Andreas Penckwitt- Rotating Bose–Einstein Condensates: Vortex Lattices and Excitations

8/3/2019 Andreas Penckwitt- Rotating BoseEinstein Condensates: Vortex Lattices and Excitations

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Rotating BoseEinstein Condensates

Vortex Lattices and Excitations

Andreas Penckwitt

December 2003

A thesis submitted for the degree of

Doctor of Philosophy

Department of Physics

University of Otago

Dunedin

New Zealand


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Revision History

Original submitted for examination: December 23, 2003.

First corrected version (post examination): July 8, 2004.


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Abstract

The main theme of this thesis is an investigation of rotating BoseEinstein condensates.This area is of considerable current interest and has been stimulated by several recent

experiments where vortex lattices are created by stirring a BoseEinstein condensate

with an anisotropic trap or by growth from a rotating thermal cloud. These experiments

echo earlier experiments on superfluid 4He in a rotating bucket which first led to the

discovery of vortex lattices.

One key question, which remained unanswered from the work on 4He, is the mecha-

nism of vortex lattice formation. Dilute BoseEinstein condensates allow the possibility

of both detailed experimental studies of the dynamics and a priori theoretical treat-ments. It is well understood that some form of dissipation is necessary to drive a

condensate into a lattice state, however, this makes the standard method of describing

a condensate, the GrossPitaevskii equation, inadequate for this problem.

In this thesis, we provide a simple, unified treatment that describes the process of

vortex lattice formation consistently, from the initiation to the final lattice stabilization.

Our work is the first application of a formalism developed by Gardiner et al. [J. Phys.

B, 35, 1555 (2002)] where the dissipation is provided by an exchange of atoms between

the condensate and the thermal cloud. In its simplest form it can be reduced to a

modified GrossPitaevskii equation with growth and loss terms that provide damping

via a bath of thermal atoms.

We model the scenario of vortex lattice formation from a rotating thermal cloud

and show that the basic mechanism of the formation is growth into surface modes of

the condensate. We give an analytic treatment that provides the gain coefficients for

this growth in terms of the excitation energies of the modes, and validate it against

our simulations. In our model, the critical angular velocity is given by the condition

for positive gain and coincides with the Landau criterion.

We show that this simple analytical model can be generalized to the case of anelliptical rotating trap, using the excitation spectrum on top of a vortex free ground

state, which we calculate numerically. We provide possible explanations for the exper-

imentally observed critical frequencies above and below the Landau critical frequency.

In particular, we find the first indication for a reason that might explain a lower limit

for vortex nucleation observed by Hodby et al. [Phys. Rev. Lett. 88, 010405 (2002)].

Finally, we simulate non-equilibrium dynamics of rapidly rotating condensates as

well as the excitation of Tkachenko modes. We calculate the excitations of a vortex

lattice and identify the Tkachenko modes in the spectrum.

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Acknowledgements

A work like this thesis is not possible without the help and support of many peoplewhom I would like to thank here whole-heartedly. First and foremost, my supervisor

Rob Ballagh, whose enthusiasm for physics is unbroken by the administrative duties

that come with his job. He is one of the friendliest and most supportive supervisors one

could wish for. Many thanks go to Crispin Gardiner, who put us on the right track with

the theoretical side of this work. His insight into the physics behind all the formulae is

almost scary. I would also like to thank my second supervisor Andrew Wilson for the

marvellous job he is doing on the experimental side of BoseEinstein condensation at

the University of Otago.A surprisingly large number of international visitors find their way to Dunedin. For

some fruitful discussions and help I would like to thank in particular Sam Morgan,

Matthew Davis, Martin Rusch, Ashton Bradley and Allan Griffin, as well as Tapio

Simula who is currently doing a postdoc at Otago.

Past and present members of the theory group at Otago I would like to mention

are Ben Caradoc-Davies, who has the patience to solve any computer problem (as long

as it is Linux related), Blair Blakie, who can explain anything about physics (and is

too modest to talk much about his personal life), Jan Kruger, formerly known as Max,

who turned into our Linux guru after Ben had left, Adam Norrie, who is just Adam,

Christopher Gies, who introduced us to the beautiful game of carom, Katherine Challis,

who motivated me to run a half-marathon and James Douglas, who is a very worthy

carom opponent.

I also appreciated the exchange of ideas with the experimental BEC group (though

I havent decided yet whether I prefer the door between our two offices open or shut)

and found all staff in the department most helpful at all times.

For keeping me sane I thank all of my friends in Dunedin and around the world,

my flatmates of the last two years, who might have wondered at times whether I wasactually still living in the same flat, and everyone at Sammys which became kind of a

second home. A very big thank you goes to my family back in Germany, who just had

to get used to the fact that I am as far away as it gets. One day you might even visit

me here.

Last but not least, I thank you, Alexandra, for everything you have done for me.

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Contents

Abstract i

Acknowledgements iii

1 Introduction 1

1.1 BoseEinstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Recent Advances in BoseEinstein Condensation . . . . . . . . . . . . . 2

1.2.1 Experimental systems . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Atom lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.3 Non-linear and quantum atom optics . . . . . . . . . . . . . . . 6

1.2.4 Atomic collisions and molecular BoseEinstein condensation . . 61.2.5 Superfluidity and vortices . . . . . . . . . . . . . . . . . . . . . 7

1.3 This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 Peer-reviewed publications . . . . . . . . . . . . . . . . . . . . . 10

2 Mean Field Theory 11

2.1 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Many-body Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Pseudo-potential approximation . . . . . . . . . . . . . . . . . . 12

2.1.3 Definition of BoseEinstein condensation . . . . . . . . . . . . . 13

2.1.4 Bogoliubov approximation . . . . . . . . . . . . . . . . . . . . . 14

2.1.5 Time-dependent GrossPitaevskii equation . . . . . . . . . . . . 14

2.1.6 Time-independent GrossPitaevskii equation . . . . . . . . . . . 15

2.2 Elementary Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Linear response theory . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Computational Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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Contents

2.4 ThomasFermi Approximation . . . . . . . . . . . . . . . . . . . . . . . 23

3 Elementary Excitation Families 253.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Classification of Excitations . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 ThomasFermi limit . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.2 Families in the isotropic case . . . . . . . . . . . . . . . . . . . . 27

3.2.3 Anisotropic case . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Ordering of Quasiparticle Eigenfrequencies . . . . . . . . . . . . . . . . 33

3.3.1 Full solutions of Bogoliubov-de Gennes equations . . . . . . . . 33

3.3.2 Comparison with harmonic oscillator solutions . . . . . . . . . . 35

4 Background on Vortices 39

4.1 Quantization of Circulation . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Characteristics of Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Healing Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Energy of Vortex Line . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Critical Frequency of Vortex Nucleation . . . . . . . . . . . . . . . . . 43

4.5.1 Thermodynamic critical frequency . . . . . . . . . . . . . . . . . 43

4.5.2 Landau criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5.3 Stability and energy barrier . . . . . . . . . . . . . . . . . . . . 45

4.5.4 Anomalous mode and vortex dynamics . . . . . . . . . . . . . . 45

4.5.5 Nucleation of vortices . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6 Vortex Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Theory of Growth from Rotating Thermal Cloud 49

5.1 Quantum Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.2 Phase space density . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.3 Master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1.4 Growth processes . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1.5 Simple growth equation . . . . . . . . . . . . . . . . . . . . . . 53

5.1.6 Evaluation of transition probability W+ . . . . . . . . . . . . . 55

5.2 Phenomenological Growth Equation . . . . . . . . . . . . . . . . . . . . 56

5.2.1 Growth and loss in GrossPitaevskii equation . . . . . . . . . . 56

5.2.2 Master equation approach . . . . . . . . . . . . . . . . . . . . . 57

5.2.2.1 GrossPitaevskii equation in hydrodynamic form . . . 57

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5.2.2.2 Local energy conservation in hydrodynamic approxima-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2.2.3 Application to quantum kinetic theory . . . . . . . . . 58

5.2.2.4 Phenomenological mean value equations . . . . . . . . 60

5.2.3 Rotating thermal cloud . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.4 Stationary solution . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Comparison with Other Theories . . . . . . . . . . . . . . . . . . . . . 63

