Manifestations of the Roton in Dipolar Bose-Einstein
Condensatesby
Ryan M. Wilson
B.S., Saint Louis University, 2006
M.S., University of Colorado, 2010

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
2011This thesis entitled:
Manifestations of the Roton in Dipolar Bose-Einstein Condensates
written by Ryan M. Wilson
has been approved for the Department of Physics

John L. Bohn

Chris H. Greene

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.

Wilson, Ryan M. (Ph.D., Physics)
Manifestations of the Roton in Dipolar Bose-Einstein Condensates
Thesis directed by Prof. John L. Bohn

Today, sixteen years after the realization of the first Bose-Einstein condensate (BEC), thefield of ultracold many-body physics is booming. In particular, much excitement has been generated
by the prospect of creating a degenerate quantum gas of dipolar atoms or molecules. Already, someexperimental groups have succeeded in Bose-condensing atomic 52 Cr and 164 Dy, while other groups
have made significant progress towards achieving degeneracy of heteronuclear molecules, such as
fermionic 40 K87 Rb and bosonic 87 Rb133 Cs, where the strength of the dipolar interaction promises
to be much greater than that of the already rich 52 Cr condensate. Just as the creation of BEC
launched a whole new field of research, dipolar BECs are likely to do the same. However, such
systems present a theoretical challenge due to the long-range, anisotropic nature of the dipolarinteraction. In this thesis, I present a theoretical investigation of ultracold Bose gases with dipolarinteractions.
The first part of this thesis is dedicated to the field theoretical treatment of a quantum Bose
fluid with dipolar interactions in the ultracold, dilute regime, where the system is well-described bya classical condensate field with quasiparticle excitations. The set of nonlinear integrodifferential
equations that describe these objects are derived and novel methods for solving them are presented
that, in general, require intricate numerical treatment. Of particular importance is the emergenceof a roton mode, reminiscent of that in superfluid 4 He. In the second part of this thesis, I show how
the roton plays a critical role in the ground state structure and dynamics of a dipolar BEC. Fullnumerical simulations show that the roton can, for example, be seen in the radial density profile ofa quantized vortex state or in the angular collapse and explosion of a dipolar BEC. Additionally,
I show the crucial role that this roton plays in determining the transition to superfluidity in thesesystems. Thus, a set of novel phenomena in ultracold dipolar Bose gases is explained by the presence

ivof the roton, and experimental signatures of these phenomena are made clear.

Dedication

To my parents, for their endless dedication and encouragement.

Acknowledgements

First and foremost, I would like to acknowledge and thank my advisor, John. If I have any
success as a theorist, I owe it to him. Among a million other things, he has taught me how to thinkcritically about physical problems and how put my work into a context that is beneficial both to me
and to the scientific community at large. Perhaps more importantly, he has made my experience asa graduate student a great one. Not only is he an incredible scientist, but he is a good person. In
five years, I never heard a negative word leave his mouth. He inspires with encouragement insteadof criticism. For that, his place as a role model in my life transcends the realm of scientific research.
I would also like to extend a deep thanks to Shai Ronen. Shai’s mentoring in the early daysof my career as a graduate student was indispensable. His mind is full of creativity and brilliance.
If I managed to absorb any of that during my time working with him, I have benefited greatly.
Of course, the rest of the “Bohn group” deserves a hearty thanks for being productive
colleagues and good friends. Thanks to Danielle Bortolotti, Manual Lara, Ed Meyer, GoulvenQuéméner, Michael Mayle, Brandon Ruzic and John Corson. I hope that I have made a sufficiently
good impression so that one of them will hire me some day.
I owe a special thanks to Chris Ticknor, Seth Rittenhouse, Hossein Sadeghpour and Eddy
Timmermans for extending generous invitations and hosting me on scholarly visits. These visits
have been incredibly beneficial to me and have highlighted my (hopefully) young career. Lobster
rolls and blue enchiladas were a big plus. Also, thanks to Han Pu for his fruitful collaboration andopen communication. I was lucky to have the opportunity to collaborate with such an intuitiveand intelligent researcher, and I hope to have the opportunity to do so again in the future. In thisviivein, I would also like to thank Ben Lev and his talented group for their open communication andcollaboration.
Thanks to Tilman Pfau, Jonas Metz and Juliette Billy from the Stuttgart 52 Cr experiment
for all of their useful input and discourse regarding their work.
Also, I feel incredibly lucky to have spent my graduate years at JILA and the University ofColorado. Outside of my research group, I would like to thank a number of other students, postdocs
and faculty who have enlightened and inspired me. This includes, but is certainly not limited to,Victor Gurarie, Chris Greene, Murray Holland, Ana Maria Rey, Eric Cornell, Debbie Jin, Dominic
Meiser, Jami Kinnunen, Kaden Hazzard, Jia Wang, Shu-Ming Li, Charlie Sievers, Adam Scheer,
Brian Neyenhuis, Zhaochuan Shen, and Ron Pepino.
Thanks to Philippe Verkerk and the other Les Houches INTERCAN and IFRAF predoctoral
school organizers for two memorable and fruitful weeks of study.
Before coming to graduate school, I received a great deal of support and inspiration as anundergraduate at Saint Louis University and as an REU student at Columbia University. From
Columbia, I would like to thank Rafeal Galea, Jeremy Dodd and my fellow students Eli Visbal, Colin
Beal and Stephen Poprocki. From Saint Louis University, I would like to thank Ian Redmount, BillThacker, Greg Comer, Thalanayar Santhanam, Larry Stacey, Les Benofy, Vijai Dixit, Jean Potvin,
Fr. Mike May, Kent Staley, Brody Johnson and my partner in chips, salsa and E&M, Ben Hurst.
Outside of the Physics world, I have developed an interest in the ceramic arts. Thanks to
Scott Chamberlin and Jeanne Quinn for supporting my clay habit.
Thanks to my family. Though I know that I am inspired to learn and research out of pureinterest, the entirety of my inspiration has much deeper roots. My family has always made a point
to tell me that they were proud of me. The impact of that on my career and life is immeasurable.
Thanks to my brother, for, whether he knows it or not, teaching me to be good.
Finally, I would like thank Lindsay Pichaske for her love and support. She inspires a joy anda confidence that penetrates deeply into my life.

Contents

Chapter
1 Introduction

1

2 Background: Theory and Experiment8

2.1

A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.1

Phonons and Rotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2

Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3

Superfluidity and Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . 13

2.2

The Ideal Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3

Bose-Einstein Condensation of Trapped Gases . . . . . . . . . . . . . . . . . . . . . . 18

2.4

2.3.1

Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2

Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Dipolar Interactions in Ultracold Bose Gases . . . . . . . . . . . . . . . . . . . . . . 25

3 Zero-Temperature Field Theory for Bosons
3.1

Second-Quantized Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1

3.2

29

Many-Body Hamiltonian in Second-Quantization . . . . . . . . . . . . . . . . 32

The Bogoliubov Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1

Long-Range Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.2

Symmetry in the Bogoliubov de Gennes Equations . . . . . . . . . . . . . . . 38

3.2.3

Quantum Depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

ix
3.3

Time-dependent formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1

3.4

First-Quantized Theory - Alternative Derivation . . . . . . . . . . . . . . . . 42

Two-body Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1

Pseudopotential for Short-Range Interactions . . . . . . . . . . . . . . . . . . 44

3.4.2

Dipole-Dipole Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.3

Mean-Field Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Homogeneous Dipolar Bose-Einstein Condensates

49

4.1

Three-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2

Quasi-Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1

Contact Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.2

Dipole-Dipole Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.3

Bogoliubov Spectrum: Emergence of the Roton . . . . . . . . . . . . . . . . . 61

4.2.4

Quantum Depletion and Roton Instability . . . . . . . . . . . . . . . . . . . . 65

4.2.5

The Other Roton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Dipolar Bose-Einstein Condensate in a Cylindrically Symmetric Trap
5.1

68

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1.1

Modified Momentum-Space Dipole-Dipole Interaction . . . . . . . . . . . . . 74

5.2

Rotationless (s = 0) Dipolar Bose-Einstein Condensate . . . . . . . . . . . . . . . . . 76

5.3

Singly-Quantized Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4

Doubly-Quantized Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5

“Perturbed” Dipolar Bose-Einstein Condensate . . . . . . . . . . . . . . . . . . . . . 88
5.5.1

5.6

Perturbation Theory for the Gross-Pitaevskii Equation . . . . . . . . . . . . . 92

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 Collapse of a Dipolar Bose-Einstein Condensate
6.1

96

Fano-Feshbach Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

x
6.2

Local collapse: Evidence from the Stability Diagram . . . . . . . . . . . . . . . . . . 99

6.3

Local Collapse: Evidence From the Collapsed Cloud . . . . . . . . . . . . . . . . . . 102

6.4

6.3.1

Modes of Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3.2

Numerics and the “Ideal Experiment” . . . . . . . . . . . . . . . . . . . . . . 104

6.3.3

A More Realistic Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7 Superfluidity in a Dipolar Bose-Einstein Condensate

113

7.1

Landau Critical Velocity for Superfluid Flow

7.2

Discrete Dipolar Superfluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.3

Anisotropic Dipolar Superfluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.4. . . . . . . . . . . . . . . . . . . . . . 115

7.3.1

Quasiparticle Production (Weak Laser) . . . . . . . . . . . . . . . . . . . . . 127

7.3.2

Vortex Production (Strong Laser) . . . . . . . . . . . . . . . . . . . . . . . . 128

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8 Dipolar Bose-Einstein Condensate on a One-Dimensional Lattice

134

8.1

Formalism for the One-Dimensional Lattice . . . . . . . . . . . . . . . . . . . . . . . 135

8.2

Wave Function Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.3

Infinite lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.4

Finite Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.5

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9 Summary

148

Bibliography

152

Appendix
A The Convolution Theorem

165

xi
B Momentum-Space Dipole-Dipole Interaction Potential

167

C Quasi-2D Dipolar Interaction Potential

169

D Discrete Hankel Transform

171

E Energy-Functional Minimization via Conjugate Gradients173

F Radial Grid Interpolation

176

G Calculation of the Mean-Field Potential in Reduced Dimensions

178

H Modified GPE Using 0th and 2nd Harmonic Oscillator Wave Functions

180

xii

TablesTable
2.1

Dipole moments and characteristic dipole lengths of relevant bosonic atomic andmolecular species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1

Orders of Hankel transforms necessary to calculate the dipole-dipole interaction
terms in the cylindrically symmetric Bogoliubov de Gennes equations. . . . . . . . . 74

Figures

Figure
2.1

Phonon-roton quasiparticle dispersion of 4 He below its lambda-point, measured via
neutron scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Condensate fraction of non-interacting Bose gas in a spherical harmonic trap. . . . . 17

2.3

Images of emergent Bose-Einstein condensates as a function of temperature from the1995 alkali atom experiments at JILA and MIT . . . . . . . . . . . . . . . . . . . . . 23

2.4

Experimental images demonstrating the presence of quantized vortex states in a
Bose-Einstein condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1

Diagrammatic representation of the grand canonical Hamiltonian for a dilute, ultracold Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2

Geometry of the dipole-dipole interaction for dipoles polarized by an external fieldin the z-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1

Quasi-2D momentum-space interaction potential for various dipole-dipole interaction
strengths ǫdd

4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Quasiparticle dispersion of the quasi-2D BEC with repulsive contact interactions andǫdd = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3

Unstable quasiparticle dispersion of the quasi-2D dipolar BEC with repulsive contact
interactions and ǫdd = −1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xiv
4.4

Quasiparticle dispersion of the quasi-2D dipolar BEC with repulsive contact interactions and ǫdd = 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5

Quasiparticle dispersion of the quasi-2D dipolar BEC with repulsive contact interactions and ǫdd = 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6

Stability diagram of the quasi-2D dipolar BEC. . . . . . . . . . . . . . . . . . . . . . 64

4.7

Quantum depletion of the purely dipolar quasi-2D BEC. . . . . . . . . . . . . . . . . 66

5.1

Structure/stability diagram of a rotationless dipolar BEC in a cylindrically symmetric harmonic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2

Integrated densities of an initially biconcave dipolar BEC, showing the biconcave
structure vanish in the ballistic expansion of the condensate (expansion in free space). 78

5.3

Schematic representation of the collapse of a dipolar BEC in prolate and oblate traps. 79

5.4

The Bogoliubov de Gennes spectrum of the rotationless dipolar BEC in a trap withaspect ratio λ = 17, and the radial profile of the discrete roton mode in this system
at D = 180.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.5The structure/stability diagram of the dipolar BEC with a singly-quantized vortex. . 83

5.6

Imaginary parts of the Bogoliubov de Gennes spectrum for a dipolar BEC with asingly-quantized vortex in various trap geometries. . . . . . . . . . . . . . . . . . . . 84

5.7

Real parts of the Bogoliubov de Gennes spectrum of a dipolar BEC in a trap withaspect ratio λ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.8

Angular quantum number m responsible for dynamic instability in a dipolar BEC
with a singly-quantized vortex as a function of trap geometry. . . . . . . . . . . . . . 87

5.9

Imaginary part of the Bogoliubov de Gennes spectrum of a dipolar BEC with adoubly-quantized vortex in various trap geometries. . . . . . . . . . . . . . . . . . . . 89

5.10 Radial profiles of a perturbed dipolar BEC in an oblate trap. . . . . . . . . . . . . . 91
5.11 Profile of radial roton and perturbative correction to dipolar BEC wave function. . . 93
6.1

Schematic of the magnetic Fano-Feshbach resonance in a two-state model. . . . . . . 98

xv
6.2

Structure/stability diagram for the 52 Cr dipolar BEC showing experimental andtheoretical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3

A comparison between the theoretical results for the stability of the 52 Cr dipolar
BEC. Shown are the results from the full numerical solution of the GPE and theGaussian ansatz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4

Real and imaginary parts of the low-lying Bogoliubov de Gennes modes for thebiconcave 52 Cr BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.5

Images of collapsed biconcave dipolar BECs from real-time simulations, showing thatthe finite grid does not effect the collapse results. . . . . . . . . . . . . . . . . . . . . 105

6.6Structure/stability diagram of the 52 Cr BEC, showing s-wave scattering length ramps
used in the simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.7

Images of a “normal” (not biconcave) dipolar BEC collapsing and expanding withpure radial character, and images of a biconcave dipolar BEC collapsing and expanding with angular nodal structure, signifying the angular collapse of the biconcave state.110

7.1

Examples of the response of a dipolar BEC in an oblate trap to a weak blue-detuned
laser moving through it at speeds slower and faster than the superfluid critical velocity.117

7.2

Some examples of discrete dispersion relations of a dipolar BEC in an oblate trap,
showing discrete rotons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.3

Averaged quasiparticle occupations of a dipolar BEC in an oblate trap as a functionof blue-detuned laser velocity after a laser has been translated through it. . . . . . . 121

7.4

Superfluid critical velocities of a dipolar BEC in an oblate trap as calculated from thediscrete dispersion relations and from direct numerical simulation of a blue-detuned
laser moving through the gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.5

Anisotropic quasiparticle dispersion of a quasi-2D dipolar BEC with a tilted polarization and anisotropic in-plane dipolar interactions. Also shown are density profilesof the perturbed stationary BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xvi
7.6

Schematic of the quasi-2D dipolar BEC with anisotropic interactions.

. . . . . . . . 127

7.7

Results from numeric simulations of a weak and a strong blue-detuned laser moving
through a quasi-2D dipolar BEC in directions perpendicular and parallel to thepolarization tilt, showing drag force on the laser and maximum vortex number as afunction of laser velocity, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.8

Examples of anisotropic vortex nucleation in a quasi-2D dipolar BEC with anisotropicinteractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.9

Example of anisotropic superfluidity in a radially trapped quasi-2D dipolar BEC
with anisotropic interactions, showing vortex formation for motion of a strong bluedetuned laser perpendicular to the polarization tilt, and no vortex formation in theparallel direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.1

The energy differences of a dipolar BEC in an oblate trap as calculated exactly on anumeric grid and as calculated using a separable ansatz with the 0th and 2nd orderharmonic oscillator wave functions for the axial part. . . . . . . . . . . . . . . . . . . 138

8.2

The values of the axial wave function parameters for the 0th and 2nd order harmonic
oscillator wave function ansatz that minimize the energy of a single dipolar BEC in
an oblate trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.3

Comparison between the structure/stability diagrams of a dipolar BEC as calculated exactly and with a separable wave function ansatz using the 0th and 2nd orderharmonic oscillator wave functions variationally. . . . . . . . . . . . . . . . . . . . . . 140

8.4

Structure/stability diagram for an infinite lattice of dipolar BECs in a trap withaspect ratio λ = 10.

8.5

Structure/stability diagram for an infinite lattice of dipolar BECs in a trap withaspect ratio λ = 20.

8.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Stability diagram for an infinite lattice of dipolar BECs with very oblate lattice sites. 144

xvii
8.7

Radial densities for a dipolar BEC in a finite lattice as compared to a single dipolar
BEC, showing the emergence of biconcave structure in an experimentally realizable
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Chapter 1
Introduction

The experimental realization of a Bose-Einstein condensate (BEC) of alkali atoms at JILA,MIT and Rice University in 1995 [1, 2, 3] opened the door to a vast, interdisciplinary field fullof opportunity and potential. The BEC, first theorized by A. Einstein in 1925 [4], was the coldest sample of matter known in the universe, and was the fruitful result of years of experimental
and theoretical progress in the field of optical and magnetic cooling and trapping [5]. The BEC
did, and still does, offer a tool with which to study a plethora of ultracold phenomena, including
superfluidity and its manifestations in quantum matter. Additionally, the realization of ultracold
temperatures allows for atoms and molecules to be trapped by purely optical means, which facilitates experimental control over the magnetic substates of these systems and allows for trappingin optical lattice potentials. Atoms and molecules in optical lattices can be used, for example, forquantum computing purposes or to study more complicated condensed matter systems in a clean,controllable environment [6, 7, 8].
The presence of dipolar interactions in the Bose-Einstein condensate enhances much of thephysics in ultracold quantum systems. The dipole-dipole interaction (ddi) is long-range, proportional to the inverse cube of the distance between two dipoles, and anisotropic. As such, the ddi canintroduce inter-site couplings in optical lattice systems in one- and two-dimensions (1D and 2D),
and anisotropic couplings in three-dimensional (3D) lattices [8]. In fermionic systems, this feature
can result in a transition to superfluidity as the attractive part of the ddi leads to pairing betweensites [9, 10, 11]. Additionally, in 2D geometries, the presence of the ddi in a Bose gas is predicted2
to lead to a self-ordered crystalline state, or Wigner crystal, for sufficiently large densities [12, 13].
Experimentally, BECs of atomic 52 Cr [14, 15, 16] and 164 Dy [17] have been achieved, where theatoms possess significant permanent magnetic dipole moments, being 6 and 10 Bohr magnetons,
respectively. By comparison, the magnetic dipole moment of 87 Rb is only 1 Bohr magneton. While
comparatively small, however, the ddi has been shown to play an crucial role in the physics of the
87 Rb F = 1 spinor BEC [18, 19, 20].

While the first report from the 164 Dy BEC experiment has already demonstrated strong
dipolar effects, the 52 Cr experiments in the group of Tilman Pfau in Stuttgart have demonstrated
that the ddi plays a strong role in the stability [21] and dynamics [22] of a BEC. Additionally,
this group demonstrated that the s-wave scattering length of the 52 Cr atoms could be tuned tozero, thus creating a purely dipolar BEC [23]. While the dipole moments of these atoms are indeed
sufficiently large to observe (and predict) some interesting dipolar effects, recent experimental
advances in the production, trapping and cooling of heteronuclear molecules inspires great promisethat such molecules will be brought to quantum degeneracy in the near future. Such molecules canpossess very large, tunable electric dipole moments when polarized in an external field, on the orderof a Debye, which is about two orders of magnitude larger than a Bohr magneton when the twoquantities are expressed in the same system of units. Already, experimentalists have managed to
produce cold samples of heteronuclear molecules in their rovibrational ground state [24, 25, 26], andthe JILA group recently demonstrated long-lifetime trapping of fermionic KRb molecules in a 3D
optical lattice geometry [27]. Such progress inspires encouragement that a BEC of polar molecules
is realizable in the near future.
In addition to the novel physics that has been predicted in, for example, optical lattice andspinor systems, the ddi has been predicted to lead to the rotonization of a dipolar BEC in a trapped
geometry [28, 29, 30]. The roton, being a local minimum at finite wave number in the quasiparticle
dispersion relation of an ultracold Bose gas, was first predicted and seen in the superfluid 4 He
system [31, 32, 33], though the origin of the two rotons are very different. The 4 He system isvery dense, and the roton therein is related to the structure factor of the liquid at the interatomic

3level, signifying a tendency for crystalline ordering in the system. The roton in the dipolar BEC,
however, is present even in the dilute, gaseous state and derives from the momentum dependenceof the ddi in a trapped geometry. The most transparent example of this is the so-called quasi-2D
dipolar BEC, where the system is harmonically trapped in the direction of the dipole polarizationand the dipoles exhibit zero-point motion in this direction. Indeed, the demonstrated control thatexperimentalists have over the trapping geometry and interactions in a dipolar BEC suggests that
this system is ideal for studying the physics of the roton. In this dissertation, we tackle this idea
head on and present a comprehensive, detailed theoretical account of the role that the roton playsin the physics of the dipolar BEC.
Because the dipolar BECs that have been created in the laboratory setting are quite dilute, they are well-described by a mean-field theory that provides a relatively simple theoreticaltreatment of these systems. The mean-field theory of dipoles, however, is not without its own
set of challenges. Whereas short-range interactions of ultracold atoms and molecules can be welldescribed by a delta-function pseudopotential, the ddi admits no such simplification and must be
handled explicitly. For example, the mean-field theory that we employ in this work presents a seriesof direct and exchange interaction terms that require the calculation of convolution integrals (seechapter 5). While the delta-function pseudo-potential trivializes these integrals, the ddi does notand the convolutions must be calculated as given. Additionally, we consider fully-trapped systems
in this work that generate hard numerical problems, both when calculating the condensate field andits set of quantum fluctuations. In this dissertation, we develop and present methods for overcoming
these difficulties. The key results of this work include a set of methods and algorithms that turnthe theoretical treatment of a fully trapped dipolar BEC into a tractable one. We then apply theseresults and predict a set of novel phenomena related to the roton in the trapped dipolar BEC. To
make the results presented in this dissertation as relevant as possible to the scientific community,
we have made a point to associate all of our results with current experiments, or experiments thatare realizable in the foreseeable future.
In chapter 2 of this thesis, we give a short background of the history of low-temperature

4physics. This includes a discussion of the early experiments and thoughts on superfluid 4 He.
Indeed, it was this early scientific discourse that laid the groundwork for our understanding ofsuperfluidity and its manifestation in matter through, for example, quantized vortices and, mostfundamentally, long-range order. We also discuss some of the more recent experimental advances in
the field, including the basic physics behind the optical and magnetic cooling and trapping methods
that led to the first experimental realization of a BEC and the first, most fundamental results thatlaid the foundation for the modern study of the ultracold physics of bosons. To give the reader anidea of how the Bose-Einstein condensate phase emerges statistically as a function of temperature,
we also discuss the phenomenon of BEC in a trapped, non-interacting (ideal) gas of bosons.
In chapter 3, we start from a second-quantized description of a quantum many-body system ofinteracting bosons and systematically derive the set of mean-field equations that describe the condensate field and the quantum fluctuations of the dilute, interacting Bose gas at zero-temperature,
being the Gross-Pitaevskii equation and the Bogoliubov de Gennes equations, respectively. Additionally, we motivate the use of a pseudo-potential for the short-range two-body interactions in theultracold gas and discuss the treatment of the ddi, where non-trivial convolution integrals must
be calculated. To treat the ddi, we employ the convolution theorem and handle the integrals in
momentum-space, moving to and from real space via Fourier transformation.
In chapter 4, we apply the mean-field theory to the homogeneous 3D and quasi-2D dipolar
BECs. We investigate the energetics and quantum fluctuations of these systems, where the quantum
fluctuations take the form of quasiparticles in the Bogoliubov theory, and thereby map their stability
in parameter space. In the quasi-2D case, an effective ddi is derived, which leads to the emergenceof the roton quasiparticle in Bogoliubov theory. Interestingly, the roton can lead the quasi-2D
dipolar BEC to collapse that is both density dependent and local, having character that opposes
the usual phonon, or energetic instability in the 3D dipolar BEC or the BEC with attractive contact
interactions. Original work from this chapter is published in [34].
We move on to treat the fully-trapped dipolar BEC in chapter 5. To simplify the problem
at hand, we consider a cylindrically symmetric harmonic trap with the dipoles polarized along

5
the trap axis of symmetry, so the system as a whole possesses such symmetry. In this case,the problem is reduced from a 3D to a 2D problem in the axial and radial coordinates where
the angular dependence of the relevant functions, being the condensate wave function and thequasiparticle modes, is included in an angular factor eikϕ . A discrete Hankel transform is usedto handle the transforms in the radial direction (see appendix D), where the Hankel transform
expands the relevant function in terms of Bessel functions of order k. Thus, condensate modes andquasiparticle modes with arbitrary vorticity are handled by simply choosing a Hankel transformof the appropriate order. We use this algorithm to study rotationless dipolar BECs and dipolar
BECs with singly- and doubly-quantized vortices by employing a conjugate gradient algorithm for
efficient minimization of the Gross-Pitaevskii energy functional. We calculate the quasiparticle
modes by solving the Bogoliubov de Gennes equations via an iterative Arnoldi diagonalization
scheme. Our results reveal that dipolar BECs with maximum densities in a ring about the centerof the trap, such as dipolar BECs with singly-quantized vortices and rotationless dipolar BECs withbiconcave structure [30], become dynamically unstable due to the softening of discrete roton-likemodes with angular nodal structure. Thus, the roton manifests with angular character in thesesystems. Additionally, we find regions in parameter space where the dipolar BEC with a singlyquantized vortex exhibits radial density oscillations. We attribute such structure to the static
manifestation of a discrete radial roton mode in the ground state due to the “perturbation” of thevortex core by applying a perturbation theory to the Gross-Pitaevskii equation. Original work fromthis chapter is published in [35] and [36].
In chapter 6, we apply a 4th order Runge-Kutta algorithm to the time-dependent GrossPitaevskii equation to show that the angular roton instability of the rotationless dipolar BEC withbiconcave structure results in an angular collapse and subsequent angular expansion when the trap
is turned off and the condensate is allowed to expand in free space. Imaging of the expanded cloudwith angular nodal structure would then provide a measurement of the angular collapse and, thus,
an indirect measurement of the presence of biconcave structure in the stable ground state of thesystem. Original work from this chapter is published in [37].

6
We move on to study the superfluid properties of the dipolar BEC in chapter 7. For the fully
trapped system, we calculate a “discrete” dispersion relation, or quasiparticle energy as a function of
momentum, which allows us to apply the Landau criterion for superfluidity to the trapped system toget an estimate of its superfluid critical velocity, or flow velocity below which flow is dissipationless.
The presence of the discrete roton serves to lower the Landau critical velocity as a function of ddi
strength or condensate density, which is confirmed via direct numeric simulation of a weak bluedetuned laser moving through the condensate with varying velocity. Indeed, these results support
the Landau criterion, but reveal finite size effects. These effects grow as the strength and size ofthe laser are increased. Indeed, if the laser is sufficiently strong so as to create a hard boundary
on a length scale on the order of the healing length of the condensate, vortices are nucleated in thegas instead of quasiparticles being produced above the critical velocity. The critical velocity forvortex nucleation, however, is much lower than the critical velocity for quasiparticle production.
We proceed by considering a quasi-2D dipolar BEC where the polarization is now allowed to point
in any direction, not just in the direction of the axial confinement. In this case, the interactions
take on anisotropic character and, for a certain ddi strength and “tilt” angle, the dispersion relationof the system possesses a roton in the direction perpendicular to the dipole tilt and only phonon
character in the parallel direction. This, in turn, predicts an anisotropic critical velocity for thesystem via the Landau criterion. We perform numeric simulations of both weak and strong bluedetuned lasers moving through this quasi-2D system and find that the superfluid critical velocity forboth quasiparticle production and vortex nucleation is anisotropic, and the quasi-2D dipolar BEC
with a tilted polarization field is thus an anisotropic superfluid. Original work from this chapter ispublished in [38] and [39].
In chapter 8, we consider again a dipolar BEC with cylindrical symmetry, but now loaded in
a 1D lattice. For the case of an infinite lattice, we find a significant simplification of the mean-fieldinteraction terms, as long as the axial wave function has an analytic form. We thus employ aseparable ansatz to the BECs at each site where the radial part of the condensate wave function
is sampled on a numeric grid, as before, and the axial part of the wave function (in the lattice

7
direction) is given by a linear combination of the 0th and 2nd order Hermite polynomials. For thecase of a single dipolar BEC, this ansatz gives excellent qualitative agreement and good quantitative
agreement with the results of the full numeric treatment given in chapter 5. Thus, the GrossPitaevskii equation for the infinite 1D lattice becomes an equation for a single dipolar BEC but
with a modified interaction potential. We study the structure and stability of this system as afunction of lattice spacing, lattice site geometry and ddi strength. We find wildly modified roton
stability in the lattice, where the system is highly destabilized for small lattice spacings due tothe attractive part of the ddi. We also find “islands” in the parameter space where biconcave
structure is present that would not be present in the absence of the lattice. Thus, we predict
emergent biconcave structure in the dipolar BEC in the infinite 1D lattice. As a check, we treat anexperimentally realistic system of nine lattice sites with varying condensate number exactly on avery large numeric grid, and find that the emergent biconcave structure persists in the finite lattice.
Original work from this chapter is published in [40].
We summarize this dissertation in chapter 9.

Chapter 2
Background: Theory and Experiment

In this chapter, we discuss some of the important points in the history of low-temperaturephysics that lead up to the discovery of Bose-Einstein condensation in a dilute alkali vapor, wherein
there are some excellent demonstrations of the advancement of scientific knowledge through theinterplay of experiment and theory. Regarding the more recent history, we discuss the experimental
advances that have occurred in the past two decades, as these are key not only to understanding
the work presented in this thesis, but also to understanding the advances that went into making
the “ultracold” regime an experimental reality. Additionally, we discuss some of the more relevant
experimental results on dilute Bose-Einstein condensates, and motivate the exploration of the rolethat the dipole-dipole interaction (ddi) plays in these systems.

2.1

A Brief History
Motivated by Satyendra Nath Bose’s work on the statistics of photons, Albert Einstein for-mulated the first theory for the statistics of massive bosons in 1924 [4]. He predicted that, below a
critical temperature Tc , the lowest energy state of a quantum many-body system of bosons would
become macroscopically occupied. This idea stemmed from two basic concepts, one being the indistinguishability of quantum particles and the other being the simple fact that bosons, as opposedto fermions, obey statistical laws such that two or more identical bosons can occupy the same
quantum mechanical state, whereas identical fermions are forbidden to do so. As it turns out, thisbehavior of fermions is responsible for, among other things, the structure of electronic orbitals in9atoms and the quantum degeneracy pressure that results in the stabilization of neutron stars, as
electrons and neutrons are both fermions.
As we will see, Einstein’s prediction was correct and the phenomenon that is now known as
Bose-Einstein condensation does indeed occur in a system of bosons at sufficiently low temperature (as long as the bosons do not solidify). Additionally, the scientific community has come tounderstand that there are many interesting physical phenomena associated with this novel stateof matter, the Bose-Einstein condensate (BEC). Perhaps the most important consequence of BoseEinstein condensation is the emergence of superfluidity, though the connection between BEC andsuperfluidity was not immediately drawn in the earlier days of its study. In fact, this connection isstill being investigated today, as we discuss further in section 2.3.2.
The word “superfluid” was first used by P. Kapitza in [41] to describe the non-classical natureof liquid 4 He that was observed at temperatures below ∼ 2.2 K [42], where the use of the prefix“super” was inspired by the already observed phenomenon of superconductivity in solid mercury
in 1911 [43]. The strange, non-classical, “super” behavior to which Kapitza referred was theobservation of a discontinuity of the specific heat of liquid 4 He around this temperature, the graphof which resembled the Greek character “λ” and was thus termed the “lambda-point.” This was
not the first time, however, that non-classical behavior was observed in liquid 4 He. For example,
experiments using a torsion pendulum showed that the viscosity of liquid 4 He drops significantly
when its temperature is dropped below the lambda-point [44], that is, the flow in liquid helium wasobserved to be non-dissipative. Inspired by the accumulating body of experimental evidence for thesuperfluid behavior of 4 He below a critical temperature, by the earlier theoretical work of Einstein,
and by the fact that such phenomena were not observed in 3 He (a fermionic isotope) at the same
temperatures, Fritz London proposed in 1938 that the unusual behavior of liquid 4 He was due tothe phenomenon of BEC manifesting in the cold fluid [45]. Not long thereafter, the work of othertalented theorists, namely L. Tisza and L. Landau, showed that a BEC-like superfluid fraction of
the system was likely present and responsible for the unique non-dissipative behavior of liquid 4 He,
supporting F. London’s earlier hypothesis.

10
2.1.1

Phonons and Rotons

Both Tisza [46] and Landau [31, 47] proposed two-fluid models to describe liquid 4 He, where
one fluid corresponded to the superfluid component and the other to the “normal,” or non-superfluid
component. Landau’s insight was particularly brilliant in that he interpreted the normal component
as a set of occupied excited states consisting of phonons and localized quantized vortices, dubbed“rotons” due to the rotational nature of such vortices. While the phonons disperse linearly, Landau
predicted that the rotons experience a quadratic dispersion,
ω(k) = ∆ +2k − kroton,
2Mroton(2.1)

where ~kroton is the roton momentum, on the order of the inverse atomic spacing in the liquid, Mrotonis the effective roton mass and ∆ is the roton energy gap. Landau was able to estimate the values ofthese parameters by matching his theory to the observed thermodynamical behavior of liquid 4 He.
From this fitted dispersion, Landau developed a hard criterion for the existence of superfluidity in
4 He, being that superfluid, or dissipationless flow only exists below a critical velocity, the so-called“Landau critical velocity,” or superfluid critical velocity. The Landau criterion for superfluidity
can be derived simply by applying arguments for the conservation of energy and momentum ofa phonon or roton excitation in a Galilean frame of reference, which we present in section 7.1.
For 4 He, the predicted Landau critical velocity is set by the roton minimum, giving a velocity ofvc ≃ ∆/kroton ≈ 60 m/s.
In 1957, Cohen and Feynman proposed that the Landau phonon-roton dispersion could bemeasured by inelastically scattering neutrons off of a liquid 4 He sample [32]. This experiment
was carried out soon thereafter, where excellent agreement was found with Landau’s theory [33,48]. Thus, strong evidence was obtained in support of Landau’s prediction for the phonon-roton
dispersion, though the explicit connection between superfluidity and a critical velocity was still not
made. It is worth noting, though, that whether rotons are associated with vorticity is inconclusive
in these experiments, and as a result rotons should not necessarily be thought to have a vorticalnature. The measured phonon-roton dispersion from Ref. [33] is shown in figure 2.1. Indeed, we11Figure 2.1: Measured quasiparticle dispersion of liquid 4 He at a temperature of 1.12 K. The circles
show the neutron-scattering data from Ref. [33] and the solid line shows the free-particle dispersion.
The phonons are seen in the linear dispersion at small momentum, and the roton is seen at the
local minimum at the momentum ∼ 2.0Å−1 . Figure taken from Ref. [33].

return with strong interest to this roton feature in the 4 He dispersion in chapter 4, where we showthat a similar feature emerges in the trapped dipolar BEC.

2.1.2

Vortices

While the neutron scattering experiments were able to measure the phonon-roton dispersion
in superfluid 4 He, such techniques could not be applied to test the Landau criterion for superfluidity,
or to measure the superfluid critical velocity in 4 He. For such measurements, a relative macroscopic
flow velocity between the superfluid component and a “perturber” is required, the magnitude ofwhich must be at least as great as the critical velocity of the superfluid. Such flow was realized
in 1985 by forcing superfluid 4 He through a small aperture (less than 1 µm) [49]. The observed
critical velocity in this experiment, however, was much lower than that predicted by the Landaucriterion, suggesting that the excitation of rotons is not the relevant mechanism for dissipation in12this superfluid system. Instead, the observed critical velocity is associated with dissipation intoquantized vortex lines, where the vortices have quantized circulation 2πn~/M , where M is the massof a particle in the fluid and n is an integer. Thus, having n = 0 implies no vorticity, while n = 1
corresponds to a singly quantized vortex, n = 2 corresponds to a doubly-quantized vortex, andso on. The existence of such vortex lines was first predicted by Feynman in 1955 [50], where heproposed that a critical flow velocity is necessary to nucleate such a quantized vortex state, just
as is the case for a phonon or roton. He predicted that the critical velocity for the formation of asingly-quantized vortex of radius a in a cylinder of radius d should be given by
 
~
d
vc =
ln
.
Md
a

(2.2)

In the experiment [49], a series of critical velocities were measured, corresponding to dissipation
into vortex lines of various quantization, the lowest of which is in good agreement with Feynman’s
prediction (2.2). We note that the breaking of superfluid flow due to the excitation of rotons wasobserved, as well, by drifting negative ions through superfluid 4 He at sub-critical and super-criticalvelocities. The ions, unlike the hard wall of the aperture, were not sufficiently intrusive so as tonucleate free vortices. The mechanism for dissipation, though, is believed to be the excitation of apair of rotons instead of a single roton above the Landau critical velocity [51].
While quantized vorticity does not exist in classical fluids, it is known to exist in singleparticle quantum mechanical systems, for example, in the atom where the electrons have quantizedangular momentum. The existence of quantized vortices in a superfluid suggests that the superfluid
state may indeed be intimately connected with the phenomenon of Bose-Einstein condensation,
where a macroscopic number of bosons occupy the ground single particle state. As we will see,quantized vortices manifest in BECs due to the single-particle nature of the condensed state, andare intimately related to the presence of superfluidity in cold Bose gases. Indeed, such phenomena
are used as a “smoking gun” of superfluidity in these systems.

13
2.1.3

Superfluidity and Bose-Einstein Condensation

In 1956, Roger Penrose and Lars Onsager devised what remains today as perhaps the mostfundamental theoretical criterion for superfluidity and Bose-Einstein condensation, linking the twoin a before unseen way. They noticed that every such system must possess long-range order [52], thatis, the single-particle density matrix or one-body correlation function ρ(1) (x, x′ ) of the superfluid
must not vanish in the limit |x − x′ | → ∞, and instead approach a finite valuelim|x−x′ |→∞

ρ(1) (x, x′ ) = ρc ,(2.3)

where ρc is the condensate number density of the system. This criterion is equivalent to saying that amacroscopic number of bosons in the fluid occupy the momentum state with ~k = 0, corresponding
to the lowest energy state of homogeneous space. Thus, the criterion for superfluidity proposedby Penrose and Onsager is simply that a finite fraction of the fluid be Bose-condensed. Whenintroducing the methods we use to treat the BEC in this thesis (in chapters 3 and 4), we return tothis point and show that the criterion for long-range order is satisfied by these methods.
Today, much of the physics of superfluid 4 He remains elusive, due primarily to its very highdensities (ρ ∼ 2 × 1022 cm−3 ) and strong interactions, resulting in very small condensate fractions
ρc /ρ ∼ 0.1 and large depleted fractions. However, significant scientific advances in the more recent
decades have provided the scientific community with a clean, reproducible and controllable toolwith which to study superfluidity and other phenomena in ultracold matter. Specifically, the adventof laser and magnetic cooling and trapping, together with other cooling techniques (evaporative
cooling) allowed scientists at JILA at the University of Colorado and NIST [1] and at MIT [2] torealize Bose-Einstein condensation of dilute alkali atom vapors for the first time in 1995. Whileinteresting in and of themselves, as they were the coldest known samples of matter to exist in theuniverse, these dilute BECs have since proven to be useful tools from which much can be gained
regarding the knowledge of cold matter. In section 2.3, we discuss some of the basic physics behindsuch cooling and trapping techniques, and present some key results that are relevant for the workin this thesis. First, we discuss in more detail the phenomenon of Bose-Einstein condensation in an14ideal, non-interacting Bose gas, pointing out some finite-size effects that are associated with putting
the system with finite particle number into a trap. While the theory formulated in chapter 3 ismeant to describe a zero-temperature Bose gas, the following treatment of an ideal gas at finite
temperatures gives insight into the nature of the phenomenon of Bose-Einstein condensation andmotivates the pursuit of the ultracold regime by showing clear, analytic results for the temperaturedependence of this phenomenon.

