rigorous

quantum-mechanical treatment, can be found in [11]. We will now focus on the dipole

force, following the semiclassical approach presented in [12], that gives a simple insight

into the origin of this force. We consider a simple model in which the atom, treated as

a two-level system, is subject to a classical radiation field oscillating at frequency ω:

(1) E(r, t) = eE(r)e−iωt + c.c.

This field induces a polarization of the atom, i.e. a deformation of the electron charge

distribution, which can described by an electric dipole moment

(2) p(r, t) = ep(r)e−iωt + c.c.

oscillating at the same frequency of the driving field and proportional to the latter

through the complex polarizability α:

(3) p(r) = αE(r).

This relation holds only in the linear regime, when saturation effects can be neglected

and the atomic population is almost entirely in the ground state. The real part of α,

describing the component of p oscillating in phase with E, is responsible for the disper-

sive properties of the interaction, while the imaginary part, describing the out-of-phase


component of p, is connected with the absorptive properties (radiation pressure). Both

these components strongly depend on ω as the frequency of the laser is scanned across

the atomic resonance ω0.

In analogy with classical electrodynamics, we define the optical dipole potential as the

interaction energy of the induced dipole p interacting with the driving electric field E

(4) Udip(r) = −1

2〈p · E〉 = −

1

2ε0cRe(α)I(r),

where the angle brackets indicate the time average over the optical oscillations and I(r)

is the average field intensity. Equation (4) describes the potential energy associated to a

conservative optical dipole force

(5) Fdip(r) = −∇Udip(r) =1

2ε0cRe(α)∇I(r),

which is nonzero when the intensity of the radiation field I(r) is not uniform. Starting

with just focusing a single laser beam, by engineering the geometry of the beams in such

a way to produce configurations with non-uniform intensity, it is possible to create almost

any kind of potential and to change its strength by tuning the laser parameters.

Solving the optical Bloch equations for a two-level system [13], one can easily derive

an analytical expression for the atomic polarizability α(ω). In the far-off resonant regime,

in which the detuning Δ = ω−ω0 from the atomic resonance ω0 is much larger than the

radiative linewidth of the excited state Γ and of the Rabi frequency Ω, eq. (4) becomes [12]

(6) Udip =3πc2

2ω30

(Γ

Δ

)I.

From eq. (6) we note that the sign of the dipole potential depends on the sign of the

detuning. More precisely, if the light is red-detuned (Δ < 0) the dipole potential is

negative, hence maxima of intensity correspond to minima of the potential: as a con-

sequence, the atoms tend to localize in regions of high field intensity. On the contrary,

if the light is blue-detuned (Δ > 0) the dipole potential is positive, hence maxima of

intensity correspond to maxima of the potential and the atoms tend to localize in regions

of low field intensity.

2.3. Crystals made of light . – A periodic potential for cold neutral atoms can be

easily produced by using laser light. In the previous section we have discussed how the

mechanical effects of the non-resonant interaction between radiation and matter can be

used to produce conservative optical potentials. What is needed to produce a periodic

potential is a periodic modulation of the light intensity, which is produced whenever two

laser beams with the same optical frequency cross and interfere. If two beams cross at

an angle θ, as sketched in fig. 1, the distance between two adjacent maxima (or minima)


Fig. 1. – Schematics of the periodically modulated intensity resulting from the intersection oftwo coherent laser beams propagating at an angle θ.

of the resulting interference pattern is

(7) d =λ

2 sin (θ/2).

The simplest and most common experimental setting is provided by two counterpropagat-

ing beams, which form a standing wave with an intensity modulation of period d = λ/2.

In this counterpropagating configuration, the resulting optical dipole potential can be

written in the form

(8) V (x) = V0 cos2(kx),

where k = 2π/λ is the laser wave number and V0 = 3πc2ΓI0/2ω3Δ is the depth of the

periodic potential, being I0 the maximum intensity of the standing-wave pattern, ω the

optical frequency, Γ the linewidth of the atomic transition and Δ the detuning of the

laser from the resonance. The height of the optical lattice V0 is often measured in units

of the recoil energy ER = �2k2/2m = h2/8md2, which physically corresponds to the

kinetic energy an atom at rest acquires after absorption of one lattice photon.

This optical way to produce periodic potentials for neutral atoms offers several unique

features. The first is the extreme tunability of the potential: the height of the optical

barriers V0 can be manipulated by changing laser intensity and detuning from the atomic

resonance, while the period d can be tuned by changing the wavelength or the angle

formed by the beams. A second important feature is the intrinsic periodicity of the

pattern: approximating the laser beams with plane monochromatic waves, the intensity

modulation turns out to be exactly periodic, with no inhomogeneities or defects.

A third important property of optical periodic potentials is the absence of phonons.

In a real crystal the atoms are arranged “in average” according to the lattice structure

(because this is the configuration which minimizes the total energy of the crystal), but

they can oscillate around this equilibrium configuration. The coupled oscillations of the


atoms in the lattice result in a vibration of the whole crystal, which can be quantum-

mechanically described in terms of phonons. The resulting vibration of the lattice po-

tential then affects the motion of the electrons and this can be modeled with an effective

electron-phonon interaction. In an optical lattice, instead, vibrations of the lattice can

be neglected (as far as instrumental sources of noise, as the laser phase noise and mirror

vibrations, are suppressed) and cold atoms can be used to study the physics of a gas of

particles in an “ideal” static crystal.

The absence of vibrations and defects, as well as the possibility of controlling the

interactions between the particles, makes ultracold atoms in optical lattices a wonderful

system for the investigation of ideal solid-state phenomena. A remarkable example has

been given by the investigation of Bloch oscillations [14-17], which is a purely quantum

effect that is hardly observable in real solids, where electrons strongly interact one with

each other and decoherence is very fast due to scattering of the electrons with phonons

and imperfections.

A further example of this possibility is represented by the observation of Anderson

localization [18], which will be discussed thoroughly in the following sections. Ultracold

atoms have recently allowed a clear observation of this phenomenon [19, 3], that was

originally predicted by P. W. Anderson in 1958 in the context of electronic transport

as a complete cancellation of conductivity induced by the presence of disorder. This

observation has been made possible by the production of strongly inhomogeneous optical

potentials, which allow the realization of atomic systems with a controlled amount of

disorder

3. – Monochromatic lattices

In this section we will briefly review the theory of quantum particles in a periodic

potential, to the aim of introducing concepts and notations that will be extensively used

throughout the paper.

3.1. Energy bands . – Periodic potentials are at the heart of the description of electric

conduction in crystalline solids, where the atoms are arranged in lattices with well-defined

symmetries. The most energetic electrons of the crystal, instead of forming a bound state

with individual atoms, experience the combined attraction of all the atoms of the lattice,

which can be modeled with a periodic potential having the same discrete translational

symmetries of the lattice.

In the one-dimensional case the periodicity condition for the potential V (x) takes the

form

(9) V (x) = V (x+ d),

where d is the distance between different lattice sites. The time-independent Schrodinger

equation for the wave function Ψ(x) of a particle of mass m moving in a periodic potential


V (x) reads

(10) HΨ(x) =

[−

�2

2m

∂2

∂x2+ V (x)

]Ψ(x) = EΨ(x).

A fundamental contribution to the solution to this problem was formulated by F. Bloch

in 1928 [20]. The Bloch theorem states that, when the symmetry in eq. (9) is applied, no

matter what is the particular expression of the potential V (x), the solutions of eq. (10)

take the general form

Ψn,q(x) = eiqxun,q(x),(11)

un,q(x) = un,q(x+ d),(12)

which describes plane waves eiqx modulated by functions un,q(x) having the same peri-

odicity of the lattice. These stationary solutions are labelled by two quantum numbers:

the band index n and the quasimomentum (or crystal momentum) q.

The quantum number q is called quasimomentum because it presents some analogies

with the momentum p, which is the good quantum number to describe the eigenstates

of the Schrodinger equation in the absence of any external potential. However, since

the potential V (x) does not present a complete translational invariance, the solutions

in eqs. (11)-(12) are not eigenstates of the momentum operator and �q is not the ex-

pectation value of the momentum. We note that, because of the discrete invariance of

the Hamiltonian under translations x → x+ nd (with n integer), the quasimomentum is

defined modulus 2π/d, that is the period of the reciprocal lattice. As a matter of fact, the

periodicity of the problem in real space induces a periodic structure also in momentum

space, in which the elementary cells are the so-called Brillouin zones(1).

For a given quasimomentum q many different solutions Ψn,q(x) with different energies

En(q) exist. These solutions are identified with the band index n. The term band refers

to the fact that the periodic potential causes a segmentation of the energy spectrum

into forbidden zones(2) and allowed zones, the so called energy bands, which in solid-

state physics are at the basis of the conduction properties of metals and insulators.

Figure 2 shows a plot of En(q) as a function of the quasimomentum q for the first

three energy bands of the periodic potential in eq. (8). The solid thick lines correspond

to a representation of the energy spectrum in the so called extended zone scheme, in

which different energy bands are plotted in different Brillouin zones. However, because

the quasimomentum is defined modulus a vector of the reciprocal lattice, the spectrum

En(q) can be more generally represented in the repeated zone scheme, in which all the

(1) In higher dimensions the periodicity in momentum space is defined by the vectors of thereciprocal lattice which can be constructed starting from the base vectors of the Bravais latticein real space [21].(2) With the term “forbidden zones” we refer to intervals of energies in which the density ofstates ρ(E) vanishes, while in the “allowed zones” ρ(E) > 0.


0 0.5 1 1.5 2 2.5 3

quasimomentum [q/k]

0

2

4

6

8

10

en

erg

y[E

/E]

R

E (q)1

E (q)2

E (q)3

E (q)0

1 Brillouin zonest 2 Brillouin zonend 3 Brillouin zonerd

Fig. 2. – Energy spectrum of a particle in a periodic potential. Solid curves: lowest three energybands for a particle of mass m in the periodic potential of eq. (8) with V0 = 4ER. Dashed curve:energy spectrum of the free particle. The energies are expressed in natural units ER = �

2k

2/2m.

energy bands are plotted in all the Brillouin zones (both thin and thick solid lines). For

comparison, we plot in the same graph also the parabolic spectrum E0(q) = �2q2/2m of

the particle in free space (dashed curve).

Some general features of the band structure already emerge from this particular case.

1) At low energies (En � V0) the bands are almost flat and, for increasing height of

the periodic potential, asymptotically tend to the energies of the bound states in the

single lattice well. 2) At high energies (En � V0) the bands are pretty similar to the

free particle spectrum (except for a zero-point energy shift) and differ from the latter

only close to the boundaries of the Brillouin zones. 3) Near the zone boundaries, in

correspondence with the appearance of the energy gap, the bands have null derivative.

We now consider the shape of the eigenstates of eq. (10) in the case of the sinusoidal

potential in eq. (8). In fig. 3 we plot the squared modulus of the ground-state wave

function Ψ0,0(x) for three different heights of the periodic potential. For small potential

heights (dotted line) the Bloch states Ψ0,q(x) are similar to the plane waves eiqx, except

for a small amplitude modulation with the periodicity of the lattice. Increasing the

lattice height (solid line), when the energy of the state becomes much smaller than the

maximum height of the potential, the Bloch wave functions are strongly modulated. In

any case, the Bloch wave functions are extended states, i.e. their amplitude is nonzero in

any position of the lattice, similarly to plane waves which are defined in the entire space.

In other words, the stationary states of a particle in a periodic potential are states in

which the particle is delocalized.


position [x/d]

1 2 3 4

0.5

1

1.5

2

00

2[a

u]

(x)

V ER0

0 R

0 R

V 5E

V 25E

E 0 47ER

E 1 82E

E 4 73E

R

R

Fig. 3. – Squared modulus of the ground-state wave function Ψ0,0(x) for different heights ofthe sinusoidal potential in eq. (8): V0 = ER (dotted), V0 = 5ER (dashed), V0 = 25ER (solid).Increasing the lattice height, the wave function changes from a weakly modulated plane wave toa function that is strongly localized at the lattice sites. The energy of these states is reportedin the inset.

Before concluding this section, we note that, in the case of a finite-sized system of

size L, boundary conditions have to be set. A natural choice is taking the wave function

to be periodic at the boundaries: Ψ(0) = Ψ(L). This is the common choice when solving

the quantum-mechnical problem of a particle in free space. Imposing such boundary

conditions leads to a quantization of the momentum of the particle (i.e. the wave function

has to be a standing wave of the system volume). The same effect happens also in the

case of the Bloch waves of a periodic potential: by imposing periodic boundary conditions

at the edges, the quasimomentum gets quantized according to the relation

(13) qi =2π

Li

with i integer. If the length L contains N lattice sites, i.e. L = Nd, it is easy to realize

that the first Brillouin zone is spanned by quasimomentum states with i ∈ [−N/2,+N/2].

This means that only N individual quasimomentum states can be defined.

3.2. Tight-binding model . – The problem of a quantum particle in a periodic potential

can also be treated in a simplified tight-binding approach, which has the advantage of

providing simple analytic results. This approach is inspired by the observation that for

large lattice depths the wave functions are strongly modulated by the periodic potential

(i.e. the particles are almost trapped in the lattice sites), and can be more conveniently

written as the sum of many wave functions localized at the lattice wells. A generic state


Fig. 4. – Tight-binding approximation to the motion of one particle in a periodic potential. Theprobability of hopping from one site |j〉 to a next-neighbor |j± 1〉 is expressed by the tunnellingmatrix element J .

|Ψ〉 (for simplicity, in the lowest energy band) can be written as a linear combination

(14) |Ψ〉 =∑

j

aj |j〉

of Wannier states |j〉 centered at the lattice site j with complex coefficients aj . When

the lattice depth is sufficiently large, the wave functions of the Wannier states become

strongly localized around single lattice sites(3).

The Hamiltonian can be expanded on the basis of maximally localized Wannier states

(which provide an alternative basis to the one provided by Bloch waves). By making

the tight-binding approximation of small overlap between the different Wannier states,

a simple equation can be derived:

(15) H = −J∑

j

(|j〉〈j + 1| + |j + 1〉〈j|) .

The physical meaning of eq. (15) is clear: a particle that is in state |j〉 has a finite prob-

ability of tunnelling through the potential barrier separating two lattice sites, hopping

into state |j − 1〉 or |j + 1〉, as schematically shown in fig. 4.

The introduction of this tight-binding approach allows us to obtain a simple expression

for the spectrum. It is easy to show that the stationary Schrodinger equation H|Ψ〉 =

E|Ψ〉 for the Hamiltonian in eq. (15) is equivalent to the set of equations

(16) −J(aj+1 + aj−1) = Eaj , ∀j.

Owing to the discrete translational invariance of the problem (all the sites are identical)

the coefficients aj must have the same modulus, i.e. the particle occupies each site of

(3) In the limit of an infinitely high lattice the Wannier functions reduce to the Gaussian groundstate of the harmonic oscillators obtained expanding the lattice potential around the potentialminima.


the lattice with equal probability, as prescribed by the Bloch theorem. What can be

different from site to site is the phase: again, owing to the translational invariance, the

phase difference from site to site should be the same all across the lattice. Therefore we

can write

(17) aj = eiqdj ,

which means that the amplitudes aj in next-neighboring sites have a phase difference

δφ = qd, where q can be clearly identified with the quasimomentum. With this definition,

eq. (16) can be rewritten as

(18) E = −2J cos qd,

which is the spectrum of the lowest band of the lattice in the tight-binding approximation.

We note that the width of the lowest band, i.e. the range of admitted energies, is 4J .

This tight-binding approach is particularly suited in the case of deep lattices, when

the total wave function can be expanded into the sum of many strongly localized states.

In the case of low lattice heights, when the wave function is just weakly modulated

and we cannot neglect the overlap integral between more distant neighboring sites, the

tunnelling rate J loses most of its physical meaning. In this case other energy scales J ′,

J ′′, etc. describing longer-range couplings should be considered and the simple form of

the energy spectrum in eq. (18) gets more complicated.

The phase coherence relation provided by eq. (17) can be verified experimentally with

ultracold atoms in time-of-flight experiments. Let us consider the simplified case of a non-

interacting Bose-Einstein condensate, where all the bosons share the same single-particle

wave function. If the BEC atoms occupy the ground state of an optical lattice, q = 0 and

the phase of each Wannier state is the same. If the lattice potential is suddenly switched

off, the atomic wave function undergoes a free-space evolution, in which the different

Wannier wavepackets expand, overlap and eventually interfere, as skematically shown

in fig. 5. The phase coherence relation in eq. (17) manifests as an ordered interference

pattern, with constructive interference peaks at position 0, ±2�ktexp/m ,±4�ktexp/m, . . .

(where texp is the expansion time and k = π/d), resembling the diffraction of light

from a material grating [22]. This pattern also provides an “image” of the initial wave

function in momentum space. As a matter of fact, for a sufficiently long time-of-flight

texp, the density distribution after free expansion corresponds to the initial momentum

distribution in the trap. We will further discuss this detection technique in sect. 5,

devoted to the experimental observation of the transition from extended (Bloch-like) to

localized (Anderson) states in the presence of an incommensurate bichromatic lattice.

3.3. Adding interactions . – Until now we have considered the physics of a single particle

in a periodic potential. However, ultracold gases are not ideal quantum gases: there

are interactions between the atoms, and these atom-atom interactions turn the single-

particle physics of the Bloch bands into a more complex many-body problem. In neutral


Fig. 5. – Expansion of a Bose-Einstein condensate from an optical lattice. The atoms aretrapped in a periodic potential generated with laser light (top) and then released from this trap.The density distribution after expansion (bottom) exhibits resolved peaks originated from theinterference of the condensates initially located at the lattice sites. Adapted from [22].

atomic gases no long-range Coulomb force is present and dipole-dipole interactions can

be neglected in most of the cases, the dominant interaction term coming from short-

range van der Waals interactions. Since the range of these interactions is much smaller

than the average inter-particle distance, in a cold dilute sample one can approximate

such interactions with collision events of two atoms occasionally coming into contact one

with the other. At ultralow temperatures the collisional channels reduce to only s-wave

collisions, whose cross-section is parametrized by just one scalar parameter, the scattering

length a (positive for repulsive interactions, negative for attractive interactions).

The quantum statistics of the atoms, i.e. their bosonic or fermionic nature, also make

a fundamental difference. As a matter of fact, ultracold identical fermions cannot undergo

s-wave collisions, which are forbidden by the Pauli exclusion principle. Bosons, instead,

do interact, and interactions are fundamental to describe static and dynamic properties

of atomic Bose-Einstein condensates [9]. For not too large atomic densities a mean-field

approach can be used, and interactions between the atoms of a Bose-Einstein condensate

can be described with a nonlinear term in the wave equation for the BEC wave function

Ψ(r, t), which is described by the following Gross-Pitaevskii equation [9]:

(19) i�∂

∂tΨ(r, t) =

[−

�2∇2

2m+ V (r) +

4π�2a

m|Ψ(r, t)|2

]Ψ(r, t).

The nonlinear term in Ψ, being analogous to the nonlinear Kerr effect for light propa-

gation, is responsible for the self-nonlinear behavior of matter waves and is at the basis

of many intriguing phenomena which are well known in optics, such as solitonic prop-

agation [23] and four-wave mixing [24]. In the context of optical lattices, the presence


Fig. 6. – Sketch of interacting bosons in a periodic potential. Hopping of particles from one siteto the next neighbor is described by the tunnelling matrix element J , while repulsive short-rangetwo-body interactions are taken into account by the on-site interaction energy U .

of such nonlinearities leads to a number of new effects deviating from the single-particle

Bloch theory, including the observation of different kinds of instabilities [25, 26].

In cold-atoms experiments the strength of interactions a can be modified by using

a static magnetic field adjusted around a Feshbach resonance [27]. The magnetic field

allows tuning the energy of a colliding atom pair in proximity of a bound molecular

state: as a result, the scattering of the two atoms is modified and the scattering length

can be resonantly tuned. This technique can even allow to cancel atom-atom interactions

or switch repulsive interactions into attractive ones. We will discuss the importance of

this tool in the following sections devoted to the experimental observation of Anderson

localization with non-interacting Bose-Einstein condensates.

3.4. Mott insulators. – An optical lattice can dramatically increase the effects of

interactions between the atoms and change a weakly interacting gas intro a strongly-

correlated many-body state. As a matter of fact, increasing the depth of the optical

lattice results in a lower tunnelling rate, i.e. a lower mobility of the atoms in the lattice,

which amplifies the effects of interactions. At the same time, the atoms get more squeezed

in the lattice potential wells, thereby increasing their interaction energy.

The physics of a gas of interacting ultracold bosons in a deep optical lattice is well cap-

tured by the Bose-Hubbard model [28], originally introduced to describe the superfluid-

insulator transition observed in condensed-matter systems [29]. The Bose-Hubbard

Hamiltonian is the second-quantization generalization of the tight-binding Hamiltonian

in eq. (15) to the case of interacting particles. It takes the form

(20) H = −J∑〈j,j′

〉

b†j bj′ +U

2

∑j

nj(nj − 1),

where bj (b†j) is the annihilation (creation) operator of one particle in the j-th site,

nj = b†j bj is the number operator, and 〈j, j′〉 indicates the sum on nearest neighbors.

The two terms on the right-hand side of eq. (20) account for different contributions to

the total energy of the system. The first term, proportional to the same hopping energy

J already introduced in sect. 3.2, describes the tunnelling of bosons from one site to an


Fig. 7. – Two interacting bosons in a double-well potential. The three possible states of thesystem are shown.

adjacent site. The second term, proportional to the interaction energy U , arises from

atom-atom on-site interactions and gives a nonzero contribution only if more than one

particle occupies the same site (as schematically sketched in fig. 6).

In the presence of repulsive interactions U > 0, this model supports a quantum phase

transition from a superfluid state (U � J) in which atoms are delocalized occupying ex-

tended Bloch states, to an insulating state (U � J) where the atom wave functions are

localized in individual lattice sites. This behavior can be studied in the simplified setting

represented in fig. 7, where one considers only two lattice sites, which are coupled by a

tunnelling probability J . This double-well problem is paradigmatic of a large number

of effects where quantum tunnelling is important, e.g. molecular spectra (inversion spec-

trum of ammonia molecule) and superconducting Josephson junctions. The Hamiltonian

describing this situation is the following:

(21) H = −J(b†1b2 + b†

2b1

)+U

2[n1(n1 − 1) + n2(n2 − 1)] .

We can use an occupation-number representation and describe the state of the system

with the notation |Ψ〉 = |mn〉, which means that m particles are present in the left

well and n particles are present in the right well(4). In the simplest case of 2 identical

bosons the state of the system can be written as |Ψ〉 = 1/√

2 sinα|20〉 + cosα|11〉 +

1/√

2 sinα|02〉, which is parametrized by the angle α and automatically satisfies the

normalization condition 〈Ψ|Ψ〉 = 1. One could easily work out that the energy of such

state is

(22) E = 〈Ψ|H|Ψ〉 = U sin2 α− 2J sin(2α).

We consider now two interesting limiting cases. When J � U the energy can be approx-

imated with E � −2J sin(2α), which is minimized by α = π/4, yielding

(23) ΨJ�U �1

2|20〉 +

1√

2|11〉 +

1

2|02〉,

(4) We note that this kind of representation automatically takes into account the indistinguisha-bility of the paricles.


which means that each particle is delocalized in the two wells and the number of particles

in each well has nonzero fluctuations around unity. When instead U � J the energy

becomes E � U sin2 α, which is minimized by α = 0, yielding

(24) ΨU�J � |11〉,

which means that no number fluctuations are present: one particle is present in the left

well and one particle is present in the right well. This happens because the energy U

saved by localizing each particle in a different well is larger than the energy J that would

have been gained if the particles were delocalized.

This localization behavior, induced by repulsive interaction between the atoms, is

amplified in a lattice, which can be thought as an infinite array of identical potential

wells linked by a tunneling coupling between next-neighboring sites. In the case of unit

filling (i.e. number of sites = number of atoms) the two limits are a superfluid state for

J � U , with each particle being delocalized in an extended Bloch wave

(25) ΨSFJ�U ∝

(b†1

+ b†2

+ . . .+ b†N

)|00 . . . 0〉,

and a Mott-insulating state for U � J in which each atom is localized in a single site:

(26) ΨMIU�J ∝ b†

1b†2. . . b†N |00 . . . 0〉.

These two states exhibit remarkably different properties. The transition from a su-

perfluid to a Mott insulator when the Hamiltonian parameters are changed is example

of quantum phase transition [30]. In this kind of transitions the control parameter is not

temperature, as in the case of classical phase transitions (e.g. the ferromagnetic tran-

sition in the Ising model [31]), but a different coupling parameter entering the system

Hamiltonian, in this case the ratio U/J .

The first experimental observation of the quantum phase transition from a superfluid

to a Mott insulator has been reported by M. Greiner and coworkers in [32], following the

initial proposal by [28]. The experiment was performed by trapping a BEC in a three-

dimensional cubic optical lattice. The control parameter for the Mott transition, i.e. the

ratio U/J between interaction energy and hopping energy, was tuned by adjusting the

depth V0 of the optical lattice. As a matter of fact, in a deep 3D optical lattice, the

following relations hold [33]:

U =

√8

πkaER

(V0

ER

)3/4

,(27)

J =4√πER

(V0

ER

)3/4

exp

(−2

√V0

ER

),(28)

and the ratio U/J scales exponentially with the lattice depth V0.


Fig. 8. – Interference pattern of an ensemble of interacting ultracold bosons released from a3D optical lattice for different lattice heights V0 = sER. Around V0 = 15ER one observes thedisappearance of the interference peaks, indication of entering the Mott Insulator state.

The Mott transition was investigated by monitoring the interference pattern of the

atomic cloud released from the 3D lattice as a function of the lattice height. When the

lattice confining the atoms is suddenly switched off, the atom wave functions expand

in free space and overlap: if long-range phase coherence exists in the system, the in-

terference of the overlapping wave functions builds up in a regular diffraction pattern,

similar to the one shown in fig. 5 for a 1D optical lattice. This pattern, resembling the

diffraction of light from a grating or the diffraction pattern of X-rays scattered from

a crystalline structure, can be taken as a measure of the degree of coherence in the

many-body system.

The interference patterns shown in fig. 8, recorded in similar more recent LENS ex-

periments, correspond to different lattice depths V0, going from zero to 22.5ER. For

small lattice depths the intensity of the side peaks increases with V0: the atoms get

more tightly confined in the lattice sites and, following the optical analogy, this means

that the lattice acts as a grating with narrower slits, resulting in a stronger “diffrac-

tion” effect(5). At a lattice height V0 � 15ER the visibility of the interference peaks

suddenly decreases, eventually going to zero, reflecting the loss of coherence of the sys-

tem after entering the Mott insulator state, where the atoms are localized and the

number fluctuations δni vanish. This coherence loss is reversible: by ramping down

the height of the optical lattice one can recover the peaked interference pattern pecu-

liar of the superfluid state, where atoms are delocalized and on-site coherent states are

created.

An important property characterizing Mott insulators is the existance of a gap in the

excitation spectrum. A Mott insulator is a “rigid” insulating state, in which no current

(5) We observe that the free-space expansion maps the initial momentum distribution in realspace. In the case of phase coherence (superfluid) the interference pattern detected is nothingbut the momentum distribution of the initial extended Bloch states.


Fig. 9. – Sketch of a bichromatic lattice.

flow can be sustained and no excitations can be produced, unless an energy gap is broken.

This gap coincides with the energy ε ∼ U that is required for moving a particle from

a site to a next-neighbor already occupied site, i.e. for the creation of a particle-hole

excitation. A superfluid, instead, is characterized by a gapless excitation spectrum, with

the lowest branch of excitations corresponding to the creation of phonons inside the

superfluid: long-wavelength excitations can be created at very low energies ε, according

to the linear dispersion relation of phonons ε = c�k [9].

The existance of a gap in the Mott state was already evidenced in [32], where the

authors applied a magnetic potential gradient in order to produce tunnelling excitations.

A marked resonance centered at a site-to-site energy shift ∼ U signaled the presence

of the Mott gap. More recently, the measurement of the excitation spectrum across

the superfluid-Mott transition has been carried out by using a lattice modulation tech-

nique [34] and inelastic light scattering (Bragg scattering) [35]. We will further discuss

these issues in sect. 6 devoted to the study of strongly interacting disordered bosons.

4. – Bichromatic lattices

In this section we will review the physics of bichromatic lattices, i.e. potentials ob-

tained from the superposition of two periodic potentials with different periods. From

an experimental point of view, bichromatic optical lattices can be realized quite easily:

starting from an already existing optical lattice, it is sufficient to add a second pair of

laser beams with different wavelength, as sketched in fig. 9. In the one-dimensional case

the resulting potential can be written as

(29) V (x) = V1 sin2(k1x) + V2 sin2(k2x).


Fig. 10. – One of the simplest examples of 2D quasicrystals is offered by the Penrose tiling,where the non-periodic combination of two elementary cells (light-grey and dark-grey rhomboids)provides a gapless tiling of the plane.

As we will see in the following, an important quantity characterizing this class of

potentials is the ratio between the two lattice periods

(30) β = k2/k1,

which quantifies the degree of commensurability between the two lattices. If this number

is rational the resulting potential is periodic (commensurate case), if it is irrational there

is no periodicity (incommensurate case). The latter case is particularly interesting, since

this broken translational invariance has important consequences on the properties of the

eigenstates of this potential.

Bichromatic incommensurate lattices are a beautiful example of a mathematical struc-

ture that can be non-periodic and ordered at the same time. Systems of this kind are

extremely interesting since their properties interpolate between the properties of crys-

talline structures and the ones of disordered systems. This behavior is intimately related

to the properties of quasicrystals [36, 37].

4.1. Quasicrystals. – The history of quasicrystals is quite recent. Their existence was

suggested in the ’60s when mathematical structures showing long-range order but no

periodicity were discovered. Quasicrystals are typically composed by a finite number of

elementary cells forming a complete set of tiles that can fill the space with no gaps (as in

the case of ordinary crystals) but in a non-periodic way. One of the simplest example of

quasicrystal is provided by the Penrose tiling [38] showed in fig. 10. In this 2-dimensional


structure only two kinds of tiles exist (light-gray and dark-gray rhomboids) which are able

to completely fill the plane with no gaps. This arrangement does not follow any discrete

translational invariance, i.e. no translation of the lattice ends up locally in the same

configuration. Quasicrystals can also possess extended classes of rotational symmetry in

addition to the ones allowed for periodic crystals, e.g. the tiling showed in fig. 10 has

5-fold rotational symmetry, which is forbidden to ordinary crystals.

The existence of quasicrystals was demonstrated in [39], where sharp peaks with for-

bidden rotational symmetry were recorded in electron diffraction experiments for rapidly-

cooled Al-Mn alloys. One of the features of quasicrystals is indeed their ability to produce

sharp diffraction peaks as crystals do, which is related to the presence of long-range order-

ing. Very recently, quasicrystalline order has been observed in a mineral sample coming

from the Museum of Natural History of the University of Florence [40]: this discovery

demonstrates that quasicrystals can spontaneously form in Nature and remain stable

under geologic conditions.

Similarly to periodic optical lattices, also optical lattices with quasicrystalline order

can be created by using suitable arrangements of laser beams. The first investigation of

cold atoms in optical quasicrystals was reported in laser cooling experiments [41] with

the realization of 5-fold rotationally symmetric potentials like the Penrose tiling. Going

to the quantum degenerate regime, the investigation of a Bose-Einstein condensate in

these quasicrystalline 2D potentials has been proposed theoretically in [42].

Incommensurate 1D bichromatic lattices can be considered as the simplest examples

of quasicrystals. In the next sections we will describe their properties, with a partic-

ular focus to the energy spectrum and to the localization transition exhibited by their

eigenstates.

4.2. General notations . – Throughout the paper we will mostly consider the limiting

case in which one lattice is much weaker than the other, i.e. V2 � V1. With clear meaning

of the notation we will refer to these lattices as main lattice and perturbing lattice, as

also indicated in fig. 9. With this assumption the resulting quasiperiodicity is mostly

compositional (and not topological as in the case of a real quasicrystal), this meaning that

the sites of the lattice are still almost periodically displaced as in the case of a periodic

lattice and the perturbation only affects the properties (the energy) of the sites.

It is easy to realize that in this limit the minima of the potential in eq. (29) sit in the

same position xj as the minima of the main lattice

(31) xj = jπ/k1

with j integer. We can also assume that the heights of the potential barriers are not

significantly changed by the addition of the perturbing lattice (see [43] for more details).

Therefore we can still assume that the system can be described by a hopping term J which

is uniform across the whole bichromatic lattice. At small energy scales, the bichromatic

lattice can be characterized by the values εj taken by the potential energy at the lattice


sites, which are not uniform but change according to

(32) εj = V (xj) = V2 sin2(k2xj) = V2 sin2(βπj).

From the above expression we note that different values of β could give rise to the same

set of energies εj . As a matter of fact, the energies εj are the same for any substitution

β → ±(β+m), with m integer. This is a consequence of the symmetry of the sin2 function

and of the fact that {εj} arise from a uniform sampling of a sinusoidal curve. Aliasing

effects result in the same sampling for sinusoids whose spatial frequencies differ by a phase

2π on the distance of the main lattice period. It is convenient to consider only one of

these equivalent samplings, i.e. the one with the longest-wavelength/smallest-frequency

modulation:

(33) γ = minm

|β −m|.

As we will see in the following, when discussing the general properties of bichromatic

potentials, this definition is particularly helpful. This choice of γ is also natural since it

is the only one satisfying the Nyquist-Shannon sampling theorem, and it automatically

implies

(34) γ < 1/2 ;

this means that among all the possible sinusoids that yield the same sampling we choose

to consider the longest-wavelength one, which has a modulation period larger than 2 sites

(and can be reconstructed by interpolation starting from the {εj} without aliasing). We

can refer to the period of this modulation as bichromatic period.

This situation is exemplified in fig. 11, showing a zoom on the lowest-energy region

of a bichromatic lattice (black line): three different choices of the perturbing lattice with

different β yield the same set of energies {εj} at the bottom of the potential wells.

4.2.1. Harper and Aubry-Andre model. The problem of a particle in a potential with two

periodicities has a long history. It has been studied by P. G. Harper in 1955, who derived

an equation describing the states of a crystal electron in a uniform magnetic field [44].

Harper considered a 2D square lattice in the plane (x, y) with a static uniform magnetic

field B = Bz, directed out of the plane, described by a vector potential A = Bxy.

By applying the canonical momentum substitution p → p − eA/c into the Schrodinger

equation and using a tight-binding approach, the following Harper model can be derived:

(35) uj+1 − uj−1 + 2 cos (2πβj + φ)uj = εuj ,

where uj is proportional to the amplitude of the wave function in site j along direction

x. In this equation two periodicities are present: the periodicity of the lattice (which is

intrinsically present in the tight-binding discretization) and an effective periodic potential

2 cos (2πβj + φ). The ratio between the two periodicities is given by β = (d2B)/(hc/e),


Fig. 11. – The solid line shows a zoom of the lowest-energy region of the bichromatic potentialin eq. (29). The other lines show perturbing lattices with different β which yield the same setof energies {εj} : β =

√3 (dotted), β =

√3 − 1 (dashed), β = 1 −

√3 (dot-dashed). The latter

choice coincides with the value of γ associated to the lowest-wavelength modulation giving riseto the same energy sampling {εj}.

which corresponds to the ratio between the magnetic flux through one square lattice cell

to the magnetic flux quantum.

Despite the simplicity of the physical system, this problem shows a very intriguing

phenomenology related to the biperiodic structure. As D. R. Hofstadter wrote in the

introduction of [45] (the paper where he discovered the fractal structure known as Hofs-

tadter butterfly, see sect. 4.6), “the problem of Bloch electrons in magnetic fields is a very

peculiar problem, because it is one of the very few places in physics where the difference

between rational numbers and irrational numbers makes itself felt”.

The case of irrational numbers is in fact particularly interesting. An extension of

the Harper model has been studied in 1980 by S. Aubry and G. Andre, who focused

on the properties of the wave functions in the incommensurate case. The Aubry-Andre

model [46] is a tight-binding Hamiltonian describing a quantum particle hopping in a 1D

bichromatic lattice:

(36) H =∑

j

|j〉〈j + 1| + |j + 1〉〈j| + λ∑

j

cos (2πβj)|j〉〈j|.

This Hamiltonian is closely related to the tight-binding Hamiltonian of eq. (15) for a single

lattice, with the presence of an additional term proportional to λ, which parametrizes the

strength of the quasiperiodic modulation (in our notations λ = V2/2J). The derivation

of this tight-binding model starting from the analytic form of the bichromatic potential

is worked out and thoroughly discussed in [47].


Fig. 12. – Sketch of the biperiodic tight-binding system described by the Aubry-Andre model.Hopping of particles from one site of the main lattice to the next-neighbor is described bythe tunnelling matrix element J , while the amplitude of the perturbing lattice is denoted with2Δ = V2.

The Aubry-Andre model can be thought as an extension of the Harper model to the

case in which the amplitude of the spatially dependent perturbation is free to change.

Aubry and Andre showed that, for incommensurate β this model supports a metal-

insulator transition at a critical value of the amplitude of the secondary lattice(6). For

small strength λ of the perturbing lattice the eigenstates are extended, similarly to the

Bloch waves of a periodic potential. For larger depths of the perturbing lattice, above

the critical strength λ = 2, the eigenstates undergo Anderson localization and become

exponentially localized.

The realization of the Aubry-Andre model (fig. 12) with ultracold atoms has been

proposed in [48] and theoretically studied in [49]. More recently, a complete theoreti-

cal investigation on the localization mechanism in optical bichromatic lattices has been

presented in [47].

4.3. Superlattices. – In order to illustrate some general features of bichromatic lattices,

we start considering the commensurate case, often called superlattice, which is defined

by a rational value of γ. Under this condition the overall potential is periodic and the

situation is not much different from what we have shown in sect. 3. The system still has

a discrete translational invariance and the Bloch theorem can be applied. Of course, the

periodicity of the system now shows up on a different length scale, which corresponds to

γ−1 times the lattice constant of the main lattice(7).

Let us consider a simple situation in which γ = 1/4, corresponding to a superlattice

period of 4 sites. An example of bichromatic potential for this choice of γ is shown in

fig. 13a for V2 = 2.5J . It is evident that the resulting potential is strictly periodic, with

a periodicity in reciprocal space (e.g. width of the Brillouin zones) which is γ = 1/4 of

(6) As we will see in the following, the Aubry-Andre model reduces to the Harper model exactlyat the critical point λ = 2.(7) In the commensurate case γ = m

′

/m′′ with m′ and m′′ integer numbers. Generally speaking,

the superlattice period amounts to g lattice sites, where g = LCM(m′

,m′′) is the least common

multiple of m′ and m′′, and can be different from γ

−1. However, in order to simplify thepresentation, we illustrate a simpler case where m

′ = 1 and γ = g−1.


Fig. 13. – A bichromatic commensurate lattice with γ = 1/4 and V2 = 2.5J . The plots show:a) the real potential (continuous curve) and the quasiperiodic energy modulation (dotted curve);b) the energy spectrum of the bichromatic lattice (black) and of the main lattice only (gray);c) the square modulus of the ground-state wave function in real space; d) the square modulusof the ground-state wave function in momentum space. The spectrum and the wave functionsare calculated numerically on a system length of L = 1000 sites.

the periodicity of the main lattice. According to the Bloch theorem, the eigeinstates are

extended plane-wave–like functions with a spectrum that shows the presence of energy

bands, as in the case of a single lattice. The spectrum of this potential is shown in fig. 13b

(thick dark lines), where for comparison we also show the spectrum for the main lattice

only (light thin line), which features a gap at q = π/d = k1. The introduction of the

perturbing lattice modifies the translational symmetry of the system and is responsible

for the opening of extra energy gaps at multiples of the quasimomentum γk1 = γπ/d,

with the larger gaps opening at γk1 and (1 − γ)k1. The lowest band of the main lattice

is therefore split into a finer structure of γ−1 bands: we can refer to these bands as

sub-bands of the main lattice fundamental band.

The squared modulus of the eigenfunctions has the same periodicity of the lattice

potential, as prescribed by the Bloch theorem eq. (12). This is evident from fig. 13c,

where we plot the amplitude of the ground state wave function |Ψj |2 in different lattice

sites j. Correspondingly, the eigenstate wave functions in momentum space are char-

acterized by δ-peaks occurring at submultiples nγk1 = nγπ/d of the main lattice wave

vector, reflecting the change in periodicity and the emergence of the new energy gaps.

The momentum distribution of the ground state wave function for the same choice of

parameters is plotted in fig. 13d.


4.4. Incommensurate lattices. – In the case of an incommensurate lattice the situation

gets more subtle. As a matter of fact, since there is no periodicity in the system, the

Bloch theorem cannot be applied and the quasimomentum cannot be defined. This

lack of translational invariance makes the Schrodinger problem more complicated and no

simple theorem helps us in finding the solution. However, numerical studies show that

for weak intensities of the perturbing lattice, the situation is not substantially changed

and many concepts that we have introduced for particles in periodic potentials still

hold. In particular, we can still think in terms of energy bands and quantities related

to the quasimomentum can be introduced. As a matter of fact, the eigenstates of an

incommensurate bichromatic potential can be labelled by a progressive index i and, in

analogy with eq. (13), we can introduce a pseudo-momentum q, defined as

(37) qi =2π

Li.

The reason why we introduce such quantity is evident if we calculate the spectrum of

the incommensurate potential for V2 < 4J (the reason for this choice will be clear in

the following sections). In fig. 14a,b we show the potential energy modulation and the

spectrum for γ = 2 −√

3 and V2 = 2.5J .

On a coarse-grained view, the spectrum still shows the emergence of energy bands,

similarly to what happens in a periodic lattice, even if the system is not periodic. When

the perturbation V2 is introduced, clear gaps open at pseudo-momenta q = γk1 and

q = (1− γ)k1 and no major differences can be detected in this regime from the commen-

surate case. Looking at the spectrum with more detail, however, one could recognize the

appearance of many weaker gaps that open across the entire band following a self-similar

fractal structure induced by the absence of periodicity. The appearance of some (the

strongest) of these gaps can be detected in the insets of fig. 14b(8).

Even if no periodicity is present, for small values of the perturbation V2 the wave

functions are extended across the entire system size, similarly to Bloch waves. The

ground state wave function for the potential in fig. 14a is plotted in fig. 14c: despite the

absence of periodicity, the wave function spreads across the whole lattice and it shows

an evident modulation at the bichromatic period γ−1d, similarly to the commensurate

case. The momentum distribution of the ground state for the same potential is shown in

fig. 14d, where several δ-peaks are observed, the strongest appearing at γk1 and (1−γ)k1

and the weaker ones following the same dense structure of the energy gaps emerging in

the spectrum. The presence of sharp peaks in the momentum distribution (reminding the

observation of sharp peaks in the diffraction of X-rays or electrons from quasicrystals [37])

is simply related to the fact that the eigenstates are extended, i.e. they occupy all the

sites of the lattice, and long-range order is present. Therefore, as prescribed by the

(8) This structure, related to the Hofstadter butterfly discussed in sect. 4.6, shows up clearly

only as the spectrum is observed on a small energy scale and the system size L is increased (seeboundary condition in eq. (37)).


Fig. 14. – A bichromatic incommensurate lattice with γ = 2 −√

3 and V2 = 2.5J . The insets inb) show the appearance of a structure of weaker energy gaps owing to the absence of periodicity.See caption of fig. 13 for more information on the figure.

uncertainty relation, the momentum distribution of these states should be made up of

δ-peaks, the position of which reflects the wavelength of the modulation imposed by the

perturbing lattice.

4.5. Localization in bichromatic lattices. – For increasing height of the quasiperiodic

perturbation the situation changes drastically. At a critical perturbation strength a

quantum phase transition from extended states to localized states takes place. This

localization transition is a manifestation of a much more general phenomenon, known as

Anderson localization, that we are going to discuss in the following.

4.5.1. Localized states. We have seen in sect. 3 that in a periodic potential all the

eigenstates are extended states. This means that a perfectly periodic crystal is able to

support a stationary current which flows from one side of the crystal to the other. This,

in turn, means that the system behaves as a conductor. Fifty years ago, P. W. Anderson

conjectured that the presence of disorder in a crystal could turn a metal into a perfect

insulator with zero conductivity [18]. The microscopic mechanism for this insulating

behavior is the localization of the wave functions: the eigenstates of the system change

from extended to localized. Spatially-localized eigenstates imply that no current can flow

in the system under stationary conditions.

Anderson formulated his localization theory for a tight-binding model of electrons

hopping in a disordered lattice where each site has a random potential energy. In


incommensurate bichromatic lattices the presence of inhomogeneities can also lead to

an exponential localization of the eigenstates, as in the case of a truly random potential.

A bichromatic potential is not a random potential, because the energy modulation is

characterized by long-range correlations and a purely deterministic form of the potential

can be written out. These correlations have important consequences on the possibility to

observe a direct localization transition. As a matter of fact, while a localization transi-

tion does not exist in 1D in a δ-correlated random potential (since all the eigenstates are

localized, as demonstrated with scaling arguments in [50]), in a 1D bichromatic potential

a localization transition takes place at a well-defined value of the perturbation strength.

Anderson localization of electrons in bichromatic lattices was first studied by S. Aubry

and G. Andre [46], who predicted a universal transition for λ = V2/2J = 2. The existence

of a phase transition was conjuctered with an elegant reasoning, inspired by the self-

duality of the model in position and reciprocal space. As a matter of fact, eq. (36) can

be recasted in momentum space by applying the transformation

(38) |k〉 =1

√L

∑j

ei2πkβj |j〉,

that connects the Wannier states |j〉 to the eigenstates of the momentum operator |k〉.