6 Numerical Propagation Method 67

6.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2.1 General points of consideration for numerical simulations . . . . 69

6.2.2 Choice of grid size and number of points . . . . . . . . . . . . . 70

6.2.3 Temporal step size . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2.4 Accuracy of angular momentum operator in GrossPitaevskii

equation propagation . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2.5 Accuracy of phenomelogical growth equation . . . . . . . . . . . 80

7 Vortex Lattice Formation in Cylindrically Symmetric Trap 85

7.1 Two Dimensional System . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2 Simple Growth and Normalization of Wave Function . . . . . . . . . . 87

7.3 Initial State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.4 General Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.5 Initiation Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.5.1 Gain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.5.2 Comparison with simulation . . . . . . . . . . . . . . . . . . . . 93

7.5.3 Critical angular velocity . . . . . . . . . . . . . . . . . . . . . . 95

7.5.4 Dominant angular momentum component . . . . . . . . . . . . 977.5.5 Role of initial seed . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.6 Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.7 Properties of Equilibrium Lattice . . . . . . . . . . . . . . . . . . . . . 103

8 Vortex Lattice Formation in Elliptical Rotating Trap 109

8.1 No Thermal Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2 With Thermal Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.3 Gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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Contents

8.3.1 Vortex free solutions of the GrossPitaevskii equation in the ro-

tating frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.3.2 Excited states on vortex free solutions in the rotating frame . . 120

8.3.3 Comparison with simulation . . . . . . . . . . . . . . . . . . . . 122

8.3.4 Critical velocity and importance of quadrupole mode . . . . . . 126

8.4 Experiments on Vortex Nucleation and Lattice Formation . . . . . . . . 130

9 Vortex Lattice Decay 137

9.1 General Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.2 Angular Velocity and Radial Velocity . . . . . . . . . . . . . . . . . . . 141

9.3 Single Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469.4 Experiments on Vortex Lattice Decay . . . . . . . . . . . . . . . . . . . 148

10 Non-equilibrium Dynamics and Lattice Excitations 151

10.1 Simple Non-Equilibrium Lattice . . . . . . . . . . . . . . . . . . . . . . 151

10.2 Deformation of Rapidly Rotating Condensates . . . . . . . . . . . . . . 153

10.2.1 Frequency splitting ofm = 2 modes . . . . . . . . . . . . . . . 15410.2.2 Excitation ofm = 2 mode . . . . . . . . . . . . . . . . . . . . 15510.2.3 Excitation ofm = 2 mode . . . . . . . . . . . . . . . . . . . . . 159

10.3 Excitations on Vortex Lattices . . . . . . . . . . . . . . . . . . . . . . . 16110.3.1 Excitations on single vortex . . . . . . . . . . . . . . . . . . . . 161

10.3.2 Excitations on vortex lattice . . . . . . . . . . . . . . . . . . . . 164

10.3.2.1 Frequency spectrum . . . . . . . . . . . . . . . . . . . 164

10.3.2.2 Tkachenko modes . . . . . . . . . . . . . . . . . . . . . 165

11 Conclusion 169

11.1 T his Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

11.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

A Optimization Methods 173

A.1 Partial Differential Equations Solver . . . . . . . . . . . . . . . . . . . . 174

A.1.1 Cylindrically symmetric case in three dimensions . . . . . . . . 174

A.1.2 Two-dimensional system in the rotating frame . . . . . . . . . . 175

A.2 Conjugate-Gradient Method . . . . . . . . . . . . . . . . . . . . . . . . 176

A.2.1 Optimality function . . . . . . . . . . . . . . . . . . . . . . . . . 176

A.2.2 Conjugate-gradient optimization . . . . . . . . . . . . . . . . . . 179

A.3 Basis State Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

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A.3.1 Basis state expansion . . . . . . . . . . . . . . . . . . . . . . . . 180

A.3.2 Construction of harmonic oscillator states . . . . . . . . . . . . 182

A.3.3 Angular momentum operator . . . . . . . . . . . . . . . . . . . 183

A.3.4 Gaussian quadrature . . . . . . . . . . . . . . . . . . . . . . . . 184

A.3.5 Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

A.3.6 Bogoliubovde Gennes equations . . . . . . . . . . . . . . . . . 186

B Propagation Methods For Dynamical Simulations 189

B.1 GrossPitaevskii Equation Propagation . . . . . . . . . . . . . . . . . . 189

B.1.1 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . 189

B.1.2 Transformation into the interaction picture . . . . . . . . . . . . 190B.1.3 Fourth-order Runge Kutta . . . . . . . . . . . . . . . . . . . . . 191

B.2 Phenomenological Growth Equation with Rotating Thermal Cloud . . . 192

References 194

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Chapter 1

Introduction

1.1 BoseEinstein Condensation

In classical mechanics a system of particles is described in the phase space of the canon-

ical variables of position and momentum. The evolution of the system is completely

determined once the position and momentum of all particles are known at some initial

time. Hence, it is in principle possible to uniquely label identical particles by specifying

their position and momentum. In a quantum mechanical many-body system, however,

the Heisenberg uncertainty principle forbids the simultaneous measurement of the ex-

act position and momentum of any particle. Instead, the evolution of the system is

described by a wavefunction that represents the probability of any of the particles be-

ing at a certain position. Identical particles become fundamentally indistinguishable.

However, certain symmetry conditions have to be imposed on the wavefunction. The

exchange of two identical particles does not change any physical observables. That

means that such a particle exchange can only introduce a phase factor in the wave-

function. Additionally, after a second exchange of the same two particles the original

wavefunction has to be recovered. It follows that the phase change is either -1 or+1, and the wavefunction is symmetric or antisymmetric under particle exchange, re-

spectively. Hence, all elementary particles are classed into two categories: fermions

that are described by an antisymmetric wavefunction and obey FermiDirac statistics,

and bosons that are described by a symmetric wavefunction and obey BoseEinstein

statistics. Remarkably, in quantum field theory it has been shown that this statistical

property is linked to an internal property of the particle, namely its intrinsic angular

momentum or spin. While this applies rigorously only to elementary particles, of which

very few are bosons, composite particles can also be regarded as bosons if their total

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Chapter 1. Introduction

spin is integer as long as their internal structure is not apparent in the collisions, i.e.

their internal energy spacing is much larger than their interaction energy.

While at high temperatures the difference between fermion and boson gases is slight,

sufficiently cooled down the different statistical behaviour leads to dramatically different

effects. Fermions, as a consequence of their antisymmetric wave function, obey the Pauli

exclusion principle which forbids any two fermions to occupy the same single-particle

quantum mechanical state. In contrast, an arbitrarily large number of bosons can

occupy the same state, and indeed, the more bosons occupy the same single-particle

state, the more particles are scattered into it. This Bose-enhanced scattering can lead

to an avalanche effect at very low temperatures and a certain phase space density,

where suddenly the ground state will be macroscopically occupied by atoms. Thiseffect is called BoseEinstein condensation (BEC). It can be considered as a phase

transition, where the atoms lose their individual identity. The condensate acts as one

entity exhibiting quantum mechanical properties on a macroscopic scale.

BEC was first predicted by Einstein in 1924 as a consequence of Bose statistics,

which was introduced in an earlier paper by Bose. Bose developed the Bose statistics

for the case of photons, and Einstein generalized the idea to the case of indistinguish-

able particles. In 1961 an important equation was derived for the treatment of weakly

interacting dilute Bose condensed gases the GrossPitaevskii equation, which con-siders the mean field of the quantum system. In 1995 the field of BEC took a huge

step forward when Anderson et al. [1] achieved BEC for the first time in a dilute

weakly-interacting gas of trapped 87Rb atoms.

1.2 Recent Advances in BoseEinstein Condensa-

tion

The first experimental observation of BEC in a dilute alkali gas [1] in 1995 initiated

an enormous renewed interest in the field of ultra-cold atoms and degenerate gases. It

has brought together researchers from fields as different as atomic physics, condensed

matter and quantum optics. The importance of this fast growing field has been rec-

ognized by two Nobel prizes the first one awarded in 1997 to Steven Chu, Claude

Cohen-Tannoudji and William D. Phillips for development of methods to cool and trap

atoms with laser light [2]. These developments paved the way for the experimental

realization of BoseEinstein condensates, which was rewarded with a Nobel prize in

2001 to Eric A. Cornell, Wolfgang Ketterle and Carl E. Wieman for the achievement

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1.2. Recent Advances in BoseEinstein Condensation

of BoseEinstein condensation in dilute gases of alkali atoms, and for early fundamental

studies of the properties of the condensates [3].

