Heat Death of an Ultracold Dipolar Gas
byReuben R. W. Wang
B.Eng., Singapore University of Technology and Design, 2019
M.S., University of Colorado Boulder, 2022

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
2024

Committee Members:
John L. Bohn, Chair
Andreas Becker
Josef Michl
Ana Maria Rey
Jun YeWang, Reuben R. W. (Ph.D., Physics)
Heat Death of an Ultracold Dipolar Gas
Thesis directed by Prof. John L. Bohn

Optically trapping bulk gases of atoms and molecules at sub-microKelvin temperatures has
become commonplace in several labs around the world. In particular, realizing low temperaturesamples of dipolar atoms and diatomic molecules has garnered great interest from the ultracold
community, due to their long-range and anisotropic nature. At temperatures not yet low enough toachieve macroscopic quantum degeneracy, the gas constituents move about more or less classically,
but experience collisions that can only be described accurately with quantum mechanics. By aligningthese dipoles with an external field, collisions inherit highly anisotropic cross sections from theirdipole-dipole interactions, leading to a wealth of tunable anisotropic collective dynamics.
I focus on characterizing anisotropic thermalization dynamics, which bears importance inmany experimental applications. In fact, progressing in parallel with my graduate work has been
exciting experiments with ultracold dipolar atoms and molecules. I tell the story of several collaborations in which my work was used to determine the scattering length of erbium atoms, characterize
universal dipolar scattering, and open opportunities for optimal evaporative cooling of moleculargases1 . This thesis lays out the development of several theoretical tools for investigating the relaxation of a nondegenerate dipolar gas, tracking its intricate journey to an eventual heat death.

P.S. If you bore easily, well, this thesis might just do nothing at all to help with that.
"There will be some math but just endure it for a while and very soon, it will all go away."– Umesh Vazirani (CU Boulder Physics Colloquium), 2023.
1

All accounts here are based on true events.

Dedication

For my parents.
To Andrew.

iv

Acknowledgements

Thank you, John. You’ve taught me so much both in science and outside of it. How someone
with such scientific authority can exude the humility and radical candor you do, truly eludes me.
A large part of who I am and want to be has been shaped by your mentorship and for that, I am
immeasurably grateful. I don’t know how many graduate students would willingly say this by thetime they graduate, but I’m genuinely going to miss being your student.
To the wonderful Bohn group members, Eli and Hui, the lovely atmosphere you have creatednever ceased to foster a deep scientific curiosity in me, day in and day out. Eli (a.k.a. the Bob Rossof physics), your passion for learning and warmth as a senior graduate student has been paramountto my growth as a budding researcher. May you be blessed with all the Illegal Pete’s your heart
will ever desire. And as though this godsent bunch wasn’t enough, the greater JILA ecosystemsurrounding our little corner in the S-wing made my PhD a truly unparalleled experience. To thepassionate faculty, staff and students, many of whom I owe this thesis to, you have made JILA theworld leading scientific institute it is. It’s been a privilege being part of it.
Despite what pop culture had me believing as a kid, physics is not done in isolation. Turns
out, "evil genius hell-bent on world domination" is not the best model of a scientist to take from. I
certainly could not have done most of my PhD work without the many amazing collaborators I was
lucky enough to work with. To the wonderful theorists, Bo Xing, Eli Halperin, Hui Li, Anthony Polloreno, Ana Maria Rey, David Wellnitz and Daniel Messenger, you’ve all made a student out of me.
I must also express exceptional gratitude to the experimentalists in the Ye, Ferlaino and Bloch/Luogroups. These include Jun Ye, Jun-Ru Li, Kyle Matsuda, Annette Carroll, Calder Miller, Junyu

vLin, Henrik Hirzler; Francesca Ferlaino, Alex Patscheider, Manfred Mark; Xin-Yu Luo, Andreas
Schindewolf, Shrestha Biswas, Xing-Yan Chen and Immanuel Bloch. I’m humbled by the incredible
patience you’ve had in answering the many questions I posed as a naive theorist. Collaborating withyou has forever arrested my passion for the intoxicating pas de deux of experimental and theoreticalphysics. It’s an honor to be part of the atomic and molecular physics world you are shaping.
Before my foray into physics (formally), I had first pursued an undergraduate degree in engineering because, well, the robotic arms at open day looked pretty cool. Naturally, my eventual
transition into theoretical physics was not without invaluable guidance from my prior research mentors, professors Cheah Chin Wei, Riccardo Comin and Dario Poletti. To Dario, what an introduction
to theoretical physics research that was. I owe much of my love and continued interest in the field of
quantum science to you. In fact, I probably would not be writing this thesis if not for that summer
UROP you offered.
There are also so many people I have to thank outside the context of research. As a first, I
would be remiss not to give a massive shout-out to the consistent problem set crew that kept me
sane through the nightmare that was classes. Sun, Benton, Luca and Jacob, I could not have done
it without any one of you. To Jacob in particular, dude, we finally made it out. I sure hope you’re
still feeling peckish after all of it! Then there’s all you other marvelous Boulderites: housemates,
physics and non-physics friends. You’ve made Boulder feel like a home that I’ll always want to comeback to.
To my beloved family and friends in Singapore, for your unwavering care, love and support.
Somehow, you managed to make a global pandemic seem like a time to look fondly back upon. Mom
and Dad, you’ve provided me with more than a son could ever ask for. All of my accomplishments,
now and in the future, are yours. HO5, TGIT, CB, LDG, GG, GD, AHAB: acronyms of the groupchats with people who have taught me what friendship means. Tricia, for teaching me how to andmaking me want to be a better person. And Andrew, for always being my inspiration, confidant
and guide. Thank you.
Last but certainly not least, I extend my most sincere gratitude to one of my most trea-vi
sured intellectual mentors, Mr Buddy (pictured in Fig. 1). The lessons in waking early, correcting
magically appearing code bugs, and patience you’ve imparted have been invaluable to my PhD.

Figure 1: A true intellectual mentor. The puss in books, Mr Tesla Buddy.

If nothing else, I’d like to pass on the inspiring words a wise professor once said to us:“There are as many ways to be a successful physicist as there are successful physicists."— Prof. Jesse Thaler (MIT 8.033 lecture), 2017.
I’ve held on to those words ever since and found them to be true, time and time again.

Contents
Chapter
1 Introduction1

1.1

Why ultracold?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Why dipoles?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

What’s in this thesis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32 Crash, Bam, Pow! Ultracold Dipolar Collisions
2.1

2.2

5

Ultracold collisions of magnetic atoms . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.1

Lanthanides: Little atomic magnets . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.2

Ultracold scattering: A primer . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.3

Dipolar interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.1.4

Close-to-threshold dipolar scattering . . . . . . . . . . . . . . . . . . . . . . .

11

2.1.5

s-wave scattering in bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14Shields up! Turning reactive molecules inert. . . . . . . . . . . . . . . . . . . . . .

16

2.2.1

Scattering of static field shielded KRb molecules . . . . . . . . . . . . . . . .

17

2.2.2

Scattering of microwave shielded NaK molecules . . . . . . . . . . . . . . . .

22

3 The Jostle and Bustle of Dipoles

28

3.1

The Boltzmann transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.2

Variational solutions: Gaussians, Gaussians, Gaussians!

. . . . . . . . . . . . . . . .

30Taming the collision integral . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Direct simulation Monte Carlo solutions . . . . . . . . . . . . . . . . . . . . . . . . .
vii

37

3.2.1
3.3

viii
3.3.1

Evolution in classical phase space. . . . . . . . . . . . . . . . . . . . . . . .

38

3.3.2

The two-body collision integral . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.3.3

Some considerations and comments

46. . . . . . . . . . . . . . . . . . . . . . .

4 Return to Equilibrium I: Dilute Gases
4.1

4.2

Anisotropic thermalization of dilute dipolar gases

48. . . . . . . . . . . . . . . . . . .

4.1.1

How many collisions before one is thermalizing?

. . . . . . . . . . . . . . . .

49

4.1.2

Universal signatures of dipolar thermalization in KRb . . . . . . . . . . . . .

55

Turn up the heat: Dipolar thermalization far from threshold. . . . . . . . . . . . .

Gaussian process interpolation of differential cross sections. . . . . . . . . .

58

4.2.2

Energy dependent collisional efficiencies . . . . . . . . . . . . . . . . . . . . .

60

5.1

How do fluids flow?

5.2

Surfaces this way, fluxes that way
5.2.1

5.4

63

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63. . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Microscopic collisions to macroscopic transport . . . . . . . . . . . . . . . . .

66

And whispered in the sound... of dipoles. . . . . . . . . . . . . . . . . . . . . . . .

68

5.3.1

The speed of dipolar sound . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.3.2

Undulating sound waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5.3.3

Shear silence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Viscous welter of a trapped dipolar fluid
5.4.1. . . . . . . . . . . . . . . . . . . . . . . .

78

An effective model for damped welter . . . . . . . . . . . . . . . . . . . . . .

81

6 Theory and Experiment Collide
6.1

57

4.2.1

5 Return to Equilibrium II: Hydrodynamic Gases

5.3

48

89

Innsbruck: Cold mountains, ultracold atoms, hot science!

. . . . . . . . . . . . . . .

89

6.1.1

Extracting scattering lengths from thermalization. . . . . . . . . . . . . . .

90

6.1.2

Scattering lengths galore! . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

6.1.3

Mass scaling of the background scattering length . . . . . . . . . . . . . . . .

95

6.2

6.3

MPQ: The road to deeper Fermi degeneracy with molecules . . . . . . . . . . . . . .

96

6.2.1

Let it flow, let it flow! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

6.2.2

In-simulation thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2.3

Numerical results and prospects

Curtain call

. . . . . . . . . . . . . . . . . . . . . . . . . 102. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

105

Bibliography

Appendix
A Matrix elements of the effective microwave shielded potential

119

B Optical dipole traps for atoms and molecules

123

C The method of averages

125

D Equilibrium dipolar mean-field per particle

127

E Deriving the collisional efficiency toward thermalization

129

F The equilibrium Boltzmann equation

131

G Transport tensors from microscopic collisions

133

G.1 Anisotropic thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
G.2 Anisotropic viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
H Spatial averaging for an effective welter model

140

I

143

Quasi-empirical determination of the hydrodynamic volumeix

TablesTable
1.1

A table of the notation used for constants and variables that will appear in this thesis.
Bolded symbols are used to denote vectors, matrices and tensors.

. . . . . . . . . .

4

2.1

A system of dipole units based on the space-fixed dipole moment d.

. . . . . . . . .

11

6.1

Fano-Feshbach resonance positions and widths included into the fit of Eq. (6.5) to
as for 164 Er (left) and 166 Er (right). The error denotes the fit error of one standard
deviation. Values without error are fixed in the fitting procedure.

6.2. . . . . . . . . .

92

Fano-Feshbach resonance positions and widths included into the fit of Eq. (6.5) to
as for 168 Er (left) and 170 Er (right). The error denotes the fit error of one standard
deviation. Values without error are fixed in the fitting procedure.

6.3. . . . . . . . . .

92

Table of parameter values for the potential confining a gas of fermionic 23 Na40 K
molecules. h denotes Planck’s constant. . . . . . . . . . . . . . . . . . . . . . . . . .

98

FiguresFigure1

A true intellectual mentor. The puss in books, Mr Tesla Buddy.
x. . . . . . . . . . .

vi

xi
2.1

Plot of the Zeeman energy levels as a function of applied magnetic field for bosonic
166 Er (a), and fermionic 167 Er (b) isotopes of erbium.

2.2. . . . . . . . . . . . . . . . .

6

Cartoon of quantum scattering. The red curve denotes the incident eik·r planewave,
light blue the eikr /r spherical wave and dark blue the f (r̂)eikr /r scattered wave. The
red and blue wave fronts are not to scale. . . . . . . . . . . . . . . . . . . . . . . . .

2.3

9Plot of the integral cross section σ, as a function of collision energy E (black curve).
The axes are both on a logarithmic scale in dipole units (2.1). For comparison, thelow energy Born (σ ∼ 1) and high energy Eikonal (σ ∼ E −1/2 ) approximations arealso plotted in blue and red respectively. The plot is adapted from Ref. [1]. . . . . .

2.4

12

Cartoon of the s-wave scattering phase shift. The atoms are asymptotically freeparticles, approaching as planewaves (ψin , blue curve) before seeing the interaction
potential between them. Conservation of probability flux mandates that the asymptotic scattered wavefunction ψout (red curve), can only differ from ψin in the s-wave
partial wave channel by a phase shift δs .

2.5

. . . . . . . . . . . . . . . . . . . . . . . .

15

A y = 0 slice of the effective adiabatic potential energy surface on which colliding
molecules would be shielded. The axes are scaled by molecular-frame dipole units(2.1).

2.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

Subplot (a) plots the anisotropic components of the scattering amplitude fd + fh.p. ,between static field shielded molecules, for partial-waves up to ℓmax = 2 (solid black
curve) and ℓmax = 100 (dashed red curve), as a function of θq . The scattering
amplitudes are plotted in units of adβ . Subplot (b) plots the coefficients (red circles)
multiplying each partial-wave spherical harmonic Yℓ,0 in fh.p. , as a function of theangular momentum quantum number ℓ. A horizontal zero-line is also plotted forreference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7

22

A y = 0 plane slice of the effective microwave shielded potential between 23 Na40 K
molecules, in units of Edd (see Tab. 2.1). Circularly polarized microwaves are appliedwith Rabi frequency Ω = 2π × 15 MHz and a detuning of ∆ = 2π × 9.5 MHz.

. . .

23

xii
2.8

Energy dependence of the angular averaged total cross section σ between microwave
shielded 23 Na40 K (dashed black curve). The energy dependence clearly differs fromthe total cross section between fermionic point dipoles (dotted blue curve). For
comparison, I plot the low energy Born and high energy Eikonal approximations withdashed red lines. The inset shows a y = 0 slice of the effective microwave shielding
interaction potential, with Rabi frequency Ω = 2π ×15 MHz and microwave detuning∆ = 2π × 9.5 MHz. The shielding core is depicted as a white patch surrounding thecoordinate origin which saturates the colorbar at Veff > 200Edd . Coordinate axes areplotted in units of 103 Bohr radii a0 .

. . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.1

Illustration of the Gaussian time-dependent ansatz for c(p, t) along x. . . . . . . . .

31

3.2

Visualization of 2 colliding partners (red spheres) relative to the collision-frame (CF)and lab-frame (LF). The collision frame (black) is defined in the laboratory frame(blue) with the relative collision momenta. The angle η is that between the vectors
pr and Ê. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

34

The numerical time integration scheme. Two time steps are utilized: a) ∆t for
Hamiltonian evolution of the particles under V (r), and b) δt for collisional processes(m, n ∈ N). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

38

Visualization of the octree algorithm refinement (b), applied to the master-grid (a),projected in the x, y-plane. There are initially 8 master-grid cells with N = 200, andmax = 3, with Gaussian distributed points in arbitrary units. . . . . . . . . . . . .
Ncell

3.5

40

The probability distribution pF (θ, ϕ, η) with η = π/4 (d), and its approximations asobtained from the accept-reject algorithm with (a) 105 , (b) 106 and (c) 107 random
samples.

3.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

A discrete ensemble of 1000 Gaussian distributed particles (a), smeared against a
Gaussian kernel to generate a smooth distribution (b). The plot is a projection of
the distributions in the y = 0 plane in arbitrary units. . . . . . . . . . . . . . . . . .

45

xiii
4.1

Cartoon of a cross-dimensional rethermalization experiment. Collisions that scatterprimarily into the transverse directions promote thermalization, whereas forward
scattering hinders it. Red indicates pre-collision colliders, while post-collision ones
are blue.

4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Experimentally measured momentum space temperatures Tz (blue circles) and Ty(red diamonds) after a trap frequency quench of the confining potential along y. The
measurement was performed at = 1 G and Θ = 0 for 166 Er. The red dashed line isan exponential+sinusoidal fit to the data. The black solid line denotes the results ofthe Enskog theory (3.8) for this specific data set. The error bars denote the standard
error for three repetitions. The inset shows a schematic of the experimental setup.
This plot is adapted from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

51

B on Θ and a , for a /a = 0, 0.05, 0.1, 0.4
Subplot (a) shows the dependence of Nxzss d
B on a at Θ = 0. The
and 0.8. Subplot (b) shows the functional dependence of Nxzsblack dashed lines show the values of N for s-wave and p-wave scattering, labeledrespectively.
4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The analytically derived number of collisions per rethermalization for fermions NijF ,as a function of Θ for all nine excitation-rethermalization configurations.

4.5

53

. . . . . .

54

Time evolution of the temperature for Θ = 45◦ (a) and Θ = 90◦ (b) after parametricheating along y. The solid lines are fits to the thermalization model in Eq. (4.16).
The thermalization is faster at Θ = 45◦ . The molecular gas is heated parametricallyalong the y direction to create an initial condition of Ty (0) ≈ 2.5Tx (0), Tx (0) = Tz (0).
Figure adapted from Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

56

F

Comparison of the averaged number of collisions per thermalization N z (Θ). The
solid (grey) curve shows the theoretical results from Eqs. (4.13) and (4.17), whereas
the (black) circles with error bars are results from the JILA KRb experiment.

. . .

57

xiv
4.7

The central plot shows the total cross section as a function of the incident collision
angle, obtained from (a) the Born approximation (red dashed curve), and from GP
interpolation (solid curves) for 3 different collision energies: (b) E = 0.2Edd (black),(c) E = 2Edd (gray) and (d) E = 20Edd (light gray). In alphabetical correspondence,
are angular plots of the differential cross section (in units of a2d ) in subplots with therespective collision energies, assuming dipoles pointing along Ê = ẑ and incident
collision angle η = 45◦ lying in the x, z-plane. Subplot (d) uses a smaller domain forclarity of presentation.

4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

εij as a function of the dipole tilt angle Θ, for all 6 unique configurations (subplots
a to f) of the excitation axis i, and measured thermalization axis j. The dashed red
curves are the analytic εij results derived with the Born approximated cross section
at threshold, whereas the solid curves are those from Monte Carlo integration using
the GP interpolated cross sections, at temperatures T = 10 nK (black), T = 100 nK(dark gray), T = 400 nK (gray), and T = 1 µK (light gray). The dashed-dotted bluelines are the efficiency for purely p-wave collisions, εp = 1/4.1.

5.1. . . . . . . . . . . .

62Single body number density function projected in x, y-plane, comparing the N = 104
dilute (a) and N = 107 hydrodynamic (b) regimes. The spatial distributions areboth the same and assumed to be a sinusoid along x, but uniform along y (chosen
arbitrarily for illustrative purposes). . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Unit-free versions of the non-trivial Γ and Λ rate coefficients with identical dipolar
fermions, as a function of Θ.

5.3

64

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

A schematic diagram over anisotropic radial sound wave propagation from a point
source, following the evolution χ(t) = χ0 cos[K(r − cdd t)]/r with η = 0.8. There are
3 time slices that showcase the growing anisotropy of wavefronts due to the speed ofsound dependence on Θ.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

xv
5.4

Imaginary part of the propagating mode frequency solutions Im[ω± (Θ)] in Hertz (Hz),as a function of the dipole tilt angle Θ in degrees (deg) at η ≈ 0.04. The figure inset
plots the absolute relative angle between K̂ and ξ̂, | cos−1 (K̂ · ξ̂)|, also as a function
of Θ.

5.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

The imaginary parts of the mode frequency solutions, Im[ωκ ] (solid black curve),
Im[ωµ,1 ] (dashed blue curve) and Im[ωµ,2 ] (dotted red curve), as a function of Θ in agas of 23 Na40 K molecules with n0 = 1012 cm−3 and d = 0.75 D (η ≈ 0.04).

5.6. . . . .

76

Relative fluid variable fluctuation amplitudes ξx (t) (dashed-dotted black curve), χ(t)(solid blue curve), ξz (t) (dashed green curve), ϵ(t) (dotted red curve) as a function oftime t at z/L = 10, after an impulse shear flow perturbation along x̂. The dipoles areoriented with Θ = π/4. The relative amplitudes for χ(t), ϵ(t) and ξz (t) are rescaled
by a factor of 103 for clarity.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

5.7

Cartoon of a Gaussian distributed gas with a hydrodynamic center and dilute corona. 79

5.8

Pseudotemperatures (4.3) obtained from Monte Carlo simulations in the dilute (upper
panel, a) and hydrodynamic (lower panel, b) regimes. The gas consists of microwave
shielded 23 Na40 K molecules with dipole moment d = 0.75 D, oriented along x̂, attemperature T = 700 nK. The gas is initially excited along z by an instantaneous
trap frequency ramp to ωz = 2π × 147 Hz, while ωx = ωy = 2π × 82.5 Hz remain
constant. The regimes are differentiated by the number of molecules N , which are
N = 104 in panel (a), and N = 2 × 105 in panel (b). . . . . . . . . . . . . . . . . . .

5.9

80

Mij matrix elements as a function of Θ. The diagonal elements are plotted on theleft in subplot (a), whereas the negated (multiplied by a minus sign) off-diagonal
elements are plotted on the right in subplot (b).

. . . . . . . . . . . . . . . . . . . .

83

xvi
5.10 Times traces of Tx (t) (solid green curves), Ty (t) (dashed blue curves) and Tz (t) (dotted
red curves) for 3 values of Θ = 0◦ , 45◦ , 90◦ , in subplots (a, d), (b, e) and (c, f)
respectively. The 2 rows are differentiated by the number of molecules, with theupper row (subplots a, b, c) having N = 2 × 103 (Kn ≈ 11.10), while the lower row(subplots d, e, f) has N = 3 × 105 (Kn ≈ 0.07). Note that the simulation times aredifferent between the upper (t = 0 to 0.1s) and lower (t = 0 to 0.04s) rows.

. . . . .

83

5.11 The points with errorbars (standard errors) give η as obtained from DSMC simulations, performed over different values of N, λ and Θ.

The linear trend of ηvs N is provided in subplot (a), being largely consistent over various values of
λ = 0.13, 0.20, 0.32, 0.50, all of which are prolate (cigar) geometries. Subplot (b)shows how η varies with Θ, closely following the angle dependence in Eq. (5.40). The
data for this latter trend is obtained with λ = 0.2, for N = 4 × 105 (black data,Kn ≈ 0.06), N = 3 × 105 (gray data, Kn ≈ 0.07) and N = 2 × 105 (light gray data,Kn ≈ 0.11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.12 Visualization of the dominant dynamics in oscillatory modes. Arrows that point in
opposing directions relative to the gas cloud (ellipsoid) indicate dynamics that areout-of-phase. These figures are for illustrative purposes. . . . . . . . . . . . . . . . .
6.1

88

Atom-loss spectroscopy (orange circles) as a function of B for a fixed hold time of
250 ms. For each value of B, each data point is an average of 3 repetitions andnormalized to the maximum averaged atom number recorded. The values of as arethose extracted from cross-dimensional thermalization measurements using both, theEnskog equations (red squares) and the analytic formula of Eq. (6.2) (blue diamonds),
for 164 Er (a), 166 Er (b), 168 Er (c) and 170 Er (d). The black triangles are as values
obtained from lattice modulation spectroscopy measurements (see Ref. [2]). The solid
black lines represent a fit of Eq. (6.5) to as . Error bars and the shaded area of thefitting results denote the standard error. The figure is adapted from Ref. [2].

. . . .

93

xvii
6.2

Background scattering length abg
s for four bosonic isotopes (red circles). The solidline represents the best fit with ϕs /π = 144(1); see text. The shaded area, enclosedby the dotted lines, represents the fitting function for ϕs = 143 and ϕs = 145. The
errorbars denote the standard error of the fit of Eq. (6.5) to the experimental data.

6.3

The optical trap potential energy surface V (r), plotted as a function of coordinates
x and z, along y = 0.

6.4

96

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Simulation ensemble energy distribution with N = 10, 000 molecules (gray histogram), after collisional thermalization for t = 0.5 s from an initial MaxwellBoltzmann distribution at 50 nK (dotted red curve). The simulation achieves a
Fermi-Dirac distribution (solid black curve) at T ≈ 42 nK with a chemical potentialof µ/kB ≈ 87 nK.

6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Plot of T /TF values after evaporating down to N molecules in the experiment at the
Max-Planck-Institut für Quantenoptik [4] (black circles with error bars) with different
final trap depths, compared to those obtained from our numerical simulations (redcrosses).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

G.1 Cartoon of the stresses (black arrows) on a differential fluid volume element (light
red cube) due to thermodynamic pressure and velocity field gradients. . . . . . . . . 136
I.1

Comparison of the momentum space temperature Tp (5.36) vs time t, obtained from
DSMC simulations (black solid curves) and our theory (red dashed curves) with
N = 5 × 105 (Kn ≈ 0.04), Θ = 90◦ . The subplots (a) to (h) correspond to various
values of trapping anisotropy with λ = 0.13 to 8.0 as labeled in the subplot headers.
The fitted values of η are also provided in the subplot headers with their fitting
standard uncertainties.

I.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Comparison of the position space pseudotemperatures Tr vs time t, obtained from
DSMC simulations (upper subplot a) and our theory (lower subplot b) with Θ = 90◦ ,N = 5 × 105 (Kn ≈ 0.04) and λ = 0.32.

. . . . . . . . . . . . . . . . . . . . . . . . . 145

Chapter 1

Introduction

Atoms: the very building blocks of observable matter in our universe. Consisting of a positively charged nucleus encased in a probabilistic cloud of electrons, individual atoms host a treasure
trove of physics we have likely only scratched the surface of. Their complexities are only amplified in
the bulk of many other such partners, whether they are chemically bonded or free to roam around.
These are the ruminations of an atomic physicist, endeavoring a pursuit of understanding and control. Although lofty in its conception, great strides have already been made with the developmentof powerful tools like the laser and its associated technologies. Spectroscopy, optical trapping andquantum state manipulation, these are just some of the vast capabilities opened by our growing
prowess in light and matter interaction.
Built off the many monumental works of others before, my work in this field has thus far focused on the collective aspects of atoms and molecules in their gaseous phase. Two crucial properties
have made such systems interesting yet tractable for me to study: 1) their cooling down to ultracold
temperatures, and 2) the dipolar character of the constituent particles. My job for the remainderof this thesis will be to motivate and communicate my contributions to these topics through the...
captivating... medium of print.

1.1

Why ultracold?
Ever since the earliest inklings of its necessity [5], quantum mechanics has continued to inspireand yet, oftentimes confuse our scientific sensibilities. It is inevitable that to better understand2
quantum mechanics, we need to be able to see more of it at play.
"As usual, nature’s imagination far surpasses our own..."—- Richard P. Feynman (The Character of Physical Law), 1965.
Sometimes, nature gives us that almost for free, showcasing quantum properties like semiconductivity in naturally occurring materials such as silicon–"just" needing us to look. Other times, humaningenuity is required to piece together specially designed systems, to probe the faint hum of ouruniverse’s quantum symphony. One such way to "hear the music", is to freeze out all thermal
tumult, until just a few fundamental modes of the quantum wave are left singing.
This serene simplicity is what ultracold temperatures provide, being just about 10-millionthsof a degree above absolute zero (that’s a decimal point with 6 zeros behind it before a 1!). In thisregime, most aspects of an atom come to a standstill, allowing the manipulation and measurement ofeach quantum energy level in isolation. Such exact addressability of quantum states is what bolsters
the desire for ultracold atomic and molecular systems, but getting there is no easy feat. 1995 was
the year that saw the first ever creation of a Bose-Einstein condensate [6], an ultracold state ofquantum matter. That Nobel prize winning accomplishment was achieved right here at JILA with
Rubidium atoms [7], then also at MIT with Sodium [8], trailblazing a new era of ultracold quantum
science.
And so, we are now left to stand on the shoulders of these giants, amongst many others. To
continue their legacy and forge a new tomorrow in ultracold physics. From artificial materials madeof light [9], to mind numbingly precise measurement devices [10], the immense quantum control thatultracold temperatures bring will undoubtedly continue its unprecedented streak in technologicaladvancement, and push the boundaries of scientific discovery.

1.2

Why dipoles?
More than 2000 years ago, some dude in ancient Greece said something like "The whole isgreater than the sum of its parts". Point is, although single quantum particles might already be3interesting in and of themselves, what more harmonious melodies might a whole ensemble vocalize
in chorus? Therein lies another key ingredient of desire, interactions. When 2 quantum bodies areallowed to interact, they can share energy and quantum information, enriching the physics of thesystem as a whole. The nature of these interactions plays an important role in this enrichment,with each kind allowing only specific quantum states to talk to one another, and only within certain
distances.
Dipole-dipole interactions, in particular, support a variety of appealing properties. For one,they engage at pretty large distances, with the dipole interaction potential falling off only as theinverse cube of inter-particle spacing. Correlations thus develop without having to get too close,facilitating the spread of quantum information throughout a large many-body system of dipoles[11, 12]. For another, the interaction is highly anisotropic, depending on the orientation of each
dipole relative to their inter-particle displacement. With rotational symmetry broken, the angularmomentum of one dipole around the other can now couple to the internal spins of each dipole. How
these dipoles are flung off from one another after a collision might therefore be tunable via theirinternal quantum spins [13]. Certainly, the allures are many, but most highlighted in this thesis willbe the interaction anisotropies, segueing nicely into the next section.

1.3

What’s in this thesis?
The remainder of the thesis is partitioned into 3 main parts. The first addresses microscopic

collisional physics (Chap. 2), where I provide a brief primer on 2-body dipolar scattering in theultracold regime, leaving in only the relevant details. The second considers the emergent effects ofthese collisions on macroscopic gas dynamics, both in the dilute and hydrodynamic regimes (Chaps. 3to 5). The overarching theme here will be a gas’ return to equilibrium, showcasing the theoretical
tools I employ. The final icing on this thesis cake comes in Chap. 6, which discusses 2 immediate
applications that, in many ways, provide motivation to the physics understood from the preceding
chapters. Namely, the accurate determination of atomic scattering lengths via rethermalization
experiments, and the evaporative cooling of polar molecules to quantum degeneracy.

4
For convenience, Tab. 1.1 presents a collated list of all the constants and variables that will
be used consistently throughout this thesis. If a symbol crops up somewhere down the line thathas not been defined within the chapter, don’t set this thesis aside just yet! It will have probably
been defined in this table. It should also be noted that all the equations used will be written
in SI units, as one is inclined to when sitting in an office 2 kilometers away from the National
Institute of Standards and Technology. On the off chance that one simply wants to go straight tothe journal-formatted sources, here are all my publications relevant to the discussions of this thesis:[2, 3, 14–21]. Although I must say, you will necessarily miss out on the fun (if I don’t say so myself)
narrative this thesis weaves by jumping straight to those.