2.2

The Ideal Bose Gas
An ideal gas of non-interacting bosons is just an ensemble of non-interacting one-body sys-tems. For the case at hand, we consider an ideal gas of N bosons with mass M in a harmonic
trapping, or external potential U (x),
1
U (x) = M ωx2 x2 + ωy2 y 2 + ωz2 z 2 .
2

(2.4)

We discuss how such a potential can be realized for a sample of atoms or molecules in section 2.3.1.
The energy spectrum of a single-particle in this harmonic potential is well-known to beǫnx ny nz = ~ (nx ωx + ny ωy + nz ωz ) + ǫ0 ,(2.5)

where ni are integers specifying the energy level in the ith coordinate and ǫ0 = 12 ~(ωx + ωy + ωz )is the ground state energy. For simplicity, we take ωx = ωy = ωz = ω, so the trap is spherical, anddefine and state vector ~l = (nx , ny , nz ), so ~l describes a direction and magnitude in the discrete
Hilbert space of a single particle in a three-dimensional (3D) harmonic oscillator. Now, the energy
eigenvalues for this system can be written as ǫ~l = ~ωTr[~l] + ǫ0 . With the degeneracy factor
gl = 12 (l + 1)(l + 2) of the spherical harmonic trap, meaning that there are gl ways that a single
particle can achieve the energy ǫl , the canonical partition function for this system can be writtenas
ZN (T, N ) =

X
~l

"

exp −βXl#gl ǫl nl ,

(2.6)

15
where β = 1/kB T , kB is the Boltzmann constant and the total particle number is given by N =Pl nl , where nl is the state occupation number, corresponding to nl bosons occupying a statewith energy ǫl . This restriction on the total particle number N makes calculating any physicalobservables or thermodynamic quantities with (2.6) very difficult, and motivates the introductionof the grand canonical ensemble, where the constraint of fixed particle number is replaced by theconstraint of fixed chemical potential, µ. This means that the system under consideration stays inthermal equilibrium with a surrounding environment at the cost of exchanging particles with theenvironment [53], that is, the system is in chemical equilibrium.
The grand canonical partition function is calculated by taking the Laplace transform of thecanonical partition function (2.6),
Ξ(T, µ) =

∞
X

βµN

e

N =0

∞ 
−gl
Y
,
1 − eβ(µ−ǫl )ZN (T, N ) =

(2.7)

l=0

from which the grand canonical potential can be calculated,
Π(T, µ) = −kB T ln Ξ(T, µ) = kB T

∞
X
l=0

i
h
gl ln 1 − eβ(µ−ǫl ) .

(2.8)

From this grand canonical potential, the average particle number in the non-interacting thermal
Bose gas can be calculated by taking the partial derivative of (2.8) with respect to the chemical
potential at fixed temperature,hN i =

∂Π(T, µ)
∂µ



=
T

∞
X
l=0

gl
.
β(ǫ
−µ)
l
e
−1

(2.9)

It is easy to identify the occupation number hnl i = gl (eβ(ǫl −µ) − 1)−1 from this result, which isjust the Bose-Einstein distribution [4] weighted by the degeneracy factor gl . For this result to bephysical, we restrict µ ≤ ǫ0 , so that the occupation numbers can not take on negative values. Thus,
the chemical potential must be less than or equal to the ground state energy in the non-interacting
Bose gas. Also, notice that as µ → ǫ0 , the ground state occupancy n0 becomes arbitrarily large,
implying that this is suitable criteria for the emergence of a condensate. So, we expect that µ → ǫ0corresponds to T → Tc , where Tc is the critical temperature for Bose-Einstein condensation.

16
We can calculate this critical temperature Tc by considering the number of excited, noncondensed bosons in the system, given by taking the sum over all l > 0 in Eq. (2.9). To simplify
this process, we go to the thermodynamic limit where the energy level spacing becomes very small
and the degeneracy becomes large and can thus be approximated by gl ≃ l2 /2. Additionally, we
rescale the ground state energy to be zero. Transforming the sum in Eq. (2.9) into an integral, we
see that the number of excited bosons is given by1
N − N0 = Nex =
2

Z

l2 dl
eβ(ǫl −µ) − 1.

(2.10)

From our criteria discussed above, the critical temperature is determined by setting µ = 0 and
Nex = N , giving
kB Tc
=
~ω



N
ζ(3)

1

3

,

(2.11)

where ζ(x) is the Riemann-Zeta function [54] and ζ(3) ≈ 1.2. This result (2.11) can be used in
Eq. (2.10) to show how the condensate fraction scales as a function of temperature [55]
 3
N0
T
.
=1−
N
Tc

(2.12)

Thus, we see that the condensate fraction grows with an inverse cubic behavior as a function oftemperature below the critical temperature Tc . The condensate fraction is plotted in figure 2.2
as a function of temperature, shown by the black dashed line. The thermodynamic limit result
tells us that for T > Tc , there is a negligible fraction of the bosons occupying the ground state,
and no condensate exists. However, for T < Tc there is a macroscopic, non-negligible ground state
occupation corresponding to the presence of a condensate. As forementioned, this was precisely
the prediction that Einstein made in 1925 [4].
The effect of indistinguishability in Bose-Einstein condensation can be seen in a clever way.
Recall that the thermal de Broglie wavelength of any massive body at temperature T is given by
s
2π~2
λdB =
.
(2.13)
M kB T
We can define the phase space density ν as the number of bosons occupying the volume element
λ3dB , ν = nλ3dB , where n is the real-space number density. For a thermal gas, this phase space

17
0.8
0.7

N = 100

0.6

N = 500
N = 1000N 0 /N0.5

N = 2000
Thermodynamic Limit

0.4
0.3
0.2
0.1
0

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

T /Tc

Figure 2.2: The condensate fraction N0 /N as a function of temperature for the ideal Bose gas ina spherically symmetric harmonic trap. The black dashed line is the thermodynamic limit result,and the dots are results from the Metropolis Monte Carlo algorithm for various particle numbers,
as indicated in the legend. Notice that larger particle numbers in the Monte Carlo simulations
exhibit better agreement with the thermodynamic limit result.

density is small as the characteristic de Broglie wavelengths of the bosons are much smaller thantheir average spacing. However, for an ultracold gas, we expect this phase space density to becomelarge and correspond to the BEC transition when ν ∼ 1. Indeed, for a homogeneous Bose gas in abox, the BEC transition occurs when ν = ζ(3/2) ≈ 2.612 [55]. Thus, below the critical temperaturefor BEC the wave functions of the bosons in the thermal system are sufficiently large that they
become comparable to the average interparticle spacing, corresponding to the formation of a BEC.
This perspective on the BEC phase transition clarifies why the critical temperature Tc is greater for
higher densities. It is also interesting to note that the phase transition to BEC is purely statisticaland not energetic, like the superfluid to Mott insulator transition of atoms on an optical lattice [56].
To obtain an estimate for the critical temperature of a dilute BEC, consider N = 50 × 103
bosons in a spherical trap with frequencies ω = 2π × 200 Hz. These are numbers that, as we will
see in section 2.3.1, are typical for modern BEC experiments. Such system parameters result in,from Eq. (2.11), a critical temperature of Tc ≃ 330 nK, which is about a factor of 6 × 10−8 smaller
than the critical temperature (lambda-point) of liquid helium.

18
Before proceeding to discuss the techniques that led to the realization of the ultracold regime
and the Bose-Einstein condensation of a dilute Bose gas, it is important to note the “finite size”
corrections that are present in the real physical system, which is not well-represented by the thermodynamic limit. To investigate the effects that finite size has on the condensate fraction and thecritical temperature of a Bose gas, we compute the harmonic oscillator state occupations exactly inthe canonical ensemble (Eq. (2.6)) using the Metropolis Monte Carlo method for particle numbersof N = 100, 500, 1000, 2000. For details on this Monte Carlo algorithm, we refer the reader to [57].
As one expects, the finite size effects are more pronounced for smaller particle numbers, for example,
N = 100, though the condensate fraction for N = 2000 is very close to the analytic thermodynamic
limit results. Finite size effects were also studied in [58], where a first order correction in finite
size predicts precisely what the Monte Carlo results show, that the condensate fraction and thecritical temperature are decreased in finite systems. The role of interactions in the Bose-Einsteincondensation of a trapped, finite sample was first considered in [59] where repulsive (attractive)
interactions were found to decrease (increase) the condensate fraction and the critical temperature
for condensation. Indeed, the presence of a trapping potential and repulsive interactions makes therealization of a BEC more difficult, as lower temperatures must be reached for these cases.

2.3

Bose-Einstein Condensation of Trapped Gases
To reach the ultracold nK regime that is necessary for the Bose-Einstein condensation ofdilute gases, a variety of experimental techniques were developed and employed that utilized thenature of the atom’s interactions with magnetic and optical fields. In this section, we discuss someof these techniques and the underlying physics that is involved. For a more detailed account, werefer the reader to [60] and [61]. Additionally, we discuss a few early experimental and theoreticalresults that are important for the work presented in this thesis, particularly, the realization of
Bose-Einstein condensation in a trap and the demonstration of quantized vortex states in BECs.

19
2.3.1

Experimental Techniques

The cooling of a gaseous atomic sample typically begins in a Zeeman slower, where coolingfrom hundreds of Kelvin down to the µ-Kelvin range is possible. The Zeeman slower relies on the
Doppler effect that is present in a sample of atoms moving relative to a beam of laser light. If thelaser frequency ω is tuned properly, the atoms moving opposite the direction of the laser light ina certain range of velocities will absorb more photons, as the light is more “blue” for these atoms.
By conservation of momentum, this slows the atoms down and, upon spontaneous emission, theatoms end up slower, and thus cooler, on average. A problem encountered with this technique isthat the cooling is limited as the atoms slow to a certain velocity, below which photon absorption is
critically suppressed. To counter this problem, a Zeeman slower uses an inhomogeneous magnetic
field along the direction of the laser propagation in order to shift the resonant frequency of theatom as a function of space via the linear Zeeman effect, making the laser cooling process more
efficient. This allows the atoms to slow to very small velocities and still “see” laser light that isresonant with the atomic transition. The laser cooling technique was developed and demonstrated
successfully by William D. Phillips and others, for which they shared the Nobel Prize in Physics in1997 [5, 62].
While laser cooling can produce a very cold, µK sample of atoms, sub-µK temperatures arenecessary to achieve BEC, as was discussed in the previous section. To achieve these temperatures,experimentalists developed evaporative cooling methods for trapped atoms. The basic idea of
evaporative cooling is to effectively lower the walls of the trap at higher energies that correspondto atoms at super-critical temperatures, so that these atoms can leave the trap and only the cooler
atoms remain. This can be achieved, for example, in magnetic traps by flipping the spin of highenergy atoms with an RF pulse. Indeed, evaporative cooling allowed experimentalists to lower
the temperature of their atomic samples sufficiently to achieve BEC. For a review of evaporative
cooling, see [63].
As suggested, the linear Zeeman effect describes the interaction of an atom with an applied20
magnetic field and the linear Hamiltonian of the interaction can be written as
Hi′ (x) = −µi · B(x)(2.14)

where µi is the magnetic moment of the atom in state i and B(x) is the applied magnetic field
that, in general, can vary throughout space. The magnetic moment can be nontrivial to calculate
and depends on both electronic and nuclear structure. We leave out a detailed discussion here,
and instead direct the reader to [64] or [60], but note that the result of such an interaction is the
splitting of the hyperfine levels of the atom (used in the Zeeman slower). Additionally, if B(x) is
not homogeneous in space, the linear Zeeman shift results in a spatially varying potential for theatoms. Such a potential can be used to trap atoms if the magnetic field B(x) possesses a potential
maximum or minimum in space. However, there are two important considerations related to suchtrapping, one being that the achievable magnetic fields in a laboratory are typically much less thana Tesla and the magnetic moment of an atom is typically on the order of 1-10 Bohr magnetons,µB = e~/2Me , resulting in maximum trap depths of less than a Kelvin. The other importantconsideration is that a magnetic field can not possess a local maximum in a current-free region [65],so only “low field seeking states” with µi > 0 can be magnetically trapped.
Magnetic trapping has other limitations as well. For example, magnetic traps, by theirnature, distinguish between magnetic substates and shift their energy levels, effectively trapping
the different states in different potentials. However, it is possible to trap atoms (and molecules) inpurely optical fields where all Zeeman substates “feel” the same trapping potential. Such opticalpotentials allow for the investigation of the spin or magnetic degrees of freedom in a quantum gas
because the potential is effectively the same for all corresponding substates. The optical trappingof atoms utilizes the AC Stark shift, which describes the energy shift of an atom in an oscillatingtime-dependent field. In the dipole approximation (valid when the wavelength of the laser is muchgreater than the size of the atom, which is automatic for optical transitions), the AC Stark shift of
the atom in its ground state is given in second-order perturbation theory by
1
Vg (x) = − Re [α(ω)] |hE(x, t)it |22(2.15)

21
where ω is the laser frequency, E(x, t) is the magnitude of the electric field as a function of space
and time and α(ω) is the electric polarizability [55],α(ω) = 2Xe

Ee − Eg|he|d · ǫ̂|gi|2 .
(Ee − Eg )2 − (~ω)2(2.16)

Here, the sum is over all excited states e with energies Ee , he|d · ǫ̂|gi are the dipole matrix elements
between the ground and excited states and ǫ̂ is a unit vector in the direction of the electric field E.
If the laser frequency ω is tuned near the resonance Ee − Eg , all other excited states in the sumin (2.16) can be neglected to a good approximation. In this case, we can define the dipole matrix
element d = he|d · ǫ̂|gi and the splitting ∆ = (Ee − Eg ) − ~ω, where the term 1/∆ dominates the
expansion of the energy-dependent coefficient in (2.16) when the laser is tuned near resonance, and
write
α≈

d2
.
∆

(2.17)For ∆ < 0, corresponding to a laser that is “blue” detuned from the dipole transition, the potential
energy shift (2.15) has a maximum where the optical field has a maximum intensity. For ∆ > 0,
corresponding to a laser that is “red” detuned from the dipole transition, the potential energy shift
has a minimum where the optical field has a maximum intensity. Thus, focusing laser light toachieve an intensity maximum can be used to attract (red-detuned) or repel (blue-detuned) atoms
from the high intensity region.
Regarding the time averaging of the field in Eq. (2.15), there are two important cases toconsider, one where there is a single propagating laser and one where there is a laser reflected backonto itself, or a retro-reflected laser. Without loss of generality, we consider a laser that propagatesin the z-direction. The electric field of the single propagating laser can be written as
E(x, t) = E0 (ρ)eikz e−iωt ,(2.18)

which has the time averaged intensity |hE(x, t)it |2 = |E0 (ρ)|2 . If we assume that the laser has aGaussian beam profile in the radial direction, then |E0 (ρ)|2 is Gaussian, as well. Thus, we see thatproper focusing of laser light can result in intensity maxima with Gaussian profiles that can trap

22
or repel atoms. We can write the field for the retro-reflected laser asE(x, t) = E0 (ρ)eikz e−iωt + E0⋆ (ρ)e−ikz eiωt ,(2.19)

This field has the time-averaged intensity |hE(x, t)it |2 = E02 (ρ) cos2 kx, taking the form of a standing
wave pattern, or a one-dimensional (1D) lattice with spacing 2π/k. This result is easily extended
to both two-dimensions (2D) and three-dimensions (3D), correspond to 2D and 3D optical latticepotentials. While the relevant optical lattice geometry for the work in this thesis is 1D (see chapter 8), there is much interest in the 2D and 3D geometries, as well, for studying quantum degenerate
systems. For a review of the physics of ultracold bosons in optical lattices, see [6] and for a review
of dipolar bosons in optical lattices, see [8].
The typical depths of optical traps that utilize the AC Stark shift are ∼ µK, and are thus
much more shallow than typical magnetic traps. However, this depth is still much greater than thecharacteristic critical temperatures for BEC, so such optical traps can still hold condensates andatoms and molecules that are pre-cooled to the sub-µK range. As a result, optical traps for dilute
BECs are well-approximated by harmonic traps of the form (2.4). Trapping by purely optical means
has proven particularly useful in recent experiments on “spinor” BECs, where homogeneous appliedmagnetic fields split the degeneracy of the magnetic sublevels and spin-exchange interactions leadto novel spin-density phases of these quantum gases. Such experiments have been performed on
F = 1 87 Rb and F = 1 23 Na (both with 3 spin components), and more recently on F = 3 52 Cr in
Paris [66] (with 7 spin components). Additionally, purely optical traps are important when using
Fano-Feshbach resonances to control the strength of the short-range interactions in BECs, as theseresonances can exhibit strong magnetic field dependence. We discuss such resonances further insection 6.1, as they are directly relevant to the results presented therein.

2.3.2

Key Results

As mentioned in section 2.1.3, the cooling and trapping techniques described above allowed
researchers at JILA [1] and MIT [2] to achieve nearly pure Bose-Einstein condensates for the first23

Figure 2.3: Images from the first BEC experiments of 87 Rb at JILA (left) and 23 Na at MIT (right).
From left to right in each image, surface plots are shown of velocity distributions and real-space
distrubutions, respectively, at temperatures just above, at, and below the critical temperature for
BEC. The sharpening of the distribution below the critical temperature provides evidence BEC
formation. Figures used with permission of E. A. Cornell from [67] (left) and W. Ketterle from [68](right).

time in 1995. Images of the emergence of the BECs at JILA and MIT are shown in figure 2.3. The
JILA image on the left shows the velocity distribution of the Bose gas as a function of temperature,and the MIT image on the right shows the real-space distribution of the Bose gas as a functionof temperature. One clearly sees a peak in the right-most image in both cases, signifying themacroscopically occupied Bose-Einstein condensate.
The realization of BEC, together with novel trapping techniques and methods to control
two-body interactions, which we discuss in detail later, gave both experimentalists and theorists a
tool with which to study a seemingly endless field of ultracold phenomena and quantum matter.
While this field is still growing today as researchers are working towards, for example, the creationof degenerate molecular gases and degenerate quantum gases with novel interactions, such as the
dipole-dipole interaction discussed for the remainder of this thesis, it is important to first note a
couple of very important results that demonstrated, for the first time, the novel superfluid natureof the dilute BEC.
Recall from the discussion in section 2.1.2 that quantized vortices are direct signatures ofsuperfluidity. Thus, the observation of quantized vortex states in a dilute BEC provides direct
evidence of superfluidity in this system. This is precisely what was done, for the first time, by the

24
JILA group in 1999 [69]. By clever spatial and temporal control of the optical pumping of 87 Rb
between two spin states, angular momentum was imparted to the BEC in order to nucleate a vortex
state. Images from this experiment are shown in the top part of figure 2.4, where the vortex ispresent in one spin component in (a) and has been transfered to the other spin component, and isthus no longer seen in (c). The presence of vorticity implies the presence of angular momentum, or
circulation, which is characterized by a phase wrapping of a quantum mechanical wave function.
For quantized vorticity, the phase wrapping ∆φ must occur in integer units of 2π, ∆φ = 2πn where
n is an integer. In the experiment [69], phase interference was used to confirm the presence of thesingly-quantized vortex
In 2001, the MIT group succeeded in realizing and imaging multiple vortex states in a dilute
BEC, where the vortex density was sufficiently high to create a vortex lattice, showing for the firsttime the presence of bulk vortex matter in a quantum degenerate system, with a lifetime of tensof seconds [70]. To impart angular momentum to their BEC, the group used two blue-detuned
lasers, which form repulsive Gaussian potentials via the AC Stark shift, and rotated them throughthe cloud. Images from this experiment are shown in the bottom part of figure 2.4, where theimages from left to right show an increase in the laser precession frequency and thus more vortices
in the BEC. Beyond the fact that the vortices (with the same circulation) form a lattice, it isinteresting that many singly-quantized vortices form instead of one or a few multiply-quantizedvortices. Indeed, multiply-quantized vortices can be, depending on the shape and interaction
strength in the BEC, dynamically unstable to the formation of multiple singly-quantized vortices.
This point was demonstrated by the JILA group in [71], where a blue-detuned laser was used
to create a density minimum in a BEC with vortex matter, wherein the vortices combined to a
multiply-quantized state, then decayed back into the lattice of singly-quantized vortices.
We have just pointed out a couple of the more interesting, relevant results from the early
BEC experiments here, though many more exist [72, 73, 67, 68]. Having seen that nearly pureBose-Einstein condensates of dilute atomic vapors are realizable in the laboratory setting and that
their superfluid properties have been demonstrated, we now turn our attention to the presence of25

Figure 2.4: Top image: Image of a singly-quantized vortex in a single spin component of a 87 Rb
BEC at JILA. In (a), the vortex is present, in (b) it is being transfered to the other spin component,
and in (c) it absent, having been transfered to the other spin component. Figure reprinted withpermission from [69]. Bottom image: Images of vortex lattice structures in the 23 Na BEC at MIT.
The leftmost (rightmost) images correspond to less (more) angular momentum transfered to the
BEC, resulting in fewer (more) vortices in the BEC. Figure reprinted with permission from [70].
Both images are from experiments that played a key role in demonstrating the superfluid natureof the dilute BEC.

interactions in these systems, in particular, the dipole-dipole interaction.

2.4

Dipolar Interactions in Ultracold Bose Gases
Since the first BEC experiments, many other atomic species have been Bose condensed,

including hydrogen [74], lithium [3], potassium [75], cesium [76], ytterbium [77], calcium [78],strontium [79], chromium [14] and most recently, dysprosium [17]. The latter two species, unlike
the others, have significant permanent magnetic dipole moments, as indicated in table 2.1. While
there is a plethora of rich physics that exists, and presumably has yet to be discovered, in quantum
degenerate gases of the other atomic species, the presence of large dipole moments in Cr and Dy
sparks particular interest, as interactions between such dipoles have been predicted to lead to new
physics related to, for example, novel states of quantum matter and quantum phase transitions [8].
Additionally, there has recently been a strong push towards the realization of a quantum
degenerate gas of heteronuclear molecules. A significant achievement in this line of research was26marked when the Jin/Ye collaboration at JILA managed to create a motionally ultracold gas offermionic KRb molecules in their ground rovibrational state via the novel STIRAP process [24],though quantum degeneracy (marked by the Fermi temperature) has remained elusive. For more
information on the STIRAP process, we refer the reader to [80]. Other groups have adopted thistechnology, and researchers now have a cold sample of bosonic KRb molecules [26] in their rovibrational ground state, and bosonic RbCs is promisingly close [81]. Heteronuclear polar molecules,
unlike dipolar atoms with permanent magnetic dipole moments, are electrically polarizable and candevelop very large electric dipole moments when polarized in an applied field. This polarizabilty,
actually, is given by the zero-frequency value of the AC polarizability α(0) from Equation (2.16).
The zero-frequency behavior of the AC Stark shift is referred to, appropriately, as the DC Stark
shift. These molecules, however, saturate at sufficiently large fields to a maximally attainable dipole
moment dmax . Maximum dipole moments for some heteronuclear molecules are given in table 2.1.
For a review of the physics related to ultracold polar molecules, see [7].
Besides having potentially large dipole moments, which can induce, for example, long-range
coupling in optical lattice models, polar molecules have other important characteristics that allow
them to be used for studying other rich, more complicated systems with a high level of control. For
example, recent work has proposed using polar molecules to simulate quantum magnetism, where
the rotational degrees of freedom map into spin degrees of freedom [82]. Additionally, it has beentheorized that, at large enough densities and in reduced dimensions, dipolar quantum gases canbe used to realize and study self-assembled crystals [83] (see section 4.2.5 for more details). Also,
because some heteronuclear polar molecules are chemically reactive, the high level of quantum state
control and ultracold temperatures allow for the study of ultracold, highly controlled atom-moleculeand molecule-molecule chemical reactions [7]. It has also been proposed that cold polar molecules
could be used as qubits in quantum computing algorithms [84].
In this work, we are most concerned with the effect of the dipole-dipole interaction in a BoseEinstein condensate of dipolar constituents. For two permanent dipoles with dipole moments d1

Species
87 Rb
52 Cr
164 Dy41 K87 Rb
87 Rb133 Cs
232 Th16 O

Dipole Moment
1 µB
6 µB
10 µB
0.57 Debye
1.25 Debye
3.89 Debye

add (a0 )
0.71
15.36
134.573.940 × 103
3.257 × 104
3.556 × 105

Reactive?
N
N
N
Y
N
N

Reference
[85]
[85]
[85]
[86]
[87, 88][89, 90]27

Table 2.1: Dipole moments, characteristic dipole lengths of candidate atomic and molecular species.
For the molecules, the dipole moment specifies the maximum achievable dipole moment. Because
41 K87 Rb is chemically reactive, it is labeled as such.

and d2 at positions x1 and x2 , respectively, the interaction potential is given by (in cgs units) [91]Vd (x1 , x2 ) = Vd (x1 − x2 ) =d1 · d2 − 3(n · d1 )(n · d2 ),|x1 − x2 |3(2.20)

where n is the unit vector in the direction of x1 − x2 . The dipole-dipole interaction (ddi) is
anisotropic and long-range, the explicit meaning of which is explained in section 3.4.2. Thus,
the interaction potential changes magnitude and sign depending on the relative orientation of thedipoles. As we will see, this feature plays a very important role in the physics of BECs with
dipolar interactions, or dipolar BECs, leading to new, novel physics even when the BEC is dilute.
Though the interaction potential (2.20) is proportional to the product of the dipole moments, thetheory that we formulate in the next two chapters shows that the strength of the ddi is actuallycharacterized by the dipole “coupling,” gd = 4π~2 add /M , where M is the mass of a dipolar atom
or molecule and add is the characteristic dipole length of a species with dipole moment d,
add =

M d2
.
3~2

(2.21)

Dipole lengths for various atomic and molecular species are given in table 2.1. For highly dipolar
molecules, such as RbCs, the maximum achievable dipole lengths are on the order of tens of
thousands of Bohr radii. We motivate the definition of this dipole length (with the factor of 3 inthe denominator) in chapter 4.
As forementioned, experimental groups have succeeded in creating Bose-Einstein condensatesof atomic Cr (in Tilman Pfau’s group in Stuttgart and in Oliver Gorceix’s group in Paris) and atomic

28
Dy (in Benjamin Lev’s group at the University of Illinois and Stanford), and strong dipolar effects
have been demonstrated in all experiments. In the following two chapters, we develop a working
theory for dipolar BECs, and motivate the introduction of relevant physical parameters therein,such as the condensate wave function, or order parameter of the ultracold Bose gas. Once thisobject, and others, are better motivated and understood, we return to discuss recent experimental
results on dipolar BECs in detail. Now, having gained an appreciation for the physics of ultracold
Bose gases and BECs, as well and the motivation for understanding the role that the ddi plays in
the Bose condensed state, we turn our attention to the characterization of an ultracold Bose gas in
the presence of interparticle interactions, in particular, dipole-dipole interactions.

Chapter 3
Zero-Temperature Field Theory for Bosons

3.1

Second-Quantized Field Theory
The goal of this chapter is to derive a working theory for a gas of interacting dipolar bosons atzero temperature (T = 0). A common and fruitful approach to such an end (for finite temperature,
as well), involves the language of second-quantization. In this framework, one describes a manybody system by keeping track of single-particle state occupations instead of a full many-body wave
function ΦN ({xi }). Thus, the theory is quantized “twice” if the full many-body wave function,
which obeys the many-body Schrödinger equation ĤN ΦN ({xi }) = EΦN ({xi }), is considered toP
be quantized “once”. Here, ĤN is the Hamiltonian describing N = i Ni interacting particles atcoordinates {xi } = {x0 , x1 , . . .}.

To motivate the definition of a quantized field, consider the separable coordinate representation of the many-body wave function Φ for a system of bosons,ΦN ({xi }) =r

N0 !N1 ! . . . X
φ0 (x0 )φ1 (x1 ) . . .
N!

(3.1)

P

where there are Ni bosons occupying the state φi (x) andP

P indicates that the sum should betaken over all possible permutations of the product of single-particle wave functions φi (x) [92].
The set of single-particle wave functions {φi } is assumed to be complete and orthonormal so thatRP
dxφ⋆i (x)φj (x) = δij and i φi (x)φ⋆i (x′ ) = δ(x − x′ ). Because we are dealing with bosons, thewave function (3.1) is symmetric with respect to all permutations P. Now, consider the matrixPelements of the operator Ô = i ôi where ôi operates only on functions of the coordinate xi . The

30
only non-zero matrix elements of this operator are those corresponding to all Ni → Ni (diagonal) or
those corresponding to the occupation of some state i being decreased by unity while the occupationof some state j is increased by unity, or Ni → Ni − 1 while simultaneously Nj → Nj + 1 so thePtotal particle number N = i Ni is conserved [93].

Such behavior can be captured by introducing the Bose creation and annihilation operators,â†i and âi , respectively. These operators are defined by their their action on a many-body statein the occupation number, or Fock representation, |ΦN i = |N0 , N1 , . . . , Ni , . . .i, corresponding
to N0 particles occupying state φ0 (x), N1 particles in the state φ1 (x), and so on, and by theircommutation algebra. These many-body occupation number states are defined to be orthonormalso that hΦN ′ |ΦN i = δN ′ N . The action of the annihilation operator âi on |ΦN i reduces the occupationof the state φi (x) by unity,âi |N0 , N1 , . . . , Ni , . . .i =p

Ni |N0 , N1 , . . . , Ni − 1, . . .i,(3.2)

and the action of the creation operator â†i on |ΦN i increases the occupation of the state φi (x) byunity,â†i |N0 , N1 , . . . , Ni , . . .i =The factors ofp

Ni + 1|N0 , N1 , . . . , Ni + 1, . . .i.

(3.3)

√
√
Ni and Ni + 1 are defined so that the annihilation of a particle from state φi (x)with occupation number Ni = 0 gives the result âi | . . . , Ni = 0, . . .i = 0 and does not allow for
unphysical negative occupation numbers. Additionally, it is straightforward to see that the number
operator N̂i is diagonal in the occupation number basis and has eigenvalue Ni . The number operator
is expressed in terms of the creation and annihilation operators as N̂i = â†i âi ,â†i âi |N0 , N1 , . . . , Ni , . . .i = â†i=p

Ni |N0 , N1 , . . . , Ni − 1, . . .ipp
(Ni − 1) + 1 Ni |N0 , N1 , . . . , (Ni − 1) + 1, . . .i

= Ni |N0 , N1 , . . . , Ni , . . .i,so the occupation number of the state φi (x) is just given by hΦN |N̂i |ΦN i = Ni .

(3.4)

31
The Bose creation and annihilation operators satisfy the commutation relationshiâi , â†j = δijih[âi , âj ] = â†i , â†j = 0,

(3.5)
(3.6)

so the many-body states are properly symmetrized. At this point, we introduce the vacuum fieldof our occupation number basis, |0i, which is defined so that âi |0i = 0 for all i. Starting withthis vacuum state, we can create a state with a given configuration by successive application of
appropriate creation operators,|ΦN i = √ N0  N1
 Ni
1
a†0
a†1
. . . |0i.
. . . a†i
N0 !N1 ! . . . Ni ! . . .

(3.7)

While the use of creation and annihilation operators in the occupation number representation
provides a clean and natural toolset to work with many-body systems, it is often advantageous towork in the coordinate representation when the objects of interest are related to, for example, thespatial density of the system. A direct connection between the occupation number representationand the coordinate representation can be made by defining the Bose field operators,ψ̂(x) =Xφi (x)âi(3.8)

φ⋆i (x)â†i ,

(3.9)

i

ψ̂ † (x) =X
iwhere, again, {φi (x)} is a complete, orthonormal set of single particle wave functions. By multiplying Eq. (3.8) by φ⋆j (x) and Eq. (3.9) by φj (x) and integrating over all x, we arrive at the inverse
relationsâi =â†i =ZZ

dx ψ̂(x)φ⋆i (x)(3.10)

dx ψ̂ † (x)φi (x).

(3.11)

Using these inverse relations, the commutators of the field operators are found to behiψ̂(x), ψ̂ (x ) = δ(x − x′ )†′(3.12)

32
h

i h
i
ψ̂(x), ψ̂(x′ ) = ψ̂ † (x), ψ̂ † (x′ ) = 0.

The number operator for the total number of particles in the system, N̂ =terms of the field operators,N̂ =Z(3.13)Pi N̂i can be written indxψ̂ † (x)ψ̂(x),(3.14)

where we have used the completeness of the set {φi (x)} to arrive at this result. Unlike the annihilation and creation operators which annihilate and create a particle in a well defined single-particle
state, the field operators (3.8) and (3.9) annihilate and create, respectively, a particle at position
x.

3.1.1

Many-Body Hamiltonian in Second-Quantization

Recall that the system under consideration is an ultracold, dilute gas of interacting bosons.
The dilute character of the gas allows us to truncate the interactions at the two-body, or binarylevel due to the fact that the probability of two particles interacting in a dilute gas overwhelms theprobably of three particles doing so. Thus, in terms of the Bose field operators, the many-body
Hamiltonian can be expressed asĤ =Zdxψ̂ † (x)Ĥ (1) (x)ψ̂(x) +

1
2

Z

dx

Z

dx′ ψ̂ † (x)ψ̂ † (x′ )V (x − x′ )ψ̂(x′ )ψ̂(x),(3.15)

where V (x−x′ ) is the two-body interaction potential and Ĥ (1) (x) is the single particle HamiltonianĤ (1) (x) = −~2 2∇ + U (x),2M

(3.16)

where M is the mass of a single boson and U (x) is the external, or trapping potential. The factor of
1/2 in the interaction term corrects for a double counting that is inherent in the integration. This
Hamiltonian operator (3.15) can be expressed in terms of the creation and annihilation operators
by substituting Eqs. (3.8) and (3.9) in for the Bose field operators, givingĤ =X † (1)
1 X † †âi âj Vijkl âk âl ,âi Hij âj +
2i,ji,j,k,l(3.17)

33(1)

where Hij are the matrix elements of the single particle HamiltonianZ(1)
Hij = dxφ⋆i (x)Ĥ (1) (x)φj (x)and Vijkl are the interaction matrix elements
Z
Z
Vijkl = dx dx′ φ⋆i (x)φ⋆j (x′ )V (x − x′ )φk (x′ )φl (x).

3.2

(3.18)(3.19)The Bogoliubov Approximation
The diagonalization of the Hamiltonians (3.15) and (3.17) is quite difficult as they are written

in full generality. However, a significant simplification can be made by restricting the systemof bosons that these Hamiltonians describe to the ultracold regime. Here, ultracold refers totemperatures T ≪ Tc , where Tc is the critical temperature for Bose-Einstein condensation. In thisregime, the number of bosons occupying the condensed state φ0 (x) is macroscopic and overwhelms
the occupation of any excited states corresponding to φi (x) with i 6= 0, that is, the condensatePfraction n0 ≡ N0 /N ∼ 1 and the excited fraction nex ≡ i6=0 Ni /N ≪ 1. To a good approximation,

we can treat the condensate part of the field operator as a c-number and write the field operator asΨ̂(x) ≃ hΨ̂(x)i + ϕ̂(x) =pN0 φ0 (x) + ϕ̂(x)(3.20)

where hΨ̂(x)i is a low-temperature ensemble average of the field operator and ϕ̂(x) corresponds tothe excited, non-condensed states, or the so-called quantum fluctuations
ϕ̂(x) =Xφi (x)âi .

(3.21)

i6=0This decomposition of the field operator was first proposed by Bogoliubov [94], and amounts to replacing the creation and annihilation operators of the condensate field by the root of the condensate√
occupation number â0 = â†0 = N0 . An important consequence of this Bogoliubov approximationis that the number operator N̂ no longer commutes with the Hamiltonian (3.15) and particle number is no longer conserved. This motivates the introduction of the grand-canonical HamiltonianK̂ = Ĥ − µN̂ where µ is the chemical potential of the system and acts as a Lagrange multiplier toconserve particle number on average [53].

34
Using the Bogoliubov decomposition (3.20), we express the grand-canonical Hamiltonian K̂
perturbatively in orders of the condensate occupation N0 as K̂ = K0 + K̂1 + K̂2 + . . . where K̂icontains the terms that are ith order in the quantum fluctuations ϕ̂(x). The zeroth order term K0
is given by
K0
=
N0

Z

dxφ⋆0 (x)

n

Ĥ

(1)

o

N0
(x) − µ φ0 (x) +
2

Z

dx

Z

dx′ φ⋆0 (x)φ⋆0 (x′ )V (x − x′ )φ0 (x′ )φ0 (x). (3.22)

An equation governing the chemical potential and the condensate field φ0 (x) is derived by enforcingthat K0 be a minimized with respect to small variations in the condensate field φ⋆0 (x). This ensuresthat K̂1 vanishes and results in the equation


Z
′ ⋆ ′
′
′
(1)
µφ0 (x) = Ĥ (x) + N0 dx φ0 (x )V (x − x )φ0 (x ) φ0 (x).

(3.23)

This Eq. (3.23) is the non-local Gross-Pitaevskii equation (GPE) and is the governing equation forthe condensate field φ0 (x), commonly referred to as the condensate wave function. The second termon the RHS of the GPE is the mean-field potential, Umf (x). It emerges by taking the mean-fieldhΨ̂(x)i in the Bogoliubov approximation (3.20), and has the form of a classical potential that is felt
by all particles in the system, both condensed and excited, due to the presence of the condensate.
The GPE (3.23) was derived independently by Gross [95] and Pitaevskii [96] in 1961. It
provides a self-contained and fruitful description for the fully condensed, zero temperature stateof a system of dilute bosons. While the chemical potential µ is given by the condensate wave
function φ0 (x) through the GPE, the stationary condensate wave function itself must be calculated
by minimizing the zeroth order part of the Hamiltonian (3.15),
E0
=
N

Z

dxφ⋆0 (x)Ĥ (1) (x)φ0 (x) +

N0
2

Z

dx

Z

dx′ φ⋆0 (x)φ⋆0 (x′ )V (x − x′ )φ0 (x′ )φ0 (x).

