The Preparation of Topological Modes in a
Strongly-Coupled Two-Component
Bose-Einstein Condensate
byJames Edgar Williams
B.S., University of Iowa, 1994

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
1999This thesis entitled:
The Preparation of Topological Modes in a Strongly-Coupled Two-Component
Bose-Einstein Condensate
written by James Edgar Williams
has been approved for the Department of Physics

Prof. Murray Holland

Prof. John Cooper

Date

The final copy of this thesis has been examined by the signatories, and we find that boththe content and the form meet acceptable presentation standards of scholarly work in theabove mentioned discipline.

iii

Williams, James Edgar (Ph.D., Physics)
The Preparation of Topological Modes in a Strongly-Coupled Two-Component Bose-Einstein
Condensate
Thesis directed by Prof. Murray Holland

In this thesis, we present a detailed theoretical study of a coupled two-component BoseEinstein condensate in a magnetic trap. We first present a quantum kinetic theory describing the Bose-condensed gas, that applies to general finite-temperature and nonequilibrium
situations. We then treat the coupled, two-component condensate at zero-temperature bysolving the Gross-Pitaevskii equation, in which the fluctuations are neglected. We showthat in the weak-coupling limit, the system behaves like a nonlinear Josephson-junction,
analogous to two condensates in a double-well potential that are coupled due to quantum
tunneling. In the opposite limit of strong coupling between internal states, we show thatthe condensate can be prepared in a variety of new topological states. In particular, wepredict a scheme for generating a quantized vortex in this two-component system, where
one component sits in the center with a uniform phase while the other circulates around it.
Subsequent related experimental work at JILA by the group of Eric Cornell and Carl Wieman has demonstrated these predictions in the laboratory —this is the first observation of avortex in a dilute-gas Bose-Einstein condensate. Finally, we study the kinetic evolution ofa single-component gas above the critical temperature by solving the Boltzmann equationand investigate the possibility of achieving a steady-state condensation, which can occur if
atoms are injected into the trap during evaporative cooling.

v

Acknowledgements
This thesis work is really the culmination of many years of learning, and I speak not only of
my studies here in graduate school at the University of Colorado, but also of my undergraduate studies at the University of Iowa and my time at Shenandoah High School. Looking
back over the years, I recall with heartfelt gratitude the many people who have inspired andencouraged me along my journey that began in a rural town in Iowa. I am very fortunate tohave been guided along the way by teachers, family, friends, and colleagues, who have all
played a role in helping me to realize my potential. As I close this chapter of my life andturn the page to the next, it is important for me to express my sincere appreciation of theirsupport.
I have been very lucky to have not only one, but two amazing advisors. It has beena great pleasure working with Murray Holland and Jinx Cooper, who have given me theopportunity to work on such an exciting and rewarding project; I deeply appreciate theirconstant guidance and encouragement throughout my research. Their thoughtful choices ofprojects have allowed me to learn many different aspects of BEC and to establish a broad
knowledge base upon which to build further understanding. Furthermore, they have givenme the freedom to explore each problem and to pose new questions myself, so that I could
develop the creative problem-solving skills needed to do original research. For this I am
forever grateful. From them I have also developed an appreciation for the need to link
theory to experiment, and the question “What’s the observable?” will forever be imprinted
on my psyche. I am also glad to have developed a friendship with both Murray and Jinx
during my time at JILA, which has made my experience here all the more memorable. I
look forward to new collaborations with them in the future.
I am also grateful for the opportunity to have collaborated with Eric Cornell and Carl
Wieman, which has made my research experience at JILA immensely more meaningful.
I thank them for inviting me to their weekly experimental meetings, where I gained an
appreciation for their craft. It was Eric who provided the impetus for our theoretical investigations of the coupled, two-component condensate and gave us continual feedback, which
helped us to stay closely tied to the experiments. I feel very lucky to have been a part of theexciting discoveries leading up to the observation of a vortex in the two-component system
and it is an experience I will never forget (for such an opportunity is certainly rare). I havealso benefitted from discussions with Carl, who has always taken the time to understandour theoretical work. It was he who planted the seed for our study of steady-state BEC, anidea I believe that will someday be realized in the laboratory.
JILA has been such a stimulating environment to do research, for I have benefitted
greatly from my frequent interactions with professors, post-docs, and fellow graduate students, who are all part of a fertile scientific community here. I would especially like toacknowledge the generosity of Reinhold Walser in our theory group, who is one of themost helpful persons I have ever met. I have thoroughly enjoyed our frequent discussions
about physics and life in general. I am indebted to him for his work on quantum kinetic theory, without which Chapter 2 of this thesis could not have been written. I have also enjoyedinteracting with Jae Park, whose curiosity and questions (that I can rarely answer) always
make me realize that I still have much to learn. I also want to express my deepest gratitude
to the BEC experimentalists, especially Mike Matthews, Brian Anderson, Paul Haljan, andDavid Hall, who have played a major role in the development of our work on the coupled
two-component condensate. Their expertise in the lab has been a real source of inspiration
and I have enjoyed learning from them. I also thank them for sharing experimental datawith me. I have also benefitted from discussions with Chris Greene, Brett Esry, Jim Burke,
and John Bohn over the years.
An important part of a graduate student’s education is traveling and interacting with fellow colleagues from other institutions. During my time at JILA, I have had the chance tomeet many scientists from around the world, who make up part of an international BEC
community. I have especially enjoyed my interactions with Allan Griffin and Eugene
Zaremba, who were visiting fellows at JILA this year. I was also very fortunate to havebeen given the opportunity to travel to Europe on two occasions, and I owe Murray mygratitude for sending me. In the summer of 1997 I spent two months at the Institute for
Theoretical Physics in Innsbruck with the group of Peter Zoller and Ignacio Cirac. It wasa memorable experience for me and I sincerely appreciate Peter’s gracious hospitality. In
the summer of 1998, I traveled to Italy to attend the Enrico Fermi summer school on BEC,
which was an extremely inspiring learning experience. That trip would not have been possible without the help of Eric and Jinx. I especially would like to thank Marilu’ Chiofalo
for her invitation for me to speak at SNS in Pisa during my visit to Italy. I will not forget theweek I spent there and the genuine hospitality of Marilu’ and Massimilliano. I have also
enjoyed the several visits Marilu’ made to JILA, for we had many stimulating discussions
on BEC while she was here.
I would also like to thank the staff at JILA for their assistance over the years, especially
Pat Mcinerny for providing me with a private office to write my thesis. I don’t think I
would have ever finished on time otherwise. I also would like to thank Linda Frueh andSusan Thompson in the physics office, who have always been very helpful.
I know for sure that my success in graduate school would not have been possible without the preparation I received at the University of Iowa as an undergraduate. I especiallywould like to thank Bill Klink, who deeply inspired me to take my studies of physics to ahigher level. I still have my lecture notes from quantum mechanics and a lingering affinity
for group theory that he instilled in me. I also would like to thank Paul Kleiber for givingme my first taste of atomic physics and Karl Lonngren for letting me do experiments in his
lab on ion-acoustic waves in a plasma, which I will always remember.
I also can not forget my two favorite high school teachers, who were the first teachers
to really inspire me over a decade ago. I am grateful to my science teacher, Larry Oiler,
whose passion for science was contagious. It was in his custom-made course “Science andSociety” that I got my first taste of the wonderment of discovery. I especially want to thankmy art teacher, Julie Gee, for always believing in me and encouraging me. She has always
been a real inspiration to me and a true friend.
I owe my deepest, most solemn thanks to my parents, who have always supported mewithout question and who brought me into a loving and caring family. All of the educationin the world can not teach me what they have mastered, and what I will always struggle toachieve: kindness and generosity toward other people. I also thank my two older brothers,viiiJohn and Michael, for always being there for me and for encouraging me to go all the way.
I thank my entire family, including my sister-in-law, Heidi, and my little niece Lexi, for
putting up with my neglect over the last five years with my head in my books.
Finally, I would like to express my sincere thanks to my best friend, Tanya Ramond,
who has been my companion throughout graduate school. She is the most giving and caring
person I have ever met. She has always been there for me when I needed encouragement
and support to get through the tough times. She has also put up with my increasing selfabsorption with my studies and has always listened to me ramble on excitedly about myresearch. Her courage and perseverance through some of the most challenging times in
graduate school are an inspiration to me and I look forward to watching her continue togrow, both as a physicist and a person.

ix

Contents
1

Introduction

1

2

Underlying Theory
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Conceptual ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Separation of timescales . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Reduced description . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Master variables f q (t)g . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Single-particle basis . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Derivation of the quantum kinetic equations . . . . . . . . . . . . . . . . .
2.3.1 Second quantized description . . . . . . . . . . . . . . . . . . . .
2.3.2 Statistical description . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Coarse-grained density operator . . . . . . . . . . . . . . . . . . .
2.3.4 Mean-field shifted energies . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Interaction representation of the integral equation . . . . . . . . . .
2.3.6 Generalized quantum kinetic equations . . . . . . . . . . . . . . .
2.3.7 Kinetic equations for mean fields and fluctuations . . . . . . . . . .
2.4 Comparison to previous work . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Low temperature limit . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 High temperature limit . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
7
8
8
9
11
12
13
14
16
17
19
19
20
24
27
27
32
33

3

Mean-Field Theory of a Single Component
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Ground state properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Thomas-Fermi limit . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Lower-dimensional modeling . . . . . . . . . . . . . . . . . . . .
3.3.3 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Non-ground-state condensates . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Vortex solution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Non-rotating states . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Condensate response to a mechanical drive . . . . . . . . . . . . . . . . .
3.5.1 Linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Excitation of topological modes . . . . . . . . . . . . . . . . . . .
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
35
35
36
37
38
39
42
42
44
46
47
48
49

CONTENTS
4

Mean-Field Theory of a Coupled Two-Component System
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The physical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Derivation of the coupled two-component GP equation . . . . . . . . . . .
4.4 Uncoupled two-component system . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Ground state properties . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Non-ground-state two-component condensates . . . . . . . . . . .
4.5 Coupled two-component system . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 External internal representation . . . . . . . . . . . . . . . . . .
4.5.2 Dressed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51
51
51
53
54
55
58
62
62
63
66

5

Weak Coupling—Nonlinear Josephson-Type Oscillations
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The standard Josephson junction . . . . . . . . . . . . . . . . . . . . . . .
5.3 Nonlinear Josephson-like equations . . . . . . . . . . . . . . . . . . . . .
5.4 Hydrodynamic-like Bloch equations . . . . . . . . . . . . . . . . . . . . .
5.5 Preparation of the initial state . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 One-dimensional model . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Comparison to the three-dimensional case . . . . . . . . . . . . . .
5.7 Closed-form solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67
67
67
69
70
71
72
73
74
76
79

6

Strong Coupling—Quantum State Engineering81
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Derivation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.1 Simplifying the mean-field terms . . . . . . . . . . . . . . . . . . 88
6.2.2 Interaction picture representation . . . . . . . . . . . . . . . . . . 89
6.2.3 Coarse graining approximation . . . . . . . . . . . . . . . . . . . . 90
6.3 Static drive due to offset traps . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3.1 Two-mode model . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3.2 Results of calculations . . . . . . . . . . . . . . . . . . . . . . . . 94
6.4 Harmonic drive: A refined state preparation scheme . . . . . . . . . . . . . 101
6.4.1 Two-mode model . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4.2 Results of calculations . . . . . . . . . . . . . . . . . . . . . . . . 104
6.5 Experimental observation of a vortex . . . . . . . . . . . . . . . . . . . . . 108
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7

Trajectory Simulation of the Quantum Boltzmann Equation
113
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2 Ergodic quantum Boltzmann equation . . . . . . . . . . . . . . . . . . . . 114
7.2.1 Quantum Boltzmann equation . . . . . . . . . . . . . . . . . . . . 114
7.2.2 Ergodic assumption . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2.3 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3 Trajectory simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3.1 Trajectory decomposition of the QBE . . . . . . . . . . . . . . . . 118
7.3.2 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . 120

xii

CONTENTS
7.4

7.5
8

Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.4.1 Finite number effects on equilibrium . . . . . . . . . . . . . . . . . 122
7.4.2 Dynamic buildup of the condensate . . . . . . . . . . . . . . . . . 123
7.4.3 Evaporative cooling simulation . . . . . . . . . . . . . . . . . . . . 126
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Achieving Steady-State Bose-Einstein Condensation
129
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.2.1 Description of the loading procedure . . . . . . . . . . . . . . . . . 130
8.2.2 Description of evaporative cooling . . . . . . . . . . . . . . . . . . 132
8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.3.1 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.3.2 Steady state solution . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A Wick’s Theorem

143

B Collisional Terms

145

C Numerical solution of the GP equation
149
C.1 Time evolution using the Crank-Nicholson method . . . . . . . . . . . . . 149
C.1.1 One spatial dimension . . . . . . . . . . . . . . . . . . . . . . . . 149
C.1.2 Two spatial dimensions—the ADI method . . . . . . . . . . . . . . 151
C.1.3 Treating the internal-state dynamics . . . . . . . . . . . . . . . . . 153
C.1.4 Diagnostics of numerical stability . . . . . . . . . . . . . . . . . . 154
C.2 Stationary state solutions using the ITP method . . . . . . . . . . . . . . . 155
C.2.1 Ground state solution . . . . . . . . . . . . . . . . . . . . . . . . . 155
C.2.2 Solution of topological modes . . . . . . . . . . . . . . . . . . . . 156

xiii

List of Figures
1.1

Flow chart of the chapters in the thesis. . . . . . . . . . . . . . . . . . . . .

2.1

One dimensional illustration of the kinetic stage of evolution for a trapped
gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Potentials for the isotropic harmonic oscillator. . . . . . . . . . . . . . . . 13

2.2
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
4.1
4.2
4.3
4.4
4.5

4.6
4.7

4.8
4.9

The density of the condensate ground state in the one-dimensional model
for increasing population. . . . . . . . . . . . . . . . . . . . . . . . . . . .
The density of the condensate assuming an axially symmetric trap. . . . . .
The column-integrated density profile for the solution of the condensate
ground state shown in Figure 3.2. . . . . . . . . . . . . . . . . . . . . . . .
A vortex solution to the two-dimensional GP equation, for the case of unit
angular momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The 2 phase wrap of the vortex solution displayed in Figure 4. . . . . . .
The approximate value of the critical frequency c . . . . . . . . . . . . . .
The densities of the first three excited states. . . . . . . . . . . . . . . . . .
The energies of the states plotted in Figure 3.7 vs the population. . . . . . .

4

39
41
41
43
43
45
45
46

Illustration of the ground-state hyperfine levels of 87 Rb. . . . . . . . . . . . 52
The density of the ground state solution for the two component system forzero trap displacement z0 = 0. . . . . . . . . . . . . . . . . . . . . . . . . 56
The ground state solution of the two-component system as a function oftrap separation z0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
The solution with the trap separation fixed at z0 = 0:15 while varying the
population difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
The wavefunction overlap  of the two components in the ground state (topgraph) and the difference in chemical potentials, vs the population fractionin the j1i state N1 =N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
The column-integrated densities of each component for the full three-dimensional
ground state solution of the two-component system. . . . . . . . . . . . . . 59
The solution to the time-independent GP equation for the case where oneof the components is in a vortex with unit angular momentum per atom,
while the other component is in the nodeless ground state. . . . . . . . . . 60
The chemical potentials of each component as a function of the population
fraction in the ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
The solution of the case where one component is in the ground state, while
the other has a dipole symmetry. . . . . . . . . . . . . . . . . . . . . . . . 61

LIST OF FIGURES
4.10 The chemical potentials of the two components vs. the population fraction
N0 =N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.11 The condensate lower dressed state amplitudes in the weak and strong coupling limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9

Illustration of the Josephson junction. . . . . . . . . . . . . . . . . . . . .
An example of using adiabatic passage to prepare the initial state for theJosephson-like equations. . . . . . . . . . . . . . . . . . . . . . . . . . . .
The time evolution of the relative population  (t) for several different cases.
A snapshot evolution of the densities of the components. . . . . . . . . . .
Time evolution of  (t) for additional cases. . . . . . . . . . . . . . . . . .
A comparison between the exact solutions of Eq. (4.11) for the one dimensional model and the full three dimensional problem. . . . . . . . . . . . .
The period of oscillations as a function of the initial relative phase (0)using expression Eq. (5.19). . . . . . . . . . . . . . . . . . . . . . . . . .
The amplitude of oscillation as a function of the initial relative phase (0). .
Amplitude of the frequency components obtained from a discrete FFT of
(t) for two different cases of (0) = =2 and =16. . . . . . . . . . . . .

68
72
73
74
75
75
78
78
79

6.1
6.2
6.3

Modulation of the fractional population in the (1,-1) state. . . . . . . . . . . 83
The revival time vs. the detuning. . . . . . . . . . . . . . . . . . . . . . . 84
Double-peaked structures in the column-integrated density of the (2; 1)
atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.4 Dipole topological mode for a two-component condensate. . . . . . . . . . 86
6.5 Modulation of the Rabi oscillations for the one dimensional model. . . . . . 94
6.6 Snapshots of the density of each state for three different Rabi cycles corresponding to the three circled numbers in Figure 6.5. . . . . . . . . . . . . . 95
6.7 The fractional population of the j1i state in the rotating frame. . . . . . . . 96
6.8 Ground and dipole states. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.9 The fractional population in the j1i state for four different values of thedetuning, obtained from a numerical solution of Eq (6.10). . . . . . . . . . 98
6.10 The period of modulation as a function of detuning Æ 0 . . . . . . . . . . . . 99
6.11 The fractional population in the j1i state for four different values of the trap
displacement z0 , obtained from a numerical solution of Eq (6.10). . . . . . 100
6.12 State preparation scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.13 Dynamical evolution to a vortex. . . . . . . . . . . . . . . . . . . . . . . . 105
6.14 Densities and phases of the two states j1i and j2i, at time t = 200ms. . . . 106
6.15 Double and triple vortex preparation. . . . . . . . . . . . . . . . . . . . . . 107
6.16 Dipole and quadrupole preparation. . . . . . . . . . . . . . . . . . . . . . 108
6.17 Experimental production of a vortex. . . . . . . . . . . . . . . . . . . . . . 109
6.18 Snapshot of a vortex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.19 Interference between the ground state and vortex. . . . . . . . . . . . . . . 111
7.1
7.2
7.3
7.4
xvi

Plot of g (emin ; em ; eq ; ep ) vs. (eq ; ep ). . . . . . . . . . . . . . . . . . . . . 117
An illustration of the distribution of populations fe over the discrete andcontinuous regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Evolution to equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Ground state fraction versus temperature for 500 atoms. . . . . . . . . . . . 123

LIST OF FIGURES
7.5
7.6
7.7
7.8
7.9
8.1
8.2
8.3

8.4
8.5
8.6
8.7

8.8

Same procedure as in Figure 7.4, but with 2  104 atoms. . . . . . . . . . . 124
Mean energy versus temperature for 500 atoms. . . . . . . . . . . . . . . . 124
Same procedure as in Figure 7.6, but with 2  104 atoms. . . . . . . . . . . 125
Population in the ground state versus time. . . . . . . . . . . . . . . . . . . 125
The final number in the ground state N0 (tf ) as a function of the cut rate cut .126
Illustration of the transfer process described in section 8.2.1. . . . . . . . . 131
Time evolution of the total number N (t) and total energy E (t) for thevalues of the parameters listed in Table 8.1. . . . . . . . . . . . . . . . . . 135
Two overlaying contours of the steady state value of the phase space density
and the total number vs. the ratio of trap frequencies and RF cut thresholdfor an adiabatic transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Same as described in the caption of Figure 8.3 except for a sudden transfer
of atoms from the MOT to the magnetic trap, instead of an adiabatic one. . 137
Values of T1 and !2 one must achieve in order to reach 0 = 2:612 in thecase of an adiabatic transfer. . . . . . . . . . . . . . . . . . . . . . . . . . 138
Same as described in the caption of Figure 8.5 but for the case of a suddentransfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Corresponds to the three lines in both Figure 8.5 and Figure 8.6, showing
the total number of atoms in steady state Nss as a function of the transfer
rate t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Curves correspond to those in Figures 8.5-8.7, showing the ratio of the RF
cut to the temperature of atoms injected into the trap, ecut =T2 as a functionof the transfer rate t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

xvii

List of Tables
6.1

Values used for the various physical parameters appearing in our calculations.100

8.1

Values used for the various physical parameters needed in the model. . . . . 134

Chapter 1

Introduction
Experimental observation of Bose-Einstein condensation
The first experimental observation of Bose-Einstein condensation (BEC) in a dilute atomicgas occurred in the summer of 1995 at the University of Colorado by a team of scientists
at JILA, led by Professors Carl Wieman and Eric Cornell [1], and soon after by the groupof Professor Wolfgang Ketterle [2] at the Massachusetts Institute of Technology (MIT). Ina short span of four years, the number of experimental groups that have observed BEC
has increased from two in the United States to over twenty worldwide [3]. Because theseatomic systems are relatively easy to probe and manipulate in the laboratory, a wealth ofexperimental data is becoming available for comparison to theory and many new questions
are being asked that are pushing the theory of low-temperature, many-particle physics inexciting new directions. This cutting-edge research has opened up a thriving new field ofphysics, as indicated by the assignment of its own section, entitled Matter Waves, in thejournal Physical Review A. It has caught the interest of some of the most talented scientists
from across a broad range of fields, such as atomic and molecular physics, quantum optics,
and condensed matter physics.
This experimental breakthrough has had such a tremendous impact on low-temperaturephysics because these dilute atomic gases offer a long list of new properties that makethem very attractive to study in the lab and extremely interesting to consider theoretically.
The ability to control and probe these atomic systems with electromagnetic radiation has
played a key role throughout this exciting development. The internal structure of the alkali
atoms used in these experiments is very well known and a large array of transitions ranging
from optical to radio frequency can be accessed with inexpensive lasers and synthesizers.
Innovative new technologies, such as laser cooling and trapping, are based on a detailed
understanding of atom-light interaction and have provided physicists with the tools needed
to cool the atomic gas to the ultra-low temperatures required to reach BEC [4, 5, 6]. Laserlight is also used to image the gas sample to obtain either the density or the momentum
distribution of the gas, from which many useful properties of the condensate can be ascertained. The condensate can also be probed by varying the magnetic fields of the confining
potential in a controlled manner in order to investigate the collective modes of the system,
for example.
With so many different ways to probe and excite the condensate, it is not surprising thatmany new properties are being explored that had never been considered previously in thevast literature on Bose-condensed systems. One such novel system is that of a driven, two-

Chapter 1 Introduction
component condensate, where an external electromagnetic field is applied that coherently
couples two internal states of the atoms. Recent experiments at JILA by the group of
Eric Cornell and Carl Wieman [7, 8, 9, 10, 11, 12, 13, 14] have explored many interesting
properties of this system; this experimental work provides the inspiration for several related
projects presented in Chapters 4 through 6 of this thesis. As a result of this productive
experiment-theory collaboration at JILA, a surprising new development has occurred thatis pushing our understanding of BEC in an exciting new direction: we have discovered
a way to coherently transfer atoms in the condensate from the usual mean-field groundstate of the trap to a variety of different macroscopic excited states, such as a vortex. This
technique exploits a rather non-intuitive effect arising from the subtle interplay between theinternal and motional dynamics of the condensate, which we describe in detail in Chapter 6.
This collaborative effort has lead to the first experimental observation of a vortex in a diluteBose-condensed gas [14].
What is Bose-Einstein condensation?
In order to appreciate the remarkable new developments in this growing field, one should
first understand the basic properties of BEC in a dilute atomic gas. The most striking feature of BEC is that the wave-like behavior of matter predicted by quantum mechanics is
exhibited on a macroscopic scale due to the condensation of, typically, millions of identicalatoms into the same quantum state. Just as a classical electromagnetic field is described byan amplitude and a phase, so too is this quantum gas. One of the most visually stunningobservations of quantum coherence on a macroscopic scale was made by allowing two initially isolated condensates to overlap; the resulting density pattern displayed interference
fringes, analogous to the interference between coherent light waves emanating from a double slit [15]. Another intriguing property of Bose-condensed systems is the unrestrictedflow of particles in the sample, such as persistent currents in super-fluid helium that flow
without observable viscosity, and electric currents in superconductors that flow withoutobservable resistance. These “super” properties of Bose-condensed systems occur because
the macroscopic occupation of a quantized mode, such as a vortex, can provide a stabilizing
mechanism that inhibits decay due to thermal relaxation [16]. The recent observation of avortex in the dilute Bose-condensed gas has opened the door to the study of super-fluidity
in these systems [14].
The idea of a macroscopic number of identical particles acting in unison to exhibit
wave-like behavior is counter-intuitive to our daily experience of the world, where objects
are distinguishable and behave like particles that follow classical trajectories described by
Newton’s laws of motion. In the BEC experiments, this classical description of the system
is initially valid when the atomic sample is loaded into a vacuum cell and laser cooling
begins, followed by evaporative cooling in a second stage of the experiment. Imagine thisgas of particles in the container at room temperature. Based on our intuition, we can labela particle and track its motion in the gas: it travels the mean free path l of the gas before
passing by another particle within some interaction radius a. This results in a momentumchanging collision, and then it continues until the next collision and so on. This evolutionof the gas is given by the classical Boltzmann kinetic equation, which predicts that inequilibrium the system is described by the Maxwell-Boltzmann distribution.
Quantum mechanics, however, predicts that as the temperature of the gas approaches
some critical value Tc , the wave-like nature of the particles becomes important to consider.
As the gas is cooled down, the particles become smeared out and each one can best be2

described as a wave-packet, with a characteristic wavelength dB = (2 
h2 =mkB T )1=2,
called the thermal de Broglie wavelength, where m is the mass of the atom and T is thetemperature. As the temperature is lowered close to Tc , the effect of smearing the positionof a single particle becomes so enhanced that the de Broglie wavelength becomes comparable to the mean-free path, dB  l and the wave-packets begin to overlap. At thispoint it becomes impossible to distinguish one particle from another. Due to this indistinguishability, the particles become correlated in a particular way, depending on whether
they are bosons or fermions. For bosons, this correlation causes all of the particles to “condense” into a single wave-packet with the longest wavelength permitted by the size of thebox. Since photons are bosons, they too exhibit this phase transition, which takes the formof the ubiquitous laser, which produces a phase-coherent beam of light. One obvious application of BEC, then, would be to produce a phase-coherent beam of atoms—an atom
laser [17, 18, 19].
Of course this description is overly simplified. In fact, one has to incorporate both ofthe particle and wave concepts described above into a proper treatment of BEC. In otherwords, one must unify the notion of collision-induced fluctuations with the idea of having a single coherent wave packet. If the interactions between particles in the system arevery strong, then the strong correlations due to collisions will dominate the system andobscure the coherence properties of the condensate, which typically had been the situation for Bose-condensed systems studied in condensed matter physics, such as super-fluid
helium. In sharp contrast, interactions between particles in a dilute atomic gas have aweak effect, so that the wave-like condensate dominates the system and collisions can betreated perturbatively. In this case, the interaction of all of the particles on a single particle can be summed to give an averaged effect, as a first approximation. This approach iscalled mean-field theory and with suitable approximations gives rise to the Gross-Pitaevskii
equation that describes the time-evolution of the condensate, in which the effect of interactions gives rise to a density dependent effective potential that makes the dynamics ofthe condensate nonlinear. This simple description does not include the fluctuations due
to collisions, but just treats their averaged effect. This approximation is appropriate in the
T = 0 limit, which we consider in Chapter 3 for a single component BEC. How to properly
treat the collisional fluctuations in these dilute systems is an active area of research in thisfield [20, 21, 22, 23, 24, 25, 26, 27]. In Chapter 2 we will present our own version of theunderlying theory for this system that describes the condensate and fluctuations about it
and includes the dissipative effect of collisions.
Overview of the thesis
This thesis is divided into three main parts, as illustrated schematically in the flow chart
shown in Figure 1.1. In Chapter 2 we present a quantum kinetic theory describing the dilute Bose-condensed gas at finite temperature. The results of this theory extend the standard
treatment of the system by including the dissipative effect of collisions. The main result of
the chapter is the closed, coupled set of kinetic equations Eq. (2.70) to Eq. (2.72) describing the mean-field, the normal fluctuations, and the anomalous fluctuations, respectively.
By dropping the collision terms from the kinetic equations in section 2.4.1, we recover theusual Hartree-Fock-Bogoliubov result, and if we make the further simplification of neglecting the fluctuations altogether, we recover the Gross-Pitaevskii (GP) equation. This will be
the starting point for Chapter 3. We also consider the high temperature limit in section 2.4.2
by completely neglecting the mean-field and anomalous fluctuations. There we derive the3Chapter 1 Introduction

Ch. 2 Underlying Theory

T=0

Ch. 3 Single Component

T>T c

Ch. 7 Quantum Boltzmann Equation

Ch. 4 Two Components

Ch. 5 Weak Coupling

Ch. 8 Steady-State BEC

Ch. 6 Strong Coupling

Figure 1.1: Flow chart of the chapters in the thesis. The structure of the thesis consists ofthree main parts: the underlying theory of BEC, the coupled two-component condensate at
T = 0, and solutions of the kinetic equation at T > Tc.
quantum Boltzmann equation, which is the starting point for Chapter 7.
The first generation of experiments in BEC, and correspondingly most theoretical investigations, dealt with a single-component condensate, where only one atomic species ispresent. In Chapter 3 we consider this basic system and outline some of the main concepts, such as the condensate ground state, topological modes, and elementary excitations,in order to get an overview of zero temperature mean-field theory for a single-component
system. In these systems, condensation occurs into the lowest-energy state of the trap.
In the absence of interactions, this would just be the Gaussian-shaped ground state of theharmonic trapping potential. However, the condensate ground state is actually much bigger than this and is shaped like an inverted parabola due to the effect of interactions. In
section 3.3 we show results of numerical calculations of the condensate ground state anddiscuss some of its basic properties.
A great deal of theoretical work has appeared in the literature over the last few years
investigating vortices in a single-component dilute-gas BEC. One proposed scheme forpreparing the condensate in a vortex mode is to distort the confining potential and mechanically rotate the trap during the cooling process. In this way, the lowest energy mode may beengineered to be circulating about the axis of symmetry. Such an approach is in direct analogy with experiments on vortices in super-fluid helium—the asymmetry of the harmonic
trap for the atomic gas plays the role of surface roughness of a rotating vessel. Althoughconceptually this method appears promising for vortex generation in a trapped gas, so far
technical difficulties have precluded its successful implementation. In section 3.4 we outline some of the basic properties of vortices and we also consider other non-ground-state
condensates, such as a dipole-mode; we refer to these macroscopic excited states as topological modes1 in order to distinguish them from the concept of an elementary excitation
1 This is somewhat of a misnomer, for we apply “topological” not only to vortex solutions, but also to noncir-

4

or linearized collective mode, which we discuss briefly in section 3.5. In an elementary
excitation, a small fraction of atoms are excited out of the condensate on average; the linear response of the condensate to a weak mechanical drive can be expressed in terms ofelementary excitations, for example. In contrast, in a topological mode a large fraction of
the atoms occupy the same excited state.
One of the most interesting new developments in this field is the creation of multicomponent condensates, where different atomic species are Bose-condensed in the same
system. The earliest work in this area occurred at JILA in 1996, where two different hyperfine internal states were populated in the condensate [7]. Later work at MIT studied atoms
trapped in an optical potential, where three different Zeeman sub-levels were populated
in the condensate [28]. These initial experiments on multi-component condensates, alongwith many more that followed, have spawned a wealth of theoretical studies on these spinor
condensates. Early work on these systems was concerned with the ground state properties
and elementary excitations of multiple components trapped together [29, 30, 31, 32].
A natural extension of the work on mixtures of different internal states in a condensate
is to consider the application of an external electromagnetic field that couples different
internal states of the atoms in the condensate. Most of the recent BEC experiments deal
with this basic system in some form or another. A useful tool for coupling atoms out of thecondensate, which is a basic ingredient of an atom laser, is to apply an external field to thecondensate that drives transitions to an untrapped internal level [17, 18, 19]. Alternatively,
one can apply an external field to couple internal states that are both trapped. A series ofexperiments at JILA have explored this situation [8, 9, 10, 11].
In Chapters 4 to 6 we investigate a driven two-component condensate, where two internal hyperfine states are trapped, though each state experiences a slightly different confining
potential. In Chapter 4, we derive the coupled two-component Gross-Pitaevskii equation
Eq. (4.11), which is a key result for the studies presented in Chapters 4 to 6. Our main focus in Chapter 4 is on the stationary solutions of the two-component system, both with andwithout coupling. For the case of no coupling, we calculate ground states, and topologicalmodes, the results of which are needed in Chapter 5 and Chapter 6. We also consider thelowest-energy stationary solution of the coupled system, which we call the dressed states.
In Chapter 5 we consider the case of weak coupling between internal states that are indisplaced traps, so that the two components are separated along the vertical axis with a finite
overlap. This system resembles a different system considered recently in the BEC literature: two condensates in a double well that can tunnel through the central barrier so that thetwo condensates are weakly coupled, as in the standard Josephson junction in condensed
matter physics. Due to the mean-field interaction, the corresponding Josephson-junction
equations are nonlinear. We show results of numerical calculations, and give a closed-form
approximate solution as well. In Chapter 6 we treat the opposite limit of strong coupling
between internal states and show that the condensate can be prepared in a variety of new
topological states. In particular, we predict a scheme for generating a quantized vortex
in this two-component system, where one component sits in the center with a uniform
phase while the other circulates around it. Subsequent related experimental work at JILA
by the group of Eric Cornell and Carl Wieman has demonstrated these predictions in thelaboratory—this is the first observation of a vortex in a dilute-gas Bose-Einstein condensate [14].
In Chapter 7 we shift our attention to the high temperature limit and study the quantum
culating modes, such as a dipole or quadrupole.

5Chapter 1 Introduction
Boltzmann equation, which was derived in section 2.4.2. We first derive the quantum
version of the ergodic Boltzmann equation by assuming detailed balance in each degenerate
subspace. This results in a distribution function that depends only on energy. We thenpresent a simulation procedure for numerically solving the quantum Boltzmann equation,
which is based on the quantum trajectory methods first developed in quantum optics tosimulate master equations. We then present explicit calculations of finite number effects
on equilibrium, the dynamic build-up of the ground state, and simulations of evaporative
cooling.
In Chapter 8 we investigate the possibility of obtaining Bose-Einstein condensation
in a steady state by continuously loading atoms into a magnetic trap while keeping thefrequency of the RF field fixed. A steady state is obtained when the gain of atoms due
to loading is balanced with the three dominant loss mechanisms due to: elastic collisions
with hot atoms from the background gas, inelastic 3-body collisions, and evaporation. We
describe our model of this system and present results of calculations of the peak phasespace density 0 in order to investigate the conditions under which one can reach the regime
0  2:612 and attain BEC in steady state.

6Chapter 2

Underlying Theory
2.1 Introduction
In this thesis we are interested in the particular system of a dilute, finite Bose-Einstein condensed gas of trapped atoms. The theoretical study of this interesting system is motivatedby the recent experimental progress in Bose-Einstein condensation [33, 34]. Conceptually,
this system resides somewhere between the laser, which can be thought of as a steady-state
Bose-condensation of photons into a single cavity mode, and superfluid helium, which is a
Bose-condensed liquid. In these two limiting cases, the constituent particles are either uncorrelated in the case of the laser since practically there is no interaction between photons,
or highly correlated in the case of superfluid helium due to the strong interactions between
particles in such a dense system. The underlying theories of these two limiting cases arewell established, yet the approaches taken toward and the questions asked about each system are very different. It is not surprising, then, that a standard theoretical framework forthe dilute Bose-Einstein condensed gas, which is somewhat of a conceptual hybrid of thesetwo cases, has not yet been established and is currently a very active area of research in thisfield [20, 21, 22, 23, 24, 25, 26, 27]. Such a theory must embrace the notions of quantum
coherence and irreversible dynamics as envisioned in quantum optics, and the idea of thecollective behavior of a system of interacting particles familiar to the condensed matterphysicist.
In this chapter we develop a theory for the dilute Bose-condensed gas that describes
a coherent mean-field and the fluctuations about it. The main result of our theory is a
closed set of coupled equations describing the condensate and the normal and anomalous
fluctuations at finite temperature; these equations go beyond the standard Hartree-FockBogoliubov theories [35, 36] by including second-order collisional terms describing incoherent processes that lead to damping in the system. In the collisionless limit, our equations
go over to the Hartree-Fock-Bogoliubov equations, while in the large temperature limit, werecover the quantum Boltzmann equation for the populations of a thermal cloud. We formulate our kinetic equations for the condensate and fluctuations in a basis-independent,
operator form that allows us to extend the usual treatment of a single component system
to the case where the atoms have internal structure, so that multi-component systems withinternal coupling can be investigated.
We begin the chapter with an overview of our approach by discussing the main ideaof the attenuation of correlations that leads to a reduced description of the system in terms

Chapter 2 Underlying Theoryof a relevant set of observables. This approach is essentially the quantum version of theclassical Chapman-Enskog procedure [37, 38, 39, 40] that has been extended to treat a
Bose-condensed system (see also Kadanoff and Baym [41]). We then derive the kinetic
equations for the mean field and fluctuations in the next section, making a serious effort toclarify the crucial physical assumptions made along the way. After presenting the kinetic
equations in section 2.3.7, we then consider the collisionless and high temperature limitsof our theory.

2.2 Conceptual ingredients
In general, an isolated macroscopic system of interacting particles will evolve irreversibly
to a steady state or equilibrium situation so as to maximize entropy. This is a profoundand universal fact of nature that has intrigued scientists and philosophers throughout thetwentieth century due to the following paradox: one is faced with the task of deriving
irreversible equations of motion for macroscopic observables starting from the reversible
microscopic description. One must inevitably address this question in order to construct
a theory of Bose-Einstein condensation. We take a rather pragmatic approach to solving
this problem by postulating a reduced description of the system, which is essentially acoarse-graining of the quantum Liouville equation [38, 39].

2.2.1 Separation of timescales
In a dilute gas, the interaction radius a given by the scattering length of a two-particle
collision is assumed to be much smaller than the mean distance between particles,

na3  1;

(2.1)

where n is the average density. Based on this inequality, we can identify four distinct
timescales that allow for a simplified description. The shortest timescale is the durationof a collision, 0 = a=v , where v is the average velocity. The time between collisions
c = 1=(na2v) is much longer than 0 in a dilute gas, and gives us the second timescale. In
general, there will be a third timescale h during which the system relaxes to a state of local
equilibrium, which requires multiple collisions per atom. And finally, there is the time eq
it takes the system to relax to global equilibrium. We summarize these timescales as0  c  h  eq :(2.2)

In other words, the system passes through three stages of evolution during its approach toequilibrium.
Let us imagine that we have complete knowledge of the initial state of the system, givenby the N -particle density operator (0). Suppose we allow the dilute gas of interacting
particles to evolve for some time interval t. During the first very short stage of evolution
t  0 , correlations between particles due to collisions have not had time to die out, so if
we want to track the evolution of the system on such a fine timescale, we have to include allof the correlations between particles. This means that we must solve for the full N -bodydensity matrix (t), which is certainly an intractable problem.
The situation becomes much easier if we are not interested in tracking the system on
the very fine time-scale of the duration of a collision 0 . Instead, suppose we consider a82.2 Conceptual ingredientsS

U ( 0) S

t0tc

x (t )
t0t

Figure 2.1: One dimensional illustration of the kinetic stage of evolution for a trapped gas.
A particle oscillates freely in the harmonic trap between the collision events shown in red.
larger time interval 0  t  h . During this time, any pairs of atoms that were initially
undergoing a collision have separated along their asymptotic trajectories, which consist of
free oscillations within the external potential, before experiencing a second collision, as illustrated schematically in Figure 2.1. In other words, the initial correlations between atoms
have died out and correlations between new pairs of atoms have formed. This timescale oft  c is called the kinetic stage of evolution, and the rapid decay of correlations is often
referred to as molecular chaos or the attenuation of correlations.
The third stage of evolution occurs for the time interval c  t  eq and is called
the hydrodynamic stage of evolution, when the system has reached local equilibrium. For
an extremely dilute gas we have h  eq , so that the hydrodynamic stage loses its meaning.
In the opposite limit of a very dense system, for which the time between collisions is nearly
the same as the duration of a collision c  0 , the kinetic stage loses its meaning. This
is often called the hydrodynamic regime and is typically applied to the study of superfluid
helium, for example. We will not consider this case in any detail.

