Observation of a Persistent Non-Equilibrium State in an
Extremely Isotropic Harmonic Potential
by
D. S. Lobser
B.A., University of Colorado, Boulder, 2009
M.S., University of Colorado, Boulder, 2012

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
2015This thesis entitled:
Observation of a Persistent Non-Equilibrium State in an Extremely Isotropic Harmonic
Potential
written by D. S. Lobser
has been approved for the Department of Physics

Eric A. Cornell

Heather J. Lewandowski

Date

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

iii
Lobser, D. S. (Ph.D., Physics)
Observation of a Persistent Non-Equilibrium State in an Extremely Isotropic Harmonic Potential
Thesis directed by Prof. Eric A. Cornell

Ludwig Boltzmann made tremendously important contributions to the problem of connecting macroscopic, empirical phenomena with microscopic, atomistic dynamics. At the endof the nineteenth century, Boltzmann was confronted with various strong objections to his
work. For example, Boltzmann’s atomistic explanations presuppose the reality of atoms, anotion that was vigorously rejected in some circles [14, 38]. Then too, there was the critique
by Loschmidt that Boltzmann’s H-theorem, put forth as a microscopic explanation for theSecond Law of Thermodynamics, could hardly account for irreversible physics when the individual two-atom collisions were each reversible [18, 42]. Still intriguing today is the existenceof special cases of the Boltzmann equation in which time-varying distributions of atoms resist the imperative of equilibration, even in the presence of collisions. Boltzmann discussed
such situations in a paper dedicated to responding to Loschmidt’s critique [7, 4]. Perhaps
Boltzmann’s motivation was to enumerate special cases where his famous H value does notrelax as it should, and by enumerating them, point out their nonnaturalness, their artificiality. Damping, or relaxation to equilibrium, of a time-invariant phase-space distribution, isan all-but universal result predicted by the Boltzmann equation.
Such improbable systems of atoms have only very recently been realized experimentally.
Kinoshita et al. [36] experimentally confirmed that atoms constrained to move in a quasi
one-dimensional potential, an atomistic Newtons cradle, exhibit vastly suppressed relaxation.
Chevy et al. [15] observed long-lived breathe-mode oscillations in highly elongated but still
3D geometries. Perhaps one of the more interesting cases is the vanishing damping of themonopole breathe-mode oscillation in a spherically symmetric harmonic oscillator [29], whereiv
a cloud of atoms experiences undamped temperature oscillations, causing the cloud to expand
and contract as if it were breathing. Until now, this phenomenon has been experimentally
inaccessible due to the difficulty in generating isotropic harmonic confinement. This thesis
discusses a new magnetic trap capable of producing spherical confinement and presents
the first experimental realization of this historically significant oddity using a magneticallytrapped gas of 87 Rb atoms.

Dedication

To my family.

vi

Acknowledgements

First and foremost I want to thank my family for their constant support. Regardlessof what I was interested in when I was growing up, my family always did whatever they
could to support me. Mom’s enthusiasm and positive outlook is infectious. She never takes
no for an answer and always finds a way to accomplish something once she sets her mind
to it. When I wanted to try fencing, she did a bunch of research and found the best place
to go for fencing lessons and drove me across town a few times a week for years so that I
could get the best training around. When I got into photography, she helped me enroll inthe best photography class in the area so I could have access to one of the few color darkrooms in the state. If it wasn’t for her, I wouldn’t have experienced so many of the things
that have defined who I am today, and I’d still probably be working at a restaurant! Dad’s
excitement about math and programming was invaluable when I was growing up. He would
always challenge me with problems and surprise me with the simplicity of his solutions tothem. Often, he told me that you only really understand how something works if you canprogram a computer to solve it. I remember getting my first graphing calculator in middle
school, a TI-86, and asking him what the PRGM button was. Of course, I got a long winded
answer that I didn’t really care about. But then he had me write a program and helped meline by line. The goal was to calculate the prime factors of an arbitrary number. He’d letme struggle through it for a while, then he’d look at the program and tell me where I’d gonewrong and why and suggest a better alternative until later that night when I finally had
a working program. You could put in a number and it would slowly churn out one prime

vii
factor at a time until it finished. I remember thinking it was a great program because it
took so long, which must have meant that it was solving a very difficult problem. After that
I started moving on to other problems and soon forgot about that first program. But a few
years later I discovered that the very same program was being used by a lot of students atthe nearby high school to cheat on their tests (I knew this because I heard people referencing
my program by it’s unfortunate and arbitrary name ”BBB”). Looking back on that firstprogramming experience, I couldn’t have chosen a better problem to solve. Ironically, thefirst algorithm to fully takes advantage of a quantum computer is Schor’s algorithm forcomputing the prime factors of a number. A concept which is extremely relevant for mytransition into quantum information research! I also have my brother—David even madesome of the figures in this thesis—and sister to thank, as well as my aunts, uncles, cousins,and grandparents, who’ve all been so supportive over the years.
I’ve worked with a number of great students during my 10 years at JILA. Paul Parazzoli
always set time aside to help me out with problems when I was an undergrad, and I’m looking
forward to working with him again during my postdoc. Along with Noah Fitch and Carrie
Weidner, I think we made an excellent team and I couldn’t ask for a better first group ofpeople to work with in the lab. Shih-Kuang Tung was a great student and gave me a lot ofadvice while he taught me about the TOP trap and many of the details of the experiment.
It was a pleasure working in the lab next door to the EDM group over the years. I’ll missall the interesting things Matt Grau used to mention to me, and I have him to thank for mylearning about the Julia language. I also want to thank Brandon Ruzic for putting up withme these last couple months. And the same goes for Cam and Victor, who’ve been very
helpful and patient with my getting them data. It has been a pleasure hanging out with somany people in my class and I’m confident that they will all go on to be very successful.
Finally, I consider myself extremely lucky to have had such great advisors over the last
10 years. As an undergraduate, I worked for Prof. Heather Lewandowski. When I started,
Heather didn’t have any lab space and I was building electronics out of her office. This gave

viii
me the chance to see how a lab is put together from the ground up, especially since the lab wasconverted from an office! Heather has been instrumental, even during my graduate career,
in my understanding and approach to experimental systems. Her knowledge and insight, as
well as her ability to impart that knowledge, has given me a strong foundation that helped
immensely in my graduate work. Heather treats her students with a great deal of respect.
Even when I was an undergraduate, who wasn’t yet well-versed in the technical aspects ofthe experiment, she encouraged me to participate in troubleshooting difficult problems with
her and the other graduate students on the experiment. My input was treated seriously,
and, even if I wasn’t always able to contribute, it was during these times that I learned themost.
Eric never ceases to amaze me with his incredible approach to physics. His ideas andway of looking at problems extremely are inspiring, which never fails to give me a new perspective and a renewed excitement about science. Working on this experiment was a totallydifferent experience from my undergraduate work. The experiment was well established andhad been for well over decade, but working on a mature experiment also has its difficulties.
Approaching problems on the TOP trap made me think about problems in a new way, andoften involved repurposing some preexisting hardware for an entirely different task. Eric’s
creativity and his ability to turn a difficult problem on its side introduced me to an entirely
new and invaluable skill set that I will continue to use in all aspects of my life. I greatly
admire his tact and his wisdom and will look back on these years with fond memories.

ix

Contents

Chapter
1 Introduction
1.0.1

1Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Theoretical Background
2.1

2.2

2.3

5
7

Collision Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.1.1

Equilibrium Distributions . . . . . . . . . . . . . . . . . . . . . . . .

12Collective Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2.1

Monopole Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2.2

Dipole Mode

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

17

2.2.3

Quadrupole Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.2.4

Mode Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Degenerate Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3 Generating a Spherical Harmonic Potential

23

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.2

Basic TOP Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.3

Controlling Ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.4

Anharmonic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

x
4 Experimental Details
4.1

4.2

4.3

Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

4.1.1

TOP Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.2.1

Expected Phase Shift . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.2.2

Expected Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Trap Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.3.1

4.4

4.5

36

Direct Determination of Trap Frequencies and Orientation of Principal
Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

Driving Collective Excitations . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.4.1

Dipole Mode

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

57

4.4.2

Monopole Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.4.3

Quadrupole Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.5.1

60

Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Observation of Suppressed Damping of Breathe Mode Oscillations in a Thermal Cloud 66
5.1

Effects of Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.2

Effects of Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

6 Monopole Mode Measurements in a Finite-Temperature BEC

77

7 Anharmonic Shearing

89

7.1

Quasi-2D Condensate Formation. . . . . . . . . . . . . . . . . . . . . . . .

90

7.2

Single-Site Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

7.2.1

98

Spatial Separation Between Lattice Sites . . . . . . . . . . . . . . . .

xi
Bibliography

106

Appendix
A Finite Temperature Data

111

xii

Tables

Table
3.1

The six fields used to control the potential, separated into cartesian components. 30

3.2

Full list of coefficients for the TOP potential, arranged by the orders of r andz which they multiply. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

34

List of full TOP expansion coefficients for a spherical trap. R0 is the radiusof the trajectory of the quadrupole zero, known as the “circle of death.” . . .

35

xiii

FiguresFigure1.1

Diagram representing Loschmidt’s argument about how time-reversal symmetry can lead to unlikely results. Initially, all of the atoms are at rest at thebottom of a vessel with the exception of a single atom placed at the very
top of the vessel. The atom at the top of the vessel will fall to the bottom,
colliding with other atoms. Eventually, the energy from the dropped atom
will distribute to all of the other atoms and they will reach a state of equilibrium. By reversing time, the time-reversal symmetry inherent to mechanicalsystems implies that the atoms will approach their initial state with one atom
at the very top of the vessel and all of the other atoms resting at the bottom. Loschmidt’s argument was that an equilibrium distribution could thus
be constructed in such a way that, as time advances, the distribution will approach the unlikely state used in his example. Because entropy was observed
to have no time-reversal symmetry, Loschmidt claimed that the mechanicalarguments used by Boltzmann to describe how a gas approaches equilibrium
were flawed.

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

3

xiv
2.1

Frequency dependence on collision rate for monopole and quadrupole modes.
√
As the γcoll → ∞ the frequency of the quadrupole mode approaches 2ωtrap
while the monopole mode frequency is unchanged. These values are calculated from the real part of the roots of Eq. (??), where the roots correspond to the angular momentum projections lz = 2 (quadrupole) and lz = 1(monopole) [29]. The trap frequency used in this calculation is ωtrap = 2π(9Hz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

19

Damping of the quadrupole mode as a function of collision rate. In the collisionless limit, the quadrupole damping has a linear dependence on collision
rate given by ΓQ ' γcoll /5. The actual slope is closer to 1/5.06328 for γcoll ≤ 1,
which is calculated from Eq. (2.46). The trap frequency used in this calculation is ωtrap = 2π(9 Hz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

20

Shifts in the damping rates for the monopole and quadrupole modes due tomode coupling in an anisotropic trap as calculated from the real part of thelz = 2 and lz = 1 roots of Eq. (2.46). Anisotropy is defined by 2δ whereωz = ωtrap (1 + δ), ωr = ωtrap (1 − δ), and ωtrap = 2π(9 Hz) is used as an
approximate mean frequency. The collision rate is set to γcoll = 5.06 (s−1 ), but
the relative shift in damping rate for large anisotropies approaches the samevalues. The final values for the quadrupole and monopole mode damping rates
sum to the quadrupole damping rate in an isotropic trap. For very anisotropic
traps the monopole (quadrupole) modes approach a rate of approximately 1/3(2/3) of the isotropic quadrupole damping result. Reducing γcoll reduces thelevel of anisotropy at which the damping rates flatten out. This model appears
to break down in the limit of large anisotropy, where both modes are populated
for roughly equal times and one would expect that the damping rates shouldboth converge to 1/2 the value of the quadrupole damping rate in a sphericaltrap.

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

21

xv
3.1

(a) Diagram of the TOP potential along the ẑ axis, the atomic sample is shownin red and the anharmonic shifts are exaggerated to show how the potentiallybecomes asymptotically linear at large displacements from the center of thepotential. (b) Representation of the effective tilting of the potential causedby gravity. The location of the trap minimum is shifted, and the cloud moves
closer to the linear region of the potential leading to a relaxation in the confinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

26

Each panel shows a representation of the rotating TOP field as well as anadditional counter rotating field in the latter 2 panels. The path traced out
by the quadrupole zero is represented on the right, and is characteristic of theellipticity induced in the trapping potential in the xy plane. . . . . . . . . .

3.3

28

Measurement of the yz cross term for different strengths of the Byz field.
The slope of the theory line is determined from Eq. 3.18 but is offset by themeasured yz cross term at Byz = 0 to account for the residual ellipticity inthe trap.

4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33Layout of the vacuum chamber. The transfer tube is 37 cm long with adiameter of 1.3 cm and has a spiral-wound wire resting against its interior
to prevent scattered light from the collection MOT beams from reaching thescience MOT. Two additional connections are made to the transfer tube, one
is attached to a 20 L/s ion pump, while the other houses a magnet encasedin glass used to block the tube so as to limit vacuum conductance when theexperiment is not in use. A portion of the vacuum system is not shown, whichconnects to a 30 L/s ion pump, a titanium sublimation pump, and a roughing
valve that was used for initially pumping out the chamber. Artistic rendering
courtesy of David Lobser. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

xvi
4.2

Configuration of the Rb getters. The collection MOT cell contains two arms
each with 3 getters arranged in this configuration. The current supply is
attached to two of the three available terminals so that the current through thegetter directly between the two active terminals is twice the current throughthe other two. This keeps the getters warm and prevents any contaminants
or extra Rb from sticking to them. . . . . . . . . . . . . . . . . . . . . . . .

4.3

39

Artistic representation of the science cell and field coils. The larger coils
above and beneath the cell generate the quadrupole field, and the remaining
coils are used for the TOP fields. The original TOP coils are mounted in on
the horizontal windows and are actually slightly oblong. Artistic rendering
courtesy of David Lobser. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

42

Layout of the phase-contrast imaging setup. The example shown has a magnification of unity, but can be enlarged by increasing the focal length of thesecond lens. It is necessary that the phase dot is larger than the waist of theprobe beam, while remaining smaller than the scattered light. In the limit
where the diameter of the phase dot is large enough that all the scattered lightpasses through it, the signal is reduced and equivalent to having no phase dot
at all. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.5

Expected phase shift of the probe as a function of detuning. . . . . . . . . .

47

4.6

Expected scattering rate as a function of detuning. . . . . . . . . . . . . . .

47

4.7

Signal vs. probe detuning, as calculated from Eq. 4.9 . . . . . . . . . . . . .

48

4.8

Comparison of expected signal to data where the signal has been scaled by afactor of 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

xvii
4.9

These are the raw data that are taken during each experimental cycle. Eachset of eight images is taken simultaneously using phase-contrast microscopy
from two orthogonal directions. The difference in cloud size between the two
sets of images arises from differences in pixel size and magnification of theseparate imaging systems. The images are taken in 15 ms intervals. . . . . .

50

4.10 Sample data and fits of the slosh mode in the yz plane. Both sets of data are
identical but viewed from a different angle. A beat frequency is visible in thefirst figure, while the amplitude appears constant in the figure on the bottom,
indicating that the viewing angle is colinear with a principal axis. The redlines are the fits, and viewed from the top, trace out a Lissajous figure. . . .

51

4.11 Simulated data measured along an axis other than the principal axes, where
beating between the three principal frequencies can be seen. The left column shows data along 3 axes that are arbitrarily rotated from the principal
axes. The right column shows the same data viewed along each principal
axis. Data in the second column results from applying a rotation matrix thatis determined from the methods discussed in this section. . . . . . . . . . . .

54

4.12 Radial, axial, and mean trap frequencies over the full range of field modulations used to drive the quadrupole mode. The maximal change in the fieldstrength is 4% for the quadrupole field and 17.7% for the TOP bias field,
which correspond to ∆ = 1. In the experiment, we also use the vertical bias
field to minimize vertical slosh and end up changing the TOP bias field byonly 13%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.13 Quadrupole and TOP field modulation for driving monopole and quadrupolemodes. Amplitudes are relative shifts in the overall quadrupole and TOP
fields.

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

62

4.14 An example of raw quadrupole mode data and the associated surface fits taken
during one cycle of the experiment.

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

64

xviii4.15 Sample data for a driven quadrupole mode in a spherical trap with residual
asphericity less than 0.002 and a collision rate of 7.4(3) s−1 . Solid lines on
the quadrupole data indicate a typical fitting procedure where individual periods taken in a single run are fit with an undamped sine wave to extract an
instantaneous amplitude. The oscillation amplitudes at various cycles are inturn fit to an exponential decay to extract the damping rates. Random observable scatter in these points is predominantly due to small, irreproducible
fluctuations in initial conditions.
5.1. . . . . . . . . . . . . . . . . . . . . . . .

65

Sample data for a driven quadrupole mode and monopole mode in a spherical
trap with residual asphericity less than 0.002. The normalized distortions
plotted are calculated using Eqs. (4.31) and (4.32). Dashed lines indicate
an exponentially damped sine-wave fit. Solid lines on the quadrupole data
indicate a typical fitting procedure where individual periods taken in a single
run are fit with an undamped sine wave to extract an instantaneous amplitude.
The amplitudes at various times are in turn fit to an exponential decay to
extract the damping rates shown in Fig. 5.2 . . . . . . . . . . . . . . . . . .

5.2

68

Monopole and quadrupole damping rates as a function of interatomic collision rate in a near-spherical trap. Residual asphericity was typically around0.001, but always less than 0.0026. Cloud full width at half maximum wasapproximately 115 µm.

5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Monopole and quadrupole motion in an anisotropic trap where the residual
asphericity is approximately 0.02. A monopole mode is initially driven andoscillations between the monopole and quadrupole modes can be seen. The
fitting functions are given by e−γt (cos2 (∆ω + πδQ ) + C), but should be usedjust as a visual guide.

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

70

xix
5.4

Monopole and quadrupole motion in traps with anisotropies of 0.0222 (top),
0.0133 (middle), and 0.0028 (bottom). Data on the left show driven quadrupole
motion, while data on the right show driven monopole motion. As the levelon anisotropy decreases, the mode coupling turns off and the decay of eachmode becomes exponential.

5.5. . . . . . . . . . . . . . . . . . . . . . . . . . .

Monopole damping rates in very isotropic traps as a function of residual asphericity. The data shown are the monopole damping data from Fig. 5.2.

5.6

72

.

73

Our same monopole data from Fig. 5.2 are revisited, but this time plottedagainst the average FWHM of the clouds. The FWHM is calculated from theaverage of all of the breathe data after the clouds have been driven in orderto account for the small amount of heating induced by the drive.

5.7. . . . . .

75

Damping of the monopole mode in a near-spherical trap as a function of thespatial extent of the atom cloud. The fractional amplitude of the excitationis the same for all points. Consistent with anharmonicity-induced damping,
we see the observed damping rate decreases rapidly for smaller clouds, but
the trend of the data suggests that even for the smallest clouds (such as thoseused in Fig. 5.2) the residual damping observed may be due to the onset ofanharmonic effects.

6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Profile of a simulated finite-temperature condensate in a harmonic trap. The
condensate atoms are shown in blue and the thermal atoms in red. Mean-fieldinteractions modify the potential, pushing the thermal atoms away from thecondensate. This particular simulation uses the ZNG model for a condensate
at a temperature T = 0.75Tc [1].

