Ultracold Gas Theory from the Top-Down and Bottom-Up
byVictor E. Colussi
B.A., Physics and Mathematics, Grinnell College, 2009
M.S., Physics, University of Colorado Boulder, 2013

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
2017This thesis entitled:
Ultracold Gas Theory from the Top-Down and Bottom-Up
written by Victor E. Colussi
has been approved for the Department of Physics

Prof. Murray J. Holland

Asst. Prof. José D’Incao

Date

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

Colussi, Victor E. (Ph.D., Physics)
Ultracold Gas Theory from the Top-Down and Bottom-Up
Thesis directed by Prof. Murray J. Holland

Advances in trapping and cooling of ultracold gases over the last several decades have madeit possible to test many formerly outstanding predictions from disparate branches of physics. This
thesis touches on three historical problems that have found new life recently in the context of ultracold Bose gases of alkali atoms. The first problem revolves around an outstanding prediction
from Boltzmann over a century and half old that the breathing mode of a isotropically trappedclassical gas should oscillate indefinitely. I analyze recent experimental results [Nat. Phys. 11,1009 (2015)], and attribute observed damping sources to trap imperfections. The second question
is about the analogue of first and second sound modes from liquid helium in trapped dilute gases.
I present the results of a joint theoretical/experimental investigation of the breathing mode of afinite temperature Bose-Einstein condensate (BEC), attributing a striking collapse revival behaviorof the resultant oscillation to in-phase and out-of-phase normal modes of the thermal cloud andcondensate. The third problem is that of the formation of Borromean ring-like three-body boundstates, referred to as Efimov trimers, in strongly-interacting few-body systems. I extend the predicted spectrum of Efimov states into the realm of many degenerate internal levels, and investigate
the difficult three-body elastic scattering problem.
These questions are part of the broader theme of this thesis: How can our understanding of
few-body physics in the ultracold limit be translated into statements about the bulk behavior of an
ultracold gas? For weakly-interacting Bose gases, this translation is well-known: the many-bodyproperties of the gas are well-described by the tracking just the one and two particle correlations.
I analyze a generalization of this procedure to higher order correlations, the general connection
between few-body physics and correlations in a dilute gas, and results for the emergence of Efimov
physics in the magnetic phase of the strongly-interacting Bose gas.

Dedication

In memory of my undergraduate advisor, Sujeev Wickramasekara.

Time is an ocean but it ends at the shore.
-Bob Dylan-v

Acknowledgements

It has undeniably been an interesting journey characterized by a steady progression frominstability to stability. First and foremost, I would like to thank my advisor, Prof. Murray Holland,for his support over the last several years. From Murray, I have learned a great deal (partiallythrough osmosis) about conducting research, especially establishing and maintaing collaborations.
Throughout my stay in his group, I have always been genuinely impressed by the amount of timeeach day his door is open to visitors, and the patience he displayed with me while I learned throughtrials, errors, and false starts.
I would like to thank Asst. Prof. José D’Incao, who has been both a good friend and mentor.
Each day at JILA, I have looked forward to our chats and to hearing your perspective. It was atreat to work with Prof. Matthew Davis of University of Queensland on the analysis of the spherical
BEC experiment during his sabbatical at JILA in 2015. I have also had the pleasure of working
with Prof. Servaas Kokkelmans of Eindhoven University of Technology during the end of Summer2016 on cumulant theory and resonance superfluity. I have enjoyed many enlightening discussions
with Prof. Chris Greene from Purdue University during his Summer 2016 visit and from my timein his group. I want to thank specifically Dan Lobser from the Cornell group and Cam Straatsma
from the Anderson group who is very patient even by Canadian standards.
I want to thank my friends who have shared a rope with me, put up with my rants, andgrown accustomed to bad puns. My family and two partners in crime, Suzannah and Chief, havealways been unconditionally supportive, engaged, and curious even when I haven’t eaten recently.
Finally, I want to thank Lee Strassel who was the first to show me the beauty of logic.

Contents

Chapter
1 Introduction
1.1

1Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Preliminary Concepts
2.1

2.2

2.3

5
8

Single-Atom Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.1

Harmonic Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2

Identical Particle Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.3

Wave-Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Two-Atom Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1

Classical Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2

Quantum Collisional Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3

The Ultracold Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.4

Interatomic Potentials for Alkali Species in the Ultracold Limit . . . . . . . . 22

2.2.5

Effective Potentials and the Zero-Range Model . . . . . . . . . . . . . . . . . 25

Many-Atom Physics: Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1

Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.2

Many-Body Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4

The Noninteracting Bose Gas at Finite Temperatures. . . . . . . . . . . . . . . . . 36

2.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii
3 Anomalous Damping of the Breathing Oscillation of a Nondegenerate Bose Gas in an Extremely Isotropic Trap
3.1

Kinetic Theory Above Tc

39. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1

The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.2

Consequences of the H-Theorem: The Undamped Breathing Mode . . . . . . 47

3.1.3

Collective Modes Above Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2

The Breathing Mode Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3

Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4

Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5

3.6

3.4.1

Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.2

Drive mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5.1

Collisionless regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.2

Crossover regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Collapse and Revival of the Breathing Mode of a Degenerate Bose gas in an Extremely
Isotropic Harmonic Trap

70

4.1

Zero Temperature: The Mean-Field Theory of the Condensate . . . . . . . . . . . . 74

4.2

Kinetic Theory Below Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3

Collisionless Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1

Collective modes in an isotropic harmonic trap . . . . . . . . . . . . . . . . . 85

4.3.2

Coupled-modes analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5

Collisional dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5.1

Simulation of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.5.2

Extraction of damping rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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4.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Efimov Physics and the Three-Body Elastic Scattering Phase Shift
5.1

102

Parametrizing the Motion of Three Atoms . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1.1

Jacobi Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.1.2

Hyperspherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2

Analytic Solution of the Three-Body Problem in the Zero-Range Model . . . . . . . 110

5.3

The Spectrum of Three-Body Bound States . . . . . . . . . . . . . . . . . . . . . . . 112

5.4

The Three-Body Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4.1

Signature of Efimov Physics in the Elastic Three-Body Phase Shift . . . . . . 121

5.4.2

Long-Range Contributions to the Elastic Three-Body Phase Shift . . . . . . . 124

5.4.3

Removing Disconnected Two-Body Physics from the Elastic Three-Body Phase
Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.5

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6 Efimov Physics with Degenerate Internal Levels

137

6.1

Adiabatic hyperspherical representation for spinor systems . . . . . . . . . . . . . . . 139

6.2

Toy Model: Efimov States with Two Internal States . . . . . . . . . . . . . . . . . . 143

6.3

Spin-1 systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.4

Spin-2 and -3 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.5

Signatures of Spinor Efimov Physics in the Elastic Three-Body Phase Shift . . . . . 154

6.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7 One, Two, Three, Many: Few Body Physics in Strongly Interacting Bose Gases

157

7.1

Time and Length Scales in Strongly-Interacting Bose Gases . . . . . . . . . . . . . . 159

7.2

The Cumulant Expansion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.2.1

7.3

Structure of the Cumulant Equations of Motion . . . . . . . . . . . . . . . . . 167

Self-Consistent Solutions of the Cumulant Equations of Motion . . . . . . . . . . . . 172

ix

7.4

7.3.1

Truncation of the Cumulant Equations at First Order: The GPE . . . . . . . 172

7.3.2

Higher Powers of the Condensate Wave Function: The Three-Body GPE

Emergence of Efimov Physics in Strongly-Interacting Bose Gases . . . . . . . . . . . 176
7.4.1

7.5

Quantum Droplets and Magnetic Ordering . . . . . . . . . . . . . . . . . . . 177

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8 Conclusion
8.1. . 175

186

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

191

Bibliography

Appendix
A Numerical Simulation of Finite Temperature Bose Gases

209

A.1 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.1.1 Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

A.1.2 Rethermalization after a quench . . . . . . . . . . . . . . . . . . . . . . . . . 215

x

TablesTable
5.1

Coefficients αλ for the 1/R3 term in Eq. (5.88) and corresponding phase-shifts asgiven in Eq. (5.89). We denote the coefficients for the three-body system where only
one of the pairs interact by α̃λ while for three-body systems where all pairs interact
we denote them by αλ . The subscript in λ(n) indicates the corresponding degeneracy
for a given value of λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.1

Values of sν relevant for f =1 and 2 spinor condensates covering all possible regionsof R and for different magnitudes of the relevant scattering lengths. (The f = 1, 2
values are from Ref.

[36]) We list the lowest few values of sν for each F3b andtheir multiplicity (superscript) if greater than one. Below, we use R  |a{n1 ,n2 ,...} |(|a{n1 ,n2 ,...} |  R) as a shorthand for the condition that R must be much less (greater)
than all elements in the set {|an1 |, |an2 |, ...}.
6.2. . . . . . . . . . . . . . . . . . . . . . 150

Values of sν relevant for f = 3 spinor condensates covering all possible regions of R
and for different magnitudes of the relevant scattering lengths. (The f = 3 values arefrom Ref. [37].) We list the lowest few values of sν for each F3b and their multiplicity(superscript) if greater than one. Below, we use R  |a{n1 ,n2 ,...} | (|a{n1 ,n2 ,...} |  R)as a shorthand for the condition that R must be much less (greater) than all elementsin the set {|an1 |, |an2 |, ...}.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Figures

Figure2.1

Left panel: The single-scattering Feynman diagram for the first Born approximation.
Right panel: Multiple scattering contributions are represented by diagrams withintermediate states whose momentum must be integrated over. Here, the doublescattering diagram is shown where the virtual state’s intermediate momentum
q must be integrated over, violating energy conservation in the system within thebounds of the time-energy Heisenberg uncertain relation.

2.2. . . . . . . . . . . . . . . 18

The direct and exchange scattering pathways in the center of mass frame of the twocolliding identical particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3

At medium separations, the total electronic spin is a good quantum number and3 +the interatomic potentials are the singlet 1 Σ+
g and triplet Σu potentials, which are

shown here for 85 Rb atoms [77, 78, 79]. At large separations the energies correspondto the hyperfine levels labeled in the inset, shown here for mf1 + mf2 = −4 at
B = 600 G, which is the location of a well-known broad Feshbach resonance. . . . . 22
2.4

A schematic of a Feshbach resonance. Scattering process in the incoming channel(red) is coupled to the closed channel (blue). When a bound state (dashed lines)
is near collision threshold, the colliding atoms form a metastable state which decays over a characteristic lifetime, indicated by the hourglass, exiting out the open
channel.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

xii
2.5

A sketch of the dependence of the cutoff contact interaction on cutoff momentum,
pc . With increasing cutoff, the maximum at the origin diverges as a the cutoff
momentum cubed, and the number of oscillations increases. . . . . . . . . . . . . . . 28

3.1

A two-dimensional projection of µ-space in both it’s discrete representation in termsof individual atoms and continuum representation. . . . . . . . . . . . . . . . . . . . 43

3.2

In the modified TOP trap, the cloud shifts under the pull of gravity away from the
midpoint between the coils. The trap can be made isotropic to harmonic order withthe cost of anisotropic anharmonicities. (Adapted from [63].) . . . . . . . . . . . . . 53

3.3

Experimental results from Ref. [24] for the monopole and quadrupole damping
rates over a range of collision rates. The monopole damping is nonzero and therefore anomalous, but the quadrupole damping agrees with well-known results from
Ref. [100].

3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Monopole damping rates versus collision rate for a set of frequencies from the JILA
experiment with various anisotropy parameters and central frequencies ν0 . The
legend gives the level of anisotropy, where λ, , and ν0 = ω0 /2π correspond to
rewriting the harmonic confining potential as in Eq. (3.39). . . . . . . . . . . . . . . 56

3.5

Driving the monopole moment for A = 0.15 and initial temperature of 152nK overfour periods of oscillation. (a) FWHM of the monopole moment versus holding time.
The grey area indicates the time in between the end of the drive and switching on
the anharmonic terms in the trapping potential. (b) The % change in the totalenergy of the cloud between a trapping potential that is purely harmonic and thesame potential with anharmonic terms switched on. The vertical line indicates thepoint at which the anharmonic terms are added to the trapping potential when the% increase of the total energy is on the order of 0.001%.

. . . . . . . . . . . . . . . 62

xiii
3.6

Resolving the energy dependence of the monopole moment for a drive strength of
A = 0.15 (9 ± 1.35Hz) and an initial temperature of 152nK. (a) After binning thetracer particles into energy bins, each bin oscillates as an independent monopole.
Here, the ratio of the average period of the monopole in each bin and the zero
0 along with the standard deviation (grey region), istemperature result, T0M = 2π/ωMplotted versus the energy of each bin in units of thermal energy. (b) The statisticalweight, P (ω), of each frequency component in the cloud. The grey region is thetheoretical prediction for the shift of the monopole frequency from L (150nK) to R
(180nK). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.7(a) Damping of the monopole over a range of drive strengths A = {0.05, 0.1, 0.15, 0.20, 0.25}
and temperatures in the collisionless regime. The grey regions are the damping results along with the uncertainty from the JILA experiment [24] for (from bottom totop) 65nK, 125nK, and 152nK. (b) The colored lines are the theoretical predictionsfor ∆ωM (T ) using f0ah (blue) and using f0 (red) for the scaling ansatz versus thesettled temperature of the cloud. The points are the damping results from (a) forall simulated drive strengths and temperatures in the collisionless regime. The solid
black line is a fit to the numerical damping results using Eq. (3.54). . . . . . . . . . 65

3.8

Damping rate versus the equilibrium collision rate. The triangle data points are a
result of numerical particle simulations. The blue curves are scaling ansatz predictions using f0ah for collisional damping of the monopole mode for temperatures 60nK,
80nK, and 100nK from bottom to top, respectively.

3.9. . . . . . . . . . . . . . . . . . 67

Resolving the energy dependence of the monopole moment for A = 0.05 and initial
temperature of 60.6nK for a range of total atom numbers. Each data point is the peakcomponent of the Fourier transform of the normalized collective oscillation in thebin. From bottom to top the curves represent frequency spreads in the collisionless,dephasing crossover, and hydrodynamic crossover respectively. . . . . . . . . . . . . . 68

xiv
4.1

Temperature dependent m = 0 quadrupole mode spectrum from Ref. [21] normalized
to the radial trapping frequency ωr . The anomalous upward shift at T > 0.7Tc differs
by 10 − 20% from zero-temperature analytic result indicated by the lower dashed line. 72

4.2

Profiles of the condensate (solid lines) and thermal cloud (dashed lines in inset) densities from noninteracting ξ ≈ 10−2 (red), ξ ≈ 10−1 (blue), to the edge of the Thomas
Fermi regime ξ ≈ 1 (green). The densities and radius are rescaled in oscillator units
for N = 105 , T = 0.8Tc . As χ increases, the difference in mean-field interactions
pushes the thermal cloud further out to the wings of the condensate. . . . . . . . . . 80

4.3

A schematic of the collision processes contained in the ZNG equations. . . . . . . . . 82

4.4

Oscillation frequency of the in-phase (φ = 0) and out-of-phase (φ = π) modes as afunction of temperature from the coupled-modes analysis (solid lines). The dashed√lines represent the TF limit ( 5ω0 ) and ideal gas limit (2ω0 ) for the monopole mode
frequency of the condensate and noncondensate, respectively. Assumptions made in
the coupled-modes analysis become invalid for T . 0.2 Tc . . . . . . . . . . . . . . . . 88

4.5

Magnitude of the response of the in-phase (φ = 0) and out-of-phase (φ = π) modes
to a trap frequency perturbation as a function of temperature. The amplitude of thetrap frequency modulation is  = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6

Amplitude of the monopole mode oscillation (squares) for atom numbers of (a) N =8.9 × 105 , (b) N = 9.7 × 105 , (c) N = 6.7 × 105 , (d) N = 5.4 × 105 , and (e)N = 7.9 × 105 . Each frame is labeled with the condensate fraction (Nc /N ) andtemperature (T /Tc ). Error bars represent the statistical uncertainty of multiple
realizations of the experiment at each time point. The dashed lines are a fit of thedata to Eq. (4.50), which represents the envelope function for the superposition of
two sinusoids. From the fit it is found that the beat frequency is ∆ω/ω0 ∼ 0.13 on
average and the damping rate lies in the range Γe ∼ 1.5–3.5 s−1 . . . . . . . . . . . . 92

xv
4.7

Amplitude of the monopole mode oscillation for experimental atom numbers of (a)N = 8.9 × 105 , (b) N = 9.7 × 105 , (c) N = 6.7 × 105 , (d) N = 5.4 × 105 , and(e) N = 7.9 × 105 . Each frame is labeled with the condensate fraction (Nc /N ) andtemperature (T /Tc ), and the legend denotes the different modulation amplitudes
used in the ZNG simulations. Error bars on the experimental data represent thestatistical uncertainty of multiple realizations of the experiment at each time point.
(f) Condensed fraction vs temperature for the ideal Bose gas, Nc /N = 1 − (T /Tc )3(solid line), the equilibrium state of the ZNG simulations (blue points), and theexperimental data (red crosses). All simulations are performed with N = 8 × 105
atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.8

Simulated mean-square radius of the condensate (1a-1c) and total (2a-2c) densities
for a trap frequency modulation amplitude of  = 0.03 at (a) T = 0.4 Tc , (b) 0.6 Tc ,and (c) 0.8 Tc (black circles) and resulting fit of Eq. (4.51) (blue line). The densityof simulated points has been reduced for clarity. . . . . . . . . . . . . . . . . . . . . . 98

4.9

The coupled-mode results (solid line) from Fig. 4.4 overlaid with results of the ZNG
simulations (crosses). Crosses are the result of fitting Eq. (4.51) to the condensate
density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.10 Damping rate of the in-phase (squares) and out-of-phase (triangles) mode as a resultof fitting Eq. (4.51) to the simulated evolution of Rc2 at each temperature for a
trap frequency modulation amplitude of  = 0.03. The solid lines are guides to theeye. Representative fits are shown in Fig. 4.8. . . . . . . . . . . . . . . . . . . . . . . 99
5.1

Schematic of the mass-scaled Jacobi vectors written in the odd-man-out notation. . . 105

5.2

A plot of the transcendental equations (Eqs. 5.34–5.35) as a function of real (a.) andimaginary (b.) s in the resonant limit R/|a0 | → 0. The si are the roots labeled inincreasing order, where s0 is the root associated with the Efimov potential Eq. 5.37. 113

xvi
5.3

The canonical plot of the spectrum of three-body bound states (Efimov trimers) neara Feshbach resonance. At resonance, there is a condensation of an infinite numberof Efimov states at threshold which is not shown.

5.4. . . . . . . . . . . . . . . . . . . 115

The adiabatic three-body potentials for a0 < 0 (a.) and a > 0 (b.). The orange
shaded region is the scale-invariant region, the yellow shaded region is the transition
region, and the green shaded region is the asymptotic region. The black curves area result of solving the transcendental equations (Eqs. 5.34–5.35) and the red curves
are the three-body potentials corresponding to the indicated values. . . . . . . . . . 117

5.5

Diagrammatic representation of the three-body scattering processes which may occur
with pairwise interactions. The grey circles represent a pairwise interaction summarized by the two-body T-matrix. I have excluded the permutations of each scattering
diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.6(a) Eigenphase sum for a system with three different negative scattering lengths(a1 ≈ −10rvdW , a2 ≈ −15rvdW and a3 ≈ −20rvdW ). The solid red curve is the
eigenphase sum from S3B , and the green-dashed curves are the eigenphase sums(C)

for S1 , S2 , S3 . The blue dotted curve is ∆ep , which scales as K 2 at threshold.
(b) Same as (a) but for a system with three different positive scattering lengths(a1 ≈ +10rvdW , a2 ≈ +15rvdW and a3 ≈ +20rvdW ). . . . . . . . . . . . . . . . . . . . 134
6.1

The first few roots sν obtained for the toy model in the hyperradial region aα R  aβ . The roots are plotted versus the mixing angle θ. . . . . . . . . . . . . . . . 144

xvii
6.2

The roots sν (R) reflect that the internal state |χ3B i of the Efimov trimer is in asuperposition of internal levels. When there is no admixture, the roots are identicalto the single-level problem. When there is nonzero admixture (ci 6= 0), the roots area weighted average (w1 (θ) + w2 (θ) = 1) of the single-level roots associated with theindividual internal states comprising the superposition. Here, this behavior is shown
schematically for the lowest root is0 which tracks between i1.00624 and i0.41370
depending on the mixing angle (see the lowest curve of Fig. 6.1). . . . . . . . . . . . 145

6.3

Schematic representation of the relevant regions in R of the three-body potentials
u(R). The shaded area, R < rvdW , represents the region where the three-bodypotentials are expected to be non-universal. For all other regions, the potentials areuniversally given by the form in Eq. (??) with s(R) ≡ s.

6.4

. . . . . . . . . . . . . . . 146

F3b = 1 (red solid line), 2 (green dashed line), and 3 (blue dash-dotted line) hyperspherical adiabatic potentials for f = 1 atoms with a0 = 102 rvdW and a2 = 105 rvdW .
(a) For R ≤ {a0 , a2 } (shaded region) two attractive potentials exist (both withs0 ≈ 1.0062i), allowing for two families of Efimov states, and for R > a0 , one of thesepotentials turns into an atom-dimer channel |F2b = 0, MF2b = 0i|f = 1, mf = 0i.
(b) For a0 ≤ R ≤ a2 (shaded region), only one family of Efimov states exists(s0 ≈ 1.0062i), and for R  a2 three (asymptotically degenerate) potentials describe atom-dimer channels, |F2b = 2, MF2b = −1, 0, 1i|f = 1, mf = 1, 0, −1i. . . . . . 149

6.5

F3b = 0 (red solid lines), 1 (red dot-dashed lines), 2 (green dashed lines), 3 (bluedashed lines), 4 (purple dashed lines), 5 (cyan dot-dashed lines) and 6 (black dotdashed lines) hyperspherical adiabatic potentials for f = 2 85 Rb atoms with a0 ≈−8.97rvdW , a2 ≈ −6.91rvdW , and a4 ≈ −4.73rvdW . Note that the potentials for themixed-symmetry states, F3b = 1 and 5, do not display an Efimov potential. . . . . . 153

6.6

Schematic of the three-body potential for spin-1 with total three-body hyperfine
spin F3B = 1 and hyperfine scattering lengths |a0 |  |a2 |. The relevant roots are
s0 ≈ i10062 and s1 ≈ 0.4097 from Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . 155

xviii
7.1

Relevant length scales in the strongly-interacting Bose gas (not drawn to scale.) The
blue atoms are part of an Efimov trimer with characteristic length scale l3B . The
effective range r0 is less than the scattering length, which is less than the interparticle
spacing, r0  a0  n−1/3 , and the inter particle spacing is indicated between a blue
atom and a red atom of the background gas.

7.2

. . . . . . . . . . . . . . . . . . . . . . 161

The cumulant expansion generates equations of motion for n-atom correlations that
depend explicitly on the dynamics of order less than n + 2. This forms a hierarchyof explicit and implicit dependence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.1 The particle is rotated from its initial trajectory along ri by an angle ϕ due tothe off axis components of pi . After the particle position is updated to rf the
momentum components are rotated and realigned with the position vector. Due tospherical symmetry the azimuthal angle is not unique and is chosen randomly. Thisis represented by the area of revolution of pf about the final position vector rf . . . . 211
A.2 Cartoon depiction of the phase space binning process. Particles (black points) arebinned in position space using constant volume shells with the radial width of eachbin decreasing with r. Within each spatial bin the particles are further binned in
momentum space using a 2D grid of equal area bins based on the magnitude of their
momentum p and trajectory cos θ = p̂ · r̂. . . . . . . . . . . . . . . . . . . . . . . . . 212
A.3 Results from simulation of our spherical ZNG code for the equilibrium thermal andcondensate populations with 5 × 104 total atoms in a 9 Hz trap at 12 nK. The total
atom number stays within the fluctuation bounds (green region.) . . . . . . . . . . . 214
A.4 The collision rates per unit length as a function of position in an 187 Hz trap with
5 × 104 total atoms at 200 nK. In both plots, the data points are a result of Monte
Carlo sampling of the collision processes, and the solid curves are a result of ananalytic calculation using the Bose distribution. Excellent agreement of the analytic
and numerical rates is essential to running a stable simulation. . . . . . . . . . . . . 215

xix
A.5 (a.) The results of a quench of the thermal cloud in a 187 Hz trap with 5 × 104 total
atoms and an initial temperature of 200 nK. The three sets of data points are from a
3D cylindrical ZNG code reproduced from Ref. [294], a 3D adaptive Cartesian code
reproduced from Ref. [293], and our spherical ZNG code. (b.) The drift of the total
atom numbers for the three simulations. The green area is the fluctuation bound. . . 217

Chapter 1

Introduction

Understanding how microscopic processes materialize in the phenomenon of macroscopic
systems is one of the most challenging questions in all fields. Examples range from personal–How
does life emerge from the chemistry of the body?–to practical–How is the behavior of economies
related to the psychology of individuals making up the market? In the study of the physics of dilutegases, these questions relate how the behavior of atoms and molecules on nanometer lengthscales
emerge in observable characteristics of the bulk gas. These questions are aimed at understanding thequantum mechanical origin of both thermodynamical properties and nonequilbrium phenomenon.
In the ultracold regime, theoretical descriptions begin on the microscopic level with Hamiltonians containing simplified, effective interactions that reproduce the scattering physics in the
low-energy limit, using a finite set of parameters. These theories are derived from extremely accurate models for the interatomic potentials that have been continually developed, maintained, andrefined over decades. The ultracold dilute gas is therefore the rare system that can be modeled
beginning from a nuanced understanding of the microscopic physics.
These systems are also attractive from an experimental standpoint due to advances in trapping and cooling techniques of ultracold gases over the last several decades, allowing for experimental control over atom number and densities. Furthermore, experimenters exercise a high degree ofquantum control of the system through application of external fields, forming the basis of techniquesthat manipulate and select the internal properties of atoms.
These facets have made the field of ultracold gases a testbed for studying both few and many-

2
body quantum phenomena. One of the basic tools in the study of condensed matter systems for
extracting information about the bulk is to investigate the linear response to external perturbation.
Often, the linear response of a system can be calculated in the noninteracting limit as the physics isessentially single-particle. The observed shift of the response tells us about the underlying spectrumof the many-body Hamiltonian including interactions. Additionally, damping of the response reveals
the rate at which the microscopic physics within the gas conspire to return the system to a stateof statistical equilibrium.
These general ideas were first codified by Ludwig Boltzmann in the late 19th century in a
famously controversial equation for the dynamics of a dilute gas bearing his moniker. His theory was
founded on the idea that a dilute gas is continually in a state of molecular chaos (Stosszahlansatz)
arising from the probabilistic decay of correlations between binary collisions and free evolution.
Reflecting on these assumptions in a world post-quantum revolution masks how troubling they
were to the deterministic worldview of the late 19th century, resulting in community-wide disputes
over the reversal and recurrence paradoxes and ultimately in Boltzmann taking his own life. The
Boltzmann equation admits our ignorance of the complicated trajectories of each individual atomin the gas. It focuses instead on the dynamics of macroscopic averages, of which the evolution of thesingle-particle correlations are the most primitive. It is the paradigm for including few-atom physics
in the dynamics of macroscopic observables, and serves as a both a foundation and launching point
for this thesis.
Recently, increasing attention has been paid to the study of how isolated many-body systems
reach a state of statistical equilibrium, and old predictions for undamped nonequilibrium motions,
including integrable systems like the quantum Newton’s cradle [1], have found new life in ultracold
gas experiments. Among the set of predictions that came from the Boltzmann equation, is acurious class of undamped collective oscillations that can be probed by external perturbation of aharmonically trapped gas. This includes the dipole or Kohn center of mass mode and the monopole
or breathing mode of an isotropic harmonically confinement [2]. Over the last two decades, many
experiments with ultracold Bose gases have probed the spectrum of collective excitations both

3
above and below [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] the criticaltemperature for Bose-Einstein condensation, confirming the undamped behavior of the dipole modeover a range of geometries, which serves as a sensitive measurement of the trapping frequencies.
That the dipole mode oscillates undamped is a statement about the factorization of the centerof mass from the relative motion of the constituent particles. Verifying that the monopole mode
oscillates undamped, on the other hand, is a direct, nontrivial test of the assumptions underlying
the kinetic theory of gases and is therefore of fundamental importance. Recent experimental resultsfrom JILA [24] for the monopole mode of a nondegenerate Bose gas an extremely spherical trap
found oscillations persisting over many seconds with small but nonzero damping. It is therefore
important to understand whether this damping results from beyond-Boltzmann effects in the gas
or is consistent with the effects arising from beyond harmonic order corrections to the trap.
Whereas collective mode experiments with classical dilute gases test the ideas of molecularchaos and irreversibility, experiments with Bose-condensed gases probe the dynamics of quantum
many body systems, including the fundamental spectrum of quasiparticle excitations. One of theearliest studies tracing back to mid-20th century was the observation of first and second soundin liquid helium, which inspired two-fluid theories to account for the transport properties of thesystem and Landau’s canonical quasiparticle spectrum of photons and rotons [25, 26, 27, 28]. Thesestudies of liquid helium framed the early language of collective mode BEC experiments in ultracoldgases (see for instance Ref. [20]), however the two-fluid picture has limited application to dilutegases.
In liquid helium, the dynamical coupling between superfluid and normal components manifests in in-phase and out-of-phase oscillation of the two components when fluctuations in density
and temperature are present, respectively. For finite temperature dilute BECs, the effects of thisdynamical coupling can lead to anomalous shifts of the spectrum of collective modes, as was firstobserved in an experiment probing the m = 0 quadrupole mode [21]. The resolution of these effects is based on the collisionless analogue of first and second sound modes as postulated by Stoofand Bijlsma [29]. It turns out however, that signatures of in-phase and out-of-phase oscillations

4
are generally obfuscated by the coupled spectrum of collective excitations in a trap with broken
rotation symmetry. The analysis of the spectrum shift for an isotropic geometry presented in thisthesis is therefore important to characterizing the link between liquid helium and dilute BECs.
Advancements in the quantum control of ultracold gases has also provided a means to tuning
the effective interatomic interaction strength near a Feshbach resonance [30]. For identical ultracold bosons in the resonance limit, an infinite ladder of three-body bound states, Efimov trimers,
appears one by one with binding energies spaced by a fundamental scaling law [31, 32, 33]. These
nonperturbative states are characteristic of the sort of unexpected physics that occurs in the regimeof strong-interactions, which cannot be understood by extrapolating the behavior of one or two interacting particles. Solutions of the three-body problem in the strongly-interacting regime however
do contain the physics of two interacting atoms as a subset, and it is important to understand how,for instance, the two-body continuum and bound state spectrum is embedded in the three. The
existence of Efimov states was inferred for the first time in 2006 through giant atom loss rates inan ultracold gas of caesium atoms frozen to a single internal level [34]. The predicted [35, 36, 37]
complexity of the trimer spectrum when each atom has access to multiple, degenerate internal levels
is however unconfirmed.
In the strongly-interacting regime, nonperturbative few-body physics emerges in the bulk
properties of the gas, contributing to the stability and ground state structure of strongly-interacting
Bose gases. Although the lifetime of these systems is severely limited by inelastic losses, Efimov
physics has been predicted to stabilize the condensate against collapse due to instabilities [38, 39],
leading to the formation of novel, self-bound superfluid droplet states of matter. Recently, theobserved crystallization of a collapsing dipolar condensate in Ref. [40, 41, 42], has inspired many
theoretical works speculating on the source of the collapse stabilization and pattern [43, 44, 45,
46, 47, 48, 49]. How contributions of Efimov physics in strongly-interacting BECs with multiple
degenerate internal levels affects the magnetic ordering remains an open question in the community
and is partially addressed in this thesis.
To construct a theory of the strongly-interacting BEC and properly account for the role of5
Efimov physics in the gas, it is natural to search first for extensions of weakly-interacting models
to include three-body correlations. A general procedure was elaborated in Refs. [50, 51, 52] basedon the method of the cumulant expansion to generate a complete set of coupled equations forthe dynamics of all orders of correlations in the gas. This theory has been used, for example,
to sketch the three-body corrections to the energy density [53]. This extension, as well as thegeneral procedure of resonance superfluidity [54, 55, 56] for including bound state physics into
the correlation dynamics first applied successful to the strongly-interacting Fermi gas, have notbeen acknowledged recently as cutting edge experiments have probed into the region of stronglyinteracting nondegenerate [57, 58] and degenerate [59] Bose gases, motivating much of the discussion
in the later portions of this thesis.

1.1

OutlineThis thesis weaves together several of my published works [60, 61, 36, 37, 62], and I haveattempted to make this account as self-contained as possible, referring to the relevant references
when a discussion has been truncated. To accomplish this, there is a large amount of review
material discussed and derived here oftentimes in detail and with proper attribution. Although
this is the thesis of a theorist, there is some basic discussion of the experimental apparatus, amodified TOP trap, used to produce the results of Chapters 3–4 along with some details about
how the experimental data was extracted and fit. Finally, the latter half of Chapter 5 is composedof so far unpublished work on the three-body elastic scattering problem.
This thesis is divided roughly into two parts: weakly-interacting Bose gases (Chapters 2-4)
and strongly-interacting Bose gases (Chapters 5-7). I begin in Chapter 2 with a review of theimportant preliminary concepts. In keeping with the spirit of this thesis, I start from the relevant microscopic few-body physics before introducing the basic many-body formalism. On the
single-particle level, the physics behind the magnetic TOP trap, which is the apparatus analyzed in
Chapters 3–4, is discussed first. This is followed by the requirements of identical particle symmetry,
and the standard length scale arguments for the semi-classical approximation. On the two-body6level, the basic concepts of scattering theory are introduced including the cross-section and scattering amplitude, followed by the decomposition of the problem into partial waves and arguments forthe truncation of this expansion in the ultracold limit. To contextualize these concepts, I discuss
the typical length and energy scales of interatomic interactions between two ground state bosonicalkali atoms. In the ultracold limit, effective interactions can be used in lieu of the full interatomic
potential, the simplest of which are the so-called zero-range interactions, which are discussed at theend of the two-body section. These one and two particle concepts are the basic foundation of the
many-body formalism which is introduced in the language of second quantization. The many-body
Hamiltonian is derived including pairwise interactions for zero-range interactions. This formalism
provides the foundation of Chapters 4 and 7. Finally, I close the chapter with a discussion of theideal Bose gas, introducing many of the important concepts that form the basis of later chapters.
Chapters 3–4 focus on the analysis of results for the breathing mode oscillation of an ultracold
Bose gas under extremely spherical isotropic confinement. The classical kinetic theory of gases isfirst developed along with a discussion of Boltzmann’s H-theorem and implications for special
undamped solutions of the Boltzmann equation–of which the breathing mode is a member. An
overview of the standard techniques for analyzing collective modes of a classical dilute gas are thenpresented and applied to the results of the breathing mode experiment from the Cornell group at
JILA [24]. To analyze the breathing mode experiment below the transition temperature, I begin
with the standard zero temperature theory of the condensate followed by a semi-classical kinetic
theory for the excitation cloud. Results of this formalism applied to the experiment are thenpresented with emphasis on the in-phase and out-of-phase normal modes of the gas, which provides
an explanation for the striking collapse revival of the condensate oscillation observed experimentally.
This concludes the portion of this thesis which deals with weakly-interacting Bose gases.
Chapters 5–7 focus on the strongly-interacting Bose gas, beginning with a discussion of thethree-body problem in Chapters 5–6. Chapter 5 begins with the basic machinery required toanalyze the three-body problem for neutral bosons in the ultracold limit. I outline the problem
in the adiabatic hyperspherical representation, the resultant adiabatic three-body potentials, and7the physics of each relevant region of space. Along the way, the spectrum of Efimov states in theresonant limit are introduced. This is followed by a relatively lengthy discussion of the three-bodycontinuum, specifically contributions to the three-body elastic phase shift from Efimov physics, fromsuccessive two-body binary interactions, and from disconnected two-body scattering. Chapter 6translates much of this analysis to the case where each atom has access to multiple degenerateinternal levels as is the case for strongly-interacting spinor Bose gases. The rich spectrum offamilies of Efimov states are enumerated in this case as well as the contribution of spinor Efimov
physics to the three-body elastic phase shifts. This discussion feeds into the final chapter, Chapter 7,
which begins with a generalization of the many-body formalism presented in Chapter 4 throughthe method of cumulants to integrate the discussed three-body physics into the equation of motion
for the three-particle cumulant. As an illustration of the generality of the cumulant method, I
follow Ref. [52] and rederive the zero temperature theory of the condensate and highlight many of
the tacit assumptions that were made previously. I then retrace the work of T. Köhler in Ref. [53]to establish a zero temperature theory of the condensate including three-body physics and finish
by characterizing the emergence of Efimov physics in the superfluid (summarizing the work of A.
Bulgac in Ref. [38]) and magnetic phase of the condensate.
This thesis concludes with a brief sketch of a possible fusion of the cumulant method withthe theory of resonance superfluidity to study the unitary Bose gas. This would produce a meanfield like set of equations for the correlation dynamics of one, two, and three body cumulants withadditional mean-field equations for the condensed trimer and dimer superfluids that are formallysimilar to the zero-temperature theory of the weakly-interacting Bose gas. Construction of thistheory will rely intimately on the developments presented in Chapters 4–7, and therefore stands asa natural, challenging, and timely extension of the work presented in this thesis.

As a final note: I’ve set ~ = 1 for notational clarity at the risk of infuriating future readers.

Chapter 2

Preliminary Concepts

This chapter provides the necessary background to understand the theoretical analysis ofultracold dilute Bose gas experiments in the weakly-interacting regime in Chapters 3–4, comprising
the first part of this thesis. It also serves as a prelude to the three-body physics and generalized
many-body theory discussed in Chapters 5–7 in the context of strongly-interacting Bose gases.
Before jumping into relevant equations and mathematics, it is useful to elaborate very briefly
on the descriptors ‘ultracold’, ‘dilute’, ‘bosonic’, and ‘weakly’ and ‘strongly-interacting’, reserving
more quantitative definitions for the sections and chapters that follow.
These gases are ultracold with typically sub-microKelvin temperatures. Semiclassically, thisis a statement that the average energy of an atom in the gas is sufficiently low so as to be completely
repulsed by the centrifugal barrier of a neighboring atom for nonzero impact parameters. Quantum
mechanically, this means that interactions proceed only along the lowest angular momentum partial
wave, greatly simplifying the collisional analysis.
These gases are extremely dilute with densities of 1014 cm−3 [63] compared to ordinary
gases at around 1019 cm−3 and liquids at around 1022 cm−3 . This diluteness is typically further
quantified by comparing the typical separation between atoms of order 102 nm compared to thespatial extent of the collision, which is on the order of a few nanometers. In the experiments
considered in Chapters 3–4, these length scales are separated by at least two to three orders ofmagnitude. The associated timescales are also comparably separated: the duration of a collision ismuch shorter than the transit time between collisions. These separations are the basis of a huge

9simplification of the many-body problem in terms of correlation decay in the gas.
These gases are composed of bosonic atoms. This is a statement about the statistics ofidentical particles. For instance when two identical atoms collide, it is impossible to track theindividual atoms during the collision and to identify which atom is which afterwards. The physics
of identical particles manifests in observable effects through a factor of 2 enhancement of thescattering cross section and also globally through Bose enhancement of transition rates responsiblefor Bose-condensation.
These gases can be either weakly-interacting or strongly-interacting, which depends on
the underlying scattering physics and their relation to the many-body problem. Chapters 3–4 areconcerned with the former regime, and Chapters 5–7 with the latter. In this chapter, the two-bodyscattering problem is however presented quite generally beginning with the basic universal concepts
and a discussion of the typical form of the interatomic interaction for ultracold gases composed ofalkali atoms.
This chapter begins with a discussion of the physics on the few-atom level present in the gas.
The basics of many-body theory are then outlined beginning with the language of second quantization. Finally, the chapter ends with the basic picture of the ideal Bose gas at finite temperatures,
which is the reference system for much of the analysis in Chapters 3–4. Each subsequent chapterof this thesis builds upon these concepts, and so this chapter is considered mandatory for what
follows.

2.1

Single-Atom Physics
In this section, the few-body physics present in ultracold Bose gas is discussed in vacuum,

which is an appropriate starting point for gases that are dilute. This begins with single-particle
concepts related to the trapping potentials discussed in this thesis, identical particle symmetry, andthe quantum mechanical wave particle duality. The basic physics of the scattering of two bosons isthen discussed in the following section with emphasis on the ultracold limit. These concepts serveas the building blocks of all analysis presented in this thesis.

10
2.1.1

Harmonic Traps

The systems considered in this thesis are confined harmonically via a magnetic trap (Chapters 3–4) or an optical trap (Chapter 6.) From a theoretical point of view, often it is convenient tojust write down the trapping frequencies, extract the oscillator lengths
r
1
,
aho =
mω0

(2.1)

where ω0 = (ωx ωy ωz )1/3 is the geometric mean of the trapping frequencies and to be done with
it. However, details of the magnetic trapping potential are crucial to the analysis of Chapter 3–4,
where real-world imperfections in the confinement are central to the analysis. I therefore discuss
the magnetic TOP trap with this later analysis in mind and omit a discussion of the optical dipole
trap (see for instance Ref. [64].)2.1.1.1

Magnetic TOP Trap

Magnetic traps are based on the coupling of an applied magnetic field with the magnetic
moment of an individual atom’s intrinsic spin. The spin components are typically broken down
into the nuclear I and electronic J spins, whose interaction in the absence of fields is described by
the hyperfine HamiltonianF2 − I2 − J2
Hhf = C I · J = C
,
2

(2.2)

written in terms of the total angular momentum of the atom F = I + J where C is a species
dependent constant. For instance, the experiment considered in Chapters 3–4 utilized 87 Rb with
I = 3/2 and J = 1/2 which has a hyperfine splitting between F = 2 and F = 1 in the microwave
domain at roughly 6835 MHz [65].
When a static magnetic field is applied, the spin Hamiltonian acquires a couplinghµN i
Hhf +mag = Hhf + gµB Jz −
Iz B,
I

(2.3)

where µN is the magnetic momentum of the nucleus, µB is the Bohr magneton, and g ≈ 2 is theg-factor for the electron. The quantity in brackets is the combined magnetic moment of the nucleus

11and electron spins projected onto the direction of the applied field, which is arbitrarily chosen to
be the z-axis. This coupling splits the hyperfine states as the total angular momentum is no longera ‘good’ quantum number. When the Zeeman energy is small compared to the hyperfine splitting,the shift is well approximated by first order perturbation theory [66]∆E(F, mF ) ≈ gF µB mF B

(2.4)

F (F + 1) + J(J + 1) − I(I + 1).
2F (F + 1)

(2.5)

for a given hyperfine level, wheregF ≈ gMaxwell’s equations allow a local minima (not maxima!) of the magnetic field to be created, andtherefore the hyperfine states which are trapped are the low-field seekers which are not repulsed by
the valley of the trap. For 87 Rb, one such state is the maximally stretched state which at zero field
is in the spin state |F = 1, mF = −1i.
The energy shifts due to the magnetic field can be utilized to spatially trap the atoms byapplying a nonuniform gradient. One such scheme is the quadrupole trap, where two parallel
circular coils with opposite currents generate a magnetic field that vanishes at the midpoint where
the field takes the form





~ =
B




Bz
2 x






Bz
.
y
2


−Bz z

(2.6)

To avoid transfer to an untrapped spin state at the origin, so-called Majorana losses, a fast rotating
bias field perpendicular to the quadrupole magnetic field can be used to produce a field of the form
[67]





Bz
2 x + B0 cos(Ωt) 





 Bz
~
B=
,
 2 y + B0 sin(Ωt) 


−Bz z

(2.7)

where B0 is the field strength of the bias field, and Bz is the strength of the quadrupole field. Thisbias field rotates at angular frequency Ω, which is typically much larger than the Larmor frequency,

12
precluding the precession of the atomic spin vector about the magnetic field direction. Near theminimum of the trap the cooled atoms experience a time averaged harmonic potential
VTOP (ρ, z) = µB0 +where µ is the magnetic moment and ρ =µBz 2(ρ + 8z 2 )
16B0

(2.8)

p
x2 + y 2 .

This is the TOP (Time-Averaged Orbiting Potential) trap scheme for magnetically trapping
ultracold neutral atoms [68] with a radial trapping depth of ∼ 100 µK and was the first potential
used to achieve Bose-Einstein condensation in a dilute gas [69]. In the high-field limit (µBz  mg)√the trap is anisotropic with aspect ratio ωz /ωρ = 8. A modified version of this trap was used to
produce the experimental results analyzed in Chapter 3–4. Importantly, this trap, like all real-world
setups, is not perfectly harmonic as can be seen by truncating the time-averaged magnetic field tohigher order.

2.1.2

Identical Particle Symmetry

In a dilute Bose gas there are many atoms buzzing about each in the same internal state.
There is no feasible way in which to tag an atom. From the viewpoint of quantum mechanics, theseatoms are identical and therefore indistinguishable. A theoretical description of identical particles
should reflect this fact, which requires that a two-particle wave function satisfy
φ(r1 , r2 ) = Aφ(r2 , r1 ) = A2 φ(r1 , r2 ).

(2.9)

This equality restricts A = ±1. When A = +1, the identical particles are bosons, and whenA = −1 the identical particles are fermions. An alternate viewpoint starts with a wave function
for two distinguishable particles φD (r1 , r2 ) and enforces the identical particle symmetry1
φI (r1 , r2 ) = √ [φD (r1 , r2 ) ± φD (r2 , r1 )] ,2

(2.10)

to construct the appropriately symmetrized wave function φI where the “+” is appropriate forbosons and the “−” for fermions. The remainder of this thesis deals with bosons and so this is theassumed symmetry whenever an ambiguity arises.