2.2.2 Reduced description
An important consequence of the attenuation of correlations is that higher-order correlations between atoms can be expressed in terms of lower-order correlations, which are described by a physically relevant set of master variables f q (t)g  f 1 (t); 2 (t);   g thatare given by the trace over the N -body density operator

q (t) = Trf(t) ^q g:(2.3)

Here ^q is the corresponding relevant operator. The particular choice of variables depends
on the specific properties of the system, such as how dilute it is and whether it is in thequantum degenerate regime. An example of a relevant observable is the single-atom density
9Chapter 2 Underlying Theoryy ^ g, where a^y and a^ are creation and annihilation operators
matrix fm~ ~n (t) = Trf(t)^a~n a
m
~
m
~
m
~
~ . In a classical system, the analogous
for the single-particle state labeled by the index mquantity is the single-atom phase-space density f ( ; ).
This reduced description is equivalent to coarse-graining the quantum Liouville equation over a time interval 0  t  c . As we will show in more detail below, this leads
to a coarse-grained form f q (t)g of the N -body density operator that does not contain
explicit time dependence but is a functional of the master variables f q (t)g i.e.

xp

(t)  f q (t)g :(2.4)

Due to the attenuation of correlations, the information required to specify the state of thesystem has been greatly reduced to the small set of variables f q (t)g.
We can further simplify the problem by observing that the particular form of f q (t)gis irrelevant, so long as the trace given by Eq. (2.3) gives the correct values of the observables f q (t)g. In fact, there is an infinite set of N -body density matrices that will satisfy
Eq. (2.3) for a particular choice of operators ^q ; we can think of this set as forming an
ensemble. Using arguments familiar from the study of equilibrium statistical mechanics,
we can construct a representative form of f q (t)g that is the most probable in the ensemble
if we maximize the information entropy S (t), with the constraints given by Eq. (2.3). Thatis, we seek an extremum of the functionalS [0 (t)] = Tr 0 (t) ln[0 (t)]Xqq Trf0 (t)^q g;(2.5)

where q are Lagrange multipliers, which are the conjugate thermodynamic coordinatesof the operators ^q . Varying S [ 0 (t)] with respect to  0 and then setting the first variation
to zero, ÆS = 0, we obtain the following reference distribution

f(0)g = exp (^q q ):(2.6)

Here we have assumed the summation over repeated indices and we have simplified thenotation for the index by writing f q (t)g = f g. The exponential form given in Eq. (2.6)
will allow us to simplify averages of products of operators using Wick’s theorem [42, 43].
It is important to realize the difference between the three density operators , f g , andf(0)g we have introduced. The exact state of the N -body system is given by . However,
due to the attenuation of correlations, it is not necessary to track the evolution of the systemwith infinitesimally precise time resolution, so that we can obtain a coarse grained solution
f g that is governed by the set of relevant observables f q (t)g. We do not need to know
the detailed form of f g ; all that is required is for it to be a functional of the observables
and satisfy the self-consistency conditions

q (t) = Trf(t) ^q g = Trff g ^q g:(2.7)

Although it is not necessary to take a particular form for f g , we have constructed a ref-(0)

erence distribution f g for convenience that we obtained by maximizing the information
entropy. The reference distribution must also satisfy the following self-consistency conditions
(0)
(2.8)
q (t) = Trff g ^q g = Trff g ^q g:
10

2.2 Conceptual ingredients

2.2.3 Master variables f q (t)g
We are interested in describing a dilute Bose-condensed gas over the range of temperatures T = 0 to T  Tc , so it is useful to consider the theory for these two limiting cases;
our choice of relevant variables must be consistent with standard results in these two limits. At high temperature, we should recover the standard kinetic theory as given by the
Boltzmann transport equation describing the irreversible evolution of the single-atom distribution of populations fm~m~ (t). In equilibrium the populations are distributed according
to the Maxwell-Boltzmann distribution. As we lower the temperature close to Tc , our theory should contain the Bose-enhancement factors [1 + fm~m
~ (t)] for the output channels ofa collision, which give rise to the Bose-Einstein distribution of populations in equilibrium.
Furthermore, as the system cools even further, quantum coherences may play a role, so thatin general we must consider off-diagonal elements fm~ ~n (t) of the single-atom density matrix. It is clear, then, that our set of master variables must contain the single-atom density
matrix fm~ ~n (t).
In the opposite limit of zero temperature T = 0, the most basic description of thesystem is given by the time-dependent Gross-Pitaevskii (GP) equation for the single-atomwave function ( ; t). In this mean-field description, interactions are treated to first orderand give rise to the nonlinear term in the GP equation. The corresponding equilibrium
state of the system is one in which all of the atoms occupy the same lowest-energy singleparticle state 0 ( ). This state of equilibrium is drastically different than the MaxwellBoltzmann distribution at T > Tc , which allows only fractional populations (fm~m~ << 1).
The most striking feature of the system at T = 0 is the spatial coherence, or off-diagonal
long range order, represented by the fact that the single atom density matrix f (c) ( ; 0 ) =0 j ranging over the entire extent of the0 ( )0 ( 0 ) is non-zero for large values of jcondensate. Represented in an arbitrary single particle basis, the single-atom density matrix(c)
of the condensate is written fm~ (t), where m~ (t) is the mean-value given~ ~n (t) = ~n (t) m
1
by m^m~ g . In order to correctly describe the system in the T = 0 limit,~ (t) = Trf(t) aour set of master variables must also include the mean-field m~ (t).
Based on theories that include higher-order corrections to the GP equation, anomalous averages may be important and also must be included in our set of master variables.
The anomalous averages, also referred to as pair correlations, are given by mm~ ~n (t) =Trf(t) a^m~ a^~n g and nm~ ~n (t) = Trf(t) a^ym~ a^~yn g. These quantities have a direct analogy
to the squeezing terms found in the description of squeezed states of light encounteredin quantum optics, which occur when appropriate coherent interactions are applied in anonlinear medium.
Based on the above considerations, we now specify the set of relevant operators to use
in our theory:

r

r

r

r

rr

r r

 a^m~ , a^ym~ , f^~m~ ~n = (^a~yn~~n )(^amm~ ), me^ m~ ~n = (^am~ ne^m~ ~n = (^aym~

a~n
m
~ )(^

~n ),

y

~n ).





a~n
m
~ )(^



j i j iji

1 The index m~ refers to a single particle state m~ . If m~ is chosen to be the position vector r , then we use
the notational convention r ; tar .
r t , and correspondingly, r

( )

()

^( )  ^11

Chapter 2 Underlying Theory
The corresponding expectation values of these operators give us the set of master variables
f m~ ; m~ ; f~m~ ~n ; me m~ ~n ; nem~ ~n g. The indices m~ and ~n refer to a generic basis of singleparticle states, the particular choice of which we discuss below. Not shown but also included in our set is the identity operator . Here we have defined the normal fluctuations
f~m~ ~n and the anomalous fluctuations memem~ ~n and n~ ~n by subtracting off the mean values m~in the averages.
With this set of independent variables, i.e., the mean field and the fluctuations aroundit, we can parameterize the reference distribution as1f(0);  ;f;~ m= exp ^1e ;eng

^ 12 12 ne^12 12  :f^~12 12 m
e

(2.9)

Here, we use the implicit summation convention for repeated indices. The conjugate
thermo-dynamic coordinates f ; ; g are implicitly defined by the quantum averagesnoho^i = Tr o^ f(0);  ;f;~ me ;en g ;(2.10)

where o^ represents a relevant operator. The average of any other multiple operator product, occurring during the evaluation of the kinetic equations, is greatly simplified by theGaussian structure of the reference distribution for it allows us to utilize Wick’s theorem.
It is worth remarking that the set of relevant observables consists of single operators andpairs of operators, which gives the reference distribution its Gaussian form. This implies
that the reference distribution given in Eq. (2.9) will not violate positive definiteness, thatis, it will not generate negative probabilities. This subtle, and perhaps pedantic, point isbased on the Marcinkiewicz theorem, or M-theorem, which states that the characteristic
function of a probability distribution function can not be an exponential of a polynomial of
degree larger than two, or it will violate its positive definitness [44]. In other words, if wewere to extend our relevant set of operators to include higher-order correlation functions
that involve, for example, a product of three or four operators, then the exponential form of
the reference distribution would not be satisfactory since it may be associated with negative
probabilities for the occupation of single particle states.

2.2.4 Single-particle basis
In the section that follows, we derive the quantum kinetic equations for the mean-field andfluctuations independent of a particular single-particle basis. However, when we want tocarry out calculations, we must choose a specific basis in which to represent the equations.
Such a basis can include quantum numbers for both internal states of the atoms, such as thehyperfine level2 jF; MF i, as well as the external spatial quantum numbers determined by
the form of the confining potential. When we work in the T = 0 limit and ignore fluctuations, for which our kinetic equations reduce to the GP equation describing the condensate
wavefunction, we will represent the external part of the system in the position representation.
However, when one considers finite temperature and includes the fluctuations about themean field, a particular eigenbasis should be chosen wherever possible that minimizes computational effort and clarifies the basic dynamical structure of the solution. Typically for a
2 The quantum number F represents the total angular momentum of the atom, given according to F
~ =L~+
~ , electronic spin S~ , and nuclear spin I~. MF is the~ + I~, which is a sum over the orbital angular momentum LS~ onto a chosen axis.
projection of F

12

2.3 Derivation of the quantum kinetic equations

Potentials
50
0.4
Eigen States
40

V(r)

30

20

0.2
0
0.2

0

2

4

6

8

r
10

0

0

2

4

r

6

8

10

Figure 2.2: Potentials for the isotropic harmonic oscillator. The black curve is the bareharmonic trapping potential, the blue one is the potential seen by the condensate, and thered one is the potential seen by the non-condensate, or fluctuations. Example eigenstatesof the non-condensate potential are shown in the inset.

homogeneous system, the most useful representation is the basis of plane wave solutions.
This is not the best basis for the system we consider, which is confined in a harmonic potential. The eigenstates of the bare harmonic oscillator are also not very useful, since they
do not account for the effect of the mean-field interaction potential. A better choice is thebasis of eigenmodes of the potential seen by the fluctuations, as shown in Figure 2.2. The
mean-field interaction potential experienced by the fluctuations (red curve) is twice as large
as that felt by the condensate (blue curve) due to the bosonic enhancement of fluctuations.
Consequently, the eigenmodes of the red potential are not centered at the origin, but sit at
the edge of the condensate, as demonstrated by the eigenstates shown in the inset. Anotherpossibility is to expand the equations in the Hartree-Fock-Bogoliubov normal modes that
we discuss in section 2.4.

2.3 Derivation of the quantum kinetic equations
We now turn to the task of formulating the quantum kinetic theory for a dilute, Bose condensed gas using the basic ideas presented in the previous section. The approach we use iswell established for quantum gases above the critical temperature T > Tc and the details of
the formalism can be found in the two well written accounts given by Zubarev et al. [39, 45]
and Akhiezer et al. [38], for example. Rather than reproduce all of the mathematical details, which have appeared in a recent publication by our group (see Walser et al. [27]), we
instead focus on the most important steps in the derivation in order to understand how theconcepts discussed in the previous section are incorporated into the theory. In this way, the13Chapter 2 Underlying Theory
new aspects of the theory that arise from extending it to a Bose-condensed gas will become
apparent.

2.3.1 Second quantized description
We are interested in describing the evolution in time of a system of N interacting bosonicatoms confined in a harmonic potential. This system is described within the framework ofelementary quantum mechanics by a many-particle wavefunction (r 1 ; r2 ;    r N ; t) thatevolves in time according to the Schrödinger equationd(r ; r ;    rN ; t) = H (r1 ; r2 ;    rN ; t);
dt 1 2where the many-body Hamiltonian H for the system is
ih

H=

N 
X
i=1

 1X
h 2 2r+ U (r i ) +V (jrii2m2 i6=jrj j);(2.11)(2.12)and m is the mass of the atom. The atoms are confined by the external potential U (r ),which is typically a harmonic potential for the systems considered throughout this thesis,1
U (r) = m !x2 x2 + !y2 y2 + !z2 z 2 ;(2.13)
2
with trap oscillation frequencies !x , !y , and !z along each axis. We assume it is sufficient
to describe interactions between atoms by a two-body potential V (jr i rj j), although

in reality interactions between two particles are modified by the presence of a third particle due to their finite extent; such effects lead to three-body recombination, for example.
However, in the limit of Eq. (2.1), these effects should be negligible. The basic form ofV (jri rj j) is a short-ranged van der Waals type of interatomic potential, the detailed
structure of which depends on the particular type of atom being considered. Because weare only interested in the very low temperatures at which Bose-Einstein condensation occurs, the kinetic energy in a collision is very low, so that we need keep only the s-wave
term in a partial-wave expansion. Furthermore, a generic form for the interatomic potential
can be used that is independent of the details of the two-body potential, but depends only
on the s-wave scattering length a. A pseudo-potential, given by

V (jr r0 j) = V0 Æ3 (r r0 );(2.14)

can be used, which reproduces the correct s-wave scattering length for a collision [46]. The
parameter V0 is given asV0 =4h2 a:m

(2.15)

So far we have not included the possibility that the atoms have internal structure. Lateron in the thesis, we will consider the case where the internal structure of the atoms canbe treated as a two-level system. In particular, we will be interested in the situation wherea harmonic electromagnetic field is applied to the system that coherently couples the twointernal states. For such a system, consisting of both internal and external degrees of freedom, it is more convenient to work directly with the state vector of the system j (t)i,14

2.3 Derivation of the quantum kinetic equations
expanded in a single-particle product basis, for example,

j (t)i = SnXm~ 1 ;~m2 ;m
~N

o

C (m
~ 1; m~ 2;    m~ N ; t)jm
~ 1 ijm~ 2 i    jm
~ Ni ;(2.16)

where C (m~ 1; m~ 2;    m~ N ; t) is an expansion coefficient. The vector m~ i represents thequantum numbers that uniquely specify the single-particle state of the ith atom, determined by some appropriately chosen complete set of commuting observables. For an atom
confined in the harmonic potential U (r ) given by Eq. (2.13), an example of a possible setof quantum numbers is jm~ i = jnx ; ny ; nz ; F; MF i, where ni are the quanta of excitationin the harmonic trap along each Cartesian coordinate, F designates a particular hyperfine
manifold, and MF is a Zeeman magnetic sublevel. It is useful to keep track of the external
and internal degrees of freedom separately by writing jm~ i = ji; i, where i denotes allof the quantum numbers that specify the external degree of freedom, and signifies theinternal state of the atom.
Due to the indistinguishability of identical particles, quantum spin statistics requiresthat the state vector j (t)i be symmetric under particle exchange for a system of bosons.
We therefore explicitly symmetrize the state vector in Eq. (2.16), as indicated by the operation S f g. Carrying out this operation can be a tedious procedure, making the singleparticle product basis rather cumbersome to use in the formulation of a many-particle theory. We will instead use the second quantization formalism, also known as the occupation
number representation, because the quantum statistics are built in to the theory in a very
compact way via the commutation relations between the creation and annihilation operators.
^yi; and a^i; create or destroy an atom in the
For a system of bosons, the operators asingle particle state ji; i, and satisfy the commutation relations, given by[^ai; ; a^yj; ] = Æij Æ ;[^ai; ; a^j; ] = [^ayi; ; a^yj; ] = 0:(2.17)In this new basis, we denote a general state by

jn1 n2    n1 i;(2.18)

where n1 is the number of atoms in the single-particle basis state j1i. These states are
orthonormal, eg

hn01 n02    n01 jn1 n2    n1 i = Æn01 n1 Æn02 n2    Æn01n1 :(2.19)The operators a^i; and a^yi; have the following creation and annihilation properties whenacting on a statea^yi; a^i; jn1 n2    ni;    n1 i = ni; jn1 n2    ni;    n1 i;(2.20)
pa^i; jn1 n2    ni;    n1 i = ni; jn1 n2    (ni; 1)    n1 i;pni; + 1 jn1 n2    (ni; + 1)    n1 i:a^yi; jn1 n2    ni;    n1 i =In this representation, the many-body Hamiltonian Eq. (2.12) describing our systemtakes the form

1
H^ = hi; jH^ 0 jj; i a^yi; a^j; + hi; ; j; jV^ jk; 0 ; l; 0 i a^yi; a^yj; a^k; 0 a^l; 0 ;2(2.21)
15

Chapter 2 Underlying Theory
where summation over repeated indices is implied. The first term describes the free evolution in the harmonic trap, and possibly an additional external field that couples the twointernal states. The matrix element can be written ash 2 2r+ U (r) + Æij  Æ
2m



Z

hi; jH^ 0 (t)jj; i = d3 ri (r)i+ Æij H d (t) j (r);(2.22)
where i (r )  hr jii form a basis. The quantity  is the internal energy of the state j i.
Here we include the possibility that the external trapping potential for each state can bedifferent U . We have also included a time-dependent drive that couples the internal states
H d (t).
The second term in Eq. (2.21) describes binary interactions between atoms with thematrix element given byZhi; ; j; jV jk; 0 ; l; 0 i = d3 r d3 r0 i (r)j (r0 )V ; ; 0 0 (jr r0 j)k (r0 )l (r):(2.23)
This is of a general form that includes the possibility of internal-state changing collisions.
However, we will restrict ourselves to elastic collisions that do not allow the internal stateof an atom to change due to collisions. We generalize the pseudo-potential to allow fordifferent scattering lengths depending on the internal states of the colliding partners, sothat Eq. (2.23) can be writtenZhi; ; j; jV jk; 0 ; l; 0 i = V Æ 0 Æ 0 d3 ri (r)j (r)k (r)l (r);(2.24)

h2 a =m and the scattering length a depends on the internal states ofwhere V = 4 
the atoms. For example, for a two-level atom with internal states j1i and j2i, there are threedifferent collision processes: j1ij1i ! j1ij1i described by a11 , j2ij2i ! j2ij2i describedby a22 , and j1ij2i ! j1ij2i described by a12 . The external part of the matrix element
Eq. (2.24) is simply the overlap of the wavefunctions for the “in” and “out” channels.
The state vector describing this many-particle system then evolves in time according toihdj (t)i = H^ j (t)i;dt(2.25)c(n1 ; n2 ;    ; n1 ; t) jn1 n2    n1 i;(2.26)^ given by Eq.(2.21). In the occupation number representation for a given singlewith H
particle basis, the general form of j (t)i is

j (t)i =Xn1 n2 n1

where c(n1 ; n2 ;    ; n1 ; t) is an expansion coefficient. This is just Eq. (2.16) rewritten in
the occupation number representation. Since we are considering aP
fixed number of atoms,
the occupation numbers must add up to the total number of atoms i ni = N .

2.3.2 Statistical description
Ultimately we are interested in observable quantities, that might be measurable in the laboratory. Such observables O(t) are found by taking the average of the corresponding oper^ for the system in state j (t)iator O

O(t) = h (t)jO^ j (t)i:

16(2.27)

2.3 Derivation of the quantum kinetic equations
Expanding j(t)i as in Eq. (2.26), we getO(t) =

X

fngfn0 gc fng; t c fn0g; t hn1 n2    n1 jO^ jn01 n02    n01 i;(2.28)

where fng stands for the set of occupation numbers fn1 ; n2 ;    ; n1 g. For a system witha large number of interacting particles, we can never measure the precise quantum statej (t)i. In a general situation, we can only give a statistical description of the state of thesystem, in terms of classical probabilities w(fng; t)O(t) =Xfng



w fng; t hn1 n2    n1 jO^ jn1 n2    n1 i:(2.29)The most general description of the state of the system is given by the density operator^(t), which can describe both mixed and pure quantum states. The observable O(t) is found^by taking the trace of ^(t) times the operator OO(t) = Tr ^(t)O^(2.30)
If the system is in the pure state j (t)i, the density operator simply takes the form ^(t) =j (t)ih (t)j. The dynamical evolution of the density operator is given according to@ ^(t) 1 ^= [H; ^(t)];@tih(2.31)

which is true for both mixed and pure states. This is usually referred to as the quantum
Liouville equation, since it resembles the Liouville equation found in classical mechanics
describing the classical phase space density.

2.3.3 Coarse-grained density operator
The quantum Liouville equation can be solved formally as^(t) = e i(t t0 )L^ ^(t0 );(2.32)^ is a super-operator, which is defined as L^ O^ = [H;^ ]=h. Starting from some initialb O
where Lstate ^(t0 ) for the many-particle system, the system evolves reversibly in time according to
Eq. (2.32). However, due to the attenuation of correlations discussed in section 2.2.2, thesystem “forgets” the precise details of the initial state after some time t > 0 , and the
behavior of the system depends only on some reduced set of relevant observables f q (t)g.
^(0) (t0 ) at the initial
We can therefore describe the system by the reference distribution3 time t0
^(t) = e i(t t0 )L^ ^(0) (t0 );(2.33)
keeping in mind that we have chosen the particular form ^ (0) given in Eq. (2.6) so that it
satisfies the self-consistency conditions

q (t) = Trf(t) ^q g = Trf^3 We now drop the subscript on 

tionals of the master variables

f g

f q (t)g.

(0) ^q g:(2.34)(0)and f g , which was used to emphasize that these operators are func-

17

Chapter 2 Underlying Theory
Within the assumption of the attenuation of correlations, the macroscopic observables

f q (t)g evolve on a much slower time-scale than the rapid decay and build-up of particle

correlations described by ^(t). Therefore, we can coarse-grain over a time t t0 that is longcompared to the duration of a collision, but short enough that we do not also coarse-grainover the evolution of the macroscopic quantities f q (t)g,^ (t) =1

Z t

t t0 t00e i(t t )L^ ^(0) (t0 ) dt0 :(2.35)

Here we are making a distinction between the true solution ^(t) and the coarse-grained^ (t). After some algebra, Eq. (2.35) can be brought to the equivalent form
solution @ ^ (t)@t

1 b^ (t) ^ (0) (t)[H; ^(t)] =
;
ih
t t0

(2.36)

which is the quantum Liouville equation with a source term that provides for the boundary
condition given in Eq. (2.35). A crucial point is that by course-graining in time to wash out
the rapid fluctuation of correlations in the system, we have introduced irreversibility into
the evolution of the system.
If we take the limiting case where the initial time is in the distant past t0 ! 1,
Eq. (2.36) can be brought to the more useful form@ ^ (t)@t

1 b[H; ^ (t)] = "f^(t) ^(0) (t)g;ihwhere " ! 0 in the limit of t0 !
by [39]1
lim
T !1 T(2.37)

1. Here we have made use of Abel’s theorem, given

Z 0

Z 0

(2.38)
f (t)dt = "!
lim+0
f (t)e"t dt:
T1
Physically, the idea that the system “forgets” the details of the initial state after a longenough time is embodied in Eq. (2.37), for in the limit " ! 0, the source term goes away
and we simply recover the quantum Liouville equation. There are, however, some subtle
issues regarding taking this limit, which we defer to a discussion later in the chapter.
The integral equation form of Eq. (2.37) can be obtained and is given by^ (t) = ^ (0) (t)n10@ ^ (0) (t0 ) o b y 0
dt0 e "(t t ) Ub (t; t0 ) [^(0) (t0 ); Hb (t0 )] +U (t; t ):
ih
@t0
1

Z t

b (t; t0 ) gives the full evolution from t0 to t and is given by
The propagator U

Ub (t; t0 ) = Tb exp

n(2.39)Zoi tb
H ( )d ;h t0(2.40)

where Tb signifies the time-ordering of the exponential. Here we are allowing for a timeb (t), which might arise if we are driving the system with an exdependent Hamiltonian H
ternal harmonic force, for example. It is clear from Eq. (2.39) that ^ (t) is a functional of^(0) (t), though in general athe set of master variables f q (t)g through its dependence on very complicated one; for after a short time, ^ (t) evolves away from the particular form of
the initial state ^ (0) (t) we have chosen.
182.3 Derivation of the quantum kinetic equations

2.3.4 Mean-field shifted energiesb (t) that governs the evolution of a weakly-interacting,
The total Hamiltonian operator Hb (0) (t), that could include a
dilute gas permits a partitioning of the energy into a free part H(1)bdriving field, and a presumably “weak” interaction H

Hb = Hb (0) (t) + Hb (1) :(2.41)

One could use the bare, single-particle energy that determines the free kinetic evolution of
the gas as a starting point of a series expansion of the coarse-grained density operator interms of the interaction strength. However, it is well known that the mean-field interaction
will significantly affect the single-particle energies. Anticipating this, we will shift theb (0) (t) by an as yet undetermined single-particle energy Q
b (1) to a dressed
expansion point H
energy prior to carrying out a perturbation expansion

Hb (0)(t) = Hb (0) (t) + Qb (1):(2.42)

To conserve energy, we have to reduce the interaction energy by an equal amountHb (1) = Hb (1) Qb(1) :(2.43)

This partitioning of the Hamiltonian is reminiscent of the Bogoliubov transformationthat leads to a description of the system as consisting of fictitious, noninteracting quasiparticles, in the first order approximation, with higher order corrections accounting forthe fluctuations due to the weak interactions between quasi-particles. The explicit formb (1) will be determined in the course of thisof the single-particle renormalization energy Q
calculation. The subscripts signify a dependence on the master variables f q (t)g.

2.3.5 Interaction representation of the integral equationb
Based on this partitioning of the Hamiltonian into a free part H(1)b
H , we reorganize the terms in Eq. (2.37) as
@

@t



+ " ^ (t)(0) (t) and a weak interaction

1 b (0)1[H ; ^ (t)] = "^ (0) (t) + [Hb (1) ; ^ (t)]ihih(2.44)

It is straightforward to show that this can be rewritten as an equivalent integral equation of
the form
Z t0^(t) = ^ (0) (t)dt0 e "(t t ) Ub (0) (t; t0 )1n @^ (0) (t0 )

@t0

+

o
y
1 (0) 0 b (0) 01[^ (t ); H (t )] + [^ (t0 ); Hb (1) ] Ub (0) (t; t0 ):ihih(2.45)

There are two crucial differences compared to the equivalent integral equation in Eq. (2.39).
b (0) (t; t0 ) appearing in Eq. (2.45) evolves the system according to the free
The propagator Ub (0) (t), which is given by
Hamiltonian H

Ub (0) (t; t0 ) = Tb expnZoi t b (0)H ( )d :h t0(2.46)
19

Chapter 2 Underlying Theory
This is in contrast to Eq. (2.39), in which the full propagator is used. Also, the full coarse^ (t) appears in the third term in the integral in Eq. (2.45); that is, ^ (t0 )grained solution 
appears on both sides of Eq. (2.45), in contrast to Eq. (2.39), where only ^ (0) (t) appearson the right-hand side.
^ (0) (t)=@t.
This integral equation can be further simplified by finding the explicit form of @ (0)^ (t) satisfies a master equaDue to its exponential functional form, it can be shown that tion of the form [39]@ ^ (0) (t) 1 (0) b (0)@ ^ (0) (t)+ [^ (t); H (t)] =L (t);@tih@ q (t) q
where

Lq (t) =1 Tr [^q ; Hb (1) ]^(t) :ih(2.47)(2.48)

In obtaining Eq. (2.47), use has been made of the Lie algebra of the relevant operators ^b (0)with the free Hamiltonian H[Hb (0) ; ^q ] = Apq ( )^p :(2.49)The structure constants Apq ( ) are well-defined for our particular set of operators ^q . Inserting Eq. (2.47) into Eq. (2.45) gives us^ (t) = ^(0) (t)
Z t
o
n @
0
^ (0) (t0 )0 ) + 1 [^(t0 ); Hb (1) ] Ub (0)y (t; t0 ):L(t
dt0 e "(t t ) Ub (0) (t; t0 )q
@ q (t0 )ih1(2.50)(1)

The commutator [^(t0 ); Hb ] explicitly contains the interaction Hamiltonian Hb
is convenient for using a perturbation expansion in this weak interaction.

(1) , which

2.3.6 Generalized quantum kinetic equations
We now turn to the second stage of the derivation, of obtaining the generalized kinetic
equations for the set of master variables f q (t)g. An observable q (t) is found by taking
the statistical average over the state of the entire system ^ (t), given by

q (t) = Tr ^ (t) ^q :(2.51)In the Schrödinger picture, the time dependence is in the density operator, so that to findthe time rate of change of the of the observable q (t) we haven @
@ q (t)^ (t) o
= Tr
^ :
@t
@t q

(2.52)^ (t)=@t given in Eq. (2.37) into this equation gives us
Substituting @ oo@ q (t) 1 n b (0)1 n b (1)+ Tr [H ; ^q ]^ (t) =Tr [H ; ^q ]^(t)
@t
ih
ih

20

(2.53)

2.3 Derivation of the quantum kinetic equations
The source term appearing in Eq. (2.37) drops out due to the self-consistency condition
Eq. (2.8). The second term on the left-hand side of Eq. (2.53) can be reduced using theoperator algebra properties Eq. (2.49) and the self-consistency condition Eq. (2.8). Finally,
inserting Eq. (2.50) into Eq. (2.53) on the right-hand side, we obtainZo@ q (t)1 n b
1 t 0 "(t t0 )+Tr [H; ^i ]^(0) (t) =
dt e
@t
ih
ih 1

 (0) 0
 Tr [Hb (1) (t; t0 ); ^q (t; t0 )] @@^ ((tt0 )) Lbp (t0 ) + i1h [^(t0 ); Hb (1) ] ;p(2.54)
where we have definedO^ (t; t0 ) = Ub (0)y (t; t0 ) O^ Ub (0) (t; t0 ):(2.55)b appearing on the left-hand side is the total Hamiltonian
We note that the Hamiltonian H
independent of our partitioning.
The generalized kinetic equation Eq. (2.54) gives the full evolution of the master variables q (t); without making any further assumptions, Eq. (2.54) is exact. This may seem
surprising, given that we have “built-in” irreversibility from the start by coarse graining theevolution of 
^ (t), which gave rise to the regularizing function exp["(t t0 )] appearing in
the integral. However, as written, Eq. (2.54) still describes the rapid fluctuations due to the
decay and build-up of correlations in the system. In fact, at this stage we could still choose
an infinite set of f ^q g that describe all of the correlations in the system. It is only when we
restrict our choice of f ^q g to single-particle operators—allowing us to use Wick’s theorem
for the resulting Gaussian averages—and make the Born and Markov assumptions that thefull coarse-graining procedure has been implemented. In conjunction with these further
assumptions, the limiting procedure " ! 0 will take on a more definite physical meaning.

Perturbation expansion
In the case of a weakly interacting system, we can seek the solution of the integral equation^ (t). In the
Eq. (2.54) in the form of a power series expansion of the density operator 
extreme case where interactions can be neglected altogether, we have ^ (t) = ^ (0) (t) andLq (t) = 0. To treat the weak interactions, we therefore make the expansions^ (t) = ^(0) (t) +Lq (t) =

1
X
n=1

1
X
n=1

L(qn) (t):^ (n) (t);(2.56)

Physically, we are assuming that the actual coarse-grained density operator ^ (t) deviates
only slightly from the Gaussian form of the reference distribution ^ (0) (t) because the interactions are so weak4 . Substituting these successive approximations into Eq. (2.54) gives
4 We will argue later that “weak” can apply to strong interactions which are well separated in time when we
make appropriate modification of the Born approximation.

21

Chapter 2 Underlying Theoryb
us the generalized kinetic equations expanded in powers of the weak interaction H
keeping just the first term in each expansion, eg.

 ^ (0) (t);1 b (1) ]^Lq (t)  L(1)
(0) (t) ;q (t) = ih Tr [^q ; H(1) . By^ (t)(2.57)(1)b . Inserting thethe kinetic equations are obtained to second order in the interaction H
approximations Eq. (2.57) into Eq. (2.54) gives us the kinetic equations in the Born approximationZ@ q (t) 1 b
1 t 0 "(t t0 )t+ [H; ^q ] (0) = 2
dt e

@t

*
h

ih

h

1

+t0
 @L(1) (t; t0 )b (1) (t; t0 )
0 i
@
H
t
q
(1)
(1)
0
0
0
 Hb ; [Hb (t; t ); ^q (t; t )] + ^p ih @ (t0 ) + [ @ (t0 ) ; ^q (t; t )] (0)
;
p
p
(0)

(2.58)
where we have performed a partial differentiation of the first term in the integral from
Eq. (2.54). We have introduced the notationnoO^ t(0) = Tr O^ ^ (0) (t)(2.59)^ with respect to the reference distribution ^(0) (t).
for the statistical average of an operator O
Truncating the expansion Eq. (2.56) at first order has simplified the kinetic equations
in two ways. The assumption of weak interactions between particles in this dilute systemallows us to simultaneously 1) treat the interactions only to second order and 2) to assume
that the fluctuations of higher-order correlations are Gaussian, as dictated by the exponen^ (0) (t). This allows us to use Wick’s theorem totial form of the reference distribution 
reduce multiple operator averages into a factorized form. Consequently, we now have aclosed set of kinetic equations for the master variables f q (t)g.
Markov approximation
As written, Eq. (2.58) is the most general second-order kinetic equation for a weakly interacting dilute system, which includes memory effects, since the relevant observables depend
on t0 in the integral. In general, this non-Markovian form of the kinetic equations is difficult to solve. It can readily be simplified by observing that the correlated averages in
the collision kernel fluctuate on a rapid timescale compared to the assumed slow evolutionof the macroscopic observables f q (t)g. The Markov approximation is made by taking
q (t0 ) ! q (t) in the collision integral, which gives us the Born-Markov form of the kinetic equations@ q (t) 1 b+ [H; ^q ] t(0)
@t
ih
+
*
Z
i t
 @L(1) (t)b (1)1 0 " h b (1)
@
H
t
q
=
+ [
;^ ]
;
de
H ( ); [Hb (1) ; ^q ] + ^p ih@ p (t)@ p (t) q (0)
h2 1
(0)
(2.60)
22

2.3 Derivation of the quantum kinetic equations
The collision kernel contains multiparticle correlations that can be factorized according to
Wick’s theorem. As we will show in the next section, these factorized multiparticle correlations include the Bose-enhancement factors for scattering into populated output channels,
which are due to the correlations arising from quantum statistics.
The first term in the collision integral resembles the second-order term appearing in themaster equations found in quantum optics for a system in contact with a static reservoir.
In contrast, for the situation we are studying, the “reservoir” (eg. the N -body reference^(0) (t)) evolves in time according the the master variables, so that both the
distribution “system” (eg. the master variables f q (t)g) and the “reservoir” depend on one another.
This makes our problem inherently nonlinear. One implication of this is that the factorized
multiparticle correlations appearing in the collision integral contain products of the mastervariables, i.e. q (t) p (t). Also, the last two terms appearing in the collision integral do notappear in the master equation from quantum optics because they are due to this nonlinear
behavior. The middle term originated from the time-variation of our reference distribution@ ^ (0) (t)=@t, while the third term appears because we have shifted the energies by theb (1) , which depends on the variables f q (t)g.
mean-field interaction Q
The regularizing function exp(" ) appearing in Eq. (2.60) originated from our coarse
graining of the system’s evolution based on the assumption of the attenuation of correlations. The full meaning of this point can now be understood. If we were to calculate
the correlation function in the collision integral of the full kinetic equation Eq. (2.54) (i.e.
before we make the Born approximation), we would find that it falls off with some characteristic time, on the order of a collision time  0 . Another way to say this is thatduring a collision we can go off of the energy shell by an amount  
h=0 . However, between collisions, we have free evolution according to the commutator on the left-hand sideof Eq. (2.60), consequently the contribution of the far-off-the-energy-shell terms becomes
negligible. This would not be true if the requirement for the attenuation of correlations
was not met, that is, if the time between collisions was roughly the same as the durationof a collision so that successive collision events could not be well separated in time. The
parameter ", then, must be chosen consistently with this physical interpretation. Since wehave made the Born approximation, taking the limit " ! 0 would be equivalent to staying
on the energy shell so that only energy conserving events are considered—this would giverise to an energy conserving delta-function in Eq. (2.60). However, a careful analysis of
the collision kernel [47] in Eq. (2.54), which includes the full evolution during a collision(before the Born and Markov approximations), would show that the appropriate correlation time is the order of 0 , so that the most appropriate choice is   1=0 . With thischoice the Born approximation result most closely approximates the full non-Markovian
collision kernel. In this manner by choosing   1=0 we allow the Born approximation
to more closely correspond to what, under some circumstances, would be the full off-theenergy-hsell T-matrix. It also allows strong interactions that are well-separated in time tobe considered as “weak.”
It is worth remarking that there is an alternative approach, based on the BBGKY hierarchy and cumulant expansion method, in which the two-particle interactions can more
readily be treated to all orders, while still truncating the hierarchy of correlations [39]. This
leads to essentially the same kinetic equations, but instead of treating interactions within
the Born approximation, the full two-body T -matrix is used. Of course, we could also treat
interactions to all orders in our approach, but this would require us to include many moreterms in our perturbation series, which is cumbersome. This is because our decorrelation
assumption is strongly tied to the notion of weak interactions. Such an assumption is not23Chapter 2 Underlying Theory
necessary, for, as mentioned above, it is possible to have “strong” interactions that are well
separated in time, which also allows for a reduced description of correlations.

2.3.7 Kinetic equations for mean fields and fluctuations
The Born-Markov kinetic equations Eq. (2.60) have been derived for a general set of mastervariables f q (t)g, which must be chosen appropriately for the particular physical situation
being described. For a dilute Bose gas above the critical temperature Tc for condensation,
there is only one master variable that is relevant, the single-particle density matrix fm~ ~n (t),defined as
fm~ ~n (t) = Trf^ (t)^a~yn a^m~ g:(2.61)
This choice is consistent with the Born and Markov approximations made in the derivationof Eq. (2.60); there is no need to include the two-particle or higher reduced density matrices
describing multi-particle correlations since we have already assumed that these quantities
fluctuate rapidly on the timescale in which we are interested, so that their averages can befactorized in terms of the single-particle density matrix fm~ ~n (t). In some sense, then, wewere predisposed to choose fm~ ~n (t), based on the physical assumptions made throughout
the development of Eq. (2.60), even though we have kept the derivation general for anunspecified set f (t)g.
The situation is quite different for a Bose-condensed dilute gas at finite temperature,when the macroscopic order parameter (r ; t) for the condensate plays a dominant role.
As discussed in section 2.2.3, in this case the set of macroscopic relevant observables must
be expanded to include the coherent mean field m~ (t), given by^ (t)^am~ g;m~ (t) = Trf(2.62)

as well as the possibility for the generation of squeezing terms, or anomalous averages,

mm~ ~n (t) and nm~ ~n (t),mm~ ~n (t) = Trf^ (t)^am~ a^~n g;
nm~ ~n (t) = Trf^ (t)^aym~ a^~yn g:(2.63)

This choice of master variables does not conflict with our basic assumption of the factorization of multiparticle correlations. Furthermore, the quadratic form of the set of operators
permits the use of Wick’s theorem, which is valid for the Gaussian form of the reference
distribution Eq. (2.9). The kinetic equations Eq. (2.60) for these variables form a closed,
coupled set of equations describing the irreversible evolution of a dilute Bose-condensed
gas at finite temperature and they conserve number and energy.
In the derivation of the kinetic equations Eq. (2.60), we partitioned the Hamiltonianb (0) that included a mean-field shift Q
b (1) , and a weak interaction part Hb (1)into a free part H
describing the fluctuations about the mean field. To complete this procedure, we now makethe corresponding transformation of the relevant observables by considering the mean fieldm~ (t) and the fluctuations about it. This gives us the normal fluctuations f~m~ ~n (t)
f~m~ ~n (t) = Trf^ (t) a^~yn ~n (t) a^m~m
~ (t) g;~ m~ ~n (t) and n~m~ ~n (t)and the anomalous fluctuations mm~ m~ ~n (t) = Trf^ (t) a^m~m
~ (t) a^~n~n (t) g; (t)a^y n~ m~ ~n (t) = Trf^ (t) a^ym~m~~n (t) g:~n24(2.64)(2.65)

2.3 Derivation of the quantum kinetic equations~ m;
We can now obtain a concrete set of kinetic equations for these master variables f ; f;
~ n~ gusing the generalized kinetic equation Eq. (2.60) that was obtained for the generic set ofobservables f q (t)g.
b (1) , the general form of
So far we have not specified the renormalization potential Q
which must be specified before carrying out Wick’s theorem to obtain the kinetic equations
for our master variables. We assume it has the form1
Qb(1) = hm
~ ~njV j~l~ki a^ym~ Q~n~la^~k :(2.66)
2~
It turns out that the matrix elements Q~nl need not be specified, for the resulting kinetic
equations obtained after carrying out the averages according to Wick’s rules are indepen~b (1) chosen in Eq. (2.66).
dent of Q~nl for the single-particle, number-conserving form of Q
One could, of course, choose a different form than that given in Eq. (2.66), which could
also include products of operators like a^m~ a^~n , in which case the collision terms would get
modified.
The kinetic equations for the mean field and fluctuations are obtained by first applying the generalized kinetic equations Eq. (2.60) to the chosen set of master variables
f ; f;~ m;~ n~ g, and then making use of Wick’s theorem to carry out the averages over themultiple operator products. The details of this procedure are outlined in Appendix (A). The
large number of individual algebraic transformations ( 10000) that were necessary to obtain the final result prohibited attempts to evaluate the collision terms manually. Therefore,a symbolic algebra package was developed by Reinhold Walser to perform the required calculations. Here we do not provide the detailed analysis, but rather simply state the results.
The presentation of the final results of this calculation is greatly simplified by introducing the following single-particle Hilbert-space vectors (co-, contra-variant)ha^ i  j i = m~ jm~ i ; ha^ iy  h j = m~ hm~ j ;(2.67)

normal operators [tensor rank (1,1)]

f~ = f~m~ ~n jm~ i h~nj ; f (c) = ~n m~ jm~ i h~nj ;(2.68)

pseudo operators [tensor rank (2,0)](2.69)me =mem~ i j~ni ; m(c) = ~n m~ jm~ i j~ni ;~ ~n jmyand their Hermitian conjugates ne =me y , n(c) = m(c) . With these definitions, we present
the kinetic equations, written in a compact, basis independent, operator form.
Mean field j i
dj i + hi H0 + 1 Uf (c) + 2 Uf~ j i + hi Vme 6dt

h j = L(2) [ ; f;~ me ](2.70)

Normal fluctuations f~

i 6 y
i
d ~ i
f + [H0 + 2 Uf (c) + 2 Uf~ ; f~] + V(m(c) +me ) 6 neme V (c)(m +me )
dt
h
h
h
(2)
= Lf~ [ ; f;~ me](2.71)
25

Chapter 2 Underlying Theory
Anomalous fluctuations m
~


i 6
d
i
e + m
m
e +
H0 + 2 Uf (c) + 2 Uf~ 6 me
H0 + 2 Uf (c) + 2 Uf~
dt
h
h
i
i
~
6
e]
+ V(m(c) +m
f~6 V(m(c) +me ) = L(2)m~ [ ; f;~ me ) (1 + f ) + hh(2.72)

The free Hamiltonian H0 is defined in Eq. (2.22). The mean-field potential is defined interms of the two-particle interaction matrix elements and a single-particle density operator
f that can be either f (c) or f~

Uf =hm~ ~njV j~l~ki f~k~n jm~ i h~lj:(2.73)

Due to the Hermiticity of the two-particle interaction energy and the positivity of the single
particle density, it is also self-adjoint , i.e., Uf = Ufy . The anomalous coupling strength Vmeis given byVm =hm~ ~njV j~l~ki m~k~l jm~ i j~ni :(2.74)In here, the subscript m in Vm stands for any anomalous average, either m(c) or m
e . Fromthe definition of the anomalous coupling, it can be seen easily that Vm = Vm> is symmetric. This non-Hermitian coupling is in general mediated by an anomalous average m , andinvolves a contraction 6 . In general, we obtain the contraction of two tensor fields A6 B ,from a basis representation of the two fields and a subsequent contraction of the last index
of A with the last index of B . For example, for the case of two tensor rank (2,0) pseudo
operators, the explicit contraction isX
A6 B y = Aik (B y )jk jiihj j:(2.75)

ijk

(2)

~m
The collisional terms L ~ [ ; f;e ] on the right-hand side are rather involved and are given
explicitly in Appendix (B).
The overall structure of the kinetic equations for each of the master variables is thesame. The unitary evolution consists of a free part H0 , determined by the external trapping
potential and any other applied electro-magnetic fields, and collision-induced mean-fieldpotentials, denoted by Uf (c) and Uf~. While the first of these potentials is proportional
the mean-field density itself, the second potential Uf~ reflects the influence of the normal
fluctuation upon the mean field. It is important to note the different weighting factors 1 and2 multiplying the potentials. They arise from the different quantum statistical fluctuation
properties of a c-number mean field and a normal single particle density. Exactly the same
weighting factors are also found with the variational Hartree-Fock-Bogoliubov approach[35].
Finally, appearing on the right-hand side of the kinetic equations are the second-order(2) ~ me ] always appear in pairs where one term
collisional terms. The terms in the L ~ [ ; f;corresponds to an in-process while the sign reversed companion describes a loss out of the
field. These terms allow for the exchange of norm between the master variables; that is,
atoms in the condensate can scatter into the noncondensate and vice versa. These collisional terms generate damping and can also give rise to collisional energy shifts. Another26

2.4 Comparison to previous work
important point is that only the normal fluctuations f~ contain Bose-enhancement factors
for the out channels. This is not the case for the mean-field density f (c) and the anomalous densities m(c) and m
e . This behavior is analogous to atomic transition rates describedby the Einstein A- and B-coefficients which can be attributed to stimulated absorption andemission, as well as spontaneous emission processes. The fact that the mean-field is neverbosonically enhanced supports the interpretation that the mean-field acts as a classical driving field.