Symbol
ℏ
kB
ϵ0
µ0
µB
a0
as
m
µ
d
ad
Edd
B
E
Θ

Descriptor
Reduced Planck’s constantBoltzmann’s constantElectric constant
Magnetic constant
Bohr magneton
Bohr radiuss-wave scattering length
Atomic/molecular mass
Reduced mass
Magnetic/electric dipole moment
Dipole length
Dipole energy
Magnetic field
Electric field
Dipole tilt angle

Symboltrpf (r, p, t)n(r, t)
T
T (t)
N (Θ)
ε(Θ)
Kn
ρ(r, t)U (r, t)
κ(T )
µ(T )
Eevap

Descriptor
Time coordinate
Position coordinate vector
Momentum coordinate vector
Classical phase space distribution
Number density
Temperature
Pseudotemperature
Number for rethermalization
Collisional efficiency
Hydrodynamic Knudsen number
Local mass density
Local flow velocity
Thermal conductivity tensor
Viscosity tensor
Evaporation efficiency

Table 1.1: A table of the notation used for constants and variables that will appear in this thesis.
Bolded symbols are used to denote vectors, matrices and tensors.

Without further ado, here comes the music.

Chapter 2

Crash, Bam, Pow! Ultracold Dipolar Collisions

Ever found yourself in a vicious game of car-collision-chicken against your evil clone on a
10-lane superhighway? Eyes locked from a mile away. Pedal to the metal. This chapter speaks ofmuch the same scenario, except there are no highspeed cars, no drivers and no highways. Just 2
extremely slow-moving identical particles that interact at long range, colliding not that violently
with one another. The makings of a real Hollywood blockbuster. Crash, bam, pow!
This riveting process is exactly what occurs among the constituent particles of a nondegenerategas. Free to roam around, gaseous particles are prone to bumping into one another, exchanging
energy and momentum. These collisions can get rather complicated and are behind much of theinteresting physics we will see arise. I provide a rapid overview of ultracold collisional physics inthis chapter, setting the stage for the remainder of this thesis. The focus here will be on collisions
that primarily result from dipole-dipole forces.

2.1

Ultracold collisions of magnetic atoms

2.1.1

Lanthanides: Little atomic magnets

Technological advances in the last 2 decades have permitted the cooling and trapping ofmagnetic lanthanide atoms [22–26], a huge triumph for the field of ultracold dipolar physics [27, 28].
With lanthanide atoms cooled to their electronic ground state, their open-shell electronic structure
warrants that the application of a magnetic field results in a splitting of its Zeeman sublevels. For

6instance, bosonic (B) erbium atoms have a ground state Zeeman sublevel spectrum given by
B(B) = gJ µB BmJ ,EZeeman(2.1)

where the Landé g-factor is reported in Ref. [29] as gJ = 1.163801(1), B is the magnetic fieldstrength and mJ is the projection of the total angular momentum J = 6, onto the field axis.
Fermionic (F) isotopes of erbium, on the other hand, possess a nuclear spin that gives rise to
hyperfine structure. The energetically lowest state would reside in the total angular momentum
quantum number F = 19/2 manifold, which has the Zeeman spectrum
F
EZeeman(B) = gF µB BmF + 2πℏz2 B 2 (F 2 − m2F ),(2.2)

where mF is the projection of total angular momentum, while gF = 0.735032 and z2 = −12.76(1)Hz/G2 are the linear and quadratic Zeeman coefficients respectively [30]. The relevant Zeeman
energy splittings for both isotopes are plotted in Fig. 2.1 in units of milliKelvin.

Figure 2.1: Plot of the Zeeman energy levels as a function of applied magnetic field for bosonic
166 Er (a), and fermionic 167 Er (b) isotopes of erbium.

Because of the still mostly linear Zeeman splitting at modest magnetic fields, a magnetic
dipole moment dJ ≈ gJ mJ µB , is induced in the atoms that in their spin-stretched groundstate(i.e. the lowest mJ sublevel), gives both isotopes approximately the same value of d ≈ −6.9828µB .

7So fermionic magnetic atoms, usually collisionally invisible at ultracold temperatures with no swave scattering, can now experience collisions via their dipole-dipole interactions. You might beconcerned that such collisions could jostle these atoms out of their Zeeman groundstates, but here
in lies the benefit that ultracold temperatures bring. At temperatures on the order of T ∼ 100 nK
and below, the application of order B ∼ 10 G magnetic fields result in a Zeeman energy splitting of∆E/kB ∼ 0.1 mK as seen from Fig. 2.1. Colliding atoms prepared in their spin-stretched groundstate will therefore remain spin polarized in that state, by virtue of all other states being energeticallyforbidden.

2.1.2

Ultracold scattering: A primer

The cool (pun possibly intended) thing about ultracold experiments is that in principle, theinteractions between colliders are completely known1 . Our task as atomic theorists is therefore
to exploit this knowledge for computing scattering observables that, god willing all factors of 2
are tracked correctly, can be used to predict and understand experimental phenomena. This runs
rather contrary to say high energy physics experiments, where observed scattering data from collider
experiments are instead, used to infer the a priori unknown interacting Lagrangian. But that’s a
topic for someone else’s thesis. Ultracold scattering considers 2 particles headed on a collisioncourse with everything else in the universe being negligible to the relevant physics. In this case, theHamiltonian between the 2 bodies is given by2b = − ℏ ∇2 + b
H
h1 + bh2 + Vbint (r),
2µ r

(2.3)

where r = r 1 − r 2 is the relative position between particles, µ = m1 m2 /(m1 + m2 ) is the reduced
mass, bhi (i = 1, 2) is the single-particle Hamiltonian containing all the internal atomic/molecular
structure stuff, and Vbint is the interaction potential between the 2 particles. Wide hats are used todenote quantum operators, while normal sized hats will denote unit vectors.
As far as degrees of freedom go, there are the spatial ones relevant to r, and the internal state
1

What I mean here is that technically, you could write down all terms in the 2-particle Hamiltonian, although
actually handling it for calculations is usually impossible without some form of approximation.

8ones relevant to b
h1 + bh2 . The entire Hilbert space is thus spanned by the basis states:⟨r, θ, ϕ|E; ℓ, mℓ ; ν⟩ = r−1 uEℓ,mℓ ,ν (r)Yℓ,mℓ (θ, ϕ) |ν⟩ ,(2.4)

where Yℓ,mℓ (θ, ϕ) are spherical harmonics and |ν⟩ diagonalizes both 1-body Hamiltoniansbh1 + bh2 |ν⟩ = ϵν |ν⟩ ,(2.5)

with associated eigenenergies ϵν . When asymptotically far apart, the interaction potential Vbint (r)vanishes to leave a series of energetic thresholds ϵν . The asymptotically good quantum numbers ν,
are therefore said to label the asymptotic scattering channels. Represented in the basis of Eq. (2.4),
the time-independent Schrödinger equation results in a coupled set of radial equations:!
ℓ(ℓ + 1) 2µ(E − ϵν ′ ) E
d2−+uℓ′ ,m′ ,ν ′ (r)δℓ,ℓ′ δmℓ ,m′ℓ δν,ν ′
ℓ
dr2
r2
ℏ2
Z
2µ
∗
= 2
d2 r̂ ⟨ν| Yℓ,m(r̂)Vbint (r)Yℓ′ ,m′ℓ (r̂) ν ′ uEℓ′ ,m′ℓ ,ν ′ (r),
ℓ
ℏ

(2.6)

that, in general, has interaction matrix elements (right-hand side above) couple the various scattering channels, and possibly even the |ℓ, mℓ ⟩ angular momentum partial waves if Vbint is anisotropic.
These couplings can result in complicated solutions, mostly requiring numerical methods to obtain.
But as with many problems in physics, interpretable quantum scattering solutions are best
constructed along with a suitably chosen ansatz. A natural one emerges from the physical picture of
2 particles approaching from infinitely far away (pre-collision), interact strongly within the effective
range of their interactions, then depart infinitely far apart once more from the interaction aftermath(post-collision):|ψν (r)⟩ ∼ eikν ·r |ν⟩ +| {z }
pre-collision

where kν ′ =peikν ′ r ′fν,ν ′ (r̂)
ν ,
{zr
}
ν′ |

X

(2.7)

post-collision

2µ(E − ϵν ′ )/ℏ2 is the scattered wavenumber, and fν,ν ′ (r̂) is the scattering amplitude

associated with scattering from channel ν to ν ′ . Fig. 2.2 gives a visual reference for Eq. (2.7). Relating the inbound and outbound particle-pair wavefunctions at infinite relative distance is referred9
to as a completed collision. To connect the ansatz of Eq. (2.7) with the representation of Eq. (2.6),
a planewave expansion of the free particle solution can be performedeikν ·r|ν⟩ = 4π

∞ X
ℓ
X

∗
(k̂ν )jℓ (kν r)Yℓ,mℓ (r̂) |ν⟩ ,iℓ Yℓ,mℓ(2.8)ℓ=0 mℓ =−ℓwhich has the kν r ≫ 1 asymptotic formeikν ·r |ν⟩ →

∞
ℓ
∗
Y
2πi X X ℓ Yℓ,mℓ (k̂ν )  −i(kν r−ℓπ/2)ℓ,mℓ (r̂)√√
e− ei(kν r−ℓπ/2)
|ν⟩ .
i
r
kν
kν

(2.9)

ℓ=0 mℓ =−ℓThe expansion above identifies pieces that are incoming towards (e−i(kν r−ℓπ/2) ) and outgoing awayfrom (ei(kν r−ℓπ/2) ) the scattering center.

f (r̂)
eik·r

eikr
r

Vbint (r)
k̂

r=0
eikr
r

Figure 2.2: Cartoon of quantum scattering. The red curve denotes the incident eik·r planewave,
light blue the eikr /r spherical wave and dark blue the f (r̂)eikr /r scattered wave. The red and blue
wave fronts are not to scale.

Because probability must be conserved in quantum mechanics, the outbound flux of particles in the planewave expansion can only be modified only by a unitary transformation after the
scattering process:
∗
2πi X X X ℓ Yℓ,mℓ (k̂ν )√|ψν (r)⟩ ∼i
δℓ,ℓ′ δmℓ ,m′ℓ δν,ν ′ e−i(kν r−ℓπ/2)
r
k
ν
′
′
′
ℓ,mℓ ℓ ,mℓ ν

′

′

ν,ν
− Sℓ,m(k )ei(kν ′ r−ℓ π/2);ℓ′ ,m′ νℓℓ

Yℓ′ ,m′ℓ (r̂) ′
√
ν ,
kν ′

(2.10)

10′

ν,ν
where Sℓ,mis the unitary S-matrix that depends on the incident scattering energy. Upon;ℓ′ ,m′ℓℓcomparing Eq. (2.10) and Eq. (2.7), the scattering amplitude is therefore identified asfν,ν ′ (r̂) = − √2π X X ℓ−ℓ′ ∗ν,ν ′iYℓ,mℓ (k̂ν )Tℓ,m′ (kν )Yℓ′ ,m′ (r̂),′ℓℓ ;ℓ ,mℓkν kν ′ ℓ,m ′ ′(2.11)ℓ ℓ ,mℓwhere the T -matrix is defined as


ν,ν ′
ν,ν ′
′ δm ,m′ δν,ν ′
Tℓ,m
(k
)
=
i
S
(k
)
−
δ
.
′
′
ν
ν
′
′
ℓ,ℓ
ℓ
;ℓ ,m
ℓ,m ;ℓ ,m
ℓ
ℓ

ℓ

ℓ

(2.12)ℓ

As will be essential to the rest of this thesis, the scattering amplitude then permits us to evaluate
the differential cross sectiondσν,ν ′kν ′2fν,ν ′ (r̂) ,(r̂) =dΩkν(2.13)

which represents the ratio of differential area dσ through which particles are incident, over thedifferential solid angle dΩ into which these particles scatter.

2.1.3

Dipolar interactions

For 2 particles that interact primarily via dipole-dipole forces, the potential between them is
b1 · db2 − 3(db1 · r̂)(db2 · r̂)d
Vbint (r) ≈ Vbdd (r) =.
34πη0 r

(2.14)

where η0 = ϵ0 for electric dipoles and η0 = µ−10 for magnetic ones. In the case of identical spinpolarized magnetic atoms described in Sec. 2.1.1, the scattering problem effectively reduces to asingle-channel one, involving only the lowest Zeeman sublevel with the rest being energeticallydisallowed. Only considering 1 channel removes all the operator structure of the dipole operators,
so the potential reduces to
Vdd (r) =µ0 d2 1 − 3(B̂ · r̂)2 ,34πr(2.15)

where B̂ is the direction of the external magnetic field, along which the dipoles with dipole moment d
are polarized. With Vdd being dominant and involving only a single relevant dipole matrix element,
it is appropriate to define a system of units around d as presented in Tab. 2.1. Even with thisdimension
masslengthenergyquantity
md = µµd2
ad =4πϵ0 ℏ2
ℏ2
Edd =
µa2d

11

Table 2.1: A system of dipole units based on the space-fixed dipole moment d.

simplification, the potential retains its coordinate space anisotropy, so that angular momentum is
no longer conserved.
Full scattering solutions to Eq. (2.6) between these atoms then require the matrix elementsof Vdd (r) in the partial wave basis, which are evaluated with B̂ = ẑ as′′⟨ℓ, mℓ | Vdd (r) ℓ , m

r
d2
π
=−
⟨ℓ, mℓ | 4
Y2,0 (θ, ϕ) ℓ′ , m′ℓ34πϵ0 r
5



′
′
2
2 ℓ 
p
d
ℓ 2 ℓ   ℓ=−2(−1)mℓ (2ℓ + 1)(2ℓ′ + 1) 
 . (2.16)
3
4πϵ0 r′0 0 0−mℓ 0 mℓ

The 2-by-3 arrays in parentheses above are Wigner 3-j symbols. But before finding solutions insuch generality, simpler ones can be obtained if collisions are restricted to the regime of ultra
low energies. Although strictly approximate, these simpler solutions show incredible accuracy inreproducing experimental data under a variety of circumstances, so much so that most of this thesis
leverages this approximation.

2.1.4

Close-to-threshold dipolar scatteringAt kinetic energies that are small compared to the characteristic energy of the interaction
E ≪ Edd , the collision is considered close-to-threshold 2 [31]. Collisions in the close-to-threshold
regime are usually associated with few partial waves contributing to the scattering process. Thisis true for power law interaction potentials r−n with n > 3, where Wigner’s threshold law dictatesthat higher partial waves diminish as increasing powers of the wavenumber k [31]. But it turns
2

Threshold here refers to the scattering energetic thresholds when the atoms are asymptotically far apart, as
described in Sec. 2.1.2.

12
out that approaching zero energy, all the T -matrix elements (2.12) from dipolar scattering have the√same energy dependence Tℓ,mℓ ;ℓ′ ,m′ℓ ∼ E [1, 32, 33]. This energy independence translates to theclose-to-threshold integral cross section
σ = ∆sπ XX
2Tℓ,mℓ ;ℓ′ ,m′ℓ (k) ,k2′′(2.17)ℓ,mℓ ℓ ,mℓbeing completely independent of energy, seen to emerge in Fig. 2.3 with a plot3 of σ vs E. In
the expression above, ∆s is a symmetry factor that accounts for scattering between distinguishable(∆s = 1) or indistinguishable (∆s = 2) particles [34].

Figure 2.3: Plot of the integral cross section σ, as a function of collision energy E (black curve).
The axes are both on a logarithmic scale in dipole units (2.1). For comparison, the low energy
Born (σ ∼ 1) and high energy Eikonal (σ ∼ E −1/2 ) approximations are also plotted in blue and red
respectively. The plot is adapted from Ref. [1].

Because the threshold laws between dipolar atoms are the same for all partial waves, a large
3

The plot in Fig. 2.3 was computed for distinguishable dipoles (includes all partial waves), with the s-wave
scattering length carefully tuned to zero (see Sec. 2.1.5).

13
number of them may be required to converge the scattering calculations of f (r̂) (2.11). It is,therefore, more convenient not to expand the wavefunction in terms of partial waves, but to describe
the scattering of dipolar species directly in coordinate space [35]. Such a formulation is most easily
obtained via the Lippmann-Schwinger equationik·r

ψ(r) ∼ e

2µ 1
− 2
ℏ 4π

Z

′

d3 r ′ e−ik·r Vdd (r ′ )ψ(r ′ )

eikr
,
r

(2.18)

that is an asymptotic integral form of the Schrödinger equation [36], from which the scattering
amplitude is read off as
2µ 1
f (r̂) = − 2
ℏ 4π

Z

′

d3 r ′ e−ik·r Vdd (r ′ )ψ(r ′ ).

(2.19)

Itself dependent on the scattering solution ψ(r), this complication of evaluating Eq. (2.19) issidestepped by employing the Born approximation, commonly used for collisions at high energies.
It turns out, however, that this approximation works just as well for ultracold dipoles by virtue ofthe Born approximated f (r̂) having the correct close-to-threshold energy dependence: none. In theparlance of partial waves, it is the ℓ(ℓ + 1)/r2 angular momentum barrier that prevents low energy
scatterers from seeing much else other than the long-range tail of the dipolar r−3 potential before
they get reflected. So Vdd is but a perturbance to the incident planewave, which echoes the songusually sung about the Born approximation.
The scattering amplitude up to first-order in Born approximation is then evaluated aslim

E/Edd →0f (r̂) ≈ −

2µ 1
ℏ2 4π

Z

′

d3 re−ik ·r Vdd (r)eik·r

Z 3µd2
d r 1 − 3 cos2 θ iq·r
e , with q = k − k′ ,4πϵ0 ℏ2
2π
r3
r
Z 3
µd2
d r C2,0 (θ, ϕ) iq·r
π=e ,where C2,0 (θ, ϕ) = 2
Y2,0 (θ, ϕ),234πϵ0 ℏ
π
r
5
Z 3
µd2
d r C2,0 (θ, ϕ) X ℓ ∗=4π
i Yℓ,mℓ (q̂)jℓ (qr′ )Yℓ,mℓ (r̂ ′ )
4πϵ0 ℏ2
π
r3
ℓ,mℓ
r
Z
Z
2
X
µd
π ℓ ∗
jℓ (qr)2=
8
i Yℓ,mℓ (q̂) d ΩY2,0 (Ω)Yℓ,mℓ (Ω) dr
2
4πϵ0 ℏ
5
r
ℓ,mℓ
r
√
µd2 X
π ℓ ∗
πΓ(ℓ/2)=8
i Yℓ,mℓ (q̂)δℓ,2 δmℓ ,02
4πϵ0 ℏ
5
4Γ((3 + ℓ)/2)
=−

ℓ,mℓ

(2.20)14= ad

2 (k̂ · B̂ − k̂′ · B̂)2−
3
1 − k̂ · k̂′!
,using cos2 θq =(k̂ · B̂ − k̂′ · B̂)22(1 − k̂ · k̂′ ).

(2.21)

As written, the scattering amplitude above corresponds to the scattering of distinguishable particles,
which is not the case for the identical groundstate atoms of interest. Accurate functional forms off (r̂) therefore requires the appropriate symmetrization (+) for bosons, and antisymmetrization (-)
for fermions [37]:i1 hfF,B (k̂, k̂′ ) = √ f (k̂, k̂′ ) ± f (k̂, −k̂′ ) ,2(2.22)

finally leaving us withadfF (k̂, k̂ ) = √2′′ almostfB (k̂, k̂ )=4(k̂ · B̂)(k̂′ · B̂) − 2[(k̂ · B̂)2 + (k̂′ · B̂)2 ](k̂ · k̂′ )1 − (k̂ · k̂′ )2
a
√d
2

!
,

4 2[(k̂ · B̂)2 + (k̂′ · B̂)2 ] − 4(k̂ · B̂)(k̂′ · B̂)(k̂ · k̂′ )−3
1 − (k̂ · k̂′ )2(2.23a)!
.

(2.23b)

If not already bored stiff by this point, one might notice that the ℓ = 0 partial wave componentof the potential is barrierless, completely unable to defend itself against the imminent atomic flux
during a scattering event. Colliding atoms are therefore defenseless against the full might of theirdipole-dipole interactions, not forgetting all the shorter range physics on top of that. Surely then, theBorn approximation is no longer accurate which warrants the "almost" above equality in Eq. (2.23b).
To rectify this shortcoming, Eq. (2.25) is modified by hand with the famed s-wave scattering length,
another triumph in atomic physics briefly discussed below.

2.1.5

s-wave scattering in bosons

If a significant portion of the wavefunction is allowed to enter within r < ad , that region of
the potential and wavefunction can be really complicated and difficult to obtain accurately fromab initio calculations. An ingenious observation from Enrico Fermi in the 1930s [38] showed that
despite all its complexity, the low energy elastic scattering in the ℓ = 0 channel just modified thescattered wavefunction by a single parameter, the s-wave4 scattering length as . Put concisely,
4

The letter s used here denotes the ℓ = 0 partial wave, as inherited from atomic orbital theory in chemistrywhere every angular momentum orbital was assigned letters s, p, d, f, . . ., corresponding to the spherical harmonics
in ℓ = 0, 1, 2, 3, . . ..

15
scattering concerns itself with colliding partners as they enter from, and exit to asymptotically far
distances from one another (see Sec. 2.1.2). If the collision is elastic (entrance and exit channels
are the same), conservation of probability flux mandates that the outbound post-collision piece of
the wavefunction can only differ from the inbound pre-collision piece by a phase factor e2iδs . The
associated phase δs , which depends on the scattering energy, is referred to as the scattering phaseshift. A schematic picture of this process and the phase shift that follows is given in Fig. 2.4.
V (r)
ψ(r)

ψout (r)ψin (r)
E

//

ℓ=0
|{z}

δs ≈ −kasrasymptotic tailkr → ∞

short-range potential
kr ≪ 1Figure 2.4: Cartoon of the s-wave scattering phase shift. The atoms are asymptotically freeparticles, approaching as planewaves (ψin , blue curve) before seeing the interaction potential between
them. Conservation of probability flux mandates that the asymptotic scattered wavefunction ψout
(red curve), can only differ from ψin in the s-wave partial wave channel by a phase shift δs .

Going back to the planewave expanded solution in Eq. (2.10), one could also construct equivalent solutions in terms of spherical Bessel and Neumann functions instead. By doing so andcomparing solutions partial wave by partial wave, an s-wave scattering length is identified in termsof the low energy ℓ = 0 phase shift astan δs (k).
k→0kas = − lim(2.24)

Strictly speaking, this result is derived for the case of isotropic interaction potentials, where every
partial wave is independent of one another. Considerations of partial wave couplings from dipoles
will be treated as higher-order corrections and neglected here. Now equipped with the piece handlingbarrierless elastic scattering, the scattering amplitude for dipolar Bosons is appropriately amended

16
from Eq. (2.23b) to [35]√adfB (k̂, k̂′ ) = − 2as + √2

4 2[(k̂ · B̂)2 + (k̂′ · B̂)2 ] − 4(k̂ · B̂)(k̂′ · B̂)(k̂ · k̂′ )−3
1 − (k̂ · k̂′ )2!
,(2.25)

for a comprehensive solution.
Oftentimes, it is useful to have an estimate of the scattering length without any prior knowledge of the system. For a potential of the form Vn (r) = −Cn /rn , some statistical gargalesis can beused to derive a probability distribution over expected scattering lengths
P (as ) =as1,
π (as − as )2 + a2s(2.26)

in terms of a most-likely scattering length first derived by Gribakin and Flambaum [39]:


2
n−3

 √
 n−2
Γ
n−2
π
2µCn

,
as = cosn−2ℏ(n − 2)
Γ n−1

(2.27)

n−2

where Γ(x) is the gamma function. Moreover, the imaginary part of as responsible for inelastic
scattering is expected to have a distribution strongly peaked around zero [20]. I therefore opt toignore Im{as } for all subsequent considerations of as .

2.2

Shields up! Turning reactive molecules inert
Strong dipolar scattering is not unique to gases of magnetic atoms, but can also occur ingases of heteronuclear diatomic molecules with externally applied electric fields [40]. In fact, thedipole lengths of these polar molecules can be hundreds of times larger than those in magnetic
atoms, accentuating the effects of dipolar physics. A concern of molecular collisions, however, is thepossibility of chemical reaction which often results in molecular loss from their optical confinement(detailed in App. B). Lucky for us, several smart cookies (one of which is the chair of this very thesis
committee!) have proposed means to mitigate reactive collisions by careful application of external
electric fields. Doing so results in what is known as collisional shielding [41, 42], whereby barriers
are engineered in the potential which prevents molecules from coming too close to one another.
Excitingly, collisional shielding of polar molecules has recently been realized in several experiments, both with static electric [3, 43–45] and microwave fields [4, 46–48], permitting collisionally17stable bulk molecular gas samples. For context and their use later on, it is worthwhile to briefly discuss the effective potentials and resulting scattering cross sections in both these shielding situations.
Only the minimal models necessary to capture the essential shielding physics will be discussed here,
to prevent this thesis from unfurling down the length of the JILA tower.

2.2.1

Scattering of static field shielded KRb molecules

In static electric fields, polar molecules can experience an effective shielded potential between
them which is, albeit reminiscent of, different from the familiar dipole-dipole interaction potential[49]. Although theoretically formulated almost a decade ago [50, 51], the first demonstration ofelectric field shielding in 3-dimensions was only recently demonstrated in 2021 with a gas of 40 K87 Rb
molecules at JILA [3]. All remaining discussions contained in this subsection will refer to thismolecular species.
For all intents and purposes, ultracold KRb molecules can be treated as quantum rigid rotors
with only rotational degrees of freedom. That is to say, each molecule in isolation only has |Ni , Mi ⟩quantum numbers (i = 1, 2) corresponding to the total rotation and its projection onto the spacefixed z axis respectively. Brought closer together, such molecules interact as dipoles (2.14) with theinteraction potential matrix elements given in the |N1 , M1 ; N2 , M2 ⟩ basis as [13]


√
2
1
1
30


⟨N1 , M1 ; N2 , M2 |Vbdd (r) N1′ , M1′ ; N2′ , M2′ = −C2,−q (θ, ϕ) 
3
4πϵ0 rq −q1 −q2



′ 1 N
′
q
N
1
N
N
1  11 1× d(−1)M1 (2N1 + 1)(2N1′ + 1) 

M1′ q1 −M1
0 0 0



′
′
q
N2  N2 1 N2 
 N2 1× d(−1)M2 (2N2 + 1)(2N2′ + 1) 
,
M2′ q2 −M2
0 0 0(2.28)

18
where q1 = M1′ − M1 , q2 = M2′ − M2 , q = q1 + q2 and



1
2



2 (3 cos θ − 1), q
C2,−q (θ, ϕ) = ± 3 sin θ cos θe∓iϕ ,
2




q



 3 sin2 θe∓2iϕ ,8q = 0,
q = ±1,(2.29)

q = ±2.

Collisional shielding then leverages the rotation-coupling interactions above to engineer a barrier in
the adiabatic (ignoring the kinetic energy term) potential energy surface that prevents moleculesfrom reaching the short-range. Key to achieving this barrier is the application of an external electricfield E. Via the Stark effect, 2 asymptotic energy thresholds are tuned to be almost degenerate,
which if coupled through Eq. (2.28), would drastically repel one another within the range of dipolarinteractions. One adiabat will be pushed downward, while the other upward to form a repulsive
core around r = 0. If prepared and allowed to approach only on the upper adiabat, the 2 molecules
would be stopped from reaching close enough distances to chemically react.
A minimal, but rather accurate model of shielding assumes the colliding molecules remainpolarized along the field axis, here taken to be Ê = ẑ. The Mi quantum numbers are thus always
zero, which stands as a good approximation at large electric fields E. The good quantum numbersof each molecule, not interacting with any other, are now dressed by the electric field. These states
are obtained by diagonalizing the single-molecule Hamiltonianbi · E,c2 − db i = BR N
Hi(2.30)

which gives the field-dressed (denoted by tildes) basis statesb i |Ñi , Mi ⟩ = ϵHÑi ,Mi |Ñi , Mi ⟩,(2.31)

with associated eigenenergies ϵÑi ,Mi . The Mi quantum numbers do not have tildes as they remain
good quantum numbers in the presence of the field.
A shielding resonance as just described is then induced between the field-dressed combined
molecular states 1̃, 0 1̃, 0 and 0̃, 0 2̃, 0 , which are asymptotically degenerate at E ≈ 12.72

19
kV/cm and get dipole-coupled upon approach. The result is a barriered upper adiabat that is wellapproximated by [49]:11p
VE (r, θ) = (Va + Vb ) +(Va − Vb )2 + 4W 2 ,22(2.32)

constructed out of the diabatic piecesrχaπVa (r, θ) = −Y2,0 (θ) + E1̃1̃ ,πϵ0 r3 5rχbπ
Vb (r, θ) = −Y2,0 (θ) + E0̃2̃ ,
3
πϵ0 r
5
r
χW
2π
W (r, θ) = −Y2,0 (θ),
3
πϵ0 r
5

(2.33a)(2.33b)(2.33c)with energy offsets E1̃1̃ and E0̃2̃ corresponding to the energetic thresholds of the |1̃, 0⟩|1̃, 0⟩ and|0̃, 0⟩|2̃, 0⟩ states respectively. The surface of VE along y = 0, with E1̃1̃ ≈ E0̃2̃ , is plotted in Fig. 2.5.

Figure 2.5: A y = 0 slice of the effective adiabatic potential energy surface on which colliding
molecules would be shielded. The axes are scaled by molecular-frame dipole units (2.1).

20
Having assumed Mi = 0, effective state-dependent dipole moments can then be defined as

2
′
X
N
1
N
√
√
′


dÑ →Ñ = d⟨Ñ |N ⟩⟨N ′ |Ñ ′ ⟩ 2N + 1 2N ′ + 1 (2.34)
 ,
N,N ′
0 0 0
with molecular-frame dipole moment d, that enter the diabatic potentials via the interaction terms
χa = d1̃→1̃ d1̃→1̃ ≈ 0.023 D2 ,(2.35a)χb = d0̃→0̃ d2̃→2̃ + d0̃→2̃ d2̃→0̃ ≈ −0.037 D2 ,(2.35b)χW = d1̃→0̃ d1̃→2̃ ≈ 0.066 D2 .

