Strongly Interacting Bose-Einstein Condensates: Probes
and TechniquesbyJuan Manuel Pino II
B.S. Physics, University of Florida, 2002

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
2010This thesis entitled:
Strongly Interacting Bose-Einstein Condensates: Probes and Techniques
written by Juan Manuel Pino II
has been approved for the Department of Physics

Dr. Deborah S. Jin

Dr. Eric A. Cornell

Date

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

iii
Pino II, Juan Manuel (Ph.D., Physics)
Strongly Interacting Bose-Einstein Condensates: Probes and Techniques
Thesis directed by Dr. Deborah S. Jin

A dilute gas Bose-Einstein condensate (BEC) near a Feshbach resonance oﬀers a system thatcan be tuned from the well-understood regime of weak interactions to the complex regime of strong
interactions. Strong interactions play a central role in the phenomena of superﬂuidity in liquid He,
and theoretical treatments for this regime have existed since the 1950’s. However, these theories
have not been experimentally tested as superﬂuid He oﬀers no similar mechanism with which totune the interactions. In dilute gas condensates near a Feshbach resonance, where interactions canbe tuned, strong interactions have proven diﬃcult to study due to the condensate’s metastable
nature with respect to the formation of weakly bound molecules. In this thesis, I introduce anexperimental system and novel probes of the gas that have been speciﬁcally designed to study
strongly interacting BECs. I present Bragg spectroscopy measurements that have accessed thisregime, as well as proof-of-principle experiments using photon-counting for Bragg spectroscopy at
low-momentum. Finally, I show preliminary data using contact spectroscopy, which is a methodthat could lead to the ﬁrst measurements of the predicted interaction energies for a dilute Bose gasof atoms in the strongly interacting regime.

Dedication

Para mi Abuelita, 1910-2010.

Acknowledgements

This work not only represents my eﬀorts here at JILA over the years, but also the contributions of a great many others whom I’d like to thank here.
Carl Wieman gave me my opportunity to work with cold atoms when he hired me in 2003.
Although he was being pulled in diﬀerent directions at the end of his tenure here, I always appreciated his eﬀorts to cultivate my understanding of physics, especially through the lunches that
Kang-Kuen and I would share with him. Debbie Jin and Eric Cornell adopted me and the 85 Rb
experiment in 2006. In many ways I felt like I had left the orphanage and was brought into a loving
family. Eric Cornell played the role of my advisorial father. He deﬁned our “family values”, thestudy of strongly interacting condensates, and his enthusiasm was infectious. He also happened
to have a wit to match his intelligence, and our family dinners (i.e. group meetings) were always
lively. The best part about working with Eric was the way he always, always seemed to be able tohave a physical picture or two of how things worked. It’s really amazing that Eric can be, to me atleast, the embodiment of “Physicist”, and at the same time be very human. His professional andpersonal advice were always appreciated, and will stay with me for a long time.
Debbie Jin would then be my advisorial mother. I consider Debbie to be my mentor, advisor,
and friend. She is both brilliant and easy going, and can make the complicated seem pedestrian.
She is also the most eﬀective person I’ve ever seen at solving a complicated problem, which is what
this thesis, and really all technical work, is all about. Debbie was always accessible, and taught
me how to be eﬀective at what I do, which, in a professional sense, is the most valuable lesson
you can learn. I will be forever in debt to Debbie for taking me on as a student, putting in time

vi
with me, and believing that I could produce. I always enjoyed my interactions with her, and ouradvisor-student relationship will be the one aspect of graduate school that I miss the most.
I’d also like to thank my labmates over the years. Scott Papp and I spent many hours
together in my early years in the lab and I will always respect Scott’s knowledge of experimental
technologies and techniques. He also has one of the strongest arms of anyone I’ve ever met, andthankfully some of his knowledge was able to rub oﬀ on me while we worked together.
If Debbie and Eric are my experimental parents, that makes Rob Wild my experimental
brother. Working with Rob in our nuclear atomic family has been a true blessing. Rob was a greatcompliment to me, often thinking of a more clever solution to a problem that I would try to solve
via “brute force”. Working in the lab can be a frustrating endeavor, and often times we are notin our best light when things aren’t going well, but Rob was always a (relatively) positive person,
able to pick me up when I was frustrated and would have preferred to quit. The lens of working
with someone for 10-12 hours a day, 5 days a week in lab where things don’t seem to ever work isrevealing, but Rob’s character always was true. He’s as quality an individual as you will ever meet,and I hope that his success in physics will one day match his success as a person.
Phil Makotyn would then be our adopted brother. Although that might sound bad it would
turn out that we are all adopted here, even Rob with his Austrian blood. Phil inaugurated theritual hazing that new labmates endure, and has been a great addition to the lab, bringing not only
new perspectives, but also a diﬀerent skill set which I think will be very useful in the long-term. I
think Phil will be a great physicist one day and wish him the best of luck. I’d also like to thank
Zach Newman, an REU student who spent a summer with us and whose enthusiasm was always
welcome. I’d also like to thank Rachel Umbel, a later REU student, for some of the beam proﬁling
data I used in this thesis.
I’d also like to thank my bi-group cousins. Russ Stutz has been a great, great friend over theyears, and we’ve enjoyed (or at least gone through) experiences ranging from ﬁnishing a crib at 3
AM, to running out of gas somewhere in Utah, to epic battles as Quantum Leapers. I don’t know
if I could have made it through these last few years, least of all grad school, without him, and so,

vii
Dr. Captain Stutz, I salute you. Jayson Stewart is another one of my closest friends I’ve made inColorado. When I ﬁrst met Jayson I was really surprised to ﬁnd that we had a lot (a lot!) of thesame interests. I saw a much of myself in Jayson, and watching him succeed in the lab and as aphysicist was instrumental in me staying the course with this PhD. I really can’t ever thank him
enough for that. Shih-Kuang (Zeke) Tung was, and will always be, a wonderful friend. Zeke is as
sharp and hardworking a guy as you’ll ever meet, and he’s also one of the nicest, with a laugh thatwould always make me smile. I consider myself lucky to have made his friendship.
To be honest, I don’t know exactly where post-doc’s lie on the experimental family tree. I
think Aaron Leanhardt would be something like that uncle that taught you your ﬁrst curse words.
Aaron’s an extremely dedicated worker who’s had the unfortunate luck to both be a fan of Detroit
sports teams as well as endure my friendship. He is a quality guy, though, and I know I could always
count on him if I needed something (like a drive to the airport in under 30 minutes). Giacomo
Lamporesi is more like...a godfather, of course. Giacomo I’ll thank for being a good roomate(ICOLS, ICAP), great cook (credit: olive oil), and bellissimo friend.
I’d also like to thank John Gaebler for not impaling me with any ski equipment while he’s
had the chance, and for being a solid neighbor (as of three weeks). I’d also like to thank past groupmembers Kang-Kuen Ni, Josh Zirbel, Silke Ospelkaus, and John Perreault for always being aroundto answer questions. I look forward to the next-generation bi-group team of Laura Sinclair (bestclimbing buddy ever!), Huanqian Loh (always makes me laugh), Brian Neyenhuis, Tyler Cumby,
Tara Drake, Ruth Shewmon, Matt Grau, Dan Lobser, and Lin Xia and fully expect this group to
bring back the JILA cup to it’s rightful home in the bi-group.
Another great thing about JILA is the wonderful support we have from everyone that works
here. First I’d like to thank Krista Beck, who has always been there for me for both professional
matters and personal dilemma’s. Pam Leland should also be thanked for always having a wonderful
smile on whenever I’ve needed her help. The electronics shop staﬀ has taught me a great deal during
my time here in JILA. In particular I should thank Terry Brown, who has explained the ﬁner pointsof electronic design to me, as well as Carl Sauer, who can always be counted on to answer a quick

viii
question, as well as unlock a laser or two with his laugh. The machine shop has also been a great
help to me over the years, and I’d like to thank the shop staﬀ as well. In particular Hans Green,
who helped Russ and I late into the night designing and building Juan Michael’s ﬁrst crib. And
ﬁnally J.R. Raith, who’s helped me with countless computer (read: Windows) problems.
Outside of JILA I’ve made a number of friends here at CU, and would like to say thanks
to the incoming class of 2002. Johnny “One-Eye” Jost, the best ﬁrst-mate a pirate could ask for.
Keith Ulmer (thanks for the super-important conversations had over soda) and Kirk Ullmann forthe best road trip of my life. Marty Boyd, who’s come as close as anyone to being as funny as me.
Amanda Carpenter and Ashley Carter for putting up with me during that Stat Mech project, andMatt Sammons for all that money he gave me playing poker.
Finally, I’d like to thank my family. My brother Marito, for being my role model as a childand inspiring me to take things apart, challenge myself, and stay humble. Those lessons were neverlost on me, and as much as anybody he’s shaped me into the man I am today. To my brother Juan
Carlos, for teaching me what it means to be a Pino, reminding me about what’s truly importantin life, and always having my back. To my sister, Ameriquita, and her family Johnny, Diego andLucas, for loving my son as much as they do. Words cannot describe how much their love for Juan
Michael means to me.
To my father, Juan Pino, for making me tough enough to get through this and making me
man enough to rely on myself for whatever lies ahead. To a son, a father’s love and support is all
you need to conquer the world, and my father has always given me an ample amount of both. I’dalso like to thank my father for the perspective he provides. He has lived an incredible life, reaching
the United States on a 15 foot boat exactly 50 years from the date I defended this thesis, and has
experienced both highs and lows that I think few will ever reach. As a child I always thought that
I was the polar opposite of my father, but, as I grow older, I realize we are more similar every day.
To my mother, Tesla Fletez Pino, I owe more than I could ever say or write. She has given
me pride, humility, empathy and conviction. She has inspired me to always ask questions and herpatience is unmatched. The thing that I am most thankful for, however, is her selﬂessness, which

ix
is the single quality you would most want in a parent. She was there to help me put labels on myscience projects late into the night (even if some of them were misspelled!). She was there to spend
her Saturday taking me on hour long treks to guitar stores on the other side of town. She wasthere to buy me my ﬁrst soldering iron, and she was there reading me Alice in Wonderland (albeit
with an accent) while I wondered when this book was going to have another picture. She embodies
what an immigrant is and I am grateful to call her not only my mother, but also mi mejor amiga.
My grandmother, mi Abuelita, Rosinda Fletez gave me my spirit. Abuelita moved in with usa few days before I was born, raised me, and left when I went to college. She taught herself to read
and write and at various points in her life was a gunslinger, a matriarch, an entrepreneur, a poet,and her grandson’s playmate. Above all else she cultivated my creativity and my sense of self, andmy ﬁrst blessing in life was that she was there to watch over me.
Finally, I thank my son, Juan Michael Kort Smith-Pino, for teaching me what love is.

Contents

Chapter
1 Introduction

1

1.1

The carrot: Superﬂuid liquid He . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Theoretical treatments: But how long is the stick? . . . . . . . . . . . . . . . . . . .

5

1.2.1

Macroscopic theories: the carrot’s perspective . . . . . . . . . . . . . . . . . .

5

1.2.2

Microscopic theories: the horse’s perspective . . . . . . . . . . . . . . . . . .

5

In the cart: a spherical cow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3

1.3.1
1.4

A hurried bull: 85 Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 BEC in a weak, spherical trap

15

2.1

The ansatz: ramping a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2

Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1

Spherical cows, spherical traps, and BEC dynamics . . . . . . . . . . . . . . . 17

2.2.2

The weak shall inherit the trap . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3

Trap design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4

Making 85 Rb BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1

Laser cooling and the magnetic trap evaporation . . . . . . . . . . . . . . . . 32

2.4.2

Hybrid trap evaporation and the grooming of a 85 Rb BEC . . . . . . . . . . . 35

2.4.3

A ﬁnal cut: the 85 Rb rf knife . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xi
2.4.4

How’s it look? Imaging 85 Rb BEC . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4.5

Shaving oﬀ the slosh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4.6

The barber’s take . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Bragg spectroscopy
3.1

3.2

48

Measuring the dispersion relation of an interacting BEC . . . . . . . . . . . . . . . . 48
3.1.1

Single-particle excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.2

Collective phonon excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.3

Dispersion at large na3 and the roton . . . . . . . . . . . . . . . . . . . . . . 54

Two pictures of Bragg spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.1

A moving grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.2

A 2-photon transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.3

Calculating Ω2-photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3Structure factor: S(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4

Challenges in resolving ω0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.1

Contributions to the Bragg width, σ . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.2

σ versus ω0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4.3

Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Photon counting for Bragg spectroscopy

77

4.1

Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2

Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3

Layout and construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1

Mezzanine level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.2

Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3.3

Strong beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3.4

LO beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3.5

Weak beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

xii
4.3.6

RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4

Calibration and performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.5

Bragg experiments with photon counting . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.6

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Contact spectroscopy

105

5.1

Making contact with an interacting BEC . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2

How rf spectroscopy connects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.1

5.3

5.4

Final-state eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Contact spectroscopy experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.1

Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.3.2

Challenges

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6 Conclusion

119

6.1

A small bite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2

Staring down on the carrot patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Bibliography

122

Appendix
A Cart switch

129

B Alternative approaches to BEC in a 10 Hz trap133

B.1 The 2nd law of graduate student dynamics . . . . . . . . . . . . . . . . . . . . . . . . 133
B.2 Crossed optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
B.3 BEC loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
B.4 Direct evaporation on 85 Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Chapter 1

Introduction

I began my career as an experimental physicist working on a superﬂuid liquid He machine at
the University of Florida. The concept of superﬂuidity, that is, a liquid with zero viscosity, presented
what I thought were some of the most elegant demonstrations of physics, quantum mechanical orotherwise. It turns out that I was in good company, as this concept has been challenging physicists
for decades. The focus of this thesis is exploring this behavior experimentally with a gas of ultracold
atoms, which has its own unique challenges.
This introduction follows the analogy of a farmer tying a carrot onto the end of a stick,inspiring (some would say tricking) his horse to pull along a cart. For us the motivating carrot is
one of nature’s gifts, the liquid 4 He Bose-Einstein condensate (BEC). I will outline the historicalsigniﬁcance, as well as some of the macroscopic phenomena that has made this system so inspiring.
I will then give a sense of how “far away” the stick is to our horse (who doubles as an atomic
physicist). Since our horse sees things in units of Bohr radii, I will focus on the microscopic
theories that attempt to describe this system from the ground (state) up. Finally, I will present
the contents of the cart: about 40,000 85 Rb atoms. I will explain how 10−21 kg proves quite
burdensome, as our horse measures weight not in kilos, but in Hz. I will conclude by explaining
why both the horse and the farmer in this analogy consider this cargo to be so precious, as we hope
40,000 Bose-condensed 85 Rb atoms can be the seeds of our own carrot patch, nestled in our cornerof the atomic physics community.