6.2

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

78

Surface fits to raw data using Eq. (6.2). No drive was used for this particularset, but the fits work equally well for driven data. The surface fits are shownin the center column and the residuals are shown in the righthand column. .

79

xx
6.3

Azimuthally averaged data (blue solid line) and surface fit (red dashed line)
around the center position of the condensate. . . . . . . . . . . . . . . . . . .

80

6.4

Condensate damping for a BEC fraction of N0 /N = 0.098(2), N = 9.1(1)×105 82

6.5

Condensate damping for a BEC fraction of N0 /N = 0.177(2), N = 7.9(1)×105 83

6.6

Condensate damping for a BEC fraction of N0 /N = 0.31(2), N = 5.4(2)×105

6.7

Condensate damping for a BEC fraction of N0 /N = 0.322(3), N = 6.7(1)×105 85

6.8

Condensate damping for a BEC fraction of N0 /N = 0.446(2), N = 9.7(1)×105 86

6.9

Condensate damping for a BEC fraction of N0 /N = 0.579(3), N = 8.9(1)×105 87

6.10 Condensate damping for a BEC fraction of N0 /N = 0.58(4), N = 8.7(2)×105
7.1

84

88

Diagram of our lattice beam set up as viewed from the side. The quasi-2D
condensates are shown in red and confined to the nodes of the interference
pattern. Roughly 10 lattice sites are occupied on average. . . . . . . . . . . .

7.2

91

Diagram of our lattice beam set up as viewed from the side. The quasi-2D
condensates are shown in red and confined to the nodes of the interference
pattern. Typically about 10 lattice sites are occupied . . . . . . . . . . . . .

7.3

93

Diagram of the energy levels for the 5S 1 state of 8 7Rb. The microwave transfer
2

is shown with the dotted arrow and is centered at a frequency of approximately
6.8 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4

96

Calculation of the microwave transfer pulse (blue) in frequency space relative
to the optical lattice (green) and condensate (red) size. . . . . . . . . . . . .

96

xxi
7.5

Diagram showing the expansion process used for the momentum-space focusing technique. Mean-field energy causes a lower-dimensional condensate toexpand in the direction of tight confinement into a magnetic potential withan aspect ratio of 2. After a quarter period of an oscillation along the weakly
confined dimension, 14 Tz , condensate atoms reach the center of the potential simultaneously giving rise to the delta-function momentum distributionof a condensate. The thermal component is shown to contract with the condensate, which does happen for thermal atoms displaced by the condensate
mean-field. But for hot thermal atoms, the width of the distribution shouldhave approximately the same size in coordinate and momentum space.

7.6

. . .

98

Example of intersite dephasing for multiple quasi-2D condensates confined toa 1D optical lattice. Individual layers, viewed from the side, are not resolved.
Instead, the entire sample appears “tilted” as the separate vertical layers
become slightly offset from each other in the horizontal direction. . . . . . .

7.7

99

Side-view image of intersite dephasing for multiple quasi-2D condensates after
a two seconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.8

Side-view image of intersite dephasing for multiple quasi-2D condensates after
two seconds with a TOP field of 1 G, where layers have been fully separated.
Separate layers can be resolved, but imaging from the side causes the probe
beam to diffract when incident on a condensate, making the individual layers
difficult to resolve. The condensates are sloshing in a circular motion, and atdifferent phases of the oscillation the clouds appear to overlap when viewed
from the side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.9

Raw data from an experimental realization of the anharmonic shearing effectviewed from the TOP. These data are taken after 2 seconds of evolution in
the sheared trap, and are taken in 20 ms intervals. . . . . . . . . . . . . . . . 104

Chapter 1
Introduction

The notion of irreversibility in nature is a concept that is ingrained in us from a very
early age. Nobody expects the cream in their coffee to suddenly separate and move to one
side of the mug, or for their coffee to get hotter as it sits. If, however, the cream were to
separate from the coffee or if its temperature were to spontaneously increase, no physicallaws would be violated. We now know that it’s really the probability of cream completely
separating from coffee that is so small it is effectively an impossibility. But it wasn’t until
the late 19th century that Ludwig Boltzmann formulated a rigorous explanation of irreversible processes as time-reversible dynamics in a statistical framework. Boltzmann madevarious contributions to physics, but he was truly interested in understanding the macroscopic phenomena described by thermodynamics at the microscopic level, to which he mademonumental advances. By 1872, he formulated his transport equation, which describes how
the phase-space distribution of a gas evolves in time as a function of individual particle
collisions. Using his transport equation, more commonly referred to as the Boltzmann equation, he derived the H -theorem, which he put forth as a description of the Second Law ofthermodynamics. While nothing about this approach seems especially problematic today,
his theory met with a great deal of skepticism and, in some cases, was even viciously opposed. For example, Boltzmann assumed the existence of atoms, which were not yet fully
accepted by the scientific community. But the objections against Boltzmann’s theory hinged
primarily on two arguments. First, the notion of a mechanistic description of the way a gas

2
approaches equilibrium was considered to be fundamentally flawed because the time reversal
symmetry of mechanical systems conflicted with the irreversible nature of entropy. Second,mechanical systems undergoing elastic collisions will eventually return to their initial state.
These two arguments are usually called the “reversibility” and “recurrence” objections respectively. The recurrence objection was spurred on by Henri Poincaré and his assistant,
Ernst Zermelo, but included various other scientists, mathematicians, and philosophers even
including Friedrich Nietzsche, who first pointed out the paradox [19]. Heated arguments
about the recurrence objection spanned nearly two decades and were often punctuated withsharply worded, deep cutting comments from both sides. Unfortunately, the debate wasnever resolved1 and ultimately died out with the people involved. The recurrence objection
has a much more colorful history than the reversibility objection, but it was the reversibility
objection that lead to the result of this thesis.
Joseph Loschmidt, a friend and colleague of Boltzmann, is attributed with developmentof the reversibility objection, however this argument had been made by William Thomsonin 1874 [4] and was essentially what Maxwell was trying to motivate with his demon analogy [4, 38]. Loschmidt’s criticism was not directed at Boltzmann, but simply pointed out the
problem with using a statistical distribution of microstates in a gas as a means of describing
thermodynamic phenomena. He argued that there are certain arrangements of atoms ina vessel that were inconsistent with the barometric law, which states that if the vessel isdivided into vertically stacked layers then the mean energy of each layer must be the same.
Suppose a single atom is placed at the top of the vessel and the rest of the atoms are at restat the bottom of the vessel, as shown in Fig. 1.1. The atom at the top will eventually fallto the bottom and its kinetic energy will be distributed amongst the other atoms. For verylarge numbers of atoms, it is very likely that an atom will never be found in the top layer
at a later time because all of the atoms must come to rest at the bottom again. However,1

Boltzmann’s ideas are now widely accepted, but it the people involved in the original debate died before
reaching an agreement.

3

Initial State

ThermalEquilibrium

Time
Reversal

Final State

Figure 1.1: Diagram representing Loschmidt’s argument about how time-reversal symmetry
can lead to unlikely results. Initially, all of the atoms are at rest at the bottom of a vessel
with the exception of a single atom placed at the very top of the vessel. The atom at the
top of the vessel will fall to the bottom, colliding with other atoms. Eventually, the energy
from the dropped atom will distribute to all of the other atoms and they will reach a state ofequilibrium. By reversing time, the time-reversal symmetry inherent to mechanical systems
implies that the atoms will approach their initial state with one atom at the very top of
the vessel and all of the other atoms resting at the bottom. Loschmidt’s argument was thatan equilibrium distribution could thus be constructed in such a way that, as time advances,the distribution will approach the unlikely state used in his example. Because entropy
was observed to have no time-reversal symmetry, Loschmidt claimed that the mechanicalarguments used by Boltzmann to describe how a gas approaches equilibrium were flawed.

Loschmidt argued that [4]If, after a sufficient time τ is elapsed from the establishment of the stationary
state, we suddenly turn the velocities of all atoms in the opposite directions,then we will find ourselves at the beginning of a state to which the characterof stationarity can apparently be ascribed. This would last for a certain
time, but then the stationary state would start gradually to deteriorate andafter a time we would arrive unavoidably again at our initial state.
While Loschmidt’s argument wasn’t directed at Boltzmann, it did suggest that Maxwell’s
distribution could not sufficiently describe states in equilibrium because special arrangements of the microstates in the distribution could be constructed to drive the system out

4of equilibrium and hence the system wasn’t necessarily stationary. At the time, the lawsof thermodynamics were considered to be absolute and could not be violated. This posedserious problems for kinetic theory, which allowed for deviations from thermodynamics inspecial cases. Boltzmann had been aware of these anomalies as early as 1868, when he showedthat a gas initially constrained to lie in one dimension would never reach a homogeneous
isotropic distribution. He published an article in response to Loschmidt’s reversibility argument, which cited the 1868 example, but also derived a number of other special cases thatexhibited bizarre behavior [7]. Originally, Boltzmann had sidestepped the issue, claimingthat the bizarre behavior that arose from extremely artificial initial conditions was characteristically unnatural. Deeming these cases impossible in the limit of extremely large atom
number, he ignored them because they would never occur in a real system. This sentiment
was shared by Thomson in his 1874 paper [58]:...the greater the number of molecules, the shorter will be the time during
which the disequalizing will continue; and it is only when we regard thenumber of molecules as practically infinite that we can regard spontaneous
disequalization as practically impossible.
Despite Boltzmann’s awareness of the unusual behavior potentially exhibited by these artifically cooked-up systems, he eventually conceded that his theories were invalid in these highly
improbable cases and, thereafter, parenthetically stipulated that the Second Law “can bemechanically represented only by means of assumptions regarding initial conditions” [14].
Loschmidt’s argument did eventually cause Boltzmann to approach the problem differently,
and in 1877 he published a paper that used a purely combinatorial approach. There are still
two schools of thought regarding Boltzmann’s viewpoint during the development of kinetic
theory. Many argue that his 1876 and 1877 papers marked a sudden realization that his
original formulation of the H-theorem was not only incomplete, but incongruent with his
combinatorial approach. However, his 1877 paper did not refute any ideas from his original
formulation, but instead carefully laid out the probabilistic framework on which the trans-

5
port equation was based. This is also evidenced by a reference to his 1876 paper in his
Lectures on Gas Theory, which was published 20 years later:
Other solutions of these equations, in which t can also occur explicitly, canalso be found [7]. For example, there is one remarkable solution in which
the gas flies outward from a center equally in all directions in such a manner
that, first, there is no viscosity; and second, while the temperature of course
drops as a result of the expansion, it drops equally everywhere in space sothat no heat conduction occurs. We shall no longer concern ourselves withthese matters, however; we shall only ask what value the quantity H has inall these cases.
He then goes on to show that entropy is constant in cases where the gas is in motion, butsatisfies the conditions for equilibrium in his transport equation while resisting the imperativeof equilibration. One such case is exhibited by the absence of damping of the monopole, or“breathe,” mode of a gas in which the cloud continually expands and contracts as if it werebreathing. The condition for this exceptional behavior of the breathe mode is that the gas
be confined to an external potential that is both isotropic and harmonic.
Variations of this effect have been experimentally verified in other dimensionalities.
Kinoshita et al. [36] experimentally confirmed that atoms constrained to move in a quasi
one-dimensional potential, an atomistic Newtons cradle, exhibit vastly suppressed relaxation.
Chevy et al. [15] observed long-lived breathe-mode oscillations in highly elongated, but still
3D, geometries. However, the difficulty in generating an isotropic harmonic potential has
prevented the experimental verification of the spherical case until now.
1.0.1

Thesis Layout

The purpose of this thesis is to present experimental confirmation of Boltzmann’s very
old, yet remarkable, result. Because the foundation of the result led to over three decades ofheated debates, the foundation of the theory will be laid out carefully in chapter 2. Chapter
3 will give a detailed description and analysis of the magnetic trap, which finally madethe experimental realization of this effect possible. In chapter 4, various other details of the6experiment will be covered. Chapter 5 presents the results of our experiment on Boltzmann’s
original prediction. Chapter 6 will briefly describe some of our extensions of these studies to
Bose-Einstein Condensates. And Chapter 7 discusses another technique altogether, which isused to spatially separate a series of quasi-2D condensates confined to a 1D optical lattice.

Chapter 2Theoretical Background

Perhaps Boltzmann’s greatest achievement was connecting microscopic, atomistic dynamics to macroscopic empirical phenomena. His ideas helped establish the foundationsof kinetic theory, despite the initial resistance of various skeptical opponents of his theories [4, 19, 42]. One of the more fundamental ideas that Boltzmann developed is known as
the transport equation, or often simply as the Boltzmann equation, which describes how
a gas evolves as a function of collisions between particles in the gas. Boltzmann was intrigued by Maxwell’s distribution, which describes the distribution of velocities in a gas atthermal equilibrium. But he was particularly interested in understanding how the complex
and chaotic behavior of atoms in a gas could lead to a stable solution, even when atoms areconstantly colliding and changing velocities [38]. His idea was that if he could keep track of
the motion of each atom and each collision, he could demonstrate how the bulk system would
invariably approach the Maxwell distribution. By 1865, Loschmidt had determined the average size of molecules and effectively calculated Avogadro’s constant [11]. So Boltzmann
was well aware that even a small volume of gas contained an immense number of atoms andkeeping track of their individual motions was impossible, and tackling such a problem would
require statistical methods. From Boltzmann’s perspective, the statistical framework wasjust a convenient mathematical tool that allowed the calculation of such a complicated andrandom system managable. Ironically, it wasn’t until later that he realized the implicationsof the statistical framework to the underlying physics.

8
The statistics used in the Boltzmann equation show up in the form of a phase-space
distribution, which is used to describe the arrangement of particles in a gas. The distributionfunction, f (r, v, t), represents the probability density of finding a particle centered about agiven region of phase space at time t. This discussion will assume that the distribution isfor a single particle and is normalized so thatZf (r, v, t)d3 rd3 v = 1,

(2.1)

where the integral is carried out over the entire phase space. Standard literature on kinetic
theory will often use momentum instead of velocity as a phase-space coordinate. For amonatomic gas, the difference amounts to a different normalization constant and the twoquantities can more or less be interchanged. In the absence of collisions, the phase space
distribution of a gas will remain constant and
df
= 0.
dt

(2.2)

Because f is a function of more than one variable, the total derivative can be expanded so
that
∂f
F
df
=
+ v · ∇r f +· ∇v f = 0,
dt
∂t
m

(2.3)

where F is an external force and m is the atomic mass. Eq. (2.3) is the single particle
equivalent of the Liouville equation, which states that the phase-space density of a system
of non-interacting particles is conserved.
Boltzmann’s goal was to describe how a gas would approach equilibrium based oncollisional processes. Thus Eq. (2.3) must incorporate another term to account for collisions,
resulting in the Boltzmann equation:
df= Icoll [f ].
dt

(2.4)

The left side gives the time evolution of the distribution and the right side accounts forany changes to f due to collisions, as described by the “collision integral” Icoll [f ]. Detailed

9treatments of the collision integral can be found in various textbooks [9, 13, 33, 47, 62], but
because the equation is so fundamental to the topic of this thesis it will be briefly discussed
here.
For now, it is convenient to represent Eq. (2.4) in the following form
Icoll [f ] =

∂f
∂t




=

coll

∂f
∂t

+
coll


−

∂f
∂t

−

.

(2.5)

coll

The last terms are used to explicitly distinguish between entering and leaving1 scattering
events, which respectively increase or decrease f in a particular region of phase-space.
The following assumptions are made about collisions:• Only binary collisions are considered, which falls under the assumption that the gas
is sufficiently dilute.
• Collisions are local and instantaneous.
• The atoms lack any internal structure so that collisions are elastic.2• Velocities of two atoms directly before and after a collision are uncorrelated, but dosatisfy conservation of energy and momentum.
Boltzmann called this last assumption “Stosszahlansatz”, now known as the molecular chaos
hypothesis, which is a necessary ingredient for deriving the irreversible kinetic equation [47].
Binary collisions are often treated as the scattering of particles from a fixed center offorce. The parameters of the collision are characterized by two scattering angles, θ and φ,
and an impact parameter, b, which is the shortest possible distance between the center ofmass of both particles if no collision occurs [23, 59]. Exact knowledge of the parameters of
1

The term “entering” refers to scattering events, which result in a final distribution that increases ata particular phase-space coordinate, while the “leaving” terms lead to a decrease in the distribution at thesame phase-space coordinate.
2
Boltzmann’s treatment of this problem preceded the development of quantum mechanics and thisargument implied that atoms could be treated as hard spheres. A modern interpretation of this assumption
is that collisions have low enough energy that the internal structure in the atoms is inaccessible in this energy
regime.

10
the collision is impossible in a statistical framework, which necessitates that the collisions
be described by a differential cross section σ(Ω)dΩ. For a beam of particles incident on theforce center, the rate of at which particles are scattered into a solid angle element dΩ is givenby
R0 σ(Ω)dΩ(2.6)

where R0 is the incident particle flux. Consider a particle with a given phase-space coordinate
r and v. Using the associated statistical representation, the existence of a single particle atthis coordinate impliesZf (r0 , v0 , t)δ(r − r0 )δ(v − v0 )dr0 dv0 = 1,

(2.7)

where the shorthand notation dr0 dv0 is equivalent to the notation d3 r0 d3 v 0 used in Eq. (2.1).
The incident flux depends on the phase-space density at r and v as well as the velocity of
the incoming particles, v0 :R0 = f (r, v1 , t) |v − v1 | dv1 .

(2.8)

So the number of particles scattered into dΩ between time t and t + dt isf (r, v1 , t) |v − v1 | dv1 σ(Ω)dΩdt.

(2.9)

So far we have assumed that a particle exists at the precise phase-space coordinate (r,v) attime t, which was imposed by Eq. (2.7). We cannot assume, however, that a particle exists
at a precise phase-space coordinate and instead must consider the probability of finding aparticle at (r,v) and thus we must include the single-particle distribution of our test particle.
Incorporating the distribution of our test particle, which is multiplied by Eq. (2.9), the total
number of scattering events can be determined by integrating over all scattering angles aswell as all possible velocities of atoms in the incoming beam, v1 . Because the velocity of bothparticles will change abruptly after a collision, the distribution will decrease abruptly at agiven phase-space coordinate. The total decrease in f (r, v, t) in some particular phase-space

11
volume element is thus


∂f
∂t

−

Z
drdvdt = f (r, v, t)drdvdt

Zσ(Ω)dΩ

coll

dv1 |v − v1 | f (r, v1 , t).

(2.10)

Multiple collisions in this infinitesimal phase-space volume between t and t+dt are neglected
because of the assumption that the gas is sufficiently dilute. This assumes collision events
where {v, v1 } → {v0 , v10 }, however, particles can also scatter such that {v0 , v10 } → {v, v1 }.
These collisions will increase f , and contribute to the entering collision term in Eq. (2.5).
The entering collision term can be derived with the same arguments and we find


∂f
∂t

+

0

0

Z

drdvdt = f (r, v , t)drdv dt

Zσ(Ω)dΩ

coll

dv10 |v0 − v10 | f (r, v10 , t).