2.4 Comparison to previous work
The master equations for the mean field Eq. (2.70), normal fluctuations Eq. (2.71), andanomalous fluctuations Eq. (2.72) obtained in the previous section form a closed set of
equations describing the time evolution of a weakly interacting, Bose-condensed gas. The
full solution of these dynamical equations for a general nonequilibrium state of a Bosecondensed gas at finite temperature is a monumental task that must be carried out on thecomputer using advanced numerical techniques, but is certainly within the reach of current
technologies. Such a general description is necessary to describe nonequilibrium phenomena, such as the proposed atom laser, where a dynamical steady-state situation occurs due
to a balance of the continuous output-coupling of condensate atoms with the replenishingof thermal atoms from a supplied reservoir. Another example of a possible nonequilibrium
situation is a two-component condensate being driven by an external field that coherently
couples the two internal states. Another point is that the collisional terms are also needed
to obtain decay rates for excitations.

2.4.1 Low temperature limit
One can, of course, learn a great deal about the system by considering various limitingcases. Most of the previous work on dilute Bose gases appearing in the literature is focussed on the static properties of BEC at finite temperature, where the collisional terms
can be neglected, or condensate dynamics at zero temperature, where, in addition to dropping the collisional terms, the fluctuations about the mean field are set to zero. Morerecent studies have begun to treat the coupled dynamics of the condensate and fluctuations [20, 21, 22, 23, 24, 25, 26, 27]. For example, in [24], the authors derive a coupled
set of kinetic equations in the semiclassical approximation, where one can neglect the offdiagonal distribution functions. Their resulting hydrodynamic equations include the collisional exchange between the condensate and normal density and are equivalent to Eq. (2.70)
and Eq. (2.71) if the anomolous fluctuations in those two equations are neglected.
In order to draw a connection to results of earlier work in this field, it is useful torepresent the master equations Eq. (2.70) through Eq. (2.72) in the position representation.
Here we will not consider the internal atomic structure, but treat only a single component.
The case of a coupled two-component system will be dealt with in detail in Chapters 4 to
6. If we project Eq. (2.70) for the mean field j i onto a position vector hr j, we obtain thefollowing kinetic equation for the wavefunction (r ; t) = hr j (t)iihd (r ; t)dth 2 2r+ U (r) + V0 j (r; t)j2 + 2f~(r; t) (r; t) V0 m
~ (r; t)  (r; t)2m= hrjL(2)[ ; f;~ m(2.76)
e ]:




27

Chapter 2 Underlying Theory
Here, f~(r ; t) = hrjf~jr i is the local density of the normal fluctuations and m~ (r; t) =hrjm~ jri is the local density of the anomalous fluctuations. In the ensuing text, we do not(2) ~ mconsider the collisional terms L [ ; f;e ], therefore we have not written their explicit
form in the position representation. One can obtain similar equations for the position representation of Eq. (2.71) and Eq. (2.72) describing the normal and anomalous fluctuations.
The Gross-Pitaevskii equation
The simplest approximation for the description of the condensate is to completely neglect
the fluctuations about the mean field by setting f~ = m~ = 0, and thus also dropping thecollisional terms. Such an approximation is appropriate in the limiting case of T = 0.
The resulting dynamical equation describing the condensate is called the time-dependent
Gross-Pitaevskii (GP) equation, given byihd (r; t)  h2 2=r+ U (r) + V0 j (r; t)j2 (r; t):dt
2m(2.77)

This is also referred to as the nonlinear Schrödinger equation, due to the dependence on thelocal density j (r ; t)j2 arising from treating interactions in an averaged way in the meanfield approximation. In the absence of an external drive, the condensate will condense into
the lowest energy solution 0 (r ) of Eq. (2.77), given byh 2 2r+ U (r) + V0 j0 (r)j2 0 (r) =  0 (r);2m

(2.78)

where  is the chemical potential of the system. This is termed the time-independent GP
equation.
Although the lowest temperatures achieved experimentally are not quite at T = 0, theGP equation has proven to give a good description of many static and dynamic propertiesof the system in many different situations at the coldest temperatures. In Chapter 2 wewill investigate the basic properties of the GP equation for a single component gas in moredetail.
Hartree-Fock-Bogoliubov treatment
There is a hierarchy of standard approximations appearing in the literature for extending
the mean-field theory from the zero temperature result of the GP equation, to finite temperatures [35]. In order to elucidate the connection between our work, embodied in the masterequations for the condensate and fluctuations in Eq. (2.70) through Eq. (2.72), it is useful tooutline an alternative approach based on the work of Bogoliubov, where the field operator^ (r; t) for a Bose-condensed system is written as^ (r; t) = (r; t) + ~(r; t):(2.79)
Here, (r ; t) is the mean field (r ; t) = h ^ (r ; t)i obtained by tracing over the N -bodydensity operator ^, and the operator ~(r ; t) describes the fluctuations about the mean field~(r; t) = ^ (r; t)(r; t).
In the Heisenberg picture, the field operator obeys the following equation of motion

ih
28

i
@ ^ (r; t) h h2 2=r+ U (r) ^ (r; t) + V0 ^ y (r; t) ^ (r; t) ^ (r; t);@t2m(2.80)

2.4 Comparison to previous workwhere we have again used the pseudo-potential introduced in Eq. (2.14) to describe binaryinteractions. Taking the trace of this with ^ gives us the following equation of motion forthe mean field (r ; t)i
@ (r; t) h h2 2(2.81)=r+ U (r) (r; t) + V0 ^ y (r; t) ^ (r; t) ^ (r; t) :@t
2mThis can be simplified by substituting the expression for ^ (r ; t) from Eq. (2.79) into theihlast term, to give

ih




h2 2
@ (r; t)=r
+ U (r) + V0 j (r; t)j2 + 2f~(r; t) (r; t) + V0 m
~ (r; t)  (r; t)@t2m+ V0 ~y (r; t) ~(r; t) ~(r; t) ;(2.82)

where we recall that the averages over the fluctuations vanish, h ~i = 0. Here we havedefined the normal and anomalous densities as f~(r ; t) = h ~y (r; t) ~(r ; t)i and m~ (r; t) =h ~(r; t) ~(r; t)i respectively.
This equation is nearly identical to the kinetic equation Eq. (2.76) describing the mean
field. The last term h ~y ~ ~i in Eq. (2.82) describes correlations between atoms due to collisions and must at equivalent levels of approximation ultimately reduce to the collision term
L(2) [ ; f;~ me ] in our kinetic equation for the mean field. Within the present Bogoliubov
approach, one could get a handle on this term by ascending up the hierarchy of equations
analogously to the BBGKY approach. One would then obtain an equation of motion forh ~y ~ ~i that would link up to higher-order correlations, and then one could truncate thehierarchy according to the assumption of the attenuation of correlations. The advantage
of our kinetic approach is that these collisional terms have been calculated explicitly (seeAppendix (B)). In contrast, the standard approach is to make the so called self-consistentmean-field approximation by writing [35, 48]~y (r; t) ~(r; t) ~(r; t)  2 ~y (r; t) ~(r; t) ~(r; t) + ~(r; t) ~(r; t) ~y (r; t):(2.83)

Within this approximation, then, the last term in Eq. (2.82) goes to zero, giving us@ (r; t)
h2 2
ih
=
r
+ U (r) + V0 j (r; t)j2 + 2f~(r; t) (r; t) + V0 m
~ (r; t)  (r; t):
@t
2m




(2.84)
Similarly, this approximation also leads to the following equation of motion for the fluctuating field operator ~(r; t) [35]@ ~(r; t)h2 2=
r + U (r) + 2V0 n(r; t) ~(r; t) + V0 m(r; t) ~y (r; t); (2.85)@t2mwhere we have defined the total density as n(r ; t) = j (r ; t)j2 + f~(r ; t) and the total
anomalous average as m(r ; t) = 2 (r ; t) + m~ (r; t).
These two equations for the mean field (r ; t) and the fluctuating field operator ~(r ; t)
ih





are equivalent to the kinetic equations Eq. (2.70) through Eq. (2.72) if we drop the colli(2) ~ me ] in those equations. This approximation is sometimes referred tosional terms L [ ; f;
as treating the collisionless regime, which is assumed to be an adequate description close
29

Chapter 2 Underlying Theory
to equilibrium where the collisional terms drop out due to detailed balance (in principle
one should also include the collisional energy shifts, since detailed balance will occur only
if these are included). This description, then, is useful in treating the system in equilibrium
at finite temperature, though recent work has been done that treats the coupled dynamics ofthe system close to equilibrium [22, 23, 24, 26, 49].
If we are interested in the static properties of the system at finite temperature in thisapproximation, then we must find the stationary solution of the mean field according toh 2 2r+ U (r) + V0 j (r)j2 + 2f~(r) (r) + V0 m
~ (r)  (r) =  (r):2m
(2.86)In order to describe the system in equilibrium, we would like to know the excitation spectrum of the system, described by ~(r; t). To find the excitations about the mean field, we
subtract the chemical potential  from the left-hand side of Eq. (2.85). It can then be shownthat a general solution of Eq. (2.85), and its Hermitian conjugate, takes the form [35, 36]~(r; t) =X~y (r; t) =

X

j
j

uj (r)^bj e iEj tvj (r)^byj eiEj t ;uj (r)^byj eiEj t
vj (r)^bj e iEj t ;(2.87)

where the creation and annihilation operators ^byj and ^bj for this transformation satisfy theusual Bose commutation relations and effectively describe a Bose gas of noninteracting“quasiparticles” that have an energy spectrum Ej . Substituting these expressions into
Eq. (2.85) and its hermitian conjugate, it can be shown that the expansion coefficients u(r )and v (r ), or normal modes, and the corresponding energies Ej must satisfy the equationsH^ uj (r) V0 m(r)vj (r) = Ej uj (r);H^ vj (r) V0 m (r)uj (r) = Ej vj (r);(2.88)^ is defined aswhere the operator H

2
H^ = 2hm r2 + U (r) + 2V0 n(r) :(2.89)The transformation defined by equations Eq. (2.87) through Eq. (2.89) generalizes the standard zero-temperature Bogoliubov treatment of the elementary excitations by including theeffect the normal and anomalous fluctuations have on the system. This generalization of
the Bogoliubov approximation is referred to as the Hartree-Fock-Bogoliubov (HFB) approximation [35].
The expansion of ~(r ; t) into the normal modes Eq. (2.87) does not by itself solve the~ (r) fluctuations.
problem, for we still need to solve for the normal f~(r) and anomolous m
The fluctuations f~(r) and m~ (r) take the following form in this quasiparticle representationf~(r) =m
~ (r) =
30

X n

j

X

j



juj (r)j2 + jvj (r)j2 h^byj ^bj i + jvj (r)j2
uj (r)vj (r) 2h^byj ^bj i + 1 ;o(2.90)

2.4 Comparison to previous workwhere h^byj ^bj i is the population in the quasiparticle level j . An obvious first ansatz is to
assume that these modes are occupied thermally according to the Bose-Einstein distributionh^byj ^bj i = e Ej1 1 ;(2.91)

where= 1=kB T . This assumption then closes the coupled set of equations for thesystem, which must be solved self-consistently since the mean-field solution given by~ (r), which in turn depend on the mean
Eq. (2.86) depends on the fluctuations f~(r) and m^ given in Eq. (2.89).
field through the density n(r ) and chemical potential  appearing in H
This procedure is outlined by Griffin [35], for example.
We can now enumerate the various approximations made within this Hartree-FockBogoliubov description of the system at finite temperature.
1. Bogoliubov approximation. This corresponds to the zero temperature case discussed~ (r) = 0, so that one simply
earlier, where the fluctuations are set to zero f~(r) = m
obtains the GP equation Eq. (2.77) describing the mean field. The excitations about
the mean field are obtained from the generalized Bogoliubov transformation by setting the fluctuations to zero in Eq. (2.87) through Eq. (2.89), however in the T = 0
limit, these excitations are not thermally occupied.
2. Popov approximation. In this approximation, the anomalous fluctuations are neglected m~ (r) = 0, but the normal fluctuations f~(r) are included. The static Popovapproximation corresponds to assuming the quasiparticle spectrum is thermally populated according to Eq. (2.91) for a given total number and energy (or chemical
potential and temperature). The resulting coupled set of HFB-Popov equations mustthen be solved self-consistently. On the other hand, one can still work in the collisionless regime, but treat the mean field and the normal fluctuations dynamically if
one is interested in the excitations of the system. In this case one expands the mean
field and normal fluctuations about the equilibrium situation considered in the staticPopov approxmation by treating the linear response of the system to a weak drive.
Such an approach is called the dynamic Popov approximation and has recently beentreated by Bijlsma et al. [49], for example.
3. Hartree-Fock-Bogoliubov approximation. In this case, the normal f~(r ) and anoma~ (r) fluctuations are both included and the excitation spectrum is assumed to belous mthermally occupied according to Eq. (2.91). One must then solve the HFB equations
self-consistently.
We have outlined here the main approximations for treating the dilute Bose-condensed
gas at finite temperature. An important point is that the kinetic equations reduce to these(2) ~ me ] in Eq. (2.70) through Eq. (2.72) are nevarious limits if the collisional terms L [ ; f;glected. Our approach, then, goes far beyond these three standard approaches, which treat
the collisionless regime, and is the most general second-order theory for a dilute weaklyinteracting Bose gas—it describes the irreversible nonequilibrium evolution of the condensate and fluctuations in a dynamical, self-consistent fashion. Naturally, we can obtain
damping rates and collisional energy shifts from the kinetic equations.
Of course, such a full treatment is an extremely difficult computational task and is nottrivial to implement. A sensible approach is to look toward these standard approaches
for guidance in finding a reasonable starting point for including the collisional effects. One
31

Chapter 2 Underlying Theory
could, for example, carry out the generalized Bogoliubov transformation Eq. (2.87) throughEq. (2.89) in order to simplify the collisional terms; in this representation, the collisionalterms might have a much simpler structure. Another useful approach would be to neglect
the anomolous fluctuations in the kinetic equations, as considered recently by Zaremba et
al. [24].

2.4.2 High temperature limit
In the high temperature limit T > Tc , the mean field and anomalous averages can beneglected. In this limit, the density matrix for the fluctuations f~ just becomes f , the single
atom density matrix. We can then write the kinetic equation Eq. (2.71) as

di
f + [H0 ; f ] = L(2)~ = 0]:(2.92)
f [ = 0; f; m
dt
h
(2)
~ = 0] specified in Appendix (B) takes the much
The collisional term L [ = 0; f; mfsimpler form in this limit

L(2)f = ff (1+f )(1+f )(1+f )f + h:c:(2.93)

The free Hamiltonian H0 just describes the motion in the harmonic trap (we are not considering the internal structure of the atoms here). It is convenient to expand the operator
equation Eq. (2.92) into the eigenstates of this Hamiltonian, H0 jii = i jii. This gives us
the kinetic equations for the matrix elements fiji
f_ij + (i j )fij = hij L(2)f jj i ;h(2.94)

where the collisional term is2 1
hij L(2)(2.95)
Æ + ; + C (m; n; p; q)C (r; s; t; u)f jj i = h h!
0 rstu mnpq m n p qhi Æuj fqs fpr (Ætm + ftm )(Æin + fin ) ftm fin (Æpr + fpr )(Æqs + fqs )
X

+

X

h:c:

Here the transition amplitudes C (m; n; p; q ) are given by

C (n; m; q; p) = V0Z

d3 rn (r)m (r)q (r)p (r);(2.96)

where hrj ii = i (r ) are the eigenstates of H0 .
Although we have included here the possibility that the coherences fij might be important, in the high temperature limit we expect these off-diagonal matrix elements to be negligible. Setting fij = fij Æij in Eq. (2.94) gives us the Uehling-Uhlenbeck equation [38, 39]
for the populations fi


@fi X
=
W (i; j ; k; l) fk fl (1 + fi )(1 + fj ) fi fj (1 + fk )(1 + fl ) ;@t jkl32(2.97)

2.5 Summary
where the transition rate W (i; j ; k; l) appearing in Eq. (2.97) is essentially just Fermi’s
Golden RuleÆ
2
W (i; j ; k; l) = jC (i; j ; k; l)j2 i +j ;k +l :(2.98)

hh!0This is the quantum version of the classical Boltzmann kinetic equation; it includes theBose enhancement factors (1 + fk ) for scattering into populated output channels of a collision. It gives a valid description above the critical temperature and may describe the initialonset of condensation. Once the condensate has formed, though, it is no longer an accurate
description if the mean-field interaction is substantial. We will return to this equation in
Chapter 7 and Chapter 8, where we consider specific applications of nonequilibrium kinetic
theory in the high temperature limit.

2.5 Summary
In this chapter we have developed a quantum kinetic theory describing a dilute Bosecondensed gas at finite temperature. We showed that in the simplest approximation for low
temperatures—obtained by neglecting the normal and anomalous fluctuations—we recover
the GP equation. We also showed that in the collisionless limit, we recover the HartreeFock-Bogoliubov description. Finally, in the high temperature limit we set the mean field
and anomalous averages to zero to obtain the quantum Boltzmann equation for the singleatom density matrix. In the rest of this thesis, we will only treat either the GP equation,
which is an approximation of the system at zero temperature, in Chapters 3 through 6, orthe high temperature limit above Tc where the the Boltzmann transport equation for the
populations is an appropriate description.

33

Chapter 2 Underlying Theory

34

Chapter 3

Mean-Field Theory of a Single
Component
3.1 Introduction
In this chapter we consider the basic properties of a trapped, single-component condensate
at zero temperature. By “single-component,” we simply mean that a single atomic species ispresent, in contrast to chapters 4 though 6 where we consider a two-component condensate,in which two different hyperfine levels are populated. This chapter serves a pedagogical
purpose of outlining the well-known solutions to the Gross-Pitaevskii equation for a single
component system. We begin with a discussion of the ground-state solution, and present
the full numerical solution along with the Thomas-Fermi approximate solution that is valid
in the limit of a large number of atoms. We next consider non-ground-state condensates,or topological modes, such as a vortex. It is useful to introduce the notion of a topologicalmode for the single component condensate, before considering it for the two-component
system in Chapters 4 and 6. In the last section we consider the response of the condensate
to a mechanical drive and show that for a weak harmonic drive, the linear response of thecondensate is a superposition of the collective modes discussed in the previous chapter.

3.2 The Gross-Pitaevskii equation
In Chapter 2 we derived the Gross-Pitaevskii equation within the framework of quantum
kinetic theory by neglecting the normal and anomalous fluctuations in the kinetic equation
for the condensate Eq. (2.76), which is an approximation for T = 0. This leads to adeceivingly simple looking equation describing the condensate wavefunction (r ; t) [50,51]ih 2 2r+ U (r) + V0 N j (r; t)j2 (r; t);(3.1)
2m
where m is the mass of the atom, U (r ) is a trapping potential, and V0 = 4 
h2 a=m is aih

@ (r; t) h=@tmeasure of the strength of binary interactions, proportional to the s-wave
scattering lengthRj (r; t)j2 d3 r = 1.
The system responds to itself through the density term in Eq. (3.1), which makes the equa-a. Here we have normalized the condensate wavefunction to unity

Chapter 3 Mean-Field Theory of a Single Component
tion nonlinear. A remarkable aspect of BEC in a dilute atomic gas is that there are no
fitting parameters in the theory—the physical parameters m, a, and the spring constants ofthe trap appearing in U (r) are all well-defined microscopic quantities.
Although Eq. (3.1) resembles the usual Schrödinger equation describing a single-particle
system, we must not forget that (r ; t) characterizes a many-particle system. In Chapter 2 we obtained the time-dependent GP equation within the statistical ensemble picture ofquantum kinetic theory, where we introduced the reduced description based on the assumption that particle correlations can be factorized into products of single-particle quantities.
Within that framework, the wavefunction (r ; t) is a macroscopic mean-field amplitude(r; t)  h ^ (r; t)i, where ^ (r; t) is a field operator and the brackets h i represent a statistical average over the N -body state of the system .
An alternative approach that is useful in understanding the GP equation is to describe
the system by a many-particle wavefunction as in Eq. (2.11), instead of the many-particle
density matrix  used in the statistical approach presented in Chapter 2. In general, themany-particle wavefunction is very complicated due to particle correlations caused by theinteractions between atoms. The problem is made tractable by first neglecting these particle
correlations and assuming that all of the atoms in the condensed system occupy the same
single-particle wavefunction (r ; t)(r1 ; r2 ; r3 ; :::; rN ; t) = (r1 ; t) (r2 ; t) (r3 ; t)::: (r N ; t):(3.2)
Physically, then we can interpret the GP equation as describing a system of N particles that

all act in unison according to Eq. (3.1)1. The factorized form of Eq. (3.2) means that correlations between particles are neglected. The static ground state of the condensate can befound within this framework using the variational procedure, starting with the assumptionthat all of the atoms occupy the same state 0 (r )(r1 ; r2 ; r3 ; :::; rN ) = 0 (r1 )0 (r2 )0 (r3 ):::0 (rN ):(3.3)
The state 0 (r) is found by minimizing the total energy of the N -particle system withrespect to variations of Æ0 , using the N -body Hamiltonian in Eq. (2.12). Carrying out thisvariational ansatz leads to the time-independent GP equation Eq. (3.4) discussed in the next
section. This variational procedure can also be used to find the elementary excitations of
the condensate, to varying degrees of sophistication [52].

3.3 Ground state properties
The bosonic enhancement factors (1 + f ) appearing in the collisional terms of the kinetic
equations derived in Chapter 2 cause stimulated scattering into the lowest-energy state of
the system during condensation. In the limit of an ideal gas, where interactions can beneglected, the state into which the atoms condense is simply the bare harmonic oscillator
ground state. However, for a dilute gas, the first-order effect of interactions is to modify
the effective external potential, so that in addition to the trapping potential U (r) the atoms
also experience a net force due to interactions, which is proportional to the local density.
The lowest-energy state 0 (r) then is determined by the solution to the time-independent
Gross-Pitaevskii equationhih 2 2r+ U (r) + V0 N j0 (r)j2 0 (r) =  0 (r);2m

1 One must replace N by N

36

1 in Eq. (3.1) in the strict case of fixed particle number(3.4)

3.3 Ground state properties
where  is the chemical potential. In general, this nonlinear equation must be solved selfconsistently using numerical techniques. It should be noted that throughout this thesis, wewill only consider the case of a positive scattering length a > 0.

3.3.1 Thomas-Fermi limit
Before giving examples of numerical solutions of Eq. (3.4), it is first useful to consider two
limiting cases of either very few or many atoms. In the limit that N ! 1, the solution to
Eq. (3.4) is trivial—it is just the bare harmonic oscillator ground state. The opposite limitof many atoms N ! 1 is referred to as the Thomas-Fermi limit. In this case, the kinetic
energy term in Eq. (3.4) becomes negligible. Upon setting the kinetic energy term to zeroand solving for 0 , the ground state solution in the Thomas-Fermi limit then becomesTF (r) =r

1 
NV0 TF

U (r) :

(3.5)

It is useful to consider the specific case of the condensate in an isotropic harmonic
potential U (r) = m!02 r2 =2. The Thomas-Fermi radius is determined by solving for r in
the expression U (r) = TF , which gives
s

rTF =

2 TF
:
m!02

The chemical potential must be determined from the normalization condition
Z rTF

4

r2 drjTF j2 = 1:

0

(3.6)

(3.7)

After carrying out the integral, we find that the chemical potential in the Thomas-Fermi
limit ish! h
a i2=5
TF = 0 15 N ( ) ;
(3.8)
p

2

lsho

h=m!0 is the size of the harmonic oscillator ground state. This result
where lsho = tells us that the chemical potential scales like the power of 2=5 with the population N
and the scattering length a. For typical values of parameters in current experiments, thechemical potential is 10 100 times larger than the bare oscillator ground state energy.
Upon substituting Eq. (3.8) into Eq. (3.6), we find that the size of the condensate in theThomas-Fermi limit is given by 15 N a 1=5:
rTF = lsho
(3.9)
lsho

For typical values of the physical parameters, the condensate size is around a factor of 10
larger than the bare oscillator ground state.
In general the trapping potential is not isotropic, but has the form

1
U (r) = m(!x2 x2 + !y2 y2 + !z2 z 2):
2(3.10)In this case, the Thomas-Fermi chemical potential is given by Eq. (3.8), withpthe subl = h=m! .
 and lsho ! lsho , where ! = (!x!y !z )1=3 andstitutions !0 ! !
p sho
The extent of the condensate along each axis is given by lTF = 2T F =m!l2 , wherel 2 fx; y; z g [53]. Most experiments have at least cylindrical symmetry, with !x = !y ,although a few experiments actually have used spherically symmetric traps [13].
37Chapter 3 Mean-Field Theory of a Single Component

3.3.2 Lower-dimensional modeling
In many cases, the system possesses a particular symmetry that allows the problem to bemodeled in fewer spatial dimensions. Throughout the remainder of this thesis, we willoften exploit such a simplification. It is important to understand what we mean by “lowerdimensional modeling”. We do not mean that the actual physical system satisfies some
limit of aspect ratios of the confining potential. For example, imagine a trap with sucha tight confinement in the xy -plane that the system can be treated as a one-dimensional
system, which becomes a better approximation as the ratio !z =!xy gets smaller, where!x = !y  !xy . This is not what we mean; we will not treat systems that satisfy such a
constraint. Instead, we consider systems that possess a particular symmetry, such that themain dynamics of the system occur along the z -axis, or in the xy -plane. This point will
become clearer as we consider specific cases throughout this thesis.
For a two-dimensional model, the condensate ground state satisfies the GP equationh 2  @ 2@2  1++ m !x2 x2 + !y2y2 + 2d V0 N j (x; y)j2
22
2m @x @y2(x; y) =  (x; y);(3.11)

while in the one dimensional model the time-independent GP equation ishih 2 @ 2 1 2 2
2
+
m!
z
+
V
N
j
(
z
)
j
(z ) =  (z ):1d 0
2m @z 2 2 z(3.12)We choose the scaling factors 1d and 2d so that the chemical potential, or equivalently,
the size, of the condensate is the same in the reduced-dimension model as in the threedimensional problem. This can be achieved by solving for the chemical potentials in the
one and two dimensional models in the Thomas-Fermi limit, equating them to the chemical
potential given in Eq. (3.8) for the three dimensional case, and solving for the scaling
factors 1d and 2d . After a bit of algebra, we obtain the results
 !
154=5
 2 4 1=5(a lsho );2d = 16 N 1=5 !
2d(3.13)

and1  53 1=5 2=5  !  4 16 1=10N
(a lsho );(3.14)2 9!zwhere !
2d = p!x!y . Note that 2d has units of length 1 and 1d has units of length 2 ,1d =as they must. These coefficients must compensate for the loss of one or two dimensions
from the problem, since the interaction parameter V0 was obtained by assuming s-wave
scattering in a three-dimensional space.
In later chapters, we model the dynamical evolution of the condensate using the timedependent versions of Eq. (3.11) and Eq. (3.12). This simplified description, in essence,
makes a factorization assumption about the different axes of the system. Due to the nonlinearity of the time-dependent GP equation, we know that the different axes are coupled.
For example, if the system possesses cylindrical symmetry and the condensate is being
driven by a force with spatial variation along the z-axis but no radial variation, then aone-dimensional model can be implemented. However, due to the coupling between dimensions, radial structure can develop, which is unaccounted for in the one-dimensionalmodel. It is therefore useful to check the lower-dimensional model against the full solution.
We will address this issue later in the thesis.
383.3 Ground state properties

density

SHO

TF−8−6−4−2 0 2 4 6 8−8−6−4−2 0 2 4 6 8−8−6−4−2 0 2 4 6 8−8−6−4−2 0 2 4 6 8

position

Figure 3.1: The density of the condensate ground state in the one-dimensional model forincreasing population. Going from left to right: N = 10, 103 , 104 , and 106 . The blue
dashed line is the exact numerical solution. The top red solid line is the bare oscillator
ground state, while the bottom red solid line is the Thomas-Fermi solution. The position z
is in units of lz . The chemical potentials obtained from the exact solution in each case are = 0:56, 1:31, 3:01, and 18:59, while for the Thomas-Fermi solution TF = 0:19, 1:17,
2:94, and 18:57. These are given in units of h!z . The trap frequencies were taken to bez = 65 Hz and xy = 24 Hz, and mass and scattering length were taken for 87 Rb.

3.3.3 Numerical solution
We now present results from numerical calculations in order to illustrate the character of theground state condensate and to demonstrate the two limiting cases of small and large population. We find the ground state solution by propagating the time-dependent GP equationin imaginary time using a finite-difference, Crank-Nicholson method, which we describe
in Appendix (C).
One-dimensional caseWe first consider the solution of the one-dimensional model given by Eq. (3.12). It ish!z for our unit of energy,
useful p
to write Eq. (3.12) in dimensionless units, where we take 
lz = h=m!z for our unit of position, and 1=!z for our unit of time (needed for thetime-dependent version of Eq. (3.12)). This rescaling of energy, position, and time can beachieved by setting 
h = m = !z = 1, which yields

1 @2 1 2+ z + 1d j (z )j2 ] (z ) =  (z );2 @z 2 2
where 1d = 4N ~ 1d (a=lz ), and ~1d = 1d lz2 .
[(3.15)In Figure 3.1 we show results of a series of calculations of the condensate ground statein which the number of atoms N is varied from N = 1 to N = 106 . For each case weplot the density of the bare oscillator ground state, the Thomas-Fermi solution, and theexact numerical solution. We also indicate the chemical potentials for the Thomas-Fermi
and exact solutions in the caption. For N < 103 the Thomas Fermi solution is a poor
39

Chapter 3 Mean-Field Theory of a Single Component
approximation, while for N > 104 the agreement is very good, giving an error of only 2%
in the chemical potential for N = 104 and 0:1% for N = 106 .
The Thomas-Fermi solution is a very good representation of the wave function in thecenter of the trap, but is a poor description at the very edge of the condensate, where thedensity is so low that the kinetic energy term in the GP equation can not be neglected [54].
This region at the boundary of the condensate where the density rolls off to zero is called
the surface thickness ds . The surface thickness can be found by setting the potential energy
at the surface equal to the kinetic energy associated with the bending of the wavefunction at
the surface (i.e. we approximate the Laplacian of as 1=d2s at the surface). For a spherical
2 and h2 =(2md2s ), respectively. Setting themtrap, these energies take the form m!02 rTF
equal and solving for ds gives the ratio l 4=3
ds= 2 1=3 sho
:
rTF
rTF

(3.16)

This ratio is small when the Thomas-Fermi approximation is valid.
Two-dimensional caseIn a later chapter we consider a two-dimensional model system in which !x = !y  !xy .
As in the above one-dimensional model, it is useful to rewrite Eq. (3.11) in dimensionless
h = m = !xy = 1, which gives
units by setting 1  @2@2  1 2 2++ (x + y ) + 2d j (x; y)j2
2 @x2 @y22

where 2d = 4N ~ 2d (a=lz ), and ~2d =

(x; y) =  (x; y);(3.17)

1d lz .

Three-dimensional case
Finally, we consider a three-dimensional solution of Eq. (3.4), where we assume axial
symmetry !x = !y  !xy . The dimensionless form of Eq. (3.4), obtained by setting
h = m = !z = 1 becomeshi
1 2 1 2 2 2r+
 + z + j (r)j2 (r) =  (r);(3.18)
22
where = !xy =!z and  = 4N (a=lz ). In Figure 3.2 we show the density of the condensate ground state for this axially symmetric trap, where the population was N = 106 .

In an experiment, it is the column integrated density profile that is measured, so it isuseful to represent the solution in this way for more direct comparison to experimental data.
In both cases of absorptive or dispersive imaging, the density along the line-of-sight gets
spatially averaged in the measurement process. We can take this into account by performing
an integration along the y -axis of the solution of Eq. (3.18), for example

n~ (x; z ) =Z 1

1dy j (x; y; z )j2 :(3.19)

We will often represent the solution of either the time-dependent or time-independent GP
equation in this column-integrated form in order to compare with experiment. We show an
example in Figure 3.3, where we have integrated through the solution shown in Figure 3.2.

40

3.3 Ground state properties

Figure 3.2: The density of the condensate assuming an axially symmetric trap with z = 65
Hz and xy = 24 Hz and the population was N = 106 atoms. Position is given in units oflz . The solution reflects the tight confinement along the z -axis, compared to the xy plane.
18

12

z

6

0

−6

−12

−18
−18

−12

−6

0

6

12

18

x
Figure 3.3: The column-integrated density profile for the solution of the condensate groundstate shown in Figure 3.2. This mimics how the density would appear on a CCD array inan experiment. With the tighter confinement along the z -axis, the condensate appears as a“pancake”.
41Chapter 3 Mean-Field Theory of a Single Component

3.4 Non-ground-state condensates
The time-independent GP equation has many stationary solutions in addition to the nodeless ground state, each of which corresponds to having all of the atoms in a state with analternative symmetry than that of the ground state. We refer to these states of the condensate as topological modes. One example that is a popular topic of research in the recent
literature is a vortex solution, which has quantized angular momentum. Another possible
solution is a non-rotating state with odd parity phase behavior along the an axis of symmetry; such a dipole2 mode has been termed a dark soliton solution. The intriguing aspectof these macroscopically occupied excited modes is that they do not necessarily decay via
usual thermal relaxation processes, so that they can be very long-lived. For example, insuperfluid helium persistent currents flow unimpeded for as long as one cares to conduct
the experiment, while currents in normal liquid helium dissipate on a very short timescalein comparison [16]. In this section we consider some of the basic properties of non-groundstate condensates.

3.4.1 Vortex solution
Let us assume an alternative solution to the GP equation that has the form(r) =  (; z )ei ;(3.20)

where  is the azimuthal coordinate around the z -axis,  is an integer, and  (; z ) is areal-valued amplitude. This ansatz describes a condensate that has a phase wrap of 2
around the z -axis, and thus has angular momentum per particle of h. Substituting thisansatz into the time-independent GP equation Eq. (3.18) yields the following nonlinear
Schrödinger equation for  (; z )i
1 2 1 2 1 2 2 2 (3.21)r+ 2+
 + z + j (r)j2  (; z ) =   (; z ):222
The additional centrifugal term (1=2)(2 =2 ) requires the solution to vanish at the origin
for  6= 0.
hThere are two different possible methods for calculating the vortex solution. One way
is to solve the ground state of Eq. (3.21), which has the ansatz Eq. (3.20) built in. This gives
the amplitude of the solution  (; z ), so that the full solution is (r ) =  (; z )ei . An
alternative method is to carry out imaginary time propagation on Eq. (3.17) while forcing
the solution to have the correct phase wrap as given by the ansatz Eq. (3.20). This techniqueis the one we have used and is described in Appendix (C).
We show an example of a vortex solution for the case of one unit of circulation, where
we have used the two-dimensional model of the GP equation. In Figure 3.4 we plot thedensity j (; z )j2 and in Figure 3.5 we show the phase. The core of the vortex sits at theorigin where the solution vanishes. The size of the vortex core is determined by the healing
length  (i.e. the scale length for significant density variation), which is set by the balanceof the kinetic energy 
h2 =(2m 2 ) and the mean-field interaction energy 4h2 an=m, where
n is the density. Upon setting these two terms equal and solving for  , one obtains for thehealing length = (8na) 1=2 :(3.22)
2 Note that we are using the term dipole somewhat loosely here; we are not referring to any specific dipole
moment (eg. electric or magnetic), but rather use the term to signify the antisymmetric form of the wavefunction.

42

3.4 Non-ground-state condensates

Figure 3.4: A vortex solution to the two-dimensional GP equation, for the case of unit
angular momentum. The population is N = 106 , and the trap frequency in the xy plane isxy = 24 Hz. Position is given in units of the harmonic oscillator length lxy . The size ofthe core is very small, and is characterized by the healing length  , which is approximately
 = lxy =8 for the solution shown here.

Figure 3.5: The 2 phase wrap of the vortex solution displayed in Figure 4.
43Chapter 3 Mean-Field Theory of a Single Component
Within the Thomas-Fermi limit, an approximation of the healing length is [55]
 l 2= sho ;
rTF
rTF

(3.23)

where the approximate Thomas-Fermi forms for the central density n(0) and radius rTF
were used for the case of a spherical trap.
If the atomic gas is being rotated during the cooling process, it is possible that theenergy of the vortex state becomes lower than the ground state energy. In this case, there
will be stimulated scattering into a circulating state, so that all of the atoms end up in the
Lz ; thatvortex [56]. The energy of a system rotating at the angular velocity is E
h. When thisis, this is the energy of the vortex mode with angular momentum Lz = Nenergy becomes lower than the energy of the non-rotating ground state, condensation canoccur in the vortex. This gives us the following value for the critical frequency needed to
produce condensation into the vortexc = (h)

1 (E =N ) (E0 =N );(3.24)

where E0 is the energy of the ground state. This is just the difference in energies between
the vortex and ground state. In order for the vortex to occur, the rotation frequency mustsatisfy  c . An approximate expression for this quantity has been worked out in theThomas-Fermi limit for the case of an axially symmetric trap [57]




5h
0:671TF
ln;c=
2m2TF(3.25)

where TF is the Thomas-Fermi radius in the xy plane. For typical values of parametersin the JILA trap, we plot this approximate form of c as a function of population N in
Figure 3.6. As N increases, the splitting between the vortex and ground state decreases, so
for one million atoms, the critical frequency is a few hertz.

3.4.2 Non-rotating states
An alternative class of solutions of the time-independent GP equation are those with nocirculation. A dipole is one example, which in the Thomas-Fermi limit is referred to as adark soliton. These solutions can be generated using the method described in Appendix (C)for finding excited states using imaginary time propagation. For example, the first excitedstate is found by subtracting out the ground state solution in each time step, starting fromsome appropriately chosen initial guess. Higher excited states can also be found this way.
This is essentially the Graham-Schmidt orthogonalization procedure, carried out in eachtime step during the imaginary time propagation. In the limit of a single atom N ! 1, thesolutions just go over to the bare Harmonic oscillator eigenstates.
We show examples of these types of solution for the one-dimensional model of Eq. (3.18)
in Figure 3.7 for three different values of the principal quantum number n = 0; 1; 2 andfor N = 1; 104 ; 106 . The top row correspond to the bare harmonic oscillator eigenstates.
As N increases, the mean-field interaction energy dominates the kinetic energy, so that inregions where the density goes to zero, the size of the node is determined by the healing
length, just as in the case of a vortex.
In Figure 3.8 we plot the energy of each of the three states shown in Figure 3.7 as ah!z , corresponding tofunction the population N . As N ! 1, the energies are spaced by 
44

3.4 Non-ground-state condensates

1

0.6

c

Ω (ω

xy

)

0.8

0.4
0.2
0 3
10

4

10

5

N population

6

10

10

Figure 3.6: The approximate value of the critical frequency c given by Eq. (3.25), in
units of the trap frequency !xy . For one million atoms in a 24 Hz trap, this frequency isapproximately c  3:5 Hz.

N=1

n=0

n=1

n=2

density

N=104

6

N=10

−8

−4

0

4

8−8

−4

0

4

8−8

−4

0

4

8

position

Figure 3.7: The densities of the first three excited states, for increasing population going
down each column. In the Thomas-Fermi limit, dark-solitons form at the nodes of the
wavefunction, with a size determined by the healing length.

45

Chapter 3 Mean-Field Theory of a Single Component

25

energy

20

n=0
n=1
n=2

15

10

5

0
0

1

2

3
Log10N4

5

6

Figure 3.8: The energies of the states plotted in Figure 3.7 vs the population. As N increases, the energies grow due to the mean field interaction, while the spacing between
levels decreases, similar to the vortex solution.
the solutions of the harmonic oscillator. As N increases, the energies grow according to
Eq. (3.8), and the spacing between levels decreases, just as in the case of the vortex.
In the case of the vortex, it is clear how one can create a situation in which the cooled
sample condenses into the rotating state. This could be done by slightly distorting thetrap and rotating it at the critical frequency c during the cooling process, though so far
no reports have been made of the experimental observation of a vortex generated in thisfashion in a dilute Bose gas. There is no obvious analog for a similar scheme that leads tocondensation into a non-rotating excited state. This leads us to the following question: If
the condensate is allowed to condense into the nodeless ground state of the system, is there
a way to transfer all of the atoms in the condensate from this initial state to a topologicalmode, such as a vortex or dark-soliton? In the next section we begin to investigate thisquestion by considering the response of the condensate to a harmonic mechanical drive.

3.5 Condensate response to a mechanical drive
The condensate response to an external driving force can be studied in a controlled way
in the present BEC experiments [58, 59, 60, 61]. Once a Bose gas has condensed into theground state, the confining potential can be manipulated by adjusting the magnetic trapping
fields, or by shining a laser on the sample to create an optical potential. We first consider thecase where a weak harmonic drive is applied and show that the response of the condensate
can be decomposed into a sum over the normal modes, which were introduced in Chapter 2.
In this way, the elementary excitation spectrum of the condensate can be studied. We also46

3.5 Condensate response to a mechanical drive
briefly discuss the nonlinear response of the condensate, and address the question of how
to generate a topological mode from an initial ground state condensate.