With this substitution one obtains the Hamiltonian in reciprocal space

(39) H =λ

2

[∑k

|k〉〈k + 1| + |k + 1〉〈k| +4

λ

∑k

cos (2πβk)|k〉〈k|

],

which has the same form as eq. (36), but the amplitude of the quasiperiodic energy

modulation (compared to the tunnelling energy), is 4/λ instead of the real-space value λ.

Clearly, if a transition from extended states to localized states exists for the Hamiltonian

in eq. (36), it must exist also for the Hamiltonian in eq. (39), and the transition point

has to be the same. Hence

(40) λ =4

λ= 2.

This transition is visualized in the left panel of fig. 15. Here the grayscale represents

the amplitude of the ground-state wave function |Ψj |2, with dark regions corresponding

to larger amplitudes. The amplitude is plotted as a function of position (horizontal axis)

and strength of the perturbing lattice V2 = 2λJ (vertical axis). Above the transition

point the wave function becomes localized on shorter and shorter length scales, the more

the strength of the perturbation is increased.

For comparison, we show in the right panel what happens in the case of the Anderson

model, i.e. a tight-binding model with uncorrelated disorder in which εj = V2 Rnd[1],

where Rnd[1] is a random real number between 0 and 1. In this case no phase transition

is observed and the ground state is localized for any amount of disorder, in agreement

with the predictions of the scaling theory for localization [50]. However, the behavior of


Fig. 15. – Squared modulus of the ground-state wave function (in greyscale) for the Aubry-Andre model with β = (1 −

√5)/2 and for the 1D Anderson model with uncorrelated site

energies. This quantity is plotted as a function of position (horizontal axis) and strength of theperturbing potential V2. Clearly, a localization transition happens in the Aubry-Andre case forλ = V2/2J = 2.

.

the wave function in the localized case is pretty similar in the two cases, bichromatic and

random. In fig. 16 the ground state in the bichromatic and random potential is plotted

for four different amplitudes of perturbation: V2 = J (a), V2 = 4J (b), V2 = 7J (c),

V2 = 10J (d). While for the lowest value the different behavior is evident—exponential

localization for the random potential (gray line), almost uniform amplitudes for the

bichromatic potential (dark line)—in the localized regime the ground state for both the

potentials is exponentially localized.

4.5.2. Spectrum of the localized states. We finally examine the energy spectrum of the

localized states. This is a beautiful demonstration of how the physics of quasicrystals

nicely interpolates between the physics of ordered (periodic) and truly disordered sys-

tems. Even above the threshold for localization, the spectrum of the eigenstates shows

the presence of energy bands and gaps similarly to the spectrum of the extended states

below the critical point. An example of this band structure is reproduced in fig. 17a,

which features a sequence of gaps, the stronger ones again located at q = γπ/d and

q = (1 − γ)π/d. The spectrum reported in figure refers to γ = 2 −√

3 and V2 = 6J

slightly above the localization transition. The energy width of the spectrum increases

with the perturbation strength(9) and approaches V2 in the regime V2 � J .

(9) We recall that for a single lattice in the tight-binding limit the energy width of the lowestband is 4J .


Fig. 16. – Ground-state wave function for the Aubry-Andre model with β = (1 −√

5)/2 (blackline) and for the 1D Anderson model with uncorrelated site energies (grey line). The differentplots refer to a) V2 = J , b) V2 = 4J , c) V2 = 7J , d) V2 = 10J . Note the log scale.

It is possible to show that the collection of states in the lowest sub-band of width γπ/d

can be built up by picking, for each quasiperiod of the lattice, an exponentially localized

eigenstate centered close to the lattice site with minimum energy within the quasiperiod.

If N is the total number of sites, the lowest sub-band is formed by γN localized states,

each occupying a different lattice quasiperiod. Going towards larger energies, the gap

at q = γπ/d forms when all the quasiperiods have been “occupied” by a localized state

and the following state gets localized in a quasiperiod that already has a localized state

with lower energy within. This is where the strongest gap at q = γπ/d forms. We show

in fig. 17b-e some examples of localized states which clarify this mechanism. Here the

horizontal grey segments represent the energies of the sites in the Aubry-Andre model,

while the dark-grey histogram represents the amplitude of the wave function in the lattice

sites. The four plots refer to the four different states b)-e) labelled in fig. 17a.

The lowest-energy state b) localizes around a lattice site which coincides with the min-

imum of the quasiperiodic modulation (and hence it is the site with the lowest potential

energy of the lattice), as clear from fig. 17b. The highest-energy state in the lowest sub-

band c) is instead the one corresponding to fig. 17c, when two lattice sites are equidistant

from the minimum of the quasiperiodic modulation. The following state d) is separated

by an energy gap, as shown in fig. 17a. It is clear from fig. 17d that states c) and d)

occupy the same quasiperiod and their wave functions show an almost equal population

of the two lowest lattice sites within the quasiperiod. What is different is the phase of

the wave function. This situation is closely related to what happens in the double-well

problem, where two degenerate states are spatially separated by a potential barrier. As


Fig. 17. – a) Energy spectrum of localized states in a bichromatic potential with V2 = 6J andγ = 2 −

√3. Note the presence of energy gaps in the spectrum, even if the states are localized.

b-e) Examples of localized states: the dark grey histograms represent the amplitude of the wavefunctions in the different sites.

a matter of fact, for the perturbation strength considered in fig. 17, the eigenstates are

strongly localized and most of the wave function occupies two twin sites. The possible

eigenstates of a particle in a double-well are the symmetric and antisymmetric combi-

nations, which are separated in energy by 2J(10): this is exactly the energy gap that

separates the lowest sub-band from the higher sub-band of the bichromatic potential, as

shown in fig. 17.

(10) A beautiful example of this splitting is given by the molecular spectrum of ammonia NH3,in which the nitrogen atom can jump between two stable configurations by “tunnelling” throughthe triangular plane arrangement of hydrogen atoms [51]. This effect, described by an effectivedouble-well potential, is responsible for the existence of “inversion doublets” of lines separatedby energy 2J .


Fig. 18. – The Hofstadter butterfly. It represents the spectrum of the Aubry-Andre model atthe critical point λ = V2/2J = 2 as a function of the incommensurate parameter β. Adaptedfrom [45].

4.6. Further considerations. – The physics of biperiodic systems shows more interest-

ing features. One particurarly intriguing property manifests at criticality, when the depth

of the perturbing lattice matches the critical value for the localization transition in the

incommensurate case. At this value the Aubry-Andre model reduces to the Harper model

and the spectrum shows a characteristic structure, shown in fig. 18, known with the name

of Hofstadter butterfly. This self-similar structure, demonstrated by D. R. Hofstadter in

1976, is one of the first fractal structures discovered in physics. The figure shows a two-

dimensional representation of the spectrum of the biperiodic system at the critical point

as a function of the energy of the states and of the incommensurate parameter β(11).

More generally, an intriguing feature of biperiodic potentials is related to how the

properties of the system change when the incommensurate parameter β is made “less”

or “more” irrational. Actually, the localization transition predicted in the Aubry-Andre

model gets sharper as the ratio between the lattice periods β is made “more” irrational.

This behavior can be observed in fig. 19, taken from [47], where the degree of localization

of the states is quantified by plotting the inverse participation ratio (IPR) as a function

of Δ/J = λ = V2/2J and of β. The IPR is defined as[∫

|Ψ(x)|4dx/∫|Ψ(x)|2dx

]−1

and

measures the inverse number of sites occupied by the state: it is macroscopic for localized

(11) The figure is obtained after averaging the spectra over an arbitrary initial phase φ to beincluded in the cos term of eq. (36).


Fig. 19. – Colour-scale plot of the inverse participation ratio (see text for definition) as a functionof perturbation strength Δ/J and incommensurate parameter β. Note that the localizationtransition becomes sharper far from simple rational numbers. Taken from [47].

states, while for extended states it vanishes as N−1 (with N number of sites). Figure 19

shows that the IPR increases when Δ/J becomes larger than 2, in a sharper or softer way

depending on how far β is from a simple rational number. This behavior is particularly

evident close to β = 1 (corresponding to a quasiperiodic perturbation with very long

period), where the localization crossover becomes extremely slow and very large values

of the perturbation are needed to localize the wave function over distances on the order

of the lattice spacing.

This discussion on the degree of irrationality of the parameter β is particularly im-

portant when one considers finite-sized systems. Owing to the presence of a trapping

potential, the size of ultracold atomic samples is typically limited to few hundreds of

lattice sites. The existence of a maximum length scale can spoil the sharpness of the

localization transition and, at the same time, release the constraints on the incommen-

surability of the potential: as we will further discuss in the following section, even a

periodic superlattice potential can be “experimentally” considered as incommensurate if

the resulting period is larger than the system size.

5. – Anderson localization of matter waves in bichromatic lattices

Experiments chasing for Anderson localization with ultracold atoms have been

started in 2005 with the first realization of a Bose-Einstein condensate in a disordered

potential [52]. Two major obstacles for the observation of this phenomenon were the pro-

duction of strongly inhomogeneous potentials with a sufficiently small disorder “grain

size” and the cancellation of interactions between the BEC atoms, which lead to de-

phasing and spoil the single-particle Anderson physics [53]. The eventual observation of

Anderson localization with ultracold atoms has been recently achieved in two different ex-


Fig. 20. – Sketch of the experimental setup for the observation of Anderson localization in anoninteracting 39BEC [3]. A 1D bichromatic lattice is realized by superimposing two standingwaves with incommensurate periodicities. An additional slightly focused laser beam provides thetransverse confinement, realizing an optical waveguide along which the BEC (initially confinedto a few lattice sites) can expand. Interactions between the atoms are cancelled by using a pairof Helmoltz coils producing a magnetic field tuned in proximity of a Feshbach resonance.

periments, one carried out at LENS [3] and the other carried out in the group of A. Aspect

at Institute d’Optique [19], which were both successful in circumventing these obstacles.

In [19] these two problems were solved by using optical speckles with short correlation

length and reducing considerably the BEC density to neglect the effects of interactions.

Following a different strategy, in [3] an incommensurate bichromatic lattice was used to

create inhomogeneities at the quite small length scale of a few lattice sites, while in-

teractions were cancelled by coupling the atoms to an external magnetic field tuned in

proximity of a Feshbach resonance. For a discussion on these different, yet complemen-

tary, approaches the reader can refer to [54].

In the following, we will focus on the LENS experiment [3] where Anderson localization

in the bichromatic lattice was investigated.

5.1. Experimental setup. – The experiment described in [3] has been performed with

a Bose-Einstein condensate of 39K. Interactions between the BEC atoms are cancelled

by using a Feshbach resonance centered at a magnetic field of 403 G, which features a

remarkably broad “zero-crossing”, i.e. a region of magnetic field in which the scattering

length crosses the value a = 0 with a very small slope da/dB [55]. This small dependence

of the scattering length on the applied magnetic field allows for a precise and stable tuning

of the s-wave interactions to zero. This is the ideal situation for studying single-particle

effects as Anderson localization, since the macroscopic BEC can be used to study “in

parallel” the behavior of ≈ 105 particles all occupying the same single-particle state

Ψ(x). Imaging the atoms then provides a direct measurement of the wave function

density n(x) = |Ψ(x)|2 without need of averaging on many experimental realizations.

In the experiment, a BEC of 39K is first produced at a finite value of the scattering

length a = 180a0 obtained at a magnetic field B = 396 G [56], then interactions are


tuned to zero by changing the magnetic field to B = 350 G. The non-interacting BEC

is confined in a quasi-1D geometry by a weakly focused laser beam which creates an

“optical waveguide” where the atoms can move (see fig. 20). The bichromatic lattice,

aligned on the same direction of the optical waveguide, is realized by superimposing two

standing waves created with two different lasers operating at λ1 = 1032 nm (primary

lattice) and λ2 = 862 nm (secondary lattice). The ratio between the two wavelengths

gives an incommensurate β = k2/k1 = λ1/λ2 = 1.197 . . . (equivalent to γ = 0.197 . . .).

We indicate with Δ the half-amplitude of the quasiperiodic energy modulation: with

reference to eq. (32) the relation Δ = V2/2 holds and the localization transition in the

Aubry-Andre model is expected to occur at Δ/J = 2.

We note that, in the experimental realization of bichromatic lattices, the mathematical

definition of incommensurability (i.e. the wavelength ratio being an irrational number)

should be substituted with a more practical definition. Since the lattice wavelengths can

be measured with finite precision, the ratio λ2/λ1 is always known as a rational number.

From a theoretical point of view, it is important to consider that the finite size of the

systems under investigation releases the constraints on the incommensurability: even a

periodic potential (resulting from a commensurate ratio) does not show any periodicity

if the system size is smaller than the period. The bichromatic lattice is thus effectively

incommensurate provided that the ratio between the wavelengths is far from a ratio

between simple integer numbers. More precisely, a bichromatic lattice can be considered

incommensurate whenever the resulting periodicity (if any) is larger than the system size.

5.2. Absence of diffusion. – In his 1958 paper [18], Anderson formulated his theory of

localization by considering a tight-binding model for an electron hopping in a disordered

lattice. In particular, he moved from a relatively simple “gedanken-experiment”: he con-

sidered the initial situation in which the electron wave function is localized on a single site

of the lattice, then he derived what the asymptotic behavior of the system should be for

long evolution times. When the electron is placed in a uniform lattice, its wave function is

expected to spread across the entire lattice, with its rms width σ expanding ballistically as

σ ∝ t: asymptotically, for t → ∞, the wave function uniformly occupies the entire space,

as the eigenstates of the uniform lattice are extended Bloch waves (see sketch in fig. 21a).

Instead, when disorder is introduced (depending on disorder strength and dimensional-

ity), after a typical localization time the wave function stops expanding, reaching a sta-

tionary state in which σ is constant: the exponential decay of the wave function reflects

the underlying exponentially-localized states of the disordered potential (see fig. 21b).

In the laboratory it is possible to realize a similar situation. As sketched in fig. 20,

BEC is initially confined in a few sites of the bichromatic optical lattice by means of a

red-detuned focused laser beam (not shown in figure) which creates an additional optical

trap(12). When this confinement is removed by switching off the laser beam, the atomic

(12) The ground-state of the non-interacting BEC in this trap is a Gaussian with a typical sizeσ ≈ 5μm.


Fig. 21. – Pictorial representation of Anderson localization: the spheres represent the asymptoticprobability for an electron, initially placed in the center of the lattice, to be detected at differentsites. a) In a uniform lattice the wave function expands across the entire lattice; b) in adisordered lattice Anderson localization sets is and the wave function is exponentially localized.

cloud is free to expand along the direction of the bichromatic lattice and can be detected

after different evolution times with in-situ absorption imaging, which directly gives the

wave function density n(x) = |Ψ(x)|2.

The results are shown in fig. 22a, where snapshots of the expansion are presented for

different evolution times up to 640 ms and for different values of the perturbation strength

Δ/J ranging from 0 to 7. Three different regimes of expansion can be observed. For

Δ = 0 the condensate expands ballistically in a monochromatic optical lattice, where the

sites are all identical and the eigenstates are extended Bloch waves(13). For finite values

of Δ the expansion is initially slowed down by the presence of the quasiperiodic energy

modulation: the perturbation strength stays below the Aubry-Andre critical value for the

localization transition, hence the eigenfunctions are still extended, but the induced wave

function modulation results in a smaller “effective” tunnelling rate. Then, for Δ/J = 7,

the dynamics is completely halted and no expansion can be observed at all: this happens

when the eigenstates of the system becomes spatially localized and no transport can take

place on distances larger than the localization length.

This transition can be investigated more quantitatively by plotting the rms size of the

atomic cloud after a fixed evolution time t as a function of the perturbing lattice strength

Δ/J . The results presented in fig. 22b for t = 750 ms show that the size decreases for

increasing Δ/J up to a value Δ/J � 7 above which the system size stays constant at the

same initial value (dashed line). The different point-styles in fig. 22b refer to different

hopping amplitudes J . It is interesting to observe how the onset of localization happens

at the same value of Δ/J regardless of the different values of J for the different datasets.

(13) Actually a very weak residual harmonic confinement is always present along the latticedirection due to the axial confinment of the optical beams, but the effects of this inhomogeneitycan be observed only on much larger time and length scales.


Fig. 22. – Expansion of a non-interacting BEC in an incommensurate bichromatic optical lattice.a) In situ absorption images of the condensate. The lattice direction is horizontal and the fieldof view is 360μm along that direction. The data refer to a main lattice with hopping amplitudeJ = h× 153 Hz. b) Rms size of the BEC as a function of Δ/J for different hopping amplitudesJ . The dashed line indicates the size at the beginning of the evolution. Taken from [3].

As a matter of fact, the Hamiltonian in eq. (36) has two terms which are proportional,

respectively, to the hopping energy J and to the disorder strength Δ, therefore it is

natural that the physics should depend only on the ratio of these two energy scales(14).

The Aubry-Andre theory predicts a localization transition to occur at Δ/J = 2 (see

sect. 4.5). As already pointed out before, a sharp transition is expected only for a highly

(14) This is indeed the behavior shown in fig. 22b, the only difference between the datasets beingthe size of the cloud at small values of Δ/J , before the onset of localization. This is not strange,however, since the plot refers to a fixed evolution time t which is the same for the differentdatasets, therefore the evolution time in natural units t/J is not the same.


Fig. 23. – Exponential localization of a non-interacting BEC in an incommensurate bichromaticoptical lattice. The points show the exponent α in eq. (41) which fits the wave function tailsafter a 750 ms long expansion in the bichromatic lattice for different perturbation strength Δ/J .Taken from [3].

incommensurate value of γ, e.g. for the golden ratio γ = (3 −√

5)/2 � 0.382 . . .. Here

γ = 0.197 . . . and the transition becomes slightly smeared out to larger values of Δ/J .

At Δ/J = 7 complete localization occurs and the localization length gets down to few

lattice sites.

We can learn about the physics underlying this localization behavior by analyzing the

shape of the expanding wave function. The tails of the wave function, extracted from

fig. 22, have been fitted with a generalized exponential function

(41) Ψ(x) = A exp[−γ(x− x0)α].

In fig. 23 we show the fitted exponent α for different values of Δ/J . The data clearly show

that the exponent changes from 2 to 1 as the system scans the localization crossover.

The value α = 2 reflects the Gaussian wave function of the BEC initially trapped in a

harmonic trap, which stays Gaussian during the whole evolution. The value α = 1 in

the localized regime reflects the appearance of exponentially decreasing tails, which are

a signature of Anderson-localized states. Also here, the crossover is completed at a value

Δ/J � 7 as before.

5.3. Imaging the localized states. – The suppression of the expansion and the obser-

vation of exponentially decreasing tails in the wave function is a first proof of Anderson

localization. We can get more insight into the localization transition by observing the

eigenstates of the system. To this aim, we take advantage of a trick which is often played

in ultracold atoms experiment, i.e. observing the BEC wave function in time-of-flight af-

ter switching off all the external potentials. When the confinment is removed, the atomic

wave function undergoes a free-space evolution in the absence of any potential (except for

gravity, which only induces an uniform center-of-mass acceleration). In the asymptotic


limit of long expansion time texp → ∞, the wave function in real space Ψ(x; texp) maps

the initial wave function in momentum space Ψ0(k):

(42) Ψ(x; texp) ∝ Ψ0(mx/�texp).

We have already discussed in sect. 4.5 that the localization transition can indeed be

observed also in momentum space: while the extended states for sub-critical quasiperiodic

perturbation have a momentum distribution made by narrow δ-like peaks, the localized

states above the transition show a much broader unstructured momentum distribution.

This is a direct consequence of the Heisenberg uncertainty principle: the more the wave

function is localized in real space, the more it is delocalized in momentum space.

In the experiment, the momentum distribution P (k) = |Ψ0(k)|2 is measured by slowly

loading the BEC in the bichromatic lattice in order to populate only the ground state (or,

at most, just a few localized states) and then switching off both the lattice and the optical

trap. In fig. 24a we show the momentum distributions measured along the lattice direc-

tion for different values of Δ/J ranging from 0 (top) to 25 (bottom). It is evident how the

distribution broadens as the strength of the quasiperiodic modulation is made stronger:

from the multipeaked pattern of the monochromatic lattice (top), the momentum distri-

bution changes to a single broad curve in the localized regime (bottom). In fig. 24b we

show, for comparison, the theoretical momentum distribution calculated for the ground

state wave function, showing an excellent agreement with the experimental data.

In fig. 24c-d we study the localization crossover by plotting two quantities which

characterize the momentum distribution: the rms width of the zero-momentum peak

and the “visibility” of the pattern defined as [P (2k1) − P (k1)]/[P (2k1) + P (k1)]. While

the rms width increases with increasing Δ/J (fig. 24c), the visibility suddenly decreases

(24d). The localization transition is particularly evident in the visibility, which shows a

sharp knee around Δ/J=7.

5.4. Effect of interactions . – In the previous sections we have focused on the

localization of a non-interacting ultracold gas in a bichromatic potential. Starting

from the conceptually simple, although yet mathematically rich, Anderson scenario,

the next level of investigation naturally involves the consideration of the interacting

case. Interactions between particles are indeed a fundamental constituent of matter

and the consideration of their effect is of crucial importance to describe the physics of

condensed-matter disordered systems.

The effect of weak repulsive interactions on the localization of a bosonic gas has been

investigated in a very recent LENS experiment [57], where the interactions between the

atoms of a 39K BEC were finely controlled by means of a Feshbach resonance. Similarly to

the experiments described in sect. 5.3, detailed information on the localization properties

was extracted in [57] by analyzing the time-of-flight density distribution of the atoms after

sudden release from the trap.

Figure 25 shows two different plots, in which the width of the momentum distribu-

tion and the exponent α fitting the wave function tails are presented as a function of


Fig. 24. – Momentum-resolved study of the localization of a non-interacting BEC in a bichro-matic lattice. a) Time-of-flight measurement of the momentum distribution for different pertur-bation strength Δ/J = 0, 1.1, 7.2 and 25 (from top to bottom). b) Corresponding theoreticaldistributions. c) Rms width of the central peak of the momentum distribution (circles: exper-iment, lines: theory). d) Visibility of the peaked structure in the momentum distribution (seetext for definition, circles: experimental data, lines: theory). Taken from [3].

the quasiperiodic perturbation amplitude Δ and of the interaction strength, here de-

noted with the interaction energy Eint (see [57] for details). These colour-scale maps

can be compared with the similar plots shown in fig. 24c and fig. 23 obtained for the

non-interacting system. Figure 25 clearly shows that the localization crossover is up-

shifted towards larger values of Δ when interactions are introduced into the system.

As it could be intuitively expected, repulsive interactions fight against disorder and

contribute to delocalize the eigenstates. A simple energetic argument suggests that this

happens when the interaction energy Eint becomes larger than the average energy sep-


Fig. 25. – Effect of weak repulsive interactions on the localization of a BEC in a bichromaticlattice. a) Rms width of the momentum distribution (see fig. 24c for comparison). b) The decayexponent of the wave function defined in eq. (41) (see fig. 23). Both the quantities are extractedfrom the experimental time-of-flight images. The white dashed line indicates the region inwhich the interaction energy Eint equals the average energy separation between localized states.Adapted from [57].

aration δE between the lowest-lying single-particle localized states [57]. If the system is

delocalized by the effect of interactions, it is possible to restore localization by increas-

ing the perturbation strength Δ in order to lift the energy separation between localized

states to be larger than Eint. The region where Eint matches δE is indicated as a white

dashed line in fig. 25.

The nature of the localized phase in the presence of interactions presents some dif-

ferences with respect to the single-particle Anderson scenario, as suggested by recent

theoretical works [58-61]. For weak values of the interactions the localized state takes

the form of an Anderson glass, in which the ground state results from the occupation of

several single-particle localized states. In this regime interactions are strong enough to

push atoms away from the single-particle localized states, but not yet sufficiently strong

to connect these spatially separated states and restore superfluidity in the system. For

larger values of the interactions, phase coherence starts to be restored and a fragmented


BEC forms, in which different single-particle localized states merge together building up

local coherent fragments. The crossover between the Anderson glass and the fragmented

BEC regime is explored in [57] with additional experimental observations, including the

measurement of the first-order correlation function, which gives information on the phase

coherence.

The physics of interacting disordered systems is an extremely interesting subject of

research. Even from a theoretical point of view, the problem is difficult to address and

many questions have not yet been answered. In this section we have only introduced the

weakly interacting regime, which has been recently investigated also in different physical

systems (e.g. in the localization of light in photonic crystals with Kerr nonlinearities [62]).

The interplay between disorder and interactions gets particularly tricky in the strongly

interacting regime, when the interaction energy stops from just being a perturbation

of the atoms total energy. When this happens, many-body correlations set in and new

physics emerge: the next sections will be devoted to this regime.

6. – Strongly interacting atoms in bichromatic lattices

In the previous sections we have discussed the physics of disordered non-interacting

or weakly interacting bosonic gases. A theoretical description of the weakly interacting

regime can be given in terms of the semiclassical Gross-Pitaevskii equation (19), which

describes the propagation of nonlinear matter waves. When interactions are strong,

however, this mean-field description is not capable to fully explain the behavior of the

system. A more appropriate description is provided by a full quantum theory, taking into

account quantum correlations between the particles. The quantum state of an interacting

gas of identical bosons in a disordered lattice potential is well described by the disordered

Bose-Hubbard Hamiltonian [29]

(43) H = −J∑〈j,j′

〉

b†j bj′ +U

2

∑j

nj(nj − 1) +∑

j

εj nj ,

where the hopping energy J and the interaction energy U are defined according to the

definitions given in sect. 3.4, while εj ∈ [−Δ/2,Δ/2] is a site-dependent energy account-

ing for inhomogeneous external potentials superimposed on the lattice (see sketch in

fig. 26)(15).

In sect. 3.4 we have already discussed the superfluid-Mott transition, which takes

place in a uniform lattice when the interaction energy U exceeds the hopping energy

J , causing a localization of the atomic wave functions in single lattice sites. The phase

diagram of the system depends on the chemical potential μ (related to the atomic density)

and shows the existence of MI lobes with integer number of atoms per site. In the left

(15) In this last section of the paper we will make use of a different definition of Δ: instead ofusing it to denote the half-amplitude of the disordered distribution of site energies, it will denotethe full-amplitude (i.e. for a quasiperiodic system, Δ = V2).


Fig. 26. – Sketch of interacting bosons in a disordered lattice.

graph of fig. 27 we show a qualitative sketch of the phase diagram for a 3D system, as

first derived by M. P. A. Fisher et al. in [29].

In the presence of a disordered external potential the additional energy scale Δ enters

the description of the system and is responsible for the existence of a new quantum phase.

In the presence of weak disorder the MI lobes in the phase diagram progressively shrink

and a new Bose glass (BG) phase appears (central graph of fig. 27), eventually washing

away the MI region for Δ > U (right graph of fig. 27) [29]. In a very simplified view,

the properties of a Bose glass are half-way from a Mott insulator to a superfluid: A Bose

glass is an insulating state with no long-range phase coherence (as the Mott insulator),

although it is compressible and supports gapless excitations (as a superfluid).

The Bose glass phase has been first identified in [64] in the context of strongly in-

teracting 1D bosonic systems. In the ’90s it was widely studied in connection to the

superfluid-insulator transition observed in many condensed-matter systems, such as 4He

adsorbed on porous media [65], thin superconducting films [66], arrays of Josephson junc-

tions [67] and high-temperature superconductors [68, 69]. The possible realization of a

Fig. 27. – Qualitative zero-temperature phase diagram for a disordered system of lattice inter-acting bosons. Three phases can be identified: a superfluid (SF), a Mott insulator (MI) and aBose glass (BG). Taken from [63].


Fig. 28. – A strong 2D optical lattice along direction y and z creates a two-dimensional arrayof independent 1D Bose gases along direction x. A bichromatic optical lattice along direction x

is used to realize a quasidisordered Bose-Hubbard model [63].

Bose glass in a system of ultracold bosons in a disordered lattice has been first proposed

in [58, 70]. More recently, the phase diagram of this system has been derived in other

theoretical papers, considering also finite-temperature effects [71, 72] and the possible

realization of a Bose glass with incommensurate bichromatic lattices [73, 74].

The Bose glass is just the simplest disordered quantum phase that can be realized

in the strongly interacting regime. When atoms of different species, or different internal

(spin) states of the same species, are considered, more complicated models can be exper-

imentally realized and new disordered quantum phases can emerge. Atomic Bose/Fermi

mixtures, in particular, represent a versatile system in which many different disordered

models can be realized [75, 76], from fermionic Ising spin glasses to models of quantum

percolation.

6.1. Towards a Bose glass. – Experiments with disordered bosons in the strongly inter-

acting regime started at LENS in 2006. The system under investigation was a collection

of 87Rb 1D ultracold gases in a bichromatic optical lattice, as schematically sketched

in fig. 28. The main optical lattice was used to induce the transition from a weakly

interacting superfluid to a strongly correlated Mott insulator. The non-commensurate

perturbing lattice was then used to add controlled quasidisorder to the perfect crystalline

structure of the MI phase.

As already introduced in sect. 3.4, the excitation spectrum is an important observable

that can be measured in order to characterize the quantum state of the system. While a

SF supports the excitation of gapless long-wavelength modes, the excitation spectrum of

a MI shows the presence of an energy gap. The latter behavior can be exemplified in the


Fig. 29. – Excitations in the deep insulating phases. a) In a Mott insulator the tunneling of oneboson from a site to a neighboring one has an energy cost ΔE = U . b) In the disordered casethe excitation energy is ΔE = U ± Δj , that becomes a function of the position. In the Boseglass state, in which |Δj | > U , an infinite system could be excited at arbitrarily small energiesand the energy gap would disappear.

strong MI limit (U/J � 1), where it is easy to observe that the lowest-energy excitation—

the hopping of a particle from a site to a neighboring one, or, in other words, the creation

of a particle-hole pair—has an energy cost U , corresponding to the interaction energy of

a pair of mutually repelling atoms sitting on the same site (see fig. 29). By exploiting the

possibility of time-modulating the lattice potential, as first realized in [34], it is possible

to probe the excitation spectrum and study how it is modified by the presence of disorder.

In fig. 30a we show the excitation spectrum of a Mott insulator measured in the

LENS experiments [63]. The plot shows a well resolved resonance at energy U , which

is distinctive of the MI state, and a second resonance at energy 2U . While the physical

origin of the excitation peak at U is the tunneling of particles between sites with the same

occupancy, the second peak at 2U can be ascribed to tunneling at the boundary between

MI regions with different site occupancy (that are present due to the inhomogeneity of

the confined sample).

When increasing disorder the experimental data in fig. 30b-d show a broadening of

the resonance peaks, which eventually become undistinguishable when Δ ≈ U . As a mat-

ter of fact, the presence of disorder introduces inhomogeneous energy differences Δj =

(εj−εj−1) ∈ [−Δ,Δ] between neighboring sites (see bottom of fig. 29). As a consequence,

the tunneling of a boson through a potential barrier costs U±Δj , that becomes a function

of the position [43]. The excitation energy is not the same for all the bosons, differently

from the pure MI case, and the resonances become inhomogeneously broadened, as can be

observed in the experimental spectra at weak disorder (Δ < U) shown in fig. 30b,c [63].

This broadening is in agreement with a semi-classical model [43] and has been predicted

in theoretical works [77] studying the dynamical response of a 1D bosonic gas in a su-

perlattice potential when a periodic amplitude modulation of the lattice is applied.

Eventually, when Δ � U , one expects that an infinite system can be excited at

arbitrarily small energies and that the energy gap would shrink to zero. When this


Fig. 30. – Excitation spectra of the atomic system in a Mott insulator state for increasing heightof the disordering lattice. The resonances are lost and the excitation spectrum becomes flat.The figure is adapted from [63].

happens, nearby sites become degenerate and regions of local superfluidity with short-

range coherence appear in the system, resulting in a Bose glass state where there is no

gap but the system remains globally insulating.

From the experimental point of view, additional information on the nature of the

many-body ground state can be acquired by analyzing the density distribution of the

atoms released from the lattice after a time of flight. Long-range coherence in the sample

results in a density distribution with sharp interference peaks at a distance proportional

to the lattice wave vector, as discussed in sect. 3.4. The visibility of these peaks provides

a measurement of phase coherence. When increasing the height of the main lattice, a

progressive loss of long-range coherence has been reported in [63] indicating the transition

from a superfluid to an insulating state, also in the presence of disorder. The combination

of the excitation spectra measurements and the time of flight images indicates that, with

increasing disorder, the system realized in [63] goes from a MI to a state with vanishing-

long-range coherence and a flat density of excitations. The concurrence of these two

properties cannot be found in either a SF or an ordered MI, and strongly suggests the

formation of a Bose glass, which is indeed expected to appear for Δ � U .

From the theoretical side, numerical studies [74,78] have considered the problem of 1D

interacting bosons in quasiperiodic lattices, working out the phase diagrams (which in-

clude the presence of Bose glass and incommensurate “band insulating”/“charge density

wave” regions) and studying how the different phases affect experimentally detectable

signals. From the experimental side, for an exhaustive characterization of this novel dis-

ordered state, new detection schemes have to be implemented in order to have access to

additional observables. This necessity is not only restricted to the study of disordered

systems, being a more general issue shared by the experimental investigation of different

strongly interacting lattice systems, including, e.g., systems with magnetic ordering or

mixtures of different species.


Before discussing in sect. 6.2 and 6

.3 the application of novel detection techniques to

the study of disordered systems, we note that recently the experimental investigation of

disordered ultracold bosons in the strongly interacting regime has been performed also in

the group of B. DeMarco at University of Illinois. In [79] the authors used a speckle po-

tential with short correlation length to realize a 3D disordered Bose-Hubbard model and

investigate the reduction of condensate fraction as a function of the Hamiltonian param-

eters in the crossover from the superfluid to the insulating state. Transport properties in

the same system were investigated in [80] by exciting center-of-mass oscillations of the

trapped gas: the authors observed that the addition of disorder drives the system into

an insulating state where strong dissipation of the atomic motion sets in, in accordance

with the expected behavior for an insulating Bose-glass phase.

6.2. Noise correlations. – In a recent experiment performed at LENS [81] noise in-

terferometry has been used to study interacting 87Rb bosons in the bichromatic lattice.

This detection technique, originally proposed in [82], is based on the analysis of the

spatial density-density correlations of the atomic shot noise after time-of-flight. These

correlations are based on the Hanbury Brown & Twiss effect [83]: if two identical par-

ticles are released from two lattice sites, the joint probability of detecting them in two

separate positions (e.g., imaging them on two separate pixels of a CCD camera) depends

on the distance between the detection points. These correlations, arising from quan-

tum interference between different detection paths, were first observed for bosons in a

Mott insulator state [84] and then also for band-insulating fermions [85]. The sign of the

correlations depends on the quantum statistics: while bosons show positive correlations

(due to their tendency to bunch, i.e. to arrive together at the detectors), fermions ex-

hibit negative correlations (due to the antibunching, consequence of the Pauli exclusion

principle). In the case of a bosonic Mott insulator, one observes positive density-density

correlation peaks at a distance proportional to the lattice wave vector k1, as shown in

the first image of the bottom row of fig. 31 for the recent experiment at LENS.

In [81] noise correlations have been measured, starting from a Mott insulator state, for

increasing heights s2 of the secondary lattice. The absorption images after time of flight

do not present significative differences, as shown in the top row of fig. 31, and demon-

strate the absence of first-order (phase) coherence of the atomic system in the insulating

state, even in the presence of the secondary lattice. However, second-order (density)

correlations turn out to be significantly different with varying s2, as illustrated in the

noise correlation functions plotted in the bottom row. More precisely, with increasing s2,

one observes the appearance of additional correlation peaks at a distance proportional

to the wave vector k2 of the secondary lattice and to the beating between the two lat-

tices k1 − k2. These peaks have to be associated with the redistribution of atoms in the

lattice sites as the disordering lattice is strengthened: the MI regions characterized by

uniform filling are destroyed and atoms rearrange in the lattice giving rise to a state with

non-uniform site occupation, which follows the periodicity of the secondary lattice. The

redistribution of atoms is then quantitatively detected by measuring the height of the

additional correlation peaks.


Fig. 31. – Top: time-of-flight absorption images of atoms initially in a Mott insulator statefor increasing height of the secondary lattice s2. Bottom: density-density correlation functionscorresponding to the pictures above. The point in the center corresponds to the correlationfunction at zero distance (the black ring is an artifact of image processing). The peak markedwith k1 corresponds to density correlations on the length scale of a lattice site, while the addi-tional correlation peaks marked as k2 and k1 − k2 arise for large s2 from the destruction of theMott domains and the redistribution of the atoms in the lattice. The data is adapted from [81].

Noise correlations thus prove to be a tool to extract important information on the

lattice site occupation, which is connected to the second-order correlation function of the

many-body state. The appearance of similar correlation peaks was predicted in theo-

retical works for hard-core bosons [86] and soft-core bosons [74] in bichromatic lattices.

Future works will study the possibility to use noise interferometry to get additional in-

sight on the nature of the disordered insulating states produced in the experiment, in

particular in connection with the realization of a Bose glass phase.

6.3. Bragg spectroscopy . – The lattice modulation technique discussed in sect. 6

.1

for the measurement of the excitation spectrum of ultracold lattice gases is affected by

important limitations. In order to detect an excitation signal in the Mott insulator regime

the lattice has to be modulated by a large amount and the atomic response is out of the

linear regime. We conclude this paper discussing a different experimental method that

can be used as a cleaner probe of excitations in a strongly interacting gas.

In condensed-matter physics, scattering of particles or radiation is the most common

way to extract information on the properties of a material. Inelastic scattering is particu-

larly important to study dynamic properties. If a monochromatic beam of particles with

momentum k and energy ω hits the sample and particles are scattered with momentum

k′ and energy ω′, momentum and energy conservation require that the sample has gained

momentum δk = k−k′ and energy δω = ω−ω′. In principle, if one is able to detect the

intensities of all the scattering channels {k′, ω′}, this could provide a measurement of the


dynamic structure factor S(δk, δω), which completely characterizes the excitations of the

system. This is the quantity measured, for instance, in inelastic scattering of decelerated

neutrons, which is a valuable technique for the investigation of magnetic and structural

excitations in condensed-matter systems.

In the case of light, inelastic scattering processes are commonly known as Raman

scattering. In a Stokes (anti-Stokes) process, the frequency of the photons scattered by

the sample is lower (higher) than the frequency of the incident photons, owing to the

transition to higher- (lower-) energy configurations. Raman spectroscopy is therefore a

precious tool for investigating the internal structure and excitations of complex molecules

or aggregates. The use of light scattering as a diagnostic tool for ultracold quantum gases,

generally referred to as Bragg scattering, has been initiated ten years ago with seminal

experiments at NIST [87] and MIT [88]. In later years, it has been used to study the

excitation spectrum of weakly interacting Bose-Einstein condensates (BECs) [89] and,

very recently, experiments have started to investigate the regime of strong interactions

between the particles, as it happens in BECs close to a Feshbach resonance [90] or in

Fermi gases across the BEC/BCS crossover [91].

In a Bragg scattering experiment the atoms are illuminated by two Bragg beams with

wave vectors and frequencies (k, ω) and (k′, ω′). These two beams induce a stimulated

inelastic Raman scattering process, in which the energy and momentum transfer can

be selected by changing the relative detuning between the two beams and their relative

angle θ, according to

δω = ω − ω′,(44)

δk � 2k sin θ/2.(45)

Instead of detecting the scattered photons, it is possible to measure the amount of energy

transferred to the system after thermalization of the excitations, which in cold-atoms

systems can be easily obtained by time-of-flight imaging techniques (for more details, see

refs. [92-94]).

We now consider the application of Bragg spectroscopy to systems of strongly inter-

acting 1D bosonic atoms in the crossover from a superfluid to a Mott insulator state [32].

A monochromatic optical lattice is used in a setup similar to the one presented in fig. 28

to control the state of the individual 1D gases. In this 1D configuration the system

exhibits a crossover between a correlated superfluid and a Mott-insulating state around

a critical value U/2J � 4.5 [95]. In fig. 32 we show the evolution of the Bragg spectra

for different lattice heights ranging from s = 0 (U/2J = 0) to s = 15 (U/2J � 50) and a

momentum transfer δk = 0.96(3)kL along the lattice direction. It is evident that, as the

height of the lattice is increased and the gas is driven into the Mott insulating regime,

the overall response to the Bragg excitation is strongly suppressed and a fine structure

with multiple peaks appears. The transition point can be estimated by measuring the

rms width of these spectra, which is shown in fig. 33a. This quantity clearly exhibits a

minimum, in good agreement with the theoretical transition point [95], which is shown

as a vertical bar around s = 5. This minimum results from a combined effect: while the


Fig. 32. – Bragg spectroscopy of strongly interacting 1D Bose gases across the SF to MI tran-sition, taken for a momentum transfer δk = 0.96kL and different values of lattice height s andU/2J . Lines are guides to the eye. Note the amplitude drop and the appearance of a multipeakedstructure when increasing the lattice height (vertical scale). Taken from [93].

response in the superfluid regime is shifted to lower energies as the lattice is increased

(because the energy band gets narrower), new features at higher energies (which are not

resolved in the crossover region) emerge in the Mott insulating regime and lead to an

increase of the width.

These features are well-resolved and clearly observable in the spectrum in fig. 33b,

corresponding to U/2J = 30. Three main components are observable in this spectrum.

There is a main peak centered around frequency U/h (vertical dotted line), correspond-

ing to the creation of particle-hole excitations in the Mott insulating domains. These

excitations have an energy gap (related to the uncompressible nature of the Mott state)

which approaches U at large U/J . A smaller peak is also clearly observable at twice

the frequency (close to 2U/h), which can be related to the presence of inhomogeneities

and “defects” in the system, either due to the shell structure induced by the trapping

potential or to an imperfect adiabatic loading of the lattice. Finally, a low-frequency

component is present at energies below the Mott insulator gap. This component can be

associated to excitations within the superfluid (or normal) domains of the system, which

are not gapped and have a bandwidth 4nJ (where n is the filling factor), represented as

the vertical stripe in fig. 33b. Excitations above the upper edge of this region and below

the Mott insulator gap could be related to the finite temperature of the gas and to the

strong correlations in the superfluid component(16).

(16) A complete theoretical picture including the combined effects of finite U/J , finite tempera-ture and trapping potential is not yet available.


Fig. 33. – Bragg spectroscopy of strongly interacting 1D Bose gases across the SF to MI tran-sition. a) Rms width of the Bragg spectra for different values of lattice height: the verticalbar corresponds to the theoretical value for the Mott transition point. b) Bragg spectrum fors = 13 with the indication of the interaction energy U and of the bandwidth of the superfluidexcitation. Adapted from [93].

An important difference between Bragg spectroscopy and the lattice modulation tech-

nique described in sect. 6.1 is that the former can be performed at non-zero momentum

transfer. As a matter of fact, the response of the Mott state is strongly dependent on

the momentum transfer, vanishing at δk = 0 and being maximum at δk = kL [96, 97].

While signals from a Mott insulator could be observed with the lattice modulation tech-

nique only for large modulation amplitudes of ≈ 20%–30% out of the linear regime of

excitation, in the case of the spectra shown in fig. 33b the amplitude of the travelling

lattice formed by the Bragg beams is only < 5% of the lattice height. This means that

the parameters U and J of the system are not significantly changed during the excitation

and that the measured spectra can be directly compared to the dynamic structure factor

describing the excitations in the perturbative regime where linear-response theory holds.

These advantages, here demonstrated for strongly interacting bosons at the superfluid-

Mott crossover, could make Bragg spectroscopy an important tool to better characterize

the properties of Bose-glass and different disordered insulating states as well.

7. – Concluding remarks

Besides constituting the focus of contemporary atomic physics, ultracold atoms are

important resources for different areas of physical research, since they can be used as

“quantum simulators” to implement and investigate well-controlled Hamiltonian models

originally introduced to describe different physical systems. One spectacular example of

this possibility is given by ultracold atoms in optical lattices, i.e. perfect crystals made

of laser light, in which atoms behave as the conduction electrons in a solid and allow the

investigation of transport phenomena typical of solid-state physics.


In this paper we have described this perspective by illustrating the physics of atoms

trapped in bichromatic optical lattices. We have mostly considered the case of incom-

mensurate bichromatic potentials, which are an intriguing example of structures which

are ordered and disordered at the same time. In particular, we have focused on their

“disordered” nature, which comes from the absence of any periodicity and is responsible

for the emergence of localized states. Thanks to these properties, incommensurate lat-

tices have been used as a powerful tool to study the emergence of Anderson localization,

which we have demonstrated in recent experiments.

This quantum simulation perspective is even more attractive when one considers

interactions between the particles. The study of interacting atomic systems is indeed

particularly interesting, especially in the regime of strong interactions where highly-

correlated atomic states can be created. The interplay between disorder and interactions

has a fundamental role in determining the state of the system, with strong implications in

contemporary condensed-matter research. First investigations of this regime have been

started and much more is likely to come in the near future, thanks to new tools and

detection techniques which are currently under development.

∗ ∗ ∗

This work was financially supported by ERC DISQUA Project, EU FP7 Programs,

ENTE CRF. We acknowledge the precious effort of all the LENS colleagues contributing

to the research works presented in this paper.