(2.11)

Elastic collisions conserve both total and relative momentum so, since we are working witha monatomic gas, this implies|v − v1 | = |v0 − v10 | .

(2.12)

At a given scattering angle dvdv1 = dv0 dv10 and the differential cross section is also identicalfor both entering and leaving collisions. Using these equalities we arrive at an enteringcollision term of the form


∂f
∂t

+

Z

0

drdvdt = f (r, v , t)drdvdt
collZ
σ(Ω)dΩ

dv1 |v − v1 | f (r, v10 , t).

(2.13)

Although the distributions in the entering collision term depend on primed velocities, whereas
the integral is carried out over a single unprimed velocity, it is implied that v is fixed whilev0 and v10 are functions of v, v1 , and Ω.
Combining the entering and leaving collision terms brings us to the full form of thecollision integral
Z
Icoll [f ] =Zdv1dΩσ (Ω) |v − v1 | (f 0 f10 − f f1 ) ,(2.14)

using the shorthand notation for the distribution functions where f ≡ f (r, v, t), f10 ≡
f (r, v10 , t), etc...

12
2.1

Collision Invariants
Once a gas reaches equilibrium, the distribution becomes static and
df
= 0.
dt

(2.15)

Collisions are of course still occuring, but have no effect on the distribution as the collision
integral must equal zero. There exists a class of solutions known as collision invariants forwhich the collision integral will always vanish. These solutions have the general form [13,29, 27]log f = α + β · p + γp2 ,(2.16)

where α, β, and γ are arbitrary constants, but can also be functions of r and t. Solutionsof this form will always force the collision integral to zero because of conservation of energy,
momentum, and atom number. In other words, because
p2 + p21 = p02 + p02
1(2.17)

p + p1 = p0 + p01 ,(2.18)

f f1 = f 0 f10(2.19)

thenwhen the distributions are collision invariants. Thus, the kernel of the collision integral must
always vanish for distributions of this form.
2.1.1

Equilibrium Distributions

While collision invariants force the collision integral to zero, they don’t necessarily
satisfy Eq. (2.3) and
df
6= 0
dt

(2.20)

unless α, β, and γ are properly constrained. Generically, distributions that cause the collision integral to vanish, but don’t satisfy the Boltzmann equation, are referred to as “local

13equilibrium distributions.” These local equilibrium distributions can be understood moreintuitively by recasting Eq. (2.16) into a more familiar exponential form
Bm

2

f = Ae− 2 |v−u| ,(2.21)

where A, B, and u can be functions of r and t. In order for the collision integral to vanish,energy conservation must now satisfy22|v − u|2 + |v1 − u|2 = |v0 − u| + |v10 − u| .

(2.22)

Because u cannot depend on v, this condition is met for local collisions and u is essentiallya Galilean transformation.
The distribution (2.21) can be made to satisfy the Boltzmann equation for a generalexternal potential V (r) by setting
A = A0 e−BV (r)(2.23)

B=

1
kT(2.24)

u = 0,(2.25)

where k is the Boltzmann constant, and T is the temperature. This is the well known
Maxwell-Boltzmann distribution
1

1

2

f (r, v) = A0 e− kT ( 2 mv +V (r)) .

(2.26)

The contraints (2.23)-(2.25) provide the traditionally well-understood solution to the Boltzmann equation, however Boltzmann also explicitly derived various stationary non-equilibrium
states that exist for certain special potentials [7, 28]. One example in particular is the undamped monopole mode in an isotropic harmonic potential.

14
2.2

Collective Excitations

2.2.1

Monopole Mode

2.2.1.1

Two Particle Derivation

The undamped nature of the monopole mode is found by calculating the evolution of
the square radius of the cloud and can be derived in various ways [29]. It is instructive to
see how the monopole nondamping result can be built up starting from small atom numbers
N = 1, 2, etc. . .

Spherical symmetry simplifies the problem, which can be treated as

analogously one dimensional. The radial motion of a single particle of mass m, energy E,
and angular momentum L is governed by the effective potential [59]
Ve =

1
L2
+ mω 2 r222mr2(2.27)d2 r
d= − Ve2dt
dr(2.28)

so that the radial force ismand the kinetic energy is
1
m
2



dr
dt

2
= E − Ve .

(2.29)

We note that d2 r2 /dt2 = 2(dr/dt)2 + 2rd2 r/dt2 , and substituting (2.28) and (2.29) yields thedifferential equation for r2

d2 2
2
2
2
r
=
−Ω
r
−
r
0 ,
dt2

(2.30)

where Ω ≡ 2ω and r02 = E/(mω 2 ). It can be seen by inspection of Eq. (2.30) that the squareradius undergoes sinusoidal oscillations, or “monopole breathe”, around its mean value r02 at
a frequency of 2ω. Now, if there are two particles, 1 and 2, each with individual values of E,
L, and r2 , each particle will oscillate at 2ω and taking the sum of their respective differential
equations (2.29) yields

d2 2
2
2
2
r
=
−Ω
r
−
r
t
t
0t
dt2

(2.31)

152
where their combined square radius, rt2 ≡ r12 + r22 , oscillates around its mean value, r0t≡(E1 +E2 )/(mω 2 ). The magnitude of the collective breathe motion depends on the magnitude
and relative phase of the individual particle trajectories. These individual quantities will
abruptly change in the event of a collision. Assuming the collisions are local, r1 , r2 , andthus rt2 will not change from the instant before to the instant after the collision. Similarly,2
are unchanged by the collision.
momentum and energy conservation imply that dtd rt2 and r0t

These three continuities imply that the parameters and boundary conditions of Eq. (2.31)
are matched directly before and after a collision. This ensures that neither the magnitude
nor phase of the oscillation will change as the result of a pairwise collision. If we instead
consider N atoms where
rt2 =

N
X

ri2

(2.32)

i=1
N

2
r0t
=

1 X
Ei ,
mω 2 i=1(2.33)

one can see that the monopole mode is left unperturbed—and in particular undamped —by
local, pairwise, momentum- and energy-conserving collisions.
The above approach is consistent with a system in the hydrodynamic limit, with total
number of atoms so large that the function f (r, v, t) is essentially continuous. However, inthe hydrodynamic limit, mean-field effects can come into play in which case the monopole
frequency is shifted and perhaps damped. In our experiment, the total number of atoms,
N , is only a few hundred thousand, and the mean-free path is large compared to the spatial
extent of the sample—we are not in the hydrodynamic limit in any sense of the word—and
mean-field effects can be neglected.

16
2.2.1.2

Moments Method

Another way to calculate the behavior of the monopole mode is by calculating moments
using the Boltzmann equation. In general, the expectation value of a quantity χ(r, v) ishχi =Zχf drdv.

(2.34)The time evolution of χ can be determined using the Boltzmann equation


Z
χ


Z
F
∂
+ v · ∇r +· ∇v f drdv = χIcoll [f ]drdv.
∂tm(2.35)

By using the identity
χ

df
d
dχ
= (χf ) − f
dt
dt
dt

(2.36)

we can rewrite Eq. (2.35) asZ F∂(χf ) + v · ∇r (χf ) +· ∇v (χf )
∂t
m

Z
∂χ
F
−f
− f v · ∇r χ − f · ∇v χ drdv = χIcoll [f ]drdv.
∂tm(2.37)

Because χ is not an explicit function of time, the fourth term in Eq. (2.37) drops out and
we are left withFdhχi
− hv · ∇r χi − h · ∇v χi = hχIcoll i.
dtm(2.38)A convenient property of the collision integral is that when χ is a collision invariant of theform χ(r, v) = a(r) + b(r) · v + c(r)v 2 , the expectation hχIcoll i is zero. This is a resultof certain symmetries that let hχIcoll i = h 41 (χ + χ1 − χ0 − χ01 )Icoll i, which vanishes whenconservation laws are imposed [33, 47].
The behavior of the monopole mode is found by calculating the time evolution of hr2 i.
This quantity is a collision invariant, and using Eq. (2.38) we quickly arrive at the following

17
closed set of equations:d 2
hr i = 2hr · vi
dtdhr · vi = hv 2 i − ω 2 hr2 i
dt
d 2
hv i = −2ω 2 hr · vi.
dt(2.39)(2.40)(2.41)

Differentiating Eq. (2.39) with respect to time then gives the wave equation
d2 2
hr i = −4ω 2 hr2 i,
dt2(2.42)

which describes undamped oscillations at frequency 2ω.
2.2.2

Dipole Mode

It turns out that there is one other undamped mode, the dipole mode, which is just a
center-of-mass oscillation in the trap. The moment method can be used to also model dipole
oscillations with the following set of equationsdhri = hvidtd
hvi = −ω 2 hri.
dt(2.43)(2.44)

These give the following wave equationd2
hri = −ω 2 hri,2dt(2.45)

which describes undamped oscillations of the center-of-mass position of the cloud at frequency ω. This result is perhaps less surprising than the undamped monopole mode and isequivalent to the solution for a single particle undergoing simple harmonic oscillations in aharmonic potential. The undamped behavior of the dipole mode is a result of the generalized Kohn theorem [17], which doesn’t depend on the trap symmetry, and the result for ananisotropic trap returns a set of uncoupled wave equations along each principal axis of thepotential. We use this mode to characterize our trapping potential as it provides a simple
way of measuring trap frequencies.

18
2.2.3

Quadrupole Mode

Behavior of the axially symmetric quadrupole mode can be determined by calculatingh2vz2 − vr2 i using the moments method. The full treatment is discussed in [29], which arrives
at the following result for monopole and quadrupole mode frequencies in an anisotropic
harmonic trap with frequencies ωr and ωz :ω2

− 4ωz2ω2− 4ωr2



i
−
ωτ



2
ω − ω 2 (5ωr2 + 4ωz2 ) + 8ωr2 ωz2
3
4


= 0.

(2.46)

This equation is obtained using the “relaxation time approximation,” where the relaxation
time, τ , is defined asτ=5.
4γcoll(2.47)

n(0)vth σ02(2.48)

The collision rate γcoll is given byγcoll =

for a peak density n(0), average thermal velocity vth =p
8kT /πm, and scattering cross-section σ0 .
Solving for the roots of Eq. (2.46) gives complex monopole and quadrupole frequencies,
where we defineω = ω0 + iΓ.

(2.49)The real and imaginary parts correspond to the mode’s natural oscillation frequency anddamping rate respectively. For ωr = ωz , this method can be used to determine the complex
monopole frequency, which is undamped with frequency 2ωtrap as expected. However, thedamping rate of the quadrupole mode in the collisionless limit depends linearly on collision
rate and is approximately
1
ΓQ ' γcoll .
5(2.50)In this limit, the quadrupole frequency is identical to the monopole frequency. When collisions are neglected entirely, the imaginary part of Eq. (2.46) is zero and the first term

19
describes undamped monopole-like oscillations along each principal axis of the potential. As
the hydrodynamic limit is approached, the monopole and quadrupole frequencies are split asshown in Fig. 2.1. While the monopole frequency is unchanged, the quadrupole frequency

Mode Frequency vs. Collision Rate
2
Quadrupole
Monopoleω0 /ωtrap

1.9
1.8
1.7
1.6
1.5
1.4
0

20

40

60

80

100

120

140

γcoll (s−1 )Figure 2.1: Frequency dependence on collision rate for monopole and quadrupolemodes.
√As the γcoll → ∞ the frequency of the quadrupole mode approaches 2ωtrap while themonopole mode frequency is unchanged. These values are calculated from the real part of
the roots of Eq. (??), where the roots correspond to the angular momentum projections
lz = 2 (quadrupole) and lz = 1 (monopole) [29]. The trap frequency used in this calculationis ωtrap = 2π(9 Hz).

asymptotically approaches√

2ωtrap . In this limit, the quadrupole damping tends toward zeroand it behaves like a true normal mode. Fig. 2.2 shows the damping in the crossover regime
between the collisionless and hydrodynamic limits.

20

Quadrupole Damping vs. Collision Rate
3.5
3

ΓQ (s−1 )

2.5
2
1.5
1
0.5
0
0

20

40

60

80

100

120

140

γcoll (s−1 )Figure 2.2: Damping of the quadrupole mode as a function of collision rate. In the collisionless limit, the quadrupole damping has a linear dependence on collision rate given byΓQ ' γcoll /5. The actual slope is closer to 1/5.06328 for γcoll ≤ 1, which is calculated from
Eq. (2.46). The trap frequency used in this calculation is ωtrap = 2π(9 Hz).

2.2.4

Mode Coupling

Earlier it was mentioned that when neglecting collisions, the oscillations are essentially
1D monopole-like oscillations along each principal axis of the potential. It is important tonote that when ωr 6= ωz , these oscillations dephase. When they are in phase, the overallbehavior is that of a monopole breathe. When perfectly out of phase, it appears like aquadrupole oscillation. In the collisionless limit, this type of shape oscillation will still occur,
but the modes are coupled by collisions [12]. This coupling causes the monopole mode todamp due to the non-zero energy populating the quadrupole mode. From Eq. (2.46), thedependence of damping rate on trap anisotropy can be calculated for both modes, and isshown in Fig. 2.3 for a collision rate of γcoll = 5.06 (s−1 ).

21ΓQ,M (s−1 )Damping in an Anisotropic Potential
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

ΓQ
ΓM

0

0.05

0.1

0.15

0.2

0.25

0.3

Anisotropy
Figure 2.3: Shifts in the damping rates for the monopole and quadrupole modes due to mode
coupling in an anisotropic trap as calculated from the real part of the lz = 2 and lz = 1
roots of Eq. (2.46). Anisotropy is defined by 2δ where ωz = ωtrap (1 + δ), ωr = ωtrap (1 − δ),and ωtrap = 2π(9 Hz) is used as an approximate mean frequency. The collision rate is set toγcoll = 5.06 (s−1 ), but the relative shift in damping rate for large anisotropies approaches thesame values. The final values for the quadrupole and monopole mode damping rates sum tothe quadrupole damping rate in an isotropic trap. For very anisotropic traps the monopole(quadrupole) modes approach a rate of approximately 1/3 (2/3) of the isotropic quadrupole
damping result. Reducing γcoll reduces the level of anisotropy at which the damping rates
flatten out. This model appears to break down in the limit of large anisotropy, where bothmodes are populated for roughly equal times and one would expect that the damping rates
should both converge to 1/2 the value of the quadrupole damping rate in a spherical trap.

2.3

Degenerate Clouds
When the cloud is Bose condensed, the behavior of the monopole mode is slightly

modified. BECs are described by the Gross-Pitaevskii equation (GPE):


∂
~2 2
2
∇ + Vext (r, t) + g |Ψ0 (r, t)| Ψ0 (r, t) = i~ Ψ0 (r, t).
−2m∂t(2.51)

Closely resembling the Schrödinger equation, the GPE contains an extra mean-field interaction term, g |Ψ0 (r, t)|2 , where g = 4π~2 ascatt /m, ascatt is the interatomic scattering length,22
and m is the atomic mass. Another difference is that Ψ0 (r, t) represents the collective wave
function of the condensate, known as the order parameter, so the mean-field term can beinstead represented by gn(r, t) where n(r, t) is the condensate density. Presence of the mean
field modifies the potential, causing a shift in the frequency of the monopole mode equalto [26]
ωM = ωtrapp
4 + ξ,(2.52)

for a 3D system with an average density, hni, and temperature, T , where
ghni.
ghni + kT√
In the Thomas-Fermi limit, ξ ' 1 and ωM = 5ωtrap .
ξ=(2.53)For a pure condensate, the monopole mode is still undamped [46]. Damping, however,does occur for a finite temperature condensate. In this case, the normal and condensed
components of the cloud oscillate at different frequencies because the normal component
isn’t as strongly affected by the mean field. Both components interact and, once theiroscillations dephase, the collective monopole motion will relax. Various finite-temperature
theories have been proposed [49] but, due to the lack of experimental tests of these models,
their accuracy remains an open question.

Chapter 3
Generating a Spherical Harmonic Potential

3.1

Introduction
From a theoretical standpoint, the realization of an isotropic harmonic potential has

enormous benefits as a testbed for numerous physical models. Not only can difficult calculations be greatly simplified by imposing spherical symmetry, which in many respects can bethought of as analogously one dimensional, but interesting physical phenomena also emergein spherically symmetric systems [7, 13, 29, 53]. Experimental study of these systems has so
far been prevented by the difficulty in generating spherical potentials.
Stable magnetic traps have long been paramount to ultracold atom experiments. Gases
are typically cooled evaporatively and must be confined by a local minimum of an inhomogeneous magnetic field. Various magnetic traps have been developed for evaporative coolingexperiments, such as the Ioffe-Pritchard trap [48], QUIC trap [21], and baseball trap [5] toname a few. However, almost none of the available traps is able to generate sphericallysymmetric confinement when gravitational forces are present. One exception is the timeaveraged orbiting potential (TOP) trap [45], which uses a fast rotating bias field to create
a local minimum in a quadrupole magnetic field. The TOP trap is capable of generating asuitably isotropic potential under ideal conditions. In practice, the water-cooled coils used to
produce the magnetic fields are bulky and difficult to align. Any construction asymmetries
give rise to a residual anisotropy in the potential, which is effectively impossible to prevent.
The nature of the TOP trap allows for careful control of trap ellipticities confined to the24plane of the rotating bias field. Previous experiments performed with a TOP trap were able
to shim out asymmetries in the xy plane, allowing for cylindrical trap geometries that wereround to within 0.1%. This chapter discusses a modification to the standard TOP trap,
capable of shimming out elliptic cross terms in all three dimensions. A method of optimizing
trap sphericity is demonstrated such that the three principal trap frequencies differ by lessthan 0.1%.
3.2

Basic TOP Theory
We begin our discussion with a brief overview of the TOP potential neglecting any

gravitational effects. Tight confinement, a necessary feature of any magnetic trap used in
evaporative cooling experiments, is produced by a quadrupole field defined by
BQ =

Bz0
B0
xx̂ + z y ŷ − Bz0 z ẑ.
2
2

(3.1)

The quadrupole field alone is insufficent for ultracold atom experiments because the vanishing
field at the center of the trap allows for Majorana transitions causing atoms to fall out of thetrap faster than they can be evaporatively cooled [41]. Other magnetic traps take advantageof the quadrupole field because of its simplicity, but add extra fields to create a non-zero
local minimum [48, 5, 21]. The TOP trap does this with the addition of a rotating bias fieldof the form
BTOP (t) = B0 cos(ωt)x̂ + B0 sin(ωt)ŷ.