3.5.1 Linear response
Let us suppose that a weak mechanical drive is applied to the condensate that varies harmonically in time. The condensate equation of motion then takes the form [62]ihi@ (r; t) h h2 2=r+ U (r) + V0 N j (r; t)j2 + f+ (r)e i!d t + f (r)ei!d t (r; t);@t2m(3.26)
where we have included the mechanical drive described by the amplitudes f (r ) oscillating
at the frequency !d . The full numerical solution of this GP equation describing the driven
condensate can be calculated for arbitrary drive strengths. However, if we assume the drive
is weak, we can carry out a standard linear response analysis, where we make the ansatz
solution [62](r; t) = e it=h 0 (r) + u(r)e i!d t + v (r)ei!d t :(3.27)

Substituting this into Eq. (3.26), keeping only linear terms in u(r), v (r ), and f (r ), andequating like powers of ei!d t , we obtain the following three coupled equations [62]
H0 + V0 j0 (r)j2 0 (r) =  0 (r)(3.28)

2
2
H0 ( + h!d ) + 2V0 j0 (r)j u(r) + V0 0 (r)v(r) = f+(r) 0 (r)
H0 ( h!d) + 2V0 j0 (r)j2 v(r) + V0 0 (r) 2 u(r) = f (r) 0 (r);where H0 = p2 =2m + U (r ). The first line of Eq. (3.29) is simply the time-independent
GP equation for the condensate ground state. Upon setting the drive to zero f = 0, theother two lines in Eq. (3.29) reduce to the equations for the normal modes of the systemH^ uj (r) V0 20 (r)vj (r) = h!j uj (r);H^ vj (r) V0 0 (r) 2 uj (r) = h!j vj (r);(3.29)

2
H^ = 2hm r2 + U (r) + 2V0 j0 (r)j2 :(3.30)

^ is given by
where H

These equations are identical to the equations Eq. (2.88) and Eq. (2.89) obtained in Chapter 2 for the elementary excitations of the system in the Bogoliubov approximation, ob~ to zero. This is not a coincidence, since the Botained by setting the fluctuations f~ and m
goliubov spectrum was also obtained by treating the fluctuations to first order. Of course,
higher order corrections, such as the collisional terms discussed in Chapter 2, can not be
obtained within the present analysis, since those higher order effects are due to particle
correlations unaccounted for in the T = 0 limit case approximated by the GP equation.
To complete the analysis, one expands the condensate wavefunction (r ) and the drivef (r) in the normal modes u(r) and v(r). For an arbitrary weak harmonic drive, the condensate solution in general will consist of many normal modes. However, the drive can be47

Chapter 3 Mean-Field Theory of a Single Component
chosen with a symmetry and drive frequency that preferably excite only one normal mode.
This suggests an experimental procedure in which one distorts and modulates the magnetic
trapping potential in such a way as to excite a normal mode. By varying the modulation frequency !d and measuring the amplitude of oscillation of the condensate, a response curve
can be obtained that should be centered on the corresponding collective mode frequency.
Several such experiments have been conducted at JILA and MIT [58, 59, 60, 61]. In experiment, damping processes not accounted for in the T = 0 Bogoliubov theory give rise
to a width of the response curve. These experiments, in which the lowest few collectivemodes were studied over a range of temperatures, have generated a a large volume of work
recently on the study of the elementary excitation spectrum of a trapped Bose-condensedgas (see, for example, the review by Dalfovo et al. [53] and references therein).
Conceptually, it is important to realize that the above analysis (and the related discussion in Chapter 2) can also be applied to a vortex. That is, instead of studying a situation
where nearly all of the atoms are in the ground state and a small fraction are participating
in a collective excitation about it, one can instead consider having nearly all of the atoms ina vortex solution and studing excitations about that. The corresponding Bogoliubov spectrum can be calculated within the same framework. A great deal of work has also been
published recently on this topic and the related issue of vortex stability (see, for example,[63, 64, 65, 66, 67] and references therein).

3.5.2 Excitation of topological modes
Several recent studies have investigated the question raised earlier in section 3.4.2 on thepossibility to excite a ground state condensate into a topological mode, such as a vortex,
using a mechanical drive [67, 68, 69]. Unlike the case of collective excitations considered
above, where effectively very few atoms are excited, in order to produce a condensate in apure vortex all of the atoms must be excited into the same self-consistent mode. Based onrecent theoretical studies, as well as experimental efforts, a pure vortex seems difficult togenerate using a mechanical drive. In [69], for example, the authors investigated the use of a
far-blue-detuned Gaussian laser beam to stir the condensate by numerically solving the GP
equation. Typically, such a stirring of the condensate generates rather complex behavior.
However, the authors of [69] found a regime where a single vortex core oscillated in and outof the condensate, so that at one point in the cycle, a pure vortex could be formed where
the average angular momentum per particle is unity. Related schemes of distorting androtating the magnetic trap, either with or without an optical potential acting as a pinningmechanism, have been explored experimentally with no observations of a vortex [70, 34].
In trying to understand the process of exciting the condensate from a ground state toa vortex using a mechanical drive, a natural starting point might be to consider the drive
as weak and study the linear response of the condensate. As a concrete example, consider
the stirring mechanism studied in [69]. The harmonic drive of the stirring beam could be
expanded in normal modes, and the condensate solution could also be obtained in termsof the normal modes. Suppose the stirring drive preferentially excited a single normal
mode possessing angular momentum, due to its symmetry and rotation frequency. Is it
possible to transfer all of the atoms into this mode? An obvious problem is that for linear
response to be valid, only a small fraction of the atoms must be excited, since the effect of
excitations on the ground state is neglected. Therefore, it would seem that a full nonlinear
treatment of the problem is required, which includes the nonlinear terms left out of thelinear response analysis. Several research groups have investigated the nonlinear response483.6 Summaryof the condensate [71, 62, 72, 73], though not in the context of exciting a pure vortex.

3.6 Summary
In this chapter we have touched upon some of the major themes of zero temperature mean
field theory of a single-component Bose-condensate in a dilute trapped gas of atoms. The
basic concepts of the condensate ground state, topological excitations, and the elementary
excitations about the mean field have been studied extensively in the recent literature in thisfield; several recent review articles explore these concepts in more detail (see, for example,
Dalfovo et al. [53] and references therein). Here we have given only a glimpse of these
popular topics, in order to provide a broader context in which to place the specific original
work of this thesis. Of particular relevance to this thesis is the question of how to transfer
an initial ground-state condensate to a topological mode. In Chapter 6 we will return tothis question and present a technique for preparing such a state in a strongly coupled, twocomponent condensate—a technique that has recently been implemented experimentally at
JILA [70], resulting in the first observation of a vortex in a dilute gas BEC.

49

Chapter 3 Mean-Field Theory of a Single Component

50

Chapter 4

Mean-Field Theory of a Coupled
Two-Component System
4.1 Introduction
In this chapter we extend the mean-field theory discussed in Chapter 3 for a single component condensate to treat a coupled two component system. We begin by deriving thecoupled, two-component GP equation for this system using the formalism developed in
Chapter 2. Our main interest in this chapter are the stationary solutions to these equations.
We first consider the case of no coupling and we numerically solve for the lowest energy
solution of the two-component system. The basic ground-state properties studied will help
prepare us for the study of two weakly coupled components presented in Chapter 5. We
then present non-ground-state solutions for this system, such as a vortex, as was done in
Chapter 3 for the single component system. This topic will resurface in Chapter 6 when we
investigate a strongly coupled system. In the last part of this chapter we discuss the lowest
energy stationary states of the coupled system, which we refer to as the dressed states.

4.2 The physical system
In the next three chapters we will focus on the basic system studied in experiments conducted at JILA [8, 9, 10, 11, 12, 13] investigating a trapped, Bose-condensed dilute gas of
87 Rb atoms, in which two internal hyperfine levels are populated. The atoms in the condensate are in the ground state electronic state, with the hyperfine structure shown in Figure 1.
Typically, the jF = 1; MF = 1i  j1i is trapped and cooled in the usual manner to thecondensation point. Once the atoms in j1i have formed in the condensate ground state, a
two photon drive can be applied that couples the j1i state to the jF = 2; MF = 1i  j2istate. That is, microwave and radio-frequency (RF) electro-magnetic fields are applied tothe sample. One can describe this two-photon process with as an effective coupling strength= a b =(4Æab ), where a and b are the Rabi frequencies for the individual transitions,and Æab is the intermediate state detuning [74, 75, 76]. The frequency of the drive !d isjust the sum of the frequencies of the two photons. In principle both states could be cooled
simultaneously, so that a condensate forms in a mixture of states, however the typical lifetime of atoms in the j2i state is  300 ms in a z = 65 Hz,  = 24 Hz trap due to inelastic

Chapter 4 Mean-Field Theory of a Coupled Two-Component System

2 MHz

2, 1

F=2

6.8 GHz

1,-1
F=1Figure 4.1: Illustration of the ground-state hyperfine levels of 87 Rb shown with Zeeman
splitting due to the presence of a magnetic field. The two-photon transition is driven between the j1; 1i and j2; 1i states.

spin-exchange collisions, which makes achieving runaway evaporation very difficult. In
contrast, atoms in the j1i state have a much longer lifetime.
The condensate is confined in a time-averaged orbiting potential (TOP) trap, that consists of a static quadrupole magnetic field and a rotating uniform magnetic field, the combined effect of which generates an approximately harmonic axially symmetric potential
with trap frequencies !z and !xy [9]. Due to the slightly different magnetic momentsof the two states j1i and j2i, as well as a state-dependent force arising from the rotating
field [9], the effective potentials for the two states are offset along the vertical axis, as illustrated in 5.1. By varying the quadrupole gradient, and the intensity and frequency of
the rotating field, many different trap configurations can be achieved. There are three main
configurations we will consider, which consist of an axially symmetric trap with frequencies !z =2 = 65 Hz, !xy =2 = 24 Hz with the vertical separation being either 0:4 m orzero, or a spherically symmetric trap with frequency !z =2 = !xy =2 = 7:8 Hz and no
separation.
The scattering lengths for binary collisions depend on the internal hyperfine level ofthe atom. There are three different cases to consider, j1ij1i ! j1ij1i, j2ij2i ! j2ij2i, or
j1ij2i ! j1ij2i. We designate the s-wave scattering lengths for each case as a1 , a2 , anda12 . For 87 Rb it turns out that the values of these three quantities are nearly degenerate,with the ratios fa2 : a12 : a1 g = f0:97 : 1 : 1:03g[10]. The different scattering lengths
effect the condensate through the mean-field interaction potential, as we show below.
524.3 Derivation of the coupled two-component GP equation

4.3 Derivation of the coupled two-component GP equation
In Chapter 2 we derived the quantum kinetic equations for the mean field and fluctuations, Eq. (2.70) through Eq. (2.72), which are generally true for atoms possessing internal
structure. We now write out explicitly the generalization of the Gross-Pitaevskii equation
to treat the case of a coupled, two-component system, by setting the fluctuations to zerof~ = m~ = 0 in Eq. (2.70) and projecting j (t)i onto the position vector jri and an internal
state vector j i. Upon setting the fluctuations to zero, Eq. (2.70) becomesdj(t)i = H0 + 1 Uf (c) j (t)i(4.1)
dt
(r; t) = hr; j (t)i, with = 1; 2. The matrix elements ofih
We define the amplitude

H0 can be found from Eq. (2.22) and are



h2 2
d
0
r + U (r) +  Æ + H (t) Æ3 (r r0 ):hr; jH0 jr ; i =2m
We take the driving field H d (t) to have the specific form

H d (t) = cos(!d t) j1i h2j + j2i h1j ;

(4.2)

(4.3)

where we have assumed to be real. The matrix elements of the mean-field interaction
can be found from Eq. (2.24) and are

hr; jUf (c) jr0 ; i = V j(r)j2 Æ3 (rr0 );with the interaction strength given as V = 4 
h2 a =m.
With these matrix elements defined, we project Eq. (4.1) on to hr ;operators in this basis to obtain the two coupled equations(4.4)

j and expand theih 2 2r+ U1 (r) + 1 + V1 j 1 (r; t)j2 + V12 j 2 (r; t)j2 1 (r; t)
2mcos(!d t) 2 (r; t)
i
h
h2 2
r
+ U2 (r) + 2 + V2 j 1 (r; t)j2 + V12 j 1 (r; t)j2 2 (r; t)ih _ 2 (r; t) =2m
(4.5)
+ cos(!d t) 1 (r; t):ih _ 1 (r; t) =+hHere, V1 = V11 and V2 = V22 . Since we are focussed on the experimental situation
described in references [8, 9, 10, 11, 12, 13], we take the following specific form for thetrapping potentials
 i
1 h 2 2
m !xy  + !z2 z z0 2 ;2 hi
1
2 2 + !z2 z + z0 2 :U2 (r) = m !xy
2

U1 (r) =(4.6)

These describe cylindrically symmetric harmonic traps that are displaced by 2z0 along thevertical axis.
53

Chapter 4 Mean-Field Theory of a Coupled Two-Component System
It is useful to write Eq. (4.5) in a more compact form as!hf] + cos(!d t) 2 ;i _ 1 = [H10 + H1MF
2 1!
i _ 2 = [H20 + H2MF + hf ] 2 + cos(!d t) 1 ;
2

(4.7)

which is written in dimensionless units from setting 
h = m = !z = 1. We have defined
h!hf = 2 1 and reset the zero-point
the hyperfine splitting between the two levels as energy between the two states. Our notation has been made more compact by defining thequantities Hi0 and HiMF as

1 2 1 2 2
Hi0 =r + 2 [  + (z + i z0 )2 ]2
HiMF = ii j i j2 + ij j j j2 ;(4.8)

where 1 = 1, 2 = 1, and = !xy =!z gives the aspect ratio of the trap. We havedefined the mean-field interaction strength now as ij = 4N (aij =lz ) (with aii  ai andii  i ). This is consistent with the following normalizationZ(j 1 j2 + j 2 j2 )d3 r = 1:

(4.9)

The coupled equations in Eq. (4.7) can be simplified somewhat by making a unitarytransformation given by!
U = ei 2d t^z ;(4.10)
where ^z is a standard Pauli matrix. We make the rotating wave approximation by neglecting counter rotating terms, which gives us the following coupled equationsi_ 1   H10 + H1MF + Æ=2
=2
_2 =
=2
H20 + H2MF



Æ=2

1
2



;

(4.11)The detuning Æ is the difference in frequencies between the driving field and the hyperfine
transition Æ = !d !hf . This is the main result of our derivation, which will serve as thestarting point as we consider the particular cases of weak and strong coupling in Chapters
5 and 6, respectively.

4.4 Uncoupled two-component system
We begin by considering the case where there is no field coupling the two states, but assume
that the system has been prepared with population in both states, given by N1 and N2 forstates j1i and j2i respectively, with N = N1 + N2 . The GP equations describing thisuncoupled two-component system is obtained from Eq. (4.11) by setting the driving fieldparameters to zero, = Æ = 0, to give

1 2 1 2 2
r + 2 (  + (z z0 )2 ) + 1 j 1 j2 + 12 j 2 j2 ] 1 ;2
1
1i _ 2 = [ r2 + ( 2 2 + (z + z0 )2 ) + 2 j 2 j2 + 12 j 1 j2 ] 2 ;22i _1 = [54(4.12)

4.4 Uncoupled two-component system
where we have suppressed the time and position variables in the wavefunctions. The offset
between the two traps along the vertical axis is given by 2z0 . With two different statespresent in the condensate, an additional mean-field interaction term appears in each equation proportional to the inter-species scattering length a12 and the density of the other state.
This term provides a motional coupling between the two states.

4.4.1 Ground state properties
The time-independent GP equations describing the stationary state of the two-component
condensates are

1 2 1 2 2
r + 2 (  + (z z0 )2 ) + 1 j 1 j2 + 12 j 2 j2 ] 1 = 1 1 ;2
1
1[ r2 + ( 2 2 + (z + z0 )2 ) + 2 j 2 j2 + 12 j 1 j2 ] 2 = 2 2 ;22[(4.13)

where 1 and 2 are the chemical potentials of the two components. As already mentioned,
throughout this thesis we will focus on 87 Rb, for which the scattering lengths are nearly
degenerate. In some cases, we will make the approximation a1 = a2 = a12 . In thiscase, the mean-field interaction terms become (i j i j2 + ij j j j2 ) ! ij ntot , where
ntot = (j 1 j2 + j 2 j2 ) is the total density. In the case of concentric traps (z0 = 0), thetime-independent GP equations Eq. (4.13) become identical for both states[

1 2 1 2 2 2
r + 2 (  + z ) + 12 ntot ] i =  i ;2(4.14)

so that regardless of the population difference between states, the chemical potential, andtherefore the overall size, of each component is identical. In the opposite limit of traps
separated by a large distance z0  rTF , the two components act independently of one
another, and the two equations in Eq. (4.13) become spatially decoupled, with ntot !
ni for each component. In the following section, we illustrate these points by solving
Eq. (4.13) directly for various cases and we will show the role played by the trap separation
z0 and the population difference N1 N2 .
In order to get a better understanding of these properties, we now show results of numerical calculations of Eq. (4.13). Here we use the one-dimensional model of the system,
where we have used the interspecies scattering length a12 and the total population N in thescaling factors discussed in the Chapter 3. The one-dimensional model of Eq. (4.13) is

1 @2 1+ (z z0 )2 + ~1 j 1 j2 + ~12 j 2 j2 ] 1 = 1 1 ;2 @z 2 2
1 @2 1[+ (z + z0 )2 + ~2 j 2 j2 + ~12 j 1 j2 ] 2 = 2 2 ;2 @z 2 2[(4.15)

where we have put a tilde over the scattering length factors to remind us that these have~ij = 4N ~1d (aij =lz ), as discussed in the previous chapter. In
been rescaled according to 
what follows, we do not vary the scattering lengths, but fix them to the appropriate values
for 87 Rb. One could, on the other hand, take these as free parameters and investigate how
the ground state properties depend on a1 , a2 , and a12 [30, 31, 32].
55

Chapter 4 Mean-Field Theory of a Coupled Two-Component System

N /N = 0.1

N /N = 0.5
1

N /N = 0.9
1

density1−10 −50

5

10 −10 −5

05

10 −10 −5

0

5

10

z position
Figure 4.2: The density of the ground state solution for the two component system forzero trap displacement z0 = 0. We vary the population difference while keeping the total
population fixed. The blue solid line is the density of the j2i atoms, the red dashed line isthat of the j1i atoms, and the black dot-dashed line is the total density. Position is in unitsof the oscillator length lz . The population is N = 106 atoms and the trap frequency isz = 65 Hz.
Varying

N1 N2 with z0 = 0

In Figure 4.2 we plot the ground state solution to the two-component GP equation for threedifferent values of the population difference for the case of z0 = 0. The most obvious
feature of the solution is that the j2i atoms, which have the smaller scattering length, sit in
the center while the j1i atoms are pushed to the outside. This feature is a consequence ofthe particular values of the scattering lengths for 87 Rb: it is energetically favorable for the
j1i atoms to spread out due to their larger scattering length. Another feature is that the total
density is very nearly preserved as the population difference is varied.

z

Varying 0 with

N1 = N2In Figure 4.3 we plot the solution for four different values of the trap spacing. The key pointis that only a very small trap displacement is needed to completely change the character ofthe solution. For the case of one million atoms considered here, the Thomas-Fermi radius
is approximately rTF =lz = 6. This means that a trap displacement less than 1=100 thesize of the condensate is enough to cause the components to separate along the z -axis. The
overlap of the two components decreases as z0 is increased, until they no longer overlap at
all in the last plot where z0 =lz = 5 is nearly equal to the Thomas-Fermi radius.
Varying

N1 N2 with z0 = 0:15

Another important property of the ground state solution of the two-component system isthat the overlap between the two components is nearly constant as the population differenceis varied, with fixed trap separation z0 . In Figure 4.4 we demonstrate this by showing thedensities of the two components for three different values of the the relative population.
56

4.4 Uncoupled two-component system

z =0.01z =0.05

z =1

0

z =5

0

0

density0−10 −50

5

10 −10 −5

05

10 −10 −5

05

10 −10 −5

0

5

10

z position
Figure 4.3: The ground state solution of the two-component system as a function of trap
separation z0 , which is given in units of the oscillator length lz . In this case the populations
are equal N1 = N2 . The blue solid line is the density of the j2i atoms and the red dashedline is that of the j1i atoms.
The component with the larger population (the j1i atoms in this case) pushes the othercomponent away. The center of the region of overlap changes, but the overall area is nearly
constant. Also note that the total density is nearly preserved as the population difference isvaried.
We can demonstrate this behavior in a more quantitative manner by calculating theoverlap between components as a function of the population difference. We define theoverlap  as=Z

dz 1 (z ) 2 (z ):(4.16)In Figure 4.5 we plot the overlap  vs the population fraction in the j1i state. The overlap
is nearly constant, with the largest variation occurring for larger values of the population
fraction N1 =N .
N1/N = 0.3

N1/N = 0.5

density

N1/N = 0.1−10 −50

5

10 −10 −5

05

10 −10 −5

0

5

10

z position
Figure 4.4: The solution with the trap separation fixed at z0 = 0:15 while varying the
population difference.

57

Chapter 4 Mean-Field Theory of a Coupled Two-Component Systemoverlap

1
0.75
0.5
0.25
0
1.5

µ1 − µ2

1
0.5
0

−0.5
−1
0

0.2

0.4

N1/N

0.6

0.8

1

Figure 4.5: The wavefunction overlap  of the two components in the ground state (topgraph) and the difference in chemical potentials, vs the population fraction in the j1i state
N1 =N . The blue solid lines are obtained from the one dimensional model, and the reddashed lines are obtained from the full three dimensional solution. The trap separation isfixed at z0 = 0:15.
In the bottom graph of Figure 4.5 we plot the difference in chemical potentials 1 2
vs N1 =N . The difference 1 2 is approximately a linear function of the population
fraction. Also note that for equal populations, 1 2 > 0, which is due to the larger
scattering length of j1i. We will return to these observations in the next chapter, where we
consider the case of two coherently coupled components.
It is useful to compare our one-dimensional model to the three-dimensional solutionof Eq. (4.13). In Figure 4.6 we show the column-integrated density of the full threedimensional solution for the case of equal populations and a trap separation of z0 =lz =
0:15. In Figure 4.5 we plot the overlap

=Z

d3 r 1 (; z ) 2 (; z );(4.17)and the difference in chemical potentials 1 2 , vs the population fraction N1 =N . The
overlap  calculated for the one-dimensional model gives very good agreement, while thedifference in chemical potentials has an overall offset compared to the three-dimensional
solution. However, the slopes of the lines are nearly the same.

4.4.2 Non-ground-state two-component condensates
It is useful now to consider the situation where one of the components is not in a groundstate, but possesses an alternative symmetry, like the vortex and dark-soliton solutions considered in Chapter 3 for the single component system. As one might expect, these solutions
58

4.4 Uncoupled two-component system

18

z

9
0
−9
(1,−1)
−18
−18

−9

0
x

9

(2,1)
18 −18

−9

0
x

9

18

Figure 4.6: The column-integrated densities of each component for the full threedimensional ground state solution of the two-component system, assuming axial symmetry
with z = 65 Hz and xy = 24 Hz for one million atoms and equal populations in eachcomponent. The trap separation is z0 =lz = 0:15, which corresponds to the case shown in
the last graph of Figure 4.4.
have markedly different properties than their single-component counterpart related to theadditional mean-field interaction term between the two components.
Vortex solution
We begin by considering the case where one component j is in a vortex while the otheri remains in the nodeless ground state. We therefore make the ansatz

i (r )j (r )= 0 (; z );=  (; z )ei ;(4.18)

where  is the azimuthal coordinate around the z -axis,  is an integer, and  (; z ) is areal-valued amplitude. This ansatz describes one component that has a phase wrap of 2
around the z -axis, and thus has angular momentum per particle of h, while the other is a
nodeless ground state. With this ansatz we find the lowest energy two-component state of
the system using the imaginary-time propagation technique described in Appendix (C).
In Figure 4.7 we show the solution of this case using the two-dimensional model. We
have set the scattering lengths equal in this calculation. We will consider the effect ofdifferent scattering lengths in the next section for the case of a dipole solution. The vortex
shown in Figure 4.7 has a much larger core than that shown in Chapter 3 in Figure 3.4. For
the case of two-components, the repulsive mean-field interaction between components has
a dramatic effect on the shape of each component: the vortex core is much larger than thehealing length of a single component condensate and the ground state is “pinched” by thevortex.
In Figure 4.8 we plot the chemical potentials of each component as a function of the
population fraction in the ground state. Remarkably, the splitting between the vortex andground state remains roughly constant for quite a range of N0 =N , and increases sharply as
N0 =N goes to zero. The splitting is a mere fraction of h !xy due to the large spatial scaleof the excitation in terms of the size of the harmonic oscillator ground state.
59

Chapter 4 Mean-Field Theory of a Coupled Two-Component System

Figure 4.7: The solution to the time-independent GP equation for the case where one of
the components is in a vortex with unit angular momentum per atom, while the other component is in the nodeless ground state. Here we have set the scattering lengths all equal toa12 , and taken concentric traps z0 = 0. The components have equal population with thetotal population being N = 106 . Position is in units of the oscillator length lxy and the trap
frequency was xy = 8 Hz.

27
26.9

energy

26.8
26.7
26.6
ε
0
ε1

26.5
26.4
0

0.2

0.4 N /N 0.6

0.8

1

0

Figure 4.8: The chemical potentials of each component as a function of the population
h !xy . We have set the scattering lengths
fraction in the ground state. Energy is in units of 
all equal in this calculation.

60

4.4 Uncoupled two-component system

N0/N = 0.5

N0/N = 0.8

density

N0/N = 0.2

−8 −4

0

4

8−8 −40

48−8 −40

48

z position
Figure 4.9: The solution of the case where one component is in the ground state, while theother has a dipole symmetry. For this calculation we took equal scattering lengths. The
trap frequency was z = 65 Hz and the total population was N = 106 .
Non-rotating states
We now consider the possibility of having one of the components j in a dipole mode,
while the other i is in a ground state, which is described by the ansatz

i (r )= 0 (; z );zj (r ) =jz j 1 (; z );(4.19)In this ansatz we are forcing the component j (r ) to be an antisymmetric function along
the z-axis by multiplying the solution times z=jz j; for z < 0 the wavefunction has a phaseof  and for z > 0 the phase is 0. The other component i (r ) is just a nodeless groundstate. With this ansatz we find the lowest energy two-component state of the system using
the imaginary-time propagation technique described in Appendix (C).
In Figure 4.9 we show the solution of the one-dimensional model for three different
values of the population fraction in the ground state N0 =N . Just as in the case of thevortex, the mean-field interaction between components has a large effect on the shape ofthe densities. This effect depends on the relative population: as N0 =N decreases, the sizeof the core shrinks, as it must in the limit that all of the atoms are in the dipole.
We plot the chemical potentials of each component as a function of the population
fraction in the ground state N0 =N in Figure 4.10. We show three different cases of equal
scattering lengths, or unequal scattering lengths with the j1i or j2i atoms in the groundstate. For equal scattering lengths, the chemical potentials behave similarly as in Figure 4.8
for the vortex solution, however in this case the splitting decreases much more rapidly as
N0 =N is decreased. The second plot corresponds to the case where the j1i atoms arein the ground state. Due the the larger scattering length of the j1i atoms a1 > a2 , thisconfiguration becomes unstable, as indicated by the level crossing. In the third plot, it is
the j2i atoms that are in the ground state; this situation is energetically stable.
61

Chapter 4 Mean-Field Theory of a Coupled Two-Component Systemenergy20

18

16

φ0 −> (1,−1)equal a‘s
14
0

0.5

1 0

0.5
N0/N

φ0 −> (2,1)
1 0

0.5

1

Figure 4.10: The chemical potentials of the two components vs. the population fraction
N0 =N for three different cases: equal scattering lengths or unequal scattering lengths withthe j1i atoms in the ground state or the j2i atoms in the ground state. Energy is given inunits of 
h!z . The blue solid line is the ground state and the red dashed line is the dipolemode.

4.5 Coupled two-component system
We now consider the case where an external field is applied that couples the two internal
states of the atoms. In this section we calculate the eigenstates of the coupled system, often
referred to as the dressed states. We classify the behavior of this system into two classes,
based on the strength of the coupling and we will consider in detail the separate cases of
weak and strong coupling in the next two chapters. We consider the coupling to be weak
if the Rabi frequency is much less than the frequency of the harmonic trapping potential<< !z , so that the spatial dynamics occur on a faster time scale than the internal dynamics in this case. In other words, the densities and currents of the two components adjust
instantly to the transfer of atoms on the time scale of the Rabi oscillations. In the next
chapter we will consider this limit, which resembles the Josephson junction in condensed
matter physics. In the opposite limit of strong coupling >> !z , the Rabi oscillations of
the internal levels occur on a much shorter time than the motional dynamics. This separation of time scales will allow us to simplify the description and consider some very novel
behavior in Chapter 6.

4.5.1 Externalinternal representation

The coupled mean-field equations Eq. (4.11) can be rewritten in a more illuminating form
by making a clear separation of the external and internal degrees of freedom. The system exists in a direct-product Hilbert space H = Hex Hin , where Hex is the infinitedimensional Hilbert space describing the motional state of the system in the trap and Hin
is the two-dimensional Hilbert space describing the spin of the system. A general operator
in H can be written as a sum over the direct-product of operators from Hex and Hin . We
62

4.5 Coupled two-component system
rewrite Eq. (4.11) in this representation as^1 + ^1 ( ^x + Æ ^z ) + H^ z ^z ]j (t)i
(4.20)
22
where f^x; ^y ; ^z g are the standard Pauli spin matrices and 1^ is the identity operator. The
state of the system j (t)i in general has a nonzero projection on the internal states j1i andj2i, represented by i (r; t) = hrjhij (t)i, where i = f1; 2g. The position representations^ 0 and H^ z are local, i.e. hrjH^ 0 jr0 i = H0 (r) Æ(r r0 ) and hrjH^ z jr0 i = Hz (r) Æ(rof Hr0 ), where H0 (r) and Hz (r) are given byi@j (t)i = [H^ 0@t

1 2 1 2 2 2
r + 2 [  + z ] + h (t)jP^r ^+ j (t)i ;2
Hz (r) = z0 z + h (t)jP^r ^ j (t)i :(4.21)The operator P^r is the projector onto the position eigenstates P^r = jr ihr j, and the matrix^+ and ^ are given as
representations of 

H0 (r) =

1
^+ =
2



1
=
2



^

1 + 1200
2 + 12
1

0

12

12

0





2

;
:

(4.22)^ 0 is centered at the origin. The mean-field interaction
Note that the harmonic potential in H
has been broken into two parts, a part that acts identically on both components h jP^r^+ j i ^1, and a part that acts with the opposite sign on each state h jP^r ^ j i ^z .
The first two terms in Eq. (4.20) separately describe the external and internal dynamics
of the system, respectively. The third term in Eq. (4.20), however, couples the internal
state evolution to the condensate dynamics in the trap and can lead to interesting behavior.
^ z were identically zero, then the problem would be completely separable inIf the term H^ z would be zero if theterms of the external and internal degrees-of-freedom. The term H
trap separation z0 were zero and if the scattering lengths were all exactly equal. In fact,^ z comes
for 87 Rb the three scattering lengths are nearly degenerate, so the main effect of H
from the term z0 z , which is the difference in the shifted traps. It causes there to be a
spatially varying detuning across the condensate. In the strong coupling case considered in
Chapter 6, we will investigate this effect in more detail.
4.5.2 Dressed states
For the driven two-level atom in quantum optics the “dressed states” are a useful way to represent the system [75]. They are simply the eigenstates of the coupled Hamiltonian. For the
coupled two-component condensate described by Eq. (4.11), there exist analogous states,
however, because we not only have the internal structure but also the external motion in
the trap, the dressed states consist of an infinite manifold. We will only consider the lowest
energy dressed states in this section. Due to the entanglement of the internal and external
degrees of freedom and the nonlinear mean-field interaction term, these dressed states exhibit a much richer behavior than their counterparts in the simple two-level atom [77]. We
now present results of numerical calculations of these states and investigate the weak and63

Chapter 4 Mean-Field Theory of a Coupled Two-Component System

wavefunction

φ1
φ2 -

strong

weak
-8

-4

0

position

4

8

-8

-4

0

4

8

position

Figure 4.11: The condensate lower dressed state amplitudes in the weak and strong coupling limits. Here, 1 (z ) = hz jh1j i and 2 (z ) = hz jh2j i. The trap frequency isz = 65 Hz, the population is N = 106 and the trap separation is z0 =lz = 0:15m. The= 6:5 Hz in the weak-coupling limit and = 650 Hz in the
coupling strength wasstrong-coupling limit. The detuning is zero and the scattering lengths were all equal.

644.5 Coupled two-component systemstrong coupling limits. We make use of the one dimensional model, which is sufficient todemonstrate the main features.
We are looking for stationary solutions of the coupled mean-field equations of the form

^1 + ^1 ( ^x + Æ ^z ) + H^ z
2
2

[H^ 0

^z ]j i =  j i:

(4.23)

where  is the energy of the ground dressed state and + is that of the excited dressedstate. These states can be found numerically via the method of imaginary-time propagation
described in Appendix (C). An alternative method, which is more closely tied to recent
experiments at JILA, is based on the technique of adiabatic passage where the system istaken from the uncoupled ground state to the lower dressed state by ramping the detuningÆ from minus infinity to zero. Ramping the detuning from plus infinity to zero takes theexcited uncoupled state to the upper dressed state.
In Figure 4.11 we show the lower dressed state for both weak and strong coupling
limits for zero detuning and with a trap separation of z0 =lz = 0:15. The two cases yield
dramatically different results. In the weak coupling limit =!z  1, the two components
are pushed apart, while in the strong coupling case they overlap nearly completely. This
property of the dressed states has been observed experimentally at JILA [13, 70].
This behavior can be understood by considering each case separately. In the weakcoupling limit, we break up the Hamiltonian as H = H 0 + H 1 with

H 0 = H^ 0 ^1 + H^ z
H 1 = ^1
^ :
2 x

^z
(4.24)The Hamiltonian H 0 is diagonal in the internal-state space. The lowest energy eigenstatesof H 0 then are simply j 1 ij1i and j 2 ij2i, where the states i = hz j i i correspond to
the ground state solutions of the time-independent GP equation, plotted in the last graph of
Figure 4.4. The dressed states are then simplyj i = p1 (j 1 ij1i  j 2 ij2i):
2

In the strong coupling limit
with(4.25)=!z  1, we break up the Hamiltonian as H = H 0 + H 1

H 0 = H^ 0
H 1 = H^ z

^1 + ^1

^z :

^
2 x
(4.26)

If the size of the trap separation is small z0 =lz  1, then H 1 can be treated as a perturbation. The zeroth order solution for the dressed states is then simplyj i = p1 j0 i(j1i  j2i);
2(4.27)

where 0 = hz j0 i is the ground-state solution to[

1 @2 1 2+ z + 12 j0 (z )j2 ] 0 (z ) =  0 (z );2 @z 2 2(4.28)
65

Chapter 4 Mean-Field Theory of a Coupled Two-Component System
This can be the starting point for a perturbative treatment of H 1 . For the dressed state given
here in the strong coupling limit, the internal and external degrees are approximately decoupled, with both components having the same spatial structure, as shown in Figure 4.11.
This is in sharp contrast to the solution given in Eq. (4.25) for the weak coupling limit,
where the density profiles of the two components are very distinct, being separated along
the z -axis.
Here we have only given a qualitative discussion in order to gain some insight into the
behavior of the dressed states shown in Figure 4.11. A more thorough investigation could
be made including the effect of different scattering lengths and extending to cases with
nonzero detuning [77].

4.6 Summary
In this chapter we have studied the stationary states of a two-component condensate, withthe two components sitting in harmonic traps that are separated along the z -axis. After deriving the coupled, two-component GP equation, we then studied various properties of theground state of the uncoupled system, = 0. An important property is that as the relative
population changes between the two components, the overlap  is nearly constant and thedifference in chemical potentials varies almost linearly with the relative population. Thisbehavior will allow us to make a close connection to the double-well tunneling problem
when we consider the case of weak coupling in Chapter 5.
We then introduced the notion of a topological excitation in the two component system,
showing several examples of rotating and non-rotating solutions. Of particular interest is
the case shown in Figure 4.7, where one of the components is in a vortex wrapped aroundthe other component in the center. In Chapter 6 we will describe a method of dynamically
preparing such novel solutions of the GP equation in a strongly coupled two-component
condensate. An important property is that the energies of the two components change very
little as population is transfered between components, as shown in Figure 4.8.
We concluded the chapter by presenting a more useful representation of the coupled GP
equations, which makes a clean separation of the internal and external degrees of freedom.
We will make use of this in Chapter 6 when we consider the strong coupling case. Finally,
we discussed the stationary states of the coupled system—the dressed states—and saw thatthe behavior is drastically different in the two limiting cases of strong and weak coupling.

66

Chapter 5

Weak Coupling—Nonlinear
Josephson-Type Oscillations
5.1 Introduction
In this chapter we consider in detail the weak coupling limit of Eq. (4.11). We begin
by reviewing the standard Josephson junction from condensed matter physics and discuss
the analogous situation of a double-well BEC. Based on the weak coupling assumption,
we then show that Eq. (4.11) can be rewritten as two nonlinear coupled equations for the
relative phase and relative population. These have nearly the same form as those describing
the double-well tunneling system. Next, we show results of numerical calculations of theseequations for a one-dimensional model of the system, which gives qualitative agreement to
the full solution of the three dimensional problem. Finally, we use results from the previous
chapter to obtain a set of equations that is identical in form to those of the double-wellsystem, and we obtain a closed form solution.

5.2 The standard Josephson junction
A hallmark experiment contributing to the conceptual development of low temperaturephysics is the well known Josephson junction experiment [78, 79, 80, 81, 82], in whicha thin insulator is wedged between two superconductors, as illustrated in Figure 5.1. The
insulator provides a weak-link between the two superconductors, allowing electron-pairs totunnel through the barrier to the other side. A remarkable feature of this low-temperaturesystem is that a finite current of electron-pairs flows across the barrier in the absence ofa voltage drop across the link. This interesting behavior is rooted in the presence of amacroscopic condensate wavefuction ( ), which has an amplitude and a phase, describing
the many-particle state of the electron pairs. As the system is brought to a temperature
below Tc , a condensate forms in each superconductor, with a well defined, but random,
relative phase. This phase gradient sets up a current that flows across the weak-link aselectron-pairs tunnel through the barrier.
1=2
1=2
Let i = ni eii denote the wavefunction for each superconductor, with ni beingii
the amplitude and e the phase. This system can be described by the simple coupled setrChapter 5 Weak Coupling—Nonlinear Josephson-Type Oscillations

potential energy

insulator

superconductor

z

Ψ2

Ψ

- z0

z0

1

Figure 5.1: Illustration of the Josephson junction. On the left we show an illustrationof the standard Josephson junction, with the top image representing the physical device,and the bottom image showing the overlap of the condensate wavefunctions due to tunneling through the barrier. On the right we show an illustration of the weakly-coupled,
two-component condensate system, where the centers of the traps for each component are
separated by 2z0 .
of equationsih _1 = E1 1 + K 2ih _2 = E2 2 + K 1 ;(5.1)

where Ei is the ground-state energy of the i’th superconductor and K is a coupling constant,which depends on the geometry and material of the barrier. If we substitute i given above
into Eq. (5.1) and solve for the relative phase r = 1 2 and the current J = n_1 = n_2 ,we get the standard Josephson junction equations

J = J0 sin(r )E E2
_r = 1
;
h

(5.2)

hn and E1 E2 is proportional to the voltage drop across the barrier.
where J0 = 2K=These equations predict that, even in the absence of a voltage drop, the existence of an
initial relative phase r (0) between the superconductors results in the flow of current. Thiscase is known as the dc Josephson effect. If a finite voltage is applied across the barrier, thecurrent will oscillate and this is referred to as the ac Josephson effect. The linear geometry
chosen here can be replaced with a circular one, which leads to a vortex solution withcurrent flowing around a ring.
Several authors have proposed an analogous experiment for the case of a trapped atomic
Bose-Einstein condensate[83, 84, 85, 86, 87, 88, 89, 90]. The basic idea is to allow theatoms to condense in a double-well potential, consisting of two regions separated by a central barrier, the height of which can be varied. This trap configuration can be achieved
using the optical dipole force, for example, by shining a far-blue-detuned laser sheet oflight that divides the trap into to separate wells. If the barrier height were initially infinite,
the two condensates form independently in the separated wells. Once the condensates haveformed, lowering the barrier height to a smaller constant value allows the atoms tunnel
685.3 Nonlinear Josephson-like equations
through the barrier, so that the condensates become coupled. Characteristic equations similar to Eq. (5.2) have been obtained for this system [89, 90, 91]. Due to the the mean-fieldinteraction for each condensate, the Josephson equations take on a nonlinear form, given
by [89]_ = (1 2 )1=2 sin 
_ = E +   +  (1 2 ) 1=2 cos  ;(5.3)

where  = (N2 N1 )=N is the relative population and  = 2 1 is the relative
phase between condensates. The quantity E is the difference in ground-state energiesof the two wells, and  is proportional to the mean-field interaction. In obtaining theseequations, the authors have assumed the densities in the barrier region are so low the meanfield interaction between the condensates can be neglected.
In this chapter we investigate the case of weak coupling in the driven two-component
condensate [92, 93] introduced in the previous chapter. We illustrate this system schematically in Figure 5.1. The two components sit in spatially offset traps along the vertical axis.
The weak link in this system is not due to the tunneling of atoms through a barrier, but isprovided by the low intensity electromagnetic field that couples the internal states of theatoms. As we will show, this system obeys nearly the same set of nonlinear Josephson
equations as given in Eq. (5.3) for the double-well system.

5.3 Nonlinear Josephson-like equations
In order to make a connection to the standard Josephson effect, we must make some approximations in order to put Eq. (4.11) in a simpler form. For a very weak coupling
p=!z << 1, where !z is the trap frequency, we can make the ansatz i (z; ; t) =Ni (t) eii (t) i (z; ). Here we put the explicit time dependence into the population Ni
and the phase i of each component while putting the spatial dependence into an adiabatic
solution i (z; ) to the time-independent uncoupled GP equations Eq (4.8)(Hi0 + HiMF ) i (z; ) = i i (z; ) :(5.4)

The chemical potentials i and functions i (z; ) vary slowly in time, being “slaved” by
the populations.
If we substitute this ansatz into Eq. (4.11), we obtain the following equations of motion
for the relative population  = (N2 N1 )=N and the relative phase  = (2 1 )

_ =
_ =
R

k (1 2 )1=2 sin [(2 1 ) + Æ] + k  (1 2 ) 1=2 cos  ;(5.5)

d3 r 2 (z; )1(z; ) is proportional to the overlap of the condensates andwhere k =so also varies slowly in time. These are non-linear versions of the usual Josephson-junction
equations and are nearly identical in form to those obtained in Eq. (5.3) describing the
double-well tunneling problem. The major difference is that in the double-well trap, thecondensates are well separated, allowing the authors in [89] to neglect the mean field in
the interaction region of the barrier. In contrast, the interaction between condensates dueto their significant overlap plays an important role in the evolution of the system described
here, as we will show.
69

Chapter 5 Weak Coupling—Nonlinear Josephson-Type Oscillations
The coupling
constant k is simply the Rabi frequency times the overlap betweenRstates  = d3 r 2 (z; )1 (z; ), which is analogous to a Frank-Condon overlap factor.
As we saw in the previous chapter,  is nearly a constant function of the relative population
. Furthermore, we also showed in the previous chapter that the difference in chemical
potentials 2 1 for the two-component ground state depends linearly on the relative
population  over a reasonable range of  . In the section 5.7 we will use these facts toconstruct a simplified, close-form solution of Eq. (5.5).