REFERENCES

[1] Morsch O. and Oberthaler M., Rev. Mod. Phys., 78 (2006) 179.[2] Bloch I., Dalibard J. and Zwerger W., Rev. Mod. Phys., 80 (2008) 885.[3] Roati G., D’Errico C., Fallani L., Fattori M., Fort C., Zaccanti M., Modugno

G., Modugno M. and Inguscio M., Nature, 453 (2008) 895.[4] Chu S., Rev. Mod. Phys., 70 (1998) 685.[5] Cohen-Tannoudji C. N., Rev. Mod. Phys., 70 (1998) 707.[6] Phillips W. D., Rev. Mod. Phys., 70 (1998) 721.[7] Cornell E. A. and Wieman C. E., Rev. Mod. Phys., 74 (2002) 875.[8] Ketterle W., Rev. Mod. Phys., 74 (2002) 1131.[9] Dalfovo F., Giorgini S., Pitaevskii L. and Stringari S., Rev. Mod. Phys., 71 (1999)

463.[10] Inguscio M., Stringari S. and Wieman C. E. (Editors), Bose-Einstein Condensation in

Atomic Gases, Proceedings of the International School of Physics, “Enrico Fermi”, CourseCXL (SIF, Bologna; IOS, Amsterdam) 1999.

[11] Cohen-Tannoudji C., in Fundamental systems in quantum optics, edited by Dalibard

J., Raimond J.-M. and Zinn-Justin J. (Elsevier, Amsterdam) 1992, pp. 1-164.[12] Grimm R., Weidemuller M. and Ovchinnikov Y. B., Adv. At. Mol. Opt. Phys., 42

(2000) 95.[13] Allen L. and Eberly H. J., Optical Resonance and Two Level Atoms (Dover

Publications) 1987.[14] Ben Dahan M., Peik E., Reichel J., Castin Y. and Salomon C., Phys. Rev. Lett.,

76 (1996) 4508.


[15] Roati G., de Mirandes E., Ferlaino F., Ott H., Modugno G. and Inguscio M.,Phys. Rev. Lett., 92 (2004) 230402.

[16] Gustavsson M., Haller E., Mark M. J., Danzl J. G., Rojas-Kopeinig G. andNagerl H. C., Phys. Rev. Lett., 100 (2008) 080404.

[17] Fattori M., D’Errico C., Roati G., Zaccanti M., Jona-Lasinio M., Modugno M.,

Inguscio M. and Modugno G., Phys. Rev. Lett., 100 (2008) 080405.[18] Anderson P. W., Phys. Rev., 109 (1958) 1492.[19] Billy J., Josse V., Zuo Z., Bernard A., Hambrecht B., Lugan P., Clement D.,

Sanchez-Palencia L., Bouyer P. and Aspect A., Nature, 453 (2008) 891.[20] Bloch F., Z. Phys., 52 (1928) 555.[21] Ashcroft N. W. and Mermin N. D., Solid State Physics (Saunders College Publishing)

1976.[22] Pedri P., Pitaevskii L., Stringari S., Fort C., Burger S., Cataliotti F. S.,

Maddaloni P., Minardi F. and Inguscio M., Phys. Rev. Lett., 87 (2001) 220401.[23] Burger S., Bongs K., Dettmer S., Ertmer W., Sengstock K., Sanpera A.,

Shlyapnikov G. V. and Lewenstein M., Phys. Rev. Lett., 83 (1999) 5198.[24] Deng L., Hagley E. W., Wen J., Trippenbach M., Band Y. B., Julienne P. S.,

Simsarian J. E., Helmerson K., Rolston S. L. and Phillips W. D., Nature, 398

(1999) 218.[25] Wu B. and Niu Q., Phys. Rev. A, 64 (2001) 061603R.[26] Fallani L., De Sarlo L., Lye J. E., Modugno M., Saers R., Fort C. and Inguscio

M., Phys. Rev. Lett., 93 (2004) 140406.[27] Inouye S., Andrews M. R., Stenger J., Miesner H.-J., Stamper-Kurn D. M. and

Ketterle W., Nature, 392 (1998) 151.[28] Jaksch D., Bruder C., Cirac J. I., Gardiner C. W. and Zoller P., Phys. Rev. Lett.,

81 (1998) 3108.[29] Fisher M. P. A., Weichman P. B., Grinstein G. and Fisher D. S., Phys. Rev. B, 40

(1989) 546.[30] Sachdev S., Quantum Phase Transitions (Cambridge University Press) 2000.[31] Ising E., Z. Phys., 31 (1925) 253.[32] Greiner M., Mandel O., Hansch T. W. and Bloch I., Nature, 419 (2002) 51.[33] Zwerger W., J. Opt. B, 5 (2003) S9.[34] Stoferle T., Moritz H., Schori C., Kohl M. and Esslinger T., Phys. Rev. Lett., 92

(2004) 130403.[35] Clement D., Fabbri N., Fallani L., Fort C. and Inguscio M., Phys. Rev. Lett., 102

(2009) 155301.[36] Levine D. and Steinhardt P. J., Phys. Rev. Lett., 53 (1984) 2477.[37] Steinhardt P. J. and Ostlund S. (Editors), The Physics of Quasicrystals (World

Scientific, Singapore) 1987.[38] Penrose R., Bull. Inst. Maths. Appl., 10 (1974) 266.[39] Schechtman D., Blech I., Gratias D. and Cahn J. W., Phys. Rev. Lett., 53 (1984)

1951.[40] Bindi L., Steinhardt P. J., Yao N. and Lu P. J., Science, 324 (2009) 1306.[41] Guidoni L., Triche C., Verkerk P. and Grynberg G., Phys. Rev. Lett., 79 (1997)

3363.[42] Sanchez-Palencia L. and Santos L., Phys. Rev. A, 72 (2005) 053607.[43] Guarrera V., Fallani L., Lye J. E., Fort C. and Inguscio M., New J. Phys., 9

(2007) 107.[44] Harper P. G., Proc. R. Soc. London A, 68 (1955) 874.[45] Hofstader D. R., Phys. Rev. B, 14 (1976) 2239.


[46] Aubry S. and Andre G., Ann. Israel Phys. Soc., 3 (1980) 133.[47] Modugno M., New J. Phys., 11 (2009) 033023.[48] Drese K. and Holthaus M., Phys. Rev. Lett., 78 (1997) 2932.[49] Diener R. B., Georgakis G. A., Zhong J., Raizen M. and Niu Q., Phys. Rev. A, 64

(2001) 033416.[50] Abrahams E., Anderson P. W., Licciardello D. C. and Ramakrishnan T. V., Phys.

Rev. Lett., 42 (1979) 673.[51] Townes C. H. and Schawlow A. L., Microwave Spectroscopy (Dover, New York) 1975.[52] Lye J. E., Fallani L., Modugno M., Wiersma D., Fort C. and Inguscio M., Phys.

Rev. Lett., 92 (2005) 070401.[53] Fallani L., Fort C. and Inguscio M., Adv. At. Mol. Opt. Phys., 56 (2008) 119.[54] Aspect A. and Inguscio M., Phys. Today, 62 (8) (2009) 30.[55] D’Errico C., Zaccanti M., Fattori M., Roati G., Inguscio M., Modugno G. and

Simoni A., New J. Phys., 9 (2007) 223.[56] Roati G., Zaccanti M., D’Errico C., Catani J., Modugno M., Simoni A., Inguscio

M. and Modugno G., Phys. Rev. Lett., 99 (2007) 010403.[57] Deissler B., Zaccanti M., Roati G., D’Errico C., Fattori M., Modugno M.,

Modugno G. and Inguscio M., Nat. Phys., 6 (2010) 354.[58] Damski B., Zakrzewski J., Santos L., Zoller P. and Lewenstein M., Phys. Rev.

Lett., 91 (2003) 080403.[59] Schulte T., Drenkelforth S., Kruse J., Ertmer W., Arlt J., Sacha K.,

Zakrzewski J. and Lewenstein M., Phys. Rev. Lett., 95 (2005) 170411.[60] Lye J. E., Fallani L., Fort C., Guarrera V., Modugno M., Wiersma D. S. and

Inguscio M., Phys. Rev. A, 75 (2007) 061603.[61] Lugan P., Clement D., Bouyer P., Aspect A., Lewenstein M. and Sanchez-

Palencia L., Phys. Rev. Lett., 98 (2007) 170403.[62] Lahini Y., Avidan A., Pozzi F., Sorel M., Morandotti R., Demetrios N. C. and

Silberberg Y., Phys. Rev. Lett., 100 (2008) 013906.[63] Fallani L., Lye J. E., Guarrera V., Fort C. and Inguscio M., Phys. Rev. Lett., 98

(2007) 130404.[64] Giamarchi T. and Schulz H. J., Phys. Rev. B, 37 (1988) 325.[65] Crowell P. A., Van Keuls F. W. and Reppy J. D., Phys. Rev. Lett., 75 (1995) 1106.[66] Goldman A. M. and Markovic N., Phys. Today, 51 (11) (1998) 39.[67] van der Zant H. S. J., Fritschy F. C., Elion W. J., Geerligs L. J. and Mooij

J. E., Phys. Rev. Lett., 69 (1992) 2971.[68] Jiang W., Yeh N.-C., Reed D. S., Kriplani U., Beam D. A., Konczykowski M.,

Tombrello T. A. and Holtzberg F., Phys. Rev. Lett., 72 (1994) 550.[69] Budhani R. C., Holstein W. L. and Suenaga M.,, Phys. Rev. Lett., 72 (1994) 566.[70] Roth R. and Burnett K., Phys. Rev. A, 68 (2003) 023604.[71] Krutitsky K. V., Pelster A. and Graham R., New J. Phys., 8 (2006) 187.[72] Buonsante P., Penna V., Vezzani A. and Blakie P. B., Phys. Rev. A, 76 (2007)

011602(R).[73] Bar-Gill N., Pugatch R., Rowen E., Katz N. and Davidson N., preprint arXiv:cond-

mat/0603513.[74] Roscilde T., Phys. Rev. A, 77 (2008) 063605.[75] Sanpera A., Kantian A., Sanchez-Palencia L., Zakrzewski J. and Lewenstein M.,

Phys. Rev. Lett., 93 (2004) 040401.[76] Ahufinger V., Sanchez-Palencia L., Kantian A., Sanpera A. and Lewenstein M.,

Phys. Rev. A, 72 (2005) 063616.[77] Hild M., Schmitt F. and Roth R., J. Phys. B: At. Mol. Opt. Phys., 39 (2006) 4547.


[78] Roux G., Barthel T., McCulloch I. P., Kollath C., Schollwoeck U. andGiamarchi T., Phys. Rev. A, 78 (2008) 023628.

[79] White M., Pasienski M., McKay D., Zhou S. Q., Ceperley D. and DeMarco B.,Phys. Rev. Lett., 102 (2009) 055301.

[80] Pasienski M., McKay D., White M. and DeMarco B., Nat. Phys., 6 (2010) 677.[81] Guarrera V., Fabbri N., Fallani L., Fort C., van Der Stam K. M. R. and Inguscio

M., Phys. Rev. Lett., 100 (2008) 250403.[82] Altman E., Demler E. and Lukin M. D., Phys. Rev. A, 70 (2004) 013603.[83] Hanbury Brown R. and Twiss R. Q., Nature, 177 (1956) 27.[84] Folling S., Gerbier F., Widera A., Mandel O., Gericke T. and Bloch I., Nature,

434 (2005) 481.[85] Rom T., Best Th., van Oosten D., Schneider U., Folling S., Paredes B. and

Bloch I., Nature, 444 (2006) 733.[86] Rey A. M., Satija I. I. and Clark C. W., Phys. Rev. A, 73 (2006) 063610.[87] Kozuma M., Deng L., Hagley E. W., Wen J., Lutwak R., Helmerson K., Rolston

S. L. and Phillips W. D., Phys. Rev. Lett., 82 (1999) 871.[88] Stenger J., Inouye S., Chikkatur A. P., Stamper-Kurn D. M., Pritchard D. E.

and Ketterle W., Phys. Rev. Lett., 82 (1999) 4569.[89] Steinhauer J., Ozeri R., Katz N. and Davidson N., Phys. Rev. Lett., 88 (2002) 120407.[90] Papp S. B., Pino J. M., Wild R. J., Ronen S., Wieman C. E., Jin D. S. and Cornell

E. A., Phys. Rev. Lett., 101 (2008) 135301.[91] Veeravalli G., Kuhnle E., Dyke P. and Vale C. J., Phys. Rev. Lett., 101 (2008)

250403.[92] Fabbri N., Clement D., Fallani L., Fort C., Modugno M., van der Stam K. M. R.

and Inguscio M., Phys. Rev. A, 79 (2009) 043623.[93] Clement D., Fabbri N., Fallani L., Fort C. and Inguscio M., Phys. Rev. Lett., 102

(2009) 155301.[94] Clement D., Fabbri N., Fallani L., Fort C. and Inguscio M., New J. Phys., 11

(2009) 103030.[95] Rigol M., Batrouni G. G., Rousseau V. G. and Scalettar R. T., Phys. Rev. A, 79

(2009) 053605.[96] van Oosten D., Dickerscheid D. B. M., Farid B., van der Straten P. and Stoof

H. T. C., Phys. Rev. A, 71 (2005) 021601.[97] Rey A. M., Blair Blakie P., Pupillo G., Williams C. J. and Clark C. W., Phys.

Rev. A, 72 (2005) 023407.



Exploring strongly correlated ultracold bosonic

and fermionic quantum gases in optical lattices(∗)

I. Bloch

Fakultat fur Physik, Ludwig-Maximilians-Universitat

Schellingstr. 4/II, 80798 Munchen, Germany

Max-Planck-Institut fur Quantenoptik - Hans-Kopfermann Str. 1

85748 Garching b. Munchen, Germany

Summary. — We review several experimental aspects of ultracold bosonic andfermionic quantum gases in optical lattices. After introducing optical lattices, weuse the superfluid-Mott insulator transition of ultracold bosonic quantum gases,to highlight the physics of strongly correlated quantum systems. We discuss thecoherence properties and recent measurements of the shell structure in the Mottinsulating phase. Subsequently, dynamical interaction effects in the collapse andrevival of the matter wave field of a BEC are analyzed, followed by a discussion oninteracting fermions, superexchange-mediated spin-spin interactions and quantumnoise correlation interferometry in optical lattices.

(∗) This paper follows the presentation given by I. Bloch, Strongly correlated quantum phases of

ultracold atoms in optical lattices, in Proceedings of the International School of Physics “Enrico

Fermi”, Course CLXIV, “Ultracold Fermi Gases”, edited by M. Inguscio, W. Ketterle andC. Salomon (IOS Press, Amsterdam and Societa Italiana di Fisica, Bologna) 2007, pp. 715-749.


234 I. Bloch

1. – Introduction

Ultracold quantum gases in optical lattices form almost ideal conditions to analyze the

physics of strongly correlated quantum phases in periodic potential [1-3]. Such strongly

correlated quantum phases are of fundamental interest in condensed matter physics,

as they lie at the heart of topical quantum materials, such as high-Tc superconductors

and quantum magnets, which pose a challenge to our basic understanding of interacting

many-body systems. Quite generally, such strongly interacting quantum phases arise,

when the interaction energy between two particles dominates over the kinetic energy of

the two particles. Such a regime can either be achieved by increasing the interaction

strength between the atoms via Feshbach resonances [4], or by decreasing the kinetic

energy, such that eventually the interaction energy is the largest energy scale in the

system. The latter can for example simply be achieved by increasing the optical lattice

depth.

This paper tries to give an introduction into the field of optical lattices and the physics

of strongly interacting quantum phases. A prominent example hereof is the superfluid-to-

Mott-insulator transition [5-9], which transforms a weakly interacting quantum gas into

a strongly correlated many-body system. Dominating interactions between the particles

are in fact crucial for the Mott insulator transition.

The paper is structured as follows: in sect. 2 the basics of trapping neutral atoms

in optical lattice potentials are introduced; in sect. 3 we turn to a discussion of the

superfluid-to-Mott-insulator transition; sect. 4 analyzes the influence of the interactions

between the particles on the stability of the coherent matter wave field of a BEC, leading

to a pronounced series of collapses and revivals of its macroscopic wave function; in sect. 5

the physics of repulsively interacting fermionic atoms in optical lattices is discussed;

sect. 6 explains how spin-spin interactions arise due to superexchange processes on a

lattice. In the same section it is shown how these superexchange interactions can be

detected and controlled using ultracold atoms. Finally, sect. 7 explains how quantum

noise correlation interferometry can be used to reveal quantum phases on a lattice using

Hanbury-Brown & Twiss type noise correlation measurements.

2. – Optical lattices

2.1. Optical dipole force. – In the interaction of atoms with coherent light fields, two

fundamental forces arise. The so-called Doppler force is dissipative in nature and can be

used to efficiently laser cool a gas of atoms and relies on the radiation pressure together

with spontaneous emission. The so-called dipole force on the other hand creates a purely

conservative potential in which the atoms can move. No cooling can be realized with

this dipole force, however if the atoms are cold enough initially, they may be trapped in

such a purely optical potential [10].

How does this dipole force arise? We may grasp the essential points through a simple

classical model, in which we view the electron as harmonically bound to the nucleus with

oscillation frequency ω0. An external oscillating electric field of a laser E with frequency

Exploring strongly correlated ultracold bosonic and fermionic etc. 235

ωL can now induce an oscillation of the electron resulting in an oscillating dipole moment

d of the atom. Such an oscillating dipole moment will be in phase with the driving

oscillating electric field, for frequencies much lower than an atomic resonance frequency

and 180◦ out of phase, for frequencies much larger than the atomic resonance frequency.

The induced dipole moment again interacts with the external oscillating electric field,

resulting in a dipole potential Vdip experienced by the atom [10]

(1) Vdip = −1

2〈dE〉,

where 〈·〉 denotes a time average over fast oscillating terms at optical frequencies. From

eq. (1) it becomes immediately clear that for a red detuning (ωL < ω0), where d is in

phase with E, the potential is attractive, whereas for a blue detuning (ωL > ω0), where d

is in 180◦ out of phase with E, the potential is repulsive. By relating the dipole moment

to the polarizability α(ωL) of an atom and expressing the electric field amplitude E0 via

the intensity of the laser field I, one obtains for the dipole potential

(2) Vdip(r) = −1

2ε0cRe(α)I(r).

A spatially dependent intensity profile I(r) can therefore create a trapping potential

for neutral atoms.

For a two-level atom a more useful form of the dipole potential may be derived within

the rotating wave approximation, which is a reasonable approximation provided that the

detuning Δ = ωL − ω0 of the laser field ωL from an atomic transition frequency ω0 is

small compared to the transitions frequency itself Δ � ω0. Here one obtains [10]

(3) Vdip(r) =3πc2

2ω30

Γ

ΔI(r),

with Γ being the decay rate of the excited state. Here a red detuned laser beam (ωL < ω0)

leads to an attractive dipole potential and a blue detuned laser beam (ωL > ω0) leads

to a repulsive dipole potential. By simply focussing a Gaussian laser beam, this can be

used to attract or repel atoms from an intensity maximum in space (see fig. 1).

For such a focussed Gaussian laser beam the intensity profile I(r, z) is given by

(4) I(r, z) =2P

πw2(z)e−2r2/w2

(z),

where w(z) = w0(1 + z2/z2

R) is the 1/e2 radius depending on the z-coordinate, zR =

πw2/λ is the Rayleigh length and P is the total power of the laser beam. Around the

intensity maximum a potential depth minimum occurs for a red detuned laser beam,

leading to an approximately harmonic potential of the form

(5) Vdip(r, z) ≈ −V0

{1 − 2

(r

w0

)2

−

(z

zR

)2}.

236 I. Bloch

Fig. 1. – (a) Gaussian laser beam together with the corresponding trapping potential for a reddetuned laser beam. (b) A red detuned laser beams leads to an attractive dipole potential,whereas a blue detuned laser beam leads to a repulsive potential (c).

This harmonic confinement is characterized by radial ωr and axial ωz trapping fre-

quencies ωr = (4V0/mw20)1/2 and ωz = (2V0/mz2

R).

Great care has to be taken to minimize spontaneous scattering events, as they lead to

heating and decoherence of the trapped ultracold atom samples. For a two-level atom,

the scattering rate Γsc(r) can be estimated [10] through

(6) Γsc(r) =3πc2

2�ω30

(Γ

Δ

)2

I(r).

From eqs. (3), (6) it can be seen that the ratio of scattering rate to optical potential

depth can always be minimized by increasing the detuning of the laser field. In practice,

however, such an approach is limited by the maximum available laser power. For exper-

iments with ultracold quantum gases of alkali atoms, the detuning is typically chosen to

be large compared to the excited-state hyperfine structure splitting and in most cases

even large compared to the fine structure splitting in order to sufficiently suppress spon-

taneous scattering events. Typical detunings range from several tens of nm to optical

trapping in CO2 laser fields. A laser trap formed by a CO2 laser fields can be consid-

ered as a quasi-electrostatic trap, where the detuning is much larger than the optical

resonance frequency of an atom.

One final comment should be made about state-dependent optical potentials. For

a typical multi-level alkali atom, the dipole potential will both depend on the internal

magnetic substate mF of a hyperfine ground state with angular momentum F , and on

the polarization of the light field P = +1,−1, 0 (circular σ± and linear polarization).

One can then express the lattice potential depth through [11,10]

(7) Vdip(r) =πc2Γ

2ω30

(2 + PgFmF

Δ2,F+

1 − PgFmF

Δ1,F

)I(r).


Fig. 2. – One-dimensional optical lattice potential. By interfering two counterpropagating Gaus-sian laser beams, a periodic intensity profile is created due to the interference of the two laserfields.

Here gF is the Lande factor and Δ2,F , Δ1,F refer to the detuning relative to the

transition between the ground state with hyperfine angular momentum F and the center

of the excited-state hyperfine manifold on the D2 and D1 transition, respectively. For

large detunings relative to the fine structure splitting ΔFS, the optical potentials become

almost spin independent again. For detunings of the laser frequency between the fine

structure splitting, special spin-dependent optical potentials can be created [11-13].

2.2. Optical lattice potentials. – A periodic potential can simply be formed by over-

lapping two counterpropagating laser beams (see fig. 2). Due to the interference between

the two laser beams an optical standing wave with period λ/2 is formed, in which the

atoms can be trapped. By interfering more laser beams, one can obtain one-, two- and

three-dimensional periodic potentials [14], which in their simplest form will be discussed

below. Note that by choosing to let two laser beams interfere under an angle less than

180◦, one can also realize periodic potentials with a larger period.

2.2.1. 1D lattice potentials. The simplest possible periodic optical potentials is formed

by overlapping two counterpropagating focussed Gaussian laser beams, which results in

a trapping potential of the form

(8) V (r, z) = −Vlat · e−2r2/w2

(z) · sin2(kx) ≈ −Vlat ·

(1 − 2

r2

w2(z)

)· sin2(kz),

where w0 denotes the beam waist, k = 2π/λ is the wave vector of the laser light and Vlat

is the maximum depth of the lattice potential. Note that due to the interference of the

two laser beams Vlat is four times larger than V0 if the laser power and beam parameters

of the two interfering lasers are equal.

2.2.2. 2D lattice potentials. Periodic potentials in two dimensions can be formed by

overlapping two optical standing waves along different directions. In the simplest form

one chooses two orthogonal directions and obtains at the center of the trap an optical

potential of the form (neglecting the harmonic confinement due to the Gaussian beam

profile of the laser beams)

(9) V (y, z) = −Vlat

(cos2(kx) + cos2(ky) + 2e1 · e2 cosφ cos(kx) cos(ky)

).

238 I. Bloch

(a) (b)

Fig. 3. – 2D optical lattice potentials for a lattice with (a) orthogonal polarization vectors and(b) with parallel polarization vectors and a time phase of φ = 0.

Here e1,2 denote the polarization vectors of the laser fields, each forming one standing

wave and φ is the time between them. If the polarization vectors are chosen not to be

orthogonal to each other, then the resulting potential will not only be the sum of the

potentials created by each standing wave, but will be modified according to the time phase

φ used (see fig. 3). In such a case it is absolutely essential to stabilize the time phase

between the two standing waves [15], as small vibrations will usually lead to fluctuations

of the time phase, resulting in severe heating and decoherence effects of the ultracold

atom samples.

In such a two-dimensional optical lattice potential, the atoms are confined to arrays of

tightly confining one-dimensional tubes (see fig. 4a). For typical experimental parameters

the harmonic trapping frequencies along the tube are very weak and on the order of

10–200 Hz, while in the radial direction the trapping frequencies can become as high as

up to 100 kHz, thus allowing the atoms to effectively move only along the tube for deep

lattice depths [16-20].

2.2.3. 3D lattice potentials. For the creation of a three-dimensional lattice potential,

three orthogonal optical standing waves have to be overlapped. Here we only consider

the case of independent standing waves, with no cross interference between laser beams

of different standing waves. This can for example be realized by choosing orthogonal

polarization vectors between different standing wave light fields and also by using different

wavelengths for the three standing waves. In this case the resulting optical potential is

simply given by the sum of three standing waves

V (r) = −Vxe−2(y2

+z2)/w2

x sin2(kx)(10)

−Vye−2(x2

+z2)/w2

y sin2(ky)

−Vze−2(x2

+y2)/w2

z sin2(kz).

Here Vx,y,z are the potential depths of the individual standing waves along the different

directions. In the center of the trap, for distances much smaller than the beam waist, the

trapping potential can be approximated as the sum of a homogeneous periodic lattice

potential and an additional external harmonic confinement due to the Gaussian laser


(a)

(b)

Fig. 4. – Two-dimensional (a) and three-dimensional (b) optical lattice potentials formed bysuperimposing two or three orthogonal standing waves. For a two-dimensional optical lattice,the atoms are confined to an array of tightly confining one-dimensional potential tubes, whereasin the three-dimensional case the optical lattice can be approximated by a three-dimensionalsimple cubic array of tightly confining harmonic oscillator potentials at each lattice site.

beam profiles

(11) V (r) ≈ Vx · sin2(kx) + Vy · sin2(ky) + Vz · sin2(kz) +

m

2

(ω2

xx2 + ω2

yy2 + ω2

zz2),

where ωx,y,z are the effective trapping frequencies of the external harmonic confinement.

They can again be approximated by

(12) ω2

x =4

m

(Vy

w2y

+Vz

w2z

); ω2

y,z = (cyclic permutations).

In addition to this harmonic confinement due to the Gaussian laser beam profiles,

a confinement due to a magnetic trapping typically exists, which has to be taken into

account as well for the total harmonic confinement of the atom cloud.

For sufficiently deep optical lattice potentials, the confinement on a single lattice site

is also approximately harmonic. Here the atoms are very tightly confined with typical

trapping frequencies ωlat of up to 100 kHz. One can estimate the trapping frequencies

240 I. Bloch

at a single lattice site through a Taylor expansion of the sinusoidally varying lattice

potential at a lattice site and obtains

(13) ωlat ≈

√Vlat

Er

�2k4

m2=

2Er

�

√Vlat

Er

.

Here Er = �2k2/2m is the so called recoil energy, which is a natural measure of energy

scales in optical lattice potentials.

3. – Bose-Hubbard model of interacting bosons in optical lattices

The behavior of bosonic atoms with repulsive interactions in a periodic potential is

fully captured by the Bose-Hubbard Hamiltonian of solid state physics [5, 6], which in

the homogeneous case can be expressed through

(14) H = −J∑〈i,j〉

a†i aj +1

2U

∑i

ni(ni − 1).

Here a†i and ai describe the creation and annihilation operators for a boson on the i-th

lattice site and ni counts the number of bosons on the i-th lattice site. The tunnel cou-

pling between neighboring potential wells is characterized by the tunnel matrix element

(15) J = −

∫d3xw(x − xi)(−�

2∇2/2m+ Vlat(x))w(x − xj),

where w(x − xi) is a single-particle Wannier function localized to the i-th lattice site

and Vlat(x) indicates the optical lattice potential. The repulsion between two atoms on

a single lattice site is quantified by the on-site matrix element U

(16) U = (4π�2a/m)

∫|w(x)|4d3x,

with a being the scattering length of an atom. Due to the short range of the interactions

compared to the lattice spacing, the interaction energy is well described by the second

term of eq. (14) which characterizes a purely on-site interaction.

Both the tunneling matrix element J and the onsite interaction matrix element can

be calculated from a band structure calculation (see fig. 5). The tunnel matrix element

J is related to the width of the lowest Bloch band through:

(17) 4J = |E0(q = π/a) − E0(q = 0)|,

where a is the lattice period, such that q = π/a corresponds to the quasi-momentum

at the border of the first Brillouin zone. Care has to be taken to evaluate the tunnel

matrix element through eq. (15), when the Wannier function is approximated by the


0 5 10 15 20 25 30

10-1

10-2

10-3

V (Er)

J (E

r)

50 10 15 20 25 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

U (

Er)

V (Er)

(a) (b)

Fig. 5. – Onsite interaction matrix element U for 87Rb (a) and tunnel matrix element J (b) vs.

lattice depth. All values are given in units of the recoil energy Er.

Gaussian ground state wave function on a single lattice site. This usually results in a

severe underestimation of the tunnnel coupling between lattice sites.

The interaction matrix element can be evaluated through the Wannier function with

the help of eq. (16). In this case, however, the approximation of the wannier function

through the Gaussian ground state wave function yields a very good approximation.

Zwerger et al. [21] have carried out a more sophisticated approximation of the tunnel

matrix element, finding

(18)J

Er

≈4√πEr

(Vlat

Er

)3/4

e−2

√Vlat/Er

and for the interaction matrix element

(19) U ≈

√8

πka

(Vlat

Er

)3/4

.

The ratio U/J is crucial for determining, whether one is in a strongly interacting or a

weakly interacting regime. It can be tuned continuously by simply changing the lattice

potential depth (see fig. 6).

3.1. Ground states of the Bose-Hubbard Hamiltonian. – The Bose-Hubbard Hamil-

tonian of eq. (14) has two distinct ground states depending on the strength of the in-

teractions U relative to the tunnel coupling J . In order to gain insight into the two

limiting ground-states, let us first consider the case of a double-well system with only

two interacting neutral atoms.

3.2. Double-well case. – In the double-well system the two lowest lying states for non-

interacting particles are the symmetric |ϕS〉 = 1/√

2(|ϕL〉+|ϕR〉) and the anti-symmetric

|ϕA〉 = 1/√

2(|ϕL〉 − |ϕR〉) states, where |ϕL〉 and |ϕR〉 are the ground states of the left-

and right-hand side of the double-well potential. The energy difference between |ϕS〉 and

|ϕA〉 will be named 2J , which characterizes the tunnel coupling between the two wells

and depends strongly on the barrier height between the two potentials.

242 I. Bloch

0 5 10 15 20 25 30

1

10

100

1000

V (Er)

U/J

Fig. 6. – U/J vs. optical lattice potential depth for 87Rb. By increasing the lattice depth onecan tune the ratio U/J , which determines whether the system is strongly or weakly interacting.

In case of no interactions, the ground state of the two-body system is realized when

each atom is in the symmetric ground state of the double-well system (see fig. 7a). Such a

situation yields an average occupation of one atom per site with the single-site many-body

state actually being in a superposition of zero, one and two atoms. Let us now consider

the effects due to a repulsive interaction between the atoms. If both atoms are again

in the symmetric ground state of the double well, the total energy of such a state will

increase due to the repulsive interactions between the atoms. This higher energy cost is

a direct consequence of having contributions where both atoms occupy the same site of

the double well. This leads to an interaction energy of 1/2U for this state.

Fig. 7. – Ground state of two interacting particles in a double well. For interaction energies Usmaller than the tunnel coupling J the ground state of the two-body system is realized by the“superfluid” state (a). If, on the other hand, U is much larger than J , then the ground state ofthe two-body system is the Mott insulating state (b).


If this energy cost is much greater than the splitting 2J between the symmetric and

anti-symmetric ground states of the non-interacting system, the system can minimize its

energy when each atom is in a superposition of the symmetric and antisymmetric ground

state of the double well 1/√

2(|ϕS〉 ± |ϕA〉). The resulting many-body state can then

be written as |Ψ〉 = 1/√

2(|ϕL〉 ⊗ |ϕR〉 + |ϕR〉 ⊗ |ϕL〉). Here exactly one atom occupies

the left and right site of the double well. Now the interaction energy vanishes because

both atoms never occupy the same lattice site. The system will choose this new “Mott

insulating” ground state when the energy costs of populating the antisymmetric state of

the double well system are outweighed by the energy reduction in the interaction energy.

It is important to note that precisely the atom number fluctuations due to the delocalized

single particle wave functions make the “superfluid” state unfavorable for large U .

Such a change can be induced by adiabatically increasing the barrier height in the

double-well system, such that J decreases exponentially and the energy cost for pop-

ulating the antisymmetric state becomes smaller and smaller. Eventually it will then

be favorable for the system to change from the “superfluid” ground state, where for

U/J → ∞ each atom is delocalized over the two wells, to the “Mott insulating” state,

where each atom is localized to a single lattice site.

3.3. Multiple-well case. – The above ideas can be readily extended to the multiple-well

case of the periodic potential of an optical lattice. For U/J � 1 the tunnelling term

dominates the Hamiltonian and the ground state of the many-body system with N atoms

is given by a product of identical single-particle Bloch waves, where each atom is spread

out over the entire lattice with M lattice sites,

(20) |ΨSF〉U/J≈0 ∝

(M∑i=1

a†i

)N

|0〉.

Since the many-body state is a product over identical single-particle states, a macroscopic

wave function can be used to describe the system. Here the single site many-body wave

function |φ〉i is almost equivalent to a coherent state. The atom number per lattice site

then remains uncertain and follows a Poissonian distribution with a variance given by

the average number of atoms on this lattice site Var(ni) = 〈ni〉. The non-vanishing

expectation value of ψi = 〈φi|ai|φi〉 then characterizes the coherent matter wave field

on the i-th lattice site. This matter wave field has a fixed phase relative to all other

coherent matter wave fields on different lattice sites.

If, on the other hand, interactions dominate the behavior of the Hamiltonian, such

that U/J � 1, then fluctuations in the atom number on a single lattice site become

energetically costly and the ground state of the system will instead consist of localized

atomic wave functions that minimize the interaction energy. The many-body ground state

is then a product of local Fock states for each lattice site. In this limit the ground state of

the many-body system for a commensurate filling of n atoms per lattice site is given by

(21) |ΨMI〉J≈0 ∝

M∏i=1

(a†i

)n

|0〉.

244 I. Bloch

Fig. 8. – Absorption images of multiple matter wave interference patterns after releasing theatoms from an optical lattice potential with a potential depth of a) 0Er, b) 3Er, c) 7Er, d) 10Er,e) 13Er, f) 14Er, g) 16Er and h) 20Er. The ballistic expansion time was 15 ms.

Under such a situation the atom number on each lattice site is exactly determined

but the phase of the coherent matter wave field on a lattice site has obtained a maximum

uncertainty. This is characterized by a vanishing of the matter wave field on the i-th

lattice site ψi = 〈φi|ai|φi〉 ≈ 0.

In this regime of strong correlations, the interactions between the atoms dominate

the behavior of the system and the many-body state is not amenable anymore to a

description as a macroscopic matter wave, nor can the system be treated by the theories

for a weakly interacting Bose gas of Gross, Pitaevskii and Bogoliubov [22].

For a 3D system, the transition to a Mott insulator occurs around U/J ≈ z · 5.6 [23,

6, 24, 21], where z is the number of next neighbors to a lattice site (for a simple cubic

crystal z = 6).

3.4. Superfluid-to-Mott-insulator transition. – In the experiment the crucial parame-

ter U/J that characterizes the strength of the interactions relative to the tunnel coupling

between neighboring sites can be varied by simply changing the potential depth of the

optical lattice potential. By increasing the lattice potential depth, U increases almost lin-

early due to the tighter localization of the atomic wave packets on each lattice site and J

decreases exponentially due the decreasing tunnel coupling. The ratio U/J can therefore

be varied over a large range from U/J ≈ 0 up to values in our case of U/J ≈ 2000.

In the superfluid regime [25] phase coherence of the matter wave field across the

lattice characterizes the many-body state. This can be observed by suddenly turning

off all trapping fields, such that the individual matter wave fields on different lattice

sites expand and interfere with each other. After a fixed time-of-flight period the atomic

density distribution can then be measured by absorption imaging. Such an image directly

reveals the momentum distribution of the trapped atoms. In fig. 8b an interference

pattern can be seen after releasing the atoms from a three-dimensional lattice potential.

If, on the other hand, the optical lattice potential depth is increased such that the


Vis

ibili

ty V

U/zt 100 101 102

100

10-2

10-1

4

Fig. 9. – Visibility of the interference pattern versus U/z t, the characteristic ratio of interactionto kinetic energy. The data is shown for two atom numbers 5.9 × 105 atoms (black circles), and3.6 × 105 atoms (grey circles). The former curve has been offset vertically for clarity. The linesare fits to the data in the range 14–25Er, assuming a coherent particle hole admixture as ineq. (22) (see ref. [29]).

system is very deep in the Mott insulating regime (U/J → ∞), phase coherence is lost

between the matter wave fields on neighboring lattice sites due to the formation of Fock

states [6, 5, 26, 3]. In this case no interference pattern can be seen in the time-of-flight

images (see fig. 8h) [7, 8]. For a Mott insulator at finite U/J one expects a residual

visibility in the interference pattern [27, 28], which can be caused on the one hand by

the residual superfluid shells and on the other hand through an admixture of coherent

particle hole pairs to the ideal MI ground state.

A more detailed picture for the residual short-range coherence features beyond the SF-

MI transition is obtained by considering perturbations deep in the Mott insulating regime

at J = 0. There, the first-order correlation function G(1)(R) describing the coherence

properties vanishes beyond R = 0 and the momentum distribution is a structureless

Gaussian, reflecting the Fourier transform of the Wannier wave function. With increasing

tunneling J , the Mott state at J/U → 0 is modified by a coherent admixture of particle-

hole pairs. However, due to the presence of a gapped excitation spectrum, such particle

hole pairs cannot spread out and are rather tightly bound to close distances. They do,

however, give rise to a significant degree of short-range coherence (see fig. 9). Using

first-order perturbation theory with the tunneling operator as a perturbation on the

dominating interaction term, one finds that the amplitude of the coherent particle hole

246 I. Bloch

admixtures in a Mott insulating state is proportional to J/U ,

(22) |Ψ〉U/J ≈ |Ψ〉U/J→∞+

J

U

∑〈i,j〉

a†i aj |Ψ〉U/J→∞.

Within a local density approximation, the inhomogeneous situation in a harmonic

trap is described by a spatially varying chemical potential μR = μ(0) − εR, with εR = 0

at the trap center. Assuming, e.g., that the chemical potential μ(0) at trap center falls

into the n = 2 “Mott-lobe”, one obtains a series of MI domains separated by a SF by

moving to the boundary of the trap where μR vanishes. In this manner, all the different

phases which exist for given J/U below μ(0) are present simultaneously! The SF phase

has a finite compressiblity κ = ∂n/∂μ and a gapless excitation spectrum of the form

ω(q) = cq because there is a finite superfluid density ns. By contrast, in the MI phase

both ns and κ vanish identically. As predicted by [6], the incompressibility of the MI

phase allows to distinguish it from the SF by observing the local density distribution in

a trap. Since κ = ∂n/∂μ = 0 in the MI, the density stays constant in the Mott phases,

even though the external trapping potential is rising.

The existence of such wedding-cake–like density profiles of a Mott insulator has been

supported by accurate Monte Carlo [27, 30] and DMRG [31] calculations in one, two,

and three dimensions. Very recently in-trap density profiles have been detected exper-

imentally by [32] and [33]. In the latter case it has been possible to directly observe

the wedding-cake density profiles and thus confirm the incompressibiltiy of the Mott in-

sulating regions of the atomic gas in the trapping potential (see fig. 10). It should be

noted that the in-trap density profiles can be used as a sensitive thermometer for the

strongly interacting quantum gas. Already for small temperatures around T ≈ 0.2U/kB

the wedding-cake profiles become completely washed out.

Close to the transition point, higher-order perturbation theory or a Green’s function

analysis can account for coherence beyond nearest neighbors and the complete liberation

of the particle-hole pairs, which eventually leads to the formation of long-range coherence

in the superfluid regime. The coherent particle-hole admixture and its consequence

on the short-range coherence of the system have been investigated theoretically and

experimentally in [29,34]. It has been demonstrated that through a quantitative analysis

of the interference pattern, one can even observe traces of the shell structure formation

in the Mott insulating regime.

In addition to the fundamentally different momentum distributions in the superfluid

and Mott insulating regime, the excitation spectrum is markedly different as well in both

cases. Whereas the excitation spectrum in the superfluid regime is gapless, it is gapped

in the Mott insulating regime. This energy gap of order U (deep in the MI regime) can

be attributed to the now localized atomic wave functions of the atoms [5, 6, 24, 3].


-20 -10 0 10 20

z Position (μm)

-20 -10 0 10 20

0

5

10

15

Slic

e F

ract

ion (%

)

z Position (μm)

0

5

10

15

20

Slic

e F

ract

ion (%

) c

f

d

e

Fig. 10. – Integrated distribution of a superfluid (a) and a Mott insulating state (b) calculatedfor a lattice with harmonic confinement. Grey solid lines denote the total density profiles, blue(red) lines the density profiles from singly (doubly) occupied sites. A vertical magnetic-fieldgradient is applied which creates almost horizontal surfaces of equal Zeeman shift over the cloud(dashed lines in inset). A slice of atoms can be transferred to a different hyperfine state byusing microwave radiation only resonant on one specific surface (coloured areas in insets). Spinchanging collisions can then be used to separate singly (blue) and doubly occupied sites (red)in that plane into different hyperfine states. Experimental data: (c) 1.0 × 105 atoms in thesuperfluid regime (V0 = 3Er), (d) 1.0× 105 atoms in the Mott regime (V0 = 22Er), (e) 2.0× 105

atoms, (f) 3.5 × 105 atoms. The grey data points denote the total density distribution and thered points the distribution of doubly occupied sites. The blue points show the distribution ofsites with occupations other than n = 2. The solid lines are fits to an integrated Thomas-Fermidistribution in (c), and an integrated shell distribution for (d) to (f). The n = 2 data points areoffset vertically for clarity.

4. – Multi-orbital quantum phase diffusion

4.1. Introduction. – Next we turn to the time dynamics of many-body states when

jumping rapidly from a superfluid into a Mott insulating parameter regime. Initially,

in the superfluid state the quantum states on a lattice are close to coherent states.

Such coherent quantum states represent the most robust and stable field solutions in

physics [35]. Due to their correspondence to classical coherent fields, characterized by

a single amplitude and phase, they have found widespread use in physics ranging from

the description of laser light to coherent matter waves in superconductors, superfluids

or atomic Bose-Einstein condensates. Whenever interactions between the underlying

particles are present, or —more generally— whenever the phase of the number states

that form the coherent state evolve nonlinear in particle number over time, such coherent

states can undergo an intriguing sequence of collapses and revivals. The quantum state

first evolves into a highly correlated and entangled state where at the time of the collapse

248 I. Bloch

the classical field vanishes, whereas at a later time the entanglement is unraveled again

and the original classical field is ideally recreated.

Remarkable examples of such collapses and revivals have been observed for a coherent

light field interacting with a single atom in Cavity Quantum Electrodynamics [36, 37],

for a classical oscillation of a single ion held in a trap [38, 39] or for a matter wave

field of a Bose-Einstein condensate via the non-linear two-body interactions between the

atoms [40-42]. In the latter case, one typically assumes the atoms to occupy a single

spatial orbital of the system. Atom-atom collisions can however promote particles to

higher lying orbitals and even for the case where real occupation of excited vibrational

states can be neglected, virtual transitions can still have a profound impact on the system.

They can, e.g., modify the spatial wave function of the atoms depending on the filling,

giving rise to renormalized two-particle interactions and the generation of higher-order

effective multi-particle interactions that are induced via higher-order virtual transitions

of the atoms to excited orbital states [43]. Current experiments have only allowed to

observe few cycles of the resulting quantum phase diffusion dynamics [40-42] and therefore

were not able to reveal the striking consequences of multi-orbital effects. Here, we have

been able to observe up to 40 collapses and revivals in long time traces of the quantum

evolution of coherent matter wave fields trapped in a 3D optical lattice potential.

4.2. Theoretical model . – We consider a single site of an optical lattice filled with

n particles occupying the lowest energy state of the system. For non-interacting par-

ticles, this corresponds to the ground state vibrational wave function ψ0(r). Assum-

ing weak interactions and excluding multi-orbital effects, cold collisions between the

atoms lead to a single-orbital interaction energy given by ESOn = Un(n − 1)/2, where

U = 4π�2a/m

∫|ψ0(r)|

4 d3r denotes the two-particle interaction energy, determined by

the on-site wave function ψ0(r), the s-wave scattering length a and the mass of an atom

m. Within the restriction to the lowest vibrational state of the system, U is independent

of the filling n at the lattice site.

By taking virtual transitions to higher vibrational states into account, however, inter-

actions modify the shape of the ground-state wave function (fig. 11b) and U itself becomes

atom number dependent [44,43,45]. For the case where real occupation of excited vibra-

tional states can be neglected, but virtual transitions to these states are important, the

multi-orbital Fock state energy of a single lattice well can be approximated by [43]

EMO

n =1

2U2n(n− 1)(23)

+1

6U3n(n− 1)(n− 2)

+1

24U4n(n− 1)(n− 2)(n− 3) + . . . .