(3.24)

This energy functional (3.24) is therefore known as the Gross-Pitaevskii energy functional. Minimization of (3.24) corresponds to the minimization of (3.22), the only difference between the twobeing a global shift by the chemical potential µ. Additionally, it is important to note that the GPE
and the energy functional (3.24) remain unchanged by any global phase shift of the condensate wave
function φ0 (x), so we see that this system possesses a U (1) symmetry that is spontaneously broken

35
in the BEC phase. It was pointed out by Goldstone [97, 98] that systems with such symmetry,here corresponding to a rotation of the condensate phase, necessarily possess some long-wavelength,
arbitrarily low-energy mode known as the Goldstone boson (i.e., the theory must have a gapless excitation spectrum). We will see that the theoretical formulation of quantum fluctuations presented
here is consistent with Goldstone’s theorem and results in a gapless excitation spectrum.
The next non-zero term in the expansion of K̂ is the second order term K̂2 , given by
K̂2 =

Z

†

n

dxϕ̂ (x) Ĥ

(1)

o

N0
(x) − µ ϕ̂(x) +
2

Z

dx

Z

dx′ V (x − x′ )n× ϕ̂† (x)ϕ̂† (x′ )φ0 (x′ )φ0 (x) + ϕ̂† (x)φ⋆0 (x′ )ϕ̂(x′ )φ0 (x) + ϕ̂† (x)φ⋆0 (x′ )φ0 (x′ )ϕ̂(x)o+ φ⋆0 (x)ϕ̂† (x′ )φ0 (x′ )ϕ̂(x) + φ⋆0 (x)φ⋆ (x′ )ϕ̂(x′ )ϕ̂(x) + φ⋆0 (x)ϕ̂† (x′ )ϕ̂(x′ )φ0 (x)
Z
Z
Z
N0
†
dx dx′ V (x − x′ )= dxϕ̂ (x)ĤGP (x)ϕ̂(x) +2no× ϕ̂† (x)ϕ̂† (x′ )φ0 (x′ )φ0 (x) + 2ϕ̂† (x)ϕ̂(x′ )φ⋆0 (x′ )φ0 (x) + ϕ̂(x)ϕ̂(x′ )φ⋆0 (x′ )φ⋆0 (x) ,(3.25)

where ĤGP (x) is defined so that ĤGP (x)φ0 (x) = 0 (see Eq. (3.23)) and we have assumed thatthat the interaction potential V (x − x′ ) has the even symmetry V (x − x′ ) = V (x′ − x). Note thatK̂2 is not diagonal in the quantum fluctuation operators ϕ̂(x). However, this expression can be
diagonalized by transforming these operators [99],ϕ̂(x) =

Xhi6=0

i
ui (x)b̂i − vi⋆ (x)b̂†i .

(3.26)

To make the transformation canonical, the creation and annihilation operators b̂†i and b̂i must obey
the Bose commutation relations (3.5) and (3.6), from which the normalization condition for theui (x) and vi (x) wave functions are derived to beZdx [u⋆i (x)uj (x) − vi⋆ (x)vj (x)] = δij .

(3.27)

Interestingly, the only constraint on these functions is that they are normalized relative to eachother while their magnitudes are otherwise unconstrained. It is precisely this fact that allows the
canonical Bogoliubov transformation (3.26) to diagonalize the quantum fluctuation Hamiltonian
K̂2 .

36
Indeed, inserting the transformation (3.26) and its conjugate into (3.25) allows us to writeK̂2 asK̂2 =X~ωi b̂†i b̂i ,(3.28)

i6=0

where the energy eigenvalue ωi is given by
~ωi =

Z



dx u⋆i (x)ĤGP (x)ui (x) + vi⋆ (x)ĤGP (x)vi (x) + N0Zdx′ V (x − x′ ) φ0 (x)u⋆i (x)φ⋆0 (x′ )ui (x′ )+ φ⋆0 (x)vi⋆ (x)φ0 (x′ )vi (x′ ) + φ0 (x)u⋆i (x)φ0 (x′ )vi (x′ ) + φ⋆0 (x)vi⋆ (x)φ⋆0 (x′ )ui (x′ )

.

(3.29)

Enforcing that the off-diagonal elements of K̂2 vanish is achieved by enforcing that the creation
and annihilation operators obey the commutation relations [100]hiK̂2 , b̂i = −ωi b̂ihiK̂2 , b̂†i = ωi b̂†i .

(3.30)(3.31)

From these relations, the functions ui (x) and vi (x) are found to obey the Bogoliubov de Gennes(BdG) equations,~ωi ui (x) = ĤGP (x)ui (x) + N0−~ωi vi (x) = ĤGP (x)vi (x) + N0ZZdx′ V (x − x′ ) φ⋆0 (x′ )ui (x′ ) + φ0 (x′ )vi (x′ ) φ0 (x)dx′ V (x − x′ ) φ⋆0 (x′ )vi (x′ ) + φ0 (x′ )ui (x′ ) φ0 (x).

(3.32)(3.33)For a non-interacting system, solutions to Eqs. (3.32) and (3.33) correspond to the single-particle
system where the ωi are the single-particle energies and the ui (x) are the single-particle excitedR
states, while dxvi⋆ (x)vi (x) = 0. When interactions are present, however, the vi (x) play animportant role in this theory and the the solutions no longer correspond to single-particles, butinstead correspond to quasiparticles. The quasiparticle picture, while perhaps unintuitive, is thenatural language with which to describe dilute Bose gases at low temperatures T ∼ 0 as the
Hamiltonian for the system is diagonal in this representation. The functions ui (x) and vi (x) are
therefore referred to as quasiparticle wave functions.
Eqs. (3.32) and (3.33) are linear in the quasiparticle wave functions ui (x) and vi (x), andthus account for quasiparticle-condensate interactions while neglecting quasiparticle-quasiparticle

37

Figure 3.1: The diagrammatic representation of the interaction part of grand canonical HamiltonianK̂int . The dashed lines represents the condensate, the solid lines represent excited quasiparticles
and the squiggly lines represent interactions. The first diagram shows the 0th order contribution
to the interaction energy given by condensate-condensate interactions. The remaining diagrams(a)-(f) represent the next to leading order contribution to the interaction energy from quasiparticlecondensate interactions. (a) and (b) show quasiparticles scattering off of the condensate (directterms) and (c)-(f) show interactions that absorb and expel quasiparticles from the condensate(exchange terms).

interactions. However, in a dilute gas at approximately zero temperature, the latter interactions
may be neglected as the total quantum depletion (see section 3.2.3) is very small. The quasiparticlecondensate interactions, on the other hand, are non-negligible and are characterized by two typesof terms. One is a “direct” term that describes a quasiparticle scattering off of the condensate,
and the other term is an “exchange” term that describes a quasiparticle scattering into or outof the condensate. The interaction parts of the grand canonical Hamiltonian K̂ are represented
diagrammatically in figure 3.1 and such direct and exchange terms are identified therein.
It is convenient to write the BdG equations in the matrix form,

 
 
⋆
X̂
 ĤGP + Ĉ + X̂ u  u    = ~ω  −X̂ ⋆−ĤGP − Ĉ − X̂vv

(3.34)

where the Ĉ and X̂ operators represent the direct and exchange terms, respectively, and we haveneglected the spatial dependence of the operators and quasiparticle wave functions. The direct
operation of Ĉ on a quasiparticle wave function ui (x) is given byhiĈui (x) = N0Z

dxφ⋆0 (x′ )V (x − x′ )φ0 (x′ )ui (x),(3.35)and the exchange operation of X̂ on a quasiparticle wave function ui (x) is given byhiX̂ui (x) = N0Z

dxφ⋆0 (x′ )V (x − x′ )ui (x′ )φ0 (x).

(3.36)

38
Then, in practice the quasiparticle wave functions ui (x) and vi (x) can be obtained simultaneouslywith the energy eigenvalues ωi via diagonalization of the 2 × 2 BdG Hamiltonian in Eq. (3.34).
3.2.1

Long-Range Order

Recall that in section 2.1.3 we discussed the criterion laid out by Penrose and Onsager thatsuperfluids, and BECs, must possess long-range order, given in terms of the single-particle density
matrix in Eq. (2.3). Here, we show that long-range order is indeed present in the Bogoliubov
decomposition given by Eqs (3.20) and (3.26) by calculating the single-particle density matrix
explicitly,
ρ(1) (x, x′ ) = hΨ̂† (x)Ψ̂(x′ )i
 p
E
Dp
N0 φ⋆0 (x) + ϕ̂† (x)N0 φ(x′ ) + ϕ̂(x′ )== N0 φ⋆0 (x)φ0 (x′ ) + hϕ̂† (x)ϕ̂(x′ )i,(3.37)

where the terms that are linear in the quantum fluctuation operators ϕ̂(x) vanish. If we consider
a homogeneous system in a cubic box of volume V , we can use a plane wave basis (which we√return to in the next chapter), φk (x) = 1/ V eik·x . In this case, lim|x−x′ |→∞ hϕ̂† (x)ϕ̂(x′ )i = 0 and
lim|x−x′ |→∞ φ⋆0 (x)φ0 (x′ ) = 1/V , so we see thatlim

|x−x′ |→∞

ρ(1) (x, x′ ) = N0 /V,(3.38)

so the single-particle density matrix asymptotes to the condensate density in the Bogoliubov decomposition, thus satisfying the criterion of long-range order.

3.2.2

Symmetry in the Bogoliubov de Gennes Equations

Note that if we can write φ⋆0 (x) = φ0 (x), the BdG equations (3.34) are unchanged under theexchange of u ↔ v ⋆ and ω ↔ −ω. Because of this symmetry, the solutions come in pairs of (u, v ⋆ )Twith eigenvalue ω, the positive-norm solution, and (v ⋆ , u)T with eigenvalue −ω, the negative-norm
solution. Thus, we can transform the u and v quasiparticle wave functions u = 12 (f − g) and39
v = 12 (f +g) and end up with an equivalent form of the BdG equations. Squaring these transformed
BdG equations results in a diagonal form [101, 102],
ĤGP + Ĉ ĤGP + Ĉ + 2X̂ f = ω 2 f(3.39)


ĤGP + Ĉ + 2X̂ ĤGP + Ĉ g = ω 2 g.

(3.40)

To obtain just the energy eigenvalues ω, it is sufficient to solve just one of these equations whileboth (3.39) and (3.40) must be solved to gain information about the quasiparticle wave functions.
For example, diagonalization of (3.39) gives the eigenvalues ω and the eigenvectors f while a matrix
inversion of (3.40) gives the eigenvectors g. With knowledge of both f and g, the quasiparticle wave
functions u and v can be reconstructed.

3.2.3

Quantum Depletion

The Bogoliubov transformation (3.26) mixes the single-particle creation and annihilation operators, and it is therefore useful to define a new ground state (vacuum) in terms of the quasiparticle
annihilation operator instead of the single-particle annihilation operator,b̂i |0i = 0,(3.41)

so the state |0i is defined as a quasiparticle vacuum, or the state that is devoid of quasiparticles,
corresponding to a pure condensate. The number of particles in this ground state can be calculated by taking the expectation value of the number operator in this state. In the Bogoliubov
approximation, the number operator takes the form,N̂ = N0 φ⋆0 (x)φ0 (x) +p

N0 φ⋆0 (x)ϕ̂(x) + φ0 (x)ϕ̂† (x) + ϕ̂† (x)ϕ̂(x),(3.42)

and applying the canonical transformation (3.26) to (3.42) givesN̂ = N0 φ⋆0 (x)φ0 (x) ++Xhi,j6=0

p

N0

i
i
h
h
X
φ⋆0 (x) ui (x)b̂i − vi⋆ (x)b̂†i + φ0 (x) u⋆i (x)b̂†i − vi (x)b̂I
i6=0

u⋆i (x)uj (x)b̂†i b̂j − u⋆i (x)vj⋆ (x)b̂†i b̂†j − ui (x)vj (x)b̂i b̂j + vi (x)vj⋆ (x)b̂i b̂†ji.

(3.43)

40
When taking the vacuum expectation value, we use the fact that h0|0i = 1 and h0|b̂†i |0i = h0|b̂i |0i = 0
to arrive at the result
XZ

h0|N̂ |0i = N0 +dxvi⋆ (x)vi (x).

(3.44)

i6=0

The second term in this expression is known as the quantum depletion. This term gives the numberof particles that are forced out of the condensate at zero temperature into excited quasiparticle
states, and was first calculated by Lee, Huang and Yang for a gas of bosons at T = 0 with hardsphere interactions [103]. It is worth emphasizing that this is a zero temperature result. At finite
temperature, thermal energy manifests in the occupation of excited quasiparticle states so theannihilation of a quasiparticle from a thermal many-body state does not result in zero as is the
6 0 in a Bose gas at T > 0.
case for the ground state, defined in Eq. (3.41). Thus, terms like hb̂†i b̂i i =3.3

Time-dependent formulation
So far, we have neglected any time-dependence in the formulation of our low-temperature field

theory. In general, the field-operators can be time-dependent Ψ̂(x) → Ψ̂(x, t), and their equationof motion is given in the Heisenberg picture byi~∂t Ψ̂(x, t) = [Ψ̂(x, t), K̂(t)]


Z
(1)
′ † ′
′
′
= Ĥ − µ + dx Ψ̂ (x , t)V (x − x )Ψ̂(x , t) Ψ̂(x, t).

(3.45)The decomposition of the field operator from Eq. (3.20) can now be inserted into Eq. (3.45) to derive
an equation for the condensate field φ0 (x, t). At ultracold temperatures, however, we employ the
perturbative approach and keep only the leading order terms in N0 , which amounts to neglecting
terms that are of cubic order or greater in the quantum fluctuations. In practice, we take a T = 0
ensemble average of Eq. (3.45) and note that hϕ̂(x, t)i = hϕ̂† (x, t)i = 0 to arrive at an equation for
φ0 (x, t),i~∂t φ0 (x, t) =



~2 2
−
∇ + U (x) − µ + N0
2M

Z



dx′ φ⋆0 (x′ , t)V (x − x′ )φ0 (x′ , t)

φ0 (x, t).

(3.46)

This equation is known as the time-dependent Gross-Pitaevskii equation, and it governs the realtime evolution of the condensate field φ0 (x, t). The stationary solution to this equation corresponds

41
to the time-independent condensate field with time-dependent phase φ0 (x)e−iµt/~, where φ0 (x)obeys the GPE (3.23).
Additionally, in section 3.2 it was shown that the Hamiltonian K̂2 is diagonalized by the
introduction of quasiparticle operators. Thus, the time evolution of the quasiparticles is given
simply by their energy eigenvalues ωi ,ϕ̂(x, t) =

Xh
i6=0

i
ui (x)e−iωi t/~b̂i − vi⋆ (x)eiωi t/~b̂†i .

(3.47)

It is easy to show that the insertion of (3.47) into the equation of motion for the field operator (3.45)
results in the BdG equations (3.34). Additionally, formulating the quantum fluctuations in thistime-dependent form reveals important information regarding the stability of the dilute Bose gas.
Manipulation of the BdG equations (3.32) and (3.33) results in the condition(ωi − ωi⋆ )Zdx [u⋆i (x)ui (x) − vi⋆ (x)vi (x)] = 0.

(3.48)

However, the normalization condition (3.27) says that the integral part of this expression must
be equal to unity, so (ωi − ωi⋆ ) must be zero. This is only guaranteed when ωi is purely real. If
ωi has a non-zero imaginary part, Eq. (3.48) can not hold unless the ui (x) and vi (x) functions
are not normalizable, i.e., they diverge relative to each other. Indeed, one sees directly from thetime-dependent form of the quantum fluctuations in Eq. (3.47) that any Im[ωi ] 6= 0 results inquasiparticle wave functions whose norms diverge exponentially, signifying a dynamical instability.
Thus, solutions to the BdG equations give us a criterion for dynamic stability [104],Im[ωi ] = 0.

(3.49)

It was shown rigorously in reference [104] that the emergence of complex eigenvalues in the BdG
equations coincides with the degeneracy between a positive-norm solution and a negative-norm
solution. Ref. [105] also confirmed this claim using a two-mode approximation. At the point ofdegeneracy, the real parts of the positive-norm and negative-norm eigenvalues become identical as
their imaginary parts emerge that are identical in magnitude and opposite in sign. This is seen
explicitly in the solutions to the BdG equations that are presented in chapter 5.

42
3.3.1

First-Quantized Theory - Alternative Derivation

In this section we present an alternative derivation of the Gross-Pitaevskii equations, bothtime-independent (3.23) and time-dependent (3.46), the Gross-Pitaevskii energy functional (3.24)
and the BdG equations (3.34). First, we notice that the energy functional (3.24) describes theenergy of a pure condensate field φ0 (x) in the presence of two-body interactions. We can write thefirst-quantized Hamiltonian for a system of N such bosons as
 X
N 
X
~2 2
Ĥ ({xi }) =−∇i + U (xi ) +V (xi − xj ),2Mi=1(3.50)

i<jand we can write the symmetrized wave function of the pure condensate, where every particle
occupies the exact same wave function φ0 (x), asΨ ({xi }) =NY

φ0 (xi ).

(3.51)

i=1

Such an approximation to the zero-temperature Bose system is known as the Hartree approximation. The energy per particle of such a system is then given by the expectation value of the
Hamiltonian (3.50) in the state (3.51),
N Z
YEdxi Ψ⋆ ({xi }) Ĥ ({xi }) Ψ ({xi }) .
=N

(3.52)

i=1

Subsequent integration over the coordinates xi results in an energy functional that is identicalto (3.24), but with the N0 that multiplies the mean-field interaction term replaced by N −1 [106, 55].
Here, N0 and N refer to the same number, being the number of particles in the condensate. Sucha result is intuitive, actually, because although such a mean-field theory is meant to describe alarge number of bosons in the condensate, the limit that N → 1 should produce no interactions,as a particle can not interact with itself in the way we consider here. Thus, we adopt the factorof N − 1 for the parts of this thesis where trapped, finite Bose gases are considered. At any rate,
the difference between N and N − 1 is typically negligible for current BEC experiments, wherecondensate particle numbers are on the order of tens of thousands.
The time-independent Gross-Pitaevskii equation is derived in this first-quantized theory byintroducing the chemical potential as a Lagrange multiplier to conserve particle number on average,

43
in the grand canonical theory, and enforcing that δE − µδN = 0. The result is, again, identicalto Eq. (3.23), but with N0 → N − 1. Similarly, the time-independent Gross-Pitaevskii equation isderived by writing out the time-dependent Schrödinger for the many-body wave function Ψ, which
we now generalize to the time-dependent form Ψ({xi }, t). Projecting out all coordinates except for
one gives the time-dependent GPE, Eq. (3.46), but, again, with N0 → N − 1.
In the first-quantized approach, the elementary excitations are treated as small perturbations
to the ground stationary state, in this case, the condensate wave function φ0 (x)e−iµt . Using thetime-dependent GPE, we write
φ0 (x, t) = [φ0 (x) + δϑ(x, t)] e−iµt ,where δ ≪ 1 andϑ(x, t) =Xui (x)e−iωi t + vi⋆ (x)eiωt .

(3.53)(3.54)i

Plugging the form (3.53) into the time-dependent GPE and linearizing about the small parameterδ results in the BdG equations (3.34), but, of course, with N0 replaced with N − 1. While thisfirst-quantized method is seemingly identical to that of the second-quantized method introduced
earlier in this section, the first-quantization does not allow one to easily go beyond this perturbative
approach, whereas the second quantization allows for such treatment, for example, of self-consistenttreatments of thermal and quantum fluctuations. An example of such methods are the HartreeFock Bogoliubov methods, developed by Allan Griffin [107], which have been proven fruitful indescribing thermal Bose gases.

3.4

Two-body Interactions
To complete our theoretical description of an ultracold, dilute gas of interacting bosons, we

now treat the two-body interactions. In the theory presented here, the effect of these interactions
emerge in the interaction matrix elements Vijkl given in Eq. (3.19). While we are primarily interested
in the dipole-dipole interaction (ddi), which is discussed in section 3.4.2, the treatment of shortrange interactions in dipolar Bose gases is necessary for a complete description of this system.

44
3.4.1

Pseudopotential for Short-Range Interactions

As discussed in section 2.2, the transition to BEC occurs when the thermal de-Broglie wave1

lengths of the bosons become comparable to their mean spacing, or when n 3 λdB ∼ 1 where n is thedensity of the gas. To satisfy the diluteness criterion that is used to truncate the Hamiltonian (3.15)
at the binary level, we enforce that the average distance between the bosons is much greater thanthe characteristic length scale on which they interact. For isotropic, short-range interactions (suchas the Van der Waals interaction), an appropriate interaction length scale is set by the s-wave
1

scattering length as . Thus, the diluteness criteria takes the form n 3 as ≪ 1. Our conditions forcondensation and diluteness tell us that the ultracold, dilute Bose gas must obey the condition
as
≪ 1.
λdB

(3.55)

This condition can be used to greatly simplify the interaction matrix elements (3.19) if we enforce
that all states exist below a certain energy cutoff Ecut so the characteristic wave numbers ki of
all states are such that ki ≪ 2π/as [108]. Since the two-body interaction changes on a lengthscale much smaller than those of the condensed and low-lying quasiparticle states with energies~ωi < Ecut , which change on a length scale ∼ 2π/ki , the short-range interaction matrix elements
can be simplified by using the shape-independent approximation,(c)Zdx dx′ φ⋆i (x)φ⋆j (x′ )Vc (x − x′ )φk (x′ )φl (x)Z≈ g dxφ⋆i (x)φ⋆j (x)φk (x)φl (x),Vijkl =

Z

(3.56)

where, for isotropic interactions, the two-body potential is just Vc (|x − x′ |) andg≡Z

dxVc (x) = Ṽ (0).

(3.57)

This result motivates the introduction of a pseudopotential Vc (x − x′ ) = gδ(x − x′ ) for the shortrange interaction. The coupling constant g can be determined by identifying the zero-energy
solution of the Lippmann-Schwinger equation for the two-body wave function with the known
low energy threshold result ψ(r) = 1 − ar , where a is the s-wave scattering length. Using the Born

45
approximation in this process allows one to identify the form of Eq. (3.57), thus g is given by
4π~2 aBorn /M where aBorn is the Born approximation for the s-wave scattering length [55]. The
Born approximation, however, does not take into account the high-momentum behavior of the twobody wave function in the vicinity of |x − x′ | ∼ 0 where the two-body potential is strong [109].
Indeed, such information is automatically lost in the many-body theory when the single-particle
wave functions are assumed to have the separable form (3.1). However, this problem is overcome
by replacing the Born approximation to the zero-energy T -matrix with the full T -matrix, whichamounts to replacing aBorn with the true s-wave scattering length as to give the proper short-range
pseudopotential
Vc (x − x′ ) = gδ(x − x′ ) =4π~2 asδ(x − x′ ).
M(3.58)The true s-wave scattering length can be obtained either empirically or through some calculations that treat the close-coupling of the two-body wave function. For example, an approximate
analytical formula for the scattering length of atoms interacting via the Van der Waals force wasderived in [110], and a delta-function pseudopotential of the form (3.58) was found to reproduce
the correct scattering behavior of two bosons with hard-sphere interactions, where the scattering length is simply the radius of the hard-sphere [111, 103]. Additionally, this pseudopotential
was used therein to calculate for the first time the zeroth-order (mean-field) energy, the quantum
depletion and the first-order beyond mean-field energy (energy due to quantum fluctuations at zerotemperature) of a dilute BEC. However, to calculate this beyond mean-field energy (the so-called
LHY energy correction) the authors used a corrected form of the pseudopotential (3.58) to firstorder beyond the momentum-independent approximation (3.58), given by iterating the T -matrix
solution of the Lippman-Schwinger equation to next order. This momentum-dependent pseudopotential provides a first-order account of higher energy scattering processes above the cutoff Ecut .
It is also interesting to note that the momentum dependent pseudopotential (3.58) is automatically on-shell (k = k′ ). At the mean-field Bogoliubov level that is employed in this thesis, it issufficient to use the on-shell transition matrix elements, while higher-order theories that, for exam-

46

Figure 3.2: Geometry of the dipole-dipole interaction for dipoles that are polarized by an external
field in the z-direction, corresponding to the interaction potential in Eq. (3.59).

ple, include quasiparticle-quasiparticle interactions, necessitate the calculation of off-shell elements.
One such theory is that of Beliaev [112, 113], which uses the momentum-dependent form of thescattering amplitude to calculate the beyond mean-field corrections to the quasiparticle spectrumof a Bose gas [114]. Nevertheless, the pseudopotential (3.58) is appropriate for the dilute BEC thatis discussed here.

3.4.2

Dipole-Dipole Interactions

We now turn our attention to the ddi which, unlike the interactions considered in the previous
section 3.4.1, is anisotropic and long-range. In particular, we consider the interaction between two
dipolar bosons that are polarized in an external field. If this polarizing field points in the z-direction
and the dipole moments are given by d1 and d2 , the two-body potential for this interaction can be
written as (in cgs units)
Vd (x − x′ ) = d1 d2

1 − 3 cos2 θx−x′
C20 (θ, φ)
=
d
d
,
1
2
|x − x′ |3|x − x′ |3(3.59)

where C20 (θ, φ) is the reduced spherical harmonic [115] and θ is the angle between the polarizing
field and the vector separating the two dipoles x − x′ . This ddi potential is given by taking dˆ1 = dˆ2
in the more general expression (2.20). The geometry of the ddi given in Eq. (3.59) is shown infigure 3.2.
Unlike short-range interactions that are ∝ 1/r 6 when they are of the Van der Waals type,47
the ddi goes as ∝ 1/r 3 , which gives this interaction very special properties. This can be seen byconsidering the long-range contribution to the low-energy threshold behavior of the elastic scattering
phase shift for the lth partial wave for a potential ∝ 1/r s [116],tan δl ∝ Ak2l+1 + Bks−2 .

(3.60)The elastic cross-section is proportional to sin2 δl /k2 , so for l > 0 and s = 3 (corresponding to
the ddi) the first term vanishes at zero energy while the second term is constant. Thus, at low
energies the scattering of dipoles is independent of k and is dominated by the 1/r 3 part of thetwo-body potential for all partial waves l > 0. As a result of this long-range behavior of the ddi,
the Born approximation works well to describe low-energy dipole-dipole scattering for non-zeropartial waves [117]. For the s-wave part, a pseudopotential similar to (3.58) can be used where
the s-wave scattering length is now dipole dependent, as = as (d1 , d2 ) [118, 119, 120, 121]. It was
shown in [122] that this pseudopotential treatment of the s-wave part of the ddi is sufficient for an
accurate description of low-energy dipolar collisions in a BEC, and the Born approximation works
well for the l > 0 partial waves as long as the strength of the ddi is sufficiently weak [123].
Thus, we take our two-body interaction potential to be
V (x − x′ ) = gδ(x − x′ ) + d2

1 − 3 cos2 θx−x′|x − x′ |3(3.61)

where we have assumed that d1 = d2 = d and as is the full s-wave scattering length characterizing
the low-energy two-body collisions. This potential has a well-defined Fourier transform Ṽ (k) [124],Ṽ (k) = g +4π 2
d 3 cos2 θk − 1 ,3(3.62)

where θk is the angle between the dipole polarization and the wave vector k. We give the detailsof the calculation of this momentum-space ddi in Appendix B for dipoles with arbitrary spatial
polarization. Indeed, the second term in this momentum-space interaction potential (3.62) describes
dipole-dipole interactions for any uniform polarization, and not just polarization in the z-direction.
We refer to the first term in this potential (3.62) as a “contact” potential due to its delta-function
form in coordinate-space. The second term in this potential accounts for the long-range, anisotropic

48
ddi that is present between two polarized dipoles. These dipoles could be either magnetic or electricin nature. Because we use cgs units, the ddi “coupling” is just d2 in Eqs. (3.61) and (3.62). In SI
units, this coupling is d2 /4πǫ0 for electric dipoles and µ0 d2 /4π for magnetic dipoles, where ǫ0 is thepermittivity of free space and µ0 is the permeability of free space. In cgs units, 1/4πǫ0 = µ0 /4π =1 [91].

3.4.3

Mean-Field Potential

We can now write our mean-field potential Umf (x) introduced in section 3.2 in terms of thecontact and ddi,Z

dx′ V (x − x′ )|φ0 (x′ )|2Z
1 − 3 cos2 θx−x′′ 2= (N − 1) g|φ0 (x)|2 + d2 dx′
|φ
(x
)|
.
0
|x − x′ |3

Umf (x) = (N − 1)

(3.63)The delta-function form of the contact interaction eliminates the integral in the mean-field potential
and leaves a term that is proportional to the square of the condensate field, or just the condensate
density, while the part of the mean-field due to the ddi cannot, in general, be reduced as suchand the integral needs to be handled explicitly. This is the essence of the long-range nature of
the ddi, that to correctly account for the dipolar interactions at a point x one must convolve overthe contributions from the dipoles at all other points x′ . Such a convolution can be simplified byworking in momentum-space, as is explained in Appendix A.
The most important results from this chapter are the equations that govern the condensate
wave function φ0 (x), the time-independent (3.23) and time-dependent (3.46) Gross-Pitaveskii equations, and the Bogoliubov de Gennes equations (3.34) that govern the quantum fluctuations of thesystem that, in the ultracold dilute Bose gas, are characterized by non-interacting quasiparticles
that are excited on top of the ground condensed state. In the following chapters, we demonstrate
the power of these mean-field relations by characterizing the ground condensed state and the quasiparticle excitations of trapped condensates with dipole-dipole interactions and by showing, whereapplicable, excellent agreement with experiment.

Chapter 4
Homogeneous Dipolar Bose-Einstein Condensates

4.1

Three-Dimensional Case
To begin investigating the role that the dipole-dipole interaction (ddi) plays in the physics ofa Bose condensed gas, we first consider the simple case of an untrapped (U (x) = 0), or homogeneousgas in the presence of both the contact and the ddi. To impose boundary conditions on the system,
we consider N bosons in a cubical box of volume V = L3 . The homogeneous limit is then achievedby taking the volume V of the box to infinity while keeping the ratio n = N/V , the density of thegas, fixed. We choose a plane wave basis for our single particle states and label them by their wave
vectors k,
1
φk (x) = √ eik·x .
V

(4.1)

In this case, the two-body interactions must conserve momentum, ki + kj = kk + kl , and withthe plane-wave free-particle energy eigenvalues ~2 k2 /2M the many-body Hamiltonian (3.17) canbe written asĤ =X ~2 k2 †1â âk +
2M k
2V
k

X

k1 ,k2 ,qṼ (q)â†k1 +q â†k2 −q âk1 âk2 .

(4.2)

Here, Ṽ (q) is the two-body momentum-space interaction potential (3.62) that describes the transfer
of momentum ~q in the scattering process. This form is obtained by calculating the interaction
matrix elements Vijkl in the plane-wave basis. For the contact interactions, the matrix elements

50
are
Z
Z
g
′
′
Vc = 2 dx dx′ e−i(k1 +q)·x e−i(k2 −q)·x δ(x − x′ )eik1 ·x eik2 ·xVZg= 2 dxe−i(k1 +q)·x e−i(k2 −q)·x eik1 ·x eik2 ·x
V
g
= .
V

(4.3)

For the ddi, we use the convolution theorem (see Appendix A). The momentum-space form of theddi is taken from Eq. (3.62) and we use the fact that the momentum-space representation of aplane wave is a delta-function whose argument is just the wave number to calculate the ddi matrix
elements,1
Vd = 2
V
..
.
=

Z

dx

Z

′

′

dx′ e−i(k1 +q)·x e−i(k2 −q)·x Vd (x − x′ )eik1 ·x eik2 ·x

4π 2
d 3 cos2 θq − 1 .
3V

(4.4)

This result shows that while the matrix elements of the contact interaction are completely momentum independent, the matrix elements of the ddi are momentum dependent, but in homogeneous
space depend only on the angle between the dipole polarization and the momentum transfer q, θq ,and not the magnitude of the momentum transfer. This angular dependence is to be expected,
though, as the ddi itself is anisotropic.
The Hamiltonian (4.2) can be simplified by employing the Bogoliubov approximation (3.20)
and separating the interaction term into parts ∝ N02 and ∝ N0 , similar to the process that was
carried out in section 3.2 for the Bose field operators. In terms of the creation and annihilation
operators, the Bogoliubov approximation amounts to approximating the zero-momentum operators3√
as c-numbers, â0 = â†0 = N0 . Note that terms ∝ N02 have only one excited particle operator and√must vanish because the interactions must conserve momentum, and terms ∝ N0 and ∝ 1, whichhave three and four excited particle operators and describe interactions between excited particles,
are ignored in the perturbative process. If we approximate N0 ≈ N and define the condensate

51
density as n ≡ N/V , the Hamiltonian for this system can be written in the form
Ĥ =

n
o
X ~2 k2 †N
1 X
nṼ (0) +âk âk + nṼ (k) 2â†k âk + â†k â†−k + âk â−k .
2
2M
2
k

(4.5)

k>0

Taking only this leading term in this Hamiltonian, corresponding to a pure condensate, gives theimportant result for the ground-state, or mean-field energy per particle of a homogeneous dilute
Bose gas, corresponding to a pure condensate,E01= nṼ (0).
N
2

(4.6)

The chemical potential µ is then given by µ = ∂E0 /∂N = nṼ (0) [55].
To diagonalize the free-space Hamiltonian (4.5) we employ the Bogoliubov transformation (3.20)
to the creation and annihilation operators â†k and âk [125],⋆âk = uk b̂k + v−k
b̂†−k

(4.7)

â†k = u⋆k b̂†k + v−k b̂−k ,(4.8)

where enforcing that the b̂k operators satisfy Bose commutation relations puts the constraint on
uk and vk , u⋆k uk′ − vk⋆ ′ vk = δkk′ , just as in Eq. (3.27). Now, a process very similar to that done for
the Bose field operators in section 3.2 results in a diagonal form of the Hamiltonian (4.5)
Ĥ = E0 +X~ω(k)b̂†k b̂k ,

(4.9)

k

where ω(k) is the Bogoliubov, or quasiparticle dispersion relation of the homogeneous Bose gas,s

~2 k2 ~2 k2~ω(k) =
+ 2nṼ (k) ,(4.10)
2M
2M
and the diagonal form of the Hamiltonian is made possible by the quasiparticle amplitudes uk andvk satisfying the relations
uk , v−k = ±s~2 k 2
2M + nṼ (k)

2ω(k)

1
± .
2

(4.11)The dispersion (4.10), first derived by Bogoliubov in 1947 for purely contact (not dipolar) interactions [94], is a key result. It tells us that the ultracold, dilute Bose gas can be described as a52condensate “reservoir” beneath a system of non-interacting quasiparticles with the spectrum ω(k).
Additionally, the speed of sound in this system can be extracted from the phonon part of thequasiparticle dispersion, or from its small momentum behavior ω(k) → ~c(θk )k, where
s


ω(k)
nṼ (θk )c(θk ) = lim
=
k→0
~k
M

(4.12)

is the speed of sound in the condensate. In (4.12), the momentum-space interaction potential iswritten as a function of θk because the limit as k → 0 only affects the magnitude of k and not its
direction. Thus, we see that the homogeneous dipolar Bose-Einstein condensate (BEC) exhibits
anisotropic sound due to the anisotropic nature of the ddi [124, 16].
At this point, it is useful to define a new coupling coefficient for the ddi. We define gd ≡4π~2 add /M , where add is defined in Eq. (2.21). In terms of this coupling, the momentum-spaceinteraction for the homogeneous system can be written asṼ (k) = Ṽ (θk ) = g + gd 3 cos2 θk − 1 .

(4.13)

It is clear from the form of the dispersion (4.10) that phonons (small-k quasiparticles) will develop
imaginary frequencies if nṼ (θk ) < 0, and therefore trigger a dynamical phonon instability in thegas according to the condition (3.49). This can occur at different angles depending on the directionof phonon propagation and on the ratio of contact to ddi couplings, though to stabilize the gas at
all angles it is necessary to have g > gd . In the literature, the dipole “strength” of the gas is definedas ǫdd ≡ gd /g, so the condition for stability of the homogeneous dipolar BEC is given byǫdd < 1.

(4.14)

It is interesting to note that the most unstable direction for phonon propagation is perpendicular tothe dipolar polarization, while the most stable direction is in the direction of the dipole polarization,even though the dipoles are most attractive in this direction. This is a consequence of the shape ofthe density waves that correspond to phonon excitations. For phonon propagation in the directionof polarization, the regions of high and low density form sheets along which the ddi is most repulsive.

53
On the other hand, for phonon propagation perpendicular to the direction of polarization, thesesheets lie in the plane of the dipolar polarization along which the ddi is most attractive, thus making
sound propagation in this direction less stable.
As discussed in section 3.2.3, the presence of interactions in a BEC gives rise to a quantum
depletion, or non-zero occupation of excited quasiparticles at zero temperature. In the homogeneousgas, the quantum depletion is given by taking the sum over the square of the Bogoliubov vk
amplitudes to an integral in the homogeneous limit (see Eq. (3.44)),
1 X 2
nex =
vk →
V
k6=0

where the function Q3 (ǫdd ) =

Z

r
dk 2
8
na3s
v
=
nQ
(ǫ
)
3
dd
k
(2π)3
3
π

(4.15)

R1

2 32
0 dx(1 − ǫdd + 3ǫdd x ) comes from the integration over the angularpart of the momentum-space interaction [126] and as is the s-wave scattering length. Interestingly,
this depletion (4.15) remains finite as the dipole strength approaches the critical value for instability
ǫdd → 1, indicating that the collapse of the homogeneous dipolar BEC is in fact a coherent, energetic
collapse of the condensate and is not due to the macroscopic occupation of phonon modes. We
will return to this point when discussing the collapse of dipolar BECs in trapped geometries in
section 4.2. Additionally, it is interesting to note that the quantum correction to the pure mean-fieldpdensity n is proportional to na3s . This term, first derived by Lee, Huang and Yang in 1957 [103],

is referred to as the LHY correction, and the same proportionality governs the first-order beyond
mean-field (bmf) correction to the ground state energy.
Phonon instability is also predicted to occur in a homogeneous, non-dipolar Bose gas withattractive interactions corresponding to negative scattering lengths as < 0 [125]. This is expected,
though, as negative scattering lengths correspond to g < 0, and thus the emergence of non-zero
imaginary phonon frequencies. However, these systems are stabilized by the presence of external
confinement, where additional kinetic and trapping energy is added to the system. Such stabilization was first predicted by explicitly solving the GPE with attractive contact interactions in [127],
and confirmed experimentally at Rice University in a BEC of 7 Li with attractive interactions [128].
The stabilization of an attractive BEC motivates the presence of external confinement for the dipo-

54
lar Bose gas. Because the trap stabilizes the BEC with attractive contact interactions, we expectthat applying a trap to a dipolar BEC in the direction of the dipole polarization and on a lengthscale shorter than the phonon wavelength will suppress the phonon instability in this direction andthus stabilize the condensate against energetic collapse for values of the dipolar strength ǫdd > 1,
presenting the possibility of creating a dominantly dipolar BEC.
The trapping of an ultracold dipolar gas is advantageous for other reasons, as well. Collisional
studies of bosonic and fermionic polar molecules in reduced geometries predict that inelastic loss
rates are highly reduced when the gas is squeezed in the polarization direction [129, 130, 131, 132,133]. Such inelastic collisions correspond to chemical reactions, for example, 2KRb → K2 + Rb2 , orchanges of internal molecular states that can occur at short range and result in the molecules being
expelled from the trap. These results stem from the fact that the attractive head-to-tail orientationof the dipoles is highly suppressed in this squeezed geometry, and it is the classical “head-to-tail”
collision pathway that allows the molecules to reach the short-range at which inelastic processes
overwhelmingly occur. It is precisely this attraction that not only gives rise to the energetic collapseof the homogeneous dipolar BEC, but also results in the sharp increase of inelastic scattering of polar
molecules [134] and three-body recombination in the presence of the ddi [135]. In the next section,
we consider such a squeezed trapping geometry, though we assume that inelastic and three-bodylosses are completely absent. However, we see that the presence of such a trap significantly stabilizes
the dipolar BEC, and leads to novel emergent physics in the system, namely, the emergence of theroton mode in the quasiparticle dispersion of the trapped system.