(2.35c)Debye units are denoted with D above. At the resonant value of the electric field, E1̃1̃ ≈ E0̃2̃ , so
the effective shielding potential simplifies to a channel energy independent form
p



π/5 
2
2 1/2
[χ
+
χ
]
Y
(θ)
−
(χ
−
χ
)
+
8χ
|Y
(θ)|
.
VE (r, θ) ≈ −
a
2,0
a
2,0
b
b
W
2πϵ0 r3(2.36)

Above, I have ignored a constant energy offset of E1̃1̃ + E0̃2̃ as they are irrelevant to the elastic
scattering physics of interest here.
At low enough temperatures, the first-order Born approximation of Eq. (2.20) is once more
appropriate to study differential scattering resultant from the effective potential VE . With the sameplanewave expansion technique used to obtain Eq. (2.21), the scattering amplitude between electricfield shielded molecules is computed asr
Z
2µ(χa + χb ) π X ℓ ∗jℓ (qr)′fE,Born (k, k ) =i Yℓ,m (q̂) d3 r 3 Y2,0 (θ)Yℓ,m (θ, ϕ)(2.37)
πϵ0 ℏ25rℓ,m
1/2 rZ
2µ (χb − χa )2 + 8χ2W
πX ℓ ∗jℓ (qr)−i Yℓ,m (q̂) d3 r 3 |Y2,0 (θ)|Yℓ,m (θ, ϕ).
πϵ0 ℏ25rℓ,m

For the second term above, the integrals are evaluated as


R


Z
 Ω |Y2,0 (Ω)|Yℓ,m (Ω),|Y2,0 (Ω)|Yℓ,m (Ω) = δm,0

Ω


0,

ℓ is even,(2.38)ℓ is odd.

With this, the scattering amplitude is given by
fE,Born = fs + fd + fh.p. ,(2.39)

21
written in terms of isotropic s-wave (ℓ = 0), dipole-dipole d-wave (ℓ = 2) and higher partial-wave
components:r
1 − lim Ci(qr0 ) Y0,0 (q̂)
r0 →0
!
r
4
d2α
π
fd = −adβ1 − √ − 2 Y2,0 (q̂)
5
3 3 dβ!
∞X 3π (−1)L Γ(L) Z√
fh.p. = −adβd2 Ω|Y2,0 (Ω)|Y2L,0 (Ω) Y2L,0 (q̂),
3
Γ(
+
L)
4
5
2
L=2
fs = −adβ4π3(2.40a)(2.40b)(2.40c)respectively, having defined the effective dipole moments
d2α = χa + χb ≈ −0.026 D2 ,q
2dβ = (χb − χa )2 + 8χ2W ≈ 0.17 D2 ,(2.41a)(2.41b)and effective dipole length adβ = md2β /(3πϵ0 ℏ2 ). Ci(x) is the cosine integral which when x → 0, is
pathological but vanishes after antisymmetrization of the cross section. In the expression for fh.p. ,the summation index L is used as a proxy for the even quantum numbers ℓ, with the factor of 2
accounted for by each term in the sum explicitly.
At the resonant shielding field strength, subplot (b) of Fig. 2.6 shows that the sum over L isconvergent, and partial-waves of ℓ > 2 contribute but a small amount to the scattering amplitudeof KRb molecules. A comparison of the anisotropic components of the scattering amplitudes withpartial waves truncated at ℓmax = 2 (solid black curve) and ℓmax = 100 (dashed red curve) is
plotted in subplot (a) of Fig. 2.6 against the momentum transfer inclination angle θq . These curves
are maximally separated at θq = 0, π with an absolute relative error of ∼ 9%. Truncating the sum atℓ = 2, the scattering amplitude is simply given as fE,Born ≈ fs + fd . At this level of approximation,
the angular dependence of this scattering amplitude exactly resembles that between 2 regular point
dipoles [35]fE,Born ≈ fs + adeff(k̂ · Ê − k̂′ · Ê)2
(1 − k̂ · k̂′ )

2
−
3

!
,

(2.42)

with an effective dipole length
adeff =

md2β
8πϵ0 ℏ24d2
1 − √ − α23 3 dβ!
.

(2.43)

22
Antisymmetrizing Eq. (2.42) then results in the scattering amplitude identical to that in Eq. (2.23a),up to a substitution of the effective dipole length.

Figure 2.6: Subplot (a) plots the anisotropic components of the scattering amplitude fd + fh.p. ,between static field shielded molecules, for partial-waves up to ℓmax = 2 (solid black curve) andℓmax = 100 (dashed red curve), as a function of θq . The scattering amplitudes are plotted inunits of adβ . Subplot (b) plots the coefficients (red circles) multiplying each partial-wave sphericalharmonic Yℓ,0 in fh.p. , as a function of the angular momentum quantum number ℓ. A horizontal
zero-line is also plotted for reference.

Turns out, static field resonantly shielded 40 K87 Rb molecules collide at threshold just as point
dipoles would! Experimental evidence of this will be discussed a little down the road in Chap. 4.

2.2.2

Scattering of microwave shielded NaK molecules

Just like with static electric fields, polar molecules can also be shielded using microwave fields[4, 42, 47, 48, 52–54]. The detailed shielding mechanism with these time-dependent fields is slightly
more involved, but has much the same flavor as with static fields: i.e. apply a microwave field thatdresses rotational states, have these dressed states repel via resonant dipole-dipole interactions, pick
the incident channel that has a repulsive barrier, and tune the microwaves such that the collision
follows this repulsive adiabat. Conveniently, Ref. [55] derives an effective potential relevant to elastic

23
scattering between microwave shielded molecules:Vµ (r) =

2


C6 
d 
4
1
−
(r̂
·
Ê)
+
3(r̂ · Ê)2 − 1 ,63r
4πϵ0 rwith circularly polarized microwaves, where d = d0 /(2.44)p12(1 + (∆/Ω)2 ) is the effective molecular

dipole moment and C6 = d40 (1 + (∆/Ω)2 )−3/2 /(128π 2 ϵ20 ℏΩ). Here Ω and ∆ are the Rabi frequency
and detuning of the microwave respectively. A y = 0 plane slice of Vµ (r) is provided in Fig. 2.7,
between 23 Na40 K molecules with molecular frame dipole moment d0 = 2.72 D, subjected to microwaves of Rabi frequency Ω = 2π × 15 MHz and detuning ∆ = 2π × 9.5 MHz. Chemical reactions
are suppressed with the shielding core, depicted as a white patch surrounding the coordinate originthat saturates the color bar at Veff > 200Edd .

2

150

1.5

100

1

50

0.5
0

0

0

0.5

1

1.5

2

Figure 2.7: A y = 0 plane slice of the effective microwave shielded potential between 23 Na40 K
molecules, in units of Edd (see Tab. 2.1). Circularly polarized microwaves are applied with Rabi
frequency Ω = 2π × 15 MHz and a detuning of ∆ = 2π × 9.5 MHz.

24
With the long-ranged dipolar tail being, apart from an overall minus sign, identical to thedipole-dipole potential between points dipoles [(2.15) but with µ0 replaced with ϵ−10 ], the close-tothreshold differential cross section in microwave shielded molecules would be exactly that obtained
from Eqs. (2.23a) and (2.25) in Born approximation. In higher energy collisions, however, moregeneral solutions require close-coupling calculations. This is certainly also true for the case of static
field shielded molecules, but analysis here will focus on the case with microwaves.

2.2.2.1

Close-coupling calculations for non-threshold collisions

Close-coupling calculations are used to obtain scattering solutions of the time-independent
Schrödinger equation (2.6) with all relevant quantum degrees of freedom [56]. I compute these byutilizing a log-derivative propagation method, which avoids direct evaluation of the radial wavefunction uEℓ,mℓ ,ν (r), but instead computes its logarithmic-derivative:
Y (r) = U −1 (r)∂U (r)∂ log U (r)
=
,
∂r
∂r

(2.45)

where U (r) is the fundamental matrix of uEℓ,mℓ ,ν (r) solutions. Conveniently, the effective potentialof Eq. (2.44) only requires us to consider 1 asymptotic channel for elastic scattering, so matrix
indices only need to be considered for partial waves. Then defining the matricesℓ′ ,m′

Dℓ,mℓℓ = δℓ,ℓ′ δmℓ ,m′ℓℓ′ ,m′

d2
,
2
dr


Wℓ,mℓ ℓ = δℓ,ℓ′ δmℓ ,m′ℓ k 2 −(2.46a)ℓ(ℓ + 1)
r2


−

2µ
⟨ℓ, mℓ | Vbµ (r) ℓ′ , m′ℓ ,ℏ2(2.46b)with matrix elements of ⟨ℓ, mℓ | Vbµ (r) |ℓ′ , m′ℓ ⟩ provided in App. A, this single-channel version of
Eq. (2.6) can be recast as the compact system of equations [D + W ] U = 0, or in terms of thelog-derivative,
Y ′ (r) + Y 2 (r) + W (r) = 0.

(2.47)

In principle, the equation above can be solved numerically at a given collision energy E, bypropagating the log derivative matrix from r = 0 to r → ∞. In practice however, propagating to

25∞ is not possible so we only do so up to r = rmatch , then match Y (r) to the asymptotic solutions
where the distant colliders no longer interact. Moreover, we sidestep the issue of singularities atthe origin by imposing a short-range boundary condition by starting the propagation at a minimum
radius r = rmin . For our considerations of reactive molecules, we assume that short-range chemicalloss is universal so that the log-derivative matrix at rmin is diagonal and given as [34, 57]
q
ℓ,mℓ
ℓ,mℓ
(r
)
=
−i
Yℓ,m
(rmin ).
Wℓ,m
min
ℓ
ℓ

(2.48)

This boundary condition prevents dipolar scattering resonances [58, 59] which simplifies things. I’ve
chosen to do numerical propagation of Y (r) with an adaptive radial step size version of Johnson’salgorithm [60]. For shielded collisions, the engineered short-range barrier typically extends out to∼ 1000a0 , so numerically accurate propagation can safely commence from rmin ∼ 100a0 . On theother hand, matching at long-range is done adaptively depending on the collision energyrℏ2 ℓmax (ℓmax + 1)
rmatch =+ 20ad ,mE(2.49)

where ℓmax is the largest value of ℓ utilized in the calculation. Typically, we utilize ℓmax = 121 or as
many as is required for numerical convergence (i.e. adding more partial waves does not significantly
change the quantities of interest).
The asymptotic solutions to Eq. (2.6) arise by considering the domain where r is much larger
than the range of the potential, so that Eq. (2.6) is well approximated byd2ℓ(ℓ + 1)
2−
+ k uE,ℓ,mℓ (r) = 0.
dr2
r2

(2.50)

This asymptotic radial equation is solved with 2 independent solutions:fE,ℓ (r) = krjℓ (kr),(2.51a)gE,ℓ (r) = krnℓ (kr),(2.51b)where jℓ (kr) and nℓ (kr) are the spherical Bessel and Neumann functions respectively. Then withmatricesℓ′ ,m′

Fℓ,mℓ ℓ (r; E) = δℓ,ℓ′ δmℓ ,m′ℓ fE,ℓ (r),(2.52a)26ℓ′ ,m′

Gℓ,mℓℓ (r; E) = δℓ,ℓ′ δmℓ ,m′ℓ gE,ℓ (r),(2.52b)arbitrary solutions to Eq. (2.6) can be written as
U (r) = N [F (r) − KG(r)] ,(2.53)

where K is the reactance matrix that is responsible for matching the numerical scattering solutions
U to the asymptotic solutions in Eq. (2.52) at r = rmatch . In particular, the off-diagonal elementsof K provide information on the channel couplings that arise due to the interaction potential for agiven incident collision channel. The matrix N is relevant only for normalization. The reactancematrix can be written in terms of the logarithmic derivative via
K=∂
F (r)Y (r) − ∂r
F (r)
∂
G(r)
G(r)Y (r) − ∂r,

(2.54)

r=rmatchfrom which we can compute other scattering matrices via the relations [1]
I + iK,I − iK(2.55a)T = i(S − I).

(2.55b)S=

Carrying out the algorithm just described gives the integral cross section with Vµ (r), over abroad range of collision energies E. The resulting energy dependence is plotted in Fig. 2.8 (solid black
curve), where I compare it with the integral cross section between point dipole scatterers (dottedblue curve) with Vdd (r). For comparison, I also plot the low energy Born and high energy Eikonal[1] approximations with dashed red lines. Worth pointing out is that the collision energy in Fig. 2.8
is given in units of dipole energy Edd (see Tab. 2.1), which scales as d−4 . Larger dipole moments
would therefore require ever lower energies to reach the threshold regime, a problem faced by gasesof diatomic molecules even at sub-microKelvin temperatures. Access to the differential cross section
from close-coupling calculations is therefore important for studying ultracold molecular gases, as
will become clear in the following chapter.

27

Figure 2.8: Energy dependence of the angular averaged total cross section σ between microwave
shielded 23 Na40 K (dashed black curve). The energy dependence clearly differs from the total crosssection between fermionic point dipoles (dotted blue curve). For comparison, I plot the low energy
Born and high energy Eikonal approximations with dashed red lines. The inset shows a y = 0
slice of the effective microwave shielding interaction potential, with Rabi frequency Ω = 2π × 15
MHz and microwave detuning ∆ = 2π × 9.5 MHz. The shielding core is depicted as a white patch
surrounding the coordinate origin which saturates the colorbar at Veff > 200Edd . Coordinate axes
are plotted in units of 103 Bohr radii a0 .

Chapter 3

The Jostle and Bustle of Dipoles

Step back a little. This chapter zooms out from the microscopic collisional physics andinstead, explores the resultant collective behavior of a whole ensemble of dipolar colliders. At
ultracold but not yet quantum degenerate temperatures, the constituent particles of the gas whizaround on classical trajectories where every once in a while, two of them collide (cue flashbacks from
Chap. 2). Such interplay of classical motion with quantum scattering is most readily treated withthe celebrated Boltzmann transport equation, a cornerstone for dynamical studies of rarefied gases
in non-equilibrium statistical mechanics. What’s to come is a short introduction to the Boltzmann
equation and the approaches I use to solve it.

3.1

The Boltzmann transport equation
The state of a classical point particle of mass m is fully characterized by its position r andmomentum p. A statistical ensemble of such identical particles can then be described by a phasespace distribution f (r, p, t), that gives the differential number of particles dN within a differential
volume d3 rd3 p at position r and time t, with momentum p. Describing the gas collectively, thisR
distribution is taken to be normalized over phase space such that the integral d3 rd3 pf (r, p, t),evaluates to the total number of ensemble particles N . In general, dynamical evolution of f involves
a combination of advective transport, response to applied forces and collisions, all of which are29
encapsulated by the Boltzmann equation [61, 62]



∂
p
+
· ∇ − ∇V (r) · ∇p f (r, p, t) = Icoll [f ] ,| {z }m } |{z}|∂t {zadvection

response(3.1)

collisions

where V (r) is any external potential felt by the particles and ∇p is the gradient operator on
momentum coordinates. For most of what follows, the confinement will be taken as a harmonictrap
1 X 2 2
ωi ri ,
V (r) = m
2

(3.2)

i

with harmonic trapping frequencies ωi , which is a good approximation to the optical potentials
normally used to confine ultracold gases (see App. B for further details).
A fundamental property of classical gases is that once brought out-of-equilibrium and left
alone, they will return to thermal equilibrium as is mandated by the second law of thermodynamics[63]. A heat death foretold, although phrased slightly differently, by Boltzmann [64]. When dilute,the route to thermalization occurs primarily by means of two-body collisions, incorporated in the
Boltzmann equation via the entropy generating collision integral
Z
Icoll [f ] =d3 p1|p − p1 |
m

Z

dΩ′





dσ  ′ ′f f1 1 ± h3 f 1 ± h3 f1 − f f1 1 ± h3 f ′ 1 ± h3 f1′ .
′
dΩ
(3.3)

As is conventional, the expression above uses the shorthand notations f1 = f (r, p1 , t) and primes toindicate the post-collision distributions of collision partners. The collision integral in Eq. (3.3) above
has been modified from its purely classical form by the addition of quantum statistical factors ±h3 f(+ for collisional enhancement in Bose gases and − for Pauli blocking in Fermi gases) [65–67] with
Planck’s constant h. Although relevant to the discussions of evaporative cooling in Chap. 6, muchof this thesis is concerned with temperatures where the thermal de Broglie wavelength is far smaller
than the mean particle spacing. This regime lends to negligibly small phase space occupancies
within h3 , so we ignore these factors unless otherwise stated:Z
Icoll [f ] ≈d3 p1|p − p1 |
m

Z

dΩ′


dσ
′ ′
f
f
−
f
f
.
1
1
dΩ′

(3.4)

30
At its final thermalized state, the entropy of a gas is maximized and so requires that thecollision integral vanishes identically. This statement is a tenet of the Boltzmann H-theorem [62],which leads to the equilibrium Maxwell-Boltzmann solution
f eq (r, p) = neq (r)ceq (p),


N
V (r)
eq
where,n (r) =
exp −
,
Zr
kB T


1
p2
eq
c (p) =
exp −,Zp
2mkB T
R
with partition functions Zr = d3 r exp − VkB(r)and Zp = (2πmkB T )3/2 .
T(3.5a)(3.5b)(3.5c)

Nonequilibrium solutions to Eq. (3.1) on the other hand, are less readily available to us [68],especially so when dealing with dipoles and their anisotropic cross sections. In particular circumstances, taking the gas only perturbatively away from equilibrium permits simpler approximate
treatments of the gas dynamics. Aided by a suitable choice of a nonequilibrium ansatz for f (r, p, t),a closed set of ordinary differential equations that track ensemble averaged observables can then be
derived [62, 69]. Here’s one way to do that.

3.2

Variational solutions: Gaussians, Gaussians, Gaussians!
Following the Maxwell Boltzmann distribution of Eq. (3.5) at thermal equilibrium, it is plau-

sible that weak perturbations will leave f (r, p, t) approximately Gaussian [68]. When asserted inreal space, the even property of Gaussians restricts the types of perturbations to ones that do notchange the gas’ center of mass. But even with such a restriction, Gaussians take us a long way forthe experiments of interest to this thesis. Enacting this approximation for nonequilibrium studies
requires the spatial and momentum variances, ⟨rj2 ⟩ and ⟨p2j ⟩, to be made time-dependent:f (r, p, t) = c(p, t)n(r, t);c(p, t) ≡Yjn(r, t) ≡ N(3.6a)
!

p2j

1

q
exp −
2⟨p2j ⟩(t)2π⟨p2j ⟩(t)

Y
j

1

rj2

,

q
exp − 2
2⟨rj ⟩(t)2π⟨rj2 ⟩(t)

(3.6b)!
,(3.6c)

31
with subscripts j = x, y, z (see Fig. 3.1). Angle brackets are used to denote a phase space average:
1⟨. . .⟩ =N

ZZ

d3 pd3 rf (r, p, t)(. . .).

(3.7)

c(px , t)(())⟨p2x ⟩(t)px
Figure 3.1: Illustration of the Gaussian time-dependent ansatz for c(p, t) along x.

Utilizing the Gaussian ansatz then admits equations of motion for the ensemble averaged
values of arbitrary dynamical variables χ(r, p, t), an approach known as the method of averages [62].
The evolution equations governing ⟨χ⟩ are then derived by multiplying the Boltzmann equation byχ and integrating over all of phase-space, detailed in App. C. Picking χ to come from the set ofnine variables {rj2 , p2j , rj pj } with j = x, y, z, gives a closed set of equations [70, 71]:d⟨rj2 ⟩
2− ⟨rj pj ⟩ = C[∆rj2 ],dtmd⟨p2j ⟩
+ 2mωj2 ⟨rj pj ⟩ = C[∆p2j ],dtd⟨rj pj ⟩1 2−
p + mωj2 ⟨rj2 ⟩ = C[∆rj pj ].
dt
m j(3.8a)(3.8b)(3.8c)In these equations, collisions are incorporated through the integralC[∆χ] =

1
N

Z

d3 r

ZZ

d3 pd3 p1|p − p1 |f f1
m

Z

dΩ′

dσ
∆χ,
dΩ′

(3.9)

where ∆χ ≡ χ′ +χ′1 −χ−χ1 denotes the amount by which χ changes during a collision event. These
nonlinear, coupled differential equations are known as the Enskog equations of change. Arguably
easier to solve than Boltzmann’s partial differential equation (3.1), the simplicity of Eqs. (3.8)

32provides a ready-for-use theory of nondegenerate trapped gases. Conveniently, the momentum spacevariances ⟨p2j ⟩ solved for in Eqs. (3.8) are directly the observables with time-of-flight thermometry,a measurement protocol common to ultracold gas experiments.
Notice that, in the absence of collisions (C = 0), all three coordinate axes are decoupled in
Eqs. (3.8). In this case, the normal modes of the Enskog equations along each axis coordinate comein two varieties: a breathing mode of angular frequency 2ωj , in which ⟨rj2 ⟩ and ⟨p2j ⟩ are out of phase;and a stationary mode of frequency ωj = 0, corresponding to the equilibrium configuration [70].
These modes are naturally modified by the presence of collisions which will, of course, depend on
the dipolar properties of the cross section [14]. A beast we now turn to taming.

3.2.1

Taming the collision integral

Eqs. (3.8) are only complete upon evaluation of the collision integrals (3.9). The key toevaluating these integrals is the assertion of, you guessed it, a Gaussian ansatz for f (r, p, t) once
more. Recast into center of mass coordinates, we get a decomposition of momenta into a center ofmass component P = (p + p1 )/2 and a relative component pr = p − p1 . These coordinates granta reformulation of the collision integral from Eq. (3.9) to
Z
Z
Z
pr
dσ
1
3
2
3
d rn (r) d pr cr (pr )dΩ′ ′ ∆χC[∆χ] =
N2m
dΩ

(3.10)

where cr (pr ) takes the same form of c(p) but with the replacement p → pr and all factors of 2
converted to 4. C[∆rj2 ] vanishes since the collisions are assumed instantaneous and therefore do
not vary the particle positions (∆rj2 = 0). Additionally, the collision integral for χ = rj pj is also
observed to vanish since∆ (rj pj ) = rj ∆pj = rj Pj′ − Pj ,(3.11)

in which Pj′ = Pj during elastic collisions, so C[∆rj pj ] = 0.
As such, the only quantities left that provide a non-trivial collision integral term are χ = p2j ,for which its collisional variation can be written only in terms of relative momenta as′2
22∆p2j = p′2
j + p1,j − pj − p1,j

33

1 ′2
pr,j + Pj′2 − p2r,j − Pj2
2

1 ′2
=
pr,j − p2r,j .
2=(3.12)

This reduction renders the integrals as separable (i.e. can be evaluated separately) in position andmomentum variables
C[∆p2j ] =1NZ

d3 rn2 (r)Z

d3 pr cr (pr )

pr
2m

Z

dΩ′

dσ
∆p2 ,
dΩ′ j

(3.13)

with the integral over d3 r evaluating to
1
⟨n⟩ =
N

Z

d3 rn2 (r) =8π 3/2Np.
⟨x2 ⟩⟨y 2 ⟩⟨z 2 ⟩(3.14)

This leaves the integrals over post and pre-collision momenta.
Evaluating the momentum integrals in Eq. (3.13) is a difficult task for dipoles, predominantly
because the differential cross-section is anisotropic. Consistency therefore requires a suitable frametransformation, for which I adopt one where the lab-frame (LF) is defined such that the dipolealignment axis points relative to it as Ê = (sin Θ, 0, cos Θ)T , and a collision-frame (CF) defined by
the relative momenta of collision partners with ẑ CF = p̂r = (sin θLF cos ϕLF , sin θLF sin ϕLF , cos θLF ).
In principle, the transverse axes could be defined to your liking so long as x̂CF · ẑ CF = 0 andŷ CF = ẑ CF × x̂CF , but the ones I chose were sin ϕLF x̂CF = − cos ϕLF  ,




0





cos θLF cos ϕLF ŷ CF =  cos θLF sin ϕLF  .
− sin θLF

(3.15)

It is necessary to perform integrals over both the lab-frame coordinates {pr , θLF , ϕLF } inwhich Θ is defined, and collision-frame coordinates {θCF , ϕCF } that defines the post-collision relative
momentum (the subscript CF is used instead of primes to disambiguate frames). A visualization ofthese two frames is provided in Fig. 3.2. As such, a transformation that relates these two sets of

34
Ê

zCF

zLF
p1

yCF
η
xCF

collision
pr
p
Θ
xLF

OLF

Figure 3.2: Visualization of 2 colliding partners (red spheres) relative to the collision-frame (CF)and lab-frame (LF). The collision frame (black) is defined in the laboratory frame (blue) with therelative collision momenta. The angle η is that between the vectors pr and Ê.

variables is necessary and constructed using the method of direction cosinesx̂LF · x̂CF x̂LF · ŷ CF x̂LF · ẑ CF R(CF → LF) = ŷ LF · x̂CF ŷ LF · ŷ CF ŷ LF · ẑ CF ẑ LF · x̂CF ẑ LF · ŷ CF ẑ LF · ẑ CF sin ϕLF cos θLF cos ϕLF sin θLF cos ϕLF = − cos ϕLF cos θLF sin ϕLF sin θLF sin ϕLF  .




0
− sin θLF
cos θLF(3.16)

With this, the differential cross-section can be obtained in the CF with Eqs. (2.25) and (2.23a), forwhich the relevant unit vectors are given as


 





sin Θ sin ϕLFsin θCF cos ϕCF 
0



 





 
k̂CF = 0 , k̂′CF =  sin θCF sin ϕCF  , Ê CF = sin Θ cos θLF cos ϕLF − cos Θ sin θLF  , (3.17)



 





 
sin Θ sin θLF cos ϕLF + cos Θ cos θLF
cos θCF1
denoting k̂ = p̂r . These result in the dot product termsk̂ · k̂′ = cos θCF ,(3.18a)35k̂ · Ê = sin Θ sin θLF cos ϕLF + cos Θ cos θLF ,(3.18b)k̂′ · Ê = sin θCF cos ϕCF sin θLF sin ϕLF+ sin θCF sin ϕCF (sin Θ cos θLF cos ϕLF − cos Θ sin θLF )+ cos θCF (sin Θ sin θLF cos ϕLF + cos Θ cos θLF ) .

(3.18c)In these coordinates, the integral over post-collision angles dΩ′ can then be performed in coordinates
defined by the LF and CF:
Ip(CF) (pr , θ, ϕ) ≡ZdΩ′dσ (0) dσ (1) dσ (2)
+
+
dΩ′
dΩ′
dΩ′

!
∆p2j ,(3.19)

where we utilize the threshold dipolar differential cross-section [see Eqs. (2.25) and (2.23a)] andfactorized it into terms of various powers in ad (denoted with superscripts)
dσ (0)dσ (1)dσ (2)
dσ
(Θ)
=
(Θ)
+
(Θ)
+
(Θ).
dΩ′
dΩ′
dΩ′
dΩ′

(3.20)

to help with algebraic bookkeeping.
Now for the integral over LF coordinates, here comes an explicit perturbative approximation.
A Taylor expansion of cr (pr ) is done up to first-order around thermal equilibrium



 2 2
 2 2
pr sin θLF cos2 ϕLF 1
pr sin θLF sin2 ϕLF 1
cr (pr ) ≈ ceq
(p
)
1
+
δp
−
+
δp
−
x
y
r
r
4⟨p2z ⟩02
4⟨p2z ⟩0
2
 2


2
pr cos θ 1
+ δpz
−
,
(3.21)
4⟨p2z ⟩02
where⟨p2j ⟩−1⟨p2z ⟩0


1
p2r
eq
cr (pr ) ≡exp −,4⟨p2z ⟩0(4π⟨p2z ⟩0 )3/2
δpj ≡(3.22)(3.23)

with ceqr (pr ) being the equilibrium distribution of relative momenta. It is noted that all terms in
cr (pr ) with constant coefficients multiplying ceq
r (pr ) are trivial since the collision integral vanishes
at thermal equilibrium. Putting all these integrals with Eq. (3.14) and Eq. (3.19) together givesC[∆p2j ] = ⟨n⟩Z

d3 pr cr (pr )pr (CF)I(pr , θLF , ϕLF ).
2m p(3.24)

36
The remaining legwork of integration can be handed off to your favorite symbolic software. The one
I used was Mathematica® [72]. For the reader’s convenience, I tabulate all the analytic collision
integrals below. For bosons (B), they are:mω 3 a2eff  2
8N⟨py ⟩ + ⟨p2z ⟩ − 2⟨p2x ⟩
15π
kB T0





64N
mω 3 as ad +
⟨p2x ⟩ − ⟨p2y ⟩ cos(2Θ) − 5⟨p2x ⟩ + 2⟨p2y ⟩ + 3⟨p2z ⟩
105π
kB T0




mω 3 a2d 4N⟨p2z ⟩ − ⟨p2x ⟩ cos(4Θ)
+
315π
kB T0


− 4 ⟨p2x ⟩ − ⟨p2y ⟩ cos(2Θ) + 61⟨p2x ⟩ − 28⟨p2y ⟩ − 33⟨p2z ⟩ , (3.25a)mω 3 a2eff  2
8N2
CB [∆py ] ≈⟨px ⟩ + ⟨p2z ⟩ − 2⟨p2y ⟩
15π
kB T0





mω 3 as ad 
64N−
⟨p2x ⟩ − ⟨p2z ⟩ cos(2Θ) − 2⟨p2x ⟩ + 4⟨p2y ⟩ − 2⟨p2z ⟩
105π
kB T0





mω 3 a2d 
16N⟨p2x ⟩ − ⟨p2z ⟩ cos(2Θ) − 7⟨p2x ⟩ + 14⟨p2y ⟩ − 7⟨p2z ⟩ ,(3.25b)
+
315π
kB T0




mω 3 a2eff  2
8N2
CB [∆pz ] ≈⟨px ⟩ + ⟨p2y ⟩ − 2⟨p2z ⟩
15π
kB T0





64N
mω 3 as ad +
⟨p2y ⟩ − ⟨p2z ⟩ cos(2Θ) + 3⟨p2x ⟩ + 2⟨p2y ⟩ − 5⟨p2z ⟩
105π
kB T0




mω 3 a2d 4N⟨p2x ⟩ − ⟨p2z ⟩ cos(4Θ)
+
315π
kB T0


− 4 ⟨p2y ⟩ − ⟨p2z ⟩ cos(2Θ) − 33⟨p2x ⟩ − 28⟨p2y ⟩ + 61⟨p2z ⟩ , (3.25c)

CB [∆p2x ] ≈where a2eff = 2 a2s − 4as ad /3 + 4a2d /9 . In the absence of dipoles, the expressions above reduce to



16N
mω 3 a2s  2⟨pz ⟩ − ⟨p2⊥ ⟩ ,
15π
kB T0




16N
mω 3 a2s  2
2
Cs [∆pz ] =⟨p⊥ ⟩ − ⟨p2z ⟩ ,
15π
kB T0

Cs [∆p2⊥ ] =

(3.26a)(3.26b)with purely s-wave scattering, where p2⊥ = p2x + p2y .
Likewise, for fermions (F), the collision integrals are:CF [∆p2x ] =



4N
315π



mω 3 a2d
kB T0







4 ⟨p2x ⟩ − ⟨p2y ⟩ cos(2Θ) + 17 ⟨p2x ⟩ − ⟨p2z ⟩ cos(4Θ)

− 45⟨p2x ⟩ + 12⟨p2y ⟩ + 33⟨p2z ⟩ ,(3.27a)37mω 3 a2d  2
16N2
CF [∆py ] =3⟨px ⟩ − 6⟨p2y ⟩ + 3⟨p2z ⟩ − ⟨p2x ⟩ − ⟨p2z ⟩ cos(2Θ) ,
315π
kB T0





mω 3 a2d 4N2
4 ⟨p2y ⟩ − ⟨p2z ⟩ cos(2Θ) − 17 ⟨p2x ⟩ − ⟨p2z ⟩ cos(4Θ)
CF [∆pz ] =
315π
kB T0

+ 33⟨p2x ⟩ + 12⟨p2y ⟩ − 45⟨p2z ⟩ .
(3.27b)(3.27c)Refs. [14, 15] also have all these collision integrals listed.
I should emphasize that these analytic forms hold only in the close-to-threshold scattering
regime (see Sec. 2.1.4), otherwise, numerical integrators are required to compute C[∆p2i ]. Moreover,caution should be taken for temperatures approaching absolute zero, where quantum statistics start
to have a significant effect on the gas ensemble distributions. In particular, the equilibrium particle
distribution deviates from one described by Maxwell-Boltzmann statistics, in turn requiring that thecollision integral also be modified to satisfy the quantum Boltzmann equation (3.3). These quantum
statistical corrections, along with more general forms of out-of-equilibrium dynamics, require moregeneral methods of solutions. So as not to leave you hanging, let me tell you about a solution
method of the numerical variety I’ve used.