5.4 Hydrodynamic-like Bloch equations
By assuming a weak coupling, we have been able to reduce Eq. (4.11) to a form that
resembles the Josephson equations describing the double-well tunneling problem. We canalso write Eq. (4.11) in a form that links up with the Bloch equations describing a driven
two level atom [75]. Conceptually, this is very nice, that we are able to draw analogies
between two standard problems from different fields of physics. The coupled equations
Eq. (4.11) can be written in an equivalent form that resembles the Bloch equations fora simple driven two-level atom. We can obtain a generalized set of Bloch equations forthe densities ni ( ; t) = i ( ; t) i ( ; t) and coherences nij ( ; t) = i ( ; t) j ( ; t) by
taking the time derivatives of these quantities and substituting in Eq. (4.11) to give

r

r

r

r

r

r

n_ 1 (r; t) + r  j1 (r; t) =i [n12 (r; t) n21 (r; t))]2
n_ 2 (r; t) + r  j2 (r; t) = i [n12 (r; t) n21 (r; t))]2
n_ 12 (r; t) + r  j12 (r; t) = i(r)n12 (r; t) + i [n2 (r; t) n1 (r; t)] : (5.6)
2
The quantities ji (r; t) are current densities for each component given by ji (r; t) = 1=2i( i r i
i r i ) while the quantity j12 (r; t) is a coherence current density given by j12 (r; t) =1=2i( 1r 2
2 r 1 ). The quantity (r) = Æ 2z0z + (H1MF H2MF) is a spatiallydependent detuning. These equations are not a closed set, but rather must be augmented by
the corresponding equations of motion for the current terms i ( ; t) and ij ( ; t).
In the absence of the current terms i ( ; t) and ij ( ; t) these equations would moreclosely resemble the usual Bloch equations. Neglecting the current terms is the same as
ignoring kinetic energy, which is also equivalent to taking the infinite mass limit so as tofix the positions of the atoms. In doing this one just obtains position-dependent Blochequations. The position-dependent detuning ( ) causes the Rabi oscillations of the population at each position to dephase across the condensate. In the weak coupling limit, thisdephasing happens so slowly that the current has time to adjust, so that no damping-likebehavior is observed. However, in the strong coupling limit, the dephasing happens very
rapidly, so that an initial decay occurs, followed by a revival due to the fact that current willflow in response to the phase winding across the condensate. We will study this effect inmore detail in Chapter 6.
This form also resembles the zero-temperature hydrodynamic-like description of thecondensate found in the literature (see, for example, [24]). For a single component theseequations simply reduce to the continuity equation n_ ( ; t) + r  ( ; t) = 0. However,
due to the coherent drive that couples the two states, the feeding term Imf n12 ( ; t)g
appears, which depends on the coherence n_ 12 ( ; t) between internal states.

j r

j r

j r

j r

r

r

r

70

jr

r

5.5 Preparation of the initial state
To make an even closer connection to the Bloch equations, we can integrate over the
spatial degree of freedom
in Eq. (5.6) and obtain equations for the populations Ni and theRcoherences Nij = dz i (z ) j (z ), which gives us

N_ 1 =i

N_ 2 = +i

2

(N12

N21 )

(N
N21 )
2 12
N_ 12 = i Æ N12 + i (t) + i (N22N1 ) ;(5.7)

where we define the time-dependent term (t) as

(t) =
+

Z

2 z0

Z

d3 r z 2 (z; ) 1 (z; )d3 r (H1MF H2MF ) 2 (z; ) 1 (z; ) :(5.8)

The equations in Eq. (5.7) resemble the Bloch equations describing an undamped, driven
two-level atom. However, because the spatial degree of freedom is correlated to the internal
states of the atom, the extra term (t) appears, which includes the difference in externalpotentials between the two states.
The first term in Eq. (5.8) arises from the difference in the shifted harmonic traps, which
is just linear in z . This term acts as a time-dependent detuning. As population is transferredfrom one condensate to the other, the position of the overlap region changes due to themean-field repulsion. This will cause the system to move in and out of resonance resulting
in a suppression of the transfer of atoms. The second term comes from the difference
in mean-field interactions and would vanish if all three scattering lengths were exactly
degenerate, which can be seen from

H2MF H1MF = (22

21 )j 2 j2 (1121 )j 1 j2 :(5.9)

5.5 Preparation of the initial state
The initial state that we have assumed in writing down Eq. (5.5) is related to the dressedstates we considered in the previous chapter. For the simple case of zero detuning Æ = 0 the
dressed states correspond to equal populations  (0) = 0 and the relative phase being either
(0) = 0 or . The case of (0) =  can be prepared by using the method of adiabatic
passage; the condensate is initially in the lower internal state j1i and a strong drive is turned
on, which is initially red detuned far off resonance. The magnitude of the detuning is thendecreased slowly to zero. This prepares the system in the dressed state shown in the rightplot of Figure 4.11. One can then adiabatically decrease the strength of the drive in order tosatisfy the weak coupling criterion, which takes us to the dressed state shown in the left plotof Figure 4.11. This procedure prepares the condensate in a dressed state with  (0) = 0
and (0) =  , which is a stationary solution to Eq. (5.5). We show an example of this in
Figure 5.2, where we have solved numerically the time-dependent coupled two-component
GP equation. At t = 100 ms we jump the phase so that the system is no longer stationary
and population begins to flow back and forth between components.
71

0

325

−2000

260

−4000

195

−6000

130

−8000

65

−10000
0

25

50

75

100

25

50

75

100

150

175

200

225

0
250

125
150
time (ms)

175

200

225

250

125

2Ω(t) Rabi frequency

δ(t) detuning

Chapter 5 Weak Coupling—Nonlinear Josephson-Type Oscillations

N1(t) population

1
0.75
0.5
0.25
0
0

Figure 5.2: An example of using adiabatic passage to prepare the initial state for theJosephson-like equations. The top plot shows the detuning in blue and the Rabi frequency
in red. The bottom plot shows the population in the j1i state. The phase is jumped to 3=16
at t = 100 ms. Therefore, t = 100 ms corresponds to the origin of time in Eq. (5.5).

This technique has been implemented in experiments at JILA and can, in principle,
produce initial states with arbitrary relative population  (0) and relative phase (0). For
example, instead of ramping the detuning to zero, one could instead stop at a finite value,
so as to prepare an initial state with unequal populations.

5.6 Numerical solution
We now show results of numerical calculations of both the exact solution, found by integrating the time-dependent coupled GP equation Eq. (4.11), and the approximate solution
given by the Josephson-like equations Eq. (5.5). To prepare the initial state, we simply
find the ground state of the two-component system for a given relative population, and thenmodify the relative phase. We do not generate the initial state using the more elaborate
scheme shown in Figure 5.2. To integrate Eq. (5.5), the chemical potentials and the overlap
factor  must be found self-consistently in each time step by solving the time-independent
GP equation for i , given  (t). We circumvent this elaborate procedure in the section 5.7,
where we make further approximations to obtain a closed-form solution.
72

5.6 Numerical solution

η(t) relative population

1

0.5

0

−0.5

−1
0

50

100

150

200

time (ms)Figure 5.3: The time evolution of the relative population  (t) for several different cases.
The black dot-dashed line corresponds to the case where the mean-field interaction has
been set to zero, the initial relative phase is (0) = 0, and Æ = 0. The red dashed line
is for the same initial relative phase of =2, but with the mean-field interaction turned onand Æ = 0:49 in units of the trap frequency. The blue solid line is for (0) = 3=16 andÆ = 0:49. The three lines described are solutions of Eq. (5.5), whereas the purple dottedline corresponds to the exact solution of Eq. (4.11). In all cases z = 65 Hz, z0 = 0:2 mand N = 8  105 .

5.6.1 One-dimensional model
We first consider the one-dimensional model first introduced in Chapter 3. In Figure 5.3 weshow four curves that are described in the caption. As a point of reference, we plot the solution of Eq. (5.5) with the mean-field terms set to zero. This is the standard
R Rabi solutiondz 2 (z )1 (z ),for a driven two-level atom, but here the Rabi frequency is given bywhich includes the Frank-Condon-type overlap of the condensate wavefunctions. However,when we turn on the mean-field interaction and set Æ = 0:49 so that the system is initiallybeing driven resonantly with (2 1 + Æ ) = 0, the amplitude is suppressed and the frequency has increased (the red dashed line). As (0) is decreased, the amplitude increases,
as illustrated by the solid blue curve where (0) = 3=16. Also, as (0) is decreased, thepresence of higher harmonics becomes stronger, as one can see in the shape of the solidline.
For comparison, we also plot the exact solution given by Eq. (4.11) for the case (0) =3=16 (the purple dotted line). In this case the adiabatic solution (blue solid) agrees quite
well with the exact solution given by Eq. (4.11). The validity of the adiabatic solution
depends on the structure of the evolving spectrum of this nonlinear system. In particular,
one must compare the time rate of change of the Hamiltonian to the spacing between the
instantaneous eigenmodes of the dressed basis. These quantities willRdepend on the size ofdz 2 (z )1 (z ), andthe mean-field interaction, the strength of the coupling, given by
also on the detuning Æ .
We plot a snapshot evolution of the densities of the system in Figure 5.4 in order toshow that the effect of the mean field is to push the system out of resonance as population73

Chapter 5 Weak Coupling—Nonlinear Josephson-Type Oscillations

t = 17t = 33t = 50t = 66

t = 83t = 99t = 116t = 132

density

t=0

−8

−4

0

4

8−8

−4

0

4

8−8

−4

0

4

8

z position

Figure 5.4: A snapshot evolution of the densities of the components. The x-axis is the
position z , and the time t of each snapshot is shown in ms. This case corresponds to thepurple dotted line plotted in Figure 5.3. The detuning is Æ = 0:49, chosen so that the system
is initially being driven on resonance. As the system evolves, the j1i component pushes the
j2i component out of the center of the trap so that the region of overlap becomes displaced
from the origin and the system is no longer being driven on resonance.
is transferred between components. The case considered in Figure 5.4 corresponds to thepurple dotted line in Figure 5.3. The detuning was chosen so as to compensate for the initialvalue of the term (0) in Eq. (5.7), which represents the difference in external potentials,
so that initially the system is being driven on resonance. However, as the system evolves,the first term in Eq. (5.8) gets larger since the j1i component is pushing the j2i component
away from the center of the trap. This causes the region of overlap to be displaced fromthe origin so that the system is no longer being driven on resonance. This reduces theeffectiveness of the drive and accounts for the suppression of the amplitude of oscillationin the relative population plotted in Figure 5.3.
In Figure 5.5 we show some examples of different initial conditions. As a reference,
we plot the solution with the mean-field interaction set to zero, with  (0) = 0:5, (0) =0, and Æ = 0 in units of the trap frequency (red dashed line). This just corresponds tothe standard Rabi solution for the driven two level atom. When we turn the mean-fieldinteraction back on and set Æ = 0:97, we see again that the amplitude is reduced, but also
the shape of the oscillation changes dramatically. If we let the initial phase be (0) = =4
we get the black dot-dashed line, which looks more like the solutions shown in Figure 5.3.

5.6.2 Comparison to the three-dimensional case
It is instructive to double check our one dimensional model solution against the three dimensional solution. In Figure 5.6 we compare the one and three dimensional exact numer74

5.6 Numerical solution

η(t) relative population

1

0.5

0

−0.5

−1
0

50

100

time (ms)

150

200

Figure 5.5: Time evolution of  (t) for additional cases. This plot is similar to Figure 5.3,
but here we have  (0) = :5. We set the mean-field interaction term to zero for the reddashed line, with (0) = 0 and Æ = 0. For the solid blue line we included the meanfield interaction, keeping (0) = 0 and setting Æ = :97 so as to initially drive the system onresonance. In the black dot-dashed line we have set the initial relative phase to (0) = =4.

η(t) relative population

1

0.5

0

−0.5

−1
0

20

40

60

80

time (ms)

100

120

140

Figure 5.6: A comparison between the exact solutions of Eq. (4.11) for the one dimensionalmodel and the full three dimensional problem. The solid lines are the three dimensional
solution and the dashed correspond to the one dimensional model. The red and green
lines are for an initial relative phase of (0) = 3=16, while the blue and black are for(0) = =2.

75

Chapter 5 Weak Coupling—Nonlinear Josephson-Type Oscillations
ical solutions of Eq. (4.11). The solutions agree very well, with small differences in theamplitudes and periods of oscillations. This agreement is not surprising, if we look backat Figure 4.5 in Chapter 4. There we found that the overlap  for both cases was nearly
identical. A more significant, though small, deviation was found for the the difference inchemical potentials, where the one dimensional model has a slightly larger slope, which
will give the differences shown in Figure 5.6. The one dimensional model appears to yieldfairly good agreement with the three dimensional problem.

5.7 Closed-form solution
We can obtain a closed-form solution of Eq. (5.5) if we make use of the observation from
Chapter 4 that the overlap  is approximately a constant function of  and that the difference 2 1 varies linearly with  , as shown in Figure 4.5. This allows us to make theapproximation

2

1 =
k =

+ ;
;

(5.10)

where the parameters , , and must be determined numerically for a set of physicalparameters (N , z0 , etc.). Substituting these into Eq. (5.5) we obtain

_ = (1 2 )1=2 sin 
_ =  +   +  (1 2 ) 1=2 cos  ;(5.11)=k and  =where we have made the change of variable t ! kt, and taken  =(Æ + )=k. These equations are now identical in form to Eq. (5.3) describing the doublewell tunneling problem.
It was shown by the authors of [89, 90, 91] for the double-well problem that a very nicemechanical analog of the problem can be constructed that admits a closed form solution, if
we treat  and  as conjugate variables, whose motion is governed by the Hamiltonian1
H =  2 +  (1 2 )1=2 cos();2(5.12)

This has the physicalanalog of describing a nonrigid pendulum of tilt angle  and lengthpproportional to 1  2 that decreases with angular momentum  [89, 90, 91]. The equations Eq. (5.11) are equivalent to

@H
;
@
@H
_ =
:
@
_ =

(5.13)

We consider the special case where  = 0, which is achieved by setting the detuning
to Æ =. We then rewrite the equations of motion as an integral equation

(t)
t
d
=

2 ;
2
(0) (b +  2 )(a  2 )Z

76(5.14)

5.7 Closed-form solution
where the constants a and b are
2 2( 2H0 + 1)1=2 + (H0  1) ;2
2b = 2 (2 2H0 + 1)1=2 (H0  1) :Here H0 is the energy at t = 0, which is a constant of motion1
H0   (0)2 + (0) (1 (0)2 )1=2 cos((0)):
2

a =(5.15)(5.16)The solution to the integral equation Eq. (5.14) is found in the literature [94, 95, 96] and iswritten in terms of the Jacobian elliptic function sn(u j m), where m is the elliptic modulus(usually denoted by k )(t) = a1=2 sn(u(t) j m):(5.17)
1=
2
Here m = a=b and u(t) = b (t + t )=2, where02t0 = 1=2 F [sin 1 ((0)=a1=2 ) j m]:(5.18)b 
Rx
Here F (x j m) = 0 dx[1 m2 sin2 (x)] 1=2 is the incomplete elliptic integral of the firstkind.
The elliptic functions and integrals are standard quantities that are implemented in common math packages, such as Mathematica. Given the parameters , , and in Eq. (5.10),
we can easily generate plots of the amplitude, which is given by a1=2 , and the period T of
oscillation, given by8
T = 1=2 K (m);b 

(5.19)

where K (m)  F (=2 j m) is the complete elliptic integral of the first kind. We showplots of the period and amplitude as functions of the initial relative phase in Figures 5.7
and 5.8. We see that as the initial relative phase decreases from (0) =  , both the period
and amplitudes increase. For (0) ! 0, we see the period increase to infinity, whichsignifies that this is a stationary state. Likewise, if (0) !  , the amplitude goes to zero
because this is also a stationary state. These two cases just correspond to the dressed statesconsidered in the previous chapter.
Within the nonrigid pendulum analogy, the case of  (0) = 0 and (0) =  corresponds
to the pendulum hanging at rest. For values of (0) close to  , this corresponds to justslightly displacing the pendulum, so that it undergoes simple sinusoidal oscillations. As thetilt angle is displaced further from  , the pendulum exhibits nonlinear behavior, because
the length depends on the angular momentum. The authors of [89, 90, 91] have exploredthe different regions of phase space of the nonrigid pendulum in detail for the case of the
double-well system.
Another interesting behavior of the solution is that the presence of higher harmonics
becomes more pronounced as (0) is decreased, as already alluded to in the previous section. In Figure 5.9 we show two cases of (0) = =2 and =16. We plot the amplitudeof the frequency component vs the frequency in units of the fundamental frequency !0 ,which is just the inverse of the period T . We see that for (0) = =2, the amplitude of
the fundamental frequency is nearly unity, with the third harmonic being the next largest
with an amplitude of 0:2%. Compare this to the case of (0) = =16, where the amplitudeof the fundamental frequency is only 90% and the third multiple is almost 10%, with theamplitudes of the other odd multiple frequency components trailing off.
77

Chapter 5 Weak Coupling—Nonlinear Josephson-Type Oscillations

200
180

Period (ms)

160
140
120
100
80
60
40
20
0
0
4

2

3
4

5
4

3
2

7
4

2

initial phase
Figure 5.7: The period of oscillations as a function of the initial relative phase (0) using
expression Eq. (5.19). The parameters = 1:05, = 0:482, and = 0:375 were foundby obtaining the solution to the time-independent GP equation as a function of the relative
population. The detuning was Æ = 0:482, chosen to make  = 0.
0.4
0.35

Amplitude

0.3
0.25
0.2
0.15
0.1
0.05
0
0
4

2

3
4

5
4

3
2

7
4

2

initial phase
Figure 5.8: The amplitude of oscillation as a function of the initial relative phase (0).

78

5.8 Summary010φ(0) = π/2

φ(0) = π/16

φ(0) = π/2−110| cn |2

−2

10

−3

10

−4

10

−5

10

1

3

5
7
n ω0

9

1

3

5
7
n ω0

9

Figure 5.9: Amplitude of the frequency components obtained from a discrete FFT of  (t)for two different cases of (0) = =2 and =16. The frequency is plotted in units of thefundamental frequency !0 . Note that some of the frequency components at the bottom of
the graphs are not exactly multiples of the fundamental frequency—this is most likely due
to numerical error.

5.8 Summary
In this chapter we have investigated the weakly coupled two-component system at zerotemperature. Our main result is that this system effectively obeys the same set of nonlinear
Josephson equations as the double-well tunneling system [89, 90, 91] due to the fact thatthe overlap between components is approximately constant, and the difference in chemical
potentials is roughly a linear function of the relative population  . This gives more credence
to our interpretation that the system can be thought of as a Josephson junction type ofsystem. This also allowed us to obtain a closed form solution for the system, whose generalbehavior can be understood in terms of the nonrigid pendulum.
An interesting extension of this study would be to treat the system at finite temperature. In [91], the authors introduce a phenomenological damping term in Eq. (5.3), andthus model the system at finite temperatures. Starting with the quantum kinetic equations
obtained in Chapter 2, we could derive an effective damping term from the collision terms
in the kinetic equations, instead of postulating the term based on physical intuition.

79

Chapter 5 Weak Coupling—Nonlinear Josephson-Type Oscillations

80

Chapter 6

Strong Coupling—Quantum
State Engineering
6.1 Introduction
The work presented in this chapter is closely tied to recent experiments at JILA lead by Eric
Cornell and Carl Wieman [7, 8, 9, 10, 11, 12, 13, 14]. It is useful to give a brief history of
how our present understanding of the strongly coupled two-component condensate emerged
as we struggled to fully comprehend the interesting behavior observed in experiments. The
most striking result of these joint experimental and theory investigations is the prediction ofa vortex and the subsequent observation of it in the laboratory. It is worthwhile to describe
the various stages of research leading up to this important result. This will help motivate
the main results presented in this chapter.
The first work on a two component system in a dilute Bose condensed gas was done
by Myatt et al. [7] at JILA, who were able to sympathetically cool atoms in two different
hyperfine levels below the critical temperature and create a mixture of condensates in the(1; 1) and (2; 2) states, which had an unexpectedly long lifetime. This was an importantresult, for it demonstrated that the spin-exchange scattering rate is suppressed for 87 Rb.
This was not known at the time, and allowed theorists to place tight bounds on the singlet
scattering length, assigning it a value that is nearly degenerate with the triplet scattering
length [97, 98, 99]. This near coincidence of the singlet and triplet scattering lengths isvery fortuitous for 87 Rb and is not the case in other alkali atoms, such as 7 Li and 23 Na.
As a result, the spin exchange rate for 7 Li and 23 Na is much larger than that of 87 Rb.
A spin exchange collision can flip the atoms into untrapped states, so that they are lostfrom the trap. This makes it very difficult to study magnetically trapped multicomponent
condensates of 7 Li and 23 Na atoms. However, this problem can be overcome for theseatoms by loading the condensate into a purely optical trap [28, 100].
The knowledge that magnetically trapped, multicomponent condensates could be studied in 87 Rb lead to the next experiment, in which a mixture of (1; 1) and (2; 1) atoms
were studied [8]. In this case, sympathetic cooling was not used, but instead, the mixture
was prepared by first cooling the (1; 1) atoms below the condensation point, and thentransferring atoms suddenly into the (2; 1) by applying a two photon pulse. In the previous
experiment, the mixture consisted of (1; 1) and (2; 2) states, which experience different

Chapter 6 Strong Coupling—Quantum State Engineering
trapping potentials due to their different magnetic moments. The advantage of using the(1; 1) and (2; 1) states is that their magnetic moments are nearly the same. The trapping
fields can be adjusted in such a way to cancel the effect of gravity, so that the atoms sit inconcentric harmonic potentials with the same spring constants [9]. The first study of thissystem investigated the dynamical response of the condensate to a sudden change in theinternal state by applying a very short  pulse to the (1; 1) atoms that transfered them tothe (2; 1) state. Due to the slightly different scattering lengths of the two states, the groundstate of the (2; 1) atoms is slightly smaller in size than that of the (1; 1) atoms, so that thesudden change in state resulted in a breathing of the condensate that could be characterized
in terms of its collective excitations [8].
If a =2 pulse is applied to the (1; 1) atoms instead of a  pulse, a mixture can beprepared. A number of interesting properties of this system were studied at JILA [10, 11].
One interesting feature of this system that was observed is that it is energetically favorable
for the atoms with the larger scattering length, (1; 1) to form a lower-density shell aroundthe (2; 1) atoms, which move inward [32, 101]. This behavior was studied in Chapter 4(see Figure 4.2) for the case of concentric traps. However, if the traps are displaced only
very slightly, it is energetically favorable for the two components to separate, with a finite
overlap, as shown in Figure 4.3. In the experiment reported in [10], the magnetic trapping fields were adjusted so that the potentials for the two states were slightly displacedalong the vertical axis. After applying a =2 pulse to the (1; 1) atoms, the system is in ahighly excited state; the two components push each other apart and slosh back and forth,while undergoing breathing excitations as well. After a duration of several trap periods, therelative motion between the two components damps away with the two components separated along the vertical axis. Although some damping of the relative motion is exhibitedin a numerical solution of the two-component GP equations [102], the condensates in theexperiment damp more strongly. This discrepancy is most likely due to finite temperatureeffects not contained in the GP equation.
In an interesting extension of this experiment, the relative phase between condensates was measured using a technique based on Ramsey’s method of separated oscillating
fields [103]. A mixture of condensates was prepared as described in the preceding paragraph, and after waiting a time TR long enough to allow the components to separate andsettle down, a second =2 pulse was applied to read out the phase difference accumulated
between the two condensates [11, 104]. By varying the time TR when the second pulse wasapplied, an interference pattern was obtained. Remarkably, even after undergoing damping
processes, the visibility of the fringes was reasonably sharp, at around 50% or so.
In the experiments described so far, the coupling drive was only applied in very shortpulses, much shorter than the period of the trap. However, one can imagine applying thecoupling drive continuously; we showed in the previous chapter that in the limit of a very
weak coupling, the system resembles a weak-link Josephson junction, which would be afeasible experiment to do. In the summer and fall of 1998, experiments at JILA weremoving in this direction. A number of technical issues had to be resolved first, however,like implementing a phase-contrast imaging system, and stabilizing mechanical sloshingof the condensate. Just as the experimenters were gearing up to study the weak coupling
limit, some very intriguing behavior of the condensate was observed that warranted (a greatdeal) of further attention. Consequently, the weak coupling case has yet to be investigatedexperimentally.
We now describe the intriguing data: It was found in experiments that if the strongcoupling field is applied for a long time, instead of in short pulses, the Rabi oscillations826.1 Introduction

1

0.5
0.25
0
1
0.75

1

N (t) population fraction

0.75

0.5
0.25
0
0

25

50

75
time (ms)

100

125

150

Figure 6.1: Modulation of the fractional population in the (1,-1) state. The top line isexperimental data [13] while the bottom line is the result of a numerical calculation of thethree-dimensional, two-component Gross-Pitaevskii equation. The coupling strength anddetuning were chosen for the calculation to be = 350 Hz and Æ = 188 Hz, respectively,
while the detuning in the experiment was shifted from this.

83

Chapter 6 Strong Coupling—Quantum State Engineering

200
175

revival time (ms)

150
125
100
75
50
25

two−state modelexperiment0−400 −300 −200 −100
0
δ+δ

MF

100
(Hz)

200

300

400

Figure 6.2: The revival time vs. the detuning. This shows the trend of the revival time toincrease with decreasing detuning. The red dots are experimental data points, shown witherror bars; they have been shifted along the x-axis by an overall detuning offset. The blueline corresponds to the two-mode model described in section 6.3.1. Experimental data is
courtesy of M. Matthews, D. Hall, and P. Haljan.

between the two components appear to collapse and revive over a time long compared tothe trap period. We show an example of this behavior in Figure 6.1. We find that thesolution of the coupled GP equation also exhibits modulations of the Rabi oscillations.
In both experiment and calculations of the GP equation, it is found that the revival time,or period of modulation, increases as the detuning is decreased about some central value,
as shown in Figure 6.2. The most serious discrepancy between numerical solution of theGP equation and experiment was that the center of the peak in the revival time of theexperimental data was not centered at the same detuning as predicted by the coupled GP
equation but was shifted by about 50 Hz. We later realized this was most likely due tothe spatially-dependent energy shift of the untrapped intermediate state in the two-photontransition, an effect that gives rise to a spatially dependent bare Rabi frequency1 (z ). The
displacement 2z0 between the traps also plays a crucial role in the collapse and revival
behavior; when z0 = 0 is set to zero, so that the two components sit in concentric traps,
this effect goes away so that the Rabi oscillations are no longer modulated.
Another striking property of the system was that the densities exhibit a double-peaked

( )=
=

4 ()

( )

1 This can be seen from00zz , where the intermediate-state detuning ÆabzÆab mgz .
a b = ÆabHere, m is the mass of the atom and g is the gravitational acceleration. With Æab
mgz , a Taylor expansion
gives zz , where 
mg=Æab .

( ) +

84



1

4

2

3

5

6

z verticalz vertical

6.1 Introduction

1

4

x horizontal

x horizontal

a.

b.

2

3

5

6

Figure 6.3: Double-peaked structures in the column-integrated density of the (2; 1) atoms.
In each plot we show one full Rabi cycle divided into six shots. Plot (a) shows the very
first Rabi cycle, where the shape of the (2; 1) component is the same as the ground state. In
Plot (b) we show the density during a Rabi cycle taken during the collapse of the Rabi oscillations. The density cycles between the single- and double-peaked shapes. The (1; 1)
component exhibit the same behavior, though it is shifted by 1=2 a Rabi cycle. Experimental data is courtesy of M. Matthews, D. Hall, and P. Haljan.
structure that appeared gradually, becoming most apparent during the collapse of the Rabi
oscillations, and then gradually disappeared as the Rabi oscillations revived. An example isshown in Figure 5.4. During the first Rabi cycle after the drive is turned on, the densities ofeach component are identical, and have the shape of the ground state (plot (a)). However,a double peaked structure gradually forms in the densities of both components. In plot (b)we show an example of the density of the (2; 1) component taken from the region around25 ms, where the collapse occurs. During one Rabi cycle, the density of the (2; 1) atoms
goes from the double-peaked structure in frame 1 to being single-peaked in frame 4 andthen back again to the double-peaked structure in frame 6. By the time the Rabi oscillations
revive, the shape of the density also revives to their initial shapes shown in plot (a). Thisbehavior was very reproducible in experiments. Also, numerical solutions of the coupled
GP equations exhibit the same behavior.
Two different approaches emerged in our collaborative efforts to understand this unexpected behavior, which, at first glance, seem to have very little in common. One approach
is motivated by previous work done in condensed matter physics on the A-phase of superfluid 3 He [105]. The other approach, which will be the focus in this chapter, is based on
ideas familiar from quantum optics.
The first approach is useful in gaining an intuitive understanding of the phenomenon.
We can describe the coupled two-component system in terms of the hydrodynamic-like
Bloch equations Eq. (5.6) presented in section 5.4. There we saw that in the unphysical infinite-mass limit, where the atoms are “frozen” in place at each position, the current
terms i ( ; t) and ij ( ; t) can be neglected, leaving us with a set of position dependent
Bloch equations; atoms at each position undergo Rabi oscillations independently of theirneighbors. Due to the spatially dependent detuning (z ) caused by the displaced traps, theRabi oscillations at each position dephase across the condensate. If we wait long enough

j r

j r

85

10

10

8

8

6

6

4

4

2

2

z (µm)z (µm)

Chapter 6 Strong Coupling—Quantum State Engineering

0
−2

0
−2

−4

−4

−6

−6

φ0

−8
−10
−20

−10

0
x (µm)

10

φ1

−8
20

−10
−20

−10

0
x (µm)

10

20

Figure 6.4: Dipole topological mode for a two-component condensate. The ground state(left) and first excited state (right) have equal macroscopic populations and must be foundself-consistently (see Appendix (C)). The mean-field interaction between the states causes
the ground state to be “pinched”, so that it is much narrower than it would be if all of theatoms were in the ground state. The peculiar two-peaked structure in plot (b) of Figure 6.3
corresponds to the dipole topological mode plotted here.
we will see a decay in the integrated population of either state. In order to get a revival,
however, we must account for the spatial coherence in the condensate described by thecurrent terms. Due to the interplay of the spatial coherence and the coherent internal dynamics, the condensate can untwist its winding phase, so as to get back to its original state.
This twisting and untwisting of the condensate phase is an example of a more general phenomenon found in any system where the U (1) gauge transformation2 is mixed up withsome other continuous symmetry, in this case the SU (2) symmetry of the spin [105]. We
will not discuss this approach in any detail, but refer to the rigorous development presented
in the work of Ho [105].
In the second approach, the system is described by the following “two-mode” modelj (t)i = ( 1 (t)c0 (t) j0 i + 2 (t)d1 (t) j1 i)j1i+ ( 2 (t)c0 (t) j0 i + 1 (t)d1 (t) j1 i)j2i:(6.1)
The states j1i and j2i are the internal hyperfine states, while the states j0 i and j1 i are
spatial states; j0 i is the ground state, while j1 i is a dipole topological mode, which we

introduced in Chapter 4. In Figure 4.9 we considered such a topological excitation for the
one-dimensional solution. In Figure 6.4, we show the corresponding states for the three
dimensional problem, for the case of equal populations in each state. The coefficients i
describe the rapid Rabi cycling, while the coefficients c0 and d1 describe a slow cycling
between the ground state j0 i and the dipole mode j1 i. If we project Eq. (6.1) onto
an internal state jii, we find that each component cycles rapidly between the two modesj0 i and j1 i. For short times, the coefficient d1 is very small, so that its contribution to
the density is negligible. However, as d1 grows in magnitude due to an effective dipole
coupling z caused by the displaced traps, the two-peaked structure of the dipole becomes
2 The Gross-Pitaevskii equation is invarient with respect to a change in the global phase of the condensate.

866.2 Derivation of the model
prominent. The two mode model accounts for all of the major features of system observed
experimentally in a very quantitative manner. We will derive this model and discuss it indetail throughout the chapter.
Once we had developed a detailed model to describe the system, it was clear that the
behavior observed in the experiment was only one specific case of a more general behavior
and we soon realized that other situations could be implemented. The displacement of thetraps along the vertical axis acts as a kind of drive that couples the initial ground state condensate to a dipole mode. We generalized this driving mechanism in order to excite othermodes with alternative symmetries. In particular, we considered the hypothetical situationof displacing the traps and then rotating them about the origin, as shown in Figure 6.12.
Our two-mode model was easily extended to treat this situation, and we found that oneof the components could be prepared in a vortex with one unit of angular momentum per
atom if the trap rotation frequency was taken to be approximately equal to the effective
Rabi frequency. We have encountered such a state in Figure 4.7 of Chapter 4. With thetwo-mode model providing some guidance, an experimental implementation of the idea
was soon realized at JILA in the laboratory of Eric Cornell, and the first observations of avortex in a dilute gas BEC were made.
In the following section, we present a derivation of our model for arbitrary driving
symmetries. We then consider the specific case of static, offset traps in section 6.3 wherea dipole mode is excited. We present results of calculations that help further clarify the
behavior of the system. In section 6.4 we present a technique for preparing somewhat arbitrary topological modes and show results of calculations for various examples, including
a vortex.

6.2 Derivation of the model
We now develop a simplified model based on a series of approximations that will lead
us to the rather elegant two-mode description. For this problem, the external internal
representation presented in section 4.5 is most useful. We start with a more general GP
equation than Eq. (4.20) by replacing the linear offset z0 z of the traps by a more general^ 1 (r; t) and we will include the possibility that the bare Rabi fretime-dependent term H
quency varies linearly with z —as mentioned earlier, this is due to the spatially dependent
detuning from the intermediate state in the two photon transition.. The most general formof the GP equation describing our system, then, is






@
Æ 
j
(t)i = H^ 0 ^1 + ^1
^x + ^z + H^ 1 (t)+ H^ 2 ^z + H^ 3 ^x j (t)i: (6.2)@t2
2^ 0 ^1 acts on each internal state identically, with H0 (r) given by
The term H
1
1  2 2 2H0 (r) = r2 +
 + z + h (t)jP^r ^+ j (t)i :
(6.3)
2
2

i

The first term is the kinetic energy, and the second is a harmonic potential centered at^+ given in
the origin. The last term is a mean-field interaction term, with the matrix ^ 1 (t)+ H^ 2 ) ^z can be thought of as a time dependent
Eq. (4.22) in Chapter 4. The term (H^ 1 (t) ^z acts with opposite sign on each state.
and spatially varying detuning. The term H
In general, H1 (r ; t) is time-dependent and spatially varying with the form

H1 (r; t) =  f (r) cos(!r t) + g(r) sin(!r t) :

(6.4)
87

Chapter 6 Strong Coupling—Quantum State Engineering
We will consider specific forms for H1 (r ; t) throughout the chapter and specify , f , andg in particular cases. The term H^ 2 ^z also acts on each state with opposite sign andoriginates in the mean-field interaction with the form

H2 (r) = h (t)jP^r^ j (t)i:(6.5)^ is given in Eq. (4.22) in Chapter 4. Finally, the term H^ 3 ^x couples
The matrix 
the internal states, with H3 (z ) being a static term varying linearly with position along thevertical axis
H3 (z ) = z:
(6.6)
2This term arises due to a subtle effect related to intermediate state in the two-photon transition not being trapped, which causes an effective linear variation in the bare Rabi frequency,
where  is the slope of this variation (see footnote 1 above).

6.2.1 Simplifying the mean-field terms
Throughout the chapter, we will present numerical calculations of Eq. (6.2). However, inorder to gain an intuitive understanding of the behavior described by Eq. (6.2), we now
make a series of approximations that will lead us to a rather simple model, which provides
us with a great deal of predictive insight. The first simplification we make is to observe thatfor 87 Rb, the term H2 can be rewritten as
H2 (r) = D n(r; t);
2N

(6.7)

r

where3 D = 1 12 = 12 2 , and n( ; t) is the total density. We have found in
most of our calculations that the total density is very nearly invariant in time. We therefore
assume here that n( ) is static, and, in addition, we approximate it by the Thomas-Fermi
limiting form. This allows us finally to write H2 asr

Æ
H2 (r) = MF + fMF(r);
2

(6.8)

where ÆMF = D TF =12 is an overall mean-field shift of the internal hyperfine levels, TF is the chemical potential for the total population in the Thomas-Fermi limit, andfMF( ) = D ( 2 2 + z 2 )=(412 ) is the Thomas-Fermi form of the density. Typically, for
populations in the range N = 8  105 to 106 atoms, we find this shift is ÆMF = 30 Hz to
35 Hz in a trap with z = 65 Hz and xy = 24 Hz.
The mean field term in Eq. (6.3) can also be simplified if we make the approximations
that (12 + 1 )=2  12 and (12 + 2 )=2  12 . Within this approximation we write
H0 ! H00 asr

H00 (r) =1 2 1 2 2 2
r + 2 [  + z ] + 12 (j 1 (r; t)j2 + j 2 (r; t)j2 ):
2

(6.9)

Note that for this term we do not take the total density to be static, but allow it to evolve intime.
3 This simplification can be made for 87 Rb, and is only approximately true (i.e. within the error of the determination of the singlet and triplet scattering lengths).

886.2 Derivation of the model
Incorporating these two approximations into Eq. (6.2) gives us




@
Æ0 
j
(t)i = H^ 00 ^1+^1^x + ^z + H^ 1 (t)+f^MF ^z +H^ 3 ^x j (t)i; (6.10)@t2
2
where we have rewritten the detuning to include the mean-field shift Æ 0 = Æ + ÆMF . Looking
at Eq. (6.10), we can see that in the absence of the last three terms (eg. H1 = fMF = H3 =0), the spatial and spin degrees of freedom are uncoupled, so that the internal states simplyiundergo the usual Rabi cycling independently of the motion of the condensate. It is usefulto think of H1 as a driving term that excites a topological mode in the system. We willconsider specific examples in later sections. For the purposes of exciting a topologicalmode, the term fMF due to the mean-field interaction acts as a contamination. We willshow that the physical parameters can be chosen to suppress its effect on the system. We
will show that the term H3 helps explain an overall shift in the peak of the revival time(shown in Figure 6.2) for the case of offset traps.

6.2.2 Interaction picture representation
In this chapter we are concentrating on the situation where the coupling is strong, so thatthe frequency of the Rabi oscillations is significantly larger than the trap frequency z .
In this case, the internal spin dynamics and the motion of the condensate in the trap occur
on two different time scales. Therefore, it is useful to go to a rotating frame that eliminates the second term in Eq. (6.10) describing the fast Rabi oscillations between the twointernal states. In the rotating frame, we will be able to understand more clearly how theterms H1 (t), fMF , and H3 in Eq. (6.10), which couple the motional and spin dynamics
of the condensate, effect the system on a time scale much longer than the period of Rabi
oscillation.
We go to the rotating frame, or interaction picture, by making a unitary transformation
using the operator
Æ0
UI (t) = e i ^1 ( 2 ^x + 2 ^z ) t :(6.11)
This can be rewritten in the equivalent formisin( e =2 t) ^x + Æ0 ^z ;(6.12)ep 2 02where e =+ Æ . The state vector j (I ) (t)i in the rotating frame is related to thestate vector in the lab frame j (t)i byj (I ) (t)i = U y j (t)i :(6.13)UI (t) = ^1cos( e =2 t)^1

I

In the rotating frame, the system evolves according toi@ (I )
j (t)i = H^ (I ) (t)j (I ) (t)i ;@t(6.14)^ (I )(t) is the interaction Hamiltonian
where HH^ (I ) (t) =+H^ 00 ^1 + H^ 1 (t) + f^MF
x (t)^x + y (t)^y + z (t)^z
(6.15)
H^ 3
x (t)^x + y (t)^y + z (t)^z :89

Chapter 6 Strong Coupling—Quantum State Engineering^ 00 , H^ 1 (t), f^MF , and H^ 3 are unaffected by the unitary transformation to theNote that H
rotating frame. The time-varying coefficients x (t), y (t), and z (t) areÆ0 
1 cos( e t)e esin( e t)y (t) =e2
Æ 02z (t) = 2 + 2 cos( e t);eewhile the coefficients x (t), y (t), and z (t) are2Æ 02
x (t) = 2 + 2 cos( e t)eeÆ0sin( e t)y (t) =e
Æ0 
1 cos( e t) ;z (t) =e e

x (t) =

(6.16)(6.17)

6.2.3 Coarse graining approximation
At first sight it appears that we have actually made the problem more complicated by goingto the interaction picture representation due to the appearance of the extra time-varying
terms proportional to i (t) and i (t). However, in this rotating frame, we expect thesystem j (I ) (t)i to evolve on a much slower time scale than the period of Rabi oscillation.
We therefore time average over the rapidly oscillating terms, which is equivalent to coarsegraining Eq (6.14). The time averaging depends on the frequency !r of H1 (t), so we
consider two cases separately, a slow or static drive !r  !z , and a fast drive !r  e(remember we are considering the strong coupling limit so that !z << e ).
Slow or static drive !r  !z and small detuning Æ 0 
If the drive varies slowly on the time scale of the period of Rabi oscillation, then we canassume H1 (t) is approximately constant over one Rabi cycle. In this case, coarse graining
Eq (6.14) allows us to reduce the coefficients i (t) and i (t) to x = z = Æ 0 = 2e ,y = y = 0, z = Æ02 = 2e , and x = 2 = 2e . This leads to the simplified form of
Eq (6.15)

H^ (I ) (t) = H^ 00+ H^ 3^1 + H^ 1 (t) + f^MF 2Æ02 ^x + 2 ^z

e

 Æ0

Æ02 

^
+
2e x
2e ^ze(6.18)The interaction Hamiltonian can be brought to an even simpler form if we assume thedetuning is small Æ 0  . In this case, we can drop the term proportional to Æ 02 = 2e . We
also assume that H3 is much smaller in magnitude than H1 , which allows us to also drop
^ 3 Æ0 = 2e ^z . This finally leaves us withthe term H
h Æ0
2 i

H^ (I ) (t) = H^ 00 ^1 + 2 H^ 1 (t) + f^MF + 2 H^ 3 ^x :ee90(6.19)

6.3 Static drive due to offset traps
In the sections that follow, we will investigate this regime—of a slow or static drive and asmall detuning—in detail for specific choices of H1 (t).
Fast drive !r e and large detuning Æ0 

In the case of a fast drive we must time average the product of H1 (t) and the coefficients
i (t) and i (t). These products can be written in terms of cosines and sines of the sum( e + !r ) and difference ( e!r ). If we make the assumption that !r  e , then we
can drop all of the counter-rotating terms oscillating at the frequency ( e + !r ), but keep
the terms rotating at ( e!r ) when we time average. As for the static terms proportional
to fMF and H3 , the same time averaging we did in the previous subsection applies. The
result is the following approximate form of the interaction Hamiltonian
 Æ0
 2

Æ02 
Æ0
H^ (I )(t) = H^ 00 ^1 + f^MF
2e ^x + 2e ^z + H^ 3
2e ^x + 2e ^zn Æ0 
% g sin(t) f cos(t) x + % g cos(t) + f sin(t) y
+
2 e
e

 o
+
f cos(t) % g sin(t) z ;(6.20)ewhere  = e!r and % = !r =j!r j gives the direction of rotation.
This looks like a rather complicated result, however, it can be further simplified byassuming a very large detuning Æ 0  . This allows us to neglect terms of order ( = e )2 .
^ (I )(t) as
If we further take jÆ 0 j  e then we can write H^x + H^ 3
H^ (I ) (t) = H^ 00 ^1 + f^MF ^z +ea^ys ^+ eit + a^s ^ e it ;
2 e

e

^z
(6.21)for Æ > 0, andH^ (I ) (t) = H^ 00 ^1 + f^MF ^z^x H^ 3
^z
e
e



(6.22)a^ ^ e it + a^ys ^ eit ;+2 e s +for Æ < 0. Here we have defined a^s = f^ i g^ and a^ys = f^ + i g^, and we have taken % = 1.
Also, the matrices ^+ and ^ are defined as ^+ = (^x + i^y )=2 and ^ = (^x i^y )=2.
We will investigate these results in more detail later in this chapter when we consider thisregime of a fast drive and a large detuning.