4.2

Quasi-Two-Dimensional Case
As discussed in section 2.3.1, the interaction of laser light with an atom or molecule canresult in an optical force field that acts to trap the atom or molecule. In this section, we consider
the case of the retro-reflected laser, where a standing optical wave is created to form an opticallattice trapping geometry. If the laser light propagates in only one direction, the resulting latticegeometry is that of a series of traps, the widths of which are set by the Gaussian width of the55
laser and the spacings of which are set by the wavelength of the laser. These length scales are
typically rather disparate, so a series of very flat, “pancake” shaped optical traps results. We notethat a BEC of 52 Cr has been achieved in such a geometry [136], which we will discuss further inchapter 8. For now, we consider only one site in a deep optical lattice potential and approximate
it as infinitely wide, so the effective trapping potential can be written as a one-dimensional (1D)
harmonic potential
1
U (x) = M ωz2 z 2 ,(4.16)
2p
where ωz is the trap frequency, and lz = ~M/ωz is the corresponding harmonic oscillator length.

Such a trapping geometry creates a system that is spatially homogeneous in the x-y plane, andthus has some two-dimensional (2D) character, be it pure-2D or quasi-2D.

4.2.1

Contact Interactions

In a pure-2D system, the BEC phase, characterized by long range phase coherence, occurs
only at T = 0, as follows from Bogoliubov’s k−2 theorem [137]. However, in 1972 Popov showedthat superfluidity may exist in the absence of the pure condensate at ultracold, but non-zero temperatures due to the macroscopic occupation of finite-wavelength modes [138]. Thus, the superfluid
state of a Bose fluid in 2D is characterized not by a pure condensate but by a quasi-condensate
with finite size domains of phase coherence [139, 140]. Additionally, the interaction coupling for a
2D system is density dependent, g2D → g2D (n2D ), as was first predicted by Schick to zeroth-orderin the 2D gas parameter n2D a2c for a 2D system of hard circles [141],
g2D (n2D ) =

1
4π~2
,
M | ln n2D a2c |(4.17)

where, here, ac is the radius of the hard circle. Such a form of the 2D interaction coupling has its
roots in the fact that the 2D scattering amplitude for hard-circles has the form ∝ 1/ ln kac in thelimit kac → 0, whereas the 3D scattering amplitude for hard spheres is momentum independent.
Historically, the study of ultracold 2D Bose fluids was motivated by experiments on monolayers of 4 He [142], where the size of the 1D confinement is typically smaller than the characteristic

56
range of the interactions in the fluid. Such is not necessarily the case, however, for current experiments on dilute BECs in optical traps. Take, for example, the tight harmonic trap frequency
ωz ≈ 2π × 40 kHz that has been achieved in the 52 Cr experiment [136]. The corresponding harmonic oscillator trap size for 52 Cr is lz ≈ 1.3 × 103 a0 where a0 is the Bohr radius. This trap size ismuch larger than the dipole length of chromium, add = 15.2 a0 , and is certainly much larger thanscattering lengths that can be experimentally realized by using a Fano-Feshbach resonance [21](see section 6.1). Such length scales may result in a gas with three-dimensional (3D) character.
However, at very low yet experimentally accessible temperatures T ∼ 200 nK [14], the ratio of thetrapping energy to the temperature ~ωz /kB T ∼ 10 is fairly large, so the character of the gas in
the direction of the trap is not free and instead the particles undergo zero-point oscillations. Suchcharacter motivates the definition of, and defines the “quasi-2D” gas.
In their work [143], Petrov, Holzmann and Shlyapnikov showed that the pure BEC state isrealizable in a quasi-2D Bose gas at ultracold, but non-zero temperatures T < Tc , where Tc is thecritical temperature for BEC in the homogeneous gas. Additionally, they calculated a modified
momentum-dependent interaction coupling for the quasi-2D BEC where the two-body scatteringwave function has the asymptotic form ψ(x) = χ(z) exp [ik · ρ],√
2 2π~2
1
gq2D (k) =
.
lz
1
√
M
ln πk12 l2
a +
2π

s

(4.18)zwhere χ(z) is the ground-state wave function of the potential (4.16), given by a normalized Gaussianz2
χ(z) = √ 1 exp − 2 ,2lzlz π 41

(4.19)

and as is the 3D s-wave scattering length. In the limit as ≪ lz , the logarithmic term in (4.18)
can be neglected and the coupling acquires a momentum independent form that is proportional
to the ratio as /lz . This is a very important result. By drawing the distinction between pure-2D
and quasi-2D systems, being that the quasi-2D system permits a BEC state with long-range phasecoherence [144], we can treat the quasi-2D system on the same footing as we treat the 3D systemand therefore apply the results from section 4.1 directly to the quasi-2D system. The difference is57that, for the quasi-2D system, we treat the condensate mode as a zero-momentum plane wave inthe x-y plane with the axial wave function χ(z). Such a treatment is known as the single-mode
approximation, and is valid for a dilute Bose gas when ~ωz ≫ kB T . In this approximation, thequasi-2D interaction coupling corresponding to the s-wave pseudopotential (3.58) can be calculated
simply by integrating out the z-dependence of the interaction terms in the many-body Hamiltonian,gq2D = g
=g

Z

Z

dz

Z

dz ′ χ⋆ (z)χ⋆ (z ′ )δ(z − z ′ )χ(z ′ )χ(z)dzχ⋆ (z)χ⋆ (z)χ(z)χ(z)√2 2π~2 as=.
M lz(4.20)

Thus, the quasi-2D BEC with s-wave interactions can be treated in the same way as the homogeneous 3D BEC but as a 2D system with an effective quasi-2D interaction coupling (4.20).
In the limit as ≫ lz , the relation k2 = 2M µ/~2 [92] can be used to show that the coupling (4.18) takes the pure-2D form (4.17). Thus, in this limit the gas is effectively 2D. It isinteresting to note that the quasi-2D coupling (4.18) suggests not only that the character of thegas from 2D to quasi-2D can be controlled by tuning the trap frequency, but also that the strengthof the interactions can be controlled by tuning the trap frequency. Indeed, a transition fromattractive (negative) to repulsive (positive) occurs at a critical harmonic oscillator trap lengthsln (1/πk2 lz2 ), corresponding to the appearance of a zero-energy bound state. Such phelcrit = √a2π

nomena in ultracold gases are appropriately referred to as confinement-induced resonances, andwere first predicted in the quasi-1D geometry by Olshanii [145].

4.2.2

Dipole-Dipole Interactions

We now turn our attention to the case of the ddi. In pure-2D and if the dipoles are polarized
in the direction of tight confinement (the z-direction if we consider the trapping potential (4.16)),
the in-plane interaction is given simply by
V2D (ρ − ρ′ ) =d2.
|ρ − ρ′ |3(4.21)

58
This interaction potential has purely s-wave character and is dominated by the s-wave contribution to the scattering T -matrix at ultracold temperatures. Indeed, it was shown in [146] that thezero-energy threshold scattering behavior of two aligned dipoles interacting via the 2-body potential (4.21) is well-described by treating the dipolar interaction as a hard-wall interaction at theradius as ≃ 19 add . This idea is considered in [147] in the many-body context where diffusion
Monte Carlo calculations were performed for a gas of bosonic dipoles in the pure-2D geometry.
Therein, the authors calculate the energy per particle using the potential (4.21) and find that it
agrees well with the 2D mean-field result E0 /N = g2D (n2D )n2D /2 (see Eq. (4.6)) where g2D (n2D )is given by (4.17) for small values of the gas parameter n2D a2s , and n2D is the 2D density of thegas, although small quantitative deviations exist. In the quasi-2D regime, however, this treatmentof the ddi is no longer appropriate. In this regime, the zero-point oscillations reintroduce the 3D
character of the ddi at a length scale ∼ lz , and a proper treatment of the interactions in such asystem must take this into account.
To begin describing the quasi-2D dipolar BEC, we make the assumption that the dipole
lengths of the bosons are sufficiently smaller than the harmonic oscillator trap length, add ≪ lz so
any pure-2D character can be neglected. Then, the effective quasi-2D ddi can be calculated in thesame way as is done for the contact interactions (4.20),′

Vq2D (ρ − ρ ) =

Z

dz

Z

dz ′ χ⋆ (z)χ⋆ (z ′ )Vd (x − x′ )χ(z ′ )χ(z).

(4.22)We handle this integral in momentum-space, and give the details of the calculation in Appendix C.
For the sake of generality, we allow for the dipoles to be uniformly polarized in any direction, notjust in the z-direction, at an angle α off of the z-axis. In this case, the momentum-space ddi in thequasi-2D geometry takes the elegant form [39] (Eq. (C.8))
gd
Ṽq2D (kρ ) = √
F
2πlz



kρ lz
√
2



,

(4.23)

where the function F (q) = cos2 αF⊥ (q) + sin2 αFk (q) has contributions from the projection of
the polarization vector dˆ onto the trap z-axis and onto the x-y plane, given in Appendix C by

59
Eqs. (C.6) and (C.7), respectively. Unlike the quasi-2D contact interaction potential, the quasi-2D
ddi potential (4.23) depends both on the direction and magnitude of the 2D momentum kρ . For
the remainder of this chapter, though, we take α = 0 so the ddi is isotropic in the x-y plane andthe momentum-space form of the ddi no longer depends on the direction of kρ . However, taking
α 6= 0 is crucial for the discussion in Chapter 7 and we return to the general anisotropic form of
this quasi-2D ddi there. For now, including both the contact and ddi from Eqs. (4.20) and (4.23),
the full quasi-2D momentum-space interaction potential takes the formgṼq2D (kρ ) = √2πlz



1 + ǫdd F⊥



kρ lz
√
2



.

(4.24)

We note that this expression was first derived for the quasi-2D dipolar BEC in [29].
Exploring the limiting behavior of F⊥ (q) illuminates interesting physics for the quasi-2D
dipolar BEC,



kρ lz
√
=2
kρ lz →0
2


kρ lz
= −1,
lim F⊥ √kρ lz →∞
2
lim F⊥(4.25)(4.26)

where the function is monotonically decreasing between these limits. The limit kρ lz → 0 in (4.25)
corresponds to zero in-plane momentum. Such character describes the condensate field of the quasi2D system, the wave function of which we write as φ0 (x) =√n2D χ(z). Thus, from this limiting

behavior we can write the mean-field interaction energy of the quasi-2D dipolar BEC from Eq. (4.6),
E0
1 g
n2D (1 + 2ǫdd ) ,= √N
2 2πlz(4.27)

where n2D is the integrated 2D density of the gas, or the density obtained by integrating the fulldensity over all z. This form suggests that in the small momentum limit, when the characteristic
wavelength of a mode is much larger than the trap size, the ddi can be treated as a contact
interaction with coupling 2gd . From Eq. (4.27), we see that for the quasi-2D BEC to be energeticallystable the interaction couplings must be such that 1 + 2ǫdd > 0, or
1
ǫdd > − ,2

(4.28)

60
ǫdd = 1.5
ǫdd = 1.0
√ √Ṽq2D (kρ lz / 2) 2πlz /g

2.5

ǫdd = 0.52ǫdd = 0.0

1.5

ǫdd = −0.5
ǫdd = −1.0

1
0.5
0
−0.5
−1 Instability02√kρ lz / 2

4

6

Figure 4.1: The quasi-2D momentum-space interaction potential (4.24) for fixed contact interaction
coupling and√ various valued of the dipole interaction strength ǫdd . The black dashed line marks
Ṽq2D (kρ lz / 2) = 0, below which instabilities are possible. For ǫdd = 0, corresponding to no dipole
interactions, the interaction is independent of momentum.

that is, quasi-2D dipolar BEC is energetically stabilized for any s-wave scattering lengths greaterthan minus twice the characteristic dipole length add , or as > −2add . This criterion shows thatthe quasi-2D dipolar BEC is significantly more stable than the homogeneous dipolar BEC, whoseenergetic stability criterion is given by Eq. (4.14).
√
The limit of F⊥ (kρ lz / 2) as kρ lz → ∞ (4.26), while unphysical in the mean-field Bogoliubov
picture because of our effective energy cutoff Ecut in the two-body scattering theory (see section 3.4), corresponds to a 3D character in the quasi-2D system as the characteristic wavelengthsof such quasiparticles are much smaller than the trap size lz in this limit. Indeed, taking this limit
results in a quasi-2D momentum-space interaction that is proportional to (g−gd ), which is the same
result that is seen for quasiparticles in the 3D system that propagate perpendicular to the dipole
polarization, corresponding to θk = π/2 (see Eq. (4.13)). The quasi-2D momentum-space interaction potential (4.24) is plotted in figure (4.1) for different dipole interaction strengths ǫdd = gd /g.
√Here, phonon instability is seen as Ṽq2D (kρ lz / 2) < 0 at small kρ lz . The case of ǫdd = −0.5 is at
the threshold of phonon instability, while ǫdd = −1.0 is clearly unstable.

61
4.2.3

Bogoliubov Spectrum: Emergence of the Roton

Because the condensate is defined in the same manner in quasi-2D as in 3D (long-wavelength
phase coherence [144]), the results from the 3D field theory can be used for the quasi-2D system,
where the sums and integrals are now taken over momenta in the 2D plane and the effective quasi2D interaction potentials are used. Thus, knowledge of the momentum-space interaction in thequasi-2D system permits the calculation of the quasiparticle dispersion from Eq. (4.10). Written
explicitly, this dispersion takes the form
s




~2 kρ2 ~2 kρ2kρ lz
2g~ω(kρ ) =+√n2D 1 + ǫdd F⊥ √
.
2M 2M
2πlz
2

(4.29)

From this, we can calculate the speed of sound in the quasi-2D dipolar gas from Eq. (4.12),sr
n2D Ṽq2D (0)
1
n2D g
=
(1 + 2ǫdd ).
(4.30)
cq2D =
1M
M lz(2π) 4
Unlike the case for the 3D dipolar BEC, the speed of sound in the quasi-2D dipolar BEC (whenα = 0) is isotropic. Additionally, Eq. (4.30) reveals that the speed of sound develops a non-zero
imaginary part, corresponding to an imaginary phonon frequency in the quasiparticle dispersion,
when ǫdd < −1/2. This result is consistent with the criteria for energetic stability (4.28).
√
Quasiparticle dispersions are plotted for 2 2π(as /lz )(n2D lz2 ) = 1 and ǫdd = 0 in figure 4.2
and ǫdd = −1 in figure 4.3. For ǫdd = 0, the system is energetically stable, as is reflected by
the purely real dispersion in the corresponding figure. Therein, the free-particle (non-interacting)dispersion is plotted and the sound velocity of the interacting condensate is shown by the slope ofthe black dotted line. For ǫdd = −1 in figure 4.3, the system is energetically unstable, as is signified
by non-zero imaginary phonon frequencies. Regarding energetic stability, the criteria set by themean-field energy (4.27) and the phonon stability predict that the quasi-2D dipolar BEC is stable
for all values of ǫdd > −1/2. Further investigation of the quasiparticle dispersion (4.29), however,reveals otherwise.
In the homogeneous 3D dipolar BEC, the momentum-space interaction is independent of themagnitude of k and instead depends only on its direction. The case for the quasi-2D dipolar BEC

62
4

1
quasiparticle

3

0.8

free particle

Im[ω/ωz ]ω/ωzsound velocity
ω/ωzRe[ω/ωz ]

2

1

0.6
0.4
0.2
Phonon Instability

0
0

0.5

1

1.5
√
kρ lz / 2

2

2.5

3

Figure 4.2: The quasiparticle dispersion (blueline) of the quasi-2D√ BEC with repulsive contact interactions 2 2π(as /lz )(n2D lz2 ) = 1 andǫdd = 0, corresponding to no dipolar interactions. The free-particle dispersion is shownby the red dashed line and the sound velocity
is given by the slope of the green dotted line.

0
0

0.2

0.4
√ 0.6
kρ lz / 2

0.8

1

Figure 4.3:
The quasiparticle dispersion (blue line) of the quasi-2D dipolar
BECwith repulsive contact interactions√
2 2π(as /lz )(n2D lz2 ) = 1 and ǫdd = −1. The
red dots show the imaginary part of thedispersion at small momenta, corresponding
to phonon instability for these parameters.

considered here is just the opposite, as mentioned previously, and the momentum-space interactionṼq2D (kρ ) does not depend on the direction of kρ in the x-y plane but does depend on its magnitudekρ , as shown in figure 4.1. This fact can be understood qualitatively by considering that while there
is no external length scale in the homogeneous 3D system, the presence of the harmonic trap sets anexternal length scale in the quasi-2D system. For phonons with wavelengths that are much greaterthan the trap size, kρ lz < 1, the interactions have a purely repulsive 2D character. This is reflectedin the fact that, for ǫdd > 0, the momentum-space interaction (4.24) is positive at small wave
number. However, for quasiparticles at intermediate momenta kρ lz ∼ 1, the interactions begin totake on a 3D character and the attractive part of the ddi plays an important role. This is reflectedin the fact that, regardless of the value of ǫdd , the scaled momentum-space interaction (4.24) crosses√
through unity at kρ lz = 2. Thus, the character of the quasiparticles may depend not only on its
wavelength, but on the ratio of their wavelength to the harmonic oscillator length of the trap.
Indeed, for large ǫdd the interaction potential becomes sufficiently negative at intermediatewave number so that ω(kroton ) = 0, where kroton is a finite, non-zero wave number. To illustrate this√fact, quasiparticle dispersions are plotted for 2 2π(as /lz )(n2D lz2 ) = 1 and ǫdd = 3.7 in figure 4.4 and

63
3

3
quasiparticle
sound velocity

2.5

2.5
2
ω/ωz

ω/ωz

2
1.5
Roton Minimum

1

0.5

0.5
0.5

1

1.5
√
kρ lz / 22

2.5

Figure 4.4: Quasiparticle dispersion of thequasi-2D dipolar√ BEC with repulsive contact
interactions 2 2π(as /lz )(n2D lz2 ) = 1 andǫdd = 3.7. The sound velocity is given by theslope of the black dotted line and the roton
minimum is labeled.

Im[ω/ωz ]

1.5

1

0
0

Re[ω/ωz ]00Roton Instability

0.5

1

1.5
√
kρ lz / 22

2.5

Figure 4.5: Dynamically unstable quasiparticle dispersion of the quasi-2D dipolarcontact interactions√ BEC with repulsive22 2π(as /lz )(n2D lz ) = 1 and ǫdd = 4.0. The
red dots shows the imaginary part of the dispersion at intermediate wave numbers, corresponding to a roton instability.

ǫdd = 4 in figure 4.5. For ǫdd = 3.7, the system is dynamically stable, but the dispersion possesses
a minimum at finite wave number. This minimum is referred to as the “roton minimum,” and thecorresponding quasiparticle at the minimum is referred to as the roton. Similarly, the quasiparticle
associated with the local maximum in the dispersion is referred to as a maxon, and the spectrum
is said to have “roton-maxon” character. As ǫdd is further increased , the roton minimum drops
and, at a critical value, becomes degenerate with the condensate, corresponding to ω(kroton ) = 0.
This signifies a roton instability in the quasi-2D dipolar BEC, as is illustrated in figure 4.5, where
the roton has a non-zero imaginary frequency for dipole strengths greater than the critical value.
We plot the stability diagram of the quasi-2D dipolar BEC in figure 4.6. Therein, the phonon
instability is present for all ǫdd < −1/2, while the roton instability is a non-trivial function of ǫdd .
This stability relation was first pointed out in [148]. We note that the roton-maxon spectrum inthe trapped dipolar BEC was first pointed out in [28], and was first predicted for a dilute Bose gas
in [149] where the ddi is induced in a quasi-one-dimensional BEC irradiated by a laser.
In figure 4.6, it appears as though the system stabilizes for some arbitrarily large ǫdd andsome sufficiently small contact interaction coupling g. In fact, the purely dipolar (corresponding to64

Figure 4.6: The stability diagram of the quasi-2D dipolar BEC as a function of the contact interaction coupling and the ddi strength ǫdd . The red region indicates phonon instability for allǫdd < −1/2, and the blue region indicates roton instability. The white, uncolored region corresponds to a dynamically stable system.

g = 0 and gd 6= 0) quasi-2D BEC is dynamically stable only within a finite range of ddi couplings.
We find that the the onset of the roton instability occurs for (add /lz )(n2D lz2 ) ∼ 0.34. Thus, thethreshold for roton instability can be controlled by tuning either the dipole strength, the densityof the gas or the shape of the confining potential such that this value is achieved. For the 52 Cr
system discussed earlier in this section with ωz = 2π × 40 kHz and as =0, a limit of the scattering
length that has been achieved experimentally [21], the roton instability is predicted to occur for an
integrated 2D density of n2D ≃ 5 × 1011 cm−2 .
There are two very important observations to be made about the roton instability. One isthat the roton instability occurs at a finite density and ddi strength regardless of the size of thetrapping potential, as long as the dipole length is sufficiently smaller than the harmonic oscillator
length of the trap. The case of add > lz breaks the quasi-2D criterion, and the physics of this caseis discussed briefly in section 4.2.5. The other important observation to be made is that the roton
instability occurs at a finite, relatively small length scale set by the size of the trap, kroton ∼ 1/lz .
Thus, we find that the roton instability is density-dependent and local, as opposed to global like thephonon instability.

65
4.2.4

Quantum Depletion and Roton Instability

As was done for the 3D homogeneous dipolar BEC in section 4.1, it is instructive to calculate
the quantum depletion for the quasi-2D dipolar BEC at the thresholds for roton and phonon
instability. For simplicity, we consider the purely dipolar case with g = 0. In this case, recall thatthe depletion is given by Eq. (4.15), where we consider the system to be in a box of area A = L2 ,
nex =

1 X 2
vkρ →
A
k6=0

Z

dkρ 2
v .
(2π)2 kρ(4.31)

We evaluate this integral (4.31) numerically using adaptive Gauss-Kronrod quadrature1 and plot
the results in figure 4.7. Recall that the phonon instability in the quasi-2D dipolar BEC with nocontact interactions occurs for all add ≤ 0. Just like the case for the 3D homogeneous dipolar
BEC, the quantum depletion remains finite, and in fact goes to zero at the phonon instability
threshold. This confirms that the phonon instability corresponds to a coherent energetic collapseof the condensate, just as in the case for the 3D dipolar BEC.
The depletion near the roton instability, however, behaves very differently. In figure 4.7,
the roton instability threshold is marked by the vertical black dashed line. As add /lz is increased
towards this threshold, the quantum depletion grows and eventually diverges at the instability. The
physical interpretation of this divergence is that the roton instability is not a coherent energetic
collapse, but actually a macroscopic depletion of the condensed state that is caused by particles
occupying roton modes as the roton energy becomes degenerate with the condensate, and it is thus
energetically favorable for them to do so.

4.2.5

The Other Roton

It is important to draw the distinction between the roton in the dilute, quasi-2D dipolar BEC
that is discussed here and the roton that was introduced in section 2.1.1 that exists in superfluid
4 He.

The superfluid 4 He system exhibits roton-maxon character due to strong correlations thatinduce ordering at short length scales, on the order of the average interatomic spacing. In 4 He, the1

Available in MATLAB through the function quadgk

66
Roton Instability

0.3

nex lz2

0.25
0.2
0.15
0.1
0.05
0
0

0.1

0.2

0.3

0.4

add/lzFigure 4.7: The quantum depletion, or excited quasiparticle occupation at zero temperature of
the purely dipolar (no contact interactions) quasi-2D dipolar BEC. The phonon instability occurs
for add < 0 in this case, and the roton instability occurs for add /lz & 0.34. The divergence ofthe quantum depletion at the roton instability indicates that this instability corresponds to themacroscopic depletion of the condensate into rotons.

densities are on the order ∼ 2 × 1022 cm−3 and the scattering lengths have been calculated to beas ∼ 8 nm [150], so the “gas” parameter for this system is on the order na3s ∼ 104 . Thus, the rotonis this system is intimately related with the structure of the fluid, and represents a tendency towards
spatial ordering at the length scale ∼ 1/kroton which, in this system, is on the order of tens of Bohr
radii, which in turn is characteristic of the lattice spacing in the solid. Thus, in 4 He, the roton isisotropic and arises due to the strong isotropic repulsion between the atoms in the liquid. This
is in strong contrast to the roton in the quasi-2D dipolar BEC, which we present in the previous
section. In this case, the condensate is dilute and thus has negligible two-body correlations and theroton instead has its roots in the momentum dependence of the interaction potential (4.24), where
the momentum dependence results solely from the anisotropy of the ddi interaction at length scales∼ lz .
Such is not the case, however, when the ddi is sufficiently strong (or the trap is sufficiently
tight) so that add ≫ lz . In this case, the anisotropy of the ddi is tuned away and the interactions
become purely repulsive, taking the 2D form (4.21). This 2D system can be characterized by the67
2D gas parameter n2D a2dd . Interestingly, a dispersion with roton-maxon character was predicted forthis system for gas parameters n2D a2dd & 100 [12, 13]. Therein, quantum Monte Carlo techniques
are used to demonstrate the elegant connection between the existence of superfluidity at smaller
densities and the breakdown of superfluidity at larger densities as correlations become strong andsuperfluidity is broken, corresponding to the transition from a superfluid state to a self-ordered
crystalline state. Additionally, such rotons have been predicted to play a strong role in the BKT
superfluid transition in a 2D system of dipoles [151]. More, recently, the quasi-2D to 2D crossoverof an ultracold dipolar Bose gas was studied using the hypernetted chain Euler-Lagrange method,wherein the crossover from the strongly interacting, 4 He-like roton to the quasi-2D roton was shownas the confining trap frequency was varied continuously [152].

Chapter 5
Dipolar Bose-Einstein Condensate in a Cylindrically Symmetric Trap

In the preceding chapters, we built a motivation for and a theoretical means to describe
the zero-temperature dipolar Bose-Einstein condensate (dipolar BEC). Here, we proceed with thisdescription. We solve for the condensate wave function φ0 (x) by minimization of the energy functional (3.24) and we calculate the chemical potential µ via the Gross-Pitaevskii equation (GPE),
Eq. (3.23). Additionally, we solve the Bogoliubov de Gennes (BdG) equations (3.34) to explore thediscrete quasiparticle spectrum of the trapped dipolar BEC and to study the nature of its instability. Such calculations are, in general, numerically intensive, and we handle them by introducing
a novel algorithm that exploits the cylindrical symmetry of the system. Rotationless states are
considered, as well and singly- and doubly-quantized vortex states. Here, we use techniques thatare, within the scope of the mean-field theory, numerically exact. Thus, we are able to see novel
physics that is invisible to the more simple calculations, such as the Gaussian ansatz or the ThomasFermi methods. This physics includes structured ground states and discrete roton-like quasiparticle
excitations that play an interesting and important role in the physics of trapped dipolar BECs.
Additionally we set the s-wave scattering length as = 0 in this chapter to illuminate purely dipolar
effects. Indeed, this is a limit that has been achieved experimentally in a dipolar BEC of 52 Cr by
exploiting a Fano-Feshbach resonance. We discuss this point further and discuss the physics of the
Fano-Feshbach resonance in chapter 6. Original work presented in this chapter has been publishedin references [35], [36] and [34].
We consider a dipolar BEC in a harmonic trap with cylindrical symmetry, so ωx = ωy = ωρ .

69This trapping potential can be written as
1
U (x) = M ωρ2 ρ2 + λ2 z 2 ,
2

(5.1)

where λ = ωz /ωρ is the trap aspect ratio. Small λ, or λ ≪ 1, corresponds to a highly prolate, or“cigar” shaped trap while large λ, or λ ≫ 1, corresponds to a highly oblate, or “pancake” shaped
trap. We can consider the limit λ → ∞ (with fixed ωz ) as a purely quasi-two dimensional (quasi2D) system, as discussed in the previous chapter, and the limit λ → 0 (with fixed ωρ ) as a purely
quasi-one dimensional (quasi-1D) system. Additionally, if we enforce that the dipoles be polarized
in the z-direction, we can use the two-body dipole-dipole interaction (ddi) potential from Eq. (3.59)
to describe the dipolar interactions. In this case, the system exhibits cylindrical symmetry, and thecondensate states may be written in cylindrical coordinates as eigenstates of the orbital angularmomentum projection ~s,
φ0 (x) = ψ(ρ, z)eisϕ .

(5.2)

Here, s is the vorticity of the condensate wave function. Thus, s = 0 corresponds to a rotationlesscondensate, s = 1 corresponds to a condensate with a singly-quantized vortex, s = 2 corresponds
to a doubly-quantized vortex, and so on. Note that the form of the condensate wave function (5.2)
exhibits a phase singularity at ρ = 0 for all s 6= 0.

5.1

Methods
We calculate the condensate wave function φ0 (x) in the form given in Eq. (5.2) by enforcingthat it minimize the energy functional (3.24), which we now write explicitly in terms of the singleparticle Hamiltonian Ĥ (1) (x) and the contact and ddi potentials from Eq. (3.61),




1
1 2
N −1
g|φ0 (x)|4dx φ⋆0 (x) − ∇2 +
ρ + λ2 z 2 φ0 (x) +
2
2
2

Z
2
N −1
′
′ 2 1 − 3 cos θx−x′2
gd dx |φ0 (x )||φ0 (x)| ,+2|x − x′ |3E0 [φ0 ]
=
N

Z

(5.3)

where we have set the number of particles in the system to be the number of particles in thecondensate, N = N0 , and we have moved to dimensionless units where all energies are in units of

70~ωρ and all lengths are in units of the radial harmonic oscillator length, aho = aρ =p~/M ωρ . We

remain in these dimensionless units for the remainder of this chapter. Recall that g = 4π~2 as /M
where as is the (generally dipole-dependent) s-wave scattering length and gd = 4π~2 add /M where
add = M d2 /3~2 is the characteristic dipole length. We also write out the explicit form of thetime-independent GPE (3.23) in the dimensionless form,


1 2
1
ρ + λ2 z 2 + (N − 1)g|φ0 (x)|2µφ0 (x) = − ∇2 +
2
2

Z
2
′
′ 2 1 − 3 cos θx−x′
φ0 (x).
+ (N − 1)gd dx |φ0 (x )||x − x′ |3(5.4)

In practice, we calculate the total energy per particle (5.3) by defining the relevant functions
on a discrete spatial grid in ρ and z, with Nρ × Nz grid points. Notice that, if φ0 (x) has the angulardependence eisϕ , a centrifugal term appears in the kinetic energy term of the energy functional and
GPE that is like ~2 s2 /2M ρ2 , which diverges at ρ = 0. To avoid this complication, we treat both
the ddi mean-field terms and the kinetic energy terms in momentum-space in order to avoid any
numerical problems that are associated with the presence of real-space divergences on the discrete
numeric grid.
Recall that the kinetic energy operator has the diagonal form in momentum-space T̂ = k2 /2.
Thus, we can easily calculate the kinetic energy terms in momentum-space and then represent themin real-space via Fourier transform, − 21 ∇2 φ0 (x) = F −1 [ 21 k2 φ̃0 (k)], where F is the Fourier transform
operator defined in Eqs. (A.2) and (A.3) and φ̃0 (k) = F[φ0 (x)]. If these functions are defined on
appropriate numeric grids, such calculations may be carried out with great speed and accuracy via
the Fast Fourier Transform (FFT) algorithm. However, the three-dimensional (3D) FFT cannot,
in general, take advantage of the cylindrical symmetry of the wave function (5.2). Instead of using
Fourier transforms in the radial direction, we can use Hankel transforms in the radial direction, in
addition to a 1D FFT in the z-direction, where the Hankel transform of a function f (ρ, z)eimϕ isgiven by [153]f˜(kρ , kϕ , z) = 2πi−m eimkϕZ ∞f (ρ, z)Jm (kρ ρ)ρ dρ,

(5.5)

0

where Jm (x) is the Bessel function of order m.

The full momentum-space representation of

71
f (ρ, z)eimϕ is then given by f˜(k) = Fz [f˜(kρ , kϕ , z)], where Fz is the Fourier transform operator in the z-direction. The inverse Hankel transform is defined analogously to the direct transformin Eq. (5.5).
Interestingly, just as the FFT algorithm computes discretized Fourier transforms with thetime complexity N ln N (where here there are N grid points used to define our function), there exist
Fast Hankel transforms (FHTs) and Discrete Hankel transforms (DHTs), which operate with timecomplexities N ln N and N 2 , respectively. FHTs have been found to be numerically unstable withregards to error propagation, while DHTs have been found to be very numerically stable [102].
Thus, we employ the DHT in the radial (ρ) direction and the FFT in the axial (z) direction,
resulting in an effectively 2D problem where the angular dependence is handled simply by choosing
the Bessel function of the proper order. We describe the DHT algorithm in Appendix D.
As forementioned, we calculate φ0 (x) by enforcing that it minimize the Gross-Pitaevskii
energy functional (5.3). Such a calculation was first performed for a BEC with purely contact
interactions by the imaginary time propagation method [154]. Using this method, one evolves thetime-dependent GPE in imaginary time, that is, t → it, which turns the GPE into a dissipativeequation. Thus, the norm of the condensate wave function is not preserved and one must renormalize the condensate wave function throughout the numerical process of imaginary time evolution.
Convergence of the condensate wave function in imaginary time by such propagation corresponds
to the condensate wave function minimizing the GP energy functional. We employed this method
using the 4th order Runge-Kutta algorithm1 and found that, while robust, it was very computationally time-intensive. Instead, in this work we solve for the condensate wave function by direct
minimization of the energy functional (5.3) using the conjugate gradients algorithm [155], whichtakes typically only a few seconds on a PC, whereas the imaginary-time evolution method takes on
the order of tens of minutes, by comparison. We describe the conjugate gradients algorithm in appendix E, but note here that the DHT of a function with angular dependence eikϕ requires defining
the radial grid points ρi in terms of the zeros of the kth order Bessel function, which we refer to
1

Available in MATLAB through the function ode45

72
from here on as grid of order k. Additionally, using such a radial grid allows for the application ofa highly accurate quadrature-like numerical integration, which we discuss in appendix D.
Once the ground state condensate wave function is obtained, we use it in solving the BdG
equations (3.34). In the trapped, cylindrically symmetric system, the Bogoliubov quasiparticles
can be characterized by a definite vorticity m, just as the condensate is characterized by a vorticity
s. Thus, we writeui (x) → ui (ρ, z)eimϕvi⋆ (x) → vi⋆ (ρ, z)e−imϕ .

(5.6)

Now, we can calculate a complete set of quasiparticle excitations (up to some energy cutoff determined by the numeric grid resolution) by specifying a set of quantum numbers m and solving the
BdG equations for each of them. Written explicitly, and in the dimensionless units, these coupled
BdG equations read



1
1 2
ωi ui (x) = − ∇2 +
ρ + λ2 z 2 − µ + (N − 1)g|φ0 (x)|222
Z
1 − 3 cos2 θx−x′  ui (x)+ (N − 1)gd dx′ |φ0 (x′ )|2|x − x′ |3
|
{z
}
a

+ (N − 1)g|φ0 (x)| ui (x) + (N − 1)gd

Z

+ (N − 1)g|φ0 (x)|2 vi (x) + (N − 1)gdZ2

1 − 3 cos2 θx−x′
φ0 (x)
dx′ φ⋆0 (x′ )ui (x′ )|x − x′ |3
{z
}
|
b

1 − 3 cos2 θx−x′
φ0 (x),
dx′ vi (x′ )φ0 (x′ )|x − x′ |3
{z
}
|
c

(5.7)

73


1
1 2
−ωi vi (x) = − ∇2 +
ρ + λ2 z 2 − µ + (N − 1)g|φ0 (x)|222
Z
1 − 3 cos2 θx−x′  vi (x)+ (N − 1)gd dx′ |φ0 (x′ )|2|x − x′ |3
{z
}
|
d

+ (N − 1)g|φ0 (x)|2 vi (x) + (N − 1)gd

Z

+ (N − 1)g|φ0 (x)|2 ui (x) + (N − 1)gdZ

1 − 3 cos2 θx−x′ ⋆dx′ φ0 (x′ )vi (x′ )φ0 (x)|x − x′ |3
{z
}
|
e

1 − 3 cos2 θx−x′ ⋆dx′ ui (x′ )φ⋆0 (x′ )φ0 (x).
|x − x′ |3
{z
}
|

(5.8)

f

For a stationary condensate with s = 0, the only relevant phase is the global phase which, underU (1) symmetry, is arbitrary. Thus, as we discussed in section 3.2, Eqs. (5.7) and (5.8) can be
reduced to the form (3.34) in this case. For condensates with s 6= 1, however, no such reduction ispossible and the BdG equations must be solved as written.
The right-hand sides of Eqs. (5.7) and (5.8) take the form of a non-sparse functional matrix
operation that, for a grid of size Nρ × Nz , has (2 × Nρ × Nz )2 elements. To solve these equations,
we employ the iterative Arnoldi diagonalization algorithm.2 In this process, all functions must be
sampled on the same spatial grid. However, because the integrands in the ddi terms have, in general,
different angular dependence, interpolation to and from different radial grids must be performedin the diagonalization process. We develop a straightforward algorithm for such interpolation,
which we describe in appendix F. In Eqs. (5.7) and (5.8), the ddi terms are labeled and thecorresponding grid orders that are necessary to calculate the corresponding Hankel transforms are
given in table 5.1. Note that, for s = 0, all exchange terms can be calculated on a grid of order m.
In practice, we interpolate all quantities onto a grid of order m when solving the BdG equations,which is the order that is necessary to calculate the kinetic energy operations. The same technique
is used when minimizing the energy functional (5.3), where all terms in are sampled on a grid oforder s.
In addition to specifying quantum numbers for the projection of angular momentum of thecondensate and its elementary quasiparticle excitations, we can characterize these modes as being
2

Available in MATLAB through the functions eigs

Term
a
b
c
d
e
f

Order k (eikϕ )
0
m−s
m+s
0
m+s
m−s

74

Table 5.1: Orders of Hankel transforms necessary to calculate the dipole-dipole interaction terms
in the cylindrically symmetric Bogoliubov de Gennes equations (5.7) and (5.8).

either even or odd with respect to reflection on the z = 0 plane. While we find that, for all s
considered here, the condensate mode possesses even symmetry (odd symmetry is always higher inenergy), we can use odd and even FFT algorithms (fast sine and cosine transforms, respectively)
when computing the kinetic energy and ddi energy to enforce that the excitation modes possess
such symmetry. However, we find that the even excitations are always the relevant excitations to
consider with regards to dynamic stability of the condensate.

5.1.1

Modified Momentum-Space Dipole-Dipole Interaction

By handling the ddi terms in momentum-space, effective “image” condensates are producedby the periodic nature of the discrete transform algorithms. One can easily see this effect byconsidering the basic discrete Fourier transform of a function fn defined on a grid with N evenly
spaced points indexed by n = 1, 2, . . . , N ,
f˜m =

N
−1
X

n

fn e−2πim N .