(3.2)

The instantaneous field simply adds a linear offset to the quadrupole field, displacing the magnetic zero-point of the quadrupole field from the atoms. By setting the rotation frequency,
ω, much faster than the characteristic trap frequency, the atoms are not only prevented fromreaching the quadrupole zero, but micromotion induced by the oscillating field is negligible,
thus creating a stable potential. The potential is calculated by taking the time average of

25
the magnitude of the total magnetic field, soω
U0 = µ
2π

Z 2π
ω

0

Bz0 2
x)
2
1
B0
+(B0 sin(ωt) + z y)2 + 4Bz02 z 2 ] 22dt[(B0 cos(ωt) +(3.3)

Expanding around the center of the trap, one finds
U0 ' µB0 +µBz02 2x + y 2 + 8z 2 .
16B0

(3.4)

It can immediately be seen that the potential is harmonic with a minimum that is greaterthan zero. From Eq. (3.4), it is evident that the potential is anisotropic and the ratio√of trap frequencies in the axial and radial dimensions is ωz /ωr = 8. This anisotropy is
problematic when attempting to generate a spherical geometry. However, we take advantageof the fact that the potential is not perfectly harmonic to add a controlled distortion to thepotential. Anharmonic effects can be seen directly by calculating U0 at extremely large axialp
displacements. In the limit where Bz0 |z| >> B0 >> Bz0 x2 + y 2 , the integrand in Eq. (3.3)
is dominated by the quadrupole field and the potential reduces to
U0 ' 2µBz0 |z|,(3.5)

which is not harmonic. For smaller displacements, the crossover between the harmonic andlinear form of the potential amounts to a relaxation of the confinement that is most significant
along ẑ.
By using gravity to our advantage, we are able to control the displacement along ẑ,and thus control the aspect ratio of the trap. To calculate this effect, an extra term is added
to the Eq. (3.3) and the new potential is given by
U = U0 + mgz.

(3.6)

The additional gravitational term adds a linear slope to the potenial, causing the minimum
to move downward as shown in Fig. 3.1. Clearly, gravity always contributes to the full

26
(a)

(b)

U0

z Position

U0 + mgz

z Position

Figure 3.1: (a) Diagram of the TOP potential along the ẑ axis, the atomic sample is shown in
red and the anharmonic shifts are exaggerated to show how the potentially becomes asymptotically linear at large displacements from the center of the potential. (b) Representation of
the effective tilting of the potential caused by gravity. The location of the trap minimum isshifted, and the cloud moves closer to the linear region of the potential leading to a relaxationin the confinement.

potential, but the displacement due to gravity, or gravitational sag, depends on the ratio of
the gravitational force to the magnetic force on the atoms. We define the ratio of these twoforces as
η≡

mg
µBz0

(3.7)

In the high-field limit µBz0 >> mg, the gravitational term can be dropped from Eq. (3.6)
and U0 is an adequate approximation of the potential. By decreasing Bz0 , we can control how
much the atoms sag in the trap and preferentially relax the confinement in the z dimension.
However, the calculation of the potential must be expanded around the new minimum, which
is given by
z0 = −

B0
η
p
ẑ.
0
Bz 1 − η 2(3.8)

Carrying out the expansion around z0 , we findU ' µB0

p
p

µBz02
1 − η2 +1 + η2
1 − η 2 x2 + y 2
16B0
3/2 2µBz02+
1 − η2
z .
2B0

(3.9)

27
The ratio of trap frequencies is now determined by η, wheresωz
1 − η2= 8.
ωr
1 + η2(3.10)

Because the only adjustable parameter in η is Bz0 , the quadrupole field strength must beptuned to set the aspect ratio of the trap, which is isotropic when η = 7/9.
Using gravity as a way of controlling the symmetry of the potential requires a weak
trap with trapping frequencies on the order of 10 Hz. Another approach has been developedby Hodby et. al. known as the “zTOP” trap, which adjusts the symmetry of a TOP trapwith an additional pair of Helmholtz coils oriented along the ẑ axis [32]. Their method adds
a linear offset to the trap so that the atoms lie in the symmetric region of the potential. The
atoms will of course feel an additional force that pushes them back toward the minimumof the potential but, using the same time-averaging principle employed by the TOP trap,
the vertical offset can be reversed before the atoms can start to approach the minimum.
Theoretically, the ideal modulation of the vertical bias field has the functional form of thesquare wave, with twice the frequency of the rotating bias field, in order to maximize thetime in which the atoms are at the symmetric region of the potential. In their experiment,
Hodby et. al. modulated the vertical field with certain harmonic components of the squarewave to reduce the aspect ratio of the trap down to a level of ωz /ωr = 1.6.
3.3

Controlling Ellipticity
An ideal TOP trap construction would produce an isotropic potential without any

modifications. However, measurements of the trap frequencies in our system reveal residual
ellipticities in all dimensions. These ellipticities are most likely due to construction asymmetries that come with the difficulty in aligning bulky, water-cooled coils. Previous experiments
were able to shim out the ellipticity in the xy plane by adding a small counter rotating field
to the normal TOP field. The eccentricity and elliptic cross terms could be controlled by adjusting the amplitude and relative phase of the counter rotating field. This idea is illustrated

28

a
TOP

b

TOP

+
Elliptifier

c
TOP

+
Elliptifier

Figure 3.2: Each panel shows a representation of the rotating TOP field as well as anadditional counter rotating field in the latter 2 panels. The path traced out by the quadrupole
zero is represented on the right, and is characteristic of the ellipticity induced in the trapping
potential in the xy plane.

in Fig. 3.2, where a small counter-rotating “elliptifier” field is added to a uniform TOP field
to produce changes in the ellipticity of the contour produced by the path of the quadrupole
field minimum. Fig. 3.2a shows the ideal TOP field, which in theory should be perfectly
circular. Figs. 3.2b and 3.2c show how the elliptifier field can induce ellipticity where theprincipal axes of the ellipse rotate based on the phase of the elliptifier field. The Lissajous
figure traced out by the quadrupole zero is directly proportional to the elliptical profile of thepotential because U = µhB(t)i. Of course, the goal is to produce a perfectly round profile,such as the one in Fig. 3.2a, however precise control of the ellipticity is required in order toshim out the residual ellipticities we observe in our trapping potential. The elliptifier field
was previously used to create extremely cylindrical trap geometries—necessary for BEC vortex experiments—where the principal trap frequencies in the xy plane differed by 0.1% [30].

29Unfortunately, this control was limited to only the plane of the TOP coils and elliptic crossterms in the xz and yz planes posed a significant problem for generating a spherical potential. Because one can control the ellipticities in the xy plane with the coils that lie in thatplane, it stands to reason that one would gain control of the xz and yz planes if coils werealigned along those planes, which is possible with an additional set of TOP coils aligned withthe ẑ axis. One way to think of the oscillating ẑ field is a field that causes the TOP field totilt relative to the quadrupole field. Suppose the ẑ field is oscillating with the same phaseas the x̂ field. When the field is maximized, the bias field will be tilted upwards towards the
positive ẑ direction. When the field is minimized, the bias field will be tilted downwards,
and there will be no tilt when the field is zero. This is equivalent to mechanically tilting the
TOP coils, which moves the Lissajous figure traced out by the quadrupole zero out of thexy plane. It should be noted that the purpose of this vertical field is entirely different thanthat of the zTOP trap, though the two ideas could presumably be combined.
It is important to work with parameters that are as orthogonal as possible to minimize
the number of adjustments necessary for removing any ellipticities. An ellipsoidal potential
contains 6 quadratic terms and has the form
U ∝ Ax2 + By 2 + Cz 2 + Dxy + Exz + F yz,(3.11)

where independent control of each coefficient lets us generate a spherical harmonic potential
by setting A = B = C and D = E = F = 0. These coefficients are directly controlled by
the quadrupole and TOP fields. Because the quadrupole field must be set to control thesymmetry between the radial and axial dimensions, it can only adjust the ratio 2C/(A + B)
and the rotating bias fields must be used to control the remaining coefficients. The diagramin Fig. 3.2 demonstrates how the phase of the elliptifier field can be changed to rotate theellipse, however this does not give independent control over the A, B, and D coefficents in thexy plane. We can instead define two separate fields corresponding to the counter-rotating
fields in Figs. 3.2b and 3.2c. The first field, which we will continue to call the elliptifier

30
field, will have a phase corresponding to the one shown in Fig. 3.2b. This field controls the
A and B coefficients relative to some overall value set by the TOP field—in other words
the elliptifier field could be described by δ, where A → A + δ and B → A − δ—while notaffecting the D coefficient. The D coefficient can be controlled by a counter-rotating fieldwith a phase corresponding to Fig. 3.2c, and will not affect the A or B coefficients, giving
independent control of the xy cross term. Cross terms in the xz and yz planes can be
adjusted by including an oscillating field along the ẑ axis, which has the same frequency andphase as the field in the x̂ and ŷ dimensions respectively. We now have 6 fields to control
the 6 coefficients of the quadratic potential. The A, B, and C terms are controlled by the
Quadrupole, TOP, and Elliptifier fields, while the D, E, and F terms are controlled by three
additional “cross-term fields”, all of which are summarized in Tab. 3.1.
x̂
Quadrupole
TOP Fieldx̂ŷ Elliptifierx̂ŷ Cross Termx̂ẑ Cross Termŷẑ Cross Term

Bz0
x
2

B0 cos(ωt)
Bell cos(ωt)
Bxy sin(ωt)
0
0

ŷ

ẑ
−Bz0 zB0 sin(ωt)
0-Bell sin(ωt)
0
Bxy cos(ωt)
00Bxz cos(ωt)
0Byz sin(ωt)
Bz0y2

Table 3.1: The six fields used to control the potential, separated into cartesian components.

The orthogonality of these fields can be shown explicitly by computing the new potential. With the additional fields, it difficult to solve for the potential analytically whengravity is taken into account. However, the analytic solution can be motivated by calculating
the time-averaged magnetic field, which can be then tuned to cancel out the construction
asymmetries to realize the ideal TOP model. The new instantaneous field now takes the

31
form
Bz0 2
x)
2
B0
+((B0 − Bell ) sin(ωt) + Bxy cos(ωt) + z y)22

Bmod (t) = [((B0 + Bell ) cos(ωt) + Bxy sin(ωt) +(3.12)

1+(Bxz cos(ωt) + Byz sin(ωt) − Bz0 z)2 ] 2Expanding Eq. 3.12 around the trap center one finds
Bmod (t) ' B0 +−Bz02 2(x + y 2 + 4z 2 )8B0(3.13)

Bz02[((B0 + Bell ) cos ωt + Bxy sin ωt) x
8B03+ ((B0 − Bell ) sin ωt + Bxy cos ωt) y−2 (Bxz cos ωt + Byz sin ωt) z ]2Here we have assumed that the extra shimming fields are small compared to B0 so, at thefield origin,
ωhBmod (t)i|r=0 =
2π

Z 2π
ω

0

|Bmod (t)|r=0 = B0 .

(3.14)

This simplification is made so that any sinusoidal terms in the denominator of Eq. 3.14 fallout, as in the solution for the ideal TOP field. Evaluating the square in the final term in
Eq. 3.14 and computing the time average, one finds
hBmod (t)i ' B0 +Bz02 2(x + y 2 + 4z 2 )
8B0
 2
B 02
2
− z 3 [ (B0 + Bell )2 + Bxy
x
16B0
 2
2
+ (B0 − Bell )2 + Bxy
y
 2
2
2
+ Byz
z
+4 Bxz+4Bxy B0 xy−4 (B0 Bxz + Bell Bxz + Bxy Byz ) xz
−4 (Bxy Bxz + B0 Byz − Bell Byz ) yz ](3.15)

32
Because the shimming fields are considered to be small, coefficients that are second order inany of the shimming fields can be dropped, leaving
Bz02 2(x + y 2 + 8z 2 )(3.16)
16B0


1
Bz02
1
2
2
− Bell x + Bell y − Bxy xy + Bxz xz + Byz yz .
+
4B02
2
2

hBmod (t)i ' B0 +One can see that Eq. 3.16 is the same as Eq. 3.4 with extra terms that give full control overthe ellipticity in the xy plane plus control over the elliptic cross terms in the xz and yz
planes. Once the elliptic cross terms are shimmed out, the resulting potential matches thatof the ideal TOP, and the frequency ratio ωz /ωr can be set by tuning the strength of thequadrupole field.
Gravitational effects slightly modify the terms in Eq. (3.16). To account for thesedifferences, the full potential was computed numerically around z0 in a spherical trap andthe strength of the elliptic shimming fields was found to differ by the following numericalfactors,
ξ=

1
,
10

ζ=

1
6

(3.17)

where Eq. 3.16 is instead written as√

2Bz02 2(x + y 2 + z 2 )(3.18)
27B0√
4 2Bz02ξξ2
2− Bell x + Bell y − ξBxy xy + ζBxz xz + ζByz yz .
+
27B02
2
2

hBmod (t)i ' B0 +This result agrees very well with the measured elliptical cross terms as shown in Fig. 3.3,
which shows the dependence of the yz coefficient on the Byz field.
3.4

Anharmonic Effects
Using a TOP trap to generate a near-spherical potential comes at the cost of intro-

ducing anharmonic perturbations. It is essentially the anharmonic effects that support thespherical solution. Anharmonic terms can be computed by expanding the TOP potential to33yz Cross Term vs. B y z
2.5

TheoryData

yz Cross Term (Hz 2 )

2

1.5

1

0.5

0

−0.5
−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

B y z (G)Figure 3.3: Measurement of the yz cross term for different strengths of the Byz field. The
slope of the theory line is determined from Eq. 3.18 but is offset by the measured yz crossterm at Byz = 0 to account for the residual ellipticity in the trap.

higher order, where we assume that any asymmetries have been shimmed out. The coefficients for the potential up to 4th order in r and z is are given in table 3.2.

r0
z

0

z1
z

2

B0

p

r2

1 − η2µBz04

1−η 2 (η 2 +1)µ
16B0

03

4

3/2

Bz02 (1−η 2 )
2B0

µ

2

Bz03 η(η 2 −1) µ2B02

4Bz04 (1−η 2 ) (5η 2 −1)µ8B03(15η 4 −12η 2 +1)µ
32B03

−

2

5/2

(5η 2 (63η 4 −91η 2 +33)−9)µ128B05

2

2

4

2

0

5/2

3Bz06 (1−η 2 )(35η 2 (9η 4 −11η 2 +3)−3)µ2048B05

3

Bz05 η(η 2 −1) (35η 4 −40η 2 +9)µ32B04
Bz06 (1−η 2 )1−η 2 (15η 6 −21η 4 +5η 2 +1)µ
1024B03(35η −30η +3)µ− 3Bz η(η −1)1024B
4

3/2

Bz04 (1−η 2 )

√

05

2

0

z3
z

√

−4η +1)µ− Bz η(3η16B
2

0

5/2

Bz02

r4

5Bz07 η(η 2 −1) (7η 2 (99η 4 −153η 2 +65)−45)µ2048B06−

7/2

15Bz08 (1−η 2 )(7(429η 6 −792η 4 +450η 2 −80)η 2 +15)µ
8192B07

Table 3.2: Full list of coefficients for the TOP potential, arranged by the orders of r and z which they multiply.

34

z0
z1
z2
z3
z4

r0
r2
r4
8.157 × 10−2 R0
3.625 × 10−2 R0−1 −9.651 × 10−4 R0−302.261 × 10−2 R0−2 −5.931 × 10−4 R0−4
−2 −1
3.625 × 10 R0
6.714 × 10−3 R0−3
2.275 × 10−3 R0−5
3.014 × 10−2 R0−2 −7.071 × 10−3 R0−4 6.771 × 10−3 R0−6
2.327 × 10−2 R0−3 −1.552 × 10−2 R0−5
1.03 × 10−2 R0−7

35

Table 3.3: List of full TOP expansion coefficients for a spherical trap. R0 is the radius of
the trajectory of the quadrupole zero, known as the “circle of death.”

Symmetry in the radial plane causes all of the odd powers of r to vanish, but thissymmetry does not exist in the z dimension, giving rise to odd powers of z. Imposing
spherical symmetry constrains the value of Bz0 and η, and we get a simplified set of coefficientsshown in table 3.3. The magnetic moment, µ, has been removed and we have introduced
R0 ≡ 2B0 /Bz0 , which is the radius of the quadrupole field zero from the center of the trap,
often called the “circle of death.” Each coefficient is of roughly the same order, with the
exception of R0 . So the anharmonic terms could be reduced by either decreasing the cloudsize, or by increasing the radius of the circular trajectory, R0 . However, increasing R0 reduces
the trap frequency and requires longer evolution times for studies of collective oscillations
and presents greater difficulty when trying to drive individual modes.

Chapter 4
Experimental Details

The experiment itself has a long and colorful history, and many of its current components were used for the realization of the first Bose-Einstein condensate (BEC) in 1995 [2].
Shortly after the 1995 experiment, the group made several modifications to the apparatus,
dubbed “JILA Mark III”, including an entirely new chamber that has been continuously
under vacuum since late 1996 [20]. A perfect way to describe the current state of the experiment is from a former graduate student on this experiment, Paul Haljan, who wrote in his
2003 thesis [31]:Although practically every part of the setup has undergone modification,
the same basic apparatus underlies it all (if you can still see it under the
optics and coils!) Only now is the experiment beginning to suffer fromsome limitations, particularly in optical access, leaving room for plenty ofingenuity.
When this was written, JILA Mark III had only been in use for 7 years, however the vacuum
system is now older than the incoming freshman class and other components (including thecontrol computer) are older than the incoming first year graduate students! It is safe to saythat ingenuity has played a key role in the development of experiments conducted with JILAMark III over the last decade. It is also safe to say that each time the experiment changed
hands, it became slightly more complicated and modifications from one generation of students
were often incorporated into modifications from later generations. Despite the various layersof obfuscation involved in the experimental process, it can still consistently produce big,37
beautiful condensates. While the experiment has endured a spectacular evolution over theyears, Paul Haljan’s statement still holds true: the same basic apparatus underlies it all.
4.1

Apparatus
The experiment’s longevity is primarily attributable to the simplicity of the vacuum

system, which lies at the core of the experiment. With the exception of the Rb getters,
the vacuum chamber does not contain any feedthroughs or components that pertain directly
to the experimental sequence. Nearly any modification to the experiment can be madeexternally, removing the need to break vacuum. Traditionally, the apparatus has been used
for BEC experiments and low background pressures are imperative to ensure long BEC
lifetimes. When the system was first constructed, the group was able to reach backgroundpressures as low as 10−12 Torr. Since then, the system has been under continuous vacuum
and determining the actual pressure is difficult; BEC lifetimes are several minutes long.
The dominant source of background pressure comes from our Rb dispensers, which areevaporative getters. Atoms are first collected in a magneto-optical trap (MOT) [43], where
the filling rate of the MOT depends largely on the background Rb pressure. During the firstexperiments with BECs in this lab, low enough background pressures required extremely long
MOT load times and one experimental cycle took nearly 20 minutes. The JILA mark III
uses a double-MOT system [43] to allow for fast filling times and maintain a low backgroundpressure in the evaporation region. The apparatus, shown in Fig. 4.1, is fitted with two
MOT cells connected by a 37 cm long, stainless steel “transfer” tube.

Push BeamIon Pump

Ti Sublimation
andIon Pumps

Collection
Cell

Magnetic
Gate

Science
Cell

38

Figure 4.1: Layout of the vacuum chamber. The transfer tube is 37 cm long with a diameter of 1.3 cm and has a spiral-wound
wire resting against its interior to prevent scattered light from the collection MOT beams from reaching the science MOT. Two
additional connections are made to the transfer tube, one is attached to a 20 L/s ion pump, while the other houses a magnet
encased in glass used to block the tube so as to limit vacuum conductance when the experiment is not in use. A portion of thevacuum system is not shown, which connects to a 30 L/s ion pump, a titanium sublimation pump, and a roughing valve thatwas used for initially pumping out the chamber. Artistic rendering courtesy of David Lobser.