3.3

Direct simulation Monte Carlo solutions
Time evolution of a dynamical system can be performed numerically by advancing the equa-tions of motion with discrete time steps. Ideally, these time steps are made adaptive to efficiently
cater for the various phenomena of interest that might occur on different time scales [73]. For the
Boltzmann equation, dynamics that arise from it can be separated into two main processes: a) classical Hamiltonian evolution in phase space [left-hand side of Eq. (3.1)], and b) two-body collisional
interactions [right-hand side of Eq. (3.1)]. This partitioning implies two distinct time scales relevantto the solver, τV and τcoll , that can be related to a characteristic confinement potential time scale
and the collision rate respectively. For instance, a harmonically trapped dilute gas would have thesetime scales set by
τV = 2πω −1 ,(3.28a)38
τcoll = ⟨nσvr ⟩−1 ,(3.28b)where ω is the geometric mean of the harmonic trapping frequencies, σ =RdσdΩ′ dΩ′ is the total

cross section and vr is the relative velocity of collision partners. My implementation thus takes
Hamiltonian evolution to occur in time steps of size ∆t ≪ τV , and collisions in time steps of sizeδt ≪ τcoll . I further allow δt to change during the simulation, setting it as a function of the mean
collision rate at any given time 1 . A visual summary of the numerical time integration scheme isprovided in Fig. 3.3.
(collisions)(collisions)t0 + δt

t0 + mδt...
t0

t0 + ∆t

t0 + 2∆t...

t0 + n∆t

...

...
t

Figure 3.3:
The numerical time integration scheme. Two time steps are utilized: a) ∆t for
Hamiltonian evolution of the particles under V (r), and b) δt for collisional processes (m, n ∈ N).

3.3.1

Evolution in classical phase space

As formulated, the phase space distribution is a continuous function of particle positions andmomenta, and so requires an appropriate discretization scheme for numerical simulation. I do so
by sampling points in phase space which I refer to as “test particles", such thatf (r, p) ≈ ξNTXδ 3 (r − r (k) )δ 3 (p − p(k) ),(3.29)

k=1

where ξ = N/NT is the ratio of the actual number of particles to test particles. A continuous
distribution is then retrieved in the limit of NT → ∞. Each test particle undergoes a classicaltrajectory subject to the potential V (r), progressing in time numerically with the Störmer-Verlet
1
In studies where large variations in local density are expected, δt could be a function of local density and velocity
distributions. This would however be way more involved, so I have kept δt as defined by the mean collision rate.

39
symplectic integrator method:q (k) = r (k) (t) +∆t (k)
p (t),
2m(3.30a)p(k) (t + ∆t) = p(k) (t) + F (k) ∆t,r (k) (t + ∆t) = q (k) +(3.30b)∆t (k)p (t + ∆t),2m

(3.30c)which is constructed to preserve Hamiltonian flow [74]. Superscripts k on the phase space variables
index the various test particles, while ∆t is the numerical time step size as previously prescribed.

3.3.2

The two-body collision integral

To tackle the collision integral of Eq. (3.3), I utilize the direct simulation Monte Carlo (DSMC)
method [75, 76] as a means of numerical integration. At time intervals of δt, the DSMC recipe apportions various chunks of 3-dimensional real space volume for Monte Carlo sampling, by constructing
a discrete spatial grid. Just as is done for time, I adopt a locally adaptive space-discretization
scheme that allows the grid to account for local density variations, performed in two phases. Phase
one is the construction of a master-grid, by first establishing the relevant simulation volume fromthe ensemble of test particle positions {r (k) }:Vsim =

3 
Y
j=1

(k)
(k)
max{rj } − min{rj }
k
k



.

(3.31)

This volume is then partitioned uniformly into cells of equal volume ∆Vsim , and test particles binned
into them based on their positions (visualized in subplot a of Fig. 3.4). The number of such cells ispreferably set such that the hyperparameter2 (∆Vsim )1/3 is much larger than the molecular meanfree path, since phase two proceeds to further refine the grid with an octree algorithm [77]. By
taking each master-grid cell and recursively subdividing it into eight octants, the octree algorithm
max test particles, another hyperparameter. In my
terminates only when each octant has at most Noc

implementation, each octree division of a sub-volume is performed such that the central vertex islocated at the barycentric coordinate of simulation particles in that sub-volume. An example of2

Hyperparameters are numerical parameters tuned by the user for optimality of computational efficiency andstochastic convergence.

40
the octree refinement algorithm (with uniform subdivisions) applied to a system of NT = 100 test
max = 10, is provided in Fig. 3.4.
particles and Noc

Figure 3.4: Visualization of the octree algorithm refinement (b), applied to the master-grid (a),max = 3,projected in the x, y-plane. There are initially 8 master-grid cells with N = 200, and Ncell
with Gaussian distributed points in arbitrary units.

When sufficiently discretized, collisions can be assumed to only occur within each grid celland not between them. Collisions between particles result in them exiting a phase space volume
element d3 rd3 p, at a rate given byδf outout= d3 rd3 pIcoll
[f ],
δt
Z 3
Z


d p1
dσ
out
where Icoll[f ] = −|p − p1 | dΩ′ ′ f f1 1 ± h3 f ′ 1 ± h3 f1′ .
mdΩ
d3 rd3 p(3.32a)(3.32b)In the simulation, products of pre-scattering distributions in a differential phase space volume
d3 rd3 pf f1 , are replaced by pairs of test particles in any given grid cell along with their associatedmomenta p and p1 . In the event of a collision, the post-scattering momenta p′ and p′1 , are thensampled from dσ/dΩ′ to obtain an approximation of d3 rd3 pf ′ f1′ . Application of these steps to all
grid cells effectively results in Monte Carlo integration of the collision integral for a time step δt.

41
In practice, collision sampling described above is performed sequentially by determining a) collisionoccurrences, and then b) post-collision momenta.
In step (a), a collision occurs between test particles i and j, with probability
Pij = ξΓcoll (pij )δt(3.33)

with a local collision rate between particles i and j of relative momentum pij = pi − pj defined as:
Γcoll =

pij
(1 + Q)2σ(pij ) ,∆Vcellm(3.34)

where quantum statistics requires Q = 1 for bosons or Q = 0 for fermions. In general, the total crosssection comprises a sum of elastic and inelastic parts σ = σel + σinel , the latter of which typicallyleads to 2-body losses from the trap. If present, an additional step (a′ ) is added which removes
particle pairs from the simulation with probability Ploss = σinel /σ. Otherwise, a collision progresses
the simulation to step (b), where the elastic portion of Eq. (3.33) carves out an upper bound for its
sampling domain. Particle momenta following an elastic collision are obtained from the anisotropic
dipolar differential cross section of Ref. [35], with an accept-reject scheme [75].

3.3.2.1

Accept-reject collision sampling

An elastic collision causes a rotation of the relative momentum vector between colliders,
directing the particles along a new set of scattering angles (θ′ , ϕ′ ). Numerical instances of theseangles (θ̄′ , ϕ̄′ ) are sampled from a probability distributionp(p̂r , p̂′r ) =1 dσel(p̂ , p̂′ ) sin θ′ ,σel (p̂r ) dΩ r r(3.35)

where I adopt the convention in which the distribution is defined with the Jacobian sin θ′ . This has
a concise form for both bosons (B) and fermions (F), written explicitly in the collision-frame (CF)of Sec. 3.2.1 as [76]:22 2 − 3ã − 3 cos2 η − 3 cos2 ϕ′ sin2 η sin θ′pB (θ , ϕ ; η) =,π [72ã2 + 11 − 30 cos2 η + 27 cos4 η − 24ã (1 − 3 cos2 η)]2
6 cos θ cos2 η − cos2 ϕ′ sin2 η + cos ϕ′ sin θ′ sin(2η) sin θ′′ ′pF (θ , ϕ ; η) =,π [3 + 18 cos2 η − 13 cos4 η]′′(3.36a)(3.36b)42
where ã = as /ad and η = cos−1 (p̂r · Ê). However, sampling directly from these distributions is
not straightforward, so an accept-reject algorithm is employed as an alternative. This algorithm
proceeds as follows:(1) First, sample the variables (θ̄′ , ϕ̄′ ) and ū byθ̄′ = πrθ′ ,(3.37a)ϕ¯′ = 2πrϕ′ ,(3.37b)ū = ru ,(3.37c)where rθ′ , rϕ′ and ru are all random samples from a uniform distribution U(0, 1);(max)(2) compute a normalization M = 2π 2 pF,B (θ̄′ , ϕ̄′ ; η), where(max)
pF,B (η) = max
pF,B (θ′ , ϕ′ ; η) ;
′ ′
θ ,ϕ

(3.38)(3) check if the rejection criterion:ū <pF,B (θ̄′ , ϕ̄′ ; η),M g(θ̄′ , ϕ̄′ )(3.39)is met. If met, (θ̄, ϕ̄) is accepted as a realization of effective sampling from pF,B . Otherwise,start over from step 1 until the condition is met.
Because the scattering angles are sampled within the collision frame, it is necessary to perform aframe rotation to convert these angles back into the lab frame. The post-collision relative momentum
vector in the lab-frame (LF) is given by
′
′
CF
′
′
CF
′
p′r = êCF
1 sin θ cos ϕ + ê2 sin θ sin ϕ + ê3 cos θ ,(3.40)

as written in terms of the collision-frame unit-vectors

cos γ cos θLF cos ϕLF − sin γ sin ϕLF 




CF
ê1 = cos γ cos θLF sin ϕLF + sin γ cos ϕLF  ,− cos γ sin θLF(3.41a)

− sin γ cos θLF cos ϕLF − cos γ sin ϕLF 




êCF
=
− sin γ cos θLF sin ϕLF + cos γ cos ϕLF  ,
2




sin γ sin θLFsin θLF cos ϕLF 




êCF
=
 sin θLF sin ϕLF  ,
3




cos θLF

43(3.41b)(3.41c)where−1

γ = tansin ϕ sin Θ
sin θ cos Θ − cos θ cos ϕ sin Θ.

(3.42)

These coordinate transformations can be rather confusing, but I refer the reader back to Fig. 3.2 asa visual aid.
Convergence of the accept-reject algorithm to pF (θ, ϕ, η) with an increasing number of samples
is illustrated in Fig. 3.5. As plotted, I have set η = π/4 and utilized (a) 105 , (b) 106 and (c) 107
random samples to approximate the analytic distribution (d). The respective color bar ranges
corresponding to each subplot also show a suppression of statistical fluctuations with more samples.

3.3.2.2

Accounting for quantum statistics

Before calling it a day, quantum mechanics requires of us one additional accept-reject step[78]. The collisions we simulate here do not simply occur in a vacuum, but are in fact, surrounded
by a sea of many other quantum bodies. And these quantum bodies will have a say on the final
scattering state outcome. Bosons accept more of these final states, while fermions do quite theopposite. Consequently, the sampled post-collision momenta are only accepted with probability
1 ± h3 fi′ 1 ± h3 fj′
PijQ =,(3.43)(1 + Q)2otherwise no collision is said to have occurred in the first place. Going back to retroactively throw
away simulated collisions is a little cumbersome, but that is just what quantum mechanics requires.

44

Figure 3.5: The probability distribution pF (θ, ϕ, η) with η = π/4 (d), and its approximations asobtained from the accept-reject algorithm with (a) 105 , (b) 106 and (c) 107 random samples.

Computing f ′ in Eq. (3.43) is, however, problematic since the particle distributions fi′ and fj′
are discretized in the simulation. We resolve this issue by “smearing" the δ-functions in Eq. (3.29)
with a Gaussian convolutional kernel [78–80]:
f (r, p, t) ≈ ξNTXδ 3 (r − r (k) )δ 3 (p − p(k) )

k=1

→



3
3
−(p−pk )2 /wp2j−(r−rk )2 /wj2
Y
Y
e
e


.
q
q
ξ
2
2
πw
πw
j=1
k=1 j=1
pj
j
NT
X



(3.44)

These Gaussian kernels are taken to have spatial width wj , and momentum width wpj , such that thediscretization noise is smoothed out while the distribution function remains physically consistent.

45
These criteria are encapsulated by the conditions [79]:1/3N
wp w r ≫ h,
NsimTwi ≪ Ri ,TF
Twp ≪ pF,
TF(3.45a)(3.45b)(3.45c)where pF = ℏ(3π 2 ⟨n⟩)1/3 is the Fermi momentum, Ri = pF /(mωi ) are the Thomas-Fermi radii andbars denote geometric means. We use widths defined by the geometric means of these upper andlower bounds, multiplied by yet another hyperparameter β. With a large number of test particles,
it is sufficient for smearing to be done with Ns < NT random samples of test particles rather than
all of them. This greatly speeds up simulation time while maintaining physical accuracy, given thatNs and β are chosen appropriately. Fig. 3.6 shows an example of a Gaussian distributed ensembleof Ns = NT = 1000 particles (a), smeared by a Gaussian kernel (b) as described above and plottedin the x, y-plane. All units used for this plot are arbitrary.

Figure 3.6: A discrete ensemble of 1000 Gaussian distributed particles (a), smeared against a
Gaussian kernel to generate a smooth distribution (b). The plot is a projection of the distributionsin the y = 0 plane in arbitrary units.

Only now, does the algorithm conclude. To summarize, I have laid out an efficient implementation of the DSMC method to numerically integrate the Boltzmann equation. Combining

46
Hamiltonian evolution in discrete phase space (3.3.1) with Monte Carlo sampling of 2-body collisions (3.3.2), this tool allows for accurate and general simulations of dipolar gas dynamics. An
isolated theorist’s proxy for running real ultracold experiments!

3.3.3

Some considerations and comments

I will end this chapter with some final practical tips and remark on issues one might encounter.
For one, when simulating a gas with very large collision rates, computing collisions by iterating overall grid cells can be extremely computationally expensive. This time cost can, however, be mitigatedwith parallel processing in the far from degenerate regime. Without the need to include quantum
statistics, the DSMC algorithm only samples collisions within each cell, making collision sampling
independent between cells and thus, parallelizable. But before you snatch up all the cores you canget from your nearest Best Buy® , the computational overhead of dispatching collision computations
to multiple cores should be weighed against how much parallelization speed up you get over plain
sequential iteration. These time savings are pretty machine dependent, so the decision to parallelize
is best left to the hands of the programmer.
Upon construction of the DSMC solver, accurate simulations require you to ensure that yournumerics are appropriately converged. A good way to verify this is by convergence in distribution[81]. Knowing that the Boltzmann equation necessarily brings a nonequilibrium gas of elastic
colliders back to thermal equilibrium, one could check that the DSMC solver satisfies this property
too. That is, the steady state phase space distribution of the gas, without the inclusion of quantum
statistics, should converge closer and closer to that in Eq. (3.5) as N increases. If not, one could
consider increasing the number of simulation test particles NT , or varying the maximum number ofmax until such convergence is achieved. With quantum statisticalsimulation particles per grid cell Noc

effects included, the simulation should instead converge to the equilibrium Bose-Einstein (BE) or
Fermi-Dirac (FD) distributions:



−1
E2
E − µcfBE (r, p) =exp
−1,2(ℏω)3kB T(3.46a)

47



−1
E2
E − µcfFD (r, p) =exp
+1,2(ℏω)3kB T
where µc is the chemical potential defined such thatR

(3.46b)f (E)dE = N . The Ns and β hyperparametersmust, therefore, also be tuned accordingly to ensure convergence in these distributions.
One final remark. When dealing with a bulk gas of dipoles, one might be concerned about longrange forces modifying the particle dynamics. Conveniently though, the temperatures and densities
considered in this thesis mostly render these forces negligible. For quantitative reassurance, one
could compare the thermal energy to the average dipolar mean-field energy experienced per particle:
d2 1 − 3 cos2 (Θ),
2σ4πϵ0 σ⊥z√
1 + 2x2 3x2 arctanh 1 − x2
h(x) =−,
1 − x2(1 − x2 )3/2
N
εmf = √ h
24 π



σ⊥
σz



(3.47a)(3.47b)derived in App. D. For the most part, εmf /(kB T ) remains ≲ 0.01 for the regimes of interest, allowing
εmf to be safely ignored. But if one needed to simulate a gas with εmf ≳ kB T , say for polar moleculeswith massive dipole moments, I’d recommend a pseudospectral method like those in Refs. [82, 83],
for efficient handling of dipolar mean-field forces.

Chapter 4

Return to Equilibrium I: Dilute Gases

Macroscopia? No, this is not an illusion1 ! This chapter explores the macroscopic anisotropy
that directly results from the microscopic anisotropies in dipolar scattering. Part one of a two-partsaga. More specifically, this manifestation of anisotropy in collective gas dynamics is studied in thecontext of the gas’ route to equilibrium. I start first with the dilute case, where collisions occur
infrequently over the course of a dipole’s journey across the gaseous sample. Think offensive phasewith running backs trying to avoid tackles for a touchdown, as opposed to defensive phase with
defensive guards perpetually colliding into the opposing offense2 . The latter of which is likened tothe hydrodynamic regime of the next chapter.

4.1

Anisotropic thermalization of dilute dipolar gases
Although the Boltzmann equation preaches a certainty of thermalization, the rate at which

this equilibrium is achieved may vary in direction due to the anisotropy dipoles bring. These rates
can be probed in experiments known as cross-dimensional rethermalization, where a gas initially atequilibrium is preferentially heated along a particular coordinate axis, then left alone to rethermalize.
Rethermalization occurs by means of elastic collisions that redistribute thermal energy throughout
the gas, at a rate proportional to the integral cross section, σ [84–87]. However, not every collisioncounts the same toward rethermalization: if the differential cross section favors forward scattering,1

Unless our universe is a simulation then perhaps it is one.
I don’t even watch American football (except for the Super Bowl halftime shows), but you can’t blame a physicist
for trying to draw analogies. Sko buffs!
2

49
it is not useful, since the collision would have done nothing to redistribute energy into the other axes
not coincident with the incident one. A mantra of collisional thermalization that holds exceptionallytrue for anisotropic scattering. Fig. 4.1 shows a cartoon of the cross-dimensional rethermalization
process and a schematic of collisions that aid or evade thermalization.
rethermalization
x
gas

Ê

heating((

collisions))zheatingtransverse scattering forward scattering(good)(bad)

rethermalization
Figure 4.1: Cartoon of a cross-dimensional rethermalization experiment. Collisions that scatterprimarily into the transverse directions promote thermalization, whereas forward scattering hinders
it. Red indicates pre-collision colliders, while post-collision ones are blue.

4.1.1

How many collisions before one is thermalizing?

When dilute, a gas that is heated along the i-th coordinate will rethermalization along the j-thcoordinate at a rate γij (Θ), following the decay law of Eq. (3.8). This rate depends on the density
and temperature of the given experiment, whereby it is useful to compare the rethermalization rate
to a standard collision rate
γcoll = ⟨nσvr ⟩.

(4.1)

At threshold, σ is energy independent and permits the factorization ⟨n⟩σ⟨vr ⟩, where σ is the integralpcross section, ⟨n⟩ is the average number density of the gas (3.14) and ⟨vr ⟩ = 8kB T /(µπ) is themean collision velocity. Thus for relaxation considerations, the rethermalization rate is proportional
to the standard collision rate via
γij (Θ) =γcoll,
Nij (Θ)(4.2)

50
whose proportionality constant Nij , is known as the number of collisions per rethermalization [84].
The number for rethermalization, a truncation of its full name I will adopt for succinctness, quantifies
the average number of collision instances required before a collision useful for thermalization occurs.
In a harmonic trap, the position and momentum widths, quantified by ⟨rj2 ⟩ and ⟨p2j ⟩, will
experience more or less out-of-phase oscillations at the trap frequency en route to rethermalization.
Easing the extraction of rethermalization rates in theoretical studies then motivates the definitionof non-equilibrium pseudotemperatures:Tj =mωj2 ⟨rj 2 ⟩⟨pj 2 ⟩
+
,
2kB
2mkB

(4.3)

that suppresses trap oscillations in time-evolution. In fact, oscillations are innately suppressedin time-of-flight thermometry common to ultracold experiments, making the quantity above therelevant one to model. Then with experimental data, γij is extracted by fitting an exponential
decay curve to the measured time trace of kinetic temperature along each direction. An example ofsuch a fit is provided by the red dashed curve in Fig. 4.2, that is fitted to data (red diamonds) froma real cross-dimensional rethermalization experiment with ultracold 166 Er atoms done at Innsbruck[2]. But more on this experiment later in Chap. 6.
Assuming that the rethermalization dynamics is dominated by a single decay rate, this rate canbe determined from short-time behavior of the decay and permits a derivation of analytic expressions
for N . I refer to this scheme as the short-time approximation. To formulate this approximation, I
define the phase space averaged quantity⟨χj ⟩ = kB (Tj − Teq ),(4.4)

which quantifies the system’s deviation from its equilibration temperature Teq . From Eqs. (3.8), therelaxation of ⟨χj ⟩ would follow the differential equation d⟨χj ⟩ = C[∆χj ]dt, approximated as
C[χj ] ≈ −γ⟨χj ⟩,(4.5)

in the short-time approximation with a decay rate γ. The relation above thus identifiesγ=−C[∆p2j ]dTj (t)1≈−,(Tj (t) − Teq ) dt t=02m (Tj (0) − Teq )

(4.6)

51
as the rethermalization rate.

Figure 4.2: Experimentally measured momentum space temperatures Tz (blue circles) and Ty (reddiamonds) after a trap frequency quench of the confining potential along y. The measurement was
performed at = 1 G and Θ = 0 for 166 Er. The red dashed line is an exponential+sinusoidal fit to
the data. The black solid line denotes the results of the Enskog theory (3.8) for this specific data
set. The error bars denote the standard error for three repetitions. The inset shows a schematic of
the experimental setup. This plot is adapted from Ref. [2].

From γ and the standard collision rate ⟨n⟩σ⟨vr ⟩, we can extract the value of Nij via the relationin Eq. (4.2). As before, we consider an excitation of axis i, following which the rethermalization
rate is measured along axis j. This is modeled by taking axis i to have an initial out-of-equilibrium
pseudotemperature
Ti = T0 +

δi
,
kB

(4.7)

where δi is a perturbance to the energy, while the initial temperatures along the 2 other axes are
simply T0 . By construction of Eq. (4.6), δi stands as an auxiliary variable that cancels out in thederivation. For clarity, I present an explicit derivation for Nxx with bosons as follows.

52
An excitation along x results in the equilibration temperature2
1
Teq = T0 +
3
3



δx
δx
T0 +
= T0 +
,
kB
3kB

(4.8)

which follows from the equipartition theorem. Then measuring the rethermalization along the xaxis, the initial pseudotemperature deviation is inserted into the collision integral C[∆p2x ] of Eq. 3.25,
to giveC[∆p2x ] =

8N m2 ω 3 a2eff
15πkB T0


64N m2 ω 3 as ad[−2δx ] +[δx cos(2Θ) − 5δx ]105πkB T0
4N m2 ω 3 a2d+[61δx − 4δx cos(2Θ) − δx cos(4Θ)] ,315πkB T0
(4.9)

where, to remind the reader, ω is the geometric mean of trapping frequencies. Dividing this by 2mgives the thermalization rate
γxx = −

C[∆p2x ]2m (Tx (0) − Teq )3 C[∆p2x ]
=−
2 mδx

 

4N m2 ω 3 a2eff
16N m2 ω 3 as ad=+[5 − cos(2Θ)]
5πkB T0
35πkB T0

N m2 ω 3 a2d+[cos(4Θ) + 4 cos(2Θ) − 61] .
105πkB T0(4.10)

Finally to get N , the mean collision rate is computed as⟨n⟩σ⟨v⟩ = Nmω 2
4πkB T0
3/2 
32π 2
16kB T0 1/2 4N m2 ω 3 45a2s + 4a2d2a=8πas +,45 dπm
45πkB T0

(4.11)and divided by γ. Utilizing the procedure just described, analytic forms for all Nij are worked out
for bosons as
 2

2
4
a
+
(4/45)asdBNxx(Θ) = 2, (4.12a)aeff (4/5) + ad as (16/35) [5 − cos(2Θ)] + a2d (1/105) [cos(4Θ) + 4 cos(2Θ) − 61]
4 a2s + (4/45)a2d
B
Nxy (Θ) = 2,(4.12b)aeff (4/5) + as ad (32/35) [2 − cos(2Θ)] + a2d (8/105) [cos(2Θ) − 7]
4 a2s + (4/45)a2d
B
Nxz (Θ) = 2,(4.12c)aeff (4/5) + as ad (96/35) + a2d (2/105) [cos(4Θ) − 33]
4 a2s + (4/45)a2d
B
Nyy (Θ) = 2,(4.12d)aeff (4/5) + as ad (64/35) − a2d (56/105)

53
 2

4 as + (4/45)a2d
B,(4.12e)Nyz(Θ) = 2aeff (4/5) + as ad (32/35) [2 + cos(2Θ)] − a2d (8/105) [cos(2Θ) + 7]
4 a2s + (4/45)a2d
B
Nzz (Θ) = 2. (4.12f)
aeff (4/5) + as ad (16/35) [5 + cos(2Θ)] + a2d (1/105) [cos(4Θ) − 4 cos(2Θ) − 61]
In the absence of dipoles, bosons with purely s-wave scattering require Ns = 2.5 collisions per
rethermalizing collision [88], a result obtained by setting ad = 0 in the expressions above. Fig. 4.3
plots Nxz as an illustrative example of its functional dependence on Θ and as (in units of ad ).
Subplot (b) shows that after an initial decrease, Nxz increases for as ≳ 0.4ad – and thus thethermalization loses efficiency – moving to the regime of contact dominated interaction, eventually
retrieving the pure s-wave scattering result. A similar plot appears in Ref. [2].

B on Θ and a , for a /a = 0, 0.05, 0.1, 0.4 andFigure 4.3: Subplot (a) shows the dependence of Nxzss d
B on a at Θ = 0. The black dashed lines
0.8. Subplot (b) shows the functional dependence of Nxzsshow the values of N for s-wave and p-wave scattering, labeled respectively.

As for fermions, NijF are functions of only Θ, taking the much more concise forms:
112,
45 − 4 cos(2Θ) − 17 cos(4Θ)
14
F
Nyx
(Θ) =
,
3 − cos(2Θ)
56
F
Nzx
(Θ) =
,
33 − 17 cos(4Θ)

F
Nxx
(Θ) =

(4.13a)(4.13b)(4.13c)

54
F
Nyy
(Θ) =

14
,
3

14
,
3 + cos(2Θ)
112
F
Nzz
(Θ) =
.
45 + 4 cos(2Θ) − 17 cos(4Θ)

F
Nzy
(Θ) =

(4.13d)(4.13e)(4.13f)

All the quantities above are plotted in Fig. 4.4, where symmetries imply Nij (Θ) = Nji (Θ). Evidently, thermalization rates can vary drastically depending on how the dipoles are orientated inspace, presenting a tuning knob for thermalizing dynamics.

Figure 4.4: The analytically derived number of collisions per rethermalization for fermions NijF , asa function of Θ for all nine excitation-rethermalization configurations.

55
In certain situations like evaporative cooling (discussions to come in Chap. 6), the relevant
thermalization rate is not just that of a single axis, but one that is averaged over all of them:1γcoll  −1−1
γ i = (γix + γiy + γiz ) =Nix + Niy
+ Niz−1 ,3
3(4.14)

where i still remains the axis of excitation. The averaged number for rethermalization is therefore
extracted as

−1
−1
−1
N i (Θ) = 3 Nix(Θ) + Niy(Θ) + Niz−1 (Θ)= Nii (Θ),(4.15)

the diagonal elements of the non-averaged number of collisions. So the collective thermalization
rate is still tunable via the dipoles for any single-axis excitation.

4.1.2

Universal signatures of dipolar thermalization in KRb

In a recent groundbreaking work by the Ye group at JILA [3], a bulk gas of ultracold 40 K87 Rb
molecular fermions was made stable in three dimensions, facilitated by an electric field-induced
shielding resonance (see Sec. 2.2.1) that suppressed reactive losses by a factor of ∼ 30. The resulting favorable ratio of elastic to inelastic collisions enabled direct dipolar thermalization, with ratedepending on dipole orientation controllable by the external electric field.
Thermalization of these shielded molecules was demonstrated and characterized with exactly
the cross-dimensional thermalization experiments described above, with geometry shown in subplot(a) of Fig. 4.5. Initially heated along the tightly confined y direction, elastic collisions then aid in
redistributing excess kinetic energy from y into x and z. Temperatures along each axis were extracted
by Gaussian fits to the gas cloud from time-of-flight imaging, resulting in the Tx and Ty data points
shown in Fig. 4.5. Rethermalization rates were then obtained by fitting these temperature data
points, after parametric excitation along y, to a set of coupled differential equations [89]:n 1 dTyn 1 dTx
dn= −KL (Ty + 2Tx )n2 −−,dt
2 Ty dt
2 Tx dt(4.16a)dTyn2Γth= KL (2Tx − Ty )Ty −(Ty − Tx ) + cy ,dt
43

(4.16b)

56
dTx
n
Γth
= KL Ty Tx +(Ty − Tx ) + cx ,dt
43

(4.16c)where Γth , KL , cx , and cy are fit parameters corresponding to a rethermalization rate, two-body lossrate and background heating rates respectively. The latter 3 quantities are required because realmolecular experiments neither have perfect optical traps nor purely elastic scattering, a shocking
revelation for naive young Reuben back in 2021. Conversion of the experimental geometry into one
utilized in my theory is done by swapping the y and z axes labels.