BoseEinstein condensates are most commonly created using 87Rb and 23Na, but

have also been achieved in all other stable alkali species 7Li [4], 85Rb [5], 41K [6], and

most recently 133Cs [7] as well as in 1H [8] and meta-stable 4He [9, 10]. There is a

huge number of publications in the field of BEC every year, which makes it impossible

to give a complete summary of all developments. A good summary of the theory of

BEC can be found in [11, 12, 13]. A highly recommended collection of review articles

on ultracold matter has been published in Nature [14, 15, 16, 17, 18, 19]. There is also

a comprehensive list of references in form of a Resource Letter [20]. In this section, we

will give a general overview concentrating mainly on recent experimental advances.

1.2.1 Experimental systems

Conventional traps

To reach the critical temperature of BEC, the atomic cloud is cooled in two steps.

First, laser cooling is applied to the cloud held in a magneto-optical trap (MOT) which

will cool it down to the region of tens of micro-Kelvins. In the next step, the cloud is

typically trapped in a quadrupole field of a magnetic trap, and the hottest atoms areremoved by evaporative cooling. In this technique, a suitably tuned radio-frequency

(RF) field flips the spin of the atoms at the edge of the trap to an untrapped mF state.

Because the hottest atoms are most likely located at the outer region of the trap,

they will be removed, while the remaining atoms re-thermalize. A slowly decreasing

RF frequency will cut further into the atomic cloud because the resonance condition

is dependent on the Zeeman shift of the atoms, and therefore on the magnetic field

strength of the trap.

There is one major problem with a magnetic quadrupole trap. The magnetic field

is zero at the centre of the trap, which allows atoms to escape undergoing Majorana

spin flips to untrapped mF states. The two most commonly used trap geometries solve

this problem in different ways. The time-orbiting potential (TOP) trap uses a rapidly

rotating bias field on top of the quadrupole field so that the point of zero magnetic

field rotates constantly around the centre of the trap. Because the movement is so fast,

the atoms only experience an averaged harmonic magnetic field. A TOP trap gives

rise to a pancake-shaped condensate. The second design is a Ioffe-Pritchard (IP) type

trap where the point of zero magnetic field is removed altogether by the application of

a static bias field using a complex design of cloverleaf coils [21]. Condensates in IP

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traps are usually long cigar-shaped.

All-optical traps and spinor condensates

Magnetic traps have the disadvantage that only atoms with certain low-field-seeking

mF states can be trapped. Even though binary mixtures of atoms in different hyperfine

states have been explored in magnetic traps [22, 23, 24, 25], purely optical traps have

the advantage that they can confine all possible hyperfine states. This allows the

exploration of so called spinor condensates that consist of atoms in all possible hyperfine

states [26, 27]. Typically, the condensate is created in a magnetic trap and subsequently

transfered into the optical trap. However, the BEC transition has also been achieved

in an all optical dipole trap formed by CO2 laser beams [28].

Micro-traps

In recent years, there has been the trend to miniaturization of magnetic traps on mi-

crochips. The traps are formed by lithographically created arrays of current-carrying

wires. The BEC transition has already been achieved on microchips [29, 30]. The main

advantage of these setups is the easy accessibility to manipulate and study the con-

densates compared to conventional techniques where the optical and mechanical access

is very limited. The JILA group has demonstrated the transport of a condensate in a

magnetic waveguide build on such micro-structures, and are also making advances with

beamsplitters for cold atoms [31] towards beamsplitters for BoseEinstein condensates.

BoseEinstein condensation in lower dimensions

The reduction of a physical system to lower dimensionality can give rise to completely

new physics with formerly unknown phenomena, e.g. the Quantum Hall effect in a two-

dimensional electron gas. It is well known that BEC cannot occur in a uniform one-

dimensional or two-dimensional system1. However, a confining trap allows the BEC to

take place even in lower dimensional systems. Such (quasi)low-dimensional condensates

have been realized by confining them in highly elongated optical traps where the energy-

spacing in one or two dimensions exceeds the interaction energy between the atoms [32].

Other possibilities are the formation of quasi-condensates [33] that behave locally like

condensates, but do not show phase coherence accross the whole system due to large

phase fluctuations [34].

1Theoretically, a uniform two-dimensional BEC can exist at T = 0 K.

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BoseEinstein condensation in optical lattices

When condensates are loaded into weak optical lattices, the phase coherence of the con-densate is maintained accross the lattice sites due to tunneling between them. Equiv-

alent to the Josephson effect, atoms oscillate back and forward between the different

lattice sites. This coherence has been demonstrated by the pulsed output of a conden-

sate falling under gravity in an optical lattice [35]. As the lattice barriers are increased,

tunneling is suppressed, and the number of atoms in each well becomes more sharply

defined while the phase coherence between lattice sites is lost. This can also be de-

scribed in terms of number squeezing since condensate phase and number are conjugate

variables [36]. In a three-dimensional lattice, this effect leads to a phase transition from

a superfluid to a Mott insulator state [37] where number fluctuations are completely

suppressed.

An optical lattice can also impart momentum to a condensate via diffraction [38]

or Bragg scattering [39]. Bragg scattering has become an important experimental

tool because it allows the condensate to be put into a superposition of well-defined

momentum states.

1.2.2 Atom lasers

In analogy to an optical laser, an atom laser is a source of coherent matter waves [40].

A trapped BoseEinstein condensate is essentially a single macroscopically occupied

mode of the trap, spatially coherent over the whole condensate region [41, 42], and as

such suitable as a source for an atom laser. In early experimental realizations of atom

lasers [43, 44] the output beam was pulsed, coupling small amounts of condensate by

RF induced spin-flips or Majorana spin flips to an untrapped state. Later, a quasi-

continuous output was achieved [45, 46] with a much better collimated beam.

In all of these experiments, the lasing time of the atom lasers was limited by the

size of the condensate because there was no mechanism to replenish the condensate

while atoms were being output-coupled. Recently, the MIT group has demonstrated

a continuous source of Bose condensed atoms [47]. The condensate held in an optical

trap is reloaded with new condensates delivered using optical tweezers that allow Bose

Einstein condensates to be transported over distances of tens of centimeters [48].

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1.2.3 Non-linear and quantum atom optics

A number of non-linear effects known from non-linear photon optics has also beendemonstrated with BoseEinstein condensates. In non-linear optics, a medium is nec-

essary to couple light fields non-linearly. With matter waves there is no need for a

medium, but the non-linearity arises from the interactions between atoms in the con-

densate. In four-wave mixing experiments [49] the production of a condensate in three

carefully phase-matched momentum states yields to the population of a fourth momen-

tum state. Similarly, the interaction of light waves with a condensate can lead to effects

like superradiant scattering [50] or wave-matter amplification [51, 52].

A completely different phenomena is the propagation of solitons, which are wavepack-

ets that can travel over long distances in non-linear media without spreading. So called

dark solitons have been imprinted onto condensates [53, 54].

Very remarkable experiments have been performed by Hau et al. [55] who were able

to effectively slow down the speed of light travelling through a condensate and even

stop it completely [56] by storing the coherent information contained in the laser field

in the internal states of the atoms. These experiments made use of a quantum effect

called electromagnetically induced transparency that allows the propagation of light

through an otherwise opaque atomic medium.

1.2.4 Atomic collisions and molecular BoseEinstein conden-

sation

At low temperatures the nature of collisions between atoms is determined by the s-

wave scattering length. If the s-wave scattering length is positive, atoms repell each

other, and the confining force of the potential is balanced by the mean-field repulsion.

If the scattering length is negative as in 7Li, however, the attractive forces between the

atoms lead to a collapse of the condensate [57]. Only for a very small number of atomsa stable condensate is possible if the self-attractive forces are balanced by a repulsion

arising due to a momentum-position uncertainty in the trap [4, 58].