39
Initially atoms are loaded in the “collection” MOT cell, which contains the Rb gettersand runs a higher background pressure of around 10−9 Torr to speed up loading, then they
are transferred to the “science” MOT using a push beam. The collection MOT has two setsof getters, each set up in a parallel as shown in Fig. 4.2

Current
SupplyFigure 4.2: Configuration of the Rb getters. The collection MOT cell contains two arms
each with 3 getters arranged in this configuration. The current supply is attached to two of
the three available terminals so that the current through the getter directly between the two
active terminals is twice the current through the other two. This keeps the getters warm
and prevents any contaminants or extra Rb from sticking to them.

At the end of each day, we turn off the getter current to maximize their lifetime, which
was originally predicted to be two years per getter [20]. However, we only switched to thesecond getter arm two years ago, and the actual lifetime is closer to 6 years per getter. We
also minimize residual background Rb from collecting in the science cell by blocking thetransfer tube with a magnetic gate. In Fig. 4.1, a small arm attached to the transfer tube
contains a bar magnet encased in glass. The gate is opened or closed using another bar
magnet, which is dragged along the outside of the tube.
Under normal operating conditions, the collection MOT load time is roughly one second. The push beam is pulsed every 400 ms until the number of atoms in the science MOT(proportional to the MOT fluorescence monitored on a photodiode) is 80% of the desired

40
value, at which point the pulse separation reduces to 200 ms. To increase the stability of
the initial number, and to reduce any initial slosh, we load 5% more atoms than specified
and wait for the science MOT to decay to the correct value before loading the atoms into
the magnetic trap. Several stages are involved when loading atoms into the magnetic trap.
After enough atoms are collected in the MOT, the quadrupole field is ramped down from
17.4 G/cm to 5.2 G/cm for a compressed MOT (CMOT) stage. Then the quadrupole fieldis removed entirely, at which point the atoms are cooled to sub-doppler temperatures inan optical molasses and then optically pumped on the |F = 2, m = 2i → |F = 1, m = −1itransition. At this point, the quadrupole and TOP fields are quickly ramped on and theevaporative cooling procedure is initiated.
Forced evaporation of the atomic sample in a TOP trap is slightly different from most
BEC experiments. Radiofrequency (RF) radiation is coupled to the atoms, pumping theoutermost (and thus most energetic) atoms into an untrapped state. During the first stagesof evaporation, the RF frequency is ramped down from 100 MHz to 13 MHz, but a key
difference of evaporating in a TOP trap is that the TOP bias field is simultaneously reduced
from 50 G to 5.2 G. Initially, the TOP field must be large so that the radius of the circleof death is larger than the size of the cloud. However, the trap frequency scales inverselywith the TOP field. Evaporating in a weakly confining potential can be problematic because
elastic collision rate, which is directly proportional to rethermalization rate, scales as thesquare of the trap frequency. Efficiency of the evaporative sequence is thus maximized bymaintaining the largest allowable trap frequency at all times.1 The maximal trap frequency
is limited by the cloud size and, during the first stages of evaporation, the TOP field isramped down with the RF frequency so that the radius of the circle of death is slightly
larger than the distance of the RF knife from the center of the trap. For the last stage of
evaporation, the TOP trap settings are changed to another set of parameters that specify the
During the early stages of evaporation this is not an issue, but when the density approaches 1014 cm−3
inelastic losses start to become problematic.
1

41
desired final trap geometry. It is during this stage that all of the shimming fields, including
the vertical TOP fields are turned on. The remainder of the evaporation ramps only thefrequency, and the TOP field is fixed.
4.1.1

TOP Fields

The primary TOP coils are set up in a Helmholtz configuration and aligned with thefour horizontal windows of the science cell used for MOT beam access. In order to provide
the 50 G field, the coils must be able to dissipate 180 W and are made from 22 AWG wire
housed in a polyflo tubing for water cooling. The quadrupole coils are placed directly aboveand below the cell in an anti-Helmholtz configuration, and are made of water-cooled hollow
copper tubing in order to dissipate a maximum of 2700 W. Because the vertical TOP coils
only need to produce fields on the order of a few hundred mG, they don’t require any watercooling. The TOP and quadrupole coils were both set up as close to the cell as possible, andthe vertical TOP coils are placed directly above and below the quadrupole coils as shown in
Fig. 4.3.
The TOP coils are driven by a 4.5 kHz sine wave that is split into two channels forthe x and y coil pairs. Each channel is passed through a circuit where both the phase andamplitude of the signals are independently adjusted. A separate function generator is used
to provide the elliptifier field signal, and passed through an identical circuit used for the
TOP signal. The elliptifier signal is 45◦ out of phase with the TOP signal, which is used asa trigger, and the amplitude is adjusted by the control computer via GPIB. The elliptifierand TOP signals added together, then the outputs for the x and y coil pairs are sent to adual-channel high-powered amplifier and impedance matched to the TOP coils using a pairof home-made transformers.
The 4.5 kHz sine wave used for the TOP fields is also passed to a separate circuitthat drives the vertical TOP coils. This circuit has the same basic functionality as the oneused for the TOP fields: the channels are split into sine and cosine components that are42

Figure 4.3: Artistic representation of the science cell and field coils. The larger coils aboveand beneath the cell generate the quadrupole field, and the remaining coils are used for
the TOP fields. The original TOP coils are mounted in on the horizontal windows and are
actually slightly oblong. Artistic rendering courtesy of David Lobser.

43
independently adjusted. Because the current required for the vertical field is less than 1 A,
the output is amplified by an LM675 power op-amp contained within the same circuit.
4.2

Imaging
The experiment is equipped with two independent imaging systems, which are set up

to probe along two orthogonal axes. The older of the two imaging setups has a dedicatedimaging port that is perpendicular to the axis of the transfer tube, while the other imaging
system is aligned vertically. To speed up the data taking process, and to reduce errors
associated with initial conditions such as atom number, we image the cloud non-destructively
using the phase-contrast technique [3]. Phase-contrast imaging probes with light far-detuned
from resonance. In this case, the light passing through the cloud experiences a small phaseshift that depends on both the detuning of the probe beam and the density of the cloud. The
resulting phase-shifted light is then interfered with the light that doesn’t pass through theatoms, to resolve a signal. Phase shifts are typically only a few degrees, and the contrast of
the interference is very small, and thus difficult to resolve. The contrast can be improved by
selectively shifting the portion of the probe beam that is not incident on the cloud. Fig. 4.4
shows the layout of a phase-contrast imaging system. Light that passes through the sample isdiffracted while the rest of the probe beam remains collimated. In Fourier space, the ratio of
the widths of the collimated probe beam and the scattered light is inverted. Placing a “phasedot,” which adds a π/2 phase shift to the light, at the focus of a lens allows the unscatteredprobe light to be phase shifted while leaving the scattered light nearly unaffected.
We use a confocal imaging setup, shown in Fig. 4.4, with the phase dot placed at thefocus of the probe beam. The phase dot is made by depositing MgFl2 onto a glass substrate.
Its thickness is determined by the wavelength of the probe light and given by λ/4(n − 1) =513 nm, where n ≈ 1.38 is the refractive index of MgFl2 [31]. For the side-view imaging
system, the phase dot has a 100 µm diameter and is large enough to capture the probe
beam, which has a 100 µm waist, but smaller than the scattered light, which has a typical

44

CCD

Probe
Light

Scattered
Light

Phase
Dot

Figure 4.4: Layout of the phase-contrast imaging setup. The example shown has a magnification of unity, but can be enlarged by increasing the focal length of the second lens. It
is necessary that the phase dot is larger than the waist of the probe beam, while remaining
smaller than the scattered light. In the limit where the diameter of the phase dot is large
enough that all the scattered light passes through it, the signal is reduced and equivalent tohaving no phase dot at all.

diameter of roughly 850 µm. The top-view imaging system uses a 200 µm phase dot toaccount for slight instabilities in the probe alignment. Because the top-view probe shares anaxis with the MOT beams, two mechanical flipper mirrors are used in the beam path andsmall variations in their position cause the beam to shift slightly from shot to shot.
4.2.1

Expected Phase Shift

The phase shift imparted by the cloud as a function of detuning isφ=ωdRe [n(∆)]c(4.1)

where d is the length of the condensate along the imaging axis andRe [n(∆)] = 1 +N e2 X∆i
 Ki
8π0 mω i ∆2i + Γ 2

(4.2)

2

is the real part of the refractive index. N is the number of atoms, so that N d is a column
density, ω is the frequency of the light, and ∆i is the detuning of the probe light from aparticular quantum state (indexed with i). The sum is taken over all hyperfine states and theP
Ki terms are normalized Clebsch-Gordon coefficients so that i Ki = 1. Because the atoms

45
only populate the |F = 1, mF = −1i hyperfine state, the detuning is taken to be constantand the sum is taken only over the relevant Ki ’s. We are interested in the phase difference
between the refracted and non-refracted component of the probe beam which is∆φ 'ωd N e2∆12Γc 8π0 mω ∆2 +
2

(4.3)

2

N de2 1'8π0 mc 2∆

for ∆  Γ(4.4)

Since the atoms are in the |1, −1i state, we typically do phase contrast imaging on the
F =1→20 transition. In this case the absorption cross section is
σ0 =

~ωΓ
.
2Isat

(4.5)

Isat c2 N σ0 d
~ω 3 4∆(4.6)

In this case, the phase shift becomes∆φ = 2.85

Limitations in phase-contrast imaging are typically due to scattering when the probe is not
sufficiently detuned from resonance. However, phase-contrast imaging is feasible because thesignal goes as S ∝ ∆−1 whereas the photon scattering rate goes as R ∝ ∆−2 . The scattering
rate is well known and given by
Rsc =Γ
I/Isat
21+4 ∆ 2+ΓΓ3 I'
8∆2 Isat

4.2.2

I

(4.7)

Isat

for ∆  Γ(4.8)

Expected Signal

The actual signal depends on the parameters of the phase dot, the thickness of whichsets the sensitivity for the measured phase shift. The exact parameters of the phase dot are46
discussed in Paul Haljan’s thesis [31] where the following relations for the signal and phaseshift are presented



δ0
δ0
δ0
S0 (∆φ) = 4 sin
sin + sin ∆φ −
2
2
2
"
#
S0
δ0
δ0
∆φ = arcsin
+ .
− sin
δ0
2
2
4 sin 2(4.9)
(4.10)

Here, S0 is the measured signal, φ is the phase shift due to the atoms and δ0 = π/2 is thephase shift due to the phase dot.
The side-view probe frequency is servoed using a dichroic atomic vapor laser lock
(DAVLL) [16]. Because the DAVLL lock is able to resolve only fine-structure spectra, thefrequency is calibrated by interfering the probe with another laser locked to the resonant
hyperfine transition used for absorption imaging. The interference is detected by a fast
photodiode, and the set point on the DAVLL lock is adjusted so that the beat frequency
between the two lasers is minimized. A small fraction of the probe light is diverted intoa Fabry-Perot interferometer, which is used to measure the relative shift in frequency at
different set points for the frequency lock and set the desired detuning from resonance. We
normally run at a detuning of about 800 MHz to 1 GHz. Figs. 4.5 and 4.6 show plots of the
expected phase shift for a typical cloud and scattering rate for detunings between 100 MHz
and 1.1 GHz.
When the phase shift is large enough, the signal is no longer linear and will oscillate as
the detuning approaches zero. The signal as a function of detuning is shown in Fig. 4.7. anda comparison with experiment is shown in Fig. 4.8 For this data, the cloud size is slightly
smaller than the one used in earlier calculations and the on resonance OD was used as afitting parameter.

47

50

0.873

40

0.698

30

0.524

20

0.349

10

0.175
100

300

500

700

900

DΦ Hrad.L

DΦ Hdeg.L

Phase Shift vs. Detuning

1100

D HMHzL

Figure 4.5: Expected phase shift of the probe as a function of detuning.

1000

0.8

500

0.4

200

0.16

100

0.08

50

0.04

20

0.016
100

300

500

700

900

Fraction of Total Atoms
Scattered in 8 Shots

Rsc HHzL

Photon Scattering Rate vs. Detuning

1100

D HMHzL

Figure 4.6: Expected scattering rate as a function of detuning.

48

Signal vs. Detuning
4

3

S

2

1

0

-1
-1000

0

-500

500

1000

D HMHzL

Figure 4.7: Signal vs. probe detuning, as calculated from Eq. 4.9

Signal vs. Detuning
350
300

OD f i t = 260

Signal

250
200
150
100
50
0
0

500

1000

1500

2000

2500

3000

Detuning (MHz)Figure 4.8: Comparison of expected signal to data where the signal has been scaled by afactor of 1000.

4.3

Trap Characterization
Trap frequencies are determined by driving center-of-mass (COM) “slosh” oscillations.

The slosh mode is driven by temporarily displacing the center of the trap and measuring

49
the COM position of our cloud at various times after the initial displacement. We use two
imaging beams aligned along orthogonal axes so that the position of the cloud can be measured in all three dimensions simultaneously. Our machine is traditionally used for studies
with Bose-Einstein condensates, which have a couple advantages. The small condensate size
makes for simple and accurate fitting of the COM position of the cloud, and also allows fornon-destructive imaging with a high SNR. Using phase-contrast microscopy, we can run ourcamera in “kinetics mode,” where multiple images can be taken on a small region of the CCD
in quick succession. We are able to take eight images in two directions before running into
issues caused by the small frame size used on our CCD cameras. Acquiring multiple images
per experimental cycle significantly speeds up data taking and provides a useful calibration
between imaging axes. Sample images are shown in Fig. 4.9.
It is important to understand the artifacts that can occur when measuring the trap
frequencies. Generally, any slosh motion aligned with a principal axis will have a constant
amplitude and frequency. These oscillations are the eigenfunctions of the system, where thebasis is defined by the principal axes. Measuring the slosh along an axis that is not consistent
with the principal axes causes these eigenfunctions to couple as they are projected onto thenew basis vectors. In a 2D system, for slosh that is driven linearly along a direction 45◦
away from two principal axes, any difference in the trap frequencies will lead to a dephasingof the slosh between those two axes at a rate proportional to the difference between the trap
frequencies. This causes the motion to eventually become circular as it traces out a Lissajous
figure, which appears as a beating between the two frequencies if the motion is observed in
one dimension. In an extremely round trap, this dephasing is very slow and the motion
remains approximately linear. However, the motion will also remain linear if it is alignedwith one of the principal axes of the trap. To avoid any confusion, we drive an initiallycircular slosh motion by displacing the trap center along one axis and waiting a quarter trap
period before displacing along the second axis. Sample data for a trap that is slightly out ofround is shown in Fig. 4.10.

50

Horizontal
Horizontal
Axis
Axis
100 µm

VerticalVertical
Axis
Axis
100 µmTime

Figure 4.9: These are the raw data that are taken during each experimental cycle. Each setof eight images is taken simultaneously using phase-contrast microscopy from two orthogonal
directions. The difference in cloud size between the two sets of images arises from differencesin pixel size and magnification of the separate imaging systems. The images are taken in 15
ms intervals.

51

Slosh Measurement in the yz Plane
8000

Time (ms)

7000
6000
5000
4000
3000
2000
1000
0
−40

−20

0

20

40

y Position (px)

−20

20

0

z Position (px)

Slosh Measurement in the yz Plane

8000

Time (ms)

7000
6000
5000
4000
3000
2000
1000
0
20

0

−20

z Position (px)

0
−20
−40

20

40

y Position (px)Figure 4.10: Sample data and fits of the slosh mode in the yz plane. Both sets of data are
identical but viewed from a different angle. A beat frequency is visible in the first figure,while the amplitude appears constant in the figure on the bottom, indicating that the viewing
angle is colinear with a principal axis. The red lines are the fits, and viewed from the top,
trace out a Lissajous figure.

Our imaging beams are set up so that three of the axes of our images are approximatelyaligned with the three bias coils used for the rotating field and we simply define our lab framefrom these imaging axes. We find that the most robust method for making the trap sphericalis to detune the frequency along one of the principal axes and then round out the trap in

52
2D. Detuning one of the principal trap frequencies by 10% is sufficient to reduce coupling of
the slosh mode driven along the other two axes. From the fits with a rotation matrix, we areable to back out the orientation and characteristic frequency of the principal axes. In otherwords, our data along the two imaging axes, say dx and dy , is fit by minimizing the following

χ2 =

N
X
i=1





i
 dx 







i

2

 cos(θ) − sin(θ)   Ax sin(ωx t + φx ) 
−

 .
i
i
dy
sin(θ) cos(θ)
By sin(ωy t + φy )(4.11)The superscript i indicates an element in an array of data points, for which the x and y data
are determined from the same image taken at a time ti . Taking the frequencies ω1 and ω2 ,and the rotation angle, θ, the elliptical trap profile can be solved using
A = ω1 cos2 (θ) + ω2 sin2 (θ)(4.12)

B = (ω2 − ω1 ) cos(θ) sin(θ)(4.13)

C = ω1 sin2 (θ) + ω2 cos2 (θ)(4.14)

Ax2 + Bxy + Cy 2 .

(4.15)

where we define our ellipse as

The amplitude and phase parameters Ax , By , φx , and φy are only used to match the conditions of the drive and are not imperative in the final determination of the principal axes. We
calibrate our system by varying the fields corresponding to terms in table 3.1 and fitting theelliptical coefficients measured from slosh data. The fit was used as a way of calibrating theinduced ellipticity as a function of convenient experimental parameters, but the measured
variation in the elliptical coefficients agrees well with theory as seen in Fig. 3.3. After theellipticity in one plane was sufficiently shimmed out, we detuned the trap frequency along
another dimension and round out another plane. This process was iterated—rounding out
each plane in xy, xz, and yz individually—until the trap was spherical, which took roughly
2-3 cycles before the principal trap frequencies differed by less then 0.1%.

53
4.3.1

Direct Determination of Trap Frequencies and Orientation of Principal
Axes

The previous method can be extended to 3D for directly determining all principal
axes and their orientation without the need for detuning. When measuring slosh along adimension that is not aligned along any principal axes, more complicated beating between
the three trap frequencies is visible as shown in the simulated data in Fig. 4.11.
As with the 2D case, a least squares fit is performed using sums of sinusoidal functions.
Here we define
Fx (t) = Ax sin(ωi t + θx1 ) + Bx sin(ωj t + θx2 ) + Cx sin(ωk t + θx3 )Fy (t) = Ay sin(ωi t + θy1 ) + By sin(ωj t + θy2 ) + Cy sin(ωk t + θy3 )(4.16)Fz (t) = Az sin(ωi t + θz1 ) + Bz sin(ωj t + θz2 ) + Cz sin(ωk t + θz3 )The amplitudes A, B, and C correspond to oscillations along the principal axes with frequencies ωi , ωj , ωk . The subscript indicates the projection of the oscillation onto the imaging
axes x̂, ŷ, and ẑ, in other words Ax = A · x̂. Each function has a variable phase, θ, but thefrequencies ω1 , ω2 , and ω3 are constrained to be the same for all three functions. Data alongeach imaging axis, dx , dy , and dz is fit simultaneously using the method of least squares byminimizing the following function
χ2 =

N h
X
i=1

2

2

dix − Fx (ti ) + dy − Fy (ti ) + dz − Fz (ti )

2

i

.