13
Identical particle symmetry plays an important role in the calculation of transition rates andprobabilities in a given system. For bosons, the rate of transition into an occupied state is enhancedby the number of bosons already occupying this state. This Bose enhancement can be understoodby thinking about the time-reversal of this process: there are many possible, identical atoms thatcan make the transition out of the occupied state. In equilibrium, the principle of detailed balance
ensures that these rates are equal. Identical particle symmetry therefore manifests in correlations of
the system even when the individual atoms are noninteracting. This is important for understanding
Bose condensation.

2.1.3

Wave-Particle Duality

From the symmetrization requirement arises a natural question: is it appropriate to consider
just two bosons in isolation or rather should the wave function be symmetrized with respect to thetotality of identical bosons in all of space? This question is addressed by the Cluster Law, whichstates that it does not matter whether different groups of particles are symmetrized or not provided
they are isolated to well separated areas and only independent measurements are performed on thegroups (see pp. 600 of Ref. [70])
These questions lie at the heart of a larger theme: what aspects of an atom should be treatedquantum mechanically and what can be treated classically to good approximation? Answers tosuch queries are founded on the smallness of plank’s constant ~ to the relevant problem. Clearly,
the internal variables of an individual atom should be treated quantum mechanically to retain, forinstance, the spectrum of quantized energy, angular momentum,m and spin levels. When studying
the internal energy spectrum there are however limits in which internal dynamics can be thoughof semi-classically, which is to say there is a mix of quantum and classical treatments in the
problem, for instance in the high lying orbits of Rydberg electrons (see for instance [71]), althoughsuch interesting effects are not of relevance to this thesis. Center of mass (COM) variables are,however, often treated as classically. In a trap, this approach is justified if the oscillator length, ismuch larger than the wave packet and coherence length of the atom, summarized by the de Broglie

14wavelength λdB = k −1 which is equal inverse the center of mass momentum. When this is the case,the motion of an individual atom can be described classically as a point-particle, like a billiard ballrolling around in a parabolic well.
The above statements must be qualified in a gas: in the presence of other atoms, if the de
Broglie wavelength spans the average separation between neighboring particles, all degrees of freedom, including the center of mass motion, must be treated quantum mechanically. Overlapping de
Broglie waves signal the regime of quantum degeneracy, which requires a many-body treatment.

2.2

Two-Atom Physics
To go beyond the single-atom picture, interactions within the gas must be accounted for.

In principle, collisions between two, three, four, etc... atoms takes place inside of the gas, butwith increasing atom order, the probability of occurrence for such processes is suppressed by higher
powers of the diluteness parameter1 . In Chapter 7, this argument is made explicit by calculating
perturbative contributions to the mean-field theory of the condensate in powers of the dilutenessparameter. The diluteness parameter reflects the smallness of the spatial extent of the collision,
which for ultracold Bose gases is quantified by the s-wave scattering length a0 discussed in Sec. 2.2.3,
compared to the average inter particle spacing n−1/3 , summarized for a dilute gas as (na30  1).
When the diluteness parameter is much less than unity, two-body scattering physics is the dominant
thermalization process within a gas out of statistical equilibrium. Additionally, when a0 is much lessthan the distance to neighboring atoms so that the outgoing trajectories of the two atoms are to agood approximation asymptotic, it is sufficient to input into a many-body theory only the results of
two-body scattering surrounded by a vacuum. For most experiments with weakly-interacting Bose
gases, the spatial extent of the collision is also orders of magnitude less than the oscillator lengthof the trapping potential, and so results from two-body scattering in free space can be applied2 .
When a two-body bound state or dimer lies near the collision threshold, the scattering length can
1

To see a concrete example of how these processes are suppressed for beam experiments refer to Refs. [72, 73]
For the experiments considered in Chapters 3–4 these length scales are separated by roughly three orders ofmagnitude2

15
be tuned, and this is addressed in Sec. 2.2.4.
The quantum versus classical argument from Sec. 2.1 demarcates two scattering regimes.
When the de Broglie wavelength is less than the spatial extent of the collision, then the collision
can be described in terms of classical trajectories. This is the classical regime of two-bodyscattering. In the opposite limit, the collision must be described in the language of quantum
mechanics where identical particle symmetry becomes important. This is the quantum collisional
regime. The classical regime permits a description of collision which aligns closely with our idea ofwhat a scattering trajectory looks like and so serves as a useful launching point into the quantum
collisional regime which is the regime of practical interest for ultracold gases.

2.2.1

Classical Regime

Classically, when an object impinges on a fixed target and scatters off, the problem is deterministic. One could in principle track the position of the object throughout the complete trajectory.
For instance, this is feasible for planetary dynamics where the scattering orbit occurs over astronomical time and length scales. In the laboratory, however, what can be counted is only the incoming
and outgoing trajectories long after a collision has occured over the inaccessible length scale of
the width of an individual atom. Therefore, what is experimentally relevant are the asymptotes
of free motion before and after the collision. The scattering problem is therefore concerned withcalculating the outgoing asymptote given the incoming trajectory.
That the scattering region is inaccessible means that what’s relevant is the outgoing asymptote averaged over a range of impact parameters ρ which measure the perpendicular distance thatthe scatterer is displaced from an axis of the target–a direct hit corresponds to ρ = 0. What’s
observable is the number of atoms that scatter into a particular solid angle ∆Ω after many repeated iterations of the same experiment with a consistent beam density ninc of incoming atoms
per unit area. The total number of scattered atoms Nsc detected over all solid angles should equal
the number of incoming atoms that actually struck the target. Therefore,
Nsc = σninc ,(2.11)

16
where σ has units of area, measuring the collisional size of the target, which is the two-bodycollisional cross section. If a detector is placed over a certain solid angle ∆Ω with respect to thetarget, the cross section σ(∆Ω) of that part of target which scatters into ∆Ω is measured. When
this angle is small the differential cross section can be measuredσ(dΩ) =dσdΩ.
dΩ

(2.12)The differential cross section is the most in-depth information that can be obtained from classicalscattering experiment, and, importantly, it is the result of randomization of the impact parameters
as a result of the inaccessibility of the atomic length scales on which the collision occurs. This does
not prevent, for instance, tracking the deterministic classical trajectories of interacting atoms as
was used in the seminal molecular dynamics study by A. Rahman on liquid phase Argon [74] inthe 1960s. The Boltzmann equation discussed in Sec. 3.1.1 however takes the opposite stance ofstudying particle dynamics beginning from the measurable differential cross section and admitting
ignorance of the details of the collision event.

2.2.2

Quantum Collisional Regime

For classical scattering, the outgoing trajectory follows along a definite angle with respect
to the origin of the collision at the target. In quantum mechanics, the incoming particle is reallya wave packet, φ with a momentum distribution well-peaked about some incident momentum, p0 .
Portions of the wave packet may scatter over a range of solid angles, and the measurement processcollapses the outgoing wave packet Ψ onto a specific solid angle with probabilityZ ∞w(dΩ ← φ) = dΩp2 dp |Ψ(p)|2(2.13)

0where the magnitude of the outgoing momentum has been integrated over to yield a probability of
scattering into that solid angle regardless of the momentum components. Recall that the classicalresult is binary: either the outgoing atom is detected or not. The total number of atoms thatscatter into dΩ averaged over many impact parametersZ
Nsc (dΩ) = d2 ρ ninc w(dΩ ← φρ )(2.14)

17
where φρ = φe−iρ·p is the displaced incoming wave packet. The differential cross section follows
from Eq. 2.11, 2.12σ(dΩ ← p0 ) =Z

d2 ρ w(dΩ ← φρ ),(2.15)

where φ has been replaced by the momentum p0 parallel to the collision axis.
The scattering problem is concerned with calculating the outgoing asymptote φout and thedifferential cross section, given the incoming trajectory φ. These two states are linked through theunitary evolution of the system, summarized by the scattering operator S
S |φi = |φout i.

(2.16)

When the S operator is factored as S = 1 + R, all of the information about the interaction iscontained in the remainder operator R. Matrix elements of the remainder operator hp| R |p0 i are
directly related to the differential cross section through the scattering amplitude, f (p, p0 ),hp| R |p0 i =i
δ(Ep − Ep0 )f (p, p0 ),
2πµ2B

dσ
(p ← p0 ) = |f (p, p0 )|2 ,dΩ(2.17)(2.18)

which has units of length. The quantity µ2B is the two-body reduced mass which for identical bosons
is µ2B = m/2 in terms of the single atom mass, m. The energy delta function is a restriction of the
problem on the energy shell of the collision Ep = Ep0 as opposed to off shell Ep 6= Ep0 .
A common way of obtaining solutions for the scattering amplitude is through analysis of the
Lippman-Schwinger equation for the scattering T-operator in terms of the two-body interacting V
and the free Green’s operator G0 ,T=

V + V G0 T,
hp0 |T (Ep + i0)|pi = −where(2.19)f (p, p0 ),(2π)2 µ2B

(2.20)

where the notation ‘i0’ indicates the limit as zero is approached from above. The LippmanSchwinger equation can be iterated
T = V + V G0 V + V G0 V G0 V + ...,

(2.21)

18

Figure 2.1: Left panel: The single-scattering Feynman diagram for the first Born approximation.
Right panel: Multiple scattering contributions are represented by diagrams with intermediate states
whose momentum must be integrated over. Here, the double-scattering diagram is shown where thevirtual state’s intermediate momentum q must be integrated over, violating energy conservationin the system within the bounds of the time-energy Heisenberg uncertain relation.

to form the Born series, whose first order approximation corresponds to T ≈ V and is referred toas the first Born approximation. The operator G0 is the Green’s operator for the noninteracting
Hamiltonian, H0 ,G0 (z) = (z − H0 )−1 ,(2.22)

which is an operator function of the complex number z [75]. The Born series is expected to converge
at high energies where the denominator of the G0 is large, and for very weak interactions where
higher order terms converge as powers of the interaction strength [75]. Truncating the series in
Eq. 2.19 at the first term T ≈ V amounts to the first Born approximation, which is commonlyused in such scenarios, and the latter scenario is of particular importance to the many-body theories constructed in this chapter. In the interaction picture, the Born series has a diagrammaticinterpretation in terms of Feynman diagrams. The Born approximation is commonly referred to as
single scattering, and the corresponding Feynman diagram is shown in the left panel of Fig. 2.1.
The totality of remaining diagrams are referred to as the multiple scattering contributions.
Effects due to identical particle symmetry enter at the level of the scattering amplitude. Asa general principle of quantum mechanics, all of the amplitudes for indistinguishable pathways

19

Figure 2.2: The direct and exchange scattering pathways in the center of mass frame of the twocolliding identical particles.

must be added before taking the modulus squared and calculating observables and probabilities.
For identical particle scattering there are two indistinguishable pathways shown in Fig. 2.2. The
first pathway is the one in which the particles are distinguishable, the direct pathway. The second
pathway is the experimentally indistinguishable from the first, the exchange pathway. The correct
scattering amplitude for the scattering of two bosons is a coherent sum of the direct and exchange
amplitudesfˆ(p, p0 ) = f (p, p0 ) + f (−p, p0 ),(2.23)

where the hat indicates the symmetrized amplitude. The symmetrized amplitude for bosons is aneven function of the outgoing momentum p whereas the amplitude for fermions (not shown) is odd.
This has direct consequences in the ultracold limit discussed presently.

2.2.3

The Ultracold Limit

In a vague sense, the ultracold limit corresponds to the low-energy limit of scattering, where
the problem is considerably simplified. This statement can be made more precise by considering
the decomposition of the scattering quantities in terms of angular momentum quantum numbers.
When the colliding atoms are spinless, the orbital angular momentum operator L commutes with
the S matrix, which is diagonal in the orbital angular momentum eigenstates |E, l, mihE 0 , l0 , m0 | S |E, l, mi = δ(E 0 − E)δl0 ,l δm0 ,m e2iδl (E) ,(2.24)

20
where δl (E) is the phase shift which is a real quantity in this simple, single-channel discussion. The
states hr |E, l, mi = yl (r)Ylm (x̂)/r are eigenstates of the noninteracting Hamiltonian, H0d2l(l + 1)
2−
+ p y(r) = 0.
dr2
r2

(2.25)The states hr |E, l, m+i = ul (r)Ylm (x̂)/r are eigenstates of the full Hamiltonian, Hd2l(l + 1)
2−
+ p − V (r) u(r) = 0.
dr2
r2

(2.26)

The asymptotic solutions can be written in terms of the phase shiftu(r)l −−−→ eiδl (Ep ) sin[pr − lπ/2 + δl (Ep )],r→∞(2.27)

which elucidates the significance of the phase shift.
The scattering amplitude can also be decomposed in terms of contributions from different
orbital angular momentum quantum numbers
f (p0 , p) = 4π

X

Yl,m (p̂0 ) Yl,m (p̂)∗ fl (Ep ),(2.28)

l,m

where fl (Ep ) = (exp(2iδl (Ep )) − 1) /2ip is the partial-wave amplitude, and Eq. 2.28 is the partial
wave expansion of the scattering amplitude. The cross section can be written in terms of a sum
over partial waves
σ(Ep ) =X

σl (Ep ) = 4πlX(2l + 1)|fl (Ep )|2 .

(2.29)l

The partial wave cross-section, σl , is bounded by the unitarity of the phase shift contribution: theunitarity boundσl (Ep ) ≤ 4π

2l + 1,p2(2.30)

when the phase shift is an odd multiple of π/2.
In the low-energy limit p → 0, the leading order threshold behavior of the partial wave
amplitude isfl (Ep ) −−−→ −al p2l .
p→0(2.31)

21
When this limiting form inserted into the partial wave expansion for identical particlesfˆ(p0 , p) = −4π

X
l,m

h
i
Yl,m (p̂0 ) Yl,m (p̂)∗ + (−1)l Yl,m (p̂)∗ al p2l ,(2.32)

it is clear that for identical bosons only even-ordered partial-waves are non vanishing enhanced bya factor of 2. When identical fermions scatter, only odd-ordered particle waves contribute.
Near threshold (Ep → 0) only a finite number of partial waves contribute, and the partialwave decomposition can be truncated. The classical explanation is that for a given incoming
momentum component p0 along the collision axis and characteristic width wt of the target, theincoming atom misses for angular momentum l > po wt . In the low-energy limit, only incoming
atoms with proportionately low angular momentum contribute to the collisional cross section. This
criterion carries over to quantum scattering where the characteristic width is replaced instead by theapproximate range of the potential, and the higher partial waves are suppressed by the centrifugal
barrier of strength l(l + 1)/2mwt2 .
When p0 < wt−1 , so that the incoming wave packet is repulsed by all nonzero centrifugal
barriers, the scattering process is in the ultracold limit, and only the lowest partial wave (l = 0)s-wave contributes significantly to the cross section for bosons. To leading order, the s-wave partial
amplitude is therefore proportional to a constant a0 with units of length, which is the two-bodys-wave scattering length. In the ultracold limit, fermionic scattering is suppressed, occurring toleading order through p-wave contributions which are quadratic in the incoming momentum andtherefore vanish in the ultracold limit, which was observed experimentally in Ref. [76]. Truncating
Eq. 2.32 to include only s-wave collisions gives the cross sectionσ̂ = 8πa0(Scattering of identical bosons in the ultracold limit.)(2.33)The effective range expansion provides corrections to Eq. 2.31 beyond leading order, written here
in terms of the s-wave phase shift
p cot δ0 (Ep ) −−−→ −
p→0

1
r0
+ p2 + ...,
a02

(2.34)

22
20

0

-20

0.1

0

-40
-0.1

-0.2

-60

50

0

10

20

70

60

30

80

90

40

100

50

60

Figure 2.3: At medium separations, the total electronic spin is a good quantum number and the
3 +interatomic potentials are the singlet 1 Σ+
g and triplet Σu potentials, which are shown here for
85 Rb atoms [77, 78, 79]. At large separations the energies correspond to the hyperfine levels labeledin the inset, shown here for mf1 + mf2 = −4 at B = 600 G, which is the location of a well-known
broad Feshbach resonance.

where r0 is the effective range often used as a characteristic length scale to describe the rangeof the potential. If the expansion is truncated to include only the scattering length and effective
range, the s-wave scattering amplitude can be extended beyond the zero energy limitf0 (p) = −
2.2.4

a0
.
1 + ipa0 + r0 p2 a0 /2(2.35)

Interatomic Potentials for Alkali Species in the Ultracold Limit

At this point it is helpful to contextualize these remarks by sketching the typical form of theinteractomic potential for alkali atoms in the ultracold limit (see as example the potentials for 85 Rb
in Fig. 2.3). For neutral alkali species, which are ‘Hydrogen-like’, there is a single unpaired valence
electron which determines much of the collisional physics. At small separations on the order of a

23
few Bohr radii, a pair of interacting atoms feel the strong repulsion of the ionic core. At medium
separations, only the orbits of the valence electrons overlap, and the atoms can covalently bondwhen the valence electrons are paired in a singlet state, as shown by the deep potential well in
Fig. 2.3. They may also pair in a triplet state as shown in Fig. 2.3, however this potential is muchshallower than the deep singlet well. However, both potentials can be deep enough to support
many bound states. The minimum of these wells for alkali species is roughly at .2–.5 nm separation
between nuclei3 .
At large separations beyond the overlap of the electronic clouds van der Waals forces
become dominant as each atom feels the electrostatic potential generated by the other, and forneutral atoms, the multipole expansion of this potential begins with the dipole moment. The
dominant interaction is therefore that between the dipole moment of one atom with the dipole
field established by the other–the dipole-dipole force. The strength of this interaction for groundstate atoms can be calculated at fixed nuclear separation beginning at second order in perturbation
theory [81]UvdW (r) = −C6+ ...,
r6(2.36)

where the power r−6 arises through the square of the dipole-dipole interaction energy which goes as
r−3 , and the positive coefficient C6 gives an attractive force. In the asymptotic limit, the individual
atoms are in the one-body hyperfine states of total angular momentum, whose zero field splitting
for alkali species corresponds to temperatures on the order of Kelvins, which is still minusculecompared to the singlet and triplet well depths [65].
For ultracold collisions, the incident energy is insufficient to probe the repulsive short-range
details beyond the singlet and triplet electronic potentials. The potential can be written in thefollowing form4 which highlights the conservation of the total electronic spin [65]Uc = U0 P0 + U1 P1 =U0 + 3U1+ (U1 − U0 )S1 · S2 ,4

(2.37)

in terms of the spin Si , singlet P0 projection, and triplet P1 projection operators. The form
3
4

For a summary of important length scales for alkali atoms see pp. 351–389 of Ref. [80].
This is the ‘central’ portion of the interaction.

24

Closed Channel

0rFigure 2.4: A schematic of a Feshbach resonance. Scattering process in the incoming channel(red) is coupled to the closed channel (blue). When a bound state (dashed lines) is near collisionthreshold, the colliding atoms form a metastable state which decays over a characteristic lifetime,
indicated by the hourglass, exiting out the open channel.

of the second equality emphasizes the relevant spin-flip and spin-exchange physics, which mix
the hyperfine levels of the incoming atoms. At large distances, these potentials asymptote into
attractive van der Waals tails. If the hyperfine splitting is ignored, scattering in the ultracold limit
can be summarized by the triplet (U1 ∝ at ) and singlet scattering lengths (U0 ∝ as ). For 87 Rb,
these are as ≈ at ≈ rvdW , which are in the neighborhood of 102 Bohr radii. The proximity of thesescattering lengths leads to suppression of spin-exchange processes which depend on their differencein Eq. 2.37.
The interatomic potentials are, to a rough approximation, an attractive van der Waals tails
with a hard wall cutoff at separations where the details of the ionic core become important. For
ultracold energies, the characteristic length of the van der Waals interaction, rvdW = (mC6 )1/4 /2,
sets the scale of as and at and is also associated with the ‘range’ of the potential. For alkali species,
rvdW is typically on the order of a few nanometers.
The scattering length can display resonance effects when the collision energy is in the neigh-25borhood of a bound state above threshold5 as shown schematically in Fig. 2.4. This is known as
Fano-Feshbach6 resonance [82] where the scattering length diverges at a pole, taking on all values
between ±∞. Commonly in ultracold gas experiments, a bound state’s energy is tuned to thecollision threshold by utilizing a difference in the magnetic moments of the open and closed channels7 ∆µ and the resultant Zeeman energies under an applied magnetic field B. The well-known
expression for the magnetic field dependence of the scattering length in this case isa0 (B) = abg 1 −∆B
B − B0,(2.38)

where abg is the background scattering length far from the resonance, ∆B is the field width of theresonance, and B0 is the value of the applied field that brings the bound state degenerate withthe collision threshold energy to produce a zero-energy resonance. For ultracold gas experiments,
widths on the order of 1 Gauss or less are referred to as narrow resonances, and broad resonances
are associated with widths which greatly exceed 1 Gauss, where the Feshbach dimer has universal
characteristics independent of the atomic species over a wider range of detunings (see the review
article [30] for a detailed discussion.) The broad or narrowness is correlated with the strength ofthe coupling between the open and closed channels. The ability to tune the scattering length via a
Feshbach resonance underlies much of the discussion of Chapters 5–7.

2.2.5

Effective Potentials and the Zero-Range Model

There are in principle many potentials, perhaps of simpler form, which reproduce, up toa given order, the first few terms of the effective range expansion. Such a potential is referred
to as an effective potential as opposed to the actual potential. In ultracold gas experiments
collisions occur only over a narrow window of energies centered around the thermal energy kb T .
It is therefore inconsequential, from the viewpoint of constructing observables, whether or not
the actual or effective potential is used as long as the energy dependence of the observables over
5

This bound state is referred to as the Feshbach dimer.
Or often just ‘Feshbach resonance’.
7‘Channel’ is used to refer to a collisional pathway. ‘Open channel’ refers to pathways which are energeticallyaccessible asymptotically, and ‘closed channels’ are inaccessible.
6

26
the experimentally relevant window of energies is reproduced. For instance, the actual interatomic
potential is typically used to calculate the scattering length, effective range, etc..., and then a simpler
effective potential is introduced that simplifies the properly cut-off many-body calculations. The
importance of this approach, and the order in which the few-body problem feeds into the many-bodyproblem is of foundational importance to the study of ultracold gases and related condensed-mattersystems. In this section, I specifically discuss two effective potentials relevant to this thesis: theFermi pseudopotential, used in few-body calculations, and the contact interaction, used in manybody calculations.
For single-channel models with a van der Waals tail, the general expression [83] for the s-wave
effective range is"ā
r0 (a0 ) = r0 (∞) 1 − 2 + 2
a0



ā
a0

2 #

(2.39)

where r0 (∞) = 2Γ(1/4)2 rvdW /3π is the effective range at unitarity and ā = 4πrvdW /Γ(1/4)2 .
Interestingly, the zero scattering length scenario where a0 → 0 gives a divergent effective range.
At zero magnetic field, for species like 133 Cs and 85 Rb, the singlet and triplet scattering lengths
can exceed both the van der Waals length and the effective range by up to an order of magnitude[80]. For weakly-interacting Bose gases at ultracold energies, even when a0  r0 as is the case for
87 Rb, the effective range contribution is however still a very small correction to the leading orderscattering length term in the effective range expansion (Eq. 2.34) for ultracold gases. In this case,an effective potential which captures only the leading order scattering length term of the effective
range expansion can be used, which is a potential of zero-range. The zero-range potential comesin two common variants: the Fermi pseudopotential and the contact interaction.
In Chapters 5–6, the Fermi pseudopotential [84] is used in few-body calculations because it
has the distinct advantage that the effect of the potential can be incorporated through enforcementof boundary conditions at the origin alone. The idea is to replace the complicated actual potential
with an interaction of the form
Vpseudo (r) = gδ (3) (r)

∂
(r.)
∂r

(2.40)

27
with, as of yet, undetermined effective coupling strength g. The coupling strength must be chosen
to reproduce the leading order term in the effective range expansion. The delta function must be
evaluated at the origin giving a boundary condition of the form
4π
u(0),
gm
4π
,
p cot δ0 (p) = −gm
u0 (0) = −(2.41)(2.42)

which fixes
g = 4πao /m,(2.43)

to reproduce the effective range expansion to leading order for distinguishable particles. Identicalparticle symmetry is typically enforced, for instance, at the level of the scattering amplitude andcross section construction. The effective range expansion for a pseudopotential interaction therefore
only has one term, matching the scattering length dependence exactly. Equation 2.41 reveals how
the pseudopotential is implemented in practice through the Bethe-Peierls boundary condition [85].
The other commonly used zero-range potential is the contact interaction, used in Chapters 3 and 7 for many-body models, which is similar in form to the Fermi pseudopotential exceptthat it is not regularized at the originVδ (r) = g0 δ (3) (r).

(2.44)The mathematical difficulties that arise through an interaction of this form can be seen in its Fouriertransform, Vδ (p) ∝ g0 , which is constant in momentum space, coupling momentum components
to all orders equally. This poses problems, for instance, when performing calculations that do
not converge with higher momentum: An ultraviolet divergence. The cure for an ultraviolet
divergence is to properly introduce a cutoff pc in momentum space to the Fourier transformg0 /(2π)−3/2 ,if p ≤ pc(2.45)Vδ (p) =−3/2η(p) g0 /(2π), p > pc
where the function η(p) tends to zero past the cutoff, pc .

28

V(r)

Increasing Cutoff

1

4

r

Figure 2.5: A sketch of the dependence of the cutoff contact interaction on cutoff momentum, pc .
With increasing cutoff, the maximum at the origin diverges as a the cutoff momentum cubed, andthe number of oscillations increases.

The cutoff contact interaction can be thought of a square well potential in momentum spacewith depth proportional to g0 , which looks like a sinc function in position space (see Fig. 2.5),
and it is in this sharp sense that it will be used presently. The cutoff sets a length scale at shortdistances p−1
c , which should be chosen to be at a range shorter than that probed by typical ultracold
collisions.
When the cutoff is introduced, it’s dependence can be hidden in the coupling constant g0
through the process of renormalization. Specifically, the zero-energy limit of the scattering
amplitude obtained using the contact interaction must be matched against Eq. 2.35 to ensure thatthe low-energy limit of the scattering amplitude is reproduced with zero effective range (r0 = 0),
and an expression must be derived for g0 involving the s-wave scattering length and the cutoff
momentum. The contact interaction will be used in the many-body theories constructed later inthis chapter, and so it is helpful to outline the process of renormalization in the remainder of thissection.
Using the contact interaction the Born series, Eq. 2.21, can be summed via a geometric series,
resulting in an analytic expression for the scattering amplitude. This calculation is done in thes-wave limit where the matrix elements of the T-operator are hp0 |T (Ep + i0)|pi = t(p, p) which will

29
be written simply as t(p) on the energy shell, depending only on the magnitude of the incident
momentum. Beginning with the Born series for the matrix elements of the T-operatort(p) ==

g0+ hp|Vδ G0 Vδ |p0 i + ...,
(2π)3

Z


mg0
q2
g0
1+
+ ... ,
dq 2
(2π)3
2π 2
p − q 2 + i0

(2.46)(2.47)

where I have made use of the momentum eigenstates hr|pi = exp (ir · p) /(2π)−3/2 and basic properties of the Dirac delta function [86]. The integral in brackets has a principal-value part and asingular part [87]



q2
q2
=P 2
+ iπδ(p2 − q 2 ),p2 − q 2 + i0
p − q2(2.48)

which can be integrated up to the cutoff momentum with result

 

p
m
iπ
−1
t(p) = g0 − g0 t(p) 2 pc + tanh
+ p .
2π
pc
2

(2.49)In the zero-energy limit, the Born series in Eq. 2.47 can be rewritten as a geometric sumt(0) =


g0 
mg0
1
g0
1
−
p
+
...
=
c
(2π)3
2π 2
(2π)3 1 + g0 pc m/2π 2(2.50)and equating with the zero-energy scattering amplitude for distinguishable particles gives the result

g0 =

g
,
1 − gαc(2.51)

written in terms of the pseudopotential coupling strength g from Eq. 2.43 and a term αc = mpc /2π 2which hides the cutoff momentum.
Provided a0 pc  1, the contact interaction adds corrections to the pseudopotential couplingin powers of the small parameter, a0 pc . The effective range expansion for the contact interactionis also truncated at a single term, but now with added cutoff dependencefδ (p) = −a0,
1 + ipa0 − αc g0(2.52)

which complements the idea that adding a cutoff adds a range to the potential. The major advantageof the contact interaction in many-body problems, where it would be difficult to use the boundary

30
condition formulation of the Fermi pseudopotential, Eq. 2.41, is that it provides finite answers tointegrals that would otherwise suffer from ultraviolet divergences and results that are independentof the cutoff chosen provided it is well beyond the typical collision energies in the gas.
Even in the case of resonance scattering where the scattering length diverges at a pole, it ispossible to derive effective contact potentials which reproduce the energy dependence of the scattering amplitude. This derivation provides the foundation of the theory of resonance superfluidity[55], which was first formulated and studied successfully in the early 2000s when experiments exploited Feshbach resonances to produce the unitary Fermi gas. Recently, experiments have begun
probing the unitary Bose gas with the help of Feshbach resonances, and a candidate theory basedon the ideas of resonance superfluidity and its generalization to include bound states of higher
numbers of atoms has so far not been investigated. This discussion is reserved for the conclusionof this thesis, Chapter 8.

2.3

Many-Atom Physics: Formalism
The one and two atom physics discussed in Secs. 2.1 and 2.2 serve as the basic ingredientsfor constructing the many-body theory of the ultracold Bose gas. To these ingredients must be
added the concept of the second quantization which builds upon the eigenstates of the singleparticle Hamiltonian to provide a tidy expression for the many-body Hamiltonian in Fock space.
In the non-interacting limit, I discuss a simple picture for the minimization of the total energy
with an additional normalization constraint capturing the basic ideas of the zero-temperature idealgas BEC. This section finishes with the an effective many-body Hamiltonian written in second
quantization in position space including the effect of the contact interaction and therefore valid forlong-wavelength phenomenon in the gas. This serves as the basic ingredients for the many-bodyanalysis in this thesis.

31
2.3.1

Second Quantization

The discussion of second quantization begins with the Hilbert space of a single particle
Hamiltonian H1 spanned by the basis of single-particle orbital wave functions φα (r) = hr|αi, where
α represents a set of quantum numbers required to completely specify the state. The many-body
Hilbert space describing the quantum mechanical state of N -atoms is the tensor product of N
single-particle Hilbert spaces:
(1)

(2)

(N )

H = H1 ⊗ H1 ⊗ ... ⊗ H1 .

(2.53)The N-body basis states can be written then as a tensor product of the single-particle basis|α1 α2 ...αN i ≡ |α1 i|α2 i...|αN i,(2.54)

where the product of kets is the tensor product, and the subscript αi denotes an orbital in the ith
single-particle Hilbert space.
When the system is composed of bosons, the basis states of the many-body Hilbert spacemust by symmetrized through projection by the symmetrizing operator
S =

1 X
P,
N!

(2.55)

P

where the sum is taken over the N ! distinct permutations P of the single-particle orbitals. The
action of the symmetrizing operator on the N-particle basis states is written|α1 α2 ...αN iS ≡ S|α1 α2 ...αN i.

(2.56)

Post-symmetrization, the notation of the symmetrized ket, which labels the occupied orbitals inall of the individual single-particle Hilbert spaces, is unnecessary. Rather, it is simpler to reporta simpler quantity: The occupation numbers of single-particle orbitals. For instance, if Nα atoms
are in the single particle orbital |αi, Nβ atom are in the single particle orbital |βi, etc...
|α1 α2 ...αN iS ≡ |Nα Nβ ...iS ,(2.57)

32
for a fixed number of particles
orbitals.

P

i ni = N , where the sum runs over all possible single-particle

This account in terms of occupation numbers is generalized through the concept of a Fockspace, which is an enlarged Hilbert space F constructed from the direct sum [88] of the 0,1,2,3,...Nparticle Hilbert spaces
F = H0 ⊕ H1 ⊕ H2 ⊕ ...

(2.58)The Fock space basis is written in the occupation notation of Eq. 2.57, which includes the vacuum
state |0i, the single particle state |1α i, the two particle states |1α 1β i or |2α i etc... The Fock basis
is therefore composed of states of variable particle numbers. Importantly, states without a welldefined number of particles are allowed in Fock space such as the coherent state |zi which is aneigenstate of the raising and lowering operators aα |zi = zα |zi, where |zα |2 is the average occupancyof the α orbital with variance |zα |2 .
Basis states of different particle numbers are connected through a linear mapping in Fockspace. Specifically, arbitrary symmetric N-particle states can be constructed beginning with thevacuum state through repeated action of the creation a† and annihilation a operators

1|Nα Nβ ...i = p
Nα !Nβ !...

|1α i = a†α |0i,
 Nα  Nβa†αa†β...|0i,h1α | = h0|aα1
hNα Nβ ...| = p
h0| (aβ )Nβ (aα )Nα ...
Nα !Nβ !...

(2.59)(2.60)(2.61)(2.62)

where the particle symmetry is ensured through the canonical commutation relations [89]hi[aα , aβ ] = a†α , a†β = 0,hiaα , a†β = hα|βi = δαβ ,(2.63)(2.64)

where the subscript aα indicates the annihilation of an atom in the |αi orbital. The creation andannihilation operators can be transformed from one orbital basis to another, and the transform to

33
the position space representation is defined by the mappings|ri =ψ † (r) =Xα

X|αihα|ri ≡a†α φ∗α (r),Xφ∗α (r)|αi,(2.65)

α(2.66)

αψ(r) =

Xaα φα (r),(2.67)

αwhere ψ † , ψ are field operators defined on a continuous basis that create or destroy an atom inthe position space eigenstate |ri, whose orbital wave function is hr0 |ri = δ (3) (r − r0 ) the Dirac delta
function. The position space field operators therefore create or destroy an atom locally. This is
in contrast to the momentum space field operators ϕ(p), ϕ(p)† that create or destroy an atom inthe orbital wave function hr|pi = (2π)−3/2 exp(ir · p) which is the nonlocal plane wave state. The
position space field operators satisfy the continuous version commutation relations, Eqs. 2.63, 2.64i hψ(r), ψ(r0 ) = ψ † (r), ψ † (r0 ) = 0,hiψ(r), ψ † (r0 ) = hr|r0 i = δ (3) (r − r0 ),2.3.2(2.68)(2.69)

Many-Body Hamiltonian

Fock space is the natural setting to investigate the thermodynamics of a many-body system
for a variable number of particles, and the creation and annihilation operators ensure that totallysymmetric systems are mapped onto totally symmetric systems. To translate the many-bodyproblem into Fock space, it is first necessary to rewrite all quantities of interest including the
many-body Hamiltonian in terms of creation and annihilation operators. Let us start with thenoninteracting N-atom Schrödinger equation in a trap
H0 =


N 
X
∇2
− i + Vtrap (ri ) = T + Vtrap ,2m
i(2.70)

34
where ri is the position vector of the ith atom. The mapping [89] of these one-particle operators
onto Fock space is
Ĥ0 =

X
αβ

a†α hα|T + Vtrap |βiaβ ,(2.71)

written in terms of a basis of discrete orbitals.
The orbital basis may be transformed into the eigenbasis |λi of the single-particle Hamiltonian
with energy Eλ , yielding the many-body Hamiltonian in diagonal formĤ =X

Eλ a†λ aλ ,(2.72)

λwhere the sum now runs over all orbital eigenstates and the product a†λ aλ = N̂λ giving the orbital
occupancy. Written in the occupancy notation, it is possible to discuss the ground state structure in
Fock space. The ground state is the arrangement that minimizes the energy of the system subject
P
to the constraint that the total number of atoms N̂ = λ a†λ aλ . These constraints are incorporated

by minimizing the grand canonical Hamiltonian, K̂ = Ĥ − µN̂ with fixed µ, playing the role of the
Lagrange multiplier and conjugate variable to N̂ ,
K̂ =

X
λ

(Eλ − µ) a†λ aλ ,(2.73)

where the identification of µ with the chemical potential is made later in Sec. 4.1 for the interacting
zero temperature Bose gas. The expectation value of K̂ can be minimized, for instance, throughvariational methods [89], but for the noninteracting gas, the ground state structure can be uncovered
through simple arguments about the structure of Eq. 2.73, depending on the fixed value of µ. For
fermionic systems and a given value of µ, all orbitals with energies Eλ ≤ µ will be occupied
according to the Pauli exclusion principle, and the µ sets the number of atoms in the ground state.
For bosonic systems, if E0 is the lowest orbital energy, then K̂ has a ground state only for µ ≤ E0 ,where condensation occurs for any number of atoms when the equality is satisfied. Minimization ofK̂ with fixed µ therefore yields the ground state structure of the many-body system in Fock space.
To go beyond the non-interacting many-particle Hamiltonian, pairwise interactions between

35
the atoms must be written in terms of creation and annihilation operators [89]V̂ =1 X † †aδ aγ aβ aα hδγ|V |αβi,
2(2.74)αβγδ

where the factor of 1/2 prevents double counting, andhδγ|V |αβi =Z

d3 rd3 r0 V (r − r0 )φ∗δ (r)φ∗γ (r0 )φβ (r0 )φα (r).

(2.75)In the case of trapped ultracold Bose gases, the many-body Hamiltonian is commonly written
in terms of the position space field operators, where the mappings in Eqs. 2.65–2.67 yield the
Hamiltonian Ĥ including pairwise interactions
Ĥ =

Z




Z
∇2
1
d r ψ (r) −+ Vtrap (r) ψ(r) +d3 r d3 r0 ψ † (r)ψ † (r0 )V (r − r0 )ψ(r0 )ψ(r),
2m
2
3

†

(2.76)

and when the contact interaction V (r) = Vδ (r) (Eq. 2.44) is used
Ĥ =

Z



g0 h † i2 h † i2∇2
+ Vtrap (r) ψ(r) +ψ (r)ψ (r) .
d3 r ψ † (r) −2m
2(2.77)

It is important to reiterate that this is an effective Hamiltonian, valid only for long wavelength, low
momentum phenomenon in the gas, where the implicit dependence of g0 on the cutoff momentum
pc eliminates the influence of short-wavelength degrees of freedom.
Using the Heisenberg equation of motionhi˙iÔ = Ô, Ĥ ,(2.78)

equations of motion for arbitrary products Ô = ψ(r1 )ψ(r2 )...ψ † (rj ) of the field operators can bederived. Rather than deriving these equations presently, they will be developed over the courseof the thesis as needed and always in the context of the motivating physics. This chapter finisheswith the simplest nontrivial many-body system relevant to this thesis: the ideal Bose gas at finite
temperatures, which will serve as the contextual framework that many of the discussions in thisthesis will refer to.

36

2.4

The Noninteracting Bose Gas at Finite Temperatures
At finite temperatures, the many-body problem can be treated through a set of standard

approximations depending on the temperature regime of interest relative to the other energy scales
in the problem. The experiments analyzed in Chapters 3–4 are well described by the semi-classicalapproximation, where the center of mass motion of the noncondensed portion of the gas can betreated classically, akin to billiard balls rattling around in a trap bouncing off of one another, rather
than fully quantum mechanically. It is therefore instructive to first study the noninteracting, ideal
Bose gas at finite temperatures and highlight many important features before moving on to theweakly-interacting Bose gas in Chapters 3–4. Since the experiments analyzed in Chapters 3–4 are in
extremely spherically symmetric traps, I will assume an isotropic trapping geometry in this section
which also serves to reduce notational clutter.
When interactions have been switched off, the harmonic oscillator states |ni, (n = {nx , ny , nz })are the eigenstates of the grand canonical many-body Hamiltonian with particle number constraint,
which has the generalized diagonal form
K̂ = ω0X

n=0(nx + ny + nz + 3/2 − µ) a†n an ,(2.79)

where the factor 3ω0 /2 is the zero-point energy of the ground state. Combinatorially speaking, there
are many distinct sets {nx , ny , nz } that yield the same energy (n + 3/2)ω0 , and this degeneracy
grows with increasing n. For example, the ground state n = 0 is unique, the first excited stateis three-fold degenerate, the second excited state six-fold degenerate, the third excited state tenfold degenerate, and so on. For increasing value of n, the excited spectrum can be treated as
approximately continuous. The degeneracy of states between energies E and E + dE in a harmonic
trap isg(E) =

E2,
2ω03(2.80)

which is derived from simple arguments for the density of states in many introductory textbooks(see for instance Ref. [90].) The total occupation of the energy neighborhood around E of width

37
dE is therefore given by the product of the degeneracy with the Bose distribution
N (E + dE) = g(E)dEz,exp(βE) − z(2.81)

where β = 1/kb T , T is the temperature of the gas, and z = exp(βµ) is the fugacity. The total
population can therefore be calculated at a given temperature, with the reservation that the groundstate must be summed discretelyzN=+exp(β3ω0 /2) − z

ZdE g(E)z.
exp(βE) − z(2.82)

From the functional form of z restricted on the domain z ∈ [0, ∞), the Bose distribution approaches
its maximum value as z → 1 from below. Values of z exceeding unity give an occupation which
is negative and nonsensical. From the discussion of Sec. 2.3.2, we concluded that condensation
occurs when µ is approximately equal to the ground state energy of the system. When z ≈ 1, thenumber of atoms in excited states, Nex approaches a maximum value, which is the point at which
the excited states are saturated by the entire cloud. The temperature Tc at which this occurs canbe calculated
Nex = N =Z1
= ζ(3)
dEg(E)
exp(E/kb Tc ) − 1



kb Tc
ω0

3

,

(2.83)whereTc =ω0 N 1/3,
kb ζ(3)1/3(2.84)

where ζ is the Riemann zeta function [86]. The saturated cloud is referred to as the thermal
cloud, and is composed of excitations out of the condensate. Below the critical temperature,
the total number of atoms is split between the thermal cloud and the condensate. The remainder
N − Nex is given by the simple relation" 3 #
T
N0 = N 1 −Tc(Ground state occupation).

(2.85)

These formulas have been derived by using the approximation µ = 0, and corrections from using
the correct relation µ = 3ω0 /2 go as N −1/3 which vanishes in the thermodynamic limit, smoothing
the phase transition for finite-sized clouds [65].

38This is the theory of the finite temperature ideal Bose gas near the critical temperature for
Bose-Einstein condensation, which serves as a basis for the finite temperature analysis of Chapters 34 on the weakly interacting Bose gas. Specifically, the ideal Bose gas is the long-time limit of any
nonequilibrium dynamics for the interacting Bose gas when inelastic loss mechanisms are ignored.
A strange exception to this trend is the breathing mode discussed in Chapter 3, which was predicted
by Boltzmann to persist unsettled indefinitely. The study of the dynamics and relaxation of a dilute
gas is the subject of kinetic theory, which is the starting point for Chapter 3.

2.5

Summary
In this chapter, the background necessary to understand both the theoretical analysis andthe relevant experiments discussed in this thesis was outlined. In the following chapters, a manybody theory for weakly and strongly interacting Bose gases is developed along with analysis of
nonperturbative three-body physics, and I have chosen to develop the relevant theory along theway on an as-needed basis. For the many-body physics in this thesis, I feel that this approach has
the two advantages. Clarity: It is easier to appreciate the structure of the many-body theory if it
is built up slowly in the context of motivating experimental results. Convenience: In a sense, it isnatural for the relevant results to be discussed following the relevant theory as would occur in ajournal article.
The few-body portion of this chapter was presented in a way that provided the necessary
ingredients for the many-body theories of Chapters 3–4. The three-body physics discussed in latter
half of this thesis builds upon these ideas, and much of the intuition from the two-body partial
wave analysis is retained in the hyperspherical adiabatic representation of the three-body problem
outlined in Chapter 5.

Chapter 3

Anomalous Damping of the Breathing Oscillation of a Nondegenerate Bose Gas
in an Extremely Isotropic Trap

This chapter contains the results from Ref. [60], which is an analysis of an experiment published in Ref. [24].

The previous background chapter concluded with the thermodynamics of the ideal BEC at
finite temperature. There is, however, much to be learned about the gas from its out-of-equilibrium
dynamics and subsequent relaxation to thermal equilibrium, and an understanding of these processes requires the incorporation of interatomic interactions. In the regime of linear response, where
the gas is only slightly perturbed from equilibrium, the dynamics of the trapped gas can be decomposed into a discrete set of collective modes which have characteristic frequencies and damping
rates. In the collisionless limit where collisions within the gas can be neglected, the oscillation
frequencies are easily calculated from single-particle considerations. Therefore, the measured frequency shift for an interacting system is a sensitive probe of physics beyond the single-particle level.
Additionally, the observation of damping in an isolated system is interesting from a fundamental
standpoint. In this chapter, data on a very peculiar collective mode, the breathing mode in 3D
under isotropic confinment, from a recent experiment in the group of Eric Cornell here at JILA [24]is analyzed. What makes this mode quite odd is that it was predicted by Boltzmann over a century
ago [2] to oscillate undamped indefinitely, awaiting experimental verification (or contradiction!)
until very recently. Before jumping into the theory in Sec. 3, I provide some historical context for40this problem.
The Boltzmann equation has had a rich and interesting history in the study of dilute gases. In
the late 19th century, Maxwell and Boltzmann uncovered a path to link the Newtonian mechanics
of molecular dynamics to the hydrodynamic equations of Euler and Navier-Stokes. The resultant
kinetic theory, of which the fluid theories were limiting cases, established many important concepts
including the Maxwell distribution and Demon [91, 92], the Boltzmann equation, the H-theorem,
and the assumption of molecular chaos or, in Boltzmann’s native German, ‘Stosszahlansatz’ [93].
These fundamental ideas, in particular the H-theorem, famously stirred controversy in the scientificcommunity, which in the 1890s was centered around the “reversal” paradox and the “recurrence”
paradox based on Poincaré’s theorem [94].
Lesser known, but equally as curious was the discovery by Boltzmann of a class of exact
solutions to his kinetic theory of which the Maxwell distribution is a special case [2]. Such solutions
included the undamped, nonequilibrium oscillations of a dilute gas with classical statistics under
3D isotropic harmonic confinement: the so-called monopole or breathing mode. Whereas someof the most controversial aspects of the Boltzmann equation began to be justified in the early
20th century with the advent of quantum theory1 , experimental demonstration of the undamped
monopole mode oscillation has yet to be fully investigated.
More recent advances in the trapping of ultracold gases have allowed for the study of collectivemodes under harmonic confinement. Indeed, many experiments have probed the transition between
the collisionless and hydrodynamic regimes[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] inanisotropic traps and the resultant collisional shifts of the collective mode spectrum. Such experiments measure the frequencies and damping rates of the various multipole modes and are generally
limited by three-body losses approaching the hydrodynamic limit. Engineering an isotropic 3D trapof sufficiently perfect symmetry has remained a technological hurdle to investigating Boltzmann’s
prediction for the monopole mode.
A recent experiment from the group of Eric Cornell at JILA [24] trapped a nondegenerate
1

The Boltzmann equation famously predated the experimental verification of the atomistic view of nature.