However, some alkali elements show Feshbach resonances where the free state of

the colliding atoms couples resonantly to a quasibound molecular state. This coupling

strongly affects the scattering length in the collision. Because the free state and quasi-

bound state have different magnetic moments, a magnetic field can be used to tune the

scattering length from positive to negative values around a Feshbach resonance [59].

When the scattering length of 85Rb was tuned from positive to negative, besides the

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expected collapse of the condensate, a blast of hot atoms was observed [60]. Because

this phenomena is similar to the neutrino burst of a collapsing star during a supernova

it was named Bose nova. Using a more controlled collapse the point of instability in

a condensate with attractive interactions was determined [61].

With the control over the scattering length Feshbach resonances provide they are a

very useful experimental tool. They were used to Bose condense 133Cs [7], which has

an enormous negative scattering length of 2000 Bohr radii for zero magnetic field, and

also in the formation of bright solitons in 7Li [62].

One way to produce ultra-cold molecules is to use photo-association starting with

atoms in a BoseEinstein condensate [63, 64] where two colliding atoms collectively

absorb a photon forming a bound, excited-state molecule. Whether this process iscoherent and whether a molecular BoseEinstein condensate is formed is still subject

to further research. Donley et al. [65] have coherently coupled atoms and molecules

in a BoseEinstein condensate using a time-varying magnetic field near a Feshbach

resonance.

1.2.5 Superfluidity and vortices

Already in 1938, London suggested that the cause of the superfluid behaviour of 4He

might be related to BEC although such a system is highly depleted. With the advent of

dilute alkali BoseEinstein condensates, there is the chance to study superfluid effects

in almost fully condensed systems.

One criterion of superfluidity is that obstacles moving through the condensate ex-

perience zero or much reduced friction as long as their velocity is below some critical

velocity given by the Landau criterion

vc = minEpp , (1.1)

where Ep and p are the energy and momentum of an excitation. Below this velocity, no

excitation can be created. This effect has been observed in several experiments using

blue-detuned laser beams [66, 67] and impurity atoms [68] moving through the conden-

sate. Similarly, a condensate can move dissipationless through an optical lattice with

velocities below the critical velocity [69]. Another interesting effect is the occurence of

a certain excitation called scissors mode [70].

A fundamental property of a superfluid is the fact that it can only support irrota-

tional flow. For low rotation rates below the critical velocity for vortex formation, this

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leads to a characteristic flow pattern which has experimentally been observed directly

[71]. For higher rotation rates, angular momentum can only be aquired sustaining

an irrotational flow by the formation of vortices. The first vortex experimentally ob-

served was created in a two component condensate where the vortex core is filled by

atoms in a different spin state [72], a configuration which is sometimes referred to as

skyrmion rather than vortex. The first observation of a pure vortex in a single compo-

nent condensate [73] has stimulated a large number of experiments. Early experiments

concentrated on states with a single vortex or a small number of vortices [74, 75, 76].

Usually, a pure condensate is set into rotation by means of an anisotropic rotating po-

tential, which is typically ellipsoidal. Potentials with different symmetries show distinct

resonance frequencies for vortex formation [77]. Haljan et al. [78] demonstrated thenucleation of a vortex lattice from a rotating thermal cloud without the need of any

rotating anisotropy.

The first realization of a large vortex lattice with more than hundred vortices [79]

initiated research into the properties of vortex lattices such as their formation and decay

[80], non-equilibrium deformations under compression [81], excitations of Tkachenko

oscillations [82] and giant vortices [83]. A major part of this thesis is concerned with

vortex lattices. Chapter 4 gives an overview on the theory of vortices and vortex lattices,

and a more detailed account on experiments on vortex lattices will be presented in laterchapters when relevant.

1.3 This Work

1.3.1 Overview

In chapter 2, we review the mean-field theory of a BoseEinstein condensate. We

present a brief derivation of the time-dependent and time-independent GrossPitaevskii

equation, which provides a very good description of a BoseEinstein condensate at

zero temperature. The linear excitations of a BoseEinstein condensate are described

by the Bogoliubovde Gennes equations. We present two different derivations of the

Bogoliubovde Gennes equations that illustrate the equivalence of elementary and col-

lective excitations in a BoseEinstein condensate. And finally, we introduce our choice

of computational units for numerical calculations.

In chapter 3 we consider the excitation spectrum of a BoseEinstein condensate in

a three dimensional cylindrically symmetric trap. This work extends earlier results re-

ported in my Masters thesis, in presenting a systematic classification of the excitations

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1.3. This Work

that generalizes the concept of families first identified by Hutchinson and Zaremba [84].

The extension involves a complete revision of the procedure for assigning a family clas-

sification to any mode in the anisotropic case. We have also been able to determine

the energy ordering of the modes, and give a simple model that explains this ordering.

A large part of this thesis is concerned with the nucleation of vortices and the for-

mation of vortex lattices in rotating BoseEinstein condensates. Chapter 4 summarizes

the fundamental properties of superfluids and vortices which are well known from the

work on superfluid Helium, as well as more recent results on vortices in BoseEinstein

condensates. In particular, we discuss the question of a critical angular velocity for

vortex nucleation in a rotationally stirred BoseEinstein condensate.

In chapter 5, we present the formalism we have used to describe the process ofvortex lattice formation. Dissipation is known to be an essential element in vortex

lattice formation. Therefore, the standard GrossPitaevskii equation cannot describe

this phenomenon, and a new approach is needed. We provide a summary of the theory

developed by Gardiner et al. [85] to treat condensate dynamics in the presence of a

thermal cloud. That formalism is based on quantum kinetic theory, and shows how

a dissipation mechanism is introduced by exchange of atoms between the condensate

and the surrounding thermal cloud of atoms. In its most simplified form, Gardiners

theory reduces to a modified GrossPitaevskii equation that includes growth and lossterms, and which we call the phenomenological growth equation. This equation forms

the basis of much of our treatment of rotating condensates.

In order to simulate the phenomenological growth equation we needed to develop

a numerical method for propagating the equation, and we describe this method in

chapter 6. We also give a detailed account of the accuracy of the method, including

the optimal choice of grid size, number of points, and temporal step size. The angular

momentum operator is required in this equation to describe a rotating thermal cloud,

and we discuss its effect on the stability and reliability of the method.

In chapter 7, we present results on the formation of vortex lattices from a rotating

thermal cloud in a cylindrically symmetric trap. We analyse the initiation process

in terms of a gain process for surface modes with non-zero angular momentum, and

obtain gain coefficients in terms of the excitation energies in good agreement with the

simulations.

In chapter 8, we apply the phenomenological growth equation to the case of vortex

lattice formation in a rotating elliptical trap. Simply stirring a condensate in the

absence of a thermal cloud does not lead to a vortex lattice. However, in the presence

of a thermal cloud, the stirring can seed angular momentum components which may

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then grow from the thermal cloud by stimulated collisions. To analyse this scenario,

we numerically calculate the stationary vortex free states of a rotating elliptical trap,

and their excitations by solving the GPE and BdG equations in the rotating frame.

We also give a critical evaluation of our model in relation to experimental results for

vortex nucleation through rotational stirring.

The phenomenological growth equation can also be applied to the decay of vortex

lattices in the limiting case of a stationary thermal cloud, which we consider in chapter

9. And in chapter 10, we present simulations on non-equilibrium lattice dynamics,

in particular on the deformations of rapidly-rotating vortex lattices in the presence of

quadrupole excitations as explored in a recent experiment by Engels et al. [81]. Finally,

we calculate the excitations of a vortex lattice and identify the Tkachenko modes inthe excitation spectrum, which have recently been observed experimentally [82].

1.3.2 Peer-reviewed publications

Some of the work presented in this thesis has been published in peer-reviewed jour-

nals. The work on the excitation families of a cylindrically symmetric BoseEinstein

condensate presented in chapter 3 has appeared in Journal of Physics B [86].

Some of the main results from chapters 7 and 8 on the nucleation, formation and

stabilization of vortex lattices have been published in Physical Review Letters [87].

The methods developed in this thesis have also formed the basis for some work on

giant vortices and Tkachenko oscillations [88], which has been published in Physical

Review Letters. A small part of those results are presented in chapter 10.