(5.9)

n=0

It is clear from this expression that taking n → n + N , equivalent to shifting the function by a grid
length, leaves the transform unchanged. Thus, because the ddi is long-range, the system underconsideration “feels” not only the local mean-field but also the ddi mean-fields from the unphysicalimage condensates.
We handle this problem by introducing a spatial cutoff to the ddi. For approximately sphericalsystems, the cutoff can be imposed by multiplying the real-space interaction by Θ(r − R), where75
R is the radius of a sphere outside of which the ddi is set to zero and Θ(x) is the Heaviside step
function of x [54]. In this case, the resulting momentum-space ddi has the analytic form [102]



4π 2
sin Rkcos RkRṼd (k) =d 1+3 2 2 −3 3 3
3 cos2 θk − 1 .
(5.10)
3
R k
R k

For very oblate, or pancake-shaped systems, the spherical cutoff must be set by the radial size ofthe condensate. In this case, the axial extent of the grid must be very large to ensure there is nofalse periodicity in this direction. However, for these oblate systems, an axial cutoff can be imposedby multiplying the real-space interaction by Θ(|z| − Z), where Z is the cutoff length of the ddi in
the z-direction. The resulting momentum-space ddi in this case has the form [102]ṼdZ (k) =



4π 2
d 3 cos2 θk − 1 + 4πd2 e−Zkρ sin2 θk cos Zkz − sin θk cos θk sin Zkz .
3(5.11)

To determine the cutoff R or Z, we consider a numeric grid in cylindrical coordinates of extent [0, P ]in the radial, or r-direction and [−A, A] in the axial, or z-direction. Additionally, suppose that
our condensate density is non-zero only for r < r0 and z < z0 . To ensure that no true interactions
are lost and all false interactions (from the image condensates) are eliminated, we require thatR > 2 × max [r0 , z0 ] for the spherical cutoff and Z > 2z0 for the axial cutoff. This also enforcesthat the extent of the grid be at least A, P > 2R for the spherical case and A > 2Z for the axial
case. These rules were tested in [102], where an improvement by a factor of at least 10−2 in therelative error was shown in the calculation of the ddi energy over using the uncorrected ddi.
The spherical and axial cutoffs are particularly useful due to their analytic forms. A cylindrical cutoff may be used as well, though the momentum-space form of the interaction must be
computed numerically in this case. In terms of the radial cutoff R and the axial cutoff Z in thecylindrical coordinates ρ and z, respectively, the momentum-space ddi takes the form [156]
4π 2
d 3 cos2 θk − 1 + 4πd2 e−Zkρ sin2 θk cos Zkz − sin θk cos θk sin Zkz
3
Z Z
Z ∞
ρ2 − 2z 2
2
dz cos (kz z)J0 (kρ ρ)
− 4πd
ρdρ
5 ,
0
R
(ρ2 + z 2 ) 2ṼdC (k) =(5.12)

where J0 (x) is the zeroth-order Bessel function. We employ such a cutoff for the calculations inchapter 8.

76

5.2

Rotationless (s = 0) Dipolar Bose-Einstein Condensate
The condensate wave function and the BdG equations were first calculated exactly (in theabsence of an s-wave scattering length), in the way described in the previous section, in [102]and [30]. The key results therein are that i) for all aspect ratios, there exists a critical ddi strength
beyond which the condensate becomes dynamically unstable; and ii) for certain ddi strengths andtrap aspect ratios, the dipolar BEC exhibits biconcave structure where the region of maximum
density exists not in the center of the trap where the trapping potential is minimum, but ina ring about the center of the trap, lying in the x-y plane. These features are shown in the
structure/stability diagram in figure 5.1. The colored regions indicate dynamical stability, while
the pink “islands” indicate the parameters for which biconcave structure is present. With as = 0,
the dipolar BEC in the cylindrically symmetric trap is fully characterized by the trap aspect ratio
λ and the dimensionless parameter D,
M d2
D = (N − 1) 2.
~ aho

(5.13)

An isodensity plot of a biconcave condensate at λ = 8 and D = 35 (showing a surface of equal
density) is shown by the inset in figure 5.1. Because of its resemblance to the cellular structure,
the biconcave dipolar BEC structure is sometimes referred to as a “red blood cell.” While it may
not be surprising that such density profiles exist in dipolar BECs, as the ddi is long-range and canforce the dipoles to the edges of the trap (in the radial direction, in which the ddi is repulsive), it
is quite surprising that these biconcave structures only occur in certain islands in the parameter
space. We find that simulations of the ballistic expansion of a biconcave dipolar BEC show thatits pronounced structure is not preserved in time-of-flight (TOF); it is however observable in otherindirect ways, which we discuss in chapter 6. As an example of this, we consider a dipolar BEC with
D = 25 in a trap with aspect ratio λ = 7, where biconcave structure is present. We perform direct
numeric simulations of the time-dependent GPE (details in chapter 6), and see the biconcave featurecompletely vanish after the condensate is expanded in free space. Images from this simulation are
shown in figure 5.2.

77

Figure 5.1: Structure/stability diagram of a dipolar BEC in a cylindrically symmetric harmonic trap
as a function of trap aspect ratio λ and ddi strength D. All shaded areas are dynamically stable,
while all white (unshaded areas) are dynamically unstable. The pink “island” regions show where
dipolar BECs with biconcave structure are predicted to exist. An isodensity plot of a biconcave
condensate is shown by the inset. Figure reprinted with permission from [30].

Regarding stability, it is also interesting to note that the biconcave islands form discontinuous
cusps along the stability threshold. In fact, there exists a double instability near λ = 7, where thedipolar BEC goes unstable with increasing D, then restabilizes in the biconcave region. On a larger
scale, the nature of the dipolar BEC instability changes drastically from prolate and spherical traps
to oblate, pancake-shaped traps. For the former, the instability is induced by the quadruple surface
mode going soft as the ddi becomes sufficiently strong so that its attractive component overwhelms
all other energies in the system. The softening of this quadruple mode can be interpreted as the
dipolar BEC elongating in the direction of the dipole polarization, as this configuration minimizes
the energy of the ddi. Such an instability is shown schematically in figure 5.3(a), where the solidline represents a cross-section of the spherical trap in the x-z plane and the shaded region represents
the condensate density. Such instability is analogous to the phonon instability in the homogeneous
dipolar BEC, as discussed in section 4.1. Thus, collapse is global for these more prolate andspherical geometries. However, in this case it is interesting to note that while the contribution to78

Figure 5.2: Integrated axial density of a dipolar BEC in a trap with aspect ratio λ = 7 and ddi
strength D = 25. For the sake of example, we consider ωz = 2π × 100 Hz. The left panel shows thetrapped stationary condensate with biconcave structure, and the right panel shows the condensate
after it has expand in free space after 1.6 ms. The biconcave structure has clearly vanished in theballistic expansion.

the full energy per particle from the ddi may be negative, the presence of the trap in the z-direction
stabilizes the gas at small but finite D.
For pancake-shaped traps, the instability is induced by a completely different type of excitation. In this case, the instability corresponds to the softening of discrete quasiparticle excitations
that have nodal structure on the same length scale as the free roton in the homogeneous quasi-2Dpdipolar BEC, with wavelengths ∼ 2πaz , where az = ~/M ωz is the axial harmonic oscillator

length. Indeed, while the dipolar BEC is significantly more stabilized in pancake-shaped traps,
instability still occurs for sufficiently large ddi strength, just as in the quasi-2D geometry, due tothe roton mechanism. This discrete roton instability can be thought of in the following way. While
the tendency for the condensate to elongate in the direction of polarization is still present in oblate
traps, the strong trapping in the direction of polarization prevents such behavior. Instead, thiselongation occurs locally, as is shown schematically in figure 5.3(c), on a length scale governed by
the axial harmonic oscillator length, and in turn by the roton wavelength.
As an example of this discrete roton instability, we plot the m = 0 excitation spectrum of ans = 0 dipolar BEC in a trap with aspect ratio λ = 17 (where no biconcave structure is present at
the stability threshold) in figure 5.4(a). Here, one sees a relatively high-energy mode near D = 0
that decreases in energy as D is increased and, at the stability threshold, softens and develops a

79

Figure 5.3: A schematic representation of the collapse of a dipolar BEC in various trap geometries.
In each case, the solid black line represents the a trap isosurface, the grey shaded region represents
the condensate density and the black arrows represent the dipolar polarization. In (a), the instability is due to ddi attraction as the condensate extends to a more prolate shape. In (b), the same
collapse mechanism occurs, though higher densities are achievable with tighter axial confinement.
In (c), or very oblate traps, the instability is due to local density fluctuations brought on by thesoftening of a roton-like mode. Figure courtesy of John Bohn.

non-zero imaginary excitation frequency as it becomes degenerate with its corresponding “negativenorm” mode. In this figure, the grey region (ω < 0) corresponds to the negative norm modes
that, for the rotationless dipolar BEC, have energy eigenvalues that are equal and opposite of thepositive-norm modes. As discussed in section 3.3, it is the degeneracy of the positive-norm and thenegative-norm modes that ultimately results in the imaginary BdG energy eigenvalues, and thus
the dynamic instability. Because the symmetry discussed in section 3.3 holds for the rotationlesscondensate, the negative norm and the positive norm modes always have equal and opposite energy
eigenvalues, and can thus only become degenerate at ω = 0.
We plot the radial profile of the discrete m = 0 roton mode near the instability threshold,marked by a red circle in figure 5.4(a), in figure 5.4(b). Here, we see clear radial oscillations with awavelength ∼ 1.5aho . For a trap with aspect ratio λ, the radial harmonic oscillator length can be√
written in terms of the axial harmonic oscillator length as aho = aρ = λaz , so the discrete roton√
wavelength is ∼ 1.5 17az ≃ 2πaz , which is precisely the roton wavelength discussed in section 4.2.
In figure 5.4(a), one also sees a mode with ω = 0 for all D. This is the Goldstone mode, which is anu(ρ, z = 0) + v ⋆ (ρ, z = 0) (arb. units)

80
0.04

b)

0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
0

2

4

6

8

10

12

ρ/aho

Figure 5.4: a) The real part of the Bogoliubov de Gennes spectrum of the rotationless (s = 0)dipolar BEC in an oblate trap with aspect ratio λ = 17. The grey region marks ω < 0, where themodes have negative norm. The discrete roton is seen as the mode that decreases in energy until it
softens at D ≃ 181.2. b) The radial profile of the discrete roton mode marked by the red circle in(a). Here, the wavelength is seen to be approximately 2πaz , which is the wavelength of the rotonin the quasi-2D dipolar BEC.

unphysical, long-wavelength excitation corresponding to the condensate, and appears necessarily
due to the U (1) symmetry of the BEC [97, 98]. Because of its relationship to the condensate, theGoldstone mode always appears in the m = 0 spectrum. The radial breathing mode is also seenin this spectrum as the positive energy mode that has energy ω/ωρ = 2 at D = 0, and maintains
approximately this energy for all stable values of D. Not visible in this figure is the corresponding
axial breathing mode with corresponding energy ω = ωz at D = 0. Though we do not show them = 1 spectrum here, the so-called Kohn mode can be seen in the m = 1 spectrum, corresponding
to the “sloshing” of the center of mass of the condensate [157]. Thus, the Kohn mode frequency
does not depend on interaction strength and has ω/ωρ = 1 for all D.
Recall our result from the discussion of the quasi-2D dipolar BEC, that the roton instability
occurs at some critical ddi strength or, equivalently in the mean-field description, at some criticaldensity. Thus, we expect the roton instability to originate in the region of maximum density. For
dipolar BECs without biconcave structure, such as the one in the trap with λ = 17 discussed
previously, this region occurs at the origin r = 0, and the purely radial (m = 0) roton is responsible
for the instability. In biconcave dipolar BECs, however, the region of maximum density exists in

81a ring with a circumference that is typically on the order of many roton wavelengths. Indeed, therelevant mode for characterizing the instability of a biconcave dipolar BEC is a discrete angularroton, with m ≥ 2 [30]. In chapter 6 we show that such angular instability translates into angularcollapse in the real-time dynamics of the condensate, which in turn provides a clear experimental
signature of the biconcave structure. There, we also go into more detail regarding the angular rotonin the biconcave dipolar BEC.
At this point, it is instructive to compare the dipolar system to a BEC with attractive contact
interactions. Consider a trapped, rotationless BEC with attractive contact interactions (as < 0)and no ddi. For such a system, there always exists a critical particle number above which thecondensate goes unstable, with preference to collapse in the region of maximum density at thecenter of the trap [127, 128, 158]. Indeed, in the attractive, non-dipolar BEC, the instability has
this global character in prolate, spherical and oblate trapping geometries. This is in stark contrast
with the dipolar BEC, where the instability exhibits local character in the latter case. However,stirring the attractive condensate into a vortex state serves to stabilize the system by introducing
a kinetic energy component due to angular momentum that offsets the interparticle attraction. So,
in general, the vortex will sustain a larger number of particles than the non-vortex state, and ismore stable.

5.3

Singly-Quantized Vortex
Now, consider the effects of the ddi on a condensate with a single vortex core, or a dipolar

BEC with s = 1 [159]. The conditions for the generation of such a dipolar BEC vortex state arestudied in Ref. [160]. In the presence of a singly-quantized vortex, the region of high density isforced away from the center of the trap due to the zero-density of the vortex core. Depending on the
aspect ratio λ of the trap, this either serves to stabilize (for smaller λ) or destabilize (for larger λ)the dipolar BEC. For smaller λ, the vortex core simply breaks the prolate shape of the condensate
along the direction of polarization, eliminating much of the attractive dipole-dipole interactionin this direction and thus increasing the energy due to interactions. Conversely, for larger λ the82vortex increases the density in the periphery of the core and thus encourages local collapse. Just as
the roton wavelength is set by the the confinement length in the direction of polarization (z), thelocal density fluctuations occur at the same length scale regardless of the trap geometry. Widening
the trap radially while keeping the axial trapping frequency fixed makes more room for regions ofdensity fluctuations instead of enlarging the existing regions. This marks a clear and importantdistinction between the dipole-dipole and contact interactions. An additional consideration relevantto the stability of dipolar BECs with a vortex are the excitations of the vortex core itself [161]. As
we will see below, these excitations are unlikely to play a role in destabilizing the vortex in oblate,or even mildly prolate, traps.
The solutions of the BdG equations characterize the stability of s = 1 dipolar BECs. The
global thermodynamical instability of s = 1 dipolar BECs is seen as a negative norm BdG modewith m = 1 and positive energy for all trap aspect ratios and dipolar interaction strengths. Thismode corresponds to the system’s decay into the energetically favored rotationless ground state,
just as for BECs with purely contact interactions. The component of the mode with angulardependence e−i(m−s)ϕ = 1 is in this case rotationless, capturing the symmetry of the s = 0 groundstate. Since this mode populates the core of the vortex, it is referred to as a core mode. However, at
ultracold temperatures, thermodynamical stability is less relevant in characterizing the stability ofa condensate since there needs to be some thermal processes acting on the system to dissipitavely
drive it into a lower energy state. We therefore disregard thermodynamical instability in thefollowing.
The dynamically stable region where all of the BdG modes for the dipolar BEC with a singlyquantized vortex have purely real energy eigenvalues is shown by the colored portion of Figure 5.5.
The dashed line in this figure marks, for a given λ, the D below which we find a local minimumof the GP energy functional by using our reduced 2D algorithm. We find that, for all λ, the Dabove which the GP energy functional has no minimum corresponding to an s = 1 ground state
and the D at which the BdG spectrum begins to possess imaginary energy, denoted Dcrit , arenever equal. Indeed, dynamical instability occurs for values of D at which the GPE has a solution.

83

Figure 5.5: The black dashed line marks the maximum dipole strength D, for a given trap aspectratio λ, above which the cylindrically symmetric Gross-Pitaevskii energy functional has no minimum
corresponding to the s = 1 dipolar BEC. The solid line marks a more restrictive stability line,
determined by the onset of dynamical instability, signaled by the emergence of an imaginary energy
in an excitation mode. The pink (darker) region represents where radial oscillations with local
minima are observed on dynamically stable states. The inset is an isodensity surface plot of ans = 1 dipolar BEC at the point in parameter space indicated by the arrow. The ripples in thedensity are explained in section 5.5.

This is because in the 2D minimization of the vortex-state energy, perturbations that break thes = 1 symmetry are not allowed, and are only examined later with the BdG equations. Using afully 3D calculation, we check the accuracy of Dcrit for various trap aspect ratios by time evolving
the condensate wave function with an initial random perturbation. The Dcrit that we calculate
using the 3D algorithm, corresponding to the D at which we observe collapse under time evolution,agrees with the Dcrit that we calculate by finding imaginary energy eigenvalues in the BdG spectrum
using our 2D algorithm. The pink (darker) region in figure 5.5 represents the region where we finddynamically stable s = 1 ground states having radial ripples with local minima, as illustrated by
the inset. The origin of such a feature is explained in detail in section 5.5.
Similar to the rotationless case, we find that s = 1 dipolar BECs possess imaginary energy
eigenvalues in their BdG spectrum only when two modes of opposite norm are degenerate with each84

6Im[ω/ωρ ]

4

a)

b)

2
0
m=1
m=2
m=3
m=4
m=5

−2
−4
−6

8

12

D

16

20

110

120

130

D

Figure 5.6: a) The imaginary part of the BdG excitation spectrum for a number of m values for a
dipolar BEC in a trap with aspect ratio a) λ = 2 and b) λ = 15. For λ = 2, the m = 2 modes clearly
develop imaginary energy eigenvalues at a D smaller than the other modes, defining Dcrit ≃ 8 forthis aspect ratio. For λ = 15, the m = 4 modes develop imaginary energy at a D smaller than theother modes, defining Dcrit ≃ 108 for this aspect ratio.

other. At all aspect ratios, we observe, for some finite value of D, two modes with opposite norm
approach and then go degenerate with each other at Dcrit . At the point of degeneracy, the modes
develop equal and opposite imaginary energies, signifying dynamical instability. If two modes thathave the same norm approach each other, they undergo an avoided crossing instead of becoming
degenerate. However, because we have s = 1 in this case, the energies of the positive norm andnegative norm modes are not equal and opposite, as is the case for s = 0.
For a BEC with pure contact interactions in the s = 1 vortex state, the mode that defines theonset of dynamical instability is independent of the trap aspect ratio λ. Positive contact interactions
ensure dynamical stability while negative contact interactions (for λ & 0.3) bring about a dynamical
instability due to an m = 2 mode [162]. This holds true for these systems no matter how oblate thetrap. The case for a dipolar BEC, however, is quite different. Figure 5.6 illustrates the imaginaryparts of the BdG spectrum for m = 1 − 5 for dipolar BECs in traps with aspect ratios λ = 2and λ = 15. Where these imaginary energies are zero, from D = 0 to Dcrit , the condensates are

85

Figure 5.7: Real part of the BdG spectrum for (a) m = 1 and (b) m = 2 modes of a purely
dipolar BEC in a trap with aspect ratio λ = 2. In both cases, a negative norm mode with positiveenergy, shown by a dashed line, goes degenerate with a positive norm mode at Dcrit , above which
the modes develop non-zero imaginary parts and the condensate is dynamically unstable. For them = 1 case in (a), the negative norm mode has positive, non-zero energy for all D, signifying global
thermodynamic instability of the s = 1 state. The Kohn mode is seen as the mode with ω/ωρ = 1
for all D.

dynamically stable. Notice that for λ = 2, an m = 2 mode develops non-zero imaginary energy
at a D well below the other modes, defining Dcrit for this aspect ratio. However, an m = 4 mode
serves to define Dcrit for λ = 15.
In figure 5.7, we show the real parts of the BdG spectrum for the m = 1 (a) and m = 2 (b)modes in a trap with aspect ratio λ = 2. In both cases, we see a negative norm mode (dashed line)
go degenerate with a positive norm mode at Dcrit , exactly where these modes develop non-zero
imaginary energy eigenvalues. For the m = 1 case in figure 5.7(a), the negative norm mode has
positive, non-zero energy for all D, signifying the global thermodynamic instability of the s = 1
state. Additionally, the Kohn mode is seen in this spectrum as the mode with ω/ωρ = 1 for all
D. As forementioned, this Kohn mode corresponds to a “sloshing” of the center of mass of thecondensate.
Unlike BECs with contact interactions, modes with different m quantum numbers serve todefine Dcrit at different aspect ratios for dipolar BECs. For moderate trap aspect ratios (suchas λ = 2), an m = 2 mode defines Dcrit for the dipolar BEC, similar to the case for contact

86interactions. However, as the trap aspect ratio is increased to more oblate shapes, modes withlarger m quantum numbers develop imaginary energy eigenvalues at smaller values of D than them = 2 mode. Figure 5.8(a) illustrates this by plotting the differences between the D’s at which
the BdG modes with different angular symmetries first develop imaginary energy eigenvalues, andDcrit , as a function of λ. Thus, for a given λ the lowest curve identifies the symmetry of the unstablemode. For 6 . λ . 12, an m = 3 mode defines Dcrit while for larger aspect ratios, an m = 4 mode
defines Dcrit . Although it is not shown here, we find that at even larger aspect ratios the vortex
decays into still higher m-modes.
The relevance of the m-dependent dynamical instability is that the ddi leads a BEC toinstability locally and at a fixed length scale, the wavelength of which is determined by the axial
harmonic oscillator length. We find that, at the onset of imaginary energy, these modes have radial
nodal spacings very similar to that of the roton in the rotationless dipolar BEC, namely λ/2 ∼ πaz .
The angular dependence of these modes behaves in the same way. Increasing the trap aspect ratio
λ decreases the ratio az /aho , so more radial nodes, fixed by az , can fit into the condensate forlarger λ. In the same way, more angular nodes can fit into the condensate, therefore bringing about
dynamical instability by modes with larger m quantum number, and hence more angular nodes.
All of the previously discussed BdG modes that we identify as being responsible for dynamical
instability are axially symmetric and nodeless in z. Modes that break this axial symmetry cancorrespond to vortex excitations, where the vortex core itself may tilt or bend, and have beentermed “kelvon” modes. Ref. [161] reports that, for a singly-quantized vortex in a dipolar BEC
that is otherwise spatially homogeneous, the condensate is dynamically unstable to a kelvon mode
when an external periodic potential is applied along the direction of the vortex. We find that, ina harmonically trapped dipolar BEC, a mode with a single node at z = 0 determines Dcrit for
λ . 0.28. Modes of this type might therefore correspond to a kelvon-instability in prolate traps,
but we leave these considerations for future work.
As was done in Ref. [162] for self-attractive BECs in the singly-quantized vortex state, we
check these BdG results for the dynamic instability by performing fully three-dimensional time-

87
30
25

Dm - Dcrit

m=0
m=1
m=2
m=3
m=4

b)

20
15

1.2
1
0.8
0.6

10

0.4

5

0.2

0
0

10

λ

20 0

10

λ

h̄2 /M (acrit - am )

a)

0
20

Figure 5.8: a) The difference between the D at which BdG modes with different m quantum
numbers first develop imaginary energy eigenvalues and Dcrit , the smallest D at which any modedevelops an imaginary energy eigenvalue. Modes with larger m serve to define Dcrit for more oblate
traps. b) The same, but for negative contact interactions instead of dipole-dipole interactions. An
m = 2 BdG mode is always the first to develop an imaginary energy eigenvalue for this case, except
for trap aspect ratios λ . 0.3, for which an m = 1 mode plays this role.

dependent simulations of an s = 1 dipolar BEC where D is chosen to be just above Dcrit , enabling
us to go beyond the small deviations from the stationary vortex state and see the actual process ofcollapse. Initializing the simulations with random noise, we observe collapse, at all aspect ratios,with an angular symmetry corresponding to the m quantum number of the mode that first develops
an imaginary energy eigenvalue. Our methods for such time-dependent simulation are discussed inchapter 6.

5.4

Doubly-Quantized Vortex
The dynamical instability of condensates with doubly-quantized vortices and purely contact

interactions has been studied extensively [163, 164, 165, 105]. These studies report windows of
positive scattering length where the BECs are dynamically unstable to an m = 2 BdG core mode,
as well as dynamical instability for all values of negative scattering length due to an m = 2 mode.
Knowing that the mean-field in a dipolar BEC can be engineered to be more attractive or repulsive

88
for smaller or larger trap aspect ratios, respectively, we investigate the presence of these featuresin s = 2 dipolar BECs. When the harmonic trap is more spherical, the dipoles are free to stackvertically, resulting in an attractive interaction in this direction. However, in pancake shaped traps
the dipoles create a more repulsive mean-field. Thus, for larger trap aspect ratios dipolar BECs
are more self-repulsive than for smaller aspect ratios, mimicking the mean-field of condensates withrepulsive contact interactions. As an example, we calculate the contribution of the dipolar meanfield to the energy of a dipolar BEC in a trap with aspect ratio λ = 2 and with λ = 15 for a fixed
D = 10. In the λ = 15 trap, we find that this contribution is about five times larger than in the
λ = 2 trap.
Indeed, for a dipolar BEC in a trap with aspect ratio λ = 2 we find that there exists an
m = 2 mode with a complex energy eigenvalue for all values of D. However, for λ = 15 wefind that there are windows in D where an m = 2 BdG mode has a complex energy eigenvalue,
while this same mode has purely real energy outside of these windows, as illustrated in Figure 5.9.
For trap aspect ratios λ . 7.5, there are no windows of dynamical stability and the condensate
is dynamically unstable for all D. However, windows of dynamical stability appear for aspectratios λ & 7.5 and continue for larger λ. As is reported in Ref. [105], we find that there is an
m = 2 core mode with negative norm and positive real energy that increases monotonically as itgoes successively degenerate with positive norm modes as D is increased to create the windows ofdynamical instability. This mode represents the s = 2 condensate’s instability to splitting into a
condensate with two singly quantized vortices. The core mode is thermodynamically unstable forall values of D and is only dynamically unstable for the windows shown in Figure 5.9.

5.5“Perturbed” Dipolar Bose-Einstein Condensate
Here, we set out to explain the origin of the radial “ripples” seen in the density of the s = 1

dipolar BEC, as shown by the inset in figure 5.5. It has previously been suggested that boundaries in superfluid 4 He, including vortex cores, should give rise to radial density oscillations whose
length scale is characteristic of the roton wavelength [166, 167, 168]. More recently, calculations

89

a)

m=1
m=2
m=3
m=4
m=5

b)

Im[ω/ωρ ]

0.6
0.4
0.2
0
0

10

D

20

0

50

D

100

Figure 5.9: The positive imaginary part of the energy eigenvalues for a doubly quantized (s = 2)dipolar BEC in a trap with aspect ratio a) λ = 2 and b) λ = 15. For λ = 2, an m = 2 mode
possesses imaginary energy for all values of D, signifying a dynamical instability for all D. However,for λ = 15 there are windows of dynamical stability as the m = 2 mode alternates having and nothaving imaginary energy for different values of D, as is the case for a purely repulsive s = 2 BEC.

of vortex states in a dipolar BEC in a highly oblate trap with λ ∼ 100 have exhibited similarradial structures [159], raising the question of the relation between these structures and rotons inthis system (progress has also been made in the understanding of the vortex state in a dipolar
BEC in the Thomas-Fermi regime. However, in this regime the vortex does not exhibit a radial
ripple [169, 160]).
We first note that the radial structure of the s = 1 vortex will be the same as that of arotationless BEC in a trap with a central potential representing the centrifugal force. Indeed, byinserting the vortex form (5.2) into the GPE (5.4), one obtains


1 ∂2
s2
1
1
+
+ (ρ2 + λ2 z 2 )− ∇2ρ −
2
2 ∂z 2 2ρ2 2
Z′ ∗ ′ ′′′ ′+ (N − 1) dx ψ (ρ , z )V (x − x )ψ(ρ , z ) ψ(ρ, z),µψ(ρ, z) =(5.14)

where V (x − x′ ) is the two-body interaction potential (3.61). Here, the centrifugal potential s2 /2ρ2
is responsible for the vortex core (i.e., vanishing density at ρ = 0). It is natural to hypothesize thatthe appearance of the ripple in the vortex structure is related to a roton mode which is excited by
the centrifugal potential of Eq. (5.14). This raises the interesting question, could such a ripple also

90
be observed in the ground (non-vortex) state of a dipolar BEC perturbed by an external potential
at the center of the trap? Such a perturbation may be realized experimentally by applying a bluedetuned laser along the trap axis. The origin of such a perturbing potential, being the AC Stark
shift, is explained in section 2.3.1. We assume that the perturbing potential will take the form
U ′ (x) = A exp (−ρ2 /2ρ20 ), where A is the height of the Gaussian and ρ0 is its width. As a test case,consider a dipolar BEC of atomic 52 Cr that, as wee see in table 2.1, has a magnetic dipole momentd = 6µB . Additionally, 52 Cr has an optical dipole transition of 427.6 nm from its 7 S3 ground state
to its 7 P3 excited state [170]. A blue-detuned laser around this transition could be used to createa repulsive potential with no significant absorption and with a minimal spatial extent of about ahalf-wavelength, 210 nm [61].
In sufficiently oblate traps, s = 0 dipolar BECs exhibit radial density oscillations in thepresence of such Gaussian potentials. Fig. 5.10 illustrates the radial profiles of s = 0 dipolar BECs
in a harmonic trap with aspect ratio λ = 17 and with a Gaussian potential having A = ~ωρ andρ0 = 0.2 aho . To give a concrete example, for 52 Cr atoms in a harmonic trap with radial frequencyωρ = 2π × 100 Hz, this translates to having a beam width of ρ0 = 280 nm. In this trap, aninteraction strength of D = 181.2, very near the point on instability for a s = 0 dipolar BEC in a
trap with the above aspect ratio, may be achieved with ∼ 104, 000 52 Cr atoms. It is seen that inthis case even a small Gaussian perturbation makes a dramatic change in the dipolar BEC density
profile. The radial oscillations near Dcrit are much more pronounced than for a smaller dipolar
BEC with D = 100. This is suggestive of the roton’s presence in this structure, since the roton is
expected to emerge with increasing dipolar interaction strength (i.e, increasing density for a fixed
dipole moment) [28, 168, 102]. As was shown in the previous section, the roton mode undergoes
a significant decrease in energy with increase in D until it achieves zero energy at Dcrit , marking
the point of dynamical instability for the s = 0 condensate. Beyond this Dcrit , the roton energy
is purely imaginary. Examining the nature of the roton itself within BdG theory tightens up its
relationship with the observed structure discussed above. Additionally, we note that, for the 52 Cr
BEC, the length scale of the radial oscillations is ∼ 2µm. This is in comparison to the length scale

91
0.2

40

φ 0 (z = 0)

0.15

20

0.1

0.05

0
0

Trap Potential (h̄ωρ )D = 100.0
D = 181.2
U(x) + U ′ (x)

0
2

4

6

8

ρ/aho
Figure 5.10: Radial profiles of the k = 0 dipolar BEC subject to the perturbing potential U ′ (x) =~ωρ exp (−ρ2 /2(.2 aho )2 ) in a trap with aspect ratio λ = 17. The red dash-dotted line represents
the trapping potential at z = 0, the black solid line represents the radial profile of the dipolar BEC
at D = 100 and the blue dotted line represents the radial profile at D = 181.2, near the point ofdynamic instability for the k = 0 dipolar BEC. The “+” signs represent the perturbation theory
results and the thin dotted lines represent the unperturbed radial profiles at the corresponding
dipole-dipole interaction strengths.

of the predicted ripple in the 4 He vortex, which is of the order of 1 Å, and has not been resolved
experimentally up to now.
For an s = 0 condensate, we saw in section 3.2 that the coupled BdG equations reduce totwo uncoupled equations, the first of which we write here asG̃F̃ |f i = ω 2 |f i,(5.15)

where Ĝ = ĤGP + Ĉ and F̂ = ĤGP + Ĉ + 2X̂, and we define G̃ = P̂ ĜP̂ and F̃ = P̂ F̂ P̂ , whereP̂ = I − |φ0 ihφ0 | is the projection operator into the space orthogonal to ground condensate wave
function |φ0 i. Recall that the eigenvector |f i is given by |f i = |ui + |vi, where {u, v} are the
Bogoliubov de Gennes eigenfunctions. In Eq. (5.15), it is understood that the linear space on
which F̃ and G̃ act, and to which |f i belongs, is orthogonal to |φ0 i. Thus, by using the projection
operator P̂ we eliminate a non-physical solution to Eq. (5.15), the Goldstone mode, with eigenvalue
zero [101]. The justification for working in this reduced linear space is that all physical excitations

92
obey hf |φ0 i = 0 [125].
5.5.1

Perturbation Theory for the Gross-Pitaevskii Equation

As forementioned, it seems natural to assume that the roton mode dominates the structureof the perturbed dipolar BEC near instability because its energy is much lower than the energiesof the other BdG modes. To explicitly demonstrate this, one needs to formulate a perturbation
theory of the GPE with respect to external potential perturbation. To do so, we begin by writinga perturbation to the trapping potential as U → U + U ′ , where U ′ is the small perturbation. The
response of the condensate wave function to this perturbation is then |φ0 i → |φ0 i + |φ′0 i. We insert
these expressions into Eq. (3.23), linearize in the primed quantities, and obtain the equationF̃ |φ′0 i = −P̂ U ′ |φ0 i.

(5.16)

To solve Eq. (5.16), we introduce a basis defined by the eigenvalue equationF̃ |ϕn i = εn |ϕn i(5.17)

and use its eigenfunction solutions to expand |φ′0 i in the |ϕn i basis. Plugging these expansions backinto Eq. (5.16) and working to first order gives the expression for the wave function perturbation,|φ′0 i = −X hϕn |U ′ |φ0 inεn|ϕn i.

(5.18)

This derivation involves the use of the orthogonality condition hφ′0 |φ0 i = 0 and the fact thathϕn |φ0 i = 0. The final expression (5.18) is formally identical to that of the usual perturbation
theory of the linear Schrödinger equation.
The connection between the BdG roton mode and the perturbative modes is clear in thelimit that the roton mode becomes degenerate with the ground state. In this limit, the rotonenergy ω goes to zero. In Eq. (5.15), this means that G̃F̃ has eigenvalue zero. Now, note that theoperator Ĝ is positive semi-definite (its lowest eigenvalue is zero, with eigenfunction |φ0 i. This isindeed the ground state, since |φ0 i is nodeless). Accordingly, the operator G̃ that, by definition,acts on the linear space orthogonal to |φ0 i, is positive definite. It then follows that any solution

Amplitude at z = 0 (×10− 3)

93
2
1
0
−1
−2

|f r ot oni|ϕ 0 i

−3
−4
0

1

2

3

4

5

ρ/a hoFigure 5.11: Radial profiles of excitations on a rotationless dipolar BEC with dipole strength
D = 181.2 in a trap with aspect ratio λ = 17. The solid blue line represents the BdG roton mode
while the red marks represent the F̃ -operator eigenfunction with eigenvalue 0, |ϕ0 i.

of G̃F̃ |froton i = 0 must also satisfy F̃ |froton i = 0. Thus, |ϕ0 i = |froton i is a solution of Eq. (5.17)
with eigenvalue ε0 = 0. Since |φ′0 i is written as an expansion in |ϕn i with weights proportional
to 1/εn , the eigenfunction |ϕ0 i with eigenvalue ε0 ∼ 0 makes a contribution to |φ′0 i that strongly
overwhelms the contributions of the other eigenfunctions. Thus, in the limit that the roton energy
goes to zero, |φ′0 i is dominated by the BdG roton mode, |froton i.
To show that |ϕ0 i becomes identical to BdG roton mode |froton i when the roton energy goesto zero, Fig. 5.11 shows the radial profiles of both of these excited modes for a rotationless dipolar
BEC with dipole strength D = 181.2 in a trap with aspect ratio λ = 17, which is very near the
point of instability. Additionally, Fig. 5.10 illustrates the accuracy with which this perturbation
theory predicts the wave function of a dipolar BEC when perturbed by a Gaussian potential, as
discussed earlier in this section.
Recall that the s = 1 solution of the GPE gave rise to a centrifugal potential in the radial
part of Eq. (5.14). This potential is constant along the trap axis and decreases quickly in the radial
direction. So, just as the Gaussian potential perturbs the dipolar BEC and gives rise to ripples on
its density profile, we expect similar behavior for trapped dipolar BECs with a centrifugal potential,

94i.e., dipolar BECs with vortex structure. To treat the centrifugal potential with our perturbation
theory, we introduce a radial cutoff that is chosen to be much smaller than the spatial extent of thevortex core itself. We find that for large λ there is good agreement between our perturbation theory
and the results of our exact calculations. Just as is the case for a Gaussian perturbing potential,
the roton mode is responsible for the rich structure observed in the s = 1 vortex state of a dipolar
BEC close to instability.

5.6

Conclusion
In this chapter, we introduced novel methods for calculating the ground condensate wave

function φ0 (x) and for solving the corresponding BdG equations (3.34) for dipolar BECs in cylindrically symmetric traps and with arbitrary vorticity s. For rotationless (s = 0) condensates, we
saw that discrete roton-like modes exist in the system, and are responsible for the dynamic instability of dipolar BECs in oblate traps. This is analogous to the roton instability in the quasi-2D
system, as discussed in section 4.2. Additionally, we saw that interesting biconcave, “red bloodcell” structures exist in dipolar BECs for certain islands in the parameter space defined by thetrap aspect ratio λ and ddi strength D, as seen in figure 5.1. Because the region of maximum
density exists in a ring about the center of the trap for the biconcave condensates, the mechanism
for dynamic instability is an angular roton.
In addition to the rotationless dipolar BECs, we systematically map the stability and structure of dipolar BECs with a singly-quantized vortex, corresponding to s = 1. Because such condensates also have rings of maximum density, but here due to the zero-density vortex core, BdG modes
with angular character are responsible for the dynamic instability of these dipolar BECs, as well.
Additionally, for sufficiently large D and λ, we find that radial ripples exist in the dynamicallystable s = 1 dipolar BECs near the stability threshold. To explain the origin of such structure, wedeveloped a perturbation theory for the GPE and applied it to dipolar BECs perturbed both by
thin Gaussian potentials centered on the trap axis (as may be created with a sufficiently focusedblue-detuned laser) and centrifugal potentials. This theory allows us to relate the radial oscillations

95
observed in the exact ground state profiles of perturbed dipolar BECs to the roton mode observedin the BdG spectrum of rotationless dipolar BECs.
As mentioned in section 5.2, we performed simulations of the ballistic expansion of a biconcave
dipolar BEC, which demonstrate that the biconcave structure is not preserved in the expansion.
In the next chapter, we revisit the biconcave dipolar BEC with the goal of finding experimental
signatures of the biconcave structure. Taking the 52 Cr BEC as a test case, we show that theangular instability predicted by the BdG theory for a biconcave condensate results in angularcollapse when the s-wave scattering length is critically lowered, and ballistic expansion of thiscollapsed cloud preserves the angular character, providing an indirect experimental signature of
the biconcave dipolar BEC.

Chapter 6
Collapse of a Dipolar Bose-Einstein Condensate

In this chapter, we explore the structure and dynamics of rotationless (s = 0) dipolar BECs
near their threshold for stability, paying particular attention to the biconcave structure that was
introduced in section 5.2. Motivated to find an experimental signature of such interesting structure,
we show that the angular roton instability associated with the presence of biconcave structure canmanifest in non-trivial, time-dependent angular collapse and subsequent expansion of the dipolar
BEC. We note that the original work presented in this chapter has been published in [37].
Collapse of dipolar BECs has been studied elsewhere, both experimentally [21, 22] with the
52 Cr BEC and theoretically [171].