Figure 4.5: Time evolution of the temperature for Θ = 45◦ (a) and Θ = 90◦ (b) after parametricheating along y. The solid lines are fits to the thermalization model in Eq. (4.16). The thermalization
is faster at Θ = 45◦ . The molecular gas is heated parametrically along the y direction to create an
initial condition of Ty (0) ≈ 2.5Tx (0), Tx (0) = Tz (0). Figure adapted from Ref. [3].

With collision energies on the order of 500 nK and an electric dipole moment of d ≈ 0.08 D,
collisions are expected to remain close-to-threshold where the number for rethermalization works
out to be that in Eq. (4.13). But with a large initial temperature anisotropy of Ty (0) ≈ 2.5Tx (0),an accurate comparison with the experiment must account for γcoll changing during the relaxation
process because of thermal energy redistribution [90]. As such, N z (Θ) is multiplied by the factor
√

−1
6ϖ
ϖ tan−1 ϖ − 1√α(ϖ) =
1+,(4.17)
1 + 2ϖϖ−1where ϖ = Tx,y (0)/Tz (0) is the initial temperature anisotropy. So with ϖ = 2.5, N z (Θ) is modified
by a factor of α(2.5) ≈ 0.89. A comparison between the theory and experimentally extracted values

57Fof α(ϖ)N z (Θ) is plotted in Fig. 4.6, showing remarkable agreement. This not only showcases
the accuracy of Eq. (4.13), but the validity of assuming point-dipole threshold scattering for KRb
molecules as argued for in Sec. 2.2.1. Similar agreement has been observed in ultracold gases offermionic erbium isotopes [91], lending this dipole-angle dependence of the N to be a smoking gun
signature of universal dipolar scattering at threshold.

F

Figure 4.6: Comparison of the averaged number of collisions per thermalization N z (Θ). The solid(grey) curve shows the theoretical results from Eqs. (4.13) and (4.17), whereas the (black) circles
with error bars are results from the JILA KRb experiment.

4.2

Turn up the heat: Dipolar thermalization far from thresholdIn some ways, we got lucky that KRb has a pretty small dipole moment at its resonant

shielding field, keeping it in the collisional threshold regime at hundreds of nK. Other diatomic
molecules may not be so fortunate (or unfortunate if large dipole moments are wanted). Take for58
instance 23 Na40 K molecules, collisionally shielded with circularly polarized microwaves [4]. Sucha species could have its dipole moment up to ∼ 10 times larger than those of shielded 40 K87 Rb,
typically requiring collision energies to be ∼ 104 times lower for the threshold regime (see Sec. 2.2.2.1
for the relevant arguments). Nonegenerate gases of 23 Na40 K and other strongly polar molecules
should, therefore, have their collisional physics be treated with the energy dependent cross sections
away from threshold (Fig. 2.8). But that means no more Born approximation and unfortunately
for us, no more analytic cross sections also means no more analytic collision integrals for crossdimensional rethermalization.
Fret not! We can still compute these integrals numerically, either by direct evaluation of
Eq. (3.24) or via the DSMC algorithm of Sec. 3.3. Though in both these methods, a high resolution of the differential cross section in angular and energy space is required to accurately evaluate
Nij . Performing close-coupling calculations (Sec. 2.2.2.1) for each sampled point would be highly
impractical given the computational cost of each one. So I present an alternative solution.

4.2.1

Gaussian process interpolation of differential cross sections

At a given collision energy, the elastic differential cross section (here denoted Del for convenience of notation), is a function of the dipole alignment axis Ê, and the relative ingoing andoutgoing momentum vectors ℏk and ℏk′ , respectively. Collectively, I shall denote this set of parameters as β. By first performing close-coupling calculations at several well chosen collision energies
E = ℏ2 k 2 /(2µ), the resultant scattering data is used to infer an M -dimensional continuous hypersurface that approximates Del , with a Gaussian process (GP) model [92–95].
GP regression is a machine learning technique used to interpolate discrete data points, stitching them together to form a continuous global surface. To do so, a GP assumes that Del (β) evaluatedany 2 nearby points in its coordinate space, β i and β j , are Gaussian distributed with a covariancegiven in terms of a function K(β i , β j ), called the kernel. A parameterized functional form for the
kernel is chosen prior to the surface fitting process, reducing the task of combing through an infinite
space of possible functions that best match the data, to a minimization over the kernel parameters.

59This minimization step is referred to as training the GP model.
Several symmetries in the differential cross section help to reduce the computational load oftraining slightly. Rotated into the frame where Ê points along the z axis, which I refer to as the
dipole-frame, the unique hypersurface regions effectively live in an M = 4 dimensional space, withcoordinates β = (E, η, θs , ϕs ). As defined, η = cos−1 k̂ · Ê is the angle between the dipole andincident relative momentum directions, where it is convenient to select k̂ to lie in its x, z plane. The
angles θs and ϕs , denote the inclination and azimuthal scattering angles respectively, in this frame.
Doing so, the differential cross section possesses the symmetry
Del (E, η, θs , ϕs ) = Del (E, η, θs , −ϕs ).

(4.18)

Consequently, we only need to specify the differential cross section for angles within the domain
η, θs , ϕs ∈ [0, π], to fully describe its global structure.
To perform the interpolation with GP regression, I utilize the Matérn- 25 kernel [93]:!
!
√√5 βi − βj5(β i − β j )2
5 βi − βj+K(β i , β j ) = 1 +
exp −
,
w
3w2
w

(4.19)

which is better able to capture the sharp jumps in a non-smooth function, over higher-order differentiable kernels such as the radial basis function. This kernel contains a parameter w that sets a
length scale over which features of the data vary in coordinate space, that is optimized during themodel training process. This kernel is typically not ideal for periodic input data, so we make theperiodicity of the angles (η, θs , ϕs ) explicitly known to the GP model by training it with the cosineof these angles, instead of the angles themselves. Furthermore, log10 (E/Edd ) is fed into the GP
model in place of E, to reduce the disparity in fitting domains between each coordinate of β. The
GP model is trained over the range log10 (E/Edd ) = −6 to 2, corresponding to collision energies of
E/kB ≈ 0.36 pK to 36 µK relevant to ultracold experiments. After training on ∼ 10, 000 samples ofDel (E, η, θs , ϕs ), the resulting GP fit obtains a mean-squared error of ≈ 0.5% against close-coupling
calculations, which are taken to represent the actual cross section accurately.
For proof of principle and to ensure the accuracy of subsequently computed quantities, I
utilized more points than is usually necessary for the GP fitting. I also optimize the model’s

60
hyperparameters [96] on top of just the kernel parameters. Even so, the Gaussian process model
still has issues faithfully reproducing the differential cross section around η, θs = 90◦ , known tohave a discontinuity at threshold [35]. Fortunately, this angular segment corresponds to forward
scattering, which does not contribute to the cross-dimensional thermalization process of interest
here. Present Reuben will leave this as a worry for future Reuben.
R
Fig. 4.7 plots the total cross section σ(E, η) = Del (E, η, Ωs )dΩs , at various collision energies.
There is a marked variation in the η dependence, indicating a higher tendency for side-to-sidecollisions (η = 90◦ ) over head-to-tail ones (η = 0◦ ) at higher energies. To highlight the dominantanisotropic scattering process, Fig. 4.7 also provides plots of the differential cross section at η =45◦ , the approximate angle at which σ is maximal. As energy increases from subplots (a) to (d),the scattered angle dependence of Del becomes biased toward forward scattering, reducing theeffectiveness of collisions for thermalization as discussed in Sec. 4.1. Alphabetic labels in Fig. 4.7
consistently correspond to the collision energies: (b) E = 0.2Edd , (c) E = 2Edd and (d) E = 20Edd .
The Born approximated cross sections at threshold [35] are labeled with (a).

4.2.2Energy dependent collisional efficiencies

Earlier, the thermalization rate was related to the standard collision rate through the number of collisions per rethermalization. The inverse of this quantity, however, has perhaps a moreintuitive interpretation: the ratio εij = γij /γcoll represents the efficiency of each collision toward
thermalization of the gas. This collisional efficiency is formally cast in terms of the integral
π2
εij ≈ αij
64Z2d3 κ e−κ /4
√
(2π)3
π

Z

d2 Ω′

′ κ
Del
∆κ2i ∆κ2j ,⟨σκ⟩(4.20)

2−1/2 ,
where ∆κ2i = κ′2i −κi is the collisional change in adimensional relative momenta κ = pr (mkB T0 )αij = 3/2 if i = j, and αij = −3 otherwise. See App. E for further details on deriving εij .

61Figure 4.7: The central plot shows the total cross section as a function of the incident collision
angle, obtained from (a) the Born approximation (red dashed curve), and from GP interpolation(solid curves) for 3 different collision energies: (b) E = 0.2Edd (black), (c) E = 2Edd (gray) and (d)E = 20Edd (light gray). In alphabetical correspondence, are angular plots of the differential crosssection (in units of a2d ) in subplots with the respective collision energies, assuming dipoles pointing
along Ê = ẑ and incident collision angle η = 45◦ lying in the x, z-plane. Subplot (d) uses a smaller
domain for clarity of presentation.

Evidently from Eq. (4.20), εij is symmetric in its indices which leaves only 6 unique configurations of i and j. Asserting once more that the dipoles lie in the x, z-plane and tilted with
angle Θ = cos−1 (Ê · ẑ), I compute Eq. (4.20) with Monte Carlo integration3 and plot the resultsin Fig. 4.8. Each subplot (a to f) shows a different (i, j) configuration, within which, εij is plottedagainst the dipole tilt angle Θ as solid curves, for the temperatures T = 10 nK (black), T = 100 nK(dark gray), T = 400 nK (gray) and T = 1 µK (light gray). Interestingly, the εij terms involving
excitation or rethermalization along y essentially lose their dependence on Θ around 400 nK, beyond
which collisions are less efficient than even nondipolar p-wave scattering (dashed-dotted blue line
in Fig. 4.8) [88] for all Θ. This decrease can be intuited by looking at the differential cross section
around η = 45◦ , around which the total cross section is maximal. As evidenced from the subplotsof Del in Fig. 4.7, forward scattering is favored at higher collision energies, limiting momentum
transfer between axes and therefore, also the efficiency of collisions toward rethermalization.
3

The Monte Carlo integration gives a ≲ 1% error, which is mostly imperceptible in the log-linear plot.

62

Figure 4.8: εij as a function of the dipole tilt angle Θ, for all 6 unique configurations (subplots
a to f) of the excitation axis i, and measured thermalization axis j. The dashed red curves are
the analytic εij results derived with the Born approximated cross section at threshold, whereas thesolid curves are those from Monte Carlo integration using the GP interpolated cross sections, attemperatures T = 10 nK (black), T = 100 nK (dark gray), T = 400 nK (gray), and T = 1 µK (lightgray). The dashed-dotted blue lines are the efficiency for purely p-wave collisions, εp = 1/4.1.

Without much additional effort, GP interpolated differential cross sections are easily deployedin the DSMC solver of Sec. 3.3, allowing for a broad range of nonequilibrium molecular gas studies away from threshold. Moreover, the GP techniques developed here are broadly applicable toarbitrary collision partners, so long as their elastic scattering matrices are accurately obtainablefrom close-coupling calculations. The road forward to exciting nondegenerate physics is evermore
so pristinely paved for anyone who will take it.

Chapter 5

Return to Equilibrium II: Hydrodynamic Gases

The epic saga continues. But now, the gas is dense, collisionally thick, and a bold-cold
weltering machine. A description most appropriate to fluids.
"... the White Whale darted through the weltering ocean."– Herman Melville (Moby Dick, Chap. 135), 1851.
With much larger dipole moments, the significant collisional cross sections of molecules cause collision rates to skyrocket, sending the ultracold gas straight into its hydrodynamic regime. There,
relaxation dynamics can be drastically different from its dilute counterpart and has its own tale totell. I unpack some of that story here, keeping in mind that many hydrodynamic stones are stillleft unturned.

5.1

How do fluids flow?
A gas is said to be hydrodynamic when collisions result in fast local thermalization, so muchso that the nonequilibrium thermodynamics is overshadowed by long wavelength scale fluid motion. Subplot (a) of Fig. 5.1 illustrates the number density one might expect from a (very) dilutegas, sparse and granular. Whereas subplot (b) shows one that is hydrodynamic, jam-packed withparticles and smooth.

64

Figure 5.1: Single body number density function projected in x, y-plane, comparing the N = 104
dilute (a) and N = 107 hydrodynamic (b) regimes. The spatial distributions are both the same andassumed to be a sinusoid along x, but uniform along y (chosen arbitrarily for illustrative purposes).

Hydrodynamic gases are best described in terms of the continuous field variables of massdensity ρ, flow velocity U and kinetic temperature T [97], each of which are dependent on space
and time. Defined as the molecular-velocity-averaged quantitiesZρ(r, t) = mn(r, t) = d3 vf (r, v, t)m,Z
1
U (r, t) =d3 vf (r, v, t)v,n(r, t)Z
2
1
T (r, t) =d3 vf (r, v, t) m [v − U (r, t)]2 ,3n(r, t)kB2

(5.1a)(5.1b)(5.1c)

these variables undergo dynamics governed by the continuity [98], Navier-Stokes [99, 100] and temperature balance equations [101]:∂ρ+ ∂j (ρUj ) = 0,∂t
∂ (ρUi )ρ+ ∂j (ρUj Ui ) = ∂j τij − ∂i P − ∂i V (r, t),∂tm∂(ρT )
2m+ ∂j (ρT Uj ) =(ςij ∂j Ui − κij ∂i ∂j T ) ,∂t
3kB(5.2a)(5.2b)(5.2c)

where ςij is the fluid stress tensor. Repeated indices are assumed summed over, and will be for theremainder of this chapter. The external potential in consideration here is that from any external

65
confinement Vtrap , and the dipolar mean-field VDMF (r, t) = n(r, t) ∗ Vdd (r) 1 , where ∗ denotes
a convolution and Vdd (r) is the dipole-dipole interaction potential from Eq. (2.15). It is worth
pointing out that the local fluid kinetic temperature is related to the flow velocity via
3n(r, t)kB T (r, t) =
2

Z

1
1
d3 vf (r, v, t) mv 2 − ρU (r, t)2 ,
2
2

(5.3)

where the integral term is the local kinetic energy density. This relation will come in handy down
the road.
Studying fluid dynamics with Eqs. (5.2) requires knowledge of the rank-2 thermal conductivity
tensor κij and the rank-4 viscosity tensor µijkℓ (discussed below). These transport tensors are
dissipative coefficients that relax fluid motion, arising from finite local thermalization times. They
are, therefore, a direct consequence of dipolar collisions, a process we are rather familiar with by
now. So with ultracold molecular gas platforms now available to us and hydrodynamic physics ripe
for exploration, I turn to deriving these collision-emergent coefficients.

5.2

Surfaces this way, fluxes that way
The switch from solving Boltzmann’s equation (3.1) to solving the hydrodynamic ones, isbetter motivated by establishing a separation of scales between the phenomena of interest. Of
concern here is the regime in which macroscopic fluid dynamics is governed by length scales λ (e.g.
wavelengths), much larger than the mean-free path L = ⟨nσ⟩−1 of its constituent molecules. The
corresponding dimensionless ratio Kn = Lλ−1 , referred to as the Knudsen number, quantifies theonset of hydrodynamics when Kn ≪ 1. Furthermore, the period over which such dynamics occur
is much longer than the timescales associated with collisions. Appropriately, I will refer to fluid
dynamics as occurring on macro-scales, whereas collisions are said to occur on micro-scales.

1
One might be concerned that the DMF is already treated microscopically when deriving the transport tensors,and so will be double counted if also included in Eq. (5.2). However, I show in App. F that this is not the case since
the DMF vanishes in the first-order Chapman-Enskog expansion.

66
5.2.1

Microscopic collisions to macroscopic transport

If only weakly perturbed, the local response of a fluid system is completely described bylinear constitutive relations and the associated, medium-specific, transport coefficients of viscosity
and thermal conductivity [102]. These coefficients are derivable from the microscopic physics withmethods established by Chapman and Enskog [69, 103], which I closely follow [16–18]. Startingfrom the Boltzmann equation (3.1), the out-of, but close to, equilibrium atomic distribution can betaken to have the formf (r, u, β) ≈ f0 (u, β) [1 + Φ(r, u, β)] ,(5.4)

where u = v − U is the comoving molecular velocity, β = (kB T )−1 is the standard inverse temperature andf0 (u, β) = n0 (β)c0 (u, β) = n0 (β)

mβ
2π

3/2




mβ 2
exp −
u ,
2

(5.5)

is the steady flow local equilibrium phase space distribution. Following mass, momentum and energy
conservation, the perturbation function Φ, must respectively satisfy the relations
Z

d3 uf0 (u)Φ(r, u, β)m = 0,

(5.6a)

d3 uf0 (u)Φ(r, u, β)mu = 0,

(5.6b)

1
d3 uf0 (u)Φ(r, u, β) mu2 = 0.
2

(5.6c)ZZ

Enskog’s prescription of successive approximations then simplifies the Boltzmann equation, suchthat to leading non-trivial order, it simply becomes∂∂k V (r) ∂+ vk ∂k −
∂t
m ∂uk


f0 ≈ C[f0 Φ].

(5.7)

With the phase space distribution at equilibrium on the left but perturbed on the right, the equation
above is interpreted as making an adiabatic approximation that separates the macro and micro-scale
phenomena. In other words, the macro-scale fluid flow still looks more or less steady amid all the
micro-scale collisional events.

67
What usually follows is a rather lengthy derivation of the transport tensors, starting from
Eq. (5.7). Although, the main text of a physics thesis is probably already dry enough as is to
warrant skipping that. So, if you are willing to trust my math, here is the resulting thermal
conductivity tensor for dipolar fermions:5 + cos(2Θ) 0 − sin(2Θ) 
175µd kB 


κ=

,
0
6
0

512m − sin(2Θ) 0 5 − cos(2Θ)

(5.8)

where
5
µd =
16a2d

r

m
πβ

(5.9)

is the Chapman-Enskog viscosity assuming a hard sphere diameter of ad [69].
The 13 unique viscosity tensor elements, for fermions, are tabulated below as a function of
scattering length as , dipole length ad and dipole orientation angle Θ:µd(117 cos(4Θ) + 84 cos(2Θ) + 415),
512
3µdµ1113 = −(39 sin(4Θ) + 14 sin(2Θ)),512
7µdµ1122 = −(3 cos(2Θ) + 11),128µd
µ1133 = −(117 cos(4Θ) + 107),512
9µdµ1212 =
(5 cos(2Θ) + 9),12845µdµ1223 = −sin(2Θ),128
9µdµ1313 = −(13 cos(4Θ) − 29),512
21µdµ1322 =sin(2Θ),128
3µdµ1333 =
(39 sin(4Θ) − 14 sin(2Θ)),512µ1111 =

µ2222 = 77µd ,7µd(3 cos(2Θ) − 11),1289µdµ2323 = −(5 cos(2Θ) − 9),128µ2233 =

(5.10a)(5.10b)(5.10c)(5.10d)(5.10e)(5.10f)(5.10g)(5.10h)(5.10i)(5.10j)(5.10k)(5.10l)68µ3333 =µd(117 cos(4Θ) − 84 cos(2Θ) + 415).
512(5.10m)All other non-trivial viscosity terms are specified by the tensor symmetry identitiesµijmn = µjimn ,(5.11a)µijmn = µjinm ,(5.11b)µijmn = µmnij ,(5.11c)µijmn δij = µijmn δmn = µijmn δijmn = 0,

(5.11d)where δijmn is 1 if i = j = k = ℓ and 0 otherwise. All other unspecified tensor elements are zero.
As for the less trusting (perhaps rightfully so), the integrals and indices of derivation in all theirtedious glory are found in App. G. Knock yourself out!
By the way, κ and µ have also been derived for dipolar bosons which also includes thescattering length as . For brevity of this thesis, however, I refer the reader to Refs. [16, 17] wherethese have already been tabulated, instead of including them here.

5.3

And whispered in the sound... of dipoles
If you played Simon and Garfunkel’s hit song, "The Sound of Silence", from a point source

speaker in an ultracold dipolar gas, it would sound pretty different depending on where you stand.
A little softer, a little louder, maybe even out of sync between both ears. What’s causing this is
our old familiar friend: dipolar anisotropy, both in the speed of sound and its attenuation.
In more technical terms, I’ve already shown that anisotropy emerges in transport tensors from
dipolar collisions, but a dense ensemble of dipoles could also generate a long-ranged anisotropic
mean-field from their cumulative dipolar potential. If strong enough, the anisotropic push and pullof the dipolar mean-field can distort wave fronts that propagate through the fluid. Best used to
study such physics is a uniform density gas of ultracold polar molecules, possibly realized in opticalbox traps [104]. And let’s consider them fermionic because why not2 . If only weakly perturbed,
2

Three-body losses [105], that’s why not.

69
the dynamical fields can be written in terms of small deviations from equilibrium:ρ(r, t) = ρ0 [1 + χ(r, t)] ,(5.12a)Ui (r, t) = cξi (r, t),(5.12b)

T (r, t) = T0 [1 + ϵ(r, t)] ,(5.12c)having defined the unit-free fluctuation fields χ, ξi , ϵ ≪ 1, with c =p
5kB T0 /(3m) being the ideal

gas thermal speed of sound. Quantities at equilibrium are denoted with a naught subscript. To
linear order in the fluctuation field variables, the fluid equations (5.2) become∂χ+ c∂j ξj ≈ 0,
∂t
µijkℓ
∂ξi 3
ρ0
+ c∂i (ϵ + χ) ≈∂j ∂ℓ ξk − 2 ∂i [χ(r, t) ∗ Φdd (r)] ,
∂t
5
ρ0
m c
∂ϵ 2
2
+ c∂j ξj ≈
κij ∂i ∂j ϵ,∂t 3
3n0 kB(5.13a)(5.13b)(5.13c)supporting acoustic wave solutions [97]. Such solutions can be found by employing a plane wave
ansatz χ, ξi , ϵ ∼ exp i(K T r − ωt) , where conveniently, the dipolar mean-field potential just becomes a Fourier transform of Φdd (r):Z
χ(r, t) ∗ Φdd (r) =d3 r ′ χ(r − r ′ )Φdd (r ′ )T= χei(K r−ωt) φdd (K̂, Ê),with φdd =RT(5.14)′

d3 r ′ e−iK r Φdd (r ′ ) already computed in Ref. [106].

∂
Rendering the derivatives of fluctuation variables ∂j → iKj and ∂t→ −iω, plane waves

reduce the partial differential equations of (5.13) into an eigenvalue problem:
  
 
0
χ 
  

  
ω  ξ  =  3 c + ρ0 φ2dd Km c   5
  
ϵ
0

cK T
iΛ

T
2
3 cK

0  χ
 
 
3
 ,
5 cK   ξ 
 
ϵ
iΓ

(5.15)

having defined the thermal conductivity and viscosity associated rates
Γ=−2
κij Ki Kj ,3n0 kB(5.16a)70
Λik = −

1
µijkℓ Kj Kℓ ,
ρ0(5.16b)respectively. With no other processes to break the symmetry of the system, the anisotropy thatarises in the mode solutions only depends on the relative angle Θ, between the dipole orientation Ê
and plane wave propagation direction K̂. Thus, all essential physics is captured by setting K̂ = ẑbut allowing Θ to vary. In these coordinates, which I assert for the remainder of this section, thetransport associated rate functions have the forms
Γ(Θ) =
Λ(Θ) =

875K 2√(cos(2Θ) − 5) ,12288a2d n0 πmβ0(5.17a)

5K 2√
2048a2d n0 πmβ0(5.17b)





×



9
4 (13 cos(4Θ) − 29)

03
4 (14 sin(2Θ) − 39 sin(4Θ))

09(5 cos(2Θ) − 9)3
4 (14 sin(2Θ) − 39 sin(4Θ))

0




,
0


1
(84
cos(2Θ)−117
cos(4Θ)−415)
4

and φdd becomesφdd (Θ) =d2(3 cos2 Θ − 1).
3ϵ0(5.18)√The Θ angle dependence of Γ and Λ is showcased in unit-free versions (multiplied by a2d n0 mβ0 /K 2 )of them plotted in Fig. 5.2.
Any fluid dynamics resultant from Eq. (5.15) can be described by normal mode solutions,comprising of mode frequencies ωa , and their corresponding mode amplitudes ψ a = (χa , ξ a , ϵa )Twith a = 1 to 5. These normal modes can be obtained analytically by considering only long wavelength excitations such that δ = KL ≪ 1. Since the transport coefficients scale as µijkℓ , κij ∼ δ,long wavelengths permit a series expansion of the mode solutions in increasing powers of the transport tensors via Taylor expansions in δ. Diagonalizing Eq. (5.15) then gives the mode frequenciespi
ω± = ±K c2 + Edd (Θ) +2
ωµ,1 = iΛ11 ,


Λ33 +

2Γc2
5(c2 + Edd ),(5.19a)(5.19b)71ωµ,2 = iΛ22 ,(5.19c)3c2 + 5Edd,
5 (c2 + Edd )ωκ = iΓ(5.19d)to first order in δ, where Edd (Θ) = n0 φdd (Θ)/m is the specific dipolar mean-field energy. Now for
the reason we’re all here, interpreting the physics of these modes.

Figure 5.2: Unit-free versions of the non-trivial Γ and Λ rate coefficients with identical dipolar
fermions, as a function of Θ.

5.3.1

The speed of dipolar soundThe first two modes, with frequencies ω± , represent propagating sound waves and identify thepanisotropic speed of sound cdd (Θ) = limK→0 ω+ (K)/K = c2 + Edd (Θ). This is usefully written
in terms of the dimensionless parameter
2πρ0
η=
5

 2

d
λth 2
,
ϵ0
h

(5.20)

72√
presented as a function of the thermal de Broglie wavelength λth = h/ 2πmkB T0 with h as Planck’s
constant. The speed of sound is then written in terms of η asp
cdd (Θ) = c 1 + η (3 cos2 Θ − 1).

(5.21)

Look, Θ appears in the formula. That’s the dipolar mean-field at play, changing the speed of soundat different Θ! In fact, with a modest value of η = 0.8, you can see in Fig. 5.3 that wave fronts of aspherical wave get distorted exactly because sound travels faster in certain directions over others.

Figure 5.3: A schematic diagram over anisotropic radial sound wave propagation from a point
source, following the evolution χ(t) = χ0 cos[K(r − cdd t)]/r with η = 0.8. There are 3 time slicesthat showcase the growing anisotropy of wavefronts due to the speed of sound dependence on Θ.

In a classical gas, quantum statistics enters only in the collision cross sections. Hence, the mean-field

73
result in (5.21) applies equally to both bosons and fermions. As written, the speed of sound in anormal dipolar gas has an anisotropy similar to that for a dipolar Bose-Einstein condensate (DBEC)[107–109], but with temperature replacing the role of quantum fluctuations. The quantity η compares the magnitude of the dipolar mean-field with thermal energies, which at a fixed temperature
T0 , varies by means of the background density ρ0 and dipole moment d [3, 4, 46, 49].
For propagation of sound waves along the direction of dipole polarization, Θ ∼ 0, this propagation is stable. That is, the value of cdd remains real-valued. However, for sufficiently large η,there is a critical value of ηc (Θ) ≡ −(3 cos2 Θ − 1)−1 , past which the dipolar mean-field interactions
overcomes the thermal kinetic energy, causing cdd (Θ) to become imaginary 3 . An imaginary speedof sound indicates a dipolar instability, also predicted and observed in DBEC [107, 110–114]. Notably, ηc is only well defined within the interval bounded by the dipolar magic angles Θmagic ≈ 54.7◦
and 125.3◦ , at which φdd (Θmagic ) = 0 4 . Within the range of dipolar magic angles, ηc has a minimaof 1 at Θ = 90◦ .

5.3.2

Undulating sound waves

The physics of sound gets more interesting at finite K, where transport tensors now enter
the dynamical arena. To ground discussions, I’ll envision an experiment with a box-trapped [104]uniform density sample of microwave shielded 23 Na40 K molecules cooled to T0 = 250 nK [4]. Whenη ≪ 1, say at n0 = 1012 cm−3 and d = 0.75 D (η ≈ 0.04), the dipolar mean-field, and therefore
Edd , becomes negligible compared to kinetic processes. In this regime, the speed of sound reverts
to that of an ideal gas cdd ≈ c, while the imaginary part of ω± is strictly negative, leading to soundattenuation. Resulting directly from dipolar collisions, the observed attenuation for an excitation
with δ = 0.1 is anisotropic, varying by a factor of ∼ 2 with Θ as shown in Fig. 5.4.

Even at this higher density of n0 = 2 × 1012 cm−3 , the transport coefficients presented in Eqs. (5.8) and (5.10)
are accurate to 10% in higher density corrections [69].
4
The critical value ηc appears to be a pole of certain solutions in Eqs. (5.19). This divergence is, however, only
a feature of the series expansion in δ. Exact solutions to Eq. (5.15) remain finite valued across η = ηc .
3

74

Figure 5.4: Imaginary part of the propagating mode frequency solutions Im[ω± (Θ)] in Hertz (Hz),as a function of the dipole tilt angle Θ in degrees (deg) at η ≈ 0.04. The figure inset plots theabsolute relative angle between K̂ and ξ̂, | cos−1 (K̂ · ξ̂)|, also as a function of Θ.