2

1.1

The carrot: Superfluid liquid He
Helium was ﬁrst liquiﬁed in 1908 by Heike Kamerlingh-Onnes [1], where it was quickly applied

as a coolant and largely forgotten about by Onnes, as he went on to discover superﬂuidity’s aﬀable
cousin, superconductivity in 1911. This is a trend we would see some 100 years later in the atomic
physics community, as the “hot topic” of late has been strongly interacting Fermi gases, with theoccasional bosons again serving as coolants.
In 1932 interest in liquid 4 He again picked up, as a phase transition at 2.18 K was consideredby Keesom and Wolfke [2], in light of work done by Onnes [3]. This was further investigated by
Keesom and Clusius as they heated liquid Helium with a small, resistive wire and observed the ensuing temperature change [4]. They found the speciﬁc heat versus temperature to be discontinuous
about what they termed the “λ” transition, named for the characteristic shape of the speciﬁc heatversus temperature near 2.18 K. This new phase of liquid He, below the λ-point, was referred toas He II. These were the ﬁrst of many macroscopic observations manifested by a quantum state ofmatter.
Although anecdotal evidence had existed, the discovery of He II’s superﬂuid nature did
not come until 1938, when Kapitza and the team of Allen and Misener both independently andsimultaneously measured the viscous ﬂow of He II through a thin capillary. They found that at
temperatures below the λ-point, the viscosity dropped several orders of magnitude, and became
unmeasurable by their respective experiment’s resolution [5, 6]. Kapitza would coin the term
superﬂuid, and later receive a Nobel prize for his work.1
Superﬂuidity itself would end up being only one slice of the proverbial carrot, as experimentalists would measure a new kind of wave propagation, termed second sound. Instead of theusual matter wave propagation traveling at the speed of sound, second sound is the propagationof a thermal wave, where the non-condensed fraction of He II transmits this energy at its own
1
This is a somewhat controversial area of Nobel history, as Allen and Misener were conspicuously left oﬀ of
the prize. For a bit more in depth historical analysis I refer the reader to Griﬃn’s wonderfully written review ofsuperﬂuidity, complete with a timeline to further understand the Kapitza/Allen-Misener debate [7].

3characteristic speed. A number of macroscopic theories for He II predicted such a phenomenon by(correctly) assuming that the non-condensed fraction of He II could be thought of as fundamental
excitations within the system. These predictions were conﬁrmed in a set of experiments by Peshkov
in 1944 and 1946 [8, 9], and provided another dramatic example of macroscopic phenomena unseen
in conventional liquids at the time.
Later, quantized vortex lines were experimentally veriﬁed by Hall and Vinen in 1956 [10], conﬁrming the prediction of Onsager made in 1949 [11]. He argued that circulation about a closed loop
would be quantized in units of h/m, owing to the single quantum state of the system. Quantizedvortices are one of the cleanest examples of macroscopic phenomena fully explained by quantum
mechanics.
Experiments aimed at understanding the microscopics of He II, speciﬁcally the elementary
excitations of the system, came to fruition with inelastic neutron-scattering measurements. He
II would not disappoint in its quirkiness, as experiments by Palevsky, Otnes and Larsson [12]conﬁrmed the predicted existence of a local minimum in the dispersion. This minimum had been
predicted by Landau, based on an elementary vortex excitation, and termed a roton. The “rotonic
minimum”, along with the complete dispersion for He II is shown in Fig. 1.1, as measured byHenshaw and Woods [13]. This particular feature of 4 He has been baﬄing physicists for years.
The aforementioned Landau, as well as Feynman, Pitaevskii and Bardeen have all taken a crackat understanding the origins of the roton excitation [14, 15, 16], but a microscopic basis for such aminimum in He II has yet to be established.2
What we have discussed here are a number of macroscopic phenomena, that we know must
be driven by the microscopics of the system. Also presented was the microscopic excitation spectrum, which had been qualitatively predicted by Landau’s macroscopic theory. What is missing
is a holistic approach that predicts these phenomological carrots, but is rooted in the quantum
mechanics we know to be of fundamental importance.
2

Unless you talk to a Monte Carlo theorist, in which case this is a fully understood problem, albeit with noanalytical solution [17, 18, 19]

4

Figure 1.1: Reprinted from Henshaw and Woods [13]. Dispersion relation for He II, measured via
inelastic neutron-scattering. The vertical axis is the measured energy of the excitation, while thehorizontal axis corresponds to the momentum of the excitation. Note the local minimum, known
as a roton excitation, as well as the linear dependence the energy has on the momentum for small
excitations. This linear dependence is a signature of a collective, phonon excitation.

5

1.2

Theoretical treatments: But how long is the stick?
Following our analogy, the length of the stick could be thought of as how far we horses haveto go to get to this carrot. I will ﬁrst brieﬂy describe the approaches taken to understand the carrot
on the surface, in a macroscopic way, as this was the dominant line of thinking for quite some time.
I will then go into more detail on the current state of the theory of interacting Bose gases, with amicroscopic approach in mind.

1.2.1

Macroscopic theories: the carrot’s perspective

A number of diﬀerent theoretical treatments were developed to understand the superﬂuid,
either in parallel, or in some cases preceding the experimental progress. Some of these approaches,particularly the work of László Tisza [20, 21, 22], extended the BEC framework (calculated for
an ideal, non-interacting gas) put forth by Fritz London [23] to a macroscopic basis. However, analternate, independent, and highly successful approach was taken by Lev Landau, who completely
ignored the bosonic nature of 4 He, yet also managed to predict much of the phenomena laid out inthe previous section. Landau’s model relied on treating He II as being composed of both normalcomponent that resembled 4 He above the λ-point and a superﬂuid component. He made great
progress with this by quantizing the hydrodynamic equations for the superﬂuid component of
He II, and postulating that the normal component was actually thermally excited elementary
excitations. This “two-ﬂuid model” was quite successful and thus was taken as the gold standard
for He II physics, with He II’s BEC nature largely ignored in some circles. In essence, this wasa macroscopic theory that ignored the most fundamental characteristics of our carrot: its bosonicﬂavor.

1.2.2

Microscopic theories: the horse’s perspective

The ﬁrst microscopic theory taking into account the interactions in a degenerate Bose gas
was put forth by Bogoliubov in 1947 [24] to develop a deeper understanding of superﬂuidity. A

6number of key assumptions were made by Bogoliubov, and our thinking about BEC is dominatedto this day by these assumptions. He ﬁrst assumed the system was suﬃciently dilute so that only
pair-wise interactions were probable, and dismissed 3-body interactions. He also assumed that thesystem was at low temperatures, much lower than the condensation temperature, Tc . He further
asserted that collisions were at low enough energies that the scattering amplitude reduced to thelow-energy limit of the s-wave scattering length a. This last assumption lets us ignore the details of
the inter-atomic potential at short-range, provided the potential gives the correct value of a. Lastly,
the number of particles in the condensed, zero-momentum state is so large one may subtract or
add a particle to that state and not aﬀect the overall behavior of the system. This was crucial tohis work and is now famously referred to as the Bogoliubov prescription.
From his work followed two important results. First is the Bogoliubov dispersion law for acondensate
p2
ϵ =
2m
2

(

p2
+ 2gn
2m

)

(1.1)

where ϵ and p are respectively the energy and momentum of the excitation, and m the mass of theboson. The interactions are characterized by the mean-ﬁeld energy, gn, where g = 4πh̄2 a/m, andn the number density.
This dispersion law, while failing to predict the roton feature observed in He II, does a nicejob in the low-momentum limit, where ϵ ≈ cBEC p, and collective, phonon excitations travel at the
speed of sound cBEC =√gn/m. This linear dependence for low momentum excitations is shown in

Fig. 1.1.
Bogoliubov’s dispersion relation also proved correct in the high-momentum limit, where ϵ ≈p2
2m + gn.

Here, we have a typical excitation with momentum p, but with the kinetic energy

modiﬁed by the mean-ﬁeld energy, gn. These excitations are characterized as particle-like, andtheir quadratic dispersion, while not addressed in Fig. 1.1, has been conﬁrmed by later experiments
probing higher momentum [25, 26]. It is worth noting that when Bogoliubov developed these
theories, He II was presumed to be the only Bose-condensed system, but 50 years later experiments

7with weakly interacting dilute-gas BECs proved his results quantitatively correct [27].
Another important result from Bogoliubov’s work is that it introduces us to the concept
of competing length scales. In many ways, the probes presented in this thesis are probes of thediﬀerent length scales in our system, and it is a concept that will be present throughout this work.
Here, the competition shows itself as the crossover of an excitation between the phonon regime andthe free-particle regime. The length scale associated with this crossover is known as the healing
length, ξ, and is deﬁned as

√

ξ=

h̄2
=
2mgn

√

1
.
8πna

(1.2)

In the world of condensed matter physics, this is referred to as the coherence length of thesystem. In atomic physics, the term coherence length has a unique meaning, but we mention thisto give the reader a sense of its importance. This length scale also sets the minimum distance overwhich a condensate can spatially distort and, in a related sense, this length scale sets the minimum
size of a vortex. Later, in Chapter 3, we will again touch upon the importance of ξ, but it is worth
mentioning here that the wavevector associated with the inverse of this length scale is at the same
momentum at which the anomalous rotonic minimum occurs in the dispersion relation of He II.
Curiously, based on the given density [3] of He one can “back out” a scattering length for He II
equal to 13 a0 , which almost exactly agrees with the predicted eﬀective range of 4 He of 14 a0 [28].
Physicists Lee, Huang, and Yang calculated the ground-state energy of an interacting Bosegas in 1957 [29, 30]. They calculated the energy density of an interacting Bose gas to be
(

)

gn2
128 √ 31+ √na + . . .
E=
2
15 π

(1.3)

Their results were calculated using a Fermi pseudo-potential method, but can also be calculated
from within the Bogoliubov framework [31, 32, 33].
The result of Eq. 1.3 is a perturbation about the mean-ﬁeld result, gn2 /2. The perturbationis referred to as the LHY correction to the mean-ﬁeld energy, or simply the LHY correction. The√
dependence of the LHY correction on na3 shows the non-linearity of this expansion.
The quantum depletion of the condensate is deﬁned as the fraction of the system that has a8ﬁnite, non-zero momentum, even at zero temperature. This excited fraction arises from interactions
between the particles, and is given byn − n0
8 √ 3
= √
na
n
3 π

(1.4)

where n is the total density, and n0 is the density of the condensed, zero-momentum component of√
the system. The similarities between Eq. 1.4 and the LHY correction (namely the na3 dependence)illustrate the importance of these excitations to the next-order correction of the mean-ﬁeld energy.
Furthermore, as pointed out by Ref. [32], the dominant contribution to the LHY correction are
the repeated interactions between two excited particles, before they drop back into the condensed,
zero-momentum state.
The next-next-order correction to the mean-ﬁeld energy was soon investigated by Wu [34]in 1959, shortly after Lee, Huang, and Yang’s result. He calculated the coeﬃcient in front of thelogarithm below(E=√
128 √ 342πh̄2 an2
1+ √na + 8( π − 3) ln(na3 )(na3 ) + O(na3 )
m
15 π
3

)

(1.5)

where the other terms of order na3 are known to be as large as Wu’s correction, and therefore
must be taken into account. These terms are non-universal, as they depend on the details of theinteratomic potential, including 3-body eﬀects. A number of theorists have since calculated theseterms for diﬀerent model potentials [35, 36]. The logarithmic dependance on na3 further shows the
non-trivial nature of this expansion.
We are now presented with another competition amongst length scales: the scattering length aversus the interparticle spacing n−1/3 . As the corrections to the mean-ﬁeld energy are all expansionsof na3 , we will refer to this as the “diluteness parameter” of our system. This competition is ofcentral importance to the work presented in this thesis, as our ultimate goal is to probe systems
where na3 is nonnegligible.
In the case of He II, however, we know all of the above treatments must be fundamentally
ﬂawed. The biggest red ﬂag being that in liquid He, the range of the repulsive interaction, a, is9on the order of the interparticle spacing, meaning that na3 ∼ 1. Hence, a perturbative approachmakes little sense. In addition, there is an attractive part to the He potential, which plays an
unknown role [31]. Finally, because He II is a liquid, there also exist correlations not taken into
account with a dilute-gas BEC theory.
With a system that is well described by such theories, however, we aim to tune the interactions
to regimes where na3 is no longer small and test these theories. The hope is that perhaps we would
be left with something more closely resembling our carrot, superﬂuid liquid He.

1.3

In the cart: a spherical cow
Having salivated over the superﬂuid carrot for long enough (albeit held by a stick some 6orders of magnitude away), we are left asking ourselves, “What exactly is it that we are pulling
along in this cart, anyway?” The answer, which assuredly will surprise none of my former labmates,is that we have been pulling along a 2,000 pound cow. A spherical one, at that.
According to Wikipedia, a spherical cow is a metaphor for a “highly simpliﬁed scientiﬁc modelof reality.” Our strongly interacting dilute-gas BEC will be the model system, and the hope is thatwith this we can mimic the peculiar reality of superﬂuid 4 He. Our system is highly simpliﬁed inthat we will be working with a diﬀerent atomic species, in its gaseous phase, no less. The hope isthat some of the properties that have been discussed might be universal enough that they can beseen in this cold atomic system when we also approach regimes of large na3 . Furthermore, as we areable to continuously tune the interactions to such a complicated regime starting from a relatively
simple system (the weakly interacting condensate) we would leave ourselves a trail of breadcrumbs
to ﬁnd our way back home.
In the ﬁeld of cold-atom physics, this has been a highly successful model. Dilute atomic
gases oﬀer amazing amounts of control, and in many cases, a tunable amount of complexity. The
trapping potentials of these systems, for example, are completely determined by the experimenter.
Through manipulation of such potentials, cold-atom systems have been able to demonstrate suchphenomena as Anderson localization [37], Josephson junctions [38] and Mott insulator transitions

10[39], to name a few.
More recently, the push has been to tune the interactions themselves via a magnetic-ﬁeld
Feshbach resonance. This work, primarily done in fermionic systems, has been fruitful in measuring
many universal properties of Fermi systems [40, 41, 42]. Further, one of the leading areas ofresearch in ultracold Fermi gases is not only the mimicking of condensed matter systems, but themeasurement techniques employed to study them as well [43].
For bosonic systems, Feshbach resonances have primarily been used to minimize interactions[44, 45]. As I will discuss in the next section, a major motivation for this line of research is the
avoidance of the major tradeoﬀs inherent in studies of strongly interacting BECs. For stronglyinteracting Fermi systems, such compromises need not be made, as the Fermi statistics nullify such
eﬀects.

1.3.1

A hurried bull: 85 Rb

In the laboratory, we know that a spherical cow is never quite the right picture to paint, as
cows tend to be docile and stationary targets. The systems we are herding tend to be heated andfast moving. In a word, temperamental. We thus reﬁne the metaphor for laboratory work and refer
to our system as the “spherical bull.”Our bull of choice is an ultracold cloud of Bose-condensed 85 Rb atoms. Such a systemoﬀers the same magnetically tunable Feshbach resonance that has proved so useful to the study ofstrongly interacting Fermi systems, albeit with some new tradeoﬀs, unique to bosons (every bull
has its horns, so they say).
Carefully measured in Ref. [46], the s-wave scattering length is described as(a = abg∆
1−
B − Bpeak

)

(1.6)

where abg = −443(3) a0 is the background scattering length, ∆ = 10.71(2) G is the width of the
Feshbach resonance, Bpeak = 155.041(18) G the center of the resonance, and B is the aforementioned
magnetic ﬁeld we tune to manipulate a. A picture of our Feshbach resonance is shown in Fig. 1.2.