41
cloud of 87 Rb well above the transition temperature utilized a modified version of the TOP trap
described in Sec. 2.1.1 to engineer anisotropies as low as 0.02%. The monopole mode was observed
to oscillate over many trap periods; however, the damping was anomalously large given the level ofanisotropy present in the system.
This chapter revolves around an investigation of possible sources of damping through a systematic account of imperfections in the trap. Upon first reading, this analysis provides an overview
of analytic and numerical2 methods for studying the Boltzmann equation with increasingly detailed functional form for the experimental trapping potential. There is however a deeper narrativeat play: if this analysis does not account for the observed damping, the central tenants of the
Boltzmann equation and of kinetic theory in general must come under scrutiny. These tenantsalso serve as the foundation for the analysis in Chapters 4, 7, and so their verification is of central importance to this thesis. The experiment measuring any violation of nonzero damping istherefore more in the spirit of metrological studies testing the value of fundamental constants, andthis narrative is bolstered by the painstaking efforts which were taken to engineer extremely low
asphericities. Importantly, the analysis presented in this chapter pinpoints the observed damping
to effects consistent with the Boltzmann equation.
This chapter begins with a derivation of the Boltzmann equation and the H-theorem, andthen explores the inherent limitations on the observation of an undamped, nonequilibrium monopolemode given a realistic trapping scenario, using the JILA experiment as illustration. The derivationof the Boltzmann equation presented in this chapter is the traditional path beginning from classicalphase space considerations. In Chapter 4, the alternate path will be presented, which begins withthe many-body Hamiltonian and the field operators.

3.1

Kinetic Theory Above Tc
Before delving into the Boltzmann equation, it is instructive to set the scene. Far above thetransition temperature, the center of mass motion of individual atoms can be treated classically2

Details of the numerical algorithms used in this and the following chapter are including in Appendix A.

42provided λdB is much smaller than aho . For finite temperature dilute gases, this comparison ismade between the average (thermal) de Broglie wavelength of atoms in the cloud
λth =

r

1
,
2πmkb T

(3.1)

and the oscillator length. The classical treatment is therefore justified provided ω0  kb T . When
this requirement is met3 the energy spacing of neighboring oscillator states is difficult to resolve,
and the continuum treatment of the excited states which was the basis of Sec. 2.4 is well-justified.
This is a semi-classical picture of the gas having both classical and quantum mechanical degrees of
freedom.
The gas is entirely composed of atoms with energies
E(r, p) = p2 /2m + Vext (r),(3.2)

and the local occupation is given by the Bose distribution4
f (r, p) =z,
exp (βE(r, p)) − z(3.3)

from which the equilibrium density of atoms in the gas can be calculated from the general formulanex (r) =Z

g3/2 (exp(β(µ − Vext (r)))
d3 p
.
f
(r,
p)
=
(2π)3
λ3th

(3.4)

Equation 3.3 is valid in thermal equilibrium when it is possible to define global thermodynamicalquantities, and it is also valid in local equilibrium when the rate of collisions is sufficiently highthat a local thermodynamic quantities can be defined.
There is a convenient framework in which to visualize the dynamics of the gas when theclassical trajectories of the atoms can be tracked. In Fig. 3.1 a projection of the µ-space [95] isshown, which is a six-dimensional space tracking the coordinates each atom in the gas. This spacecan be divided into cells of volume d3 rd3 p. The volume must be chosen large enough to hold many
particles, while simultaneously not violating the the Heisenberg uncertainty relation, ∆x∆p ≥ 1/2.
3

For the experiments considered in Chapters 3–4, kb T /ω0 ≈ 101 –102 .
Note the abuse of the notation. I’ll use ‘f ’ for both the scattering amplitude and the distribution function. The
meaning should be clear from context whenever ambiguity arises.
4

43

p

f˙(r, p, t) = I

r

[f ]

Figure 3.1: A two-dimensional projection of µ-spacecoll
in both it’s discrete representation in terms ofindividual atoms and continuum representation.

And chosen to be small compared to the characteristic scale of density fluctuations in the system,
then the discrete representation (Eq. 3.5) can be thought of as a continuum5 . In the continuum
limit, the number of atoms in a µ-space cell centered about the position r and momentum p is
d3 rd3 pf (r, p).
The µ-space and the continuum limit provide a complementary view of the dynamics of theatoms in the cloud, mapping the problem onto a study of the distribution function. Kinetic theory
is concerned with how an arbitrary distribution function, which can be written in terms of thediscrete coordinates (ri (t), pi (t)) of each atom in the cloudf (r, p, t) = (2π)3

Ne x
X
i

δ (3) (r − ri (t))δ (3) (p − pi (t)),(3.5)

evolves in time, relaxing to the equilibrium distribution (Eq 3.3.) In the remainder of this subsection, the kinetic theory for the interacting Bose gas above Tc is detailed. The theory below Tc isthe subject of Chapter 4, where it will become necessary to connect the distribution function withthe field operators and many-body Hamiltonian.
5

See Appendix A for cell requirements for numerical computations.

44
3.1.1

The Boltzmann Equation

In this subsection, I will sketch the traditional derivation of the Boltzmann equation, which
is an equation governing the dynamics of f (r, p), and motivate its generalization to include theeffects of Bose enhancement.6
Generally, f (r, p, t) evolves as atoms pass between cells in µ-space, changing the occupation
numbers. This can be as a result of free evolution or as a result of collisions. Importantly, theseprocesses occur on characteristic timescales separated by orders of magnitude for a dilute gas.
To derive the free evolution portion of the Boltzmann equation it is sufficient consider the
motion of a individual points in µ-space, which manifests as a coordinate transformation. Under theaction of an external force F, the point (r, p) moves to (r, p) → (r0 , p0 ) = (r + (p/m) dt, p + Fdt)
over the infinitesimal time window dt. If the external force depends only on position, f (r, p, t) =f (r0 , p0 , t + dt) and the single-particle distribution function at that point can expanded in a Taylor
series∂p+· ∇r + F · ∇p f (r, p, t) = 0∂t m(Collisionless Boltzmann equation) ,(3.6)

where the subscript of the gradient operator indicates differentiation with respect to momentum
or position. Free evolution is therefore analogous to the single-particle dynamics of each point inphase-space.
Collisions however intrinsically involve multiple phase space points, and occur over timescales
where the free space evolution is negligible. During the evolution t → t + dt, the number of atoms
in a cell may decrease as they participate in a scattering process with an atom in a neighboring
cell, and these collisions are referred to as out collisions. The number of atoms in a cell may also
increase as the output of a collision elsewhere in µ-space ends up in the cell, and these collisions
are referred to as in collisions. Higher order than two-body collisions are increasingly unlikely inthe weakly-interacting dilute gas, and it is therefore sufficient to consider only the effect of binary
collisions.
6

For more details, including complete derivations, I typically refer to the seminal texts Refs [95, 96].

45
The cumulant effect of collisions serves only to locally reorganize µ-space with no net loss of
atoms, which can be summarized in the time dependence of the occupation of each cell∂f (r, p, t)= Rin [f ] − Rout [f ].
∂t
coll

(3.7)

Let’s sketch the form of the in and out coefficients7 . Rin [f (r, p)] represents the sum of all collisions within the gas that wind up contributing an atom to the phase space cell containing thepoint (r, p). The rate at which these collisions occur is proportional to the quantum-mechanicalscattering amplitude fˆ(p0 , q0 → p, q), written in such a way as to highlight the incoming atoms
with momentum p0 and q0 and collisional products p and q. For cells which are more occupied thanothers, this process is increasingly likely, and therefore Rin should depend on the two-atom distribution function F (r, p0 , q0 , t) which is the correct measure of the likelihood of finding a particularincoming collisional pair in the cells (r, p0 ) and (r, q0 ). Rin should also reflect the conservation of
total momentum and kinetic energy for each collision. This motivates the expression
Rin −Rout ∝Z

d3 qd3 p0 d3 q 0 δ (3) (p+q−p0 −q0 )δ(E−E 0 )|fˆ(p0 , q0 → p, q)|2 F (r, p0 , q0 , t)−{p, q ↔ p0 , q0 }(3.8)

where the brackets denotes the exchange of coordinates, reflecting the construction of Rout as thetime-reversal of Rin .
In the ultracold limit, fˆ can be replaced by the constant s-wave contribution and can bepulled form the integrals. After collapsing the delta functions, the combined expression isσ̂
Rin − Rout =4πmZd3 q dΩ |p − q|(F (r, p0 , q0 , t) − F (r, p, q, t)),(3.9)

where σ̂ = 8πa20 is the s-wave cross section for identical boson scattering from Sec. 2.2.3.
To reduce Eq. 3.9 further, we follow Boltzmann and assumed that the dilute gas is in a stateof molecular chaos. The incoming atoms are not correlated beyond the single-atom level
F (r, p, q, t)) ≈ f (r, p, t)f (r, q, t),
7

R[f ]d3 rd3 q is the associated local collision rate.

(3.10)

46
which is also required to formulate a closed form expression. If this approximation is not made,the dynamical equation for the two-atom correlation function must be produced which depends
on the three-particle correlation function, etc... This chain of equations is known as the BBGKY
hierarchy [97], and will be addressed further in Chapter 7.
This truncation gives the canonical form of the Boltzmann equation, including the effect ofbinary collisions through the collision integral Icoll that depends on the cross-section for identicalparticle scattering



∂
p
+
· ∇r + F · ∇p f (r, p, t) = Icoll ,
∂t m
Z
σ̂
Icoll =
d3 q dΩ |p − q|(f (r, p0 , t)f (r, q0 , t) − f (r, p, t)f (r, q, t)),4πm(3.11)

which is a nonlinear integro-differential equation for the single-particle distribution. The lefthand side describes the single-particle evolution of the thermal cloud and the right-hand side is
the collision integral which drives the system toward a state of statistical equilibrium at a rate
proportional to the cross-section. For binary elastic collisions, both the center of mass (COM)momentum and the magnitude of the relative momentum are conserved, and the solid angle integral
is over the full range of possible final directions for the relative momentum.
The Boltzmann equation contains the effects of identical particle scattering through thesymmetrized scattering amplitude fˆ. Near Tc , the transition rates Rin and Rout can be enhanced byscattering into µ-space cells that have a significant occupation. Bose enhancement of the scattering
rates occurs also in the more familiar setting of the stimulated output of coherent light in a laser[98]. The relative importance of Bose enhancement can be gauged on a cell by cell basis by thecomparison of f (r, p, t) to unity, or in an average sense by evaluating the phase-space density
Ξ = nex λ3th ,(3.12)

where nex is typically the density of atoms evaluated at the center of the trap or averaged overthe entire trap. The phase space density is the ratio of atoms per unit volume to the number ofavailable states that can be occupied at a given temperature per unit volume. This ratio yields the47average number of atoms per state, and Ξ ≈ 1 provides a ballpark estimate of both the transition
temperature Tc and the importance of Bose enhancement [65]. The formal procedure for including
Bose-enhancement in the Boltzmann equation is described in [96], however, it is clear that it should
appear through the substitution→ fp0 fq0 (1 + fp )(1 + fq ),(3.13)

fp fq → fp fq (1 + fp0 )(1 + fq0 ),(3.14)

fp0 fq0

to reflect enhancement in the output channels. This upgraded form of the Boltzmann equation is
known as a quantum Boltzmann equation or QBE for short.

3.1.2

Consequences of the H-Theorem: The Undamped Breathing Mode

Basic postulates of thermodynamics can be recovered from conclusions about the long-time
behavior of the Boltzmann equation. In particular, the second law of thermodynamics, whichstates that the entropy of an isolated gas never decreases, was proved by Boltzmann in his famous
H-theorem, which I sketch now. The conclusion of the H-theorem leads to a special set of solutionsof the Boltzmann equation of which the breathing mode is a member.
In an isolated gas, consider the behavior of the functional
h(t) ≡

Z

d3 vf (r, v, t) log f (r, v, t),(3.15)

which seems rather random, but is precisely the quantity which vanishes when the system is inequilibrium, and is decreasing for any initial state. To see this, the time derivative can be written
in the following form [95] using properties of the Boltzmann equationσ̂ḣ(t) = −
4

Z

3

d v1

Z

3

d v2

Z


h
i
dΩ |v2 − v1 | fv20 fv10 − fv2 fv1 log(fv10 fv20 − log(fv1 fv2 ) .

(3.16)

Now, the distribution function is generally much less than unity and therefore when fv20 fv10 > fv2 fv1 ,the logarithms satisfy the opposite relation, log(fv20 fv10 ) < log(fv2 fv1 ). Provided the assumption ofmolecular chaos is valid so that the factorization in Eq. 3.10 holds, ḣ(t) < 0 indicates a definitive

48
direction in time (the arrow of time) and vanishes in the long-time limit when the system hasreached equilibrium. The entropy, S, is therefore directly proportional
h=−

S
,
V kb

(3.17)and increases as the system settles towards equilibrium in a volume V . This is the H-theorem.
The necessary condition for Eq. 3.11 to vanish islog(fv20 fv10 ) = log(fv2 fv1 ),(3.18)

log(fv20 ) + log(fv10 ) = log(fv2 ) + log(fv1 ),(3.19)

which has the form of a collisional constraint: the sum before the collision is equal to the postcollision sum. Namely, it restricts log(f ) to a sum of quantities which are collisional invariants{χ1 , χ2 , ...}. This defines a class of possible solutions of the form
log(f (r, v, t)) = χ1 + χ2 + ...

(3.20)For a binary elastic collision, this set is spanned by the kinetic energy, momentum, and any velocity
independent function, givinglog (f (r, v, t)) = a + b · v + cv2 ,(3.21)

where a, b, c are possibly functions of position and/or time.
The familiar Maxwell-Boltzmann distribution of a stationary dilute gas follows from thisexpression for the choice |b| = 0, c = −1/kb T and a = log n (m/2πkb T )3/2 . Generally, thesecoefficients satisfy a set of coupled differential equations [99]∇r b = 0,(3.22)

v 2 ḃ + v · ∇r (c · v) = 0,(3.23)ċ + ∇r a + 2b F = 0,(3.24)ȧ + F · c = 0,(3.25)

49
which can be obtained by inserting the general distribution function in Eq. 3.5 into the collisionlessBoltzmann equation (Eq. 3.11) where the collision integral vanishes by construction. When theforce, F, is due to an isotropic harmonic trapping potential, b(t) executes simple harmonic motion
at frequency 2ω0 , which is the so-called breathing or monopole mode. When b(t) is oscillatory,
what is usually identified as the temperature of the cloud oscillates in time in the absence of heattransfer, which is out of phase with the spatial relaxation and contraction of the density profile.
It is therefore useful to think of a ‘kinetic’ temperature associated with b(t) and a ‘potential’
temperature associated with the density variation. From these definitions, an analogy with aswing-like mechanism can be made as the average of the two temperatures maintains a constantvalue as the potential and kinetic temperatures oscillate out of phase over a single period [24]. This
picture is independent of the amplitude of the oscillation and strength of the binary interactions.
It is therefore of fundamental importance to measure the damping rate of this mode as a violation
indicate a breakdown of the assumptions underlying the Boltzmann equation.

3.1.3

Collective Modes Above Tc .

The collective modes of a 3D isotropic trap with trapping frequency ω0 reflect the underlying
spherical symmetry and the quadratic form of the trapping potential. Any mode of the trap can be
expressed in terms of irreducible spherical tensors, in this case they are multipole modes denotedby the labels (l, m). The l = 0 mode is the monopole mode of the system, and is characterized byan undamped oscillation at 2ω0 . The l = 1 dipole mode is a COM oscillation independent of thecharacter of the interaction and therefore undamped. There are two limiting regimes of oscillation:
the collisionless and hydrodynamic regimes. In the collisionless regime, the average duration
between subsequent binary collisions is much longer than the trap period. This is summarized
by the product of the average collision rate γcoll = n(0)vth σ̂/2 with the trap period ω0 /γcoll  1.
pThe thermal velocity is given by vth = 8kb T /πm. In the hydrodynamic limit, an average

atom undergoes many binary collisions in a trap period ω0 /γcoll  1. The l = 2 quadrupole mode√
oscillates at 2ω0 in the collisionless regime and 2ω0 in the hydrodynamic regime. It is collisionally50
damped in the hydrodynamic crossover between the two extremes. In the following, I discuss two
common methods for deriving the collective mode spectrum including damping effects in the regimeof linear response.

3.1.3.1

Moment method

By taking moments of the Boltzmann equation (Eq. 3.11), a set of coupled equations of
motion can be derived that describe the dynamical evolution of various collective modes [95]. For
an arbitrary quantity χ(r, v), the equation of motion is


dhχi
F
− hv · ∇r i −· ∇v χ = hχIcoll i,dtm(3.26)

where Icoll is shorthand for the collision integral in the Boltzmann equation and the average quantity
or moment is defined as
1
hχi =
N

Z

d3 rd3 vf (r, v, t)χ(r, v), .

The collisional average of this quantity is given byZ
1hχIcoll i =d3 rd3 v1 ∆χIcoll [f ],4N(3.27)(3.28)

where ∆χ = χ1 + χ2 − χ10 − χ20 with χi = χ(r, vi ).
From Eq. (3.26), it is straightforward to form a closed set of coupled equations for thequadratic moments, recalling that moments of the form in Eq. (3.21) have no collisional contribution. For an isotropic trap, taking the moment hr2 i yields a closed set of equations describing theevolution of the monopole mode:
dhr2 i− 2hv · ri = 0,dt
dhv · ri
− hv2 i + ω02 hr2 i = 0,dt
dhv2 i+ 2ω02 hv · ri = 0.
dt

(3.29)

Solutions to Eqn. (3.29), where the moments oscillate around their equilibrium values as eiωt ,confirm Boltzmann’s general result as previously discussed in 3.1.2 for an undamped monopolemode at oscillation frequency 2ω0 .

51
One can also take (l = 2, m = 0) quadrupole moments of Eq. (3.26) to derive a closed setof six coupled equations for an isotropic trap. However, for the quadrupole mode the moment,
h3vz2 − v2 i cannot be expressed in the form of Eq. (3.21), which leads to a nontrivial collisional
contribution h(3vz2 − v2 )Icoll i in its equation of motion. For small amplitude oscillations about
equilibrium, collisional contributions of this form can be approximately rewritten in the relaxation
time approximation [95, 100, 101] which uses the Chapman-Enskog procedure ashχIcoll i = −

χ
τcoll

,

(3.30)

where the relaxation time τcoll is given byτcoll =

5,.
4γcoll(3.31)

As with the monopole mode, the set of coupled equations for the quadrupole mode can also be√
solved algebraically, giving frequencies of 2ω0 and 2ω0 in the collisionless and hydrodynamic
regimes respectively. However, in the hydrodynamic crossover the quadrupole mode damps due tothe nonvanishing collisional contribution.

3.1.3.2

Scaling Ansatz Method

The scaling ansatz method has been used for Bose gases above and below the transition
temperature [102, 103, 104, 105, 106, 29, 107, 108, 109]. In the nondegenerate regime, the ansatz
method has been used to evaluate the effects of the mean field interaction [107, 108] and to estimate
the effects of anharmonic corrections to the trapping potential on the collective mode frequencies
and damping rates [109]. Here this method is outlined.
When the thermal cloud is in statistical equilibrium under harmonic confinement, the Boltzmann equation for the equilibrium distribution f0 (r, v) is just the Poisson bracket:
!
3
X
∂f0
2 ∂f0
vi
− ωi ri
= 0,
∂ri
∂vi

(3.32)iwhere the subscript denotes components of a Cartesian vector in 3D and the possibly differing
frequency components in each direction. Moments of the equilibrium distribution are the same as

52
those obtained in the previous Subsection when the temporal derivatives are set to zero.
To describe time-dependent collective oscillations, a scaling ansatz on the form of the
distribution function f (r, v, t) can be made:f (r, v, t) = Γf0 (R(r, t), V(r, v, t)).

(3.33)

This ansatz utilizes the symmetry of the problem and the nature of the collective modes in theforms of the renormalized position, R(r, t), and velocity, V(r, v, t), with an additional factor Γ toenforce the normalization.
For the dipole mode, the form of the ansatz is Ri = ri − ηi (t) and Vi = vi − η̇i (t) with Γ equal
to unity. The details of the motion are contained in the time-dependent vector of free parameters,~η . Such a scaling clearly describes a translation of the COM of the equilibrium cloud.
For the monopole and quadrupole mode, the ansatz mimics the form of Eq. 3.21:
Ri =
Vi =
Γ =

ri
,
bi (t)
1
1/2

θiḃi (t)vi −ribi (t)!

,

1
1/2 ),j (bj (t)θj (t)(3.34)

Q3

where there are two time-dependent vectors of free parameters θ~ and ~b. Such a scaling describes
the stretching and compression of the equilibrium cloud in µ-space along with a space-dependentand time-dependent translation in the local velocity, while maintaining a stationary COM.
Substituting Eq. (3.33) into Eq. (3.11) and following [107, 108], a Newton-like set of equationsof motion for the free parameters can be derived
θi
=0
bi

ḃi
1 
θ̇i + 2 θi = −θi − θ̄ ,bi
τcollb̈i + ωi2 bi − ωi2where the quantity θ̄ =Pi θi /3 is the average temperature.

(3.35)
To obtain information about thecollective oscillations, the free parameters are linearized about the equilibrium position for small

53
U0

U 0 + mgzsag

z Positionz Position

Figure 3.2: In the modified TOP trap, the cloud shifts under the pull of gravity away from the
midpoint between the coils. The trap can be made isotropic to harmonic order with the cost ofanisotropic anharmonicities. (Adapted from [63].)

amplitude oscillation, bi ≈ 1 and θi ≈ 1, and assumed to have time-dependence of the form eiωt .
For a harmonic trap, the results obtained agree with the moment method. The advantage of thescaling ansatz method is that it can produce closed form expressions when anharmonic corrections
to the trapping potential are included, which will shortly be found necessary.

3.2

The Breathing Mode Experiment
Although this is a theory thesis, it is useful to give a theorists account of the experimental

setup used to produce extremely isotropic harmonic confinement because the imperfections of thistrap play a dominant role in understanding the observed damping of the breathing mode oscillation.
The trapped state of 87 Rb is the maximally-stretched spin state discussed in Sec. 2.2.4, and to adecent approximation, we take as ≈ at ≈ a0 ≈ 100 Bohr radii in the cross section formulas.
To produce isotropic confinement, the quadrupole field strength of the TOP trap (see Sec. 2.1.1.1)
is decreased, the effects of gravity become important as the atoms in the trap sag away from thehigh-field limit trap minimum. The extent of the sag is characterized by the dimensionless quantity √
Λ = mg/µBz and shifted trap minimum z0 = − (B0 Λ) / Bz 1 − Λ2 . The potential expanded

54
about the sag position z0 is√√µBz22 )r 2 1 − Λ2
Usag (ρ, z) = µB0 1 − Λ2 + 16B
(1
+
Λ
0
2

2 3/2 .
z 2
+ µB
2B0 z (1 − Λ )The ratio of the trap frequencies is
ωz
=
ωρ

r

8

1 − Λ2
1 + Λ2(3.36)(3.37)

with Λ = 7/9 giving an isotropic trapping potential. For further details of the modified TOP trap
used in the JILA experiment, we refer the reader to Ref. [63].
Utilizing gravity to symmetrize the trap results in trapping frequencies on the order of 10 Hz.
In such a loose trap anharmonic corrections become important and must be included in calculations.
The potential used in calculations involving the anharmonic corrections has the form
Uah (ρ, z) =

m
2


mβ 2
3
ωz2 z 2 + ωρ2 ρ2 + mα
3 z + 3 ρ zmδ 2 2m 4
4+ mκ
4 z + 4 ρ z + 4 ρ ,(3.38)

and for general formulation of the cubic and quartic terms see Ref. [63]. The values of the anharmonic coefficients in SI units used in this chapter were quoted from the JILA experiment (Ref. [63])as α = 5.58 × 106 s−2 m−1 , β = 4.17 × 106 s−2 m−1 , κ = 5.94 × 109 s−2 m−2 , δ = 1.716 × 109 s−2 m−2 ,and  = 2.469 × 108 s−2 m−2 .
The monopole mode is driven experimentally by applying a sinusoidal variation in the trapping frequency over four periods of monopole oscillation. The strength of the TOP field is changed
along with a vertical bias which fixes the minimum of the potential (see Ref. [63]). The net result
is a roughly 25% increase in the mean cloud size. The main results of the experiment [24] areshown in Fig 3.3 for the damping rates, where the data was taken over a range of collision rates attemperatures 2Tc < T < 3Tc to avoid mean-field effects and to ensure that the gas is close to theregime described by the original formulation of the Boltzmann equation. The monopole damping
rate in Fig. 3.3 is small compared to the quadrupole rate and the trapping frequency but nevertheless nonzero, and it should re-emphasized that any damping is anomalous in light of Boltzmann’s55

Figure 3.3: Experimental results from Ref. [24] for the monopole and quadrupole damping rates
over a range of collision rates. The monopole damping is nonzero and therefore anomalous, but
the quadrupole damping agrees with well-known results from Ref. [100].

prediction and therefore potentially relevant to the underpinnings of kinetic theory. The quadruple
damping rate is nonzero, agrees with well-known expressions, and is therefore not focused on inthis analysis.

3.3

Theoretical Analysis
Slight anisotropies in the harmonic confinement and anharmonic corrections to the trapping

potential Eq. (3.38) present possible sources of damping. Modeling these imperfections is discussed
in this section using trapping data from the JILA experiment.

3.3.0.3

Damping due to trap anisotropies

Anisotropies in the confining potential lead to coupling between the monopole and quadrupolemodes, which are damped in the hydrodynamic crossover regime. The moment method can account56ν0Γ(s−1 )

0.08

λ



9.020 1.0028 0.0000
9.044 1.0019 −0.0014

9.029 0.9989

0.06

9.035 0.9998
9.065 0.9990
9.070 0.9996

0.00020.00020.00030.0002

0.04
0.02

10

-2

-1

10

10 1

1

1

γcoll (s

−1

)

10

10

2

Figure 3.4: Monopole damping rates versus collision rate for a set of frequencies from the JILA
experiment with various anisotropy parameters and central frequencies ν0 . The legend gives thelevel of anisotropy, where λ, , and ν0 = ω0 /2π correspond to rewriting the harmonic confining
potential as in Eq. (3.39).

for this coupling in a straightforward way given a potential of the form
U (x, y, z) =mω02
(1 + )x2 + (1 − )y 2 + λ2 z 2 ,2(3.39)√√
where λ = ωz /ω0 , ω0 1 +  = ωx , and ω0 1 −  = ωy .

Starting with hr2 i, the moment method yields a set of nine coupled equations for the monopoleand quadrupole modes (see Ref. [110]). In Fig. 3.4, frequency measurements from the JILA experiment with various anisotropies were used to estimate the damping rate for small oscillations. For
nonzero  and λ different from unity, the monopole couples to multiple quadrupole modes and theshape of the curve departs from that expected for the typical transition region shape when  = 0.
The monopole damping measured in the JILA experiment is on the order of 0.1 s−1 or larger.
Therefore, the trap can roughly be treated as isotropic with λ = 1 and  = 0 for the lowest
four curves in Fig. 3.4. Thus, the observed monopole damping must then be due to some othermechanism that is present even when the harmonic piece of the trapping potential is virtuallyisotropic.

57
3.3.0.4

Damping due to anharmonic corrections

Computing the monopole damping using the moment method for nonzero anharmonic coefficients generates an infinite set of coupled equations for successively higher order moments and so
is not a viable method. However, the scaling ansatz method, which takes as an input momentsof f0 , can provide an estimate for the effects of the anharmonic coefficients which enter throughhigher order moments of f0 . The usefulness of such an approach hinges on being able to neglect
deformations of the equilibrium distribution due to the anharmonic corrections.
Using the scaling ansatz, Eq. (3.35) can be rewritten in a more general form for an arbitrary
trapping force, Fi :
b̈i −

θi 2
1
hv i0 − hFi (bj rj )ri i0 = 0,
bi i
m

1 
ḃi
θi − θ̄ ,θ˙i + 2 θi = −
bi
τcoll

(3.40)

where the 0 subscript indicates a moment as in Eq. (3.27) over the equilibrium distribution f0 ,and bj rj is a short hand notation for the set {bx rx , by ry , bz rz }. From Eq. (3.40), the procedure forincluding even order corrections to the trapping potential is straightforward. However, to include
odd order corrections, the equilibrium distribution, f0 , must be deformed. A simple deformation is

f0ah (r, v) =





mβ
mα 3
2
2
1−
x +y z−z f0 (r, v),
3kb T
3kb T

(3.41)

which is a perturbative expansion to first order in the odd order anharmonic corrections from
Eq. 3.38. The deformation does not change the overall normalization nor the value of the evenorder moments, and the added terms are collisional invariants of the form Eq. (3.21).
In Sec. 6.5, the validity of this deformation compared to numerical and experimental results
for the damping of the monopole mode is addressed. However, in the remainder of this section, thegeneral solution of Eq. 3.40 is discussed for small amplitude oscillations, following the derivation in
Ref. [109] and noting that the derivation is the same for f0ah .
Equation (3.40) is solved in both the collisionless and hydrodynamic regimes. Making the

58
assumption that the monopole damping is much less than the collective mode frequency (ΓM  ωM )leads to a generally applicable formula for the damping [111]Γ≈

2 − ω2τ̃coll ωCLHD
2 τ̃ 2 .
2 1 + ωCLcoll

(3.42)In the JILA experiment, ΓM /ωM is on the order of 10−3 to 10−2 .
In the collisionless regime, τcoll → ∞, which gives a power law relation between the two scaling
parameters: θi = b12 . Using this power law relation, Eq. (3.40) collapses into three differentialiequations for each component of the parameter bi . For small amplitude oscillations bi ≈ 1, and thesubstitution ηi = bi − 1 linearizes the three differential equations for each component of ηi :
η̈i +3 hUi ri i0
1 X hUij ri rj i0η+ηj = 0,im hri2 i0mhri2 i0(3.43)jwhere Ui = ∂U/∂ri and Uij = ∂ 2 U/∂ri ∂rj . Equation (3.43) can be treated with matrix methods
by defining
3 hri Ui i0δij ,m hri2 i0
1 hri rj Uij i0
,
m
ri2

ACL
=
ij
CL
Bij
=

(3.44)and solving the secular equation ACL + B CL − ω 2 I = 0 for the collective mode frequencies.
In the hydrodynamic regime, τcoll → 0, and thus the system is in local equilibrium everywhere.
Furthermore, the components of the θi ’s must be equal to their average, θi = θ̄. From the normalQ 2/3
ization of the ansatz then follows a power law relation between the parameters: θi = 1/ j bj .

This implies that Eq. (3.40) collapses to a set of three differential equations for each component of
the parameter bi that couple when the power law relation for θi is substituted. Making the small

amplitude assumption leads to the following equation:
1 hUi ri i0  5
2X 
1 X hUij ri rj i0
η¨i +
ηi +
ηj +
= 0.
2
2i
m hri i0
3
3
m
hr
0
i
j

(3.45)

i6=j

As in the collisionless case, we define matrices 5 hUi2ri i0
3m hri i0
HD
Aij
=

 2 hUj ri i0
3m hri2 i0

HD
Bij

=

CL
Bij
,



i=j 

i 6= j ,(3.46)(3.47)

59
and solve the secular equation AHD + B HD − ω 2 I = 0 for the collective mode frequencies.
Solving the secular equation for the hydrodynamic and collisionless regimes yields frequencies
for the monopole mode and the (l=2,m=0,2) quadrupole modes. The frequencies have simple
expressions in terms of the A and B matrices
2
ωM
2
ωQ0

p
δ 2 − ∆,p= δ − δ 2 − ∆,
=δ+

2
ωQ2
= Axx − Axy + Bxx − Bxy ,(3.48)whereδ = (Axx + Axy + Azz + Bxx + Bxy + Bzz ) /2,∆ = (Axx + Axy + Bxx + Bxy ) (Azz + Bzz )− 2 (Axz + Bxz ) (Azx + Bzx ) .

(3.49)

The calculated mode frequencies have temperature dependence through higher-order moments coming from the anharmonic corrections. For a given temperature, the scaling ansatz theory
gives the shift in the frequency ∆ωM and damping Γ from that anticipated by the harmonic limit
0 = 2ω . The damping Γ is from collisional relaxation in the system.
ωM00 − ω P , measures the gap between the result in the
The shift in the frequency ∆ωM = ωMM
0 and the peak frequency component ω P present in the cloud. Centered about
harmonic limit ωMM

the peak component is a spread of frequencies whose width δωM is not predicted from the scaling
ansatz method. To obtain full information about the width, one can perform a numerical simulation
as discussed in Sec. 3.4. The width provides a measure of the dephasing-induced damping Γdephase ,which is responsible for the appearance of actual relaxation in the system and is separate from thecollisional damping estimate from the scaling ansatz theory. In Ref. [109] it was argued that fora given width δωM , the dephasing time δt is given approximately by δωM δt ≈ 2π, and thereforeΓdephase ≈ 2π/δt ≈ δωM . For an anisotropic trap, the frequency shift was used in Ref. [109] in lieu
of the actual width to obtain dephasing-induced damping rates which agreed with experimental

60
results within an order of magnitude. This approximation is reasonable only in the collisionlessregime where the spread of frequencies in the cloud begins with the collective oscillation of the
0 . As will be made precise in Sec. 6.5, when the collision rate increases,
lowest energy atoms at ωM
P and decreases proportionately with ΓδωM narrows about the peak value ωMdephase . The shift∆ωM becomes then a poor estimate for the width δωM . As Γdephase decreases a transition regime
exists where dephasing effects in the cloud are destroyed through collisions, and collisional damping
becomes the dominant effect. Such an effect can account for anomalous damping of the monopolemode in regimes where ωM τcoll  1, whereas collisional damping dominates when ωM τcoll ≈ 1. We
refer to the regime where δωM τcoll ≈ 1 as the dephasing crossover, and in Sec. 6.5 this crossover isstudied in detail with the aid of numerical simulation.

3.4

Numerical Methods
In this section the numerical algorithm used to simulate the dynamical evolution of thethermal cloud is described, with further details provided in Appendix A. Numerical simulationof the system isolates the roles of various damping mechanisms and permits quantification of thedephasing effects in the cloud. The thermal cloud algorithm from Ref. [112] is adopted, working
in the quantum collisional regime [113] where two-body collisions are s-wave. However, the manyparticle statistics are still classical and well-described by the Boltzmann equation. The simulationconsists of a swarm of tracer particles that act as a coarse-grained distribution function [114]:
Ntp

Nth 3 Xf (r, p, t) ≈hδ(r − ri )δ(p − pi ),Ntp(3.50)iwhere Nth /Ntp is a weighting factor. The sum is over the entire set of tracer particles, where each
is uniquely described by their position and momentum (ri , pi ). The tracer particles first undergo
collisionless evolution via a second order sympletic integrator [115, 116]. Following free evolution,
the tracer particles are binned in space and tested for collisions.

61
3.4.1

Collisions

The collision algorithm, which is a procedure for Monte Carlo sampling the collision integral[117], follows that described in Ref. [112]. After binning the particles in space, random pairs ineach bin are selected and their collision probability is calculated. If a collision is successful, theparticle velocities are updated according to the differential cross-section for an s-wave collision.
For each set of simulation parameters we check that the equilibrium collision rate averaged
over the entire cloud matches with the analytic result:
γeq = Nthσ̂ω03 m,2π 2 kb T(3.51)

to < 2%.

3.4.2

Drive mechanism

To mimic the experimental monopole drive scheme, the trap frequency, ω0 , is modulated overfour periods of monopole oscillation ω0 (1 + A sin(2ω0 t)) t ≤ 4π/ω0 ω0 (t) =, ω0t > 4π/ω0 (3.52)

where A is a unitless measure of the strength of the drive. During the drive, the anharmoniccorrections are neglected, and are switched on when the next oscillation minimum occurs after thedrive is turned off. Figure 3.5 illustrates the increase of the monopole amplitude during the drive
and the effect of turning on the anharmonic shifts on the total cloud energy. For A = 0.15 anda starting temperature of 152nK, the percent increase in the total energy when the anharmoniccorrections are switched on is on the order of 0.001%. Neglecting the anharmonic terms during
the drive effectively captures the experimental scenario without including the modulation of thebias, quadrupole, and all of the shimming fields. Results are presented then over a range of drive
strengths that produce an increase in the mean cloud size at the end of the drive phase on theorder with that experimentally measured. Driving the cloud ideally emphasizes the impact of theanharmonic shifts on the monopole mode and eliminates any residual driving of the quadrupole

% Error

FWHM(µm)

62

(a)

6
4
2

(b)

1
10

-1

10

-2

0

0.1

0.3

0.2

0.4

time(s)Figure 3.5: Driving the monopole moment for A = 0.15 and initial temperature of 152nK over four
periods of oscillation. (a) FWHM of the monopole moment versus holding time. The grey area
indicates the time in between the end of the drive and switching on the anharmonic terms in thetrapping potential. (b) The % change in the total energy of the cloud between a trapping potentialthat is purely harmonic and the same potential with anharmonic terms switched on. The verticalline indicates the point at which the anharmonic terms are added to the trapping potential when
the % increase of the total energy is on the order of 0.001%.

and dipole modes which are not the subject of the current study. For the full details of the drive,refer to Ref. [63].
During the drive there is also a shift in the average temperature and full width at half
maximum (FWHM) of the oscillation due to an increase in the energy of the cloud from the work
done on it by pumping of the trap.

3.5

Results
We are now in a position to compare and contrast the results of the JILA experiment against

the theoretical model and numerical simulation. Here, only data with the monopole drive is considered. Moreover, our analysis is focused on experimental data that is not dominated by effects
due to trap anisotropy, as this is well-understood [100, 101]. Characterizing the sensitivity of the63monopole mode to anharmonic corrections around the dephasing crossover is the main result of thischapter. This sensitivity stands as a general issue for undamped, nonequilibrium collective modes,
such as the monopole oscillation. The quadrupole modes must also damp around the dephasing
crossover; however, these modes are not explored further as they do not fall into the category ofundamped, nonequilibrium modes.
In the JILA experiment, the monopole data was taken for a range of atom numbers and temperatures. The small-cloud data (N ≈ 104 atoms) which lies in the collisionless regime is analyzed
first. Finally the large-cloud data (N ≈ 105 − 106 atoms) which lies between the collisionless andhydrodynamic regimes is analyzed.

3.5.1

Collisionless regime

In the collisionless regime, the many-body dynamics of the thermal cloud are dictated by thesingle-particle trajectories. Here, the system behaves in analogy with a simple pendulum. For small
amplitude oscillations the pendulum executes simple-harmonic motion. However, as the amplitude
grows, the small-angle approximation breaks down and anharmonic corrections become important.
The pendulum traces out a repeating trajectory but with an energy-dependent period.
The procedure for replacing anharmonic corrections by an energy-dependent harmonic potential in 1D is well-known [111]. The renormalized trapping potential for small amplitude motion
along the z-axis is:
U (0, 0, z) ≈mω̃z2 (E) 2z ,2ω̃z = ωz (1 + ξz E),
ξz = −

5α2
3κ
+
.
6
3
6m ω0
4m2 ω04(3.53)In Fig. 3.6a, atoms in the anharmonic trap have been binned according to their energy and thecollective motion in each bin averaged to obtain the monopole period in each bin. For the first
several bins the period of collective oscillation increases linearly with the bin energy. For higher
energy bins the scaling is still monotonic; however, the deviation increases as the difference in the

64
(a)

1.02

T/T0M

1.01
1
0

2

4

6

Bin energy (kb T )

0.2

8
(b)

P ( ω)
0.1

0

0.01

0

0.02

0
1 − ω/ωM

0.03

Figure 3.6: Resolving the energy dependence of the monopole moment for a drive strength of
A = 0.15 (9 ± 1.35Hz) and an initial temperature of 152nK. (a) After binning the tracer particles
into energy bins, each bin oscillates as an independent monopole. Here, the ratio of the average
0 along with theperiod of the monopole in each bin and the zero temperature result, T0M = 2π/ωMstandard deviation (grey region), is plotted versus the energy of each bin in units of thermal energy.
(b) The statistical weight, P (ω), of each frequency component in the cloud. The grey region is thetheoretical prediction for the shift of the monopole frequency from L (150nK) to R (180nK).

linear correction between the different axes becomes more apparent. At sufficiently high binningenergy, the linear correction Eq. (3.53) breaks down as higher-order terms become important, andthe atom number in each bin decreases, spoiling the appearance of an undamped collective mode.
In the absence of collisions, the population of atoms in each bin is static over the entire simulation. The statistical weight of each frequency component, as shown in the normalized histogramof Fig. 3.6b, is then also static. The determining characteristics of the frequency distribution are
0 = 2ω . The theoretical prediction of ∆ω from the scaling
its width and shift of the peak from ωM0M

ansatz using f0ah is for the shift of the peak (grey region of Fig. 3.6b) and agrees with the numericalresult. However, the width, denoted δωM , of the frequency distribution determines the relative importance of dephasing effects. If the frequency distribution is shifted and sharply peaked, dephasing
induced damping is minimal, mimicking the large amplitude oscillation of a single pendulum. Oth-

65
65nK
125nK

Γ (s − 1 )

3

( a)

152nK

2

1

 
Damping and Frequency Shifts s−1

.05

.10

.15

Drive strength

.20

.25

( b)

2
1
0

Γdephase from simulations-1

Γdephase from fit Eq. (32)∆ωM from scaling ansatz with f0-2∆ωM from scaling ansatz with f0ah

400

T (nK)

800

1200

Figure 3.7:(a) Damping of the monopole over a range of drive strengths A ={0.05, 0.1, 0.15, 0.20, 0.25} and temperatures in the collisionless regime. The grey regions are thedamping results along with the uncertainty from the JILA experiment [24] for (from bottom to top)
65nK, 125nK, and 152nK. (b) The colored lines are the theoretical predictions for ∆ωM (T ) using
f0ah (blue) and using f0 (red) for the scaling ansatz versus the settled temperature of the cloud. The
points are the damping results from (a) for all simulated drive strengths and temperatures in thecollisionless regime. The solid black line is a fit to the numerical damping results using Eq. (3.54).

erwise, if the distribution is broad, as in Fig. 3.6a, the net sum of many ‘pendulums’ oscillating at
shifted frequencies is dephasing induced damping through interference. The scaling ansatz theory
also gives a prediction for the damping due to collisional relaxation, and in the collisionless regime,
this prediction is vanishingly small as expected (see Fig. 3.8).
We now compare directly to the small-cloud data from the JILA experiment. Starting with

66
the temperature and atom number quoted in the experiment, the drive strength is varied and theamplitude of the resulting oscillation is fit to a simple exponential decay to compare directly tothe fitting function used in the experiment. As shown in Fig. 3.7a, a drive strength of A = 0.15
matches the experimental data, suggesting that the anomalous damping seen in the experiment
for the small-cloud data is mainly due to dephasing effects. Such effects depend on the settled
temperature of the cloud as shown in Fig. 3.7b.
Although the width of the frequency distribution determines the strength of the dephasing
induced damping, in the collisionless regime where the distribution is broad, the frequency shift∆ωM and the width scale proportionately (see discussion at the end of Section 3.3.2) and can beused as a fitting function, looking then for a scaling fit of the form
Γdephase (T ) = ∆ωM (ζT ),(3.54)

where ζ is a free parameter independent of the temperature. For the experimental trap parameters,
ζ = 0.4 from a fit to the data shown in Fig. 3.7b.
The scaling ansatz result using f0 instead of f0ah predicts a decrease of the period with increasing bin energy, which disagrees directly with the numerical result and Eq. (3.53). Additionally,
from the red and blue lines in Fig. 3.7b, as the settled temperature increases the deformed gaussianpredicts that the peak frequency decreases (∆ωM (T ) > 0).

3.5.2

Crossover regime

The collisionless regime picture of a collection of uncoupled monopole modes oscillating at
shifted frequencies begins to break down when the dephasing period τdph = 2π/δωM exceeds τcoll .
This is typically in a different collisional regime than the hydrodynamic crossover where ω0 τcoll ≈ 1,
and an average atom suffers a collision on a timescale faster than the oscillation period. Therefore,
as the collision rate increases there are two important regimes: the dephasing crossover and the67Γ (s − 1 )

0.4
0.3
0.2
0.1
10-3

10-2

10-1

1
101
1 )
γcoll (s−−1

102

103

Figure 3.8: Damping rate versus the equilibrium collision rate. The triangle data points are a
result of numerical particle simulations. The blue curves are scaling ansatz predictions using f0ah
for collisional damping of the monopole mode for temperatures 60nK, 80nK, and 100nK frombottom to top, respectively.

hydrodynamic crossover.
δωM τcoll ≈ 1
ω0 τcoll ≈ 1(Dephasing Crossover),(3.55)(Hydrodynamic Crossover).