Because the (magnetic) dipole moment of 52 Cr is effectively

fixed, stability can be controlled by changing the trap geometry or by tuning the s-wave scattering
length of the atoms by using a magnetic Fano-Feshbach resonance, which we discuss briefly insection 6.1. Recall from chapter 4 that the stability of the polarized dipolar BEC depends both on
the magnitude of the dipole moment, the s-wave scattering length and the geometry of the trapping
potential relative to the dipole polarization. Thus, for the fixed dipole system, the scattering lengthmust be tunable to as < −2add and as > add to fully explore the stability in both prolate andoblate geometries. Indeed, this level of control was achieved in the experiment [22], and the 52 Cr
condensate was stabilized for a variety of trap aspect ratios by tuning the s-wave scattering length toa sufficiently large value. In this experiment, instability of the condensate was triggered by reducing
this scattering length below a critical value. Strikingly, the experimentally observed anisotropic
density distribution of the collapsed cloud was reproduced well within mean-field theory and the97
Gross-Pitaevskii equation [22]. Further, Ref. [171] has explored circumstances of local collapse invarious trap geometries, contrasting approaches where the collapse is initiated by either rapid or
else adiabatic changes in the scattering length. While collapse of the 52 Cr BEC has been observed,
thus far there is no direct experimental evidence for the roton or for the local collapse of a dipolar
BEC.
In this chapter we tackle head-on the prospects for observing local collapse in a dipolar BEC.
In section 6.2 we analyze and extend the experimental result from Ref. [21]. We argue that, foroblate traps, the scattering length required to stabilize the condensate can be explained within
mean-field theory, but only if the the theory is sensitive to local collapse phenomena. Further,
the distinction between local and global collapse becomes more clear if the number of dipoles isincreased. In Sec. 6.3 we develop an understanding of a more direct measurement of condensate
collapse, following the experimental procedure of Ref. [22], which includes an expansion that allows
for imaging of the cloud. We show that angular structure in the expanded image is a direct
signature of biconcave structure. Before we proceed, we briefly discuss the physics of a FanoFeshbach resonance, as this is the method that is used to control the s-wave scattering length in
the 52 Cr experiments.

6.1

Fano-Feshbach Resonances
Unlike potential resonances and shape resonances, which are inherently single-channel phe-nomena, the Fano-Feshbach resonance is intrinsically a multi-channel phenomenon [60]. The basic
idea of a Fano-Feshbach resonance can be expressed by a simplistic two-channel model. Imagine
two scalar particles scattering in an open channel 1, where a closed channel 2 also exists and ishigher in energy that 1. For now, let the incident scattering energy be E = 0 and let there be abound state in channel 2 at energy ǫ. Such a scenario is shown schematically in figure 6.1(a). In
the scattering process, this bound state may be “visited” and, if ǫ is sufficiently small, the amountof time that the particles take to exit the scattering process can be quite large.
It turns out, this phenomenon can be directly related to the s-wave scattering length of the

98
2.5

a)

Energy (arb. units)

2

2

1.5
1

ǫ

0.5
0

1

−0.5
−1

−1.5
−2
−2.5
1

1.5

2

2.5

3

3.5

ρ (arb. units)

Figure 6.1: a) Schematic two-state system that is necessary for the appearance of a Fano-Feshbachresonance. Here, channel 1 is the incoming channel and a bound state at energy ǫ exists in channel
2. b) The s-wave scattering length in units of the background scattering length as a function of
the applied magnetic field in units of the resonance width for a magnetic Fano-Feshbach resonance,
given by Eq. (6.1).

particles. A particularly relevant case here is the magnetic Fano-Feshbach resonance (or set ofresonances) in 52 Cr. In this case, the energy ǫ can be controlled by tuning an applied magnetic
field that shifts the Zeeman levels of the atom. Thus, this magnetic Fano-Feshbach resonance allows
us to map the s-wave scattering length as in terms of an applied field B. Where B0 corresponds
to ǫ0 and ∆ is the width of the resonance, we can write a single resonance asas = abg 1 −∆B − B0



,

(6.1)

where abg is the “background” scattering length in the absence of an applied magnetic field. This
relation is plotted in figure 6.1(b). The magnetic fields for which negative scattering lengths canbe obtained are shaded in this figure. For the 52 Cr system in the experiment [22], the magnetic
Fano-Feshbach resonance that is used occurs at B0 = 589 G and has a width of ∆ = 1.7 G. While
this width is relatively narrow, it permits experimental control over the scattering length with anuncertainty ∆as ∼ 2a0 [16].

99

6.2

Local collapse: Evidence from the Stability Diagram
Thus far only one experiment has explored the stability of a dipolar BEC as a function of thetrap aspect ratio [21]. The resulting experimental stability diagram (reproduced with permission
from [21]) is presented in figure 6.2 as a plot of the critical scattering length acrit versus aspectratio λ for the 52 Cr BEC. These results represent the measurement performed on a condensate of
N = 2 × 104 52 Cr atoms. We note that the experimental results shown here are actually quite
remarkable, as they demonstrate that the relatively small dipole moments of the 52 Cr atoms havean astounding effect on the BEC stability. In this figure, we see that the dipolar BEC in a prolate
trap requires a comparatively large scattering length to achieve stability. As λ is increased, thezero-point energy in the axial direction stabilizes the dipolar BEC, and stable condensates are
possible with a smaller critical scattering length. Additionally, we see that a purely dipolar BEC(with as = 0) is achieved near λ = 10.
This figure also shows the results of two alternative numerical calculations of the criticalscattering length acrit . In one, the theoretical division between stable (shaded) and unstable (unshaded) regions of parameter space is determined by solving the GPE, Eq. (3.23), and the BdG
equations (5.7) and (5.8) exactly on a numeric grid, using the methods described in section 5.1.
A second approach, already employed as an approximation in the experimental paper [22], shows
the division between the stable and unstable regions as a dashed line. This approximation posits a
Gaussian ansatz for the condensate wave function (normalized to unity),
!2



1
−1 ρ2
z2
φ0 (ρ, z) =exp+
2ā2ho σρ2 σz2
π 3/2 σρ2 σz ā3howhere σρ and σz are the variational parameters and āho =p~/M ω̄, where ω̄ =

(6.2)
q
3

ωρ2 ωz is thegeometric mean trap frequency. Using this ansatz, the GP energy functional (5.3) is calculated
for a given āho to determine whether the energy E[σρ , σz ] has a minimum, and thus to determine
if the Gaussian condensate is energetically stable. The presence of a minimum, local or global,
corresponds to the presence of a stable ground state. A key feature of the Gaussian trial wave
function is that it always places the maximum density in the condensate’s center, i.e., it is incapable

100

Figure 6.2: The stability diagram of a dipolar BEC of N = 2 × 104 52 Cr atoms plotted as criticalscattering length versus trap aspect ratio. The points show the experimental results of [21], theshaded regions show the results of solving the GPE exactly and the dashed line shows the resultsof the Gaussian ansatz. The theoretical methods disagree as trap aspect ratio λ increases, and theexact results fit the experimental data with great accuracy. The pink (darker) regions are wherebiconcave structure is predicted on the condensate profile.

of describing local collapse. For prolate traps, the maximum density is in the center. In this casethe Gaussian ansatz and the numerical solution to the GPE agree with each other on the criticalscattering length, and they both are in good agreement with the experimental result.
Care must be taken, however, using this approximation for oblate condensates. This can beseen in the λ > 1 region of the stability diagram in figure 6.2, where the Gaussian ansatz predicts alower critical scattering length than does the exact numeric solution of the GPE. We attribute thisdifference to the ability of the methods described in section 5.1 to model local collapse. Indeed, forlarger aspect ratios we observe local collapse into roton-like modes, as we will discuss in the next
section, and as has been reported in Ref. [171]. Further, the experimental determination of acrit
tends to show better agreement with the GPE prediction than with that of the Gaussian ansatz. Weinterpret this as experimental support for the occurrence of local collapse, albeit somewhat indirect
evidence. However, the roton modes involved in collapse might have either radial or angular nodal
structure. This experiment does not make this distinction.
Within the uncertainty in the experiment, the data in figure 6.2 discriminates between the

101

add
10

as /a00−10

N = 104
N = 105
N = 106

−20

−2add
−2

10

0

2

10

10

λ
Figure 6.3: For a dipolar BEC of atomic 52 Cr, this figure illustrates the critical s-wave scattering
length (below which the dipolar BEC is unstable) as a function of trap aspect ratio λ in units of
the Bohr radius for ω̄ = 2π × 700 Hz. The blue (dotted), black (dot-dashed) and red (solid) lines
correspond to N = 104 , N = 105 and N = 106 , respectively. The lines without crossed symbols
are the results of the Gaussian ansatz and lines with crossed symbols are the results of the exact
solution of the GPE. Notice how, as N is increased, the Gaussian ansatz predicts a more stablecondensate while the exact solution predicts a less stable condensate.

two methods, but one may wish for a clearer discrimination. We therefore consider cases withvarying atom number N . The critical scattering length acrit is shown in figure 6.3 for dipolar
BECs with atom numbers of N = 104 , 105 , and 106 . For a given trap, increasing the number ofdipoles increases the relative importance of the dipole-dipole interaction (ddi), which acts to further
destabilize the condensate. Thus, as predicted by our exact numeric solutions to the GPE, acrit
increases with increasing atom number. Vice versa, the Gaussian ansatz predicts a more stablecondensate with increasing atom number. The difference between the two theoretical approaches
could then be clearly distinguished in such an experiment.
Although figure 6.3 plots only the domain of aspect ratios 10−2 < λ < 103 , it is straightforward to obtain the stability thresholds in the λ ≪ 1 and the λ ≫ 1 limits for fixed ω̄. Theselimits are usefully described in terms of the characteristic dipole length add , and correspond to
the stability threshold for the 3D homogeneous dipolar BEC for λ ≪ 1 and the stability threshold

102
for the quasi-2D dipolar BEC for λ ≫ 1. We see this by noting that in the limit of large ddi’s,
N add /āho ≫ 1, the interaction-dependent term in E[φ0 ] dominates over the other terms since it
scales with a factor of N . Thus, in this limit, the condensate is unstable if this interaction-dependent
term is negative.
Taking the limit λ → 0 corresponds to an infinitely prolate, or cigar-shaped trap. In thisgeometry, the dipolar mean-field term reduces to a simple coupling to the condensate density,R
cos2 θ
d2 φ⋆0 (ρ′ ) 1−3
φ0 (ρ′ )dx′ = −4π~2 add |φ0 (ρ)|2 /M because the ddi reduces to a delta-function|x−x′ |3in ρ for this geometry.

Thus, in the quasi-1D geometry, the total mean-field term becomes

4π~2 (as − add ) |φ0 (ρ)|2 . Similarly for the limit λ → ∞, corresponding to an infinitely oblate trap,
the dipolar mean-field term reduces to 8π~2 add |φ0 (z)|2 /M (similar to the quasi-2D case discussed
in section 4.2), giving a total mean-field term of 4π~2 (as + 2add ) |φ0 (z)|2 for this geometry. Thus,in the limit N add /āho ≫ 1, we find that the dipolar BEC is unstable when as < add for λ ≪ 1 andthe dipolar BEC is unstable when as < −2add for λ ≫ 1, in agreement with the analysis given in
section 4.2. These limits are indicated in figure 6.3, and the mean-field calculations are detailed inAppendix G.

6.3

Local Collapse: Evidence From the Collapsed Cloud
To take a closer look at the nature of collapse, it is necessary to track the collapse itself as afunction of time. This, too, has been achieved in the 52 Cr BEC experiments. These experiments
have not, however, focused directly on observing the consequences of local collapse. Here we discuss
the prospects of making such a measurement.
Briefly, in such an experiment the s-wave scattering length is altered from a value where thecondensate is stable against collapse to a somewhat lower value a < acrit . After this transition, theatoms begin their collapse into high density regions where three-body recombination takes over,ejecting atoms from the trap. The trap is generally released after some hold time, to expand thecloud for imaging. The resulting density patterns show intricate shapes and depend on details suchas whether the passage from stable to unstable is adiabatic or diabatic [171].

103
6.3.1

Modes of Instability

As forementioned, the underlying physics of the instability and collapse of oblate dipolar
BECs is determined by the softening of a roton mode. For the case at hand, however, the roton
does not emerge with increasing D, but instead emerges with decreasing as . Figure 6.4 illustrates
the mode softening for a dipolar BEC containing N = 104 52 Cr atoms in a trap with aspect ratio
λ = 8. Plotted is the energy of the excitation as a function of the scattering length as , labeledby its azimuthal angular momentum quantum number m. The solid lines depict the real parts ofthese energies, while the symbols represent their imaginary parts. As as diminishes, the energiesof these modes drop to zero, and thereafter become purely imaginary. The first such transition, at
as ∼ −0.9a0 , identifies the scattering length at which the dipolar BEC goes dynamically unstable,
since any small perturbation is capable of exciting this mode, which then grows exponentially intime. Thus an unstable condensate quickly grows high-density peaks in regions defined by theantinodes of these modes.
Figure 6.4 is a particular example illustrating the modes that contribute to instability at aparticular aspect ratio λ = 8. At this aspect ratio the condensate’s density exhibits a biconcave
shape, and so decay into angular rotons is expected. We reiterate, from section 5.2, that at all
aspect ratios where the maximum density lies at the center rather than at the periphery, the rotons
responsible for instability are always m = 0 modes that do not exhibit an angular structure. Thisconnection is essential to connecting observed angular decay circumstantially to the existence ofbiconcave structure.
However, regardless of whether the roton is purely radial or angular in nature, it leads a
dipolar BEC to instability at a fixed length scale with a wavelength ∼ 2πaz . As the trap aspectratio λ is increased, the ratio of the axial to the radial harmonic oscillator lengths, az /aρ , isdecreased, so more roton wavelengths can fit around the circumference of the condensate for larger
λ. For biconcave condensates, this results in angular rotons with larger m quantum number being
responsible for instability for larger λ, since more angular nodes can fit into the condensate in this

104

2

ω/ωρ

1.5
1

m=2
m=3
m=4
m=5

0.5
0
−1.5

−1

−0.5

0

as /a0Figure 6.4: The real and imaginary parts of the low-lying BdG modes for a condensate of N = 104
52 Cr atoms with mean trap frequency ω̄ = 2π × 700 Hz and trap aspect ratio λ = 8, plotted asa function of the s-wave scattering length as . The real parts are represented by solid lines andthe imaginary parts, developing where the real parts go to zero, are represented by markers. The
m = 3 mode, being the first to develop a nonzero imaginary energy, serves to define acrit for this
aspect ratio.

regime. For N = 104 52 Cr atoms in a trap with λ = 8 and ω̄ = 2π × 700 Hz, this mode happens tohave m = 3. Indeed, the circumference of the region of maximum particle density in this biconcave
condensate is ∼ 6πaz , or three roton wavelengths.
6.3.2

Numerics and the “Ideal Experiment”

The mode that brings about the dynamical instability determines not only the scattering
length at which the condensate will collapse, but also how the condensate will collapse as thestability threshold is crossed. Consider preparing a dipolar BEC of N = 104 52 Cr atoms just above
the stability threshold in a trap with aspect ratio λ = 8. These are the collective modes whoseenergies are shown in figure 6.4. A small jump in scattering length to a value just below acrit would
cause the condensate to go unstable by a macroscopic occupation of the m = 3 mode that has
a nonzero imaginary energy at this scattering length. The density of the condensate during thecollapse would change, on a time scale τ ∼ 2π/Im[ω], as the atoms macroscopically occupy three

x 10

-4

1

y/aho

y/aho

α=0
0

-1
-1

0

x/aho x 10 -4

1

x 10

-4

√

1

α = π/ 13
y/aho

1

0

-1
-1

0

x/aho x 10 -4

1

x 10

105

-4

√

α = π/ 3

0

-1
-1

0

x/aho x 10 -4

1

Figure 6.5: Collapsed condensate in a trap with mean frequency ω̄ = 2π × 700 Hz and aspect ratio
λ = 8 after 10.5 ms. The perturbation for the collapse is controlled to have m = 3 symmetry andthe global phases as shown in the frames. Each frame corresponds to a different value of the initial
phase α. The collapsed condensates are rotated by α/3, ensuring that the finite grid size does not
influence the small length scale dynamics of the condensate collapse.

clumps that self-attract in the z-direction.
Decay of the condensate into a roton mode with m > 0 requires breaking the condensate’s
initial cylindrical symmetry by introducing fluctuations into the mode. In an experiment this iscaused by thermal fluctuations, but in our calculation we must make this happen artificially. To
do this, we seed the condensate wave function by adding to it a small contribution of the excitedstate mode:
φ0 (~r) → φ0 (ρ, z) + 0.01e2πiα e3iφ ū3 (ρ, z),(6.3)

where e3iφ describes the basic angular variation of the m = 3 roton mode, and α is an additional
phase that will determine the overall rotation of the collapsed condensate. In the absence of a seed
like this, the numerical solution remains at its unstable equilibrium for a time long compared to
the natural lifetime 2π/Im[ω]. The apparent lifetime in this case is determined by the time before
roundoff error starts to affect the numerical time evolution of the GPE. However, once the wave
function is seeded as above, the decay occurs on the expected time scale.
After the collapse is triggered, the condensate indeed forms the three “clumps” as expected,
as seen in figure 6.5. Shown is the density of particles, as viewed in the x-y plane, i.e., looking downfrom the axis of the dipoles’ polarization. Each peak was initially seeded by a density fluctuation
at the antinode of the m = 3 excited state roton wave function. Thus the three peaks are uniformly

106
equally spaced in angle, as befits the symmetry of the mode. An angular display of this sort would
provide unambiguous evidence for nonlocal collapse. Moreover, the fact that the collapse occurredin an angular coordinate provides indirect evidence for the biconcave structure of the initial state.
While the relative positions of the three peaks in this experiment are well-defined by thesymmetry of the roton mode, there is still an overall undetermined angle of rotation of the whole
pattern. Numerically, this is set by the angle α in (6.3). Since the angular dependence of thecondensate density, with this wave function, is proportional to cos (3ϕ + α), we expect that, if
there is no unphysical dependence on the numerical grid, the collapse will occur rotated by anangle α/3 for any initial phase α. Indeed, we find that the collapse dynamics are unaffected by the√√grid, as is illustrated in figure 6.5. Here, we input the initial phases α = π/ 13 and α = π/ 3
and find that the collapsed wave function is rotated by exactly these phases times 1/3. Although
not shown here, simulations for other initial phases give the same results. Based on this ability
to reproduce the same angular pattern, but rotated in a predictable way, we conclude that theunderlying Cartesian grid is adequate to describe this collapse.

6.3.3

A More Realistic Experiment

Figure 6.5 illustrates the kind of clean angular distribution that might be expected in the ideal
experiment, where an infinitesimal change in scattering length is possible, and where only a single
roton mode is excited. Thus far, neither of these circumstances is true in the 52 Cr experiments.
Whereas figure 6.4 shows the difference in scattering length at which modes with different m
quantum numbers develop imaginary energies to be a fraction of a Bohr, experimental uncertainties
in the Feshbach-tuned scattering length of 52 Cr are ±2a0 [21]. Additionally, imaging of a 52 Cr
dipolar BEC was done after a time of free-expansion in this experiment, not in-trap as is describedin the scenario above. We propose, with slight modification, an experiment similar to the one
described by [21] that presents us with the possibility of observing angular structure in the collapse
and expansion of a dipolar BEC.
Instead of making a very small jump in scattering length across the stability threshold,

107

Figure 6.6: Stability diagram for N = 104 52 Cr atoms. The white region is dynamically unstable,
while the colored regions are stable. The pink (darker) islands are where biconcave structure isfound in the ground state of the condensate. The mean trap frequency is ω̄ = 2π × 700 Hz for
all aspect ratios λ. The arrows illustrate the initial and final values of scattering length in theexperiment proposed in the text.

consider making a jump of ∆as = −14a0 . For a 52 Cr dipolar BEC with N = 104 atoms, we
numerically prepare, for ω̄ = 2π × 700 Hz, a condensate in a trap with λ = 2 and scattering lengthas = 20a0 and a condensate in a trap with λ = 8 and scattering length as = 10a0 , where both
scattering lengths are about 10a0 above acrit for their respective aspect ratios. We then ramp
the scattering length from its initial value to its final value over a time period of 8 ms. These
scattering length ramps are illustrated by the arrows in figure 6.6. Although an 8 ms ramp time
is not sufficiently slow to make the change completely adiabatic (the characteristic trap period is
2π/ω̄ = 1.4 ms), it is sufficiently slow to allow a biconcave shape to form during the ramp. Once
this ramp has been made, we hold the collapsing condensate in the trap for thold = 2 ms and thenturn off the trap to let the collapsed condensate propagate in free space. In an actual experiment,
the expanded cloud could then be imaged to determine its density profile after expansion.
To ensure that we accurately simulate an experimental scenario and to break the cylindricalsymmetry of the condensate in a physically consistent way, we seed the condensate prior to the time
evolution with numerical noise. Although the non-projected GPE can only model the condensate

108
dynamics at zero temperature, an experiment will unavoidably have a very small but finite temperature present in the gas. This means that the condensate fraction will not be exactly unity, butsomething slightly less. To account for this, we correct our condensate by adding excited modes(quasiparticles) with weights determined by the Bose-Einstein distribution [172], ωj −µ−1kB T
nj = e−1(6.4)

where nj is the number of particles occupying the quasiparticle state with energy ωj , T is thetemperature of the Bose gas, µ is the chemical potential of the condensate and kB is the Boltzmann
constant.
Using the quasiparticle spectrum given by solving the BdG equations (5.7) and (5.8) and atemperature of T = 100 nK, we then perturb our initial condensate by
φ0 (x) → φ0 (ρ, z) +

X
j

r


nj 2πiαj ⋆eum,j (ρ, z)eimϕ + vm,j(ρ, z)e−imϕ ,
N

(6.5)

where {αj } are random numbers between 0 and 1, nj is given by Eq. (6.4) and um,j (ρ, z) and⋆ (ρ, z) are BdG modes with quantum number m and energy ω . Also, we include the factorvm,jjp
1/N in the weighting because the condensate wave function φ0 (x) is normalized to unity insteadof being normalized to N . We impose a cutoff on the sum in Eq. (6.5) of ωj < 2kB T , where T = 100nK, simplifying the problem by ignoring higher energy modes that contribute little to the thermal
excitations of the system. Indeed, T = 100 nK is an experimentally accessible temperature [14].
Additionally, because the condensate density becomes very large during the collapse process,a three-body loss term is required to accurately model the collapse and expansion dynamics [21].
The rate constant for three-body recombination was experimentally determined to be L3 = 2 ×
10−40 m6 /s for 52 Cr. We account for this loss in our simulations by including the term −i~N (N −1)L3 |φ0 (x)|4 /2 in the time-dependent GPE, Eq. (3.46), now given byZ~2 2∂φ0 (x, t) n
= −∇ + U (x) + (N − 1) dx′ V (x − x′ )|φ0 (x′ , t)|2
i~
∂t
2M
o
i~L3
|φ0 (x, t)|4 φ0 (x, t),− N (N − 1)
2

(6.6)

109
where V (x − x′ ) is given in Eq. (3.61). To solve Eq. (6.6), we employ the 4th order Runge-Kutta
algorithm with adaptive step size.1
Figure 6.7 illustrates the numerical time evolution of these condensates through the collapse
and expansion described above. As before, these images represent integrated density profiles inthe x-y plane, as viewed from the polarization axis. The top four frames illustrate the collapse
and expansion of a condensate in a trap with aspect ratio λ = 2, in which there is no biconcave
shape and in which, consequently, there should be no collapse to angular roton modes. During its
collapse, the condensate maintains its peak density in the center. After the trap is removed andthe gas is allowed to expand, its cylindrical symmetry is preserved.
By contrast, the lower four panels of figure 6.7 illustrate a representative time evolution for acondensate in a trap with aspect ratio λ = 8. In this case, by 7.5 ms the condensate has establishedits biconcave structure. When the condensate collapses, it does so into roton modes with angular
nodal structure, leading to local collapse with angular nature. After the trap is turned off andthe condensate expands for 4.5 ms, the angular structure remains in the density of the expanded
cloud. The collapse is clearly dominated by a roton with m = 3 in this simulation. However,
because several angular modes are involved, the angular pattern no longer experiences pure m = 3angular symmetry. Moreover, each mode arrives with a random initial phase, meaning that there
is a random asymmetry due to the interference between the unstable modes. In the experiment
this will imply non-repeatability of the observed density peaks from shot to shot.
Nevertheless, once the angular pattern is established, its vestiges remain in the expanded
cloud. In the final expanded picture, the clear break from cylindrical symmetry indicates thatthe decay modes have angular dependence, hence that the condensate went through a biconcave
phase. We note here that the collapse and expansion experiment that was performed with a dipolar
BEC of 52 Cr [173] did not probe the parameter regime for biconcave structure formation that weinvestigate here.
The results of simulations that are very similar to the ones described above are presented
1

Available in MATLAB through the function ode45.

110

Figure 6.7: Collapse dynamics of dipolar BECs, both in harmonic traps with mean frequencyω̄ = 2π × 700 Hz, corresponding to the scattering length ramps illustrated in Figure 6.6. (a) A
dipolar BEC in a trap with aspect ratio λ = 2, at t = 0 ms the condensate has as = 20a0 , thescattering length is ramped down to as = 6a0 over 8 ms, the collapsed condensate is held in thetrap for thold = 2 ms and then expanded until t = 16 ms. The collapse and expansion is purely
radial. (b) A dipolar BEC in a trap with aspect ratio λ = 8, at t = 0 ms the condensate has
as = 10a0 , the scattering length is ramped down to as = −4a0 over 8 ms, the collapsed condensate
is held in the trap for thold = 2 ms and then expanded until t = 14.5 ms. The condensate becomesbiconcave during the ramp in scattering length and thus collapses with angular structure, preserving
an angular character during expansion.

111
in [171], where dipolar BEC collapse is modeled in-trap and not through the expansion process
and not including a three-body loss term in the simulation. Ref. [171] performs simulations ofdipolar BEC collapse for both adiabatic and non-adiabatic (instantaneous) jumps in scattering
length, and find very interesting results regarding the presence of global and local collapse in thecondensate dynamics. For adiabatic collapse, where the change in scattering length is sufficiently
slow to track the condensate across the roton softening in the BdG spectrum, Ref. [171] presentsa critical trap aspect ratio above which local collapse occurs. We confirm these results, but pointout that while local collapse is very interesting (and can be evidence for the presence of the rotonin these systems), its manifestation in a dipolar BEC is much richer than has been discussed
in previous work. A mapping of dipolar BEC collapse via the experiment proposed above candetermine not only whether collapse was global or local, but whether collapse was radial or angular(including which m quantum number is responsible for the collapse) and thus provide evidence forthe underlying biconcave structure.
We point out that the experiment proposed above is just one of many experimental methods
that would demonstrate the angular nature of dipolar BEC collapse. Certainly, taking data for anumber of additional trap aspect ratios would assist in mapping out the regions where biconcavity
exists. Also, we expect that smaller and slower jumps in scattering length, which may be had withless uncertainty in the Feshbach-induced scattering length, would assist in understanding how andwhere collapse occurs. Slower ramping of the scattering length allows the condensate to be tracked
more adiabatically and thus allows for collapse to begin when only one BdG mode has a nonzero
imaginary energy, making the mapping of the collapse much more clear. Instantaneous or very fast
jumps in scattering lengths across a biconcave region will miss this structure completely and thus
result in a purely radial collapse. For an angular collapse to occur, the biconcave structure must
manifest itself in the condensate prior to collapse.

112

6.4

Conclusion
In conclusion, we have shown that in order to correctly map out the stability of a dipolar

BEC, a computational method that is sensitive to the local nature of dipolar BEC collapse must
be used. Methods such as the gaussian ansatz that are not sensitive to such phenomenon will
incorrectly predict the qualitative features of the stability. Also, we draw a connection between
the BdG spectrum of a dipolar BEC and the nature of the dipolar BEC collapse. Not only can
the BdG quasiparticles predict where a dipolar BEC will collapse in parameter space, they canalso predict how a dipolar BEC will collapse. For dipolar BECs without biconcave structure, thiscollapse is purely radial while for dipolar BECs with biconcave structure, this collapse has angularstructure. Performing collapse and expansion experiments on a 52 Cr dipolar BEC can reveal thisangular structure and thus provide an experimental method for mapping biconcave structure in
dipolar BECs.
Having studied the static properties of dipolar BECs in harmonic traps, including a proposed
experiment that could map the rich structure seen in the parameter space of the rotationlesscondensate, we now proceed to investigate the superfluid properties of dipolar BECs.

Chapter 7
Superfluidity in a Dipolar Bose-Einstein Condensate

Recall from our discussion in section 2.1 that a fundamental property of a superfluid is its
ability to support dissipationless flow. That is, a superfluid can move through slits or capillaries,
or an object can translate through the superfluid without exciting any quasiparticle or topological(vortex) modes in the system. In this case, no energy is transferred to the fluid and the flow isdissipationless, provided the flow is sufficiently slow.
As discussed in section 2.1, liquid 4 He was the first experimentally accessible system toexhibit dissipationless flow at low temperature, i.e., to demonstrate the existence of superfluidity
in a quantum system. However, the Bose-Einstein condensates (BECs) that have been createdmore recently have a distinct advantage over liquid helium in that they are dilute and hence easily
characterized in terms of microscopic interactions. In particular, their superfluid critical velocity
is nominally given by the speed of sound in the center of the gas, which can be easily calculated
from the density and the s-wave scattering length of the constituent atoms (see section 4.1). Earlyexperiments at MIT sought to measure vL in a BEC of sodium atoms by stirring the condensate
with a blue-detuned laser [174, 175]. However, these experiments measured a critical velocity
for spinning off vortices rather than the critical velocity for shedding energy into quasiparticle
excitations. This is a generic feature of such experiments in which the size of the object (in thiscase, the blue-detuned laser) is large compared to the healing length of the gas [176, 177, 178, 179].
In this chapter, we are concerned with the superfluid properties of dipolar BECs, in particular,
those that have been produced experimentally with 52 Cr [14, 23] and 164 Dy [17]. These gases present

114
a middle ground between atomic BECs and dense superfluid helium. Namely, the dipolar BEC isdilute enough to be understood in detail, yet its spectrum may exhibit roton features in prolate
traps, as we have discussed in previous chapters. The characteristic momentum of such a roton isset by the geometry of the trap in which it is held, whereas its energy is controlled by the density of
dipoles, as well as the magnitude of the dipole moment [16]. Thus, the superfluid critical velocity
is completely under the control of the experimentalist. In contrast, the superfluid critical velocity
in 4 He can be only weakly modified by changing the pressure of the liquid [48]. Thus, the dipolar
BEC provides an unprecedented opportunity to explore the fundamental relationship between theroton dispersion and superfluidity.
This chapter is composed of three sections. In section 7.1, we reproduce Landau’s original
argument for the derivation of the superfluid critical velocity, the so-called “Landau critical velocity”
vL , in terms of the elementary quasiparticle spectrum of the superfluid. In section 7.2, we modelan experiment on a purely dipolar BEC, similar to the MIT experiments. We consider a relatively
weak blue-detuned laser sweeping through a dipolar BEC at a constant velocity, and compute theresulting condensate depletion due to the excitation of quasiparticles (not vortices). We find an
onset of depletion at a critical velocity that is near the Landau critical velocity at low densities.
At higher densities, where the roton determines vL , the critical velocity is a decreasing function ofdensity, a behavior unique to a dipolar BEC. Moreover, the simulations show a critical velocity thatis somewhat smaller than vL at higher densities. We attribute this to the role that the roton plays inthe mechanical stability of a dipolar BEC. In section 7.3, we exploit the asymmetry of the dipoledipole interaction (ddi) to model an experimentally realizable system that exhibits anisotropicinteractions, and thus anisotropic superfluid character. By performing simulations of both weak andstrong blue-detuned lasers translating through both the homogeneous quasi-two dimensional (quasi2D) and trapped quasi-2D systems, we calculate the critical velocities for quasiparticle production
and vortex formation, demonstrating the anisotropic nature of superfluidity in the dipolar BEC
and illuminating the crucial role that the roton plays in this physics. Original work in this chapter
has been published in [38] and [39].

115

7.1

Landau Critical Velocity for Superfluid Flow
Here, we derive a expression for the critical velocity for superfluid flow in terms of thequasiparticle dispersion of the superfluid, ω(p), as was first done by Landau in 1941 [180, 47]. To
begin, we consider two inertial frames of reference, F and F ′ , where F ′ is moving relative to F
with velocity V. If the superfluid has mass M , then a Galilean transformation of the momentum
P and energy E from frame F to the frame F ′ is given by
P′ = P − M V

P′ 2
1
1
E =
=
|P − M V|2 = E − P · V + M V 2 ,
2M
2M
2
′

(7.1)
(7.2)

where E = P 2 /2M and P is the momentum of the superfluid in frame F .
We now consider the scenario where the fluid is moving at velocity v relative to an object. If
the object is at rest in the frame F ′ and the fluid is at rest in the frame F , then the object moves
with velocity −v relative to the fluid. If the flow dissipates, it is possible to excite a quasiparticle
with, say, energy ω(p) and momentum p. Then, the total energy of the fluid in the frame F is
E + ω(p), which is just the rest energy of the fluid plus the energy of the quasiparticle, and themomentum in frame F is just P = p. Now, in the frame F ′ , the momentum and energy of the fluid
plus quasiparticle are just given by replacing V with −v and P with p,P′ = p + M v

(7.3)

1
E ′ = E + ω(p) + p · v + M v 2 .
2

(7.4)

From the results (7.3) and (7.4), we see that the quasiparticle contribution to the momentum inframe F ′ is just p and the the contribution to the energy in frame F ′ is ω(p) + p · v. For such a
process to be energetically favorable, it must then be that ω(p) + p · v < 0. This is of course mosteasily achieved when p and v have opposing directions, which gives ω(p) < pv, or v > ω(p)/p. It
is the very threshold of this inequality that defines the Landau critical velocity, or the minimum
relative flow velocity beyond which such a dissipative process is possible. The Landau critical

116
velocity is thus given by
vL = min
p

ω(p)
.
|p|

(7.5)

In a dispersion relation plot of the quasiparticle energy ω(k) as a function of momentum k,
as we see in figures 4.2 and 4.4, the Landau critical velocity is given by the minimum slope of a linethat intersects both the origin ω(0) = 0 and a point on the dispersion curve. For a gas with purely
isotropic contact interactions, such as a dilute alkali atom BEC, vL is set by the speed of sound in
the gas. However, in superfluid 4 He and in trapped BECs with dipolar interactions, where rotons
are present in the dispersion, vL is set by the roton mode. While this critical velocity has beenmeasured in the 4 He system [51], no such measurement has been made for a dipolar BEC. Indeed,
proposing an experimentally realizable scenario in which such a measurement can be made is themain objective of the work presented in the next section.