11
For all of the work presented in this thesis, we will be studying our condensate in the regimes of
positive scattering length. The parameters for the Feshbach resonance in 85 Rb are listed in Table
1.1. Also included in the Table are the other popular Feshbach resonances in the diﬀerent alkalisthat have been Bose condensed.
Table 1.1: Feshbach resonance parameters for the most popular resonances in the diﬀerent bosonicspecies that have achieved degeneracy. The ratio |∆/Bpeak |, is useful in assessing the feasibility of
studying a resonance, in terms of both the necessary magnetic ﬁeld stability, as well as the range
over which one must alter the magnetic ﬁeld to access large na3 . Much of this information has
been compiled in Ref. [47]. It is interesting to note that, aside from 85 Rb, every other Feshbachresonance listed applies to atoms in a high-ﬁeld seeking state. In other words, ours is the only
resonance occurring in a system that is magnetically trapable. This perhaps speaks to the lackof popularity of magnetic traps in the 21st century. We will later see that it is only through theunique advantages provided by magnetic traps that we are able to probe regimes of large na3 .

abg (a0 )∆(G)
Bpeak (G)|∆/Bpeak |

85 Rb

7 Li

23 Na

39 K

87 Rb

133 Cs-443
10.7
155.040.07-25-192.3
736.8
0.26

62
-1.4
1195
0.001

-29
-52
402.4
0.13

100
0.21
1007.40.0002

1720
28.7
-11.7
2.5

Now cast as experimental matadors, we must be mindful of our bull’s horns. In the case ofstrongly interacting 85 Rb, both horns have to do with the limited timescales available to probe oursystem, with one horn being three-body recombination, and the other rapid density oscillations.
When 85 Rb was ﬁrst considered for evaporation, the large abg listed in Table 1.1 was seen as
promising. It was thought that evaporation would be straight forward, with the rethermalization
rate going as the collision rate, and thus proportional to a2bg . The hidden truth, however, is that in
85 Rb inelastic collisions tend to dominate, with both two-body and three-body mechanisms playing

a prominent role. Eventually, experimenters were able to overcome inelastic loss, either with morestages of laser cooling prior to evaporation [48] or via sympathetic cooling [49, 50], and achieved
temperatures low enough for condensation.
The problem of inelastic, three-body recombination would prove to be a stubborn one, however. The rate for this process, by which two atoms form a bound molecule and a third atom carries

120scattering length (a )

10000

5000

0

-5000
150

155

160

165

magnetic field (G)Figure 1.2: The donkey pulling the experimenter’s cart: the 85 Rb Feshbach resonance at 155 G.
Using this scattering resonance, we tune the scattering length from small a, corresponding to weakinteractions (where the physics is well understood), to large a, corresponding to strong interactions.
We probe the strongly interacting regime to better understand the beyond-mean-ﬁeld physics ofstrongly interacting superﬂuids. The resonance is 11 G wide, allowing for a precise level of control,
which makes 85 Rb ideally suited for studying such physics.

13
away the necessary energy and momentum, scales as a4 [51]. The eﬀects of this are both loss, as
the atoms are no longer in their initial internal state, as well as heating due to the large amount ofenergy given to the third atom; this makes studying a BEC with large na3 diﬃcult.
An example of just how sharp the three-body horn is can be seen in the experiments done on
the ﬁrst 85 Rb BECs by Roberts et al., where they were trying to measure beyond-mean-ﬁeld physics[52]. These spectroscopic experiments measured the breathing modes, or density oscillations, ofa trapped gas of 85 Rb near the Feshbach resonance. The oscillation frequency was known to be
modiﬁed by the mean-ﬁeld energy [53], and it was thought that the additional LHY correction could
be measured in this same way. The experiment’s undoing, however, was the aforementioned threebody recombination. Because the timescale for three-body events were faster than the timescale
for the spectroscopy measurements (essentially the period of the trap) too much loss and heating
was incurred and an LHY measurement proved impossible.
Further experiments of strongly interacting 85 Rb were performed in our group, using a secondgeneration BEC machine, and these met with success, having dodged the three-body horn by taking
advantage of Bragg spectroscopy, a relatively quick measurement compared to 3-body rates [54].
These experiments, however, were not able to get out of the way of the second horn, and were
ultimately limited by the same density oscillations that were originally being measured by Roberts
et al.. Probes of high na3 , after all, need a relatively constant n. This horn is unavoidable as well,as the size of the condensate in the trap is set by the interactions in the gas. Hence, when we use
the Feshbach resonance to change a, the BEC is no longer at its equilibrium size and a breathe
oscillation ensues.
Such is life for our experimental bull ﬁghter. In order to have success in his encounter with85 Rb, he must not only probe the beast quickly (before being poked by the horn of three-bodyloss), but also distract the bull long enough so as to avoid the horn of rapid density oscillations.
Playing the role of the red cape used to distract the bull will be the relatively weak, spherical traps
made possible with magnetic conﬁnement. This suits our matador just ﬁne, as he wasn’t planningon winning any popularity contests in the 21st century anyway.

14

1.4

Contents
This thesis, then, describes the design, construction and execution of both apparatus andmeasurements probing the diﬀerent lengthscales of an interacting Bose gas, while maximizing thelimited timescales available for such measurements.
In Chapter 2 I describe the implementation of our spherical magnetic trap, which was the
major experimental change made to circumvent the limitations of rapid density oscillations. The
third chapter lays the conceptual and analytical groundwork of Bragg spectroscopy, which is oneof the few workable probes we have for these quickly decaying systems. In Chapter 4 I present
a novel photon-counting technique for Bragg spectroscopy whereby the measurement is made on
the Bragg light rather than the atoms. This chapter, together with Chapter 3, is intended to bea self-contained “cookbook” for the experimenter that would want to set up a similar system. In
Chapter 5 I present our latest experiments using a diﬀerent kind of spectroscopy based on a shortrange quantity called the contact. Contained therein is preliminary data of what we hope will be
our most accurate probe of the LHY correction. Finally, in Chapter 6 I touch upon the future
directions of the strongly interacting BEC machine.

Chapter 2

BEC in a weak, spherical trap

In this chapter, I present our development of a trap and cooling scheme that has allowed
us to conﬁne 85 Rb BECs in a weak, spherical trap, which is necessary for the study of stronglyinteracting condensates. I will show that the dynamics of BEC systems are dominated by themean-ﬁeld energy. We model these dynamics, allowing us to characterize the breathing modes
induced from changes in the scattering length. Through this modeling we can see the beneﬁts ofa weak, spherical geometry. The trap design will be presented, as will the evaporation scheme wehave developed, and which we believe to be novel in atomic physics experiments.

2.1

The ansatz: ramping aCentral to the work of this thesis is the concept of establishing equilibrium after ramping

the scattering length in time. This ramp is necessary, as we cannot create BECs at large na3 dueto large inelastic loss rates. Hence, to probe these strongly interacting regimes we must do so
dynamically. A key to this approach, then, is the necessary time to establish such equilibrium. In
other words, we will need to evaluate the “quickness”, or conversely, the adiabaticity of our ramp
in a.
The timescale associated with our ramp rate is set by (ȧ/a)−1 , to which we can compare
diﬀerent criteria for adiabaticity. In the past, we have argued [54] that the timescale to establish
local, many-body equilibrium is set by ma2 /h̄. For cases of a > 0, this is the timescale associatedwith the binding energy of a molecule. One could argue that the lengthscale, a, associated with

16
this timescale, is the lengthscale that sets the interactions of the many-body physics we study. This
timescale for establishing equilibrium also happens to be the shortest and least demanding of thevarious arguments presented here.
A stricter criteria would be the timescale associated with the interparticle spacing, n−1/3 .
From the Thomas-Fermi approximation, we can write the peak density of a trapped BEC as [55]mωhonpk = µ/g =
8πh̄a

(

15N a
aho

)2/5

(2.1)

where ωho = (ωρ2 ωz )1/3 is the geometric mean angular trap frequency, aho =√h̄/mωho the har-monic oscillator length, and N the number of atoms in the condensate. This criteria for achieving−2/3

equilibrium is given by mnpk/h̄.

Finally, one might be a true “stickler”, and argue that the most important lengthscale (in−1/3

terms of being locally adiabatic) of the system is not a or npk , but rather the healing length,√
ξ, where ξ = 1/ 8πna. This is also the timescale of the chemical potential, or h̄/µ. For theexperiments presented in this thesis, our system will always be in the Thomas-Fermi limit, where
the dominant energy scale of the system is the interaction energy. We can thus write the chemical
potential, µ, as [55]µ = gn(2.2)

where gn will be referred to as the interaction energy, set by g = 4πh̄2 a/m (as deﬁned in Ch. 1)
and the density n. Ignoring prefactors, this timescale also goes as m(na)−1 /h̄. One could argue
that this is in fact the relevant timescale, since the dominant energy of the system is, by deﬁnition,
the chemical potential. We consider this the strictest criteria for achieving equilibrium.
How slowly one needs to ramp a in order to be adiabatic with respect to local, many-bodyequilibrium is an open question that is not answered in this thesis work. The hope is that future
experiments, taking advantage of the previously unattainable time resolution oﬀered by the trap
geometry presented in this chapter will shed light on this outstanding question.

17

2.2

Motivation
As discussed, creating BEC of 85 Rb atoms has historically been a challenging endeavor. Aftercompleting the 2008 Bragg experiments [54], we had a system capable of studying strongly interacting condensates, yet we elected to essentially scrap that design in favor of something new, which
would add further complications to an already diﬃcult experiment. The most obvious question is,“why?”

2.2.1

Spherical cows, spherical traps, and BEC dynamics

Just as the spherical cow provided a simpliﬁed model of reality, the spherical trap, provides
a simpliﬁed realization for our model. In other words, working with such a trap should make theexperiments performed therein easier for the concerned theorist. In our 2008 publication [54], we
presented no fewer than ﬁve diﬀerent theoretical interpretations, knowing all ﬁve to be wrong in
one way or another. I mention this to illustrate the theoretical complexities associated with BEC
in the strongly interacting regime. In simplifying the trap geometry, we garner achieve a systemthat is that much easier to interpret for the theorists.
The second, and driving, motivation for implementing a spherical geometry is the need toprobe our system for longer times. As discussed, we are only able to produce condensates atreasonably small values of a ∼ 100 a0 , and so must then quickly change a to probe large values ofna3 . The equilibrium condensate size, however, is set by the interaction energy, gn, in the ThomasFermi limit. We express this as the radius of the BEC in the x direction, in terms of N and a, fora harmonic conﬁning potential
1
Rx =
ωx

√

√

1
2µ
=
m
ωx

h̄ωho
m

(

15N a
aho

)1/5

.

(2.3)

This sets another timescale, which is the timescale to establish an equilibrium density proﬁlein the trap. This time can be deﬁned as Rx /cBEC , where cBEC =√gn/m. If we again ignore

prefactors Rx /cBEC goes as 1/ωx . However, a quick (relative to 1/ωx ) ramp in a is unavoidable forthe reasons discussed. The result is an induced expansion to accommodate the extra interaction

18
energy that was just dumped in. As we will see, the subsequent BEC dynamics, in particulardensity oscillations induced by a sudden change in a, give rise to a density that is rapidly changing
in time. This time-varying density will ultimately limit the time over which we can probe the large
na3 condensate.
We take, for example, our Bragg experiments of 2008 where the measurements were completed
before the density fell to 30% of its original value. At the largest values of na3 , this criteria allowed
for interrogation times of 90 µs. Using square pulses of light, this corresponds to a Fourier-limited
rms width of 4 kHz ([49]). This makes observations of beyond-mean-ﬁeld LHY corrections to the
BEC energy, which are predicted to be on the order of 400 Hz, experimentally quite diﬃcult. Our
aim in designing a new trap is to be able to probe the condensate for longer times, as well as suﬀer
less change in our overall density. This was our primary motivation for developing a spherical trap.

2.2.1.1

The Pérez-Garcia model

To better understand the breathe mode excited by a rapid change in a, we use a model
developed by Pérez-Garcia et al. [56] to simulate the BEC dynamics. This is a variational modelthat describes the condensate with a simple gaussian density proﬁle that has a characteristic rms
width of wρ (wz ) in the radial (axial) direction. The results are given in the form of these coupled
diﬀerential equations [56]ẅρ + ωρ2 wρ =h̄2m2ẅz + ωz2 wz =

h̄2
m2

(

(

1
+
w3ρ
1
+
w3z

√

√

2 Na
π w3ρ wz
2 Na
π w2ρ w2z

)

(2.4)
)

(2.5)

The stationary solutions for the widths (wρ0 , wz0 ) are given by the real solutions of [56]√42
2/3 wz0 wρ0= w z0 +
Na
λ
4
π
aho

w4z0 wρ0= wρ0 +
λ4/3 a4ho

√

2
wz
Na
π
wρ

(2.6)

(2.7)

19
where λ = ωρ /ωz is the aspect ratio of the trap. For non-critical calculations, where a 10% error in
the size is of no consequence, we can approximate these stationary solutions with analytic solutions
for wρ0 and wz0 , given by [56](√wρ0 = λ

1/45

2/π
15

(√

w z0 = λ−28/45

)1/5

2/π
15

Rρ

(2.8)

)1/5

Rz .

(2.9)
√

These solutions are valid in the Thomas-Fermi limit, which is deﬁned here as

2
π N a ≫ ahoρ . Thismodel is most useful for scaling arguments, because although the condensate density proﬁle is nota simple gaussian, its radius is linear with w.
Using the Pérez-Garcia model, we can simulate the predicted breathing modes induced by a
rapid change in a for condensates. For example, we consider 4 × 104 BEC atoms and a mean trap
frequency of ωho = 2π × 40 Hz, which are conditions similar to our 2008 experiments in Fig. 2.1.
We plot the normalized density of the BEC versus the time after a 10 µs ramp from a = 100 a0 to
a = 500 a0 . As one can see, a breathe oscillation is induced, and the density decreases versus time.
The diﬀerent colored lines in Fig. 2.1 correspond to traps with diﬀerent aspect ratios λ =ωρ /ωz . The simulation makes apparent the beneﬁts of working with an aspect ratio of 1 (red line),
which corresponds to a spherical geometry. With other aspect ratios, with the same geometricmean trap frequency, there are actually two breathe modes induced corresponding (in the noninteracting case) to twice the radial and axial trapping frequency. This is most easily seen in thetime dependence of the sizes of the clouds, as shown in Fig. 2.2. The radial frequency corresponds
to the faster oscillation (Fig. 2.2(b)) while a slower breathe (Fig. 2.2(c)) occurs due to the axial
conﬁnement. It is the shorter timescale radial breathe that dominates the initial density evolutionof the condensate. This motivates us to implement a spherically symmetric trap which has nopreferred radial direction.

normalized density

20

1.0

= 1
= 5
= 50

0.8

0.6

0.4

0.2

0.0
0

1

2

time (ms)Figure 2.1: Simulated breathe from a quick jump in a. The density (n), normalized to the initial
density, is displayed on the vertical axis. On the horizontal axis is time, measured after the jumpfrom a = 100 a0 to a = 500 a0 . Diﬀerent colored curves represent traps of diﬀerent aspect ratios
λ = ωρ /ωz . A spherical aspect ratio (λ = 1, red curve) gives the slowest breathe and hence thelongest interrogation times at a relatively constant n. This simulation is done with a BEC of 4×104
atoms and a mean trap frequency of ωho = 2π × 40 Hz.

n
n

(a)

21

1.0
0.8
0.6
0.4
0.2

wwr r

(b)

2.0
1.5
1.0
0.5

wwz z

(c)

1.5

1.0

0.5

Figure 2.2: Illustrating how the larger trap frequency (ωρ ) sets the timescales for breathe. This
simulation is done with a trap whose radial conﬁnement (ωρ ) is ﬁve times larger than the axial
conﬁnement (ωz ), giving an aspect ratio of λ = 5. In (a) we plot the normalized density (verticalaxis) against time (horizontal axis). The density oscillations correspond to breathes of the radialsize, shown in (b), which in turn correspond to ωρ . The overall slope of the density oscillations(which can be considered the density time-averaged over 1/ωρ ) correspond to the slow increase in
the axial size of the condensate, which likewise corresponds to ωz .