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Chapter 2

Mean Field Theory

In this chapter the basic equations describing a single-component BoseEinstein con-

densate will be derived. There are many different ways to derive these equations,

e.g. via Greens functions [89] or variational principles [90, 91]. We have chosen to

follow the mean field approach because it provides a particularly straightforward pro-

cess for obtaining the basic equations. The underlying idea is that in a BoseEinstein

condensate all atoms occupy the same single-particle quantum state and can, therefore,

be described by the same wave function. A single atom is not aware of the individual

behaviour of the others and loses its individuality moving through the condensate. Toa good level of approximation it only sees the mean field generated by the condensate

as a whole. The condensate can be thought of as acting coherently on a single atom.

In the final sections of this chapter, we briefly show how to transform the Gross

Pitaevskii equation into a form suitable for numerical solution. We introduce our choice

of dimensionless units and also introduce the ThomasFermi approximation, which is

a very good description of a condensate in the hydrodynamic limit.

2.1 Second Quantization

2.1.1 Many-body Hamiltonian

The starting point of a theoretical treatment of a single-component BoseEinstein con-

densate is the exact many-body Hamiltonian for a system of identical, structureless

bosons. If the system is sufficiently dilute only pairwise interactions have to be taken

into account. This condition is well satisfied in current experiments on alkali atoms

with lifetimes of the order of seconds where three-body recombinations are the main

loss factor that limits the lifetime. The many-body Hamiltonian can be written in

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Chapter 2. Mean Field Theory

terms of the boson field operator (r, t) in second quantization as

H =

dr (r, t)H0(r, t)

+1

2

dr dr (r, t)(r, t)V(r r)(r, t)(r, t), (2.1)

where the single particle Hamiltonian is given by

H0 =

2

2m2 + Vext(r, t)

. (2.2)

Here, Vext

(r, t) represents an external potential, which may consist of a trapping poten-

tial and other parts, e.g. from an external laser, Vext(r, t) = VT(r) + Vother(r, t). If the

external potential is set to zero, we retrieve the equation for a homogeneous interacting

Boson gas. The field operator (r, t) creates a particle of mass m at position r at time

t. It satisfies the boson commutation relation(r, t), (r, t)

= (r r)

(r, t), (r, t)

= [(r, t), (r, t)] = 0.(2.3)

The interatomic potential V(r r) represents the interaction strength between theatoms in a binary collision. While this Hamiltonian is exact it is intractable to analytical

or numerical solutions except in the case of only a few particles.

2.1.2 Pseudo-potential approximation

At low temperatures s-wave scattering is the dominant collision process in dilute gases.

If the s-wave scattering length a is small compared to the de-Broglie wavelength dB =

22/mkBT, a good approximation for the interatomic potential is the replacementby a pseudo-potential [92]

V(r r) = U0(r r), (2.4)

where U0 represents the effective interaction strength which is related to the s-wave

scattering length by

U0 =42a

m. (2.5)

Effectively, the collisions are now treated as hard sphere collisions. We note that this is

a low momentum approximation. For high momentum collisions the pseudo-potential

approximation gives rise to ultra-violet divergences because it scatters high momentum

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2.1. Second Quantization

particles just as effectively as low momentum ones. This assumption is unphysical since

in real collisions the energy transfer is less effective for higher momenta. Hence, care

has to be taken if high momentum collisions are included. Morgan [93] has shown that

a high energy renormalization can be achieved in a natural way if the pseudo-potential

approximation is done on the scattering T matrix rather then V (which is simply

the first term of the T matrix series). In this thesis, we will consider only very low

temperatures so that the pseudo-potential approximation of equation (2.4) can safely

be used.

2.1.3 Definition of BoseEinstein condensation

BoseEinstein condensation (BEC) is the macroscopic occupation of a single quantum

state. Following Legget [12], we can specify this by considering the one-particle reduced

density matrix

(r, r

, t)

(r, t)(r

, t)

, (2.6)

where the average indicated by the angled brackets is in general statistical as well as

quantum mechanical. It is always possible to write the boson field operator in terms

of an orthonormal set of single-particle wave funtions i(r, t) [91]

(r, t) =i

i(r, t)ai(t), (2.7)

where ai are the corresponding boson annihilation operators. These annihilation oper-

ators and their creation counterparts ai are defined in Fock space by

ai |n0, n1, . . . , ni, . . . =

ni + 1|n0, n1, . . . , ni + 1, . . .ai|n0, n1, . . . , ni, . . . = ni|n0, n1, . . . , ni 1, . . ., (2.8)

where {nk} are the occupation numbers of the single-particle states. The creation andannihilation operators obey the usual boson commutation rules at the same time

ai, a

j

= ij

[ai, aj] = [ai , a

j] = 0.

(2.9)

In this notation the reduced density matrix (2.6) takes the form

(r, r

, t) = i

ni(t)i (r, t)i(r

, t). (2.10)

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Here, ni(t) ai(t)ai(t) are the the expectation values of the number operator. Weshall say that the system shows BEC at any given time t if one or more of the eigenvalues

ni(t) are of the order of the total number of particles N, and in particular simple BEC

if only one of the eigenvalues is of the order N, while all others are of order 1. In the

case of simple BEC we will use the index zero to indicate the state of macroscopical

occupation with N0 n0 N. We shall call the single-particle state 0(r, t) thecondensate wave function and N0 the (mean) number of particles in the condensate.

The simplest and most direct choice of an order parameter for the BEC phase transition

is then (r, t) =

N0(t)0(r, t), which is simply the single-particle wave function into

which condensation occurs scaled by the number of atoms in this state. It may be

stressed that the overall phase of this order parameter has no physical significance.

2.1.4 Bogoliubov approximation

An approximation commonly used in the BoseEinstein literature is the Bogoliubov

approximation in which the operators a0 and a0 are replaced by

N0. It is based on

the idea that states with N0 and N0 + 1 N0 atoms in the condensate correspondto essentially the same physical configuration. The boson field operator can then be

separated as

(r, t) = (r, t) + (r, t) (2.11)

into a condensate part described by the order parameter (r, t) defined in this context

as (r, t) = (r, t) and an operator (r, t) representing fluctuations, whose expec-tation value is zero by definition. However, this approach is not consistent with the

conservation of number of atoms because the boson field operator has a non-zero expec-

tation value, which is only possible if the condensate wave function is a superposition

of states with different numbers of atoms N. The same difficulty is often referred to as

the problem of spontaneously broken symmetry which occurs due to the definite phase

associated with the condensate part. Phase and particle numbers are conjugate vari-

ables, i.e. it is not possible to have a definite condensate phase and simultaneously a

definite number of condensate atoms.

2.1.5 Time-dependent GrossPitaevskii equation

Despite the conceptual difficulties associated with the Bogoliubov approximation we

will use it here to derive the GrossPitaevskii equation (GPE), the central equation in

the description of a dilute BoseEinstein condensate. Ultimately, its use is justified by

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2.1. Second Quantization

more complex number-conserving approaches which lead to essentially the same results

[94].

The Heisenberg equation of motion for the boson field operator is

i(r, t)

t=

(r, t), H

=

H0 + U0(r, t)(r, t)

(r, t). (2.12)

Inserting the decomposition (2.11) into (2.12) and taking the expectation value yields

i

t=

H0 + U0||2

+ U0

2 + +

. (2.13)

The term

is identified with the non-condensate density n acting back on thecondensate, and the term m is known as the anomalous average which modifiesthe interaction strength between condensate atoms due to virtual processes. However,

a careful consideration shows that the dominant part of this term has already been

included by the introduction of the contact potential [93]. Finally, the term represents collisions of two thermal atoms in which one of them enters the condensate.

In the limit of zero temperature all these terms can be neglected because the fraction

of thermal atoms is very small. Then, we are left with the famous GPE

i(r, t)

t=

H0 + U0|(r, t)|2

(r, t). (2.14)

The GPE is strictly valid only when all atoms are in the condensate and can,

therefore, be used only for temperatures near to T = 0 K. Actually, even at T = 0 K

some atoms are depleted from the condensate due to interactions. In liquid 4He this

depletion is about 90% so that the mean field approach is not useful to obtain qualitative

results. In the case of condensed alkali gases, however, the depletion is of the order of

0.5% [95] and the GPE enables accurate quantitative predictions.