(3.56)In the remainder of this subsection an individual experimental run from the large-cloud data set at
60.6nK with 3.463 × 105 atoms with measured trap frequencies (fx = 9.036Hz, fy = 9.034Hz, fz =9.034Hz) is analyzed. From the criteria Eq. (3.56) can be obtained crude estimates for the dephasing
crossover γcoll ≈ 10−1 − 1 s−1 and the hydrodynamic crossover γcoll ≈ 102 s−1 . This data set
corresponds to the green curve in Figure 1 for a quoted collision rate γcoll = 8.88 s−1 which lies
between the two regimes. The maximum damping rate due to anisotropies from Figure 1 is on theorder of 10−3 s−1 compared to the experimentally measured rate 0.14 ± 0.02 s−1 . Therefore, theharmonic part of the trapping potential is treated as isotropic.
To quantify the effect of the anharmonic corrections in the crossover regime, numerical simulations with A = 0.05 over a range of collision rates using the trapping frequency ν0 = 9.035Hz andinitial temperature 60nK were performed. Figure 3.8 contains the results along with the prediction

6801 − ω/ωM

0.005

0.003
γcoll (s−1 )

113
11.3
0.011

0.001
0

2

4

6

8

Bin energy (kb T )Figure 3.9: Resolving the energy dependence of the monopole moment for A = 0.05 and initial
temperature of 60.6nK for a range of total atom numbers. Each data point is the peak componentof the Fourier transform of the normalized collective oscillation in the bin. From bottom to top
the curves represent frequency spreads in the collisionless, dephasing crossover, and hydrodynamic
crossover respectively.

for the collisional damping from the scaling ansatz theory using f0ah . The scaling ansatz predictionis plotted for several temperatures, illustrating the effect of the increase in temperature of the cloudpost-drive on the damping.
The estimates for the hydrodynamic and dephasing crossover regimes also agree with thequalitative structure of Fig. 3.8. The dephasing crossover is marked by a decrease in the dephasing
induced damping; whereas the hydrodynamic crossover is marked by an increase in the collisional
damping. The region in between the two crossovers is characterized by a local minimum in thedamping rate. Comparing the 65nK damping rate from Fig. 3.7 with the 60.6nK damping ratefrom this subsection draws experimental support for this result. That the collisionless regime
is characterized by higher damping than the dephasing crossover is a common feature of all of
the large-cloud data compared for similar temperatures to the small data [24]. The intermediate
regime between dephasing and hydrodynamic crossovers then provides an experimental window
where damping from trap anharmonicities can be minimized.

69
Whereas the physics of dephasing in the collisionless regime is analogous to the uncoupledoscillation of a collection of pendulums with different oscillation energies, in the dephasing crossoverthe pendulums begin to couple, and, as illustrated in Fig. 3.9, the width of the spread in frequency
components narrows. The coupling effectively synchronizes the different pendulums, which leads toa local minima in the damping rate as seen in Fig. 3.8. As the coupling increases, the system nears
the hydrodynamic crossover, and the spread in the frequencies is minimal. The damping is thendue to the appearance of a temperature dependent anisotropic trap, where the monopole mode
damps naturally through coupling with the quadrupole modes whose moments are not collisionallyinvariant.

3.6

Conclusion
The Boltzmann equation is based on the assumption that the chain of correlations describing

the dilute gas can be broken through the assumption of molecular chaos. The resultant theory
has a special class of undamped oscillations, which includes the breathing mode in an 3D isotropic
trap. In the JILA experiment [24], this prediction was tested for the first time, finding small but
nonetheless anomalous damping rates. These rates can be explained by numerical and analytic
techniques for solving and simulating the Boltzmann equation when imperfections in the trap
beyond harmonic order are included, and therefore the assumptions underlying the Boltzmann
equation are not falsified by the experiment.
It is an open question what the damping contributions to the breathing mode are frombeyond-Boltzmann effects which include higher order correlation dynamics beyond single-atomlevel. These effects are evidently swamped by damping due to anharmonicities in the JILA experiment. In the next chapter, the theory of the interacting Bose gas is developed under thesemi-classical approximation and compared to the data taken from the same experimental apparatus. The Boltzmann equation will be seen to be a limiting case of the more general theory beginningfrom the many-body Hamiltonian.

Chapter 4

Collapse and Revival of the Breathing Mode of a Degenerate Bose gas in an
Extremely Isotropic Harmonic Trap

This chapter contains the results from a joint theory/experimental collaboration in JILA
published in Ref. [61], using the same apparatus as described in Sec. 3.2 and Ref. [24].

As the temperature approaches and is reduced below the critical temperature for BoseEinstein condensation, Boltzmann’s prediction for the breathing mode is stretched beyond therange of validity. This is due to the presence of density dependent mean-field effects, which arise
when the de Broglie wavelength becomes comparable to the inter particle spacing, signaling theregime of quantum degeneracy. This is also due to the addition of Bose enhancement in the transition rates of the system as the identical particle symmetry must be accounted for. And, importantly,
it is due to the formation of a macroscopically occupied condensate, which introduces a gap to thelow-energy spectrum of excitations characteristic of the emergence of superfluidity and establishes
a coherent phase within the system. In the semi-classical limit, where the noncondensed portion
can be thought of as the thermal cloud of particle-like excitations, the breathing mode can be√visualized as the oscillation of a condensate at a frequency 5ω0 coupled to a thermal component
oscillating at the classical result of 2ω0 , and this is the subject of this chapter.
It is well-understood that the energy and atom number exchange of these components introduces nonzero damping rates, however there is another intriguing possibility unique to “twocomponent” systems composed of identical atoms which originates from the study of liquid helium.

71
In supercooled helium II, a change in the pressure leads to density fluctuations traveling with asound velocity; however, changes in temperature also lead to fluctuations in entropy propagating with a ‘sound’ velocity [118]. To distinguish these diffierent propagation modes, the density
fluctuations are label first sound or ordinary sound and are described by the superfluid and normal component (thermal cloud) compressing and rarefying in-phase with each other, mimicking
familiar pressure waves. The entropy fluctuations are named second sound and are associatedhowever with out-of-phase oscillations of the superfluid and normal components at fixed density.
Spatially, this appears as a swapping of components superfluid→normal→superfluid→... at a fixedlocation in space. Importantly, these sound modes result from the bulk coupling of the condensed
and noncondensed portions.
It is an intriguing question to what extent these in-phase and out-of-phase oscillations can beobserved in degenerate dilute ultracold Bose gases. One obstacle is the difference in the collectivemodes for a trapped system compared to homogenous systems, and the second difference is ordersof magnitude difference in diluteness of the two systems. It might be expected then that ultracold
dilute gases display some collisionless analogue of the in-phase and out-of-phase oscillations, andto understand this question requires an account of the recent history of collective mode studies fordilute ultracold Bose gases below the transition temperature.
Experiments probing the collective modes of ultracold gases were carried out shortly after
the demonstration of Bose-Einstein condensation in a dilute atomic vapor [69, 119]. In earlyexperiments at JILA [19] and MIT [20], the low-lying quadrupole modes of a nearly pure BoseEinstein condensate (BEC) were excited, and the observed oscillation frequencies showed good
agreement with the Bogoliubov spectrum [120, 121]. Experiments were then conducted over arange of temperatures below the critical point, and temperature-dependent shifts in the oscillation
frequencies and damping rates were observed [21, 6].
One mode in particular, the m = 0 quadrupole mode reproduced in Fig. 4.1 from Ref. [21], defied early attempts at explanation. At temperatures nearing the transition point where a significant
portion of the cloud is in the noncondensed component, the frequency unexpectedly shifted upwards

72

2.1

ω/ωr

2.0
1.9
1.8
0.4

0.6
0.8
T /Tc

Figure 4.1: Temperature dependent m = 0 quadrupole mode spectrum from Ref. [21] normalized
to the radial trapping frequency ωr . The anomalous upward shift at T > 0.7Tc differs by 10 − 20%from zero-temperature analytic result indicated by the lower dashed line.

away from the zero-temperature result. Further exploration of collective-mode behavior at finite
temperatures yielded additional puzzling temperature-dependent shifts [22, 23]. The resolution ofthese difficulties required the inclusion of the dynamics of the noncondensate in analogy with helium
II. Using a semi-classical coupled-modes model, Stoof, Bijlsma, and Al Khawaja [29, 122] described
the coupled dynamics of the condensate and noncondensate in terms of in-phase and out-of-phaseeigenmodes, which were exactly the collisionless analogs of first and second sound hydrodynamic
modes [123, 124, 102]. They concluded that the anomalous behavior found in Ref. [21] was theresult of simultaneous excitation of both eigenmodes of the system. Numerical simulation of theZaremba-Nikuni-Griffin (ZNG) equations by Jackson and Zaremba [125, 126, 127] confirmed thispicture, and Morgan, Rusch, Hutchinson, and Burnett provided additional analysis in an extensionof their previous work [128, 129, 130, 131, 132]. These efforts highlighted the important role of thenoncondensate dynamics in the behavior of collective modes at finite temperature.
Experiments to date have operated with anisotropic trapping geometries, which lead to anincreased degree of complexity in the collective-mode spectrum and mask effects due to the dynamical coupling of condensed and noncondensed portions of the gas. The signature of this coupling in

73
Ref. [21] shown in Fig. 4.1 was an upward shift of the m = 0 quadrupole mode frequency aroundT ∼ 0.7Tc indicative of a crossover between two distinct branches. Two mode fits of the condensate oscillation revealed the presence of the natural oscillation frequencies of both the condensate
and thermal clouds in the condensate oscillation [125]. The upper branch, which was likely theone predominately excited experimentally, corresponds to an in-phase oscillation of the condensate
and thermal cloud, as the condensate is being driven by the thermal cloud. In Ref. [133], Geddes, Morgan, and Hutchinson analyzed the breathing mode under isotropic harmonic confinement,
predicting strong dynamic coupling between condensate and thermal cloud as a result of mode
matching between the condensate and a strong resonance of the thermal cloud near the breathing mode frequency of the classical gas at 2ω0 . Importantly, they surmised that the condensate
oscillations should be out-of-phase as the cloud warms nearing the transition temperature. In thebreathing mode experimental results for an isotropic trap presented in this chapter, the observed
striking collapse and revival behavior of the condensate oscillation provides a smoking-gun signatureof dynamic coupling between the thermal cloud and condensate.
In this chapter, joint experimental/theoretical results for the breathing mode oscillation below
Tc are presented. First, the semi-classical theory of the finite temperature BEC must be developed
beginning from the zero-temperature limit and the many-body Hamiltonian. The collisionlesscoupled-modes model of Stoof and Bijlsma is then presented with results for the temperaturedependence of the mode spectrum over the experimental range of interest. The experimental
results are then detailed along with a description of the relevant observables. The striking collapse
and revival timescales observed in the experiment are then analyzed from the framework of thecoupled-modes analysis and from numerical simulations using the ZNG methodology [134] which
includes collision processes and can account for the observed damping1 . The results of this chapter
play a dual role. The immediate role is that of extending the many-body theory for ultracold
weakly-interacting Bose gases developed thus far and benchmarking it against experiment. Thereis however also the broader role which is historical context of the results of this chapter with respect
1

The details of the algorithm are described in Appendix A.

74
to the study of liquid superfluidity and Bose-Einstein condensation of a dilute atomic cloud.

4.1

Zero Temperature: The Mean-Field Theory of the Condensate
This section picks up from the discussion in Sec. 2.3.2 for the noninteracting Bose gas, where it

was concluded that formation of BEC is characterized by a macroscopic population of the groundstate orbital of the system. When an orbital becomes macroscopically occupied N0  1 =⇒N0 + 1 ≈ N0 , the added effect of an additional atom become negligiblea†0 |N0 , ...i =and therefore a†0 ≈√pp
N0 + 1|N0 + 1, ...i ≈ N0 |N0 , ...i,(4.1)

N0 appropriate for a coherent state. This approximation can be carried overto the field operators through Eq. 2.66, yieldingψ † (r) =Xp
N0 φ∗0 (r) +a†α φ∗α (r),

(4.2)

λ6=0

which is commonly written in the formψ † (r) = Ψ∗ (r) + δψ † (r),

(4.3)

p
N0 φ∗0 (r) = Ψ∗ (r)(4.4)

with the identificationsXa†α φ∗α (r) = δψ † (r)

(4.5)

λ6=0

of the complex-valued field Ψ(r) describing the condensate wave function. The remainderδψ † (r) is the fluctuation field, which is typically small compared to the condensate occupation
for zero temperature theories, and near the transition temperature is composed of the thermal
cloud discussed in Sec. 2.4.
When an orbital is not macroscopically occupied, the expectation value of its associatedcreation or annihilation operator is zero:h..., Nα , ...|a†α |...., Nα , ...i =pNα + 1h..., Nα , ...|...., Nα + 1, ...i = 0.

(4.6)

75
However, the macroscopic orbital expectation value returns√

N0 . Therefore, the expectation valueof the field operator is nonzero hψ † (r)i = Ψ∗ (r), indicating an underlying broken symmetry,
which is the U(1)-gauge symmetry, associated with the global alignment of the phase of individual atoms in the ground state orbital [120]. The condensate wave function Ψ∗ (r) is the orderparameter associated with the symmetry broken phase.
When the gas is interacting, the condensate is deformed from the shape of ground state
orbital, as we will see shortly. To allow for greater flexibility in the condensate wave function, it is
written in the general form
Ψ(r, t) =p
nc (r, t)eiθ(r,t) ,(4.7)

where θ(r) is the coherent phase, and nc is the condensate density. Furthermore, we denote theRcondensate normalization d3 r |Ψ(r)|2 = Nc rather than the orbital occupation, N0 . In thenoninteracting limit, Nc = N0 .

Let’s now derive the equation of motion for the condensate wave function from the Heisenberg
equation of motion for the field operator ψ(r) including interactions. The Heisenberg equation ofhimotion iψ̇ = ψ, Ĥ gives the time evolution for a single field operator
∇2†+ Vtrap (r) + g0 ψ (r)ψ(r) ψ(r),iψ̇(r) = −
2m


(4.8)

where the commutation relations, Eqs. 2.68, 2.69 for the field operators have been utilized. The
contact strength g0 therefore plays the role of an effective coupling strength. When a0 > 0, thismanifests as a repulsive coupling, and when a0 < 0 it manifests as attractive. To construct anequation of motion for the condensate wave function, the field operator can be replaced by its
symmetry broken form (Eq. 4.3) and fluctuation term is ignored in the zero temperature limit



∇2
2
iΨ̇(r) = −+ Vtrap (r) + g0 |Ψ(r)| Ψ(r),
2m

(4.9)

which is the time-dependent Gross-Pitaevskii equation (GPE) for the dynamics of the condensate wave function. The rightmost term is a nonlinear term of coupling strength g0 that takes into
account the mean-field produced by the surrounding condensed bosons.

76
To understand the ground state structure of the condensate at zero temperature, a stationary
form of the GPE must be formulated. Generally, the ground state of the many-body system must
minimize the expectation value of the many-body Hamiltonian, Eq. 2.77, and when a condensate
is present, the substitution ψ(r) = Ψ(r) yields the energy functional
Etot [Ψ] =

Z

3

d r




|∇Ψ(r)|2
g0
2
4+ Vtrap (r)Ψ(r)| + |Ψ(r)| ,2m
2(4.10)

where it is convenient to denote the respective contributions Etot [Ψ] = Ekin [Ψ] + Epot [Ψ] + Eint [Ψ].
The factor of 1/2 in the mean-field contribution prevents double counting of interaction energy
Eint between all distinct pairs of condensate atoms of which there are Nc (Nc − 1)/2 ≈ Nc2 /2.
The ground state condensate wave function should therefore minimize this functional along withR
the normalization constraint Nc = d3 r|Ψ(r)|2 . The method of Lagrange multipliers recasts thisproblem as a functional minimization of Etot [Ψ] − µN [Ψ] for fixed Lagrange multiplier µ. This
minimization is achieved by setting the functional derivative δ (Etot [Ψ] − µN [Ψ]) = 0, which results
in a stationary equation for the ground state wave-function [65]



∇2
2
µΨ(r) = −+ Vtrap (r) + g0 |Ψ(r)| Ψ(r),2m(4.11)

which is the time-independent GPE. Alternatively, the time-independent GPE can be reproducedpfrom the time-dependent GPE by looking for stationary solutions of the form Ψ(r) = nc (r)e−iµt .
The Lagrange multiplier µ is therefore also the eigenvalue of the condensate wave-function. In

the remainder of this section, µ is connected with the chemical potential from thermodynamics byanalyzing the solutions of the time-independent GPE.
In the absence of interactions, the condensate wave function is simply the harmonic oscillator
ground state scaled by the number of atoms
1/2

nc (r) =

2
2
2
2
2
2
Nc
e−(x /2ax +y /2ay +z /2az )π 3/4 (ax ay az )1/2

(Non-Interacting gas.)(4.12)The quantities ai are the oscillator lengths along a particular axis of the trap with frequencypωi , ai = 1/mωi . When this solution is plugged into the time-dependent GPE, assuming anPisotropic geometry, the result is µ = i ωi /2, which agrees with the condensation criterion µ =77
E0 from Sec. 2.3.2 for noninteracting Bose gases. Additionally, this is just the average energy
per particle, Etot /Nc , which agrees in the non-interacting limit with the general definition of thechemical potentialµ = Etot [N0 ] − Etot [N0 − 1]

(General definition) ,(4.13)

as the variation of the total energy when the total number of atoms is varied.
In the limit where the mean-field interaction term term is large compared to the kinetic
energy, captured by the dimensionless parameter ξ = (Nc a0 /aho )  1, the gradient operator in
Eqs. 4.11 can be ignored. This is referred to as Thomas-Fermi approximation [65], which is a fairapproximation in many weakly-interacting Bose gas experiments (na30  1). In the Thomas-Fermi
approximation, the time-independent GPE can be solved algebraically for the condensate density
nc (r) = [µ − Vtrap (r)] /g0(Thomas-Fermi approximation.)(4.14)

This is a parabolic profile with a sharp boundary at the position where Vtrap (r) = µ. Normalizing
the Thomas-Fermi density distribution gives an expression for the chemical potentialµT F =

152/5 2/5
ξ ω0 ,
2(4.15)

which is no longer equal to the average energy per particle. This can be seen clearly by comparing
the result from Eq. 4.10 for Etot with the general expression for µ from Eq. 4.11 which differ by afactor of 2 in the interaction energyµ=Ekin + Epot + 2Eint
Etot + Eint
=
,
Nc
Nc

(4.16)

reflecting the fact that the average interaction energy changes with increasing particles, which
compliments the scaling of the mean-field interaction with the density of condensate atoms [135].
This also agree with the general definition Eq. 4.13, which completes the identification of µ withthe chemical potential even when interactions are included.
The stationary form of the GPE therefore describes the ground state of the condensate
including the effects of two-body interactions. Higher order effects can also be incorporated in

78
a systematic way described in the final chapter of this thesis, Chapter 7. From this viewpoint,
the energy functional, Eq. 4.10, is therefore a low-density expansion based on the smallness of thediluteness parameter (see Refs [136, 137] for the lowest order ‘LHY’ corrections.) In the following
section, the finite temperature Bose gas, where the fluctuations δψ are non negligible, is discussed
through a kinetic theory for the fluctuation field.

4.2

Kinetic Theory Below Tc
Below the transition temperature, the excitation spectrum is more complex than the particle-

like energies of Chapter 3. Namely, there is a transition between particle-like excitations at highenergies and long wavelength phonon sound modes, which are number preserving density oscillationsm, at low energies which is captured by the Bogoliubov dispersion relation for the uniform
Bose gas with repulsive interactions (g0 > 0)E(p) =
χp =q
χ2p − (ng0 )2 ,p2
+ ng0 ,
2m

(4.17)(4.18)

written in the form emphasizing the gap, ng0 , at zero energies which is an indicator of superfluidity[65]. In a trapped Bose gas, this excitation spectrum is valid over a window which is small comparedto the spatial variation of the trap, which generally holds provided the collective oscillation varies
sufficiently slowly. Therefore, the short wavelength, particle-like picture holds locally over the entire
cloud. The low-momentum modes become collective modes of the trap when viewed over the lengthscale of the entire gas. In extremely elongated traps, it is possible to produce collective oscillations
that are well-approximated by phonon sound modes [138].
The dividing temperature below which particles become phonon-like is kb T∗ ≈ ng0 . This
result applies approximately to a trapped gas using the density at the center of the trap, which
for the experiment considered in this chapter gives T∗ < 0.1Tc . This is well below the temperature
range considered in this chapter (0.7Tc < T < Tc ) and therefore it is appropriate to consider the
cloud of excitations as particle-like in constructing a semi-classical kinetic theory.

79
To formally construct a particle-like expectation spectrum, I make the Hartree-Fock (HF)
approximation and consider expectation values of the orbital interaction, Eq. 2.74,
hN0 , N1 , N2 , ...|V̂ |N0 , N1 , N2 , ...i =1 X
hN0 , N1 , ...|a†δ a†γ aβ aα |N0 , N1 , ...ihδγ|V |αβi,
2(4.19)αβγδ

where hN0 , N1 , N2 , ...| is a symmetrized Fock-space ket including the number of atoms, Ni , in the
ith orbital. Non-vanishing expectation values correspond to δ = α, γ = β, the direct or Hartreecontribution, and δ = β, γ = α, the exchange or Fock contribution.
For particle-like excitations, it is appropriate to take the expectation value in Eq. 4.19 withrespect to plane waves in a cubic volume of side L small compared to aho . In this case with thezero-range contact interaction (Eq. 2.44), the coupling matrix is a constant g0 /L3 and the excitationenergies are given byẼ(p) =

p2
+ ng0 + (n − np )g0 ,2m(4.20)

where np is the density of a plane-waves with momentum p. The tilde has been added to denote
energies under the Hartree-Fock approximation. The second and third term in Eq. 4.20 are themean-field energies associated with the Hartree and Fock contributions, respectively. This gives thegeneralized expressions including the trapping potential for the local energy of a (p = 0) excitationin the condensateẼ(r, 0) = [nc (r) + 2nex (r)] g0 + Vext (r),(4.21)

and also gives the energy of the excitations out of the condensate asẼ(r, p 6= 0) =p2
+ 2 [nc (r) + nex (r)] g0 + Vext (r).
2m

(4.22)

The factor of two difference reflects the fact that the excited states are not macroscopically occupied
and therefore n − np6=0 ≈ n.
Equation 4.22 is the energy of an atom moving in the combined mean-fields of the condensate
and excitations. The factor of two difference in the condensate and thermal cloud energies has
ramifications for the the ground state structure. In Fig. 4.2, the thermal cloud and condensate
ground state densities are shown over a range of the dimensionless parameter ξ, from noninteracting

80

Density8000ξ ≈ 10− 2
80

6000
4000

40

ξ ≈ 10− 1

Repulsion of thermal cloud `wing.’

0

2000

4

8

3

4

ξ ≈ 100

0

1

2

r

5

Figure 4.2: Profiles of the condensate (solid lines) and thermal cloud (dashed lines in inset) densities
from noninteracting ξ ≈ 10−2 (red), ξ ≈ 10−1 (blue), to the edge of the Thomas Fermi regime ξ ≈ 1(green). The densities and radius are rescaled in oscillator units for N = 105 , T = 0.8Tc . As χ
increases, the difference in mean-field interactions pushes the thermal cloud further out to the wingsof the condensate.

ξ  1 to the Thomas Fermi regime ξ  1, where the profile of the thermal cloud on the ‘wings’ of
the condensate becomes pronounced due to the additional repulsion by a factor of nc g0 in the meanfield energy. These equilibrium profiles are the result of a self-consistent numerical calculation for
the ground state of the thermal and condensate clouds using the algorithm described in Appendix A
and Ref. [139].

4.2.0.1

The ZNG Equations

At temperatures approaching Tc where the noncondensed portion of the cloud is significantly
populated, it becomes increasingly important to include interactions between excitations. Under
the Hartree-Fock approximation, the excitations are particle-like and a kinetic theory can be formulated to describe their dynamics in terms of a quasi-probability distribution function, the Wigner

81
distribution, which plays a role analogous to the phase space distribution f (r, p, t) from Sec. 3.1,
and therefore I use identical notation. Importantly, the Wigner distribution can be derived froman expectation value of the field operators ψ, ψ †f (r, p, t) ≡

Z

3 0

d r e

ip·r0

†

0

0



ψ (r + r /2, t)ψ(r − r /2, t) ,(4.23)

where r and r0 are the center of mass and relative coordinates, respectively, and p is the momentumof a single excitation. Therefore, a kinetic theory for the Wigner distribution function can be derived
beginning from the many-body Hamiltonian. Technically, the Wigner distribution function can takeon negative values. However, if we are concerned only with it’s coarse-grained evaluation on thecells of µ-space, which effectively averages over quantum fluctuations on the order of the uncertainty
principle, the resultant distribution function is positive definite and fits within our semi-classicalpicture [96].
From the Heisenberg equation of motion, it is possible to derive the equation of motion for
the Wigner distribution function, which is a quantum Boltzmann equation (QBE) given by[96, 112, 139]




∂
p
∂f
+
· ∇r + F · ∇p f (r, p, t) =,∂t m∂t coll(4.24)

where F = −∇r [2g0 n(r + Vext (r)] is the local force due to the potential energy of an individual
excitation in the Hartree-Fock approximation (Eq. 4.22.) In this chapter, I follow the approach due
to Zaremba, Nikuni, and Griffin (ZNG) [139] where the collisional contribution on the right handside of this equation is the sum of two terms∂f= C12 [f, Ψ] + C22 [f ],∂t coll(4.25)

where C22 is the Bose-enhanced collision integral in the Boltzmann equation, Eq. 3.11, and C12 isa new contribution which involves both the condensate wave function and the Wigner distributionfunction.
The graphical interpretation of these two terms is shown in Fig. 4.3. The C22 term represents
the effect of binary collisions within the thermal cloud, which serve to bring the excitations into

82
(a.)

C22

in
C12

out
C12

Figure 4.3: A schematic of the collision processes contained in the ZNG equations.

a state of statistical equilibrium. This is precisely the Bose-enhanced version of the collisional
integral discussed in Sec. 3.1.1 and has the form
C22 [f ] =σ̂
πh3 m2Z
dp2 dp3 dp4 δ (p + p2 − p3 − p4 ) δ Ẽ + Ẽ2 − Ẽ3 − Ẽ4× [(1 + f )(1 + f2 )f3 f4 − f f2 (1 + f3 )(1 + f4 )] .

(4.26)The C12 term represents the scattering of excitations into and out of the condensate, leads to growthor decay of the condensate, and acts to bring both clouds into diffusive2 and global equilibriumZσ̂ |Ψ|2
dp2 dp3 dp4 δ (mvc + p2 − p3 − p4 )πm2
× δ Ẽc + Ẽ2 − Ẽ3 − Ẽ4 [δ (p − p2 ) − δ (p − p3 ) − δ (p − p4 )]C12 [f, Ψ] =× [(1 + f2 )f3 f4 − f2 (1 + f3 )(1 + f4 )] .

(4.27)

Importantly, the condensate must already be seeded for growth and decay to occur. The ZNG
approach does not describe spontaneous emission.
In Eqs. (4.27) and (4.26) the delta functions ensure conservation of energy and momentum
in a collision, and fi is shorthand for f (ri , pi , t). Furthermore, Eq. (4.27) depends on the local
2

Diffusive equilibrium is reached when the chemical potentials of the condensate and thermal cloud are matched.
A chemical potential gradient drives a transport of the conjugate thermodynamical variable, which is atom number.

83
condensate velocity, energy, and chemical potential given by [139]1∇θ(r, t),m1
mv 2 + µc ,
Ẽc =
2 c
√
1 ∇2 nc+ V + gnc + 2gñ,
µ = −
√
2m
nc
vc =

(4.28)(4.29)(4.30)

where the dependence of these quantities on r and t has been omitted for brevity. The condensate
velocity can be extracted from the GPE using the form of the condensate given in Eq. 4.7, andthe energy Ẽc is the Hartree-Fock energy for the zero-momentum mode (Eq. 4.21) in the ThomasFermi approximation. The lengthy derivation of the collision integrals C22 and C12 can be foundin Refs. [134, 139] and is not reproduced in this thesis.
The C12 term does not conserve the total number of atoms in the thermal cloud, and so inequilibrium there must be a corresponding non-Hermitian contribution to the GPE which maintains
detailed balance within the system. The zero-temperatue theory developed in Sec. 4.1 can begeneralized to include the effects of the fluctuation field. Namely, from the Heisenberg equation of
motion for the field operator ψ, Eq. 4.8, the expectation value for δψ 6= 0 gives∇2iΨ̇ = −
+ Vtrap + g0 (nc + 2nex ) Ψ + g0 hδψδψiΨ∗ + g0 hδψ † δψδψi,2m(4.31)

where the position and time dependence have been suppressed for clarity. Neglect of the pairing
term hδψδψi is consistent with the Hartree-Fock approximation. Retention of the triplet termhδψ † δψδψi adds a non-Hermitian contribution which can change the norm of Ψ. In the ZNG
approach, the triplet term is expanded in powers of g0 about it’s equilibrium value and truncated at leading non vanishing order which happens to be an imaginary contribution hδψ † δψδψi ≈−iR(r, t)Ψ(r, t). This contribution is typically written in a form which makes explicit the link
between the C12 process
1
R(r, t) =2 |Ψ|2Z

dp
C12 [f, Ψ] .
(2π)3(4.32)

When Eq. 4.31 is rewritten with −iR(r, t) in place of the triplet term, the resultant equation is a

84
generalized Gross-Pitaevskii equation (GGPE)∂Ψ(r, t)
i
=
∂t




∇2
−
+ V (r, t) + g[nc (r, t) + 2ñ(r, t)] − iR(r, t) Ψ(r, t),2m(4.33)The ZNG equations are comprised of the two coupled equations, Eqs 4.24, 4.33, for the
dynamics of the condensate and thermal cloud. The form of Eq. 4.32 makes the number conservationof the equations explicit. In Appendix A a numerical procedure for simulating these equations isoutlined.
From a general standpoint, the ZNG formalism is a prescription for describing a partiallycondensed Bose gas by breaking the Bose field operator into a condensed part and a noncondensed
part. It couples a GGPE for the condensate with a QBE for the noncondensate. It has previously
been utilized to study collective oscillations at finite temperature [125, 126, 140, 127], as well as finite
temperature effects on solitons [141], vortices [142, 143], and turbulence [144]. In addition, recent
work by Lee and Proukakis [145] applies the ZNG method to study collective modes, condensate
growth, and thermalization dynamics for both single and multicomponent condensates. It has been
particularly successful in describing the temperature range T∗ < T < Tc which is within the rangeof the experiment considered in this chapter.

4.3

Collisionless Dynamics
Here, we provide an overview of the collisionless dynamics of a trapped Bose gas, beginningwith a discussion of the limiting cases for collective modes in an isotropic trap. We then discuss themonopole mode of a finite temperature BEC through application of the semi-classical collisionlessmodel from Ref. [29, 122] to a spherically symmetric trapping geometry, and show how the monopolemode response can be cast in terms of two eigenmodes of the system. This analysis provides
a framework for understanding the collapse and revival behavior of the condensate oscillation
observed in the experimental results.

85
4.3.1

Collective modes in an isotropic harmonic trap

In an isotropic harmonic trap, the collective modes of a Bose gas are well understood in two
limits. In the Thomas-Fermi (TF) limit at zero temperature the ratio of the kinetic to interactionenergy is small when the number of atoms in the BEC is large; thus, the kinetic energy can beneglected. The collective-mode frequencies of a BEC in a three-dimensional isotropic harmonic
trap in this limit can be estimated using a hydrodynamic approach [121]. The mode frequencies
depend on the principle quantum number n and angular quantum number l according toω 2 = ω02 l + 3n + 2nl + 2n2 ,(4.34)

where ω0 is the harmonic trap frequency and ω is the frequency of the collective mode. For the
spherically symmetric monopole, or breathing mode (n = 1, l = 0), the mean-square radius of√
the condensate oscillates at ω = 5ω0 , and the motion is undamped. Above the BEC criticaltemperature, Tc , mean-field effects can be neglected and the gas can be described by a classicalBoltzmann equation. In this case, the mode oscillates at ω = 2ω0 in both the collisionless andhydrodynamic regimes [100], and the motion is undamped. In the collisionless regime individual
atoms may undergo many oscillations before experiencing a collision while the hydrodynamic regime
implies the gas is in local statistical equilibrium.

4.3.2

Coupled-modes analysis

To obtain insight into the behavior of the monopole mode at finite temperature in an isotropic
trap we apply a model previously developed by Bijlsma and Stoof [29]. This methodology introduces
a dynamical scaling ansatz, similar in spirit to the method of Sec. 3.1.3.2, for the condensate andnoncondensate that successfully reproduces the limiting cases of the monopole mode behavior
described in the previous section. The condensate and noncondensate are described by the QBE(Eq. 4.24) and GGPE (Eq. 4.33) from the the ZNG equations in the collisionless limit, retaining
the mean-field couplings. The analysis here assumes a small amplitude perturbation of the system,
and a linear response such that the effects of damping are absent.

86
In the following calculation, a scaling ansatz is made for the time evolution of the condensate
density,nc (r, t) =1 0r
n,
λ3 c λ

(4.35)and the Wigner distribution function of the noncondensate,1f (r, p, t) = 6 f 0
ᾱ





r α
mα̇
p−
,
,
αᾱ ᾱ
α(4.36)

which are written in terms of a Gaussian density profile n0c for the condensate and a saturated
Bose-Einstein distribution f 0 for the noncondensate: mω 1/2
20n0c (r) = Nce−mω0 r ,
 π

ω0
0
f (r, p) = ÑkB T ζ(3)

#−1
"  2
×

e

p
+ 12 mω02 r2
2m

/kB T

−1

.

(4.37)The number of atoms in the condensate and noncondensate are denoted by Nc and Ñ , respectively,and ζ(s) is the Riemann zeta function. The scaling parameters, λ(t) and α(t), capture the oscillationof the widths of the two components, and the bar denotes the equilibrium value. Inserting the scaling
ansatz into the GPE and QBE results in a set of coupled equations of motion for the condensate
and noncondensate characteristic widths:ü + ω02 u = v(u),where the vector u contains the scaling parameters


λ


u=
,
α

(4.38)(4.39)and v(u) is a nonlinear vector function describing the spreading of the cloud due to kinetic energy
and the effects of nonlinear interactions (see Ref. [29] for details).
In the limit of a small amplitude oscillation, the total density of the system can be writtenasn(r, t) = n̄(r) + δn(r)eiωt ,(4.40)

87
where the perturbation is generated by modulating the trap frequency with amplitude :ω0 (t) = 1 + eiωt ω0 .

(4.41)In this limit Eq. (4.38) can be linearized:−ω 2 δu + ω02 δu = [∇u v] ū · δu − 2ω02 ū,(4.42)and the eigenfrequencies ωn and eigenmodes u(n) of the homogeneous part of Eq. (4.42) can beextracted with a solution of the form
δu = 2ω02

X u(n) · ūn

ω 2 − ωn2u(n) .

(4.43)

From this solution, we find two eigenmodes that we refer to as the in-phase and out-of-phase modesof the system. The in-phase mode corresponds to the condensate and noncondensate monopolemodes oscillating together (no phase difference φ = 0), and the out-of-phase mode corresponds tothe condensate and noncondensate monopole modes oscillating opposed (phase difference φ = π).
In Fig. 4.4 the frequencies of the in-phase and out-of-phase modes as a function of temperature areshown.
Given the eigenmodes of the system, the time-averaged work done by a perturbation of thetrap frequency can be used to characterize the response of the system:
W =

X
n

bn
,
2
ω − ωn2(4.44)

where the bn are a measure of the magnitude that each eigenmode responds with when the system
is perturbed. Figure 4.5 shows the bn as a function of temperature for the two modes discussed
above. For temperatures T > 0.2 Tc both modes of the system will be excited by a perturbation of
the trap frequency.
The coupled-modes analysis suggests that the oscillation of a single component of the gas(e.g., condensate) is described by a superposition of two eigenmodes oscillating at slightly different
frequencies. Therefore, we expect measurements of the condensate width as a function of time fortemperatures T > 0.2 Tc to beat at a frequency corresponding to the frequency difference between

88

2.3
Thomas-Fermi

!=!0

2.2

?=0

2.1

1.9

?=:

Ideal gas

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T =Tc

Figure 4.4: Oscillation frequency of the in-phase (φ = 0) and out-of-phase (φ = π) modes as afunction of temperaturefrom the coupled-modes analysis (solid lines). The dashed lines represent√
the TF limit ( 5ω0 ) and ideal gas limit (2ω0 ) for the monopole mode frequency of the condensate
and noncondensate, respectively. Assumptions made in the coupled-modes analysis become invalid
for T . 0.2 Tc .

Mode Response (arb. units)

30
?=0
?=:

25
20
15
10
5
0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T =Tc

Figure 4.5: Magnitude of the response of the in-phase (φ = 0) and out-of-phase (φ = π) modes toa trap frequency perturbation as a function of temperature. The amplitude of the trap frequency
modulation is  = 0.01.

the two eigenmodes, ∆ω/ω0 ∼ 0.2–0.25 (see Fig. 4.4). Given this result, we present experimental
observations of the monopole mode in an isotropic trap in the next section.

89

4.4

Experiment
The experimental system is a Bose gas of 87 Rb atoms cooled to quantum degeneracy via forced

radio-frequency evaporation in a time-averaged, orbiting potential (TOP) trap [68]. A standard√
TOP trap configuration results in an oblate harmonic trap with an aspect ratio of ωz /ωr = 8,
where ωz (ωr ) is the axial (radial) trapping frequency. Here, the overall harmonic confinement of
the trap is reduced and the trap minimum is allowed to sag under the force of gravity. This causes
the curvature of the magnetic field along the z axis to decrease, which effectively decreases theratio ωz /ωr . The end result is an isotropic harmonic trap with ω0 ≡ ωr = ωz = 2π × (9.03(2) Hz)
with a residual asphericity of less than 0.2%. This system was used in previous work to study themonopole mode of a Bose gas above the BEC critical temperature [24], and a detailed descriptionof the apparatus can be found in Ref. [63].
The monopole mode was excited below the BEC critical temperature in the range of approximately 0.75–0.9 Tc . The experimental procedure parallels that of Ref. [24]—beginning from a systemat equilibrium, the trap frequency is sinusoidally modulated at a driving frequency ωD ≈ 2π × 18–
19 Hz for four periods with an amplitude  ≈ 0.1:
ω(t) = [1 +  sin (ωD t)] ω0 .

(4.45)

After driving, we find that the peak TF radius of the condensate is 10–15% larger than the equilibrium value for all of the experimental data sets. The system is then allowed to freely evolve inthe static isotropic trap for a time t before six nondestructive phase-contrast images record theintegrated column density of the cloud at intervals of 10 ms or 17 ms, sampling between 1 and 1.5
oscillation periods of the monopole mode. This experimental procedure is repeated between 2–4
times for each t, and for times up to t ≈ 1.5 s.
Each phase-contrast image is analyzed using a 2D bimodal fit to the atomic column density.

90
The fitting function is the sum of a Gaussian and integrated TF function [146]:
" 

 #

x − xc 2z − zc 2
ncol (x, z) = AG exp −
−
σG,x
σG,z
"

 #3/2
z − zc 2x − xc 2−+ AT F 1 −σT F,xσT F,z+ Ccol ,(4.46)

where AG and AT F are the amplitudes of the Gaussian and TF functions, respectively, xc and zc
are the center points of the cloud, σG,i are the Gaussian widths, σT F,i are the TF widths, and Ccol
is a constant offset. Note that the TF function is defined to be zero if the argument in brackets isnegative.
The dynamics of the condensate monopole mode are captured by the spherically symmetricquantity2
σM= σT2 F,x + σT2 F,y + σT2 F,z /3.

(4.47)

During the data runs for this experiment, images were consistently taken in the xz plane. In
earlier measurements described in Ref. [24], data were also taken along the xy plane, but technicaldifficulties were encountered with the imaging system along this axis during the course of theexperiments discussed here. However, the limited data available from the xy plane suggests thatthe cloud was highly symmetric [63]. Therefore, we set σT F,y = σT F,z in Eq. (4.47) when calculating
the amplitude of the condensate monopole mode. Although we observe excitation of other collectivemodes (dipole and quadrupole), we find that the key features of the experimental results for themonopole mode are independent of whether this assumption is made or σT F,y is simply excluded
from Eq. (4.47).
We determine the instantaneous amplitude of the condensate monopole mode by fitting afixed frequency sine wave to each set of six consecutive time points. The fitting function is of theformgσ (t) = Aσ cos (2πνt) + Bσ sin (2πνt) + Cσ ,(4.48)

where ν = 19 Hz, and Aσ , Bσ , and Cσ are fit parameters. This functional form is chosen because we91
are concerned with the amplitude of the mode, not the frequency, which enables a straightforward
linear regression analysis for computing Aσ , Bσ , and Cσ . Finally, we present the data in the formof a fractional amplitude given by
AM =A2σ + Bσ2,
Cσ2(4.49)

where Aσ , Bσ , and Cσ correspond to the fit parameters of Eq. (4.48). The results of this analysis
are shown in Fig. 4.6, where time t = 0 is defined as the point at which the modulation of the trap
frequency ceases.
A central feature of the data is that the amplitude of the condensate monopole mode does
not decay exponentially. Across the five data sets, there is a consistent collapse in the amplitude of
the monopole mode between t = 0.3–0.5 s, and a partial revival around t = 0.7–0.9 s. Motivated by
the results of the coupled-modes analysis, we fit the data to an envelope function that represents
the superposition of two sinusoidal modes, and we include an overall exponential damping factor
to represent the loss of amplitude with time:ge (t) = Ae cos∆ω 2 −Γe tt e,2(4.50)

where Ae is the initial amplitude, ∆ω is the beat frequency, and Γe is the damping rate of theenvelope. The results of this fit are overlaid with the data in Fig. 4.6. From the fit, we find∆ω/ω0 ∼ 0.13 on average, and damping rates in the range Γe ∼ 1.5–3.5 s−1 . The observed beatfrequency is less than ∆ω/ω0 ∼ 0.2 as expected from the coupled-modes analysis. We attribute
this disagreement to the naive form of the fitting function, which assumes that the two sinusoidal
modes damp at the same rate, respond equally to the trap frequency perturbation, and have nophase difference between them. These assumptions are investigated further in the next section
where the two sinusoidal modes are identified with the in-phase and out-of-phase modes predictedby the coupled-modes analysis.
Before moving on, it is important to note that in Ref. [60] it was shown that anharmoniccorrections to the trap geometry were likely responsible for the anomalous exponential dampingof the monopole mode observed above the critical temperature [24]. As shown in Ref. [24], this92

Figure 4.6: Amplitude of the monopole mode oscillation (squares) for atom numbers of (a) N =8.9 × 105 , (b) N = 9.7 × 105 , (c) N = 6.7 × 105 , (d) N = 5.4 × 105 , and (e) N = 7.9 × 105 .
Each frame is labeled with the condensate fraction (Nc /N ) and temperature (T /Tc ). Error bars
represent the statistical uncertainty of multiple realizations of the experiment at each time point.
The dashed lines are a fit of the data to Eq. (4.50), which represents the envelope function for thesuperposition of two sinusoids. From the fit it is found that the beat frequency is ∆ω/ω0 ∼ 0.13
on average and the damping rate lies in the range Γe ∼ 1.5–3.5 s−1 .

93damping is < 0.2 s−1 for clouds with a full width at half maximum (FWHM) of < 125 µm. In
this work, the FWHM of the cloud below the critical temperature satisfies this criterion; thus, we
neglect anharmonic corrections to the trap geometry as the observed damping rate is approximately
an order of magnitude larger.

4.5

Collisional dynamics
We now investigate the damping observed in the experimental data through numerical simu-

lations within the semi-classical ZNG formalism. The coupled-modes analysis ignores collisions andexchange of particles between the two components, as well as nonlinear mean-field effects. However,
below the critical temperature, these interactions between the condensate and noncondensate canshift the frequencies of collective modes and cause damping.
In Sec. 3.1.3.1, the time τcoll was given which was the characteristic relaxation time due tobinary collisions between atoms in the cloud. There are now two collision rates, τ22 and τ12 . The
former timescale is the Bose-enhanced relaxation time for the thermal cloud, and the latter is therelaxation rate of the condensate and thermal cloud to a state of diffusive equilibrium. These rates
typically peak on the wings of the thermal cloud where the overlap between the components is at
its greatest (see Appendix A for a discussion of equilibrium collision rates.)
There are also additional processes which arise through the mean-field couplings: Landau
damping (see Refs. [147, 148, 149, 150, 151, 152, 153, 154] for further discussion) and Beliaev damping [155]. Landau damping describes a process where a collective mode decays due to its interaction
with a thermal excitation, and it is expected to dominate at higher temperatures approaching thecritical temperature. In the semi-classical picture, this is manifested through a thermal excitation
catching the wave of the collective mode, much as surfer would, taking energy from the wave in theprocess. On the other hand, Beliaev damping is a process where a collective mode decays into twolower energy excitations, which is suppressed for the lowest energy collective modes of a trappedgas due to the discretization of energy levels. Thus, the Beliaev process is absent for the monopolemode, and is therefore excluded from our analysis in this chapter. Landau damping dominates the

94
decay of the monopole mode when the collisional processes are excluded. In the remainder of thissection, we discuss the results of numerical simulation of the ZNG equations in the context of thecoupled-modes analysis and experimental data already presented.