22
2.2.2

The weak shall inherit the trap

If we are going to all the trouble, then, of redesigning the trap for a spherical geometry, we
would have some ﬂexibility as to the mean trap frequency as well. From Eq. 2.1, we see that alower mean trap frequency, while perhaps allowing us to probe longer, would also cost us in the
6/5

initial density, which scales as ωho . Hence, to probe the same value of na3 in a weaker trap, wemust ramp to a higher ﬁnal value of a. We have simulated the dynamics that ensue for ramps tona3 = 5.4 × 10−4 and plot these density oscillations in Fig. 2.3(a). This value of na3 corresponds
to an LHY correction to the energy density of 11%. As one might guess, the weaker traps always
win this gambit and allow for the longest possible probes of BECs with large na3 .
While weaker trap conﬁnements will make work more accessible from an experimental standpoint (e.g. computer control of ms pulses is much easier compared to µs pulses), there is no beneﬁt
in terms of resolving the interaction energy, as shown in Fig. 2.3(b). Here we see density oscillations
plotted against the normalized time after the ramp. When considering direct probes of the interaction energy, the relevant normalization is the timescale associated with the chemical potential,h̄/µ. What Fig. 2.3(b) illustrates is that regardless of ωho , we suﬀer the same amount density dropwhen comparing the breathes at constant tµ/h̄. We nonetheless set out for a weak geometry as
earlier attempts to create 85 Rb BEC in traps of large ωho proved unsuccessful [49].

2.3

Trap design
Having laid out the gauntlet, our challenge loomed: design a weak, spherical trap to workwith 85 Rb BECs near the 155 G Feshbach resonance. Magnetic traps oﬀer the advantage of being
extremely ﬂexible in their design, with the trap aspect ratio being completely controllable via therelative amounts of currents in the magnetic trap coils. In addition, weak geometries imply large
amounts of gravitational sag. This sag introduces undesirable anharmonic components in an opticaltrap design (the other standard for cold atom experiments) that magnetic traps are inherently lesssensitive to, due to the large coil spacings that deﬁne the trapping potential.

normalized density

23

(a)

1.0

0.8

0.6

0.4
ho

0.2

ho

ho

= 2

x 10 Hz= 2

x 20 Hz= 2

x 40 Hz

0.0
0

1

2

normalized densityt (ms)

(b)

1.0

0.8

0.6

0.4
ho

0.2

ho

ho

0.0
0

= 2

x 10 Hz= 2

x 20 Hz= 2

x 40 Hz

2

4

t

/

Figure 2.3: Breathing mode induced by a quick jump of a to achieve a large na3 = 5.4 × 10−4 .
In (a) the vertical axis is the normalized density, and on the horizontal axis is the time after thejump to large a. Diﬀerent colored curves represent traps with diﬀerent mean trap frequencies.
This illustrates the advantage of working with a relatively weak trap, which gives the longest
interrogation times possible at constant density. In (b) we plot the same normalized density breathe,
only now versus the time after the jump normalized to h̄/µ, the timescale associated with thechemical potential. This demonstrates how even for the slower breathes at small ωho (seen in (a)),
we are still subject to the same drop in density for the same tµ/h̄ .

24
Our magnetic trap is in the Ioﬀe-Pritchard style; more in-depth discussions of the IoﬀePritchard design can be found in [57, 52], while details particular of our geometry are in Ref. [49].
What we have done in this work is reconﬁgure the currents and provide an additional vertical ﬁeld
(supplied by the cart coils) to achieve our desired trapping potentials and bias ﬁeld.
The relevant parameters for the coils used in our magnetic trap are listed below in Table
2.1, including the currents we presently use to make spherically symmetric BECs. The bias ﬁelds,
gradients and curvatures are all taken from Scott Papp’s thesis [49], except for the cart parameters,
which are new to this trap.
Table 2.1: Relevant parameters measured for the coils used in the 10 Hz spherical trap. Many ofthese can be found in Scott Papp’s thesis [49]. The currents we list here for the trap might change
as much as a few percent (in particular the cart coil) for slightly diﬀerent optical trap alignments,
as the currents are used to align the magnetic trap to the optical trap. Currents were measured
using the in-loop Hall current sensors, September 2010.

Bias ﬁeld (G/A)
Gradient (G/cm/A)
quadratic (G/cm2 /A)
current (A)Bias (ẑ)
1.61

Pinch (ẑ)
0.515

Ioﬀe bars (ρ̂)

cart (ŷ)
1.02

1.04
0.097
47.3

0.695
167.2

167.2

30.2

To give a conceptual picture of the Ioﬀe-Pritchard design, we present a cartoon of the coils
involved in the trap in Fig. 2.4. What this geometry gives you is a large bias ﬁeld in the ẑ direction,with the magnitude of the magnetic ﬁeld having a local minimum in the center of the trap. The⃗ Because it ispotential felt by our atoms is due to the Zeeman energy, where UZeeman = µ|B|.
only this magnitude of the magnetic ﬁeld we care about, the ﬁelds orthogonal to ẑ contribute to
the trapping potential only by adding to the large bias ﬁeld in quadrature.
Roughly half of the large bias ﬁeld (B0 ) in ẑ is provided by the so-called bias coils, depictedin blue in Fig. 2.4. These are in a Helmholtz conﬁguration, and function only to produce thisuniform ﬁeld. For the previous, all-optical trap, which was used for our 2008 experiments, thesecoils provided the entirety of the bias ﬁeld, as well as the axial conﬁnement due to a slight curvature,

25

y
x
z

Figure 2.4: Cartoon picture of coils used to make the 10 Hz spherical trap. Bias coils in blue,
arranged in a Helmholtz conﬁguration, produce a uniform ﬁeld and provide roughly half the bias
ﬁeld along ẑ. The smaller loops of coil in orange are the pinch coils, providing the other half of thebias ﬁeld as well as conﬁnement along z. The straight, orange wires are the Ioﬀe bars, and provide
radial conﬁnement when their linear, quadrupole gradient is added to the large bias ﬁeld along z.
Large black loops seen at the top and bottom are referred to as the cart coils. These provide asmall bias ﬁeld (∼ 30 G) in the vertical direction and are used to adjust the center of the magnetic
potential in y.

26(see Table 2.1). This curvature, with a minimum at the center of the coil pair, results from thespacing between the coils being slightly larger than in the standard Helmholtz conﬁguration.
Also seen in Fig. 2.4 is the pinch coil pair in orange. The current ﬂows through these coils inthe same direction as in the bias coils. This is not standard for the Ioﬀee-Pritchard trap, but wefound it necessary to achieve our desired bias ﬁeld with weak conﬁnement. The function of thesecoils is two-fold. First, they add to the bias ﬁeld, providing the other half of the needed 160 G.
The ﬁeld from the pinch coils also has a curvature along ẑ, providing parabolic conﬁnement. This
curvature is the dominant contributor to the axial conﬁnement of the atoms.
Also seen in orange are the Ioﬀe bars, depicted by straight lines in Fig. 2.4. These bars sharethe orange color of the pinch coils to illustrate that they are electrically in series with the pinch
coils. The function of these coils is to provide radial (ρ̂) conﬁnement to our atoms. Although
the gradient they produce is linear (βρ ), the resultant trap is parabolic in ρ. The key here is thelarge bias ﬁeld (B0 ) in the ẑ direction. Again, the Zeeman energy of our weak-ﬁeld-seeking stateis proportional to the magnitude of the magnetic ﬁeld because the magnetic moment of the atom
adiabatically follows the ﬁeld direction. When we add the bias ﬁeld to the linear gradient producedby the Ioﬀe we are left |B| =√B02 + βρ2 ρ2 . For a large enough bias ﬁeld (B0 ≫ βρ ρ), this can beapproximated by |B| ≈ B0 + 12 (βρ2 /B0 )ρ2 , giving us a trap harmonic in ρ.
Finally, in black is another pair of Helmholtz coils, which we refer to as the cart coils. These
work in the same way as the bias coils, providing a constant ﬁeld, Bc , only this time in ŷ. The roleof the cart coils is to shift the center of the magnetic trap position vertically, by eﬀectively shifting
the zero of the magnetic ﬁeld produced by the Ioﬀe bars. This is illustrated in Fig. 2.5. If we focus
on the ﬁelds in ŷ, then By = βy (y − y0 ), where y0 = −Bc /βy is the new center of the Ioﬀe gradientin y. This trap center shift will prove necessary to overlap this trap with the trap used for earlier
stages of the experiment. This is required because the ﬁnal conﬁning potential in the new trap isso weak that gravitational sag is substantial (roughly 2.5 mm). Without the addition of the cartﬁeld, the atoms could not be loaded cleanly from earlier cooling stages of the experiment. This, wewill see, is a key design consideration.

27

(a)
y
x
z

(b)
y
x
z

Figure 2.5: Cart coils lifting the center of the quadrupole potential for alignment of the spherical
trap in y. In (a) we see both the direction and magnitude of the ﬁeld generated by the Ioﬀe bars
in the orange vectors. The black vectors represent the uniform ﬁeld from the cart coils. In (b) we
see the resultant ﬁeld in blue. Notice this looks just like the Ioﬀe ﬁeld (orange) in (a), only now
oﬀset in y, eﬀectively shifting the center.

28
The resultant trap potential is modeled and shown in Fig. 2.6. An unexpected aspect of ourspherical trap is the rotation of the primary axes of the trap by roughly 45 degrees, as seen in
Fig. 2.6(c). We have found that this symmetry breaking of the primary axes of the trap is dueto the pinch coils. Because they provide conﬁnement along the ẑ direction, we know that there
must also be some curvature along the other directions to fulﬁll the requirements of Gauss’ law,
∇ · B = 0. Invoking the known symmetry of the pinch coil geometry, we can assume that the ﬁeld
goes as Bpinch = (B0 + B ′′ z 2 )ẑ − B ′′ zxx̂ − B ′′ zy ŷ. We notice there are two crossterms here, B ′′ zxx̂and −B ′′ zy ŷ. Because our trap is always centered in x, but oﬀset in y due to gravitational sag(ysag = βmg
2 B ), it is the last term, in ŷ, whose crossterm features both z and y, that breaks the0
symmetry for a non-zero value of y. We only return to the primary axes of the trap if we eliminate
both the eﬀects of gravity as well as the cart ﬁelds we use to center the trap.

(a)

(b)

(c)
or
in
m

or
aj

m

Figure 2.6: Contour plots for the 10 Hz spherical trap, as seen along the ŷ (a), ẑ (b), and x̂ (c)direction. Darker colors represent lower potential energy. By eye it is clear that the trap is fairly
spherical, except for the plots in (c). The reason for this are discussed in the text. We will deﬁnethe “long” axis of the trap, with the weaker trap frequency as the minor axis (y = z), and theorthogonal direction, with the tighter trap frequency, as the major axis (y = −z).

Finally, we can ask ourselves how good we have done as trap-builders by measuring the trap
frequencies directly with the atoms. The results of these measurements are shown in Fig. 2.7. Here,
we have rotated the axes of the ﬁts by 45 degrees with respect to the primary axes of the trap(the vertical, ŷ, and axial, ẑ, direction) so that the y ′ and z ′ positions correspond to the major
and minor axes, respectively. We then ﬁt the data to a sinusoidal oscillation. We measure trap

29
frequencies of 10.21(5) Hz in the x̂ direction, and frequencies of 10.41(4) Hz and 9.39(7) Hz for the
major and minor axes of our trap. This ultimately gives us an aspect ratio of 10.41/9.39 = 1.1.
Considering that our previous optical trap had an aspect ratio of 50, we consider this a successful
endeavor.

2.4

Making 85 Rb BEC
In this section, I outline the cooling scheme we have developed for 85 Rb. The ideas presented

here were new to us, and exploring these ideas represented some of the most exciting times in thelab. The basic concept is the same as that of many of the dual-species machines: use 87 Rb to
sympathetically cool the more stubborn species, which in our case is 85 Rb. Sympathetic coolingprovides the advantage of avoiding the scattering processes of 85 Rb that proved so painful for theoriginal 85 Rb machine [58, 59] by outnumbering and overwhelming them with the good elastic
collisions provided by another species. What we have developed is a new twist, with a coolingscheme that is truly sympathetic, empathetic even, all the way to quantum degeneracy. The
results of our work are large, round, stable condensates and a scheme that could be readily applied
to other systems as well.
We will breakdown each stage of the experiment in the following sections, but begin witha general overview diagrammed in Fig. 2.8 to help facilitate an understanding of both the entire
experimental cycle, as well as the evaporation scheme used with the hybrid trap (a combination of
the weak spherical trap and the optical trap), which is being presented for the ﬁrst time. Muchof the cooling process is standard in cold-atom experiments. The atoms are cooled from room
temperature via optical cooling performed in a magneto-optical trap (MOT). From the MOT, weload atoms into a quadrupole trap (QT) to transfer them to a high-vacuum region of the experiment
where they will be further cooled. It is here that 85 Rb is ﬁrst sympathetically cooled, with direct
radio frequency (rf) evaporation performed on 87 Rb in the traditional sense [60].
From the magnetic trap, we load the 85 Rb and 87 Rb gases into our hybrid trap. This isdone in two stages as ﬁrst we conﬁne the atoms optically. We then turn on the 10 Hz spherical

30

(a)

x position (pixels)

690

680

670

660
0

100

200

300

400

500

time (ms)z' position (pixels)

0

(b)

-5

-10

-15

0

100

200

300

400

500

time (ms)y' position (pixels)

155

(c)

150

145

140

0

100

200

300

400

500

time (ms)Figure 2.7: Slosh data for each of the principal axes of the trap, obtained from images of a thermal
gas taken after time-of-ﬂight with a 28 ms expansion time. This data gives trap frequencies of 10.21
× 9.37 × 10.41 Hz for the x̂ (a), minor (b) and major (c) axes. For the minor and major axis sloshdata, we have had to take our normal images, oriented in ŷ and ẑ, and rotate them by 45 degrees.

20

a

10

10

8

9

15 20

|3,mF>

optical|2,mF>

opticaltransfer
5

quadrupole
trap

quadrupole
trap

.001

25

|2,-2>|1,-1>

opticalpumping

10 mK
107

55

80

magnetic: 250 Hz
magnetic: 14 Hz

10 mK
6
10

sympathetic

magnetic: 215 Hz
magnetic: 13 Hz

rf evap

MT evap

rf removal

evap

10

opticalmagnetic: 10 Hz

sympathetic

opticalmagnetic: 9 Hz85crf evap<< Tc
4
NBEC = 4x10

rf evap

Rb evap2

92

bmagnetic: 10 Hz
magnetic: 10 Hz

90

optical

< Tc
5
10

< Tc

evap out of optical confinement,
rf removal from magnetic confinement

Hybrid trap evap

Figure 2.8: Big picture view of the duty cycle and cooling of our 85 Rb-87 Rb machine. Time is outlined on the horizontal axis (not to scale),with the respective parameters for 85 Rb and 87 Rb detailed in the ﬁgure. The hybrid trap evaporation is what we will be introducing in thissection, as many of the concepts are new, and have not been discussed prior to this thesis. As we see from the ﬁgure, the conﬁnement isdue to the combination of a magnetic trap and an optical trap. The optical trap serves to both provide a mechanism to preferentially cool
87 Rb as well as to spatially overlap 87 Rb and 85 Rb, for eﬃcient sympathetic cooling. The circled letters in the ﬁgure are further clariﬁed
as follows: (a) Note the 85 Rb MOT begins at 15 s. (b) During the 85 Rb evaporation the optical potential is giving way to the sphericalmagnetic trap, both cooling 85 Rb (via rf evaporation) as well as loading the weak, spherical trap. (c) This rf evaporation is diﬀerent thanmost in that the rf knife is at a constant frequency versus time, while the aforementioned optical trap is weakened. This will be discussed
more in Section 2.4.3.

duration (s)elapsed time (s)opticalconfinement(power)cooling methodtemperature
numberinternal stateradial confinement
axial confinementRb85

cooling methodtemperature
numberinternal stateradial confinement
axial confinement

Rb

87

MOT

31

32
magnetic trap described previously. The optical trap is aligned so that its position coincides withthe equilibrium position of the 85 Rb potential. This optical trap position, however, is verticallyoﬀset from the equilibrium position of the 87 Rb potential due to the diﬀerential sag experienced
by each species due to their diﬀerent magnetic moments. Now lowering the optical potential has
the eﬀect of normal optical trap evaporation for the 87 Rb gas. The 85 Rb gas, on the other hand,is never directly evaporated on, but is spatially overlapped with 87 Rb, giving prime conditions forcooling all the way to quantum degeneracy sympathetically.