2.1.6 Time-independent GrossPitaevskii equation

If we use the usual ansatz (r, t) = (r)eit/ for a stationary solution of the GPE

(2.14) we arrive at the time-independent GPE

H0 + U0|(r)|2

(r) = (r). (2.15)

The lowest energy state of this equation is the ground state into which condensation

occurs. At T = 0 K the eigenvalue can be identified as the chemical potential of the

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system.

To include thermal effects, terms of higher order in from equation (2.13) must be

accounted for. In the Popov approximation only the term n = is kept

H0(r) + U0[nc(r) + 2n(r)](r) = (r), (2.16)

where nc(r) = |(r)|2 denotes the condensate density. If the anomalous average m(r) = is also kept we obtain the Hartree-Fock-Bogoliubov (HFB) equation

H0(r) + U0[nc(r) + 2n(r)](r) + U0m(r)(r) = (r). (2.17)

We used the symbol for the eigenvalue of these latter two equations because at finite

temperatures it differs from the thermodynamic quantity of the chemical potential

by = + kBT ln(1 + 1/N0) [96]. Whereas the HFB-Popov equation (2.16) is

self-consistent, the HFB equation (2.17) yields a gap in the excitation spectrum [97].

Hutchinson et al. [84] have introduced a gapless HFB approximation by modifying the

interaction strength U0. This is justified by an approximation for the many-body T

matrix [98].

2.2 Elementary Excitations

In this section we will derive the so called Bogoliubovde Gennes equations (BdG equa-

tions) at zero temperature which describe the excitations of a BoseEinstein condensate.

In an ideal gas the only excitations possible are single-particle excitations, i.e. a single

particle occupies an energy state above the macroscopically occupied ground state. In

the homogeneous case, these excitations are simply plane waves, while in a confined

system they are the eigenstates of the trapping potential. If the gas is interacting, the

nature of these elementary excitations changes because as the excited particle movesthrough the sytem it interacts with the neighbouring atoms. However, often it is pos-

sible to describe the combined system of the single particle and the surrounding cloud

of atoms it interacts with in terms of a fictitious quasiparticle, similar to the concept

of a dressed state of an atom in an electromagnetic field. Quasiparticles represent the

excited energy levels of an interacting many-body system.

Interacting many-body systems also have a completely different type of excitations:

collective excitations. They are associated with density fluctuations and involve the

collective wave-like motion of all particles. Since interactions are crucial for those kind

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2.2. Elementary Excitations

of collective excitations, there is no equivalent in a non-interacting gas.

A peculiar property of a BoseEinstein condensate is that its elementary and col-

lective excitations are identical. This can be understood by the fact that all atoms in a

condensate are described by the same single-particle wave function. Thus, any excita-

tion involving one particle (or quasiparticle) automatically involves all others leading

to a collective response. To illustrate this property we will derive the BdG equa-

tions in two distinctively different ways. The first approach uses the grand-canonical

Hamiltonian, which is diagonalized by a Bogoliubov transformation into a collection of

non-interacting quasiparticles [99]. In this pure quantum mechanical approach the exci-

tations are necessarily orthogonal to the condensate since only excited non-condensate

atoms take part in the collective modes [100]. The second derivation uses linear re-sponse theory around the time-dependent GPE. Thus, the excitations are considered

as collective motions of condensate atoms. These modes are not necessarily orthogonal

to the ground state wave function. We shall outline both derivations for zero temper-

ature following closely Ref. [100] and [101] and make the differences between the two

approaches clear.

2.2.1 Bogoliubov transformation

The Bogoliubov transformation is a well known method to diagonalize a quadratic

Hamiltonian and gives a transformed set of bosonic operators, which are called quasi-

particle operators due to their particle-like character.

A many atom system can be described by the grand canonical, many-body Hamil-

tonian K = H N, where H is the many-body Hamiltonian, N the number operatorand the chemical potential. Within the pseudo-potential approximation described in

section 2.1.2 this is written in terms of field operators as

K = H N = dr (r)[H0 ](r) + 12U0 dr (r)(r)(r)(r). (2.18)Inserting the decomposition of the field operator (2.11) into (2.18) and neglecting terms

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in (r) higher than quadratic yields

K =

dr [H0 + 12

U0||2 ]

+

dr [H0 + U0||2 ]

+

dr [H0 + U0||2 ] (2.19)

+

dr [H0 + 2U0||2 ]

+1

2U0

{dr 2 + 2}.

The first term in the above equation is just a c-number. The second and third term

vanish if satisfies the time-independent GPE (2.29). The remaining Hamiltonian can

be diagonalized by a linear canonical transformation, i.e. a transformation of creation

and annihilation operators that preserves the commutation relations. This is done by

the Bogoliubov transformation

(r) =

i[ui(r)i + vi (r)

i ]

(r) = i[ui (r)i + vi(r)i],(2.20)

which expresses the fluctuation operator in terms of the quasiparticle creation and

annihilation operators i and i that are required to fulfill the usual boson commutation

relations[i,

j ] = ij

[i, j ] = [i ,

j] = 0.

(2.21)

The Hamiltonian is diagonalized if the functions ui(r) and vi(r) are chosen to satisfy

the following equations

dr {ui [Luj + U02vj] + vi [Lvj + U02uj]} = iijdr {ui[Lvj + U02uj] + vi[Luj + U02vj ]} = 0

(2.22)

where L is defined asL = H0 + 2U0||2 . (2.23)

From the equations (2.22) it can be shown [102] that the uis and vis obey the

orthogonality relation

dr {uiu

j

viv

j

}= ij (2.24)

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2.2. Elementary Excitations

and the symmetry relation

dr {uivj viuj} = 0. (2.25)The normalization for i = j in the orthogonality condition is forced to be one. This is

a consequence of the commutation rule of the operator (r)

[(r), (r)] = (r r) (r)(r), (2.26)

which follows from the decomposition (2.11) and the commutation rules of the boson

field operator (2.3).

The quasiparticle amplitudes ui and vi must be orthogonal to the ground state

wave function , which can be understood as follows. In the expansion of the field

operator (2.7) the single particle wave functions i have to be orthonormal to preserve

the commutation relations for the boson field operators (2.3). This implies in particular

that the coefficients i(r, t), i = {1, 2, . . .}, of the operator (r, t) =

i=1 i(r, t)ai(t) are

orthogonal to the ground state wave function (r, t) =

N0(t)0(r, t). The Bogoliubov

transformation (2.20) simply casts the operator into a collection of non-interacting

quasiparticles represented by creation and annihilation operators i , i, but preserves

this orthogonality to the ground state. Hence, ui and vi are orthogonal to the ground

state wave function.

2.2.2 Linear response theory

At very low temperatures a Bose condensed gas is described by the time-dependent

GPE

i(r, t)

t=

H0 + U0|(r, t)|2

(r, t). (2.27)

Assuming that all atoms are in the condensate the wave function is normalized to the

total number of particles in the trap

dr |(r, t)|2

= N. The lowest energy eigen-state solution of equation (2.27) is of the form (r, t) = g(r)e

it/, where is the

eigenvalue of the system. To find the linear excitations consider a small harmonic dis-

turbance of frequency . Assuming that the excitations are only weakly occupied so

that they do not affect the condensate ground state and do not couple to each other,

we can look for solutions of the form

(r, t) = eit

g(r) + u(r)eit + v(r)eit

, (2.28)

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where g(r) is a solution of the time-independent GPE (2.15). If this is substituted

into equation (2.27) and only terms linear in u(r) and v(r) are retained one obtains the

BdG equations by equating terms of eit,

H0 + U0|g(r)|2

g(r) = g(r), (2.29)

Lui(r) + U02g(r)vi(r) = iui(r),Lvi(r) + U02g (r)ui(r) = ivi(r),

(2.30)

where L is defined as before L = H0 + 2U0|g(r)|2 .The first equation is just the time-independent GPE, which does not contain the

functions ui(r) and vi(r). Hence, it can be first solved independently. The following

two equations are a set of coupled equations, which are also dependent on the solution

of the GPE through g(r) and . They determine the shape and frequencies of the

linear excitations completely.

The BdG equations (2.30) imply the orthogonality and symmetry relations (2.24)

and (2.25), respectively [99]. In this approach, however, the normalization in the or-

thogonality relation, when i = j, is not forced to unity, but could be chosen arbitrarily.

