Anderson Anderson Localization Localization 1958 1958 - - 2008 2008 From From electrons electrons , , towards towards ultrasound ultrasound and and … … cold cold atoms atoms Bart van Tiggelen Bart van Tiggelen Laboratoire de Physique et Mod Laboratoire de Physique et Mod é é lisation des Milieux lisation des Milieux Condens Condens é é s s Universit Universit é é Joseph Fourier /CNRS Joseph Fourier /CNRS Scattering Scattering Systems Systems with with complex complex Dynamics FOR760 2008: Dynamics FOR760 2008: October October 6 2008 6 2008
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Anderson Localization 1958-2008 · Anderson Localization 1958-2008 From electrons, towards ultrasound and … cold atoms Bart van Tiggelen Laboratoire de Physique et Modélisation
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Anderson Anderson LocalizationLocalization19581958--20082008
FromFrom electronselectrons, , towardstowards ultrasoundultrasoundand and …… cold cold atomsatoms
Bart van TiggelenBart van TiggelenLaboratoire de Physique et ModLaboratoire de Physique et Modéélisation des Milieux lisation des Milieux
CondensCondensééssUniversitUniversitéé Joseph Fourier /CNRSJoseph Fourier /CNRS
ScatteringScattering Systems Systems withwith complexcomplex Dynamics FOR760 2008: Dynamics FOR760 2008: OctoberOctober 6 20086 2008
Localization [..] very few believed it at the time, and even fewer saw itsimportance, among those who failed was certainly its author.
It has yet to receive adequate mathematical treatment, and one has to resort to the indignity of numerical simulations to settle even the simplest questions about it.
P.W. Anderson, Nobel lecture, 1977
1958 Anderson « vanishing of diffusion »
1960 Mott/Ioffe-Regel
1965 Mott Minimum conductivityVariable range hopping
1972 Thouless Sensitivity to BC: g < 1(Thouless criterion)
>1982 Sharvin, Lagendijk,Maret, Maynard,…
Weak localizationMesoscopic physics!
1977 Anderson/Mott Nobel Prize
1980 « gang of four » Scaling theory
1980 Götze, Vollhardt, Wölfle Self-consistent theory
1982 Halperin, Pruisken Scaling theory of Quantum Hall effect
πλ2≤l
)log()(log
Lg
∂∂
Half a century…….
1983 Fröhlich & Spencer Mathematical proof for 3D Anderson model
1984 Anderson 25 years localization« unrecognizable monster »
1986 Anderson « Theory of white paint »
1987 Papanicolaou, Sheng Prediction of Localization of Seismic Waves in layered Earth
Crust1987 Souillard Localization of Gravitational Waves
in Universe?
1986 Kramer, Mackinnon, Economou,
Soukoulis, Schreiber
Tight binding model
1990 Dorokhorov, Melloetal, Beenakker
Altschuler
DMPK equation for wire
interactions
1988 John Prediction of Localization of light in Photonic crystals
>1991
Bell-labsWeaverGenack
Exxon (Sheng etal)
2D localization microwaves2D localization of ultrasoundQ1D localization microwaves
2D localization of bending waves>1995 BEC community Localization of light in BEC cold atomes
gazes or the contrary?
2000 GenackBeenakker, …Van Tiggelen/
Lagendijk/Wiersma
Statistics in localized regime (exp)Idem (theo)
D(r) in localized regime
2006 Maret Anomalous dynamic transmission of lightnear mobility edge
2007 Fishman/Segev Transverse Light Localization in 2D lattices
1997 Wiersma, Lagendijk 3D localizationof infrared light
>1998
Cao, Wiersma etal, Soukoulis, Sebbah, …
Random lasering from(pre) localized states
2008 Page/Skipetrov/Van Tiggelen 3D localization of ultrasound
•• complexcomplex dynamicsdynamics (open media): T(t) (open media): T(t) ~ exp(~ exp(--D(tD(t) t /L) t /L22))((anomalousanomalous leakingleaking))
•• T(L), G(L) T(L), G(L) ~ exp(~ exp(--L/L/ξξ))
Mesurable?
Mesurable?