2.4.1

Laser cooling and the magnetic trap evaporation

The ﬁrst step of bringing atoms from room temperature to quantum degeneracy is achieved
via laser cooling in a magneto-optical trap (MOT). We ﬁnd that we have better overall performanceby loading more atoms from the MOT into the quadrupole trap (QT) by loading the atoms into atight trap, which can conﬁne atoms in the |F = 3, mF = 2⟩ and |F = 3, mF = 1⟩ states in addition
to the |F = 3, mF = 3⟩ stretched state (previously only the stretched state was loaded). This eﬀect
can be seen in Fig. 2.9. Because we later optically pump the atoms into the desired ﬁnal spin state,|F = 2, mF = −2⟩, (after transferring the gas from the QT to the magnetic trap), we are not as
worried about atoms in other, non-stretched, spin states. In the end, higher 87 Rb number is betterfor our evaporation performance, since most of our cooling comes from the evaporation of 87 Rb
atoms.
We load a much smaller number of 85 Rb atoms than 87 Rb atoms in the MOT so as to notload down the later stages of evaporation with the additional heat capacity. The reduction in thenumber of 85 Rb atoms is achieved with a couple neutral density ﬁlters that attenuate the 85 Rb MOT
light to 50% of its previous value. We estimate that we now load an 85 Rb to 87 Rb number ratioof ∼ 1:10 in the two-species MOT. The atoms are further optically cooled (compressed MOT, see
Ref. [49]) and subsequently loaded into a quadrupole trap and transferred to the science chamber.
This leads us to the ﬁrst stage of evaporation, which is performed in the magnetic trap
(MT). This trap is described in Ref. [49] and provides trapping for both 85 Rb and 87 Rb atoms in

33

6

20
0.5

10
numbers taken after MT evap

was at -0.85 V for spin state filtering
leave at -1.5 V for maximum number

00

Rb number (x10 )

30

-1

-2

-3

0.0
-4

85

87

6

Rb number (x10 )1.0

QT load control voltage (V)Figure 2.9: Quadrupole trap (QT) load after the MOT. In black, is the 87 Rb number after magnetic
trap evaporation (left hand axis). In red is the 85 Rb number at the same stage of cooling, with thecorresponding number on the right hand axis. Previously we worked with a weaker initial QT load
(23 G/cm), which corresponds to a smaller current (horizontal axis), because of the additional spinﬁltering this scheme provided. We later found this unnecessary for our new evaporation scheme,
and now we load more atoms into the QT at a larger current (40 G/cm).

the |F = 2, mF = −2⟩ and |F = 1, mF = −1⟩ states, respectively, with measured trap frequencies of√

215 × 12.5 Hz for 87 Rb. The corresponding trap frequencies for 85 Rb should be ω87 µ85 /µ87 , with
µ85 and µ87 the corresponding magnetic moments of the 85 Rb and 87 Rb species being trapped, andω87 the corresponding angular trap frequency for 87 Rb. In this trap, the 87 Rb gas is evaporatively
cooled using rf transitions driving the 87 Rb atoms to the |F = 1, mF = 0⟩ state. The 85 Rb
gas is cooled sympathetically through its thermal contact with the 87 Rb gas. We evaporate to atemperature of 10 µK, ﬁnding that this gives the best conditions for 85 Rb later on in the experiment.
After the rf evaporation in the magnetic trap, we load the gas into the optical trap. It should
be noted here that the laser for the OT is no longer the 1030 nm ELS laser mentioned in Ref.
[49]. We now use a multi-mode ﬁber laser at 1090 nm manufactured by SPI Lasers. After the ELS
laser died, we tried a 1064 nm multi-mode ﬁber laser, but found that this light was resonant witha 85 Rb molecular transition, and we were not able to make condensates. We turn on the opticaltrap while the MT is still on, and then turn oﬀ the MT conﬁnement. In Fig. 2.10, we look at the

87

Rb scaled number

34

1.0

0.5

0.0
-5

0

5

10

15

20

Horizontal OT alignment (ticks)

Figure 2.10: Optical trap loading techniques. On the vertical scale is the 87 Rb number, normalized
to the peak number, after one stage of optical trap evaporation. On the horizontal axis is the
position of the optical trap at the atoms, in the experimental units of “ticks” on our knob used foralignment. We ﬁnd that by loading the optical trap earlier, while the magnetic conﬁnement is stillon (green points) we are less sensitive to the optical trap position than if we had used the techniqueof “catching” the atoms after turning oﬀ the magnetic trap (black points). This will prove crucial
for optimizing both the load from the MT into the OT as well as the subsequent load from the OT
in to the weak, spherical trap.

35
sensitivity of the OT load to the relative alignment of the MT and the OT. This is an importantconsideration since later on we will see that the alignment between the OT potential and that of
the weak, spherical trap is crucial, but that the spherical trap center is not necessarily aligned withthat of the MT. In the ﬁgure, we see that the current OT loading procedure is less sensitive to the
OT alignment than a sudden transfer technique that we used previously.

2.4.2

Hybrid trap evaporation and the grooming of a 85 Rb BEC

We would like to take advantage of the sympathetic cooling aﬀorded in an optical trap, but
still be left with the spherical trap geometry we carefully designed for our new magnetic trap. At
the end of the day, our solution seems somewhat straightforward: physically combine the two. We
refer to this scheme as our “hybrid trap”, and the evaporation and sympathetic cooling of atoms
trapped in this conﬁguration will be discussed in this section.
To develop a conceptual picture, we model the trap potentials for both 85 Rb and 87 Rb, inFig. 2.11(a) and (b) respectively, at an early stage of the hybrid trap evaporation. On the verticalaxis we show the total potential energy provided by the optical and magnetic traps, including theeﬀect of gravity. The magnetic trap provides a broad, parabolic conﬁnement, while the deep opticaltrap gives rise to a small dimple. The magnetic trap potential is diﬀerent for 85 Rb and 87 Rb atoms
due to the slightly larger magnetic moment of 85 Rb (µ85 = 23 µB , µ87 = 12 µB ) for the spin states
we cool. While in Fig. 2.11 one might be able to tell that the 87 Rb trap is√

3/4 times weaker, themost striking eﬀect of the magnetic moment diﬀerence is on the equilibrium position of the broad
conﬁnement when including gravity. This diﬀerence in sag is given by∆ysag =

g
2B ′′

(

m85 m87
−
µ85
µ87

)

(2.10)

where B ′′ is the curvature of the total magnetic ﬁeld. In the 10 Hz spherically symmetric trap
∆ysag = 1.7 mm, and this is a primary reason why sympathetic cooling all the way to quantum
degeneracy is not feasible with this trap.
We choose the position of the optical trap potential to coincide with the center that of the

36

(a)

85

Rb

(b)

87

Rb
evaporation

Figure 2.11: Modeled potentials during the hybrid trap evaporation for both 85 Rb (a) and 87 Rb(b). The vertical axis is potential energy, in units of µK, while the horizontal axis is the positionin the vertical, y direction, in units of µm. The broad conﬁnement is due to the magnetic trap,
while the tight potential is due to the optical trap. During the hybrid trap evaporation the depthof the optical trap potential is lowered. For 85 Rb (a) the result is a trap frequency that is getting
smaller and smaller as the optical trap potential is lowered. For 87 Rb the eﬀect of lowering theoptical potential is the same as how normal OT evaporation works, with the most energetic of the
87 Rb atoms spilling over the lip of the potential towards −y. These freshly evaporated atoms arelater hit with an rf knife, resonant with the |F = 1, mF = −1⟩ to |F = 1, mF = 0⟩ transition,leaving the newly untrapped atoms free to fall under the force of gravity. With this hybrid trap
evaporation, we are able to cool 85 Rb with minimal number loss, as 87 Rb is the species directly
being evaporated.

37
broad trapping potential for 85 Rb atoms. At the start of the hybrid trap evaporation, there is nomagnetic conﬁnement at all, and the deep optical trap is the only potential the atoms know. Thismeans that the 85 Rb and 87 Rb gases are well overlapped. When the magnetic trap is ﬁrst ramped
on, the optical potential is so deep that, for the most part, the atoms are oblivious.
Of course, we are not in the business of loading optical traps, but rather evaporating towards
degeneracy. In the hybrid trap evaporation scheme, this is achieved in exactly the same way as witha conventional optical trapping scheme [61, 62], whereby we lower the laser intensity and hence
lower the depth of the optical conﬁnement. During this hybrid trap evaporation we also manipulate
the magnetic ﬁeld, setting it at 168.5 G to minimize 85 Rb losses, as was done similarly in Ref. [49]and Ref. [52]. For the 87 Rb gas, lowering the depth of the optical conﬁnement has the desired eﬀectof preferentially “spilling” out the most energetic of the atoms. The key to sympathetic coolingwith the hybrid trap evaporation is that as the trap depth is lowered, 85 Rb atoms don’t spill out
but are instead loaded into the ﬁnal, weak magnetic trap.
The energetic 87 Rb atoms actually “spill” into a conservative potential and will reach theirturning point and eventually redeposit their energy back into the ensemble. Therefore, we zap themwith rf, driving them to a state with zero magnetic moment, so that they fulﬁll their destiny ofgoing splat on the bottom of our science chamber. In the last stages of the hybrid trap evaporation
scheme, we change our bias ﬁeld to a value that gives the soon-to-be condensate a positive, 100 a0
scattering length. Eventually, the optical trap can no longer support 87 Rb and we lose all the 87 Rb
atoms. At this point a signiﬁcant 85 Rb BEC has been born.

2.4.3

A ﬁnal cut: the 85 Rb rf knife

Once the last bits of the 87 Rb gas have been evaporated away, we are left with a cloudof 85 Rb atoms that has reached the onset of quantum degeneracy with a signiﬁcant condensed
fraction. However, this gas is still not fully in the 10 Hz spherical trap. What is left is to reduce
the remaining optical potential until it is completely gone, that is, remove the last elements of thehybrid trap.

38
The transition from the hybrid conﬁnement to a purely magnetic trap presents an interesting
regime in terms of trapping potentials for our condensate. We have found that at this stage ofthe experiment, we must introduce a 85 Rb rf knife, otherwise the BEC that 87 Rb has so selﬂessly
sacriﬁced itself for will not survive the transition to the 10 Hz trap. This seems counter-intuitiveat ﬁrst, considering that our ramping down of the optical trap potential is adiabatic. One would
naively suspect the phase-space density would remain constant and hence, so would the condensate
fraction. One mechanism for destroying the 85 Rb condensates could be heating from inelastic
collisions, but these timescales are much longer than the time it takes to go from the last stage ofthe hybrid trap to our spherical trap. So what gives?
A similar phenomena was observed by the Ketterle group [63]. In a conﬁguration very similar
to ours (an optical trap overlapped with a much weaker magnetic trap) they were able to create
BECs by increasing the optical potential until the trap depth was equal to the chemical potential.
They then observed the reversible destruction and revival of the condensate as they lowered theoptical potential and then increased it again. A very similar mechanism is at work in our system.
In an experiment quite similar to the Ketterle group’s, we see both the destruction of 85 Rb BEC
and subsequent revival as we lower and raise the optical trap depth, shown in Fig. 2.12. Here, wehave plotted the BEC number (vertical axis) for a decreasing optical trap depth (bottom and topaxes) as we either move towards a purely magnetic conﬁnement (black points) or eventually ramp
up the optical trap depth to where we started (red points).
The easiest way to think about this phenomena is by going to the limit of two diﬀerent traps
that are in thermal equilibrium, with the number of atoms in the broad, magnetic trap greatly
outnumbering the atoms in the deep optical potential. In this limit, the gas of atoms containedin the magnetic trap can be thought of as a thermal reservoir. As we lower and raise the opticalpotential, the overall temperature of sample is ﬁxed by this reservoir temperature (TR ). The factthat this conﬁguration can seemingly spontaneously create (and destroy) a condensate comes about
opticalbecause of the much higher critical temperature for the deep optical potential (Tcmagto that of the weak magnetic trap (Tc), compared) which has a higher trap frequency (Tc ∼ ωho ). With the39optical trap depth (nK)
35

100

160

220

20k

N

BEC

30k

10k
ramp down only
recompress to 30 mV

0

15

20

25

30

optical trap control voltage (mV)

Figure 2.12: Evidence of the “85 Rb thermal reservoir” in the ﬁnal transfer from the optical trap
to the 10 Hz trap. Condensate number on the vertical axis is plotted versus the optical trap
control voltage on the horizontal axis. The traps are modeled and the optical trap depth is thencalculated from this model using the optical trap voltage. The optical trap power is given by
0.55 mW
mV (V − 12 mV). This optical trap depth is shown on the upper horizontal axis. The black
points show the BEC being melted as the trap depth is lowered. We attribute this to 85 Rb atoms
at a much smaller phase-space density (compared to those 85 Rb atoms in the optical dimple) witha signiﬁcant population in the broader magnetic trap. The red points correspond to the measured
BEC number when the optical trap depth is again lowered to the value shown on the x-axis, butthen adiabatically ramped back to 220 nK (30 mV of control voltage). We see for these experimentsthat the BEC number has partially restored. This seemingly “reversible” process of BEC creation
supports the proposed mechanism describing this process in the text. We attribute the fact thatthe BEC is not fully recovered by this process to inelastic collisions.

40
opticaltwo systems in thermal equilibrium, all one needs for local condensation is for Tc> TR .

Another way of thinking about this is that while the two systems are in thermal equilibrium,
the phase-space density is a local quantity. By lowering the potential of the optical trap, we are,in-eﬀect, increasing the local phase-space density for those atoms due to a corresponding increasein density. This “adiabatic increase in phase-space density” was ﬁrst put forth by Pinkse et al.
[64].
To avoid melting our newborn condensate in going from the hybrid trap to the purely magnetic trap, we remove the most energetic 85 Rb atoms from the magnetic conﬁnement with a 85 Rb
rf knife. We apply this knife after the optical conﬁnement can no longer support 87 Rb, roughly 2s before the entire evaporation is over. We found that bringing the rf knife in closer to the trap
bottom serves to ﬁne tune the condensate number and fraction, as shown in Fig. 2.13. We keep
the knife at a ﬁxed frequency of roughly 120 kHz (6 µK) from the trap bottom to remove roughly
35,000 of the most energetic atoms. In understanding this unique evaporation, it is helpful to keep
in mind that the 1/e2 waist of our focussed optical trap beam is roughly 50 µm, and the rf knife
frequency corresponds to a position roughly 60 µm from the trap center. On a day-to-day basis
this, along with the 85 Rb/87 Rb ratio controlled by the MOT ﬁll, the ﬁnal 85 Rb rf knife provides
the experimenter with the two knobs needed to achieve large 85 Rb BECs from the hybrid trap
evaporation.