In this description, the coherent multi-particle interactions become explicitly visible,

where characteristic strengths of the n-particle interactions are given by Un.


n = 2 n = 3 n = 4

c0 + c1 + c2 + c3 + ... a

b

c

1.0

0 0 5 10

Co

he

ren

ce

t (trev h/U)

Fig. 11. – (Colour on-line) Quantum phase diffusion and multi-orbital effects. (a) A Bose-Einstein condensate loaded to a weak optical lattice forms a superfluid with each atom beingdelocalized over several lattice sites. The quantum state on each site can be expressed as a super-position of Fock states |n〉 with amplitudes cn. (b) For repulsive onsite interactions virtual tran-sitions to higher lattice orbitals broaden the ground-state wave function of the non-interactingsystem (grey dashed line), giving rise to coherent multi-particle interactions. (c) A coherentstate confined to a deep lattice well undergoes multi-orbital quantum phase diffusion (blue solidline, see text). The dynamics are markedly different from the monochromatic evolution expectedin a single-orbital model with a single two-body interaction energy scale U (grey solid line).

4.3. Probing the energy scales via Fock state heterodyning . – An efficient way to ex-

perimentally probe the eigenenergies of a Hamiltonian is to monitor the non-equilibrium

dynamics of a quantum state prepared in a superposition of different eigenstates. In

our case, such a superposition state consists of different atomic Fock states {|n〉} and

forms an atomic matter wave field of the general form |ψ(t)〉 =∑

∞

n=0cne

−iEnt/�|n〉.

Experimentally, we create a 3D array of such matter wave fields by loading a BEC into

a shallow 3D lattice potential. Their time evolution can be probed by analyzing the vis-

ibility of the atomic interference pattern as observed after rapid switch-off of the lattice

potential and subsequent time-of-flight expansion [29]. For an array of identical states

|ψ〉, the visibility of the interference pattern is proportional to |〈ψ(t)|a|ψ(t)〉|2/n, where

a denotes the annihilation operator on a lattice site. Hence, the dynamical evolution of

250 I. Bloch

the matter wave field, the quantum phase diffusion of |ψ(t)〉 is given by

|〈ψ(t)|a|ψ(t)〉|2 =

∞∑m,n=1

√n+ 1

√m+ 1 cm c∗n c

∗

m+1cn+1(24)

×ei(En−En+1−Em+Em+1)t/�

≡ |〈a〉|2.

Multi-orbital effects, however, reach beyond the picture of monochromatic collapses

and revivals: the time evolution of |〈a〉|2 contains multiple frequency components since

Fock state energies are no longer integer multiples of the U (fig. 11c (blue solid line)).

Detection of multi-orbital quantum phase diffusion over sufficiently long times allows for

a precise measurement of the individual Fock state energies EMOn .

4.4. Experimental setup and results. – Our experiments begin with an atomic Bose-

Einstein condensate of 87Rb atoms in the |F = 1,mF = +1〉 state, with variable atom

numbers between 1.2×105 and 4.5×105. The atoms are initially held in a pancake-shaped

crossed optical dipole trap. Subsequently a 3D blue-detuned optical lattice (λ = 738 nm)

of simple cubic type is ramped up to lattice depths VL between 3Er and 13Er.

A sudden increase of the lattice depth from VL to VH ranging between 25Er and 41Er

then essentially freezes out the equilibrium atom number distribution at VL on each site

through a strong suppression of the tunnel coupling. In this regime, the time evolution

of each site is governed by the Hamiltonian of eq. (23) and the quantum phase diffusion

process is initiated. After letting the system evolve for hold times t in the deep lattice

VH , we have monitored the phase coherence by simultaneously switching-off all trapping

potentials and recording an absorption image of the matter wave interference pattern

after 10 ms time of flight. The visibility of the interference pattern is used as a measure

of the phase coherence of the system [29].

A typical time trace (VL = 8Er) of the quantum phase diffusion is shown in fig. 12a

displaying up to 40 revivals. On top of a fast series of collapse and revivals, we observe a

slower modulation of the envelope indicating a beat note between at least two different

energy scales in the system. In a Fourier analysis of the corresponding time trace, we

find clear evidence for three distinct frequency components present in the time trace,

the smallest one originating from sites occupied by up to four atoms. In general, the

measurement allows one to reveal even very small Fock state amplitudes cn due to a

heterodyning effect with other Fock states |n−1〉 and |n−2〉 of typically larger amplitude

cn−1 and cn−2 (see eq. (24)).

5. – Compressible and incompressible quantum phases of fermionic spin

mixtures in optical lattices

Next to bosonic particles, interacting fermions in periodic potentials lie at the heart of

condensed matter physics. They present, however, some of the most challenging problems

to quantum many-body theory. A prominent example is high-Tc superconductivity in


0 2000 60004000 8000

0.2

0.4

0.6

0.8

0

1.0

Holdtime (μs )

3.0

Frequency ( kHz )

3.5 4.0 4.5

1.0

0.5

0

3 53 25

0

0 5

Vis

ibili

tyP

ow

er

sp

ectr

al

density (

au

)

40μs 2720μs 5120μs 7600μs a

b c

time

lattice

dipole

trap

time100ms

50μs

VH

VL

Fig. 12. – Collapse and revival dynamics of an atom number superposition state. (a) Timetrace of observed collapses and revivals in the phase coherence of the system. A BEC of about1.5 × 105 Rb atoms has been adiabatically loaded to a VL = 8Erec lattice within 100 ms. Phaseevolution is induced by a non-adiabatic jump (50 μs) into a VH = 40Erec deep lattice, pre-serving superposition states with finite number fluctuations and an average filling of aboutn = 1.5 (c). The evolution of phase coherence shows a beat-note signature resulting from co-herent multi-particle interactions with different interaction strengths. (b) Spectral analysis ofthe time traces using a numerical Fourier transform of the data (a) reveals the contributingfrequencies.

cuprate compounds [46]. An essential part of the physics in these systems is described by

the fermionic Hubbard Hamiltonian [47], which models interacting electrons in a periodic

potential [46, 48]. Probing this Hamiltonian in a controllable and clean experimental

setting is therefore of great importance.

For the case of bosonic particles [5, 6], the significance of ultracold quantum gases in

this respect has been discussed in the previous sections. For both bosonic and fermionic

systems, the entrance into a Mott insulating state is signaled by a vanishing compress-

ibility, which can in principle be probed experimentally by testing the response of the

system to a change in external confinement. This is a straightforward way to identify the

interaction-induced Mott insulator and to distinguish it, e.g., from a disorder-induced

Anderson insulator [49-51]. In a solid, however, the corresponding compressibility can

usually not be measured directly, as a compression of the crystalline lattice by an external

force does not change the number of electrons per unit cell.

In a recent work, non-interacting and repulsively interacting spin mixtures of fermionic

atoms deep in the degenerate regime were studied in a three-dimensional optical lattice,

252 I. Bloch

Fig. 13. – Relevant phases of the Hubbard model with an inhomogeneous trapping potential fora spin mixture at T = 0. A schematic is shown in the left column. The center column displaysthe corresponding in-trap density profiles and the right column outlines the distribution of singlyand doubly occupied lattice sites after a rapid projection into the zero tunneling limit, with p

denoting the total fraction of atoms on doubly occupied lattice sites.

where the interaction strength, the lattice depth and in one experiment the external

harmonic confinement of the quantum gas could be varied independently [52, 53]. By

monitoring the in-trap density distribution for increasing harmonic confinement [53], we

could directly probe the compressibility of the many-body system. This measurement

allowed us to distinguish compressible metallic phases from globally incompressible states

and revealed the strong influence of interactions on the density distribution.

In previous experiments, a suppression of the number of doubly occupied sites was

demonstrated for increasing interaction strength for bosons [54] and fermions [52] at fixed

harmonic confinement, signaling the entrance into a strongly interacting regime.

Here we compare the experimentally observed density distributions and fractions of

doubly occupied sites to numerical calculations using Dynamical Mean Field Theory

(DMFT) [55-58]. DMFT is a central method of solid state theory being widely used

to obtain ab initio descriptions of strongly correlated materials [56]. The comparison of

DMFT predictions with experiments on ultracold fermions in optical lattices constitutes a

parameter-free experimental test of the validity of DMFT in a three-dimensional system.

5.1. Hubbard Hamiltonian in a trap. – Restricting our discussion to the lowest en-

ergy band of a simple cubic 3D optical lattice, the fermionic quantum gas mixture can

be modeled via the Hubbard-Hamiltonian [47] with an additional term describing the


U/12J 1 5

a

b

c

de

0 0.5 1 1.5 2

U/12J 0

U/12J 0 5

U/12J 1

0 40 80 120

Ramp time (ms)

fEt /12J=0.4

R (d

)

50

55

45

Et /12JCompression

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Re

no

rma

lize

d C

lou

d S

ize

Rsc

(d)

Fig. 14. – Cloud sizes of the interacting spin mixture versus compression. Measured cloud sizeRsc in a Vlat = 8Er deep lattice as a function of the external trapping potential for various inter-actions U/12J = 0 . . . 1.5. Dots denote single experimental shots, lines the theoretical expecta-tion from DMFT for T/TF = 0.15 prior to loading. The insets (a-e) show the quasi-momentumdistribution of the non-interacting clouds (averaged over several shots). (f) Resulting cloud sizefor different lattice ramp times at Et/12J = 0.4 for a non-interacting and an interacting Fermigas. The arrow marks the ramp time of 50 ms used in the experiment.

underlying harmonic potential:

H = −J∑

〈i,j〉,σ

c†i,σ cj,σ + U∑

i

ni,↓ni,↑(25)

+Vt

∑i

(i2x + i2y + γ2i2z) (ni,↓ + ni,↑) .

Here the indices i, j denote different lattice sites in the three-dimensional system (i =

(ix, iy, iz)), 〈i, j〉 neighboring lattice sites, σ ∈ {↓, ↑} the two different spin states, J the

tunneling matrix element and U the effective on-site interaction. The operators ci,σ(c†i,σ)

are the annihilation (creation) operators of a fermion in spin state σ on the i-th lattice

site and ni,σ measures the corresponding atom number. The strength of the harmonic

confinement is parameterized by the energy offset Vt = 1

2mω2

⊥d2 between two adjacent

lattice sites at the trap center, with ω⊥ = ωx = ωy �= ωz being the horizontal trap

frequency, d the lattice constant and m the mass of a single atom. The constant aspect

ratio of the trap is denoted by γ = ωz/ω⊥.

The quantum phases of the Hubbard model with harmonic confinement are governed

by the interplay between three energy scales: kinetic energy, whose scale is given by

the lattice bandwidth 12J , interaction energy U , and the strength of the harmonic

254 I. Bloch

f

0

20

4001

2

0

1

r (d)E

t /12J

n i,σ

e

01

2

0

20

40

0

1

r (d)E

t /12J

n i,σ

Et /12J

0

1

2

3

0

1

2

3

0

1

2

3

0

0.5

1

1.5

0 1 2.00.5 1.5 0 1 2.00.5 1.5

0 1 2.00.5 1.50 1 2.00.5 1.5

a b

c d

T/TF 0.07

T/TF 0.15

0.07

0.10 0.15

0.170.20

T/TF T/TF

Et /12J

0.5 1 1.5

1

0.6

0.2

κR

sc (

d -2

)C

om

pre

ss

bty

Fig. 15. – (a-c) Global compressibility κRsc of the atom cloud for various interactions((a) U/12J = 0, (b) U/12J = 1, (c) U/12J = 1.5). Dots denote the result of linear fitson the measured data, the error bars represent the fit uncertainty. Solid lines display thetheoretically expected results for an initial temperature of T/TF = 0.15. The influence of theinitial temperature on the calculated compressibility is shown in (d) for U/12J = 1.5. Thecorresponding density distributions are plotted in (e, f) with r denoting the distance to thetrap center. The red lines mark the region where a Mott-insulating core has formed and theglobal compressibility is reduced.

confinement, which can conveniently be expressed by the characteristic trap energy

Et = Vt(γNσ/(4π/3))2/3, which denotes the Fermi energy of a non-interacting cloud in

the zero-tunneling limit, with Nσ being the number of atoms per spin state (N↓ = N↑).

The characteristic trap energy depends both on atom number and trap frequency via

Et ∝ ω2

⊥N

2/3

σ and describes the effective compression of the quantum gas, controlled by

the trapping potential in the experiment.

Depending on which term in the Hamiltonian dominates, different kinds of many-

body ground states can occur in the trap center (fig. 13). For weak interactions in

a shallow trap U � Et � 12J the Fermi energy is smaller than the lattice bandwidth

(EF < 12J) and the atoms are delocalized in order to minimize their kinetic energy. This

leads to compressible metallic states with central filling n0,σ < 1, where the local filling

factor ni,σ = 〈ni,σ〉 denotes the average occupation per spin state of a given lattice site. A

dominating repulsive interaction U � 12J and U � Et suppresses the double occupation

of lattice sites and can lead to Fermi-liquid (n0,σ < 1/2) or Mott-insulating (n0,σ = 1/2)

states at the trap center, depending on the ratio of kinetic to characteristic trap energy.

Stronger compressions lead to higher filling factors, ultimately (Et � 12J,Et � U)

resulting in an incompressible band insulator with unity central filling at T = 0.


Finite temperature reduces all filling factors and enlarges the cloud size, as the sys-

tem needs to accommodate the corresponding entropy. Furthermore, in the trap the

filling always varies smoothly from a maximum at the center to zero at the edges of

the cloud. For a dominating trap and strong repulsive interaction at low temperature

(Et > U > 12J), the interplay between the different terms in the Hamiltonian gives

rise to a wedding-cake-like structure (fig. 15e, f) consisting of a band-insulating core

(n0,σ ≈ 1) surrounded by a metallic shell (1/2 < ni,σ < 1), a Mott-insulating shell

(ni,σ = 1/2) and a further metallic shell (ni,σ < 1/2) [57]. The outermost shell remains

always metallic, independent of interaction and confinement, only its thickness varies.

5.2. Experimental setup. – In the experiment, we used an equal mixture of quantum

degenerate fermionic 40K atoms in the two hyperfine states |F,mF 〉 = |92,− 9

2〉 ≡ |↓〉

and |92,− 7

2〉 ≡ |↑〉 in a pancake-shaped optical dipole trap (aspect ratio γ ≈ 4), which

is formed by overlapping two elliptical laser beams (λ = 1030 nm) travelling in the

horizontal plane. Applying evaporative cooling, final temperatures of T/TF = 0.15(3)

with 1.5–2.5 × 105 potassium atoms are reached. The temperature is extracted from

time-of-flight images using Fermi fits. A Feshbach resonance located at 202.1 G [59] is

used to tune the scattering length between the two spin states and thereby control the

on-site interaction U . The creation of the spin mixture and the last evaporation step are

performed either above the resonance (220 G), giving access to non-interacting (209.9 G)

and repulsively interacting clouds with a ≤ 150 a0 (B ≤ 260 G), or below the resonance

(165 G), where larger scattering lengths up to a = 300 a0 (191.3 G) can be reached. A

further approach to the Feshbach resonance is hindered by enhanced losses and heating

in the lattice.

After evaporation, the magnetic field is tuned to the desired value. Subsequently, a

blue-detuned 3D optical lattice (λlat = 738 nm) with simple cubic symmetry is increased

to a potential depth of Vlat = 1Er, where Er = h2/(2mλ2

lat) denotes the recoil energy.

The combination of a red-detuned dipole trap and a blue-detuned lattice potential

allows us to vary lattice depth and external confinement independently. In this way

a wide range of horizontal trap frequencies can be accessed (ω⊥ � 2π × (20–120) Hz),

especially enabling metallic states with high atom numbers. To monitor the in situ

density distribution for different external confinements, we ramp the dipole trap depth

in 100 ms to the desired external harmonic confinement, followed by a linear increase

of the optical lattice depth to Vlat = 8Er within 50 ms. An in situ image of the cloud

is taken along the short (vertical) axis of the trap using phase-contrast imaging [60]

at detunings of Δ = 2π × (200–330) MHz after a hold time of 12 ms in the lattice.

From this picture the cloud size R =√

〈r2⊥〉 is extracted using adapted 2D Fermi fits.

As phase-contrast imaging modifies the state of the atoms only marginally, the quasi-

momentum distribution can be measured in the same experimental run using a band-

mapping technique [61, 16, 62]. For this, the lattice is ramped down in 200μs and a

standard absorption image is taken after 10 ms time of flight.

All experimental data are compared to numerical calculations, in which the DMFT

equations of the homogeneous model are solved for a wide range of temperatures and

256 I. Bloch

chemical potentials using a numerical renormalization group approach [63, 64]. The

trapped system can be approximated to very high accuracy by the uniform system

through a local density approximation (LDA) even close to the boundary between a

metal and an insulator [65,57]. For a comparison with the experimental results it is con-

venient to express the cloud size R in rescaled units Rsc = R/(γNσ)1/3, along with the

dimensionless compression Et/12J (see fig. 14). In these units, the cloud size depends

only on the interaction strength U/12J and the entropy. In all calculations, we use the

entropy determined from a non-interacting Fermi gas in a harmonic trap at an initial

temperature T/TF and assume adiabatic lattice loading.

5.3. Cloud compression. – The numerically calculated density distributions, the cor-

responding column densities and the experimentally measured ones are presented in

fig. 13. While for low compression all distributions are metallic (first row), we find a

Mott-insulating core with half-filling at intermediate compression and strong repulsion

(second row). For high compression the non-interacting curve shows a band insulating

core and the repulsive curves display a metallic core. In order to compare experiment

and theory quantitatively, the measured cloud size Rsc and the numerically calculated

one are plotted in fig. 14 as a function of the characteristic trap energy Et. Addition-

ally, the global compressibility κRsc= − 1

R3sc

∂Rsc

∂(Et/12J)(fig. 15) of the system can be

extracted from these measurements using linear fits to four consecutive data points to

determine the derivative. In the non-interacting case we find the cloud sizes (fig. 15,

black dots) to decrease continuously with compression until the characteristic trap en-

ergy roughly equals the lattice bandwidth (Et/12J ∼ 1). For stronger confinement the

compressibility approaches zero (fig. 15a), as almost all atoms are in the band insulat-

ing regime while the surrounding metallic shell becomes negligible. The corresponding

quasi-momentum distribution (fig. 14a-e) changes gradually from a partially filled first

Brillouin zone, characteristic for a metal, to an almost evenly filled first Brillouin zone for

increasing compressions, as expected for a band insulator. Note, however, that a band-

mapping technique reveals only the relative occupations of the extended Bloch states.

For an inhomogeneous system, it therefore provides no information about the real-space

density and especially cannot distinguish insulating from compressible states, e.g. non-

equilibrium states in which the atomic wave functions are localized to single lattice sites.

In contrast, the measurements shown here directly demonstrate the incompressibility of

the fermionic band insulator, in excellent agreement with the theoretical expectation for

a non-interacting Fermi gas (black line).

For moderately repulsive interactions (U/12J = 0.5, 1) (green, blue) the cloud size is

clearly bigger than in the non-interacting case but eventually reaches the size of the band

insulator. For stronger repulsive interactions (U/12J = 1.5) (red) we find the onset of a

region (0.5 < Et/12J < 0.7) where the cloud size decreases only slightly with increasing

harmonic confinement, whereas for stronger confinements the compressibility increases

again. This is consistent with the formation of an incompressible Mott-insulating core,

surrounded by a compressible metallic shell, as can be seen in the corresponding in-

trap density profiles (fig. 15e, f). For higher confinements an additional metallic core


(1/2 < ni,σ < 1) starts to form in the center of the trap and the cloud size decreases

again. A local minimum in the global compressibility is in fact a genuine characteristic of

a Mott insulator for large U and low temperature. The experimental data, indeed, show

an indication of this behavior (fig. 15c) for increasing interactions. For Et/12J � 0.5 a

minimum in the compressibility is observed, followed by an increase of the compressibility

around Et/12J � 0.8, slightly earlier than predicted by theory.

When the system is compressed even further, all cloud sizes approach that of a band

insulating state and all compressibilities tend to zero. In the theory predictions, the

repulsively interacting clouds can even become slightly smaller than the non-interacting

ones due to Pomeranchuk cooling [66]: At the same average entropy per particle, the

interacting system has a considerably lower temperature in the lattice, as the spin entropy

is enhanced due to interactions. In the experiment this feature is barely visible, as

a second effect becomes important: At very high compressions (Et/12J � 2) the second

Bloch band gets slightly populated during the lattice ramp up, which leads to smaller

cloud sizes for all interactions, as a small number of atoms in a nearly empty band can

carry a considerable amount of entropy.

Overall, we find the measured cloud sizes to be in very good quantitative agreement

with the theoretical calculations up to U/12J = 1.5 (B = 175 G). Nevertheless, for

repulsive interactions and medium compression (Et/12J ≈ 0.5) the cloud size is slightly

bigger than the theoretical expectation. The discrepancies become more prominent for

stronger interactions, i.e. on further increasing the scattering length. This could be

caused by non-equilibrium dynamics in the formation of a Mott-insulating state for strong

interactions or may be an effect not covered by the simple single-band Hubbard model [66]

or the DMFT calculations and requires further investigation.

To ensure that the used lattice loading time of 50 ms is adiabatic, we have mea-

sured the resulting in situ cloud size as a function of ramp time (fig. 14f) in the regime

around Et/12J ≈ 0.5, where the differences between experiment and theory are most

pronounced. In this regime a too fast loading would result in a larger cloud and our

measurement therefore indicates adiabaticity for the used ramp time. However, a sec-

ond longer timescale, which could become more relevant for stronger interactions [67],

cannot be ruled out. In addition, the temperatures before loading into the lattice and

after a return to the dipole trap with a reversed sequence are compared. We find a rise

in temperature between 0.010(5)T/TF for a non-interacting cloud and 0.05(2)T/TF for

a medium repulsion of U/12J = 1 at compressions around Et/12J ≈ 0.5. The good

agreement between the experimental data and the numerical calculations, which assume

adiabatic loading and an initial temperature of T/TF = 0.15, indicates that our ac-

tual initial temperatures lie rather at the lower end of the measured temperature range

T/TF = 0.15(3).

The theoretical calculations of the compressibility shown in fig. 15d demonstrate that

the minimum in the local compressibility, which signals the Mott-insulating state, starts

to form at initial temperatures in the range of 0.15 � T/TF � 0.2. At these tempera-

tures, the entropy per particle is much higher than possible in a Mott insulator in the

homogeneous case (< kB ln 2), and even exceeds the maximum possible entropy per par-

258 I. Bloch

0

0.5

ln(2)

1

2ln(2)

0

0.5

ln(2)

1Entr

opy (

k B)

Distance from trap center r (d)

0 20 40

0

20 40

Et /12J = 0.1

Et /12J = 1 Et /12J = 1.8

ba

c d

Et /12J = 0.5

T / TF = 0.15

T / TF = 0.07

Fig. 16. – Calculated entropy distributions in the lattice. Solid (dashed) lines show the entropyper lattice site (per particle) for initial temperatures of T/TF = 0.07 (black) and T/TF = 0.15(red) and strong repulsive interaction U/12J = 1.5 (see refs. [53,70]).

ticle of a half-filled homogeneous Hubbard model (kB ln 4). In the trap, however, a large

fraction of the entropy is accumulated in the metallic shells at the edges of the atomic

cloud where the diluted atoms have a large configurational entropy. Therefore, the tem-

perature remains on the order of kBTlat � J � U in the Mott insulating regime. This is

similar to the results obtained in recent calculations and experiments on the melting of

incompressible bosonic Mott insulating shells for increasing temperatures [68, 33].

5.4. Entropy distribution. – In fig. 16a the entropy distribution of a purely metallic

state with less than half-filling in the center of the trap is shown. At higher compression

(b) a Mott-insulator with unity filling and kB ln 2 entropy per particle has formed in

the center of the trap even in the case of T/TF = 0.15, for which the average entropy

per particle is above kB 2 ln 2. This is possible only due to the inhomogeneity of the

system, as most of the entropy is carried by the metallic shells where the entropy per

particle can diverge. For high compressions (d) a band-insulating core has formed and

for low enough temperatures (black) nearly all entropy is carried by the surrounding

shells. The small dip at r = 20 is a remnant of the Mott-insulating shell between the

two metallic shells. Note that entropy per lattice site and entropy per particle are equal

at half-filling. The inhomogeneous entropy distribution could allow for efficient cooling

schemes, by removing entropic metallic layers followed by a subsequent equilibration of

the system [69].


6. – Controlled superexchange interactions

Quantum spin systems on a lattice have served for decades as paradigms for condensed

matter and statistical physics, elucidating fundamental properties of phase transitions

and acting as models for the emergence of quantum magnetism in strongly correlated

electronic media. In all these cases the underlying systems rely on a spin-spin inter-

action between particles on neighboring lattice sites, such as in the Ising or Heisenberg

model [71-73]. As initially proposed for electrons by Dirac [74,75] and Heisenberg [72,76],

effective spin-spin interactions can already arise due to the interplay between the spin-

independent Coulomb repulsion and exchange symmetry and do not require any direct

coupling between the spins of the particles. The nature of such spin exchange inter-

actions is typically short-ranged, since it is governed by the wave function overlap of

the underlying electronic orbitals. In several topical insulators, such as ionic solids like,

e.g., CuO and MnO, however, antiferromagnetic ordering arises even though the wave

function overlap between the magnetic ions is practically zero. In this case a “superex-

change” interaction can be effective over large distance, as introduced in the seminal

works of Kramers and Anderson [77, 78]. Here, the spin-spin interactions are mediated

by higher-order virtual hopping processes, which in the case of bosons (fermions) leads

to an (anti)-ferromagnetic coupling between particles on neighboring lattice sites [73].

Such superexchange interactions are believed to play an important role in the context

of high-Tc superconductivity [46]. Furthermore, they can form the basis for the gener-

ation of robust quantum gates similar to recent work in electronic double quantum dot

systems [79], and can be employed for the efficient generation of multi-particle entangled

states [80], as well as for the production of many-body quantum phases with topological

order [81, 82].

6.1. Theoretical model . – An isolated system of two coupled potential wells consti-

tutes the simplest conceptual setup for the investigation of superexchange-mediated spin

dynamics between neighboring atoms. In the following, we consider a single double-well

potential occupied by a pair of bosonic atoms with two different spin states denoted by

|↑〉 and |↓〉. If the vibrational level splitting in each well is much larger than all other rel-

evant energy scales and intersite interactions are neglected, the system can be described

in a two-mode approximation by the Hubbard-type Hamiltonian

(26) HBH =∑

σ=↑,↓

[−J

(a†σL

aσR+a†σRaσL

)−

1

2Δ (nσL−nσR)

]+U (n↑Ln↓L+n↑Rn↓R) ,

where the operators a†σL,R and aσL,R create and annihilate an atom with spin σ in the

left and right well, respectively, nσL,R count the number of atoms per spin-state and

well, J is the tunnel matrix element, Δ the potential bias or tilt along the double-well

axis and U = U↑↓ = g ×∫w4

L,R(x)dx the onsite interaction energy between two atoms

in |↑〉 and |↓〉. Here, g = (4π�2a↑↓

s)/mRb is the effective interaction strength with a↑↓s

being the (positive) scattering length for the spin states used in the experiment and mRb

260 I. Bloch

4J2

U

+ α t+

t α+ +s

4J2

U

U

Fig. 17. – Schematics of superexchange interactions. a) Second-order hopping via |↑↓, 0〉 and|0, ↑↓〉 mediates the spin-spin interactions between atoms on different sides of the double well.b) Energy levels for Δ = 0 and U � J . The evolution in the upper doublet of states correspondsto the correlated tunneling of atom pairs [83], while the superexchange takes place in the lowerone. Both doublets are coupled by first-order tunneling processes.

the rubidium mass, and wL,R(x) denote the wave functions for a particle localized on

the left or right side of the double well. The state of the system can be described as

a superposition of the Fock states |↑, ↓〉, |↓, ↑〉, |↑↓, 0〉 and |0, ↑↓〉, where the left and

right side in the notation represent the occupation of the left and right well, respectively,

and the states |↑↓, 0〉 and |0, ↑↓〉 are spin triplets. In the following, we will focus on the

dynamical evolution of the population imbalance x = nL − nR and the spin imbalance

m = (n↑L+n↓R−n↑R−n↓L)/2 starting with double wells initially prepared in |↑, ↓〉. Here

n↑,↓;L,R = 〈n↑,↓;L,R〉 denote the corresponding quantum-mechanical expectation values

and nL,R = n↑L,R + n↓L,R.

In the limit of strong interactions (U � J), when starting in the subspace of singly

occupied wells spanned by |↑, ↓〉 and |↓, ↑〉, the energetically high lying states |↑↓, 0〉 and

|0, ↑↓〉 can only be reached as “virtual” intermediate states in second-order tunneling

processes. Such processes lead to a long-ranged (super) spin-exchange interaction, which

couples the states |↑, ↓〉 and |↓, ↑〉 (see fig. 17a). The effective coupling strength for this

superexchange can readily be evaluated by perturbation theory up to quadratic order in

the tunneling operation and yields Jex = 2J2/U . More generally, for an arbitrary spin

configuration with equal interaction energies U↑↑ = U↑↓ = U↓↓, the second-order hopping

events are described by an isotropic Heisenberg-type effective spin Hamiltonian

(27) Heff = −2JexSL · SR = −Jex

(S+

LS−

R+ S−

LS+

R

)− 2JexS

zLSz

R,

where S+

L,R = |↑〉〈↓|L,R and S−

L,R = |↓〉〈↑|L,R and SzL,R = (n↑L,R − n↓L,R)/2 denote the

corresponding spin operators of the system, with S±

L,R = Sx ± iSy.

When a potential bias Δ > 0 is applied, the degeneracy of the two intermediate

states in the superexchange process is lifted (see fig. 17a). For J , Δ � U this leads to

a modification of the effective superexchange coupling with now Jex = J2/(U + Δ) +

J2/(U − Δ) = 2J2U/(U2 − Δ2). By tuning the bias to Δ > U , it is possible to change


the sign of Jex and therefore to switch between ferromagnetic and antiferromagnetic

exchange interactions. For J � |U−Δ| the picture of an effective coupling via two virtual

intermediate states is again valid and the full reversal of the sign to Jex = −2J2/U is

found to be reached for Δ =√

2U .

For symmetric double wells (Δ = 0), the Hamiltonian eq. (26) can be diagonalized

analytically to give a valid picture for all values of J and U within the single band

Hubbard model. A convenient basis to express the eigenstates is given by the spin triplet

and singlet state |t/s〉 = (|↑, ↓〉 ± |↓, ↑〉)/√

2 and the states |±〉 ≡ (|↑↓, 0〉 ± |0, ↑↓〉)/√

2.

Two of the eigenstates are linear combinations of |t〉 and |+〉, where the one having the

larger overlap with |t〉 is the ground state. The spin singlet |s〉 and the state |−〉 are

already eigenstates themselves with energy 0 and U , respectively (see fig. 17b). As a

direct consequence, |−〉 cannot be reached with the initial state |↑, ↓〉 = |s〉+ |t〉 and the

dynamical evolution in this case is therefore given by only two frequencies

(28) �ω1,2 =U

2

⎛⎝√(

4J

U

)2

+ 1 ± 1

⎞⎠ ,

which allow for the extraction of J = �√ω1ω2/2 and U = �(ω1 − ω2).

6.2. Time-resolved observation of superexchange interactions. – In order to investigate

the spin dynamics across a single quantum link, we initially prepare a sample of ultracold

neutral atoms with two relevant internal states |↑〉 and |↓〉 in a 3D array of double wells

with Neel type antiferromagnetic ordering.

The spin dynamics is initiated by rapidly ramping down the short lattice and thereby

the double-well barrier in 200μs, thus significantly increasing the tunneling and superex-

change couplings. After letting the system evolve for a hold time t, we freeze out the

spin-configuration by ramping up the barrier in 200μs, quenching both J and Jex again.

The measurement of the mean values 〈x(t)〉 and 〈m(t)〉 is carried out as described above.

Three typical time traces obtained by this procedure are shown in fig. 18. For low

barrier depths (J/U > 1), we observe a pronounced time evolution of the spin imbalance

m(t) consisting of two frequency components with comparable amplitudes and frequen-

cies (fig. 18a). With increasing interaction energy U relative to J , the frequency ra-

tio increases, leaving a slow component with almost full amplitude and an additional

high-frequency but small-amplitude modulation (fig. 18b). For J/U � 1, the fast os-

cillation is completely suppressed and the only process visible is the superexchange os-

cillation (fig. 18c). For all barrier heights, the population imbalance 〈x(t)〉 stays flat,

emphasizing that even though strong spin-currents are present in the system, there is no

net mass flow for our initial state. Since the energies of the states |↑, ↓〉 and |↓, ↑〉 are

hardly affected by small deviations of the tilt from Δ = 0, the damping of the signal due

to dephasing is only small, thus allowing us to tune the system far into the regime of

strong interactions.

The comparison of the results with the theoretical predictions by the simple Hubbard

model shows significant deviations toward low barriers which cannot be explained by our

262 I. Bloch

0 1 2 3

-0.5

0.0

0.5

t (ms)

Po

pu

latio

n a

nd

sp

in im

ba

lan

ce

0 2 4 6 8

-0.5

0.0

0.5

0 50 100 150 200

-0.5

0.0

0.5

PopulationSpin

a

b

c

Fig. 18. – (Colour on-line) Spin and population dynamics in symmetric double wells. Thetime evolution of the mean spin 〈m(t)〉 (blue circles) and population imbalance 〈x(t)〉 (browncircles) are shown for three barrier depths within the double-well potential: (a) Vshort = 6Er,J/U = 1.25, (b) Vshort = 11Er, J/U = 0.26 and (c) Vshort = 17Er, J/U = 0.048. The measuredtraces for the spin imbalance are fitted to the sum of two damped sine waves (blue lines). Thepopulation imbalance x(t) stays flat for all traces.

uncertainties in the lattice depths. In this region, we find an extended Hubbard model

to describe the experimental data much more accurately, which can be understood by

the fact that the inter-well interaction energy increases with decreasing barrier and thus

has to be included in the model [84].

7. – Quantum noise correlations

For now almost 15 years, absorption imaging of released ultracold quantum gases

has been a standard detection method for revealing information on the macroscopic

quantum state of the atoms in the trapping potential. For strongly correlated quantum

states in optical lattices, however, the average signal in the momentum distribution that

one usually observes, e.g. for a Mott insulating state of matter, is a featureless Gaussian

wave packet. From this Gaussian wave packet one cannot deduce anything about the

strongly correlated quantum states in the lattice potential apart from the fact that phase

coherence has been lost. Recently, however, the widespread interest in strongly correlated

quantum gases in optical lattices as quantum simulators has lead to the prediction of

intriguing quantum phases for ultracold atoms, e.g. with anti-ferromagnetic structure,

spin waves or charge density waves. So far it has not been clear how one could detect


Fig. 19. – Hanbury-Brown-Twiss correlations in expanding quantum gases from an optical lattice.For bosonic particles that are detected at distances r (e.g. on a CCD camera), an enhanceddetection probability exists due to the two indistinguishable paths the particles can take to thedetector. This leads to enhanced fluctuations at special detection distances r, depending on theordering of the atoms in the lattice. Detection of the noise correlation can therefore yield novelinformation on the quantum phases in an optical lattice.

those states. Recently a theoretical proposal by Altman et al. [85] has shown that noise

correlation interferometry could be a powerful tool to directly visualize such quantum

states. Noise correlation in expanding ultracold atom clouds can in fact be seen as

a powerful way to read out the quantum states of an optical-lattice–based quantum

simulator.

The basic effect relies on the fundamental Hanbury-Brown-Twiss correlations [86, 87]

in the fluctuation signal of an atomic cloud. For bosons, e.g., a bunching effect of the

fluctuations is predicted to occur at special momenta of the expanding cloud, which di-

rectly reflect the ordering of the atoms in the lattice. Such bunching effects in momentum

space can be directly revealed as spatial correlations in the expanding atom cloud. Our

goal therefore is to reveal correlations in the fluctuations of the expanding atomic gas

after it has been released from the trap. Such correlations in the expanding cloud at

distance d can be quantified through the second-order correlation function

(29) C(d) =

∫〈n(x + d/2)n(x − d/2)〉d2x∫〈n(x + d/2)〉〈n(x − d/2)〉d2x

.

Here n(x) is the density distribution of a single expanding atom cloud and the angular

brackets 〈·〉 denote a statistical averaging over several individual images taken for different

experimental runs.

264 I. Bloch

Spatial correlations in the noise of expanding atom clouds arise here due to a funda-

mental indistinguishability of the particles as is well known from the foundational exper-

iments in quantum optics of Hanbury, Brown and Twiss [86, 87]. Let us for simplicity

consider two detectors spaced at a distance d below our trapped atoms and furthermore

restrict the discussion two only two atoms trapped in the lattice potential (see fig. 19).

As the trapping potential is removed and the particles propagate to the detectors, there

are two possibilities for the particles to reach these detectors, such that one particle is

detected at each detector. First, the particles can propagate along path A in fig. 19

to achieve this. However, another propagation path exists, which is equally probable,

path B in fig. 19. If we fundamentally cannot distinguish which way the particles have

been propagating to our detectors, we have to form the sum for bosons or difference

for fermions of the two propagation amplitudes and square the resulting value to obtain

the two particle detection probability at the detectors. As one increases the separation

between the detectors, the phase difference between the two propagation paths increases,

leading to constructive and destructive interference effects in the two-particle detection

probability. The length scale of this modulation in the two-particle detection probability

of the expanding atom clouds depends on the original separation of the trapped particles

at a distance alat and is given by the characteristic length scale

(30) l =h

malat

t,

where t is the time of flight.

Such Hanbury-Brown-Twiss correlations in the shot noise of an expanding atom cloud

from a Mott insulating state of matter have recently been observed for 3D and 2D Mott

insulating states [88, 9]. Similar pair correlations at the shot noise level have also been

obtained in the group of D. Jin [89] for dissociated molecular fragments. In the following

we will discuss how these noise correlations can be described formally and how they are

different from typical time-of-flight absorption images.

7.1. Time-of-flight versus noise correlations. – Let us begin by considering a quantum

gas released from a trapping potential. After a finite time-of-flight time t, the resulting

density distribution yields a three-dimensional density distribution n3D(x). If interac-

tions can be neglected during time-of-flight, the average density distribution is related

to the in-trap quantum state via

〈n3D(x)〉tof = 〈a†tof

(x)atof(x)〉tof(31)

≈ 〈a†(k)a(k)〉trap = 〈n3D(k)〉trap,

where k and x are related by the ballistic expansion condition k = Mx/�t (a factor

(M/�t)3 from the transformation of the volume elements d3x → d3k is omitted in the

equation). Here we have used the fact that for long time-of-flight times, the initial

size of the atom cloud in the trap can be neglected. It is important to realize that in

each experimental image, a single realization of the density is observed, not an average.


Moreover, each pixel in the image records on average a substantial number Nσ of atoms.

For each of those pixels, however, the number of atoms recorded in a single realization

of an experiment will exhibit shot noise fluctuations of relative order 1/√Nσ which will

be discussed below. As shown in eq. (31), the density distribution after time of flight

represents a momentum distribution reflecting the first-order coherence properties of the

in-trap quantum state. This assumption is however only correct, if during the expansion

process interactions between the atoms do not modify the initial momentum distribution,

which we will assume throughout the text. When the interactions between the atoms

have been enhanced, e.g. by a Feshbach resonance, or a high-density sample is prepared,

such an assumption is not always valid. Near Feshbach resonances one therefore often

ramps back to the zero crossing of the scattering length before expansion.

Density-density correlations in time-of-flight images. – Let us now turn to the ob-

servation of density-density correlations in the expanding atom clouds [85]. These are

characterized by the density-density correlation function

(32) 〈n(x)n(x′)〉 = 〈n(x)〉〈n(x′)〉g(2)(x,x′) + δ(x − x′)〈n(x)〉,

which contains the normalized pair distribution g(2)(x,x′) and a self-correlation term.

Relating the operators after time-of-flight expansion to the in-trap momentum operators,

using eq. (31), one obtains

〈n3D(x)n3D(x′)〉tof ≈ 〈a†(k)a(k)a†(k′)a(k′)〉trap =(33)

〈a†(k)a†(k′)a(k′)a(k)〉trap + δkk′〈a†(k)a(k)〉trap.

The last term on the r.h.s. of the above equation is the autocorrelation term and will

be dropped in the subsequent discussion, as it only contributes to the signal for x = x′

and contains no more information about the initial quantum state than the momentum

distribution itself. The first term, however, shows that for x �= x′, subtle momentum-

momentum correlations of the in-trap quantum states are present in the noise-correlation

signal of the expanding atom clouds.

Let us discuss the obtained results for two cases that have been analyzed in the

experiment: 1) Ultracold atoms in a Mott insulating state or a fermionic band insulating

state released from a 3D optical lattice and 2) two interfering one-dimensional quantum

gases separated by a distance d.

7.2. Noise correlations in bosonic Mott and fermionic band insulators. – Consider

a bosonic Mott insulating state or a fermionic band insulator in a three-dimensional

simple cubic lattice. In both cases, each lattice site R is occupied by a fixed atom

number nR. Such a quantum gas is released from the lattice potential and the resulting

density distribution is detected after a time of flight t. In a deep optical lattice, the

(in-trap) field operator ψ(r) can be expressed as a sum over destruction operators aR of

266 I. Bloch

localized Wannier states and neglecting all but the lowest band. The field operator for

destroying a particle with momentum k is therefore given by

(34) a(k) =

∫e−ikrψ(r)d3r � w(k)

∑R

e−ikRaR,

where w(k) denotes the Wannier function in momentum space.

For the two states considered here, the expectation value in eq. (33) factorizes into

one-particle density matrices 〈a†RaR′〉 = nR δR,R′ with vanishing off-diagonal order. The

density-density correlation function after a time of flight is then given by (omitting the

autocorrelation term of order 1/N)

〈n3D(x)n3D(x′)〉 = |w(Mx/�t)|2|w(Mx′/�t)|2N2(35)

×

⎡⎣1 ±

1

N2

∣∣∣∣∣∑R

ei(x−x′)·R(M/�t)nR

∣∣∣∣∣2⎤⎦ .

The plus sign in the above equation corresponds to the case of bosonic particles and

the minus sign to the case of fermionic particles in a lattice. Both in a Mott state

of bosons and in a filled band of fermions, the local occupation numbers nR are fixed

integers. The above equation then shows that correlations or anticorrelations in the

density-density expectation value appear for bosons or fermions, whenever the difference

k − k′ is equal to a reciprocal lattice vector G of the underlying lattice. In real space,

where the images are actually taken, this corresponds to spatial separations for which

(36) |x − x′| = � =2ht

λM.

Such spatial correlations or anticorrelations in the quantum noise of the density dis-

tribution of expanding atom clouds can in fact be traced back to the famous Hanbury,

Brown and Twiss effect [90, 86, 87] and its analogue for fermionic particles [91-96]. For

the case of two atoms localized at two lattice sites, this can be readily understood in the

following way: there are two possible ways for the particles to reach two detectors at

positions x and x′ which differ by exchange. A constructive interference for the case of

bosons or a destructive interference for the case of fermions then leads to correlated or

anticorrelated quantum fluctuations that are registered in the density-density correlation

function [85,87].

The correlations for the case of a bosonic Mott insulating state and anticorrelations

for the case of a fermionic band insulating state have recently been observed experimen-

tally [88, 95, 9]. In these experiments several single images of the desired quantum state

are recorded after releasing the atoms from the optical trapping potential and observing

them after a finite time-of-flight time (for a single of these images see, e.g., fig. 20a or

fig. 21a). These individually recorded images only differ in the atomic shot noise from


x10-4

x (μm)-400 -200 0 200 400

0.2

0.1

0

x (μm)-400 -200 0 200 400

-2

0

2

4

6

2

4

6

0

a

-2

db

c

Co

rr.

Am

p.

(x1

0-4

)

Co

um

n D

en

sty

(a

.u.)

Fig. 20. – Single-shot absorption image including quantum fluctuations and associated spatialcorrelation function. a) 2D column density distribution of a Mott insulating atomic cloudcontaining 6 × 105 atoms, released from a 3D optical lattice potential with a lattice depth of50Er. The white bars indicate the reciprocal lattice scale l defined in eq. (30). b) Horizontal cut(black line) through the centre of the image in a) and Gaussian fit (red line) to the average over43 independent images each one similar to a). c) Spatial noise correlation function obtainedby analyzing the same set of images, which shows a regular pattern revealing the lattice orderof the particles in the trap. d) Horizontal profile through the centre of pattern, containing thepeaks separated by integer multiples of l. The width of the individual peaks is determined bythe optical resolution of our imaging system.

each other. A set of such absorption images is then processed to yield the spatially

averaged second-order correlation function g(2)

exp(b)

(37) g(2)

exp(b) =

∫〈n(x + b/2) · n(x − b/2)〉d2x∫〈n(x + b/2)〉〈n(x − b/2)〉d2x

.

As shown in fig. 20, the Mott insulating state exhibits long-range order in the pair

correlation function g(2)(b). This order is not connected with the trivial periodic mod-

ulation of the average density imposed by the optical lattice after time of flight, which

is factored out in g(2)(x,x′) (see eq. (32)). Therefore, in the superfluid regime, one ex-

pects g(2)(x,x′) ≡ 1, despite the periodic density modulation in the interference pattern

after time of flight. It is interesting to note that the correlations or anticorrelations can

268 I. Bloch

Fig. 21. – Noise correlations of a band insulating Fermi gas. Instead of the correlation bunchingpeaks observed in fig. 20, the fermionic quantum gas shows an HBT type antibunching effect,with dips in the observed correlation function [95].

also be traced back to the enhanced fluctuations in the population of the Bloch waves

with quasi-momentum q for the case of the bosonic particles and the vanishing fluctua-

tions in the population of Bloch waves with quasi-momentum q for the case of fermionic

particles [95].