Only the quantum mechanical approach from the previous section reveals that the

bosonic character of the quasiparticles forces the normalization to unity. A particularsolution of the BdG equations is the Goldstone mode u0(r) = g(r) and v0(r) = g(r)with 0 = 0. Both the orthogonality and symmetry relation for this solution take the

form dr {gui + gvi} = 0. (2.31)

It can be easily verified that the BdG relations (2.30) satisfy equations (2.22) and

are, therefore, sufficient to diagonalize the Hamiltonian. However, they do not nec-

essarily preserve the orthogonality to the ground state whereas the relations (2.22)

allow one to choose the excitations orthogonal to the ground state. This can be done

by solving the BdG equations (2.30) first and then projecting out the overlap with the

condensate [103, 100]. The corresponding projection operator P acting on any function

f(r) is defined as

P f(r) = f(r) g(r)

drg(r)f(r). (2.32)

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2.3. Computational Units

Thus, the orthogonal excitations become

ui(r) = P ui(r) = ui(r) gig(r) (2.33)vi (r) = P v

i (r) = v

i (r) + g

i g(r), (2.34)

where gi

dr g(r)ui(r

) = drg(r)vi(r). The latter equality follows fromequation (2.31). The orthogonal excitations ui and vi satisfy the following modified

BdG relationsLui + U02gvi = i(ui + gig)Lvi + U02g ui = i(vi gig)

(2.35)

giving the same eigenvalues i, and they still diagonalize the Hamiltonian. They alsofulfill the orthogonality and symmetry relations.

Following the nomenclature of reference [100] we will call the excitations satisfy-

ing the ordinary BdG equations (2.30) linear excitations and the ones orthogonal to

the ground state satisfying (2.22) orthogonal excitations. Because the orthogonal ex-

citations can always be obtained from the linear ones by the projection method just

described, we will numerically solve for the linear excitations if not explicitly stated

otherwise.

2.3 Computational Units

We now briefly introduce some important quantities for the numerical treatment of the

GrossPitaevskii and Bogoliubovde Gennes equations. In computational physics it is

customary to use dimensionless units for notational simplicity and numerical accuracy.

In most experimental realizations of BEC, the confining trap is cylindrically symmetric

with a harmonic trapping potential, which can be written as

VT = 12

m2r [x2 + y2 + (z)2], (2.36)

where we have chosen the z-axis of the coordinate system parallel to the symmetry

axis of the trap. Here, r is the radial trapping frequency. The anisotropy of the trap

is given by = z/r, the ratio between the trapping frequency in z-direction z and

r. In terms of the radial trapping frequency, we define the following units for distance

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and time

r0 =

2mr

, (2.37)

t0 =1

r. (2.38)

The unit r0 is known as the harmonic oscillator length. From those units, we can obtain

derivative units for momentum, angular momentum and energy

p0 =

2mr (2.39)

L0 = , (2.40)

E0 = r, (2.41)

respectively.

As an example, we will transform the time-independent GPE (2.15) to computa-

tional units. We introduce dimensionless quantities, which we will indicate by a tilde,

using

r = rr0, = r. (2.42)

It is worth noting that the wave function itself has units of [length]

d/2

, where d is thenumber of spatial dimensions. If the wave function is expressed as

(r) =

N

rd0(r), (2.43)

(r) is dimensionless and normalized to unity

dr|(r)| = 1. Substituting the trans-formations (2.42) and (2.43) in the GPE (2.15) gives the GPE in dimensionless form

2r + VT(r) + C| (r)|2 (r) = (r), (2.44)where the non-linearity parameter C is defined as

C =NU0rrd0

. (2.45)

The trapping potential (2.36) takes the form

VT(r) =1

4[x2 + y2 + (z)2]. (2.46)

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2.4. ThomasFermi Approximation

In all chapters of this thesis where numerical results are discussed we will use compu-

tational units exclusively, and for convenience we henceforth omit the tilde denoting

the dimensionless units. It will be clear from the form of the equations and the context

whether SI or computational units are used.

2.4 ThomasFermi Approximation

When there is a large number of atoms in the condensate (large C) the non-linear term

in the GPE (2.44) will dominate the kinetic contribution. In that case, it is a good

approximation to neglect the kinetic energy completely and write the GPE (2.44) in

the ThomasFermi limit as VT + C||2

= . (2.47)

This has the analytical solution

TF(r) =

VTC

for VT > 00 otherwise

(2.48)

The value of the chemical potential is determined by the normalization condition of

the wave function and depends on the dimensionality of the equation.

Three dimensions

In three dimensions with a trapping potential of the form (2.46), the chemical potential

is given by

3D =

15C

64

25

. (2.49)

Within the ThomasFermi approximation the radial and axial extent of the wave func-

tion are

Rr = 2

3D, (2.50)

Rz =2

3D, (2.51)

and the peak density at the centre of the condensate is

np =3D

C. (2.52)

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Two dimensions

In two dimensions, we are mainly interested in elliptical harmonic traps of the form

VT =1

4[(1 )x2 + (1 + )y2]. (2.53)

The chemical potential becomes

2D = [(1 )(1 + )] 14

C

2, (2.54)

and the extent of the wave function is

Rx = 2(1 ) 14

2D, (2.55)

Ry = 2(1 + ) 14

2D. (2.56)

One dimension

Any one dimensional harmonic trap can be cast into the form

VT =1

4x2. (2.57)

Then, the chemical potential is given by

1D =

3C

8

23

. (2.58)

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Chapter 3

Elementary Excitation Families

In this chapter we consider the excitation spectrum of a BoseEinstein condensate in

a three dimensional cylindrically symmetric trap. We will present a systematic classi-

fication of these excitations that generalizes the concept of families first identified by

Hutchinson and Zaremba [84]. We will also relate the energy ordering of the modes to

their family classification and provide a simple model that explains this relationship.

The work presented in this chapter extends initial results obtained earlier for my Mas-

ters thesis [102]. The new elements of the work carried out for this PhD thesis are (i)

a complete revision of the procedure to assign a family classification to any mode inthe anisotropic case (section 3.2.3) and (ii) the determination of the energy ordering

of the modes, along with a simple model that give the basis for the explanation of the

ordering (section 3.3). These two significant extensions of the earlier work have led to

it being published [86].

3.1 Symmetries

In the case of an isotropic harmonic trap, the time-independent GrossPitaevskii equa-tion (GPE) is completely separable and reduces to a one-dimensional problem in the

radial coordinate r. However, for a cylindrically symmetric harmonic trap of the form

VT(x) =1

2m2r [x

2 + y2 + (z)2], (3.1)

where the anisotropy parameter of the trap is defined as the ratio between the axial

and radial trapping frequencies = z/r, a complete separation of variables is not

possible. Although the ordinary Schrodinger equation is separable in Cartesian or

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Chapter 3. Elementary Excitation Families

cylindrical coordinates [104], this separation is not possible for the GPE due to the

non-linear term. Nevertheless, solutions of the form

(x) = (, z)eimc (3.2)

can be found, where , and z denote the usual cylindrical coordinates. The magnetic

quantum number mc is a good quantum number, and the ground state solution of the

GPE corresponds to mc = 0. Since no further separation is possible, the equation has

to be solved in the two variables and z. In non-dimensional units, the GPE takes

then the form

1

+

2

z 2 m2c

2

+ 1

4

2 + (z)2

+ C|(, z)|2

(, z) = (, z).

(3.3)

Correspondingly the normal modes of the Bogoliubovde Gennes (BdG) equations

will also have specific angular momentum compositions if (x) is given by (3.2). If

ui(x) is an eigenfunction of Lz with eigenvalue m, then vi(x) will be an eigenfunction

with eigenvalue (m 2mc) [105]. Thus, the BdG equations can be written as

1

m2

2+

2

z 2 + 1

42 + (z)2 (3.4)

+2C||2

ui(, z) + C2vi(, z) = iui(, z)

1

(m 2mc)

2

2+

2

z 2

+

1

4

2 + (z)2

(3.5)

+2C||2

vi(, z) + C2ui(, z) = ivi(, z).