The ω± modes have associated eigenvectors of the form
 


Γ
∓i cdd K 1  
 


 


Λ13

 0  
i cK
 


 




,

ψ± =  0  + 
0


 


 c    Γ 

Edd
Λ33 
± dd  −i
4
+
+
2
5cK
2cK 
 c  
cdd

 

2
0
3

(5.22)

defined up to an overall scale factor. The first term of the sum in the expression above has nonzero
and comparable amplitudes in the fractional density shift χ, fractional z-velocity ξz , and fractionaltemperature shift ϵ, as expected for a longitudinal wave propagating in the z direction. The second
term in ψ ± shows the additional effects introduced by viscous and thermal damping. Specifically,terms in the density and z-velocity are explicitly damped, while the temperature is not yet damped
at this level of approximation. Along with these effects, a new one appears, namely, a damped

75
motion in the x-velocity ξx .
Therefore, despite initiating sound along K̂ = ẑ, the fluctuations in flow velocity could
occur in a slightly different direction depending on the dipole orientation. That is, along withregular sound propagation along z that alternately compresses and rarefies the gas, the fluid velocity
simultaneously alternates between flow along the ±x directions. The general fluid motion is thereforeof a slightly undulatory nature. This effect, albeit small, is one unique to anisotropic transport
which I illustrate with a plot of the absolute relative angle between K̂ and ξ̂, | cos−1 (K̂ · ξ̂)|, against
Θ in the inset of Fig. 5.4. These weak transverse motions are potentially observable in Doppler
spectroscopy of the undulating molecules.

5.3.3

Shear silence

Even in the absence of sound, a silent hydrodynamic gas of dipoles still has a story to tell.
This narrative is populated by the latter 3 modes of Eq. (5.19), all of which have purely imaginary
frequencies for any value of η and Θ. Once again with η ≈ 0.04 and δ = 0.1, suppressing the dipolar
mean-field, anisotropic damping is accentuated in these silent modes with plots of the imaginaryparts of their mode frequencies in Fig. 5.5.
One of these modes, with frequency ωµ,2 , has a particularly simple form:Tψ µ,2 = 0 0 1 0 0.

(5.23)

This mode consists exclusively of flow velocity in the ±y directions, the velocity being sinusoidally
modulated along z with wavelength 2π/K. If one were to “grab” the z = 0 layer of the fluid
and shake it with frequency |ωµ,2 |, a shear wave would thus develop. This is an overdamped mode,
hence its amplitude reaches only to approximately a certain penetration depth, defined as the inverse
absolute imaginary part of the wave-number [17, 97]:
s
r
µ2323
3 5(9 − 5 cos(4Θ))√
rµ,2 =≈.
2ωρ064
ωa2d n0 πmβ0(5.24)The expression above is obtained from Eq. (5.19), by instead solving for K in terms of a fixed
driving frequency ω. Such waves are, of course, already familiar in ordinary, isotropic fluids.

76

Figure 5.5: The imaginary parts of the mode frequency solutions, Im[ωκ ] (solid black curve),
Im[ωµ,1 ] (dashed blue curve) and Im[ωµ,2 ] (dotted red curve), as a function of Θ in a gas of 23 Na40 K
molecules with n0 = 1012 cm−3 and d = 0.75 D (η ≈ 0.04).

A shear mode with fluid flow in the ±x direction is, however, affected by the anisotropy of
dipolar collisions, given by








Γ c2 Λ13
i cK c2dd Λ11 

0



 

2


5c
Γ 
 1 − ΛΓ11 2cdd
 
+
0
2
Λ
11

 

 
+
ψ µ,1 = 
0
0

 

 

 

 
0
0

 

 
13
0
−i 5Λ
3cK






.







(5.25)

Here, once more, the first term accounts for the dominant motion, namely, oscillations in the ±x
directions induced by shear. The penetration depth for this x-shear mode under an oscillatory shear
drive is, not surprisingly, also dipole angle dependent, and given by the relation
s
r
µ1313
3
5(29 − 13 cos(4Θ))√
rµ,1 =≈.
2ωρ0128
ωa2d n0 πmβ0(5.26)

77
The second term in ψ µ.1 denotes additional accompanying effects associated with this shear
mode, which in this case are damped modulations in the density and temperature fields. In thiscircumstance, where the dipoles are oriented somewhere in the x, z plane, and the shear flow in thex direction, the anisotropy of the collision cross section is capable of shoveling both matter andkinetic energy preferentially into the ±z directions, the same as it ordinarily does for momentum.
Because shear modes are a direct consequence of viscosity in the gas, they present themselves as anexperimental means to measure the viscosity coefficients 5 [18].

Figure 5.6: Relative fluid variable fluctuation amplitudes ξx (t) (dashed-dotted black curve), χ(t)(solid blue curve), ξz (t) (dashed green curve), ϵ(t) (dotted red curve) as a function of time t atz/L = 10, after an impulse shear flow perturbation along x̂. The dipoles are oriented with Θ = π/4.
The relative amplitudes for χ(t), ϵ(t) and ξz (t) are rescaled by a factor of 103 for clarity.
5

The shear viscosity has also been measured in harmonically confined unitary Fermi gases with breathing mode
damping experiments [115]. These measurements were, however, performed for non-dipolar gases with a single shear
viscosity coefficient.

78
The time evolution of relative mode amplitudes ψ(t) in Fig. 5.6 demonstrate this anomalous
shear excitation, following an impulse shear flow perturbation along x̂ with Θ = π/4. Along with χand ϵ, ξz is also subsequently excited. In the plot, the relative amplitudes for χ(t), ϵ(t) and ξz (t) arerescaled by a factor of 103 for ease of visualization, but indicate that this effect is indeed a small one.
Temperature field variations via the introduction of heat into a fluid are commonly referred to as
entropy waves [116, 117], which if initiated by laminar shear flow, motivates the title shear-entropy
waves. As far as I know, this phenomenon is not present in existing shear flow literature, making
for a brand new scientific discovery!
Not forgetting the fifth and final mode ψ κ , within the range of Θ where Edd < 0, ωκ canvanish identically at suitably large values of n0 and ad such that Edd = −3c2 /5 (i.e. η = 3ηc /5).
Satisfying this condition would lead to a mode where ϵ = ξx = ξy = ξz = 0 but χ ̸= 0, as madeevident from its functional form






0

13
i ΛcK




 

 
 




2 +5E
3


5Edd 
dd
 5 1 − ΛΓ11 3c 5c



1
+
0
2
3c2 



dd

 



.

ψκ = 
0
0
+


 


 


 

0
0

 


 


13
dd
−i ΛcK
1 + 5E03c2(5.27)Such a mode implies the existence of long-lived density modulations due to a balance between
thermal and dipolar mean-field energies. This could be rather cool to observe in an experiment, but
I cater further analysis to some other future publication.

5.4

Viscous welter of a trapped dipolar fluid
What happens when a dipolar gas gets really collisionally dense (i.e. more collisions perunit volume per second), while confined in a harmonic trap? Maybe the central region of the gas ishydrodynamic, but surely its Gaussian density profile makes the peripheries dilute (see Figs. 3.1 and
5.7). Would hydrodynamics fail and cause relaxation to revert to that seen in Sec. 4.1? How do webest describe the dynamics then? Well, previous works have reported cross-dimensional relaxation

79
rates being close to the trapping frequency in the hydrodynamic regime [4, 118, 119], unlike thedilute regime being closely related to the collision rate. So surely the faster collision rates result indynamics different from the dilute limit. The gist of my investigations into these questions thus far,is that harmonically trapped gases can be hydrodynamic, well described by equations derived from
(5.2). These equations do work pretty well, but only up to ad hoc corrections which beckon further
study. The following will get you up to speed on the status of things.
x
Ê
fluid

z

gas
Figure 5.7: Cartoon of a Gaussian distributed gas with a hydrodynamic center and dilute corona.

Right from the get go, the disparity between dilute and hydrodynamic trapped gases is clearly
revealed in cross-dimensional rethermalization experiments (Sec. 4.1), illustrated with time traceplots of the pseudotemperatures (4.3) in Fig. 5.8 (obtained from the Monte Carlo solver of Sec. 3.3).
In both panels, a collection of 23 Na40 K molecules is subjected to the same harmonic trapping
potential (3.2), excited along z, then subsequently left to relax. The only systemic difference is themolecule number: for fewer molecules in the upper panel (a), the gas is dilute, while for a greaternumber of molecules in the lower panel (b), it is hydrodynamic. As opposed to the gradual meltingof pseudotemperatures back to equilibrium seen in (a), the behavior in (b) is distinctly fluid-like,resembling the wobble of a water balloon after squishing it along some direction. This latter typeof motion has been christened weltering dynamics [19], and done so quite proudly I might add.
"Welter, my boy – welter like the wind! "– John L. Bohn (random email thread), 2023.

80

Figure 5.8: Pseudotemperatures (4.3) obtained from Monte Carlo simulations in the dilute (upper
panel, a) and hydrodynamic (lower panel, b) regimes. The gas consists of microwave shielded
23 Na40 K molecules with dipole moment d = 0.75 D, oriented along x̂, at temperature T = 700 nK.
The gas is initially excited along z by an instantaneous trap frequency ramp to ωz = 2π × 147 Hz,
while ωx = ωy = 2π × 82.5 Hz remain constant. The regimes are differentiated by the number ofmolecules N , which are N = 104 in panel (a), and N = 2 × 105 in panel (b).

For a more quantitative delineation, the change in N between subplots (a) and (b) results in asignificant difference in the Knudsen number. For these trapped gases, Kn is appropriately defined

81
by the ratio of the mean-free path L = ⟨nσ⟩−1 , to the thermal width of the gas Rth =Kn =pkB T /mω 2 :L
8π 3/2 kB T=.
Rth
N mω 2 σ(5.28)In this definition, subplot (a) has the gas with Kn ≈ 2.2, while in subplot (b), Kn ≈ 0.1 6 .

5.4.1

An effective model for damped welter

Affecting its response and route toward equilibrium, the quest for new hydrodynamic physics
motivates constructing a model for the welter in these gases. Yes I know, Eq. (5.2) is already sucha model, but look at how hard it is to solve, even numerically. Instead, I adopt a variational ansatz
approach to solving these partial differential equations much like that done in Sec. 3.27 . External
confinement from a harmonic potential (3.2) results in the equilibrium (denoted by subscript 0)
density distribution followingρ0 (r) =where Z =

R

d3 re

V (r)
B T0

−k



mN
V (r)
exp −
,
Z
kB T0

(5.29)

gives the appropriate normalization. Considering collective oscillations anddamping from long wavelength excitations that do not induce center-of-mass sloshing, Eq. (5.29)
motivates a Gaussian variational ansatz for the local density:


3
Y
1
ri2
q
ρ(r, t) = mN
exp − 2
,
2σ
(t)
2 (t)
i
2πσ
i=1
i

(5.30)

where σi (t) is the distribution width along each axis i, that varies in time (depicted in Fig. 3.1).
Plugging the ansatz of Eq. (5.30) into the continuity equation (5.2a) gives
  2



3 
X
ri
σ̇i (t)
ri
+
−1
= 0,
∂i Ui (r) − Ui (r)σi (t)σi2 (t)σi2 (t)(5.31)

i=1

which admits the velocity field solutionUi (r) =6σ̇i (t)σi (t)

ri .

(5.32)The relations above provide an approximate mean Knudsen number. In practice, the thermal width can differ indirections with different trap frequencies, while the cross section, for dipolar scattering, can depend on the directionof the collision axis. Thus the boundary between hydrodynamic and dilute flow can be anisotropic, a topic to be
dealt with below.
7
Our method has similarities to the scaling ansatz employed in Ref. [120–122], but formulated slightly differently
and with the inclusion of transport tensors.

82
For clarity, I’ll revert to treating repeated indices as unsummed in this section, unless following an
explicit summation symbol Σ. The ρ and U above then render the Navier-Stokes equation (5.2b)kB
σ̈i (t) + ωi2 σi (t) =mX ∂j µijkℓ (T )1σi (t)σ̇k∂i T (r, t) + σi−δk,ℓ ,σi (t)riri ρ(r)σℓ(5.33)

j,k,ℓwhich bears no dependence on thermal conductivity. Since σi (t) does not depend on spatial coordinates, consistency requires taking a spatial average to suppress local fluctuations of the temperaturefield in Eq. (5.33). This average is taken by multiplying Eq. (5.33) and the temperature balance
equation (5.2c), by n(r, t), then integrating over d3 r. App. H gives further details of the spatial
averaging procedure, which results inσ̈i (t) + ωi2 σi (t) + 2kB T0
1 X 2 2
2 Vhy X µiijj (T (t)) σ̇j (t)≈−.
ωj σj (t) + σ̇j2 (t) −3σi (t)mσi (t)
5 Nmσi (t) σj (t)j(5.34)j

The relevant viscosity matrix elements can be recast in terms of a unit-free matrix
Mij (Θ) ≡µiijj (T ; Θ)µd (T )

(5.35)


117
cos(4Θ)+84
cos(2Θ)+415−28(3
cos(2Θ)+11)−(117
cos(4Θ)
+
107)



1 


=

,
−28(3
cos(2Θ)
+
11)
616
28(3
cos(2Θ)
−
11)

512 


−(117 cos(4Θ) + 107)28(3 cos(2Θ) − 11) 117 cos(4Θ) − 84 cos(2Θ) + 415

as is read off from Eq. (5.10), where µd was that defined in Eq. (5.9). For a gas of 23 Na40 K molecules
microwave shielded to have an effective dipole moment d = 0.75 D, its isotropic viscosity at T0 = 700nK takes a value of µ0 ≈ 2.5 × 10−15 Pa·s, around 1010 times less than air at room temperatureand pressure [123]. These parameters are assumed for the rest of what follows. The Mij (Θ) matrix
elements are plotted in Fig. 5.9, with components coupled to the x and z axes showcasing a significant
variation with Θ. The magnitude of off-diagonal matrix elements M13 = Mxz and M23 = Myz
become maximally separated around Θ ≈ 45◦ , explaining the slight separation of Tx (t) and Ty (t) in
Fig. 5.10, otherwise negligible when Θ = 0◦ , 90◦ .

83

Figure 5.9: Mij matrix elements as a function of Θ. The diagonal elements are plotted on the left
in subplot (a), whereas the negated (multiplied by a minus sign) off-diagonal elements are plottedon the right in subplot (b).

Figure 5.10: Times traces of Tx (t) (solid green curves), Ty (t) (dashed blue curves) and Tz (t) (dotted
red curves) for 3 values of Θ = 0◦ , 45◦ , 90◦ , in subplots (a, d), (b, e) and (c, f) respectively. The
2 rows are differentiated by the number of molecules, with the upper row (subplots a, b, c) having
N = 2 × 103 (Kn ≈ 11.10), while the lower row (subplots d, e, f) has N = 3 × 105 (Kn ≈ 0.07).
Note that the simulation times are different between the upper (t = 0 to 0.1s) and lower (t = 0 to
0.04s) rows.

84
For the relevance of time-of-flight imaging, the momentum space temperature, which differsfrom the local temperature of Eq. (5.1c), can also be obtained from solutions to Eq. (5.34) via therelation1
kB Tp (t) =3NZd3 rd3 vf (r, v, t)mv 2= 2kB T0 −1X
mωi2 σi2 (t),3(5.36)

i

as follows from Eqs. (5.3) and (5.32), along with the total energy relation Etotal = 3kB T0 from theequipartition theorem.
Eq. (5.34) above treats the temperature field appearing in µijkℓ (T ) to be spatially uniform
over the region where the gas is hydrodynamic. Such an approximation follows from the form ofcollective oscillations implied by the density (5.30) and flow velocity fields (5.32) in an initially
isothermal gas, disallowing a spatial temperature variation on the order of the gas spatial widths[124, 125]. Hence, temperature as appears in the viscosity is simply treated as T ≈ T (t). Doing so,Rrequires defining an effective hydrodynamic volume Vhy = d3 r 8 , over which viscosity is relevant.
Proper identification of this volume, including its dependence on aspect ratio, density, and dipole
tilt, is essential to the performance of the model (5.34). This is going to be a real undertaking, but
I’ll leave most of the details to App. I. Here are the just highlights.
First, I define this volume to be the spheroidal volume bounded by the outer classical turning
radius of the trap, multiplied by an empirical factor η. The outer turning radius is obtained byequating Etotal = V (RHD , θ, ϕ), to give (see App. H)
2
RHD
(θ) =

−1
6kB T (t)  2
sin θ + λ cos2 θ,2mω⊥(5.37)

where λ = (ωz /ω⊥ )2 quantifies the trapping anisotropy. The effective hydrodynamic volume is thencomputed asη(λ, Kn)Vhy (λ, Kn) =
3
8

Z

4π
3
RHD
(Ω)dΩ =
3



6kB T (t)
2
mω⊥

3/2

η(λ, Kn)√.
λ(5.38)

Refs. [124, 126] provides another estimate for the effective hydrodynamic volume which, however, we findsystematically underestimates the results from our numerical Monte Carlo simulations.

85
As written, I have assumed that η could depend on the trapping geometry through λ and on the
Knudsen number, which in turn, also implicitly depends on N and the dipole angle Θ. Such generality allows η to act as a coarse-graining parameter which accounts for all non-hydrodynamic effects
excluded from the current theoretical treatment. Additionally, Eq. (5.9) implies the temperature√dependence of viscosity goes as µiijj (T ) ∝ T , for which I will simply approximate as T ≈ T0 for
all times 9 .
Then by careful comparison of Eq. (5.34) to a large batch of numerical simulations (detailedin App. I), I find that the effective volume in a prolate (i.e. cigar shaped) trap is best described by
the functional form
Vhy (λ, N, Θ) ≈

4π
3



6kB T0
2
mω⊥

3/2





1
σ(Θ) N√ 2.21 + 0.67 1 + 0.26
,
σ
105
λ

(5.39)

whereσ(Θ) =


πa2d 
3 + 18 cos2 Θ − 13 cos4 Θ ,3(5.40)

is the total cross section. With the parametrization above, Eq. (5.34) can now be used to reliably
determine the cross-dimensional relaxation dynamics of a hydrodynamic dipolar Fermi gas, subject
to excitation along the long axis of the prolate trap. As presented, Eq. (5.39) shows that thehydrodynamic volume, at a fixed temperature and harmonic confinement, grows linearly with thenumber of molecules N . This trend is inferred from the numerical simulation reflected in subplot(a) of Fig. 5.11. Notably, subplot (b) of Fig. 5.11 also endows Eq. (5.39) with a dipolar angle
dependence, which can be intuited as follows. In a prolate trap, the gas has a larger thermal widthalong the weak trapping axis z. As a result, the mean-free path along that axis is relatively smaller
compared to the sample size, and consequently more hydrodynamic. Collisions that occur withrelative momentum directed along the long axis, are then most able to keep molecules behaving
collectively as hydrodynamic. So by attributing collisions incident along ẑ as most effective towardviscous damping, the hydrodynamic volume can be taken as dependent on dipolar scattering via
the total cross section with k̂ = ẑ.
9Incorporating time-dependence in the temperature T (t), requires more sophisticated treatments such as secondorder hydrodynamics [127], that I cater to a future study.

86
(a)

(b)

Figure 5.11: The points with errorbars (standard errors) give η as obtained from DSMC simulations,performed over different values of N, λ and Θ. The linear trend of η vs N is provided in subplot(a), being largely consistent over various values of λ = 0.13, 0.20, 0.32, 0.50, all of which are prolate(cigar) geometries. Subplot (b) shows how η varies with Θ, closely following the angle dependence
in Eq. (5.40). The data for this latter trend is obtained with λ = 0.2, for N = 4 × 105 (black data,Kn ≈ 0.06), N = 3 × 105 (gray data, Kn ≈ 0.07) and N = 2 × 105 (light gray data, Kn ≈ 0.11)

Evidently from Eq. (5.39), Vhy remains a stagnant parameter through the gas’ dynamical
evolution. There is, however, no physical reason why such a volume has to be so. A new avenueof research thus presents itself. Can Vhy be formulated self-consistently from first principles? Orperhaps a machine might better be able to help with this task [128, 129]. All terribly captivating,these prospects must surely be pursued with haste. Though it is not this thesis that I will answer
to the cry of these scientific curiosities. Perchance a work to come soon just might.
The physics of Eq. (5.34) is made more accessible in the language of normal modes, motivating
a linear analysis. If only taken perturbatively out-of-equilibrium, the time variation of the ensemble
widths can be written in terms of small deviations away from its equilibrium values σi (t) = σ0,i +δσi (t). Then expanding to first-order in δσi (t), Eq. (5.34) becomes¨ i (t) + 2
δσXj˙ j (t) +Γij δσXjOij δσj (t) ≈ 0,

(5.41)

87with squared-frequency and damping matrices
2
Oij = 2ωi2 δi,j + ωi ωj ,3
µ0 Vhyωi Mij (Θ)ωj .
Γij =5N kB T0

(5.42a)(5.42b)

The matrices above encode the anisotropies from both the trap and anisotropic collisions. Well what
do you know, Eq. (5.41) tells us that weltering perturbations are simply the motion of damped
˙ i , it iscoupled harmonic oscillators! With Γ multiplying the first-order time derivative terms δσmade clear that damping of weltering oscillations results from the trap frequency weighted viscosities
within the hydrodynamic volume. A factor 2 multiplies Γ in Eq. (5.41) as is convention in damped
harmonic oscillators.
Diagonalizing the squared-frequency matrix O gives the eigenfrequencies2,ω02 = 2ω⊥

p
1
2
2
ω±
=
,
4λ + 5 ± 16λ2 − 32λ + 25 ω⊥
3(5.43a)(5.43b)which are exactly those obtained for inviscid Euler flow in Refs. [124, 130], and correspond to therespective eigenmodes (up to arbitrary normalization)
 
1
 
 
o0 = −1 ,
 
 
0


(5.44a)p5 − 4λ ± 25 + 16λ(λ − 2)


p


o± = 5 − 4λ ± 25 + 16λ(λ − 2) .




√
4 λ

(5.44b)The eigenmode o0 is a strictly radial quadrupole mode, while o− and o+ are 3-dimensional quadrupole
and breathing modes respectively. See Fig. 5.12 for a visual reference of these modes.

88radial quadrupolemode

3D breathingmode

3D quadrupole
mode

Figure 5.12: Visualization of the dominant dynamics in oscillatory modes. Arrows that point in
opposing directions relative to the gas cloud (ellipsoid) indicate dynamics that are out-of-phase.
These figures are for illustrative purposes.

Similarly, Γ results in two nontrivial eigenvalues γ± , that constitute the eigen rates of Γ. Although it is tempting to assign one of these eigenrates as the overall relaxation rate, the eigenmodes
associated with each γ± , are in general, not the eigenmodes of O. Consequently, coupling between
the eigenmodes of Γ is inevitable during dynamical evolution, enforcing that accurate relaxation
trajectories are best obtained from full solutions to Eq. (5.41), if not Eq. (5.34).
The physics predicted by our weltering theory has yet to be validated by experiments. Ifyou, the reader, are an experimentalist looking for fun hydrodynamics to do with your ultracold
molecular setup, my digital door remains open for the welter wonderland we could pursue.

Chapter 6

Theory and Experiment Collide

An allure to working in atomic and molecular physics is the bevy of incredible experiments
happening around the globe. At the very cutting edge of scientific discovery, these experiments areoften pushing the boundaries of what theorists have to propose and can predict. Like any healthy
competition, I’ve found that this exhilarating scientific back-and-forth drives much ingenuity andprogress. Still, world leading experimentalists will every so often look to theorists for modeling,understanding and interpreting the physics observed in their experiments. Such collaboration also
provides opportunities to actually validate the otherwise purely intellectual musings of how theuniverse truly behaves. Here are two experiments that utilized some of my musings.
“Theory will take you only so far."— J. Robert Oppenheimer (Oppenheimer), 2023.

6.1

Innsbruck: Cold mountains, ultracold atoms, hot science!
By this point, Sec. 2.1.5 has already explained all the details one needs to know about scat-tering lengths to follow along here. Although, knowing its numerical value, outside the context
of thermalization, has thus far gone rather unmotivated. It turns out, the scattering length lends
itself to parameterizing interactions in multiple models of quantum many-body physics. Like theGross-Pitaevskii equation for studying Bose-Einstein condensates [131, 132], and the Bose-Hubbard
model for describing the superfluid to Mott insulator transition [133], amongst others [9].

90More recently, interest has grown in ultracold atom platforms for studying many-body physics
with dipole-dipole interactions [27, 28, 134]. Such systems utilizing bosonic isotopes still require
knowledge of the scattering length, which is really difficult to obtain from ab initio calculations.
An alternative is to extract it from experimental measurements, which is what I’ve done with the
Ferlaino group at Innsbruck. Together, we accurately determined as in much the same scheme asthat for Dysprosium atoms by the Lev group [87], but now supercharged with analytic expressions
for thermalization. Here’s the inside scoop of how we did it.

6.1.1

Extracting scattering lengths from thermalization

At Innsbruck, spin-polarized thermal clouds of erbium (Er) atoms were produced in theirlowest Zeeman sublevel [2], primed for rethermalization experiments. To achieve this, Er atomic
ensembles are first cooled and trapped in a narrow-line magneto-optical trap [135], then transferred
into a crossed optical dipole trap (see App. B). The gas temperature is further lowered via evaporative cooling, following which the optical confinement is re-tightened to prevent further atom loss
from residual evaporation at final trap frequencies (ωx , ωy , ωz ) ≈ 2π × (65, 19, 300) Hz. Simultaneously, B is ramped to its desired value, which for all those considered (up to B = 5 G), the Zeeman
splitting is linear and gives ad = 98.2a0 (see Sec. 2.1.1). At this stage, the gas is typically at atemperature between T = 250 to 300 nK with a population of N ≈ 105 atoms. The calculated
critical temperature for the onset of Bose-Einstein condensation Tc [136] lies between 150 nK and
200 nK, around 1.5 times less than T , ensuring a nondegenerate sample. The orientation of themagnetic dipoles is controlled by the direction of the polarizing magnetic field and is parameterized
by the angle Θ between B and the vertical direction z (see inset Fig. 4.2). Careful now, notice thatthe Θ is defined in the y, z-plane for the experiment, but in the x, z-plane in Chap. 4. That’s going
to be important when characterizing anisotropy.
With the thermal sample prepared, cross-dimensional thermalization can commence [91]. The
cloud is heated almost instantaneously along y, after which thermalization dynamics is probed alongz. This excitation scheme relies on a rapid increase in power of one trapping beam, leading to a

91
60% increase of the y trapping frequency. Thermometry of the effective temperatures Tz (Ty ) overa time interval t is obtained from widths of the momentum distribution σz (t) (σy (t)) after time offlight expansion for tToF = 25 ms (20 ms). This scheme leads to an out-of-equilibrium cloud with aneffective temperature increase along y from about 300 nK to 600 nK. Fig. 4.2 plots the subsequentrelaxation time traces of Tz and Ty to the same equilibrium temperature, at B = 1 G. Despite some
oscillations attributed to a breathing mode induced by the finite-time excitation, the observed Tz
displays a dominant exponential behavior of the form:Tz (t) = Teq + ∆Tz e−γEr t ,(6.1)

where Teq is the equilibration temperature and ∆Tz denotes the change in temperature due to theadded energy. However, applying this simple fit does not immediately give as unless we know the
explicit functional dependence of γEr (as , Θ) on as . Well, aren’t we fortunate, for this dependence
is exactly what Eqs. (4.2) and (4.12) give us with
14 45a2s + 4a2d
NEr (as , Θ) =.
252a2s + 96as ad + (3 cos(4Θ) + 13)a2d(6.2)

As presented, NEr corresponds to Nxz in theory-land axes, and Nyz in the real-world experimental
setup.
A competition between the anisotropic character of NEr (as , Θ) is clearly seen in Fig. 4.3,
with plots of it as a function of Θ (a) and as (b). Although a pronounced dipole angle dependencewith maxima at 45◦ is observed at small as (≲ 10a0 ), such behavior progressively washes out forincreasing as . Around as ≈ 70a0 , the thermalization behavior becomes basically independent of Θ
but NEr (as , Θ) is still smaller than the case of purely s-wave collisions. Much faster thermalizationis thus expected, arising from a more efficient distribution of momenta from dipolar scattering
scattering. The experiment only measures rethermalization for relatively large values of as ≳ 30a0
with little Θ-dependence in NEr (as , Θ). So Θ is simply fixed at 0◦ which simplifies Eq. (6.2) to
14 45a2s + 4a2d
NEr (as , 0 ) =.
252a2s + 96as ad + 16a2d
◦

(6.3)

92
Notably at Θ = 0◦ , the number for rethermalization is minimized at ad /as ≈ 2.7 withvalue NEr ≈ 1.65, indicating highly efficient collisional thermalization. This efficiency is directly
attributed to the innate anisotropic differential cross section in dipolar bosons [35].

6.1.2

Scattering lengths galore!

Before taking cross-dimensional thermalization measurements for 166 Er, a high resolution
scan of the atom number as a function of the magnetic field is performed in order to record thedense spectrum of Fano-Feshbach resonances [137, 138]. The Fano-Feshbach spectra is tabulated in
Tabs. 6.1 and 6.2, with B oriented along z, and taking values from B = 0 G to 5 G (see Fig. 6.1).

Table 6.1: Fano-Feshbach resonance positions and widths included into the fit of Eq. (6.5) to as
for 164 Er (left) and 166 Er (right). The error denotes the fit error of one standard deviation. Values
without error are fixed in the fitting procedure.
Position Bi (G)
1.52
2.67
2.83
3.26

Width ∆Bi (G)0.22(3)0.005
0.005
0.10(3)

Position Bi (G)
0.02(5)3.04(5)
4.208
4.96

Width ∆Bi (G)0.05(2)0.15(2)0.01
0.005

Table 6.2: Fano-Feshbach resonance positions and widths included into the fit of Eq. (6.5) to as
for 168 Er (left) and 170 Er (right). The error denotes the fit error of one standard deviation. Values
without error are fixed in the fitting procedure.
Position Bi (G)0.49
0.911(6)1.51
2.174(4)2.471(9)
2.86
3.79
4.23
4.5

Width ∆Bi (G)0.005
0.032(2)0.01
0.038(2)0.19(1)0.005
0.006(5)
0.005
0.005

Position Bi (G)
0.35
0.86
1.12
1.62
2.17
2.74
3.3
3.57
4.38
4.49
6.91

Width ∆Bi (G)0.005
0.028(12)
0.005
0.01
0.067(7)0.134(9)0.01(1)
0.01
0.005
0.01
0.8(7)

93

Figure 6.1: Atom-loss spectroscopy (orange circles) as a function of B for a fixed hold time of
250 ms. For each value of B, each data point is an average of 3 repetitions and normalized tothe maximum averaged atom number recorded. The values of as are those extracted from crossdimensional thermalization measurements using both, the Enskog equations (red squares) and theanalytic formula of Eq. (6.2) (blue diamonds), for 164 Er (a), 166 Er (b), 168 Er (c) and 170 Er (d).
The black triangles are as values obtained from lattice modulation spectroscopy measurements (seeRef. [2]). The solid black lines represent a fit of Eq. (6.5) to as . Error bars and the shaded area ofthe fitting results denote the standard error. The figure is adapted from Ref. [2].