4.5.1

Simulation of the experiment

To model the experiment, we simulate a gas of N = 8 × 105 87 Rb atoms in a sphericallysymmetric harmonic trap with ω0 = 2π × 9 Hz. Using the algorithm outlined in Ref. [139], we
generate equilibrium initial states of the condensate and noncondensate for temperatures ranging
from 0.1–0.9 Tc 3 . We then directly simulate the excitation of the monopole mode as in theexperiment by sinusoidally modulating the frequency of the trapping potential at ωD = 2ω0 for√
four periods. We find that our results are essentially unchanged for drive frequencies of (1+ 5/2)ω0√or 5ω0 . We use trap frequency modulation amplitudes of  = 0.02, 0.03, or 0.04, and then allow
the system to evolve freely for t = 2 s. We find that this range of  excites the monopole modeof the condensate with an amplitude comparable to that observed in the experiment (i.e. 10–15%
peak increase in the TF radius of the condensate from equilibrium). We note that these values are
somewhat less than the quoted experimental value of  ≈ 0.1, and speculate that this discrepancy isa result of multiple collective modes being excited in the experiment due to the difficulty of driving
the trap perfectly spherically. This is in contrast to the simulations where only the monopole modeis excited, and therefore less energy is required to be added to the system to achieve the same levelof excitation of the condensate.
We record the mean-square radius of the condensate as a function of time, along with snapshots of the individual density profiles. Although the experimental data sets have total atom
numbers that range between about 6 × 105 and 1 × 106 , we find simulations for 8 × 105 atoms
represent the features of interest, namely the collapse and revival behavior and damping rate. In
order to compare directly with the experimental data, we generate 2D column densities from the3

As T → Tc the numerical method for calculating the equilibrium state of the gas becomes unstable, and 0.9 Tc
is an empirical upper bound.

95
simulation results, and determine the TF radii using the same bimodal fitting routine describedin Sec. 4.4. Equation (4.47) is used to calculate the amplitude of the condensate monopole mode,and Eq. (4.48) is fit to single periods of the oscillation corresponding to a window of approximately
53 ms. The results of this analysis are overlaid with the experimental data in Fig. 4.7 for thethree different values of . The timescale of the first collapse and revival observed in the simulation
results show good agreement with the experiment.

4.5.2

Extraction of damping rates

The prediction of the coupled-modes analysis and results of the ZNG simulations show good
agreement with the collapse and revival behavior observed in the experimental data (see Figs. 4.6
and 4.7). In addition, the damping observed in the results of the ZNG simulations agrees well withexperimental observations. Therefore, due to the limited and noisy experimental data available,
we use the results of the ZNG simulations instead of experimental data to get an estimate ofthe damping rates for the in-phase and out-of-phase eigenmodes predicted by the coupled-modes
analysis.
We fit the simulated evolution of the condensate mean-square radius, Rc2 =by the sum of two sine waves with decaying amplitudes

R

dr r2 nc (r),gc (t) = A1 sin (2πν1 t + φ1 )e−Γ1 t+A2 sin (2πν2 t + φ2 )e−Γ2 t + Cc ,(4.51)

where Ai , νi , φi , and Γi are the amplitudes, frequencies, phases, and damping rates, respectively, ofthe two eigenmodes, and Cc is a constant offset. In the left column of Fig. 4.8, typical results of thisfitting procedure for simulation results with a trap frequency modulation amplitude of  = 0.03. In
the right column of Fig. 4.8, the oscillation of the total density is shown for reference. Time t = 0
is defined as the point at which the modulation of the trap frequency ceases. We choose to fit to
the mean-square radius of the condensate as its time evolution is most sensitive to the presence ofboth eigenmodes across the temperature range investigated. The mean-square radius of the total

96

Figure 4.7: Amplitude of the monopole mode oscillation for experimental atom numbers of (a)N = 8.9 × 105 , (b) N = 9.7 × 105 , (c) N = 6.7 × 105 , (d) N = 5.4 × 105 , and (e) N = 7.9 × 105 .
Each frame is labeled with the condensate fraction (Nc /N ) and temperature (T /Tc ), and the legend
denotes the different modulation amplitudes used in the ZNG simulations. Error bars on theexperimental data represent the statistical uncertainty of multiple realizations of the experiment at
each time point. (f) Condensed fraction vs temperature for the ideal Bose gas, Nc /N = 1 − (T /Tc )3(solid line), the equilibrium state of the ZNG simulations (blue points), and the experimental data
(red crosses). All simulations are performed with N = 8 × 105 atoms.

97
density becomes dominated by the noncondensate at higher temperatures, and any signature ofa second eigenmode is lost. Similar behavior is observed in the evolution of the noncondensate
mean-square radius.
The mode frequencies extracted from this fitting procedure show excellent agreement withthe results of the coupled-modes analysis across the temperature range simulated as shown in
Fig. 4.9, and add weight to the identification of the beating in the total density oscillation shown in
Fig. 4.8 as arising due to the in-phase and out-of-phase modes. Simulation results at temperaturesof T = 0.1 Tc and 0.2 Tc are fit with a single decaying sinusoid due to the absence of a second
mode. This also agrees with the prediction of the coupled-modes analysis, where only a single
mode responds to a trap frequency perturbation for T ≤ 0.2 Tc (see Fig. 4.5). A notable feature in
Fig. 4.8 is a downward shift in the carrier frequency of Rc2 with increasing temperature, an effectalso observed in Ref. [112] for the monopole mode. As the temperature of the system increases theout-of-phase mode begins to get excited in conjunction with the in-phase mode, and the carrier
frequency shifts to a lower, out-of-phase frequency because it represents a weighted average of thetwo independent mode frequencies.
It is instructive to put these results in a larger context. In the absence of driving the thermal
cloud, Ref. [133] observed that the condensate breathing mode oscillation frequency is roughly
constant as a function of temperature. This is due in part to both strong coupling and resultant
level repulsion from the thermal cloud resonance at 2ω0 forcing the condensate frequency upwards,
which is balanced by the downward shift one obtains from the cumulative effect of coupling to allother modes in the problem. This is in contrast to the upward shift of the condensate oscillation
frequency obtained from single mode fits in Refs. [21, 125, 130] for the m = 0 quadrupole mode
above 0.7 Tc . The observed m = 0 quadrupole shift was from the zero temperature result 1.8ωr ,where ωr is the radial frequency in a TOP trap, upwards towards thermal cloud resonances at 2ωr .
In the language of eigenmodes this was instead a shift towards the in-phase mode at 2ωr (seeRef. [29]).
The damping rate of each mode determined from the fitting procedure is shown in Fig. 4.10

98
480

(1a)

(2a)

460

(μm2)

(μm 2 )

440
420
400
380
360
340
320

(1b)

0

0.15

0.3

0.45

Time (s)

0.6

1020

0.75

(2b)

960

(μm2)

(μm 2 )

900
840
780
720
660
600
540

(1c)

0

0.15

0.3

0.45

Time (s)

0.6

2400

(2c)

2200
2000

(μm 2 )

(μm2)

0.75

1800
1600
1400
1200
1000

0

0.15

0.3

0.45

Time (s)

0.6

0.75

Figure 4.8: Simulated mean-square radius of the condensate (1a-1c) and total (2a-2c) densities fora trap frequency modulation amplitude of  = 0.03 at (a) T = 0.4 Tc , (b) 0.6 Tc , and (c) 0.8 Tc(black circles) and resulting fit of Eq. (4.51) (blue line). The density of simulated points has been
reduced for clarity.

99

2.3
Thomas-Fermi

!=!0

2.2

?=0

2.1

1.9

?=:

Ideal gas

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T =Tc

Figure 4.9: The coupled-mode results (solid line) from Fig. 4.4 overlaid with results of the ZNG
simulations (crosses). Crosses are the result of fitting Eq. (4.51) to the condensate density.

Damping Rate (s -1)

5
?=0
?=:

4

3

2

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T =Tc

Figure 4.10: Damping rate of the in-phase (squares) and out-of-phase (triangles) mode as a resultof fitting Eq. (4.51) to the simulated evolution of Rc2 at each temperature for a trap frequency
modulation amplitude of  = 0.03. The solid lines are guides to the eye. Representative fits areshown in Fig. 4.8.

as a function of temperature. In the temperature range of the experiment, there is a mismatch of
the damping rates between the two eigenmodes. This mismatch, along with the beating between
the two modes, captures the behavior seen experimentally of strong collapse and subsequent revivalof the condensate oscillation. At lower temperatures, the in-phase mode dominates and the out-ofphase mode is strongly damped, and the inverse is true at higher temperatures. Thus, the in-phase

100mode appears to be dominated by the condensate while the out-of-phase mode consists primarilyof the noncondensate.
Based on the results of the coupled-modes analysis, one may suspect that the particulardrive frequency used to excite the system has a large effect on the nature of the response due to thepresence of two resonant excitation frequencies. However, results from ZNG simulations showed
little dependence on the drive frequency, which can be attributed to the presence of damping.
Damping in the system effectively broadens the resonances such that both modes are appreciably√
excited when the system is driven in the range 2ω0 to 5ω0 . Thus, the main characteristics of thecondensate collapse-revival behavior are relatively insensitive to the particular drive frequency.

4.6

Conclusion
In conclusion, the experimentally observed non-exponential collapse and subsequent revival ofthe monopole mode was analyzed via a coupled-modes analysis and through simulations of the ZNG
equations. The coupled-modes analysis identified two eigenmodes of the system corresponding toin-phase and out-of-phase oscillations of the condensate and noncondensate. These modes appear
to be collisionless analogs to the first and second sound modes as previously discussed in Ref. [29].
Simultaneous excitation of these two modes results in the observed collapse and partial revivalof the condensate monopole mode, which has a timescale compatible with the mismatch in the
eigenfrequencies. Damping of the oscillations was also observed experimentally, and simulations
within the ZNG formalism resulted in good agreement with the data.
These results hinge on a proper account of the dynamics of the thermal cloud including
coupling with the condensed portion of the cloud. Consequently, there are many parallels between
the model presented in this chapter and the two-fluid theory of liquid helium, and indeed the Landau
two-fluid equations can be derived from the ZNG equations in the non-dissipative limit [139]. A
discussion of sound modes in a BEC is sensible only in the hydrodynamic limit where an atom
undergoes many collisions over a trap period, and only when the trap is extremely elongated andapproximately uniform over a considerable length. This has been investigated experimentally in101
Ref. [156], and modeled using a one-dimensional version of the ZNG equations in Ref. [157]. To myknowledge, the ZNG equations have not been simulated in the hydrodynamic regime in a trapped
3D geometry mainly due to computational limitations. It would therefore be interesting to study
the transition of the in-phase and out-of-phase modes from the collisionless to the hydrodynamical
regimes taking advantage of the simplifications described in Appendix A due to the sphericalsymmetry for an isotropic trap and sketching the transition into two-fluid behavior.
This chapter concludes the discussion of weakly-interacting Bose gases. The remaining chapters deal with regime where scattering lengths are many times the van der Waals length and presumably enhanced through a Feshbach resonance where nonperturbative effects beyond the two-atomlevel become important. The formalism established in this and the previous chapter however serves
as a basis for their extension in Chapter 7 to include higher order correlation dynamics as would
be appropriate when interactions are strong and the gas becomes highly-correlated.

Chapter 5

Efimov Physics and the Three-Body Elastic Scattering Phase ShiftThis is the first chapter in a series of three revolving around the physics of strongly-interacting
Bose gases. Chapters 5–6 are devoted to the nonperturbative vacuum three-body physics that arisesin this regime. Chapter 7 focuses on how few-body physics beyond the level of two atoms can besystemically incorporated into the many-body problem.
Generally, for weakly-interacting Bose gases, the important role of three-body physics is to
reduce the lifetime of the gas through recombination of three atoms into deeplya bound moleculeand free atom, releasing enough kinetic energy to both be ejected from the trap. The rate equationthat determines the time evolution of the atom loss rate scales with the density asṅ = −L3 n3 ,(5.1)

where L3 is the loss rate coefficient, which is proportional to the recombination rate K3 = L3 /3that can be extracted from transition rates in the three-body problem. The factor of 3 reflects
the ejection of both the dimer and the free atom from the trap after recombination, and K3 scales
as a40 for three identical ultracold bosons [158, 159]1 . For 87 Rb, L3 = 4 × 10−30 cm6 s−1 , which
multiplied by typical densities of 1014 cm−3 yields a loss rate on the order of 1016 cm−3 s−1 ,which can be significant [160], however for a pure condensate this rate is reduced by a factor of
6 [161] which was experimentally confirmed in an experiment with Bose-condensed 85 Rb [162].
As a Feshbach resonance is approached from the side of negative scattering length, a remarkablephenomenon occurs at roughly |a0 | ∼ 10rvdW : the formation of a ladder of three-body bound1

Note that Ref [158] is off by a factor of 6.

103
states (Efimov trimers) starts, and each formation is characterized by a resonant enhancement in
the three-body recombination rate, as first observed in Ref. [34]. Exactly on resonance (|a0 | = ∞),an infinite number of Efimov trimers form, accumulating at zero energy from below [31, 32, 33].
Currently, there is an ongoing effort in the community to understand how these nonperturbative
states impact collective behavior of the strongly-interacting Bose gas. The next three chapterstherefore revolve around understanding the impact of Efimov physics on the scattering observables
and the thermodynamics of the strongly-interacting Bose gas.
Before proceeding further, it is useful to solidify the terminology used in the discussion ofstrongly-interacting Bose gases in this thesis. In what follows, the scattering length is treated as afree parameter used to explore the range of possible phenomenon around the unitarity limit whichis really a bound on the value of partial cross-sections, and it is assumed that the enhancement isdue to resonance effects for instance from a Feshbach resonance. The unitarity limit, discussed
in Sec. 2.2.3, is the limit in which the two-body partial wave cross sections are maximal, which
occurs as the phase shift passes through an odd multiple of π/2 as a bound state is formed nearthreshold. There is also the unitary regime, which for ultracold gases occurs when the scattering
length exceeds the inter particle spacing, (na30  1), which also exceeds the van der Waals lengthand the effective range of the interaction as is the case for a broad Feshbach resonance. To describea unitary gas considering only two-body physics, there remains only a single remaining finite lengthscale n−1/3 , and therefore the experimental results are considered universal and independent of
the experimental particulars and of the atomic species. In this thesis, the criterion for a gas to bestrongly-interacting is that the scattering length exceed the van der Waals length a0  rvdW ,which is coincidentally the point at which Efimov physics becomes important. It is not appropriate
to apply the solution of three-body problem in vacuum to the unitary Bose gas, and this point is
discussed further in Chapter 7.
This chapter is focused on the three-body problem for identical Bosons in vacuum where each
atom is restricted to a single internal level–the multi-level case relevant to spinor condensates isreserved for Chapter 6. To begin with, it is necessary to outline a convenient coordinate system for104parametrizing the motion of three atoms, which is done in Sec. 5.1. I then outline the adiabatic
hyperspherical approach for three identical bosons, the three-body Schrödinger equation in thisrepresentation, and analytic solutions of the three-body problem using the Fermi pseudopotential(Eq. 2.40) and boundary conditions from Sec. 2.2.5. Solving this in the unitarity limit yields theinfinite spectrum of three-body bound Efimov states studied in Sec. 5.3. The adiabatic hyperspherical representation also has the advantage that it naturally discretizes the three-body continuum,
and allows a huge simplification in the ultracold limit similar to the partial wave analysis fromSec. 2.2.3. In Sec. 5.4, I discuss the three-body continuum, the emergence of Efimov physics inthe scattering observables, and some of the difficulties associated with calculating the three-bodyelastic phase shift, which is a work in progress as of the writing of this thesis.

5.1

Parametrizing the Motion of Three Atoms
There is in general no unique parametrization of the three-body problem. In the lab frame,each atom is specified by an position vector ri , and therefore the problem is nine-dimensional.
In this thesis, I only consider the three-body problem for pairwise interactions. The three-body
Hamiltonian written in this coordinate system is therefore
H=

3
X
p2
i

i

2m

+

X
i<j

U (ri − rj ).

(5.2)

In the COM frame, the problem is reduced to six dimensions, and can be parametrized a variety
of ways. Here I begin with Jacobi coordinates which are two vectors connecting a pair of atoms
and the third atom with the COM of the pair. From the Jacobi coordinates, the hypersphericalcoordinates can be defined as a generalization of spherical coordinates to higher dimension.

5.1.1

Jacobi Coordinates
(k)

(k)

In the COM frame, it is possible to define two Jacobi vectors ρ~1 , ρ
~2

shown in Fig. 5.1.

The superscript (k) indicates the kth atom as the ‘odd-man-out’. The parametrization of the systemin terms of Jacobi vectors is therefore not unique as there are three possible choices for the odd-

105

Figure 5.1: Schematic of the mass-scaled Jacobi vectors written in the odd-man-out notation.

man-out. The Jacobi vectors can each be parametrized by a magnitude and two spherical angles:ρ~i = (|ρi |, θi , φi ). The Jacobi vectors are defined in terms of the lab frame coordinates as follows(k)ρ1 = d−1
k (ri − rj ) ,mi ri + mj rj(k)ρ2 = dk rk −,mi + mj
mk 
ml 
1−
,
d2k =
µ3B
M
m1 m2 m3
µ23B =
,
M
X
M=
mi .

(5.3)
(5.4)
(5.5)
(5.6)
(5.7)

i

The mass-scaling factor dk ensures that the six-dimensional volume element has the same functional
form in each set:
(k)

(k)

dV = d3 ρ1 d3 ρ2 .

(5.8)

Each set of Jacobi coordinates is related by the so-called kinematic rotations:
(j)

(i)

(i)

(5.9)

(j)

(i)

(i)

(5.10)

ρ1 = −ρ1 cos(γk ) + ρ2 sin(γk ),ρ2 = −ρ1 sin(γk ) − ρ2 cos(γk ),


mk −1
(This is ijk π/3 for identical masses.)γk = ijk tanµ3B
The conjugate momentum are labeled k1 (k) , k2 (k) .

(5.11)

106
An advantage of the mass-scalings is that the kinetic energy operator has the same form forall permutations, which is simply the sum of the kinetic energies for the two Jacobi vectors
Hrel = −

X
1 X 2
∇ρi +
U (ri − rj ),2µ3Bi(5.12)

i<jwhich is the Hamiltonian for the relative motion of identical mass particles.
The Jacobi coordinates suffer from two main difficulties. First, when the system is composedof identical particles, it becomes laborious to perform many kinematic rotations. Often, the wave
function is written using the Faddeev decomposition [163]
Ψ3B =

X

(k)

(k)

(k)

Ψ3B (ρ1 , ρ2 ),(5.13)

l(k)

where each function Ψ3B is a Fadeev wave function. The flexibility of the Faddeev wave function
allows description of various three-body structures in terms of few two-body angular momentum
in each component. These wave functions satisfy the Faddeev equations [164]


(1)
(2)
(3)
(1)
(T − E)Ψ3B + U (1) Ψ3B + Ψ3B + Ψ3B = 0,


(2)
(1)
(2)
(3)
(T − E)Ψ3B + U (2) Ψ3B + Ψ3B + Ψ3B = 0,


(1)
(2)
(3)
(3)
(T − E)Ψ3B + U (3) Ψ3B + Ψ3B + Ψ3B = 0,(5.14)(5.15)(5.16)

where E is the total three-body energy, T is the kinetic energy operator, and U (k) = U (ri −rj ) is the
pairwise interaction between particles i and j. The second difficulty that the Jacobi coordinates
suffer from is a lack of a discretization of the three-body continuum: there is a continuum ofenergies that can be shared between the Jacobi pairs. For two bodies, this is achieved through thedecomposition of the continuum into partial waves. This was first remedied by Delves [165, 166] byswapping the magnitude of the Jacobi vectors for the hyperradius R2 = ρ21 +ρ22 and a hyperrangle
ρ1 = R sin ω, ρ2 = R cos ω. It is also convenient to define the hypermomentum K 2 = k12 + k22 .
(k)The Faddeev wave function can be written in Delves coordinates as Ψ3B (R, ω, θ1 , φ1 , θ2 , φ2 ), andcan be decomposed into a discrete sum of spherical harmonics for the spherical angles of theJacobi vectors and a discrete sum of hyperspherical harmonics for the hyperangle. Delves showed107that the in the hyperspherical representation of the three-body continuum the scattering S-matrix
is symmetric and unitary and can be represented by a countable set of channels rather than acontinuum [165, 166]. In the following section, I outline the related Smith and Witten democratic
hyperspherical coordinates [167].

5.1.2

Hyperspherical Coordinates

In this section, the Smith and Witten democratic hyperspherical coordinates are presented[167, 168], which are the coordinates used in the numerical results presented later in this chapter.
In the COM frame, the molecular plane of the three atoms defines a body-fixed frame whoseorientation in the space-fixed COM frame is specified by the three Euler angles (αβγ) as is typicalfor describing the motion of a rigid body [169]. The orientation of the three atoms within themolecular plane is described by the hyperradius and two hyperspherical ‘kinematic’ angles θ and φ,which capture the spatial extent and relative motion of the three atoms, respectively. The problem
can then be reduced entirely to the body-fixed frame where all atoms are treated on equal footing(democratically) rather than requiring kinematic rotations or the Faddeev decomposition.
Let’s first make a connection with the mass-scaled Jacobi coordinates:(ρ1 )x = R cos(θ/2 − π/4) sin(φ/2 + π/6),(ρ1 )y = R sin(θ/2 − π/4) cos(φ/2 + π/6),(ρ1 )z = 0,
(ρ2 )x = R cos(θ/2 − π/4) cos(φ/2 + π/6),(ρ2 )y = R sin(θ/2 − π/4) sin(φ/2 + π/6),(ρ2 )z = 0.

(5.17)The domain of the hyperrangles is restricted to θ ∈ [0, π/2] and φ ∈ [0, 2π/3] to ensure that thewave-function is single valued for three indistinguishable particles. The z-axis is defined by the
direction of ρ1 × ρ2 , which is perpendicular to the plane containing the three atoms. The x-axis is

108
the direction of the smallest moment of inertia. Under this construction, identical particle symmetryis easily imposed [168].
The wave function is rescaled Ψ3B = R5/2 ψ to simplify the kinetic energy operator in the
Schrödinger equation



1 ∂2
Λ2
−
+
+ U (R, θ, φ) ψ = Eψ,
2µ3B ∂R2 2µ3B R2(5.18)√
where µ3B = m/ 3 is the three-body reduced mass for identical masses. Λ is the grand angularmomentum operator
Λ2= Tθ + Tφ + Tr
2µ3B R2∂∂
2
sin 2θ ,Tθ = −µ3B R2 sin 2θ ∂θ
∂θ


∂
1
Jz 2
i
Tφ =
− cos θ,2
µ3B R2 sin2 θ ∂φ
Jy2
Jx2
Jz2
Tr =
+
+
.
µ3B R2 (1 − sin θ) µ3B R2 (1 + sin θ) 2µR2

(5.19)The operators (Jx , Jy , Jz ) are the body-frame components of the total angular momentum J. The
volume element is
dV = 2dR sin 2θdθdφdα sin βdβdγ.

5.1.2.1(5.20)

Adiabatic Hyperspherical Wave Function

A common assumption made is that the hyperradius is a slow variable relative to the variationof the hyperrangles Ω = {θ, φ, α, β, γ} in the wave function. This leads to the formally exact
expansion of the wave function ψ(R, Ω) in terms of the complete, orthonormal set of angularchannel wave functions Φν and radial wave functions Fν ,ψ(R, Ω) =XFν (R)Φν (R; Ω).

(5.21)

νThe channel functions satisfy the fixed-R, hyperangular partial differential equation
Λ2 + 15/4+ U (R, θ, φ) Φν (R; Ω) = uν (R)Φν (R; Ω),2µ3B R2(5.22)

109
where the eigenvalue uν (R) depends parametrically on R and is referred to as the adiabatic threebody potential. Typically, the channel function is expanded on the Wigner D functions of theEuler angles, which are the basis functions for the rotation group [88], asΠ
ΦJM(R; Ω) =νXJ
ΦKν (R; θ, φ)DKM(α, β, γ).

(5.23)K

The quantum numbers K and M denote the projections of J onto the body-fixed and space-fixed
z-axes, respectively, and Π denotes the parity quantum number (+ or −). For three free particles,
the solutions of Eq. 5.22 are the hyperspherical harmonics [170, 171]
Λ2 Yλµ (Ω) = λ(λ + 4)Yλµ (Ω),(5.24)

where λ is the hyperangular momentum quantum number, and µ labels the set of degenerate states.
In this thesis, only the s-wave interactions with the total symmetry state J Π = 0+ are considered.
For free particles, ν = λ, and the adiabatic three-body potential uλ (R) therefore has the form of acentrifugal potential
uλ (R) =λ(λ + 4) + 15/4,
2µ3B R2(5.25)

which can be rewritten in the form of the an effective angular momentum lef f (lef f + 1) = λ(λ +4) + 15/4 in which case lef f = 3/2 + λ.
For the general interacting problem, the adiabatic three-body potential is often written interms of the parameter s(R) asuν (R) =s(R)2 − 1/4,
2µ3B R2(5.26)

with the identification ν = s − 2. There are often other symbols used, and their meaning is madeclear by comparison with the form of the hyperangular eigenvalue, either Eq. 5.25 or 5.26.
Inserting the adiabatic decomposition into the full Schrödinger equation and integratingout the hyperangular eigenfunctions yields a set of coupled ordinary differential equations, in thehyperradius R, as





1 d2
1 X
d
−
+ uν (R) Fν (R) −2Pνν 0 (R)+ Qνν 0 (R) Fν 0 (R) = EFν (R).
2µ3B dR22µ3B 0dRν(5.27)

110
The coupling elements Pνν 0 (R) and Qνν 0 (R) involve partial derivatives of the channel functions asfollows:∂Pνν 0 (R) = Φν (R; Ω)Φν 0 (R; Ω)
∂R


∂2
0
Qνν 0 (R) = Φν (R; Ω)
Φ
(R;
Ω)
.
ν
∂R2

(5.28)The P ’s consequently have no diagonal contribution. For a system of identical bosons with scattering length a0 , they have a particularly simple form in terms of ε = s(R)2 [172, 173]:p
ε0m ε0nPmn =,εm − εn
s #
#
"
"
p
 
0
00
0 ε0
2ε
ε
ε
1 ε00n 2 1 ε000
ε0m
n
n m
n
n
+
−
Qmn − δmn −+ (1 − δmn )
.
0
0
2
4 εn
6 εn
(εm − εn )(εm − εn ) ε0n(5.29)In the adiabatic limit, the system is completely uncoupled as a function of the hyperradius andthe off-diagonal P ’s and Q’s are set to zero. This is the starting point for the analysis of thefollowing section, which deals with analytic solutions to the hyperspherical Hamiltonian for zerorange, pairwise pseudopotential interactions. For the remainder of this and the following chapter,
I consider only equal mass atoms.

5.2

Analytic Solution of the Three-Body Problem in the Zero-Range Model
In general, analytic solutions of the three-body problem are scarce, however, in the zero-range model discussed in Sec. 2.2.5, the problem can be solved with the Fermi pseudopotential
by enforcing the Bethe-Peierls boundary condition at the origin. For two bodies, this boundary
condition needs to be only applied once in the limit of zero separation, however, for three bodies,
the boundary condition must be applied three times–once for each pairwise interaction.
!
a0
lim Φν (R; Ω) = 1 −Cν(k) ,
(k)
(k)
dk ρ1
ρ1 →0
(k)

where Cν(5.30)

is a constant determined by the Bethe-Peierls boundary condition and the overall

normalization of the hyperangular channel function, |hΦν |Φν i|2 = 1. The hyperrangular differential

111
equation, Eq. 5.22, can be rewritten as a Lippman-Schwinger equation and the channel function
solution can be phrased implicitly through integral equationΦν (R; Ω) = −2µ3B R2Z

dΩ0 Gν (Ω, Ω0 )3
X

Vpseudo (dk ρ1 (k) )Φν (R; Ω0 )(5.31)kwhere Gν is the hyperangular Green’s function given in Ref. [172]. This integral can be evaluateddirectly due to the delta-functions in the pairwise potentials, and the result is a matrix equation
(1)

(2)

(3)

that can be solved for the vector of coefficients Cν = (Cν , Cν , Cν )Cν(k) =X00Mνk ,k Cν(k ) ,(5.32)

k0

where the matrix M is given by [172]

Mk0 ,k =




1/4

 3√ aR0 s cot(sπ/2) k = k 0 ,
2
√



− 321/42 aR0 sin(sπ/6)
sin(sπ/2) .

(5.33)

k 6= k 0 .

We would expect an infinite number of solutions s(R) at each hyperradius, which may be obtainedby solving for the roots of the transcendental equation det(Mν − 1) = 0. Insertion of these roots
into Eq. 5.32 yields the the solution for Cν , from which the channel functions can be constructed.
Often however, only the s(R) are of immediate interest as they allow the adiabatic three-bodypotentials to be constructed from which the bound state spectrum can be studied.
(1)

When considering three identical bosons (Cν

(2)

= Cν

(3)

= Cν

= Cν ), the M matrix has onlya single element. The transcendental equation generated in this case is√31/4 s cos(sπ/2) − (8/ 3) sin(sπ/6)
R
= √
,
a0
sin(sπ/2)2(5.34)

which is the transcendental equation for real roots. Making the substitution s(R) → is(R) gives
the transcendental equation for imaginary values√R
31/4 s cosh(sπ/2) − (8/ 3) sinh(sπ/6)= √.
a0sinh(sπ/2)2(5.35)In the following section, the solutions, s(R), of these equations are discussed as a function of thehyperradius, highlighting the important physics.

112

5.3

The Spectrum of Three-Body Bound States
The roots s of the transcendental equations for fixed R can be labeled beginning with any

imaginary roots in the zeroth place and then counting the real roots of increasing value {is0 , s1 , ...}.
As R is increased, the transcendental equations must be iteratively solved. This generates anarray si (R) from which an array for the three-body adiabatic three-body potential uνi (R) given
by Eq. 5.26 can be constructed. The set {uν0 (R), uν1 (R)...} comprises the network of hyperradialpotential curves extending from the region of validity of the zero-range interaction R  r0 to beyond
the asymptotic region R  a0 . It is convenient to refine this further into regimes which contain
universal and non-universal physics for which the ‘range’ rvdW is the appropriate demarcator [174]:• Non-universal Region: (R . |rvdW |) sensitive to the details of the interatomic interaction beyond the scattering length.
• Universal Region: (|rvdW |  R) depends only on the scattering length.
In this section, I discuss only the adiabatic three-body potentials in the non-universal and universal
regions in the resonant limit R/|a0 | → 0, and obtain the bound state spectrum. Away fromresonance, when the scattering length is finite, it is useful to distinguish additional regions in theuniversal region [174]:(1) Scale-Invariant Region: (|rvdW |  R  |a0 |).
(2) Transition Region: (R ∼ |a0 |).
(3) Asymptotic Region: (|a0 |  R).
I make this distinction in the following section (Sec. 5.4), which is focused on the three-bodycontinuum.
Non-Universal Region: Although the zero-range model does not extend into the nonuniversal region, the effects of propagation through this region can be summarized by the logarithmic derivative Γ that arises from matching the wave function at the boundary (R ∼ R0 ) of the

113
20

0

2

4

6

8

-20

0.5
0
-0.5

Figure 5.2: A plot of the transcendental equations (Eqs. 5.34–5.35) as a function of real (a.) andimaginary (b.) s in the resonant limit R/|a0 | → 0. The si are the roots labeled in increasing order,
where s0 is the root associated with the Efimov potential Eq. 5.37.

universal and non-universal regions
Φ0ν (R; Ω)= Λ,Φν (R; Ω) R=R−

(5.36)

0where the prime indicates a derivative w.r.t. the hyperradius and the subscript ‘-’ indicates that
this is the logarithmic derivative of the unknown wave function for the non-universal region. As
we will see shortly, Λ plays an important role in setting the spectrum of three-body bound states
and also appear in the scattering observables.
Universal Region: When the hyperradius is less than the scattering length, each of thethree atoms are within the two-body scattering cross section of the others, and the physics is114
truly three-body. It is perhaps no surprise then that this region hosts nonperturbative physics.
In Fig. 5.2, the transcendental equations are plotted over imaginary and real values of s in theresonant limit R/a0 → 0. There is a single imaginary root is0 ≈ 1.00624i independent of R in thislimiting case, which when inserted into the adiabatic three-body potential (Eq. 5.26) produces anattractive potentialuν0 (R) =−s20 − 1/4.
2µ3B R2(5.37)

It is well known that an attractive 1/R2 potential supports and infinite number of bound states
which accumulate at zero energy. It is however not physical to imagine the critical limit R = 0
which extends beyond the range of validity of the zero-range model. Generally, there is some cutoff
distance R0 ∼ rvdW beyond which the interaction becomes strongly repulsive, which can be seen for
instance by performing calculations with a Lennard-Jones potential or other models of finite range.
The three-body bound states that form are referred to as Efimov states, and their properties werefirst discussed by V. Efimov (see Refs. [31, 32, 33]). There are also an infinite number of real roots{s1 , s2 , ...}, (s1 ≈ 4.46529) which correspond to repulsive adiabatic three-body potentials, and thefirst few are shown in Fig. 5.2a.
In the resonant limit, there is not an asymptotic region in the sense characterized above.
The spectrum of Efimov states can be obtained by matching the non-universal wave function in
the lowest channel with a Bessel function with short-range phase shift ∆ [31]Fν0 (R) ∝ sin(|s0 | ln(KR) + ∆).

(5.38)At the boundary of the two regions, equating the log-derivatives gives
Λ=

|s0 |
cot (|s0 | log(KR) + ∆) ,R0

(5.39)and evaluating when the phase shift passes through π/2 gives the trimer spectrum at unitarity
E

(N )



ΛR0
1
2
arctan− ∆ e−2πN/|s0 | .
=−
exp
|s0 |
|s0 |
2m0 R02

Successive levels are separated by a geometric scaling E (N ) /E (N +1) = exp(2π/|s0 |).

(5.40)

115

a
hb

s
Fe
ch
er

m

di

Efimov trimer

Figure 5.3: The canonical plot of the spectrum of three-body bound states (Efimov trimers) near a
Feshbach resonance. At resonance, there is a condensation of an infinite number of Efimov statesat threshold which is not shown.

The cumulative effect of Λ and ∆ in Eq. 5.40 is to set the spectrum of Efimov states. Often
this spectrum is reformulated
iN κ2
h
∗
E (N ) = − e−2π/|s0 |,m(5.41)

in terms of the three-body parameter κ∗ , which is the wave number associated with the energyof the first Efimov state E (0) that appears as the Feshbach resonance is approached from negative
scattering lengths. The scattering length a−3B at which the first Efimov state is formed is also often
referred to as a three-body parameter. Recently, it was observed experimentally and confirmed
theoretically that for ultracold alkali species, both κ∗ and a−
3B have a universal value when rescaledby the van der Waals scattering length, which seem to lie near κ∗ ≈ 0.226/rvdW and a−
3B ≈−9.73rvdW [175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192,
193, 194, 195, 196, 197, 198, 199, 200]. This is due to a semiclassical suppression at short distancesof the wave function as it traverses the sharp cliff created by deep two-body potentials supportinga large number of bound states.
In the next section, I move from the bound state spectrum to address the three-body continuum, focusing on the threshold behavior of the elastic three-body phase shift.

116

5.4

The Three-Body Continuum
The asymptotic region has a comparatively clearer meaning for the two-body problem com-pared to the three-body problem. Two atoms are asymptotically separated in the limit that the
distance between their nuclei approaches infinity. For three atoms, there are four limits: the limit
in which all three atoms are infinitely separated, and the three limits in which one of the atoms
is infinitely separated from the remaining pair. This underscores a general feature which is thatthe two-body continuum is embedded in the three. In the resonance limit R/a0 → ∞, this does
not pose a problem because each atom is within the scattering cross section of every other atom
over the entire hyperradial range, and therefore the physics is truly three-body all the way out toinfinity. When the scattering length is finite, from general length scale considerations, the physics
of the transition region must be that of on-shell and off-shell two-body scattering events. In thehyperspherical picture, it is difficult to connect this diagrammatic view of the hyperradial regionswith the adiabatic three-body potentials themselves, and this is the main obstacle against calculating observables for three-body elastic scattering, beginning at the level of the three-body elastic
phase shift as we shall see in this section. First, through a WKB model in Sec. 5.4.1, we will see
how Efimov physics functionally enters into the threshold behavior of the three-body elastic phaseshift. The final two sections (Sec. 5.4.2–5.4.3) are concerned with connecting the diagrammaticinterpretation with the adiabatic three-body potentials and separating the two-body continuum
contributions to the phase shift from the three, respectively.
Let’s begin by resuming our discussion of the regions began in the previous section, but now
for a finite scattering length. In the scale-invariant region (|rvdW |  R  |a0 |), the roots behave
as in the universal region in the previous section, namely they take on the resonant values. Thisregion is invariant of the length scale set by the scattering length. As the hyperradius approaches
R ∼ |a0 |, the scattering length reenters the problem as the asymptotic region is approached.
Transition Region: Beyond the scale-invariant region, the adiabatic three-body potentials
begin a changeover to their asymptotic forms. For a0 < 0, the Efimov potential transitions into a

117

0.5

0

-0.5

0

5

10

15

20

25

30

25

30

0.5

0

-0.5
Atom-Dimer Channel

-1

-1.5

0

5

10

15

20

Figure 5.4: The adiabatic three-body potentials for a0 < 0 (a.) and a > 0 (b.). The orange shaded
region is the scale-invariant region, the yellow shaded region is the transition region, and the green
shaded region is the asymptotic region. The black curves are a result of solving the transcendental
equations (Eqs. 5.34–5.35) and the red curves are the three-body potentials corresponding to theindicated values.

118
repulsive centrifugal barrier, characterized by a hump whose height depends on a0 , which appears
near the boundary between the transition and scale-invariant regions as shown in Fig. 5.4a. For
incoming energies below this barrier, the wave function must tunnel to reach the universal region
and the well of the Efimov potential. Scattering in this channel should therefore display a shaperesonance when each new Efimov trimer is formed. The higher adiabatic three-body potentials
corresponding to real roots transition from repulsive universal potentials to repulsive centrifugal
barriers. For a0 > 0, the Efimov potential transitions into the shallow s-wave Feshbach atomdimer potential with bound-state energy −1/ma20 shown in Fig. 5.4b. The Efimov effect therefore
manifests itself in the scattering of three free atoms through transition to the atom-dimer channel
and vice versa. This is the process of three-body recombination mentioned in the introduction ofthis chapter and of the time-reversed process three-body dissociation.
In the transition region the roots s(R) and adiabatic three-body potentials are not constant.
Rather, to characterize the potentials in this region, the roots can be expanded in powers of thesmall parameter a0 /R about their asymptotic values λ + 2
sn (R) = (λ + 2) +X  a 0 i
R

i=1

(i)

βλ .

(5.42)

The coefficients β (i) of this power series can be calculated by insertion of sn (R) into the transcendental equation (Eq. 5.34) and equating each power of a0 /R. When inserted into the adiabatic
three-body potential (Eq. 5.26) the result is the asymptotic expansion of the adiabatic potential: a iλ(λ + 4) + 15/4 X αλ0+.
2µ3B R2
2µ3B R2 R∞

uνn (R) =(i+2)(5.43)

i=1

The corrections therefore come in the form of long-range power law tails beginning at cubicorder. I’ll list here the coefficients associated with increasing order tails for λ = 0, and it is119
illustrative to do this for distinct scattering lengths ai between the pairs
1
8 X ai,(5.44)−→
R3 λ=0
dπµ3B
R3
i
√
 
X a2
1
16(9 + 2 3π) X ai aj
24
i+,(5.45)−→
R4 λ=0d2 π 2 µ3BR49d2 π 2 µ3B
R4
i
i6=j
√
 
2
3
X
1
32(π − 6)ai32(27 + 12 3π − 2π 2 ) X ai a2j−→−+
R5 λ=03d3 π 3 µ3BR527d3 π 3 µ3B
R5
i
i6=j
√
64(3 3 + 4π) a1 a2 a3+,(5.46)
9d3 π 2 µ3B
R5
√
√
 
1
160(2π 2 − 3) X a4i
64(−243 − 162 3π + 297π 2 + 22 3π 3 ) X ai a3j−→ −−R6 λ=03d4 π 4 µ3BR6
243d4 π 4 µ3B
R6
i
i6=j
√
√
64(81 + 54 3π − 72π 2 + 2 3π 3 ) X a2i a2j+
81d4 π 4 µ3B
R6
i6=j
√
√
128(81 3 + 135π + 32 3π 2 ) X ai aj a2k+,
243d4 π 3 µ3BR6
i6=j6=k(5.47)

where d = dk for equal mass atoms.
These coefficients have an interesting structure. Beginning with the cubic potential, we seethat the coefficient has a two-body scaling, which is to say that it is a sum of the pair interaction
strengths. Therefore, this term must represent purely two-body physics, which is to say physics
where only two of the atoms interact while the third is unaffected, embedded in the transition region.
Alternatively, this coefficient can be derived easily through first order stationary perturbation
theoryhΦ0λ (R; Ω)|X
i>jv(rij )|Φ0λ i ∼

P

i ai
,
R3

(5.48)

where Φ0λ (R; Ω) are the free hyperangular solutions (hyperspherical harmonics). This long-range
tail is therefore the hyperangular mean of the pairwise interactions at a fixed hyperradius. In
Sec. 5.4.2, we will see how this term serves to ‘spoil’ calculations of the three-body phase shift.
The coefficients of the quartic potential arise from second order perturbation theory. The
first term quadratic in each of the scattering lengths is purely two-body, however the second term

120
involves a product of two pairs of scattering lengths. This term represents the interaction of multiple
pairs of atoms. In general, higher order tails contain all permutations of the pair scattering lengths
to a given order.
It is tempting to interpret these terms diagrammatically in analogy with Feynman diagramsfor the Born series in the time domain mentioned in Sec. 2.2.2 (see Fig. 2.1). It must be remembered however that these results are for fixed R. To connect with a diagrammatic approachfor calculating an observable, the full expectation value over the entire adiabatic wave functionP
ψ(R, Ω) = ν F (R)Φν (R; Ω) must be taken so that all variables are integrated over. In Sec. 5.4.2,

I connect this approach to some relatively old results from Amado and Rubin [201] for the threshold
behavior of the three-body T-matrix.
Asymptotic Region: The long-range tails decay faster than the centrifugal barrier, andbeyond a certain point these corrections become numerically insignificant, and the potential ispurely centrifugal to a good approximation. The solution of the transcendental equation in thisregion gives only real values for s, and in the limit a0 /R → 0, {s1 = 2, s2 = 6, s3 = 8, ...}. Thisis the set of roots for three identical bosons, which does not include the root s = 4 for the 0+
symmetry. In the asymptotic region, the adiabatic three-body potentials therefore correspond to
repulsive centrifugal barriers, and the wave functions are phase-shifted spherical Bessel functions
with angular momentum lef f = 3/2 + λ multiplied by hyperspherical harmonics
Fνi (R)Φνi (R; Ω) =eiδλ,µ sin(KR − lef f π/2 + δλ,µ )Yλ,µ (Ω),
KR(5.49)

where δλ,µ is the three-body elastic scattering phase shift for the channel (λ, µ). This phase shift ispart of the larger unitary multichannel three-body scattering S-matrix Sλ,µ,λ0 ,µ0 whose off-diagonal
elements describe transitions between channels. When a0 > 0, the asymptotic region also contains
the shallow Feshbach dimer channel (see Fig. 5.4b). The asymptotic form of the atom-dimer
potential isl(l + 1)+ E2B ,
2µ2B R2(5.50)

where l is the relative angular momentum of the dimer with respect to the other atom. In the

121asymptotic limits discussed in the beginning of this section, the limit where all three atoms become
well separated are dominated by the continuum channels whereas the limit where a single atom isseparated from a pair are dominated by the atom-dimer channels [202].
In the following section, I formulate a WKB model which captures the functional dependence of the threshold elastic phase shift on Efimov physics. Sections 5.4.2–5.4.3 are devoted tounderstanding the contributions of the long-range power law tails to the elastic phase shift.

5.4.1

Signature of Efimov Physics in the Elastic Three-Body Phase Shift

The WKB model is a semi-classical approach to solving the Schrödinger equation, relying
on the relative smallness of ~ (see Ref. [203] for an overview). The approximate form for the wave
function in the WKB model is1
FW KB (R) = p
exp (±iS(R)) ,K(R)(5.51)

written in terms of the action integral

S=

Z R

K(R0 )dR0 .

(5.52)

Ri

where the local classical momentum K(R) = 2µ3B

λ(R) =p

E − uν (R) defines a local de Broglie wavelength

2π
.
K(R)

(5.53)The usefulness of the WKB wave function can be gauged by the smallness of the local value of thequantality function
Q(R) =

K 00
3 (K 0 )2
−
4 K4
3K 3



(5.54)

compared to unity. Where is this violated? An obvious point is the classical turning point Rt
where uν (Rt ) = E, K(Rt ) vanishes, and Q(Rt ) diverges making FW KB (Rt ) singular. Even inregions where the WKB method cannot be expected to give numerically accurate results, it canoften provide at least the general functional form of the wave function. It is in this latter sense that

122
I employ the WKB model in this section to obtain general results for the scaling of the three-bodyelastic phase shift including contributions from Efimov physics.
Under favorable conditions, the WKB approximation may be accurate away from Rt on one
or both sides of the turning point. On the classically allowed side
1

FW KB (R) = p
cos
K(R)

Z R

φ
K(R )dR −
2
Rt
0

0



,

(5.55)

where φ is the phase difference (loss) between the incoming and outgoing waves due to reflection
at the barrier. The wave function is matched at the turning point according to the connection
formula
N



p
exp −
|K(R)|

Z Rt
R

0

K(R )dR

0



2

=p
cos
|K(R)|Z R

φK(R )dR −
2
Rt
0

0



.