(4.17)

Minimizing χ2 is enough to determine the three principal trap frequencies, but in order tofigure out the orientation of the principal axes relative to the imaging axes another step must
be taken. The projection of the principal axes onto the imaging axes can be encoded by thefollowing matrix î · x̂ ĵ · x̂ k̂ · x̂  î · ŷ ĵ · ŷ k̂ · ŷ  ,î · ẑ ĵ · ẑ k̂ · ẑ

54

15

5
4

10

x Position (arb.)

x’ Position (arb.)

3

5

0

−5

2
1
0
−1
−2
−3

−10
−4
−15

−5
0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

6000

7000

8000

9000

6000

7000

8000

9000

Time (ms)

20

20

15

15

10

10

y Position (arb.)

y’ Position (arb.)

Time (ms)

5

0

−5

5

0

−5

−10

−10

−15

−15

−20

−20
0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0

1000

2000

3000

Time (ms)

4000

5000

Time (ms)

20
8

15

z Position (arb.)z’ Position (arb.)

6

10

5

0

−5

4
2
0
−2
−4

−10
−6

−15
−8

−20
0

1000

2000

3000

4000

5000

Time (ms)

6000

7000

8000

9000

0

1000

2000

3000

4000

5000

Time (ms)Figure 4.11: Simulated data measured along an axis other than the principal axes, where
beating between the three principal frequencies can be seen. The left column shows data
along 3 axes that are arbitrarily rotated from the principal axes. The right column shows
the same data viewed along each principal axis. Data in the second column results fromapplying a rotation matrix that is determined from the methods discussed in this section.

55
which reduces to the identity matrix when the trap and imaging axes are aligned. Incorporating the results from the fit, the amplitudes can be written in the form
 Ax Bx Cx 
Mimaging =  Ay By Cy  .
Az Bz Cz

(4.18)

Here our basis is set by the imaging axes, if instead the basis is set by the trap axes thematrix becomes A 0 0 .
Mtrap = 
0
B
0




0 0 C

(4.19)The transform between bases is simply given by
T = Mtrap M−1
imaging .

(4.20)

Because we are not equipped with the actual values that comprise Mtrap , we need to diagonalize Mimaging . Numerically, the most stable solution of T is computed using a singularvalue decomposition (SVD). The SVD returns three matrices such thatUW V † = M(4.21)

where U and V † are unitary matrices and W is a diagonal matrix that is positive andreal-valued. The diagonal elements in W are known as the “singular values,” and can beinterpreted as eigenvalues. This is one of the main benefits of using an SVD instead of justfinding the eigenvectors of the system; the SVD diagonalizes a matrix and also returns thetransformation that rotates into the original basis. The SVD has various uses, but is often
described geometrically as a set of matrix transformations that produce an ellipsoid. In
this interpretation, V † is used to rotate a unit sphere into a new basis, W scales the spherealong the axes of the new basis to form an ellipsoid, and U rotates into the final basis. So,
if we were to find the singular value decomposition of Mimaging , then we would find that

56
W = Mtrap with the exception that the values in W are sorted in decreasing order along
the diagonal. But the singular values in W just correspond to the amplitude of the slosh
oscillations, which are set by the drive amplitude and are irrelevant to the problem at hand.
By replacing W with an identity matrix, we can back out the rotation between the imaging
and trap frames with
R = U V †.

(4.22)

R is also unitary, meaning that the matrix that rotates Mimaging into Mtrap is simply R† inwhich caseR† Mimaging = Mtrap .

(4.23)

Although the values in the M matrices are defined by the amplitude of oscillations, they
are only a way of determining the projection of the principal axes onto our imaging axes.
Once the rotation matrix has been determined, the principal axes should be instead definedin terms of the trap frequencies determined by minimizing Eq. (4.17).
The benefit of this approach lies in the experimental simplicity because individual trap
axes don’t need to be changed and the process can ideally be used to reduce trap anisotropy
with a single set of data. In a near-spherical trap, the method is less stable when there is any
noise in the data. However, when the trap frequencies differ by more than 1%, the method
can be used to quickly and reliably map out the principal ellipsoid.
4.4

Driving Collective Excitations
To study the monopole and quadrupole modes, it is imperative that we can selectively

excite them with minimal coupling into other modes. A particular mode is often parametrically driven by modulating the strength of the trap at the resonant frequency of themode. However, in the collisionless limit, the monopole and quadrupole mode frequencies
are degenerate and careful consideration must be taken in order to drive only one mode.
Anharmonicities in the trap pose other issues, especially if the cloud has any vertical slosh.

57
The condition for a spherical trap depends strongly on the sag position of the atoms, andthe ratio of radial to axial trap frequencies changes based on the position of the atoms along
the ẑ axis. This means that vertical slosh will couple to the quadrupole mode because of
the anisotropic modulation of the trap strength. The TOP bias field, quadrupole field, andshim fields are the three tools we have for driving excitations, and adjustment of each field
will couple to all three modes.
4.4.1

Dipole Mode

Driving the dipole slosh mode for characterizing trap frequencies is very straightforward. Because we care about only the COM position of the cloud, we work with small condensates for maximum signal-to-noise, so that any inadvertant excitation of the monopoleor quadrupole modes doesn’t affect the measurement. Instead of parametrically squeezing
the trap, we excite the dipole mode by abruptly displacing the center of the trap using fields
generated by large shim coils. Our iterative process of optimizing trap sphericity requires aninitially circular slosh motion in either the xy, xz, or yz planes. Circular motion is excitedby first shifting the trap along one axis. After a quarter period of the trap frequency the
first shim field is returned to its initial value and a second shim field is applied along an
orthogonal axis. The shim coils are roughly aligned along the same axes as the TOP coils so
the shim fields can be adjusted more or less independently, but the fact that they are large
and bulky makes them difficult to align and not perfectly orthogonal. Final values for the
shim fields are roughly optimized by using the resulting slosh measurements immediately
after the drive.
4.4.2

Monopole Mode

Excitation of the monopole and quadrupole modes require a less abrupt drive. The
monopole mode is driven by sinusoidally modulating the strength of the trap symmetricallyat twice the trap frequency. Changing the strength of the trap symmetrically can be done

58
using only the TOP bias field. Symmetry is preserved only around the minimum of thetrap, which unfortunately depends on both the TOP bias and quadrupole fields as found in
Eq. (3.8). Modulation of the TOP bias field strength alone will not only drive a quadrupolemode due to the asymmetry of the trap at the position of the atoms, the atoms will also
undergo vertical slosh. Sloshing can be minimized by using a very long drive, but this doesn’t
prevent the coupling of the drive to the quadrupole mode. The best solution is to shift thetrap minimum using a vertical bias field so that it remains stationary and aligned with thecloud. This will remove any residual slosh and preserve the symmetry of the drive.
The modulation of the TOP field can be described with a small time-dependent perturbation ξ(t):B0 (t) = (1 + ξ(t))B0 .
For a spherical trap, the sag position of the atoms is
r
B0
7
ẑ,
z0 = − sph
2
Bz(4.24)(4.25)

where Bzsph is the quadrupole field gradient required for a spherical trap, and is defined byrG
9 mg
Bzsph ≡
≈ 34.61 .
(4.26)
7 µcm
where mg is the gravitational force and µ is the magnetic dipole moment of the atoms. z0
is directly proportional to B0 , so the time-dependent shift in the trap minimum is ξ(t)z0 .
The vertical bias field needed to shim out the vertical slosh is given by the shift in the trap
minimum multiplied by the quadrupole field gradient:
Bv (t) = ξ(t)Bzsph z0r
7=
ξ(t)B0 ẑ
2(4.27)In the experiment, ξ(t) is a sine wave driven for 4 periods at a frequency of 18 Hz with
an amplitude of 12%. The extrema of the drive roughly correspond to trap frequencies of
8.55 Hz and 9.64 Hz for a stable trap frequency of 9.05 Hz. Ideally, shimming out the

59slosh with the vertical bias field should be sufficient to drive a pure monopole mode. In
practice, however, a small amount of energy is put into the quadrupole mode. We reduce
this coupling by modulating the quadrupole field in phase with the TOP field by roughly
5%. This value is optimized by alternately tweaking the strength of the quadrupole andvertical bias fields and minimizing the quadrupole mode and slosh amplitudes respectively.
Assuming the modulation is small, the extra TOP fields used for shimming out ellipticity
don’t need to be changed and the resulting monopole amplitude is about 25% of the mean
cloud size.
4.4.3

Quadrupole Mode

When exciting the quadrupole mode, it is important that the geometric mean of thetrap frequencies is constant. Calculating the mean frequency change analytically requires
another Taylor expansion of the potential. The expansion about the trap minimum summarized in table 3.2 won’t give the correct results, because these terms automatically accountfor the new sag position of the atoms. Because the excitation frequency is faster than thetrap frequency, the atoms aren’t able to follow the trap minimum, and the correct form of thepotential needs to be expanded around a stationary position. The instantaneous change in
the average trap frequency can be calculated numerically. For a typical quadrupole drive, we
dither the quadrupole field with a maximal relative shift of α = 4%. To maintain a constantaverage frequency, the TOP field must be modulated with a relative shift that is 4.424 times
larger than that of the quadrupole field. Fig. 4.12 shows the change in the radial, axial, andmean trap frequencies as a function of the change in field strengths where the quadrupole
and TOP bias fields shift by
δBq = ∆αBq

(4.28)

δB0 = −4.424∆αB0(4.29)

and60
respectively. The parameter ∆ varies from -1 to 1 during the drive, where the fields are
modulated sinusoidally at a frequency of 18 Hz for 4 periods such that ∆ = sin(2ωtrap t).
Simultaneous modulation of the quadrupole and TOP bias fields is necessary for both themonopole and quadrupole fields in practice. While the relative drive strength of each field isdifferent in each case, the main difference is the overall sign of the modulation of the TOP
bias field as shown in Fig. 4.13.
4.5

Detection

4.5.1

Fitting

To extract the quadrupole and monopole oscillation amplitudes, we measure the widthsof the clouds using gaussian surface fits. Our fitting functions are defined by(x − xc )2 (y − yc )2
A exp −
−
+C
2σx2
2σy2

(4.30)

where xc and yc are the center coordinates and σx and σy are the cloud widths along each
imaging axis. There is a small angle between our side-view image axes and our trap axes.
This angle is most obvious when the quadrupole distortion is maximized, but when the cloudis nearly spherical the angle is not well defined. Using the rotation angle as a fit parameter,
we were able to determine the actual rotation from quadrupole data with maximal distortion,
and we fix the rotation to the resulting 5◦ tilt for our normal fitting routine. The fringes in
our images don’t cause any differences in the fits, and can be seen in the residual images inFig. 4.14.
Phase-contrast imaging not only has the benefit of increasing our data acquisition rate,
but also allows us to measure the relative monopole and quadrupole distortion in a single
experimental cycle. Small variations in the initial conditions, such as temperature or atom
number, would normally add noise to our measurement of the monopole mode. We use atime step of 17 ms between images to capture roughly 1.5 oscillations for both the monopoleand quadrupole modes. Using a fixed frequency sine wave to fit the distortion using the six

61

RadialAxial
Mean

Trap Frequency (Hz)

10
9.5
9
8.5
8
-1

-0.5

0

0.5

1

∆
Figure 4.12: Radial, axial, and mean trap frequencies over the full range of field modulations
used to drive the quadrupole mode. The maximal change in the field strength is 4% for
the quadrupole field and 17.7% for the TOP bias field, which correspond to ∆ = 1. In theexperiment, we also use the vertical bias field to minimize vertical slosh and end up changing
the TOP bias field by only 13%.

62

0.2

B0
Bz!

Monopole Drive

0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
0.2
Quadrupole Drive

0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
−100 −50

0

50

100

150

200

250

300

Time (ms)Figure 4.13: Quadrupole and TOP field modulation for driving monopole and quadrupolemodes. Amplitudes are relative shifts in the overall quadrupole and TOP fields.

63
frames from a phase-constrast image, we can calculate the relative instantaneous amplitudeof an oscillation by looking at the ratio of the amplitude to the mean. Amplitudes of themonopole and quadrupole distortion are scaled by the average width of the cloud during one
cycle, given by the following relations
σi2 + σj2 + σk2
AM = 2−1
hσi + σj2 + σk2 i
AQ =2σk2 − σi2 − σj2.
hσi2 + σj2 + σk2 i(4.31)(4.32)

Oscillation amplitudes are determined by fitting a cycle of oscillation in cloud width, obtained
from each experimental run, with a fixed frequency sine wave as indicated by the solid lines
in figure 5.1.

64

Data

Residuals

Fit

50 µm

0

50

100

150

200

0

50

100

150

200

−50

0

50

Figure 4.14: An example of raw quadrupole mode data and the associated surface fits taken
during one cycle of the experiment.

65

Quadrupole Distortion (µm 2 )

50
40
30
20
10
0
−10
−20
−30
−40
−50

0

200

400

600

800

1000

1200

1400

1600

1200

1400

1600

Time (ms)Relative Quadrupole
Oscillation Amplitude

1

.8

.6

.4

.2

0

0

200

400

600

800

1000

Time (ms)Figure 4.15: Sample data for a driven quadrupole mode in a spherical trap with residual
asphericity less than 0.002 and a collision rate of 7.4(3) s−1 . Solid lines on the quadrupole
data indicate a typical fitting procedure where individual periods taken in a single run are fit
with an undamped sine wave to extract an instantaneous amplitude. The oscillation amplitudes at various cycles are in turn fit to an exponential decay to extract the damping rates.
Random observable scatter in these points is predominantly due to small, irreproducible
fluctuations in initial conditions.

Chapter 5
Observation of Suppressed Damping of Breathe Mode Oscillations in a
Thermal Cloud

Boltzmann’s prediction of the absence of damping of the monopole mode in an isotropic
harmonic potential was made nearly 140 years ago. His result depended on the laws of mechanics and motivated his rigorous development of the statistical aspects of his theories. For
over a century, experimental studies of gases had essentially no control over the confining
potential, but were restricted to hard wall vessels. Even if experimentalists could create
harmonic confinement, the depth of the potential would have to be enormous to account for
temperatures of hundreds of Kelvin. Experimental study of the monopole mode in a harmonic potential has only been feasible with the advent of ultracold quantum gas techniquesin the 1980s, but generating isotropic harmonic confinement hasn’t been possible until now.
Although Boltzmann’s prediction is a fully classical result, treating the atoms as hard
spheres, an ultracold monatomic gas provides an ideal system to test his theory. As weapproach the Bose-Einstein condensate transition temperature, Tc , mean-field effects can
add an undesirable shift to the potential. We work at temperatures between 2Tc < T < 3Tc
to avoid mean-field effects and also to ensure that the gas is sufficiently dilute in order that
we are consistent with Boltzmann’s original formulation of the transport equation. Ourinteratomic collision rate ranges from 2 s−1 to 9 s−1 , which means that an atom in our2π(9 Hz) trap will typically undergo several oscillations before colliding with another atom.
These short-range s-wave collisions also approximate the hard-sphere collisions used in the67
development of the Boltzmann equation.
Of course, we expect some damping as a result of our system not being totally ideal.
Quadrupole damping rates serve as a baseline for comparison with any damping observedin the monopole mode. Damping of the quadrupole mode is well understood and, in thecollisionless limit, the quadrupole damping rate scales linearly with interatomic collision rateas1
ΓQ ' γcoll .
5

(5.1)

We vary the initial atom number and final evaporation frequency to adjust the collision rate
in our gas, then selectively drive monopole or quadrupole motion. The cloud is then allowed
to evolve freely in the spherical trap before imaging the cloud. Six images are taken of thecloud along two orthogonal axes with an interval of 17 ms in order to sample roughly 1.5
oscillation periods.
The residual asphericity of the trap drifts with time, and we periodically retune thetrap to keep the asphericity small, typically less than 0.002. Each day, we characterize
the trap by taking dipole slosh measurements at the beginning, middle, and end of taking
damping data. We also alternate between the monopole and quadrupole driven data after
every few shots to reduce any systematic errors. Monopole and quadrupole data are takenover a range of evolution times, and three sets of data are taken at each evolution timethroughout the day. Instantaneous oscillation amplitudes are measured from a single image,and the resulting amplitudes are averaged over the three different sets of data.
Suppressed damping of the monopole mode can be seen in the sample data in Fig. 5.1,
and although the damping rate is small, it is nonzero. Fig. 5.1 shows only a small subset
of our typical monopole data to show the contrast between the amount of damping between
the two modes, and actual damping rates are acquired for up to 30 seconds of free evolution
in the potential. A direct comparison of quadrupole and monopole damping rates in a nearspherical trap is shown in Fig. 5.2. The measured dependence of quadrupole on collision

68

Monopole Distortion

0.3
0.2
0.1
0
-0.1
-0.2
-0.3
Quadrupole Distortion

1
0.5
0
-0.5
-1
0

200

400

600

800 1000 1200 1400 1600
Time (ms)Figure 5.1: Sample data for a driven quadrupole mode and monopole mode in a spherical trapwith residual asphericity less than 0.002. The normalized distortions plotted are calculatedusing Eqs. (4.31) and (4.32). Dashed lines indicate an exponentially damped sine-wave
fit. Solid lines on the quadrupole data indicate a typical fitting procedure where individual
periods taken in a single run are fit with an undamped sine wave to extract an instantaneous
amplitude. The amplitudes at various times are in turn fit to an exponential decay to extract
the damping rates shown in Fig. 5.2

rate is ΓQ = (4.9(1))−1 γcoll , which is in good agreement with Eq. (5.1). There is a small
amount of damping in the monopole mode with an average rate of ΓM = 0.14(3) s−1 , but
the damping rate is independent of collision rate as expected.
In order to understand the source of monopole damping, we note that Boltzmann’s

69

Damping Rate (s−1 )2Monopole
Quadrupole

1.5

1

0.5

0
0

2

4

6

8

10

Collision Rate (s−1 )Figure 5.2: Monopole and quadrupole damping rates as a function of interatomic collision
rate in a near-spherical trap. Residual asphericity was typically around 0.001, but always
less than 0.0026. Cloud full width at half maximum was approximately 115 µm.

result hinges on the assumption that the potential is both isotropic and harmonic. Naturally,a real system will never satisfy both of these conditions perfectly, and the remainder of thischapter is a devoted to a discussion of the effects that small anisotropies and anharmonicities
have on the monopole damping rate.
5.1

Effects of Anisotropy
Certain subtleties arise for gases in the collisionless limit when anisotropies in thepotential are small enough that trap frequencies differ by less than a few percent. In a totally
collisionless system, oscillations along the principal axes of the trap are fully decoupled andmonopole- or quadrupole-like oscillations are undamped. If the principal trap frequencies
differ such that ωi = ωj 6= ωk , dephasing occurs between oscillations along different principal

70
axes and oscillations between pure monopole motion and pure quadrupole motion occurs
with a period given by
TM Q =π.
|ωk − ωi |(5.2)

When collisions are included, the two modes become coupled and, as the population in thequadrupole mode increases, so does the damping. This effect can be seen when TM Q < 1/ΓQ ,where multiple oscillations between monopole and quadrupole modes occur. Data in Fig. 5.3

Monopole Oscillation
Amplitude

0.5
0.4
0.3
0.2
0.1

Quadrupole Oscillation
Amplitude

0
0.8
0.6
0.4
0.2
0
0

1

2

3

4

5

Time (s)Figure 5.3: Monopole and quadrupole motion in an anisotropic trap where the residual
asphericity is approximately 0.02. A monopole mode is initially driven and oscillations
between the monopole and quadrupole modes can be seen. The fitting functions are givenby e−γt (cos2 (∆ω + πδQ ) + C), but should be used just as a visual guide.