94
Thermalization measurements are then performed at values of B where the system is not
dominated by resonant atom loss. For each thermalization curve, we extract as using two different
approaches, one numerical and one semi-analytical. The first, constitutes a direct fit of the full
Enskog solutions to the experimental data, leaving as as a float parameter of the theory. That is,
we numerically evolve Eq. 3.8 for various values of as until the one that minimizes
min
ω,as


tend 
X
T (t) − TE [T (0); as ] (t) 2
t=t0

δT (t),(6.4)

is obtained, where the sum runs over measurement time instances t, T (t) is the temperature datafrom the experiment, δT (t) is the temperature measurement uncertainty, and TE [T (0); as ] is thesolution to the Enskog equations with initial condition T (0).
The second method is based on the exponential decay rate γ from Eq. (4.2), using the analytic
expression for NEr (as , 0◦ ) in Eq. (6.3). Since as is a priori unknown, we use an iterative approachto determine NEr (as , 0◦ ) starting from the Gribakin-Flambaum mean scattering length value ofEq. (2.27), as ≈ 100a0 . We use the calculated as and the analytic formula (6.3) to obtain a newvalue for NEr (as , 0◦ ). We stop the iteration once the relative change of NEr (as , 0◦ ) is ≤ 10−7 .
Fig. 6.1 summarizes as for 164 Er (subplot a), 166 Er (subplot b), 168 Er (subplot c) and 170 Er (subplotd) in the region from 0 G to 5 G. The values of as extracted from both the Enskog model andthe analytic formulas are in very good agreement with one another, reflecting the strength of theanalytic formula of Eq. (6.3). A triumph for theory!
As an added assurance, lattice modulation spectroscopy was also performed to extract as forthe 166 Er and 168 Er isotopes [139–141] (see the black triangles in subplots b and c of Fig. 6.1). This
technique is rather involved but gives highly accurate measurements, serving as a useful benchmark
for the thermalization methods of focus here.

95
6.1.3

Mass scaling of the background scattering length

More broadly, the global behavior of as with B can be described by generalizing the well-known
formula in Ref. [142], toas (B) =abgs + sB

res 
N
Y

i=1

∆Bi
1−
B − Bi,(6.5)

allowing for Nres overlapping resonances at positions Bi with widths ∆Bi . Fitting Eq. (6.5) to thethermalization measured values of as (see the solid lines in Fig. 6.1) then allows us to map out
its complete functional dependence on B, for all 4 bosonic isotopes 164 Er, 166 Er, 168 Er, and 170 Er.
Knowledge of as as a function of the magnetic field then allows us to extract an effective backgroundscattering length abg
s for each isotope. Various mechanisms could lead to an off-resonant variationof as , such as broad Fano-Feshbach resonances not within the measurement range [143] or dipole
couplings to higher Zeeman sublevels [144]. We thus allow for a smooth off-resonant variation of aswith B, by including a linear slope parameter s in Eq. (6.5) that already reproduces the data well.
The zero field value of as (0) is then identified as the effective background scattering length.
Fig. 6.2 shows the value of abg
s from the fit as a function of the isotope mass. We observea monotonic rise of abg
s with increasing m, compatible with a variety of functional forms. For
convenience, we pick the one consistent with ytterbium and cesium [39, 145, 146], which assumes
only a Van der Waals potential V6 (r) = −C6 /r6 , with C6 being the Van der Waals coefficient. Thismight not be a great approximation for magnetic erbium, but in the absence of alternative models,
this will have to do for the time being.
Starting from Eq. (2.27), we write as ashπ i
as = ā 1 − tan ϕs −,8
√ Z ∞ pm
ϕs =dr −Vint (r),
ℏ R0

(6.6a)(6.6b)

with the classical turning points of Vint (r) denoted as R0 . Although the exact shape of Vint (r) is√
unknown, Eq. (6.6a) can be employed to extract a mass-scaling due to the dependence of ϕs ∝ m[145]. Such a scaling is valid, as long as the mass-dependent modification of Vint (r) is negligible.

96
Furthermore, ϕs allows for the calculation of the number of bound states NB , via Levinson’s theorem
NB = ⌊ϕs /π − 5/8⌋ [147], where ⌊. . .⌋ denotes the integer floor function. Applied to our case of
erbium, we fit Eq. (6.6a) to the experimental data and obtain Fig. 6.2. The optimal fit gives
ϕs /π = 144(1), implying NB = 143(1) boundstates for 168 Er, a factor of 2 larger than ytterbium
despite their similar C6 coefficients [145].

140
120
100
80
60
40
163

164

165

166

167

168

169

170

171

Figure 6.2: Background scattering length abg
s for four bosonic isotopes (red circles). The solid line
represents the best fit with ϕs /π = 144(1); see text. The shaded area, enclosed by the dotted lines,
represents the fitting function for ϕs = 143 and ϕs = 145. The errorbars denote the standard errorof the fit of Eq. (6.5) to the experimental data.

6.2

MPQ: The road to deeper Fermi degeneracy with molecules
If you’ve ever left the lid off your morning cup of coffee, you would have likely found it

already cooled off by the time you’re done checking your emails. This disappointment is driven bya process known as evaporative cooling, where the hot water molecules are free to evaporate away,
transporting thermal energy to the surrounding room temperature air. The remaining coffee thenquickly thermalizes its now less energetic distribution, leaving behind a sad lukewarm beverage but
a spanking new thermoflask in your Amazon cart.

97
In very much the same way, this continuous cycle of evaporation and rethermalization is whatenabled the 2001 Nobel laureates to achieve Bose-Einstein condensates of atoms [7, 8]. More recently, the advent of collision shielding (see Sec. 2.2) has also allowed diatomic molecules to see thelikes of Bose-Einstein condensation [54] and Fermi degeneracy [4, 44], ushering us into a new age ofmolecular quantum gases. Toward explorations of many-body quantum physics with deeply degenerate molecular fermions, I have been collaborating with the group at the Max-Planck-Institut für
Quantenoptik (MPQ), in the hopes of reaching T /TF < 0.1. However, evaporating these molecularfermions to such low temperatures is problematic for several reasons: 1) Fermi statistics become
significant at low temperatures, Pauli-blocking elastic collisions from thermalizing the sample during
evaporation, while 2) 2-body losses continue to occur even with microwave shielding applied, which
are not Pauli-blocked. Nevertheless, we would still like to optimize the evaporative cooling procedure as much as possible, which would minimally provide us with an extremely low temperaturesample to then commence further cooling protocols down to T /TF < 0.1.

6.2.1

Let it flow, let it flow!

The harmonic approximation of Eq. (3.2) is only valid in deep optical dipole traps, wherelaser powers are large enough to keep most of the molecules close to the trap’s energetic minimum.
In actuality, optical dipole traps have a Gaussian shape due to the profile of the laser beams, justlike what you see in Fig. 6.3. In the figure, I’ve plotted a realistic model of the optical dipole trap
(see App. B) along the y = 0 plane, utilizing the laser parameters in Tab. 6.3. These parameters
are adopted as the starting point for all evaporation experiments in our current investigations.
This finite energetic height of the trap, controllable by varying the laser beam power, isexactly how hot molecules are forced to evaporate out of the trap. In the process of doing sohowever, a significant fraction of molecules will now see a large region of this potential energy
landscape, rendering the anharmonicities of this potential, especially the sag along z due to gravity,
highly relevant to the evaporation experiments of interest here. A faithful representation of the trap
should therefore be adopted for accurate evaporation simulations, which is exactly what we employ.

98

Figure 6.3: The optical trap potential energy surface V (r), plotted as a function of coordinates x
and z, along y = 0.

Table 6.3:
Table of parameter values for the potential confining a gas of fermionic 23 Na40 K
molecules. h denotes Planck’s constant.
Parameter
Beam 1 vertical width
Beam 1 horizontal width
Beam 1 wavelength
Beam 1 power
Polarizability in beam 1
Beam 2 vertical width
Beam 2 horizontal width
Beam 2 wavelength
Beam 2 power
Polarizability in beam 2

Symbol
W1,z
W1,⊥
λ1
P1
α1
W1,z
W1,⊥
λ2
P2
α2

Value
57.5
113
1064
0.242
2.79 × 10−3 h
45
156
1064
0.253
2.79 × 10−3 h

Unit
µm
µm
nm
W
m2 Hz/Wµmµmnm
Wm2 J/WTo effectively simulate the evaporation of molecules out of the trap, I have opted to use aposition space cutoff scheme. That is, a molecule is taken as evaporated if it falls past the outer99
turning point along z, or goes past a position that is 6 times the thermal width of the initial cloudfrom the trap minimum:s|ri − ri,min | > 6

kB T0,mωi2 (0)(6.7a)and z < zmax ,(6.7b)

where T0 is the initial equilibrium temperature, ri,min is the position of the trap minimum along
axis i and zmax < 0 is the trap local maxima along z. The criteria above allow us to account for
the anisotropic molecular loss in space, resulting from the gravitational trap sag. Furthermore, theGaussian trap profiles in the transverse directions imply that if molecules are too far away from thetrap minima, they will no longer experience a large enough restorative potential, nor collisions, toreturn toward the trap center and are thus effectively evaporated.
Following the experiment, evaporation is forced by gradually reducing the trapping laser
power, in turn lowering the trap depth and promoting the loss of hot molecules. During thisprocess, the trap powers follow an exponential time-dependence
Pi (t) = Pi (0) − ∆Pi

1 − e−2t/τ
1 − e−2

!
,

(6.8)

where ∆Pi is the change in laser power along axis i, and τ is the forced evaporation time. During
this ramp, the efficiency of evaporation Eevap , is measured by the slope of log phase space density(PSD) vs the log number decrease [148]:
Eevap = −∂ log10 ρPSD,
∂ log10 N(6.9)

where PSD is defined as
ρPSD = ⟨n⟩λ3th ,(6.10)

comparing the ensemble averaged number density ⟨n⟩ against the cubed thermal de Broglie wave√
length λth = h/ 2πmkB T . In practice, Eevap is extracted from a linear fit to the evaporation
trajectory data. The PSD is extracted from temperature measurements of the simulation ensemble

100through the relation

Li3

ρPSD
ρPSD − 1



1
=−
6



T
TF

−3
,

(6.11)

where TF = ℏω(6N )1/3 /kB is the Fermi temperature confined with geometric mean trapping frequencies ω [149] and Li3 (z) is the trilogarithmic function.
I’ll point out that although Eevap provides a useful guide for experiments, it does not guarantee
to achieve the highest phase space density amongst other schemes with possibly lower predictedefficiencies. For instance, a rapid decrease in the trap depth would still allow favorable evacuationof hot molecules and a seemingly efficient decrease in total energy. Unfortunately, the subsequent
sample would have had no time to thermalize during the fast quench, disallowing the thermal
tails of the distribution from being re-populated for further evaporative cooling beyond the initial
evacuation. It is therefore useful to also track the final T /TF and ρPSD achieved, toward the goalof deeply degenerate Fermi gases.

6.2.2

In-simulation thermometry

As the gas enters into deeper quantum degeneracy from evaporation, Fermi statistics become significant. Consequently, Pauli blocking −h3 f factors in the collision integral (3.3) become
significant at PSD of order ρPSD ≳ 0.1, requiring its inclusion in the simulation which shifts theequilibrium distribution from Maxwell-Boltzmann to Fermi-Dirac, illustrated in Fig. 6.4 (description
in caption). So to accurately extract the values of T , we utilize a Fermi-Dirac fit to the y-integrated
simulation ensemble, likened to the optical density (OD) from absorption imaging of the molecular
cloud [4, 150, 151]:

2
2 
ODmax
− x2− z 22σ2σxzOD(x, z) =Li2 −ζe.
Li2 (−ζ)(6.12)

Above, ODmax is the peak optical depth, σi are the distribution widths, ζ is the fugacity and Li2 (z)is the dilogarithmic function. In time-of-flight imaging, the distribution widths will evolve in time

101
for every time-of-flight instance as
σi =

q

s
1 + ωi2 t2TOF

kB Ti
,
mωi2

(6.13)

over the time interval tTOF . In the long time limit, this time-dependence changes the densitypimages from position to momentum space distributions, since ri → vi tTOF and σi → tTOF kB T /m,rendering
OD(x, z)|tTOF →∞ = OD(vx , vz )(6.14)

m(v 2 +v 2 )ODmax− 2kx T z
B,=
Li2 −ζe
Li2 (−ζ)where vi is the molecular velocity along axis i.

Figure 6.4: Simulation ensemble energy distribution with N = 10, 000 molecules (gray histogram),
after collisional thermalization for t = 0.5 s from an initial Maxwell-Boltzmann distribution at 50
nK (dotted red curve). The simulation achieves a Fermi-Dirac distribution (solid black curve) atT ≈ 42 nK with a chemical potential of µ/kB ≈ 87 nK.

102
Leaving T and ζ as float parameters, OD(vx , vz ) is then fitted to the simulation distribution,
obtained by constructing an appropriately normalized 2D histogram from the simulated particle
ensemble, projected into the x, z plane. In practice, obtaining the fugacity by fitting to the shapeof the distribution results in large errors with noisy data. So we opt to utilize the relation
1
Li3 (−ζ) = −
6



T
TF

−3
,

(6.15)

to infer the fugacity, floating only T . If T > TF , we simply revert to assuming a Boltzmann
distributed gas, with temperature related to the mean-squared momenta T = ⟨p2 ⟩/(3mkB ).

6.2.3

Numerical results and prospects

Implementing the components for forced evaporation above into my DSMC solver (see Sec. 3.3),
I was able to reproduce the trends in evaporative cooling observed in the experiment at MPQ [4].
In that study, the final trap depth was varied over several experimental instances with forced evaporation occurring over 150 ms, resulting in various molecule numbers and gas temperatures attainedat the end of each evaporation trajectory 1 . Fig. 6.5 showcases the favorable agreement of ournumerically simulated data points (red crosses) with the experimental measurements (black circles
with error bars), except for disagreement at low molecule numbers attributed to experimental trap
jitter at low laser powers.
With the DSMC solver benchmarked against experimental data, we are now assured thatfurther explorations with it will be informative of actual experimental observation. Moreover, theversatility of our solver makes incorporating other relevant aspects such as accurate 2-body loss
rates or energy dependent elastic cross sections a simple modification. However, since utilizing thistool for attaining T /TF < 0.1 is still an ongoing pursuit, I will leave all further details to upcoming
works that will soon be published.
1

Reported to commence in an excited sloshing mode of the molecular distribution, our simulated evaporationaccounts for resultant heating by simply using a constant 2-body loss rate of βL = 10−12 cm3 /s, and adds anestimated h = 100 nK/s background heating rate. Background heating was included with momentum kicks during
our simulated time evolution, taking the momentum of particle k, and increasing it by pk → pk 1 + 2p−2
k mkB h∆t ,after the second Verlet integration step of Eq. (3.30) in Sec. 3.3.1.

103

Figure 6.5: Plot of T /TF values after evaporating down to N molecules in the experiment at the
Max-Planck-Institut für Quantenoptik [4] (black circles with error bars) with different final trap
depths, compared to those obtained from our numerical simulations (red crosses).

6.3

Curtain call
Before I let the credits roll (i.e. the bibliography section), I want to quickly recount several

key highlights of this thesis to cap it all off.
Setting the foundation for most of my work here was an understanding of ultracold dipolar
collisions, provided in Chap. 2 for both magnetic atoms and field-shielded polar molecules. The
anisotropy of dipole-dipole interactions gives rise not only to anisotropic 2-body differential scattering, but also manifests in the collective dynamics of a nondegenerate gas. When dilute, thesegases thermalize at rates that are highly dependent on the direction of excitation, measurement,
and dipole orientation (Chap. 3). This sensitivity to direction allowed experimentalists to observe

104
universal signatures of threshold dipolar scattering in potassium-rubidium molecules (Sec. 4.1.2),
and accurately extract the s-wave scattering length in dipolar erbium atoms (Sec. 6.1). My firstencounters with experimental collaboration.
As with the mentioned examples, a deeper understanding of dipolar collisional thermalization
can better inform experimental applications such as evaporative cooling (Sec. 6.2), and unexploredphenomena like hydrodynamic excitations. So came my works in Chap. 5, exploring exactly thesehydrodynamic gases that support a rich tapestry of fluid dynamics such as acoustics (Sec. 5.3),
and weltering excitations (Sec. 5.4). The latter arises in harmonically trapped gases, where dipolar
collisions serve, once more, to return the gas to equilibrium. The manner in which equilibration
occurs, however, does not immediately follow from our intuition on dilute gases, so much so that I
had to derive an entirely new set of equations to capture it. Although initiated here, more work is stillrequired to fully grasp and characterize nonegenerate dipolar fluids in all their weltering intricacies(Sec. 5.4.1): the inklings of evermore pertinent questions, awaiting exciting new discoveries.
And with that, I’ll declare this the end of my thesis and PhD journey.
“Those were the days of our lives."— Freddie Mercury (Innuendo), 1991.

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org/article/10.1088/1367-2630/11/5/055039/meta.
[2] A. Patscheider, L. Chomaz, G. Natale, D. Petter, M. J. Mark, S. Baier, B. Yang, R. R. W.
Wang, J. L. Bohn, and F. Ferlaino. Determination of the scattering length of erbium atoms.
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De Marco, Reuben RW Wang, Lucas Lassablière, Goulven Quéméner, John L Bohn, et al.
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Appendix A

Matrix elements of the effective microwave shielded potential

To perform the scattering calculations on the effective single-channel microwave shielded potential energy surface in Sec. 2.2.2, we are required to compute the matrix elements of the effective
microwave shielded potential ⟨ℓ, m| Vµ (r) |ℓ′ , m′ℓ ⟩. We list these elements explicitly in this section.
In all generality for those who might want to use this in future, I will consider the case of arbitrary
microwave ellipticity, parameterized by the angle ξ, but with the dipoles aligned along z such that:Vµ (r; ξ) = V6 (r; ξ) + Vdd (r; ξ),(A.1)

having defined


2

2

2

V6 (r; ξ) = (1 − Fξ (ϕ)) cos θ + 1 − Fξ (ϕ)
Vdd (r; ξ) =

sin2 θ,

2


d
2
2
3
cos
θ
−
1
+
3F
(ϕ)
sin
θ
,
ξ
4πϵ0 r3(A.2a)(A.2b)and the function Fξ (ϕ) = cos(2ϕ) sin(2ξ) [55].
The associated matrix elements of these potentials are then listed as follows. For the dipoledipole potential Vµ,dd , we have:⟨ℓ, m| Vµ,dd (r; ξ) ℓ′ , m′ℓ

" r
#
r
2
d
π
2π
Y2,0 (θ, ϕ) + 12
Re[Y2,2 (θ, ϕ)] sin(2ξ) ℓ′ , m′ℓ=⟨ℓ, m| 44πϵ0 r3
5
15
" r Z
2
d
π
∗
=
4
dΩYℓ,m(Ω)Y2,0 (Ω)Yℓ′ ,m′ℓ (Ω)
34πϵ0 r
5
#
r
Z
2π
∗
+6
sin(2ξ) dΩYℓ,m (Ω) [Y2,2 (Ω) + Y2,−2 (Ω)] Yℓ′ ,m′ℓ (Ω)
15

120


2

ℓ′



ℓ 2
p
d

m
′ + 1) 
2(−1)
(2ℓ
+
1)(2ℓ


3
4πϵ0 r
0 0 0

 


r

′
′
′
2 ℓ   ℓ
2 ℓ 
2
ℓ 
3
 ℓ
sin(2ξ) 
+
+
 ,

2

−mℓ 2 m′ℓ0 m′ℓ−mℓ −2 m′ℓ
=




 ℓ
× 

 −m

ℓ

while the matrix elements for Vµ,6 (r) are given as⟨ℓ, mℓ | Vµ,6 (r; ξ) ℓ′ , m′ℓ =

h
C6
⟨ℓ,
m
|
1 − cos2 (2ϕ) sin2 (2ξ)ℓ
r6i
+ (1 − cos(2ϕ) sin(2ξ))2 cos2 θ sin2 θ ℓ′ , m′ℓ√ C61= 4 π 6 ⟨ℓ, mℓ |(cos(4ξ) + 5)Y0,0 (θ, ϕ)
r
15
r
1 2
sin(2ξ)Y2,−2 (θ, ϕ)
−
7 15
r
2 1
−
(cos(4ξ) + 2)Y2,0 (θ, ϕ)
21 5
r
1 2
−
sin(2ξ)Y2,2 (θ, ϕ)
7 15
r
1 2
sin2 (2ξ)Y4,−4 (θ, ϕ)
−
3 35
r
2 2
−
sin(2ξ)Y4,−2 (θ, ϕ)21 5
1+(cos(4ξ) − 5)Y4,0 (θ, ϕ)
105r
2 2
−
sin(2ξ)Y4,2 (θ, ϕ)
21 5
r

1 2
2
−
sin (2ξ)Y4,4 (θ, ϕ) ℓ′ , m′ℓ
3 35
p
C6
= 6 2(−1)mℓ (2ℓ + 1)(2ℓ′ + 1)
r




′
′
0 ℓ 
1
ℓ 0 ℓ   ℓ×  (cos(4ξ) + 5) 


15
′
0 0 0
−mℓ 0 mℓ



r
′
′
2
ℓ 
1 2
ℓ 2 ℓ   ℓ−sin(2ξ) 


7 3
′
0 0 0
−mℓ −2 mℓ



′
′
2 ℓ 
2
ℓ 2 ℓ   ℓ− (cos(4ξ) + 2) 


21
′
0 0 0
−mℓ 0 mℓ

(A.3)

121

r

ℓ′



ℓ′



2
2
ℓ 2
 ℓ

sin(2ξ) 


3
0 0 0
−m 2 m′



r
′
′
ℓ
4
ℓ
ℓ
4
ℓ
2



sin2 (2ξ) 
−


35
0 0 0
−mℓ −4 m′ℓ



r
′
′
ℓ
4
ℓ
ℓ
4
ℓ
2 2



sin(2ξ) 
−


7 5
0 0 0
−mℓ −2 m′ℓ



′
′
ℓ
4
ℓ
ℓ
4
ℓ
1



+ (cos(4ξ) − 5) 


35
0 0 0
−mℓ 0 m′ℓ



r
′
′
ℓ
4
ℓ
ℓ
4
ℓ
2 2



−
sin(2ξ) 


7 5
0 0 0
−mℓ 2 m′ℓ



r
′
′
ℓ
4
ℓ
ℓ
4
ℓ
2



−
sin2 (2ξ) 
 .

35
0 0 0
−mℓ 4 m′ℓ
−

1
7

(A.4)

Even more conveniently for the reader, I will also list the simplified matrix elements forcircularly polarized microwaves (ξ = 0) explicitly below:′′⟨ℓ, mℓ | Vµ,dd (r; 0) ℓ , m

 r

d2eff
π
=
⟨ℓ, mℓ | 4
Y2,0 (θ, ϕ) ℓ′ , m′ℓ
4πϵ0 r3
5
#
" r Z
d2eff
π
∗
=
dΩYℓ,m(Ω)Y2,0 (Ω)Yℓ′ ,m′ℓ (Ω)
4ℓ4πϵ0 r3
5



′
′
2
2 ℓ 
p
deff
ℓ 2 ℓ   ℓ=2(−1)mℓ (2ℓ + 1)(2ℓ′ + 1) ,
3
4πϵ0 r′0 0 0−mℓ 0 mℓ(A.5a)
C6⟨ℓ, mℓ | 1 + cos2 θ sin2 θ ℓ′ , m′ℓ
6
r
r


√ C6
2
2 1
4
= 4 π 6 ⟨ℓ, mℓ | Y0,0 (θ, ϕ) −Y2,0 (θ, ϕ) −Y4,0 (θ, ϕ) ℓ′ , m′ℓ
r
5
7 5
105
p
C6
= 6 2(−1)m (2ℓ + 1)(2ℓ′ + 1)
r
 





′
′
′
′
0 ℓ  2 ℓ 2 ℓ   ℓ
2 ℓ  2 ℓ 0 ℓ   ℓ
× 

− 


5
7
0 0 0
−mℓ 0 m′ℓ0 0 0
−mℓ 0 m′ℓ⟨ℓ, mℓ | Vµ,6 (r; 0) ℓ′ , m′ℓ =

122

−

ℓ′



ℓ′



4
4 ℓ 4
 ℓ



 .
35
0 0 0
−mℓ 0 m′ℓ

There you go, matrix elements on a silver platter.

(A.5b)

Appendix B

Optical dipole traps for atoms and molecules

In the parlance of classical physics, an oscillating electric field results in the separation of
positive and negative charges within a neutral atom, inducing an electric dipole moment. Thisdipole can in turn interact with its very maker, the oscillating field, either resonantly or not. Below resonance (red detuned), the dipole-field interaction energy is negative, pulling the atom into
the region of highest electric field intensity. The converse process occurs above resonance (bluedetuned). In real quantum mechanical atoms, a Gaussian laser beam could act as the oscillating
field, which dresses the atom’s internal electronic states to induce a dipole transition. With theintensity maximal at the Gaussian peak, a red detuned laser is best used to trap the atoms close to
the energetic minima of the beam center. And that folks, is how you get optical confinement [152].
More rigorously, the external laser field can be modeled by the electric field E(r)eiωt , oscillating at a frequency ω. A proportional dipole moment
dind (r) = α̃E(r),(B.1)

is then induced in the atom, with an atomic species, laser frequency and polarization dependent
proportional constant α̃, known as the complex atomic polarizability. The interaction of d(r) with
E(r) then results in the potential energy felt by the atom as
1
Vtrap (r) = − ⟨dind (r, t) · E(r, t)⟩t = − Re{α̃}|E(r)|2 ,
2

(B.2)

where ⟨. . .⟩t is a time average over the rapid oscillating terms. For an elliptic Gaussian beam

124
propagating along z, its intensity profile is given as

  2
P
x
y2
|E(r)| =,exp −2+πcϵ0 wx (z)wy (z)wx2 (z) wy2 (z)
2

(B.3)

with laser power P , beam width along axis i:swi (z) = Wi

1+

z2
,
Ri2

(B.4)

Rayleigh length Ri = πWi2 /λ, minimum beam width Wi and laser wavelength λ.
For optical trapping in 3-dimensions, 2 of such Gaussian beams are generally utilized in acrossed configuration, so that the electric fields are orthogonal to each other and add linearly inintensity. As a result, the crossed optical dipole trap resultant from lasers propagating along x andy is described by the potential:
2y 2
2α1 P1
2z 2
VODT (r) = −exp − 2− 2
πw1,y (x)w1,z (x)w1,y (x) w1,z (x)!

2z 22α2 P22x2− 2−
exp − 2πw2,x (y)w2,z (y)w2,x (y) w2,z (y)!
,(B.5a)

where αi = Re{α̃i }/(2cϵ0 ) is the polarizability constant. Close to the minima of this potential,
VODT can be expanded up to second order in spatial coordinates as
Vharm (r) =XAi ri2 ,(B.6)

idropping all constants as they simply amount to an energy offset, with harmonic coefficients
4 + W4
α1 P1 λ21 W1,y1,z

4α2 P2
,
3 W
πW2,x
2,z

4 + W4
α2 P2 λ22 W2,x
4α1 P1
2,z
Ay =
+
,
3 W
5 W5
πW1,y
π 3 W2,x
1,z
2,z

Ax =

Ay =

5 W5
π 3 W1,y1,z+4α1 P1
4α2 P2+.
3
3 W
πW1,y W1,z
πW2,x
2,z

(B.7a)(B.7b)(B.7c)

Comparing Eq. (B.6) to Eq. (3.2) finally identifies the harmonic trap frequencies as
ωi2 =

2
Ai .
m

(B.8)

Appendix C

The method of averages

I utilize the method of averages to derive Enskog equations in this appendix chapter. For someobservable quantity of interest χ(r, p, t), the method of averages multiplies it with the Boltzmann
equation (3.1) and takes an integral over phase space to give
1
N

Z

3

3

d pd r χ(r, p, t)!
X pi ∂X ∂V (r) ∂∂+−f (r, p, t)∂tm ∂ri∂ri ∂piiiZ
1=d3 pd3 r χ(r, p, t)Icoll [f ].
N

(C.1)

Each term on the left hand side of Eq. (C.1) is evaluated as
 Z



Z
∂f
∂χ
d
= d3 p d3 r
(χf ) − fd3 r χ
∂t
dt
∂t
 
d⟨χ⟩
∂χ
=N
−N
,
dt
∂t
Z
Z
Z
Z
X pj ∂fX  ∂  pj  pj ∂χ 33
33d p d rχ= d p d r
χf −
f
m ∂rj
∂rj m
m ∂rj
j
j


Z
Z
I
X pj
X pj ∂χ= d3 p χfn̂j − d3 r
f
m
m ∂rj
∂S
j
j
Z
Z
X pj ∂χ= − d3 p d3 r
f
m ∂rj
Z

d3 p

Z

(C.2a)

j

Dp

Z

d3 p

Z

E

· ∇r χ ,(C.2b)
= −N
m




Z
Z
X ∂V (r) ∂f
X ∂∂V (r)∂V (r) ∂χ
d3 rχ
= d3 p d3 r
χf−f
∂rj ∂pj∂pj∂rj∂rj ∂pj
j
j


Z
I
Z
X ∂V (r)X ∂V (r) ∂χ= d3 r χfn̂j − d3 pf∂rj∂rj ∂pj
∂Sp
j

j

126
Z
=−

d3 r

Z

d3 p

X ∂V (r) ∂χj∂rj ∂pjf= −N ⟨∇V (r) · ∇p χ⟩,(C.2c)

where n̂j is the j-th unit vector component normal to the surface ∂S.
Putting these terms together, the phase-space averaged substantial derivative evaluates to!
ZZ
X ∂V (r) ∂X pi ∂
1∂
3
3
d p d r χ(r, p, t)−f (r, p, t)
+
N
∂t
m ∂ri
∂ri ∂pi
i
i
  D
E
d⟨χ⟩
p
∂χ
=
−
−
· ∇r χ − ⟨F · ∇p χ⟩.
(C.3)
dt
∂t
m
Then having χ by χ = ri2 , p2i , ri pi , performing the averaging procedure described above results inthe system of equations in (3.8).

Appendix D

Equilibrium dipolar mean-field per particle

The dipolar mean-field energy per particle εmf , used in Sec. 3.3.3 is derived here. First, εmf
is written in terms of the integral
1
εmf =
2N

Z

Z

3

d rn(r)

d3 r ′ n(r ′ )Φdd (r − r ′ ).