(5.56)

If the potential is approximately linear at the turning point then the amplitude N and phase φ aregiven by
N = 1,

φ=π.
2(5.57)

This is the general result of the semiclassical or short-wave limit and is the basis of conventional
WKB applications involving a classical turning point.
An important case is when the potential is the centrifugal barrieruν (R) =1 lef f (lef f + 1)
.
2µ3B
R2

The Schödinger equation can be solved exactly in this caselπ
cos KR − ef2f
F (R) =,
KR
where K =√2µ3B E. The asymptotic form of the WKB wave function ispcos KR − lef f (lef f + 1) π2 − φ2
FW KB (R) =.
KR(5.58)(5.59)(5.60)

Using the phase and normalization from Eq. 5.57 gives φ = π/2 disagreeing with the exact result.
This discrepancy can be repaired using the Langer correction where the cenrifugal potential in the123
WKB approximation is ‘corrected’ by the mapping
1 2
lef f | (lef f + 1) → lef f +,2(5.61)uW KB (R) = u(R) +(5.62)

ulanger (R) =1 14,
2µ3B R2

1 14.
2µ3B R2(5.63)The Langer correction is essential to obtain the correct low-energy dependence of the wave function.
We can now extract the functional form of the elastic phase shift for the lowest hypersphericalchannel. In this thesis, I do this only for a0 < 0, where the ladder of Efimov states appear as aseries of shape resonances. The adiabatic three-body potential in this section is simplified to provide
analytic results: the Efimov potential in the universal region is flanked at smaller hyperadii by abarrier in the non-universal region, and a hump where the wave function must tunnel under toreach the asymptotic region. This problem is very similar to that analyzed by M. V. Berry in
Ref. [204], where the WKB wave function was constructed for the case of three turning points.
The first pathway is to tunnel from the classical turning point R ∼ 1/K and reflect off of thehump at the edge of the scale-invariant region which is at a distance R ∼ |a0 | where the potential
begins changing form. This pathway contributes a tunneling phase to the total phase shift!
Z 1/K rlef f (lef f + 1) − 1/4tan(δI ) ≈ exp −2E−
dR = (K|a0 |)2lef f +1
(5.64)R2|a0 |In the threshold limit, the collision energy contribution provides comes in as higher order corrections
and is therefore discarded in this simple calculation. The second pathway is to tunnel from theclassical turning point into the well of the scale-invariant region and back off the potential humpon the way out (see Fig. 5.4a). This process acquires the phase
!




Z R0 r 2s0|a0 |2lef f +1
2lef f +1
tan(δII ) ≈ (K|a0 |)tandR = (K|a0 |)tan s0 ln+ ∆ . (5.65)
R2
R0
|a0 |
The final term is the unknown phase shift due to the non-universal region. The poles of this function
correspond to three-body resonances associated with Efimov trimer formation.

124
In the threshold limit, the total phase shift corresponds to a sum of these two contributions,and the first term (lef f = 3/2) of the three-body effective range expansion istan(δI + δII )
−
−−−→
K→0
K4






|a0 |
A + C tan s0 ln+∆
a40 = A3B
R0

(5.66)

where A and C are proportionality constants. A3B is the three-body elastic scattering volume
having units of length4 , due to the three additional degrees of freedom compared to the two-bodyproblem, and was first derived by Efimov in Ref. [32]. Near the formation of an Efimov trimer,A3B diverges.
This is the leading order threshold contribution of Efimov physics to the three-body elastic
phase shift. The K 4 dependence reflects the fact that this is a short range contribution, which canbe checked by inputting lef f in Eq. 2.31 from Chapter 2, which is the Wigner threshold law for
short-range potentials. In a numerical calculation of the three-body elastic phase shift for a finite
range potential, there are however contributions to the phase shift that arise at lower orders of K.
It is well-known in the scattering community that long-range power law potentials can modify theusual threshold laws. In the next section, we discuss how the power law tails in the transition region
contribute lower orders of K to the elastic phase shift. These contributions however do not containthe effects of three-body resonance physics from the inner well, which is described in Eq. 5.66 andtherefore this section and the following can be viewed as complimentary.

5.4.2

Long-Range Contributions to the Elastic Three-Body Phase Shift

There is some old evidence that the threshold behavior of the three-body phase shift does
not begin at order O(K 4 ), coming from a paper of Amado and Rubin [201], who showed that
S3B (K) − 1|Connected ≈ AK 2 + BK 3 + CK 4 ln(K) + K 4 D + O(K 5 ) + · · ·(5.67)

for proportionality constants A, B, C, D where S3B (K) is the elastic three-body S-matrix2 . The
Efimov contributions derived in the previous section enter as nonperturbative contributions to the2

The coefficients A and B are unknown. C was calculated analytically in Ref. [205]. The nonperturbative portionof D was calculated in Ref. [206].

(a) disconnected-scattering

125

Figure 5.5: Diagrammatic representation of the three-body scattering processes which may occur
with pairwise interactions. The grey circles represent a pairwise interaction summarized by thetwo-body T-matrix. I have excluded the permutations of each scattering diagram.

D = Dpert + Dnonpert coefficient [174], where the dependence on the three-body parameter κ∗ ishidden in the nonperturbative portion.
Equation 5.67 is the result of extracting the energy dependence at each order of the equivalentof the Born series3 for three-body elastic scattering, including only the ‘connected’ contributions,which is to say all contributions that did not arise from disconnected scattering of a pair of atoms
while the third is noninteracting, so-called spectator scattering. Each power in this expansion
results from a certain order in the Born series and has a diagrammatic interpretation in the senseof Feynman diagrams (see Fig. 2.1). The first term is the so-called double-scattering contribution,
the second is triple-scattering, etc... which are summarized in Fig. 5.5. The double scattering
3

For three-body the corresponding series is the Faddeev series [163].

126
contribution results from an on-shell or off-shell binary collision followed by a binary collision
between a distinct pair. There is a well-known re-scattering singularity when both collisions indouble-scattering process happen on shell, and this causes difficulties when looking at, for instance,
the scattering amplitude in the space of final momentum [72, 73]. Generally, any scattering event
which occurs entirely on the shell of a subset of the total number of atoms appears as a singularcontribution in the scattering amplitude for the total system4 . This diagrammatic calculation in
the momentum basis for the threshold behavior of the elastic three-body S-matrix has however
not been connected with an approach which discretizes the continuum, such as the hypersphericaladiabatic representation. It is hoped that this connection can introduce power numerical techniques
into the challenging three-body elastic scattering problem.
In the adiabatic hyperspherical representation, the transition region spans the length scale
where one would expect the on-shell and off-shell binary collisions in Fig. 5.5 to occur. In what
follows, I will sketch out some partial results connecting the results of Amado and Rubin with thephase shift due to the long-range power law tails in the transition region.
Scattering from long-range potentials can modify the usual Wigner threshold laws. The
expansion in Eq. 5.43 contains a leading order cubic tail, and the threshold behavior for thispotential is linear in K, which can be seen, for instance, using the WKB analysis from Ref. [207].
Let’s detail the threshold behavior of each of the long-range tails separately, to show how the a tailof power i + 2 modifies the threshold laws for general λ
K i + · · ·,if i < 2λ + 4,





(i+2)
δλ
∼ K 2λ+4 ln K + · · ·, if i = 2λ + 4,







K 2λ+4 + · · ·if i > 2λ + 4.

(5.68)

where the dependence on µ has been suppressed in the phase shift for notational clarity, and thesubscript of the phase shift indicates the order of the corresponding long-range potential. Power
law tails of order i = 2λ + 4 are therefore at the critical value between short-range and long-range,
4This is due to the presence of momentum-restricting delta functions describing the process.

127
and the logarithm generally complicates the effective range expansion and threshold laws [207]. At
order K 4 , the phase shift for the full problem therefore contains both long-range and short-range
contributions, and there is an analogous separation of the K 4 coefficients in terms of power-law tailand short range Efimov physics complementing the perturbative and nonperturbative separationof the D coefficient in Eq. 5.67.
The coefficient of the leading order threshold behavior for a long-range potential depends
only on the form of the potential and was derived by Landau in Ref. [208] for dispersion forces
related to the scattering of slow particles. For the K i (i 6= 2λ + 4) term this is
(i+2)
δλ
≈−

√

(i+2) i3
i−1
a0 Γ( i+12 )Γ(λ + 2 − 2 ).
3i+3
Γ( i+22 )Γ(λ + 2 + 2 )παλ4(5.69)

Therefore, for λ = 0, the leading order expansions of the phase-shift from each power-law tail in
Eq. 5.43 up to the critical value are(3)(3) 15δλ=0 = α0

4

(4)

(4) π

K + · · ·, δ0 ∼ α0

48

(5)

(5) 16

K 2 + · · ·, δ0 ∼ α0315

K 3 + · · ·.
(5.70)In the Born approximation, the total phase shift due to long-range power law tail in Eq. 5.43 is
equivalent to a sum of these threshold results. For λ = 0, this gives
δλ=0 = A0 K + B 0 K 2 + C 0 K 3 + D0 K 4 ln(K) + E 0 K 4 + · · ·,(5.71)

which is consistent with the energy dependence of the phase-shift in Eq. (5.67), predicted by Amado
and Rubin [201]. It should be noted that beyond leading order contributions to the threshold laws
are also contained in the coefficients B 0 , C 0 , D0 , ... and therefore the dependence is not quite as
simple as what Eq. 5.70 suggests.
The coefficient A0 corresponding to the cubic tail is however the only term that is left ‘uncontaminated’ by the threshold behavior of the higher ordered tails. This tail has a purely two-bodyscaling, which was addressed in Sec. 5.4. We can therefore identify it directly with disconnected
spectator scattering by matching with Eq. 5.67. It is however erroneous to extend this analogy

128
between the higher order power law tails and Amado and Rubin’s double, triplet, etc.. scattering
contributions shown diagrammatically in Fig. 5.5. This is due to terms with purely two-body scaling inside of the coefficients listed in Eq. 5.45–5.47. These are the terms which would survive in the
spectator problem where only one of the pairs interact. To match with results for the coefficients in
Eq. 5.67, the disconnected scattering contributions must therefore be entirely removed. In the next
section we will see that this is also necessary to obtain numerically converged elastic three-bodyphase shifts, and efforts to remove these disconnected contributions are discussed.

5.4.3

Removing Disconnected Two-Body Physics from the Elastic Three-Body
Phase ShiftIn Sec. 5.4.1, I investigated the signature of Efimov physics to the three-body elastic phaseshift, which arise as O(K 4 ) log-periodic contributions at threshold. In Sec. 5.4.2, contributions
from long-range power law tails in the transition region to the elastic phase shift were considered
where it was observed that the usual Wigner threshold laws require modification with contributions
beginning at order K, K 2 , K 3 , K 4 ln K in addition to terms of order K 4 and higher. The resultant
energy dependence was compared to the results of Amado and Rubin’s diagrammatic calculationof the connected contributions. To quantitatively match adiabatic hyperspherical representation
calculations to these results, a method for removing the disconnected contributions must be formulated. In this section, I discuss a factorization method due to R. G. Newton [209] and present
some numerical evidence. I finish this section by pinpointing how this removal is also required toobtain phase shifts that are converged with higher values of λ.
Let’s start by defining the scattering objects for the disconnected scattering problem, where
only one pair of atoms is interacting. We first define the three-particle disconnected S-matrix as
Si = 1 − 2πiδ(E − EK )Ti ,(5.72)

which is appropriate for the spectator scattering of particles j and k while i is noninteracting. Ti

129
is the disconnected three-body T-matrix, satisfying the Lippmann-Schwinger equation
Ti = Vi − Vi G3B
0 (z)Ti ,(5.73)

where G3B
0 is the Green’s operator for the three-particle non-interacting Hamiltonian (see Eq. 2.22),
and Vi is the pairwise interaction between atoms j and k.
Let’s see how the disconnected T-matrix arises in the Lippmann Schwinger equation5 forthe three-body T-matrix of the full problem. The three-body S-matrix can be written in terms of
the T-matrix
S3B = 1 − 2πiδE T3B ,(5.74)

where δE = δ(E − EK ) ensures the collision happens on the three-particle energy shell, and
T3B =

X

Ti +

i

X
i6=j

Ti G3B
0 Tj + · · · .

(5.75)

Therefore, the entire disconnected contribution is isolated in the first term.
In Refs. [72, 73], the S-matrix was decomposed as a sum S3B = [1 − 2πiδEP

i Ti ] + S̃, whichisolates the disconnected portion from the remainder S̃. In this factorization, a generalized expression was derived for the scattering cross section for two beams impinging on a fixed target,
highlighting the physical meaning of the disconnected and rescattering contributions. In this section, however, I pursue another factorization
S3B = S1 S2 S3 S(C) ,(5.76)

which produces the connected operator S(C) . This decomposition is adopted here because it has
been proven that S(C) is compact [209]. This means that it enjoys the same important properties as
the two-particle S-matrix: A discrete spectrum of eigenphases exp(2iδn ) associated with a complete set of eigenvectors. Eigenvectors of the multichannel scattering problem are generally linear
combinations of the scattering wave functions of multiple channels. Additionally, the eigenphases
are the eigenvalues of the unitary S-matrix and therefore lie on the unit circle. Furthermore, the5Technically these are the Faddeev equations [163].

130
eigenphase sum,

P

(C)
(C)
n δn (E) = ∆ep (E), should change by π whenever a three-body bound stateis formed, as well as giving a Briet-Wigner form for the resonance away from the background value(C)∆ep (0) [210], satisfying the three-body Levinson’s theorem [211]
N3B =

1 (C)
(∆ )(0) − ∆(C)
ep (∞)),
π ep(5.77)

where N3B is the total number of supported three-body bound states. The properties of theeigenphase sum also do not depend on the order in which the S matrix is factorized in Eq. 5.76,(C)which can be seen from properties of the determinant, ∆ep = log [det(S3 S2 S1 S3B )] /2i.
First, I follow Ref. [209] to derive and interpret the explicit form of S(C) that results fromthe factorization in Eq. 5.76. Then, I demonstrate the numerical convergence of S(C) in the adiabatic hyperspherical representation. Finally, I investigate whether the threshold behavior of theconnected S-matrix is quadratic to leading order in K, bolstering the identification of the cubic
tail with disconnected physics made in the previous section.
The derivation that Eq. 5.76 removes the disconnected contributions proceeds by requiringthat S(C) have the following form
S(C) = 1 − 2πiδE T(C) .

(5.78)

Substituting this expression into Eq. 5.76 yields
1 − 2πiδE T = S1 S2 S3 − 2πiS1 S2 S3 δE T(C) .
Both sides can be expanded to


X
X

1 − (2πi)δE 
Ti +Ti G3B
0 Tj + · · ·= 1 − 2πiδE

i

i6=j

X
i

Ti + (2πi)2 δE

Xi<jTi δE Tj − (2πi)3 δE

Yi(5.79)Ti δE Ti δE Ti  − 2πiS1 S2 S3 δE T(C) ,(5.80)

131
and the spectator piece subtracted. After sorting out6 all of pieces, we get
D
T(C) = S3† S2† S1† (T(C)+ · · ·),

(5.81)

D=
T(C)(5.82)

X
i<j


+
Ti G−
0 Tj + Tj G0 Ti .

Therefore, T(C) contains double scattering events at leading order. However, the factorization
has done more than just subtract off the spectator collisions, scrambling the original expressionP
T D = i6=j Ti G3B
0 Tj for the double-scattering T-operator in the process. It is therefore not correct
D to the double scattering results of Amado and Rubin.
to compare T(C)

Newton’s factorization therefore removes the disconnected scattering contributions at thecost of producing the altered connected S(C) . With this in mind, we check the threshold behaviorof the eigenphase shifts and whether or not removing the disconnected contributions leads to aconvergent calculation of S(C) in the adiabatic hyperspherical basis.
In the previous section, I surmized that the 1/R3 contribution to the long-range power law tailin the transition region was purely two-body in nature and was responsible for the O(K) thresholddependence of the elastic phase shift (see Eq. 5.70.) The connected eigenphases therefore should bequadratic to leading order, and it is possible to prove this threshold result in the adiabatic limit7
where the channel couplings can be ignored. To do this, I will deal instead with the connected
eigenphase sums that should also be quadratic to leading order and are also a natural candidate
for assessing the converge of a calculation of S(C) with increasing channel number. It is possible to6
The relation δE = −2πi(G+ − G− ) and the + or − indicates the limit from above or below of G3B
0 (z + i0) is
useful.
7
In fact, the zero-range analysis shows that the P and Q couplings among three-body continuum states are
asymptotically proportional to 1/R2 and 1/R3 , respectively, while they are 1/R5/2 and 1/R7/2 between boundchannels and continuum channels (see, for instance, Ref. [212].)132
derive a simple relation between the sums beginning with the decomposition in Eq. 5.76
det(S3B ) = det S1 S2 S3 S(C) ,(5.83)

det(S3B ) = det(S1 ) det(S2 ) det(S3 ) det(S(C) ),
!
3


X
i
exp (2i∆ep ) = exp 2i∆ep exp ∆(C)
,
ep
"

i

log [exp (2i∆ep )] = log exp 2i
∆ep =

3
X

3
X

∆iep

i

!



exp ∆(C)ep

(5.84)(5.85)#,∆iep + ∆(C)ep ,(5.86)(5.87)iwhere ∆ep is the eigenphase sum for S3B , ∆iep is the eigenphase sum for Si , and I have utilized
some standard properties of the determinant. In the adiabatic limit, the eigenphase sums are justPthe sum of the phase shifts in each channel (∆ = λ δλ ) which were discussed in the previous

section, Sec. 5.4.2.

Let’s work out the cancelation of O(K)-dependence of the connected eigenphase sum firstwhen the potential in Eq. 5.43 is truncated to include only the cubic tail and imagined to extend
from R ∈ [0, ∞)a αλλ(λ + 4) + 15/4
0
+
,
2
2
2µ3B R
2µ3B RR(3)uν (R) =(5.88)

with coefficients αλ given in Table 5.1. From Eq. (5.69) the analytic expression for the phase shift
Table 5.1: Coefficients αλ for the 1/R3 term in Eq. (5.88) and corresponding phase-shifts as given
in Eq. (5.89). We denote the coefficients for the three-body system where only one of the pairsinteract by α̃λ while for three-body systems where all pairs interact we denote them by αλ . The
subscript in λ(n) indicates the corresponding degeneracy for a given value of λ.
S-matrix: Sk
λ(n)
0(1)
2(2)
4(3)
6(3)
8(3)

(3)
α̃λ
16
dπ
64
dπ ,0
144
dπ ,0,0256dπ ,0,0400
dπ ,0,0(3)δ̃λ0- 32Ka15dπ
0- 128Ka
63dπ ,00- 288Ka
143dπ ,0,00
- 512Ka
255dπ ,0,00- 800Ka
399dπ ,0,0

S-matrix: S3B
(3)
αλ
48
dπ
96 96
dπ , dπ
144 144 144
dπ , dπ , dπ
192 192 384
dπ , dπ , dπ
240 480 480
dπ , dπ , dπ

(3)

δλ

0
- 32Ka5dπ64Ka00- 64Ka21dπ ,- 21dπ
288Ka0 288Ka00- 288Ka
143dπ ,- 143dπ ,- 143dπ
128Ka0 256Ka00- 128Ka
85dπ ,- 85dπ ,- 85dπ
320Ka0 320Ka00- 160Ka
133dπ ,- 133dπ ,- 133dπ

133
for this potential for a given λ is
(3)
αλ
(3)
δλ = −(Ka0 ) + O(K 2 ).
(λ + 5/2)(λ + 3/2)

(5.89)

Table 5.1 lists the phase-shifts for the disconnected problem where only one of the pairs interact
with identical scattering lengths (δ̃λ ), and the phase shifts where all pairs interact (δλ ). Inserting
values from Table 5.1, it is easy to check that
∆(C)
ep =

X
λn


δλn − 3δ̃λn = O(K 2 ),(5.90)and therefore the threshold behavior of the eigenphases are shifted beyond O(K) when disconnected
scattering is factored from S3B .
Going beyond the adiabatic limit and model potential (Eq. 5.88), this shift of thresholddependence to quadratic order can be verified numerically (see Ref. [168] for a description of thecode) using Lennard-Jones interactions, including all couplings between channels. The results ofthis calculation are shown in Fig. 5.6. In Fig. 5.6(a), the results for a system with three different
negative scattering lengths8 (a1 ≈ −10rvdW , a2 ≈ −15rvdW , and a3 ≈ −20rvdW ) are shown. The
solid red curve is the eigenphase sum from S3B and the green-dashed curves are the eigenphasesums for S1 , S2 , S3 . It is apparent that ∆ep , ∆1ep , ∆2ep , and ∆3ep are all linear in K near threshold,which was expected from the analysis of the transition region in Sec. 5.4.2. The blue dotted curve(C)

is the connected eigenphase sum ∆ep , which confirms the shift to quadratic order at threshold.
In Fig. 5.6(b), the results for a system with three different positive scattering lengths9 (a1 ≈+10rvdW , a2 ≈ +15rvdW and a3 ≈ +20rvdW ). The eigenphase sum is defined excluding the
eigenphases associated with the bound channels in this case. Again, the connected eigenphase sum(C)∆ep is shifted to quadratic order at threshold.
To finish this section, I discuss the convergence of the phase shift for the long-range powerlaw tail (Eq. 5.69) in the limit of increasing λ. For the cubic tail, the coefficient to the 1/R3 term of
the potential, αλ , is proportional to λ2 , and therefore the phase-shift (Eq. (5.89)) converges in the
8
9

In the absence of any two-body bound states.
In the presence of two-body bound states.

134|∆ep (K)||∆ep (K)|

(C)
|∆ep
(K)|

(C)
(K)|
|∆ep

0.001

0.01

0.1

0.001

0.01

0.1

Figure 5.6: (a) Eigenphase sum for a system with three different negative scattering lengths (a1 ≈−10rvdW , a2 ≈ −15rvdW and a3 ≈ −20rvdW ). The solid red curve is the eigenphase sum from S3B ,(C)and the green-dashed curves are the eigenphase sums for S1 , S2 , S3 . The blue dotted curve is ∆ep ,which scales as K 2 at threshold. (b) Same as (a) but for a system with three different positive
scattering lengths (a1 ≈ +10rvdW , a2 ≈ +15rvdW and a3 ≈ +20rvdW ).

limit of large λ to a constant. Therefore, the contribution of the cubic potential to the eigenphasesum diverges as one includes more channels. Curiously, the phase-shifts due to the higher ordertails do converge. For the quartic tail
(4)

(4)

δλ = −

αλ π/8(Ka0 )2 .,(λ + 3)(λ + 2)(λ + 1)

(5.91)

(4)
where αλ −−−→ O λ2 and the denominator scales as λ3 , and the ratio converges when λ  1.
λ1The connected S-matrix, S(C) is therefore convergent in the adiabatic hyperspherical rep-resentation. In the language of adiabatic three-body potentials, the divergence of the eigenphasesum in the full calculation of S3B is due solely to the 1/R3 tail, which can be associated entirely
with disconnected scattering. Disconnected scattering also contributes to the higher order long-135
range tails (see Eq. 5.44) although these contributions do not lead to divergent phase shifts in thissimplified picture, and this ‘contamination’ spoils the connection with the diagrammatic approach.
Unfortunately, S(C) is altered and the results obtained in this section cannot be matched with
Amado and Rubin’s low-energy expansion in Eq. 5.67.
It remains an open question how to properly decompose the three-body S-matrix into connected and disconnected components in a representation that discretizes the continuum without
spoiling double, triple, and higher order scattering contributions. Formally, the decompositionP
S3B = [1 + 2πδE i Ti ] + S̃ mentioned at the beginning of this section in fact does this separation

without spoiling the higher order contributions
"
#
X


S̃ = 2πiδE T −Ti = 2πiδE T D + · · · .

(5.92)i

The remainder S̃ is therefore exactly the physical connected T-matrix containing only contributions
from double-scattering upwards. Calculating the matrix elements of S̃ in the adiabatic hyperspherical representation10 is therefore a promising avenue towards matching with Amado and Rubin’s
low-energy expansion in Eq. 5.67.

5.5

Conclusion
This chapter is the first of three devoted to the physics of strongly-interacing Bose gases.

After introducing some standard coordinate space parametrizations of the three-body problem, I
discussed the adiabatic hyperspherical representation, which treated the hyperradius as an adiabatic
coordinate. Importantly, this representation discretizes the continuum, retaining the utility of thepartial wave expansion discussed in Ch. 2, and introduces adiabatic three-body potentials whichplay the role of effective three-body potentials. In the unitarity limit, an infinite number of Efimov
trimers form. Through a WKB model, I discussed the signature of Efimov physics in the threshold
behavior of the elastic phase shift. When the scattering length is finite, this short range contribution
is complemented by long-range contributions from the long-range power law tails in the transition
10

The matrix elements of Si should be in principle possible to calculate analytically from two-body results.

136
region. These long-range contributions modify the usual Wigner threshold laws for the phaseshift, and I attempted to match them against well-known results from Amado and Rubin [201]obtained from a diagrammatic approach. Unfortunately, the effects of the two-body continuum
embedded in the three serve to spoil this comparison as well as producing divergent three-bodyelastic phase shifts. To remove these disconnected scattering contributions, a decomposition due toR. G. Newton [209] was discussed, which was analytically and numerically confirmed to produce a
convergent calculation of the connected three-body S-matrix. This matrix is however altered, andan alternative, promising factorization of the S-matrix into connected and disconnected components
was proposed, which remains the subject of future work.

Chapter 6

Efimov Physics with Degenerate Internal Levels

This chapter contains results from Refs. [36, 37, 62].

In the previous chapter, I addressed the three-body Efimov physics that becomes importantwhen interactions between atoms in a Bose gas become strong. When the individual atoms haveaccess to multiple degenerate internal levels, the Bose-condensed gas is referred to as a spinor
BEC. In this chapter, I will demonstrate how the three-body bound state spectrum of Efimov
states is modified with the introduction of degenerate internal levels and discuss the signature ofEfimov physics in the three-body elastic phase shift. The long-range power law tail modificationof the Wigner threshold laws discussion which bookended the previous chapter will however not be
extended in this thesis to spinor systems. The results of this chapter and Chapter 5, serve as the
few-body foundation of the final chapter, Chapter 7, which combines these results with a manybody formalism for describing the strongly-interacting Bose gas. I begin with some brief context
and overview of the spinor BEC.
In spinor condensates the atomic spin degrees-of-freedom are energetically accessible leading
to a unique scenario in which both superfluity and magnetism can exist [64]. The first spinor
condensates were made with 87 Rb in a double magnetic trap [213, 214] and with 23 Na in an opticaltrap [215]. There has been a subsequent explosion of experimental [215, 216, 217, 218, 219, 220,
221, 222, 223, 224, 225, 226], and theoretical [227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237]
studies in the area. Studying the interplay between superfluidity and magnetism has proven to be

138
a rich area for probing the many-body dynamics of the static and nonequilibrium behavior of thesesystems leading to novel phenomena such as spin textures, spin domains, and complex spin mixing
dynamics [64].
Many of the interesting phenomena in spinor condensates hinge on extreme sensitivity to
the relative strengths of two-body interactions between the internal states; those interactions are
typically weak for alkali atoms (with the exception of 85 Rb [238, 239], and possibly also 133 Cs
and 7 Li) and as a consequence the spinor physics with strong interactions has largely remained
unexplored. Although the usual Feshbach resonance technique for tuning the interactions [30]cannot immediately be applied to spinor condensates, several proposals for achieving such control
exist [240, 241, 242, 243, 244, 245, 246], which enable possible ways to study strongly correlated
spinor condensates. Evidently, as the interactions become resonant, three-body contributions need
to be considered. In the strongly correlated regime, i.e., when the s-wave two-body scattering
lengths associated with the hyperfine spin states exceed the typical range of interatomic interactions(the van der Waals length, rvdW ), Efimov physics becomes important [32, 174, 247], and several new
aspects have to be considered. For instance, the existence of an infinity of weakly bound Efimov
trimers strongly affects the scattering observables at ultracold energies and can potentially impact
the spin dynamics in spinor condensates.
One of the major differences between the usual “spinless” Efimov physics and the effect forspinor systems is the presence of multiple length scales in the problem. In spinor condensates theatomic energy levels are (2f + 1)-fold degenerate (f is the atomic hyperfine angular momentum and
mf = −f...f its azimuthal component), and there exist f +1 rotationally-invariant s-wave scattering
lengths [64]. In the context of nuclear physics, where isospin symmetry plays an important role,
the work of Bulgac and Efimov [35] demonstrated a much richer structure for Efimov physics when
the isospin degree of freedom was considered.
This chapter begins with a summary of the multichannel generalization of the adiabatic
hyperspherical representation presented in Chapter 5 appropriate for the three-body problem withmultiple internal levels. I then begin with a simple toy model for the three-body problem with

139
two degenerate internal levels to illustrate how the spectrum of Efimov state shifts as the mixing
between three-body internal states is adjusted. Then, the Efimov physics for spin-1, -2 and -3
systems are analyzed by first determining the complex topology of the three-body hypersphericalpotentials and by enumerating the possible families of Efimov states present in each case. Special
attention is given to the spin-2 85 Rb condensate due to the naturally large values for the atomic
scattering lengths found in Ref. [238, 239]. Finally, I derive the 3-body scattering length using
the WKB model from Sec. 5.4.1 for an individual case to demonstrate the general procedure whenspin degrees of freedom have been introduced. This provides the fundamental building blocks
for studying the effect of three-body physics on the ground state magnetic ordering of spinor
condensates in Chapter 7.

6.1

Adiabatic hyperspherical representation for spinor systems
Our study of few-body physics in spinor condensates begins from the multichannel general-

ization of the zero-range Fermi pseudopotential for s-wave interactions. In Refs. [248, 172, 249,
250, 251], a suitable generalization was proposed to bev̂(r) =

4π Â 3∂δ (~r) r,
m
∂r

(6.1)

where Â is a scattering length operator which incorporates all the important multichannel
structure of the two-body interactions [248]. For spinor condensates we assume the scattering
length operator in Eq. (6.1) can be defined for each pair of atoms asÂ =X

F2b MF2b|F2b MF2b iaF2b hF2b MF2b |,

(6.2)

where
|F2b MF2b i ≡ |(f1 f2 )F2b MF2b i =X

mf1 mf2

hf1 mf1 f2 mf2 |F2b MF2b i|f1 mf1 i|f2 mf2 i.

(6.3)

140
is the two-body spin functions of total angular momentum |f1 − f2 | ≤ F2b ≤ f1 + f2 and projection
MF2b = mf1 + mf2 , expressed in terms of the Clebsch-Gordan coefficients. Therefore, the twobody interaction in Eq. (6.1) is diagonal in the spin basis {|σi} = {|F2b MF2b i}. We note that,
due to bosonic symmetry only the symmetric spin states (F2b even) are allowed to interact withrotationally-invariant scattering lengths aF2b ≡ {a0 , a2 , ..., a2f }1 . These scattering lengths set
important length scales in the system. Many-body properties of the system such as the miscibilityof spin components are sensitive to the relative strength and sign of the scattering lengths [64]. Their
strength also signifies the appearance of universal three-body physics in the scattering observables
as we will see in Chapter 7.
The three-body problem is solved using a multichannel generalization of the adiabatic hyperspherical method via a Green’s function approach developed in Ref. [248, 172, 249]. The three-bodywave-function can be generalized for the multichannel problem asψ(R, Ω) =XFν (R)

νXΦΣ
ν (R; Ω)|Σi

(6.4)

Σ

where {|Σi} = {|mf1 , mf2 , mf3 i} are the three-body spin functions, formed by the combination of
all possible product states, chosen to simplify the formulation. The adiabatic channel functions are
eigenstates of the fixed-R hyperangular Schrödinger equation
X h Λ̂2 (Ω) + 15/4
Σ0δΣΣ0 + hΣ0 |V̂ (R, Ω)|Σi
2µ3B R2iΣ+ EΣ δΣΣ0 ΦΣ
ν (R; Ω) = uν (R)Φν (R; Ω),(6.5)

with corresponding eigenvalues given by the adiabatic potentials u(R) which determine the hyperradial motion and describe (in conjunction with the nonadiabatic couplings) the possible boundstates and scattering properties of the system. V̂ is the sum of all possible pairwise interactions inspin space [Eq. (6.1)], given by
"
#
4π X
∂ (k)
(k)
(k) 3
V̂pseudo =|fk mfk i Â δ (d~ρ1 ) (k) ρ1 hfk mfk |.
m
∂ρ
i<j

1

1

Note the unavoidable abuse of the notation ‘a0 ’. The meaning should be clear contextually.

(6.6)

141
In the present study the three-body energy levels, EΣ , which is the sum of the energies of the threeatoms, are degenerate and set to zero.
Equation (6.5) is solved using a hyperangular Green’s function and the corresponding LippmannSchwinger equation [248] for each component of the channel function,ΦΣ (R; Ω) = −2µ3B R

2XZ

dΩ0 GΣΣ (Ω, Ω0 )

Σ0 ,k
(k)

×vΣΣ0 (R, Ω0 )ΦΣ0 (R; Ω0 ),

(6.7)

where
(k)

vΣΣ0 (R, Ω) = hΣ|V (rij )|Σ0 i.

(6.8)

(k)

Evaluating the integral over vΣΣ0 , considering only states of total orbital angular momentum L = 0,
and solving the Lippmann-Schwinger equation reduces to determining values of s for which thedeterminant of the matrix"
31/4  (1)Q̂ = 1/2
M + M (2) P− + M (3) P+ − 1
2 R

#

(6.9)

vanishes [250, 251, 248]. In the equation above the matrix element for the M matrices are given by




A(i)
i = 1,
ΣΣ0 s cot(sπ/2),(i)MΣΣ0 =(6.10)

(i)
4
sin(sπ/6)−A 0 √
, i = 2, 3,ΣΣ

3 sin(sπ/2)(i)where AΣΣ0 is the matrix element for the two-body scattering matrix written in the three-bodyspin basis,(i)AΣΣ0 = hΣ|Â(i) |Σ0 i=

X

F2b ,MF2b

aF2b hmfj mfk |F2b MF2b i

× hF2b MF2b |m0fj m0fk iδmf m0f
i(6.11)iwith the ith particle as spectator to the interaction. The P+ and P− matrices represent cyclic and142
anti-cyclic permutations of the the three-body spin basis(P+ )ΣΣ0= hΣ|P123 |Σ0 i= hmf1 , mf2 , mf3 |P123 |m0f1 , m0f2 , m0f3 i= δmf ,m0f δmf ,m0f δmf ,m0f
1(P− )ΣΣ0

2

2

3

3

(6.12)
1= hΣ|P132 |Σ0 i= hmf1 , mf2 , mf3 |P132 |m0f1 , m0f2 , m0f3 i= δmf ,m0f δmf ,m0f δmf ,m0f .
1

3

2

1

3

(6.13)

2

The spin basis {|Σi} was chosen to to give simple expressions for the components of Q̂ whichwere written in the odd-man-out notation. In this form, however, Q̂ does not account for any
symmetry property in the system. This is fixed by the unitary transformation Ŝ Q̂Ŝ T withX(S)ΣΣ0 = hΣ| |F3b MF3b (F2b )ihF3b MF3b (F2b )| |Σ0 i,(6.14)
F2b

where the three-body spin functions of total angular momentum |F2b − f | ≤ F3b ≤ |F2b + f | andprojection MF3b = MF2b + mf are given by|F3b MF3b (F2b )i ≡ |(f1 f2 f3 )F3b MF3b (F2b )i =X

MF2b mf3

hF2b MF2b f3 mf3 |F3b MF3b i|F2b MF2b i|f3 mf3 i.

(6.15)

Therefore, evaluating the det[Ŝ Q̂Ŝ T ]=0 results in a transcendental equation whose roots s(R)determine the channel functions Φ(R; Ω) and the three-body potentials u(R) with well defined
hyperfine angular momentum (F3b and MF3b ) and permutation symmetry. In the spinless problem
discussed in Chapter 5, the solution for three identical bosons in the resonant limit R/a0 → 0
gives the lone imaginary root s0 ≈ 1.00624i, which produces the attractive three-body potential.
This potential supports an infinite number of bound trimers. In the spinor case elaborated above,the degeneracy of the hyperfine spin-manifold yields fundamentally different three-body physics as
demonstrated through a toy model in the following section.

143

6.2

Toy Model: Efimov States with Two Internal States
Before jumping into the three-body problem on the hyperfine spin manifold, we will first

explores the parameter space of the matrix Â through a toy model where there are only two degenerate internal levels labeled |1i and |2i. The toy model will then be generalized and a connection
with the hyperfine spinor three-body problem and the appearance of novel Efimov roots made before the full solutions are discussed. In our two-level toy model, Â is a real-valued symmetric 3 × 3
matrix with eigenvalues aα , aβ , and aγ . The full parameter space of Â is large, but by considering
a simple case the structure becomes more apparent. Consider the scenario where A1,3 = A2,3 = 0
with eigenvalues
2 #1/2and aγ = A3,3 ,(6.16)|σα i = cos θ|11i + sin θ|12iS , |σβ i = sin θ|11i − cos θ|12iS ,and |σγ i = |22i,(6.17)

aα/β =

A1,1 + A2,2
2



±

"

A21,2 +

A1,1 − A2,22

and eigenvectors

parameterized by the mixing angle
2A1,2
tan θ = hi1/2  .
2
2(A1,1 − A2,2 ) + 4A1,2 + (A1,1 − A2,2 )(6.18)

By construction, variation of the angle θ ∈ [0, π/2] ensures that the scattering lengths aα , aβ , andaγ remain constant while the eigenvectors |σα i and |σβ i describe admixtures of the basis states |11i
and |12iS .
Let’s discuss the results for the first two roots s0 and s0 , shown in Fig. 6.1 for the hyperradial
region aα  R  aβ . When θ = 0, there are two imaginary roots s0 = s1 ≈ 0.41370i, associatedwith the three-body internal states |221i, |122i (plus all other permutations). The root 0.41370i
is the result for two identical bosons and one dissimilar atom from the single-level problem2 .
When θ = π/2, there is only one imaginary root s0 ≈ 1.00624i associated with the three-bodyconfiguration |111i, which is the single-level root for three identical bosons discussed previously. In
2This root can be obtained by setting C1 = C2 6= C3 in the matrix equation (Eq. 5.32).

144

4
3
2
1
0
-1
-2
0.001

0.01

0.1

1

Figure 6.1: The first few roots sν obtained for the toy model in the hyperradial region aα  R  aβ .
The roots are plotted versus the mixing angle θ.

between these limiting results, the Efimov states are in a superposition of internal levels describedby the mixing angle (see Fig. 6.2). It is in this mixing region that the roots s0 and s1 departfrom their single-level results and take on novel roots for the problem of equal mass bosons which
are a mix of the limiting case results from the single-level problem.
The toy model is a result of a unitary transformation, Â → Û † ÂÛ . Equivalently, it is a resultof a transformation (rotation) of the basis {|11i, |12iS , |22i} → {|σα i, |σβ i, |22i}. What is then theclass of unitary transformations which produce novel families of Efimov states? Importantly, theeigenvector of Â attached to the resonant channel must mix two-body product states under thetransformation, which cannot arise from a unitary transformation on the one-body level. Clearly
this restricts the set of unitary transformations that produce novel Efimov states, although I don’t
go into this further here.
There is a simple connection between these results and those from for spin-1, 2, and 3
presented in the following two sections. When the internal levels are composed of the hyperfine
spin multiplet there is the additional requirement that the two-body eigenstates be simultaneous
eigenstates of both the symmetrization operator and spin operator F̂ 2 . The unitary matrix that
diagonalizes the matrix F̂ 2 carries the product basis |mf1 i ⊗ |mf2 i into the |F2B , mF 2b i basis, and145

Figure 6.2: The roots sν (R) reflect that the internal state |χ3B i of the Efimov trimer is in asuperposition of internal levels. When there is no admixture, the roots are identical to the singlelevel problem. When there is nonzero admixture (ci 6= 0), the roots are a weighted average (w1 (θ)+
w2 (θ) = 1) of the single-level roots associated with the individual internal states comprising thesuperposition. Here, this behavior is shown schematically for the lowest root is0 which tracks
between i1.00624 and i0.41370 depending on the mixing angle (see the lowest curve of Fig. 6.1).

the elements of this matrix are Clebsch-Gordan coefficients, which is also equivalent to a unitarytransformation of the matrix Â. In the toy model, the admixture was set by the mixing angle θand treated as a free parameter for the purpose of exploring the behavior of the model. In the
spinor case, this admixture is set by the values of the Clebsch-Gordan coefficients. As in the toymodel, this admixture produces roots sν which are associated with three-body spin configurations
that can be composed of a linear combination of product state spin functions and are capable ofproducing novel families of Efimov states.

6.3

Spin-1 systems
This section analyzes the spin-1 three-body problem, focusing on the structure of the three-body potentials and the families of Efimov states. In order to explore the roots of det[Ŝ Q̂Ŝ T ]=0
and the physics resulting from them, we assume the relevant scattering lengths for the problem,uν (R)

146

...
...
Figure 6.3: Schematic representation of the relevant regions in R of the three-body potentials u(R).
The shaded area, R < rvdW , represents the region where the three-body potentials are expected to
be non-universal. For all other regions, the potentials are universally given by the form in Eq. (??)
with s(R) ≡ s.

aF2b ≡ {a0 , a2 , ..., a2f }, to differ greatly in magnitude from each other. By doing so, the values fors with a range, say, |aα |  R  |aβ | is constant and the corresponding values can be easily listed
and emphasize the attractive (s imaginary) or repulsive (s real) 1/R2 interaction that characterizes
the Efimov physics. When this condition of well-separated scattering length magnitudes is notsatisfied, one must in general return to the exact solutions of the above transcendental equation. In
the regimes considered here, there will exist typically various regions in which a detailed analysis
must be performed (see Fig. 6.3). Here this analysis is carried out for every possible case.
Symmetry considerations that allow us to disregard some three-body spin states are lessevident than in the two-body case and we begin by analysing the three-body spin functions forf = 1 atoms for each value of F3b separately. Here we will consider only states with MF3b = 0 but
the same considerations also applies for MF3b 6= 0.
For F3b = 0 one can, by inspection, determine that the corresponding spin function√√√|00(1)i = |1−1i|1i− |10i|0i+ |11i|−1i,333(6.19)

is antisymmetric under permutations of any two spins. Similar to the two-body case, a three-bodyantisymmetric spin state requires an antisymmetric spacial wave function in order to form a symmetric total wave function. Since in our model only s-wave interactions are allowed, antisymmetric

147
three-body states are noninteracting, and we neglect them from our analysis and calculations.
For F3b = 1, the analysis is more complicated. Now, there exist three spin states with
F3b = 1, each one corresponding to the allowed values for F2b . They are given by,|10(0)i = |00i|0i,(6.20)√√+ |11i|−1i,|10(1)i = − |1−1i|1i
2
2(6.21)√√√|10(2)i = |2−1i|1i− |20i|0i+ |21i|−1i
.
10/3

5/2

10/3

(6.22)

Besides the fact that permutations of spins 1 and 2 are symmetric (F2b =even) or antisymmetric(F2b =odd), no clear symmetry property can be derived by inspection. The three spin functions for
F3b = 1 can be symmetryzed to form a pair of mixed symmetry states (one symmetric and other
antisymmetric with respect to permutations of spins 1 and 2) and a fully symmetric spin state [88],given by√|10(0, 2)i = 32 |10(2)i + 35 |10(0)i.

(6.23)

Although in our calculations it is crucial include all these states, we determined that collision
processes involving three atoms in mixed symmetry states are suppressed at low energies due tostronger centrifugal barriers in the three-body potentials. Therefore, in this case, the F3b = 1
totally symmetric state is dominant. For F3b = 2 states, the corresponding spin functions√√√|20(1)i = |1−1i|1i+ |10i|0i+ |11i|−1i,66(6.24)√√|20(2)i = − |2−1i|1i+ |21i|−1i,22(6.25)

3/2

form a pair of mixed symmetry states. Similar to F3b = 1, we determined that for F3b = 2, collision
processes involving three atoms in such states are also suppressed at low energies due to stronger
centrifugal barriers in the three-body potentials. Consequently, they are neglected for the analysis
of collisions between three atoms. For F3b = 3, there exists only one spin function√√√|30(2)i = |2−1i|1i+ |20i|0i+ |21i|−1i
,
5
5
5/3

(6.26)

148
that forms a totally symmetric spin state and, therefore, it is included in our analysis.
Let’s now derive the transcendental equations that determine sν , and consequently the threebody potentials uν (R). For F3b = 1, the spin functions are fully symmetric, as discussed above,and the values for s are determined by solving the transcendental equation,1

3 4 (a0 + a2 )s cot( π2 s)
1

−
+

22 R
3
2 2 (2a0 + a2 ) sin( π6 s)
5

3 4 sin( π2 s)R
16 a0 a2 sin( π6 s)21

3 2 sin( π2 s)2 R21

3 2 (a0 a2 )s2 cot( π2 s)2−2R2
2 a0 a2 s cot( π2 s) sin( π6 s)+sin( π2 s)R2

= 1.