Note that in general the signal amplitude obtained in the experiments for the cor-

relation function deviates significantly from the theoretically expected value of 1. In

fact, one typically observes signal levels of 10−4–10−3 (see figs. 20 and 21). This can

be explained by the finite optical resolution when imaging the expanding atomic clouds,

thus leading to a broadening of the detected correlation peaks and thereby a decreased

amplitude, as the signal weight in each correlation peak is preserved in the detection

process. Using single-atom detectors with higher spatial and temporal resolution such as

the ones used in refs. [97] and [96], one can overcome such limitations and thereby also

evaluate higher-order correlation functions.


8. – Outlook

Ultracold atoms in optical lattices have proven to be versatile model systems for the

investigation of strongly correlated many-body systems with ultracold atoms. For the

future, one of the focus areas of this research will be directed towards the realization

of quantum magnetism with ultracold atoms. Both the prospect of reaching low-energy

magnetically ordered quantum phases, such as an antiferromagnetically ordered phase

in the fermionic Hubbard model or the non-equilbirium spin dynamics of 1D and 2D

systems, seem within reach. To observe several of these pheneomena, the thermal energy

of the many-body system will have to be lowered below the superexchange interaction

energy scale, which currently requires further cooling of the atomic systems by a factor

of 2–3. If such antiferromagnets for fermionic particles could eventually be doped or ana-

lyzed at non-integer filling, one could hope to solve one of the long-standing problems of

condensed matter physics, namely whether the fermionic Hubbard model with repulsive

interactions contains a superconducting phase [98, 46] and what the mechanism behind

such a superconducting phase could be.

∗ ∗ ∗

The author would like to acknowledge stimulating discussions with U. Schnei-

der, S. Will, S. Trotzky, B. Paredes, A. Rosch, E. Demler, M. Lukin,

A. Polkovnikov and W. Zwerger. This work was supported by the Deutsche

Forschungsgemeinschaft, the European Union, EuroQUAM, the US Army Research Of-

fice with funding from the Defense Advanced Research Projects Agency (Optical Lattice

Emulator programme), and the US Air Force Office of Scientific Research.

REFERENCES

[1] Jaksch D. and Zoller P., Ann. Phys. (N.Y.), 315 (2005) 52.[2] Lewenstein M., Sanpera A., Ahufinger V., Damski B., Sen De A. and Sen U., Adv.

Phys., 56 (2007) 243.[3] Bloch I., Dalibard J. and Zwerger W., Rev. Mod. Phys., 80 (2008) 885.[4] Chin C., Grimm R., Julienne P. and Tiesinga E., Rev. Mod. Phys., 82 (2010) 1225.[5] Fisher M. P. A., Weichman P. B., Grinstein G. and Fisher D. S., Phys. Rev. B, 40

(1989) 546.[6] Jaksch D., Bruder C., Cirac J. I., Gardiner C. W. and Zoller P., Phys. Rev. Lett.,

81 (1998) 3108.[7] Greiner M., Mandel M. O., Esslinger T., Hansch T. and Bloch I., Nature, 415

(2002) 39.[8] Stoferle T., Moritz H., Schori C., Kohl M. and Esslinger T., Phys. Rev. Lett., 92

(2004) 130403.[9] Spielman I. B., Phillips W. and Porto J., Phys. Rev. Lett., 98 (2007) 080404.

[10] Grimm R., Weidemuller M. and Ovchinnikov Y. B., Adv. At. Mol. Opt. Phys., 42

(2000) 95.[11] Jessen P. S. and Deutsch I. H., Adv. At. Mol. Opt. Phys., 37 (1996) 95.[12] Jaksch D., Briegel H. J., Cirac J. I., Gardiner C. W. and Zoller P., Phys. Rev.

Lett., 82 (1999) 1975.

270 I. Bloch

[13] Mandel O., Greiner M., Widera A., Rom T., Hansch T. W. and Bloch I., Phys.

Rev. Lett., 91 (2003) 10407.

[14] Petsas K., Coates A. and Grynberg G., Phys. Rev. A, 50 (1994) 5173.

[15] Hemmerich A. and Hansch T., Phys. Rev. Lett., 68 (1992) 1492.

[16] Greiner M., Bloch I., Mandel M. O., Hansch T. and Esslinger T., Phys. Rev. Lett.,87 (2001) 160405.

[17] Moritz H., Stoferle T., Kohl M. and Esslinger T., Phys. Rev. Lett., 91 (2003)250402.

[18] Tolra B., OHara K., Huckans J., Phillips W., Rolston S. and Porto J., Phys.

Rev. Lett., 92 (2004) 190401.

[19] Paredes B., Widera A., Murg V., Mandel O., Folling S., Cirac J. I., Shlyapnikov

G. V., Hansch T. W. and Bloch I., Nature, 429 (2004) 277.

[20] Kinoshita T., Wenger T. and Weiss D. S., Science, 305 (2004) 1125.

[21] Zwerger W., J. Opt. B, 5 (2003) S9.

[22] Pitaevskii L. and Stringari S., Bose-Einstein Condensation (Oxford University Press,Oxford) 2003.

[23] Sheshadri K., Krishnamurty H. R., Pandit R. and Ramakrishnan T. V., Europhys.

Lett., 22 (1993) 257.

[24] van Oosten D., van Der Straten P. and Stoof H., Phys. Rev. A, 63 (2001) 1.

[25] Pedri P., Pitaevskii L., Stringari S., Fort C., Burger S., Cataliotti F. S.,

Maddaloni P., Minardi F. and Inguscio M., Phys. Rev. Lett., 87 (2001) 220401.

[26] Sachdev S., Quantum Phase Transitions (Cambridge University Press, Cambridge) 1999.

[27] Kashurnikov V. A., Prokof’ev N. V. and Svistunov B. V., Phys. Rev. A, 66 (2002)31601.

[28] Roth R. and Burnett K., Phys. Rev. A, 66 (2003) 031602(R).

[29] Gerbier F., Widera A., Folling S., Mandel O., Gericke T. and Bloch I., Phys.

Rev. Lett., 95 (2005) 50404.

[30] Wessel S., Alet F., Troyer M. and Batrouni G. G., Phys. Rev. A, 70 (2004) 53615.

[31] Kollath C., Schollwock U., von Delft J. and Zwerger W., Phys. Rev. A, 69 (2004)031601(R).

[32] Campbell G. K., Mun J., Boyd M., Medley P., Leanhardt A. E., Marcassa L. G.,

Pritchard D. E. and Ketterle W., Science, 313 (2006) 5787.

[33] Folling Simon, Widera Artur, Muller Torben, Gerbier Fabrice and Bloch

Immanuel, Phys. Rev. Lett., 97 (2006) 60403.

[34] Sengupta P., Rigol M., Batrouni G. G., Denteneer P. J. H. and Scalettar R. T.,Phys. Rev. Lett., 95 (2005) 220402.

[35] Glauber R. J., Phys. Rev., 131 (1963) 2766.

[36] Brune M., Schmidt-Kaler F., Maali A., Dreyer J., Hagley E., Raimond J. M.

and Haroche S., Phys. Rev. Lett., 76 (1996) 1800.

[37] Haroche S. and Raimond J.-M., Exploring the Quantum, Oxford Graduate Texts(Oxford University Press) 2006.

[38] Meekhof D. M., Monroe C., King B. E., Itano W. M. and Wineland D. J., Phys.

Rev. Lett., 76 (1996) 1796.

[39] Poschinger U. G., Huber G., Ziesel F., Deiß M., Hettrich M., Schulz S. A.,

Singer K., Poulsen G., Drewsen M., Hendricks R. J. and Schmidt-Kaler F., J.

Phys. B, 42 (2009) 154013.

[40] Greiner M., Mandel M. O., Hansch T. W. and Bloch I., Nature, 419 (2002) 51.

[41] Anderlini M., Sebby-Strabley J., Kruse J., Porto J. V. and Phillips W. D., J.

Phys. B, 39 (2006) S199.


[42] Sebby-Strabley J., Brown B. L., Anderlini M., Lee P. J., Johnson P. R., Phillips

W. D. and Porto J. V., Phys. Rev. Lett., 98 (2007) 200405.[43] Johnson P. R., Tiesinga E., Porto J. V. and Williams C. J., New J. Phys., 11 (2009)

093022.[44] Luhmann D.-S., Bongs K., Sengstock K. and Pfannkuche D., Phys. Rev. Lett., 101

(2008) 50402.[45] Lutchyn R. M., Tewari S. and Das Sarma S., Phys. Rev. A, 79 (2009) 11606.[46] Lee P. A., Nagaosa N. and Wen X.-G., Rev. Mod. Phys., 78 (2006) 17.[47] Hubbard J., Proc. R. Soc. London, Ser. A, 276 (1963) 238.[48] Gebhard F., The Mott metal-insulator transition - models and methods (Springer, New

York) 1997.[49] Anderson P. W., Phys. Rev., 109 (1958) 1492.[50] Billy J., Josse V., Zuo Z., Bernard A., Hambrecht B., Lugan P., Clement D.,

Sanchez-Palencia L., Bouyer Ph. and Aspect A., Nature, 453 (2008) 891.[51] Roati G., D/’Errico Ch., Fallani L., Fattori M., Fort Ch., Zaccanti M.,

Modugno G., Modugno M. and Inguscio M., Nature, 453 (2008) 895.[52] Jordens R., Strohmaier N., Gunter K., Moritz H. and Esslinger T., Nature, 455

(2008) 204.[53] Schneider U., Hackermuller L., Will S., Bloch I., Best Th., Costi T. A., Helmes

R. W., Rasch D. and Rosch A., Science, 322 (2008) 1520.[54] Gerbier F., Folling S., Widera A., Mandel O. and Bloch I., Phys. Rev. Lett., 96

(2006) 90401.[55] Georges A., Kotliar G., Krauth W. and Rozenberg M. J., Rev. Mod. Phys., 68

(1996) 13.[56] Kotliar G. and Vollhardt D., Phys. Today, 57 (2004) 53.[57] Helmes R. W., Costi T. A. and Rosch A., Phys. Rev. Lett., 100 (2008) 56403.[58] DeLeo L., Kollath C., Georges A., Ferrero M. and Parcollet O., Phys. Rev.

Lett., 101 (2008) 210403.[59] Regal C. A. and Jin D. S., Phys. Rev. Lett., 90 (2003) 230404.[60] Andrews M. R., Mewes M.-O., van Druten N. J., Durfee D. S., Kurn D. M. and

Ketterle W., Science, 273 (1996) 84.[61] Kastberg A., Phillips W. D., Rolston S. L., Spreeuw R. J. C. and Jessen P. S.,

Phys. Rev. Lett., 74 (1995) 1542.[62] Kohl M., Moritz H., Stoferle T., Gunter K. and Esslinger T., Phys. Rev. Lett.,

94 (2005) 80403.[63] Bulla R., Costi T. A. and Vollhardt D., Phys. Rev. B, 64 (2001) 45103.[64] Bulla R., Costi T. A. and Pruschke T., Rev. Mod. Phys., 80 (2008) 395.[65] Helmes R. W., Costi T. A. and Rosch A., Phys. Rev. Lett., 101 (2008) 66802.[66] Werner F., Parcollet O., Georges A. and Hassan S. R., Phys. Rev. Lett., 95 (2005)

56401.[67] Winkler K., Thalhammer G., Lang F., Grimm R., Hecker-Denschlag J., Daley

A. J., Kantian A., Buchler H. P. and Zoller P., Nature, 441 (2006) 853.[68] Gerbier F., Phys. Rev. Lett., 99 (2007) 120405.[69] Bernier J.-S., Kollath C., Georges A., De Leo L., Gerbier F., Salomon Ch. and

Kohl M., Phys. Rev. A, 79 (2009) 061601.[70] Jordens R., Tarruell L., Greif D., Uehlinger Th., Strohmaier N., Moritz

H., Esslinger T., De Leo L., Kollath C., Georges A., Scarola V., Pollet L.,

Burovski E., Kozik E. and Troyer M., Phys. Rev. Lett., 104 (2010) 180401.[71] Ising E., Z. Phys. A, 31 (1925) 253.[72] Heisenberg W., Z. Phys. A, 38 (1926) 411.

272 I. Bloch

[73] Auerbach A., Interacting Electrons and Quantum Magnetism (Springer) 2006.[74] Dirac P. A. M., Proc. R. Soc. London, Ser. A, 112 (1926) 661.[75] Dirac P. A. M., Proc. R. Soc. London, Ser. A, 123 (1929) 714.[76] Heisenberg W., Z. Phys., 49 (1928) 619.[77] Kramers H. A., Physica, 1 (1934) 182.[78] Anderson P. W., Phys. Rev., 79 (1950) 350.[79] Petta J. R., Johnson A. C., Taylor J. M., Laird E. A., Yacoby A., Lukin M. D.,

Marcus C. M., Hanson M. P. and Gossard A. C., Science, 309 (2005) 2180.[80] Briegel H. J. and Raussendorf R., Phys. Rev. Lett., 86 (2001) 910.[81] Kitaev A., Ann. Phys. (N.Y.), 321 (2006) 2.[82] Duan L.-M., Demler E. and Lukin M. D., Phys. Rev. Lett., 91 (2003) 90402.[83] Folling S., Trotzky S., Cheinet P., Feld M., Saers R., Widera A., Muller T.

and Bloch I., Nature, 448 (2007) 1029.[84] Trotzky S., Cheinet P., Folling S., Feld M., Schnorrberger U., Rey A. M.,

Polkovnikov A., Demler E. A., Lukin M. D. and Bloch I., Science, 319 (2008) 295.[85] Altman E., Demler E. and Lukin M. D., Phys. Rev. A, 70 (2004) 13603.[86] Brown R. Hanbury and Twiss R. Q., Nature, 178 (1956) 1447.[87] Baym G., Act. Phys. Pol. B, 29 (1998) 1839.[88] Folling S., Gerbier F., Widera A., Mandel O., Gericke T. and Bloch I., Nature,

434 (2005) 481.[89] Greiner M., Regal C., Stewart J. and Jin D., Phys. Rev. Lett., 94 (2005) 070403.[90] Brown R. Hanbury and Twiss R. Q., Nature, 177 (1956) 27.[91] Henny M., Oberholzer S., Strunk C., Heinzel T., Ensslin K., Holland M. and

Schonenberger C., Science, 284 (1999) 296.[92] Oliver W. D., Kim J., Liu R. C. and Yamamoto Y., Science, 284 (1999) 299.[93] Kiesel H., Renz A. and Hasselbach F., Nature, 418 (2002) 392.[94] Iannuzzi M., Orecchini A., Sacchetti F., Facchi P. and Pascazio S., Phys. Rev.

Lett., 96 (2006) 80402.[95] Rom T., Best Th., van Oosten D., Schneider U., Folling S., Paredes B. and

Bloch I., Nature, 444 (2006) 733.[96] Jeltes T., McNamara J. M., Hogervorst W., Vassen W., Krachmalnicoff V.,

Schellekens M., Perrin A., Chang H., Boiron D., Aspect A. and Westbrook C.,Nature, 445 (2007) 402.

[97] Schellekens M., Hoppeler R., Perrin A., Gomes J. V., Boiron D., Aspect A. andWestbrook C. I., Science, 310 (2005) 648.

[98] Hofstetter W., Cirac J. I., Zoller P., Demler E. and Lukin M. D., Phys. Rev.

Lett., 89 (2002) 220407.


Two-dimensional Bose fluids: An atomic physics

perspective

Z. Hadzibabic

Cavendish Laboratory, University of Cambridge

JJ Thomson Avenue, Cambridge CB3 0HE, UK

J. Dalibard

Laboratoire Kastler Brossel, CNRS, UPMC, Ecole Normale Superieure

24 rue Lhomond, 75005 Paris, France

Summary. — We give in this lecture an introduction to the physics of two-dimensional (2d) Bose gases. We first discuss the properties of uniform, infinite2d Bose fluids at non-zero temperature T . We explain why thermal fluctuationsare strong enough to destroy the fully ordered state associated with Bose-Einsteincondensation, but are not strong enough to suppress superfluidity in an interact-ing system at low T . We present the basics of the Berezinskii-Kosterlitz-Thoulesstheory, which provides the general framework for understanding 2d superfluidity.We then turn to experimentally relevant finite-size systems, in which the presenceof residual “quasi–long-range” order at low temperatures leads to an interesting in-terplay between superfluidity and condensation. Finally we summarize the recentprogress in theoretical understanding and experimental investigation of ultracoldatomic gases confined to a quasi-2d geometry.


274 Z. Hadzibabic and J. Dalibard

1. – Introduction

The properties of phase transitions and the types of order present in the low-tempera-

ture states of matter are fundamentally dependent on the dimensionality of physical

systems [1]. Generally, highly ordered states are more robust in higher dimensions, while

thermal and quantum fluctuations, which favor disordered states, play a more important

role in lower dimensions.

The case of a two-dimensional (2d) Bose fluid is particularly fascinating because of

its “marginal” behavior. In an infinite uniform 2d fluid thermal fluctuations at any non-

zero temperature are strong enough to destroy the fully ordered state associated with

Bose-Einstein condensation, but are not strong enough to suppress superfluidity in an

interacting system at low, but non-zero temperatures. Further, the presence of residual

“quasi–long-range” order at low temperatures leads to an interesting interplay between

superfluidity and condensation in all experimentally relevant finite-size systems.

This behavior is characteristic of a wide range of physical systems which share some

generic properties such as dimensionality, form of interactions, and Hamiltonian sym-

metries. These include liquid-helium films [2], spin-polarized hydrogen [3], Coulomb

gases [4], ultracold atomic gases [5, 6], exciton [7, 8] and polariton [9, 10] systems. More-

over, the Berezinskii-Kosterlitz-Thouless theory which provides the general framework

for understanding 2d superfluidity is also applicable to a range of physical phenomena

in discrete systems, such as ordering of spins on a 2d lattice [11] and melting of 2d

crystals [12, 13].

In this paper we give an introduction to the physics of 2d Bose fluids from an atomic

physics perspective. Our goal is to summarize the recent progress in theoretical under-

standing and experimental investigation of ultracold atomic gases confined to 2d geom-

etry, and we also hope to provide a useful introduction to these systems for researchers

working on related topics in other fields of physics.

1.1. Absence of true long-range order in 2d . – The most familiar phase transitions

in three spatial dimensions (3d), such as freezing of water, ferromagnetic ordering in

spin systems, or Bose-Einstein condensation (BEC), are all associated with emergence

of true long-range order (LRO) below some non-zero critical temperature. Such order is

embedded in a spatially uniform order parameter, e.g. magnetization in a ferromagnet

or the macroscopic wave function ψ describing a BEC. Further, in all the above cases

emergence of true LRO corresponds to spontaneous breaking of some continuous sym-

metry of the Hamiltonian. In case of crystallization (freezing), translational symmetry is

spontaneously broken. For Heisenberg spins on a (fixed) lattice, the Hamiltonian has a

continuous spin rotational symmetry which is spontaneously broken in the ferromagnetic

state. In case of BEC, the phase of ψ is arbitrarily spontaneously chosen at the transition.

As we will discuss in more detail later, under certain conditions this makes a Bose gas for-

mally equivalent to a system of two-component spins on a lattice, the so-called XY model.

Already in 1934 Peierls pointed out that the possibility for a physical system to

exhibit true LRO can crucially depend on its dimensionality [14, 15]. Peierls considered

Two-dimensional Bose fluids: An atomic physics perspective 275

a 2d crystal at finite temperature T , and studied the effects of thermal vibrations of

atoms around their equilibrium positions in the lattice (i.e. phonons). He found that

the uncertainty in the relative position of two atoms diverges with the distance between

their equilibrium positions:

(1)⟨(u(r) − u(0))2

⟩∝ T ln

( ra

)

for r = |r| � a, where u(r) is the atom displacement from its equilibrium position r, a is

the lattice spacing, and 〈. . .〉 denotes a thermal average. In contrast, the corresponding

result in 3d is finite, and small compared to a if T is below the melting temperature.

The result in eq. (1) is in direct contradiction with the starting hypothesis of long-range

crystalline order, since it implies that, based on the positions of atoms in one part of the

system, we cannot predict with any certainty the positions of atoms at large distances.

The Peierls result (1) is a simple example of the absence of spontaneous symmetry

breaking at non-zero T in 2d. The absence of LRO in low-dimensional systems was later

more generally and formally studied by Bogoliubov [16], Hohenberg [17], and Mermin

and Wagner [18]. The general statement is that LRO is impossible in the thermodynamic

limit at any non-zero T in all 1d and 2d systems with short-ranged interactions and a

continuous Hamiltonian symmetry. This is now most commonly known as the Mermin-

Wagner theorem. In all such systems Hamiltonian symmetry is always restored by low-

energy long-wavelength thermal fluctuations, the so-called Goldstone modes. In the case

of an interacting Bose gas Goldstone modes are phonons, while in the case of the XY

model on a lattice they are spin-waves. As a direct consequence of the functional form of

the density of states in low dimensionality, such modes always have a diverging infrared

contribution and destroy LRO. It is however important to stress that the absence of true

LRO does not preclude the possibility of any phase transitions in 2d systems, just the

symmetry-breaking ones. As we will see later, a phase transition associated with the

apparition of a topological order does take place in a uniform 2d gas.

1.2. Outline of the paper . – In sect. 2 we will explicitly see that Bose-Einstein con-

densation is impossible in both the ideal and a repulsively interacting infinite uniform

2d Bose gas. The absence of true LRO in these systems is seen in the fact that the

first-order correlation function

(2) g1(r) ≡⟨Ψ†(r)Ψ(0)

⟩,

where Ψ(r) is the annihilation operator for a particle at position r, always tends to zero

for r → ∞. Note that these two statements are equivalent under the Penrose-Onsager

definition of the condensate density [19]:

(3) n0 ≡ limr→∞

g1(r).


However, the weak logarithmic divergence in eq. (1) suggests that the destruction

of LRO is only marginal in 2d. The consequence of this weak divergence is that an

interacting Bose gas at low T exhibits “quasi–long-range order”, corresponding to g1(r)

which decays only algebraically with distance.

This low-T state is also superfluid, and the phase transition between the superfluid

and the high-T normal state is described by the Berezinskii-Kosterlitz-Thouless (BKT)

theory [20,21], which we discuss in sect. 3. Further, the slow decay of g1(r) at low T has

important consequences for condensation and symmetry breaking in the experimentally

relevant finite-size systems. We address this issue in sects. 4 and 5, first for a finite

box potential, and then for the experimentally most pertinent case of a harmonically

trapped gas.

In sects. 6 and 7 we introduce the experimental methods of atomic physics used in

the current studies of 2d Bose gases. In sect. 6 we give an overview of experimental

systems in which (quasi-)2d Bose gases have been realized, and in sect. 7 we discuss

the experimental probes of coherence in these systems. We conclude by outlining some

research directions which are likely to be of interest in the near future in sect. 8.

2. – The infinite uniform 2d Bose gas at low temperature

In this section we discuss thermal fluctuations and the absence of Bose-Einstein con-

densation in an infinite uniform 2d gas. In subsect. 2.1 we consider the ideal gas, in which

no phase transition occurs. In this case the first order correlation function g1(r) gradu-

ally changes from a gaussian function in the high-temperature, non-degenerate regime,

to an exponentially decaying function in the degenerate regime.

In the repulsively interacting 2d Bose gas the BKT phase transition to a superfluid

state occurs at a non-zero critical temperature, but the conclusion that BEC transition

does not occur remains true. We first introduce the description of interactions in an

atomic 2d gas in subsect. 2.2. In subsect. 2

.3 we qualitatively discuss why interactions

lead to a strong suppression of density fluctuations in a degenerate gas, so that the low-

energy long-wavelength excitations (phonons) in this system are almost purely phase

fluctuations. The Bogoliubov analysis presented in subsect. 2.4 provides a more quan-

titative justification for this conclusion and also indicates why we expect an interacting

2d gas to be superfluid at very low T . As we show in subsect. 2.5, the long-wavelength

phonons still lead to a vanishing g1(r) at r → ∞. Therefore, in accordance with the

Mermin-Wagner theorem, these “soft” Goldstone modes still destroy the long-range or-

der and restore the Hamiltonian symmetry. However, the decay of g1 at large distances

is only algebraic with r at very low T . This low temperature, superfluid state is said to

exhibit “quasi–long-range order”. The BKT phase transition from the superfluid state

with algebraic correlations to the normal state with exponentially decaying correlations

is discussed in the following sect. 3.

2.1. The ideal 2d Bose gas. – The absence of Bose-Einstein condensation in the ideal

2d Bose gas can straightforwardly be seen by following the standard Einstein’s argument


which associates condensation with the saturation of the excited single-particle states at

some non-zero temperature.

In 2d, the density of states for spinless bosons mL2/(2π�2) is constant, where m is

the particle mass and L → ∞ is the linear size of the system. Assuming no condensation,

the total number of particles is then

(4) N =mL2

2π�2

∫∞

0

dε

eβ(ε−μ) − 1,

where β = 1/(kBT ), and μ ≤ 0 is the chemical potential. Equivalently, the phase-space

density D is given by

(5) D ≡ nλ2 =

∫∞

0

dx1

Z ex − 1

= − ln(1 − Z),

where n = N/L2 is the 2d number density, λ = h/√

2πmkBT is the thermal wavelength,

and Z = eβμ is the fugacity.

In 3d, the signature of BEC is that the analogous relationship between the phase

space density and fugacity has no solutions for Z when the phase space density is larger

than the critical value n3λ3 ≈ 2.612, where n3 is the 3d number density. Below the

condensation temperature, chemical potential is fixed at μ = 0 and the phase space

density of particles in the excited states is saturated at ≈ 2.612. However, we see that

in 2d a valid solution,

(6) eβμ = 1 − e−nλ2

,

always exists. In other words, for any non-infinite phase space density there exists a

negative value of μ which allows normalization of the thermal distribution of particles

in the excited states to the total number of particles in the system N . This shows that

BEC does not occur in the ideal infinite uniform 2d Bose gas.

We next look at the first-order correlation function g1(r), which we can write as the

Fourier transform of the momentum space distribution function nk:

(7) g1(r) =1

(2π)2

∫∞

0

nk eik·r d2k, with nk =

1

eβ(εk−μ) − 1, εk =

�2k2

2m.

In the absence of condensation g1(r) always vanishes at r → ∞. However, it still shows

qualitatively different behavior at high and low temperature:

– In a non-degenerate gas, eq. (6) gives Z ≈ nλ2 � 1. In this regime |μ| � kBT and

all momentum states are weakly occupied:

(8) nk ≈ Z e−βεk ≈ nλ2 e−k2λ2/4π � 1 (∀k).


In this case g1(r) is Gaussian, showing only short-range correlations which decay

on the length scale λ/√π:

(9) g1(r) ≈ n e−πr2/λ2

.

– In a degenerate gas with nλ2 > 1, from eq. (6) we get Z ≈ 1 and β|μ| ≈ e−nλ2

� 1,

so that D = nλ2 ≈ ln(kBT/|μ|). In this regime the occupation of high-energy

states, with βεk � 1, is still small and given by the Boltzmann factor

(10) nk ≈ e−βεk = e−k2λ2/4π � 1, for k2 � 4π/λ2.

However, the low-energy states with βεk � 1 are strongly occupied:

(11) nk ≈kBT

εk + |μ|=

4π

λ2

1

k2 + k2c

� 1, for k2 � 4π/λ2,

where kc =√

2m|μ|/�. In this case g1(r) is bimodal. At short distances, up to

r ∼ λ, correlations are still Gaussian as in eq. (9). However, the Lorentzian form

in eq. (11) corresponds to approximately(1) exponential decay of g1(r) at larger

distances, r � λ:

(12) g1(r) ≈ e−r/ , with � = k−1

c ≈ λ enλ2/2/√

4π.

We can also estimate the partial phase space densities corresponding to the Gaus-

sian and the Lorentzian parts of the momentum distribution, and see that most

particles accumulate in the Lorentzian part corresponding to low momentum states:

DG ≈λ2

(2π)2

∫∞

√4π/λ2

nk d2k ≈ 1/e � D,(13)

DL ≈λ2

(2π)2

∫ √4π/λ2

0

nk d2k ≈ D.(14)

We thus see that even though there is no phase transition in this system, the first-order

correlation function gradually changes from a Gaussian with short-range correlations

in the non-degenerate regime, to an exponential in the degenerate regime. Further,

for nλ2 > 1, the correlation length � ∝ enλ2/2 grows exponentially. Therefore, while

g1(r) formally vanishes at r → ∞ at any non-zero T , the length scale over which it

decays is exponentially large in the deeply degenerate ideal Bose gas. This has important

(1) More precisely the Fourier transform of the 2d Lorentzian distribution 1/(k2 +k2

c ) is definedfor r = 0 and is proportional to the Bessel function of imaginary argument K0(kcr), whoseasymptotic behaviour is e−kcr

/√r.


consequences for the finite-sized experimental systems, since for any arbitrarily large fixed

system size L there exists a low enough non-zero temperature T for which � > L, and

correlations span the entire system. This issue will be addressed in sects. 4 and 5.

2.2. Interactions in a 2d Bose gas at low T . – The interaction between two atoms

at positions ri and rj in a 3d Bose gas at low temperature is well characterized by the

contact potential(2)

(15) V (ri − rj) =4π�

2

mas δ

(3d)(ri − rj),

where as is the 3d s-wave scattering length, and mechanical stability requires repulsive

interactions as > 0. In 2d the two-body scattering problem is in general more complicated

and the scattering amplitude is energy-dependent [25]. We will address this issue in

more detail in subsect. 6.2. However, in all experimentally relevant situations so far, the

analysis of interactions is simplified by the fact that while the gas is kinematically 2d,

the interactions can be described by 3d scattering. The condition for this simplification

is that the thickness of the sample �0 is much larger than the 3d scattering length as. In

all current atomic experiments the ratio �0/as is larger than 30. In this case we can to

a very good approximation write the interaction energy as

(16) Eint =g

2

∫n2(r) d2r,

where g is the energy-independent interaction strength and n(r) is the local density.

Note that here and in the following we treat the density n(r)—and later the phase

θ(r)—as a classical function. This will notably simplify the mathematical aspects of our

approach, while capturing all important physical consequences related for example to

quasi–long-range order and to the normal-superfluid transition.

In 3d, the interaction strength g(3d) = (4π�2/m)as explicitly depends on the scatter-

ing length. However, we see on dimensional grounds that in 2d we can write

(17) g =�

2

mg,

so that g is a dimensionless coupling constant. The 2d healing length, which gives the

characteristic length scale corresponding to the interaction energy, is then given by

(18) ξ = �/√mgn = 1/

√gn .

(2) Strictly speaking, the delta-function in 3d must be regularized by using for example thepseudo-potential [22]. The extension of the notion of zero-range potential to the 2d case isdiscussed in [23] (see also [24] for a discussion in terms of many-body T -matrix).


We can anticipate on dimensional grounds that g ∼ as/�0, corresponding to n = n3�0.

Specifically, if a gas is harmonically confined to 2d, say in the x-y plane, we will see in

subsect. 6.2 that the expression for g is

(19) g =√

8πas

az,

where in this case �0 = az is the oscillator length along the kinematically frozen direc-

tion z.

We can qualitatively define the strongly interacting limit by the value of g for which

the interaction energy of N particles, Eint, reaches the kinetic energy EK of N non-

interacting particles equally distributed over the lowest N single-particle states. In this

limit, we expect the many-body ground state to be strongly correlated. We can estimate

the interaction energy with a mean-field approximation by setting n(r) = n in eq. (16),

which corresponds to neglecting density fluctuations:

(20) Eint =�

2

2mgNn.

Using the 2d density of states, mL2/(2π�2), the energy of the N -th excited single-particle

state is EN = (2π�2/m)n, and so

(21) EK =1

2NEN =

π�2

mNn.

The strongly interacting limit then corresponds to

(22) Eint = EK ⇒ g = 2π.

For comparison, the value of g in the current experiments on atomic 2d Bose gases varies

between ∼ 10−1 [26, 27] and ∼ 10−2 [28], while in the more strongly interacting 4He

films [2] it is estimated to be of order 1 [29].

The fact that result (22) is independent of the density of the gas n is a natural conse-

quence of the fact that g is dimensionless. This is in contrast to the 3d case, where the

relative importance of interactions is characterized by the dimensionless parameter n3a3s.

2.3. Suppression of density fluctuations and the low-energy Hamiltonian. – At strictly

T = 0, a weakly interacting 2d Bose gas is condensed and described by a constant macro-

scopic wave function (a uniform order parameter) ψ =√neiθ, where n and θ are classical

fields. At any finite T , both the amplitude and the phase of ψ show thermal fluctuations.

However, repulsive interactions will always lead to a reduction of density fluctuations in a

low-temperature gas. The interaction energy is (g/2)∫n2(r) d2r = (g/2)L2〈n2(r)〉, and

so keeping the average density n = 〈n(r)〉 fixed, we see that minimizing the interaction

energy is equivalent to minimizing the density fluctuations:

(23) (Δn)2 = 〈n2(r)〉 − n2 = (g2(0) − 1)n2,


where g2(r) = 〈n(r)n(0)〉/n2 is the normalized second-order (density-density) correla-

tion function(3). In the ideal Bose gas g2(0) = 2, while if the density fluctuations are

completely suppressed g2(0) = 1.

We can estimate the minimum cost of density fluctuations by equating it with the

increase in interaction energy from adding a single particle to the system:

(24)∂Eint

∂N= gn =

�2

mgn,

where we have set g2(0) = 1, which is consistent with the density fluctuations being

significantly suppressed. Comparing this with the thermal energy kBT , we get

(25)gn

kBT=

g

2πD.

This strongly suggests that at sufficiently low temperature, given by D � 2π/g, any

density fluctuations must be very strongly suppressed. Numerical calculations [29] show

that they can be significantly suppressed already for D � 1, with the exact extent of the

suppression depending on the strength of interactions g.

In the limit of strong suppression of density fluctuations, the interaction energy be-

comes just an additive constant ((1/2)gn2L2) in the Hamiltonian, and the kinetic energy

arises only from the variations of θ, the phase of ψ. The system is then often described

by an effective low-energy Hamiltonian:

(26) Hθ =�

2

2mns

∫(∇θ)2 d2r,

where one heuristically replaces the total density n with the (uniform) superfluid density

ns ≤ n. This is physically motivated because one expects only the superfluid component

to exhibit phase stiffness and to flow under an imposed variation of θ, with local velocity

of the superfluid given by vs = (�/m)∇θ. Also note that at T = 0 superfluid density is

equal to the total density, and at very low T they are similar. In essence, renormalizing n

to the lower ns is an effective way of absorbing all the short distance physics, including any

residual density fluctuations, and Hθ provides a good description of long-range physics,

at distances r � ξ, λ.

The effective low-energy Hamiltonian Hθ is the continuous version of the Hamiltonian

of the XY model of spins on a lattice. It can be used to derive the correct long-range

algebraic decay of g1(r) in the low-T superfluid state (see subsect. 2.5). However, it is

important to stress some caveats:

1) Even though at low temperature ns ≈ n, Hθ fundamentally cannot be the correct

microscopic Hamiltonian. We can see this on very general grounds since the proper

(3) Note that following the conventions in the literature on different topics we normalize g2 sothat it is dimensionless, while g1 has units of density.


Hamiltonian is by definition a temperature-independent entity, while Hθ depends on the

temperature through ns. More accurately Hθ represents the increase of the free energy

of the gas if one imposes the superfluid current (�/m)∇θ, for instance by setting the gas

in slow rotation (see the appendix in [6] for more details).

2) If and only if the density fluctuations are completely absent can the 2d Bose gas be

formally mapped onto the XY model. This condition is essentially fulfilled for D � 2π/g,

but it is not satisfied at the BKT critical point. As discussed in sect. 3, BKT transition

occurs at a critical phase space density Dc = ln(380/g), which for the experimentally

relevant values of g corresponds to Dc ∼ 6–10. At that point density fluctuations are sig-

nificantly suppressed, but cannot be completely neglected. This is one of the key reasons

that makes the proper microscopic theory of the BKT transition in the Bose gas difficult.

In summary, Hθ can be used as an effective quantum Hamiltonian for the derivation

of some key features of long-range physics at low temperature, where ns ∼ n. However

Hθ alone is not sufficient for the proper derivation of the BKT transition.

2.4. Bogoliubov analysis. – By performing Bogoliubov analysis near T = 0 we can

see more explicitly why the density fluctuations are suppressed at sufficiently low T , but

the phase fluctuations are not, and also why it is natural to expect the low-temperature

state to be superfluid.

The basic idea is that we start with the assumption that the T = 0 state of a weakly

interacting gas is described by the uniform order parameter ψ =√n eiθ, find the excita-

tion spectrum, and then consider the effects of the thermal occupation of the excitation

modes. This approach may not seem justified in 2d, because in the end we find that

thermal fluctuations destroy the order parameter, and thus invalidate our starting as-

sumption. However we can qualitatively argue that it still works well as long as we have

a well-defined local order parameter, which is true if the order parameter is destroyed

only at very large distances by the long-wavelength phase fluctuations. The applicability

of the Bogoliubov approach to 2d quasi-condensates was formally justified in [30,31].

Assuming contact interactions, the classical field Hamiltonian is given by

(27) H =�

2

2m

∫(∇ψ∗(r))(∇ψ(r)) d2r +

g

2

∫(ψ∗(r))2(ψ(r))2 d2r.

The dynamics of ψ(r, t) are governed by the Gross-Pitaevskii equation [32,33]:

(28)

(−

�2

2m∇2 + g|ψ|2

)ψ = i�

∂ψ

∂t,

which we can derive from eq. (27) by treating ψ and ψ∗ as canonical variables.

We now define the local phase θ(r, t) from ψ(r, t) = |ψ(r, t)| eiθ(r,t). Assuming that

the density fluctuations are small, we write |ψ(r, t)|2 = n (1 + 2η(r, t)), with η � 1 and∫η d2r = 0. Within this approximation we get, up to the additive constant gn2L2/2,

(29) H =�

2

2mn

∫(∇θ(r))2 d2r +

∫ [�

2

2mn (∇η(r))2 + 2gn2(η(r))2

]d2r,


and a set of coupled linear equations for the time evolution of θ(r, t) and η(r, t):

∂θ

∂t=

�

2m∇2η −

gn

�(1 + 2η),(30)

∂η

∂t= −

�

2m∇2θ.(31)

It is now convenient to Fourier expand the phase and the density fields:

(32) θ(r, t) =∑

k

ck(t) eik·r, η(r, t) =∑

k

dk(t) eik·r,

where k = 2π(jx, jy)/L, with jx, jy integers, is a discrete variable since we consider for

the moment a sample of finite size L2. We will let L → ∞ at the end of the calculation.

The functions θ and η are real, which implies c∗k

= c−k and d∗k

= d−k. In addition the

conservation of particle number∫η d2r = 0 leads to d0 = 0. This yields the Hamiltonian

(33) H = nL2∑

k

[�

2k2

2m|ck|

2 +

(�

2k2

2m+ 2gn

)|dk|

2

],

and the coupled equations of motion (for k �= 0):

ck = −

(�k2

2m+

2gn

�

)dk,(34)

dk =�k2

2mck.(35)

For k = 0 the equation of motion deduced from eq. (30), c0 = −gn/�, simply gives the

time evolution of the global phase of the gas.

In this formulation ck and dk are conjugate dimensionless quadratic degrees of freedom

corresponding to the phase and the density fluctuations, respectively. At each k we have

a harmonic-oscillator–like Hamiltonian, and from the equations of motion we can read

off the eigenfrequencies

(36) ωk =

√�k2

2m

(�k2

2m+

2gn

�

),

which is the well-known Bogoliubov result. At low k, the eigenmodes are phonons with

ωk = ck, where c =√gn/m. At high k we have free particle modes with ωk = �k2/(2m)+

gn/�. The crossover between the two regimes is at k ∼ 1/ξ =√gn.

From this analysis, and specifically the results in eqs. (33) and (36), we can draw

several conclusions:

1) From the dispersion relation ωk we see that the excitation modes in this system

have a non-zero minimal speed c. From the Landau criterion, we thus expect it to be


superfluid with a critical velocity vc = c. This argument is identical to the 3d case, and

relies just on the existence of a (reasonably) well-defined local order parameter, rather

than true LRO. We note however that the Landau criterion is a necessary, but not a

sufficient condition for the existence of a superfluid state, because it does not address

the question of metastability of the superfluid flow (see e.g. [34]).

2) Both in the classical and in the quantum regime the harmonic oscillator at thermal

equilibrium obeys the virial theorem, 〈mω2x2〉 = 〈p2/m〉, where x and p are position

and momentum, respectively. Reading-off the coefficients in front of |ck|2 and |dk|

2 in

eq. (33), we see that in our case this corresponds to

(37)〈|dk|

2〉

〈|ck|2〉=

�2k2/2m

�2k2/2m+ 2gn.

We thus explicitly see that long-wavelength phonons involve only phase fluctuations,

since 〈|dk|2〉 � 〈|ck|

2〉 for k → 0. On the other hand, the high-k free particles involve

both phase and density fluctuations in equal parts, since 〈|dk|2〉 = 〈|ck|

2〉 for k → ∞.

3) We also explicitly see that density fluctuations are not “soft” Goldstone modes,

because their energy cost does not vanish in the k → 0 limit. The effect of interactions is

to suppress (compared to the ideal gas) the density fluctuations at length scales > ξ. The

fluctuations on short length scales (< ξ, corresponding to k > 1/ξ) are not suppressed

by interactions, but those in any case do not have a diverging contribution.

4) On the other hand, the energy cost of phase fluctuations (phonons) vanishes for

k → 0. This conclusion would be the same in 3d, but the crucial difference is that in

2d the density of states at low k leads to a diverging effect of these fluctuations and the

destruction of true LRO, as shown in the next subsect. 2.5.

5) We can use this Bogoliubov analysis to provide an estimate of the relative den-

sity fluctuations Δn2/n2 = 4〈η2〉. In principle the thermal equilibrium distribution of

the Bogoliubov modes is given by the Bose-Einstein distribution. For simplicity we ap-

proximate it in the following way; we suppose that the modes with frequency ωk lower

than kBT/� have an average energy kBT/2 (classical equipartition theorem) and that

the higher-energy modes have a negligible population. Therefore we take(4)

(38) nL2

(�

2k2

2m+ 2gn

)〈|dk|

2〉 =

{kBT/2, if �ωk < kBT,

0, if �ωk > kBT.

In addition we assume that kBT is notably higher than the interaction energy gn, which

is the case in most experiments realized with cold atoms so far. The cutoff �ωk = kBT

then lies in the free-particle part of the Bogoliubov spectrum, at the wave vector kT ≈

(4) We use here that ck and dk are complex amplitudes, and that the modes k and −k arecorrelated, so that d−k = d

∗

k .


√mkBT/�. We can now estimate the relative density fluctuations

(39)Δn2

n2= 4

∑k

〈|dk|2〉 ≈

L2

4π2

∫k<kT

4

nL2

kBT/2

�2k2/2m+ 2gnd2k

and we find

(40)Δn2

n2≈

2

nλ2ln

(kBT

2gn

).

For realistic values of the ratio kBT/gn (i.e. not exponentially large), we recover here

the previously announced result that density fluctuations are notably suppressed when

nλ2 � 1.

Based on these arguments, with some more quantitative justification we arrive at the

same conclusions as before. First, for effectively describing the low-temperature state of

the gas, the most important part of the Hamiltonian in eq. (33) is the term corresponding

to the phase fluctuations. Second, if we keep only that term, we also have to keep in mind

that we are neglecting short-distance physics and effectively introducing a momentum

cutoff at kmax = 1/ξ. As before we can heuristically incorporate short-distance physics

by replacing n → ns to obtain

(41) H ≈ nsL2

∑k

k<ξ−1

�2k2

2m|ck|

2 =�

2

2mns

∫(∇θ(r))2 d2r,

which coincides with the Hamiltonian Hθ given in eq. (26).

2.5. Algebraic decay of correlations. – To derive the low-T behavior of the one-body

correlation function g1(r) = 〈ψ∗(r)ψ(0)〉 at large distances, r � ξ, λ, we start with the

Hamiltonian Hθ, and the wave function with no density fluctuations ψ(r) =√nse

iθ(r).

Note that this normalization of ψ leads in r = 0 to the incorrect value g1(0) = ns, whereas

it should be g1(0) = n > ns. However, as discussed earlier, density fluctuations at short

distances lead to a more complicated decay of g1, and at large r replacing n → ns is the

appropriate normalization.

The long-distance (r � ξ) behavior of g1(r) essentially depends on the population of

phonon modes with wave vector k � ξ−1, which coincides with the momentum cutoff

introduced in Hθ. The occupation of eigenmodes is in general given by the standard Bose

result (exp(β�ωk)−1)−1. Assuming as in eq. (40) that kBT > gns, the phonon modes are

in the regime �ωk � kBT and the occupation number simplifies into kBT/(�ωk) which

leads to (classical equipartition theorem)

(42) phonon modes: nsL2�

2k2

2m〈|ck|

2〉 =kBT

2.