6.3 Static drive due to offset traps
As described in Chapter 4, a typical situation for a condensate of 87 Rb atoms in a TOP
trap is for there to be a static offset between the traps for the two hyperfine states j1i andj2i. This offset was a primary ingredient in drawing an analogy to the Josephson junctionin the weak-coupling limit described in the previous chapter. By adjusting the quadrupole
gradient and the intensity and frequency of the rotating field, various trap configurations
91

Chapter 6 Strong Coupling—Quantum State Engineering
can be attained. In this section, we consider the case where z = 65,  = 24 Hz, and thetraps for each state are separated by 2z0 = 0:4 m along the vertical axis (see Figure 5.1).
^ (t) = z z in Eq. (6.19), so that H^ (I )(t)This corresponds to setting the drive term to H

1

can be written as0H^ (I )(t) = H^ 00 ^1 + C1 z + C2 ( 2 2 + z 2 ) ^x ;(6.23)
where C1 = (z0 Æ 0  =2)( = 2e ) and C2 = [D =(412 )](Æ 0 = 2e ). The parameterC1 varies linearly with detuning Æ0 , with an overall offset given by (=2) 2 = 2e . For
now, we will neglect this offset in the development that follows, by taking  ! 0, so thatC1 ! z0 Æ0 = 2e . Later in this section we will return to this issue and briefly discuss

how this offset modifies our results. For typical values of the physical parameters, the third
term in Eq. (6.23) arising from the mean-field interaction due to the difference in scattering
lengths is small, with C2  0:002, compared to C1  0:04. Therefore, we also neglect thisterm by taking C2 ! 0.
With the approximations   0 and C2  0, Eq. (6.23) becomesH^ (I ) (t) = H^ 00 ^1 C1 z ^x ;(6.24)We refer to this result Eq. (6.23) as the coarse-grained, small detuning (CGSD) model todistinguish it from the two-mode model presented below, which makes further assumptions.

6.3.1 Two-mode model
It is useful to define a basis of motional states with which to describe the system in the^ 00 , which satisfy
rotating frame. A natural choice is the set of instantaneous eigenstates of Hhi
1 2 1 2 2 2r+
 + z + 12 n~ (r; t) i (r) = i i (r)22
Z 1
i (r) j (r) d3 r = Æi;j ;1(6.25)

where the index i refers to all of the relevant quantum numbers that uniquely specify eacheigenstate, i = fnz ; n ; n g, given the cylindrical symmetry of the system. In general,many modes can be occupied and the state vector is writtenj (I ) (t)i =

r

Xh

i

r

i

ci (t)ji ij1i + di (t)ji ij2i ;(6.26)

where i ( ) = h ji i. The density appearing in Eq. (6.25) is thenn~ (r; t) =

X

i

2 X
2
ci (t)i (r) +di (t)i (r) :(6.27)i

It is clear that the set of coupled eigenvalue equations given in Eq. (6.25) is nonlinear andrequires a numerical procedure that will converge upon the solution in a self-consistentmanner. The eigenstates i ( ) and eigenenergies i depend on time implicitly throughthe coefficients ci (t) and di (t), however we do not show this time dependence in order
to simplify the notation. We assume that the eigenbasis evolves slowly in time so thatthe adiabatic condition is satisfied [106]. These topological modes i ( ) are similar to the

r

r

92

6.3 Static drive due to offset traps
states plotted in Figure 3.7 for the single component and in Figure 4.9 in the two-component
case.
Based on the experiment reported in [13] the initial motional state of the system is(I) ( ; 0) = 0 ( z0 z^); the system is in the ground state of H^ 0 , but displaced from the0
1
origin along the vertical axis by z0 . This displacement is small compared to the width wzof the condensate z0 =wz  0:01. We therefore approximate the initial state of the system
as j (I ) (t)i = j0 ij1i.
The system in the rotating frame evolves according to the Hamiltonian described by^x couples the internal states j1i and j2i via ^x . It also
Eq. (6.24). The term C1 z drives transitions between motional states via the dipole operator z^. The dipole matrix
element hz iij = hi jz^jj i is the largest between neighboring states and falls off quickly
as ji j j increases. For a small coupling parameter C1 , we expect the coupling to thefirst excited state j1 i to dominate the other transitions, making the evolution of the system
predominantly a two state evolution. We therefore make the approximation that the system
occupies only two modesrr

j (I ) (t)i = c0 (t) j0 ij1i + d1 (t) j1 ij2i ;(6.28)
where j0 i is the ground state i = f0; 0; 0g and j1 i is the first excited state with odd parity
along the z-axis i = f1; 0; 0g.
If we substitute this ansatz into Eq. (6.14), using the Hamiltonian described by Eq. (6.24),
we get the equation of motion for the coefficients c0 (t) and d1 (t)


c_
i d_0
1



=



0
C1 hz i01

C1 hz i01
1



c0
d1



;

(6.29)

where we have neglected the time rate-of-change of the slowly-varying adiabatic eigenbasis. This coupled pair of equations must be solved numerically by updating the energies
i and the dipole matrix element hz i01 from solving Eq. (6.25) at each time step. However, in order to see how the behavior depends on the various physical parameters, one
can obtain a simple estimate of the solution by fixing i and hz i01 to their initial values.
In this case the solution of Eq. (6.29) is trivial and is given by c0 (t) = cos( 01 =2 t)i(01 = 01 ) sin( 01p=2 t) and d1 (t) = i(2C1hz i= 01 ) sin( 01 =2 t), where 01 =1 0 and 01 = 4C12 hz i2 + 201 . In the rotating frame, the system oscillates between the two states at a frequency of 01 , which is much slower than the effective Rabi
frequency e .
The oscillation frequency 01 increases with increasing detuning Æ 0 and increasing trap
separation z0 through the coupling parameter C1 . The amplitude of oscillation depends
h!z due toon the energy spacing between modes 01 . This splitting is a fraction of 
the effect of the mean-field interaction: 01 decreases with increasing population N . In
addition, the dipole matrix element hz i increases with increasing N , since the width of thecondensate increases with increasing population. The effect of the mean-field, then, is toenhance the coupling between the ground and dipole modes.
The solution in the lab frame can be obtained by applying UI (t) from Eq (6.12) to j (I) i
in Eq. (6.28) to yieldj (t)i = ( 1 (t)c0 (t) j0 i + 2 (t)d1 (t) j1 i)j1i+ ( 2 (t)c0 (t) j0 i + 1 (t)d1 (t) j1 i)j2i ;(6.30)
93

Chapter 6 Strong Coupling—Quantum State Engineering

1N1(t) population

0.8

0.6
1

2

3

0.4

0.2

0
0

20

40
60
time (ms)80

100Figure 6.5: Modulation of the Rabi oscillations. The fractional population in state j1i isplotted as a function of time, obtained from a numerical solution of the one dimensionalversion of Eq. (6.10). The values of the various parameters are given in the text. In Figure 6.6, the densities for both states are shown for three different Rabi cycles designated by
the circled numbers in this plot.
where the rapidly varying coefficients are 1 (t) = cos( e =2 t) i(Æ 0 = e ) sin( e =2 t)
and 2 (t) = i( = e ) sin( e =2 t). Eq. (6.30) is the main result of this section, withwhich we can explain the essential properties of the system. During the first few Rabi
cycles t  1= e , the coefficient d1 (t)  0, so that the solution for short times isj (t)i = ( 1 (t)j1i + 2 (t)j2i) j0 i. That is, for short times, the internal and external
degrees of freedom appear to be decoupled and the system simply oscillates rapidly between internal states. However, for longer times, the coefficient d1 (t) grows in magnitude
as c0 (t) correspondingly decreases. This results in a modulation of the Rabi oscillations.
Furthermore, a two-peaked structure in the density appears, associated with the first-excitedstate j1 i.

6.3.2 Results of calculations
The main goal of this section is to illustrate the behavior of the system by showing resultsof numerical calculations. For this purpose, it is useful to treat the system in only one
dimension—along the vertical axis. Values of most of the physical parameters are given in
Table 6.3.2. Values of the remaining parameters are stated for each case considered in the
text.
94

6.3 Static drive due to offset traps

density (arbitrary units)

1

2

3

1

1

1

2

2

2

3

3

3

4

4

4

5

5

5

6

6

6

position (arbitrary units)

Figure 6.6: Snapshots of the density of each state for three different Rabi cycles corresponding to the three circled numbers in Figure 6.5. The first two strips resemble the two
strips of snapshots taken from experimental data shown in Figure 6.3. The solid line is thedensity of the j1i state, while the dashed line is that of the j2i state. Each snapshot within
a set is numbered in sequential order. The first set starts at t = 0 ms, and runs for a fullRabi cycle 1.41 ms. The second and third sets begin at t = 45:2 ms and t = 90:3 ms,
respectively. The time increment between snapshots is t = 0:28 ms for all three sets.
Understanding the dual dynamicsR

In Figure 6.5 we plot the fractional population of state j1i, given by N1 (t) = jhz jh1j (t)ij2 dz ,for the case of = 700 Hz and Æ 0 = 100 Hz. This is a numerical solution of Eq. (6.10)
with the terms fMF and H3 set to zero. The population is cycling rapidly at the effective
Rabi frequency e = 707 Hz, while simultaneously being modulated at a much lower
frequency of about 11 Hz.
In order to visualize how the spin and motional dynamics become entangled over atime long compared to the Rabi period, we show snapshots of the density of each state in
Figure 6.6. Three different sets of snapshots are shown, corresponding to the three circled
numbers in Figure 6.5. A full Rabi cycle is shown for each set. The first two strips resemble
the two strips of snapshots taken from experimental data shown in Figure 6.3. The first set
begins at t = 0 with all of the atoms in the j1i internal state and in the mean-field groundstate of the trap j0 i. During this first Rabi cycle, the shape of the density profile foreach internal state does not change much—only the height changes. That is, the motional
state remains the ground state while population cycles rapidly between internal states, as
95

Chapter 6 Strong Coupling—Quantum State Engineering
1N(I)(t) population
1

0.8

0.6

0.4

0.2

0
0

20

40
60
time (ms)80

100Figure 6.7: The fractional population of the j1i state in the rotating frame. The solid lineis the solution given by the CGSD model, while the dot-dashed line corresponds to thesolution of the two-mode model. If the two-mode model is extended to include coupling
to the first even-parity excited mode, then we get better agreement to the CGSD model, as
shown by the dashed line.
discussed below Eq. (6.30).
The second set of snapshots in Figure 6.6 is taken at around t = 45 ms, which is halfway
through the modulation. The density profiles for each spin state cycle rapidly between asingle-peaked and a double-peaked structure. For example, in the first snapshot, the j1istate is in the single-peaked structure, while the j2i state is in the double-peaked structure,but halfway through the Rabi cycle the situation is reversed, as shown in the third andfourth snapshots. Finally, at about t = 90 ms when the amplitude of the Rabi oscillations
has revived, the third set shows that the motional and spin degrees of freedom appear to be
decoupled again, with the density profile of each spin state appearing as it did during thefirst Rabi cycle.
This peculiar behavior is most easily understood by going to the rotating frame. In(I )Figure 6.7, we plot the fractional population in the j1i state in the rotating frame N1 (t) =R
jhz jh1j (I ) (t)ij2 dz . The solid line corresponds to the CGSD model. In the rotating^ z0 ^xframe, population is slowly transferred out of the j1i state due to the coupling from H
in Eq. (6.18).
In the rotating frame, the system is being excited out of the ground state j0 i due to thedipole coupling Hz0 . This can be seen in the top strip of snapshots in Figure 6.8, where thedensity of each spin state in the rotating frame is shown, corresponding to the solid line in
Figure 6.7. Initially, all of the atoms are in the j1i internal state and the mean-field groundstate of the trap j0 i. Due to the dipole coupling, population is transferred out of the groundstate. The strongest coupling is between the ground j0 i and the first excited j1 i modes.
These eigenmodes are shown in the bottom strip of Figure 6.8. They evolve slowly in timeas the coefficients c0 (t) and d1 (t) change. For example, initially the ground state is justthe Thomas-Fermi-like ground state, since all of the population is in that state. However, at96

density

6.3 Static drive due to offset traps

t = 23t = 45

t = 68t = 90φi(z)t=0z position (arbitrary units)

Figure 6.8: Ground and dipole states. The top strip of this plot shows snapshots of the density of each state corresponding to the solution of the CGSD model given by the solid line
in Figure 6.7. The bottom strip shows the corresponding two self-consistent eigenmodes
given by the solution of Eq. (6.25). The times of each snapshot are shown in the region
between the two strips, in units of milliseconds. The solid line corresponds to the densityof the j1i state, while the dashed is that of the j2i state.

t = 45 ms, about one-third of the population is in the first excited mode, which pinches theground state due to the mean-field interaction term. That is why the self-consistent groundstate at t = 45 ms is narrower than at t = 0.
It is clear from Figure 6.7 that the low-frequency modulation of the rapid Rabi oscillations in the lab frame is just the frequency of oscillation in the rotating frame between thestates j0 ij1i and j1 ij2i. This is reflected in the two-mode solution given by Eq. (6.30),
which also helps explain the peculiar behavior of the densities shown in Figure 6.6. In
the lab frame the system is cycling rapidly between the two modes shown in Figure 6.8.
h!z and 1 = 13:7 h!z , which makes
The initial values of the energies are 0 = 13:6 
01 = 0:1 h!z . This small energy splitting is due to the effect of the mean field, since inthe limit N ! 1 these energies move apart by a factor of ten, which greatly reduces thecoupling between the modes and thus greatly reduces the modulation effect.
If we make the two-mode ansatz and solve Eq. (6.29), we get the dot-dashed line in
Figure 6.7. The discrepancy from the solid line arises due to a weak coupling between thefirst j1 i and second j2 i excited modes. If we extend our two-state model to include thisthird mode, we get the dashed line in Figure 6.7, which nearly sits on top of the solid line.
In this case, the second excited mode j2 i gains less than 5% of the total population.
Dependence on detuning
In Figure 6.9, we show how the behavior of the system depends on the detuning Æ 0 . The
Rabi frequency = 700 Hz was held fixed for each plot while the detuning was varied
from zero at the top Æ 0 = 0 to Æ 0 = 200 Hz in the bottom plot. As predicted by the coupling
parameter C1 = z0 Æ 0 = 2e in the CGSD model, no coupling between motional states
97Chapter 6 Strong Coupling—Quantum State Engineering

1
0.5
0
1

N1(t) population

0.5
0
1

0.5
0
1
0.5
0
0

20

40

60

80

100 120time (ms)

140

160

180

200

Figure 6.9: The fractional population in the j1i state for four different values of the detuning, obtained from a numerical solution of Eq (6.10). Starting from the top, the detuning isÆ0 = 0, Æ0 = 50 Hz, Æ0 = 100 Hz, and Æ0 = 200 Hz. The values of the other parameters are
given in the text.

986.3 Static drive due to offset traps
200

period of modulation

175
150
125
100
75
50
25
0
−400 −300 −200 −100
0100
δ detuning

200

300

400

Figure 6.10: The period of modulation as a function of detuning Æ 0 . The dashed line corresponds to the numerical solution of the one-dimensional version of Eq. (6.10), while thesolid line was obtained from a numerical solution of the two-mode model Eq. (6.29). The
Rabi frequency was = 700 Hz.
occurs if Æ 0 = 0, and thus the Rabi oscillations experience no modulation. As Æ 0 is increased
the motional-state coupling becomes stronger and we expect the modulation frequency toincrease. The amplitude of modulation also increases as the detuning is increased.
We show the dependence of the period of modulation on detuning more explicitly
in Figure 6.10. The dashed line is the numerical solution of the full problem given by
Eq. (6.10), while the solid line is the numerical solution of the two-mode model given by
Eq. (6.29). Here we would like to point out that, had we included the spatial variation of
the bare Rabi frequency described by H3 , the peak of the curve in Figure 6.9 would not sit
at zero detuning, but would be shifted by (=2) 2 = 2e , where  is the slope of the spatial
variation of the bare Rabi frequency. (see Eq. (6.23)). In the experiments at JILA, a shift
was also seen, and we attribute it to this effect.
Dependence on trap displacement
In Figure 6.11, we show how the behavior of the system depends on the trap displacement

z0 . The Rabi frequency = 700 Hz and the detuning Æ0 = 100 Hz were held fixed, while
the trap displacement was varied from zero z0 = 0 in the top plot to z0 = 1 m in thebottom plot. Again, the coupling parameter C1 predicts no modulation if z0 = 0. As z0 isincreased, the frequency of modulation increases as the system is driven harder. However,for the large separation in the bottom plot, the modulation becomes highly irregular and thetwo-mode model most certainly breaks down. This behavior may be chaotic and warrants
further investigation.
99

Chapter 6 Strong Coupling—Quantum State Engineering

1
0.5
0
1

N1(t) population

0.5
0
1

0.5
0
1
0.5
0
0

20

40

60

80

100 120time (ms)

140

160

180

200

Figure 6.11: The fractional population in the j1i state for four different values of the trap
displacement z0 , obtained from a numerical solution of Eq (6.10). Starting from the top,
the displacement is z0 = 0, z0 = 0:1 m, z0 = 0:4 m, and z0 = 1:0 m. The values ofthe other parameters are given in the text.

Table 6.1: Values used for the various physical parameters appearing in our calculations.
The scattering lengths are taken from [10].

N
a21
a22
a11

100

8  105
5:5(3) nm
0:97 a21
1:03 a21

z

zsho
z0

65 Hz
24 Hz
1:3 m0:2 m6.4 Harmonic drive: A refined state preparation scheme

V
V
eff

r0
r
(a)

(b)

Figure 6.12: State preparation scheme. (a) If the two traps are rotated in the xy -plane about
the z axis at a frequency !r , (b) while simultaneously driving transitions between the twointernal states at the effective Rabi frequency e , a vortex mode possessing one unit ofangular momentum can be prepared if !r  e .

6.4 Harmonic drive: A refined state preparation scheme
In the previous section we considered the case of a static drive H1 = z0 z that resulted
in the excitation of a dipole mode due to the symmetry of the dipole operator z^. Suppose
we want to excite a vortex mode; what particular form for H1 (r; t) should we take? Areasonable hypothesis is to displace the traps by some distance r0 in the x y plane andthen rotate the traps around the origin at some frequency !r , as illustrated in Figure 6.12.
2 r0 for the drive H1 (r; t) inThis corresponds to taking f (r ) = x, g (r) = y , and  = m!xy
Eq. (6.4). So, the centers of the two traps rotate about the origin, while at the same time theinternal states of the atoms undergo Rabi oscillations. We will show that this configuration
does indeed excite a unit vortex in one of the components in a very controlled way.
To achieve this experimentally , one can shine a laser beam along the z axis into thetrap, so that the cloud sits in the middle of the Gaussian beam waist where the gradient of
the beam intensity is approximately linear; this gives rise to a constant force on the atoms.
If the frequency of the laser is tuned between the two hyperfine states, the optical dipole^z for the internal part,force acts in opposite directions for each state—this gives the form and the linear potential for the external part. If the beam is then rotated continuously, we
obtain the desired result shown in Figure 6.12. There is one drawback, however: there will
be heating due to spontaneous emission, which can limit the lifetime of the trap. For oursystem, however, this turns out not to be a limiting factor.
Now that we have the basic form of the drive H1 , we must decide on a frequency !rat which to rotate the traps. Looking back to section 6.2, we see there are basically two
101

Chapter 6 Strong Coupling—Quantum State Engineering
choices, corresponding to the two main frequencies in the problem: we can either rotate
the traps slowly at a frequency close to the trap frequency !r  !xy , or we can rotate
them close to the effective Rabi frequency !r  e . We have considered both cases, andalthough both work, the fast drive is much more effective. The main problem with the slow
drive is that the time scale for coupling to the vortex is very long, on the order of seconds
in a xy = 10 Hz trap. In sharp contrast, for a fast rotation !r  e , a vortex can beprepared in less than 100 ms. If we wanted to show the details for the slow drive, we would
start with Eq. (6.19) and derive a two-mode model, just as we did in the previous section
for the static drive. Instead, we will focus on the fast drive case.

6.4.1 Two-mode model
A two-mode model can be derived, similarly to the case of static offset traps that we considered in section 6.3. We start with the coarse-grained, large detuning model derived in
section 6.2.3 for the case of a fast drive
(6.31)a^ys ^+ eit + a^s ^ e it ;2 ewhere we have dropped the terms proportional to H3 and fMF from Eq. (6.21), assuming
they are so small they can be neglected. This equation corresponds to a positive detuning
Æ0 > 0; we will later discuss the effect the sign of the detuning has on the behavior of^ 00 ^1 acts
the system. This is the Hamiltonian in the interaction picture. The term H0^ 0 given in Eq. (6.3) describing a harmonic trap centered at
identically on both states, with H^ys ^+
the origin with a mean-field potential proportional to the total density. The operator asimultaneously raises the spin and creates a motional excitation according to a^ys = f^ + i g^.
y^s = x + iy. Instead of coupling to a dipole
For the drive shown in Figure 6.12, this is aH^ (I ) (t) = H^ 00 ^1

mode as in the previous section, the symmetry of the drive shown in Figure 6.12 couples toa state with one unit of angular momentum.
We therefore make the two-mode ansatz as in section 6.3.1 by assuming that the statevector in the interaction picture has the simple formj (I ) (t)i = c0 (t) j0 ij1i + cn (t) jn ij2i ;(6.32)

where the coefficients evolve in time according to

ih



c_0
c_n



=



1 hn jf + igj0 ieit   c0 0
1 h0 jf igjn ie it 2
cn :
n
2

(6.33)
Here =  =jÆ 0 j is the coupling coefficient, and 0 and n are the energy eigenvalues ofj0 i and jn i, respectively. For the case of a unit vortex (n=1), these states 0 (r) and 1 (r)are plotted in Figure 4.7 of Chapter 4 and the corresponding eigenenergies 0 and 1 areplotted in Figure 4.8. As shown in Figure 4.8, the splitting between these energies is verysmall (a fraction of 
h!xy ) and is nearly constant as a function of the relative population,
except for when most of the atoms are in the vortex. If we make a unitary transformationaccording to
U = eit[^z (0 +n)]=2 ;(6.34)
Eq. (6.33) can be brought to the form

ih
102



c_0
c_n



= 21



(n 0 )h0 jf igjn i

hn jf + igj0 i

(n

0 )



c0
cn



:

(6.35)

6.4 Harmonic drive: A refined state preparation scheme
This is the main result; we now summarize the crucial information contained in it:
1. In order to get this result, we had to make the following assumptions(6.36)e  !0 ; !r  e ; Æ0  :We also made the further approximation Æ 0  e , which follows from the lastinequality. The first inequality e  !0 allowed us to separate out the internaland external evolution by going to the interaction picture and coarse-graining overrapidly oscillating terms, which also required the second inequality !r  e . The
third inequality allowed us to drop extra terms that would otherwise contaminate thepure coupling given by a^s .

2. Eq. (6.35) tells us the timescale for coupling to the vortex, given by =  =jÆ 0 j.
2 r0 , so that = m!xy
2 r0 =jÆ0 j. ThisFor the drive shown in Figure 6.12,  = m!xy
tells us that we can couple more strongly to the vortex by increasing the displacement
r0 , or by increasing the ratio =jÆ0 j—which is constrained by the third inequality in
Eq. (6.36).
3. We can also see from Eq. (6.35) that if  is chosen to be  = n 0 , we can drive
the system on resonance. For a fixed e , this means that the trap rotation frequency!r should be adjusted to compensate for the splitting n 0 . However, this is almosta moot point, since the splitting n 0 is so small, as shown in in Figure 4.8. Inother words, in practice we can take !r  as the resonance.
4. In obtaining Eq. (6.35), we have neglected the mean-field term fMF arising fromthe scattering lengths a1 ; a2 ; a12 being different. We are very fortunate that for
87 Rb these scattering lengths are nearly degenerate, so that the size of the terms
in Eq. (6.21) proportional to fMF are small. By making  large, the relative importance of the fMF terms diminish. If the scattering lengths were drastically different,
then the mean-field terms would dominate and generating a pure vortex would bevery difficult.
Equation 6.35 is valid only for positive detuning, for we based our two-mode model on
Eq. (6.21); for negative detuning, we should use Eq. (6.22). It is straightforward to showthat in the case of negative detuning Æ 0 < 0, we get essentially the same results, but a^ys =f ig, instead of f +ig. In other words, changing the sign of the detuning, with the directionof rotation of the trap centers fixed, causes the vortex to rotate in the opposite direction.
This is a very intriguing property of our state preparation scheme because the direction of
circulation of the vortex can be opposite to that of the rotating trap centers. This emphasizes
the fact that we are not simply stirring angular momentum into the condensate, as wasconsidered by the authors of [69] for the case of a stirring beam in a single component, forexample. A further remarkable feature is that a vortex containing a large fraction of theatoms can be generated when !r is much larger than the trap oscillation frequency !0 , sothat in our scheme only a fraction of a trap period is required to generate a macroscopic
population of the vortex state.
The solution in the lab frame can be obtained by applying UI (t) from Eq (6.12) to j (I) i
in Eq. (6.32) to yieldj (t)i = ( 1 (t)c0 (t) j0 i + 2 (t)cn (t) jn i)j1i+ ( 2 (t)c0 (t) j0 i + 1 (t)cn (t) jn i)j2i ;(6.37)
103

Chapter 6 Strong Coupling—Quantum State Engineering
where the rapidly varying coefficients are 1 (t) = cos( e =2 t) i(Æ 0 = e ) sin( e =2 t)
and 2 (t) = i( = e ) sin( e =2 t). This resembles the analogous equation Eq. (6.30)
in section 6.3.1. However, in the present situation, we have the limit of a large detuning
Æ0  , so that the Rabi oscillations will look quite distinct from the case of the static
drive, where a small detuning was assumed. In the large detuning limit, the coefficients i
become1 (t)  cos( e =2 t) i sin( e =2 t)
2 (t)  0;

(6.38)

so that Eq. (6.37) can be written

j (t)i  e i eff t=2 c0 (t) jo i j1i + cn (t) jn i j2i :(6.39)

This is rather spectacular: even though we are driving the system off resonance Æ 0  ,it is possible to transfer a significant fraction of the population from the j1i state to thej2i state due to the subtle interplay between the external drive H1 (r; t) and the internal
Rabi coupling. We also see that the cycling between spatial modes for each internal stateis minimal, so that there is little contamination from the other mode during the evolution;that is, on average the j1i state is almost purely in the ground state j0 i and the j2i state isalmost purely in the excited mode jn i.
Intuitively, we can see why our scheme couples a non-rotating condensate to a vortex
state if we consider the frame co-rotating with the trap centers at angular frequency !r so^ 1 becomes time-independent. The free Hamiltonian in the co-rotating frame is giventhat H^^ z . The energy of the vortex with one unit of angular momentum is therefore
by H0 !r L
shifted by 
h!r in the rotating frame relative to its value in the lab frame. When this energyhÆ of the internal coupling field and theshift compensates for both the energy mismatch 
small chemical potential difference between the vortex and non-rotating condensate, resonant transfer of population may take place. Within this picture, we can also understandintuitively how changing the sign of the detuning while keeping the trap rotation direction
fixed couples to a vortex with opposite circulation. Vortices with opposite circulations experience opposite energy shifts in transforming to the rotating frame and therefore require
opposite signs of detuning in order to achieve resonant coupling.

6.4.2 Results of calculations
We now show results of numerical calculations of the coupled GP equation. In our calculations, we are solving a two-dimensional version of Eq. (6.10), with fMF and H3 set tozero; we are not solving the two-mode model Eq. (6.35). We found in section 6.3 that thetwo-mode model for that system gave good agreement to the full solution. Here, we alsofind that all of the predictions embedded in the two-mode model are exhibited in the full
solution of Eq. (6.10). The two-mode model was indispensable in pinpointing the specificregion in parameter space where a pure vortex could be generated.
In Figures 6.13 and 6.14 we show an example of the dynamical state preparation of aunit vortex. Initially all of the atoms are in the j1i internal level and the mean-field groundstate j0 i. The two traps are displaced by r0 = 1:7m and rotated around at !r = 205:4
Hz, so that we have taken f = x and g = y . The two internal levels are being driven, withthe bare Rabi frequency = 50 Hz, and the detuning is Æ 0 = 200 Hz. The trap frequency104

N2(t) population

6.4 Harmonic drive: A refined state preparation scheme

1

0.5

1
1
amplitudeangular momentum

0

0.5

0.5
0
30

0
0

50

t s 100

15

0
15∆ (Hz)

150

30

200

time (ms)Figure 6.13: Dynamical evolution to a vortex. The top graph shows the fractional population of atoms in the j2i internal state. The bottom graph shows the angular momentum
h. The inset shows the maximum amplitudeof the j2i state, in units of Planck’s constant of population transfer to the j2i state as a function of the trap rotation frequency !r , with
 = e !r .
is !xy = 10 Hz and the number of atoms is N = 8  105 atoms. Both internal and external
drives are turned off at t = ts , i.e. = 0 and r0 =0, after which the system evolves freely.
In Figure 6.13 we plot the fractional population (top) and the angular momentum per
atom (bottom) of the j2i state as a function of time. The small-amplitude rapid oscillationsof the top line correspond to the cycling between internal levels due to the off-resonant
coupling. The gradual rise of this line is due to coupling from the ground state to the vortex
mode caused by the drive H1 in Eq. (6.4). This is just what we expect from the two-mode
model. The rise and fall of the angular momentum in the bottom curve corresponds to a
rapid cycling of the j2i atoms between the non-rotating condensate and the vortex. Once
during each Rabi cycle, the angular momentum approaches unity and at that time the j2istate wave function approaches a pure vortex mode.
The maximum population transfer between the two states obeys a Lorentzian response
curve as !r is varied near e exhibiting a narrow resonance. This is shown in the inset of
Figure 6.13, where  = e!r . The peak of this curve sits at  = 1 0, which isaround 0:05
h!xy , which corresponds to the splitting shown in Figure 4.8. We find that the
maximum population that can be transferred to the vortex is around 80%. This makes sensefrom looking at Figure 4.7, since the energy splitting 1 0 is roughly constant until mostof the atoms are in the vortex, when the energy splitting starts to increase rapidly. With
105

Chapter 6 Strong Coupling—Quantum State Engineering|22πdensity

phase

|1

y

y

x

x

Figure 6.14: Densities and phases of the two states j1i and j2i, at time t = 200ms. At thistime, one third of the atoms are in the j2i state, which is in a pure vortex mode with unit
angular momentum. A characteristic feature of a vortex mode is the 2n phase wrap about
the core, where n is an integer that is equal to unity in this case.

1066.4 Harmonic drive: A refined state preparation scheme
L z = 3 -h

Lz = 2 -hdensity

phase

2π

y

y

x

x

Figure 6.15: Double and triple vortex preparation. Shown are the densities and phases
modulo 2 of the j2i state after a dynamical evolution similar to that shown in Figure 6.13,
h, f = x2 y2 andbut with different symmetries of the drive H1 . In the case of Lz = 23223
g = 2xy, while for Lz = 3h, f = x 3xy and g = 3x y y . In both cases, thesystem was evolved from the same initial condition as that for the calculation described in
Figure 6.13, with one third of the atoms in the j2i state at the time ts . The values taken
for the various parameters were the same, except for the trap rotation frequency, which was!r = 204:3 Hz for the Lz = 2h case, and !r = 200:2 Hz for the Lz = 3h case.

!r fixed, this means that the system pushes itself out of resonance, a behavior also seen in

Chapter 5 in the weak-coupling case.
Our key idea is that by turning off the coupling at a precise time, ts , on a given Rabi
cycle, the j2i state can be prepared to have unit angular momentum. In Figure 6.14 weshow a snapshot of the numerically calculated densities and phases of the two components
at time 200 ms. The snapshot illustrates the preparation of a high-quality vortex in state j2istate, with the j1i state providing a natural “pinning” mechanism that stabilizes the vortex
core due to a repulsive mean-field barrier along the symmetry axis.
The symmetry of the coupling is determined by f + ig , which has the form x + iy as
previously stated for the case of the unit vortex. In order to produce a vortex with n units^ z i = nh), f + ig should take on the form (x + iy)n . So forof angular momentum (i.e. hLn = 2, we must construct a H^ 1 in which f = x2 y2 and g = 2xy, corresponding to arotating saddle potential. In Figure 6.15, we illustrate vortex generation with two and threeunits of angular momentum.
The generalization of our scheme for the preparation of new states in a trapped Bose
condensed gas with arbitrary symmetry is straightforward. For example, to generate amode with a dipole symmetry one superimposes a left and right circulating vortex so thatf = Refx + iyg and g = 0. A quadrapole symmetry would be generated by f = Imf(x +iy)2 g and g = 0. The dynamical state preparation of these two examples is illustrated in107Chapter 6 Strong Coupling—Quantum State Engineering
quadrupoleπdensityphase

dipoleyyxx

Figure 6.16: Dipole and quadrupole preparation. Different symmetries were constructedusing specific forms for the drive H1 in order to prepare the j2i state in non-circulating
modes. Instead of having a 2n phase wrap corresponding to a current flowing around acentral core, these modes have regions of constant phase separated by a discontinuous jumpof  where the wave function changes sign. To generate the dipole mode, we used f = x
and g = 0 with !r = 205:4 Hz; while for the quadrupole, we used f = xy and g = 0 with
!r = 204:3 Hz.
Figure 6.16.

6.5 Experimental observation of a vortex
The proposed scheme for generating a vortex presented in section 6.4 has been implemented in the laboratory of Eric Cornell and Carl Wieman here at JILA, resulting in thefirst experimental report of a vortex in a dilute Bose-condensed gas [14]. The experimental
data is remarkably clean with excellent shot-to-shot reproducibility, showing clearly thatthe prepared state has the required vortex topology. The basic experimental scheme described earlier on page 101 was implemented; the details of the experimental apparatus are
given in [14].
In Figure 6.17 we show the dynamical evolution of a vortex over a 80 ms time period.
Shown are phase contrast images for the j2i atoms, with each frame corresponding to atime-averaging of the signal for 9 ms. The evolution shown in Figure 6.17 corresponds tothe first 80 ms of the evolution of the similar case considered in Figure 6.13. The time
averaging washes out the spatial structure from the interference between the ground j0 i
and vortex j1 i states that occurs during the rapid Rabi cycling. In Figure 6.18 we show asingle snapshot of the j2i atoms, taken approximately when 2 (t) = 0 in Eq. (6.37) duringa Rabi cycle, so that the j2i atoms are nearly in a pure vortex mode.
108

6.6 Summary

Figure 6.17: Experimental production of a vortex. Shown are phase contrast images of
the j2i atoms, with dark regions corresponding to higher density in the cloud. Each frame
corresponds to a 9 ms signal integration and 10 ms between frames, so that by the endof the sequence 80 ms have elapsed. Initially, all of the atoms are in the j1i state andgradually population is built up into the j2i state into a vortex, similar to the case shown in
Figure 6.13. Experimental data is courtesy of M. Matthews, B. Anderson, and P. Haljan.
In order to show that the prepared state is a vortex that has a 2 phase wrap, andis not simply a uniform-phase ring, an interference of the two states j1i and j2i can bemeasured and the sine of the phase of the vortex can be calculated. In Figure 6.19 weshow experimental data taken for this case. A vortex is prepared in the j2i atoms, shownin the first frame. A  pulse is applied (the two-photon drive is applied for half of a Rabi
cycle), which transfers the atoms to the j1i state, which is in the ground state j0 i. Halfway
through the pulse, the two states interfere, with the distinctive density pattern shown thatindicates a spatially-nonuniform relative phase. The data from the three plots was used to
calculate the cosine of the phase, which matched the expectations for a vortex solution [14].

6.6 Summary
In this chapter we have studied a strongly-coupled two-component condensate and havefound that topological modes can be excited if the traps of the two states are spatially
offset. We have presented a novel state-preparation scheme based on this mechanism thatallows for the generation of pure vortices in this two-component system, as well as othertypes of non-ground-state modes, such as dipole and quadrupole modes. Our approachgrew out of a collaborative effort at JILA, between our group and the laboratory of Eric
Cornell and Carl Wieman, to understand the initially unexplained data (examples shown in
Figures 6.1 through 6.3.) of the first observations of this system. The culmination of these
efforts is the first observation of a vortex in a dilute Bose-condensed gas [14].
It should be mentioned that prior to our work, there had been several similar proposals
for the dynamical production of a vortex through the use of the internal structure of theatoms [107, 108, 74]. For example, Dum et al. [74] discuss the preparation of dark solitonsand vortices using a Raman coupling scheme in which the light beams have a particularspatial intensity pattern chosen with the right symmetry for coupling to the desired mode.
As discussed in [107] and [108], such an intensity pattern could be attained through the useof a laser beam with a higher-order Laguerre-Gaussian mode profile. The spirit of the basic
approach described in [74], for example, is much the same as in our work, however, thespecific underlying mechanism for coupling to a vortex that we consider is quite distinct
109

Chapter 6 Strong Coupling—Quantum State Engineering

Figure 6.18: Snapshot of a vortex. Shown is the density profile of atoms in the j2i state fora single snapshot taken during the dynamical evolution shown in Figure 6.17. The snapshot
was taken approximately when the j2i atoms were in a pure vortex. Experimental data is
courtesy of M. Matthews, B. Anderson, and P. Haljan.
from these earlier works. In our system it is the combined effect of spatially homogeneous
coupling fields and time-varying spatially offset traps that gives rise to the required symmetry for coupling to a topological mode. This effect is rather nonintuitive and only occurred
to us while trying to interpret the experimental data described in section 6.1.
The experimental realization of the state-preparation scheme presented in this thesis
has opened the door to a whole new class of problems related to topological modes in a
two-component system. This is an extremely rich system that has been almost untapped in
the the literature, since most of the recent efforts have been on the study of vortices in single component systems. Issues like vortex stability, the role of different scattering lengths,
collective modes of the system, and the relation to the phenomenon of superfluidity are essentially open problems waiting to be investigated, both theoretically and experimentally.
Similar properties can be investigated for the dipole and quadrupole modes as well. One
could also study the preparation scheme in more detail to find the optimal path for generating a pure vortex with no remaining j1i atoms. This might entail chirping the trap rotation
frequency !r in a specified time-dependent way so as to adiabatically follow the mean-field
shifted energies between the two modes [74]. A further extension would be to treat finite
temperature effects by including the collision terms in the kinetic equation for the mean
field and fluctuations, given in Chapter 2.

1106.6 Summary

Figure 6.19: Interference between the ground state and vortex. Shown are phase contrast
images with darker regions corresponding to higher density. Initially all of the vortex atoms
are in the j2i internal state. A  pulse is applied that transfers the atoms into the j1i state.
The middle frame is taken halfway through the pulse, and shows the interference between
the vortex and ground states. The third frame shows the atoms at the end of the  pulse,when they are in the ground state. The top frame and bottom frame correspond to the twostates shown in Figure 6.14. Experimental data is courtesy of M. Matthews, B. Anderson,
and P. Haljan.

111

Chapter 6 Strong Coupling—Quantum State Engineering

112Chapter 7

Trajectory Simulation of the
Quantum Boltzmann Equation
7.1 Introduction
In experiments producing Bose-Einstein condensation of a dilute alkali vapor, the temperature of a gas is reduced by several orders of magnitude using the crucial technique of evaporative cooling [33, 34]. The theoretical study of this important technique and the descriptionof condensate formation requires a kinetic theory that treats non-equilibrium, open systems
in both the classical and quantum degenerate regimes. It is also necessary to consider asystem of finite size as determined by the form of the confining potential [109, 110].
We present a novel approach to describing quantum kinetics which is motivated by thequantum trajectory methods developed in quantum optics to describe the dissipative evolution of open systems [111, 112, 113, 114, 115, 116, 117]. The basic principle on whichthese models are built is that the evolution of the open system, described by a density operator master equation, can be obtained by accumulating an infinite number of stochastic
realizations of a wave-function trajectory. A simple example of an open system is an excited atom coupled to the radiation vacuum reservoir. A trajectory in this case consistsof the interruption of the continuous evolution of the atomic wave function by a quantum
jump to the ground state when a spontaneous photon is emitted.
We have applied this trajectory method to a fundamentally different problem. In ourcase, we wish to describe the evolution of a gas of atoms which are not coupled to a reservoir at all. The jumps which occur in the single-atom trajectory that lead to the irreversible
evolution of the system are caused by atomic collisions with other atoms in the gas. Thus,
the role of the reservoir in our problem is played by the system itself. This inherent nonlinearity is illustrated by the kinetic equation we are trying to simulate. We have previously
applied this approach to treat the classical Boltzmann equation in order to describe the evaporative cooling process [118]. In this chapter, we extend the theory to treat Bose-Einsteincondensation.
This chapter is divided into three parts. We first present the theory for kinetic evolution
in section7.2. In section7.3, we describe our trajectory approach and outline in detail thecorresponding simulation procedure. Finally, in section7.4, we present applications of themethod for thermodynamics, condensate growth, and evaporative cooling.