(6.27)(Imaginary roots can be obtained by mapping s → is.) The above transcendental equation depends
on both two-body scattering lengths, a0 and a2 . For F3b = 2 and 3 the spin states are symmetric andmixed-symmetry states [36], respectively, and the interaction strength s is obtained, respectively,
through,
31/4 a2 s cot( π2 s) 2 21/2 a2 s sin( π6 s)= 1,+
21/2 R
31/4 sin( π2 s)R(6.28)

31/4 a2 s cot( π2 s) 4 21/2 a2 s sin( π6 s)+= 1.
21/2 R
31/4 sin( π2 s)R(6.29)

andNote that for F3b = 2 and 3 the transcendental equations depend only on a2 . Table 6.1 lists thesolutions of Eqs. (6.27)–(6.29) for the regions in R in which the value of s is constant —in Table 6.2
we use R  |a{n1 ,n2 ,...} | (|a{n1 ,n2 ,...} |  R) as a shorthand for the condition that R must be much less(greater) than all elements in the set {|an1 |, |an2 |, ...}. Note that we also list the relevant scattering
lengths for each value of F3b . For the cases where an imaginary root exists, i.e., when Efimov
states are allowed, we list both the imaginary root and the lowest real root. For cases in which
an imaginary root does not exist, we list only the lowest real root. Interestly, note that when R ismuch smaller than all relevant scattering lengths, R  |a{0,2} |, the imaginary root, s0 ≈ 1.0062i,
is the same than the usual three identical boson problem. For the regions |a0 |  R  |a2 | and|a2 |  R  |a0 |, i.e., when the a0 and a2 are not effectively resonant, respectively, the possible

149
2
1

1

0

0

-1

-1

-2

-2

-3

-3
-4

-4
0

2

4

6

8

10

1000

2000

3000

4000

5000

Figure 6.4: F3b = 1 (red solid line), 2 (green dashed line), and 3 (blue dash-dotted line) hyperspherical adiabatic potentials for f = 1 atoms with a0 = 102 rvdW and a2 = 105 rvdW . (a) For
R ≤ {a0 , a2 } (shaded region) two attractive potentials exist (both with s0 ≈ 1.0062i), allowing for
two families of Efimov states, and for R > a0 , one of these potentials turns into an atom-dimer
channel |F2b = 0, MF2b = 0i|f = 1, mf = 0i. (b) For a0 ≤ R ≤ a2 (shaded region), only one familyof Efimov states exists (s0 ≈ 1.0062i), and for R  a2 three (asymptotically degenerate) potentials
describe atom-dimer channels, |F2b = 2, MF2b = −1, 0, 1i|f = 1, mf = 1, 0, −1i.

roots can be different than the ones obtained for the usual bosonic problem. These new roots result
from the fact that the spin functions (Eq. (6.15))are superpositions of single particle product states
as discussed in the toy model problem, Sec. 6.2 (see Fig. 6.2.)
Figure 6.4 shows the three-body potentials for f = 1 as a result of solving Eq. (6.5) for theallowed values of F3b . The scattering lengths chosen illustrate only one of a pair of possibilities forthe relative strengths of the scattering lengths. The hyperradius is written in terms of the typicalrange of interatomic interactions, rvdW . The totally symmetric F3b = 1, 3 states are associatedwith a pair of attractive potentials for R  {a0 , a2 }. This pair is associated with s0 ≈ 1.0062i, andallow for the coexistence of two families of Efimov states (represented in Fig. 6.4 by the horizontal
solid and dash-dotted lines.) As the hyperradius exceeds a0 , the F3b = 1 attractive potential
becomes an atom-dimer channel for collisions between the dimer |F2b = 0, MF2b = 0i with energy−1/ma20 and a free atom in the |mf = 0i state. In the intermediate region a0  R  a2 , only theEfimov potential in the F3b = 3 state remains and is associated with s0 ≈ 1.0062i. For a2  R,the F3b = 3 Efimov potential joins repulsive barriers from F3b = 1, 2 in an atom-dimer channel

150

Table 6.1: Values of sν relevant for f =1 and 2 spinor condensates covering all possible regions of R
and for different magnitudes of the relevant scattering lengths. (The f = 1, 2 values are from Ref.
[36]) We list the lowest few values of sν for each F3b and their multiplicity (superscript) if greaterthan one. Below, we use R  |a{n1 ,n2 ,...} | (|a{n1 ,n2 ,...} |  R) as a shorthand for the condition that
R must be much less (greater) than all elements in the set {|an1 |, |an2 |, ...}.
(f = 1)
Relevant aF2bF3b = 1
{a0 , a2 }F3b = 2{a2 }F3b = 3{a2 }R|a{0,2} ||a0 |R|a2 ||a2 |R|a0 |R|a{0,2} |1.0062i, 2.1662
0.74290.409722.16622.16624
41.0062i, 4.4653
1.0062i, 4.4653
22(f = 2)
Relevant aF2bF3b = 0{a2 }F3b = 1{a2 }F3b = 2{a0 , a2 , a4 }F3b = 3{a2 , a4 }1.0062i, 4.4653
1.0062i, 4.4653
2
1.0062i, 4.4653
2
1.0062i, 4.4653
22

2.16622.16624
2.16624
2.16624
41.0062i, 2.1662(2)0.49050
0.7473i, 2.1662
0.3788i, 2.1662
0.97895
1.31730.686092

1.0062i, 2.1662
1.0062i, 2.1662
1.10440.5528i, 3.5151
1.10440.5528i, 3.5151
22

F3b = 4
{a2 , a4 }F3b = 5{a4 }F3b = 6{a4 }1.0062i, 2.1662
1.0062i, 2.1662
0.660800.521860.660800.5218622

2.16622.16622.16624
2.16624
4
41.0062i, 4.4653
1.0062i, 4.4653
1.0062i, 4.4653
2
1.0062i, 4.4653
2
2
2

R|a{0,2,4} ||a0 |R|a{2,4} ||a2 |R|a{0,4} ||a4 |R|a{0,2} ||a{0,2} |R|a4 ||a{0,4} |R|a2 ||a{2,4} |R|a0 |R|a{0,2,4} |(f = 2)
RelevantR|a{0,2,4} ||a0 |R|a{2,4} ||a2 |R|a{0,4} ||a4 |R|a{0,2} ||a{0,2} |R|a4 ||a{0,4} |R|a2 ||a{2,4} |R|a0 |R|a{0,2,4} |151

Table 6.2: Values of sν relevant for f = 3 spinor condensates covering all possible regions of R andfor different magnitudes of the relevant scattering lengths. (The f = 3 values are from Ref. [37].)
We list the lowest few values of sν for each F3b and their multiplicity (superscript) if greater thanone. Below, we use R  |a{n1 ,n2 ,...} | (|a{n1 ,n2 ,...} |  R) as a shorthand for the condition that R
must be much less (greater) than all elements in the set {|an1 |, |an2 |, ...}.
(f = 3)
Relevant aF2bR|a{0,2,4,6} ||a0 |R|a{2,4,6} ||a2 |R|a{0,4,6} ||a4 |R|a{0,2,6} ||a6 |R|a{0,2,4} ||a{0,2} |R|a{4,6} ||a{0,4} |R|a{2,6} ||a{0,6} |R|a{2,4} ||a{2,4} |R|a{0,6} ||a{2,6} |R|a{0,4} ||a{4,6} |R|a{0,2} ||a{0,2,4} |R|a6 ||a{0,2,6} |R|a4 ||a{0,4,6} |R|a2 ||a{2,4,6} |R|a0 |R|a{0,2,4,6} |(f = 3)
Relevant aF2bR|a{0,2,4,6} ||a0 |R|a{2,4,6} ||a2 |R|a{0,4,6} ||a4 |R|a{0,2,6} ||a6 |R|a{0,2,4} ||a{0,2} |R|a{4,6} ||a{0,4} |R|a{2,6} ||a{0,6} |R|a{2,4} ||a{2,4} |R|a{0,6} ||a{2,6} |R|a{0,4} ||a{4,6} |R|a{0,2} ||a{0,2,4} |R|a6 ||a{0,2,6} |R|a4 ||a{0,4,6} |R|a2 ||a{2,4,6} |R|a0 |R|a{0,2,4,6} |F3b = 1
{a2 , a4 }F3b = 2{a2 , a4 }F3b = 3{a0 , a2 , a4 , a6 }F3b = 4
{a2 , a4 , a6 }F3b = 5{a2 , a4 , a6 }1.0062i, 2.1662
1.0062i, 2.1662
0.66080.52191.0062i, 2.1662
0.66080.52191.0062i, 2.1662
2
0.66080.521920.66080.521922

2.1662(2)2.1662(2)2.16622.16622.1662(2)2.16622.16622.16624(2)2.16622.16624
2.16622.166244(2)1.0062i(2) , 2.1662(2)1.0062i, 0.3420
1.0062i, 0.5469
1.0062i, 1.1901
1.0062i, 0.4112i
0.6521i, 1.0098
0.6080i, 1.6045
0.9666i, 1.1343
0.5858i, 1.7691
0.9100i, 1.1558
0.4289i, 1.2015
0.99840.6415i, 20.63920.78192(2)1.0062i, 2.1662(2)1.0062i, 2.1662(2)0.87540.9352i, 2.1662
0.4309i, 2.1662
0.87540.9352i, 2.1662
0.4309i, 2.1662
1.01111.66930.0803i, 3.3774
1.01111.66930.0803i, 3.3774
22

1.0062i, 2.1662(2)1.0062i, 2.1662(2)0.25880.6678i, 2.1662
0.3351i
0.25880.6678i, 2.1662
0.3351i, 2.1662
0.9552
1.21890.81990.9552
1.21890.81992
2

F3b = 6
{a4 , a6 }F3b = 7
{a4 , a6 }F3b = 8{a6 }F3b = 9{a6 }1.0062i, 2.1662
1.0062i, 2.1662
1.0062i, 2.1662
1.13290.5842i, 3.5329
1.0062i, 2.1662
1.13290.5842i, 3.5329
1.13290.5842i, 3.5329
2
1.13290.5842i, 3.5329
2
2
2

1.0062i, 2.1662
1.0062i, 2.1662
1.0062i, 2.1662
0.63720.54911.0062i, 2.1662
0.63720.54910.63720.5491
2.6372
0.54912
2
22.11622.11622.11622.11624
2.11622.11624
2.11624
42.11624
4
4
4

1.0062i, 4.4653
1.0062i, 4.4653
1.0062i, 4.4653
1.0062i, 4.4653
2
1.0062i, 4.4653
1.0062i, 4.4653
2
1.0062i, 4.4653
22
1.0062i, 4.4653
2
2
2
2

152
|F2b = 2, mf = −1, 0, 1i|mf = −1, 0, 1i, which introduces the interesting possibility of studying
atom-dimer spin mixtures [36].
As a final note, if a small (but finite) magnetic field is applied, the Efimov states in F3b = 1
and 3 can interact creating an overlapping series of states. The physics we study in this chapter
is that of zero applied magnetic field, which implies that the hyperfine angular momentum isconserved. We make no claim about the mechanism by which the scattering lengths are controlled;
rather we explore the parameter space of all permutations of the relative strengths of the scattering
lengths. In the case of a small applied magnetic field, Efimov states associated with different threebody hyperfine angular momentum states can be coupled, thus creating an overlapping series.
Such controllability can produce ultra-long-lived states [252] whose presence can affect the spin
dynamics of the condensate depending on the short-range physics. In the following section, thecase of higher spin is analyzed where many families of Efimov states with additional novel roots
appear, along with the presence of overlapping series of Efimov states in the absence of an appliedfield.

6.4

Spin-2 and -3 Systems
In this section, the three-body potentials and Efimov states for f = 2 and 3 spinor condensates

are detailed. For spin-2 there are three rotationally invariant scattering lengths {a0 , a2 , a4 } andF3b = 0...6. The three-body spin states [Eq. (6.15)] for F3b = 0, 2, 3, 4, and 6 are symmetric while
for F3b = 1 and 5 they are of mixed-symmetry. Shown in Table 6.1 are the relevant roots sν andrelevant scattering lengths for each F3b state. Similar to the f = 1 case, for f = 2 the roots forthe regions in which R is smaller than all relevant scattering lengths correspond to the roots forthe usual identical bosons problem. All other regions display new roots. For instance, within theregion |a4 |  R  |a{0,2} | imaginary roots s0 ≈ 0.3788i (F3b = 2) and s0 ≈ 0.5528i (F3b = 3) can
be found. These values are novel in the context of the three-body problem with identical masses.
Although typically the scattering lengths for most alkali species are small (comparable to
rvdW ), one notable exception is 85 Rb with f = 2. For that system a0 ≈ −8.97rvdW , a2 ≈ −6.91rvdW ,

153
0.3

0.2

0.1

0

-0.1

0

5

10

15

20

25

30

35

40

Figure 6.5: F3b = 0 (red solid lines), 1 (red dot-dashed lines), 2 (green dashed lines), 3 (bluedashed lines), 4 (purple dashed lines), 5 (cyan dot-dashed lines) and 6 (black dot-dashed lines)
hyperspherical adiabatic potentials for f = 2 85 Rb atoms with a0 ≈ −8.97rvdW , a2 ≈ −6.91rvdW ,and a4 ≈ −4.73rvdW . Note that the potentials for the mixed-symmetry states, F3b = 1 and 5, do
not display an Efimov potential.

and a4 ≈ −4.73rvdW [238, 239]. Recent experimental and theoretical advances with alkali atoms
have determined that the value of the scattering length at which the first Efimov state appears isapproximately −10rvdW [175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189,
190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200]. Therefore, for f = 2 85 Rb spinor condensates,
it is not unreasonable to expect that the Efimov physics can strongly impact the spin dynamics andmany-body physics. In Fig. 6.5, the hyperspherical adiabatic three-body potentials for spin-2 85 Rb
are shown. As one can see, the potentials for F3b = 0, 2, 3, 4 and 6 states support an attractive
Efimov potentials for distances smaller than and comparable to the relevant scattering lengths,
but forming a repulsive barrier as the hyperradius exceeds those distances. The potentials for themixed-symmetry states, F3b = 1 and 5, are completely repulsive and support no Efimov states
Finally, for three identical spin-3 atoms there are four rotationally invariant scattering lengths{a0 , a2 , a4 , a6 } and F3b = 0, ..., 9. For spin-3, the F3b = 0 spin state is fully anti-symmetric, while
F3b = 2 and 8 states are mixed-symmetry states and all others being fully symmetric states.

154
Candidates for interesting Efimov physics (i.e., some of the relevant scattering lengths are large)
are 85 Rb, 133 Cs and 7 Li. In Table 6.2, the spin-3 values of sν for all values of F3b and all hyperradial
ranges are presented in addition to the spin-2 and spin-1 results from Ref. [36]. Where the spin-2
analysis produced only a few novel imaginary values of sν , there are many more families Efimov
states for spin-3. Of particular note is the appearance of overlapping Efimov series for F3b = 3
when R  |a{0,2,4,6} | and |a{6}  R  |a{0,2,4} |. Even at zero applied field these overlapping states
have the potential of being ultra-long lived [252] depending on the short-range physics. Analysis ofthese overlapping states is beyond the scope of the present work. The final section concludes withan analysis of the signature of spinor Efimov physics in the three-body elastic phase shift.

6.5

Signatures of Spinor Efimov Physics in the Elastic Three-Body Phase
Shift
In this section, I extend the WKB model from Sec. 5.4.1 to the spinor three-body problem

to derive the functional dependence of the elastic three-body phase shift for a given value of total
hyperfine spin on Efimov physics. This is done here for only one example as the procedure can beeasily generalized (see Ref. [36] for tabulated expressions.) The main new feature is the presenceof scattering lengths associated with two-body collisions in a total hyperfine state. There are thenmany possible reflections as the potential changes at hyperadii comparable to the magnitude of therelevant scattering lengths. For spin-1 atoms with F3B = 1, a0 < 0, a2 < 0, and |a0 |  |a2 | there
are three hyperradial regions as shown in Fig. 6.6 and therefore three possible pathways. Pathway
one reflects after tunneling from the classical turning point to R ∼ |a0 |tan(δI ) ≈ K|a0 |2l+1 .

(6.30)The second pathway tunnels to the region R ∼ |a2 | and reflects before entering the inner well(Z)!
r
Z 1/K r|a0 |lef f (lef f + 1) 1/4
s21tan(δII ) ≈ −2dR ++ 2 dR
,
R2
R2
R
|a2 |
|a0 |
 2s1
a2≈(K|a0 |2lef f +1 ).
(6.31)a0

uν ( R )155

Figure 6.6: Schematic of the three-body potential for spin-1 with total three-body hyperfine spin
F3B = 1 and hyperfine scattering lengths |a0 |  |a2 |. The relevant roots are s0 ≈ i10062 ands1 ≈ 0.4097 from Table 6.1.

And the third pathway tunnels all of the way to the inner well and is reflected at R0 ∼ rvdW
tan(δIII ) ≈



a2
a0

2s1





|a2 |
(K|a0 |2lef f +1 ) tan s0 ln+∆ .
R0(6.32)The threshold behavior of the contribution of Efimov physics to the elastic three-body phase shift
for lef f = 3/2 is therefore





 
tan(δI + δII + δIII )|a2 ||a2 | 2s1 4(1)4−−−−→ γa0 + α + β tan s0 ln+∆
a0 = A3B
K→0
k4
R0
|a0 |

(6.33)

where the subscript denotes the three-body scattering length for F3B = 1, and α, β, and γ are
undetermined proportionality constants. When compared to the single-level result (Eq. 5.66),
there is now additional dependence on the ratio of hyperfine scattering lengths. This ratio arises
from the part of the calculation which accounted for the tunneling between R ∼ |a0 | and R ∼ |a2 |,and in the limiting case |a0 |  |a2 | is responsible for the attenuation of the flux that reaches theinner well.

6.6

Conclusion
When individual atoms have access to degenerate internal levels, the spectrum of Efimov

states is modified from the single-level results of Chapter 5. The strongly-interacting spinor Bose156
gas is a regime which has yet to be experimentally accessed, and so the results from this chapter
for the Efimov physics in this many-body system are of a speculative nature. There are however
many exciting features on the few-body level absent from the single-level problem including possibly
long-lived Efimov states and novel geometric scalings. In Chapter 7, I detail how the existence ofEfimov states can impact both the superfluid phase of the condensate and also the ground statemagnetic structure.
There are still many unknown aspects of the relatively unstudied spinor three-body problem
including the Efimov physics for heterogenous mixtures, the Efimov spectrum in the infinite spin
limit 3 , and the solution for finite-range potentials where care must be taken when enforcing aparticular symmetrization through boundary conditions of the wave function. These interesting
avenues await future investigation.

3

This is a classical limit in the sense that the angular momentum can be thought of as a continuous variable.

Chapter 7

One, Two, Three, Many: Few Body Physics in Strongly Interacting Bose Gases

This chapter contains results published in Refs. [36, 37]. I review some known results fromT. Köhler and K. Burnett in Refs. [52, 53] in some detail to attempt to make this chapter as selfcontained as possible.

This is the final chapter in a series of three focused on strongly-interacting Bose gases. In
Chapters 5–6, I discussed the three-body problem for ultracold bosons, the bound state spectrumof Efimov states for single and multiple internal levels, and the signature of Efimov physics in thecontinuum along with the general problem of understanding the three-body elastic phase shift. In
this chapter, I generalize the many-body theory developed in Chapter 4 to include few-body physics
beyond the two-atom level. This is done systematically in this chapter through introduction of themethod of cumulants. Along the way, the GPE from Chapter 4 is rederived, and many of thetacit assumptions of the derivation in Sec. 4.1 become exposed in this more general framework.
This chapter finishes with a discussion of the emergence of Efimov physics in the superfluid andmagnetic ordering of a strongly-interacting Bose gas. Before jumping in, however, it is instructive
to provide some general context from the field of strongly-interacting systems, which is provided in
the remainder of this introduction. Sec. 7.1 adds to the length scale discussion fro the beginningof Chapter 5 to include the length scale of the inter particle separation n−1/3 .
Nobel-prize winning physicist P. W. Anderson famously quipped: “More is different” [253] inattack of the reductionist line of thinking that the only scientists studying anything fundamental are158
those working on fundamental laws. Strongly-correlated1 systems are a sharp rebuke to this reductionist philosophy, including such important examples as quark-gluon plasmas, high-temperatureand organic superconductors, and clouds of strongly-interacting ultracold atoms [254]. A common
theme of these systems is that they cannot be understood by extrapolating the behavior of one or
two non-interacting particles to understand the bulk properties of the system. To understand thesematerials and their potential applications, the interactions between particles must be taken into
account through novel approaches. However, successful efforts in this area are frustratingly scarce.
For ultracold strongly-interacting Bose gases in the unitary regime, where individual atoms
are interacting as strongly as possible and are therefore highly correlated, even the physics of threeatoms is different: When two atoms wont bind, an infinite number of Efimov trimers appear in thegas, as discussed in Chapters 5–6. This regime is fraught by the disastrous scaling of the threebody inelastic loss rate, discussed in the beginning of Chapter 5, growing as the scattering length
raised to the fourth power. The strongly-interacting Bose gas should be considered therefore as aquasi-stable system which will eventually decay over the characteristic time scale for three-bodyrecombination. Nevertheless, recent results from JILA using 85 Rb [59] indicate that through aquench of the interaction strength to the unitary limit, this timescale is exceeded by the relaxation
rate to local thermodynamic equilibrium, and so the thermodynamics of the unitary Bose gas are
currently of considerable interest.
This chapter aims at the question of how knowing the correlations of three atoms can act
as a platform to improve our understanding of ultracold strongly-interacting many-body systems,
and so we begin in the neighborhood of a Feshbach resonance where the scattering length is manytimes rvdW but still smaller than the interaparticle spacing n−1/3 , so that much of the machineryof mean-field theory from Chapter 4 can be retained and extended. These length scale matters are
addressed in the following section.
1

In this chapter, the ‘strongly-correlated’ is used interchangeably with ‘strongly-interacting.’

159

7.1

Time and Length Scales in Strongly-Interacting Bose Gases
Referring to a many-body system as strongly-interacting or strongly-correlated requires qual-

ification. This distinction is characterized by three length scales: the scattering length a0 , the rangeof the potential given by the van der Waals length rvdW , and the inter particle spacing n−1/3 . I
discuss appropriate theories for different relevant magnitudes of these length scales.
The limit a0  n−1/3 is the regime where the theories of Chapter 3–4 are appropriate as the
dynamics of the mean-field are temporally separated from the duration of a single collision. Here,interactions between atoms can be well approximated by scattering results in vacuum. Additionally,
atoms are spaced at distances which are larger than the range of the potential rvdW  n−1/3 andso short range correlations are unimportant.
In the limit a0  rvdW , Efimov trimers begin to form past |a0 | ≈ 10rvdW , introducing
universal physics into the problem. In this thesis, this limit signals the regime of strongly-interacting
Bose gases as discussed earlier in Chapter 5. In the unitary limit na30  1, the basis of the meanfield theories of Chapter 2 breaks down as the free flight time of individual atoms is not temporally
separated from the duration of a collision. It is no longer appropriate to consider the collisions in
the condensate as taking place in vacuum, and medium effects must be incorporated somehow; thishas been qualitatively accounted for on the few-body level by introducing an artificial harmonic
trap with oscillator length proportional to the inter particle spacing (see Refs [255, 256, 257, 258]).
There are (at least) three viewpoints on how to address the unitary Bose gas:
(1) Feign ignorance and proceed with the weakly-interacting theory of Chapter 2 by tuning thescattering length to arbitrarily large values. This theory diverges with increasing scattering
length.
(2) Formulate a beyond mean-field theory of the gas.
(3) Utilize a two-channel many-body model that includes the closed-channel bound state physics
responsible for the energy dependence of the scattering observables and large scattering

160
length near resonance. Such a theory has been formulated for bosons and fermions near a
Feshbach resonance, referred to as the theory of resonance superfluidity (see for instanceRefs. [54, 55, 56]) and a two-channel mean-field-like set of equations established based on
the smallness of the effective range, r0 and the background scattering length, abg comparedto the inter particle spacing n−1/3 . This theory was highly successful over a decade ago
studying the unitary Fermi gas and transitions smoothly through the resonance.
In the outlook portion of the conclusion of this thesis, I discuss a blueprint of the third viewpoint
appropriate for the unitary Bose gas. This theory is attractive as it is both divergenceless, meanfield like, and directly marries multichannel scattering calculations including bound state physics
with many-body theories of strongly-interacting gases.
It is however important to first discuss strongly interacting Bose gases when na30 is still lessthan unity and sketch the impact of universal three-body physics on the thermodynamics of the
many-body state. In this regime, interactions are strong but still separated in time. This regime isthe subject of this chapter, and I refer to it simply as the strongly-interacting Bose gas, reserving
the classification of unitary Bose gas for the case where the diluteness parameter exceeds unity.
This regime allows the retention of mean-field concepts from Chapter 2, but now the theory must
be augmented to include the effects of three-body physics. Furthermore, an additional length scale,l3B , associated with the size of the Efimov trimer and additional timescales associated with the
lifetime of the trimer and the timescale for inelastic loss from three-body recombination must enter
the theory. These relevant length scales are summarized in Fig. 7.1. I discuss how the extensionof the mean-field theory from Chapter 4 can be extended to include three-body physics in thefollowing section.
The final limit, r0 ≈ n−1/3 , is appropriate to fluid states like liquid Helium. For these systems, the intricate details of the interactions at short distances where the effects of electronic cloud
repulsion are relevant, and zero-range models are not appropriate. In this regime, hydrodynamical theories can be formulated based on bulk flow of a fluid consistent with the symmetries and161

Figure 7.1: Relevant length scales in the strongly-interacting Bose gas (not drawn to scale.) The
blue atoms are part of an Efimov trimer with characteristic length scale l3B . The effective range
r0 is less than the scattering length, which is less than the interparticle spacing, r0  a0  n−1/3 ,and the inter particle spacing is indicated between a blue atom and a red atom of the backgroundgas.

conservation laws of the system (see the early seminal theoretical works [25, 26, 28] and a recent
historical perspective [118]).

7.2

The Cumulant Expansion Method
In this section, I generalize the theory of the weakly interacting Bose gases, by outlining acomplete set of coupled equations for the dynamics correlations to all orders, which is based on the
Bogoliubov-Born-Green-Yvon-Kirkwood (BBGYK) hierarchy and cumulant expansion method (seeFig. 7.2 for the chain of dependence) [51, 259, 97]. I then discuss the structure and various limitsof these equations in Sec. 7.2.1. These equations must be truncated to formulated a theory which
is closed, and I discuss self-consistent solutions from first order truncation in Sec. 7.3, rederiving
the GPE and its extension including quartic nonlinearities with coupling strength depending on
three-body scattering quantities, tracing the work of T. Köhler in Refs. [52, 53]. This three-body
GPE forms the basis for the third section, Sec. 7.4, focused on the emergence of three-body physics
in the superfluid and magnetic ground state phases of the strongly-interacting Bose gas.

162

Figure 7.2: The cumulant expansion generates equations of motion for n-atom correlations that
depend explicitly on the dynamics of order less than n + 2. This forms a hierarchy of explicit andimplicit dependence.

All of the information contained in the many-body wave function can be gleaned in principle
from the dynamics of correlations taken to all orders. This is equivalent to reconstructing a generalprobability distribution through its associated cumulants that to lowest order are the mean, variance, and skewness shape parameters. In statistics, often only the lowest order cumulants suffice
to model weakly correlated data, and this is also true in weakly interacting Bose gases where stateof-the-art theories, including those discussed in Chapters 3–4, track only the dynamics of one andtwo particle correlations. One possible approach using the master equation (see Ref. [260]) coarse
grains the system and constructs a distribution function composed of a set of master variables. If
master variables beyond quadratic order are included, the Marcinkiewicz theorem, or M-theorem
states that the positive-definitness of the probability distribution function is violated [261]. The
method of cumulants does not suffer from these difficulties.
It is natural to start with a definition of cumulants in terms of expectation values in thecontext of quantum mechanical operators. Given a set of operators {B1 , ..., BN }, expectation

163
values can be factored in terms of cumulants as follows
hB1 i = hB1 ic
hB1 B2 i = hB1 B2 ic + hB1 ic hB2 ic
hB1 B2 B3 i = hB2 B2 B3 ic + hB1 B2 ic hB3 ic + hB2 ic hB1 B3 ic+ hB1 ic hB2 B3 ic + hB1 ic hB2 ic hB3 ic
..
.

(7.1)

The bracket hB1 ....Bj−1 Bj ic indicates a cumulant of jth order. Eq. 7.1 is the cumulant expansion of
expectation values up to third order. Alternatively, at each order cumulants can be written in termsof expectation values by inverting the cumulant expansion. The form of Eq. 7.1, however, highlights
the meaning of the cumulant: the jth order cumulant contains only j-body physics whereas the
expectation value of the same order contains the physics of j, j-1,...,1 atoms. For bosonic operators,
which will be the only type of operators considered in this chapter, the canonical commutation
relations state that
hBi Bj† ic = δij + hBj† Bi ic .

(7.2)

Wick’s theorem, which is appropriate, for instance, when using grand-canonical equilibrium
density matrix, is a special case of the above remarks where the cumulant expansion has beentruncated to second order [262, 263, 89]. When the cumulant expansion is truncated at second
order, Eq. 7.1 can be simplified
hB1 i = hB1 ic
hB1 B2 i = hB1 B2 ic + hB1 ic hB2 ic
hB1 B2 B3 i = hB1 B2 ic hB3 ic + hB2 ic hB1 B3 ic+ hB1 ic hB2 B3 ic + hB1 ic hB2 ic hB3 ic... = ...

(7.3)

Truncating the cumulant expansion at second order is equivalent to taking expectation values withrespect to a gaussian distribution.

164
Provided the system is not too far away from equilibrium, cumulants of increasing orderdecrease unlike higher order expectation values. For the ideal Bose gas in equilibrium discussed
in Sec. 2.4, cumulants beyond second order vanish and the system is well described by a gaussian
density matrix. Higher order cumulants therefore provide a measure of the extent to which thesystem is away from the interaction-free equilibrium [51, 52, 50].
From the Heisenberg equation of motion, a linked set of equations of motion for expectation
values from the many-body Hamiltonian can be constructedi∂hB1 ...Bi i = h[B1 ...Bi , H]i .
∂t

(7.4)

Through the cumulant expansion, these equations of motion can be reformulated into an equivalenthierarchy of equations of motion for the cumulants, and the chain can be broken unambiguously at
any desired order [51]. Arguments for closing the cumulant equations of motion at a given order are
discussed further in Sec. 7.3, and I focus now on the reformulation of expectation value equationsof motion into cumulant equations of motion up to the third order borrowing the notation andterminology of Ref. [52].
Let’s start with the many-body Hamiltonian for pairwise interactions2H=Z1dxψ (x)H1B (x)ψ(x) +
2
†

Z

dx1

Z

dx2 ψ † (x1 )ψ † (x2 )V (x1 − x2 )ψ(x2 )ψ(x1 ),(7.5)

where V (x1 − x2 ) is left general purposefully and not taken be contact interaction. In Section 7.3,
I will connect with the GPE from Chapter 4 where the link between two-body scattering physics
and the condensate equation of motion is justified explicitly. The first, second, and third order2

Introduced first in Sec. 2.3.2 (Eq. 2.76).

165
cumulants areΨ(x, t) = hψ(x)ic (t),(7.6)

Φ(x1 , x2 , t) = hψ(x1 )ψ(x2 )ic (t),(7.7)

Γ(x1 , x2 , t) = hψ † (x1 )ψ(x2 )ic (t)(7.8)

Λ(x1 , x2 , x3 , t) = hψ † (x1 )ψ(x2 )ψ(x3 )ic (t)(7.9)

χ(x1 , x2 , x3 , t) = hψ(x1 )ψ(x2 )ψ(x3 )ic (t)(7.10)

where the first-order cumulant Ψ is the condensate wave function, the second order cumulant Φ isthe pairing field, Γ with the one-body density matrix for the cloud of excitations out of
the condensate, and Λ and χ are the triplet cumulants, following the language of F. Lalöe [264].
For reference, the one-body correlation function isρ(1) (x1 , x2 , t) = hψ † (x1 )ψ(x1 )ic (t) = Γ(x1 , x2 , t) + Ψ(x1 , t)Ψ(x2 , t),(7.11)

associated with the total density of the gas ρ(1) (x, x, t).
The first order equation of motion for condensate wave function Ψ is∂i Ψ(x, t) = H1B (x)Ψ(x, t) +
∂t
+
+
+

Z

Z

Z

Z

dyV (x − y)|Ψ(y, t)|2 Ψ(x, t)dyV (x − y)Φ(x, y, t)Ψ∗ (y, t)dyV (x − y) [Γ(x, y, t)Ψ(y, t) + Γ(y, y, t)Ψ(x, t)]dyV (x − y)Λ(y, y, x, t).

(7.12)

This illustrates a general feature of all cumulant equations of motion: the nth order cumulant
equation includes cumulants up to order n+2 explicitly and terms which contain in total n and
n+2 operators . The structure of Eq. 7.12 and the following equations of motion for the second
and third order cumulants will be discussed in Sec. 7.2.1

166
The second order equation of motion for the pairing field Φ isi∂
Φ(x1 , x2 , t) = H2B (x1 , x2 )Φ(x1 , x2 , t) + V (x1 − x2 )Ψ(x1 , t)Ψ(x2 , t)∂tZ+dyV (x1 − y) [Φ(x1 , x2 , t)Ψ(y, t) + Φ(x2 , y, t)Ψ(x1 , t)]Z+
dyV (x1 − y) [Γ(y, y, t)Φ(x1 , x2 , t) + Γ(y, y, t)Φ(x1 , x2 , t)]Z+
dyV (x1 − y) [Λ(y, x2 , x1 , t)Ψ(y, t) + Λ(y, y, x2 , t)Ψ(x1 , t)]Z+
dyV (x1 − y)χ(y, x2 , x1 , t)Ψ∗ (y, t)+ {x1 ↔ x2 },(7.13)

where fourth order cumulants have not been included mainly for sake of clarity, the arrow ↔ denotes
the exchange of coordinates in the bracket, and H2B (x1 , x2 ) = H1B (x1 ) + H1B (x2 ) + V (x1x2 ) is
the two-body Hamiltonian.
The second order equation of motion for the excitation field Γ isi∂
Γ(x1 , x2 , t) = [H1B (x1 ) − H1B (x2 )] Γ(x1 , x2 , t)∂tZ+dyV (x1 − y) [Φ∗ (y, x2 , t)Φ(x1 , y, t) + Φ∗ (y, x2 , t)Ψ(x1 , t)Ψ(y, t)]Z+dyV (x1 − y) Γ(y, x1 , t)Ψ(x1 , t)Ψ∗ (y, t) + Γ(x1 , x2 , t)|Ψ(y, t)|2Z+
dyV (x1 − y) [Γ(y, x2 , t)Γ(x1 , y, t) + Γ(x1 , x2 , t)Γ(y, y, t)]Z+
dyV (x1 − y) [Λ∗ (x2 , y, x1 , t)∗ Ψ(y, t) + Λ∗ (x2 , y, y, t)Ψ(x1 , t)]Z+
dyV (x1 − y)Λ(x2 , y, x1 , t)Ψ∗ (y, t)− {x1 ↔ x2 }∗ ,(7.14)

where again the fourth order cumulants have not been included.
The third order equations of motion for the triplet cumulants are lengthy, and for the purposesof this thesis, it is sufficient to include only terms which include up to three operator products.

167
The equation of motion for Λ isi∂
Λ(x1 , x2 , x3 , t) = [H2B (x1 , x2 ) − H1B (x3 )] Λ(x1 , x2 , x3 , t)∂t
+ V (x1 − x2 ) [Ψ(x2 , t)Γ(x1 , x3 , t) + Ψ(x1 , t)Γ(x2 , x3 , t)] ,(7.15)and the equation of motion for χ isi∂
χ(x1 , x2 , x3 , t) = H3B χ(x1 , x2 , x3 , t)∂t
+ V3B [Φ(x1 , x2 , t)Ψ(x3 , t) + {x1 → x2 → x3 } + {x1 → x3 → x2 }] ,(7.16)

where H3B =P

i H1B (xi ) + V3B is the three-body Hamiltonian, and V3B =sum of all possible pairwise interactions.

7.2.1

P

i<j V (xi − xj ) is the

Structure of the Cumulant Equations of Motion

The complicated structure of the cumulant equations of motions Eqs. 7.12–7.16 can be unraveled by investigating their free and vacuum limits which I discuss presently.
There are two main approaches to truncating the cumulant equations of motion. For example,
choosing to truncate at first order, all cumulants of higher order could be ignored [51, 50]. This
results in the GPE in the first Born approximation∂i Ψ(x, t) = H1B (x)Ψ(x, t) +∂tZdyV (x − y)|Ψ(y, t)|2 Ψ(x, t).

(7.17)

Truncating at second order gives the Hartree-Fock Bogoliubov equations of motion for the condensate, pairing, and excitation fields [89].
The second approach requires that for truncation at nth order, the free equations for then+1 and n+2 order cumulants must solved self-consistently [52] and inserted into the nth order
cumulant equation. For the jth order cumulant equation of motion, the free equation is obtainedby neglecting all terms that involve products of operators higher than order n. For instance,

168
Eqs. 7.15–7.16 are already in their free form. When the self-consistent solutions for the n+1 andn+2 order cumulants are inserted into the nth order cumulant equation of motion, they containthe n+1 and n+2 body T-matrices from the few-body scattering problem in vacuum. Therefore,
they significantly extend the validity of the first approach beyond the first Born approximation to
include multiple scattering, in the sense of the Born series. This is the truncation scheme used inthis chapter, specifically in Sec. 7.3 to re-derive the GPE. Importantly, this scheme explicitly links
the few-body scattering problem with the cumulants of the many-body problem establishing thetheme of this chapter.
The free equations are different from the cumulant equations of motion in the vacuum limit,
the vacuum equations. What is meant by the jth order cumulant describing the physics of jatoms becomes precise in the vacuum limit, corresponding to setting cumulants containing both
daggered and undaggered operators to zero along with eliminating the condensate wave function–
there is no broken phase in vacuum. The vacuum limit of the noncondensate field, for instance, canbe understood from the equilibrium result for an ideal gas of bosons in a box in the grand-canonicalensembleΓ(r, r0 ) ≈N −π(r−r0 )/λ2
th ,
e
V

(7.18)

which is a gaussian decaying with a width proportional to the thermal de Broglie wavelength,
corresponding to the coherence length of a classical gas. This expression vanishes in the vacuum
limit where the atom number, N , approaches zero while the volume, V , is held fixed.
The vacuum limit of Eqs. 7.12–7.16 yields two nonzero cumulant equations of motion for thepairing Φ and triplet χ fields∂
Φ(x1 , x2 , t) = H2B Φ(x1 , x2 , t),∂t(7.19)∂
χ(x1 , x2 , x3 , t) = H3B χ(x1 , x2 , x3 , t),∂t(7.20)

ii

which are formally identical to the two and three-body Schrödinger equations. These two equations

169
can be solved using the Green’s function methodΦ(x1 , x2 , t) = iG2B (t, t0 )Φ(x1 , x2 , t0 ),(+)(7.21)(+)(7.22)

χ(x1 , x2 , x3 , t) = iG3B (t, t0 )χ(x1 , x2 , x3 , t0 ),(+)

where GN B (t, t0 ) is the N-body retarded Green’s function [265], which propagates the wave function
from the initial time, t0 , to the present time, t, with defining equation


∂
(+)
i~ − HN B GN B (t, t0 ) = δ(t − t0 ),∂t(7.23)

where HN B is the N-body Hamiltonian. The advantage of the Green’s function method is that theadditional source terms in the free equations can be handled naturally. For the pairing field, thesolution to the free equation is(+)Φ(x1 , x2 , t) = iG2B (t, t0 )Φ(x1 , x2 , t0 ) +

Z ∞
t0

(+)

dt1 G2B (t, t1 )V (x1 − x2 )Ψ(x1 , t1 )Ψ(x2 , t1 ), (7.24)

and an easy check using Eq. 7.23 confirms that the original free equation is satisfied. For the triplet
χ field, the solution to the free equation is therefore(+)χ(x1 , x2 , x3 , t) = iG3B (t, t0 )χ(x1 , x2 , x3 , t0 )
Z ∞
(+)
+
dt1 G3B (t, t1 )V3B [Φ(x1 , x2 , t1 )Ψ(x3 , t1 )t0

+ {x1 → x2 → x3 } + {x1 → x3 → x2 }]
Z ∞
(+)
+
dt1 G3B (t, t1 )V3B [Ψ(x1 , t1 )Ψ(x2 , t1 )Ψ(x3 , t1 )] .

(7.25)

t0

It should be emphasized that knowledge of the N-body Green’s function is equivalent to knowledgeof the complete set of eigenstates of the N-body problem including the bound and continuous
spectrum. Therefore, obtaining for the Green’s function is just as difficult as solving the N-body
Schrödinger equation. The main advantage to the Green’s function formalism is that it allows theself-consistent solutions of Sec. 7.3 to be neatly formulated. Alternatively, the vacuum equations
can be written in a partial wave decomposition for the spherical and hyperspherical dependence
truncated in the ultracold limit.

170
From the above arguments, it is clear that the N-body Schrödinger equation is a limitingcase of the nth order cumulant equation of motion when medium effects are negligible. At shortdistance, the triplet and singlet potential wells are much deeper, on the order of hundreds of kelvins(see Fig. 2.3), compared to the energy scale of background medium contributions3 ∼ ng0 . The
nth order cumulant is therefore well described by the vacuum N-body scattering physics at shortrange. It is erroneous to extend this picture to long ranges where the interatomic interactions
become negligible and density effects persist. This agrees with the picture of the gas when na30  1
discussed in Sec. 7.1, namely that collision physics can be separated from the evolution of the meanfields. From these length scale considerations come the inverse argument for relevant energy scalesof the problem, namely that the high momentum portion of the momentum density is dominated byshort-range vacuum scattering physics. This is the essence of Tan’s relations [266, 267, 268], which
relate the short-range few-body wave function to the bulk observables including the tail of the
momentum distribution and the energy of the gas through a set of universal parameters, called the“contacts.” The N-body contact measures the probability for N particles to be very close together
and is extracted from the N-body vacuum scattering wave function. From the above arguments, it
is clear that, for instance, the two-body contact should appear in the the pair correlation function,g (2) (r, r0 ) ≈ Φ(r, r0 ) + Ψ(r)Ψ(r0 ). In the zero-range model, valid when the range of the potential
is much smaller than all other length scales in the system, this is indeed the case, and the paircorrelation function and two-body wave function both scale as r−2 directly proportional to thetwo-body contact, C2 . These ideas also relate the triplet cumulant χ at short distances with thethree-body wave function through the definition of the three-body contact, C3 [269, 270]. As an
example of how the contact enters into the large-momentum tail of the distribution, I’ve summarized
the following result without proofn(k) →

C2 F (k)+ 5 C3 ,k4k(7.26)

where C2 and C3 depend on the two-body scattering length and a three-body parameter associated
3

In Chapter 4, we estimated this for the weakly-interacting breathing mode experiment, finding ng0 < 0.1Tc .

171with Efimov physics [270]. Although interesting from the standpoint of the signatures of few-body
physics emerging in global properties of the many-body system, the contact is not the focus of thischapter, and so I only briefly mention it here contextually.
In a medium, the density dependent terms in the cumulant equations of motion for thepairing and triplet fields become important on the order of the average interparticle spacing. Equations 7.13 and 7.16 can therefore be viewed as two and three-body Schödinger equations that include
the dressing effect of the background gas. In the limit of a static background gas, which is appropriate for short-range processes occurring over shorter timescales than the mean-field evolution,
the solution of these nonlinear Schrd̈inger equations yields the complete spectrum of eigenstatesin the presence of a medium and also gives the general, short-time dynamics as the simple oscillatory evolution of a superposition of the dressed eigenstates. This viewpoint was implemented
approximately in the past to study coherent atom-molecule oscillations by introducing an artificial
harmonic trapping potential with oscillator length proportional to n−1/3 and trapping frequency asa fit parameter [256, 257]. This model was recently applied to the unitary Bose gas on the two andthree-body level to calculate the dressed eigenstates, momentum distribution, and obtain roughestimates for the lifetime of the gas [255].
On longer timescales comparable to the period of the artificial trapping potential or thetimescale for mean-field evolution, these approximate methods break down. A full dynamicalsolution of the coupled equations of motion, Eqs. 7.12–7.16, is required, and this is obviously a very
computationally demanding undertaking, beyond the scope of this thesis.
These remarks illustrate the general structure of the cumulant equations of motion andhighlight the link with the few-body problem in vacuum. They are therefore a powerful tool forunderstanding Bose gases beyond the level of one and two-body correlations and provide a natural
avenue to study signatures of the Efimov effect in the strongly-interacting Bose gas, pursued further
in the following section.

172

7.3

Self-Consistent Solutions of the Cumulant Equations of Motion
This section is devoted to constructing equations of motion for the condensate wave function

using the cumulant equations, Eqs. 7.12–7.16. Truncating the cumulant equations at first order,
the GPE can be derived by solving the second and third order cumulant free equations of motion
self-consistently. This provides an alternate derivation to that of Sec. 4.1, illuminating some of thetacit assumptions made earlier. Using these self-consistent solutions, powers of the condensate wave
function in Eqs. 7.12–7.16 can be collected and fed back into the first order cumulant equation. In
this way, terms of quartic order in the condensate wave function can be added to the GPE andthe three-body GPE constructed. This equation allows three-body physics to be included into thecondensate equation of motion, which is the starting point for an analysis of the impact of Efimov
physics on the strongly-interacting Bose gas pursued in the following section, Sec. 7.4.

7.3.1

Truncation of the Cumulant Equations at First Order: The GPE

When the cumulant equations are truncated at first order, the relevant equations are Eq. 7.12
and the free equations for the cumulants Φ, Γ and , Λ on which the first order cumulant depends
explicitly. The free equations for Φ and Γ are the first lines of Eqs. 7.13 and 7.14, and the freeequation for Λ is Eq. 7.15 which has already been written in free form. The first order cumulant
equation does not depend explicitly on the other triplet cumulant, χ, and so it is not part of thisself-consistent solution although it will play a pivotal role in the following section, Sec. 7.3.2 as asource of higher order contributions to the GPE, containing three-body physics.
From the free equations for Γ and Λ it is clear that if both cumulants vanish initially at timet = t0 , they do not evolve, and their contributions can be neglected. Additionally, if the initialstate of the gas is the a ideal gas ground state, the many-body wave function is a coherent state andall expectation values factor into products of the condensate wave function. Therefore, cumulants
higher than first order vanish initially, which includes the pairing field. The self-consistent solution

173of the pairing field is therefore a special case of Eq. 7.24
Z ∞(+)
dt1 G2B (t, t1 )V (x1 − x2 )Ψ(x1 , t1 )Ψ(x2 , t1 ),Φ(x1 , x2 , t) =(7.27)

t0

from which it is clear that the evolution of Φ is seeded by the condensate source terms. Inserting
this result into the first order cumulant equation yields the following equation for the condensate
wave functioni∂Ψ(x, t) = H1B (x)Ψ(x, t) +∂tZdyV [Ψ(y, t)Ψ(x, t) + Φ(x, y, t)] Ψ∗ (y, t)
Z
Z ∞
h
i
(+)
= H1B (x)Ψ(x, t) + dy
dt1 V δ(t − t1 ) + V G2B (t, t1 )V Ψ(x, t1 )Ψ(y, t1 )Ψ∗ (y, t)t0(7.28)

where the argument of V (x−y) has been suppressed for reasons of clarity. The quantity in brackets(+)

is the definition of the retarded two-body T-matrix [75] in the time domain, T2B (t, t1 ), and Eq. 7.28
can be written concisely∂i Ψ(x, t) = H1B (x)Ψ(x, t) +
∂t

Z

dy

Z ∞
t0

(+)

dt1 T2B (t, t1 )Ψ(x, t1 )Ψ(y, t1 )Ψ∗ (y, t).