Introducing the real and imaginary part of the Fourier coefficients ck = c′k+ ic′′

k, we have

(43)⟨|c′k|

2⟩

=⟨|c′′k|

2⟩

=π

nsλ2

1

L2k2.

We recall that c′k

and c′′k

are independently fluctuating variables (〈c′kc′′k〉 = 0) and that

the mode k and −k are correlated because θ is real: c′k

= c′−k

and c′′k

= −c′′−k

.

We want to calculate

(44) g1(r) = 〈ψ∗(r)ψ(0)〉 = ns

⟨ei(θ(r)−θ(0))

⟩,

where

(45) θ(r) − θ(0) =∑

k

c′k(cos(k · r) − 1) − c′′k sin(k · r).

For each independent Gaussian variable u, 〈eiu〉 = e−12〈u2

〉. Using eq. (43), and trans-

forming the discrete sum over k into L2/(4π2)∫

d2k we obtain:

(46) g1(r) = ns exp

(−

1

2πnsλ2

∫1 − cos(k · r)

k2d2k

).

The integral in the exponent has significant contributions only from modes k > 1/r so

that 1 − cos(k · r) ∼ 1. Since we restrict our analysis to r � λ, this is not inconsistent

with the classical field approximation which requires k < 1/λ. The upper limit of the

integral is set by the short-distance cutoff kmax = 1/ξ. We thus expect the integral to

be ∼ ln(r/ξ). More formally, we can note that ∇2∫

(1 − cos(k · r))k−2 d2k = (2π)2δ(r),

from which we infer

(47)

∫1 − cos(k · r)

k2d2k = 2π ln

(r

ξ

).

This leads to

(48) g1(r) = ns

(ξ

r

)1/(nsλ2)

.

Note that depending on the relative size of λ and ξ we can also set the upper limit of the

integral to 1/λ, but this difference in the short-distance cutoff does not affect the main

conclusion about the power law decay of correlations at large distances.

To summarize, we have shown that in an interacting 2d Bose gas at low T , the

first-order correlation function g1(r) decays algebraically with r at large distances. The

conclusion that g1(r) vanishes for r → ∞ is consistent with the Mermin-Wagner theorem,

i.e. the absence of BEC and true LRO at any non-zero T . However the decay of g1(r)

is very slow and the system exhibits a “quasi–long-range order”. Further, as discussed


in the following sect. 3, the exponent 1/(nsλ2) is never larger than 1/4 in the superfluid

state, making the decay of g1(r) extremely slow. The superfluid state with suppressed

density fluctuations can be viewed as a superfluid “quasi-condensate”, i.e. a condensate

with a fluctuating phase [35-37].

3. – The Berezinskii-Kosterlitz-Thouless (BKT) transition in a 2d Bose gas

Our analysis so far does not explain how the phase transition from the low-

temperature superfluid state to the high-temperature normal state takes place. This

transition is unusual because it does not involve any spontaneous symmetry breaking in

the superfluid state, and in the usual classification of classical phase transition is termed

“infinite order”, suggesting that most thermodynamic quantities (except for example su-

perfluid density) vary smoothly at the transition. There is no true LRO on either side

of the transition, but the functional form of the decay of g1(r) changes from algebraic

in the superfluid state (corresponding to quasi-LRO) to exponential in the normal state

(corresponding to no LRO).

The microscopic theory of the 2d superfluid transition was developed by Berezin-

skii [20] and Kosterlitz and Thouless [21] (see [4] for a more recent review). The transition

takes place in the degenerate regime, where the density fluctuations in an interacting gas

are significantly suppressed. We therefore expect that the transition can still be at least

qualitatively explained by considering only phase fluctuations. However, a sudden tran-

sition with a well-defined critical point cannot be explained by considering only phonons,

since we have seen in eq. (48) that, while they destroy true LRO at any non-zero T , their

effect grows smoothly with temperature.

3.1. The role of vortices and topological order . – The key conceptual ingredient of the

BKT theory is that in addition to phonons described by the Hamiltonian (26), another

natural source of phase fluctuations are vortices, points at which the superfluid density

vanishes, and around which the phase θ varies by a multiple of 2π. For our purposes we

can consider only “singly-charged” vortices with phase winding ±2π, which are energet-

ically stable. Around an isolated single vortex, centered at the origin, the velocity field

�∇θ/m varies as �/(mr), corresponding to angular momentum � per particle. The two

signs of the vortex charge correspond to the two senses of rotation around the vortex.

The size of the vortex core (hole in the superfluid density) is set by the healing length

ξ, so their presence is not inconsistent with the picture that the density fluctuations are

suppressed at length scales r � ξ. In fact, it makes sense to speak of well-defined indi-

vidual vortices only if away from the vortex cores the density fluctuations are suppressed

on the length scale ξ; otherwise we simply have a fully fluctuating thermal gas.

As we will illustrate below, once one considers vortices as another source of phase

fluctuations, one can explain the microscopic mechanism behind the superfluid-to-normal

phase transition. Below a well-defined critical temperature TBKT, vortices can exist only

in the form of bound (“dipole”) pairs of vortices with opposite circulations ±2π. Since

they do not have any net charge, such pairs do not create any net circulation along closed


Fig. 1. – The BKT mechanism at the origin of the superfluid transition. Below the transitiontemperature (left figure), vortices exist only in the form of bound pairs formed with vortices ofopposite circulation. When approaching the transition point the density of pairs grows and theaverage size of a pair diverges. Just above the transition point (right figure), a plasma of freevortices is formed and the superfluid density vanishes.

contours larger than the pair size, which for a tightly bound pair is also of order ξ(5).

Such pairs therefore have only a short-range effect on the phase θ and the velocity field,

and do not have a large effect on the behavior of g1(r) at large distances. They fall under

“short-range physics” of the system and together with the residual density fluctuations

they lead to some renormalization of ns, but do not qualitatively alter the phenomenology

of the long-range physics discussed in sect. 2. On the other hand, above TBKT unbinding

of vortex pairs and proliferation of free vortices becomes energetically favorable. Free

vortices then form a disordered gas of phase defects and completely “scramble” the

phase θ (see fig. 1). This destroys the quasi-LRO and suppresses superfluidity. At

even higher temperature where density fluctuations are strong, the notion of individual

vortices becomes physically irrelevant.

In hindsight, we can associate superfluidity with presence of a “topological order”.

Long-wavelength phase fluctuations (phonons) destroy true LRO, but do not alter the

topology of the system. In other words, phonons lead to smooth local variations of the

field ψ which can be eliminated (or “ironed out”) by continuous deformations. The same

argument holds for bound vortex pairs which can be annihilated. Therefore, the super-

fluid quasi-condensate with no free vortices is topologically identical to the BEC with

true LRO. On the other hand, an isolated free vortex cannot be unwound and eliminated

from the system by continuous deformations of ψ; it affects the phase θ non-locally, at

arbitrary large distances. The annihilation argument also does not work for a plasma of

free vortices, because if we consider a closed contour of arbitrary large size it will in gen-

eral not contain equal number of vortices with opposite charges. Therefore at any length

scale the free vortex plasma is topologically different from an ordered BEC. Although we

could not have necessarily anticipated this, we can deduce that topological order, rather

than true LRO, is a sufficient condition for superfluidity in an interacting 2d Bose gas.

(5) For a dipole field |∇θ| ∼ 1/r2, so the circulationH

∇θ · dr vanishes for large contours.


3.2. A simple physical picture. – The full thermodynamic description of the 2d gas,

including the role of vortices, is a difficuly task. It first requires to introduce the velocity

fields [38] or the mass-current densities [39] in the fluid. The normal and superfluid

components can then be extracted from the spatial correlation functions of this mass-

current density [39]. Finally an analysis using renormalization group arguments leads

to the so-called universal jump for the superfluid density: This density takes the value

ns = 4/λ2 on the low-temperature side of the transition point, and ns = 0 on the high

temperature side. The existence and the value of the universal jump in the superfluid

density was formally derived by Nelson and Kosterlitz [38], and it was first confirmed to

high accuracy in experiments with liquid He films by Bishop and Reppy [2].

This full description is outside the scope of this set of lectures. Here we simply

illustrate how vortices drive the BKT transition, starting from a superfluid with finite

ns, and then considering the free energy associated with spontaneous creation of a single

free vortex. Without any loss of generality, in order to simplify the calculations we

consider a circular geometry, with R → ∞ the radius of the system. The kinetic energy

cost of a single vortex placed at the origin is simply given by

(49) E =

∫ R

ξ

1

2ns

(�

mr

)2

d2r =�

2π

mns ln

(R

ξ

),

where as always we assume that only the superfluid component rotates under the influence

of the vortex. The normal component does not have any phase stiffness and its motion

is not affected by the presence of the vortex.

The entropy associated with a single vortex core is given by the number of distinct

positions where a vortex of radius ξ can be placed in a disc of radius R:

(50) S = kB ln

(R2π

ξ2π

)= 2kB ln

(R

ξ

).

Note that in the above calculations we ignore the “edge effects” such as the correction

to the energy for an off-centered vortex. One can check that these effects are negligible

for R � ξ. Combining eqs. (49) and (50), we get for the free energy F = E − TS

(51) βF =1

2(nsλ

2 − 4) ln

(R

ξ

).

We thus see that the free energy associated with a free vortex changes sign at nsλ2 = 4.

Since ln(R/ξ) diverges with the size of the system, this point separates two qualitatively

different regimes. For nsλ2 > 4, F is very large and positive, so the superfluid is stable

against spontaneous creation of a free vortex. On the other hand, for nsλ2 < 4, the large

and negative F signals the instability against proliferation of free vortices. Appearance

of first free vortices reduces ns and makes the appearance of further free vortices even

easier, and this avalanche effect renormalizes the superfluid density to zero. We thus find


that in contrast to 3d, where below the BEC critical temperature the superfluid density

grows smoothly, in 2d the superfluid density cannot have any value between 4/λ2 and 0.

Even though it does not explicitly address the microscopic origin of the vortex (the

breaking of a vortex pair), this simple calculation correctly predicts the result for the

universal jump in the superfluid density which takes place at the transition temperature

TBKT:

(52) nsλ2 = 4.

This success relies on the fact that the above derivation is a powerful self-consistency

argument. Whatever the origin of the vortex, and the relation between ns and total

density n, it shows that it is inconsistent to suppose that we have a system with superfluid

density which is non-zero, but smaller than 4/λ2. Remarkably, this result also does not

depend on the strength of interactions g, even though we know that the phase transition

is mediated by interactions, since it does not occur in the ideal gas. Recent classical field

Monte Carlo calculations performed with parameters relevant for atomic gases [40-42]

have confirmed the proliferation of vortices around the critical point characterized by

eq. (52), although the transition was rounded off by finite-size effects.

If we repeat the above arguments for tightly bound vortex pairs, we find that a finite

density of pairs is present in the gas at any non-zero temperature. The energy of a pair

is finite since the velocity field decays as v ∝ 1/r2 and the integral∫v2 d2r is convergent.

On the other hand, the entropy is still divergent and essentially identical to the result of

eq. (50), since the size of a tightly bound pair is of the same order as the size of a single

vortex. The free energy for vortex pairs is therefore always negative. At any non-zero T

pairs are continuously created and annihilated through thermal fluctuations.

As the temperature is increased, but still kept below TBKT, the density of pairs grows

and also thermal fluctuations result in pairs of increasing size. As the average size of the

pairs becomes comparable to the distance between the pairs, they start to overlap and

this leads to effective screening of the attraction between two bound vortices, making

it easier for fluctuating pairs to grow to even larger sizes. We can use an analogy with

a Coulomb gas: Two nominally paired but well-separated vortices create a field which

polarizes the more tightly bound vortex dipoles between them. This results in an effective

dielectric constant which reduces the attraction between two oppositely charged vortices.

As TBKT is approached from below, this creates an avalanche effect which eventually leads

to breaking up of pairs and creation of a plasma of free vortices. Within the Coulomb

gas analogy, this plasma provides a perfect screening at the transition point: A charge

added at a given point does not change the flux of the electric field across a large radius

circle centred on this point. Proper BKT calculation identifies the phase transition with

the temperature at which the average size of the pairs, or equivalently the screening

dielectric constant, diverges.

3.3. Results of the microscopic theory. – The relation (52) between the superfluid

density and the temperature TBKT at the critical point is elegant and universal, in the


sense that it does not depend on the interaction strength g. However this self-consistent

result alone does not allow us to predict the value of TBKT in a given system. It just

tells us that whatever TBKT is, the superfluid density jumps accordingly to 4/λ2 at the

transition.

Calculating the actual value of TBKT in terms of the bare system properties n and g

is a difficult problem, because it depends on the short-distance physics, such as density

fluctuations which control the relationship between ns and n at the transition. In the

weak coupling limit, g � 1, a combination of analytical [43] and numerical [29,44] efforts

gives the value for the critical phase-space density:

(53) Dc = (nλ2)c = ln(C/g),

where the dimensionless constant C = 380 ± 3 is obtained by a classical field Monte

Carlo simulation [29]. The calculation leading to eq. (53) is formally valid only in the

weak-coupling limit g � 1. We can set one obvious bound on its validity by noting that

at the transition n ≥ ns. Setting Dc > 4 we obtain g ≤ 7. This result is remarkably

close to our estimate of the strong-coupling limit g = 2π (eq. (22)).

The numerical simulation of [29] provides another quantity of interest, which char-

acterizes the reduction of density fluctuations due to interactions. The authors of [29]

introduce the quasi-condensate density defined as

(54) nqc ≡(2n2 −

⟨n2(r)

⟩)1/2

.

When interactions are negligible, 〈n2〉 = 2〈n〉2 and nqc = 0. On the other hand if density

fluctuations are completely suppressed, 〈n2〉 = 〈n〉2 and nqc = n. According to [29], at

the critical point

(55)nqc

n=

7.16

ln(C/g).

This result indicates that nqc is of the order of the total density n at the transition

point, unless the interaction strength g is exponentially small. In other words, for realistic

parameters density fluctuations are notably reduced in the vicinity of the BKT transition,

which justifies the simplified Hamiltonian (26) used above. Actually this result sets a

stronger constraint than eq. (53) on the applicability of the classical Monte Carlo analysis:

the condition n ≥ nqc requires g ≤ 0.3.

Note that the terminology quasi-condensate density can sometimes be misleading.

The quantity nqc takes a non-zero value even above the critical temperature for the

BKT transition. It refers only to the properties of the density distribution in the gas,

and not to the phase distribution as the word condensate might suggest. For example,

we can have nqc ∼ n at T > TBKT, but this does not imply that a large contrast

interference would be observed if one would superpose a pair of 2d gases prepared in the

non-superfluid regime (see sect. 7).


Finally, we note that as the transition temperature is approached from above, the

length scale � characterizing the exponential decay of correlations g1(r) ∼ e−r/ in the

normal state diverges as

(56) � = λ exp

( √a TBKT

√T − TBKT

),

where a is a model-dependent dimensionless constant. Diverging correlation length is a

very general property of phase transitions, but while in the case of most conventional (3d)

second-order phase transitions the divergence of the correlation length is polynomial, in

the case of the BKT transition it is exponential. This makes the critical region above

TBKT larger, and has implications for the broadening of the transition in finite size

systems (see, e.g., subsect. 4.3).

4. – The 2d Bose gas in a finite box

It is well known that finite-size effects can play a significant role in the quantitative

analysis of the phase transitions that are observed experimentally. In the two-dimensional

situation of interest here, this is even more the case because the thermodynamic limit is

reached only when ln(R/ξ) � 1 (see for example eq. (51)), which can be only marginally

true for realistic systems. The analysis of these finite-size effects is therefore crucial for

understanding the observed phenomena, and we will do so for the cases of a flat box

potential (this section) and a harmonic trap (next section).

We consider in this section a gas of N particles confined in a flat box of area L2. The

box is supposed to be square (Lx = Ly = L) except in the last subsection (subsect. 4.5)

where we discuss possible effects due to an anistropic confinement (Lx �= Ly). The

confinement introduces a natural energy scale E0 = �2/(mL2) in the problem and makes

it possible to reach a true Bose-Einstein condensate at non-zero temperature, in contrast

to the infinite case. In this section we first review the results that can be derived for the

ideal gas case, and then discuss what happens for an interacting system. We will assume

that the size L is much larger than the thermal wavelength λ so that E0 � kBT .

4.1. The ideal Bose gas. – In the absence of interactions, the statistical description of a

Bose gas in a square box of size L is straightforward. Let us choose for simplicity periodic

boundary conditions so that the single-particle eigenstates are plane waves eik·r/L, of

energy εk = �2k2/2m, where the momentum k = (jx, jy)(2π/L), with jx,y positive, zero

or negative integers. The chemical potential μ is always negative so that the fugacity

Z = eβμ lies in the interval 0 < Z < 1. Three regimes can be identified, the first two

being identical to what we have met for the infinite case:

– The non-degenerate, high-temperature regime with a phase space density D =

nλ2 � 1. This corresponds to a negative chemical potential such that |μ| � kBT

(Z � 1). The one-body correlation function is a Gaussian function in this regime,

g1(r) = n e−2πr2/λ2

, and is vanishingly small for distances r � λ, which are much


smaller than the box size L. The confinement has no significant consequence on

the coherence of the gas in this regime.

– The degenerate, but non-condensed regime, where the momentum distribution is

bimodal, with a Lorentzian shape for small k (kλ �√

4π) and a Gaussian shape

for large k. The one-body correlation function decays exponentially at large r, with

a characteristic decay length � = λeD/2/√

4π. No significant condensed fraction

appears as long as � is small compared to the size L of the sample, i.e. when

D < ln(4πL2/λ2) or equivalently |μ| � E0/2.

– The condensed regime, which occurs when the characteristic decay length � of g1 is

larger than the system size L. This occurs when the phase space density D reaches

the value ln(4πL2/λ2) (or equivalently |μ| ≤ E0/2). A significant phase coherence

then exists between any two points in the gas.

4.2. The interacting case. – We now turn to the interacting case with repulsive in-

teractions and discuss what can be expected in the vicinity of the BKT transition. For

now we assume that the size of the sample is large enough so that D � ln(4πL2/λ2)

at the point where the phase space density D is equal to Dc, the critical phase space

density for the BKT transition in an infinite system (see eq. (53)). This condition has

the following physical meaning: suppose that we increase the density of particles n at

fixed temperature; the point where the superfluid transition occurs is reached well before

the point at which the Bose-Einstein condensation due to the finite system size would

occur in the absence of interactions (see subsect. 4.1 above).

We first recall the nature of the superfluid transition in an infinite, homogenous sam-

ple. When D is notably below Dc, but larger than 1, one expects that no superfluid

component is present and g1(r) decays exponentially. When D reaches Dc the super-

fluid transition occurs and g1(r) decays algebraically (eq. (48)): g1(r) ≈ ns(ξ/r)α for

r > ξ, with α = 1/(nsλ2) and nsλ

2 ≥ 4. According to the Penrose-Onsager criterion,

no condensate is expected in an infinite system since g1(r) vanishes at infinity for any

non-zero temperature. In sharp contrast with the infinite case, we now show that the

BKT transition in a realistic finite system is always accompanied by the appearance of

a significant condensed fraction, defined as the largest eigenvalue Π0 of the one-body

density matrix. The basic reason for this effect is that the algebraic decay of g1(r) is

extremely slow. To prove this result we proceed in two steps: first we give a general

relation between Π0 and the value of g1(r) for distances r comparable to the size L of

the box; then we discuss what a realistic value of Π0 can be for a typical atomic gas.

Let us denote by Πj and φj(r) the eigenvalues and eigenstates of the one-body density

matrix. The condensed fraction Π0 is associated with the eigenstate φ0(r) = 1/L. We

now consider the general expansion of g1

(57) g1(r) = N∑

j

Πjφ∗

j (0)φj(r)


and integrate this expression over the area L2 centered on the origin. For simplicity we

integrate the left-hand side over a disk of radius R = L/√π and the right-hand side over

a square of side L. This simplification cannot significantly affect our conclusions. The

left-hand side gives

(58)

∫g1(r) d2r = 2π ns

∫ R

ξ

(ξ/r)α r dr � L22

2 − απα/2 g1(L).

On the right-hand side, only the contribution of j = 0 is non zero and gives NΠ0. All

the φj ’s with j �= 0 are orthogonal to φ0, so their integral over L2 is zero. We thus get

g1(L)/n ∼ Π0. Using eq. (18) we can also write this result for the condensed fraction as

Π0 ∼ (ns/n)g−α/2N−α/2.

For α ≤ 1/4, in practice we have g−α/2 ∼ 1 and ns ∼ n, so just below the transition

temperature Π0 ∼ N−1/8. Taking N = 105 as a typical value for cold atom experiments,

we get Π0 ∼ 0.25.(6). We therefore meet here a paradoxical situation: the appearance

of a non-zero condensed fraction may be used as a signature of the BKT transition,

whereas the BKT mechanism was presented (for an infinite system) as a feature that

takes place in a 2d interacting gas instead of the usual BEC of 3d Bose fluids. Note that

in a “true” BEC the condensed fraction Π0 should not explicitly depend on N . However,

for α ≤ 1/4 this distinction becomes experimentally irrelevant, and in order to observe a

BKT transition with no significant BEC one would need to consider unrealistically large

samples. This was pointed out by the authors of [45] who wrote the famous statement (in

the context of 2d magnetism): “With a magnetization at the BKT critical point smaller

than 0.01 as a reasonable estimate for the thermodynamic limit, the sample would need

to be bigger than the state of Texas for the Mermin-Wagner theorem to be relevant!”.

A similar remark holds in the context of superfluid helium films [2].

4.3. Width of the critical region and crossover . – The intricate mixing between the

BKT mechanism and the emergence of a significant degree of coherence exists even at

temperatures slightly above the BKT transition point. We mentioned in subsect. 3.3

that in the normal state the characteristic decay length � of g1(r) diverges exponentially

in the vicinity of the critical point: ln(�/λ) ≈ (aTBKT/(T − TBKT))1/2, where a is a

model-dependent coefficient (see eq. (56)). The critical region where � becomes larger

than λ as a precursor of the BKT transition is therefore very broad, (T −TBKT) ∼ TBKT.

Further, there clearly exists a temperature close to (but still above) TBKT for which �

exceeds the system size. At this temperature a significant condensed fraction appears in

the system. Because of the exponential variation of � with T − TBKT, the temperature

(6) We get an equivalent estimate from Π0 ∼ g1(L)/n and ns ∼ n in the algebraic decay regime.In cold atom gases, the typical values of λ and ξ are 0.1–1μm, while the maximal system sizeis L ∼ 100μm. At the transition point we have Π0 ∼ (10−3)1/4 to (10−2)1/4 ∼ 0.2 to 0.3.


range where � � L, and condensation gradually sets in, can be significant(7):

(59)ΔT

TBKT

=ΔD

Dc∼

a

(ln(L/λ))2.

Although cold atom systems are so far commonly confined in harmonic rather than

box-like potentials, it is interesting to provide an estimate for this case. Taking a = 1

and reasonable values for L/λ between 10 and 100, the BKT transition is expected to

become a crossover with a relative width ΔT/TBKT ranging from 5% to 20%.

The above analysis explicitly concerns only the emergence of a non-zero condensed

fraction, but it also suggests broadening of the universal jump in the superfluid density.

For example, using standard Bogoliubov argument (see subsect. 2.4) we can deduce that

finite condensed fraction in an interacting system also implies a finite superfluid density.

Also, if we take the definition of superfluid density which associates it with the energy

cost of twisting the phase of the wave function at the edge of the system [46,47], we again

conclude that � � L implies a finite superfluid density. In the critical region, quantitative

conclusions might actually depend on what theoretical definition of superfluid density we

accept, but the qualitative conclusions will not change.

4.4. What comes first: BEC or BKT? – This is an often raised and subtle question.

We have so far discussed the case of a large system such that Dc � ln(4πL2/λ2), where

Dc = ln(380/g) is the critical phase space density for the BKT transition in an infinite

system. For such large systems, the first relevant mechanism that occurs when increasing

the phase space density is a BKT transition. However, we have seen that the approach of

the BKT threshold always results in the appearance of a significant condensed fraction.

We can qualify this “BKT-driven” condensation as “interaction-enhanced”, since it would

not take place in an ideal gas with the same density and temperature. Experimentally,

the strength of interactions in cold atom systems can be dynamically controlled using a

Feshbach resonance [48, 49]. One can therefore imagine preparing a non-interacting gas

at phase space density where no condensation occurs and then driving the condensation

via the BKT mechanism by turning on the interactions. In the opposite regime of a small

system where the BKT transition would require Dc > ln(4πL2/λ2), the first phenomenon

that is encountered when the phase space density is increased is “conventional” Bose-

Einstein condensation as for an ideal Bose gas. As in the 3d case, in the presence of weak

repulsive interactions the formation of a condensate is accompanied by the apparition of

a superfluid fraction with a comparable value.

It would be very interesting to study cold atomic gases in a (quasi-)uniform potential

and vary the experimental parameters so as to explore both regimes. However we point

out that this will not be an easy task if we require that the criteria for the two types of

transitions are well separated, for example by more than the crossover width discussed in

(7) Here we can assume that λ under the logarithm on the right-hand side is constant over therange ΔT .


subsect. 4.3. To analyze the system requirements for reaching the two different regimes, it

is convenient to fix the ratio L/λ and write ln(4πL2/λ2) = γDc. Now γ is a dimensionless

parameter such that γ > 1 means that (as particle number is increased) condensation

occurs via the BKT mechanism. The critical number for the BKT transition is then

(60) Nc =L2

λ2Dc =

1

4πDce

γDc ,

while the critical number for condensation in an ideal gas is γNc. For illustration purposes

we may define the BKT regime by γ ≥ 1.5, and the BEC regime by γ ≤ 0.5. (For values of

γ close to 1, the two effects are difficult to disentangle experimentally.) The experiments

with cold atoms have so far been performed at coupling strength g ∼ 10−2–10−1. Taking

g = 0.1, γ = 1 corresponds to Nc ≈ 2.5 × 103, and the BKT regime γ = 1.5 corresponds

to Nc ≈ 1.5 × 105, which is easily achievable. However the opposite BEC regime of

γ = 0.5 corresponds to Nc ≈ 40; studying such a small particle number is experimentally

very challenging, although it might become feasible with the development of single-atom

detection [50-52]. For a more weakly interacting gas with g = 0.01, γ = 1 corresponds to

Nc ≈ 3 × 104, and γ = 0.5 to Nc ≈ 160, which might be easier to explore. On the other

hand, γ = 1.5 corresponds to Nc ≈ 6 × 106, which would be experimentally challenging.

It would therefore generally be difficult to explore both the large (BKT) and the small

(BEC) system regime using the same value of g, and reaching the BEC regime may

require a more weakly interacting quasi-2d atomic gas than has so far been studied.

4.5. The case of anisotropic samples. – So far we have assumed that for phase space

densities D larger than the critical value Dc for the BKT transition, the functional form

of g1 found in the infinite case (algebraic decay) remained valid for a finite-size system.

This assumption is reasonable for square samples (Lx = Ly), but may not be valid for

anisotropic samples, with a width along one direction (say x) much larger than the other

one: Lx � Ly. We briefly review the expected properties in this regime, which is relevant

for several of the previous or current experimental setups. Since we are interested in the

regime D > Dc, we assume that a superfluid component is present in the sample and

we use the Hamiltonian Hθ given in (41) to estimate the amplitude of phase fluctuations

and their consequence on the one-body correlation function g1.

We start from the result (46) obtained in the infinite case. In a finite-size system

the integral over k is replaced by a discrete sum over k = 2π(jx/Lx, jy/Ly) times the

constant prefactor 4π2/(LxLy):

(61) ln(g1(r)/ns) = −2π

nsλ2LxLy

∑k

1 − cos(k · r)

k2x + k2

y

.

We are interested here in the decay of g1 along the long axis of the sample and we choose

r = xux, where ux is the unit vector along the x direction. We take x � Lx so that

the finiteness of the sample along x has no relevance here. On the contrary the size Ly


(� Lx) will play an important role since we can choose x either small or large compared

to Ly. We now show that these two cases lead to different decaying regimes for g1.

Since we assume |x| � Lx, the discrete sum over kx can always be replaced by an

integral and we get

(62) ln(g1(x)/ns) = −1

nsλ2Ly

∑ky

∫1 − cos(kxx)

k2x + k2

y

dkx = −π

nsλ2Ly

∑ky

1 − e−|xky|

|ky|.

We single out the contribution of ky = 0 in the sum, introduce a cutoff kmax for large ky

and use∑jmax

j=11/j ≈ ln jmax and

∑∞

j=1ζj/j = − ln(1 − ζ) for 0 < ζ < 1. We then get

(63) ln(g1(x)/ns) = −1

nsλ2

{π|x|

Ly+ ln

[kmaxLy

2π

(1 − e−2π |x|/Ly

)]}.

Two regimes clearly appear in this expression. If π|x| � Ly then the logarithm on the

right-hand side is the dominant term, and we recover the algebraic decay that holds for

an infinite system:

(64) k−1

max� |x| � Ly : g1(x) ≈

ns

(|x|kmax)α, with α =

1

nsλ2.

This result is intuitive: as long as we probe the coherence of the system on a distance

shorter than the smallest size of the sample, the anisotropy introduces no significant

deviation with respect to an infinite system.

The situation is dramatically different when |x| � Ly. In this case the dominant

contribution on the right-hand side of (63) is the linear term π|x|/Ly. This term, which

originates from the contribution of the ky = 0 mode to the sum (61), leads to an expo-

nential decay of g1:

(65) Ly � |x| : g1(x) ≈ ns e−|x|/d, with d = nsλ

2Ly/π.

It is quite remarkable that when we probe the coherence of this anisotropic system on

distances x ≥ d > Ly, we obtain an exponential decay as if the system was not superfluid.

At the same time, the characteristic distance over which g1 decays, d, explicitly depends

on the superfluid density. The physical interpretation of this counterintuitive result is

that over such distances the system acquires a quasi–one-dimensional character: the

phase stiffness between the origin and the point at coordinate x is decreased with respect

to an infinite plane because there is a severe reduction in the number of independent

paths connecting these two points. Ultimately, for very large |x|, only the channel ky = 0

contributes significantly to the connection between these two points. This explains why,

although the system is superfluid, the decay of g1 turns to an exponentially decaying

function, that is characteristic of 1d degenerate gases (see also [53] for a similar discussion

for elongated atomic 3d gases).


5. – The 2d Bose gas in a harmonic trap

Up to now, experiments performed with (quasi-)2d atomic gases used harmonic trap-

ping in the xy-plane. We discuss in this section how the presence of the harmonic

trapping potential modifies the above results. We shall see that it can lead to a dra-

matic change of the properties of the system, through the modification of the density

of states of the single-particle Hamiltonian. In particular, in this case “conventional”

Bose-Einstein condensation, as defined through the saturation of excited states at some

non-zero temperature Tc, can occur in the ideal Bose gas even in the thermodynamic

limit. However, in the presence of repulsive interactions, and in large enough systems,

this type of condensation is suppressed and replaced by the BKT normal to superfluid

transition.

5.1. The ideal case. – Consider for simplicity an isotropic 2d harmonic potential

V (r) = mω2r2/2. The single-particle energy levels are Ej = (j + 1)�ω, with j positive

or zero integer, and each level having a degeneracy gj = j + 1. The maximum number

of atoms Nc that can be placed in all excited states (j > 0) at a given temperature T is

obtained by choosing a chemical potential μ equal to the ground-state energy:

(66) N (id)

c(T ) =

+∞∑j=1

gj

eζj − 1,

where ζ = �ω/(kBT ). Assuming ζ � 1, the discrete sum can be replaced by an integral

and one obtains

(67) N (id)

c(T ) ≈

π2

6

(kBT

�ω

)2

.

For an atom number N > Nc(T ) there must be at least N − Nc atoms occupying the

single-particle ground state j = 0. Equivalently, for a given atom number N placed in

the trap, there must be a significant fraction of the atoms that occupy the ground state

j = 0 if the temperature is reduced below the critical value

(68) kBTc =

√6

π�ω

√N.

Since the ground state is separated from the first-excited state by a non-zero gap �ω,

this Bose-Einstein condensation can be viewed as a natural consequence of the finite size

of the system [54], similar to the condensation of the ideal gas in a finite box. However,

one can see that the condensation of the ideal gas in a 2d harmonic trap is a more

interesting phenomenon by considering the appropriately defined thermodynamic limit

for the harmonic confinement. This limit is obtained by taking N → ∞ and ω → 0,

while keeping T and Nω2 constant. For a gas described by Boltzmann statistics, this

ensures that the central density n0 = Nmω2/(2πkBT ) remains constant. Equation (68)


leads to a non-zero critical temperature in the thermodynamic limit, contrarily to what

happens in the uniform case. This result can be understood by noticing that the density

of states has a different functional form in a box (ρ(E) constant) and in a 2d harmonic

potential (ρ(E) ∝ E) [55]. The vanishing density of states at E = 0 for a 2d harmonic

potential leads to a similar situation to the 3d uniform case, hence the possibility for a

genuine Bose-Einstein condensation in the ideal gas (for a discussion of small logarithmic

anomalies in the compressibility of the trapped ideal 2d gas, see [56]).

Quite remarkably the result (67) can be recovered by starting from the uniform result

given in eq. (6): n = −λ−2 ln(1 − eβμ) and using a local density approximation (LDA).

This approximation amounts to replacing the uniform chemical potential μ by the local

one μ− V (r), which gives the following expression for the total atom number:

N = −λ−2

∫ln(1 − eβ(μ−V (r))) 2πr dr(69)

= −

(kBT

�ω

)2 ∫ +∞

0

ln(1 − Ze−R2/2

)R dR,

where we set R = r/rT with r2T = kBT/mω2. For μ = 0, the result coincides with

eq. (67). Therefore in spite of the fact that LDA leads to a diverging spatial density at

the center of the trap for μ = 0 [n(r) ∝ − ln(r)], it provides the same upper bound Nc

as eq. (66) for the total number of atoms, assuming no macroscopic occupation of the

single-particle ground state.

5.2. LDA for an interacting gas . – In order to take into account repulsive interactions

for a trapped gas, we use again the local density approximation. We will start with an

analytical mean-field treatment based on the Hartree-Fock approximation. We will then

use the numerical results of a classical field Monte Carlo approach [44] that will provide

a more precise determination of the BKT transition.

In the mean-field Hartree-Fock approach when no condensate is present, interactions

are taken into account by adding the energy 2gn(r) to the external potential [57,58]. The

local chemical potential is now μ − V (r) − 2gn(r) so that the local phase space density

D(r) = n(r)λ2 is the solution of the implicit equation

(70) D(r) = − ln {1 − Z exp[−βV (r) − gD(r)/π]} .

Putting R = r/rT as above, we can write the total atom number as

(71)N

N(id)

c

=6

π2

∫+∞

0

D(R)R dR,

where D is solution of

(72) D(R) = − ln{1 − Z exp

[−R2/2 − gD(R)/π

]}.


0 1 2 3 4 5 60

2

4

6

8

10

12

14

(a)

0.3

0.1

0.03

0.01C

en

tra

ph

ase

sp

ace

de

ns

ty

N / Nc

(id)

0 1 2 30

2

4

6

8

10

12

14

(b)

0.03 0.05 0.1

0.2

0.3

N / N c

(id)

Fig. 2. – Variation of the central phase space density as a function of the atom number, nor-malized by the critical atom number N

(id)

c for the ideal gas in the same potential and at thesame T . The value of the interaction strength g is given for each curve. The black squaresindicate the values of N at which the BKT criterion of eq. (53) is met at the center of the trap.(a) Results obtained using the mean-field Hartree-Fock approach. (b) Results obtained usingthe bulk results of [44] and the local density approximation.

Note that the solution D(R) depends only on the fugacity Z and the interaction

strength g. The trap frequency and the temperature do not appear explicitly so that the

scaling of the atom number with ω and T (at fixed Z) is identical to the result (69) for

the ideal gas.

Interactions, when treated at the mean-field level, dramatically change the nature

of the solution of eqs. (71)-(72). For a given trapping frequency ω and temperature

T , and for any non-zero g, the atom number N obtained from eq. (71) can be made

arbitrarily large by choosing properly the fugacity Z. The condensation phenomenon

that was obtained in the ideal gas case does not occur anymore. This can be understood

qualitatively. For an ideal gas, the saturation of the atom number occurs when the

central density in the trap becomes infinite. In the presence of repulsive interactions,

this singular point cannot be reached and the mean-field treatment provides a solution

for any atom number [59].

We have plotted in fig. 2a the prediction of the mean-field approach for the central

phase space density D(0) as a function of the total number of atoms in the trap, for

various values of g. As expected, D(0) is a monotonically increasing function of the atom

number N : more atoms in the trap lead to a larger central density. The other expected

feature is that, for a given atom number, D(0) decreases as the repulsion between atoms

is increased. In particular the divergence of D(0) that is found in the ideal case for

N = N(id)

c does not show up anymore in the presence of repulsive interactions. One

could try to push further the mean-field analysis of the equilibrium state, and look for

dynamical or thermodynamical instabilities that could appear above some critical atom

number [60-63]. However we will rather follow the spirit of LDA and assume that the

normal to superfluid BKT transition occurs at the center of the trap when the phase


space density at this point exceeds the critical value Dc (eq. (53)) predicted for the

uniform system [64].

Within the mean-field Hartree-Fock analysis, one can show that, to a very good

approximation, the number of atoms that have to be placed in the trap so that the

central phase space density reaches the critical value Dc is [65]

(73)N

(mf)

c

N(id)

c

= 1 +3g

π3D2

c.

An obvious consequence of this result is that for a given trap and a given temperature,

the BKT threshold in the presence of interactions requires a larger atom number than the

BEC of the ideal gas. Equivalently, for a given atom number, the superfluid transition

temperature in the presence of interaction is lower than the ideal gas condensation tem-

perature. In fig. 2a we have indicated with black squares the value of N(mf)

c for various

interaction strengths.

So far we have relied on the mean-field approximation to obtain the relationship

between the density n(r) and the local chemical potential μ− V (r) − 2gn(r). Although

this gives a good feeling for the scaling laws that appear in the problem, it cannot

provide a very accurate description of the transition. Indeed the mean-field expression

2gn(r) for the interaction energy can only be valid at relatively low density, where the

density fluctuations are important, so that 〈n2〉 = 2〈n〉2. When the density increases

and/or the temperature decreases, density fluctuations are reduced and one eventually

reaches a situation at very low temperature where those fluctuations are frozen out and

〈n2〉 = 〈n〉2. At zero temperature, one expects a quasi-pure condensate in the trap

with a density profile given by the Thomas-Fermi law gn(r) = μ − V (r) (whereas the

Hartree-Fock approximation would lead to replacing g by 2g in this equation).

To capture the reduction of density fluctuations as the phase space density increases,

we now use the numerical results of [44] obtained using a classical field Monte Carlo

analysis. They provide the value of the phase space density D as a function of μ/kT

in the vicinity of the BKT critical point for a uniform system. Injecting this numerical

prediction in the LDA scheme, we obtain the results shown in fig. 2b for the central

density as a function of the total atom number. As expected this figure is qualitatively

similar to the one obtained using the mean-field Hartree-Fock approach. However the

classical Monte Carlo results lead to a noticeable reduction of the critical atom number

with respect to the mean-field treatment. For example, for g = 0.15 (as in the ENS

experiment [27], see below) the critical atom number for reaching the BKT threshold

is expected to be ∼ 1.4N(id)

c using the numerical predictions of [44] instead of N(mf)

c ∼

1.9N(id)

c using the Hartree-Fock approximation. One might wonder if the classical Monte

Carlo simulations of [44], which assume g � 1, remain accurate for the relatively large

interaction strength g = 0.15. The (positive) answer was given in [96], which provides a

detailed comparison between the predictions of [44] and those of a quantum Monte Carlo

simulation of an assembly of trapped bosons for the interaction strength and trapping

geometry of the ENS experiment.


5.3. What comes first: BEC or BKT? – In the previous section devoted to the study

of a square potential we have explained that the distinction between a conventional BEC

transition and a BKT transition is subtle, and we introduced two related concepts:

– “BKT-driven condensation”, meaning that if the phase space density is increased

at constant g the first many-body mechanism encountered is the BKT transition,

but due to the resulting slow decay of g1 this transition is accompanied by the

appearance of a finite condensed fraction.

– “Interaction-enhanced condensation”, meaning that there exists a range of phase

space densities for which no condensation occurs in an ideal gas but condensation

via the BKT mechanism can be induced by increasing the interaction strength from

0 to g.

In the case of a box potential these two concepts are equivalent. We identified a range of

parameters, such that Dc � ln(4πL2/λ2), for which both effects occur. In the opposite

regime of a small system and/or small g neither of the two effects occurs.

The question “what comes first” is even more subtle in the case of a harmonically

trapped gas because of the inhomogeneous density profile. In this case the notions of

BKT-driven and interaction-enhanced condensation are not equivalent, and the answer

depends on what we keep constant in an experiment, i.e. which path we follow in the

phase diagram. Since in a harmonic trap ideal gas condensation occurs even in the ther-

modynamic limit (i.e. if we neglect the discreteness of single-particle energy levels in the

trap), we start by analyzing that case, and separately consider two different experimental

paths:

– The critical phase space density Dc for a BKT transition (at a fixed non-zero

g) is finite, while the critical phase space density for ideal gas condensation is

infinite. Therefore in a standard experiment where g is kept constant and phase

space density is increased, BKT-driven condensation always occurs. In this sense,

in practice “BKT always comes first”.

– While the critical phase space density for the BKT transition (at non-zero g) is

lower than for the ideal gas condensation, the critical atom number at fixed T

is higher. The critical atom numbers N(id)

c and N(BKT)

c scale similarly with the

temperature and the trap frequency (∝ (kBT/�ω)2), and the ratio N(BKT)

c /N(id)

c is

always larger than 1. This can be seen from the mean-field result of eq. (73) or from

the Monte Carlo data shown in fig. 2b. This means that, at fixed N , interactions

always reduce the transition temperature. Therefore, contrary to the case of a

square box potential, we can never have interaction-enhanced condensation.

Note that it is not inconsistent that the BKT transition occurs at a lower critical

density but higher critical number than the ideal gas BEC, because in a harmonic trap

with fixed N and T the peak (phase space) density in a repulsively interacting gas is lower

than in an ideal gas. Also note that if we work within the BKT theory and then formally


take the g → 0 limit, we exactly recover the criterion for ideal-gas condensation, which

is usually derived from a conceptually completely different viewpoint of the saturation of

single-particle excited states. We can therefore think of the BEC transition as a special

non-interacting limit of the more general BKT theory. This connection naturally emerges

when analyzing the case of a harmonically trapped gas, but it could not be made in a

uniform system, where the critical temperature for both transitions vanishes in the g → 0

limit.

Finally, we briefly comment on the case of a realistic experimental harmonic trap,

where the spacing of the single-particle energy levels is non-zero. The results for the

critical atom numbers for the BKT and the ideal gas BEC transition are essentially

unaffected by the non-zero level-spacing. It therefore remains true that interaction-

enhanced condensation is not possible. However, the ideal gas BEC in this case occurs

at a finite phase space density DBEC in the trap center, which can in principle be lower

than Dc for some values of g. In this case BKT-driven condensation would also no longer

occur, and “BEC would come first” no matter what path we take in the phase diagram.

This scenario is however not relevant for the currently realistic experiments. The value

of DBEC is not universal and depends on the details of the trapping potential, but we

have evaluated it numerically for a typical trap used in the ENS experiments [26,27], and

obtained DBEC ≈ 13 [66]. This means that the condition DBEC < Dc can be fulfilled

only in an extremely weakly interacting gas with g < 10−3. Experimentally, this regime

is essentially indistinguishable from the g → 0 limit, where the BKT and the BEC

transition are no longer distinct.

5.4. Width of the crossover . – The divergence of the correlation length that we already

discussed in the case of a square box potential (see subsect. 4.3), must also be taken into

account. Suppose that one lowers the temperature of a cold gas in a trap until the BKT

threshold density is reached right at the center of the trap. Since the density is everywhere

lower than the critical threshold for BKT, LDA would imply that no significant superfluid

fraction is present in the gas at this stage. However a significant part of the gas may

exhibit a certain degree of coherence as we show now. The critical length (56) is now

position dependent, since the critical temperature is a function of the local density. Since

�(r) is a monotonically decaying function of the distance from the trap center r, we can

self-consistently assume that the gas is coherent over a region of radius rc such that

�(rc) = rc. We can also provide a crude estimate of rc: rc ≈ rT (ln(rT /λ))−1/2. For

practical parameters (rT /λ ∼ 10–100), we find rc ∼ rT , which means that this coherence

actually extends over a significant fraction of the cloud when D = Dc at the center of

the trap. We can also estimate the width of the cross-over over which the condensed

fraction becomes significant. If instead of taking D = Dc at the center of the trap, we

take D = 0.7Dc then � ≈ 5λ at the center of the cloud and no significant coherence

exists at this point. The above analysis is confirmed at least qualitatively by numerical

simulations performed using a classical field Monte Carlo analysis. These simulations

indeed indicate the emergence of an extended coherence over the cloud at temperatures

10% to 20% above the one for which the bulk BKT criterion is met at the trap center [42].