(D.1)

I assume a cylindrically symmetric trap, which simplifies the derivation and asserts a number density
distribution n(r) given by


z2
ρ2
n(r) =
exp − 2 − 2 ,2σρ2σz(2π)3/2 σρ2 σz
N

(D.2)

where ρ2 = x2 + y 2 and σj is the standard standard deviation of particles along axis j. To evaluate
this, I will utilize the convolution theorem:Z

d3 r ′ n(r ′ )Φdd (r − r ′ ) = n(r) ∗ Φdd (r)= F −1 [F [n(r)] · F [Φdd (r)]]hi= F −1 ñ(k) · Φ̃dd (k)Z1=pd3 k ñ(k)Φ̃dd (k)eik·r .
3(2π)

where F[. . .] = √ 1

R

(2π)3

(D.3)

d3 r(. . .)e−ik·r is a Fourier transform and tildes denote Fourier transformed

functions. Putting this back into the full integral gives
1
p
εmf =
2N (2π)3Z3d r n(r)Z

d3 k ñ(k)Φ̃dd (k)eik·r

128
1
2N

Z

d3 k ñ(k)Φ̃dd (k)

1
=
2N

Z

d3 k ñ2 (k)Φ̃dd (k).

=

1
p
(2π)3

Z

!
d3 r n(r)eik·r(D.4)

The Fourier transforms of the integrand functions are given by



1
N
2 2
2 2
exp − ⟨ρ ⟩kρ + ⟨z ⟩kz ,ñ(k) =
2(2π)3/2h2dΦ̃dd (k) =2 sin2 α sin2 θk cos(2ϕk ) + 2 sin(2α) sin(2θk ) cos ϕk
4ϵ0i
1+ (3 cos(2α) + 1)(3 cos(2θk ) + 1) ,6

(D.5a)(D.5b)

with the result for Φ̃dd (k) taken from R. Wilson’s thesis in Ref. [153]. Plugging these in andevaluating the integral gives εmf as written in Eq. (3.47).

Appendix E

Deriving the collisional efficiency toward thermalization

Obtaining the form of εij in Eq. (4.20) follows a very similar procedure to that in Sec. 4.1,
since after all, εij is just the reciprocal of Nij . So once more with ⟨χi ⟩ = kB (Ti − Teq ), the Enskog
equations that govern the relaxation of ⟨χj ⟩ is derived to be:d⟨χi ⟩= C[χi ],
dt
Z 3
Z
⟨n⟩
d pr
C[χi ] =pr cr (pr , t) d2 Ω′ Del ∆χi .
2m(E.1a)(E.1b)

Then Taylor expanding the relative momentum distribution with respect to the temperature perturbation δi /kB T0 , gives!
p2r,icr (pr , 0) =exp −4mkB Ti
i
!
#
p2r,i
1
δi
(0)
≈ cr (pr ) 1 +−,4mkB T0 2 kB T0


1
p2r
(0)
cr (pr ) =exp −.
4mkB T0(4πmkB T0 )3/2
Y

1
4πmkB Ti"

1/2

The expressions above render the collision integral
"
!
#Z
Z 3
p2r,i
⟨n⟩
d pr
1
δi
(0)
C[χj ] ≈pr cr (pr ) 1 +−d2 Ω′ Del
2m
4mkB T0 2 kB T0
Z 3
Z

δi
⟨n⟩
d pr (0)′ 22cr (pr )pr d2 Ω′ Del
pr,i p′2=r,j − pr,j ,2
16(mkB T0 ) 2m(E.2a)(E.2b)

2p′2
r,j − pr,j
4m

(E.3)

which upon utilizing the time-reversal symmetry of elastic collisions
C[χj ] ≈δi⟨n⟩216(mkB T0 ) 2Z

p2r dpr c(0)r (pr )

pr
m

Z

′ 2
2
d2 Ωd2 Ω′ Del
pr,i p′2r,j − pr,j

!



130
=

δi
⟨n⟩
2
16(mkB T0 ) 2Z

p2r dpr c(0)r (pr )

pr
m

Z


′ ′2
d2 Ω′ d2 ΩDel
pr,i p2r,j − p′2
r,j ,

(E.4)

the expression above can also be written in a form that is explicit in the symmetry under exchange
of indices i and j:
C[χj ] = −⟨n⟩
δi32(mkB T0 )2 2Z

d3 pr (0)c (pr )pr
m r

Z

2
′
p′2
d2 Ω′ Delr,i − pr,i




2
p′2
r,j − pr,j .

(E.5)′ = D (p , Ω′ ). Plugging C[χ ] as written into Eq. (E.1a) andI have used the suggestive notation Deljel rtaking Tj (t) − Teq = ϵj /kB , we obtain
γij = −

C[Tj ](Tj (t) − Teq )kB
C[Tj ]ϵj ZZ
δi ⟨n⟩
p2r dprp2r
pr′=d2 Ωd2 Ω′ Del
exp −
3/2
ϵj 5124mkB T0 m(πmkB T0 )
=−

2
p′2
r,i − pr,imkB T0!

2p′2r,j − pr,jmkB T0(E.6)!
.

Finally, taking the limit of δi /(kB T0 ) → 0, we obtain εij in Sec. 4.2.2, having defined αij = δi /ϵjand using the equipartition theorem (Teq = T0 + δi /3kB ) to get




3/2,
i = j,
δi
=

ϵj


−3,
i ̸= j.

(E.7)

Appendix F

The equilibrium Boltzmann equation

This appendix evaluates the Boltzmann equation at thermal, but not necessarily hydrodynamic equilibrium. The treatment here extends the derivation of the equilibrium Boltzmann Equation found in [103], to include arbitrary external potentials.
At thermal equilibrium, the left-hand side of the Boltzmann equation is given as∂∂i V (r) ∂+ vi ∂i −
f0
∂t
m ∂ui


∂i V (r) ∂∂+ vi ∂i −ln f0 ,
= f0
∂t
m ∂ui

D
f0 =
Dt

(F.1)
(F.2)

where f0 is of the form in Eq. (3.5) but with v replaced with u, and n0 is determined by the formof V (r), so thatln f0 =


 1
3 m
ln
+ ln n0 β 3/2 − βmu2 .
2
2π
2

(F.3)

The material derivative is defined asD∂=+ Ui ∂i ,
Dt
∂t

(F.4)

so that the D/Dt operator can be rewritten asD
D∂i V (r) ∂=+ ui ∂i −
.
Dt
Dt
m ∂ui
We now treat the derivatives term by term.
First considering ui ∂i ln f0 , we haveui ∂i ln f0 = ui ∂i3
mu2
ln(n0 ) + ln(β) − β
2
2



(F.5)

132


mu2 3
= ui ∂i ln(n0 ) + β
ui ∂i ln T + βmui uj ∂i Uj .
−
2
2

(F.6)

As for D ln f0 /Dt, we first consider the equations of conservation at thermal equilibrium which readD
ln n0 = −∂j Uj ,Dt
D2ln T = − ∂j Uj ,
Dt
3
D
kB T
1
Ui = −∂i ln(n0 T ) − ∂i V (r).
Dtmm

(F.7a)(F.7b)(F.7c)The material derivative of ln f0 then becomes


D
D
3
mu2
ln f0 =
ln(n0 ) + ln(β) − β
Dt
Dt
2
2
β
= − mu2 ∂j Uj − βui ∂i V (r) − ui ∂i ln(n0 ) − ui ∂i ln T.
3

(F.8)

Finally, the term explicit in the potential is−∂i V (r) ∂ ln f0
β∂i V (r) ∂u2== βui ∂i V (r).
m
∂ui
2
∂ui

(F.9)

Putting all these terms together, we get
D
D∂i V (r) ∂ln f0 =
ln f0 + ui ∂i ln f0 −
ln f0
Dt
Dt
m ∂ui


β
2
= − mu ∂j Uj − ui ∂i ln T − βui ∂i V (r) − ui ∂i ln(n0 )
3




mu2 3
+ βmui uj ∂i Uj + ui ∂i ln(n0 ) + β−
ui ∂i ln T22+ βui ∂i V (r) ,(F.10)

which gives the final resultD
ln f0 =
Dt



mu2 5
β
−
2
2





1
2
ui ∂i ln T + βm ui uj − δij u ∂i Uj .
3

(F.11)

This result is exactly that derived by Chapman and Enskog in the absence of an external potential,
showing that V (r) does not affect the derivation of the transport tensors.

Appendix G

Transport tensors from microscopic collisions

Deriving the transport tensors in the Chapman-Enskog fashion can be pretty gnarly [69],especially when anisotropic cross sections are involved. I’ll do my best to make all the steps clear
in this appendix chapter, for anyone who wants to utilize this method.
As detailed in App. F, the left-hand side of Eq. (5.7) evaluates to∂∂k V (r) ∂+ vk ∂k −
∂t
m ∂uk





f0 = f0 Vk ∂k (ln T ) + mβWkℓ Dkℓ .

(G.1)

where

mβu2 5Vi (u) ≡
−
ui ,
2
2
1
Wij (u) ≡ ui uj − δij u2 ,31
1
Dij (U ) ≡ (∂j Ui + ∂i Uj ) − δij ∂k Uk .
2
3


(G.2a)(G.2b)(G.2c)

The collision integral (ignoring quantum statistical effects) on the right-hand side of Eq. (5.7) isthenZC[f ] ≈

d3 u1 |u − u1 |f0 (u)f0 (u1 )

Z

dΩ′

dσ
∆Φ,
dΩ′

(G.3)

where ∆Φ = Φ′ + Φ′1 − Φ − Φ1 as in Sec. 3.2. Since Eq. (G.3) is linear in Φ, and Eq. (G.1) is linear
in the quantities ∂i ln T and ∂j Ui , one can infer an ansatz for the scalar function Φ, of the form
Φ(u, β) = Bk ∂k (ln T ) + mβAkℓ Dkℓ ,

(G.4)

134
where B (vector) and A (rank-2 tensor) are functions of u and β. The ansatz above allows separationof Eq. (5.7) into an equation in velocity gradients, and those in temperature gradients:
f0 Wkℓ Dkℓ ≈ C[f0 Akℓ ]Dkℓ ,(G.5a)

f0 Vk ∂k (ln T ) ≈ C[f0 Bk ]∂k (ln T ).

(G.5b)

Upon further comparison of terms, it is only suitable to write B and A as
Aij (u, n0 , β) = Wkℓ (u)akℓij (u, n0 , β),(G.6a)Bi (u, n0 , β) = Vj (u)bji (u, n0 , β),(G.6b)

where u = |u|, while akℓmn (u, n0 , β) and bkℓ (u, n0 , β) are introduced as variational coefficientsthat can, in general, be expanded into a basis of Sonine polynomials (a.k.a. associated Laguerre
polynomials). The assumption of a low temperature gas however, allows us to approximate a andb with only the first basis term, which is independent of u. This truncation leaves us withΦ(u, β) = Vℓ (u)bℓk (n0 , β)∂k (ln T ) + mβWij (u)aijkℓ (n0 , β)Dkℓ ,(G.7)

which has shown to give good accuracy (relative errors of ∼ 1%) in computing transport coefficientsfor gases of isotropic scatterers [62, 154, 155] Determining a and b will eventually allow us toevaluate κ and µ, but requires consideration of the micro-scale collisional physics. So, I’ll do just
that.

G.1

Anisotropic thermal conductivity
Thermal conduction in a collisional gas arises through a transfer of kinetic energy by kinetictransport of the gaseous atoms, out of a region of fluid, resulting in a heat flux [101]Z
1
Ji (r, t) = d3 uf (r, u, t) mu2 ui .
2

(G.8)

In the first-order approximation with Eq. (G.4), the A associated term does not contribute to theheat flux integral, leaving
m
Ji =
2

Z

d3 uf0 (u) [1 + Φ(u)] u2 ui

135

=

kB mβ
2

Z



d3 u f0 (u)u2 ui Vk bkj ∂j T ,(G.9)

where the local temperature T (r, t) is written in terms of its kinetic definition,
3
1
kB T =
2n(r, t)Z

1
d3 u f (r, u, t) mu2 .
2(G.10)

Across a temperature gradient ∂j T , Fourier’s law of heat conduction dictates that the flow ofheat across it follows the relation
Ji = −κij ∂j T.

(G.11)

A comparison of Eq. (G.9) and Eq. (G.11), then relates the thermal conductivity to the integral

Z
kB mβ
3
2
d u f0 (u)u ui Vk bkj
κij = −
2
5n0 kB
=−
bij ,
2mβ


with the coefficients bkj introduced in Eq. (G.7).

(G.12)

Referring back to Eq. (G.5b), multiplying

Eq. (G.5b) by Vi (u) and integrating over u gives
Z

Z

3
3
d u f0 (u)Vi (u)Vj (u) ∂j (ln T ) ≈d u Vi (u)C[f0 Vk ] bkj ∂j (ln T ),(G.13)

imposing that the coefficients of ∂j (ln T ) satisfy
Kik bkj = δij ,where Kik ≡

2mβ
5n0

(G.14a)Z

d3 u Vi C[f0 Vk ].

(G.14b)

The integrals above are made complicated by the highly anisotropic differential cross section
for dipoles, but that’s not going to stop me from computing them! In fact, its evaluation is pretty
identical to that already described in Sec. 3.2.1. So with much the same Mathematica® finagling
as in Sec. 3.2.1, I arrived at the thermal conductivities in Eq. (5.8) of the main text.

G.2

Anisotropic viscosity
Now comes the other transport tensor, viscosity. With twice as many indices, it’s a real treat.

On each differential parcel of fluid, the stress tensor ς describes the forces in direction r̂ j , acting

136
on its surface defined by normal vectors r̂ i . These forces are illustrated schematically in Figure
G.1, where all vectors appear to have the same length, although this is not always true in the fluid.
Quite generally, the stress tensor can be written asςij = −P δij + τij ,(G.15)

where P is the thermodynamic pressure which, in a sufficiently dilute collisional gas, is related tothe density and temperature by the ideal gas law. The remaining part, τ , is the viscous stresstensor, i.e., the part arising from viscosity. In a Newtonian fluid, the viscous shear is assumed to
be a linear function of the velocity gradients:
τij = µijkℓ ∂ℓ Uk ,(G.16)

where µ is the rank-4 viscosity tensor and the flow velocity gradients ∂ℓ Uk , characterize the rate ofstrain on differential fluid volumes. It turns out, however, that ultracold elastic collisions leave only
a symmetrized portion of ∂ℓ Uk relevant, as will be shown soon enough.
x

ςxx
ςxz
ςxy

ςzx

ςyx

ςzz
ςyz

ςyy

ςzy

z

y
Figure G.1: Cartoon of the stresses (black arrows) on a differential fluid volume element (light redcube) due to thermodynamic pressure and velocity field gradients.

Microscopically, the fluid stress tensor arises from the decrease in flux of molecular momentum
directed along j, across the plane oriented along axis i:Z
ςij = −

d3 uf (u)ui pj

137Z

d3 uf0 (u) [1 + Φ(u)] ui ujZn0= − δij − 2 m2 β d3 uf0 (u)ui uj Wmn amnkℓ Dkℓ .
β= −m(G.17)

This expression identifies the thermodynamic pressure as P = n0 /β, and the quantity in parentheses
as related to the shear viscosity. Notice, the viscosity tensor in Eq. (G.16) gives the stresses in termsof the unsymmetrized rank-2 tensor ∂l Uk , while the microscopic evaluation of stresses (G.17) relates
these stresses only to the symmetrized tensor D. The difference is telling: generally, this tensor
can be reduced in the usual way into the traceless D tensor, along with an antisymmetric tensor
R, and a scalar:1∂ℓ Uk = Dkℓ + Rkℓ + δkℓ ∇ · U ,3(G.18)

where
Rkℓ =

1
(∂ℓ Uk − ∂k Uℓ ) .
2(G.19)

The absence of the antisymmetric tensor and the scalar from the expression connotes that there are
no rotational viscosities, nor bulk viscosities in a dilute gas of particles with no internal degrees of
freedom [69, 156, 157]. Without loss of generality, the viscous stress tensor can now be written asτij = 2µijkℓ Dkℓ .

(G.20)

Note that this conclusion is independent of the form of the collision cross section of the molecules.
The relation between the two forms of the symmetrized tensors is conveniently handled via acontraction,
Wij (u) = Iijkℓ uk uℓ = uk uℓ Ikℓij ,(G.21a)Dij (U ) = Iijkℓ ∂ℓ Uk = ∂ℓ Uk Ikℓij ,(G.21b)with the traceless symmetric tensor
Iijmn =

δim δjn + δin δjm 1− δij δmn .
2
3(G.22)

138
Written in these terms, the expression for the shear stress tensor in Eq. (G.17) is2

τij = −2m βZ

d3 uf0 ui uj um un Imnop aopkℓ Dkℓ .

(G.23)The integrand now consists of products of components of the peculiar velocity u, multiplied by the
known equilibrium velocity distribution. All such integrals are readily evaluated (many leading to
Kronecker delta functions), whereby the viscosity tensor ultimately becomesµijkℓ = −2n0
Iijmn amnkℓ .
β(G.24)

To obtain explicit forms for the variational coefficients aijkℓ and hence the viscosities by Eq. (G.24),
we rewrite the right-hand side of Eq. (G.5a) as
2C[f0 Wmn ]amnkℓ Dkℓ = 2C[f0 Wmn ] (Imnrs arskℓ ) Dkℓ=−β
C[f0 Wmn ]µmnkℓ Dkℓ .
n0(G.25)

Multiplying both sides of Eq. (G.5a) by Wij and integrating over u then gives
Tijkℓ = Mijmn µmnkℓ .

(G.26)

where
Z

2n0
d3 uf0 (u)Wij Wkℓ =Iijkℓ ,(mβ)2Z
βd3 uWij C[f0 Wmn ].
Mijmn = −n0
Tijkℓ =

(G.27a)(G.27b)

The collision integrals of Eq. (G.27b) are evaluated with the collision-varied quantity∆Wij = ∆(ui uj ) =
1 ′ ′ur,i ur,j − ur,i ur,j ,2(G.28)and differential cross sections of Ref. [35], aided once more by Mathematica® [16]. Even after doingso, obtaining µ requires M to be inverted (in the matrix sense), which is most easily performed byconverting M into its matrix representation denoted by an overhead circle, M̊ . This representationis constructed by mapping index pairs to single indices (i, j) → (i′ ), using the map
i′ = 3(j − 1) + i,

(G.29)

139
rendering Mijkℓ → M̊i′ k′ . In its 9 × 9 matrix representation, the inherent symmetries of M reduces
its matrix rank from 9 to 5. This prevents us from inverting the matrix in its current representation,so we are now required to perform a change of basis transformation which decomposes the 9 × 9matrix into a block-diagonal matrix with a 5 × 5 irreducible block. The desired change of basis
matrix C̊, is obtained by diagonalizing the isotropic tensor I in its matrix representation,I̊ = C̊ (I 5×5 ⊕ 04×4 ) C̊ −1 ,(G.30)

where I and 0 are the identity and zero matrices respectively, with dimensions specified by theirsubscripts, and ⊕ denotes a direct sum. Applying the transformation C̊ to Eq. (G.26) givesC̊ −1 T̊ C̊ = C̊ −1 M̊µ̊ C̊= C̊ −1 M̊ C̊ −1 C̊µ̊C̊ ,(G.31)

which leaves both sides of the equation above to only have a 5 × 5 non-trivial matrix block. The
structure of these matrices is shown more explicitly by writinghC̊ −1 T̊ C̊

i
5×5⊕ 04×4 =ihC̊ −1 M̊ C̊ C̊ −1 µ̊C̊5×5

⊕ 04×4 .

(G.32)The direct sum with 04×4 is trivial, leaving only the 5 × 5 irreducible subspace to handle. This now
allows M̊ to be effectively inverted byhC̊ −1 µ̊C̊

i
5×5

=

h
i−1 hiC̊ −1 M̊ C̊
C̊ −1 T̊ C̊
5×5

5×5

,

(G.33)and taking the direct sum of the expression above with 04×4 , to giveµ̊ = C̊ C̊ −1 M̊ + C̊ C̊ −1 T̊ C̊ C̊ −1 = M̊ + T̊ ,(G.34)

where M̊ + is a pseudo-inverse of M̊ defined by the procedure above, satisfying M̊ + M̊µ̊ = µ̊.
Finally, I apply the inverse mapping of Eq. (G.29) to attain the rank-4 tensor form of µ, presented
in Eq. (5.10) of the main text.

Appendix H

Spatial averaging for an effective welter model

To obtain the spatially averaged equations of motion in Sec. 5.4, I start by defining a notation
for spatially averaged quantities:
Z

1⟨. . .⟩r =Nn(r, t) (. . .) d3 r.

(H.1)

This renders the density averaged equation for σi (t) as


Z
X σ̇k
⟨ri2 T ⟩r
d3 r
m σ̈i (t)
2
2
+
ω
⟨r
⟩
−
δ
ri ∂j µijkℓ (T )
−
⟨r
∂
T
⟩
=
r
i
i
r
k,ℓ
i
i
kB σi (t)σℓN kB
σi2 (t)j,k,ℓX σ̇k Z d3 rm σ̈i (t)22ri ∂j µijkℓ (T )δk,ℓ .
=+ ωi σi (t) −kB σi (t)
σℓ
N kB
j,k,ℓ

As for the temperature balance equation:∂T (r, t) X
2X+
Ui ∂i T (r, t) +∂i Ui T (r, t)
∂t
3
i

i

X
X
2
2
=
(∂j Ui )(∂ℓ Uk )µijkℓ (T ) +∂i [κij ∂j T (r, t)] ,3n(r, t)kB
3n(r, t)kB

(H.2)

i,j

i,j,k,ℓthe product rule gives
d3 r∂T (r, t)∂n(r, t)n(r, t)+ T (r, t)
N
∂t
∂t


 2

X
∂T
σ̇i (t) ⟨ri T ⟩r+=
− ⟨T ⟩r ,∂t r
σi (t) σi2 (t)d⟨T ⟩r
=
dt

Z

(H.3)

i

where I’ve utilized the continuity equation. Then multiplying the temperature balance equation byn(r, t)/N and integrating over d3 r givesX σ̇i (t)  ⟨r2 T ⟩rd⟨T ⟩r
5 X σ̇i (t)i+⟨T ⟩r −
− ⟨ri ∂i T ⟩rdt
3
σi (t)σi (t) σi2 (t)
i
i

141
=

X σ̇i (t)
2
δi,j
3N kB
σi (t)Z

d3 rµijkℓ δk,ℓi,j,k,ℓσ̇ℓ (t)2+
σℓ (t) 3N kB

XZ

d3 r [∂i (κij ∂j T )] .

(H.4)

i,j

Combining equations (H.2) and (H.4), we getd⟨T ⟩r
5 X σ̇i (t)m X+⟨T ⟩r −σ̇i (t) σ̈i (t) + ωi2 σi (t)dt3
σi (t)
kB
i
i
Z

Z

X σ̇i (t)σ̇ℓ (t)σ̇k
1 X σ̇i (t)233d rµijkℓ δk,ℓd rri ∂j µijkℓ δk,ℓ
δi,j
−
≈
3N kB
σi (t)σℓ (t) N kB
σi (t)
σℓi,j,k,ℓi,j,k,ℓZ
2 X+
d3 r [∂i (κij ∂j T )] .
(H.5)
3N kB
i,j

At this point, conservation of energy has that


Z 3 3
mX
d rd vf (r, v, t)vi2
ωi2 ⟨ri2 ⟩r +
2
N
i


Z 3 3
mX
d rd v
22 2=
f (r, v, t)vi ,ωi σi +
2
N

Etotal =(H.6)

iwhere Etotal is the total energy of the hydrodynamic system. Therefore, the relation above alongwith Eqs. (5.3) and (5.32) motivates the form for ⟨T ⟩r as
2Etotalm X 2 2−ωi σi (t) + σ̇i2 (t) ,
3kB
3kB

(H.7)


2m X  2d⟨T ⟩r=−ωi σ̇i (t)σi (t) + σ̈i (t)σ̇i (t) .
dt
3kB

(H.8)

⟨T ⟩r =iand its time-derivativei

Plugging these relations into Eq. (H.5) and assuming each axis can be solved independently, we
obtain1X

 σ̇i (t)σ̇i (t) σ̈i (t) + ωi2 σi (t) +σi (t) 3

 2Etotalωj2 σj2 (t) + σ̇j2 (t) −
3m
j
Z

σ̇k
3 X σ̇i (t)3≈d rri ∂j µijkℓ δk,ℓ5N mσi (t)
σℓ
j,k,ℓ
Z

2 X σ̇i (t)σ̇ℓ (t)−
δi,j
d3 rµijkℓ δk,ℓ5N mσi (t)σℓ (t)
j,k,ℓ
Z
2 X
d3 r [∂i (κij ∂j T )] .
−
5N m
j

(H.9)

142
Finally, the conserved total energy Etotal , is made up of the potential energy and thermal
equilibrium temperature T0 :3
mX 2 2
Etotal = kB T0 +ωi σ0,i = 3kB T0 ,22iqwhere we utilized that σ0,i = kB T0 /mωi2 .

(H.10)

Appendix I

Quasi-empirical determination of the hydrodynamic volume

The quasi-empirical rigmarole I used to infer a functional form for the effective hydrodynamic
volume parameter η, is a little involved. That’s why I tucked it away here in the appendix, where
sleepful nights go to die. But if you’ve stuck around till this point, I’ll give you the details you
asked for. Otherwise, Ref. [19] has them too.
Returning to the regularly scheduled program of Chap. 5, Eq. (5.34) is expected to be areasonable representation of dynamics, provided the shape of the gas remains nearly Gaussian. To
employ these equations, a value of the effective hydrodynamic volume must first be established. A
first guess at this volume is given in Eq. (5.38), which left available a free parameter η, that may
depend on λ and Kn. As noted in Sec. 5.4, Kn is implicitly dependent on N and Θ, which are taken
as the relevant independent variables for this study.
To extract η, I perform a bunch of DSMC (see Sec. 3.3) runs while varying λ, N and Θ,
providing me time traces of Tp (t) (5.36) for each combination of parameter values. I then fit Tp (t)as computed from Eq. (5.34) to those from the DSMC simulations while floating η, such that it
minimizes the relative root-mean-squared errorv!
uuX T DMSC (t) − Tptheory (t; η) 2pt
ε(η) =.
TpDMSC (t)
t

(I.1)

In these numerical experiments, we tune the trap anisotropy in a manner that does not affect Kn,
by setting ω⊥ = ω/λ1/6 and ωz = ωλ1/3 . This construction ensures that ω, and therefore Kn, both
remain independent of λ. The dipoles are taken to point along x̂ (Θ = 90◦ ) for the data shown.

144
Dependence on dipole orientation will be included below.
Results of several such fits are shown in Fig. I.1, which compares the Tp time traces for aseries of cross-dimensional rethermalization experiments with N = 5 × 105 (Kn ≈ 0.04) over a rangeof λ = 0.13 to 8.0, as obtained from DSMC simulations (solid black curves) and my fitted theory(dashed red curves). Noticeably, there is a clear beating of various modes with different frequencies
which our theory is able to describe, showing favorable agreement in both the amplitude and phaseof oscillations. A representative comparison plot of Tr (t) as obtained from DSMC and Eq. (5.34) isalso provided in Fig. I.2, with N = 5 × 105 (Kn ≈ 0.04) and λ = 0.32. Good agreement is seen in
all Tri (t) time traces as well. We note that temperature time traces tend to show better agreementwith the DSMC ones for excitation along the long axis of a prolate trap, even for larger Knudsen
numbers (Kn ≈ 0.1). So, I’ll stick to this excitation geometry for a more focused study.

Figure I.1: Comparison of the momentum space temperature Tp (5.36) vs time t, obtained from
DSMC simulations (black solid curves) and our theory (red dashed curves) with N = 5 × 105(Kn ≈ 0.04), Θ = 90◦ . The subplots (a) to (h) correspond to various values of trapping anisotropy
with λ = 0.13 to 8.0 as labeled in the subplot headers. The fitted values of η are also provided in
the subplot headers with their fitting standard uncertainties.

145

Figure I.2: Comparison of the position space pseudotemperatures Tr vs time t, obtained from
DSMC simulations (upper subplot a) and our theory (lower subplot b) with Θ = 90◦ , N = 5 × 105(Kn ≈ 0.04) and λ = 0.32.

For a given orientation of the dipoles, it may be expected that η depends on both the trap
aspect ratio λ and the number of molecules N . Increasing N , ceteris paribus, evidently increases thedensity and hence likely the hydrodynamic volume. As for aspect ratio, a tentative λ dependenceof Vhy is already taken into account by (5.38), whereby the scaling parameter η may depend only

146
weakly on λ. This hypothesis is supported by the numerics shown in subplot (a) of Fig. 5.11,
where η is linearly dependent on N , but mostly independent of λ for the range of these parameters
explored.
Finally, for a given λ and N , it remains to resolve the dependence of η on the dipole orientation
Ê. In this context, recall that the dilute and hydrodynamic regimes are distinguished by the Knudsen
number, which is inversely proportional to the collision cross section, Eq. (5.28). Sec. 5.4 detailed
how this cross section results in anisotropic viscosities, that work to bring local thermodynamic
fluctuations back to equilibrium. Having accounted for this aspect of differential scattering, I
posit that η should only depend on the cross section that’s averaged over post-collision anglesRdσ
σ = dΩ′ dΩ′ , which still preserves an incoming-collision angle dependence [35]. As to how so, I
present the following argument. Prolate traps have a weak trapping axis z, along which the gas
has a larger thermal width. As a result, the mean-free path along that axis is relatively smaller
compared to the sample size, and consequently more hydrodynamic. Collisions that occur withrelative momentum directed along the long axis, are then most able to keep molecules behaving
collectively as hydrodynamic. The bulk total cross section is, therefore, most simply taken to be
σ = a2dπ
3 + 18 cos2 (Ê · êhy ) − 13 cos4 (Ê · êhy ) ,
3

(I.2)

where êhy = ẑ denotes the most hydrodynamic axis (weakest trap frequency) so that Ê · êhy = Θ.
Sure enough, I find that η follows a Θ dependence very similar to that of Eq. (5.40), whencomparing η as obtained from DSMC experiments, to a fitting function of the form (σ/σ)α + β inR
subplot (b) of Fig. 5.11, where σ = (4π)−1 σ(êhy )dêhy = 32πa2d /15 is the angular averaged total
cross section. The observations above motivate the functional form
η ≈a+b

N
105




σ(Θ)
1+c
,
σ

(I.3)

for some constants a, b and c, which I determine from fits to be a ≈ 2.21 ± 0.017, b ≈ 0.67 ± 0.020
and c ≈ 0.26 ± 0.015. The empirically determined hydrodynamic volume is therefore that presented
in Eq. (5.39) of the main text.

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