••NeedNeed goodgood observable observable withwith continuouscontinuous control control parameterparameter: L,t, f, E ..: L,t, f, E ..••NeedNeed a a goodgood theorytheory thatthat allowsallows engineeringengineering
3D Transverse localisation 3D Transverse localisation ofof ultrasoundultrasoundPage (Winnipeg), Page (Winnipeg), SkipetrovSkipetrov, Van Tiggelen, , Van Tiggelen, CherroretCherroret
diffuse localized
10 mm
8 – 23 mm
Diffuse: <ρ2> = 4Dttransition: <ρ2>~ L2, not t2/3
Localized: <ρ2>~ Lξ
3D Transverse localisation 3D Transverse localisation ofof ultrasoundultrasoundPage (Winnipeg), Page (Winnipeg), SkipetrovSkipetrov, Van Tiggelen, , Van Tiggelen, CherroretCherroret
ρ
L=15 mm
3D Transverse localisation 3D Transverse localisation ofof ultrasoundultrasoundPage , Page , SkipetrovSkipetrov, Van Tiggelen, , Van Tiggelen,
SpeckleSpeckle distribution of transmissiondistribution of transmission
Rayleigh Random matrixg=0.8 +/- 0.02
Inhibition of transport of Q1D BEC in Inhibition of transport of Q1D BEC in randomrandom potentialpotential
Orsay group, Firenze groupOrsay group, Firenze groupPRL PRL octoct 20052005
Time after trap extinction
expansion
1<μδV
n(x,t)V(x)
mobility edge
Diffusive regime Localizationwith ξ > ℓ
Localizationwith ξ < ℓ
LocalizationLocalization of of noninteractingnoninteracting cold cold atomsatoms in 3D white noisein 3D white noise
⎟⎟⎠
⎞⎜⎜⎝
⎛−∝=
mkt
2)0,(
22hμθψ k
band edge
TrapTrap stagestage
µchemical potential
2)(rψ
expansion stage (t=0)expansion stage (t=0)
gVet ti )(),( rr −
= − μψ μ
∞→
==
tttn ??),(),( 2rr ψ
RandomRandom potentialpotential
εε ∝)(D
with Sergey Skipetrov, Anna Minguzzi and Boris Shapiro
DensityDensity profile of profile of atomsatoms atat large timeslarge times
),(),()()2(
),( 23
3
tPEAdEdtn E rkkkr ∫∫∞
∞−= φ
π
Distribution of velocitiesmv = ћk
Spectral function
Probability of quantum diffusion
EiUUkE +−−−
4/1Im122π
)'(4)'()( rrrr −= δπUVV
Mean free path: U1
=l 2UE < localized
DensityDensity profile of profile of atomsatoms atat large timeslarge times
Potential fluctuation/chemical potential
FractionOf
Localized atoms
DensityDensity profile of profile of atomsatoms atat large timeslarge times
-2 -1 0 1 2 3 4-17.5
-15
-12.5
-10
-7.5
-5
-2.5
0
),()(),( loc tnntn AD rrr Δ+=
ssADs
trtnD c /1/23
1),()()( −∝Δ⇒−∝ rεεεννεε
εξ /13loc1)(
)(1)( +∝⇒−
∝r
nc
r
localized anomalous diffusion
nloc(r)
Selfconsistent theory (ν=s=1)
50 50 yearsyears of Anderson of Anderson LocalizationLocalizationParis, Institut Henry PoincarParis, Institut Henry Poincaréé
4/5 4/5 decemberdecember 20082008
http://www.andersonlocalization.com
SpeakersSpeakers: P.: P.W.AndersonW.Anderson, D.J. , D.J. ThoulessThoulessA. Aspect, S. John, P. A. Aspect, S. John, P. WWöölflelfle, , G. Maret, R. Weaver, A.Z. G. Maret, R. Weaver, A.Z. GenackGenack, , A. A. LagendijkLagendijk, A. , A. MirlinMirlin, M. Schreiber, M. Schreiber
TheoryTheory and and experimentsexperiments, , oldold and new and new thingsthings
ClichClichéé, clich, clichéé !!LL’’article darticle d’’AndersonAnderson’’s en 1958 a s en 1958 a ééttéé «« oftenoften quotedquoted but but nevernever readread »»Il y a autant de dIl y a autant de dééfinitions et dfinitions et d’’approches quapproches qu’’il y a de publications.il y a de publications.La moitiLa moitiéé des publications est des publications est ……....
•• …….. soit fausse.. soit fausse•• …….. soit triviale .. soit triviale •• …….. soit d.. soit dééjjàà contenue dans le papier dcontenue dans le papier d’’Anderson en 1958 Anderson en 1958 •• …….. soit trop compliqu.. soit trop compliquéée ou trop phe ou trop phéénomnoméénologiquenologique
pour comprendre les exppour comprendre les expéériencesriences
ProtectProtect me me fromfrom knowingknowing whatwhat I I dondon’’tt needneed to knowto knowProtectProtect me me eveneven fromfrom knowingknowing thatthat therethere are are thingsthings to know to know thatthat I I dondon’’tt knowknowProtectProtect fromfrom knowingknowing thatthat I I decideddecided not to know not to know
about the about the thingsthings thatthat I I decideddecided not to know aboutnot to know about
Douglas Adams, Douglas Adams, MostlyMostly HarmlessHarmless, 1992, 1992