Chapter 7 Trajectory Simulation of the Quantum Boltzmann Equation

7.2 Ergodic quantum Boltzmann equation
In this section, we outline our mathematical description of kinetic theory and highlight thephysical assumptions made in its derivation. We present the quantum Boltzmann equation(QBE), which is the starting point for our treatment. An ergodic assumption is made,which simplifies the problem by assuming that the population of a state only depends on its
energy. We show how the ergodic QBE goes to the ergodic classical Boltzmann equation in
the appropriate limit, a result which greatly simplifies the trajectory simulation presented
in section 7.3.

7.2.1 Quantum Boltzmann equation
We consider a dilute system of atoms confined in an isotropic harmonic oscillator of frequency ! . The Hamiltonian H = H0 + HI consists of a free part H0 , and an interaction
term HI due to binary collisions

H0 =

X

~n

E~n a~yn a~n ;1 X
C (~n; m~ ; ~q; ~p) a~yn aym~ a~qap~ ;HI =2 ~nm~ ~qp~

(7.1)

where E~n = (nx+ ny + nz+ 23 )
h! is the energy eigenvalue of H0 with quantum number~n = (nx ; ny ; nz ). Here a~n is the annihilation operator which removes an atom from thesingle-particle eigenstate ~n . The transition amplitude isZ

C (~n; m~ ; ~q; p~)= d3 xd3 x0 ~n (~x)m~ (~x0 )V (~x; ~x0 )~q (~x)p~ (~x0 );(7.2)

where V (~x; ~x0 ) is a two-particle potential. In the temperature range of interest, s-wave
scattering predominates and collisions are characterized by a contact potential

4h2 a 3
V (~x; ~x0 ) =Æ (~x ~x0 ) ;
m

(7.3)

where a is the scattering length. This gives the quantum mechanical cross section for thecollision  = 8a2 .
The correlation time arising from the duration of a collision in Bose-Einstein condensation experiments is typically much shorter than the time scale on which the system relaxes
to equilibrium. The time scale for pair correlations is given by cor = a=v, where v is themean velocity . The relaxation time is determined by the time between elastic collisions
col in the gas . For example, in the experiments described in Ref. [1], at the critical temperature cor  1 s, compared to col  0:1 s. In this regime, Wick’s theorem [119, 120, 38]
may be applied to give the evolution under the Born and Markov approximations of theatomic population
f~n = Trfa~yna~n g ;(7.4)
where  is the N -particle density matrix for the system. In the representation of the bareharmonic oscillator states, off-diagonal elements f~n~n0 = Trfa~yn a~n0 g, ~n 6= ~n0 , [38] are114

7.2 Ergodic quantum Boltzmann equation
negligible when !col  1. This gives the QBE [38, 39], derived in Chapter 2 in section 2.4.2

@f~n X=W (~n; m~ ; ~q; ~p) f~qfp~(1 + f~n )(1 + fm~ ) f~n fm~ (1 + f~q)(1 + fp~) ;@t m~ q~p~(7.5)

where the transition rate W (~n; m~ ; ~q; ~p) appearing in Eq. (7.5) is obtained from Fermi’s
Golden RuleÆ2W (~n; m~ ; ~q; p~) = jC (~n; m~ ; ~q; ~p)j2 E~n+Em~ ;E~q+Ep~ ;
(7.6)

h

h!

and Æ is the Kronecker delta function giving energy conservation.
P The total number ofparticles in the system N determines the normalization of f~n by ~n f~n = N .

7.2.2 Ergodic assumption
A practical problem for simulating the QBE is that the degeneracy of states increases
rapidly with increasing energy. The degeneracy of a level for the isotropic harmonic oscillator is proportional to the square of the energy so that even for very low temperatures, thenumber of states whose populations must be calculated may be very large. For example,
h!, we would have to consider approximately 105 states. This is a severe
if kB T = 10
limitation on the computation speed.
We resolve this difficulty by assuming that the population within a degenerate subspaceis uniformly distributed among the degenerate states. In other words, in the spherical basis
we assume that the population f~n depends only on the principle quantum number n and noton the angular momentum quantum numbers l and m. This assumption is true at equilibrium since then the distribution function is purely a function of the Hamiltonian. In manycases of interest, the system remains close to equilibrium and we expect the approximation
to be valid. In Ref. [121], it was shown that, for a homogeneous system originally in equilibrium, if one of the three degenerate states in the first excited level is depleted, and thesystem is allowed to evolve back to equilibrium, the population gets redistributed equally
among the three degenerate states in a time on the order of the mean collision time in thegas. Essentially, the ergodic approximation corresponds to detailed balance amongst theangular momentum levels, i.e. l-changing collisions are rapid compared to rates in and outof n.
This ergodic assumption is defined as

f~n =

X

n

Æn;E~n fn :(7.7)

Each state ~n in the degenerate subspace has the same population fn . Therefore, a sum
over all of the degenerate states is justgn fn =X~nÆn ;E~n f~n ;(7.8)

where gn is the degeneracy of energy level n , which is given by1gn = (n + 1)(n + 2):
2

(7.9)
115

Chapter 7 Trajectory Simulation of the Quantum Boltzmann Equation
With the ergodic assumption, Eq. (7.5) can be reduced to

gn

X


@fn
=
W (n ; m ; q ; p ) fq fp (1+ fn )(1+ fm ) fn fm (1+ fq )(1+ fp ) :@tm q p(7.10)
The collision kernel W (n ; m ; q ; p ) is now a sum over all of the rates corresponding to
the possible degenerate states which could participate in the collision

W (n ; m ; q ; p ) =Y2 Æn+m ;q +p X
j
C (~n; m~ ; ~q; p~)j2 Æj ;E~j ;
h
h!
j
~nm
~ q~p~(7.11)

2 fn; m; q; pg in the product. The normalization of the distribution becomesgfn n n = N .

where jP

We have made a great simplification by reducing the QBE to Eq. (7.10). Considering
our previous example of kB T = 10
h!, the number of levels occupied is approximately
100, which is much less than the corresponding number of states 105.

7.2.3 Classical limit
Although the ergodic assumption presented in section 7.2.2 greatly simplifies the problem
of simulating the QBE, it is desirable to reduce the problem further in order to describe theevaporative cooling technique used to reach the critical density for Bose-Einstein condensation. During the evaporation process, the temperature of the gas is reduced by several
orders of magnitude. Even with the ergodic assumption, a very large number of energy
levels must be considered to describe the system in this entire range. The process can be
described by the classical Boltzmann equation for most of this range, down to some pointclose to the critical temperature when quantum statistics becomes important. In this criticalregion, the QBE must be used.
The problem may be simplified if a smooth transition connecting these two regions
can be found, which would allow energy levels above some cutoff to be treated classically.
Taking the classical limit of Eq. (7.10) corresponds to taking n ! 1, and assuming that
T > Tc so that the (1 + fn ) Bose-enhancement factors go to unity.
Eq. (7.10) can be written in dimensionless form

gen

X
@fen
=
Æen +em ;eq +ep g(en ; em; eq ; ep)(7.12)@em eq ep
 feq fep (1 + fen )(1 + fem ) fen fem (1 + feq )(1 + fep ) ;2
where time and energy are in the natural units of the problem,  = m!
2 h t and en = n =h! .
The collision kernel given in Eq. (7.11) is now dimensionless and given by
Y

4 X Y
g(en ; em ; eq ; ep ) = 2
jNni Nmi Nqi Npi Ini mi qi pi j2 Æej ;E~j ; ~nm~ ~qp~ ij(7.13)

where i 2 fx; y; z g in the first product and Nni = (2ni ni !) 1=2 is the normalization factor
for component i of the state ~n . The overlap integral of the four Hermite polynomials
Hni (ui ) isZ
2
Ini mi qi pi = dui Hni Hmi Hqi Hpi e 2ui :(7.14)
116

7.2 Ergodic quantum Boltzmann equationg( e0 , em ; eq , ep )

2
1.5
1
0.5
0
0

2

4

ep

6

8

10 10

8

6

4

2

0

eq

Figure 7.1: Plot of g (emin ; em ; eq ; ep ) vs. (eq ; ep ) for the minimum energy being in the
ground state emin = e0 , where em is determined by energy conservation. It is a very flat
function, differing from its asymptotic form, ge0 = 1, only when e is very small.
In taking the limit en ! 1, fen is replaced by a continuous function of energy f (en ),h! iseach sum over en is replaced by an integral, and the Kronecker delta Æn +m ;q +p =
replaced by a Dirac delta function Æ (n + m q p ). For temperatures well above
Tc, the Bose-enhancement factors (1 + fen ) can be put to unity. The collision kernelg(en ; em; eq ; ep) has a rather simple limiting form which may be found by computing numerically the quantity on the RHS of Eq. (7.13). We use the following expression for theoverlap integral [122]Inmqp =

min(m;n)m!n! X
12t
1
1
1
2k 2 (k p+ ) (k m n+2t+ ) (k q+ ) ; t=0 t!(m t)!(n t)!
2
2
2(7.15)
where 2k = n 2t and n = n + m + p + q must be even. The integral is zero if n is
odd.
We calculated the kernel g (en ; em ; eq ; ep ) numerically for e0  ej  e20 , where ej 2fen; em ; eq ; ep g, and obtained a remarkable result. This collision kernel converges very
quickly to the degeneracy gen , given by Eq. (7.9), as the differences e  ej eminbetween the minimum energy and the other three energies increase!1 gg(emin; em ; eq ; ep ) e!
emin ;(7.16)

where emin = minfen; em ; eq ; ep g is the minimum energy. Figure 7.1 shows a plot ofg(emin; em; eq ; ep) vs. (ep ; eq ) for emin = e0 , with em determined by energy conservation.
It is a very flat function differing from its asymptotic form only when e is very small.
From our calculations, we found that the convergence in Eq. (7.16) becomes faster as emin
117

Chapter 7 Trajectory Simulation of the Quantum Boltzmann Equation
increases. Furthermore, as ej increases, the degeneracy factor gej converges to the densityof states (ej )  21 e2j
e !1gej j! (ej ) :(7.17)
In our simulation procedure described below, we use the limiting form Eq. (7.16) forg(en ; em; eq ; ep) when any one of the energies is greater than the tenth level e10 , and use
the limiting form Eq. (7.17) when all of the energies are greater than a cutoff energy ec
chosen such that (1 + fec )  1.

The comparison with the ergodic classical Boltzmann equation can now be made. The
classical limit of Eq. (7.12) is(en )@f (en )=@Zdemdeq dep Æ(en + em eq ep )(emin )(f (eq )f (ep ) f (en )f (em )) ;(7.18)
where the same natural units given below Eq. (7.12) are used here. This agrees with theclassical Boltzmann equation given by Eq. (14) in Ref. [109].

7.3 Trajectory simulation
In this section, we explain our novel approach to simulating the QBE using simulated
single-particle trajectories. We first show that the time evolution of fen given by Eq. (7.12)
can be described by a sum over all possible single-particle trajectories. We then outline thespecific simulation procedure.

7.3.1 Trajectory decomposition of the QBE
We can now incorporate quantum statistics into the trajectory method using the results ofsection 7.2. We can not expect the simulation to be valid far below the critical temperaturewhen a large proportion of atoms are in the ground state since we neglect the effect of themean field on the system and work in the representation of the bare harmonic oscillator
basis. However, in the regime where !col  1 holds, the buildup of the condensate canstill be investigated. This simulation method is now ideal for describing the evaporative
cooling of atoms all the way down to temperatures below Tc .
The trajectory decomposition of the ergodic QBE closely resembles that given in Ref. [118]for the classical case. Still working in the natural units of time and energy given below
Eq. (7.12), we begin by defining a trajectory function f (en ; t j t ; e ; : : : ; t1 ; e1 ), which describes a specific collision history with energy en at time t: The trajectory is labeled by its
history of  collisions occurring at times t1 ; : : : ; t , with t > t > : : : > t1 , and with theenergy before each collision given by e1 ; : : : ; e .
Our task is to correctly describe the time evolution of this trajectory function so thatupon accumulating all possible realizations of trajectories, fen (t) is attained with the correct time evolution governed by Eq. (7.12). To obtain fen (t) from the accumulated trajectories, we form a distribution of the energies collected from M trajectories at the particulartime t. Then in the limit M ! 1, the distribution should converge upon fen (t). Realistically, the number of trajectories we accumulate is in the range 104 105 . Of course, thetrajectories must be weighted so that the final distribution obtained from the accumulated
trajectories is normalized to N .
118

7.3 Trajectory simulation
This sum over trajectories can be written explicitly as

gen fen (t) =1 X ZtX
=0 e :::e1 t0

dt

Z t

t0

dt 1   

Z t2

t0

dt1 f (en ; t j t ; e ; : : : ; t1 ; e1 ) ; (7.19)

where we sum over the number  of possible collisions, sum over all possible energies
occurring before each collision, and integrate over the possible times at which each collision can occur. Each trajectory has a weight given by N=M , where M goes to infinity in
Eq. (7.19).
The time evolution of fen (t) in terms of the trajectory decomposition is found by differentiating Eq. (7.19)
Z t2
1 X nZ t
X@f (t)@f (en; t j t ; e ; : : :)gen en=dt    dt1@t@tt0
=0e :::e1 t0
Z t

Z t2

t0

t0

o

+ dt 1    dt1 f (en ; t j t; e ; : : :) :(7.20)

We must find the correct time evolution of the trajectory function f (en ; t j t ; e ; : : : ; t1 ; e1 )so that the time evolution given by this decomposition is equivalent to the QBE. The firstterm on the RHS of Eq. (7.20) gives the time evolution of the trajectory between collisions,while the second term is related to the instantaneous change in the trajectory’s energy whena collision occurs.
Between collisions, the trajectory’s norm will decay due to the probability for a collision to occur with an atom from the rest of the gas@f (en ; t j t ; e ; : : :)= (en; t)f (en ; t j t ; e ; : : :) :(7.21)@t
The rate of decay (en ; t) is equal to the collision rate. A particle in the system with energy
en has a rate of colliding with any other particle in the system given byXg (e ; e ; e ; e )(en ; t) =Æen +ep ;eq +ep n m p q fem (t)(1 + feq (t))(1 + fep (t)) : (7.22)
gen
em eq ep
Because of Bose statistics, some of the collisions will be enhanced by the factors (1 +feq (t)) if the populations in those output channels are large. This dependence on the popu-

lation in summing over the output channels is absent in the trajectory method presented forthe classical Boltzmann equation in Ref. [118]. In that case, the integrals over the output
channels can be done analytically, which makes the problem scale linearly with the numberof energy bins used to store f (e): We can not make that simplification here.
The function f (en ; t j t; e ; : : :) indicates that a collision has occurred at time t, changing the energy from e to en . We interpret the function f (en ; t j t; e ; : : :) as the rate thata particle with energy e will collide with any atom in the system and attain the energy en
afterwards. Thus, f (en ; t j t; e ; : : :) can be obtained by omitting the sum over the output
channel en in Eq. (7.22) and weighting the rate by the norm of the trajectory before thecollisionf (en ; t j t; e ; : : :) =g(e ; ep ; en ; em )(7.23)
ge
em ep
fep (t)(1 + fen (t))(1 + fem (t))f (e ; t j t 1 ; e 1 : : :) :XÆe +ep ;en +em

119Chapter 7 Trajectory Simulation of the Quantum Boltzmann Equation
With the above description of the time evolution of a trajectory, one can easily verify
that the decomposition in Eq. (7.20) is equivalent to Eq. (7.12) by substituting Eq. (7.21),
Eq. (7.22), and Eq. (7.24) into Eq. (7.20) and using Eq. (7.19) to reduce the expression.
One crucial point, however, has not been addressed in our proof. Because the rates
given in Eq. (7.22) and Eq. (7.24) depend on fen (t), which is the quantity being calculated,
the problem is nonlinear and must be solved self-consistently. A self-consistent solution
can be found if we make the incremental time of evolution dt much smaller than the mean
collision time col , so that fen (t) does not change appreciably during the coarse-grained
time steps.

7.3.2 Simulation procedure
We now describe the procedure for simulating the QBE using the trajectory method. One
begins by creating the initial distribution fen (t0 ), the time evolution of which is desired
over a chosen time interval. This distribution is evolved incrementally, by adding up M
trajectories over a time interval dt that must be chosen smaller than the average singleparticle collision time col . Then, the simulated distribution at the end of the time t0 + dt
is used as the starting distribution for the next time step, and so on, until the desired time isreached. In more detail, this procedure is as follows:
1. Create an empty distribution function fen (t0 + dt) with zero population throughout
the levels.
2. Choose an initial energy e1 for the trajectory from the initial distribution gen fen (t0 ).
3. Calculate the initial collision rate(e1; t0 ) =X

em eq ep

Æe1 +em ;eq +ep(e1; t0 ) using fen (t0 ) for the distributiong(e1 ; em; ep ; eq )fem (t0 )(1+feq (t0 ))(1+fep (t0 )) :ge1
(7.24)

4. Simulate a realization of a uniform random variable R1 2 [0; 1] and find the time theparticle will next collide given by tc = t0 ln(R1 )= (e1 ; t0 ).
5. If tc > t0 + dt, record the atom in fen (t0 + dt) by incrementing the level corresponding to its energy by the amount N=M .
6. If t < t0 + dt, a collision occurs. Simulate a second random variable R2 2 [0; 1].
The energy after the collision is found from the solution of esim in

esim XX
en em ep

Æe1 +ep ;en +em

g(e1; ep ; en ; em )fep (t0 )(1+fen (t0 ))(1+fem (t0 )) = R2 (e1 ; t0 )ge1
(7.25)

7. Continue steps 3 - 6 until the end of the interval t0 + dt is reached. Because dt mustbe much smaller than col , multiple collisions should be rare.
8. Calculate the next trajectory by choosing another initial energy in step 2 and carryingout steps 3 - 7. Continue the process for M trajectories.
120

7.3 Trajectory simulation

1
0.8

fe

0.6
0.4
ec
0.2
0
0

10

20
30
40
Energy e/ -h ω

50

60

70

Figure 7.2: An illustration of the distribution of populations fe over the discrete and continuous regions. Above the cutoff ec , we drop the (1 + fen ) factors and the levels are put
in bins as though they formed a continuous spectrum.
9. When all of the trajectories have been accumulated, a good approximation to fen (t0 +dt) has been determined. One can then move on to the next time step, and repeat steps
1 - 8 by using fen (t0 + dt) as the initial distribution. This coarse-grained time evolution can be continued until fen (t) has been obtained for the desired time duration.
This trajectory simulation scales quadratically with the number of levels whose populations must be stored, which is an improvement over the cubic scaling of a direct numericalintegration of Eq. (7.12). However, as already pointed out, the trajectory simulation of theclassical Boltzmann equation scales linearly with the number of bins used to store f (e). It
is now very clear that making the ergodic assumption and using the classical, limiting formof the ergodic, QBE will increase the speed of the simulation enormously by decreasing thenumber of discrete energy levels whose populations must be simulated. By treating mostof the levels above some cutoff ec classically according to Eq. (7.18), the linear scaling of
the method described in Ref. [118] can almost be restored.
In order to speed up the simulation by using the smooth transition to the ergodic, classical Boltzmann equation described in section 7.2.3, we use a distribution that has discretelevels below a cutoff energy ec and a continuous spectrum of energies above this point, asshown in Figure 7.2. Below ec , we retain the (1+fen ) factors. We also use g (en ; em ; eq ; ep )if all four energies are less than or equal to e10 , and use its limiting form gemin if any of
the four energies is greater than e10 . Above ec , we drop the (1 + fen ) factors, and use thedensity of states (e) as the limiting form of the degeneracy factor. When all four energies are above ec , the simplifications made on the integrals in the collision rate, shown in
Ref. [118], can be used.
121

Chapter 7 Trajectory Simulation of the Quantum Boltzmann Equationlog(1/f e + 1)

5
4
3

120
Population

6

90
60
30
0
0 10 20 30 40 50
Energy

2
1
0
0

10

20
30
Energy e/ -h ω

40

50

Figure 7.3: The system evolves to equilibrium from the distribution shown in the sub-plot,
where a single trajectory has on average 100 collisions. The final, stationary distribution(circles) agrees with the Bose-Einstein equilibrium distribution (dashes).

7.4 Simulation results
In this section, we carry out the simulation procedure described in section 7.3 to investigate physical properties of a condensing gas of atoms trapped in an isotropic, harmonic
potential. We show results demonstrating the equilibrium properties of a finite system,
the build-up of the ground state population starting from zero, and evaporative cooling ofatoms in the trap.
A simple test of the trajectory simulation is to start the distribution fen (t0 ) in a nonequilibrium state and allow it to evolve to equilibrium. In the sub-plot of Figure 7.3, theh!.
initial non-equilibrium distribution is shown for N = 103 , and a mean energy of 20:5 
It is allowed to evolve to the stationary state shown in Figure 7.3, where each trajectory
had, on average, 100 collisions. The simulation data is compared with the Bose-Einsteinequilibrium distribution

fen =

1
;
e (en ) 1

(7.26)

where = 1=kB T and  is the chemical potential. Both and  are chosen in the plot
so that gen fen is normalized to N and the mean energy matches that of the simulated
distribution. As Figure 7.3 shows, the trajectory simulation evolves fen (t0 ) to the correctequilibrium distribution for a finite number of atoms without a mean field interaction.

7.4.1 Finite number effects on equilibrium
Finite number effects can be studied by allowing the distribution fen (t) to evolve to equilibrium. In Figure 7.4, the ground state fraction is plotted vs. the temperature for the case122

7.4 Simulation results

ground state fraction No /N

1
0.8
0.6
0.4
0.2
0
0

trajectory model
Jaksch model
thermodynamic limit
0.2

0.4
0.6
0.8
temperature T/Tc

1

1.2

Figure 7.4: Ground state fraction versus temperature for 500 atoms. Each point is obtainedby allowing a system of 500 atoms with a known mean energy to evolve to equilibrium.
Once equilibrium is obtained, the fraction in the ground state is recorded. The plot shows
our data, data from Ref. [121], and the thermodynamic limit.
of N = 500. The graph shows three different sets of data: the trajectory simulation, resultsfrom Ref. [121], and the thermodynamic limit. The trajectory data agrees with the resultsof Ref. [121], where a different approach to simulating the QBE is used. The line for a
finite number of atoms has the same qualitative shape as in the thermodynamic limit, but it
is shifted toward lower temperatures [121, 123, 124]. In Figure 7.5, the same plot is shownfor the case of 2  104 atoms. As expected, the line is shifted less from the thermodynamic
limit. The effect of finite size on the mean energy of the system can also be studied. In
Figure 7.6, the mean energy is plotted vs. temperature for the case of N = 500. Again,
the trajectory simulation agrees with the results of Ref. [121]. The same plot is shown in
Figure 7.7 for the case of 2  104 atoms. The mean energy for a finite number of atoms
is larger than that in the thermodynamic limit below Tc . In the thermodynamic limit, thenumber and volume are taken to infinity, with the local density held fixed. For the case of
atoms in a trap, taking the volume to infinity is achieved by allowing ! to go to zero. Fora finite system, the effect of the potential remains, thus giving the system a higher meanenergy than it would in the limit of ! ! 0.

7.4.2 Dynamic buildup of the condensate
The buildup of the condensate can be investigated by starting in a non-equilibrium distribution fen with no atoms in the ground state initially, and allowing the distribution to
evolve to equilibrium. One can monitor the occupation of the ground state over time. As
Figure 7.8 shows, the time dependence of the population in the ground state is given by
N0 (t) = N0 (1)(1 e t=0 ), where the time constant 0 is determined by fitting the data.
123

Chapter 7 Trajectory Simulation of the Quantum Boltzmann Equation

ground state fraction No /N

1
0.8
0.6
0.4
trajectory model
thermodynamic limit

0.2
0
0

0.2

0.4
0.6
0.8
temperature T/T c

1

1.2

Figure 7.5: Same procedure as in Figure 7.4, but with 2  104 atoms.

4

mean energy ε /k BTC

3.5
3

trajectory model
Jaksch model
thermodynamic limit

2.5
2

1.5
1

0.5
0
0

0.5
1
temperature T/TC

Figure 7.6: Mean energy versus temperature for 500 atoms. Each point is obtained byallowing a system of 500 atoms with a known mean energy to evolve to equilibrium. The
plot shows our data, data from Ref. [121], and the thermodynamic limit.

124

7.4 Simulation results

4

mean energy

ε /k BTC

3.5
3

trajectory model
thermodynamic limit

2.5
2
1.5
1
0.5
0
0

0.2

0.4

0.6
0.8
1
temperature T/TC

1.2

Figure 7.7: Same procedure as in Figure 7.6, but with 2  104 atoms.

1

No /No ( )

0.8
o
o

20
population

0.6
0.4

15
10
5

0.2
0
0

0
0

0.5

1

1.5
t/τ o

3

6
9
energy

2

2.5

12

3

Figure 7.8: The population in the ground state increases as the system of 100 atoms evolves
to equilibrium, starting in the initial distribution shown in the sub-plot (where energy is
given in units of h!). The ground state fraction increases according to N0 (t) = N0 (1)(1e t=0 ).

125Chapter 7 Trajectory Simulation of the Quantum Boltzmann Equation200

N 0 (t f )

160
120
80
40
0
−3
10

γcol (t i )

γbl

10

−2

γcut

10

−1

0

10

Figure 7.9: The final number in the ground state N0 (tf ) as a function of the cut rate cut ,with (circled-cross) and without (circled-dot) background loss. The cut rate must be slower
than the initial single particle collision rate col (ti ) and faster than the background loss rate
bl for the evaporation to be successful.
This result also agrees with that given in Ref. [121]. The initial distribution, shown in the
sub-plot in Figure 7.8, had 100 atoms with 10% in the ground state after it had reachedequilibrium.
It was found that 0 depends slightly on the initial distribution: With the mean energy
and total number fixed, the further the atoms are from e0 , the longer it will take to reachthe ground state. This is also why the time constant 0 decreases as N0 (1) increases while
keeping N fixed, since the mean energy decreases, requiring atoms to reside in levels closer
to e0 . As N gets large, more energy levels will be occupied and one might expect there to
be a delay time for the atoms to begin filling the ground state [121]. Finally, it was foundthat 0 =col increases with increasing N , while keeping N0 (1)=N fixed [125]. For thecase of 100 atoms shown in Figure 7.8, 0 =col  25, compared to a separate case for 50
atoms, where 0 =col  10.

7.4.3 Evaporative cooling simulation
A practical use of our simulation method is to study the evaporative cooling of a gas of
atoms in an isotropic harmonic trap. Evaporative cooling may be described by allowing
trajectories with an energy above a time-dependent energy threshold ecut (t) to be lost fromthe trap. We can also allow there to be a finite probability for trajectories to be lost due tocollisions with background atoms, which occur at a rate bl .
In Figure 7.9, we show data points collected from simulations of a particular evaporative cooling scenario. We began with N = 104 atoms in thermal equilibrium at a temperature T = 15 Tc. We then allowed the system to evolve while lowering the energy threshold126

7.5 Summaryecut(t) exponentially in time at a rate cut , with no background losses. As the temperatureapproached Tc , when there was one atom in the ground state with 500 atoms left in the trap,
we stopped the simulation. We then proceeded cutting exponentially to ecut (tf ) = e2 , thesecond energy level, at varying rates cut , as well as with and without background loss.
Figure 7.9 shows a semi-log plot of the final number in the ground state N0 (tf ) vs. cut .
The result is intuitive: With no background loss, N0 (tf ) decreases as the cut rate increases. If we cut faster than the collision rate col , then the evaporation completely fails
because the gas does not have time to equilibrate as the threshold is lowered. When background loss is included, one can see that the lower limit on cut is determined by the
background loss rate bl . If we cut slower than the background loss rate, all of the atoms
are lost from the trap before the evaporation process is finished. Thus, there is an optimum
cut rate cut bracketed by these two physical properties.

7.5 Summary
We have presented a novel approach to treating quantum kinetics which is based on adecomposition of the ergodic QBE into single-particle trajectories. We presented the underlying physical theory, explained our trajectory approach to simulating quantum kinetics,
and displayed results of our method applied to some contemporary problems concerning
Bose-Einstein condensation. As a test of the validity of our method, it agrees well withindependent studies on the processes studied in section 7.4 [121]. Our approach gives anefficient simulation of quantum kinetics and is valid for the entire range of phase-spacedensities, excluding at this stage the region close to T = 0 when the mean field effect on
the system must be considered.
The trajectory approach of quantum kinetics described in this chapter is applicableto many problems of interest. One such problem is that of finding the optimum way tolower ecut(t) during the evaporative cooling process, while taking into account all of thevarious loss mechanisms, such as loss due to collisions with background atoms and heating
due to two-body and three-body inelastic collisions. Another interesting problem is that ofincluding the mean field effect on the system during the kinetic evolution close to T = 0. To
address this problem using the QBE, we will have to work in the representation of the meanfield states, which requires finding the mean field eigenstates self-consistently after eachtime-step in the simulation. Finally, it may also be interesting to use the trajectory approach
to treat Fermi-Dirac statistics and describe the time-evolution of a gas of fermions.

127

Chapter 7 Trajectory Simulation of the Quantum Boltzmann Equation

128

Chapter 8

Achieving Steady-State
Bose-Einstein Condensation
8.1 Introduction
In the usual method of evaporative cooling used so far in BEC experiments [109, 110,
118, 126, 127, 128], a finite number of atoms are collected in a magnetic trap after being
laser cooled to a phase space density at least five orders of magnitude below the criticaldensity needed for BEC. The frequency of an external RF radiation field, which spin-flips
the atoms to an un-trapped state, is then lowered continuously. This further cools the gas
by removing high energy atoms from the tail of the distribution. This evaporative coolingprocedure increases the phase space density above the critical point needed to reach BEC.
The success of this method is well established experimentally, allowing many fundamental
properties of Bose-Einstein condensation to be investigated [33, 34].
This standard method of achieving BEC has one critical drawback: once a condensate
has been obtained, it has a finite lifetime in the trap determined by various loss mechanisms,
such as collisions with hot atoms from the background gas, and inelastic collisions between
the trapped atoms. Although the finite lifetime of the condensate does not prevent many
crucial properties of the system to be studied, it is still very desirable to achieve a steadystate situation so that a condensate can be sustained for an indefinite period of time. Sucha situation is essential for the continuous output of a coherent beam of atoms in an atom
laser [17, 129, 130, 131, 132, 133]. To date, no experiment has demonstrated a steady state
condensation.
We address this problem by constructing an intuitive model describing the two aspects
to such an experiment: The continuous loading of atoms into the magnetic trap and theclassical kinetic evolution of the trapped atoms toward a steady state during the evaporation.
Our description of the loading procedure is based on the experimental setup describedin [134], where the authors loaded a magnetic trap with atoms which had been cooled in a
separate MOT. This allows us to estimate the rate f that atoms enter the trap below the RF
cut, as well as the mean energy ef of the injected atoms.
To model the classical kinetic evolution, we assume a truncated Boltzmann distribution
for the trapped atoms and obtain rate equations for the total number N (t) and energy E (t)of the system [109, 118, 126, 127, 128]. These rate equations include the loss of atoms

Chapter 8 Achieving Steady-State Bose-Einstein Condensation
due to elastic collisions with the background-gas atoms, inelastic 3-body collisions, andevaporation, as well as the gain of atoms due to loading. We then numerically calculate thesteady state solution of these equations and show plots of the peak phase space density 0
as a function of the various physical parameters of the system. We show that the criticalregime 0  2:612 may be reached in order to obtain BEC in steady state.

8.2 Description of the model
In constructing a model of steady state evaporative cooling, there are several experimental
schemes one could consider for describing the loading of atoms into the magnetic trap, as
well as several layers of approximation in describing the kinetic evolution of the trapped gas
toward steady state. However, we consider only one realization of the loading procedure,
assuming the atoms are first trapped and cooled in a MOT and then transferred to a separate
magnetic trap [134, 135, 7]. Furthermore, we consider a simplified model of evaporative
cooling that assumes classical statistics, and is therefore valid only for phase space densities
below the critical point 0 = 2:612; one would have to include quantum statistics in orderto properly model the system above this point. These two parts to our model are describedin the following subsections.

8.2.1 Description of the loading procedure
In a real experiment, irreversibility is introduced at each step of the transfer of the atoms
from the MOT to the magnetic trap; the atoms are first pushed out of the MOT, they thentravel through a magnetically-confining tube, and finally must be caught in the magnetic
trap and optically pumped into a trapped hyperfine state. In order not to get lost in thedetails of modeling all of these heating and loss mechanisms, we consider two extreme
idealizations of the transfer: an adiabatic transfer which preserves the phase space density
0 and a sudden, irreversible transfer which decreases 0 .
We assume the atoms feel an isotropic, linear restoring force in both the MOT and themagnetic trap, neglecting the possibility of a radiation pressure in the MOT, which would
distort the effective harmonic trapping potential [136]. Then the free Hamiltonian of an
atom in either trap can be written

Hi (r; p) =p2 1 2 2+ m! r ;2m 2 i(8.1)

where m is the mass of the atom, and !i is the effective radial frequency of the trapping
potential. The index i = 1 indicates the MOT, while i = 2 indicates the magnetic trap.
We model the transfer of atoms in order to obtain a reasonable estimate of the feed rate
f and the mean energy ef of atoms injected into the trap below the RF cut. We treat thistransfer process as a succession of discrete transfers each consisting of a finite number ofatoms. We only need to consider a snapshot of this transfer process: a finite number ofatoms N1 are collected in the MOT at a temperature T1 in equilibrium, they are then either
adiabatically or suddenly transfered to the magnetic trap. In our model, we allow these N1
atoms to come to an equilibrium in the magnetic trap, characterized by a new temperature
T2 . We then place the RF cut ecut and calculate the fraction of atoms f which remain in
the magnetic trap below ecut , as well as the mean energy per atom ef of these atoms
1308.2 Description of the model

T1 ω1

MOT
T2 ω2
magnetic trap

ecut

Figure 8.1: Illustration of the transfer process described in section 8.2.1. A finite numberof atoms are cooled in the MOT to a temperature T1 in equilibrium. We approximate thepotential in the MOT as an isotropic harmonic oscillator at frequency !1 . They are thentransferred to the magnetic trap, either suddenly, or adiabatically. We also approximate
the magnetic trap as forming an isotropic harmonic oscillator potential, with a different
frequency !2 . In equilibrium, the atoms have a temperature T2 in the magnetic trap. Then,the RF energy threshold ecut is applied and only a portion of the original atoms from the
MOT remain. This transfer can be repeated many times in order to obtain a piecewise
continuous transfer of atoms.
R ecute2 e e=kB T2 de
f = R0 1 e2 e e=kB T2 de ;0
R ecut 3 e=k T
e e B 2 deef = R0ecut 2 e=kB T2 :
de
0 ee

(8.2)
(8.3)

The e2 factor appears due to the density of states for an isotropic harmonic oscillator potential. A schematic diagram in Figure 8.1 illustrates the transfer process.
This process can be repeated many times each second so that atoms are transfered tothe magnetic trap at a rate t . The rate that atoms enter below the RF threshold ecut is thengiven by f = f t . We estimate an upper limit on the number of these transfers eachsecond to be on the order of 100.
The equilibrium temperature T2 which the atoms attain after a sudden transfer can beobtained by considering the sudden change in the energy of the atoms after the instantaneous change in trapping frequencies !1 ! !2 . Then for a sudden transfer, the temperature
T2 is related to the temperature T1 in the MOT according toT!2
T2 = 1 (1 + 22 ) (sudden) :
2
!1

(8.4)
131

Chapter 8 Achieving Steady-State Bose-Einstein Condensation
The adiabatic case can be treated as a succession of infinitesimal steps !1 ! !1 + Æ! , eachtreated as a sudden transfer. This yields the relationship!
T2 = T1 2 (adiabatic) :(8.5)!1Note that both cases give T2 = T1 when !2 = !1 as they must. With the peak phase spacedensity 0 = n0 3 of the trapped atoms given byh !
0 = N1 ( i )3 ;
(8.6)
kB Ti
it is clear that 0 is invariant through an adiabatic transfer, while it decreases after a suddentransfer. Here,  is the deBroigle wavelength and n0 is the peak spatial density.
The two quantities f and ef depend on the frequency in the lower trap !2 , as well as
the RF field threshold ecut ; as the trap is made looser, more atoms will make it into the trap
below the cut so that f increases. The feed rate is also increased as ecut is raised, however
the mean energy ef of those atoms increases as well.

8.2.2 Description of evaporative coolingWith the feeding rate f and mean energy per atom ef of the injected atoms given by
the above model of the loading procedure, it remains to describe the kinetic evolutionof the atoms in the magnetic trap during evaporation. Our model can be constructed on
phenomenological considerations, with the goal of characterizing the steady state of thesystem.
We characterize the trapped atoms by a single-particle distribution over energy (e) f (e; t)instead of retaining the more detailed description in phase space using f (~x; ~p; t) [109].
Here (e) is the density of states for an isotropic harmonic potential. We also make anassumption that the non-equilibrium distribution f (e; t) of the system can be well approximated by a truncated Boltzmann distribution [109, 118, 126, 127, 128]f (e; t) =(t)e (t)e e < ecut0e  ecut ;(8.7)

where  (t) and (t) = 1=kb T (t) are functions of time. They are related to the total number
N (t) and total energy E (t) of the atoms according to

N (t) =
E (t) =

Z ecut

0
Z
0

ecut

de (e) f (e; t) ;(8.8)

de (e) e f (e; t) :(8.9)

With the assumption of the truncated Boltzmann form for f (e; t), the description of thesystem can be reduced to finding the equations of motion for the total number and energy.
The equations of motion for N (t) and E (t) will be written in terms of the various
gain and loss processes which occur. There are four competing processes which take place
during the evaporation: the constant feeding of atoms into the trap at a rate f with a meanenergy per atom ef , the loss of atoms from the trap due to collisions with the atoms from the
hot background gas, characterized by a constant rate bl , the loss of atoms and heating due
132

8.3 Results
to 3-body inelastic collisions, given by the rate 30 , and the rethermalization due to elastic
collisions which will eject atoms from the trap which obtain an energy above ecut after acollision. We can include all of these effects in the kinetic equation for f (e; t)(e)@f (e; t)= f gf (e)@bl (e)f (e; t)

30 ((e)f (e; t))3 + col (t) ;(8.10)

where the distribution of atoms injected into the trap is gf (e) and the density of states is(e) = 21 e2 =(h!2 )3 . col (t) is the collision integral given by [109]col (t) = 0

Z

der de0 de0r Æ(e + er e0e0r )(emin)[f (e0 )f (e0r ) f (e)f (er)] ; (8.11)

where 0 = m=( 2 h3 ) and emin = minfe; er; e0; e0r g is the minimum energy.
By substituting Eq. (8.7) into Eq. (8.10), and using Eq. (8.8) and Eq. (8.9), we obtainthe following equations of motion for the total number and total energy(8.12)
f bl N 3 N 3 N ;2(8.13)
f ef bl E (t) 3 3 N 2 E E ;where the 3-body loss rate for the total number is 3 = 31:5 K3 (m!22 =2kB T )3 . K3 is

N_ =
E_ =

an experimentally determined constant to be specified [137]. In obtaining the 3-body loss
terms, an approximation has been made that ecut  kB T (t) in order to simplify the terms.
Initially during the evolution, this assumption may not hold, but the density is low enoughinitially that the 3-body loss terms are negligible in any case. By the time the density
has increased enough so that 3-body losses are significant, the assumption does hold. The
factor of 2=3 in Eq. (8.13) signifies that the energy will decrease at a slower rate than thenumber due to 3-body losses, which gives rise to an effective heating.
The two terms N and E represent the loss of number and energy due to evaporation
and are given by

N =
E =

Z ecutZ ecut Z e+er ecut

0 de der de0r (e0r )f (e; t)f (er ; t) ;0 ecut e 0
Z ecutZ ecut Z e+er ecut
0 de der de0r e0 (e0r )f (e; t)f (er ; t) :0 ecut e 0(8.14)(8.15)

The fourth atom in these equations is lost from the trap since its energy is always greaterthan the RF cut e0 > ecut . Due to energy conservation and the truncated form of f (e), thismeans that emin = e0r , as indicated in Eq. (8.14) and Eq. (8.15). Also, the energy whichappears in the term E (t) is that of the escaping atom e0 = e + er e0r .

8.3 Results
In order to carry out explicit calculations, we choose realistic values of the various physicalparameters needed in our model. These are listed in Table 8.1 for a gas of 87 Rb atoms.
The parameters !2 and ecut are not listed in the table but are variables to be specified in
the following calculations. We have specified a reference point for the MOT parameterswhich yields a phase space density in the MOT of 0 = 6:9  10 6 , if one assumes that
N1 = 5  105 at 20 transfers per second [136].
133

Chapter 8 Achieving Steady-State Bose-Einstein Condensation

Rbbl  1= bl
K3
T1(ref)!1(ref)=2
(ref)t

7:5  10 16m2

200 s

4:9  10 29cm6 =s
20 K
100 Hz107 atoms/s

Table 8.1: Values used for the various physical parameters needed in the model. Rb isthe s-wave scattering cross section for 87 Rb. The explanations for the other parameters are
given in the text.