6. – Achieving a quasi-2d gas with cold atoms

The experimental realization of a 2d atomic Bose gas is based on a strongly anisotropic

trap with one very tightly confining direction, say z, and two more loosely confined

degrees of freedom, x and y. The z degree of freedom can be considered as frozen from

the thermodynamic point of view if the energy gap Δz between the ground state and

the first-excited state of the z motion is much larger than both kBT and the interaction

energy gn (both being typically on the order of one to a few kHz). Since the confinement

along z is usually harmonic, with frequency ωz, the gap is Δz = �ωz. The z degree

of freedom is thermodynamically frozen when the extension of the ground state of the

z-motion, az =√

�/mωz, is such that az � ξ, λ/√

2π.

6.1. Experimental implementations. – Conceptually, the simplest scheme to produce

a 2d gas is to use a single Gaussian light beam that is red-detuned with respect to the

atomic resonance. The beam propagates along the x-direction, with waists along y and

z such that wy � wz, so that it forms a horizontal light sheet. The dipole potential

created by this light sheet attracts the atoms towards the focal point, and ensures a

strong confinement in the z-direction. This technique was used at MIT to produce the

first atomic gas (of sodium atoms) in a quasi-2d regime [67]. More recently it has been

implemented at NIST to study the coherence properties of the 2d gas [28].

One can also produce a 2d gas by using an evanescent light wave at the surface of a

glass prism [68, 69], so that the atoms are trapped at a distance of a few micrometers

from the horizontal glass surface. The confinement in the horizontal xy-plane is provided

by an additional laser beam or by a magnetic field gradient. The fact that the confine-

ments in the xy-plane and along the z-axis have different origins is an interesting feature

because it offers the possibility, by releasing only the planar confinement, to study the

ballistic expansion of the atoms in the xy-plane only (see subsect. 7.2). Another experi-

mental system providing independent confinement in the xy-plane and along z has been

investigated at Oxford, where a blue detuned, single node, Hermite Gaussian laser beam

traps atoms along the z-direction, and the confinement in the xy-plane is provided by a

magnetic-field gradient [70].

Two-dimensional confining potentials that are not based on light beams have also

been investigated. One possibility discussed in [71] consists in trapping paramagnetic

atoms just above the surface of a magnetized material that produces an exponentially

decaying field. The advantage of this technique lies in the very large achievable frequency

ωz, typically in the MHz range. One drawback is that the optical access in the vicinity

of the magnetic material is not as good as with optically generated trapping potentials.

Another appealing technique to produce a single 2d sheet of atoms uses the so-called

radio-frequency dressed state potentials [72-74].

A 1d optical lattice setup, formed by the superposition of two running laser waves,

is a very convenient way to prepare stacks of 2d gases [75-79]. The 1d lattice provides

a periodic potential along z with an oscillation frequency ωz that can easily exceed the

typical scale for interaction energy and temperature. The simplest lattice geometry is


formed by two counter-propagating laser waves, and provides the largest ωz for a given

laser intensity. One drawback is that the lattice period is small (λ0/2, where λ0 is

the laser wavelength) so that many planes are generally populated and the addressing

of a single plane is difficult. Therefore practical measurements only provide averaged

quantities. Another interesting geometry consists in forming a lattice with two beams

crossing at an angle θ smaller than 180◦ [80]. In this case the distance λ0/(2 sin(θ/2))

between adjacent planes is adjustable, and each plane can be individually addressable if

this distance is large enough [50,81]. Furthermore the tunneling matrix element between

planes can be made completely negligible, which is important if one wants to achieve a

truly 2d geometry and not a periodically modulated 3d system.

Finally, while here we are primarily interested in continuous 2d gases of spinless

Bosons, two related experiments on 2d physics also need to be mentioned:

First, an experiment performed in Boulder constitutes a direct implementation of

the XY model [82]. There, an array of parallel elongated (quasi-)condensates is created

in a 2d optical lattice, and tunneling matrix element J provides a Josephson-type cou-

pling between the neighboring lattice sites. In this system proliferation of vortices is

observed when the temperature is increased. Vortices are detected by turning off the

optical lattice and allowing the quasi-condensates trapped on different sites to overlap

and interfere. The measured surface density of vortices as a function of the ratio J/T is

in good agreement with the BKT theory applied to this system.

Second, in an experiment at Berkeley 2d physics was studied in a spinor Bose-Einstein

condensate of Rb atoms with total spin F = 1 and weak ferromagnetic spin-dependent

interactions [83]. This system is anisotropic, but still 3d with respect to the density

degrees of freedom, i.e. the healing length ξ is shorter than the shortest extension of

the cloud, along z. However, weak spin-dependent interactions correspond to a longer

healing length ξs, so that the system is 2d with respect to the spin degrees of freedom.

In this case the magnetization transverse to the quantization axis has a role analogous to

the phase of the wave function in a spinless Bose gas. At low T ferromagnetic interactions

favor spontaneous symmetry breaking but spin-vortex structures are also observed.

6.2. Interactions in a 2d atomic gas . – To address the role of interactions in these

gases, we start with some considerations concerning the quantum scattering of two atoms

when the z motion is strongly confined. In a strictly 2d problem and at low energy, the

scattering state between two identical bosonic particles with relative wave vector k is [25]

(74) ψk(r) ∼ eik·r −

√i

8πf(k)

eikr

√kr

, f(k) ≈4π

− ln(k2a22) + iπ

,

where a2 is the 2d scattering length. One should notice that contrarily to the 3d case,

the scattering amplitude f(k) does not tend to a non-zero finite value when k tends

to 0. In the experimental implementations of 2d gases that have been achieved so far,

the confinement along z was still relatively weak from a collisional point of view, in

the sense that the thickness az of the gas remained notably larger than the 3d scattering


length as. The scattering problem in this confined geometry has been discussed in [84,85]

(see also [86]); the general expression (74) for the scattering state ψk remains valid and

the scattering amplitude can be written

(75) f(k) ≈4π

√2πaz/as − ln(κ k2a2

z) + iπ

with κ ≈ 3.5, corresponding to the 2d scattering length

(76) a2 = az

√κ exp

(−

√π

2

az

as

).

Now for all experiments realized so far, the first term√

2πaz/as in the denominator of

eq. (75) is large compared to 1, and dominates over the logarithmic term ln(κ k2a2z) and

the imaginary term iπ. We can then take a constant scattering amplitude (as in 3d) to

describe the collisions in the gas: f(k) ≡ g ≈√

8πas/az. With this approximation the

interaction energy of the gas with density n(r) in the xy plane is

(77) Eint =�

2g

2m

∫n2(r) d2r.

The corresponding values for the 2d scattering length are extremely small, due to the

exponential factor in eq. (76). Taking for example az = 200 nm and as = 5 nm (87Rb

atoms), we find a2 = 6 10−29 m. Typical surface densities are in the range 1013 m−2,

and the dimensionless parameter na22

that is relevant for perturbative expansions of the

equation of state of the 2d Bose gas [87-93] is also extremely small: na22∼ 4 10−44 for

the numbers given above.

The expression (77) can also be obtained by starting from the 3d interaction energy

(78) Eint,3d =2π�

2as

m

∫n2

3(r) d3r,

in which we plug directly n3(x, y, z) = n(x, y) exp(−z2/a2z)/

√πa2

z. However the validity

condition as � az remains hidden in this procedure.

To summarize, the collision dynamics in the experiments performed so far is still

dominated by 3d physics. The 3d scattering length as is much smaller than the thickness

of the gas and the scattering amplitude is nearly k-independent. This regime is often

referred to as “quasi-2d”. It is important to note that the term “quasi-2d” is also used

to describe another aspect of the 2d gases: very often the temperature of the gas (and

possibly the interaction energy) is not small compared to �ωz, but comparable or even a

bit larger. We discuss in the next subsection how to handle this problem.


6.3. Residual excitation of the z-degree of freedom. – For a quantitative analysis of

experiments performed with 2d gases, in particular for the determination of the tem-

perature, it is important to take into account the residual excitation of the z-degree

of freedom. This was first pointed out in [94], where a quantum Monte Carlo sim-

ulation gave an estimate for the distortion of the density profile due to this residual

excitation. Several possible ways were subsequently proposed to take this excitation into

account [94,65,66,95,96]. The simplest method consists in renormalizing the interaction

strength g to account for the density profile of the gas along the z direction [94]. The

predictions derived with this method were compared with quantum Monte Carlo results

in [65] and later analyzed in detail in [95]. In the following we outline the slightly more

elaborate treatment of [66] which has the advantage of taking into account not only the

thermal excitation of the z degree of freedom, but also the possible deformation of the

ground state of the z-motion due to atomic interactions.

The method used in [66] is a direct implementation of the Hartree-Fock approximation

(see, e.g., [97]) and we first present it for a gas which is uniform in the xy-plane. We

choose a 3d trial density profile n3(z) uniform in the xy-plane and varying along the

strongly confined z direction. We then consider the Hamiltonian with the mean-field

energy

(79) H = −�

2

2m∇2 +

1

2mω2

zz2 + 2g(3d)n3(z).

The single-particle eigenfunctions of this Hamiltonian can be written ψk,j(x, y, z) =

ϕj(z) ei(kxx+kyy)/2π, with energy Ek,j = �

2k2/(2m) + εj , where k2 = k2x + k2

y. The

normalized functions ϕj(z) and the energies εj of the z-motion of course depend on

the choice of the trial density profile n3(z). In the Hartree-Fock approximation the

average occupation of the single particle level ψk,j is given by the Bose factor f(Ek,j) =

(exp(β(Ek,j − μ)) − 1)−1. We calculate the corresponding 3d density profile, which is

still uniform in xy and has the following z-dependence:

(80) n′

3(z) =

∑j

∫d2k |ψk,j |

2 f(Ek,j) = −1

λ2

∑j

|ϕj(z)|2 ln

(1 − Ze−βεj

).

The self-consistency of the Hartree-Fock approximation requires that n3(z) and n′

3(z)

coincide, which can be achieved by iterating the solution of the above set of equations

until a fixed point is reached. With this method, we fulfill two goals: i) We take into

account the residual thermal excitation of the levels in the z-direction. ii) Even at

zero temperature we take into account the deformation of the z ground state due to

interactions. When interactions can be neglected, the eigenstates ϕj(z) are the Hermite

functions and εj = �ωz(j + 1/2).

Using the local density approximation, the above method can be straightforwardly

adapted to the case where a trapping potential V⊥ is present in the xy-plane. The trial

density distribution n3(r) is now a function of all three spatial coordinates. At any point


(x, y), we treat quantum mechanically the z motion and solve the eigenvalue problem for

the z variable

(81)

[−�

2

2m

d2

dz2+ Veff(r)

]ϕj(z|x, y) = εj(x, y)ϕj(z|x, y),

where Veff(r) = V⊥(x, y) + mω2zz

2/2 + 2g(3d)n3(r) and∫|ϕj(z|x, y)|

2 dz = 1. Treating

semiclassically the xy degrees of freedom, we obtain a new spatial density

(82) n′

3(r) = −

1

λ2

∑j

|ϕj(z|x, y)|2 ln

(1 − Ze−βεj(x,y)

).

Again the Hartree-Fock prediction is obtained by iterating this calculation until the

spatial density n3(r) reaches a fixed point. Since ϕj is a normalized function of z at any

point (x, y), the total 2d density is

(83) n(x, y) =

∫n3(r) dz = −

1

λ2

∑j

ln(1 − Ze−βεj(x,y)

).

In the limit where only the ground state j = 0 of the z motion is populated, the result of

this Hartree-Fock approach coincides with the solution of (70). The method used in [95]

is similar to this approach, but the deformation of the eigenstates due to mean-field

interaction was neglected.

The density profiles predicted by this method have been compared with the results

of a quantum Monte Carlo simulation [96]. An important outcome for the analysis

of experimental data is the excellent agreement between the two approaches as long as

nλ2 < 2. This agreement holds for the temperature regime (kBT ≤ 2�ωz) and interaction

strength (g � 0.15) relevant for current experiments. The Hartree-Fock approach is

therefore well suited for fitting the wings of the experimental density profiles of a quasi-

2d gas to extract the temperature and chemical potential.

7. – Probing 2d atomic gases

This section is devoted to the presentation of some methods that have been used

for the experimental study of 2d Bose gases. We start with the conceptually simplest

approach, which consists in the measurement of the steady-state distribution of atoms

in a trap. We then turn to the information that can be acquired in a Time-of-Flight

expansion. Finally we discuss two schemes that give access to the phase coherence of

the gas.

7.1. In situ density distribution. – Conceptually the simplest information that can

be obtained on a 2d gas is a picture of the sample along the direction that is strongly

confined. Since this degree of freedom is supposed to be frozen out, there is no loss of

information due to integration along the line-of-sight. This is in contrast to what happens


in 3d, where one has to resort to a non trivial transformation to reconstruct the spatial

distribution [98] (see also [99] for a review).

Assuming that local density approximation (LDA) is valid, the density distribution

in the trap n(r) can be obtained from the equation of state of the homogeneous system.

The general form of this equation of state is

(84) nλ2 = F (μ, kBT, a2),

where F is at this stage an unknown function and a2 is the 2d scattering length. Within

LDA the density n(r) in the trap is calculated by replacing μ by μ− V (r), where V (r)

is the trapping potential.

In the quasi-2d regime that is of practical interest (as � az), we have seen in sub-

sect. 6.2 that the interactions in the gas are characterized to a good approximation by

the dimensionless number g =√

8π as/az � 1. In this case eq. (84) can be simplified

using dimensional analysis; the expression of the phase space density D = nλ2 must take

the functional form

(85) D = G(α, g) with α =μ

kBT.

For a gas that is trapped in a harmonic potential mω2r2/2, the in situ density profile is

then given by

(86) n(r)λ2 = G

(α−

r2

2r2T, g

),

where we set as above mω2r2T = kBT . This expression clearly shows a scale invariance for

a given interaction strength g. Suppose that different density profiles n(r) are recorded

for various temperatures T and various atom numbers N (hence different chemical po-

tentials μ). According to eq. (86) the profiles can all be superimposed on the same

curve G(α, g), provided they are plotted as a function of r2/r2T and translated along the

x-axis by the dimensionless quantity α = μ/kBT . This scale invariance behaviour has

been checked with excellent accuracy by M. Holzmann and W. Krauth using quantum

Monte Carlo simulations [100]. These simulations were performed for g = 0.15, which

corresponds to the interacting strength in ENS experiments with Rb atoms.

For g � 1, various asymptotic forms of the function G(α, g) have been given earlier.

When interactions can be neglected (g = 0), the equation of state is (cf. eq. (6)): D =

− ln(1 − eα). In the presence of interactions and for small phase space densities, the

mean-field Hartree-Fock method amounts to replace μ by μ − 2gn into the ideal-gas

result, which leads to the implicit equation D = − ln(1 − eα−gD/π), from which one

can extract D as a function of α and g (see also eq. (70)). In the strongly degenerate

limit, where μ � kBT and D � 1, density fluctuations are strongly reduced and one

expects μ = gn, which can be written as D = (2π/g)α. In the intermediate regime,

in particular close to the BKT transition point, one can use the results of the classical


0

5

10

15

20

0 +0.5-0.5-1.0 0.0 0.5 1.0 1.5

0

5

10

15

20

Ph

ase

sp

ace

de

ns

ty

μ/kBT

g = 0 .15

r/rT

(a) (b)

g = 0 .15

Fig. 3. – (a): Phase space density as a function of α = μ/kBT for g = 0.15. Continuous line:Total phase space density D = nλ

2; dashed line: superfluid phase space density nsλ2. The

dotted and dash-dotted lines represent the asymptotic regimes for low and high phase spacedensities, respectively. (b) In situ density profiles in a trap deduced from the left panel usingthe local density approximation. The plot is made for μ/kBT = 0.5 so that the Thomas-Fermiradius rTF =

p2μ/mω2 is equal to rT .

field Monte Carlo analysis of [44]. The resulting function D = F (α, g) is represented in

fig. 3 for g = 0.15. The two asymptotic regimes that we just described are indicated

with dotted and dash-dotted lines. A remarkable characteristic of the function F (α, g) is

precisely the absence of significant features at the critical point for the BKT transition

(corresponding to α ≈ 0.2 and D ≈ 8 for g = 0.15). This is due to the infinite order of

the BKT transition, that does not cause any singularity in the dependance of the total

density n on T or μ. On the other hand, the superfluid density ns (plotted as a dashed

line in fig. 3) is discontinuous at the transition point, but this quantity is not directly

accessible from an in situ measurement. For a detailed comparison between the results of

the mean-field Hartree-Fock approach and those obtained from a quantum Monte Carlo

simulation and from a renormalization group treatment, see [96] and [63], respectively.

Finally we note that the analysis of individual images requires a proper knowledge of

the temperature and the chemical potential. These are usually obtained by fitting the

wings of the distribution with the appropriate function for the quasi non-degenerate gas

(see the discussion following eq. (83)). An interesting alternative consists in using in situ

density fluctuations to determine these thermodynamic quantities [101]. This promising

method that relies on the fluctuation-dissipation theorem for a non-uniform system has

not yet been implemented experimentally for a 2d Bose gas.

7.2. Two-dimensional Time-of-Flight expansion. – Generally speaking, a Time-of-

Flight (TOF) procedure consists in switching off abruptly the potential confining the

atoms, letting the cloud expand for an adjustable time, and then measuring the den-

sity profile. If the role of interactions is negligible during the expansion, the density

profile after a long TOF is proportional to the in-trap momentum distribution. For a

two-dimensional system, two types of TOF can be considered. One can switch off the


potential confining the atoms in the xy-plane, while keeping the strong confinement along

the frozen direction z; we will call this procedure a “2d TOF”. Alternatively, one can

switch off simultaneously the potential in the xy-plane and the confinement along z,

corresponding to a “3d TOF”. We discuss 2d TOF in this section, and 3d TOF in the

following one.

We consider here the case of an isotropic harmonic trap in the xy plane V (r) =

mω2r2/2. A 2d gas is initially at thermal equilibrium in this trap, with a density profile

neq(r). Suppose that this potential is suddenly switched off at time t = 0 whereas the

confinement along the z direction remains unchanged. Using the Bogoliubov approach,

it was predicted in [102] that the subsequent evolution of the density distribution is given

by the scaling law

(87) n(r, t) = η2

t neq(ηtr), ηt = (1 + ω2t2)−1/2.

This means that the global form of the spatial distribution is preserved during the TOF.

As the expansion proceeds, the interaction energy that was initially present in the gas is

converted into kinetic energy in such a way that the density profile at time t is obtained

using a scaling transform of the initial one. We emphasize that this remarkable result is

stronger than its 3d counterpart [103,102] which holds only in the Thomas-Fermi regime:

In the 2d case the scaling behavior is valid both for the superfluid component and for

the thermal Bogoliubov excitations. This scaling behavior has been recently observed by

the ENS group [104].

The scaling invariance in the expansion of a 2d interacting gas has been explained

in terms of the SO(2, 1) symmetry group for a whole class of interaction potentials

U(r) between two particles [105]: it is sufficient that U is a homogeneous function of

degree 2, U(αr) = U(r)/α2. When this is the case, the result of eq. (87) holds for an

arbitrary initial state of the 2d gas, irrespective of its temperature. The 2d contact

interaction potential, which is implicitly assumed in eq. (16), belongs to this class of

functions. We note however that a true contact interaction is singular in 2d because it

leads to ultraviolet divergences at the level of quantum field equations. A real interatomic

potential has a finite range which provides a UV cut-off that eliminates the divergences.

This regularization will occur if one uses the more precise treatment of atomic interactions

given in eq. (75). It will lead to deviations with respect to the universal law (87), which

remain to be evaluated and characterized.

7.3. Three-dimensional Time of Flight. – In a 3d TOF both the trapping potential in

the xy-plane and the strong confinement along the z-direction are switched off simulta-

neously. The physics is then very different from that of a 2d TOF. Along the initially

strongly confined direction z, the atom cloud expands very fast since the momentum

width Δpz ∼ �/Δz is large. If the atoms are initially in the ground state of a harmonic

potential with frequency ωz along this axis, the extension of the cloud is multiplied by√2 in a time t = ω−1

z . The time scale for the expansion of the gas in the xy-plane

is much longer; it is given by ω−1, where ω � ωz is the trapping frequency in this


plane. Therefore it is a good approximation to decompose a 3d TOF into two phases.

During the first phase, whose duration is a few ω−1z (typically 1 ms if ωz/2π = 3 kHz),

the thickness of the gas along z increases by a factor much larger than 1, but the xy

spatial distribution is nearly not modified. At the end of this first phase, the interactions

between atoms have become negligible. During the subsequent phase the expansion in

the xy-plane becomes significant, but on a much longer time scale. It corresponds to the

expansion of an ideal gas, whose initial state is equal to the state of the system in the

xy-plane before the beginning of the TOF.

We now focus on the evolution of the xy degrees of freedom during the second phase,

which is essentially governed by single-particle physics. The evolution of the density

distribution in the xy-plane can be determined from the initial one-body density matrix

g1(r, r′) = 〈r|ρ(1)|r′〉, or from its Fourier transform Π(p) with respect to the variable

r − r′, which represents the momentum distribution in the xy-plane.

In the absence of any extended coherence in the gas, g1(r, r′) tends to zero when

|r − r′| increases, with a characteristic decay length given by the thermal wavelength

λ. The corresponding momentum width is Δp ∼ �/λ and the spatial distribution after

TOF will reflect the initial momentum distribution if the TOF duration t is such that

Δp t/m � rT , where rT is the initial size of the gas. For a harmonic confinement in the

xy-plane, this “far field” regime corresponds to ωt � 1. Taking ω/2π = 30 Hz as a typical

value, the far field regime (say ωt > 3) is reached for t > 15 ms. This corresponds to a

typical value for TOF experiments, which thus give access to the momentum distribution

in this non (strongly) degenerate regime.

The situation is very different if a significant condensed fraction is present in the gas,

as expected in the vicinity and below the BKT transition temperature. In this case we

have seen in subsect. 5.4 that the size rc of the coherent region of the cloud is rc ∼ rT .

The momentum width Δpc = �/rc of this coherent component is then extremely narrow,

and it would require a very long TOF to reach the “far field” regime for this coherent

component. Taking rc = rT as a typical value, we find that the time t required for a

significant expansion of this component, i.e. Δpc t/m = rc, is such that ωt = kBT/�ω.

For ω/2π = 30 Hz and T = 100 nK, this gives t > 300 ms, which is too long in practice

for a TOF.

Therefore in the regime where a relatively strong coherence of the gas is present, a

3d TOF of a realistic duration gives access to a hybrid information. The high-energy

fraction of the gas is in the far-field regime and the wings of the density profile after TOF

give access to the large momentum part of the initial state. On the contrary the central

feature corresponding to the condensed, superfluid fraction, has not yet undergone a

significant expansion. The detailed study of the border between these two components

is still a matter of debate. In experiments with rubidium atoms [27, 106], the density

profile after a 3d TOF is well modeled by a two-component distribution and fits with

the line of reasoning we just presented. In contrast, in the experiments performed at

NIST with a sodium gas [28], an intermediate third component was introduced in order

to obtain a good description of the density profiles after a 3d TOF. This component

corresponds to a phase with a spatial scale of coherence that is intermediate between the


microscopic length λ and the macroscopic one rT , and it is qualified as a “non-supefluid

quasi-condensate” in [28].

It is interesting to note that 3d TOF is the most common and natural experimental

method used in the studies of 3d atomic gases. However, in hindsight, its availability is a

non-trivial and rather serendipitous feature of atomic systems for studies of BKT physics.

In combination with the finite-size induced condensation, the ability to suddenly turn off

the interactions through the fast z-expansion provides a much more striking signature of

the BKT transition [27, 28, 106] than one might have theoretically expected. Thinking

strictly in 2d, the transition is extremely smooth and one would not naturally expect

to see such a dramatic signature in any quantity except the superfluid density. As we

discussed in subsect. 7.2, in 2d TOF the observed density distribution indeed varies

smoothly across the transition.

So far we have discussed the “average” density profile in 3d TOF, which theoretically

corresponds to the average of a large number of images obtained under same conditions. It

is also interesting to consider the density noise in individual images, which can be related

to the phase noise of the gas before expansion. This connection has been exploited for

quasi-1d gases since 2001 [107]. For the 2d case, it has been shown theoretically in [108]

that the two-point density correlation function after TOF can provide information on

the in situ g1 function, at least in the superfluid regime. This method is also specific

to 3d TOF, where the phase noise evolves into density noise in interaction-free ballistic

expansion.

7.4. Interference between independent planes . – Since an important aspect of the

physics of 2d Bose gases is related to phase properties, it is natural to investigate mea-

surement schemes based on interferometry. We start with the proposal by Polkovnikov

et al. [109] which showed how a single experimental procedure could characterize both

the normal regime (exponential decay of g1) and the superfluid regime (algebraic decay

of g1) (see also [110] for a more complete review). Consider two independent, infinite

planar gases located at za = +dz/2 and zb = −dz/2. They are prepared in identical

conditions, i.e. they have the same temperature and the same density. We perform a 3d

time-of-flight of duration t, that is chosen such that the final extension along z of each

cloud is large compared to the initial separation dz between the planes. The two clouds

thus overlap and we want to extract information about the one-body correlation function

g1 from their interference pattern (fig. 4a).

The state of each plane is described by the wave function ψa/b(x, y). After expansion,

the spatial atomic density n is modulated along any line parallel to the z axis with the

period Dz = ht/mdz [111]:

(88) n ∝ |ψa|2 + |ψb|

2 +(ψaψ

∗

b ei2πz/Dz + c.c.

).

For simplicity we have omitted in the above equation a global envelope factor giving

the variation of the density along the z-axis. Also we have neglected the expansion in

the xy-plane during the TOF. As explained above there exists a range of TOF duration


Fig. 4. – (a) Principle of an experiment giving access to the interference between two independentplanar gases, observed after time-of-flight. (b-d): Examples of interference patterns measuredwith the experimental setup described in [26]. The imaging beam is propagating along they-axis. The pattern (b) is obtained with very cold gases, whereas (c) corresponds to a largertemperature. The dislocation in (d) is the signature for the presence of a vortex in one of thetwo gases.

where this is valid, if the trapping frequency ω in this plane is much smaller than ωz. We

see from eq. (88) that the local (complex) contrast of the density modulation is ψaψ∗

b .

Experimentally one cannot measure this quantity along a single line, and one rather has

access to the average contrast over a region of finite area A in the xy-plane. In particular

if one performs absorption imaging along the y-axis (fig. 4b-d), the image involves an

integration of the local contrast ψaψ∗

b along the y-direction(8). Averaging the result of

this contrast measurement over a large number of realizations, one can define the average

contrast C(A):

(89) C2(A) =1

A2

⟨∣∣∣∣∫

A

ψa(r)ψ∗

b (r) d2r

∣∣∣∣2⟩.

Using the fact that the fluctuations of the wave functions ψa and ψb are uncorrelated

and taking advantage of the translational symmetry of the system, we find

(90) C2(A) =1

A

∫A

|g1(r)|2 d2r,

where g1(r) = 〈ψ∗

j (r)ψj(0)〉 for j = a, b. Suppose for simplicity that the area A is a

square and consider the two cases of an exponentially decaying g1(r) ∝ e−r/ , with a

characteristic length � �√A (normal fluid), and an algebraically decaying g1(r) ∝ r−α,

with an exponent α < 1/4 (superfluid regime). In the first case, the integral is nearly

independent of A and C2(A) ∝ A−1. In the second case we find C2(A) ∝ A−2α which

corresponds to a decay always slower than A−1/2. This method is very appealing in the

sense that the measurement of a single number, i.e. the exponent η characterizing the

variation of C ∝ A−η, is sufficient to identify the two possible regimes of a 2d Bose gas,

and obtain the value of nsλ2 = 1/η in the superfluid case.

(8) The length over which the line-of-sight integration occurs can be adjusted by a proper“slicing” of the cloud just before the imaging process, as in [111].


A measurement scheme inspired by this method was implemented at ENS [26], and it

indeed revealed a relatively rapid variation of the exponent η, in qualitative agreement

with what is expected near the BKT transition point in the center of the trap. However

some notable deviations with respect to the original proposal must be stressed. First,

the measurement was performed with anisotropic samples, with lengths Ly � Lx. The

imaging beam was propagating along y and the measured contrast involved a line-of-sight

integration over the full length Ly(9), which formally breaks the translational invariance

that we used to prove eq. (90). Also the presence of a trapping potential in the ex-

periment causes an additional softening of the transition, because of the inhomogeneity

of the density along the line-of-sight of the imaging beam. Finally we note that even

deep in the superfluid regime where nsλ2 � 1, the anisotropy of the sample adds some

complexity as discussed in subsect. 4.5. At large distances (Δx > nsλ

2 Ly), g1 starts to

decay exponentially, which complicates the analysis of the dependence of C2 on Δx. In

summary the rapid increase in coherence that occurs in the vicinity of the BKT tran-

sition point is sufficiently robust to be revealed experimentally in the average contrast

of the interference pattern, but it is difficult to provide a quantitative analysis of the

experimental measurements for the variations of C2(A) over a large range of Δx.

A subsequent experiment at ENS has compared the conditions for observing a signif-

icant interference contrast between the planes and for measuring a clear bimodal density

profile after a 3d TOF [27]. The onsets of the two phenomena were found to coincide

within experimental error. Furthermore the spatial part of the gas that gives rise to a

visible interference signal coincides with the central, “non expanding” component of the

TOF profile.

An important outcome of the experiments on the interference between two planes is

a direct evidence for thermally activated vortices. At low temperatures, long-wavelength

phase fluctuations (phonons) result in smooth variations of the phase of the interference

fringes, such as seen in fig. 4c. However, if a single isolated vortex is present in one of the

two planes while the phase profile of the other plane is smooth, the interference pattern

exhibits a sharp dislocation at the coordinate x of the vortex core. Such dislocations

have been observed experimentally [81, 26, 69] and an example is shown in fig. 4d. The

occurrence probability of these dislocations has been measured as a function of tempera-

ture [26]. The number of dislocations increases with T , until one reaches the temperature

at which no interference is visible anymore. Moreover, the relatively sharp increase in the

probability of dislocations experimentally coincides with the increase in the exponent η

characterizing the decay of g1 [26]. Such dislocations also appear in a classical field sim-

ulation mimicking the interference between two planar gases [40]. They result from the

thermal activation of a vortex pair for which the two members are sufficiently separated

from each other. In principle one should also observe in the interference patterns tightly

bound vortex pairs where the two members are separated by ∼ ξ. However these pairs

(9) The area A was varied by changing the integration distance Δx along x. In this case, theBKT transition causes a crossover from η = 1/2 for an exponentially decaying g1 function (witha decay length � � Δx), to η = 1/4 for a superfluid state.


only shift the fringe pattern by a small fraction of the fringe spacing, which is below the

current sensitivity of the experiments.

7.5. Interfering a single plane with itself . – An interesting alternative to the two-plane

interference described above consists in preparing a single plane of atoms and looking at

its “self-interference”, using a Ramsey-like method [28]. The gas is initially prepared in

an internal state |1〉. Half of the atoms are coherently transferred into another internal

state |2〉 by a stimulated laser Raman process (π/2 pulse) that also provides a momentum

kick k0 to the atoms. After this process, the part of the cloud in |1〉 is still globally at

rest and the part in |2〉 moves with the global velocity v0 = �k0/m. After an adjustable

time t a second π/2 Raman pulse remixes the amplitudes of |1〉 and |2〉 and provides

a momentum kick k0 − k1. Immediately after this second Raman pulse, one measures

the spatial density distribution in |2〉. This distribution exhibits a modulation along the

direction k1, resulting from the interference between the initial state of the cloud and

the state displaced by the distance R = v0t:

(91) n(r) ∝ |ψ(r)|2 + |ψ(r − R)|2 +(ψ(r)ψ∗(r − R)eik1·R + c.c.

).

Note that we assume here that no collision occurred during the time t between the part

of the cloud at rest in |1〉 and the part moving at velocity v0 in state |2〉. This is a valid

assumption for the weakly interacting sodium gas of [28]. The modulated density profile

in eq. (91) gives a direct access to the function g1(r, r −R). It can be observed with an

imaging beam along the z-direction, so that its measurement does not involve any line-

of-sight integration. This method can then reveal finer details than the one presented

in subsect. 7.4. In particular the authors of [28] could observe a gradual increase of

the coherence length � of the cloud, as expected from eq. (56). For small phase space

densities the measurement gives � ∼ λ, and � increases to much larger values when

the temperature decreases towards the critical temperature TBKT. When T < TBKT

a significant interference contrast is observed for all values of R within the size of the

central superfluid region.

8. – Conclusions and outlook

We have reviewed in these notes the theoretical basis for the understanding of the

physics of 2d quantum fluids, and discussed some recent experiments performed with

atomic gases. These experiments have given access to some aspects of 2d physics that

were previously hidden or not measurable in other physical systems, such as the exis-

tence of thermally activated individual vortices or the spatial variation of the first-order

correlation function g1(r). However a number of issues is still open in the physics of 2d

quantum gases, and we outline below some topics which are likely to be of experimental

and theoretical interest in the future.

Higher-order correlation functions. – The matter-wave interference between two sta-

tistically similar, but independent quasi-condensates (such as shown in fig. 4), can reveal


a wealth of information on the correlations within each individual 2d gas. So far only a

fraction of this information has been harnessed, with the study of the average contrast

of the interference pattern integrated over some area of interest. A convenient tool for

extracting more complete information on g1 as well as higher-order correlation functions

is the full statistical distribution of interference contrasts. Two limiting cases can easily

be characterized: i) If the two independent fluids are fully condensed, each image shows

a 100% contrast, with the position of the fringes fluctuating randomly from shot to shot.

ii) If each cloud exhibits only short-ranged correlations, the observed interference results

from many uncorrelated fringe patterns along the light of sight, and the distribution of

contrasts is an exponential function. For 1d gases, it is possible to describe quantitatively

the transition between these two limiting cases [112], and the experimental results [113]

are in good agreement with the predictions. In the 2d case, the evolution of the contrast

distribution through the BKT transition is still an open problem.

Out-of-equilibrium dynamical effects. – Throughout this paper we restricted our dis-

cussion to the equilibrium properties of a 2d Bose fluid. The study of dynamical effects,

such as transient regimes, can reveal additional information about the system. For ex-

ample Burkov et al. [114] have studied the dynamics of decoherence between two planar

Bose gases, assuming that their local phases are initially locked together, and then the

two gases are allowed to evolve independently. This can be achieved experimentally by

having a weak potential barrier and hence large tunnel coupling between the two planes

for t < 0, and then suddenly raising the barrier at t = 0. The contrast of the interference

between the two gases gives access to the evolution of the phase distribution under the

influence of thermal fluctuations. In [114] this contrast was shown to decay algebraically

at long time, as t−ζ , with the exponent ζ proportional to the ratio T/TBKT. Therefore,

in addition to being a stringent test of thermal decoherence in a quantum many-body

system, this out-of-equilibrium study could constitute a novel thermometry method. A

related phenomenon occurs in 1d systems, where the interference contrast is predicted to

decay as exp(−(t/t0)2/3) (with t0 constant) [114], and this prediction is nicely confirmed

in the experiments by the Vienna group [115].

Transition from 2d to 3d behavior . – The possibility to vary the tunnel coupling be-

tween two or more planar gases can also be used to study the so-called “deconfinement

transition” [116], corresponding to a gradual evolution from 2d to 3d behavior. The phase

coherence between the planes will build up as the strength of the coupling is increased,

creating a situation that is reminiscent of the high-Tc cuprate superconductors. For a

large number of parallel planes, the deconfinement transition should give rise to a true

Bose-Einstein condensate [116]. The two-plane situation is also very interesting, and can

lead to the observation of the Kibble-Zurek mechanism [117]: the superfluid transition

temperature is higher for two coupled planes than for a single one, so that sudden switch-

ing on of the coupling between the planes (initially in the normal state but close to the

single-plane critical temperature) constitutes a quench of the system, and one could ob-

serve the subsequent dynamical apparition of a macroscopic quantum (quasi-)coherence.


Tunable interactions. – As we have seen throughout the paper, interactions between

particles play a crucial role in our understanding of the superfluid phase transition and

condensation in 2d fluids. In contrast to the conventional 3d BEC of an atomic gas,

where the critical temperature can to a good approximation be predicted using the ideal

gas model, the BKT transition is fundamentally interaction-driven. The strength of

interactions also affects a variety of other phenomena such as the suppression of density

fluctuations in the normal state and the connection between the 2d Bose fluid and the

XY model. It would therefore be interesting to revisit the various effects described

in these notes while continuously tuning the strength of interactions with a Feshbach

resonance [48, 49]. In the weak coupling regime (g < 10−1) we expect a gradual change

from the BKT-dominated to the BEC-dominated behavior, as discussed in sects. 4 and 5.

Further, it would be very interesting to explore the strong-coupling regime (g > 1), which

is closer to liquid helium films. This regime, which is outside the domain of validity of

the Monte Carlo results [29, 44], corresponds to the case where the scattering length as

becomes comparable the thickness of the sample along the kinematically frozen direction,

az (see 6.2). There the very nature of two-body interactions is expected to change from

3d to 2d [84, 85, 118, 119]. Therefore, experimentally reaching the condition as ≥ az

would correspond to producing a “truly 2d” as opposed to a quasi-2d Bose gas.

Superfluid density . – Generally speaking, studies of coherence and correlation func-

tions in a 2d fluid, which are well suited to experimental tools of atomic physics, are a

natural complement to the “traditional” studies of superfluidity based on transport mea-

surements, which are well suited to other physical systems such as liquid helium films [2].

For example, we have so far assumed that the two types of measurements probe the same

superfluid density (see, e.g., subsect. 7.4). However this correspondence may in fact de-

pend on the theoretical model and the exact definition of the superfluid density, and be

valid only within the effective low-energy theories. It is therefore important to stress

that superfluidity in the traditional transport sense has not yet been directly observed

in atomic 2d Bose gases (see, e.g., [116]). Establishing atomic 2d gases as experimental

systems in which both coherence and transport measurements of superfluidity could be

performed would be an important advance, as it would allow experimental scrutiny of the

theoretical connections between the two types of probes, and a direct comparison of the

different definitions of superfluidity. Two promising schemes for a direct measurement of

the superfluid density (as traditionally defined through transport properties [46]) in an

atomic gas have recently been proposed [120,121]. The first scheme [120] is based on ex-

tracting the superfluid density from the in situ density profiles of a rotating 2d gas. The

second scheme [121] is based on using a vector potential generated by Raman laser beams

to simulate slow rotation of a gas [122], and allows direct spectroscopic measurement of

the superfluid density.

Note added in proofs.

A measurement of the equation of state of a 2d Bose gas for various interaction strengths

has just been reported in [123].


∗ ∗ ∗

We warmly thank the directors of the school R. Kaiser and D. Wiersma, as well

as the scientific secretary L. Fallani, for organizing this very successful meeting. Many

colleagues helped us with discussions and interactions and the list of those we would like

to thank is too long to fit here, but we mention in particular E. Altman, N. Cooper, E.

Cornell, E. Demler, B. Doucot, T. Giamarchi, T.-L. Ho, M. Holzmann, M.

Kohl, W. Krauth, W. Phillips, A. Polkovnikov, G. Shlyapnikov, D. Stamper-

Kurn, W. Zwerger, as well as the past and present members of the ENS cold atoms

group. ZH is supported by EPSRC Grant No. EP/G026823/1. JD is supported by

Region Ile de France IFRAF, CNRS, the French Ministry of Research, ANR (Grant

ANR-08-BLAN-65 BOFL), and the E.U. project SCALA. Laboratoire Kastler Brossel is

a mixed research unit n◦ 8552 of CNRS, Ecole normale superieure, and Universite Pierre

et Marie Curie.

REFERENCES

[1] Peierls R. E., Surprises in Theoretical Physics (Princeton University Press) 1979.[2] Bishop D. J. and Reppy J. D., Phys. Rev. Lett., 40 (1978) 1727.[3] Safonov A. I., Vasilyev S. A., Yasnikov I. S., Lukashevich I. I. and Jaakkola S.,

Phys. Rev. Lett., 81 (1998) 4545.[4] Minnhagen P., Rev. Mod. Phys., 59 (1987) 1001.[5] Posazhennikova A., Rev. Mod. Phys., 78 (2006) 1111.[6] Bloch I., Dalibard J. and Zwerger W., Rev. Mod. Phys., 80 (2008) 885.[7] Snoke D., Science, 298 (2002) 1368.[8] Butov L. V., J. Phys. Condens. Matter, 16 (2004) R1577.[9] Kasprzak J., Richard M., Kundermann S., Baas A., Jeambrun P., Keeling

J. M. J., Marchetti F. M., Szymanska M. H., Andre R., Staehli J. L., Savona

V., Littlewood P. B., Deveaud B. and Dang L. S., Nature, 443 (2006) 409.[10] Amo A., Lefrere J., Pigeon S., Adrados C., Ciuti C., Carusotto I., Houdre R.,

Giacobino E. and Bramati A., Nature Phys., 5 (2009) 805.[11] Kosterlitz J. M., J. Phys. C: Solid State Physics, 7 (1974) 1046.[12] Nelson D. R. and Halperin B. I., Phys. Rev. B, 19 (1979) 2457.[13] Strandburg K. J., Rev. Mod. Phys., 60 (1988) 161.[14] Peierls R. E., Helv. Phys. Acta, 7 (1934) 81.[15] Peierls R. E., Ann. Inst. Henri Poincare, 5 (1935) 177.[16] Bogoliubov N. N., Physica, 26 (1960) S1.[17] Hohenberg P. C., Phys. Rev., 158 (1967) 383.[18] Mermin N. D. and Wagner H., Phys. Rev. Lett., 17 (1966) 1307.[19] Penrose O. and Onsager L., Phys. Rev., 104 (1956) 576.[20] Berezinskii V. L., Sov. Phys. JETP, 34 (1971) 610.[21] Kosterlitz J. M. and Thouless D. J., J. Phys. C: Solid State Physics, 6 (1973) 1181.[22] Huang K., Statistical Mechanics (Wiley, New York) 1987.[23] Olshanii M. and Pricoupenko L., Phys. Rev. Lett., 88 (2002) 010402.[24] Al Khawaja U., Andersen J. O., Proukakis N. P. and Stoof H. T. C., Phys. Rev.

A, 66 (2002) 013615.[25] Adhikari S. K., Am. J. Phys., 54 (1986) 362.


[26] Hadzibabic Z., Kruger P., Cheneau M., Battelier B. and Dalibard J., Nature,441 (2006) 1118.

[27] Kruger P., Hadzibabic Z. and Dalibard J., Phys. Rev. Lett., 99 (2007) 040402.

[28] Clade P., Ryu C., Ramanathan A., Helmerson K. and Phillips W. D., Phys. Rev.

Lett., 102 (2009) 170401.

[29] Prokof’ev N. V., Ruebenacker O. and Svistunov B. V., Phys. Rev. Lett., 87 (2001)270402.

[30] Mora C. and Castin Y., Phys. Rev. A, 67 (2003) 053615.

[31] Castin Y., J. Phys. IV, 116 (2004) 87.

[32] Gross E. P., Il Nuovo Cimento, 20 (1961) 454.

[33] Pitaevskii L. P., Sov. Phys. JETP, 13 (1961) 451.

[34] Ma S.-K., Statistical Mechanics, Chapter 30 (World Scientific, Singapore) 1985.

[35] Kagan Y., Svistunov B. V. and Shlyapnikov G. V., Sov. Phys. JETP, 66 (1987)314.

[36] Popov V. N., Functional Integrals and Collective Modes (Cambridge University Press,Cambridge) 1987.

[37] Petrov D. S., Gangardt D. M. and Shlyapnikov G. V., J. Phys. IV, 116 (2004) 5.

[38] Nelson D. R. and Kosterlitz J. M., Phys. Rev. Lett., 39 (1977) 1201.

[39] Minnhagen P. and Warren G. G., Phys. Rev. B, 24 (1981) 2526.

[40] Simula T. P. and Blakie P. B., Phys. Rev. Lett., 96 (2006) 020404.

[41] Giorgetti L., Carusotto I. and Castin Y., Phys. Rev. A, 76 (2007) 013613.

[42] Bisset R. N., Davis M. J., Simula T. P. and Blakie P. B., Phys. Rev. A, 79 (2009)033626.

[43] Fisher D. S. and Hohenberg P. C., Phys. Rev. B, 37 (1988) 4936.

[44] Prokof’ev N. V. and Svistunov B. V., Phys. Rev. A, 66 (2002) 043608.

[45] Bramwell S. T. and Holdsworth P. C. W., Phys. Rev. B, 49 (1994) 8811.

[46] Leggett A. J., Physica Fennica, 8 (1973) 125.

[47] Fisher M. E., Barber M. N. and Jasnow D., Phys. Rev. A, 8 (1973) 1111.

[48] Tiesinga E., Verhaar B. J. and Stoof H. T. C., Phys. Rev. A, 47 (1993) 4114.

[49] Inouye S., Andrews M., Stenger J., Miesner H. J., Stamper-Kurn D. M. andKetterle W., Nature, 392 (1998) 151.

[50] Schrader D., Dotsenko I., Khudaverdyan M., Miroshnychenko Y.,

Rauschenbeutel A. and Meschede D., Phys. Rev. Lett., 93 (2004) 150501.

[51] Nelson K. D., Li X. and Weiss D. S., Nature Phys., 3 (2007) 556.

[52] Bakr W. S., Peng A., Folling S. and Greiner M., Nature, 462 (2009) 74.

[53] Gerbier F., Europhys. Lett., 66 (2004) 771.

[54] Bagnato V., Pritchard D. E. and Kleppner D., Phys. Rev. A, 35 (1987) 4354.