(7.29)

Comparison of this equation with Eq. 7.17 reveals the Born approximation. This equation is
non-Markovian, meaning that it has memory of the past behavior of the system and is moregeneral than the GPE. Self-consistent solutions of the cumulant equation to a given order therefore
contain effects from higher order correlations through non-Markovian contributions and few-body
T-matrices beyond the first Born approximation.
It is of value to briefly trace how the Markov limit of Eq. 7.29 yields the familiar form of
the GPE for gases not far from the ideal gas equilibrium. The first step is to go to the interaction
picture, where the fast oscillation associated with the non-interacting portion of the Hamiltonian
can be separated from the oscillation due to the mean-field interaction . To do this, the cumulants
are rewritten in the eigenbasis of the one-body Hamiltonian, H1B with single-particle orbitals φiand energy Ei . The first order cumulant projected onto this basis is written as Ψi (t), and in theinteraction picture
Ψi (t) = ΨIi (t)e−iEi t ,(7.30)

174
which removes the fast oscillation due to the one-body orbital energy, Ei . The temporal evolutionof ΨIi (t) then depends on the mean-field interaction term which is weak when the gas is not far from(+)

the interaction free equilibrium. T2B (t, t1 ) is however strongly peaked about t1 = t with width
proportional to the duration of the collision. In the integrand of the non-Markovian contribution, allfirst-order cumulants can be evaluated then at time t, and the equation of motion in the interaction
picture isX∂
i ΨIi (t) = eiEi tei(Ej +Ek +El )t ΨIj (t)ΨIk (t)(ΨIl (t))∗
∂t
j,k,l

Z t−t0−∞(+)
dτs hi, l|T2B (t, t − τ )|j, kis e−i(Ej +Ek )τ



(7.31)

where the ‘s’ subscript indicates a symmetrized ket and where the quantity in brackets is a Fouriertransform to the energy domain in the limit t − t0 → ∞, which can be taken without penalty due tothe vanishing of the retarded T-matrix for times before t0 , yielding the familiar two-body T-matrix
for vacuum scattering provided the length scale of the collision is much smaller than the oscillator
length of the trap. After this transformation, Eq. 7.31 can be cast into the GPE in the orbital basis
in the Schödinger pictureiX∂∗Ψi (t) = Ei Ψi (t) +s hi, l|T2B (Ej + Ek + i0)|j, kis Ψj (t)Ψk (t)Ψl (t) ,∂t(7.32)

j,k,lfrom which the GPE of Chapter 2 can be recovered using a contact interaction in the Born approximation.
It is clear then how the Markov approximation can break down for instance with resonancescattering where the collision duration can in principle extend indefinitely exactly on resonance,
rendering the GPE meaningless. The weak time dependence argument for the mean-field interactionis also not appropriate for nonequilibrium states far from the ideal gas limit. These scenarios
highlight wider range of applicability of self-consistent solutions of the cumulant equations and theimportance of understanding the underlying few-body scattering physics in vacuum.

,

175
7.3.2

Higher Powers of the Condensate Wave Function: The Three-Body GPE

The Markovian GPE derived in the previous section has a mean-field interaction quadratic in
the condensate wave function. If instead of using the free equation for Φ, the full equation, Eq. 7.13
is considered, it is clear that a condensate equation of motion be derived that includes higher order
powers of Ψi∂
Ψ = H1B Ψ + g2 |Ψ|2 Ψ + g3 |Ψ|4 Ψ + ...
∂t(7.33)The reason that the next order correction with coupling strength g3 is quartic instead of say cubic
is because the cumulant equation for Φ contains contributions which are second and fourth order
operator products. The cumulant equation for Λ contains third and fifth order operator products(not shown), but this term is not multiplied by the condensate wave function in the first order
cumulant equation of motion, Eq. 7.12.
The full functional form of g3 was derived by T. Köhler in Ref. [53] using the cumulant
formalism, and it is sufficient for present purposes to just sketching how three-body physics enters
into this coupling constant. Let’s follow one of the paths that contributes three-body physics
starting with the solution of Eq. 7.13 using the Green’s function formalism
Φ(x1 , x2 , t) =
+

Z ∞

Zt0∞
t0

(+)

dt1 G2B (t, t1 )V (x1 − x2 )Ψ(x1 , t1 )Ψ(x2 , t1 )
Z ∞
Z
(+)
(+)
dt1
dt2 dy G2B (t, t1 )V (x1 − y)G3B (t1 , t2 )V3B Φ(y, x2 , t2 )Ψ(x1 , t2 ) + ...,
t0

(7.34)
where the first line contributes the retarded two-body T-matrix to the quadratic mean-field termin the GPE as was shown in the previous subsection. The second line contains the leading ordercontribution in Ψ from the self-consistent solution of the free equation for χ, Eq. 7.25. The dots on
the right hand side indicate that there many additional terms in the exact expression not consideredthat may end up in the final result for g3 but not included here for sake of clarity.
The remaining dependence on Φ in Eq. 7.33 is replaced by the free solution, Eq. 7.27, whichyields a final expression that is quadratic and quartic in the condensate wave function. Inserting176
this into the condensate equation of motion yields a non-Markovian GPE that includes the retarded
three-body T-matrix in the quartic term. Just as in the previous subsection, the Markov approximation can be made assuming the two and three-body collision duration is much faster than themean-field evolution of Ψ. The two and three-body T-matrices can be separated from the quadratic
and quartic products of Ψ and act as coupling functions. The final result for g2 and g3 is [53]
g2 =
g3 =

4π
a0
m
(2π~)6 (5)T3B (E → 0) + ...,
2

(7.35)(5)

where T3B (E → 0) is the threshold result for fifth order multiple scattering diagrams and upwards,
which includes the contributions from the ‘true’ three-body scattering in the universal region discussed in Sec. 5.4.1. The terms not listed in the dots are logarithmic terms to fourth order in the
scattering length, reflecting contributions that are third and fourth order in the two-body scattering
T-matrix. In the following section, the significance Efimov physics in the three-body contribution
to g3 is discussed for strongly-interacting Bose gases.

7.4

Emergence of Efimov Physics in Strongly-Interacting Bose Gases
In Sec. 5.4.1 and Sec. 6.5, the threshold behavior of the contribution of Efimov physics to(5)

the elastic scattering phase shift [∼ Re(T3B ] was sketched using a WKB model. The imaginary(5)part of T3B is proportional to the the three-body recombination rate, which determines the lifetime(5)and stability of the gas. The real part of T3B , is proportional to the three-body elastic scattering
volume, A3B , which can be positive, negative, and larger that the two-body scattering length. It
sets additional length scale in the problem and can be grouped with the class of three-body lengthscales, including l3B in Fig. 7.1 which indicated the characteristic size of trimers in the gas.
In this section, only the case where a0 < 0 is considered, and Efimov physics appears in A3B
through a set of poles at location of trimer formation, analogous with the structure of the two-bodyscattering length near Feshbach resonances discussed in Sec. 5.4.1. Curiously, near one of thesepoles, the gas can in principle still be dilute with respect to two-body interactions but not dilute177with respect to three. A Bose gas can therefore be in the unitary regime of three-body resonancescattering physics and be considered strongly-interacting. The thermodynamics of this state andof the unitary Bose gas at a Feshbach resonance are extremely intriguing but beyond the scopeof this thesis, however, I touch upon these states in the conclusion of this thesis. This section isconcerned with the case where the quartic nonlinearities in the GPE are larger than the quadratic
nonlinearities near a three-body resonance.

7.4.1

Quantum Droplets and Magnetic Ordering

The ground state structure of a weakly-interacting Bose gas can be understood by solving the
GPE for the stationary wave function Ψ(t) = Ψ exp(−iµt) as was done in Sec. 4.1. For instance, in
the Thomas Fermi limit, where the interaction energy exceeds the kinetic energy, the condensate
has the parabolic form
|Ψ(r)|2 =µ − Vext (r)g2(7.36)

where Vext is the harmonic trapping potential. When the quartic interaction is introduced into theGPE, the ground state shape changes
2|Ψ(r)| =s2(µ − Vext (r))
+
g2



g2
g3

2

−

g2
.
g3

(7.37)

When g2 < 0, the effective attraction forces the collapse of the condensate beyond a critical value
Nc a0 /aho . −0.67 [271]. This collapse was observed in Ref. [272] where the burst of bright matterwave solitons during the collapse inspired the ‘Bosenova’ label.
When an unstable condensate collapses, the density of the gas increases, and corrections
to the GPE that are higher order in the density such as the beyond mean-field LHY correction[137, 136] due to quantum fluctuations, or the quartic nonlinearity dictate the dynamics of thecollapse. Collapse experiments are therefore one probe of higher-order density corrections to theweakly-interacting theory of the Bose gas.
On the repulsive side of a three-body resonance, just as a new trimer has been formed, g3 > 0,(5)

and it is an open question whether the effective three-body repulsion, due to enhancements to T3B

178
from Efimov physics, can stabilize the collapsing gas, although a recent experiment Ref. [273]suggests that such terms do play a role. Work by A. Bulgac in Refs. [38, 39] suggest that in thisregime, the system could become self-bound forming bosonic droplet states. This state of matteris one possible signature of three-body correlations and Efimov physics in the system. The radius,
Rd , of such droplet states is predicted to scale with the number of atoms, N , in the droplet andthe ratio g2 /g3
Rd =



g3
N
2π|g2 |

1/3.

(7.38)

Analysis in Refs. [274, 275, 276] concluded that the quartic nonlinearity in the GPE introduces afirst-order phase transition between the condensate and droplet state.
Recent theoretical works [36, 37] postulate that g3 can not only take the system into novel
phases of superfluidity, it can also impact the magnetic ordering when spin-degrees of freedom are
unfrozen. For strongly-interacting spinor Bose gases, the three-body GPE must be reformulated toaccount for spin degrees of freedom in the system. The position space field annihilation operator
which can create or destroy an atom in a particular spin level is decomposed on the one-body spin~basis, using the one-body hyperfine spin states, as ψ(r)= (ψ−f (r), ..., ψ0 (r), ..., ψf (r)), where thesubscript corresponds to the magnetic sublevels. The many-body Hamiltonian for spinor gases isH=Zdxψa† (x)H1B (x)ψa (x) +

1
2

Z

dx1

Z

dx2 ψa† (x1 )ψb† (x2 )ha|hb|V̂ (x1 − x2 )|b0 i|a0 iψb0 (x2 )ψa0 (x1 ),(7.39)

where the repeated indices are to be summed per the Einstein convention. The pairwise interaction
projects onto two-body total spin states, and in the zero-range model it can be rewritten in operator
formV̂ (r) =where PF2b =

4π
4π (3) X
Â2B δ (3) (r) =δ (r)aF2b PF2b ,mm(7.40)

F2b

P

MF2b |F2b MF2b ihF2b MF2b | are projection operators into a two-body total hyperfine

state. In Ref. [227] it was shown that the relation,

f~1 · f~2

n=

X  F2b (F2b + 1)
F2b2

− f (f + 1)

n

PF2b

(7.41)

179
allows us to rewrite Â2B as
Â2B =

f
X

(n)

α2b

n=0



f~1 · f~2n,(7.42)

where the α2b ’s are linear combinations of the scattering lengths {a0 , a2 , ..., a2f }.
The above formulation allows us to rewrite the scattering length operator Â2B in Eq. (7.42)
for f = 1 atoms in terms of two parameters in complete analogy with the reformulation of thesinglet and triplet potentials in Sec. 2.2.4 Eq. 2.37,


(0)
(1)
Â2B = α2b + α2b f~1 · f~2 ,(7.43)

where
(0)

α2b =

a0 + 2a2
3(1)

and α2b =a2 − a0,3(7.44)

representing a direct interaction terms and an spin-exchange term, respectively. It has been shown
in Refs. [227, 228] that these two parameters do characterize important phases of the gas. For(1)

instance, depending on the sign of α2b , the ground state of the spinor condensate is antiferromag(1)(1)

netic (α2b < 0) or ferromagnetic (α2b > 0) ordered [227, 228, 64]. For f = 2, the scattering length
operator Â2B in Eq. (7.42) can be rewritten in terms of three parameters
(0)

α2b

(1)

α2b

(2)

α2b

2a0 8a2 9a4
+
+
,
5
7
35
a0
2a2 9a4
+
,
= − −
30
21
70
a0
a2
a4
=
−
+ .
30 21 70

= −(7.45)

We note that in the literature the Â2B operator for f = 2 has been rewritten in terms of the projector
P0 allowing for a structure similar to d-wave BCS superfluids [277]. For f = 2, there are three
categories of ground state magnetic orderings depending on the relative strengths of the scattering
lengths: cyclic, polar, and ferromagnetic (see Ref. [277].) The cyclic phase arises from the additional
P0 term which describes scattering of a pair of atoms into the F2b = 0 singlet state introducing an
additional order parameter describing the formation of singlet pairs in the system. For f = 3, the(0)four parameters that characterize Â2B [Eq. (7.42)] are: α2b = 9a0 /35−4a2 /7+486a4 /385+4a6 /77,

180
(2)

(4)

α2b = 9a0 /70 − 17a2 /63 + 81a4 /770 + 25a6 /693, α2b = −a0 /315 + 5a2 /378 − 6a4 /385 + 23a6 /4158,(6)and α2b = −a0 /630 + a2 /378 − a4 /770 + a6 /4158. Â2B was rewritten in Ref. [278] including P0
and rewritten in Ref. [279] including a nematic tensor for the dipolar spinor condensate 52 Cr using
the terminology of liquid crystals for the many ground state magnetic phases of the system at zeroand nonzero fields which we do not go into further
There are, however, many ways to rewrite Â2B in the literature that better suggest treatments
similar, for instance, to magnetism, BCS theory, and liquid crystal theory. For the three-body meanfield interactions, however, we choose to simply extend the above analysis and in an analogous form
to Eq. (7.42), in light of the lack of suggestive three-body treatments from other theories to ourknowledge. Therefore, we define the three-body scattering volume operator as
X

A3B3b PF3b ,)(7.46)|F3b MF3b (F2b )ihF3b MF3b (F2b )|.

(7.47)

Â3b =

F3b

(F

where
PF3b =

X

MF3b F2b

In the equation above, A3B is the usual three-body scattering volume (units of length4 ) as definedin Refs. [31, 32, 205, 206, 39, 38] for the allowed values of F3b . Note that our usage of the term
“three-body scattering volume” is the same as that in Efimov’s original work in Refs. [31, 32], andit corresponds, in the adiabatic hyperspherical representation, to elastic processes involving only
the lowest three-body continuum channel.
We now proceed in a similar fashion to the two-body case and rewrite Â3b in Eq. (7.46) toemphasize spin-exchange terms. Using the usual relations for angular momentum addition and theorthogonality of the projection operators, one can obtain the following relation,
nX
X  F3b (F3b + 1) 3f (f + 1) n
f~i · f~j  =−PF3b ,
2
2
i<j(7.48)

F3b

which allows us to write the three-body scattering volume operator Â3b as

n
NX
3b −1
X
(n)
Â3b =
α3b 
f~i · f~j  ,
n=0

i<j

(7.49)

181
where N3b is the number of relevant three-body scattering volumes, i.e., the three-body scattering
volume for the values of F3b whose spin functions are fully symmetric. (Anti-symmetric and mixedsymmetry states correspond to higher partial waves and are not considered in the present work.)
Therefore, the three-body mean-field contributions to the spin dynamics is now determined in termsof the parameters α3b , which are linear combination of the physical three-body scattering volumes
for all allowed values of F3b .
For f = 1 atoms, only the three-body scattering volumes for F3b = 1 and 3 contribute to theinteraction (see Table 6.2), allowing us to write the three-body scattering volume operator Â3b in
Eq. (7.49) in terms of only two parameters,
(0)

(1)



Â3b = α3b + α3b 

where

(1)

(0)

α3b =

X
i<j

f~i · f~j  ,(3)

3A3B + 2A3B
,
5



(3)

(1)

and α3b =(7.50)(1)

A3B − A3B,
5

(7.51)

representing a direct interaction term and an spin-exchange term, respectively. This is in close
analogy to the form of the two-body scattering length operator for f = 1 atoms [see Eq. (7.43)].
For f = 2 atoms, only the states with F3b = 0, 2, 3, 4, and 6 contribute in the interaction, resulting
in the following parameters for Eq. (7.49),
(0)

(0)
α3b

=

α3b

(1)

=

α3b

(2)

=

α3b

(3)

=

(4)

=

α3b

(2)

(3)

(4)

(6)

2a
3a
243a3ba2a3b− 3b + 3b +− 3b ,
35
7
5
385
385
(0)
(2)
(3)
(4)
(6)
a
23a3b
29a3b
513a3ba− 3b +−++ 3b ,
30
126
60
1540
990
(0)
(2)
(3)
(4)
(6)
29a3b
13a3b
43a3b
117a3ba−+−++ 3b ,
1260
126
360
3080
770
(0)
(2)
(3)
(4)
(6)
a
a
a
3a
17a3b
,
− 3b + 3b + 3b − 3b +945 1134 540
1540
62370
(0)
(2)
(3)
(4)
(6)
a3b
a3b
a3b
a3b
a3b
−
+
−
+
.
3780 1134 1080 3080 62370(7.52)For f = 3 atoms, the degree of complexity increases rapidly where now the states with F3b = 1, 3,
4, 5, 6, 7, and 9 contribute in the interaction (see Table 6.2) and analog expressions for α3b can beobtained.

182
In order to estimate the two- and three-body mean-field contributions and their relative
importance, we analyze the two- and three-body coupling constants
(i)

g2b =

4π (i)αm 2b(i)and g3b = 31/2

12π (i)α ,m 3b(7.53)

respectively [38]. This way, we can estimate the two- and three-body mean-field energies simply
as ng2b and n2 g3b , respectively, with n being the atomic density. This makes clear that relative
importance of three-body effects will depend on the density (the denser the gas the more important
three-body contributions are.) For instance, as pointed out in Ref. [36], the f = 1 ferromagnetic
and anti-ferromagnetic phases can be affected by three-body physics whenever the three-bodymean-field energy exceeds the two-body mean-field energy, i.e.,
(1)

(1)

n2 |g3b | > n|g2b |,(7.54)

and are of opposite sign. It is interesting noting that for most of the alkali species the two-bodyscattering lengths are typically small, implying that the two-body direct and spin-exchange meanfield energies are also small opening up ways to explore the importance of three-body effects for thesystem. The fact that the two-body scattering lengths are small does not necessarily imply thatthe three-body scattering volumes, A3B , are also small. A notable case is the f = 1 87 Rb spinor
condensate where not only are the scattering lengths small, but they are also approximately thesame. For this case, a0 ≈ 1.23rvdW and a2 = 1.21rvdW [238, 239] implying an extremely small spin(1)exchange energy term α2b ≈ −5.6 × 10−3 rvdW [see Eq. (7.44)], and the three-body spin-exchange
can be important. Determining the three-body contributions (the actual value of the three-bodyscattering volumes) for such cases is extremely challenging since it would require a full numericalcalculation including realistic two- and three-body interactions.
In the strongly interacting regime, not only do the three-body interactions become important,
but they also become universal [36]. The Efimov physics for the spin problem displays many featuresthat can allow for an independent control of both two- and three-body physics opening up ways
to strongly modify the spin dynamics in the condensate. From the analysis above, it is clear that

183
the impact of Efimov physics to the spin dynamics is made through its role in three-body elastic
processes. In general, Efimov states are manifested in the three-body scattering observables throughlog-periodic interference and resonant effects.
For f = 1, an interesting case emerges when both a0 and a2 assume large and negative values.
In the case where |a0 |  |a2 | an analytical expression for the relevant three-body scattering volume
for F3b = 1 was derived in Sec. 6.5, and the other relevant three-body scattering volume for F3b = 3
can also be derived using similar pathway arguments
(1)
A3B
(3)

A3B




  0.82
a2
a2= α − β tan s0 ln −a40 + γa40 ,
a0
a1



a2
a42 ,
= α − β tan s0 ln −a3

(7.55)(7.56)

where s0 ≈ 1.0062, and α, β and γ are universal constants that can be determined from numericalcalculations for each value of F3b . The factor (a2 /a0 ) above originates from the regions where thethree-body potentials are repulsive–see Table 6.2. For negative values of a0 and a2 , the Efimov
physics is manifested in elastic processes when a Efimov state becomes bound. In the above
equations, a−
F3b are the values where a F3b = 1 and 3 Efimov resonances occur. Near such valuesof a−F3b , the three-body scattering volume A3B can be extremely large, assuming both positive andnegative values, allowing for control of the spin dynamics via the spin-exchange term in Eq. (7.51).
In reality, since three-body losses are present and Efimov states have a finite lifetime, A3B does
not strictly diverge and, in fact, is a complex quantity whose real and imaginary parts describe
elastic and inelastic collisions—one can introduce loss effects in the expressions above (and below)
by adding an imaginary phase term iη in the argument of the tangent, where η is the so-called
three-body inelasticity parameter [174, 247]. Nevertheless, for typical values of η [174, 247] one canstill expect substantial tunability of the three-body spin-exchange dynamics.
For f = 2 atoms, evidently, the complexity in determining the three-body spin-dynamics
increases. As we mentioned above, an interesting case is presented for f = 2 85 Rb atoms, where all
the relevant two-body scattering lengths are large and negative (a0 ≈ −8.97rvdW , a2 ≈ −6.91rvdW ,and a4 ≈ −4.73rvdW [238, 239]) and close to the value in which identical bosons display an Efimov

184
resonance, −10rvdW [175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189,
190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200]. This implies that Efimov resonances mightenhance three-body spin-exchange. Since the magnitude of the scattering lengths in this case arecomparable, it makes it difficult to obtain an accurate universal expression for the three–bodyscattering lengths. Nevertheless, it is instructive to analyze the results obtained by assuming|a0 |  |a2 |  |a4 |. In this case, the relevant three-body scattering volumes are given by
(0)
A3B

=

(2)

A3B =
(3)

A3B =
(4)

A3B =
(6)

A3B =





a2
α − β tan s0 ln −
a42
a0


  1.37a4a2a2
α − β tan s0 ln − + s00 ln
a40 + γa40
a4
a0
a2



a4
a2
0
α − β tan s0 ln − + s0 ln
a42
a4
a3


  1.04
a4
a4
α − β tan s0 ln −a42 + γa42
a
a4
2



a4
a44
α − β tan s0 ln −a6

(7.57)(7.58)(7.59)(7.60)(7.61)

where s0 ≈ 1.0062, s00 ≈ 0.3788, and a−F3b < 0 are the values of the two-body scattering lengths
where a F3b Efimov resonance occur. Note that now two distinct families of Efimov states (associated to s0 and s00 ) can affect the spin dynamics. Although we expect the values for a−F3b to be close
to -10rvdW an analysis similar to the one for identical bosons [197, 198] is necessary to precisely
determine such values.
For higher spin the three-body contribution introduces additional complexity on the manifoldof possible ground state phases of the spinor condensate. The full study of the impact of three-bodycontributions on the many-body phases of spinor condensates remains however the subject of future
investigation.

7.5

Conclusion
The cumulant expansion method generates a linked set of equations for all orders of cor-

relation dynamics in the many-body system. For ultracold strongly-interacting Bose gases, thevacuum limit of these equations is formally identical to the one, two, three, etc... atom Schrödinger

185
equation. This framework is therefore a natural way to incorporate few-body physics into the
many-body problem. As a demonstration, the GPE was rederived by making the Markov and first
Born approximations. The generality of the framework allowed higher order terms in the dilutenessparameter to be included into the GPE to produce the three-body GPE. This upgraded GPE depends on the three-body T-matrix, whose real part is proportional to the elastic three-body phaseshift, which was analyzed in Chapters 3–4. When an Efimov trimer is near threshold, the real partof the three-body T-matrix can display resonant enhancement similar to the Feshbach resonancediscussed in Sec. 2.2.4. This chapter closed with an analysis of how this resonant enhancement can
impact the ground state superfluid and magnetic structure of the strongly-interacting Bose gas.
Although the cumulant expansion method was employed in this section to derive the GPE andhigher order corrections in the diluteness parameter, neither the chain of dependence nor complete
studies of the underlying dynamics of three-particle correlations have been explored in detail todate. This is largely due to the crux requirement: three-particle correlation dynamics dependon the scattering physics of three atoms, including the effect of Efimov trimers. The discussionof the elastic three-body phase shift in Chapter 5 highlighted how this necessary ingredient for
constructing the three-particle T-matrix is notoriously difficult to calculate.
The cumulant expansion method provides an avenue to insert the few-body scattering solutions into the many-body problem, however, near a scattering resonance, the bound spectrum
becomes crucial to the problem and must be included in the many-body problem somehow. In
the conclusion chapter of this thesis, Chapter 8, I propose the union of the cumulant expansion
method and resonance superfluidity method [55], which is a set of procedures for building boundstate physics into the many-body problem. This union could potentially yield a mean-field-like setof equations for the unitary Bose gas, and so stands as an important future extension of the workin this thesis.

Chapter 8

Conclusion

The pace of scientific advancement is nonlinear, as was famously codified by Thomas Kuhn
in his famous book on scientific revolutions [280]. There is normal science which consists of fillingin the defects and gaps of existing theories, and then there is the crisis and resultant paradigm
shift due to the violation of our expectations from normal science. Normal science is paradigm
based. As a practicing theorist, the immediate avenue taken with any new problem or result is to
first reflect on what worked in the past. And it is precisely the intermittent failure of this approach
which leads to the reshaping of our understanding of the natural world.
The work presented in Chapter 3 drew from established methodology for analyzing the Boltzmann equation, which is by now a very mature subject more than a century after its first formulation. The Boltzmann equation and underlying assumptions predated Einstein’s work on Brownian
motion and the experimental verification of the atomistic view of nature and was extremely controversial. Demonstrating agreement between experiment and theory for the undamped breathingmode in an isotropic harmonic trap is therefore an important piece to understanding the inherent
limitations on the observation of an undamped, nonequilibrium monopole mode given a realistic
trapping scenario. This work reaffirms and refines the foundation of the kinetic theory of gases.
The results of an experimental and theoretical collaboration presented Chapter 4 falls under
the set of collective mode experiments on finite temperature BECs initiated almost two decadesago. These are the first such results for an isotropic trapping geometry which has only recently
been technologically achieved. The unexpected temperature dependence of the m = 0 quadrupole

187
mode in Ref. [21] (see Fig. 4.1), inspired theoretical analysis that explicitly included the dynamics
of the thermal cloud. From the set of successful theories, we applied the coupled-modes analysis
of Bijlsma and Stoof [29] and the ZNG formalism [139] to analyze the breathing mode experiment
for an extremely spherical trap, drawing an analogy between the observed collapse revival of theoscillation envelope with the hydrodynamic theory of the first and second sound modes in liquid
helium. This is an application of theoretical frameworks that have worked in the past combinedwith ideas about the transport properties of helium II originating from Landau, Tisza, and London
more than a half century ago, and the results feed into the ongoing construction of consensus in theultracold gas community about the range of validity of various theories of the finite temperature
BEC.
The results of these chapters underscored the relatively firm foundational footing of our understanding of weakly-interacting Bose gases. In the regime of strong interactions, nonperturbative
effects beyond the level of one and two atoms become important. Chapter 5 introduced the nonperturbative Efimov states that arise on the three-body level in this regime, characterized theirbound state spectrum at unitarity, and presented results for the three-body elastic scattering problem; Chapter 6 discussed the Efimov physics of three-atoms with access to multiple degenerateinternal levels, finding novel families of (spinor) Efimov states. The spectrum of Efimov states
was probed experimentally for the first time using an ultracold gas of 133 Cs in Ref. [34]. This andsubsequent demonstrations with a range of atomic species, provided the unexpected universality ofthe three-body parameter for ultracold alkali species. The predicted spectrum for spinor Efimov
states remains unconfirmed.
It remains an open question how Efimov physics emerges in the thermodynamics of stronglyinteracting Bose gases. In Chapter 7, one framework for understanding these effects, the methodof cumulant expansion, was outlined. Within this framework, a natural connection between the
few-body Schrödinger equation can be established in the vacuum limit, and medium effects serve todress the few-body problem. This framework was used to establish the leading order contribution ofelastic three-body scattering to the condensate equation of motion, which permitted a study of the188
impact Efimov physics on the superfluid and magnetic phases in spinor condensates. Quantitative
predictions for the role and onset of these effects however remains elusive due primarily to thedifficulty of extracting scattering observables from the elastic three-body scattering problem.
Experiments in this regime are also plagued by the disastrous scaling of inelastic three-bodyloss processes as the scattering length approaches the resonance limit. Cutting-edge experiments
representing the state-of-the-art in the field of ultracold gases have recently made progress ininvestigating the unitary Bose gas. Using nondegenerate clouds, experiments at the Cavendish
Laboratory [57] and at the École Normale Supérieure [58] uncovered regimes where loss rates aresuppressed and accessed the strongly-interacting regime. For the quantum degenerate unitary
Bose gas, a remarkable experiment [59] at JILA in Boulder Colorado quenched to the center ofthe resonance and observed a quasistable state with universal dynamics saturating on a shorter
timescale than the expected three-body loss rate. It has been proposed that the JILA measured
momentum distribution contains the effect of Efimov physics [281, 282], and an interferometry
experiment from the Cavendish laboratory has shown effects that cannot be explained by twobody correlations in the gas [283]. Although many questions remain open, these developments
point toward a deeper understanding of strongly-correlated and strongly-interacting systems on
the horizon. In the following section, I discuss briefly how the results of Chapters 5–7 might beused in conjunction with lessons learned from the unitary Fermi gas to construct a mean-field liketheory to describe the unitary Bose gas.

8.1

OutlookIn the conclusion portion of each chapter of this thesis, avenues of future work have beenhighlighted. Perhaps none of these possibilities touch on an area producing such startling results
as the unitary Bose gas, and so I devote this final outlook to discussion of a candidate theory.
In Sec. 7.1, I discussed three possible avenues towards describing the unitary Bose gas. One ofthese candidates stems from the resonance superfluidity model [54, 55, 56], which is a two-channel
many-body model that deals with the closed channel Feshbach dimer explicitly. The addition of the189second channel is accounted for through the addition of auxiliary molecular field operator ψm (x)which obeys Bose statistics. Just as was done in Sec. 2.2.5, an effective theory can be derived [55]which reproduces the energy dependence of the scattering observables at the Feshbach resonanceand has been renomarlized to hide the cutoff dependence in the detuning, channel coupling, andstrength of the contact interaction. Importantly, this model was very successful in studying therelated unitary Fermi gas over a decade ago, which is stable due to the long lifetime of the Feshbachdimer and statistical suppression of three-body losses.
This model has yet to be applied to the recent results for the unitary Bose gas, nor hasa generalization to include bound state physics of higher numbers of atoms been derived. Thisis partly due to the difficulty and relative importance of three-body correlation dynamics for theunitary Bose gas compared to the unitary Fermi gas where Efimov physics is suppressed by identicalparticle symmetry in the ultracold limit. The theory of resonance superfluidity outlines a generalprocedure for installing bound states of any number of atoms, including, in principle, trimers.
The cumulant theory outlined in Chapter 7, which was originally extended to the level of threebody correlations in Ref. [53] to analyze higher order contributions to the GPE in the dilutenessparameter, is the logical candidate for studying the coupled dynamics of one, two, and three-bodycorrelations in the gas as discussed in Chapter 7. There is therefore a hierarchy of candidate theories
for the unitary Bose gas based on the union of the cumulant expansion method and the resonancesuperfluidity model, which increasingly reflect the underlying few-body bound state physics:(1) The coupled equations of one, two, and three-body correlations, excluding bound statephysics as generated in Chapter 7.
(2) The coupled equations of one, two, and three-body correlations, including the Feshbachdimer as a molecular BEC. This requires the introduction of an auxiliary dimer field operator ψm (x).
(3) The coupled equations of one, two, and three-body correlations, including the Feshbachdimer as a molecular BEC, and the Efimov states as a trimer BEC. Additionally, this

190
requires the introduction of an auxiliary trimer field operator ψt (x).
These are effective theories, and must reproduce the energy dependence of the scattering observables
over the experimental energy range using a set of fitting parameters. Therefore it is essential tounderstand the behavior of the phase shift discussed in Chapter 5. Importantly, the finite size of
the Efimov trimer provides a new length scale to the third candidate theory in addition to theinterparticle spacing, breaking the universal scaling of observables as powers of the inter particle
spacing n−1/3 .

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Appendix A

Numerical Simulation of Finite Temperature Bose Gases

The ZNG equations describe the evolution of a degenerate Bose gas in a six-dimensional phasespace (r, p), which is a computationally demanding problem. However, making use of symmetries
to reduce the number of degrees of freedom needed to describe the system can lead to significant
numerical advantages. Here, we make use of the spherical symmetry of the trap which reduces
the dimensionality of the problem from six to three, leaving a radial displacement r, a momentum
magnitude p, and an angular variable, cos θ, describing the orientation of the vector p with respect
to r. The main algorithm for numerically solving the ZNG equations is discussed in detail in
Refs. [112, 139], and in this Appendix we describe the additional details needed to apply thisalgorithm to solve the GGPE and QBE in a spherically symmetric geometry.
With spherical symmetry the condensate wavefunction depends only on r, and a 1D GGPE
can be used to describe its evolution. Furthermore, rewriting the GGPE in terms of the variable
φ(r) = rΦ(r) eliminates the first derivative term in the Laplacian, allowing for application of
simple Dirichlet boundary conditions where φ(r) → 0 as r → 0, ∞. We employ the Crank-Nicolson
method [284] to solve the GGPE in this form.
As in Ref. [112], a tracer particle method is used to evolve the noncondensate distributionfunction in phase space such that a Monte Carlo sampling method can be employed to simulate theeffects of collisions. We use 2×105 tracer particles for all simulations presented in Chapter 3. At eachtime step the tracer particle positions and momenta are updated based on Newton’s equations of
motion. Following the method outlined by Bird [285], we take advantage of the spherical symmetry

210
by only storing the radial coordinate of each tracer particle. However, the complete motion of eachparticle in 3D space must be tracked such that three momentum components are stored for eachparticle. At the beginning of each time step we utilize the rotational symmetry of the problem
and arbitrarily align the position vector of each particle with the x axis. The action of the y andz directed momentum components is to then push the particle off this axis. It is straightforward
to calculate the new radial position of the particle; however, the off-axis motion causes a rotationof the particle trajectory and the momentum components must be rotated accordingly. The newparticle position on the x axis is
x = ri +

px
∆t,
m

(A.1)

where ri is the initial radial position of the particle, px its momentum along the x axis, and ∆tis the length of the current time step. The action of py and pz moves the particle off axis by adistanced=rp2  p
2
z
∆t +
∆t ,
m
m
y

(A.2)

such that the new radial position rf of the particle isrf =p
x2 + d2 .

(A.3)

The sine and cosine of the rotation angle are then given bysin ϕ = d/rf ,(A.4)

cos ϕ = x/rf ,(A.5)

and an azimuthal angle is chosen at random such that φ ∈ [0, 2π]. Finally, the new momentum
components are calculated,
(A.6)

py,f

q
= px cos ϕ + p2y + p2z sin ϕ,= pc,f sin φ,

(A.7)

pz,f

= pc,f cos φ,

(A.8)

px,f

211

ϕ
pf
rf

d

pi

ϕ
ri

px
∆t
m

Figure A.1: The particle is rotated from its initial trajectory along ri by an angle ϕ due to the off
axis components of pi . After the particle position is updated to rf the momentum components arerotated and realigned with the position vector. Due to spherical symmetry the azimuthal angle is
not unique and is chosen randomly. This is represented by the area of revolution of pf about the
final position vector rf .

where pc,f = −px sin ϕ +q
p2y + p2z cos ϕ. Figure A.1 provides a graphical representation of theparticle movement algorithm. Note that although three momentum components are stored foreach particle in addition to the position, the algorithm is effectively three-dimensional since theazimuthal angle is randomized at each time step.
After the tracer particles are moved they are binned in phase space to get an estimate of
the local noncondensate density and collision rates. The particles are first binned in radial shells
using a constant volume binning scheme. Given the size of the simulation domain, lr , and the total
number of bins, Nb , the position of each bin edge is given by
rb,i = lr



i
Nb

1/3

,

(A.9)

where i ∈ [0, Nb ] is an integer representing the bin index. The simulations performed here use lr =60aho and Nb = 8×104 . This scheme results in wider bins near the origin, and progressively narrowerbins as r increases, which we find reproduces the equilibrium collision rates more accurately thana scheme with equal width bins in r. Once the particles are binned in space a 2D scheme isimplemented for binning the particles in momentum space based on p and cos θ, where p is themagnitude of the particle momentum. The momentum space bins are equally spaced, and we use

212
p
r

cos θ

Figure A.2: Cartoon depiction of the phase space binning process. Particles (black points) arebinned in position space using constant volume shells with the radial width of each bin decreasing
with r. Within each spatial bin the particles are further binned in momentum space using a 2D
grid of equal area bins based on the magnitude of their momentum p and trajectory cos θ = p̂ · r̂.

20 bins in p and 10 bins in cos θ for the simulations performed here. Figure A.2 provides a graphicalrepresentation of the binning procedure.
After binning, the tracer particles are used to reconstruct the noncondensate density andphase space distribution function on the discrete numerical grid defined for evolution of the condensate. The density term is required for updating the condensate wave function as well as themomentum of the tracer particles, whereas the phase-space distribution function is necessary forcomputing the collision rates. Typically, a cloud-in-cell method [286] is employed to reconstruct
a discrete function (e.g., density) from the tracer particle distribution by linearly weighting eachparticle to the nearest grid points defined by the binning process (i.e., the edges of each bin).
Following this weighting step, the reconstructed function can be interpolated from the binninggrid to another numerical grid if necessary. However, in spherical coordinates a linear weighting
scheme results in errors, particularly near the grid boundaries [287]. Therefore, we employ a volume
weighting scheme where the particles are weighted to grid points in proportion to the volume ofspace between the particle and a given grid point [288, 289]. We find this technique improves theaccuracy of function reconstruction from the tracer particle distribution, thus allowing fewer tracer
particles to be used, which results in improved computation speed. Although not explored in this213
thesis, this technique can be extended to cylindrical geometries, which might provide a modest
speed-up compared to simulations which assume no symmetries.
Binning with respect to r, p, and cos θ suggests an alternate solution method where the
distribution function is expanded in the basis of Legendre polynomials, and the QBE can be decomposed into a set of coupled differential equations for each partial wave [290]. These equations
can be solved without recourse to Monte Carlo integration as is commonly done in 1D for semiconductor and nuclear reactor transport theories (for example see Refs. [291, 292]). This point ishowever not taken further here, although it would be an intriguing method the avoids the use of
tracer particles to simulate the anomalous density.
The algorithm utilized in Sec. 3.4 utilizes the Monte Carlo procedure of the ZNG approachwithout the C12 terms and the condensate, and is therefore similar to the Direct Simulation MonteCarlo (DSMC) due to Bird [285].

A.1

Benchmarks
In this section, benchmarks of the spherical ZNG code used in this thesis against ZNG codesfrom previous works are presented, including: the 3D cylindrical code used in Ref. [139, 112], andthe adaptive 3D Cartesian code from Ref. [293]. Our code is limited in the benchmarks that canbe made due to the spherical symmetry (i.e. no dipole/Kohn mode etc...) Our benchmarks consistof demonstrating that the code is stable in equilibrium and re-equilibrates properly after a quenchof the thermal cloud to a lower temperature. Although not discussed, the first sanity check we
performed was to ensure that individual collision events left the total energy invariant.

A.1.1

Equilibrium

Achieving a stable equilibrium populations in a ZNG simulation is a nontrivial affair. Despite
the appearance of stable thermal and condensate population in Fig. A.3, both clouds are exchanging
atoms and energy continually. The stability of the populations is ensured through detailed balance
in process is equal to it’s time-reversed partner of the C outof the exchange collisions, i.e. that the C12
12

214
3.4

Atom Number (x104)

3.2

Nth

3
2.8

Ntot /2

2.6
2.4
2.2
2

Nc

1.8
1.6
50

100

150

200

Time [ms]

250

300

350

Figure A.3: Results from simulation of our spherical ZNG code for the equilibrium thermal andcondensate populations with 5 × 104 total atoms in a 9 Hz trap at 12 nK. The total atom number
stays within the fluctuation bounds (green region.)process1 . In Fig. A.4, an analytic calculation using the equilibrium Bose distributiont to calculate
the spatial dependence of the C22 and C12 collision rates is compared to the rates extracted fromnumerical simulation of the equilibrium state.
In this simulation, fixed total atom number is not an enforced constraint on the system. In
general it will have small fluctuations around the starting value due to the different ways in which
the condensate and thermal atom number are updated in the simulation. In Fig. A.3, the total
atom number fluctuates about its initial value showing no longterm drift outside of the fluctuation
bound denoted by the green region.
As a final remark, that the populations in Fig. A.3 do not change over time should becontrasted with the ZNG code in Ref. [293]2 whose initial populations adjust by as much as a few
percent. This initial adjustment is a result of the simulation finding a new numerical equilibrium
as a result of discretization effects in the problem. Our spherical ZNG code appears to capture theactual equilibrium state with minimum discretization effects given the comparative fineness of the
grid.
1
2

The terminology ‘in’ and ‘out’ is in reference to Fig. 4.3.
See Fig. 2.16 in this reference.

215

3000

TheoryZNG Simulation−1]
4πr2Γ22(1/ω) [aho

2500
2000
1500
1000
500
0
0

5

r [aho]

10

15

1500

ho

4/r2K (1/t) [a−1 ]TheorySimulated "In"
Simulated "Out"

12

1000

500

0

1

2

3

4

5

6

r [aho]Figure A.4: The collision rates per unit length as a function of position in an 187 Hz trap with
5 × 104 total atoms at 200 nK. In both plots, the data points are a result of Monte Carlo samplingof the collision processes, and the solid curves are a result of an analytic calculation using theBose distribution. Excellent agreement of the analytic and numerical rates is essential to runninga stable simulation.

A.1.2

Rethermalization after a quench

Achieving a stable equilibrium is an important test of detailed balance in the system. Additionally, our spherical code must be benchmarked in a dynamical, out-of-equilibrium situation
before being applied to the study of collective oscillations. In the ZNG literature [139, 294], a216
quench of the thermal cloud temperature in an isotropic trap was investigated, and the results ofthis simulation serve as an excellent benchmark of our code. Taking an equilibrium distribution,
the quench is accomplished by adjusting the momentum distribution of the thermal atoms to atemperature which is half of the equilibrium temperature. The lower velocities of the thermal
atoms result in a collapse of the thermal cloud which contracts towards the center of the cloud
and the entire system undergoes oscillations which are then damped as the system rethermalizes.
These oscillations are visible in the populations shown in the upper panel of Fig. A.5 where thetwo timescales in the problem are apparent: a shorter timescale for the balancing of populations≈10 ms and a longer timescale for the damping of the oscillations ≈30 ms.
In Fig. A.5, the result of three simulations are shown: a simulation using 3D cylindricalbinning geometry (reproduced from Ref. [294]), a simulation using a 3D adaptive Cartesian binning geometry (reproduced from Ref. [293]), and a simulation of our spherical ZNG code. This
comparison raises two important points. The first of which is that these simulations all begin withdifferent starting populations and end up displaced by roughly the same amount. This is a resultof discretization effects in the simulations where the numerical equilibrium state is different fromthe actual equilibrium state. This difference is minimized in codes with finer spatial meshes which
are adapted to the symmetry of the problem, and whose equilibrium populations do not drift after being initialized. The discretization effects are most pronounced in the 3D cylindrical and 3D
adaptive Cartesian and minimized in our spherical code. The second important point is the drift of
the total atom number shown in Fig. A.5 (lower panel). As mentioned in the previous subsection,√
fixed total atom number is not a constraint on the system and undergoes Ntot fluctuations as aresult of the noise from the Monte Carlo sampling. A long term drift of the total atom number
outside of these bounds indicates usually poor reconstruction of the distribution function and canarise from a suboptimal choice in the binning geometry of the problem. The 3D cylindrical code
violates these bounds, however the 3D adaptive Cartesian and spherical ZNG codes do not.

217
3.2

Nc

3

3D Cylindrical

Atom Number (x104)

2.8

3D Adaptive Cartesian
Spherical Symmetry

2.6

Ntot /2

2.4
2.2

Nth

2
1.8
0

5

10

15

Time [ms]

20

25

30

2.54
2.53

4

Atom Number (x10 )2.52

Ntot /2 +

2.51

√

N /2

2.5
2.49

Ntot /2 −

√

N /2

2.48
2.47
2.46

0

5

10

15

20

25

30

Time [ms]Figure A.5: (a.) The results of a quench of the thermal cloud in a 187 Hz trap with 5 × 104 total
atoms and an initial temperature of 200 nK. The three sets of data points are from a 3D cylindrical
ZNG code reproduced from Ref. [294], a 3D adaptive Cartesian code reproduced from Ref. [293],and our spherical ZNG code. (b.) The drift of the total atom numbers for the three simulations.
The green area is the fluctuation bound.