8.3.1 Time evolution
We first consider the dynamical evolution of the system toward steady state. In Figure 8.2
we show results of a numerical integration of the rate equations in Eq. (8.12) and Eq. (8.13)
for the total number N (t) and energy E (t). Since the magnetic trap frequency !2 ismatched to the MOT frequency !1 in this calculation, the adiabatic and sudden transfers
are equivalent. For case 1 in the figure, we chose the optimum value of ecut to yield thehighest phase space density 0 , while in case 2 the value chosen for ecut is ten times higher
than that in case 1. There are some interesting features to consider from this plot.
It is instructive to take a simple limiting case of Eq. (8.12) and Eq. (8.13) in order tolearn something about the build-up time for steady state to occur. If we let ecut ! 1 and
3 = 0, then the solution to the rate equations for N (t) and E (t) is given by

N (t) =
E (t) =t (1 e bl t ) ;blt e (1 e bl t ) :f
bl(8.16)The time-scale for steady state to occur in this simple case is just the lifetime of the trap
as determined by background losses, bl . In the case where the RF cut is present andevaporation is occurring, while still neglecting 3-body losses, the build-up time for steadystate will be on the order of magnitude of bl , although it will be shorter, based on resultsof numerical calculations. We define this build-up time to be the time at which N (t) =(1 e 1 )Nss . When 3-body losses are included, the build up time can be very shortcompared to bl if the density is high enough for 3-body losses to dominate. So this gives
us an upper limit of the build-up time to be bl , and if steady state occurs on a much
shorter time scale than this, it indicates that 3-body losses are dominating the other loss
mechanisms.
In Figure 8.2, the build-up time in case 1 is slightly less than bl , which is 200 seconds.
This indicates that the choice of ecut in case 1 minimizes 3-body losses. In case 2, on theother hand, where ecut is ten times larger than that in case 1, the build-up time is much
shorter at roughly 25 seconds. This is because in case 2, f is larger, causing the density
to build-up more quickly which allows 3-body losses to dominate. This also stops theevaporative cooling quickly and so one does not obtain as high of a phase space density 0
as in case 1. It should be noted that when we calculated case 2 with 3 = 0, the build-up
time was approximately equal to bl , and the steady state value of the phase space density
was close to being optimized at that value of ecut , with 0 = 3:9 in steady state.
1348.3 Results

1
0.9
2

1

1

0.7

10

0.6

0

1

10

0.5
ρ0

fraction

0.8

0.4

2

−1

10

−2

0.3

10

0.2

−3

10
0.1

0

200

400
600
time (s)

800

0
0

100

200

300

400 500
time (s)

600

700

800

Figure 8.2: Time evolution of the total number N (t) and total energy E (t) for the valuesof the parameters listed in Table 8.1. The magnetic trap frequency is equal to the MOT
frequency !2 = !1 in this calculation. Two values of ecut were chosen: 1:1 K, labeled by
1, and 11 K, labeled by 2. Each of the curves is normalized by its final steady state value.
The solid curve is the total number and reaches a steady state value of Nss = 2:0  104 forcase 1, and Nss = 2:8  106 for case 2. The dashed curve is the total energy and reaches asteady state value of Ess = 0:33 K  Nss for case 1 (case 2 is not shown). The evolutionof the peak phase space density 0 is shown in the inset for the two cases.

135Chapter 8 Achieving Steady-State Bose-Einstein Condensation
1.5

0

1.3

1

1.1

2

0

ω2 / ω1

10

0.9

3

0.7

4

0.5

5
9
6

−1

10

0.3

7
−1

10

8
0

10

1

ecut (µ K)

10

2

0.1

10

Figure 8.3: Two overlaying contours of the steady state value of the phase space density
and the total number vs. the ratio of trap frequencies and RF cut threshold for an adiabatic
transfer. The shaded contours represent the steady state value of 0 , with the gray-scale bar
shown to the right. The numbered lines represent log10 Nss (i.e. a value of 6 for the line in
the center corresponds to Nss = 106 ). It is !2 which is varied in the ratio, while !1 is fixedat 2 100 Hz. The values used for the other parameters are displayed in Table 8.1.

8.3.2 Steady state solution
Now that we have characterized the time scale for steady state to occur, it is useful to solve
Eq. (8.12) and Eq. (8.13) directly for the steady state values of Nss and Ess by setting theleft-hand sides equal to zero. We were not able to solve the resulting coupled algebraic
equations analytically, since they are transcendental in form. However, they are straightforward to solve numerically. In the following sections, we present calculations of thesteady state value of 0 while varying some of the physical parameters in order to discern
what values of the parameters yield 0 = 2:612 so that BEC can be achieved in steadystate.
Varying ecut and !2
In trying to understand what it takes to reach a steady state BEC, it is useful to look at how0 varies with !2 and ecut. In Figure 8.3 and Figure 8.4, we show shaded contour plotsof the steady state value of 0 , for both an adiabatic and a sudden transfer. Also shown
are contours of the total number Nss overlaying the shaded contours. Again, we use thereference point of parameters displayed in Table 8.1. The two different idealizations of thetransfer process yield quite distinct shapes for the surfaces of 0 and Nss .
For the adiabatic case shown in Figure 8.3, 0 increases with increasing !2 , keeping
ecut fixed. However, it levels off quite quickly, varying from 1.1 to 1.5 with an order ofmagnitude increase in !2 =!1 from 0.1 to 1.0 at ecut = 1 K. Also, with !2 fixed, the136

8.3 Results
1.5
0
1.3
1
1.1
0

10

2

ω2 / ω1

0.9
3
0.7
4
0.5

5

6

7

8

9

−1

10

0.3

−1

10

0

10

1

ecut (µ K)

10

2

0.1

10

Figure 8.4: Same as described in the caption of Figure 8.3 except for a sudden transfer ofatoms from the MOT to the magnetic trap, instead of an adiabatic one.
optimum value of ecut which yields the highest 0 does not depend much on !2 , but isroughly a straight line at ecut = 1 K. Perhaps the most interesting and crucial feature
exhibited in the plot is that Nss decreases very rapidly as !2 is increased, going from 107
down to 104 as !2 =!1 goes from 0.1 to 1.0. This is because 3-body losses increase as thetrap is tightened, since the density increases. Therefore, one will gain a lot in number bykeeping the magnetic trap shallow, while losing only a small amount in phase space density.
The results of a sudden transfer are shown in Figure 8.4. The most striking difference
between this and the plot shown in Figure 8.3 for an adiabatic transfer is a strong peakwhich occurs at !2 =!1 = 1. This can be attributed to the fact that the phase space density
always decreases in a sudden transfer, with a peak occurring at !2 = !1 where the sudden
and adiabatic transfers are equivalent. Notice also that 0 drops off much more rapidly as!2 =!1 is varied from unity, compared to the adiabatic case. Another difference between
the two cases is that the optimum value for ecut increases as !2 =!1 is varied from unity.
Finally, it can be seen also that one does not gain that much in number as !2 is decreased,in sharp contrast to the adiabatic case.
Varying T1 andt

We now have an understanding of how the steady state values of 0 and Nss vary withecut and !2 . Another useful calculation is to see how 0 depends on the MOT temperature
T1 and the transfer rate t . In the plots below, ecut is chosen so as to maximize 0 , fora given T1 , t , and !2 . Then, given t and !2 , T1 is chosen so as to reach 0 = 2:612.
This is done for 106  t  108 , as well as three values of the trap frequency ratio!2 =!1 2 f0:1; 0:5; 1g, with !1 = 2 100 Hz.
The results of an adiabatic transfer are shown in Figure 8.5. Along each of the three137

Chapter 8 Achieving Steady-State Bose-Einstein Condensation
1

T1 / T(ref)
1

10

0

10

1
0.5
0.1
−1

10 −1
10

0

10
γ / γ(ref)
t

1

10

t

Figure 8.5: Values of T1 and !2 one must achieve in order to reach 0 = 2:612 in the case ofan adiabatic transfer. Three different values of !2 =!1 are shown: !2 =!1 2 f0:1; 0:5; 1g,
with !1 = 2 100 Hz. For each line, ecut was chosen so as to maximize 0 . The reference(ref) = 20 K and (ref) = 107 atoms/s.
values are T1tlines 0 = 2:612. The most important feature of this plot is that the three lines lie nearly
on top of each other. This agrees with Figure 8.3 in that 0 decreases vary little as !2 islowered. The plot also shows that 0 depends more critically on T1 than on t . Startingfrom the reference point in the center, one has to either decrease T1 by 20%, or increase t
by 100% in order to get to the 0 = 2:612 line.
The sudden transfer is shown in Figure 8.6. In contrast to the adiabatic case, the threelines are separated, so that as !2 is decreased, one has to try much harder to reach 0 =
2:612, which is also consistent with Figure 8.4.
The total number Nss curves corresponding to the 0 = 2:612 lines in Figure 8.5
and Figure 8.6 are shown in Figure 8.7. The results are the same in both the sudden andadiabatic cases (thus there are only three lines instead of six). For the adiabatic case, byloosening the magnetic trap, one does not have to vary T1 and t much at all in order tostay at 0 = 2:612 while increasing the number Nss by orders of magnitude. On the other
hand, for the sudden transfer, one has to decrease T1 and increase t a lot in order to stayat 0 = 2:612 as !2 is decreased. However, one will achieve the same increase in number
as in the adiabatic case.
Finally, in Figure 8.8 we show a plot of the ratio ecut =T2 corresponding to the 0 =
2:612 lines shown in Figures 8.5-8.7. This ratio of the optimum cut to the temperature T2

of atoms being injected into the trap is the same in both the adiabatic and sudden transfers.
As !2 is decreased, one does not have to exclude as much of the distribution from the trap.
Also, as t is increased, one has to cut further into the injected distribution in order toprevent 3-body losses from dominating.
138

8.3 Results

110

T1 / T(ref)
1

1
0.5
0.1

0

10

−1

10 −1
10

0

10
γ / γ(ref)
t

1

10

t

Figure 8.6: Same as described in the caption of Figure 8.5 but for the case of a suddentransfer.
7

10

6

Nss

10

5

10

4

10

3

10 −1
10

0

10
γt / γ(ref)
t

1

10

Figure 8.7: Corresponds to the three lines in both Figure 8.5 and Figure 8.6, showing thetotal number of atoms in steady state Nss as a function of the transfer rate t . Along eachof these curves, 0 = 2:612. The legend in Figure 8.5 and Figure 8.6 applies to this plot
also.

139

Chapter 8 Achieving Steady-State Bose-Einstein Condensation
010

ecut / T2

1
0.5
0.1

−1

10

−2

10 −1
10

0

10
γ / γ(ref)
t

1

10

t

Figure 8.8: Curves correspond to those in Figures 8.5-8.7, showing the ratio of the RF cut
to the temperature of atoms injected into the trap, ecut =T2 as a function of the transfer ratet . Along each of these curves, 0 = 2:612.

8.4 Summary
In this chapter we have addressed the problem of achieving a steady state condensationby continuously feeding atoms into the magnetic trap below a fixed RF threshold. We
have included losses due to elastic collisions with atoms from the background gas, as wellas inelastic 3-body collisions. Our model of the loading of atoms into the magnetic trap
treats two idealizations of transferring atoms from a separate MOT; either an adiabatic or asudden transfer. The description of the kinetic evolution to steady state assumes a truncatedBoltzmann form for the non-equilibrium distribution f (e; t), reducing the problem to thatof solving coupled rate equations for the total number N (t) and total energy E (t) of thegas. Our calculations show that it is possible to achieve a steady state condensation using
optimistic values of the relevant physical parameters.
We have shown several results of numerical solutions of the rate equations in Eq. (8.12)
and Eq. (8.13). First, we addressed the build-up time for steady state to occur and determined that an upper limit on the build-up time is given by the background loss lifetime bl .
If 3-body losses are dominating due to a high density, then the build-up time will be much
shorter than this. We next looked at how the steady state value of the peak phase spacedensity 0 depends on the magnetic trap frequency !2 and the RF cut ecut . We found thatin the adiabatic case, one can gain a large increase in the total number in steady state Nss byloosening the magnetic trap, while only losing a small amount in 0 . This is not true for asudden transfer. Finally, we looked at how one must vary the transfer rate t and the MOT
temperature T1 in order to reach 0 = 2:612. We found that 0 depends more critically on
T1 than t . Also, it was shown that one must try much harder to reach the critical point
while achieving a large Nss in the sudden case compared to the adiabatic case.
There are several shortcomings of our model which might be improved, however, we
140

8.4 Summary
believe that the present calculations are qualitatively correct and are sufficient for experimental guidance. An obvious extension to our model would be to include the effect of
the growth of the condensate which will make the evaporation more efficient but at thesame time increasing 3-body losses due to the increase in density in the center of the
trap. [126, 121, 138, 21] Another improvement would be to construct a more accurate
model of the transfer process by understanding the relationship between T1 , !1 and t ,since these can not be varied independently in an experiment. Alternatively, one could construct a model of the loading procedure based on an entirely different experimental methodthan that described in [134, 135, 7].

141

Chapter 8 Achieving Steady-State Bose-Einstein Condensation

142

Appendix A

Wick’s Theorem
The quantum kinetic equations in Chapter 2 for the mean field Eq. (2.70) and fluctuations
Eq. (2.71) and Eq. (2.72) were derived from the Born-Markov form of the generalized
kinetic equation Eq. (2.60) for a general relevant observable q . We do not provide thedetails in getting from Eq. (2.60) to Eq. (2.70) through Eq. (2.72), however it is importantto understand the main steps, which involve evaluating the averages of multiple operator
products. Due to the Gaussian structure of our reference distribution  (0) (t) in Eq. (2.9),
we can utilize Wick’s theorem to simplify the calculation [42].
We are interested in calculating the average of a product of s operatorshA^1 A^2    A^s it(0) = Trf(0) (t) A^1 A^2    A^s g;(A.1)^yi or a lowering operator a^i . These
where we are denoting A^i to represent either a raising aaverages arise in the kinetic equations. For example, in the kinetic equation for the normal
fluctuations, the second order collisional terms involve a product of ten operators due toterms of the form  hV 2 f~iy(0) , since the interaction V contains a product of four operators

V  a^yi a^yj a^k a^l and the normal fluctuations have the form f~  a^yj a^i .

Wick’s theorem is valid for averages taken over a Gaussian distribution that can involve
both normal a^yj a^i and anomalous pairs a^i a^j in the exponent, as our reference distribution
does in Eq. (2.9). We do not derive Wick’s theorem here, but simply state the results. Theinterested reader can find derivations of Wick’s theorem in Louisell [119] (p. 182) for thecase of bosons, and in Zubarev [39] (p. 172) for fermions. Both of these derivations assume^yj a^i in the exponent; one can find a more generala Gaussian form with only normal pairs aderivation including anomalous pairs (for a squeezed vacuum) in Vaglica [43]. To include
the anomalous pairs in the proof, one can simply make a canonical transformation of the^i and a^yi so that only normal pairs appear in the exponent for the new set ofoperators aoperators; doing this allows one to use the standard proof, which treats only normal pairs.
Wick’s theorem states that the average value of a product of creation and annihilation
operators is equal to the sum of all complete systems of pairings, which can be stated moreformally as

hA^1 A^2    A^s it(0) =X

PdhA^1 A^2 it(0) hA^3 A^4 it(0)    hA^s 1 A^s it(0) ;(A.2)

where the sum runs over all Pd distinct permutations of the s indices. An alternative way

Appendix A Wick’s Theorem
to write Eq. (A.2) ishA^1 A^2    A^s it(0) = hA^1 A^2 it(0) hA^3 A^4    A^s it(0) + hA^1 A^3 it(0) hA^2 A^4    A^s it(0)+ hA^1 A^s it(0) hA^2 A^3    A^s 1 it(0) ;(A.3)
and then applying this relation recursively to all of the multiple operator averages until only
pairs of operators remain. If the number of operators s is odd, Wick’s theorem reduces tothe simple resulthA^1 A^2    A^s it(0) = 0 if s is odd:
(A.4)
Wick’s theorem is augmented by the further rules for averages of pairs of operators, whichhold for bosonsha^1 a^y2 it(0) = Æ12 + ha^y1 a^2 it(0) ;ha^1 a^2 it(0) = ha^2 a^1 it(0) ;ha^y1 a^y2 it(0) = ha^y2 a^y1 it(0) :(A.5)

As an example, we consider the product of four operators hA^1 A^2 A^3 A^4 it(0) , which can be
reduced according to Wick’s theorem tohA^1 A^2 A^3 A^4 it(0) = hA^1 A^2 it(0) hA^3 A^4 it(0) + hA^1 A^3 it(0) hA^2 A^4 it(0) + hA^1 A^4 it(0) hA^2 A^3 it(0)(A.6)

The application of Wick’s theorem to the kinetic equation Eq. (2.60) for our set of
relevant operators is complicated by the fact that the operators a^i are shifted by the cnumber mean-field i . The evaluation of the average of a product of ten shifted operators
involves 210 terms, each of which must be reduced according to Wick’s theorem. Even ademonstration on the example in Eq. (A.6) for the case four operators would involve on
the order of  3  24 = 48 terms. A symbolic algebra package using Mathematica wasdeveloped by Reinhold Walser in our group to carry out all of these steps.

144

Appendix B

Collisional Terms
In Chapter 2 we derived the quantum kinetic equations for the mean field and fluctuations,which lead to Eq. (2.70) through Eq. (2.72), however we did not display the explicit formsof the second order collisional terms since they are rather involved, and furthermore arenever used in the calculations of this thesis. However, it is the full inclusion of collisional
effects embodied in these terms that gives our theory its real power. Future studies of
the coupled, two-component system, which we treated at zero temperature in this thesis,
will involve extending the calculations to finite temperature, an endeavor that will benefit
greatly from having derived the full quantum kinetic equations.
(2) ~ me ] for the mean
On page 146 we write the second-order collision terms: L [ ; f;

(2)

(2)

~m
field, Lf~ [ ; f;e ] for the normal fluctuations, and Lme ] for the anomalous fluc~ [ ; f;~ mtuations.

146






f~m
e (1+f~)(1+f~) me f~ 6 h j

(1+f~)(1+f~) f~ + 2 f~me en 2 (1+f~) me en j i +~~ (c) + 2 f~(m(c) +me )en + 2 f~me n(c) + 2 f (c) me en (1 + f )f~f~(1+f~) + 2 f (c) f~(1+f~) + f~ff



f~f~(1+f~)



+

L(2)
e] =
m
~ [ ; f;~ me~ (c) + 2 ff~ (c) (1+f~) + 2 f~me (n(c) +en ) + 2 fm
~ (c)en 6 m
f~f~(1+f~) + f~ff





6 (1 + f~) + transp:
2 (1+f~)(m(c) +m
e )f~ + 2 (1+f~) me f (c) + 2 f (c) me f~ + meme (n(c) +e
n) + 2 me m(c) e
n





e +
(1+f~)(1+f~) f~ + (1+f~)(1+f~) f (c) + 2 (1+f~) f (c) f~ + 2 (1+f~) me (n(c) +en ) + 2 (1+f~) m(c)en 6 m

6 f~ +
2 f~(m(c) +m
e )(1+f~) + 2 f~m
e f (c) + 2 f (c) me (1+f~) + meme (n(c) +e
n) + 2 me m(c) en

(1+f~)(1+f~) f~ + 2 f (c) (1+f~) f~ + (1+f~)(1+f~) f (c) + 2 (1+f~)(m(c) +me )en + 2 (1+f~) me n(c) + 2 f (c) me en f~
+ 2 f~(m(c) +m
+
+
(
c
)
(
c
)
(
c
)
(
c
)
(
c
)
~
~
~
~
~
~
~
e )(1+f )f m
e (1+f )fm
ef(1+f )(m +me )f f me f (1+f ) me f 6 ne + h:c:

~ e] =
L(2)f~ [ ; f; m

+ 2

L(2)[ ; f;~ m
e] =

(B.3)

(B.2)

(B.1)

Appendix B Collisional Terms

In these expressions we have defined the following collision operators and pseudo operators

fff
fmf
fmn
mmn

=
=
=
=

2 C (1; 20; 30 ; 40) C" (100 ; 200; 300 ; 400 ) f30 100 f40 200 f400 20 j1i h300 j ; (B.4)
2 C (1; 20; 30 ; 40) C" (100 ; 200; 300 ; 400 ) f30 100 m40 300 f400 20 j1i j200 i ; (B.5)
2 C (1; 20; 30 ; 40) C" (100 ; 200; 300 ; 400 ) f30 100 m40 300 n200 20 j1i h400 j ; (B.6)
2 C (1; 20; 30 ; 40)C" (100 ; 200 ; 300; 400 ) m30 400 m40 300 n200 20 j1i j100 i : (B.7)

The transition amplitudes C (1; 2; 3; 4) are given by

C (1; 2; 3; 4) = V0Z

d3 r1 (r)2 (r)3 (r)4 (r):(B.8)

The amplitude C" (1; 2; 3; 4) arose from evaluating the time-integral over the free propagator. The details of obtaining this term can be found in [27]; here we give the resultC" (1; 2; 3; 4) = C (1; 2; 3; 4) Æ" (1234 ) + iP"

1

1234

This is non-zero only if the energy difference 1234 = 1 (t) + 2 (t)smaller than ", where we define11=  Æ" () + i P" :
" i


!

:
3 (t)(B.9)

4 (t) is(B.10)

Here we have expanded in terms of the mean-field states j1i and their energies, defined by

H0 + Q

j1i = 1 (t) j1i :(B.11)

147

Appendix B Collisional Terms

148

Appendix C

Numerical solution of the GP
equationIn this appendix we give the details of the numerical procedure used to solve the GrossPitaevskii equation. In the first part, we consider evolving the wavefunction in time, startingwith some initial condition (0), which we do by using the Crank-Nicholson procedure.
We describe this procedure for the case of one spatial dimension, (z; t), and then briefly
discuss the extension of the Crank-Nicholson method to two spatial dimensions, (x; y; t)or (; z; t), which is called the alternating-direction implicit (ADI) method. We also
describe the numerical procedure for treating the coupled, two-component system using
the split-operator approximation.
In the second part, we describe the technique of propagating the wavefunciton in imaginary time in order to find the ground state of the system, and we discuss the extensionof this to also obtain excited eigenstates. One can also use imaginary-time propagation(ITP) to obtain topological solutions of the GP equation, which we discuss. Finally, we
briefly describe the application of this method to find the dressed states for a coupled, twocomponent system.

C.1

Time evolution using the Crank-Nicholson method

C.1.1 One spatial dimension
We are interested in propagating the wavefunction (z; t) forward in time, starting withsome initial state (z; 0). The GP equation in dimensionless units (i.e. taking 
h = !z =m = 1) is writteni@ (z; t)1 @2 1 2=[+ z + 1d j (z; t)j2 ] (z; t);@t
2 @z 2 2(C.1)

where 1d is proportional to the s-wave scattering length and is given explicitly below
Eq. (3.15) in Chapter 3. We represent this equation numerically as a difference equation,with position z broken up into a grid of points zj space by a distance h, and time t as
discrete steps of length  . The wavefunction is defined on this grid as as vector of lengthAppendix C Numerical solution of the GP equation

M



n
j
zj
tn

(zj ; tn );= (j 1)h;= (n 1);(C.2)

where j = 1; 2;    ; M is an integer, and n is an integer. The GP equation can be written
in this discrete representation as

n+1

i j



n
j

=

1 jn+1
2

2 jn + jn 1 h 1
+ (j
h2
2

i

1)2 h2 + 1d j jn j2 jn ;(C.3)

where we have represented the time derivative as a forward difference, and the Laplacianas a central difference [139]. It is convenient to write this as the matrix equation

n+1 =



I i H n



n;

(C.4)

where I is the identity matrix and the matrix elements of H n are

Hijn =

1Æ
2h2 i;j+12Æi;j + Æi;j 1 +

h1

2

i

(i 1)2 h2 + 1d j in j2 Æij :(C.5)

The result Eq. (C.4) is called the explicit form of the discrete representation of Eq. (C.1).
It is numerically unstable if the time step  is too large [139, 140]. We can obtain analternative form if we notice that Eq. (C.4) is the expansion of the unitary matrix U =exp[ i H n ], keeping only the first two terms. That is, the exact solution n+1 is givenby
n+1 = e iH n n :(C.6)
If  is small enough, then Eq. (C.4) is a reasonably good approximation to Eq. (C.6).
However, a better solution can be obtained by writing Eq. (C.6) as

e+iH

n

n+1 =

n:

(C.7)

Here, we are propagating the, as yet unknown, solution n+1 backward in time to
Expanding U y to first order in  , this can be rewritten as [139, 140]

n+1 =



I + i H n

 1

n:

n.

(C.8)

This is known as the implicit discrete representation of Eq. (C.1). Although it seems likea subtle change from Eq. (C.4), it is actually unconditionally stable, whereas Eq. (C.4) is
unstable.
There is a problem with both the explicit and implicit methods, in that they do notpreserve the norm of . That is, the approximate forms of the unitary operator used in
Eq. (C.4) and Eq. (C.8) are not unitary. This can be remedied if we use the Cayley formof U , which is essentially an average between the explicit and implicit forms, so as toconstruct an approximation that is unitary [140, 141, 139]

n+1 =
150



I + i 2 H n 1I i 2 H n



n:

(C.9)

C.1 Time evolution using the Crank-Nicholson method
This result is known as the Crank-Nicholson form of the discrete representation of Eq. (C.1).
It is stable, unitary, and accurate to second order in space and time [141].
We have not discussed the fact that H n is a nonlinear Hamiltonian that depends on thestate through the mean-field interaction (this is the source of the superscript n, whichindicates that H n depends Ron time). Strictly speaking, one would have to use the gener
alization of U = T exp[ i 0 H (t)dt] to account for the time dependence in H n , which
involves a time-ordered integration from t = 0 to t =  . However, if is slowly varying,then we can take it as constant over the short time interval  and approximate U as given
inEq. (C.9).
In order to implement the Crank-Nicholson scheme, it is useful to rewrite Eq. (C.9) as

n+1 = 

n:(C.10)The vector  is found by solving the matrix equation

Q =

n;

(C.11)

with Q = [I + (i H n =2)]=2. Writing Eq. (C.9) in this way let us avoid taking the matrix
inverse. The matrix Q is a tridiagonal matrix, so that equation Eq. (C.11) can be solved
using a simple Gaussian elimination algorithm tailored for a tridiagonal system [139].

C.1.2 Two spatial dimensions—the ADI method
We now consider the case of having two spatial dimensions in Cartesian coordinates, with
the GP equation given by@ (x; y; t)
1 @2
@2
1
= [ ( 2 + 2 ) + (x2 + y2 ) + 2d j (x; y; t)j2 ] (x; y; t); (C.12)
@t
2 @x
@y
2
where 2d is given below Eq. (3.17) in Chapter 3. The discrete form of (x; y; t) is now
an Mx by My matrix instead of a vector, given as
i

n
i;j
xi
yj
tn



(xi ; yj ; tn );= (i 1)h;= (j 1)h;= (n 1);(C.13)

where i = 1; 2;    ; Mx , j = 1; 2;    ; My , and n are integers. We have taken the grid
spacing h to be the same along each axis. The discrete form of the GP equation now takes
the form
n + n
n
n
n
n+1
n
1 (ni+1);j 2 i;j(i 1);j 1 i;(j+1) 2 i;j + i;(j 1)
i;j
i;j

i



=

+

2
h2
h
1
(i 1)2 h2 + (j
2

i

2

n j2
1)2 h2 + 2d j i;j

!

h2

n
i;j :

(C.14)

As written, the Laplacian is a fourth rank tensor operating on the second rank tensor representation of . This is difficult to represent numerically, so we instead introduce a vectorn as [142]representation of i;jn;vrn = i;j
(C.15)
with r = (i 1)My + j:
151

Appendix C Numerical solution of the GP equation
Conversely, i = int[(r 1)=My ] + 1 and j = [(r 1)mod(My )] + 1. We have simply
n to form a vector of length Mx  My . In this representation,linked together the rows of i;j
the GP equation takes the form [142]vn+1 vrn
1 vrn My + vrn 1 4vrn + vrn+1 + vrn+Myi r(C.16)=+ Vrn vrn ;2h2

where Vrn corresponds to the harmonic and mean-field potentials. The Laplacian in thisrepresentation is a pentadiagonal Mx  My by Mx  My matrix.
At this stage we could simply carry out the Crank-Nicholson procedure to propagate
the vector v from our initial condition. However, we would have to solve a pentadiagonal
matrix equation instead of a tridiagonal one. There is a more efficient method that breaks
the Laplacian up into two tridiagonal matrices, and then makes use of the split operator
approximation. To do this, we first define an alternative vector representation of the matrixnni;j by stacking the columns up, instead of linking the rows, to obtain wr defined as [142]n;wrn = i;jwith r = (j 1)Mx + i:
(C.17)

Conversely, i = int[(r 1)=Mx ] + 1 and j = [(r 1)mod(Mx )] + 1. The vector wrn
n.
contains the same information as vrn , both of which contain the same information as i;jnWith this definition of wr , Eq. (C.16) can be written in the simpler form [142]1 vrn+1 2vrn + vrn 1 1 wrn+1 2wrn + wrn 1
vn+1 vrn=+ Vrn vrn : (C.18)
i r

2
h2
2
h2
It is convenient to write this in the matrix formhivn+1 = I i D(2) + D(2) M re + V vn ;(C.19)

where V corresponds to the harmonic and mean-field potentials and we have defined thematrix D(2) as(2) = 1 Ær;s+1 2Ær;s + Ær;s 1 :
Drs
(C.20)
22h

The matrix M re defines the reordering transformation that takes us from the vector v tothe vector w , w = M  v .
In order to make use of the split operator approximation, we rewrite Eq. (C.19) as


(2)
(2)
vn+1 = e i Dx +Dy +V vn ;(C.21)(2) and D(2) = D(2) M re. For awhere we have made the simple definitions D(2)x = Dygiven pair of operators F1 and F2 , the exponential can be written as [143]e (F1 +F2 ) = eF1=2 eF2 eF1 =2 + O( 3 ):(C.22)

This is known as the split-operator approximation and is accurate to second order in  ,which is consistent with the Crank-Nicholson procedure. This can be extended to the caseof three or more operators in the sum [144], so that applied to Eq. (C.21) we obtainvn+1 = e i Dx(2) =2(2)(2)(2)
e i Dy =2 e i V e i Dy =2 e i Dx =2 vn :(C.23)

This amounts to propagating the solution in increments of =2, alternating between the x
and y axes. Each segment can be propagated according to the Crank-Nicholson procedure152

C.1 Time evolution using the Crank-Nicholson methodby using the Cayley approximation for the exponential operator, allowing us to simply carryover the results of the previous section, where we discussed the one-dimensional case.
In the case of a cylindrical geometry, (; z; t), we must be careful with the singularity
at the origin, where the radial terms in the kinetic energy diverge. In Ref. [145], Holland etal. describe a discrete representation of the radial part of the Laplacian that carefully avoids
these difficulties through an adaptive differencing scheme: at  = 0, forward differencing isused, and as  is increased, a combination of forward and central differencing is used suchthat for large  a pure central difference is used. This does not change the ADI proceduredescribed above, for we are simply redefining the discrete representation of the Laplacian
to avoid numerical instability near the origin  = 0.

C.1.3 Treating the internal-state dynamics
In Chapters 4 to 6 we treat a two-component condensate. The basic Crank-Nicholson
and ADI procedures described above can easily be extended to treat the case of having
two internal states. For concreteness, let us consider the system defined in one spatialn , where, as
dimension z . We define a vector representation of the wavefunction as j;
before, n denotes the time index and j denotes the spatial index. The index = 1; 2ndenotes the internal state. The explicit vector form ~ is written
0

~n =

B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@

1

1nn;1
2 ;1 C
C
: C
C
: C
C
: C
n C
M;1 C
C;
1nn;2 CC
2 ;2 C
C
: C
C
: C
C
: A
n
M;2

(C.24)

which can be written in the more compact form


~n =

n 
1n :

(C.25)

2

The explicit discrete representation of the two-component GP equation Eq. (4.11) from
Chapter 4 is


n+1 
1n+1

2

=

"

I

0

0 I

i



H n1

=2
=2 H n2

# 

n 
1n :

(C.26)

2

The quantities I , H n , and are M by M matrices. Here, H n is the discrete representationof the spatial part of the Hamiltonian for state iH =1 @2 1+ (z +
2 @z 2 2h)2 +  j (z; t)j2 + 

j(z; t)j2iÆ;2(C.27)
153

Appendix C Numerical solution of the GP equation
where 1 = 1 and 2 = 1. The matrix is diagonal, given by
If we define the 2M by 2M matrices 1 , 2 , and x as


= I.


0
1 =
0 0 ;

0 0 ;
2 =
0 I


0
I
x =
I 0 ;

I

(C.28)(C.29)(C.30)

we can write Eq. (C.26) in the more compact form


~ n+1 = e i H 1 1 +H 2 2+ x=2 ~ n :(C.31)

We now invoke the split operator approximate form of Eq. (C.31) using Eq. (C.22), so that
we can write~ n+1 = e i x =4 e i H 1 1 +H 2 2 e i x =4 ~ n :(C.32)

This allows us to solve the internal and spatial parts of the evolution separately. The spatial
part of the evolution can be solved for each component in separate sequential steps using
the Crank-Nicholson or ADI method described above. The internal evolution can be solved
exactly, since the unitary operator can be written explicitlye i

x =4 =

cos( =4) I
i sin( =4) I

i sin( =4) I
cos( =4) I:(C.33)

C.1.4 Diagnostics of numerical stability
In order to check for numerical stability, there are a few diagnostic tools at our disposal.
The norm of the wavefunction should always be preserved, so that N = ( ; ) is a constantin time. In the case of a single component in one dimension, for example, the discretized
form of the inner product is justXn n
N = ( n; n) = h
j j;(C.34)

jwhich should be constant as the time step n increases. For a two component system this isnn

N = (~ ; ~ ) = h

Xh

j

i

n n
n n
j;1 j;1 + j;2 j;2 :(C.35)

If the two components are coupled, population can cycle between the two states, but the
total N is a constant in time. We monitor N in our calculations.
If the Hamiltonian does not have any explicit time dependence, then we can check the
total energy of the system, which should be a constant in time. In the case of a single
component in one dimension, for example, the matrix elements of the energy operator E
are given as

Eij =
154

1
Æ
2h2 i;j+12Æi;j + Æi;j 1 +h1i(i 1)2 h2 + 1d j in j2 Æij :2
2(C.36)

C.2 Stationary state solutions using the ITP method
The total energy is then given by E = ( n ; E n ). Note that the energy differs fromthe Hamiltonian by the factor of 1=2 in the mean-field potential energy term [53]. The
total energy E is a constant in time and serves as a very useful diagnostic for testing fornumerical instability. This can be extended also to the two-component system.
One final technique for testing for instability is to simply halve the time step  and/or
the spatial grid size h to see if the solution changes. The criterion  < 2h2 should always
be met [141], so that as the spatial grid resolution increases, the time resolution must also
increase. We find the Crank-Nicholson and ADI methods to be stable, so long as thiscriterion is met. However, in some cases, when very fine spatial structure begins to develop,
the Crank-Nicholson method becomes very ineffective, for in order to resolve such finescale features, one must crank up the number of grid points and the number of time steps,
making the algorithm very slow in these extreme cases.

C.2

Stationary state solutions using the ITP method

C.2.1 Ground state solution
In order to find the self-consistent ground state solution of the GP equation, we propagate
the wavefunction in imaginary time, starting from some initial state that is a first guess ofthe solution. To see how this works, we set ~ = i in the unitary evolution operator sothat the solution in imaginary time is given by

If we expand(r; ~) = e ~H (r; 0):(C.37)(r; 0) in the eigenbasis of H as1
X(r; 0) = ci i (r);(C.38)

i=0

then the solution in Eq. (C.37) becomes [146](r; ~) =


1
X
i=0

ci e ~i i (r):(C.39)

The coefficients ci = i ; (0) and i are the eigenenergies of H , i.e. Hi = i i . In
imaginary time, the norm of the solution decays away exponentially. The crucial point,
however, is that the lowest energy state of the decomposition Eq. (C.39) has the smallest
decay constant, so that it decays away the slowest. Our initial guess (r ; 0) for the groundstate 0 (r ) will contain some finite contribution from excited states in the decomposition.
This “contamination” in our initial guess will decay away faster than the ground state partof the decomposition. After a sufficiently long time interval T0 , the solution is then(r; T0 )  c0 e T0 0 0 (r):(C.40)The ITP method is also referred to as the method of steepest descents.
In general, when one carries out this procedure, it is a good idea to re-normalize thewavefunction (r ; ~) periodically in time to avoid trouble with round-off errors due tothe decaying norm of the solution. In the case of solving the nonlinear GP equation, onemust re-normalize the solution incrementally, since the Hamiltonian depends on the densityj (r; t)j2 of the solution. The basic steps for carrying out the ITP procedure are as follows
155

Appendix C Numerical solution of the GP equation
1. Choose an initial guess (r ; 0). If the system is in the Thomas-Fermi regime, aneffective guess is to take the Thomas-Fermi approximate solution, given by Eq. (3.5)
in Chapter 3.
2. Propagate the solution for a short time interval ~ in imaginary time. The CrankNicholson or ADI method described above can be used.
3. Re-normalize the solution to the value of the norm it had before the time evolution

0 (r; ~) =s

(r; 0); (r; 0) (r ; ~):
(r; ~); (r; ~)(C.41)

0 (r; ~); H 0 (r; ~):(C.42)

4. Calculate the chemical potential as(~ ) =As the solution converges onto the proper ground state, the chemical potential should
converge to a constant value—it serves as a useful quantitative measure of convergence.
5. Repeat steps 2 through 4, using

0 as the initial state.

6. When the chemical potential converges to a constant value within the desired accuracy, the procedure can be stopped.
As a double check on the final solution, one can perform an additional integration by propagating the solution in real time to make sure that it is a stationary solution of H . That is,
if allowed to propagate in real time, the density should be constant in time, while the phaseshould vary as exp( it). We find the ITP method to be robust and to work remarkably
well in solving for the condensate ground state 0 (r). The procedure described here fora single-component system can easily be extended to treat an uncoupled, two-component
system, as well as a coupled system to find the dressed states.

C.2.2 Solution of topological modes
Throughout Chapters 3 through 6 we discussed what we called “topological” solutions to
the Gross-Pitaevskii equation, for both single and double component condensates. This is
somewhat of a misnomer, for we apply it not only to vortex solutions, but also to noncirculating modes, such as a dipole or quadrupole. We use the term “topological” in this general
way in order to distinguish between macroscopically occupied excited modes (what we arecalling topological modes) and the elementary excitations (or equivalently collective modes
for a Bose-consensed system), which only involve a small fraction of atoms being excited.
Symmetry ansatz
In the text, we have defined the topological modes in two distinct ways. In the first approach, we force the solution to have the desired symmetry through an appropriate ansatz
solution, as was done in Chapter 3 for a single component where we made the ansatz(r) =  (; z )ei for the vortex in section 3.4.1. This method was also used in Chapter 4 for the two-component system in both cases of a vortex and a dipole mode in section 4.4.2. For the case of the dipole, we forced the solution of one of the components tohave a phase of  for z < 0 and a phase of 0 for z > 0.
156

C.2 Stationary state solutions using the ITP method
The numerical implementation of this approach can be carried out using the ITP method,
discussed in section C.2.1 for the solution of the ground state; one need only modify steps(1) and (3) in the procedure above. In step (1), obviously a different choice of initial
guess is required. For example, in the case of the unit vortex, an appropriate guess is(r; 0) = TF (; z )ei , where TF (; z ) is the Thomas-Fermi ground state solution. Secondly, in addition to renormalizing the solution in step (3), one must also “imprint” thedesired symmetry onto the solution. In the case of the vortex, this extra step simply involves multiplying the solution by ei . This was the method used to solve for the vortex
shown in Figure 3.4 on page 43.
Eigenbasis solution
In the second approach, one solves for the eigenbasis of the Hamiltonian of the GP equation. In the limit of a single atom N ! 1, this is a well defined and trivial problem: one
is simply finding the eigenbasis of the simple harmonic oscillator. However, in general,the problem of solving for the eigenbasis of the GP Hamiltonian is ill-defined due to thenonlinear term j j2 . One can overcome this difficulty by specifying the decomposition ofin the, as yet undetermined, eigenbasis

j i=

X

i

ci ji i:(C.43)In section 3.4.2, we calculated the eigenstates and energies of the ground and first two
excited eigenmodes, shown in Figure 3.7, for the one-dimensional version of the GP equation. In each case, we assumed all of the atoms in the system occupied the state shown.
So, for example, in the case of the dipole mode, we took c1 = 1 and ci = 0, for i 6= 1
in Eq. (C.43). To clarify, the modes shown in Figure 3.7 are not considered to be partof the same eigenbasis; there are three different eigenbases, one for each choice of thedecomposition Eq. (C.43).
In general, a macroscopic number of atoms in the condensate can occupy multiple
modes, rather than simply one. In section 6.3.1, we discussed a two-mode model, in which
there is a macroscopic population in two modes. Of course, for a two-component system,
the decomposition Eq. (C.43) was generalized to

j i=

Xh

i

i

ci ji ij1i + di ji ij2i :(C.44)In the two-mode problem presented in section 6.3.1, we assumed that c0 and d1 were theonly nonzero coefficients in the expansion, so that both the ground state j0 i and the dipolej1 i states were macroscopically occupied.
The numerical implementation of this method can also be carried out using the ITP
method. The basic idea is to supplement the ITP method with a Gram-Schmidt orthogonalization of eigenstates [147] in each time increment; one must do an imaginary-time
propagation for all of the eigenstates being considered. The modifications to the ITP procedure (steps (1) through (6) above) are most easily understood by considering a specificexample. Consider the case of a single component in the one dimensional model of theGP equation. Suppose we want to calculate the first excited, or dipole mode shown in Figure 3.7. In this case, we set = 1 in the nonlinear Hamiltonian and add the following
steps to the ITP procedure157

Appendix C Numerical solution of the GP equation

 Since we are interested in the first excited state, we must solve for both the ground0 and dipole 1 states simultaneously, by propagating both solutions in imaginary
time. In general, if we were looking for the nth excited mode, we would have to
do an ITP on the first n states. In the simple limiting case of N ! 1, for which
the nonlinear term in the Hamiltonian drops out, one would not have to solve for allof the states self-consistently, but could apply the ITP in sequence, starting with theground state and progressing up to higher states after having found the solutions toall of the lower states.

 In step (1), we must now specify an initial guess for the ground state 0 (r) and an
1 (r).

initial guess for the first excited mode

 In addition to renormalizing the solutions in step (3), we must now also orthogonalize
the two states 0 and 1 , using the Gram-Schmidt procedure (see, for example,
section 9.3 of Arfken [147]).

This generalization of the ITP procedure to the solution of excited states can be understood by expanding each each function 0 (r ; 0) and 1 (r ; 0) in the eigenstates i (r ) of
the HamiltonianX

0 (r; 0) =iX

1 (r; 0) =i

ci i (r);di i (r):(C.45)

A “good” initial guess means that hopefully the coefficients in the expansions are suchthat jc0 j2  1 and jd1 j2  1, with the contribution from the other modes being minimal.
Propagating these states in imaginary time by some small increment ~ gives us

0 (r; ~) =1 (r; ~) =

X

i

X

i

ci e i~ i (r);di e i ~ i (r):(C.46)We then perform the Gram-Schmidt orthogonalization, which amounts to subtracting out
the ground-state contribution to the expansion of 1 (r ; ~), so that the lowest energy statein the expansion of 1 (r ; ~) becomes 1 (r). As a result of this, 1 (r ) becomes the lowest
energy state in the expansion and will have the slowest decay constant. For long times, itwill be the only remaining term.

158

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