2.4.4

How’s it look? Imaging 85 Rb BEC

After it is all said and done, what we are left with is a 85 Rb BEC with zero measurable
remnant 87 Rb to speak of. As presently constituted, however, our apparatus does not have thecapability of absorption imaging 85 Rb atoms at high-magnetic ﬁeld. This, in turn, means that wewill have to play a couple tricks for imaging our condensates.
There are two issues to overcome when imaging the 85 Rb gas after expansion from the 10 Hz
trap. The ﬁrst is that with such a weak trap, the expansion time needed for the optical density(OD) to fall oﬀ to something reasonable (i.e. no saturation eﬀects) for resonant absorption imaging

41

60k

85

0.8
40k
0.6

0.4

20k

fraction

Rb BEC Number

1.0

0.2

0
82.15

0.0
82.20

82.25

82.30

85

Rb rf knife (MHz)Figure 2.13: Final evaporation of the 85 Rb gas. The vertical axis on the left corresponds to theblack circles, and gives the overall BEC number determined from a two-component Gaussian ﬁt
to the time-of-ﬂight absorption image. The axis on the right-hand side corresponds to the red
triangles, and gives the BEC fraction (NBEC /Ntotal ). The horizontal axis is the ﬁxed frequency of
the rf knife in question. The knife is on for 2 seconds of the evaporation and is the ﬁnal tool we
use for purifying our 85 Rb BEC.

42
is painfully long. In our science cell, we are limited to expansion times of about 30 ms before the
BEC hits the cell wall that is 5 mm away. In order for our in-trap OD (∼ 50) to fall oﬀ to something
2π2measurable, say 2, we would need to expand for at least 70 ms (NBEC = 2π
5 ( λ2 )OD R , where λis the wavelength of the resonant light, and R the measured condensate radius [65]). For 30 ms of
expansion, the BEC volume has increased by a factor of 8, but is still much too dense to image
accurately. To work around this, we transfer a small fraction of the condensate into the upper,
F=3 manifold via an ineﬃcient adiabatic rapid passage (ARP). The small fraction transferred via
the ARP is calibrated using a thermal gas (above Tc ). The shot-to-shot ﬂuctuation in the transfer
eﬃciency is signiﬁcantly smaller than the 10% shot-to-shot number ﬂuctuation in our experiment.
The second imaging issue comes from the fact that 85 Rb has a large negative backgroundscattering length of -450 a0 . Because we do not image at high-ﬁeld, we must eventually hand overcontrol of the scattering length to the background value at the moment of imaging. Condensatesin the presence of a negative scattering length are unstable, and the attractive interactions induce
a collapse [66]. This collapse leads to large, inelastic loss, which only leaves a thermal gas for thesubsequent absorption images. For the large, relatively dense condensates we have in expansion,
this attractive scattering length causes an implosion that leads to loss of atoms as well as loss of
information in the time-of-ﬂight image. This problem was solved previously in our experiment byleaving the magnetic ﬁeld on during expansion, until the density had fallen oﬀ enough that collapse
was no longer possible. In our 10 Hz conﬁnement, however, the cloud does not possess the necessary
kinetic energy to reach such a low-density regime in a reasonable amount of time. Therefore, in
addition to leaving the magnetic ﬁeld on during expansion, we also ramp the ﬁeld to a value where
the scattering length is roughly 500 a0 right before turning oﬀ the trap conﬁnement. This ramp ina gives the condensate extra energy in its expansion.
We model the expansion (and eventual implosion) of the BEC with the Pérez-Garcia modelthat was discussed in section 2.2.1. We choose a scattering length for the expansion that is well
above what the model deems necessary, to avoid the possibility of local collapse due to ﬂuctuationsin the density that give rise to small regions of higher density. However, too high a scattering

43
length for the expansion (a ∼ 1000 a0 ) can cause loss from three-body recombination, which wehave observed as loss in our absorption images.
With an imaging scheme now in place for our 85 Rb condensates, we can take a step back andevaluate the performance of the hybrid trap evaporation. In Fig. 2.14, we show the number versus
temperature for both 85 Rb and 87 Rb during the hybrid trap evaporation. Note that even though
we do no direct cooling of 85 Rb, we still see loss. It is unclear where this loss comes from, but
we speculate the cause is inelastic collisions, either 85 Rb-85 Rb collisions or, more likely, 85 Rb-87 Rb
collisions. We also observe a discrepancy between the measured temperature of the 87 Rb gas andthe 85 Rb gas near the end of the evaporation (corresponding to the coldest temperature in Fig.
2.14), even though both are thought to be in thermal equilibrium. We attribute this to the increased
sag in the weaker optical trap potentials for 87 Rb. We describe the sag of 87 Rb in the optical trap
as ysag = g ′ /ωρ2 , where ωρ is the radial trap frequency for 87 Rb due to the optical trap conﬁnement
and g ′ = g/4 is the downward acceleration experienced by 87 Rb at the optical trap position, were
there no optical conﬁnement. We remind the reader that the optical trap position has been chosen
to correspond to the equilibrium position of 85 Rb in the purely 10 Hz spherical trap, while theequilibrium position of 87 Rb in the same trap is displaced 31 g/(2π × 10 Hz)2 from that of 85 Rb,
due to 87 Rb’s weaker magnetic moment. The magnetic conﬁnement eﬀectively weakens gravity for
87 Rb (reﬂected in g ′ ) while canceling out the force of gravity for 85 Rb. The fact that 87 Rb incurs

sag eﬀectively makes the 87 Rb trap frequencies smaller than those of 85 Rb.1 The trap frequencies
used to calculate the temperature of 87 Rb are those for 85 Rb, which we know experiences zero sag,by design.
An interesting comparison between the hybrid trap evaporation and our previous optical trap
evaporation is in the relative number of 85 Rb to 87 Rb. Where as in the old evaporation, we beganwith roughly twice as much 85 Rb than 87 Rb, we are now operating on the other side of things,
1

The parabolic conﬁnement from an optical trap is an approximation valid at the center of the laser beam, which
is described with a gaussian proﬁle. We have modeled this beam proﬁle and found that as one moves away fromthe center of the beam the parabolic approximation is no longer valid, and the conﬁning potential looks eﬀectively“weaker” to the trapped atoms, making for smaller trap frequencies.

44

8

10

temperatures scaled to 10 Hz trap

7

Number10

6

10

11-19-2009

5

10
0.01

0.1

1

scaled temperature ( K)
Figure 2.14: Evaporation trajectory for both 87 Rb (black) and 85 Rb (red). On the vertical axis
we have the number of atoms at diﬀerent points in the hybrid trap evaporation. All the data
here is above the onset of condensation. The horizontal axis is the temperature of the clouds
measured using time-of-ﬂight expansion. These temperatures have been scaled (Tscaled = Tρ (2π ×10 Hz/ωρ )2/3 ) to a 10 Hz trap, assuming an adiabatic transfer, so that we can, compare the phasespace density at diﬀerent stages of the evaporation. The points here are taken at the same stages of
evaporation, with the 85 Rb and 87 Rb clouds in thermal equilibrium. We attribute the disagreement
between the temperatures of 85 Rb and 87 Rb to the increased sag distorting the optical potential, as
described in the text. The 85 Rb number loss is minimal for this type of evaporation, reﬂecting only
the inelastic losses to which we are defenseless, as no direct evaporation is done on 85 Rb during
this stage of cooling (see Fig. 2.8). The eﬃciency of the evaporation is determined by a ﬁt of the
87 Rb data to N = αT β , which gives an eﬃciency of β = 1.04 for this evaporation.

45
beginning our evaporation with roughly 30 times more 87 Rb than 85 Rb. This, I believe, more thananything illustrates the diﬀerence between the old way of simultaneous cooling in the all opticalpotential, and our new way of cooling in the hybrid geometry.

2.4.5

Shaving oﬀ the slosh

With the production of 85 Rb BEC now under control, we focus our attention to its stability
in terms of center-of-mass oscillations, or slosh. Slosh is particularly damaging to the hybrid trap
evaporation because there is relative slosh between 85 Rb and 87 Rb, owing to the diﬀerent trap
frequencies for 87 Rb and 85 Rb when conﬁned magnetically. This is a problem for the sympathetic
cooling of 85 Rb with 87 Rb, as the atom clouds will spend less time thermally overlapped when theyexperience diﬀerent frequency slosh versus when they are both at equilibrium at the center of the
trap. The most eﬀective way to minimize slosh is with a careful alignment of the 10 Hz trap onto
the optical trap. While this step is our primary method for staving oﬀ slosh in the radial direction,
axially (ẑ) slosh direction is trickier to suppress since it largely comes from the initial load into the
OT from the MT. In other words, it is here from the beginning and here to stay.
What we have found is that the proximity of the 87 Rb rf knife to the trap bottom during
the earlier stages of the hybrid trap evaporation can have a positive eﬀect on the axial slosh. We
show the eﬀect of the 87 Rb rf knife frequency (horizontal axis) on the gas in Fig. 2.15. In Fig.
2.15(a) we see the danger of bringing in the knife too close, as it eﬀectively destroys our 87 Rb
coolant and leaves us with hot 85 Rb clouds. In Fig. 2.15(b) the beneﬁcial eﬀect (the amplitude of
the center-of-mass oscillations) of the rf knife is also shown. We take advantage of this eﬀect byleaving the knife 200 kHz detuned from the trap bottom. Reduction of slosh is also beneﬁcial tothe performance of the hybrid trap evaporation.

2.4.6

The barber’s take

So, after all is said and done, having developed this cooling scheme, we are in a unique position
to analyze the pros and cons. All in all, after the magnetic trap was designed and working well,

1.0

0.8

0.6

0.4

0.2
imaged at last stage of hybrid trap evaporation

0.0
121

122

123

Rb slosh amplitude (pixels)

10

85

(b)

87

Rb rf (MHz)

85

(a)

Rb phase-space density

46

5

imaged at last stage of hybrid trap evaporation

0
121

87

122

123

Rb rf (MHz)Figure 2.15: Eﬀects of the 87 Rb rf knife applied during the hybrid trap evaporation on the ﬁnal
85 Rb phase-space density (a) and slosh of the 85 Rb cloud (b). The horizontal axis for both plots
corresponds to the frequency of the 87 Rb knife being applied. The resonant frequency at the bottomof the trap is depicted by the red dashed vertical line. In (a) we see how cutting too close to thetrap bottom has a detrimental eﬀect on 85 Rb evaporation performance, as seen by the low PSD atthe smallest rf detunings. Here, we presume that the 87 Rb is not eﬀectively evaporating, with theknife cutting farther and farther into the cloud than would be ideal. In (b), however, we see how
there is a beneﬁt to bringing the 87 Rb knife in close to the trap bottom, as the amplitude of the
85 Rb slosh (vertical axis) has been reduced as a result of the closer knife. We attribute the reduced
85 Rb cloud’s slosh to the reduced slosh of the 87 Rb cloud (the 87 Rb knife essentially damping theoscillations) who eﬀects the 85 Rb cloud through collisions. This slosh data was taken on clouds
whose rms size was 22 pixels, corresponding to a temperature of 130 nK. Through a similar set ofexperiments we have seen that a knife that is 200 kHz (10 µK) or 130 µm away from the hybrid
trap center presents the best compromise in terms of minimizing the axial slosh, while still leaving
enough 87 Rb in the trap to adequately perform its role in the hybrid trap evaporation.

47
it took two months, from inception of the hybrid trap evaporation scheme to production of 85 Rb
BECs. I think that for any graduate student, that would be, fractionally speaking, a fairly quick
turn-around. Our scheme routinely produces 85 Rb condensates of 40,000 atoms, with condensatesof 90,000 atoms on the best of days2 . More importantly, however, these condensates are in a weak,
10 Hz spherical trap that is ideal for the study of strongly interacting BEC physics, which is thegoal of this lab.
The major issue with this evaporation is that it relies heavily on the alignment between the
10 Hz trap and the optical potential. This alignment is presently a painstaking process that usuallyeats up the better part of a work day in the lab. It is also one that must be revisited every few weeks
or so, as the traps will tend to drift out of alignment. When the traps are misaligned the creationof 85 Rb BEC is nearly impossible, with the 87 Rb and 85 Rb having diﬀerent modes of oscillations,making for poor thermal overlap. Also, as mentioned before, with more degrees of freedom comes
more opportunity for things to go wrong. This seems like an unavoidable trade-oﬀ in experimental
physics, however, as we will always be pushing towards more complicated experiments.
The particular cooling scheme laid out here requires the species being cooled to be magnetically trappable. However, this idea could be extended to species that were not magneticallytrappable, as the only criteria one really needs is to apply a magnetic ﬁeld gradient that cancels
out the eﬀects of gravity of the species of interest and a “coolant” species that has a diﬀerent
magnetic moment. I believe this scheme could be quite promising for species that are diﬃcult tocool on their own because of inelastic collisions. Here, minimizing the absolute number of thisspecies is favorable, which the sympathetic cooling in the hybrid trap provides. Another beneﬁt ofthis cooling scheme is that one can ensure that there is no coolant species, 87 Rb in our case, left at
the end of the evaporation. This is important for experiments where the interactions between the
coolant and the science species are non-negligible.

2

Check it! File number 138, May 6th, 2010.

Chapter 3

Bragg spectroscopy

Having designed and implemented a trap with our scientiﬁc goals of studying strongly interacting BECs in mind, we now can look past the system of interest and begin to consider the
relevant probes. One such probe, Bragg spectroscopy, looks tailor-made for studies at large na3 ,as it is quick probe relative to three-body loss and breathing timescales that can complicate thesignal. We will present data showing the success we have had with Bragg spectroscopy investigatingcondensates at large na3 . In addition, the concepts of Bragg spectroscopy will be laid out, whichis relevant for the next chapter’s discussion of a photon-counting technique for Bragg spectroscopy.
I will present this chapter in three parts, beginning by telling the reader ﬁrst about what
Bragg spectroscopy probes: the dispersion relation of a BEC. This will serve to motivate both thischapter and the next. I will then present the theoretical underpinnings of how Bragg spectroscopy
works by giving two equivalent pictures of the Bragg process. Finally, I will give the pertinent
experimental considerations in measuring a Bragg response. This chapter will both lay the groundwork of the next chapter, as well as provide motivation for measuring the Bragg response in a new
way.

3.1

Measuring the dispersion relation of an interacting BEC
Bragg spectroscopy oﬀers the experimenter a means to measure the dispersion relation, which

is deﬁned as the energy-momentum relationship for excitations, of a BEC. In measuring the dispersion of a condensate, we probe fundamental excitations of the system. Many of these excitations

49
play a role in the macroscopic observations discussed in Chapter 1.
To measure the dispersion relation with Bragg spectroscopy, we drive an excitation with
momentum h̄k with two coherent lasers, referred to as the Bragg beams. In Fig. 3.1(a) we show
the BEC response measured in a time-of-ﬂight (ToF) image, where the horizontal and verticalaxes correspond to momentum space. Because of the condensate’s small momentum spread, theexcitations appear as a distinct atom cloud whose position corresponds to +h̄k momentum on thehorizontal axis. We determine the response of the system by counting the number of atoms at h̄kexcitations (N+k ) in the ToF image. We can later normalize N+k to the total number of atoms todetermine the fraction of atoms excited.
We determine the energy for resonant excitations spectroscopically. That is, we vary thefrequency diﬀerence of the Bragg beams, and measure the corresponding response of the system,
while keeping k constant. Note that by changing the frequency diﬀerence between the two Bragg
beams we can create and measure momentum excitations in either the +k, as well as the −kdirection. In Fig. 3.1(b) we show data for a typical Bragg spectrum, plotted as the response versus
the probe frequency. The response is given in terms of the transferred number fraction, which isdeﬁned as the net response of the system, (N+k − N−k ) /Ntotal . For example, a transferred number
fraction of 100% (-100%) would correspond to a ToF image where the entirety of the cloud is at
the +k (−k) position.
We typically ﬁt the data shown in Fig. 3.1(b) to a two-gaussian function of the form A(e−(ω−ω0 ) /2σ −2e−(ω+ω0 ) /2σ ). A ﬁt to this form is drawn as a black line in the Figure 3.1(b). The peak Bragg2

2

response is characterized by the ﬁt parameter A. The resonant frequency of the k excitation is
given by ω0 , while σ describes the rms width of the Bragg lineshape.
To map out the dispersion relation, we could measure the resonant frequency, ω0 , of excitations at diﬀerent k. We plot in Fig. 3.1(c) what such a dispersion relation would look like for botha free particle (black line) as well as for a weakly interacting BEC (red line). Bogoliubov predicted

2

50
(a)

(b)

transferred number (%)

0+ kmomentum

10

5

0

-5
k = 16

-10
-40

-20

0

/2

(c)

m

-1

20

40

(kHz)

6
weakly interacting BEC
22

k /2m)

4

0

/ g n

free particle (

2

0
0

1

k

2

exc

Figure 3.1: Three pictures of Bragg spectroscopy. (a) Time-of-ﬂight (ToF) absorption image taken
to measure Bragg excitation. The response is measured by counting the number of atoms (N+k )in the small cloud at position +h̄k. Excitations at −h̄k measured similarly. (b) An example Bragg
spectrum at ﬁxed k for 85 Rb at a = 100a0 . Transferred number is given by (N+k − N−k ) /Ntotal .
The detuning between the Bragg beams, ω, is shown on the horizontal axis. We ﬁt the data to alineshape whose functional form is described in the text. Lineshapes are modeled as two gaussianfunctions, centered at ±ω0 . The Bragg resonance, ω0 , is determined from the center frequency ofthe lineshapes, as determined by the ﬁt. (c) Dispersion relation, as could be measured throughBragg spectroscopy at diﬀerent k. On the horizontal axis is the momentum of the excitation, inunits of the inverse healing length, ξ −1 . On the vertical axis is the excitation energy, normalized
to the interaction energy. Notice that for the case of a weakly interacting condensate (red line)
there are two regimes, kξ < 1 and kξ > 1, where the dispersion goes as k (phonon regime) ork 2 (single-particle regime), respectively. In the phonon regime, the slope of the dispersion curve
corresponds to the speed of sound in a condensate. The black line is the solution for a free particle,
given by h̄ω0 = h̄2 k 2 /2m. Note that far into the single-particle regime the excitation energy for
the weakly interacting condensate diﬀers from the free particle case (black line) by gn. Thus Bragg
spectroscopy at large k can be used to determine the interaction energy of a condensate.