We note that excitations on the ground state (mc = 0) with |m| are degenerate

because m enters the BdG equations quadratically.The axially symmetric trap potential has also a reflection symmetry with respect

to the x-y plane, and thus the solutions to the BdG equations can be chosen to have

a well-defined parity [84]. In the isotropic case, where l and m are good quantum

numbers for the excitations, the parity is simply given by = (1)lm.To solve these equations we employed Matlabs partial differential equations tool-

box which uses finite element methods based on a triangular segmentation. More details

can be found in appendix A.1 and my Masters thesis [102].

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3.2. Classification of Excitations

3.2 Classification of Excitations

3.2.1 ThomasFermi limit

In the ThomasFermi limit, Fliesser et al. [106] recognized some underlying symme-

tries of the GPE equation in its hydrodynamic form [107] by identifying three operators

which commute with each other. They introduced three corresponding quantum num-

bers (n,j,m) that classify the solutions completely. An explicit separation of the wave

equation was achieved in cylindrical elliptical coordinates , and , and in terms of

these variables the quantum numbers represent

n: order of polynomial in and

j: index to label different eigenvalues for fixed n and |m| ;j runs from 0 to N = 1 + int

n2

m: z-component of angular momentum.

Although the solutions of the full BdG equations do not strictly conserve these

quantum numbers, we find that they exhibit in general the same patterns and symme-

tries, and we will show, in the appropriate regime, how the family classification scheme

we develop can be related to n,j,m.

3.2.2 Families in the isotropic case

We first consider the isotropic case because then the patterns of the different mode

families can be easily described in terms of Legendre polynomials. As the trap geom-

etry is changed from spherical to cylindrical symmetry these patterns are continously

modified, being squeezed in the direction of the stronger confinement, but the basic

character remains recognizable.

In an isotropic trap, the solutions for the excitations can either be completely sep-

arated in spherical coordinates as u(x

) = ur(r)Ylm(, ), or partially separated incylindrical coordinates as u(x) = u(, z)eim. In the latter case, u(, z) is essentially

the radial function ur(r) modulated by the Legendre polynomial Plm(cos ), where

cos = z(2 + z2)1/2. Thus, the general shape of the families is determined by the

symmetries of the Legendre polynomials. We now show that the family classification

suggested by Hutchinson and Zaremba [84] can be generalized, in the isotropic case, as

follows. First we assign a principal family number which is given by

f = l

|m

|+ 1, (3.6)

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and then an additional number characterizing the radial function is needed to complete

the classification into families. We shall introduce the nodal family number nr for

this purpose, which in the isotropic case is simply the number of nodes in the radial

function. The family is given by the pair (f, nr), which together with the magnetic

quantum number m uniquely specifies any mode. A generalization of this classifiction

to the anisotropic case is presented in section 3.2.3.

We illustrate the spatial character of the family assignment by considering first the

excitation modes with no radial node (nr = 0). We begin with the case m = 0, whichwe illustrate in figure 3.1 with contour plots of full numerical solutions of u(, z) for the

specific case of the degenerate modes of the lowest l = 3 excitation with m = 3, 2, 1.

Their principal family numbers are f = 1, 2, 3 respectively. Since the radial function

4 2 0 2 46

4

2

0

2

4

6

Family (1,0)

xposition

z

pos

ition

(a)

4 2 0 2 4

++

Family (2,0)

xposition

(b)

4 2 0 2 4

++

Family (3,0)

xposition

(c)

Figure 3.1: General shape of mode families 1 to 3 with no radial node. Contour plots inthe x-z plane of the quasiparticle amplitude u(, z) are given for the degeneratemodes l = 3, (a) m = 3, (b) m = 2, (c) m = 1 modes.

is the same for each of these modes, the relative overall shape is determined by the

Legendre polynomials. The important property of the Legendre polynomials Plm(cos )for our purposes is that they have n = l |m| nodes between 0 < < . Thus, thenumber ofangular nodal surfaces n beween 0 < < , i.e. surfaces of zero density that

are characterized by a constant value of in the isotropic case, determines the principal

family number f since f is given by equation (3.6) as f = l |m| + 1 = n + 1. Wenote also that for m = 0 all Legendre polynomials are zero along the z-axis, but this isnot a nodal surface. Because the sign of the wave function changes as it crosses a nodal

surface, family 1 members have even parity, family 2 have odd parity, and in general the

parity of the mode is related to the principal family number by = (1)f1

= (1)lm

.

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The m = 0 member of each family has a shape that derives from the Pl0 Legendre

polynomial. We call it the anomalous member of the family, since its shape differs from

other members of the family only in that it is non-zero along the symmetry axis, which

does not change the character of the excitation significantly. We illustrate the shape

of the anomalous modes of the families (2, 0) and (3, 0) in figure 3.2. The anomalous

6 4 2 0 2 4 66

4

2

0

2

4

6

+

(a)

Family (2,0)

xposition

z

pos

ition

6 4 2 0 2 4 6

+

+

(b)

Family (3,0)

xposition

Figure 3.2: Contour plots in the x-z plane of the amplitude u(, z) of the anomalous firstmembers of family 2 and 3 (m = 0).

member of family (1, 0) is the ground state, which is a solution of the BdG equations

[100].

The case where the radial function has a non-zero number of nodes (i.e. nr = 0)can now be easily visualized. The principal family number f determines the number of

angular nodal surfaces (f 1) between 0 < < , while nr determines the number ofradial nodal surfaces, which intersect the angular nodal surfaces. In the isotropic case

they are spherical and centered on the origin. In figure 3.3 we illustrate the first two

modes having one node in the radial function, which both belong to the family (1 , 1).

The mode in figure 3.3 (a) is the anomalous first member of this family (m = 0), while

all other modes of this family have the general shape shown in figure 3.3 (b), which

can be recognized as the same shape as in figure 3.1 (a), but with one radial node.

We stress that all members of the same family, apart from the anomalous one, have

the same general shape, i.e. the same number of peaks (in the contour plot) in similar

spatial distribution. The main qualitative difference between modes of the same family

is that the peaks move radially outwards and become narrower in both radial and

azimuthal direction with increasing eigenfrequency. This is illustrated in figure 3.4 for

the so-called surface modes (nr = 0) of a condensate.

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5 05

5

0

5

0

0.1

0.2

Family (1,1)

xposition

zposition

Amp

litude

u(,z

)

(a)

5 05

5

0

5

0.05

0

0.05

0.1

0.15

Family (1,1)

xposition

zposition

Amp

litude

u(,z

)

(b)

Figure 3.3: Family 1 with one radial node (nr = 1). (a) Anomalous first member (l = 0,m = 0), (b) general shape (here l = 1, m = 1).

0 2 4 6 8 100.05

0

0.05

0.1

0.15

0.2

0.25

position

radialu

Family 1

Figure 3.4: Quasiparticle amplitude u along the -axis for family 1 modes with nr = 0. Thecurves from left to right correspond to (l, m) = (1, 1), (2, 2), . . . , (5, 5) respec-tively. The non-linearity is C = 332.

To illustrate the mode classification by family and m value, we list in table 3.1 the

18 lowest lying modes for the C = 332 case in an isotropic harmonic trap.

3.2.3 Anisotropic case

The main value of the concept of families is in its extension to the anisotropic cylin-

drically symmetric case. Hutchinson and Zaremba identified the first four families by

the dependence of the eigenvalue on trap anisotropy. Here, we show that the mode

topology determines the family.

In figure 3.5 we have plotted the quasiparticle wave functions for three families: the

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Mode Family Mode Familyl m f nr l m f nr

0 0 0.000 1 0 0 0 2.193 1 11 0 1.000 2 0 4 0 2.660 5 0

1 1 0 1 4 02 0 1.526 3 0 2 3 0

1 2 0 3 2 02 1 0 4 1 0

3 0 2.065 4 0 1 0 2.872 2 11 3 0 1 1 12 2 03 1 0

Table 3.1: Lowest quasiparticle modes of a condensate in an isotropic trap for C = 332,listed by family.

anomalous member of the (3, 0) family, and the (2, 1) and (3, 1) families, for the case

of prolate, spherical and oblate traps. These graphs ill

Andreas Penckwitt- Rotating Bose–Einstein Condensates: Vortex Lattices and Excitations

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