71
show oscillations between monopole and quadrupole modes resulting from an initial monopole
drive in a trap with residual asphericity of approximately 0.02. Oscillations of the individual
modes are π out of phase and the damping rates for both modes are nearly equal, with amean value of Γ = 0.36(4) s−1 . The collision rate is roughly 3.7 s−1 leading to an expected
quadrupole damping rate of 0.74 s−1 in a spherical trap, which is twice the value of themeasured damping rate in the anisotropic trap. This is no surprise because the quadrupolemode is effectively populated only half of the time, leading to the factor of 2 decrease in
the damping rate. If we decrease the amount of anisotropy such that TM Q >> 1/ΓQ , thequadrupole mode damps before it can fully couple back into the monopole mode. One cansee this effect in Fig. 5.1, where the quadrupole mode damps before it can couple with themonopole mode. Fig. 5.4 shows a summary of driven quadrupole and monopole modes intraps with varying degrees of anisotropy. As we approach the spherical limit, mode coupling
is no longer visible and both modes damp exponentially.
The data in Fig. 5.2 were taken in traps with residual asphericities ranging from 0.0005–
0.0026, corresponding to 10.7 s< TM Q /2 < 55.6 s. The typical relaxation time for themonopole mode in these traps is τM = 7(2) s. While one could argue that this relaxation
time agrees with the most anisotropic case, where the monopole mode had enough time tocouple with the quadrupole mode, it cannot account for the data taken in more isotropic
potentials, for which the damping is nearly the same. Thus, there must be another sourceof damping. It should also be noted that the results shown in Fig. 2.3 predict a dampingthat is two orders of magnitude smaller than what we observe. There also appears to be nocorrelation between the damping rate and residual anisotropy for our monopole data in anear-spherical trap. Fig. 5.5 shows the same monopole data in Fig. 5.2, but plotted against
anisotropy. Thus, some other physical effect must provide the dominant source of residual
monopole damping.

0.4

0.8

0.3

0.6

0.2

0.4

0.1

0

0
0.5

1

0.4

Quadrupole Amplitude

0.2

1.2

0.2

0.1

0

0

1.2

0.5

1

0.4

0.8

0.3

0.6

0.2

0.4

0.8

0.3

0.6

0.2

0.4
0.2

0.1

0

0
0

1

2

3

Time (s)

4

5 0

1

2

3

4

Monopole Amplitude

1

Monopole Amplitude

0.5

Monopole Amplitude

1.2

Quadrupole Amplitude

Quadrupole Amplitude

72

5

Time (s)Figure 5.4: Monopole and quadrupole motion in traps with anisotropies of 0.0222 (top),
0.0133 (middle), and 0.0028 (bottom). Data on the left show driven quadrupole motion,
while data on the right show driven monopole motion. As the level on anisotropy decreases,
the mode coupling turns off and the decay of each mode becomes exponential.

73

Monopole Damping Rate (s−1 )

0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0

0.0005 0.001 0.0015 0.002 0.0025 0.003
Residual Asphericity

Figure 5.5: Monopole damping rates in very isotropic traps as a function of residual asphericity. The data shown are the monopole damping data from Fig. 5.2.

74
5.2

Effects of Anharmonicity
We now come to the second condition of Boltzmann’s result, which requires that thepotential be harmonic, and discuss the effect of anharmonic perturbations to our trapping
potential as a source of damping. Amplitude-dependent frequency shifts caused by anharmonic perturbations lead to dephasing of individual particle trajectories, effectively dampingof the collective monopole amplitude. Moreover, the anharmonic corrections to our potential are asymmetric, giving rise to an amplitude-dependent anisotropy. A calculation of the
expected damping rate that takes into account all of the relevant anharmonic corrections isdifficult. But the effect can be explored experimentally by measuring the monopole damping
as a function of cloud size. Fig. 5.6 again shows the same monopole damping data from
Fig. 5.2, but plotted against the average cloud FWHM. In this case, there does appear to
be a slight increase in the damping rate as a function of cloud size. However, it is difficult
to draw any conclusions from these data alone.
Just as we deliberately broke the symmetry of the trap to investigate the effects ofanisotropy, we can look at monopole damping rates for extremely large clouds. Fig. 5.7
shows data for cloud widths that are up to twice the FWHM of our typical samples. The
first point in Fig. 5.7 is calculated from the average cloud size and damping of the monopole
data in Fig. 5.2. The trend of the data in Fig. 5.7 suggests that the damping seen at a cloudsize of 115µm may be due to the onset of anharmonic effects. Unfortunately, we are unable
to work with smaller clouds due to limitations in the signal-to-noise ratio of our imaging
system.
While the damping is highly suppressed, the small but finite relaxation of the monopolemode is an artifact of small anharmonic perturbations to our trap, which decrease with cloudsize. We find that, in the limit of zero anharmonic shifts, the damping of the monopole mode
vanishes, as predicted by Boltzmann in 1876 [7].

75

Monopole Damping Rate (s−1 )

0.22
0.2
0.18
0.16
0.14
0.12
0.1
0.08
105

110

115

120

125

Cloud FWHM (µm)
Figure 5.6: Our same monopole data from Fig. 5.2 are revisited, but this time plotted against
the average FWHM of the clouds. The FWHM is calculated from the average of all of thebreathe data after the clouds have been driven in order to account for the small amount ofheating induced by the drive.

76

Damping Rate (s−1 )

1.2
1
0.8
0.6
0.4
0.2
0
0

50

100

150

200

250

Cloud FWHM (µm)
Figure 5.7: Damping of the monopole mode in a near-spherical trap as a function of thespatial extent of the atom cloud. The fractional amplitude of the excitation is the same forall points. Consistent with anharmonicity-induced damping, we see the observed damping
rate decreases rapidly for smaller clouds, but the trend of the data suggests that even forthe smallest clouds (such as those used in Fig. 5.2) the residual damping observed may be
due to the onset of anharmonic effects.

Chapter 6
Monopole Mode Measurements in a Finite-Temperature BEC

One great feature of an isotropic harmonic potential is that complex and computationally demanding theories can be greatly simplified through symmetry considerations. An
unresolved problem in the world of BEC physics is an accurate, yet accessible, theory of theinteractions between the normal and condensed components of a finite-temperature condensate. There currently exist numerous, competing finite-temperature models [49], which inmany cases are difficult to adapt to the irregularities that can often arise in experimental systems. Collective modes in a finite-temperature condensate are complicated not only by thefact that mean-field interactions modify the potential, but the effective potential is different
for the condensate and the thermal atoms. A profile of a finite-temperature condensate isshown in Fig. 6.1 where the thermal atoms (red) no longer have a gaussian profile, but arepushed away from the center because of the presence of the condensate. Monopole oscilla√tions for a condensate have a modified frequency that approaches 5ω in the Thomas-Fermi
limit due to mean-field effects. In a finite-temperature cloud, however, the thermal atoms
will still breathe at a frequency of about 2ω with a slight variation caused by the mean-field.
As the two components dephase, they start beating together leading to damping and revivalsof the mode [34].
Experimentally, distinguishing a thermal atom and a condensate atom is impossible,and correctly fitting surface data is a very difficult task. However, before decommissioning
the experiment, we decided to take some finite-temperature condensate monopole data and78

Figure 6.1: Profile of a simulated finite-temperature condensate in a harmonic trap. The
condensate atoms are shown in blue and the thermal atoms in red. Mean-field interactions
modify the potential, pushing the thermal atoms away from the condensate. This particularsimulation uses the ZNG model for a condensate at a temperature T = 0.75Tc [1].

provide our results here for anyone who wants to use them. Each set of data is taken using
the same imaging and drive parameters specified in chapter 4. We use a simplified surface
fit defined by the sum of a gaussian and an integrated Thomas-Fermi profile, defined by"3 #
(r − rc )2 (z − zc )2 2
−
,0
(6.1)
f = max ABEC 1 −
Sr2
Sz2


(r − rc )2 (z − zc )2
−
+ C.
(6.2)
+ Ath exp −
2σr2
2σz2
Despite having the wrong functional form, this function fits to our data surprisingly
well. Sample surface fits are shown in Fig. 6.2 and the residuals are small and the backgroundis relatively flat, both for undriven data and monopole data. For comparison, Fig. 6.3 shows
the first frame after both the data and fit have been azimuthally averaged about the center

79

Data

Fit

Residuals

File: 47
Figure 6.2: Surface fits to raw data using Eq. (6.2). No drive was used for this particularset, but the fits work equally well for driven data. The surface fits are shown in the center
column and the residuals are shown in the righthand column.

80
position of the cloud. The variables ABEC and Ath are the fit amplitudes for the Thomas350

300

Aver age Signal

250

200

150

100

50

0

−50
0

10

20

30

40

50

60

70

80

90

100

Radial Dis tance (µm)

Figure 6.3: Azimuthally averaged data (blue solid line) and surface fit (red dashed line)
around the center position of the condensate.

Fermi and gaussian surfaces respectively. Widths of the condensate, Sr and Sz , and thermal
cloud, σr and σz , are given in microns. The radial widths with the r subscript are measured
along an axis that is 45◦ relative to our x and y TOP coils. Images were taken along twoaxes for our thermal breathe experiments, but due to an issue with our top-view laser that
became more problematic when these sets were taken, it was not used for most of these data.
However, images that were taken from the top showed that the oscillation was extremely
symmetric in the radial plane. The final variable in Eq. (6.2) is C, and is simply added tocorrect for any offsets in the background signal on the images.
It is clear that we see damping and revivals of the monopole mode. The effect isless marked for the thermal component, and damping is most clearly seen in the condensate
oscillations. Deviations from a simple beat are in part caused by our simplified surface fitting
function. At different phases of the oscillation, our Thomas-Fermi function is presumably
fitting to some thermal atoms as well. This is obscured by the fact that our images are81
integrated along one dimension. Attempts at extracting reasonably quiet data from aninverse Abel transform of our images proved difficult, and to reduce the noise in our data
the idea was abandoned. The monopole oscillation data (for the BEC component only, givenby the average TF radius hSi) are shown on the next several pages, and compared with thefollowing fitting function
e−γt cos2 (2πf t + δ) + c



(6.3)

which was also used for our anisotropic thermal data. Individual periods are fit to sine wave
oscillations with the frequency fixed at 20 Hz, compared to a mean trap frequency of 9.05(3)Hz with a residual anisotropy less than 0.003. For each set, we list the BEC fraction N0 /N ,where total atom numbers are on the order of 105 . Actual values for the data plotted are
given in appendix A.
These data is provided “as is” and we make no claim about what sort of behavior we
expect to see or whether or not the data is in agreement with any particular theory. However,
we hope that these data will be of some use to theorists who are looking for a simplified
system that can be used to test different finite-temperature theories.

N 0 /N = 0.098(2)10f = 0.48(4) Hz
γ = 0.8(2) s − 1

98hS i (µm)

7
6
5
4
3
2
1
200

400

600

800

1000

1200

Time (ms )82

Figure 6.4: Condensate damping for a BEC fraction of N0 /N = 0.098(2), N = 9.1(1)×105

N 0 /N = 0.177(2)9f = 0.60(2) Hz
γ = 0.7(1) s − 1

87hS i (µm)

6

5

4

3

2

1
100

200

300

400

500

600

700

800

900

1000

Time (ms )83

Figure 6.5: Condensate damping for a BEC fraction of N0 /N = 0.177(2), N = 7.9(1)×105

N 0 /N = 0.31(2)6f = 0.33(4) Hz
γ = 0.6(1) s − 1

5hS i (µm)

4

3

2

1

0
200

400

600

800

1000

1200

1400

1600

1800

2000

Time (ms )84

Figure 6.6: Condensate damping for a BEC fraction of N0 /N = 0.31(2), N = 5.4(2)×105

N 0 /N = 0.322(3)9f = 0.64(4) Hz
γ = 1.4(2) s − 1

87hS i (µm)

6
5
4
3
2
1
0
100

200

300

400

500

600

700

800

900

1000

1100

Time (ms )85

Figure 6.7: Condensate damping for a BEC fraction of N0 /N = 0.322(3), N = 6.7(1)×105

N 0 /N = 0.446(2)12f = -0.75(4) Hz
γ = 1.6(2) s − 1
10hS i (µm)

8

6

4

2

0
100

200

300

400

500

600

700

800

Time (ms )86

Figure 6.8: Condensate damping for a BEC fraction of N0 /N = 0.446(2), N = 9.7(1)×105

N 0 /N = 0.579(3)12

f = 0.79(5) Hz
γ = 1.3(2) s − 1

10hS i (µm)

8

6

4

2

0
0

200

400

600

800

1000

1200

1400

1600

1800

Time (ms )87

Figure 6.9: Condensate damping for a BEC fraction of N0 /N = 0.579(3), N = 8.9(1)×105

N 0 /N = 0.58(4)9f = 0.64(4) Hz
γ = 1.3(1) s − 1

87hS i (µm)

6
5
4
3
2
1
0
0

100

200

300

400

500

600

700

800

900

1000

Time (ms )88

Figure 6.10: Condensate damping for a BEC fraction of N0 /N = 0.58(4), N = 8.7(2)×105

Chapter 7
Anharmonic Shearing

Prior to our spherical trap experiments, the apparatus was configured to generatequasi-2D BECs. These quasi-2D condensates are formed using an optical lattice where a 3D
condensate is separated into multiple 2D layers. One of the biggest obstacles with the quasi2D experiments is rooted in our detection procedure. Due to the geometry of our setup,
our imaging laser passes through multiple quasi-2D condensates. To image a single layer,
we use a special microwave pumping process that allows atoms in a single lattice site to be
selectively imaged. This detection technique suffers mainly from two limitations: inefficientstate transfer and instability between experimental cycles. Limited transfer efficiencies ofroughly 10% were suitable, if not beneficial, for some of our previous experiments [60]. But
our microwave pumping method requires that the magnetic field strength at the center of thecondensate is consistent between experimental runs. Variations in the field strengths, position of the condensate, or even the phase of the lattice often lead to unusable or inconsistent
data that pose more significant issues for some of our subsequent experiments.
Accounting for these problems with our microwave pumping proved to be an extremely
difficult task. However, we came up with a solution that circumvented these issues entirely
by using the anharmonic shifts in our trap to our advantage. By turning up the strengthof the anharmonic terms, we devised a method of spatially separating, or “anharmonicallyshearing,” quasi-2D condensates on different lattice sites. It is perhaps ironic that we wereable to address one problem in our system by enhancing another problem, but the result is

90
a tool that is potentially useful for a variety of experiments. This chapter will first discusscertain key aspects of our quasi-2D setup1 and some of the detection issues in greater detail
as well as describe the anharmonic shearing process.
7.1

Quasi-2D Condensate Formation
Magnetic traps are useful for creating smooth, stable confining potentials but are in-capable of providing the tight confinement necessary for a 2D geometry. On the other hand,
optical dipole traps are immensely useful for generating complicated and exotic trap geometries for ultracold neutral atoms [25]. Atoms in an optical dipole trap interact with
far-detuned light via an induced electric dipole moment. This interaction is attractive for
red-detuned light and repulsive for blue-detuned light. A focused, red-detuned laser is ideal
for providing quasi-1D confinement, also known as a “cigar trap.” Quasi-2D confinement,
or a “pancake trap,” can be achieved using a variety of methods. One technique uses aholographic plate to generate a blue-detuned beam with a TEM01 profile [56, 52]. Anothermethod uses an evanescent wave to create a gravito-optical surface trap [54]. In our experiment, quasi-2D confinement is provided by a 1D optical lattice in which a periodic potential
is produced by interfering two laser beams [6]. Superimposing the interference pattern onto a
3D condensate causes it to separate into a series of quasi-2D condensates occupying adjacentlattice sites.
Lattice potentials have a variety of applications, such as understanding condensed
matter phenomena [22, 35, 24] or quantum computation [10]. These applications take advantage of the fact that multiple lattice sites are occupied, but for studying a single quasi-2D
condensate it is necessary for the depth of the lattice potential to be large enough that intersite interactions are turned off. In practice, 1D optical lattices are often generated using
a retroreflected laser beam in which case the spacing between lattice sites (lattice constant)is λ/2, where λ is the wavelength of the laser. Tunneling is typically an issue if the lattice1

For a complete treatment of the quasi-2D experimental setup see [61]91
constant is small and, in order to study a single quasi-2D condensate, tunneling must be
minimized as much as possible. The lattice constant, d, can be changed by varying the angle
between two interfering laser beams, θ, whered=λ.
2 sin 2θ

(7.1)

To maintain the sensitive control of the radial confinement we get from the TOP
fields, we use a vertically oriented blue-detuned lattice as depicted in Fig. 7.1. Our lattice8

3.8 µmFigure 7.1: Diagram of our lattice beam set up as viewed from the side. The quasi-2D
condensates are shown in red and confined to the nodes of the interference pattern. Roughly
10 lattice sites are occupied on average.

beams are 532 nm and interfered at an angle of 8◦ , which gives a lattice constant of 3.8µm. It is imperative that we have sensitive control over the lattice beam power in order toadiabatically load the atoms into the lattice. We are afforded precise control of the latticeintensity by adjusting the power transmitted through the first order of an acousto-optic
modulator (AOM). This beam is then coupled into a high-power single-mode polarizationmaintaining fiber fitted with coreless end caps and cooling fins so that we can inject up to92
2W of power. On the output end of the fiber, we monitor the power using a photodiode,which provides a feedback signal for a power servo connected to the AOM. A diagram of theexperimental setup is shown in Fig. 7.2.

2

PD
PBS

8

3.8 µmShutter

2

2
AOM
2

Verdi
OutputPower Servo
Box

Figure 7.2: Diagram of our lattice beam set up as viewed from the side. The quasi-2D condensates are shown in red and confinedto the nodes of the interference pattern. Typically about 10 lattice sites are occupied
93

94
Temperature drifts cause the fiber alignment to change, so we optimize the alignment
for maximum input power of 2W, but typically work at powers closer to 1.6-1.8W whenrunning the experiment. Optimizing the fiber coupling at a higher power ensures that the
dynamic range of the the lattice beam intensity is not limited by small fluctuations in the
coupling efficiency. At the output of the fiber, a small amount of power is diverted toward
the photodetector that provides an error signal for the feedback on the input power. The
remainder of the output is split into two beams with equal power using a polarizing beam
splitter (PBS). Each beam has roughly 400 mW of power and are each focused at the centerof the trap with horizontal and vertical beam waists of 225 µm and 84 µm respectively.
For most of the experimental sequence, the lattice beam is blocked by a shutter while a
3D condensate is created. Even when the system is idle, we inject most of the available power
into the lattice fiber to maintain a stable temperature and alignment. Once the evaporation
sequence finishes, the BEC is adiabatically loaded into the lattice by ramping the set pointfrom 0 to 400 mW over the course of 500 ms. After loading the condensate into the lattice,evaporation is no longer possible, partly because our evaporative profile removes atoms fromthe bottom of the trap but also because of the slow rethermalization rate of 2D clouds.
Alignment of the lattice is also simplified since the evaporation is an entirely independent
process.
Rough alignment is performed by blocking one of the lattice beams and measuring
the condensate displacement once the power is ramped on. The single beam alignment is
optimized when the condensate appears to have expanded symmetrically and is not pushed inany particular direction. This procedure is then repeated with the second beam. The second
step is to maximize the contrast of the interference at the center of the atoms. Measuring thecontrast directly isn’t possible because the interference is centered in our vacuum system.
Instead, we maximize the trap frequency along the direction of tight confinement, which is
proportional to the contrast of the interference pattern. The axial frequency in the lattice
is roughly 1.4 kHz, and cannot be determined from conventional slosh mode measurements.