51
the dispersion of a weakly interacting condensate, which we can write as [24]h̄2 ω02 = (h̄2 k 2 /2m)(h̄2 k 2 /2m + 2gn).

(3.1)

We consider this well-understood case of weakly interacting BEC as a platform upon which to build
understanding of the strongly interacting case that interests us. As laid out in Ch. 1, and seenin Fig. 3.1(c), there are two distinct regimes for excitations, delineated by the product of kξ = 1√√(where ξ = h̄/ 2mgn = 1/ 8πna is the healing length). We will separately discuss the dispersion
relation of a BEC in these two regimes of kξ > 1 and kξ < 1 in the following sections.

3.1.1

Single-particle excitations

Single-particle, high momentum (kξ > 1) excitations in a Bose-Einstein condensate are, forme, the most straight-forward and intuitive to understand. The energy of these excitations have afamiliar, quadratic dependence on momentum that goes as h̄2 k 2 /2m + gn. This is quite similar tothat of a free particle, like a baseball, whose energy goes as h̄2 k 2 /2m (see black line in Fig. 3.1).
The fact that we are not performing Bragg spectroscopy on baseballs, but rather interacting BECs
modiﬁes things slightly, but the quadratic nature of the excitation is still present.1
One can understand how two-body interactions between the atoms in the condensate modify
the dispersion of high-k excitations by considering the energy of the system. To create an excitation
at high momentum, we are eﬀectively removing an atom from the condensate (−µBEC ) and adding
an excitation at high momentum (+h̄2 k 2 /2m + 2gn), where the high momentum excitation has
both a kinetic energy (+h̄2 k 2 /2m) and an interaction energy (2gn) to be taken into account. In
light of this, we modify the free particle dispersion
lim h̄ω0 =kξ≫1h̄2 k 2+ 2gn − µBEC
2m

(3.2)

As discussed in Ch. 2, the chemical potential for interacting condensates comes about becauseof the interactions between the atoms, and µBEC = gn. For the excited particles, with momentum
1

For review on the energetics of baseballs and softballs, I refer the reader to the annual DAMOP breakout session:
Working Group on Trajectories.

52h̄k, there are equal direct and exchange contributions to the interaction energy between the highmomentum particles and atoms in the BEC. This leaves us with an interaction energy of 2gn. This
gives a dispersion for Bragg spectroscopy in agreement with Bogoliubov’s prediction in the highmomentum limit
lim h̄ω0 =kξ≫1h̄2 k 2
+ gn
2m

(3.3)

This diﬀerence between the free particle dispersion and that of an interacting condensate is reﬂectedin the constant oﬀset at high momentum between the black and red line in Fig. 3.1(c). Because
this oﬀset, or line shift, is exactly the interaction energy, gn, Bragg spectroscopy at large kξ oﬀers a
direct probe of the interaction energy. This interaction energy, while well-understood for a weaklyinteracting BEC, presents to us one of the “carrots” we have been trying to catch for the stronglyinteracting case where the LHY term (Eq. 1.3) is expected to modify this interaction energy.

3.1.2

Collective phonon excitations

The subject of phonons in a BEC is rich enough that a graduate student could devote an
entire career to their study. What I present here will only be a few useful pictures in understanding
the physics of phonons and how they relate to Bragg spectroscopy. I refer the reader to Dr. Tung’s
thesis [67] for a more thorough discussion.
In Fig. 3.1(c), the high momentum, single-particle excitations that are quadratic in k give
way to excitations that depend linearly on k at low momentum, where the excitation energy goesas√

lim h̄ω0 = h̄

kξ≪1

gn
k
m

(3.4)

The crossover to collective, phonon excitations occurs when the wavevector of the excitation isequal to one over the healing length, k = ξ −1 . The healing length is of fundamental importanceto the physics of interacting Bose-Einstein condensates and it can be thought of as the minimum
distance over which the BEC wavefunction (or density) can vary signiﬁcantly. In the condensed
matter community, ξ is referred to as the coherence length.22

As an example, we consider a box

In atomic physics, the term coherence length has been taken to describe our lasers. As prudent researchers, we53potential where the density of a BEC would be uniform until it reached the edges of the box, where
the density must drop to zero. The healing length is the distance in space that the density takes
to go from zero to its uniform value. This fundamental length scale, where the quantum pressure(h̄2 /2mξ 2 ) balances out the interaction energy (gn), manifests itself in such phenomena as theminimum size of a vortex core.
Phonons, on the other hand, can be thought of as a manifestation of the interactions in a
Bose-Einstein condensate, since phonon modes only exist for kξ < 1. For a condensate to support
phonon modes, the interaction energy, gn, must be ﬁnite. Phonons also reﬂect the BEC interactionsin the speed at which they propagate, cBEC =√gn/m. This is the speed of sound in a BEC. This

intimate relationship between interactions and phonon excitations motivates their study in thestrongly BECs we aim to probe.
We understand the collective nature of phonons through experiments done by Vogels et al.
[68], which veriﬁed the quasiparticle basis of phonons predicted by Bogoliubov [24]. In theseexperiments, they were able to both create a phonon excitation through low momentum Bragg
spectroscopy, and then probe this momentum class using a second Bragg probe at high momentum, as suggested by Brunello et al. [69]. The experimenters demonstrated the collective nature ofa phonon excitation by ﬁrst creating Nph phonon excitations in the +k direction and then probing
them to ﬁnd that they were composed of Nph u2k excitations in the +k direction and Nph vk2 excitations in the −k direction. Furthermore, they conﬁrmed that these quasiparticle amplitudes (u2k
and vk2 ) agreed with Bogoliubov’s predictions in the phonon regime of excitations u2k − vk2 = 1 andu2k ≈ 1/2kξ > 1.
For Bragg spectroscopy, excitations in the phonon regime are typically measured in the same
way as their free-particle cousins, using ToF imaging. The quasi-particles described above areprojected onto corresponding single-particle excitations by turning oﬀ the trapping potential [70].
This projection is understood by considering that the cloud expands once the trapping potential isgone, and therefore with the density (and hence the interactions) rapidly decreases. As we know,
know better than to oﬀend our lasers.

54
a non-interacting condensate can only support single-particle excitations. This fact explains thetransfer of phonon-excitations to single-particle excitations.
However, resolving these low momentum excitations in a ToF image is not easy. Practicallyspeaking, the excitations corresponding to phonons will never cleanly escape the radius of the cloud,which is set by the interaction energy. This is easy to see if one considers that phonon energies
are almost by deﬁnition less than the chemical potential of the condensate. Otherwise, phonons
would have enough energy to “pop out” of the condensate and be distinguishable, single-particle
excitations. This makes the resultant image analysis much more diﬃcult [70, 27].

3.1.3

Dispersion at large na3 and the roton

To take inventory, there exists a fully understood dispersion relation for a weakly interacting
BEC, which has been experimentally veriﬁed [27, 70]. There also exists a measured dispersion
relation in superﬂuid He, with a few theoretical underpinnings, which was presented in Ch. 1. Our
stated experimental challenge is to connect these two wildly diﬀerent regimes.
We show in Fig. 3.2 the ﬁrst Bragg experiments accessing the strongly interacting regime.
This part of my thesis work, which I will refer to as the 2008 Bragg experiments, was published
in Ref. [54]. While we were enthusiastic about our results, and the departure from the mean-ﬁeld
physics depicted in red, we ran into unexpected complications in the data’s interpretation. For one,
although we were at a ﬁxed momentum transfer of 16 µm−1 , our data begins in the single-particle
regime (kξ = 4 at a = 150 a0 ) and ends up in the crossover between the single-particle and thephonon regime (kξ = 2 at a = 900 a0 ). Another hurdle to overcome in the interpretation of thedata was a momentum-dependent scattering length as our largest values of a meant that ka = 0.8,
which modiﬁes the scattering amplitude from the usual low energy limit. These two eﬀects aretaken into account in the red theory line in Fig. 3.2, as described in the caption. Preliminary
versions of data similar to this were presented in [49], with a brief discussion on some of the othertheoretical treatments used to analyze the data.
In the end, this data suggested the existence of beyond-mean-ﬁeld physics, as our data clearly

55

LHY correction

line shift (kHz)

5

0.04

0.1

0.22

0.34

0.48

4
3
2
1
0
0

200

400

600

800

1000

scattering length (a )0

Figure 3.2: First Bragg experiments exploring the strongly interacting regime of dilute gas BECs.
Vertical axis corresponds to the line shift: the oﬀset between the measured Bragg resonance andthe calculated free-particle result. In the single-particle regime, this would exactly correspond to
the interaction energy gn. Horizontal axis
√ represents the scattering length (lower axis) as well as
the LHY correction to the energy 3π32√8 8πna3 (upper axis). White circles correspond to the raw
data, and black circles are the same data with an empirical correction for the thermal contribution.
Green theory line is simply gn. Red theory line gives a result predicted by Beliaev [71] for a weaklyinteracting gas as the excitations go from the single-particle regime to the phonon regime (rememberkξ is not held constant as a is changing), with a momentum-dependent scattering amplitude (kais√ not small). To our knowledge there exists no theoretical treatment where a ﬁnite ka, 1/kξ andna3 is taken into account, illustrating the theoretical complexity in interpreting these initial Bragg
results.

56
disagrees with the theoretical prediction. However, Bragg data taken in this regime, at present,
oﬀers no way to test some of these theories, such as the LHY correction, due to the complications
mentioned above. This will prove to be a driving motivation to go to lower-momentum transfer,which we will focus on in the next chapter.
To illustrate the regimes of excitations that have been studied to date, we chart the “experimental phase-space” that has been so far been mapped out in Fig. 3.3, where we plot na3
versus kξ. We use this plot to both understand the regimes that have been studied in interacting
BECs, as well as the territory still uncharted in this many-body map. Depicted in black are thestudies of weakly interacting BEC, while in blue we represent the neutron and X-ray scatteringexperiments performed on He II. In magenta, we have our 2008 Bragg spectroscopy experiments,
which represent our ﬁrst eﬀorts to connect the weakly interacting BEC to the strongly interacting
world of superﬂuid He [54].
A thesis on the study of strongly interacting BECs would be remiss without some discussionof the roton, which occurs in the regime of kξ and na3 that is depicted by a hollow circle in Fig.
3.3. The roton corresponds to an anomalous local minimum in the dispersion relation of He II
that occurs near kξ = 1. One might speculate that a similar phenomena might be observed in
the dispersion relation of an atomic BEC at large na3 . Richard Feynman presented his take on aroton with his “smoke ring” interpretation [72]. This was really just a modiﬁcation to his proposalof a roton excitation consisting of a density current, where he now included some backﬂow, which
both conserved current and was thought to better agree with the experimental data [73]. Feynman
argues that the roton excitation’s lower energy is a consequence of its wavelength, k −1 , being equal
to the interparticle spacing. He postulated that density variations would be easier to arrange at
this characteristic wavelength compared to other wavelengths, and that this would be reﬂected ina lower energy needed to excite them.
One of the motivations of this work is to further explore the experimental phase-space shownin Fig. 3.3 and probe some of the rich, many-body physics that has not been studied in an atomic
Bose system. We hope that with such studies we can get a clearer understanding of the beyond

57

10
10

0

SF Helium neutron scatteringroton minimum

-1

na

3

10

1

10

-3

=

1

5

b
R
gg
ra
B

-4
dilute Bragg

10

ka
8

10

08
20

10

-2

-5

0.1

1

10

k

Figure 3.3: Charting the “experimental phase space”, or regimes, where the dispersion relation has
been experimentally tested for interacting BECs. On the vertical axis is the diluteness parameter,
na3 . On the horizontal axis is the excitation wavevector, normalized to the healing length, kξ.
The black line illustrates the experiments that have been done with dilute gas BEC’s, such as inRef. [27]. The magenta line corresponds to the experiments performed in our lab in 2008 [54].
We plot in blue the regimes covered by studying the strongly interacting BEC that motivates us,
superﬂuid liquid He, with the roton minimum depicted by the hollow circle. We also highlight ingreen another interesting competition of length scales, by plotting k = a−1 . This proved to be an
additional complication to the experiments of 2008 and illustrates another interesting regime inthis map of diﬀerent regimes for the many-body physics of bosons.

58
mean-ﬁeld physics that dominates these systems, and perhaps even knock on the door of the elusive
roton.

3.2

Two pictures of Bragg spectroscopy
In Bragg spectroscopy, we probe the sample with two incident laser beams, and throughthe beams’ collective eﬀort a k momentum excitation is produced when the diﬀerence frequency
between the two beams, ω, is equal to the excitation resonance frequency, ω0 . One of the mostbeautiful insights of quantum mechanics is the concept of wave-particle duality: two independent
ways of looking at something, each never telling the full story by themselves, but when taken
together reveal the true nature of the physics. Here, we also have two diﬀerent ways to look at
this Bragg process. These are both equivalent, and complete descriptions in and of themselves. We
present both of these concepts to give the reader a deeper insight into Bragg. Each picture has its
own merits and, when taken together, reveal the true story of Bragg spectroscopy.