[55] Bagnato V. S. and Kleppner D., Phys. Rev. A, 44 (1991) 7439.

[56] Yukalov V. I., Phys. Rev. A, 72 (2005) 033608.

[57] Pitaevskii L. and Stringari S., Bose-Einstein Condensation (Oxford University Press,Oxford) 2003.

[58] Pethick C. and Smith H., Bose-Einstein Condensation in Dilute Gases (CambridgeUniversity Press) 2002.

[59] Bhaduri R. K., Reimann S. M., Viefers S., Ghose Choudhury A. and Srivastava

M. K., J. Phys. B: At. Mol. Opt. Phys., 33 (2000) 3895.

[60] Ho T. L. and Ma M., J. Low Temp. Phys., 115 (1999) 61.

[61] Fernandez J. P. and Mullin W. J., J. Low Temp. Phys., 128 (2002) 233.

[62] Gies C. and Hutchinson D. A. W., Phys. Rev. A, 70 (2004) 043606.

[63] Lim L.-K., Smith C. M. and Stoof H. T. C., Phys. Rev. A, 78 (2008) 013634.


[64] Holzmann M., Baym G., Blaizot J. P. and Laloe F., Proc. Natl. Acad. Sci. U.S.A.,104 (2007) 1476.

[65] Holzmann M., Chevalier M. and Krauth W., EPL, 82 (2008) 30001.[66] Hadzibabic Z., Kruger P., Cheneau M., Rath S. P. and Dalibard J., New J. Phys.,

10 (2008) 045006.[67] Gorlitz A., Vogels J. M., Leanhardt A. E., Raman C., Gustavson T. L., Abo-

Shaeer J. R., Chikkatur A. P., Gupta S., Inouye S., Rosenband T. and Ketterle

W., Phys. Rev. Lett., 87 (2001) 130402.[68] Rychtarik D., Engeser B., Nagerl H.-C. and Grimm R., Phys. Rev. Lett., 92 (2004)

173003.[69] Gillen J. I., Bakr W. S., Peng A., Unterwaditzer P., Folling S. and Greiner

M., Phys. Rev. A, 80 (2009) 021602.[70] Smith N. L., Heathcote W. H., Hechenblaikner G., Nugent E. and Foot C. J.,

J. Phys. B, 38 (2005) 223.[71] Hinds E. A., Boshier M. G. and Hughes I. G., Phys. Rev. Lett., 80 (1998) 645.[72] Zobay O. and Garraway B. M., Phys. Rev. Lett., 86 (2001) 1195.[73] Colombe Y., Knyazchyan E., Morizot O., Mercier B., Lorent V. and Perrin

H., Europhys. Lett., 67 (2004) 593.[74] Hofferberth S., Lesanovsky I., Fischer B., Verdu J. and Schmiedmayer J.,

Nature Phys., 2 (2006) 710.[75] Orzel C., Tuchmann A. K., Fenselau K., Yasuda M. and Kasevich M. A., Science,

291 (2001) 2386.[76] Burger S., Cataliotti F. S., Fort C., Maddaloni P., Minardi F. and Inguscio

M., Europhys. Lett., 57 (2002) 1.[77] Kohl M., Moritz H., Stoferle T., Schori C. and Esslinger T., J. Low Temp.

Phys., 138 (2005) 635.[78] Morsch O. and Oberthaler M., Rev. Mod. Phys., 78 (2006) 179.[79] Spielman I. B., Phillips W. D. and Porto J. V., Phys. Rev. Lett., 98 (2007) 080404.[80] Hadzibabic Z., Stock S., Battelier B., Bretin V. and Dalibard J., Phys. Rev.

Lett., 93 (2004) 180403.[81] Stock S., Hadzibabic Z., Battelier B., Cheneau M. and Dalibard J., Phys. Rev.

Lett., 95 (2005) 190403.[82] Schweikhard V., Tung S. and Cornell E. A., Phys. Rev. Lett., 99 (2007) 030401.[83] Sadler L. E., Higbie J. M., Leslie S. R., Vengalattore M. and Stamper-Kurn

D. M., Nature, 443 (2006) 312.[84] Petrov D. S., Holzmann M. and Shlyapnikov G. V., Phys. Rev. Lett., 84 (2000)

2551.[85] Petrov D. S. and Shlyapnikov G. V., Phys. Rev. A, 64 (2001) 012706.[86] Naidon P., Tiesinga E., Mitchell W. F. and Julienne P. S., New J. Phys., 9 (2007)

19.[87] Schick M., Phys. Rev. A, 3 (1971) 1067.[88] Popov V. N., Theor. Math. Phys., 11 (1972) 565.[89] Andersen J., Eur. Phys. J. B, 28 (2002) 389.[90] Pricoupenko L., Phys. Rev. A, 70 (2004) 013601.[91] Pilati S., Boronat J., Casuelleras J. and Giorgini S., Phys. Rev. A, 71 (2005)

023605.[92] Astrakharchik G. E., Boronat J., Casulleras J., Kurbakov I. L. and Lozovik

Y. E., Phys. Rev. A, 79 (2009) 051602.[93] Mora C. and Castin Y., Phys. Rev. Lett., 102 (2009) 180404.[94] Holzmann M. and Krauth W., Phys. Lett. Lett., 100 (2007) 190402.


[95] Bisset R. N., Baillie D. and Blakie P. B., Phys. Rev. A, 79 (2009) 013602.[96] Holzmann M., Chevallier M. and Krauth W., Phys. Rev. A, 81 (2010) 043622.[97] Kadanoff L. and Baym G., Quantum Statistical Mechanics (Benjamin/Cummings

Publishing Company) 1963.[98] Ozeri R., Steinhauer J., Katz N. and Davidson N., Phys. Rev. Lett., 88 (2002)

220401.[99] Ketterle W. and Zwierlein M., in Ultra Cold Fermi Gases, Proceedings of the

International School of Physics “Enrico Fermi”, edited by Inguscio M., Ketterle W.

and Salomon C., Vol. CLXIV (SIF, Bologna; IOS Press, Amsterdam) 2007.[100] Holzmann M. and Krauth W., Private communication (June 2009).[101] Zhou Q. and Ho T.-L., Universal thermometry for quantum simulation, arXiv:0908.3015.[102] Kagan Y., Surkov E. L. and Shlyapnikov G. V., Phys. Rev. A, 54 (1996) R1753.[103] Castin Y. and Dum R., Phys. Rev. Lett., 77 (1996) 5315.[104] Rath S., Yefsah T., Gunter K. J., Cheneau M., Desbuquois M., Holzmann M.,

Krauth W. and Dalibard J., Phys. Rev. A, 82 (2010) 013609.[105] Pitaevskii L. P. and Rosch A., Phys. Rev. A, 55 (1997) R853.[106] Cornell E. A., Private communication (September 2009). See also: Tung S.,

Lamporesi G., Lobser D., Xia L. and Cornell E. A., Observation of the pre-

superfluid regime in a two-dimensional Bose gas, arXiv:1009.2475.[107] Dettmer S., Hellweg D., Ryyty P., Arlt J. J., Ertmer W., Sengstock K.,

Petrov D. S., Shlyapnikov G. V., Kreutzmann H., Santos L. and Lewenstein

M., Phys. Rev. Lett., 87 (2001) 160406.[108] Imambekov A., Mazets I. E., Petrov D. S., Gritsev V., Manz S., Hofferberth

S., Schumm T., Demler E. and Schmiedmayer J., Phys. Rev. A, 80 (2009) 033604.[109] Polkovnikov A., Altman E. and Demler E., Proc. Natl. Acad. Sci. U.S.A., 103

(2006) 6125.[110] Imambekov A., Gritsev V. and Demler E., in Ultra Cold Fermi Gases, Proceedings of

the International School of Physics “Enrico Fermi”, edited by Inguscio M., Ketterle

W. and Salomon C., Vol. CLXIV (SIF, Bologna; IOS Press, Amsterdam) 2007.[111] Andrews M. R., Townsend C. G., Miesner H. J., Durfee D. S., Kurn D. M. and

Ketterle W., Science, 275 (1997) 637.[112] Gritsev V., Altman E., Demler E. and Polkovnikov A., Nature Phys., 2 (2006)

705.[113] Hofferbeth S., Lesanovsky I., Schumm T., Imambekov A., Gritsev V., Demler

E. and Schmiedmayer J., Nature Phys., 4 (2008) 489.[114] Burkov A. A., Lukin M. D. and Demler E., Phys. Rev. Lett., 98 (2007) 200404.[115] Hofferberth S., Lesanovsky I., Fischer B., Schumm T. and Schmiedmayer J.,

Nature, 449 (2007) 324.[116] Cazalilla M. A., Iucci A. and Giamarchi T., Phys. Rev. A, 75 (2007) 051603.[117] Mathey L., Polkovnikov A. and Neto A. C., EPL, 81 (2008) 10008.[118] Kestner J. P. and Duan L.-M., Phys. Rev. A, 74 (2006) 053606.[119] Pricoupenko L., Phys. Rev. Lett., 100 (2008) 170404.[120] Ho T.-L. and Zhou Q., Nature Phys., 6 (2010) 131.[121] Cooper N. R. and Hadzibabic Z., Phys. Rev. Lett., 104 (2010) 030401.[122] Lin Y.-J., Compton R. L., Jimenez-Garcıa K., Porto J. V. and Spielman I. B.,

Nature, 462 (2009) 628.[123] Hung C.-L., Zhang X., Gemelke N. and Chin C., Observation of scale invariance and

universality in two-dimensional Bose gases, arXiv:1009.0016.

International School of Physics “Enrico Fermi”

Villa Monastero, Varenna

Course CLXXIII

23 June – 3 July 2009

“Nano Optics and Atomics:

Transport of Light and Matter Waves”

Directors

Diederik S. WIERSMA

European Laboratory for Non-linear

Spectroscopy (LENS)

and INFM-CNR

Via Nello Carrara 1

I-50019 Sesto Fiorentino (FI)

Italy

tel.: +39 055 457 2492 - 2452

fax: +39 055 4572451

[email protected]

Robin KAISER

Institut Non Lineaire de Nice

CNRS/UMR 6618

1361, Route des Lucioles

F-06560 Valbonne

France

tel.: +33 (0)4 92 96 73 91

fax: +33 (0)4 92 96 73 33

[email protected]

Scientific Secretary

Leonardo FALLANI


Spectroscopy (LENS)

and Universita di Firenze

Via Nello Carrara 1


Italy

tel.: +39 055 457 2458

fax: +39 055 457 2451

[email protected]

Lecturers and Seminar

Speakers

Alain ASPECT

Groupe d’Optique Atomique

Laboratoire Charles Fabry

UMR 8501 du CNRS

Institut d’Optique Theorique et Appliquee

CNRS et Universite de Paris-Sud XI

Campus Polytechnique, RD 128

F-91127 Palaiseau cedex

France

tel.: +33 1 64 533103

fax: +33 1 64 533118

[email protected]

Bill BARNES

School of Physics

University of Exeter

Stocker Road

Exeter EX4 4QL

UK

tel.: +44 1392 264135

fax: +44 1392 264111

[email protected]

Immanuel BLOCH

Institut fur Physik

Johannes Gutenberg Universitat

Staudingerweg 7

D-55128 Mainz

Germany

tel.: +49 6131 39 26234 -22279

Fax: +49 6131 39 25179

[email protected]


324 Elenco dei partecipanti

Jean DALIBARD

Laboratoire Kastler Brossel

24 rue Lhomond

F-75005 Paris

France

tel.: +33 1 44 32 25 34

fax: +33 1 44 32 34 34

[email protected]

Matthias FINK

Institut Langevin

Laboratoire Ondes et Acoustique

Ecole Superieure de Physique et de Chimie

Industrielle de la Ville de Paris

Universite D. Diderot - UMR CNRS 7587

10 Rue Vauquelin

F-75005 Paris

France

tel.: +33 1 713 348 6087

fax: +33 1 713 348 5492

[email protected]

Randall HULET

Fayez Sarofim Professor of Physics

Rice University

Physics and Astronomy Department

MS-61

6100 Main Street

Houston TX 77251

USA

tel.: +1 713-348-6087

fax: +1 713-348-5492

[email protected]

Massimo INGUSCIO


Spectroscopy (LENS)

and Universita di Firenze

Via Nello Carrara 1


Italy

tel.: +39 055 457 2465-2461-2462

fax: +39 055 457 2451

[email protected]

Jossie KLAFTER

School of Chemistry

Tel Aviv University

Tel Aviv 69978

Israel

tel.: +972 3 6408254

[email protected]

Georg MARET

Fachbereich Physik

Universitat Konstanz

D-78457 Konstanz

Germany

tel.: +49 7531 88 4151

fax: +49 7531 88 4151

[email protected]

John PAGE

Ultrasonics Research Laboratory

Department of Physics and Astronomy

University of Manitoba

Winnipeg MB

Canada R3T 2N2

tel.: +1 204 474-9852

fax: + 1204 474-7622

[email protected]

Paul STEINHARDT

Center for Theoretical Science

and Princeton University

Princeton NJ 08544-0708

USA

tel.: +1 609 258 1509

[email protected]

Peter WOLFLE

Institut fur Theorie

der Kondensierten Materie

Universitat Karlsruhe

Wolfgang-Gaede-Str. 1

D-76128 Karlsruhe

Germany

tel.: +49 721 608 3590-3367

fax: +49 721 608 7779

[email protected]

Elenco dei partecipanti 325

Arjun YODH


University of Pennsylvania

Philadelphia PA 19104-6396

USA

tel.: +1 215 898-6354

fax: +1 215 898-2010

[email protected]

Students

Holger AHLERS

Institut fur Quantenoptik

Leibniz Universitat Hannover

Welfengarten 1

D-30167 Hannover

Germany

tel.: +49 511 762 4406

fax: +49 511 762 2211

[email protected]

Julien ARMIJO


Institut d’Optique


F-91127 cedex Palaiseau

France

tel.: +33 1 6453 33 59

[email protected]

Pierre BARTHELEMY

LENS - Universita di Firenze

Via Nello Carrara 1


Italy

tel.: +39 055 4572477

fax: +39 055 4572451

[email protected]

Mathis BAUMERT

School of Physics and Astronomy

University of Birmingham

Edgbaston

B15 2TT Birmingham

UK

tel.: +44 (0)121 414 4672

fax: +44 (0)121 414 4693

[email protected]

Albert BENSENY CASES

Grup d’Optica

Universitat Autonoma de Barcelona

Edifici CC - Campus UAB

E-08193 Bellaterra

(Cerdanyola del Valles)

Barcelona

Spain

tel.: +34 935814914 / +34 615957453

fax: +34 935812155

[email protected]

Marco BERRITTA

MATIS CNR-INFM

Viale A. Doria 6

I-95125 Catania

Italy

tel.: +39 095 7382811

fax: +39 095 333231

[email protected]

Shivakiran BHAKTHA B. N.

LPMC-CNRS-UMR 6622

Universite de Nice Sophia Antopolis

Parc Valrose

F-06108 cedex02 Nice

France

tel.: +33 492076798

fax: +33 492076754

[email protected]

[email protected]


Tom BIENAIME

Institut Non Lineaire de Nice

1361 Route des Lucioles

F-06560 Valbonne

France

tel.: +33 492 967336

[email protected]

Laura CHANTADA SANTODOMINGO

Department of Physics

University of Fribourg

Perolles

CH-1700 Fribourg

Switzerland

tel.: +41 263009132

[email protected]

Nellu CIOBANU

Academy of Sciences of Moldova

Institute of Applied Physics

Str. Academiei 5

MD-2028 Chisinau

Moldova

tel.: +373 22 739907

fax: +373 22 738149

nellu [email protected]

Martin Robert DE SAINT VINCENT


Institut d’Optique


F-91127 Palaiseau cedex

France

tel.: +33 0164533329

martin.robert-de-saint-vincent

@institutoptique.fr

David ELAM jr.


One UTSA Circle

University of Texas San Antonio

TX 78249 San Antonio

USA

tel.: +1 210 216 0293

fax: +1 210 458 4919

[email protected]

Vitalie EREMEEV

LPMMC

Maison des Magisteres - CNRS

25 Avenue des Martyrs, BP166

F-38042 Grenoble

France

tel.: +33 476887984

fax: +33 476887983

[email protected]

Nicole FABBRI


Via Nello Carrara 1


Italy

tel.: +39 055 4572458

fax: +39 055 4572451

[email protected]

Berihu Teklu GEBREHIWOT

Dipartimento di Fisica

Universita di Milano

Via Celoria 16

I-20133 Milano

Italy

tel.: +39 02 50317686

fax: +39 02 50317269

[email protected]


Silvia GIUDICATTI

Dipartimento di Fisica “A. Volta”

Universita di Pavia

Via Bassi 6

I-27100 Pavia

Italy

tel.: +39 0382987693

[email protected]

Silvio GUALINI

Politecnico di Torino

Corso Duca degli Abruzzi 24

I-10129, Torino

Italy

tel.: +39 011 5644120-4125

fax: +39 0115644099

[email protected]

Timo HARTMANN

Institut fur Theoretische Physik

Universitat Regensburg

D-93040 Regensburg

Germany

tel.: +49 9419432016

fax: +49 9419434382

timo.hartmann

@physik.uni-regensburg.de

Jonas HARTWIG



Welfengarten 1

D-30167 Hannover

Germany

tel.: +49 51176219192

fax: +49 5117622211

[email protected]

William Kurt HILDEBRAND


University of Manitoba

301 Allen Bldg.

30A Sifton Rd.

R3T 2N2 MB Winnipeg

Canada

tel.: +1 2044748415

fax: +1 2044747622

[email protected]

Christoph HOFMANN

Physics Department

University of Heidelberg

Philosophenweg 12

D-69120 Heidelberg

Germany

tel.: +49 6221549205

fax: +49 6221475733

[email protected]

Sascha HOINKA


University of Otago

730 Cumberland Street

Science Building III

NZ Dunedin 9016

New Zealand

tel.: +64 34797789

fax: +64 34790964

[email protected]

Michael HOLYNSKI



Edgbaston

B15 2TT Birmingham

UK

tel.: +44 (0)1214144672

fax: +44 (0)1214144644

[email protected]


Takuya KITAGAWA

Physics Department

Harvard University

17 Oxford Street

MA 02138 Cambridge

USA

tel.: +1 6179904979

[email protected]

Priyaranga L. KOSWATTA

Birck Nanotechnology Center

Purdue University

1205 West State st.

IN 47907 West Lafayette

USA

tel.: +1 7655862920

fax: +1 7654962018

[email protected]

Eran KOT

Niels Bohr Institute

Blegdamsvej 17

DK-2100 Copenhagen

Denmark

tel.: +45 35325426

fax: +45 35325016

[email protected]

Siarhei KURYLCHYK

Istitute for Optical Materials

and Technologies

Belarus National Technical University

65 Nezavisimosti Avenuer

Building 17

BY-220013 Minsk

Belarus

tel.: +37 5297535774

fax: +37 5172926286

[email protected]

Yoav LAHINI


of Complex Systems

The Weizmann Institute of Science

Rehovot 76100

Israel

tel. 972-8-9342058

fax. 972-8-9344109

[email protected]

Marco LARCHER

CNR-INFM BEC Center

and Dipartimento di Fisica

Universita di Trento

Via Sommarive 14

I-38100 Povo (TN)

Italy

[email protected]

Ricard MENCHON ENRICH

Grup d’Optica

Departament de Fısica


Edifici CC - Campus UAB

Cerdanyola del Valles

E-08193 Bellaterra (Barcelona)

Spain

tel.: +34 935814914 / +34 647155161

fax: +34 935812155

[email protected]

Nicolas MERCADIER

Institut Non Lineaire de Nice - CNRS

1361 Route des lucioles

F-06560 Valbonne

France

tel.: +33 492967374

fax: +33 492967333

[email protected]


Nadine MEYER



Edgbaston

B15 2TT Birmingham

UK

tel.: +44 (0)121 414 4672

fax: +44 (0)121 414 4693

[email protected]

[email protected]

Rajeshkumar MUPPARAPU


Via Nello Carrara 1


Italy

[email protected]

Johan Raunkjær OTT

DTU-Fotonik

Institute of Photonics Engineering

Ørsted Plads, Building 343

DK-2800 Kongens Lyngby

Denmark

tel.: +45 26176113

[email protected]

Tobias PAUL

Institut fur Theoretische Physik

University of Heidelberg

Philosophenweg 19

D-69120 Heidelberg

Germany

tel.: +49 6221549399

fax: +49 6221549331

[email protected]

Wouter PEETERS

Quantum Optics and Quantum Information

Leiden Institute of Physics

Leiden University

P.O. Box 9504

NL-2300 RA Leiden

The Netherlands

tel.: +31 (0)715275926

fax: +31 (0)715275819

[email protected]

Abe PENA


One UTSA Circle

University of Texas San Antonio

TX 78249 San Antonio

USA

tel.: +1 9563575686

fax: +1 2104584919

abee [email protected]

Simone Luca PORTALUPI

Dipartimento di Fisica “A. Volta”

Universita di Pavia

Via Bassi 6

I-27100 Pavia

Italy

tel.: +39 0382 987693

fax: +39 0382 987563

[email protected]

Ivan PRIETO

Instituto de Microelectronica de Madrid

Centro Nacional de Microelectronica

Consejo Superior de Investigaciones

Cientificas

Isaac Newton 8 - PTM

E-28760 Tres Canto, Madrid

Spain

tel.: +34 91 806 0700

fax: +34 91 806 0701

[email protected]

[email protected]


Rami PUGATCH


of Complex Systems

The Weizmann Institute of Science

Rehovot 76100

Israel

tel.: 972-8-9343543 - 9342675

[email protected]

Katrijn PUTTENEERS

TQC - Department of Physics

Campus Groenenborger

Universiteit Antwerpen

Groenenborgerlaan 171

B-2020 Antwerpen

Belgium

tel : +32 32653529

fax: +32 32653318

[email protected]

Ana RAKONJAC


University of Otago

PO Box 56

NZ Dunedin 9016

New Zealand

tel.: +64 34797789

fax: +64 34790964

[email protected]

Lothar RATSCHBACHER

AMOP Group Cavendish Laboratory

University of Cambridge

JJ Thomson Avenue

CB3 0HE Cambridge

UK

tel.: +44 1223766464

[email protected]

Stefan RIST

Departament de Fisica


E-08193 Bellaterra

Spain

tel.: +34 669263241

fax: +34 935812155

[email protected]

Marina SAMOYLOVA

Faculty of Physics

St.-Petersburg State University

Ulyanovskaya 5, Petrodvoretz

St.-Petersburg

Russia

tel.: +7 (812) 4284339

[email protected]

[email protected]

Stephan Tobias SEIDEL



Welfengarten 1

D-30167 Hannover

Germany

tel.: +49 5117624887

fax: +49 5117622211

[email protected]

Oleksii SLIUSARENKO

Akhiezer Institute for Theoretical Physics

NSC

Kharkiv Institute of Physics

and Technology

Akademichna str.1

61108, Kharkiv

Ukraine

tel.: +38 057 335 65 23

fax: +38 057 335 26 83

[email protected]


Mihai STRATICIUC

Horia Hulubei National Institute

of Physics and Nuclear Engineering

407 Atomistilor Str.

CP 077125 Magurele, jud. Ilfov

Romania

tel.: +40 214042342

fax: +40 214574440

[email protected]

Piotr SZANKOWSKI

Institute of Theoretical Physics

Faculty of Physics

University of Warsaw, Hoza 69

PL-00-681 Warsaw

Poland

tel.: +48 225532000, +48 226283396

fax: +48 226219475

[email protected]

[email protected]

Mark THORESON

Birck Nanotechnology Center

Purdue University

1205 West State st.

IN 47907 West Lafayette

USA

tel.: +1 7654963317

[email protected]

Arif ULLAH


University of Auckland

Private Bag

NZ-92019 Auckland

New Zealand

tel.: +64 93737599 ext. 88845

fax: +64 93737445

[email protected]

Francisco Javier VALDIVIA VALERO

Instituto de Ciencia de Materiales

de Madrid

Consejo Superior de Investigaciones

Cientificas

C/ Sor Juana Ines de la Cruz, 3

E-28049 Cantoblanco (Madrid)

Spain

tel.: +34 913721420

fax: +34 913720623

[email protected]

Radhalakshmi VIVEKANANTHAN


Via Nello Carrara 1


Italy

tel.: +39 055 4572477

fax: +39 055 4572473

[email protected]

Enrico VOGT

Cavendish Laboratory

University of Cambridge

JJ Thomson Avenue

CB3 0HE Cambridge

UK

tel.: +44 1223766464

[email protected]

Kevin VYNCK


Via Nello Carrara 1


Italy

tel.: +39 055 4572477

[email protected]


Tomasz WASAK

Faculty of Physics

University of Warsaw

Hoza 69

PL-00681 Warsaw

Poland

tel.: +48 663381531

[email protected]

Christian WUTTKE

Institut fur Physik - AG QUANTUM

Johannes-Gutenberg-Universitat

Staudingerweg 7

D-55128 Mainz

Germany

tel.: +49 61313925918

fax: +49 61313923428

[email protected]

[email protected]

Observers

Andrea CAMPOSEO

National Nanotechnology Laboratory

CNR-INFM

Distretto Tecnologico ISUFI

Universita del Salento

via Arnesano

73100 Lecce

Italy

tel.: +39 0832298147

fax: +39 0832298146

[email protected]

Dubravka MILOVANOVIC

Department of Physical Chemisty

VINCA Institute of Nuclear Sciences

Belgrade

Serbia

tel.: +381 11 2453 967

cell. +381 64 2184435

[email protected]

Eduardo NUNES-PEREIRA

Centro de Fisica

Escola de Ciencias

Universidade do Minho

Campus de Gualtar

P-4710-057 Braga

Portugal

tel.: +35 1253604336

fax: +35 1253604061

[email protected]

Juan Jose SAENZ

Dept. Fisica Materia Condensada

Universidad Autonoma de Madrid

E-28049 Madrid

Spain

tel.: +34 914973804

fax: +34 914973961

[email protected]

PROCEEDINGS OF THE INTERNATIONAL

SCHOOL OF PHYSICS

“ENRICO FERMI”

Course I (1953)Questioni relative alla rivelazione delleparticelle elementari, con particolareriguardo alla radiazione cosmicaedited by G. Puppi

Course II (1954)Questioni relative alla rivelazione delleparticelle elementari, e alle lorointerazioni con particolare riguardoalle particelle artificialmenteprodotte ed accelerateedited by G. Puppi

Course III (1955)Questioni di struttura nucleare e deiprocessi nucleari alle basse energieedited by C. Salvetti

Course IV (1956)Proprieta magnetiche della materiaedited by L. Giulotto

Course V (1957)Fisica dello stato solidoedited by F. Fumi

Course VI (1958)Fisica del plasma e relative applicazioniastrofisicheedited by G. Righini

Course VII (1958)Teoria della informazioneedited by E. R. Caianiello

Course VIII (1958)Problemi matematici della teoriaquantistica delle particelle e dei campiedited by A. Borsellino

Course IX (1958)Fisica dei pioniedited by B. Touschek

Course X (1959)Thermodynamics of Irreversible Processesedited by S. R. de Groot

Course XI (1959)Weak Interactionsedited by L. A. Radicati

Course XII (1959)Solar Radioastronomyedited by G. Righini

Course XIII (1959)Physics of Plasma: Experimentsand Techniques

edited by H. Alfven

Course XIV (1960)Ergodic Theoriesedited by P. Caldirola

Course XV (1960)Nuclear Spectroscopyedited by G. Racah

Course XVI (1960)Physicomathematical Aspects of Biologyedited by N. Rashevsky

Course XVII (1960)Topics of Radiofrequency Spectroscopyedited by A. Gozzini

Course XVIII (1960)Physics of Solids (Radiation Damage

in Solids)

edited by D. S. Billington

Course XIX (1961)Cosmic Rays, Solar Particles and SpaceResearchedited by B. Peters

Course XX (1961)Evidence for Gravitational Theoriesedited by C. Møller

Course XXI (1961)Liquid Heliumedited by G. Careri

Course XXII (1961)Semiconductorsedited by R. A. Smith

Course XXIII (1961)Nuclear Physicsedited by V. F. Weisskopf

Course XXIV (1962)Space Exploration and the Solar Systemedited by B. Rossi

Course XXV (1962)Advanced Plasma Theoryedited by M. N. Rosenbluth

Course XXVI (1962)Selected Topics on Elementary ParticlePhysicsedited by M. Conversi

Course XXVII (1962)Dispersion and Absorption of Soundby Molecular Processesedited by D. Sette

Course XXVIII (1962)Star Evolutionedited by L. Gratton

Course XXIX (1963)Dispersion Relations and their Connectionwith Casualityedited by E. P. Wigner

Course XXX (1963)Radiation Dosimetryedited by F. W. Spiers and G. W. Reed

Course XXXI (1963)Quantum Electronics and Coherent Lightedited by C. H. Townes and P. A. Miles

Course XXXII (1964)Weak Interactions and High-EnergyNeutrino Physicsedited by T. D. Lee

Course XXXIII (1964)Strong Interactionsedited by L. W. Alvarez

Course XXXIV (1965)The Optical Properties of Solidsedited by J. Tauc

Course XXXV (1965)High-Energy Astrophysicsedited by L. Gratton

Course XXXVI (1965)Many-body Description of NuclearStructure and Reactionsedited by C. L. Bloch

Course XXXVII (1966)Theory of Magnetism in TransitionMetalsedited by W. Marshall

Course XXXVIII (1966)Interaction of High-Energy Particleswith Nucleiedited by T. E. O. Ericson

Course XXXIX (1966)Plasma Astrophysicsedited by P. A. Sturrock

Course XL (1967)Nuclear Structure and Nuclear Reactionsedited by M. Jean and R. A. Ricci

Course XLI (1967)Selected Topics in Particle Physicsedited by J. Steinberger

Course XLII (1967)Quantum Opticsedited by R. J. Glauber

Course XLIII (1968)Processing of Optical Data by Organismsand by Machinesedited by W. Reichardt

Course XLIV (1968)Molecular Beams and Reaction Kineticsedited by Ch. Schlier

Course XLV (1968)Local Quantum Theoryedited by R. Jost

Course XLVI (1969)Physics with Intersecting Storage Ringsedited by B. Touschek

Course XLVII (1969)General Relativity and Cosmologyedited by R. K. Sachs

Course XLVIII (1969)Physics of High Energy Densityedited by P. Caldirola and H. Knoepfel

Course IL (1970)Foundations of Quantum Mechanicsedited by B. d’Espagnat

Course L (1970)Mantle and Core in Planetary Physicsedited by J. Coulomb and M. Caputo

Course LI (1970)Critical Phenomenaedited by M. S. Green

Course LII (1971)Atomic Structure and Properties of Solidsedited by E. Burstein

Course LIII (1971)Developments and Borderlines of NuclearPhysicsedited by H. Morinaga

Course LIV (1971)Developments in High-Energy Physicsedited by R. R. Gatto

Course LV (1972)Lattice Dynamics and IntermolecularForcesedited by S. Califano

Course LVI (1972)Experimental Gravitationedited by B. Bertotti

Course LVII (1972)History of 20th Century Physicsedited by C. Weiner

Course LVIII (1973)Dynamics Aspects of Surface Physicsedited by F. O. Goodman

Course LIX (1973)Local Properties at Phase Transitions

edited by K. A. Muller and A. Rigamonti

Course LX (1973)C*-Algebras and their Applicationsto Statistical Mechanics and QuantumField Theoryedited by D. Kastler

Course LXI (1974)Atomic Structure and MechanicalProperties of Metalsedited by G. Caglioti

Course LXII (1974)Nuclear Spectroscopy and NuclearReactions with Heavy Ionsedited by H. Faraggi and R. A. Ricci

Course LXIII (1974)New Directions in Physical Acousticsedited by D. Sette

Course LXIV (1975)Nonlinear Spectroscopyedited by N. Bloembergen

Course LXV (1975)Physics and Astrophysics of Neutron Starsand Black Holeedited by R. Giacconi and R. Ruffini

Course LXVI (1975)Health and Medical Physicsedited by J. Baarli

Course LXVII (1976)Isolated Gravitating Systems in GeneralRelativityedited by J. Ehlers

Course LXVIII (1976)Metrology and Fundamental Constantsedited by A. Ferro Milone, P. Giacomo

and S. Leschiutta

Course LXIX (1976)Elementary Modes of Excitation in Nucleiedited by A. Bohr and R. A. Broglia

Course LXX (1977)Physics of Magnetic Garnetsedited by A. Paoletti

Course LXXI (1977)Weak Interactionsedited by M. Baldo Ceolin

Course LXXII (1977)Problems in the Foundations of Physicsedited by G. Toraldo di Francia

Course LXXIII (1978)Early Solar System Processesand the Present Solar Systemedited by D. Lal

Course LXXIV (1978)Development of High-Power Lasersand their Applicationsedited by C. Pellegrini

Course LXXV (1978)Intermolecular Spectroscopy andDynamical Properties of Dense Systemsedited by J. Van Kranendonk

Course LXXVI (1979)Medical Physicsedited by J. R. Greening

Course LXXVII (1979)Nuclear Structure and Heavy-IonCollisionsedited by R. A. Broglia, R. A. Ricci

and C. H. Dasso

Course LXXVIII (1979)Physics of the Earth’s Interioredited by A. M. Dziewonski and E. Boschi

Course LXXIX (1980)From Nuclei to Particlesedited by A. Molinari

Course LXXX (1980)Topics in Ocean Physicsedited by A. R. Osborne and P. Malanotte

Rizzoli

Course LXXXI (1980)Theory of Fundamental Interactionsedited by G. Costa and R. R. Gatto

Course LXXXII (1981)Mechanical and Thermal Behaviourof Metallic Materialsedited by G. Caglioti and A. Ferro Milone

Course LXXXIII (1981)Positrons in Solid-State Physicsedited by W. Brandt and A. Dupasquier

Course LXXXIV (1981)Data Acquisition in High-Energy Physicsedited by G. Bologna and M. Vincelli

Course LXXXV (1982)Earthquakes: Observation, Theoryand Interpretationedited by H. Kanamori and E. Boschi

Course LXXXVI (1982)Gamow Cosmologyedited by F. Melchiorri and R. Ruffini

Course LXXXVII (1982)Nuclear Structure and Heavy-IonDynamicsedited by L. Moretto and R. A. Ricci

Course LXXXVIII (1983)Turbulence and Predictabilityin Geophysical Fluid Dynamicsand Climate Dynamicsedited by M. Ghil, R. Benzi and G. Parisi

Course LXXXIX (1983)Highlights of Condensed Matter Theoryedited by F. Bassani, F. Fumi and M. P. Tosi

Course XC (1983)Physics of Amphiphiles: Micelles, Vesiclesand Microemulsionsedited by V. Degiorgio and M. Corti

Course XCI (1984)From Nuclei to Starsedited by A. Molinari and R. A. Ricci

Course XCII (1984)Elementary Particlesedited by N. Cabibbo

Course XCIII (1984)Frontiers in Physical Acousticsedited by D. Sette

Course XCIV (1984)Theory of Reliabilityedited by A. Serra and R. E. Barlow

Course XCV (1985)Solar-Terrestrial Relationshipand the Earth Environmentin the Last Millenniaedited by G. Cini Castagnoli

Course XCVI (1985)Excited-State Spectroscopy in Solidsedited by U. M. Grassano and N. Terzi

Course XCVII (1985)Molecular-Dynamics Simulationsof Statistical-Mechanical Systemsedited by G. Ciccotti and W. G. Hoover

Course XCVIII (1985)The Evolution of Small Bodies in the SolarSystem

edited by M. Fulchignoni and L. Kresak

Course XCIX (1986)Synergetics and Dynamic Instabilitiesedited by G. Caglioti and H. Haken

Course C (1986)The Physics of NMR Spectroscopyin Biology and Medicineedited by B. Maraviglia

Course CI (1986)Evolution of Interstellar Dustand Related Topicsedited by A. Bonetti and J. M. Greenberg

Course CII (1986)Accelerated Life Testing and Experts’Opinions in Reliabilityedited by C. A. Clarotti and D. V. Lindley

Course CIII (1987)Trends in Nuclear Physicsedited by P. Kienle, R. A. Ricci

and A. Rubbino

Course CIV (1987)Frontiers and Borderlinesin Many-Particle Physicsedited by R. A. Broglia

and J. R. Schrieffer

Course CV (1987)Confrontation between Theoriesand Observations in Cosmology:Present Status and Future Programmesedited by J. Audouze and F. Melchiorri

Course CVI (1988)Current Trends in the Physicsof Materialsedited by G. F. Chiarotti, F. Fumi

and M. Tosi

Course CVII (1988)The Chemical Physics of Atomicand Molecular Clustersedited by G. Scoles

Course CVIII (1988)Photoemission and AbsorptionSpectroscopy of Solids and Interfaceswith Synchrotron Radiationedited by M. Campagna and R. Rosei

Course CIX (1988)Nonlinear Topics in Ocean Physicsedited by A. R. Osborne

Course CX (1989)Metrology at the Frontiers of Physicsand Technologyedited by L. Crovini and T. J. Quinn

Course CXI (1989)Solid-State Astrophysicsedited by E. Bussoletti and G. Strazzulla

Course CXII (1989)Nuclear Collisions from the Mean-Fieldinto the Fragmentation Regimeedited by C. Detraz and P. Kienle

Course CXIII (1989)High-Pressure Equation of State:Theory and Applicationsedited by S. Eliezer and R. A. Ricci

Course CXIV (1990)Industrial and Technological Applicationsof Neutronsedited by M. Fontana and F. Rustichelli

Course CXV (1990)The Use of EOS for Studiesof Atmospheric Physicsedited by J. C. Gille and G. Visconti

Course CXVI (1990)Status and Perspectives of NuclearEnergy: Fission and Fusionedited by R. A. Ricci, C. Salvetti

and E. Sindoni

Course CXVII (1991)Semiconductor Superlatticesand Interfacesedited by A. Stella

Course CXVIII (1991)Laser Manipulation of Atoms and Ionsedited by E. Arimondo, W. D. Phillips

and F. Strumia

Course CXIX (1991)Quantum Chaosedited by G. Casati, I. Guarneri

and U. Smilansky

Course CXX (1992)Frontiers in Laser Spectroscopy

edited by T. W. Hansch and M. Inguscio

Course CXXI (1992)Perspectives in Many-Particle Physicsedited by R. A. Broglia, J. R. Schrieffer

and P. F. Bortignon

Course CXXII (1992)Galaxy Formationedited by J. Silk and N. Vittorio

Course CXXIII (1992)Nuclear Magnetic Double Resonsonanceedited by B. Maraviglia

Course CXXIV (1993)Diagnostic Tools in Atmospheric Physicsedited by G. Fiocco and G. Visconti

Course CXXV (1993)Positron Spectroscopy of Solidsedited by A. Dupasquier and A. P. Mills jr.

Course CXXVI (1993)Nonlinear Optical Materials: Principlesand Applicationsedited by V. Degiorgio and C. Flytzanis

Course CXXVII (1994)Quantum Groups and their Applicationsin Physicsedited by L. Castellani and J. Wess

Course CXXVIII (1994)Biomedical Applications of SynchrotronRadiationedited by E. Burattini and A. Balerna

Course CXXIX1 (1994)Observation, Prediction and Simulationof Phase Transitions in Complex Fluidsedited by M. Baus, L. F. Rull

and J. P. Ryckaert

Course CXXX (1995)Selected Topics in Nonperturbative QCDedited by A. Di Giacomo and D. Diakonov

Course CXXXI (1995)Coherent and Collective Interactionsof Particles and Radiation Beamsedited by A. Aspect, W. Barletta

and R. Bonifacio

1This course belongs to the NATO ASI Series C, Vol. 460 (Kluwer Academic Publishers).

Course CXXXII (1995)Dark Matter in the Universeedited by S. Bonometto and J. Primack

Course CXXXIII (1996)Past and Present Variability of theSolar-Terrestrial System: Measurement,Data Analysis and Theoretical Modelsedited by G. Cini Castagnoli

and A. Provenzale

Course CXXXIV (1996)The Physics of Complex Systemsedited by F. Mallamace and H. E. Stanley

Course CXXXV (1996)The Physics of Diamondedited by A. Paoletti and A. Tucciarone

Course CXXXVI (1997)Models and Phenomenologyfor Conventional and High-TemperatureSuperconductivityedited by G. Iadonisi, J. R. Schrieffer

and M. L. Chiofalo

Course CXXXVII (1997)Heavy Flavour Physics: a Probeof Nature’s Grand Designedited by I. Bigi and L. Moroni

Course CXXXVIII (1997)Unfolding the Matter of Nucleiedited by A. Molinari and R. A. Ricci

Course CXXXIX (1998)Magnetic Resonance and Brain Function:Approaches from Physicsedited by B. Maraviglia

Course CXL (1998)Bose-Einstein Condensationin Atomic Gasesedited by M. Inguscio, S. Stringari

and C. E. Wieman

Course CXLI (1998)Silicon-Based Microphotonics:from Basics to Applicationsedited by O. Bisi, S. U. Campisano, L. Pavesi

and F. Priolo

Course CXLII (1999)Plasmas in the Universeedited by B. Coppi, A. Ferrari

and E. Sindoni

Course CXLIII (1999)New Directions in Quantum Chaosedited by G. Casati, I. Guarneri

and U. Smilansky

Course CXLIV (2000)Nanometer Scale Science and Technology

edited by M. Allegrini, N. Garcıa

and O. Marti

Course CXLV (2000)Protein Folding, Evolution and Designedited by R. A. Broglia, E. I. Shakhnovich

and G. Tiana

Course CXLVI (2000)Recent Advances in Metrologyand Fundamental Constantsedited by T. J. Quinn, S. Leschiutta

and P. Tavella

Course CXLVII (2001)High Pressure Phenomenaedited by R. J. Hemley, G. L. Chiarotti,

M. Bernasconi and L. Ulivi

Course CXLVIII (2001)Experimental Quantum Computationand Informationedited by F. De Martini and C. Monroe

Course CXLIX (2001)Organic Nanostructures: Scienceand Applicationsedited by V. M. Agranovich

and G. C. La Rocca

Course CL (2002)Electron and Photon Confinementin Semiconductor Nanostructuresedited by B. Deveaud-Pledran,

A. Quattropani and P. Schwendimann

Course CLI (2002)Quantum Phenomena in MesoscopicSystemsedited by B. Altshuler, A. Tagliacozzo

and V. Tognetti

Course CLII (2002)Neutrino Physicsedited by E. Bellotti, Y. Declais

and P. Strolin

Course CLIII (2002)From Nuclei and their Constituentsto Starsedited by A. Molinari, L. Riccati,

W. M. Alberico and M. Morando

Course CLIV (2003)Physics Methods in Archaeometryedited by M. Martini, M. Milazzo

and M. Piacentini

Course CLV (2003)The Physics of Complex Systems

(New Advances and Perspectives)

edited by F. Mallamace and H. E. Stanley

Course CLVI (2003)Research on Physics Educationedited by E.F. Redish and M. Vicentini

Course CLVII (2003)The Electron Liquid Model in CondensedMatter Physicsedited by G. F. Giuliani and G. Vignale

Course CLVIII (2004)Hadron Physicsedited by T. Bressani, U. Wiedner

and A. Filippi

Course CLIX (2004)Background Microwave Radiationand Intracluster Cosmologyedited by F. Melchiorri and Y. Rephaeli

Course CLX (2004)From Nanostructures to NanosensingApplicationsedited by A. D’Amico, G. Balestrino

and A. Paoletti

Course CLXI (2005)Polarons in Bulk Materials and Systemswith Reduced Dimensionalityedited by G. Iadonisi, J. Ranninger

and G. De Filippis

Course CLXII (2005)Quantum Computers, Algorithmsand Chaosedited by G. Casati, D. L. Shepelyansky,

P. Zoller and G. Benenti

Course CLXIII (2005)CP Violation: From Quarks to Leptonsedited by M. Giorgi, I. Mannelli,

A. I. Sanda, F. Costantini

and M. S. Sozzi

Course CLXIV (2006)Ultra-Cold Fermi Gasesedited by M. Inguscio, W. Ketterle

and C. Salomon

Course CLXV (2006)Protein Folding and Drug Designedited by R. A. Broglia, L. Serrano

and G. Tiana

Course CLXVI (2006)Metrology and Fundamental Constants

edited by T. W. Hansch, S. Leschiutta,

A. J. Wallard and M. L. Rastello

Course CLXVII (2007)Strangeness and Spin in FundamentalPhysicsedited by M. Anselmino, T. Bressani,

A. Feliciello and Ph. G. Ratcliffe

Course CLXVIII (2007)Atom Optics and Space Physicsedited by E. Arimondo, W. Ertmer,

W. P. Schleich and E. M. Rasel

Course CLXIX (2007)Nuclear Structure far from Stability:New Physics and New Technologyedited by A. Covello, F. Iachello,

R. A. Ricci and G. Maino

Course CLXX (2008)Measurements of Neutrino Massedited by F. Ferroni, F. Vissani

and C. Brofferio

Course CLXXI (2008)Quantum Coherence in Solid State Physics

edited by B. Deveaud-Pledran,

A. Quattropani and P. Schwendimann

Course CLXXIV (2009)Physics with Many Positronsedited by R. S. Brusa, A. Dupasquier

and A. P. Mills jr.

Course CLXXV (2009)Radiation and Particle Detectorsedited by S. Bertolucci, U. Bottigli

and P. Oliva


Nano Optics and Atomics_ Transport of Light and Matter Waves.pdf

Documents