7.2

Discrete Dipolar Superfluid
Just as we “perturbed” a dipolar BEC with a stationary blue-detuned laser in section 5.5,

here we consider perturbing a dipolar BEC with a blue-detuned laser moving at constant velocityv, which results in the time-dependent potential
U0−2(x2 + (y − yob (t))2 )Ulas (x, t) =exp
σ
σ w̃02(7.6)

where σ = 1 + (z/z0 )2 , z0 = π w̃02 /λlas is the Raleigh length, w̃0 is the beam waist of the laser,
λlas is the wavelength of the laser, yob (t) = Θ(t − t0 )[v(t − t0 )] describes the motion of the laser
in the y-direction and Θ(t) is the Heaviside step function [61]. This potential describes a laserthat is stationary until t = t0 , at which time it moves towards the edge of the condensate withvelocity ~v = v ŷ. The effect of this potential is taken into account by inserting it directly into thetime-dependent GPE, Eq. (3.46), which we integrate using a 4th order Runge-Kutta algorithm.
The effect of this blue-detuned laser on a dipolar BEC is shown in figure 7.1 for a dipolar BEC
with a trap aspect ratio λ = 20, a ddi strength D = 124 (where D is defined in Eq. (5.13)), and alaser with w̃0 = 0.4aρ and U0 = 2~ωρ where we calculate the chemical potential of the unperturbed

117
v = 0.3

a)

6

4

4

2

2

y/aρ

y/aρ

6

0

0

-2

-2

-4

-4

-6

v = 3.0

b)

-6
-5

0

5

x/aρ

-5

0

5

x/aρ

Figure 7.1: The density profiles of a dipolar BEC with D = 124 in a trap with aspect ratio λ = 20
after a blue-detuned laser with axis ẑ, beam waist w̃0 = 0.4aρ , z0 = 1.24aρ and U0 = 2~ωρ has
traveled through the dipolar BEC with velocity (a) v = 0.3 aρ ωρ (b) v = 3.0 aρ ωρ . In (a), there are
no visual excitations present in the system while in (b), excitations are clearly present, indicating
the presence of a critical velocity for the system. The 1/e2 contour of the laser is shown by the red
dotted lines at the center of the condensates.

condensate to be µ = 26.3~ωρ . We estimate the Landau critical velocity for this system to bevL ∼ 1.5 aρ ωρ . For a laser velocity less than this (figure 7.1(a)), the condensate is completely
unaffected whereas for a velocity larger than this (figure 7.1(b)), quasiparticles are excited and thefluid would produce a net force on the moving laser.
To determine the Landau critical velocity vL , we first calculate the condensate’s quasiparticle
spectrum by solving the Bogoliubov de Gennes (BdG) equations [102]. Due to cylindrical symmetryof the system, the condensate plus BdG quasiparticles can be written as
φ0 (x, t) → φ0 (ρ, z)e−iµt +Xhjicj (t)uj (ρ, z)ei(mϕ−ωj t) + c⋆j (t)vj⋆ (ρ, z)e−i(mϕ−ωj t) e−iµt(7.7)

where ωj is the quasiparticle energy, m is the projection of the quasiparticle momentum onto the zaxis and µ is the chemical potential of the ground state. Here, φ0 (ρ, z) is the stationary condensate
wave function, i.e., the solution of Eq. (3.23) with time-dependence e−iµt , and is normalized tounity. The coefficients cj (t) must be sufficiently small so that the BdG equations can be derived
by linearizing the GPE about them, as discussed in section 3.3.1. Notice that this form is identicalto the usual BdG ansatz, but now the small amplitudes cj (t) have been made time dependent

118
to describe the slowly varying quasiparticle occupations (compared to ωj−1 ) in out-of-equilibrium
states.
In this formalism, the quasiparticles are characterized by their energies ωj and m quantum
numbers. However, in order to apply the Landau criterion to this system, the quasiparticles must
be characterized by a momentum, as well. To do this, we calculate the expectation value of theqmomentum, or hkρ i ≡ hkρ2 i, of the quasiparticles. Using a Fourier-Hankel transform of the ap-

propriate order (see appendix D), we transform the modes into momentum-space and compute theexpectation value of the linear momentum of the j th quasiparticle in momentum-space representation,hkρ ij =sR

dk kρ2 (|ũj (k)|2 + |ṽj (k)|2 )R,
dk (|ũj (k)|2 + |ṽj (k)|2 )(7.8)

where we have time-averaged cross terms ∝ cos 2ωj t that oscillate on fast time scales [181]. By
associating these momenta to the excitation energies ωj , we determine a discrete dispersion relation
for this system. Whereas the Landau criteria for superfluid critical velocity presented in section 7.1
is derived for a translationally invariant fluid, we apply it to this translationally variant system toprovide a hint as to where a critical velocity for quasiparticle excitations might be and to test theapplication of this criterion to discrete systems.
Figure 7.2 shows the discrete dispersion relations of a dipolar BEC for various values of D.
For D = 0 (not shown), the dispersion is given by the well known harmonic oscillator spectrumpω = nρ ωρ with hkρ i = nρ + 1/aρ and nρ = 0, 1, 2, ... However, as D is increased, the spectrum
changes to develop a phonon character at low-momenta and a roton character at intermediate

momenta. Indeed, for D = 175.2, and more so for D = 230.0, there are some quasiparticles that√branch off from the dispersion towards lower energies and approach a momentum hkρ i ∼ 20/aρ ,pcorresponding to the characteristic roton wavelength λroton ≃ 2πaz , where az = ~/M ωz is theaxial harmonic oscillator length [28, 35]. The modes with similar momenta but larger energy,

on the upper branch of the dispersion, exist in lower-density regions of the condensate while thequasiparticles on the roton branch exist in the high density center of the condensate. Note that

119
20

20

ω/ωρ

15

a)
D = 17.5

10

10

5

5

0
0
20

4

6

c)

15

ω/ωρ

2

0
20

10

5

5
2

4

6

0

2

4

6

4

6

d)

15

D = 175.2

0

D = 70.1

0

10

0

b)

15

D = 230.00hkρ iaρ2

hkρ iaρFigure 7.2: The discrete BdG quasiparticle dispersions for a dipolar BEC in a trap with aspectratio λ = 20 for various values of D showing m = 0 (black + sign), m = 1 (teal squares) and m = 2(pink circles) quasiparticles. As D is increased, the dispersion develops a phonon-like character at
low momenta and a roton-like character at intermediate momenta. The slopes of the black dottedlines represent the corresponding Landau critical velocities for each D.

figure 7.2 includes only quasiparticles with m = 0, 1, 2.
In each case, the Landau critical velocity vL is determined according to Eq. (7.5) as the slopeof the shallowest line (or line with smallest slope) through the origin that intersects a point on
the dispersion curve. These lines are indicated in the figure. For smaller D, vL is determined by
the low-momentum phonon-like modes where ω is linear in hkρ i. By contrast, for larger D, vL isdetermined by the low-lying roton mode and becomes a decreasing function of interaction strength
in contrast to a BEC with only contact interactions, where vL grows as the square root of scattering
length [55, 125].
In evaluating vL from the discrete dispersion relation, we have ignored two excitations. One
is the unphysical m = 0 Goldstone mode [97]. A second is the m = 1 Kohn mode, which haseigenvalue ω1 = ~ωρ independent of interactions, and which corresponds to transverse sloshing of
the condensate [157]. The Kohn mode moves the condensate’s center of mass rather than exciting

120
quasiparticles relative to the center of mass, which would imply the breaking of superfluidity ina translationally invariant system. We therefore ignore it here. In any event, we find that theoccupation of the Kohn mode is very small compared to the total condensate depletion.
We now compare vL as determined from the discrete dispersion relation with the onset ofcondensate depletion due to the laser having been moved through the dipolar BEC. To quantify thebreaking of superfluidity in the simulations, we calculate the depletion of the condensate by finding
the quasiparticle occupations, or the number of particles that are excited out of the condensed
state. In practice, this is achieved by calculating the amplitudes cj (t) in Eq. (7.7) [182] via theorthogonality relations of the BdG modes [181] (see section 3.2) to givecj (t) =Zdx′ u⋆j (x′ )φ0 (x′ , t) − φ⋆0 (x′ , t)vj⋆ (x′ ) eiωj t ,(7.9)

where φ0 (x, t) is the numerical solution of the time-dependent GPE with the blue-detuned laser poR
tential. The quasiparticle occupations are then given by nj (t) = |cj (t)|2 dx′ (|uj (x′ )|2 + |vj (x′ )|2 ).
In the simulations, the system evolves for a time T after the laser has completely left the system.
We average the quasiparticle occupations for a time T after this, giving the average excited stateRT
occupations n̄j = T1 0 dt′ nj (t′ ). We find that T = 5 ωρ−1 is sufficient to converge these averages.
P
Figure 7.3 illustrates the total quasiparticle occupation ntot = j n̄j as a function of laser

velocity for various values of D using the laser parameters w̃0 = 0.3aρ , z0 = 0.7aρ and U0 = 0.4~ωρ .
For each D, ntot stays very small until, at a certain critical velocity vcrit , it begins to increasesignificantly. Operationally, vcrit is determined by the intersection of linear fits below and above
vcrit . Well above vcrit , the occupations decrease with velocity since the laser spends proportionallyless time in the system as its velocity is increased.
Notice that the overall depletion remains small with our weak laser. We have deliberately
remained in the perturbative limit with our simulations to uncover the basic physics without thecomplications of large laser size. Additionally, we have checked that these lasers are not sufficient toexcite vortex states in the dipolar BEC. In practice, larger condensate depletion would be obtained
from a repeated back-and-forth stirring, as was done in the MIT experiments, or from a wider,

121

−5

x 10
8

D = 17.5
D = 70.1
D = 175.2

ntot

6
4
2
0
0

0.5

1

1.5

2

2.5

v (aρ ωρ )Figure 7.3: The occupations of the quasiparticles excited from a dipolar BEC with aspect ratio
λ = 20 by a blue-detuned laser moving with velocity v (plotted on the horizontal axis) and withparameters w̃0 = 0.3aρ , U0 = 0.4~ωρ and z0 = 0.7aρ , for various values of D. At a critical v(indicated by the arrows), the occupations increase suddenly, indicating that the laser has excitedquasiparticles in the system and superfluidity has been broken.

stronger laser. While such a laser may spin off vortices in the condensate, thus defining a criticalvelocity smaller than vL , the roton, for large enough D, would still determine the critical velocity.
Critical velocities determined from numerical simulations are presented in figure 7.4 as afunction of D. Results are shown for the comparatively weak (U0 = 0.4~ωρ ) and strong (U0 = 2~ωρ )lasers. Also shown for comparison is vL (dashed line) as determined from the discrete dispersion
relations. At small D, the critical velocity grows slightly as the phonon modes stiffen and the speedof sound increases. This behavior is much like that of a BEC with purely contact interactions.
At higher density, the critical velocity instead decreases, due to the decreasing energy of theroton, and this is seen in both simulation and vL . The agreement is less perfect than in the phonon
regime, however, with the simulated result coming in lower. This is because the roton, being thecollapse mechanism for dipolar BECs in traps with larger aspect ratios, softens with increasingcondensate density. The presence of the laser in the dipolar BEC serves to increase the density ofthe system, softening the roton and thus decreasing the critical velocity of the condensate, just as

122
1.6

vcrit (aρ ωρ )

1.4
1.2
1
0.8
0.6
0.4

vL
w̃0 = 0.3aρ , Ulas = 0.4h̄ωρ

0.2w̃0 = 0.4aρ , Ulas = 2.0h̄ωρ

0
0

50

100

150

200

250

D
Figure 7.4: The superfluid critical velocities vcrit for dissipation due to the excitation of quasiparticles in a dipolar BEC as a function of D. The black dashed line represents the Landau criticalvelocity extracted from the discrete dispersion relations of the system. The teal circles represent theresults of numerical simulation for a laser with parameters w̃0 = 0.3aρ , z0 = 0.7aρ and U0 = 0.4~ωρand the pink squares represent the results of numerical simulation for a laser with parametersw̃0 = 0.4aρ , z0 = 1.24aρ and U0 = 2~ωρ .

a stationary laser leads a dipolar BEC to instability [183]. For vanishingly small lasers, the criticalvelocities extracted from numerical simulation show increasingly better agreement with vL .
Finally, it is worthwhile to consider measurements of critical velocities in experimentally
accessible dipolar BECs, such as the 52 Cr system in Stuttgart [14]. Consider 52 Cr atoms whose
scattering lengths have been tuned to zero in a trap with radial and axial frequencies ωρ = 2π × 100
Hz and ωz = 2π × 2000 Hz, respectively. This corresponds to a radial harmonic oscillator lengthof aρ = 1.391 µm, particle numbers of N ∼ 570D and critical velocities in the range of 0.11
cm/s. These circumstances suggest that it may be plausible to observe the decline of the superfluid
velocity with D for N & 8.5 × 104 52 Cr atoms, and hence to exhibit directly the roton’s influence onsuperfluidity. This atom number corresponds to a maximum condensate density of nmax ≃ 9.5×1014
cm−3 , which, given the measured 3-body loss coefficient L3 = 2 × 10−28 cm6 /s [16], should not
produce significant losses over the time scales considered here. Additionally, we have checked that,for sufficiently large D, the roton serves to determine vL for 52 Cr dipolar BECs with non-zero123
s-wave scattering lengths within the experimental uncertainty for 52 Cr, −3a0 ≤ as ≤ 3a0 [22],which is expected because these scattering lengths are sufficiently less than 52 Cr’s dipole length
add ≃ 15a0 [16].
Here, we have seen that the roton in the discrete dipolar BEC can play a very importantrole in the superfluid properties of the gas. Additionally, a measurement of the superfluid criticalvelocity in such a system, which is indeed within experimentally realizable limits, could provide
indirect evidence of the presence of the roton in such a system. As we have discussed, the roton
arises in a trapped dipolar BEC due to the anisotropy of the interactions. Interestingly, we can
exploit this anisotropy further by tilting the polarization field off of the symmetry axis, in orderto achieve a system where the interactions are anisotropic in-plane, and thus possess anisotropic
superfluid properties.

7.3

Anisotropic Dipolar Superfluid
To simplify the problem at hand, consider a dipolar BEC in a quasi-2D, or highly pancake-shaped trapping potential with the form U (z) = 12 M ωz2 z 2 and allow the dipole polarization dˆ to beat an arbitrary angle α off of the z-axis, so dˆ· ẑ = cos α. For sufficiently large ωz , or for sufficiently
tight trapping, we employ the separable ansatz φ0 (x, t) = χ(z)ψ(ρ, t) where χ(z) is a normalized
Gaussian with width lz , which we discuss in detail in section 4.2. In such a geometry, the problem
is reduced to an effectively 2D problem by integrating out the axial dependence of the relevant
equations (GPE, GP energy functional). By doing this, an effective quasi-2D interaction potential
is derived. For the system with dipolar and isotropic contact interactions, this potential is given
by Eq. (4.23), and we derive this form explicitly in appendix C. For completeness, we write out
the time-dependent GPE, Eq. (3.46), for the quasi-2D dipolar BEC here,1− ∇2ρ + U (ρ, t) + n2D g|ψ(ρ, t)|2
2

Z
′
′
2
+ n2D gd dρ Vq2D (ρ − ρ )|ψ(ρ, t)| ψ(ρ, t),i∂t ψ(ρ, t) =(7.10)

124√√
where, in this section, we define g = 2 2π~2 as /lz M and gd = 2 2π~2 add /lz M , where as is the
3D s-wave scattering length add is the characteristic dipole length of the constituent dipoles, given
in Eq. (2.21). Also, n2D is the integrated density of the unperturbed (homogeneous in-plane)
condensate. In (7.10), we write the trapping potential U (ρ, t) to be time-dependent to accountfor a moving blue-detuned laser, as discussed in the previous section. Like we do for the fully 3D
case, we handle the ddi mean-field term with the convolution integral over the q2D ddi potential
Vq2D (ρ − ρ′ ) in momentum-space, where the momentum-space form of this potential is given by
Eq. (C.8).
Additionally, in this section, we rescale energies in units of the chemical potential, given byµ∗ = gn2D [1 + ǫdd (3 cos(α)2 − 1)] for the unperturbed system, where ǫdd = gd /g [148]. This leads√
to characteristic units of length given by the coherence length ξ ∗ = ~/ M µ∗ ; time τ ∗ = ~/µ∗ ;p√and velocity c∗ = µ∗ /M . Additionally, we rescale the wave function ψ → ψ/ n2D . The rescaled
interaction coupling constants are then g∗ = gn2D /µ∗ and gd∗ = gd n2D /µ∗ = ǫdd g∗ .

To begin investigating the superfluid properties of the quasi-2D dipolar BEC with anisotropicinteractions, we consider the quasiparticle dispersion relation of a homogeneous quasi-2D dipolar
BEC (see section 4.2),
ω(k) =

s




klz
k4
2
∗
,
+ k g 1 + ǫdd F √42(7.11)

where the function F (q) is defined in appendix C. For α = 0 (polarization along the trap axis)
this dispersion does not depend on the direction of the quasiparticle propagation. However, forα 6= 0, or for nonzero projection of dˆ onto the x-y plane, the direction of k becomes importantin describing the quasiparticles of the system. The Landau critical velocity is determined fromthe dispersion relation via Eq. (7.5), and here depends on the direction of k and is an anisotropic
quantity when | cos (α)| < 1. We note that other anisotropic dispersions have been predicted for a
1D lattice system of quasi-2D dipolar BECs [184], periodically dressed BECs [185] and for dipolargases in a 2D lattice [186]. Additionally in this vein, anisotropic solitons have been predicted for
dipolar gases [187].

ω (µ∗ )2
1.5v L (ξ ∗ /τ ∗ )

2.5

125

1

k

0.5
0

1/4

η/π

1

1/2

⊥

0.5
0
0

0.5

1

1.5

2

k (1/ξ ∗ )

||

0.2

−5

y/ξ*
0

5

Density
0.6
1.0

⊥

b)
−5

0
x/ξ*

5

c)
−10

−5

0
5
x,y/ξ*

10

Figure 7.5: (a) The Bogoliubov dispersions for g∗ = 0.25, gd∗ = 1.50, lz /ξ ∗ = 0.5 and α = π/4
for propagation perpendicular (⊥) to and parallel (k) to the tilt of the dipoles, shown by the blue
and red lines, respectively. The dashed lines have slopes that are the Landau critical velocities(vL ) of the dispersions, while the inset shows vL as a smooth function of the angle η between thek (η/π = 0) and ⊥ (η/π = 1/2) propagation directions. (b) A contour plot of the density for astationary obstacle with amplitude U0 /µ∗ = 1.0. The shaded region indicates a density exceeding
1.05n2D , and the arrow indicates the direction of polarization. (c) Density slices of (b) along theparallel (dashed red) and perpendicular (blue) directions. The density oscillation due to the roton
is clear in the perpendicular case.

To illustrate the anisotropy in vL , we use the parameters g∗ = 0.25, gd∗ = 1.50, lz /ξ ∗ = 0.5 andα = π/4 (µ∗ = 1), which are chosen to best illustrate anisotropic effects while keeping safely awayfrom the unstable regime; we will identify them with experimental parameters below. Figure 7.5(a)shows the dispersion calculated using these parameters for quasiparticle propagation parallel to(k) and perpendicular to (⊥) the tilt of the dipoles into the plane. For parallel propagation, thedispersion resembles that of a system with contact interactions; the curve goes smoothly from the126linear phonon regime at small k to the free-particle regime at large k. For this case, vL /c∗ = 1.0(dashed red line), meaning that the critical velocity is identical to the speed of sound. In contrast,
the perpendicular dispersion curve exhibits a roton-like feature at intermediate k, setting vL /c∗ =0.50 (dashed blue line). The inset in figure 7.5(a) shows vL as a function of the azimuthal angle, η,
the angle between the polarization tilt and the direction of quasiparticle propagation. Interestingly,
the speed of sound, given by c = limk→0 [ω(k)/k], is the same for both parallel and perpendicularpropagation, c/c∗ = 1, and is in fact isotropic. Therefore, the anisotropy in the spectrum occurs
only at finite k due to the presence of an anisotropic roton.
The impact of the anisotropic roton can be directly seen in the density of the gas. In
figure 7.5(b) we show a contour plot of the density in the presence of a repulsive Gaussian potential,
or “probe,” of the form (7.6) with U0 /µ∗ = 1.0 and σ/ξ ∗ = 2.0. We calculate this density profile
by solving for the condensate wave function ψ(ρ) by evolving Eq. (7.10) in imaginary time, as we
discuss in chapter 5. The shaded regions indicate density above 1.05n2D , and the arrow indicatesthe direction of polarization. In figure 7.5(c) we plot density slices of this distribution to moreclearly show the density profile in the parallel (dashed red) and perpendicular (solid blue) directions.
Interestingly, the high-density regions occur in the direction perpendicular to the tilt of the dipoles,
the same direction that exhibits a roton feature in the dispersion. In section 5.5, we showed thata dipolar BEC in the presence of a repulsive Gaussian can exhibit density oscillations due to the
manifestation of the roton. Here, we see a manifestation of the anisotropic roton in the static
structure of the quasi-2D dipolar BEC.
We now address the question of what happens to this anisotropic dipolar BEC when theprobe is moved through it with velocity v, by numerically evolving Eq. (7.10) in real time, with therepulsive Gaussian potential U (ρ, t) = Ulas (x − vt, y). For concreteness, we consider motion paralleland perpendicular to the tilt of the dipoles by tilting dˆ into the x̂ and ŷ directions, respectively,
while fixing the direction of the probe velocity so that v = vx̂. Figure 7.6 illustrates a schematicof this system where the polarization vector dˆ is tilted in the x-direction. Here, the grey surface
represents the condensate and the blue surface represents the blue-detuned laser. The green arrow

127

Figure 7.6: A schematic of a blue-detuned laser acting as a moving object in a dipolar BEC withanisotropic interactions. Here, the grey surface represents the condensate and the blue surface
represents the blue-detuned laser. The green arrow represents the polarization vector dˆ and the redand blue arrows show the x and y directions, respectively. In this setup, the red (x) arrow defines
the parallel direction and the blue (y) arrow defines the perpendicular direction.

represents the polarization vector dˆ and the red and blue arrows show the x and y directions,
respectively. In this setup, the red (x) arrow defines the parallel direction and the blue (y) arrow
defines the perpendicular direction. In this section, we identify the color red with parallel motion
and the color blue with perpendicular motion.

7.3.1

Quasiparticle Production (Weak Laser)

Figure 7.7(a) shows the time-averaged drag force (averaged up to t = 100τ ∗ ) acting on a“weak” probe with parameters U0 /µ∗ = 0.1 and σ/ξ ∗ = 2.0. The force acting on the laser potential
Ulas at time t is given by [177]F(t) = −Z~ las (ρ, t).
d2 ρ|ψ(ρ, t)|2 ∇U(7.12)In this case, the probe is sufficiently weak so that no vortices are nucleated in the fluid, and instead
only quasiparticles are excited. The presence of a force on the probe signifies the excitation of quasiparticles, and thus the breakdown of superfluid flow. There is a clear anisotropic onset of force inthese simulations that agrees very well with the anisotropic vL given by the Bogoliubov dispersions
in figure 7.5(a), resulting in critical velocities of vc /c∗ = 0.90(0.46) for parallel (perpendicular)
motion of this probe, determined by the velocity at which the drag force suddenly rises.
It has been shown that vL is recovered as the true critical velocity only when the superfluid

128
is perturbed by a vanishingly small object [178, 182]. Additionally, while quasiparticle excitations
are a natural feature to study when considering the breakdown of superfluid flow, they may bedifficult to observe experimentally. Vortices, on the other hand, are superfluid excitations in theform of topological defects that create regions of zero density and are easier to observe experimentally than quasiparticles. As forementioned, the first measurements of vc in a BEC were fromobservations of the sudden onset of heating [174, 175], believed to be related to vortex productionin the BECs. More recently, the Anderson group at the University of Arizona used experimental
finesse to controllably create vortex pairs to observe vc [188]. Motivated by these circumstances,
we investigate the critical velocity for vortex formation in the quasi-2D dipolar BEC by using amoving probe with an amplitude that is linearly ramped from U0 /µ∗ = 0 to U0 /µ∗ = 1.0 in a time10τ ∗ with σ/ξ ∗ = 2.0. The critical velocity in this case corresponds to the probe velocity above
which vortices are formed, signaling the breakdown of superfluidity.

7.3.2

Vortex Production (Strong Laser)

We observe a significant difference in the critical velocity at which vortices are formed between
a probe moving parallel and perpendicular to the dipole polarization. In figure 7.7(b) we show the
maximum number of vortices formed within t/τ ∗ = 100. The critical velocities are vc = 0.46(0.28)
for motion parallel (perpendicular) to the dipole tilt. These values are about half the value ofthe critical velocities obtained using the weaker probe, but this is not unexpected [177, 176]. As
discussed previously in sections 2.1.2 and 5.3, vortices in a superfluid have quantized circulation:Hv · dl = 2π~n/m, where v is the velocity field of the fluid and n is an integer, corresponding tophase winding of 2πn around the vortex core. We count vortices in our simulations by finding thephase winding on a plaquette of grid points [189].
The physical mechanism that sets the critical velocity for vortex formation is not rigorously
understood. However, it is theorized that the maximum local fluid velocity about an obstacle,
being larger than the background flow velocity, sets the critical velocity via the Landau criterion.
This idea has been fruitful [177, 176], and we find qualitative agreement with this theory here, as

129

−3

F̄x (µ∗ /ξ ∗ )

20

x 10

A p = 0.1µ ∗ , σp = 2.0ξ ∗

⊥
k

15
10
5

a)

0
0.5

1

1.5

v/c∗
Maximum vortex number

25

A p = 1.0µ ∗ , σp = 2.0ξ ∗

⊥
k

20
15
10
5

b)

0
0.2

0.3

0.4

0.5

0.6

v/c

∗

Figure 7.7: (a) The mean drag force acting on the probe with U0 /µ∗ = 0.1 and σ/ξ ∗ = 2.0 calculated
up to time t/τ ∗ = 100 for motion perpendicular to (blue diamonds) and parallel to (red squares)
the dipole tilt. The dotted lines represent the corresponding vL , in excellent agreement with thenumerical simulations. (b) The maximum vortex number produced by a probe with U0 /µ∗ = 1.0 andσ/ξ ∗ = 2.0 calculated up to t/τ ∗ =100 for motion perpendicular to (blue diamonds) and parallel(⊥)
to (red squares) the dipole tilt. The corresponding critical velocities are: vc /c∗ = 0.27 and(k)vc /c∗ = 0.46.

the direction with lower vc is also the direction of flow most likely to spawn vortices. However,
we note that the ddi is anisotropic although the fully condensed (ground) state of the system iscompletely isotropic. The anisotropies only appear in the dispersion relation and in the groundstate of the system in the presence of a perturbing potential, which is intimately related to thedispersion relation (see figures 7.5(b) and 7.5(c)) [35]. Thus, the anisotropies in the critical velocity
for vortex formation are due to the anisotropy of the roton mode, just like the critical velocities for

130

Figure 7.8: (a) and (b) The densities are shown for the k (top, red) and ⊥ (bottom, blue) cases forv/c∗ =0.30 and t/τ ∗ =125. Only the ⊥ case has exceeded its vc . (c) and (d) Densities for v/c∗ =0.50
at t/τ ∗ =100. The parallel case (top, red) has exceeded its critical velocity and the perpendicularcase (bottom, blue) has been wildly excited. The obstacle is at x = y = 0 and moving in the +x
direction. The shaded regions of this plot occur when the density exceeds 1.05n2D , and the arrows
indicate the direction of dipole tilt. An image showing the data from panel (d) made the cover of
PRL 106, Issue 6.

quasiparticle excitations.
Figure 7.8 shows contour plots of the condensate density for both parallel (red contours, toprow) and perpendicular (blue contours, bottom row) motion of the probe relative to the dipole tilt
for velocities v/c∗ = 0.3 (left column) and v/c∗ = 0.5 (right column), where the probe is moving in
the x̂ direction and is located at the origin at the time shown in figure 7.8. Recall that v/c∗ = 0.3 isjust above vc for vortex formation for perpendicular motion, but well beneath vc for parallel motion.
This is reflected in the figure, where in (a) no vortices have been formed for parallel motion, whilein (b) a vortex pair has been formed for perpendicular motion for the same probe velocity.
For the case of v/c∗ = 0.5, we see that the parallel case in (c) has formed a vortex pair, and in

131

Figure 7.9: Density contours from simulations of a blue-detuned laser translating through a dipolar
BEC with velocity v/c⋆ = 0.4 in a trap with aspect ratio λ = 50, as discussed in the text. (a)and (b) are at t/τ ⋆ = 0, (c) and (d) are taken just after the laser has left the condensate and (e)and (f) are taken well after the laser has left the condensate. The red profiles (a), (c) and (e)represent motion of the laser parallel (k) to the polarization tilt and the blue profiles (b), (d) and(f) represent motion perpendicular (⊥) to this tilt. One clearly sees a vortex pair that has been
nucleated in the perpendicular case, but not in the parallel case. Image courtesy of Chris Ticknor.

(d) the perpendicular case has been wildly excited. There is an important contrast to be made in thedensity profiles when there is a single vortex pair in (b) and (c). In the parallel case (c) we see thata high density region occurs between the vortex pair and is elongated in the polarization direction.
In contrast, for the perpendicular case (b) there is a low density region between the vortex pairand high density regions on either side of the vortex pair. Both the anisotropic superfluid criticalvelocity for vortex pair production and these contrasting density profiles present means to observe
the effects of the roton in dipolar BEC directly.

132
In addition to investigating the purely quasi-2D system, we have performed simulations fora quasi-2D dipolar BEC in the presence of a radial trapping potential. For a trap aspect ratioof λ = 50, we find that critical velocities for vortex production are strongly anisotropic, andthe vc are numerically similar to the free case. In these simulations, we start the probe in thecenter of the trap and move outwards in the parallel or perpendicular direction, linearly ramping
the amplitude of the laser down to zero by the time it reaches the zero density region. Sucha simulation is experimentally realizable in a dipolar BEC of atomic 52 Cr, for example, having apermanent magnetic dipole moment of 6µB where µB is the Bohr magneton, for a dipolar BEC withparticle number N ≃ 18.5×103 , scattering length as = 5.0a0 where a0 is the Bohr radius, radial trap
frequency ωρ = 2π × 20Hz and a blue-detuned laser with width σ = 1.76µm. The speed of soundin this system is c = 0.16 cm/s in the center of the trap. We plot frames from these simulations forv/c⋆ = 0.4 in figure 7.9, where the left panel ((a), (c) and (e)) shows motion of the laser parallelto the polarization tilt and the right panel ((b), (d) and (f)) shows motion perpendicular to thistilt. The times at which these frames are sampled are t/τ ⋆ = 0 ((a) and (b)), just after the laser
leaves the condensate ((c) and (d)), and well after the laser has left the condensate ((e) and (f)).
Here, one clearly sees that a vortex pair has been nucleated in the perpendicular case, but not in
the parallel case, signifying the breaking of superfluidity for perpendicular flow and superfluid flowin the parallel direction.

7.4

Conclusion
In this chapter, we characterized the superfluid properties of dipolar BECs with regards tosuperfluid flow and the superfluid critical velocity. For the fully trapped, 3D dipolar BEC, we
showed that the discrete roton plays a crucial role in the superfluidity of this system by serving
to lower the critical velocity as a function of ddi strength D. By direct numerical simulations, we
proposed an experiment that could provide both a measurement of the superfluid critical velocity
in a dipolar BEC of 52 Cr and provide evidence of the roton in this system.
Additionally, we characterized the quasi-2D dipolar BEC as an anisotropic superfluid by

133
performing numerical simulations of a blue-detuned laser moving through the system in directions
parallel and perpendicular to the dipole polarization. We found a sudden onset of drag on thelaser at velocities that depend strongly on the direction of motion, and attribute the anisotropy incritical velocity to the anisotropic roton so that a measurement of an anisotropic critical velocity
in a dipolar BEC corresponds to a measurement of the roton in the system. Additionally, by
considering a dipolar BEC that is experimentally realizable with atomic 52 Cr, we propose a single,
stable constituent with which to study anisotropic superfluidity. As discussed in section 2.3.1, acandidate geometry for creating quasi-2D systems is a retro-reflected laser that results in a 1D
lattice trapping geometry. We go on now to study the static properties of a dipolar BEC in a 1D
lattice.

Chapter 8
Dipolar Bose-Einstein Condensate on a One-Dimensional Lattice

As we have seen in the previous chapters, the physics of a dipolar system depends strongly
on the geometry of the trap in which it is held. For example, in section 4.2 we saw that a dipolar
Bose-Einstein condensate (BEC) in a quasi-two dimensional (quasi-2D) geometry experiences a roton instability that can be suppressed by increasing the trap frequency or, equivalently, tightening
the trap in the direction of polarization. Another example of such stabilization involves the inelastic scattering processes of both bosonic and fermionic species, which are predicted to be highly
suppressed in tighter, quasi-two dimensional (quasi-2D) traps when the trap is applied along thepolarization axis of the dipoles [133, 190, 191, 131, 130]. This suppression leads to more stable,
longer-lived many-body systems of reactive molecular species. So, we see that tight trapping in the
direction of polarization is necessary to obtain stable, high density dipolar quantum fluids of both
atomic and molecular species.
Such a trap is realizable in a one-dimensional (1D) optical lattice, as discussed in section 2.3.1,
which can be created by retro-reflecting a laser back onto itself. The presence of the lattice brings
up an interesting point regarding the physics of such a system. While the dipole-dipole interaction(ddi) is anisotropic, it is also long-range, scaling as 1/r 3 , and if the lattice spacing is sufficiently small
compared to the characteristic dipole length add then the effect of the ddi is non-negligible between
the lattice sites. For example, interlayer superfluidity is predicted to exist in two adjacent layers ofpolar fermions [9, 10, 11], and scattering in the 2D plane is predicted to be significantly modified
by the presence of a weakly bound state of dipoles in adjacent layers [192]. Dramatic effects are135
predicted for layers of bosons, as well, for both quasi-2D [193, 184] and radially trapped [194, 195]
lattice sites. The presence of the lattice is predicted to significantly alter the dispersion via thesoftening of a roton-like mode in the system, and thus to alter the stability properties of the Bosegas.
In this chapter, we consider a gas of bosonic dipoles in a 1D lattice with the dipoles polarizedalong the lattice axis, so that the system is cylindrically symmetric. Assuming that the latticerecoil is sufficiently large, we model the potentials of the individual sites as cylindrically symmetricharmonic traps. At ultracold temperatures, this leads to a lattice of non-overlapping dipolar BECs
coupled by the long-range ddi. We study the stability of this system both for an infinite andfinite 1D lattice. Additionally, we find regions in parameter space where biconcave structure ispredicted to exist that is emergent in the lattice system, in other words, that does not exist in asingle condensate. To ensure the accuracy of our results, we calculate the elementary excitations of
the system and use them to determine whether our solutions are dynamically stable. In doing so,
we map the structure and stability of a 1D lattice of purely dipolar BECs. Original work in thischapter has been published in [40].

8.1

Formalism for the One-Dimensional Lattice
Consider a 1D lattice with spacing dlat between adjacent sites. If the lattice is sufficiently

deep, it can be modeled by a series of Nlat harmonic traps, where each site is described by acylindrically symmetric potential Uj (x) = 12 M ωρ2 ρ2 + λ2 (z − jdlat )2 , where M is the mass of theindividual bosons and λ = ωz /ωρ is the trap aspect ratio of each trap. We describe this systemwith a coupled set of non-local Gross-Pitaevskii equations (GPEs)


N
lat


X
′
(1)
Ĥj (x) +Udj (x) − µj φ0,j (x) = 0,


′

(8.1)

j =1

(1)

where Ĥj (x) is the non-interacting, or single-particle Hamiltonian for site j,1(1)
Ĥj (x) = − ∇2 + Uj (x),
2

(8.2)

136
φ0,j (x) is the condensate wave function at site j, j is an integer and µj is the corresponding chemical
potential. Without the presence of the long-range ddi, these Nlat equations would be independent.
The ddi couples the equations through the mean-field potentials Udj (x), given by the convolution
Udj (x) =Z

dx′ Vd (x − x′ )nj (x′ )(8.3)

where nj (x) = |φ0,j (x)|2 is the density of the condensate occupying the j th site with normR ′dx nj (x′ ) = Nj , Nj is the condensate number for site j and Vd (x − x′ ) is the two-body ddi

potential, Eq. (3.61). A description of the fully-condensed, stationary state of this system of dipo-lar BECs is then given by the set of solutions {φ0,j (x), µj } that minimize the energy functionalcorresponding to Eq. (8.1), given byE[{φ0,j (x)}] =

XZ
j





N
lat

X
′
1
(1)
dx φ⋆0,j (x) Ĥj (x) +Udj (x) φ0,j (x).


2 ′

(8.4)

j =1

As we have discussed, a full description of a dilute BEC of interacting atoms or molecules
includes a short-range s-wave part, proportional to the s-wave scattering length. However, because
this interaction is short-range, it results in the mean-field potential Ucj (x) = g|φ0,j (x)|2 . Modelinga system of non-overlapping BECs in a 1D lattice interacting only via contact interactions results
in a set of uncoupled GPEs. While the interplay of contact and ddi interactions is predicted to
produce interesting effects [156] that would likely be modified by the presence of the lattice, we set
as = 0 in this chapter to illuminate purely dipolar effects.
As we have done previously, the dipolar mean-field is calculated in momentum-space toeliminate the problems associated with the divergence of the ddi in real-space. Here, it will prove
useful to define the shifted densities νj (x) = nj (xj ) where xj = {ρ, z − jdlat }, so that all νj (x) areformally centered about the origin. Then, νj (x−j ) = nj (x), and we can write ñj (k) = F[νj (x−j )].
With some simple manipulation, this expression reduces toñj (k) = F [νj (x)] eikz dlat j .

(8.5)

So, the momentum-space density of the dipolar BEC at site j can be rewritten as the Fouriertransform of nj (x) translated into the local set of coordinates, with an additional exponential term

137
accounting for this spatial translation. Now, by the convolution theorem (see appendix A), themean-field contribution from the dipolar BEC at site j can be written ashi
Udj (x) = F −1 Ṽd (k)ñj (k)(8.6)

where Ṽd (k) is the Fourier transform of the ddi, given in Eq. (3.62). In this work, we calculate νj (x)directly by calculating the shifted condensate wave functions Φj (x) such that νj (x) = |Φj (x)|2 andaccount for the spatial separation of the dipolar BECs, or the presence of the lattice, with theexpression given in Eq. (8.5). So, the wave functions φ0,j (x) and Φj (x) are related by Φj (x) =φ0,j (xj ).

8.2

Wave Function Ansatz
As we have seen for a single dipolar BEC, calculating the mean-field energy on a full numericgrid has proven fruitful, however, this method is very computationally expensive when considering
multiple interacting dipolar BECs, both in real- and momentum-space. In real-space, the convolution integral for the dipolar mean-field must be done directly, where there is no 1/r 3 divergence if
the condensates do not overlap. In momentum-space, the grid must be large enough to resolve theentire lattice because of the eikz dlat j dependence of the momentum-space densities. To avoid these
problems, we consider solutions of the form Φj (x) = ψj (ρ)χj (z) where1χj (z) = q(χ0,j (z) + A2,j χ2,j (z)) ,1 + A22,jz2
χ0,j (z) = p
1 exp −
2lz,j 2
lz,j π 4
1

and



 
z2
z
χ2,j (z) = p,
H2
1 exp −lz,j
2lz,j 2
2 2lz,j π 4
1

(8.7)

(8.8)

(8.9)

where H2 (x) = 4x2 − 2 is the second Hermite polynomial [54]. This ansatz includes the zeroth andsecond harmonic oscillator wave functions with variable width and relative amplitude. Plugging
this ansatz into the GPE and integrating out the z-dependence results in a modified GPE in the

138

Figure 8.1: The differences in energy of a dipolar BEC in a trap with aspect ratio λ = 7 as afunction of interaction strength between that calculated using the ansatz given in Eq. (8.7) andthat calculated exactly on a full numeric grid. The blue dotted line shows the energy difference
calculated using no 2nd order h.o. wave function and a fixed axial width lz = az , the red dashedline shows the energy difference using the same wave function but with lz treated variational andthe black solid line shows the energy difference including the 2nd order h.o. wave function where
the relative amplitude A2 and lz are treated variationally.

radial coordinate ρ that also depends on the widths lz,j and the relative amplitudes A2,j of theaxial wave functions, but not the z-coordinate explicitly. We derive this modified GPE for a single
dipolar BEC, given by Eq. (H.2), in Appendix H.
To test the ansatz given in Eq. (8.7), we apply it to the well known system of a single
dipolar BEC in a harmonic trap. For this case, using the wave function (8.7) gives us an effectively Nρ + 2 dimensional numeric problem, where we use a numeric grid in ρ with Nρ grid points(see appendix D) and two variational parameters (for the single dipolar BEC), lz and A2 . To
minimize the corresponding energy functional, we employ the conjugate gradients (cg) algorithm(see appendix E) with fixed lz and A2 , then minimize the resulting functional of these variationalparameters between each cg iteration. Having achieved sufficient convergence in the energy minimization, we solve the corresponding Bogoliubov de Gennes (BdG) equations (3.34) in the single
mode approximation, where the low-lying BdG modes are assumed to occupy the same axial wave
function of the condensate.

139

Figure 8.2: The values of the axial wave function parameters that, together with the radial wave
function calculated on a grid, minimize the energy of a single dipolar BEC in a trap with aspectratio λ = 7. The blue dotted line shows the result for lz = az , the red dotted line shows the valuesof lz when it is treated variationally and A2 = 0 and the black solid line shows the values of lz when
it and A2 , the black dotted line (marked by the right vertical axis) are both treated variationally.

Figure 8.1 compares the total energies of a single dipolar BEC in a trap with λ = 7 as afunction of the ddi strength (N − 1)gd /aho , where, in this chapter, we find it convenient to redefine
√

2

add
. Plotted is the energy difference (E −Efull )/Efull , where Efull is the energy calculated
gd = 2 2π~
M

by solving the GPE exactly (within strict numerical precision) on a full numeric grid in ρ and z.
The blue dotted line shows the energy of the dipolar BEC when A2 = 0 and lz is fixed to be thepaxial harmonic oscillator length, az = ~/M ωz , the red dashed line shows the energy when A2 = 0

and lz is treated variationally, and the black line shows the energy when A2 and lz are both treatedvariationally. Clearly, the full variational treatment is much more accurate than the cases where
the second harmonic oscillator wave function is not included (A2 = 0). Indeed, it stays within 1%of the exact energy for all values of (N − 1)gd /aρ for which the dipolar BEC is stable. We find thisto hold true for larger trap aspect ratios, as well. Figure 8.2 shows the values of the variationalparameters for the same cases as in figure 8.1. In this figure, the left vertical axis labels lz /az andthe right vertical axis labels A2 , shown by the black dots.
Beyond energetics, this ansatz also predicts semi-quantitatively the structure and stability

140

a

300
250

(N − 1) agdρ

b

200

c

150
100
50
0

5

λ

10

15

Figure 8.3: Structure/stability diagram for a single dipolar BEC. The colored regions indicate adynamically stable condensate, and the pink (darker) regions indicate parameters for which the
dipolar BEC has biconcave density. (a) and (b) are calculated using the ansatz for the axial wave
function given in Eq. (8.7) and (c) is calculated using a full numeric grid. For (a), A2 = 0 andlz /az = 1, and for (b), A2 and lz are treated variationally.

of a single dipolar BEC. An interesting feature of the biconcave structure discussed in section 5.2
and chapter 6 is that it exists in “islands” of parameter space, defined here by (N − 1)gd /aρ andλ. Figure 8.3 shows this structure/stability diagram for a single dipolar BEC calculated using a)
A2 = 0 and lz /az = 1, b) A2 and lz variational and c) a full numeric grid in ρ and z. Interestingly,
the biconcave islands are present in each diagram and occur for almost exactly the same valuesof the ddi strength (N − 1)gd /aρ . They are, however, shifted in λ, moving to smaller values as
more restrictions are placed on the condensate wave function. The diagram for the full variational
ansatz (b) qualitatively matches that of the full numeric grid. Thus, we expect that this ansatz
will give physically meaningful results, if not quite quantitative ones. Of course, the diagram (c)in figure 8.3 is exactly that shown in figure 5.1.
As we discuss below, a key benefit of this ansatz for the 1D lattice system is that it is analytic
in z. Another such ansatz that has this property is that of correlated Gaussians, which have beenshown to reproduce the results of full numeric calculations for dipolar BECs quite well [196].
However, we applied this ansatz to the lattice system and found that it is numerically unstable

141
with the minimization techniques used here.