95
However, modulating the strength of the lattice potential at twice the trap frequency willparametrically heat the atoms and provides a useful tool for determining the trap frequency.
After heating is observed, we measure the width of the cloud and repeat the measurement at a slightly different modulation frequency. The final cloud widths are compared overa range of modulation frequencies and fit with a gaussian to determine the trap frequency.
Small tweaks in the alignment, focus, and power balance of the lattice beams are made,repeating the modulation procedure each time, until the trap frequency is maximized. This
process can be very time consuming for the initial alignment, but after the maximum trap
frequency has been determined it is straightforward to correct for small misalignments.
7.2

Single-Site Detection
Our standard method of imaging uses a microwave transition to pump atoms in a single

layer into a state that is resonant with our imaging laser. These experiments used absorption
imaging, which was necessary for our state-dependent detection. We take advantage ofthe quadrupole field gradient, which in these experiments is Bz0 =78 G/cm, so that the
microwave transition has a spatial dependence. Atoms in a single lattice site are pumped
from |F = 1, m = −1i → |F = 2, m = −1i, as shown in the level diagram in Fig. 7.3. Oncethe microwave pulse is applied, we quickly remove the quadrupole and TOP fields and image
on an |F = 2i → |F 0 = 3i cycling transition. We apply a microwave pulse with a gaussian
temporal profile that is matched to the width of a single layer in frequency space. The
center frequency is resonant with the |F = 1, m = −1i → |F = 2, m = −1i transition, andthe width of the envelope is 50 µs. In frequency space, the width of the microwave pulse
is approximately 4 kHz, compared to a lattice spacing of 40 kHz. The calculated frequency
width of the pulse is shown relative to the lattice spacing and condensate size in Fig. 7.4
Constraints on the width of the gaussian pulse and a maximum microwave power of 30
dBm limits the pumping efficiency to around 10%. Absorption images have a much larger
SNR than phase-contrast images, but, for optically thick clouds, the absorption signal will

96

1

F =2
1

2

0

2
⇡ 6.8 GHz
F =1

1

0
1

Figure 7.3: Diagram of the energy levels for the 5S 1 state of 8 7Rb. The microwave transfer
2
is shown with the dotted arrow and is centered at a frequency of approximately 6.8 GHz.

Figure 7.4: Calculation of the microwave transfer pulse (blue) in frequency space relative tothe optical lattice (green) and condensate (red) size.

eventually saturate causing image distortion. By transferring only a small percentage of theatoms to a detectable state, nonlinearities associated with large optical depths (ODs) were
removed [61, 51].
Small transfer efficiencies were also beneficial for a second detection technique we used97
for imaging the momentum-space distribution of atoms without normal time-of-flight (TOF)
methods. Probing the momentum distribution of a cloud typically requires that the atoms are
pumped into an untrapped state, effectively removing the external potential, and waiting
for a long time so that the atoms spatially separate by an amount proportional to theirindividual momenta. TOF methods have a couple drawbacks in practice. Long expansion
times come at the cost of image distortions such as vignetting [37], and mean-field energy
adds additional momentum for condensates or especially dense clouds. In lower dimensional
systems, the momentum-space distribution can be imaged directly by harnessing the meanfield energy. We use a 2D analog of a momentum-space “focusing” technique developed
by Shvarchuck et al. [55] for a 1D cigar condensate confined to an optical dipole potential.
In their experiment, they abruptly turned off the optical trap and let the atoms expand
into a magnetic potential. The magnetic potential, which was used to provide confinement
along the elongated dimension, was set up so that the trapping frequency in the direction of
the tight confinement produced by the optical trap was twice the trap frequency along theelongated ẑ direction. Upon releasing the atoms from the optical trap, the mean-field energy
was preferentially transferred to the transverse momentum states, allowing the longitudinal
distribution of atoms to fall toward the center of the trap once the mean-field potential had
vanished. The aspect ratio of the trap is set so that the time it takes for atoms in thecondensate to move to the center of the trap along ẑ is the same amount of time it takes
for the atoms expand radially and fall back towards the center of the trap in r̂, as shownin Fig. 7.5. Probing the condensate after a quarter period of an oscillation along ẑ, 41 Tz ,ideally provides the exact momentum distribution, which is described by a delta function,
but mean-field effects begin to turn back on as atoms approach the center of the trap. Bytransferring most of the atoms into an untrapped or antitrapped state, the mean-field effects
are vastly diminished resulting in a distribution that closely resembles a delta function.

98
Latticeturned o↵

1
Tz
4

Time
Figure 7.5: Diagram showing the expansion process used for the momentum-space focusing technique. Mean-field energy causes a lower-dimensional condensate to expand in the
direction of tight confinement into a magnetic potential with an aspect ratio of 2. After aquarter period of an oscillation along the weakly confined dimension, 14 Tz , condensate atoms
reach the center of the potential simultaneously giving rise to the delta-function momentum
distribution of a condensate. The thermal component is shown to contract with the condensate, which does happen for thermal atoms displaced by the condensate mean-field. But for
hot thermal atoms, the width of the distribution should have approximately the same sizein coordinate and momentum space.

7.2.1

Spatial Separation Between Lattice Sites

The dominant anharmonic term in our potential is of order zr2 . This means thatat different positions directly below the trap, the frequency in the radial plane is shifted.
For a 3D experiment, this effect is washed out by oscillations along ẑ, however if atoms
are confined to distinct planes, which are separated in ẑ, then the anharmonic shift vanishes
within each plane but the radial trap frequency is different between planes. Slosh oscillations
will eventually dephase between sites and eventually the condensates on each site will bespatially separated in the radial plane, as shown in Fig. 7.6. An example of this dephasing
can be seen from the data in Fig. 7.7, where small differences in the slosh phase make thetotal cloud appear tilted. COM separations that are larger than the condensate diameters
allow each condensate to be imaged separately, but require long evolution times with ournormal trap settings. Our magnetic trap is configured so that the radial trap frequency is 10

99

Time

Figure 7.6: Example of intersite dephasing for multiple quasi-2D condensates confined to a
1D optical lattice. Individual layers, viewed from the side, are not resolved. Instead, theentire sample appears “tilted” as the separate vertical layers become slightly offset from eachother in the horizontal direction.

Hz, with an intersite frequency difference of ∆f = 0.0032Hz. Waiting for the clouds to fully
separate takes nearly a minute in the standard trap configuration. However, the amountof dephasing can be turned up by enhancing the strength of the zr2 term. By decreasing
the TOP bias field strength from 7.4 G to 1 G, the radial trap frequency increases to 19 Hz
with an intersite frequency difference of ∆f = 0.074Hz. Such a significant reduction in thestrength of the TOP bias field would normally cause a number of issues, such as verticalslosh, or inadvertantly evaporating the cloud because of the small size of the circle of death.
But, when the atoms are confined to a lattice, this adjustment can be made very quickly andwithout having to worry about these problems. Because we want to anharmonically shear
the cloud and then let the condensates evolve freely with minimal intersite dephasing, we
temporarily ramp the TOP field while the cloud is sloshing, and return the field strength toits original setting before taking data. Fig. 7.8 shows data taken after 2 seconds of evolution
in a trap where the TOP field had been reduced to 1 G. In this alternate configuration of thetrap, the intersite dephasing has increased by more than a factor of 20 and separate layers
can be resolved.
Once the cloud has fully separated, we are able to increase the amount of data acquired

100With Vertical Bias
100 µmt = 2.000 s
50

2.015 s100

2.030 s150
200

2.045 s250

2.060 s300

2.075 s350
400

2.090 s450

2.105 s
500
50

100

150

200

250

300

350

400

Figure 7.7: Side-view image of intersite dephasing for multiple quasi-2D condensates after a
two seconds.

101

Slosh Dephasing Between Lattice Sites
100 µmt = 2.000 s
50

2.015 s100

2.030 s150
200

2.045 s250

2.060 s300

2.075 s350

400

2.090 s450

2.105 s
500
50

100

150

200

250

300

350

400

Figure 7.8: Side-view image of intersite dephasing for multiple quasi-2D condensates after
two seconds with a TOP field of 1 G, where layers have been fully separated. Separate
layers can be resolved, but imaging from the side causes the probe beam to diffract whenincident on a condensate, making the individual layers difficult to resolve. The condensates
are sloshing in a circular motion, and at different phases of the oscillation the clouds appear
to overlap when viewed from the side.

102
in a single shot by simultaneously imaging all of the layers in the lattice. Because we are
not limited to imaging 10% of the cloud, the signal is much larger allowing us to use phasecontrast imaging. Of course, the phase-contrast images shown from the side don’t provide anice source of data. Diffraction from the thin layers can overlap with other layers, and thephase of the oscillations can obscure the measured widths of clouds that are closely aligned
along the direction of our imaging beam. Similar issues can arise when imaging from thetop, where the individual phases of slosh in each site causes the clouds to overlap.
By driving the slosh mode in a circular motion, the individual slosh oscillations ineach lattice site would begin to dephase in such a way that their radial positions don’t
overlap at any given phase during an oscillation, as depicted in Fig. 7.6. After the cloud has
fully separated, the TOP field is ramped back to its initial value of 7.4 G and the intersite
dephasing is reduced. Returning the trap to its original parameters also lets us optimize theroundness of the trap at a setting where the intersite dephasing is small enough to allow
for much longer evolution times. One issue that can arise during the dephasing process is achange in the isotropy in the radial plane of the trap. In this case, Lissajous patterns start
to appear and the final orientation of the clouds depends on the individual phases of sloshingalong the two principal axes. For example, if dephasing along the two principal axes is suchthat the oscillations become linear, then linear slosh motion will persist once the trap has
been has been returned to its original configuration. The easiest way to account for this isby varying the evolution time during the anharmonic shearing process such that the cloud
trajectories are circular upon ramping the TOP field to the standard setting. In this case,
it is actually easier to ramp only the TOP fields without any adjustment to the elliptifier
field strength during shearing. This provides a slightly more anisotropic trap as the cloudseparates and allows for more controlled adjustement of the final slosh trajectories.
Fig. 7.9 shows some sample images of the cloud as viewed from the top, after the
cloud has been anharmonically sheared. In these data, we were measuring the monopole
frequency in a 2D cloud, which could be directly compared with the frequency of the slosh

103
mode resulting from the anharmonic shearing.

Figure 7.9: Raw data from an experimental realization of the anharmonic shearing effect viewed from the TOP. These data aretaken after 2 seconds of evolution in the sheared trap, and are taken in 20 ms intervals.

104

105
This anharmonic shearing technique provides a useful method of probing multiple
quasi-2D condensates simultaneously without having to worry about the signal being obstructed by their overlap. Data acquisition rates can be sped up dramatically because
phase-contrast imaging can be used, and measurements on multiple condensates can bemade in a single image. We developed this technique to measure a small, expected shift
in the monopole frequency in 2D from a quantum anomaly [44]. However, we did not end
up using this approach because we worried that the frequency shift of the monopole mode,
which was expected to be 0.1% in our system, would be obscured by any residual parametricdriving of the cloud due to any slight asymmetry in our lattice beams.

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Appendix A
Finite Temperature Data

112

N0 /N = 0.098(2)t (ms) hSi (µm)
234.5
7.1(7)
1034.56.0(3)
634.5
3.0(2)
42.5
7.0(2)
434.5
6.0(2)
834.5
4.0(1)
134.5
4.0(1)
1234.50.7(4)
534.5
3.0(2)
934.5
4.0(1)
334.5
5.0(2)
734.5
3.4(3)
1134.52.0(3)
134.5
5.0(3)
1234.55.0(3)
434.5
1.0(3)
1034.55.0(1)
634.5
1.6(9)
42.5
10.0(1)1134.55.0(2)
734.5
3.7(2)
334.5
3.0(3)
834.5
3.0(1)
234.5
7.4(6)
534.5
2.9(10)934.5
2.0(2)
434.5
3.0(2)
1234.53.0(2)

113
N0 /N = 0.177(2)t (ms) hSi (µm)
42.59.0(2)
434.5
2.0(1)
834.5
4.0(2)
234.5
5.0(1)
534.5
3.0(1)
734.5
5.4(3)
1034.54.0(1)
134.5
5.8(10)634.5
3.0(1)
934.5
4.0(1)
334.5
3.0(1)
234.5
5.1(4)
734.5
6.0(2)
42.5
7.0(1)
434.5
3.0(2)
1034.53.0(1)
634.5
3.6(4)
134.5
6.0(2)
834.5
4.0(2)
334.5
2.0(2)
534.5
5.0(2)
734.5
3.5(9)
934.5
3.0(3)
434.5
2.0(1)
834.5
2.0(2)
134.5
6.0(2)
534.5
3.0(1)
42.5
9.0(1)
634.5
3.1(7)
334.5
1.0(3)
934.5
4.0(1)
234.5
5.0(2)
1034.54.0(2)
534.5
3.0(2)
334.5
1.6(10)734.5
4.0(2)
1034.52.9(6)
734.5
3.8(9)

114

N0 /N = 0.31(2)t (ms) hSi (µm)
164.55.0(3)
834.5
1.0(1)
434.5
3.0(1)
64.5
5.2(7)
1034.5
1.8(8)
164.5
6.0(1)
434.5
4.0(1)
834.5
2.0(1)
64.5
4.4(10)1034.50.8(10)164.5
5.0(2)
434.5
1.7(8)
834.5
2.7(2)
64.5
6.0(1)
634.5
3.4(10)1234.53.0(8)
434.5
2.0(1)
634.5
3.4(4)
1234.52.1(9)
1434.53.4(2)
634.5
3.0(1)
1234.51.1(6)
1434.5
1.6(3)
1634.52.0(2)
1834.51.8(4)
1634.50.8(7)
2034.51.2(3)
1834.50.5(7)
1434.53.1(9)
2034.50.9(3)
634.5
4.5(7)
1634.5
1.6(6)
434.5
2.0(1)
834.5
4.0(1)
164.5
7.0(2)

115

N0 /N = 0.322(3)t (ms) hSi (µm)
42.59.0(2)
434.5
1.0(2)
1034.53.0(9)
734.5
2.4(9)
134.5
6.0(1)
534.5
2.0(2)
334.5
1.0(2)
934.5
0.9(5)
234.5
6.0(2)
634.5
3.1(8)
1134.51.0(1)
834.5
2.0(6)
434.5
2.0(2)
234.5
3.0(1)
634.5
2.0(1)
334.5
2.0(1)
1034.52.5(8)
42.5
8.0(1)
1134.53.0(1)
734.5
3.1(6)
134.5
6.0(2)
834.5
1.0(1)
934.5
2.0(1)
534.5
4.2(6)
1034.52.0(1)

N0 /N = 0.322(3) (cont’d)t (ms) hSi (µm)
134.54.4(8)
334.5
0.3(7)
734.5
3.6(7)
42.5
7.0(1)
234.5
4.0(2)
1134.53.0(2)
834.5
1.0(1)
234.5
4.6(8)
434.5
2.5(9)
434.5
1.7(10)634.5
3.0(1)
934.5
0.8(7)
534.5
1.0(2)
834.5
2.0(1)
334.5
2.0(2)
1134.52.0(1)
734.5
2.4(8)
42.5
6.7(9)
634.5
2.0(1)
534.5
3.0(1)
1034.5
1.4(8)
134.5
5.0(2)
234.5
5.0(2)

116

N0 /N = 0.446(2)t (ms) hSi (µm)
334.51.0(3)
834.5
1.6(9)
234.5
5.0(1)
434.5
3.0(1)
634.5
4.9(7)
134.5
6.0(1)
734.5
5.0(1)
534.5
2.0(2)
42.5
7.0(1)
334.5
2.0(2)
734.5
2.4(3)
134.5
7.0(3)
534.5
1.0(1)
834.5
0.0(1)
234.5
4.0(1)
634.5
3.3(9)
434.5
0.0(2)
42.5
9.0(2)
734.5
1.0(1)
334.5
1.0(2)
134.5
7.6(6)
834.5
2.0(1)
534.5
7.0(3)
234.5
5.0(1)
634.5
2.0(2)
434.5
3.0(1)
42.5
10.0(2)734.5
5.0(9)
534.5
3.0(2)
534.5
4.0(1)
634.5
4.0(1)

117

N0 /N = 0.579(3)t (ms) hSi (µm)
232
2.7(7)
624
3.0(1)
17
10.0(1)1317
0.5(8)
417
3.6(9)
917
3.3(8)
117
7.2(4)
717
2.0(1)
1117
4.0(1)
317
2.7(8)
1817
0.2(8)
517
4.8(8)
1017
2.0(1)
617
4.6(6)
1217
2.5(7)
17
11.0(2)717
3.5(6)
417
1.0(1)
1117
3.0(1)
217
4.62(6)817
3.2(5)
624
4.0(1)
117
6.6(7)
1317
3.1(5)
317
4.0(1)
917
1.0(10)517
0.6(3)
817
2.7(7)
1817
2.0(2)
1017
1.9(9)
1217
2.2(10)118

N0 /N = 0.583(4)t (ms) hSi (µm)
117
8.3(7)
717
2.0(9)
1017
1.0(1)
317
2.3(2)
17
8.0(2)
617
3.7(9)
917
3.0(1)
217
6.4(4)
817
1.0(5)
417
0.0(1)
517
3.9(6)
267
3.2(4)
967
1.5(5)
567
6.0(1)
67
6.5(10)867
2.2(4)
467
3.6(5)
167
6.0(1)
167
5.2(3)
667
4.0(1)
367
3.3(5)
767
2.2(5)
617
4.0(1)
117
8.2(8)
1017
3.4(6)
417
2.0(1)

N0 /N = 0.583(4) (cont’d)t (ms) hSi (µm)
7173.0(1)
17
9.0(2)
567
4.4(3)
217
5.0(2)
917
2.5(8)
367
3.2(7)
67
7.0(1)
817
2.0(2)
517
2.7(3)
1017
2.5(9)
317
3.0(4)
667
5.0(1)
867
2.6(9)
267
6.0(2)
967
2.0(1)
467
2.7(5)
567
2.0(1)
267
4.1(7)
417
1.8(5)
767
2.9(7)
267
5.0(1)
567
2.0(1)
1017
2.9(10)567
2.8(9)
667
4.0(8)