8.3

Infinite latticeWith confidence in the ansatz given in Eq. (8.7), we now apply it to the 1D lattice system. An

interesting example to consider is that of an infinite lattice, with Nlat → ∞. This approximation
introduces a discrete invariance to the system so that we can set Φj (x) = Φj ′ (x) for all j, j ′ . Thus,
we can neglect, for the time being, the indexing of the wave functions and let Φj (x) → Φ(x) for all
j. Then, the mean field potential at any site is, from Eq. (8.5), given by
Ud (x) =∞ Z
X

dkṼd (k)ñ(k)eikz dlat j e−ik·x(8.10)

j=−∞

where
1 2 2

e− 4 kz lzñ(k) = ñρ (kρ )1 + A22







1
1
2 2
4 4
√
+ A2 kz lz + A2 kz lz.
1 + A2 A2 −8
2Additionally, we manipulate the infinite sum in Eq. (8.10) to give [197]


∞
∞
X
2πj
2π X
ikz dlat jδ kz −
.
e
=
dlat
dlat
j=−∞

(8.11)(8.12)

j=−∞

The term that accounts for the infinite lattice can therefore be written as a Dirac comb in kz
with spacing 2π/dlat between peaks. Inserting this expression into Eq. (8.10) gives the mean-fieldpotential
Ud (x) = 2

gd −1
F [Finf (kρ )ñρ (kρ )] ,dlat 2D(8.13)

where 2gd Finf (kρ )/dlat is the effective momentum-space ddi for the infinite lattice and Finf (kρ ) is
given byFinf (kρ ) =×r−2π 2 j 2

l2
z



2


∞
d2
2
4
lat
π X e
1
2 2 lz4 4 lz
1 + A2 A2 − √ + A2 4π j 2 + 2A2 π j 4
2(1 + A22 )2
dlat
dlat
2
j=−∞
!
12π 2 j 2
−1 .
(8.14)kρ2 d2lat + 4π 2 j 2Thus, we see that the GPE for an infinite lattice of interacting dipolar BECs is reduced to a single
GPE in the radial coordinate ρ where all of the axial dependence of the wave function is capturedby the variational parameters A2 and lz .

142

Figure 8.4: Structure/stability diagram for an infinite lattice of dipolar BECs in traps with aspectratio λ = 10 as a function of lattice spacing dlat /az and interaction strength (N − 1)gd /aρ . The
colored region indicates dynamic stability, while the pink (darker) regions indicate parameters
where the dipolar BECs have biconcave density. The inset shows an isodensity plot of a dipolar
BEC with biconcave density.

We study the structure and stability of this infinite lattice of interacting dipolar BECs bysolving the modified GPE for the system (again, by applying conjugate gradients to minimize thecorresponding energy functional, see appendix E) and studying the BdG excitations in the single
mode approximation. We find that the sum in Eq. (8.14) is sufficiently converged if a cutoff jcut isapplied to the index j such that jcut ≫ dlat /2πlz .
Consistent with other results [193, 184], we find that the presence of the lattice serves todestabilize the system due to the softening of a discrete roton-like mode in the system. For a single
dipolar BEC in a trap, tight axial confinement aligns the dipoles so that they are predominately
repulsive and, for sufficiently low densities or interactions strengths, stabilizes the condensate. In
the presence of a 1D lattice, the attraction from the dipoles at other lattice sites extends thecondensate in the axial direction, increasing the integrated axial density and, ultimately, making
the system less stable. This destabilization is made less dramatic as dlat is increased.
To study the structure and stability of the infinite lattice, we choose specific trap aspectratios and explore the parameter space defined by (N − 1)gd /aρ and dlat . Figure 8.4 shows the143

Figure 8.5: Structure/stability diagram for an infinite lattice of dipolar BECs in traps with aspectratio λ = 20 as a function of lattice spacing dlat /az and interaction strength (N − 1)gd /aρ . The
inset shows a close-up of the diagram at the parameters indicated. The pink (darker) region in theinset indicated parameters where the dipolar BECs have biconcave density. An isodensity plot of
a dipolar BEC with biconcave density is shown in this inset.

region of dynamic stability for an infinite lattice of dipolar BECs in traps with λ = 10. For
lattice spacings dlat /az . 5, the condensate wave functions at adjacent sites overlap and the strong
dipole-dipole attraction leads to complete instability. In this figure, the colored regions indicate
dynamic stability and the pink (dark) regions indicate parameters at which the dipolar BECs exhibit
biconcave density. As dlat /az is increased, the diagram approaches that given by a line at λ = 10
in figure 8.3 for a single dipolar BEC. However, for smaller lattice spacings, a second biconcave
island appears. Without the presence of the lattice, biconcave structure would not exist for theseparameters. Thus, this structure is “emergent” in the lattice system. The inset in figure 8.4 shows
an isodensity plot of a dipolar BEC with biconcave density.
Figure 8.5 shows the region of dynamic stability up to lattice spacings of dlat /az = 80 for
a infinite lattice of dipolar BECs in harmonic traps with λ = 20. Here, the convergence of thestability line to (N − 1)gd /aρ ∼ 550 is clear. The inset shows a close-up view of the diagram wherea biconcave island is predicted to exist. As the aspect ratio is increased, the values of interaction
strength (N − 1)gd /aρ that the biconcave islands span becomes relatively smaller compared to theasymptotic value of the stability line. Figure 8.6 shows the stability lines for infinite lattices with

144
3

×10

(unstable)10

λ = 50λ = 100
λ = 150(N − 1) agdρ

8
6
4
2
0

(stable)

6

7

8

9

10

dlat /az
Figure 8.6: Stability lines for an infinite lattice of dipolar BECs for aspect ratios λ = 50, 100, 150
as a function of lattice spacing dlat /az and interaction strength (N − 1)gd /aρ . The parametersbeneath the lines are dynamically stable, while those above the lines are dynamically unstable.

aspect ratios λ = 50, 100, 150. We find stability islands that exist at the stability threshold within
the lattice spacings dlat /az = 6 to 10 for all of these aspect ratios. Because they are so narrow in(N − 1)gd /aρ , though, they are not included in this plot.
By working in the ρ- and z-coordinates, a cylindrical symmetry is assumed. However, weknow from chapters 5 and 6 that dipolar BECs with biconcave density profiles are dynamicallyunstable to angular modes, or quasiparticles. While the method used here is sensitive to dynamic
instabilities that are purely radial, an extra step must be taken to detect angular instabilities. To
do this, we solve the BdG equations where the BdG modes are assumed to have the form (5.6).
In doing so, we treat the system as a single dipolar BEC with a modified mean-field potential dueto the presence of BECs in other lattice sites, where the other BECs are stationary and are notexcited. Like the single dipolar BEC in section 5.2, we find that the biconcave structures in theinfinite lattice are, for some critical density or ddi, dynamically unstable to angular quasiparticles
with m ≥ 2. In chapter 6, we show that this angular instability leads to angular collapse, or
collapse with angular nodes, of the biconcave dipolar BECs. A measurement of the character of145collapse, whether it be radial or angular, then provides a tool to map the structure along thestability threshold of the system.
In our analysis, we found exotic ground state densities very close to the stability threshold,like those found for a finite lattice in Ref. [194]. These solutions host multiple radial density oscillations, however, we find they are dynamically unstable and are thus unlikely to be experimentallyobservable.

8.4

Finite Lattice
While the infinite lattice of dipolar BECs provides a clear, simple example of emergent

structure in this system, it is a difficult system to realize experimentally. In a realistic experiment,
the lattice has a finite extent and the occupations of the sites vary from site to site. To model
this more realistic lattice system, we consider an odd number of occupied lattice sites indexed byj ∈ [−jlat , jlat ] where jlat = (Nlat − 1)/2 with particle number given by a Gaussian distribution,
Nj = Nmax exp [−(j/jlat )2 ], where Nmax is the particle number in the condensate in the center ofthe lattice at site j = 0, and the outer-most sites have particle number Nmax /e [194].
Instead of using an analytic form for the axial parts of the condensate wave functions, wesolve the coupled set of GPEs given by Eq. (8.1) on a full grid (large enough to encapsulate theentire lattice) in ρ and z for each condensate. We find good convergence by using the conjugate
gradients method to minimize the full energy functional of the system [102]. Additionally, to ensure
numerical precision we apply a cutoff to the ddi in ρ and z so that a relatively small grid can beused while eliminating the effects of artificial “image” condensates that are present due to the useof the FFT algorithm in our calculation. We give the momentum-space form of the ddi with sucha cutoff in section 5.1.1.
As an example, we consider a lattice with trap aspect ratios λ = 50, jlat = 4 (corresponding
to 9 occupied lattice sites), lattice spacing dlat = 8az and (Nmax − 1)gd /aρ = 550 on a numericgrid of size [Nz , Nρ ] = [1024, 128]. Figure 8.7 shows the density at z = 0 of a dipolar BEC at thecenter of the lattice (j = 0) and, for comparison, the density of a dipolar BEC with the same trap

Density at z = 0

146

with latticewithout lattice

0.02

0.01

0

5

10

15

ρ/a ρ
Figure 8.7: Radial densities at z = 0 of a dipolar BEC with (N − 1)gd /aρ = 550 in a trap withaspect ratio λ = 50. The blue dashed line shows the density of a dipolar BEC in a single harmonic
trap, and the red solid line shows the density of a dipolar BEC in the center site of a 1D latticewith nine occupied sites (jlat = 4). This dipolar BEC exhibits biconcave structure, while the single
dipolar BEC does not, demonstrating the emergence of this structure in the lattice system. Thesedensities were calculated by solving the GPE (coupled GPEs) exactly on a full numeric grid.

aspect ratio and ddi strength (N − 1)gd /aρ = 550 but without the presence of the lattice. Whilethe dipolar BEC in the single trap does not exhibit biconcave structure, the dipolar BEC in thelattice does, showing that this emergent structure in the lattice system is present not only in theinfinite lattice system, but also in the experimentally realistic system of a finite lattice with variable
occupancy. Indeed, such a system is realizable with atomic 52 Cr, having a permanent magnetic
dipole moment of µ = 6µB where µB is the Bohr magneton, axial harmonic oscillator frequenciesof ωz = 2π × 30 kHz and a maximum condensate occupancy of Nmax ≃ 77 × 103 atoms.
The experimental observability of biconcave structure in a lattice of dipolar BECs is an
important point to address. While non-destructive phase-contrast imaging techniques of trapped
condensates have been successful [198], it is questionable whether such techniques can resolve thesmall differences in the spatial density of the biconcave condensate, especially in a 1D latticewhen the other trapped condensates do not necessarily exhibit such structure. However, it was
shown in chapter 6 that dipolar BECs with biconcave structure will, at a critical density, collapse

147
anisotropically with nodal structure about the ring of maximum density. This angular characteris predicted to be preserved in the expansion of the collapsed state, thereby providing an indirect
signature of biconcave structure in the expanded cloud. In a system of dipolar BECs with non-zero
scattering lengths, collapse of the system can be induced by decreasing the scattering length below
a critical value via a Fano-Feshbach resonance, as discussed in chapter 6 [37].

8.5

Conclusion
In conclusion, we have mapped the structure and stability of a lattice of interacting, purely

dipolar BECs. By asserting an analytic form for the axial part of the condensate wave functions(Eqs. (8.7)-(8.9)), we derive a simple, modified GPE for the radial part of the wave functions
when the lattice is infinite. We find isolated regions (“islands”) in the parameter space defined by
the lattice spacing and the ddi strength where the dipolar BECs are predicted to exhibit biconcave
densities. To model a more experimentally realistic system, we consider a finite lattice with varying
condensate number and solve the coupled set of GPEs exactly on a full numeric grid. In doing
so, we show that this emergent biconcave structure should be observable in a finite 1D lattice ofdipolar BECs of atomic 52 Cr.

Chapter 9
SummaryThis dissertation presented a number of theoretical results concerning an ultracold Bose gas,or Bose-Einstein condensate (BEC), in the presence of both isotropic short-range and dipolar interactions. This work was strongly motivated by recent experimental advances in the cooling andtrapping of atomic species that possess permanent magnetic dipole moments, such as 52 Cr [16] and
164 Dy [17], which were both recently Bose-condensed in the laboratory setting. Besides these dipo-

lar atoms, the work presented in this thesis is relevant to gases of heteronuclear polar molecules,
which can achieve even larger (electric) dipole moments when polarized in an applied field, andhave been produced in their absolute rovibrational ground state and cooled to near-degeneratetemperatures [24, 25, 26]. While an enormous body of literature, both theoretical and experimental in nature, has been generated with regards to ultracold quantum gases with dipolar interactions [199, 16], we have focused on a particular phenomenon, being the emergence of a roton-like
quasiparticle mode in the trapped dipolar BEC. This feature of dipolar BECs shares many similarities with the roton that exists in the superfluid helium system, however, the dilute BEC is mucheasier to control experimentally and describe theoretically. Thus, the dipolar BEC presents us with
the ideal testing ground to explore the role of the roton in a superfluid system.
We have made a point to present theoretical predictions that speak to current experiments,
or experiments that are realizable in the foreseeable future. Thus, much of this work is phenomenological in nature. However, after giving a short background of low-temperature physics in chapter 2,
in chapter 3 we presented a detailed account of the theory that underlies our phenomenological149
predictions. Therein, we discussed the Bogoliubov approximation to the problem of an ultracold
quantum many-body system of bosons at ultracold temperatures, out of which comes the meanfield equations that describe the condensate and the quantum fluctuations (quasiparticles) of theultracold, dilute Bose gas, being the Gross-Pitaevskii equation and the Bogoliubov de Gennes equations, respectively. Additionally, we discussed the treatment of two-body interactions in the BEC
and justified a two-body pseudopotential for both isotropic short-range and long-range, anisotropic
dipole-dipole interactions. In chapter 4, we applied these theoretical results to the homogeneous(translationally invariant) three-dimensional and quasi-two-dimensional dipolar Bose gases at zerotemperature. By studying the energetics and phonon modes of the fully three-dimensional system,
we showed that the polarized dipolar BEC is significantly stabilized when confined along the direction of polarization, however, instability can still occur in this trapped system for sufficiently
large dipole-dipole interaction strength or condensate density due to the emergence and subsequent
softening of the roton. This roton instability is thus found to be both density dependent and local, meaning that the dipolar BEC will collapse locally on a length-scale set by the size of theconfinement.
In the subsequent chapters 5-8, we proceeded to characterize the stability, quasiparticle excitations, vortex states and superfluid properties of the fully trapped dipolar BEC. Because thedipole-dipole interaction potential for polarized dipoles is cylindrically symmetric, the fully trapped
dipolar BEC possesses cylindrical symmetry for trapping geometries that are cylindrically symmetric, as well. For this case, we constructed an efficient numerical algorithm for treating the kinetic
energy and dipolar interaction terms in the Gross-Pitaevskii and Bogoliubov de Gennes equations
that can easily be adjusted to handle condensate and quasiparticle states with well-defined vorticity. We applied this algorithm, which has its footing in the discrete Hankel transform [102](see appendix D), to rotationless dipolar BECs and dipolar BECs with both singly- and doublyquantized vortices. An analysis of the discrete quasiparticle spectrum of these systems revealed
that the vortex states and the strange “biconcave” states of the rotationless dipolar BECs go unstable to discrete rotons with angular nodal structure. In chapter 6, we showed via explicit numeric

150
simulation that, for example, the 52 Cr BEC can exhibit angular collapse and subsequent angular
explosion under realistic experimental conditions. In chapter 8, we showed that a dipolar BEC
loaded on a 1D lattice exhibits wildly modified stability properties and emergent biconcave structure that is not present in a single dipolar BEC in the absence of the other occupied lattice sites.
Regarding the dipolar BEC with a singly-quantized vortex, we found regions in parameter spacenear the instability threshold where this system exhibits radial density oscillations near the vortex
core. We attributed such structure to the static manifestation of the discrete radial roton in the
ground state of the dipolar BEC by applying a linear perturbation theory to the Gross-Pitaevskii
equation that shows the equivalence of the condensate response to a small time-independent perturbation and the discrete roton mode. In chapter 7, we examined the superfluid properties of
the quasi-two-dimensional and fully trapped dipolar BECs by direct numerical simulations of an
object (blue-detuned laser, in an experiment) moving through the system at varying velocity. We
showed that the roton not only serves to decrease the critical velocity as a function of density inthe fully trapped system, but that the dipole polarization can be tilted to create an anisotropic
superfluid with anisotropic critical velocity. In both cases, we showed that a measurement of thecritical velocity in the experiments proposed therein would correspond to a measurement of theroton in a dipolar BEC.
To this date, the roton has remained elusive to experimentalists. However, motivation forsuch a measurement and rapidly advancing experimental capabilities inspire confidence that it’spresence will soon be empirically known. As forementioned, a large point in this thesis was topredict phenomena that will not only demonstrate the novel behavior of a superfluid with dipolarinteractions, but also serve as indirect evidence of the presence of the roton in a dipolar BEC. In
addition to the experimental work that is under active investigation, there remains a plethora ofphysics to explore theoretically with regards to ultracold dipolar gases. Extensions of the work
presented in this thesis include, but are certainly not limited to the possibility of macroscopicallyoccupying a roton field and the physics of roton-roton interactions. At the mean-field Bogoliubov
level of the many-body field theory used in this thesis, such physics is automatically excluded as

151
the the Bogoliubov approximation permits only perturbative occupations of quasiparticle statesand does not account for interactions between quasiparticles. Additional extentions of the work
presented in this thesis include the effect of temperature on the rotonization of a dipolar Bosegas, the miscibility-immiscibility transition in a two-component dipolar Bose gas and the role ofthe dipole-dipole interaction in spinor BECs at finite temperature. Such work is under active
investigation by the author.

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Appendix A
The Convolution Theorem

The calculation of the dipolar mean-field potential (3.63) and other convolution integrals inthis thesis require special treatment. Consider the integralZdx′ V (x − x′ )n(x′ )(A.1)

where V (x) and n(x) are both well-behaved, i.e., they have well-defined Fourier transforms. Here,V (x − x′ ) can be any function, though the relevant form for this work is the two-body dipole-dipole
interaction (ddi) potential given in (3.61). In three-dimensions, the 1/r 3 real-space behavior of the
ddi does not present a true mathematical divergence, but can be very difficult to handle numericallywhen the integrals are discritized onto a real-space grid. For example, numerical sums must be
taken over the entire spatial grid (corresponding to x′ ) for each grid point (corresponding to x),
and the divergence corresponding to the r = 0 pole of the ddi can not be handled numerically.
These complications can be overcome by moving to momentum-space. We define the Fouriertransform operator F and its inverse F −1 to move into and back from momentum-space, respectively,f˜(k) = F [f (x)] =

Z

h
i Z
f (x) = F −1 f˜(k) =dxf (x)eik·x

(A.2)

dk ˜
f (k)e−ik·x .
(2π)3

(A.3)

166
With these operations, we can rewrite the convolution in Eq. (A.1) as
Z

Z


 Z
dk
dq
−ik·(x−x′ )−iq·x′
dx V (x − x )n(x ) = dx
Ṽ (k)eñ(q)e
(2π)3
(2π)3
Z
Z
Z
dq
dk
′
−ik·x
Ṽ
(k)e
ñ(q)
dx′ e−i(q−k)·x
=
(2π)3
(2π)3
Z
Z
dk
dqṼ (k)e−ik·x ñ(q)δ(q − k)
=
(2π)3
Z
dk
=
Ṽ (k)ñ(k)e−ik·x .
(2π)3
′

′

′

Z

′

(A.4)

Thus, we see that the convolution of a function n(x′ ) with the function V (x − x′ ) is just given by
the inverse Fourier transform of the product of their individual Fourier transforms,Zhidx′ V (x − x′ )n(x′ ) = F −1 Ṽ (k)ñ(k) .

(A.5)

Appendix B
Momentum-Space Dipole-Dipole Interaction Potential

Here we calculate the momentum-space dipole-dipole interaction potential for dipoles with
arbitrary, but identical polarization. Consider the unit polarization vector dˆ = x̂ sin α cos η +ŷ sin α sin η + ẑ cos α, tilted by an angle α off of the z-axis and an angle η off of the x-axis. Then,if we say that the vector x separates the dipoles we can write dˆ · x̂ asdˆ · x̂ = cos θ cos α + cos φ sin θ sin α cos η + sin φ sin θ sin α sin η,(B.1)

thus, the ddi from Eq. (2.20) takes the form
Vd (x) = d2

1 − 3 (cos θ cos α + cos φ sin θ sin α cos η + sin φ sin θ sin α sin η)2
.
x3

(B.2)

To simplify the math that follows, we use the spherical harmonics Ylm ↔ Ylm (θ, φ) [54] to expand (B.2),
r



√
π
1
d2
2
Vd (x) = 3 sin α − 4cos2 αY20 − 4 π sin2 α cos2 η Y00 − √ Y20
x
5
5
r
r
6π
6π
−i
sin2 α sin 2η (Y2−2 − Y22 ) −sin 2α cos η (Y2−1 − Y21 )
5
5
)
r
6π
sin 2α sin η (Y2−1 − Y21 ) .
−i
5

(B.3)

We wish to compute the Fourier transform of (B.3). To do this, we use the definition of the Fouriertransform (A.2) and the expansion of the spherical plane wave [91]eik·x = 4π

X
l,m

⋆
il Ylm(k̂)jl (kx)Ylm (x̂),

(B.4)

168
where jl (kx) is the spherical Bessel function of order l, and the orthonormality relations of thespherical harmonics,ZΩYlm (θ, φ)Yl⋆′ m′ (θ, φ)dΩ = δll′ δmm′ .

(B.5)

The resulting momentum-space ddi interaction potential isṼd (k) = πd2 2 sin2 α sin2 θk cos (2φk + 2η) + 2 sin 2α sin 2θk cos (φk + η)




4
2
2
+
− 2 sin α 3 cos θk − 1 .
3Note that this expression was independently derived in [201], giving a consistent result.

(B.6)

Appendix C
Quasi-2D Dipolar Interaction Potential

Here we calculate the effective dipole-dipole interaction (ddi) potential for the quasi-2D geometry given by Eq. (4.22) in section 4.2.2,′

Vq2D (ρ − ρ ) =

Z

dz

Z

dz ′ χ⋆ (z)χ⋆ (z ′ )Vd (x − x′ )χ(z ′ )χ(z),(C.1)

where χ(z) is the axial wave function in the quasi-2D geometry given in Eq. (4.19). In this work,R
we use the quasi-2D potential to calculate terms like dρ′ f (ρ′ )Vq2D (ρ − ρ′ ). According to the
convolution theorem (A.5), we can calculate this integral by taking the inverse Fourier transform of
the product of f˜(kρ ) and Ṽq2D (kρ ), so knowledge of Ṽq2D (kρ ) is sufficient to handle this calculation.
This quasi-2D momentum-space interaction potential is given by the momentum-space equivalentof Eq. (C.1),
Ṽq2D (kρ ) =Z

dkz ñ2z (kz )Ṽd (k),

(C.2)

where


 2 
1 22
ñz (kz ) = F χ (z) = exp − kz lz .
4

(C.3)

Consider the polarization that is described in appendix B but, without loss of generality, we set
η = 0 so the the polarization is tilted into the x-axis only. At the end of this calculation, we
rotate the coordinates to generalize to any tilt into the x-y plane. We rewrite the result for the170momentum-space ddi (B.6) asṼd (k) = πd2 2 sin2 α sin2 θk cos2 φk − sin2 φk + 4 sin 2α sin θk cos θk cos φk+ (4/3 − 2 sin2 α)(3 cos 2 θk − 1)
(
 2

)
2 − k2
k
4
k
k
k
x
y
z x
= πd2 2 sin2 α.
− 2 sin2 α
3 z2 − 1+ 4 sin 2α 2 +
k2
k
3
k

(C.4)

We now compute the integral (C.2), where we leave out the second term in Eq. (C.4) because it islinear in kz and therefore integrates to zero,
(
Z


 2
)
kx2 − ky2
4
kz
2
Ṽq2D (kρ ) = πddkz e+
− 2 sin α
3 2 −1
2 sin α
k2
3
k
!
"


2 l2
2
2
kρ
z kx − ky
kρ lz
2
2
2
= πd 2 sin α πeerfc √
kρ
2





2
2
kρ lz
√
4
kρ lz
2
+
− 2 sin α
3 2πlz − 3πe 2 kρ erfc √
−1
3
2
..
.





kρ lz
kρ lz
gd
2
2
=√
cos αF⊥ √+ sin αFk √
,
2πlz
2
2
2

− 12 kz2 lz2

2

(C.5)

where gd = 4π~2 add /M , add is given by Eq. (2.21) and the dimensionless functions F⊥ and Fk
account for the projection of dˆ onto the z-axis and onto the x axis, respectively. The function
Fk depends explicitly on kx , or the direction on which the dipoles are tilted. Without loss ofgenerality, we can rotate our coordinates and replace this momentum dependence with kx → kd =q
kx2 cos2 η + ky2 sin2 η, giving
√
2
(C.6)
F⊥ (q) = 2 − 3 πqeq erfc[q]√ q2 2
Fk (q) = −1 + 3 π d eq erfc[q],
q

(C.7)

where erfc is the complimentary error function [54]. Thus, we can write the quasi-2D momentumspace ddi as
gd
Ṽq2D (kρ ) = √F
2πlz
where F (q) = cos2 αF⊥ (q) + sin2 αFk (q).



kρ lz
√
2



,

(C.8)

Appendix D
Discrete Hankel Transform

Here, we give the algorithm for the discrete Hankel transform (DHT) of a function f (ρ)eimϕ .
This algorithm is also given in [102], and is derived in detail in [205]. From Eq. (5.5) in the text,
we see that the Hankel transform of such a function is given by projecting f (ρ) onto the mth order
Bessel function Jm (kρ). Here, we assume that the function f (ρ) = 0 for all ρ > R and its transformf˜(k) = 0 for all k > K. Additionally, we sample the real- and momentum-space functions on gridswith N points that are defined in terms of the roots of the mth order Bessel function αmj , so that
Jm (αmj ) = 0 for all j,ρmj = αmj /K(D.1)

kmj = αmj /R,(D.2)

where j = 1, 2, . . . , N . Then, the DHT of the function f (ρmj ) is given byNα α 
2 X f (ρmj )mi mj.
f˜(kmi ) = 2
J
m
2
K
RK
J
(α )
j=1 m+1 mj(D.3)

Similarly, the inverse DHT of the function f˜(kmj ) is given byf (ρmi ) =Nα α 
2 X f˜(kmj )
mj mi
J
m
2
R2
RK
J
(α
)
mj
j=1 m+1(D.4)

By defining the functions
α 
R
mj
f
|Jm+1 (αmi )|
K

αmj 
K
,
f˜
F̃ (j) =|Jm+1 (αmi )|RF (j) =

(D.5)
(D.6)

172
the DHT in Eq. (D.3) reduces toF̃ (i) =NX

Tij F (j),

(D.7)

j=1

where the matrix Tij is given by
α αmj 2Jm miRK
Tij =.
|Jm+1 (αmi )||Jm+1 (αmj )|RK(D.8)

We determine the cutoffs R and K by first choosing a reasonable R based on the parameters in the
problem, and then K is given by K = αm,N +1 /R.
Interestingly, defining the radial grids in terms of the zeros of the mth order Bessel function,
as in (D.1) and (D.2), and enforcing that f˜(k) = 0 for all k > K allows us to write the radialR∞
integrals I[f ] = 0 f (ρ)ρdρ as [102]∞mi
2 X f αK,I[f ] = 22K
Jm+1
(αmi )(D.9)

i=1

which has a form very similar to that of Gaussian quadrature. To a very good approximation, we
employ this formula on the finite grid with N points, and thus truncate the sum in Eq. (D.9) at
i = N . The accuracy of this numeric integration is increased with increasing N and R, though wefind that sufficient convergence is obtained for typical grids used in this work.

Appendix E
Energy-Functional Minimization via Conjugate Gradients

In this appendix we describe the conjugate gradients (cg) algorithm for finding the functionthat minimizes the Gross-Pitaevskii energy functional, Eq. (5.3) [206, 155]. We rewrite the energyfunctional here, representing the real-space condensate wave function φ(x) by the “ket” |φi,E[φ]hφ|Ĥ (1) |φi N − 1 hφ|hφ|V̂ |φi|φi
=
+
,
N
hφ|φi
2
hφ|φi2(E.1)

where Ĥ (1) is the single particle Hamiltonian (3.16) and V̂ represents the two-body interaction
potential (3.61), so hφ|V̂ |φi is the condensate mean-field potential. In (E.1), we explicitly normalize
the terms so that the energy of a non-normalized state can be obtained.
To begin the cg algorithm, we choose a trial wave function |φ1 i and modify it with thevariational term |φ1 i → |φ1 i + λ|χ1 i, where hφ1 |χ1 i = 0 and λ is a variational scalar. This modified
wave function is then substituted into the energy functional (E.1), which then becomes a 4th orderpolynomial in λ. The minimum of this polynomial is then easily found and the λ that defines theminimum, λmin , determines the modified wave function, |φ2 i = |φ1 i + λmin |χ1 i, and this process isiterated until sufficient convergence is reached.
Let us adopt the notation that |φi i and |χi i correspond to the states used in the ith iterationof the cg algorithm. To calculate |χi i given |φi i, we simply take a gradient of the energy E[φi ]/Nand add a “preconditioning” term P (k) by hand, which gives the Gross-Pitaveskii type expression
h
h

ii
|χi i = F −1 P (k)F Ĥ (1) + (N − 1)hφi |V̂ |φi i |φi i ,

(E.2)

174
where F is the 3D Fourier transform operator, P (k) = 1/(k2 /2 + L), L = max (E, Ek ), E is thetotal energy per particle in the state |φi i and Ek is the kinetic energy of the state |φi i. The
preconditioning function P (k) is included to accelerate the convergence of the cg algorithm [207].
To ensure the orthogonality of |φi i and |χi i, we operate of |χi i with the projection operator P̂i =1 − |φi ihφi |, which projects a function into the space orthogonal to |φi i. We also normalize |χi i ineach iteration.
Now, the energy functional E[φi + λχi ]/N takes the form
4X
E[φi + λχi ]1
Cj λj ,=
N
(1 + λ2 )2

(E.3)

j=1

where the Cj polynomial coefficients are j th order in χi , and are given by
C0 = hφi |Ĥ (1) |φi i + hφi |hφi |V̂ |φi i|φi i
C1 = hφi |Ĥ (1) |χi i + hχi |Ĥ (1) |φi i + hφi |hφi |V̂ |φi i|χi i + hφi |hφi |V̂ |χi i|φi i + hφi |hχi |V̂ |φi i|φi i+ hχi |hφi |V̂ |φi i|φi i
C2 = hφi |Ĥ (1) |φi i + hχi |Ĥ (1) |χi i + hχi |hφi |V̂ |φi i|χi i + hφi |hχi |V̂ |χi i|φi i + hφi |hφi |V̂ |χi i|χi i+ hχi |hχi |V̂ |φi i|φi i + hφi |hχi |V̂ |φi i|χi i + hχi |hφi |V̂ |χi i|φi i
C3 = hφi |Ĥ (1) |χi i + hχi |Ĥ (1) |φi i + hχi |hχi |V̂ |χi i|φi i + hχi |hχi |V̂ |φi i|χi i + hχi |hφi |V̂ |χi i|χi i+ hφi |hχi |V̂ |χi i|χi i
C4 = hχi |Ĥ (1) |χi i + hχi |hχi |V̂ |χi i|χi i.

(E.4)

We calculate λmin by solving for ∂λ E[φi + λχi ]/N = 0, where


4
X
1
4λj+1
∂ E[φi + λχi ]
j−1
C
=
.
jλ
−
j
∂λ
N
(1 + λ2 )2
1 + λ2(E.5)

j=0

The idea of the cg algorithm is to sample the shape of the Gross-Pitaevskii energy manifold ona large scale, accomplished by calculated the polynomial line (E.3) through the manifold. Finding
the minimum of this line then tells us what linear combination of |φi i and its orthogonal counterpart|χi i minimizes the energy in the space spanned by these functions. For a purely dipolar BEC in175
a trap with aspect ratio λ = 10 and ddi strength D = 50, sampled on a grid of size (Nρ , Nz ) =(128, 64), we find that convergence to a part in 10−14 is reached in about 4 seconds.

Appendix F
Radial Grid Interpolation

In appendix D we introduce radial grids in real and momentum-space that are defined interms of the zeros of the mth order Bessel function, and are thus termed mth order grids. Suchgrids are constructed to properly handle the numeric integration and discrete Hankel transformsof functions with angular dependence eimϕ . In the methods described in section 5.1, we see thatfunctions must be interpolated between radial grids of different orders in order to properly handle
the integrals written therein. Here, we derive an accurate interpolation scheme based on the DHT.
As explained in Ref. [153], the function f (ρ) may be expanded in an mth order Bessel series,f (ρ) =

N
X
i=1


ρ
cmi Jm αmi
R

(F.1)

Z R

(F.2)

where the coefficients cmi are given bycmi =2R2 [Jm+1 (αmi )]2

0


ρ
ρ dρ.
f (ρ)Jm αmi
R

Note that the integral in Eq. (F.2) is just the Hankel transform (5.5) with αmi /R = kmi , giving
the transformed function f˜(ki ). If ρ is discretized in Eq. (F.1), then this prescription gives exactly
Eq. (D.4).
Consider the case where the function f (ρ) is defined on the grid ρmi but needs to be definedon the grid ρni , with n 6= m. To do this, we expand f (ρni ) in a Bessel series,f (ρni ) =

N
X
j=1


ρni 
cmj Jm αmj
R

(F.3)

177
where the coefficients cmj are given by Eq. (F.2) and are computed in terms of the zeros of the Bessel
function of order m. However, in Eq. (F.3), the function is expanded on the grid ρni , proportional
to the zeros of the Bessel function of order n. The interpolation algorithm then simply follows byinserting the expression for the coefficients,Nα α 
2 X f˜(kmj )mj ni
f (ρni ) = 2
J
m
2
R
S
Jm+1 (αmj ),j=1

where f˜(kmj ) is the discrete Hankel transform of f (ρmj ), given by Eq. (D.3).

(F.4)

Appendix G
Calculation of the Mean-Field Potential in Reduced Dimensions

We consider the calculation of the mean-field potential due to the dipole-dipole interaction(ddi) in two different geometries, one with λ → 0 (a quasi-two dimensional (2D) geometry) and one
with λ → ∞ (a quasi-one dimensional (1D) geometry). In the quasi-2D geometry, we assume thatthe condensate wave function depends only on z and is homogeneous in the ρ-direction (or in thex- and y-directions) and in the quasi-1D geometry, we assume that the condensate wave function
depends only on ρ (or on x and y) and is homogeneous in the z-direction.
We begin with the expression for the dipole-dipole interaction potential in momentum-spacefrom Eq. (3.62)
Ṽd (k) =
4π 2
d 3 cos2 θk − 1 ,
3

(G.1)

where θk is the angle between the direction of the dipole polarization (ẑ or k̂z , for the dipolar BEC
we are considering) and the vector k. Using this momentum-space representation, the coordinatespace mean-field potential due to the dipole-dipole interaction is given by the convolution of Ṽd (k)with the condensate density in momentum-space, ñ(k),hi
Ud (r) = F −1 Ṽd (k)ñ(k) ,(G.2)

where F −1 is the inverse Fourier transform operator. First, consider the quasi-2D geometry, inwhich the condensate density is homogeneous is x and y. The condensate density in momentumspace is then given by the Fourier transform,ñ2D (k) = F [n2D (z)] = ñ2D (kz )δ(kx )δ(ky ).

(G.3)

179
Substituting this result and Eq. (G.1) into Eq. (G.2) and writing cos2 θk = kz2 /(kx2 + ky2 + kz2 ) gives
an expression for the mean-field potential in the quasi-2D geometry,
Ud2D (r) = F −1






4π 2
kz2
d 3 2
− 1 ñ2D (kz )δ(kx )δ(ky ) .
3kx + ky2 + kz2(G.4)

The operation of the inverse Fourier transform on this momentum-space function gives
Ud2D (r) =8π~2 add
8π 2
d |φ0 (z)|2 =|φ0 (z)|2 ,
3
M

(G.5)

where |φ0 (z)|2 is the coordinate-space condensate density in the quasi-2D geometry. We carry out
the same calculation for the quasi-1D geometry, where the condensate density in momentum-spaceis given byñ1D (k) = F [n1D (x, y)] = ñ1D (kx , ky )δ(kz ).

(G.6)

Substituting this function into Eq. (G.2) gives
Ud1D (r) = −4π 2
4π~2 addd |φ0 (ρ)|2 = −|φ0 (ρ)|2 ,
3
M

(G.7)

where |φ0 (ρ)|2 is the coordinate-space condensate density in the quasi-1D geometry, written interms of ρ instead of x and y.

180

Appendix H
Modified GPE Using 0th and 2nd Harmonic Oscillator Wave Functions

Consider the ansatz given by Eqs. (8.7)-(8.9) for a system with a single harmonically trapped
dipolar BEC, so the indexing of the condensate wave function can be ignored and we can simply
write φ0 (x) = ψ(ρ)χ(z). We derive the modified GPE by multiplying the (dimensionless) GPE,
1 2
− ∇ + U (x) + Ud (x) − µ φ0 (x) = 0,
(H.1)
2
by χ(z) and integrating over z. In Eq. (H.1), U (x) is the trapping potential, Ud (x) is the ddi
mean-field potential and µ is the chemical potential. This operation gives the modified GPE,




gd −1
kρ lz
(1)
Ĥeff (ρ) + 2 F2D
ψ(ρ) = 0,(H.2)
ñρ (kρ )Feff √
lz
2
(1)

where Ĥeff (ρ) is the effective single-particle Hamiltonian,





1 5 2
1
A2 1
1
1 2 1 2 2
(1)
2 2
2 2
+ λ lz− λ lz+ A −√,Ĥeff (ρ) = − ∇ρ + ωρ ρ − µ +
2
2
lz2
4 4 2
1 + A22
2 lz2

(H.3)

and Feff (x) is given by

√

 3 2
2
4
2
2A
1
+
Feff (x) =
A
3
+
5x
+
6x
3x
−
1
+

2
2
2
4
1 + A2
1

2



1
1
+ √ A32 1 + 9x2 + 42x4 + 12x6 + A42 41 + 3x2 81 + 134x2 + 60x4 + 8x6
64
4 2
√ h



√


3 π
2 + 2A2 1 + A22 x2−
1 + A2 A2 2 + A22 + 2
2


√



√
1 3 8
4
2
6
2
x2
2 + 2A2 x + A2 xxe erfc [x](H.4)+ A2 3 + 4 2A2 + 5A2 x + A2
4

and erfc is the complimentary error function. The corresponding dipolar mean-field energy is then



Z
kρ lz
gd
−1
2
d ρ nρ (ρ)F2D ñρ (kρ )Feff √
.
(H.5)
Ed =
lz
2