3.2.1

A moving grating

Bragg spectroscopy owes its name to analogous experiments referred to as Bragg scattering.
In these experiments a crystal medium is probed with incident X-rays. At particular incidence
angles ϕ, constructive interference occurs and an X-ray diﬀraction pattern is seen at the corresponding reﬂectance angle ϕ. This diﬀraction pattern is intimately related to the lattice spacing of
the crystal, with constructive interference established by Bragg’s law
2d sin(ϕ) = iλ(3.5)

where d is the lattice spacing, ϕ the angle of incidence and reﬂection, i an integer and λ thewavelength of the incident light.
In Bragg spectroscopy, the crystalline lattice structure has been replaced by an optical latticecreated by the interference of two laser beams (referred to as beam 1 and beam 2). We write the59q

k1 , w 1

k2, w2}l = p/|k1,2| sin(q/2)|k| = 2|k1,2| sin(q/2)

w = w1-w2

Figure 3.4: Moving grating picture for Bragg spectroscopy. The two coherent laser beams create
an interference pattern in the condensate, shown as the periodic lattice in red. Because the beams
are at diﬀerent frequencies this optical lattice travels through the condensate with some velocity,ω, and transfers momentum h̄k = 2h̄k1,2 sin(θ/2) to the excitations. This picture is usefulc = |k|in considering the magnitude of the momentum transfer, Doppler shifts, as well as the competing
lengthscales involved (e.g. ξ vs k −1 ).

60electric ﬁeld produced by beam 1 (similarly for beam 2) as()⃗⃗ 1 (⃗r, t) = E0 cos(⃗k1 · ⃗r − ω1 t) k1
E
2
k1

(3.6)

where ⃗k1 is the wavevector and ω1 the optical frequency of beam 1. We write the magnitude of ⃗k1 ,the wavenumber, as k1 . For simplicity, we will set the phase of both ﬁelds equal to zero and assume
that the amplitude of the electric ﬁeld, E20 , is the same for both beams. The atoms experience a
potential due to the AC Stark shift, which can be expressed as)2
1 (⃗⃗ 2 (⃗r, t)U (⃗r, t) = − α Er, t) + E
1 (⃗
2

(3.7)

where α is the AC polarizability of the atom at laser frequency ω [33]. We explicitly write Eq. 3.7
asE2
U (⃗r, t) = −α 08()⃗k1 · ⃗k1
cos2 (⃗k1 · ⃗r − ω1 t) + cos2 (⃗k2 · ⃗r − ω2 t) + 2
cos(⃗k1 · ⃗r − ω1 t) cos(⃗k2 · ⃗r − ω2 t)·
k1 k2

(3.8)
We take a time average over the optical periods so that the ﬁrst two terms collective contribution
is to shift the potential by −αE02 /8. For simplicity, we will deﬁne U ′ (⃗r, t) = U (⃗r, t) + αE02 /8 and⃗ ⃗α′ = α4 kk11·kk21 and employ a trigonometric identity to write
U ′ (⃗r, t) = −α′)E02 (cos(⃗k1 · ⃗r − ω1 t + ⃗k2 · ⃗r − ω2 t) + cos(⃗k1 · ⃗r − ω1 t − ⃗k2 · ⃗r + ω2 t)
2

(3.9)

We again time average over optical periods (in this case 2π/(ω1 + ω2 )) and are left with a potential
in the form of a traveling wave
U ′ (⃗r, t) = −α′()
E02
cos ⃗k · ⃗r − ωt2

(3.10)

where ⃗k = ⃗k1 − ⃗k2 and ω = ω1 − ω2 . To ﬁnd the magnitude of |⃗k| we write|⃗k| = k =√(k1 − k2 cos θ)2 + k22 sin2 θ(3.11)

where θ is the angle between the two Bragg beams. We simplify the above knowing that |⃗k1 | ≈ |⃗k2 |(as both Bragg beams are derived from the same laser) and deﬁne k1,2 = |⃗k1 | giving√

k = 2k1,2

1 − cos θθ= 2k1,2 sin
22(3.12)

61
A cartoon of such an arrangement is shown in Fig. 3.4 to illustrate the moving grating picture.
The optical lattice travels through the condensate at velocity c = ω/|k|. This potential lattice willin turn transfer its momentum k to the gas, resulting in a momentum excitation in the gas of
wavevector ⃗k = ⃗k1 − ⃗k2 . This process is quite similar to Bragg scattering, where the stationary
crystal lattice changes the momentum of the incident X-rays, while in Bragg spectroscopy, theincident optical lattice changes the momentum of the stationary deBroglie waves of the atoms.
The moving grating picture is useful in conceptualizing Bragg excitation in a BEC. One cancompare the wavelength of the excitation, 2π/k, to the healing length. One can imagine that latticespacings longer than the healing length are allowed to modify the BEC wavefunction, and in doing
so coherently result in collective excitations. Short-wavelength lattice, on the other hand, vary ona lengthscale smaller than that which the condensate wavefunction can change. Here the only BEC
response is to promote a particle from the condensate to a single-particle excitation.
Laser stability requirements are also easily born out of the moving grating picture. Naively,
one might expect that the relevant parameter for laser stability would be the linewidth of the laser
compared to that of the Bragg spectrum (what sets this width will be discussed in Section 3.4.1).
This could mean laser linewidths of a few 100 Hz!3

In truth, it is the interference between thetwo beams that make the Bragg grating, meaning that the relevant comparison is between thecoherence length of the laser and k −1 . Because coherence lengths of diode lasers are typically on
the order of several cm, and the Bragg grating spacing at most a micron, this is rarely an issue.
This does, however, point to requirements for stability of posts and various other mounted optics
in a Bragg experiment, as signiﬁcant amounts of vibrations, coupled to a mirror for example, could
be enough to perturb the Bragg grating.
3

While this might seem trivial in the eyes of one Laser General Ye, it would be quite an undertaking for us with
our external cavity diode lasers, where our linewidths are typically on the order of 100 kHz.

62
3.2.2

A 2-photon transition

The picture of Bragg spectroscopy that connects best with the atomic physicist’s perspectiveis that of a coherent, two-photon transition process of absorption of a photon from one laser beam,and stimulated emission of a photon into the other laser beam. This process is diagramed in Fig.
3.5(a). For me, this picture is the most clear in terms of the momentum transfer, as conservation of
momentum tells us that the absorption process (+h̄⃗k1 ) and emission process (−h̄⃗k2 ) must accountfor the total momentum transferred to the excitation, (h̄⃗k = h̄⃗k1 − h̄⃗k2 ).
In the two-photon framework, we show a corresponding level diagram in Fig. 3.5(b). Here,the quantum levels we couple are the zero momentum state and the excited, h̄k momentum state.
We will use a virtual state (dashed line in Fig. 3.5), that is detuned ∆ from the electronically
excited state, to couple the two momentum states. I particularly like this picture because it shows
how the Bragg process is very much analogous to a two-photon Raman transition, where now
the traditional | ↑⟩ and | ↓⟩ states have been replaced by |k = 0⟩ and |k = k⟩ states. Indeed,
the atom optics community takes advantage of this coherence and uses Bragg excitation as an“atomic beam-splitter”, creating two states whose phase relationship is well deﬁned and set by theinterfering Bragg lasers [74].
Of course, with a coherent, two-photon process, there is an associated two-photon Rabi
frequency, which we writeΩ2-ph =

Ω1 Ω2
2∆

(3.13)

where Ω1 (Ω2 ) is the Rabi frequency associated with beam 1 (2) and the electronically excited state,and ∆ is the detuning between beams 1 and 2 and that same electronically excited state. In Eq.
3.13 we have neglected ωexc , the small frequency diﬀerence between beams 1 and 2, as ωexc ≪ ∆.
The two-photon Rabi frequency, in turn, allows us to predict the strength of our Bragg signal. In
the low-excitation, linear regime, we have∫ ∞0πΓ(ω)dω = S(k) Ω2-ph2(3.14)

where Γ(ω) is the scattering rate into the k momentum state and S(k) the static structure factor

63

k1, W1q

k2, W2(a)-hk2

k = k1 - k2

+hk1

Energy

(b)

}D
W1

W2

|k >
|0 >

}w

exc

Figure 3.5: Two-photon transition picture for Bragg spectroscopy. (a) Cartoon of absorption ofa photon from beam 1 and stimulated emission of a photon into beam 2, and the corresponding
momentum transfer. Because momentum must be conserved, this excitation has a net momentum ofk = 2k1,2 sin(θ/2). (b) Level diagram depicting the 2-photon transition. The dashed line represents
the virtual state, detuned ∆ from the electronic transition (note that ∆ ≫ ω). The two light ﬁelds
used to couple the |k = 0⟩ state and the |k = k⟩ state are described by Ω1 and Ω2 . The resultant
2-photon Rabi frequency is then Ω2-ph = Ω1 Ω2 /2∆.

64[75, 76]. We will discuss S(k) in detail later, but typically it is a number between zero and one.

3.2.3

Calculating Ω2-photon

We would now like to connect the two-photon Rabi frequency to experimental parameters,
namely, the intensities of our laser beams. We know that we can write the single-photon Rabi
frequency for a two-level system as1Ω2-level = √
2

√

I
Isat

Γ

(3.15)

where Γ is the decay rate (or natural linewidth) of the excited state and Isat = πhcΓ/3λ2 [77]. In
the case of 85 Rb, we have λ = 780 nm, Γ = 2π × 6.07 MHz and Isat = 1.67 mW/cm2 [78]. We notethat Isat is deﬁned for a “cycling” transition of 85 Rb.4
In most cases, one does not have a two-level system, as there are often many possible states.
We account for this by rewriting the two-level Rabi frequency for arbitrary initial and ﬁnal states,
essentially scaling Isat . The scaling is given by the appropriate coupling between the dipole moment(e⃗r) and the polarization of the light (ϵ̂), normalized to the cycling transition. We thus write
1
Ω= √
2

√

⟨Ff′ , mf |ϵ̂ · e⃗r|Fi , mi ⟩ΓIsat ⟨F ′ = 4, m = −4|ϵ̂(σ − ) · e⃗r|F = 3, m = −3⟩stretch
I(3.16)For laser light driving the cycling transition, we recover Eq. 3.15.
We can rewrite Eq. 3.16 in terms of Clebsch-Gordan coeﬃcients (given in references like
[78, 77]). Invoking the proper selection rules, we have
1
Ω= √
2

√

I
Isat

Γ

∑

c∆F,∆m P∆m(3.17)∆F,∆mwhere c∆F,∆m are the Clebsch-Gordan coeﬃcients that couple the initial state |Fi , mi ⟩ to the ﬁnal
state |F ′ = Fi + ∆F, mf = mi + ∆m⟩, normalized to the stretch transition. P∆m = ϵ̂ · ⃗r describes
the coupling between the photon and the atomic transition, able to drive ∆m = ±1, 0 transitions,
2 + P 2 + P 2 = 1.
and normalized so that P+1−1
0

For example, in 85 Rb a cycling transition would be perfect σ − light coupling the |F = 3, mF = −3⟩ to the|F = 4, mF = −4⟩ transition.
4

′

65
We can now extend this to the case of a coherent, two-photon transition. From Eq. 3.13 we
write
√
Ω1 Ω2
I1 I2 2 ∑
Ω2-ph ==
Γc∆F 1,∆m1 c∆F 2,∆m2 P∆m1 P∆m2 δ∆F 1,∆F 2 δ∆m1,∆m2
2∆
4∆Isat ∆F,∆m(3.18)

where we have introduced the Kronecker delta function because by deﬁnition in Bragg spectroscopy
our atoms do not change their hyperﬁne states in the process. We introduce I1 and P∆m1 as theintensity and polarization of the transition for laser beam 1, and likewise for laser beam 2. Imposing
the conditions of the Kronecker delta function leaves us with√I1 I2 2 ∑
Ω2-ph =
Γ|c∆F,∆m |2 P∆m1 P∆m2
4∆Isat ∆F,∆m(3.19)

We take, as an example, the case of 85 Rb at low magnetic ﬁeld, with our atoms in the|F = 2, mF = −2⟩ state. We will assume that both beams are launched parallel to the magnetic
ﬁeld axis, and the light is linearly polarized driving ∆m = ±1 transitions. We write the polarization√of the beams as P+1 = P−1 = 1/ 2. From the Clebsch-Gordan coeﬃcients, then√
( )I1 I2 2 2
Ω2-ph =
Γ
(3.20)
4∆Isat3This factor of 2/3 is not all that surprising, as this is the limit for far-detuned light, regardless ofpolarization; this is pointed out in Ref. [78].5
We have measured the Bragg response by counting the number of atoms transferred to the h̄kmomentum excitations and integrating the lineshape to calculate S(k), in turn giving a measured
Rabi frequency as established by Eq. 3.14. We can also predict a Rabi frequency from the measured
intensities of the laser beams at the atoms and ﬁnd that our predicted scattering rate agrees
reasonably well. We do, however, observe a systematically lower response (∼ 20 %) than Eq. 3.20
would predict, which we can attribute to a number of factors. In particular, misalignments ofthe polarization between beams 1 and 2 would systematically lower our scattering rate. Collisions
between the excited fraction and the BEC before imaging would also contribute to a lower measured
scattering rate, since this would reduce the number of atoms imaged at h̄k momentum.
5This is a very useful result, since typically we perform these experiments at high magnetic ﬁeld (∼ 160 G)
where F and mF are no longer good quantum numbers for the excited state and the Clebsch-Gordan analysis is more
complicated.

66

3.3

Structure factor: S(k)The static structure factor, S(k), and what it means in the context of condensed matterphysics can ﬁll a textbook chapter all by itself [79]. In this section, we will discuss S(k) in thecontext of Bragg measurements and interacting condensates. In general, S(k) is deﬁned as theFourier transform of the density correlation function, and is unity for an uncorrelated gas.
The proud parent of S(k) is the dynamic structure factor S(k, ω), where S(k) =∫∞0

S(k, ω)dω.

In fact, were it not for the broadening mechanisms inherent in Bragg experiments, our lineshapes
frequency dependence would exactly correspond to the dynamic structure factor. In addition, one
can take ω0 to be∫

ωS(k, ω)dω.
1.0

SHkL

0.8
0.6
0.4
0.2
0.0

0

1

2

3
kΞ

4

5

6

Figure 3.6: Predicted static structure factor from Eq. 3.21 for a weakly interacting condensate.
Vertical axis is the structure factor, S(k), while the horizontal axis corresponds to the wavevectorof the excitation, k, normalized to 1/ξ. S(k) = 1 is what one would expect for an uncorrelated gas.
Experimentally, smaller S(k) means a Bragg transition that requires more laser intensity for thesame scattering rate.

The structure factor’s importance in Bragg spectroscopy experiments is elucidated in Eq.
3.14, which gives the strength, or scattering rate, of the Bragg process. From Feynman’s relation[72], S(k) = (h̄k 2 /2m)/ω0 , we can predict S(k) for a trapped condensate and write [80]15
S(k) =
4(())

3 + β 3 + 2β − β 2
β−1−
π + 2 arctan( √ )
2
5/2
4β
2 β
16β

(3.21)

67
where

(

h̄2 k 2β = 2gnpk /2m)(3.22)We plot the predicted S(k) for a weakly interacting BEC, at zero temperature, in Fig. 3.6.
One notes that for large kξ, S(k) approaches unity, indicating a lack of correlations. On the other
hand, in the phonon regime, S(k) approaches zero. This is due to quantum depletion, whichtakes atoms from the zero momentum state and promotes them to low-energy excitations at low
k [33]. Because these phonons excitations are generated in pairs of ±h̄k with opposite phase,they tend interfere destructively and suppress density ﬂuctuations, resulting in S(k) < 1 [70, 27].
Experimentally, S(k) < 1 has the eﬀect of making the excitations harder to drive for the same
Ω2-photon , so that one has a lower Bragg scattering rate.6

Figure 3.7: Measured S(k) for He, reprinted from Ceperley [81], where the vertical axis correspondsto S(k) and the horizontal axis to the wavevector of the excitation, given in inverse angstroms.
Notice the peak in S(k), where S(k) is greater than one, occurs at the same momentum as theroton minimum. Black circles correspond to neutron scattering experiments, white circles to X-ray
scattering and the solid line is a path-integral Monte Carlo calculation.

In superﬂuid Helium, S(k) > 1 at the roton minimum. We reprint measurements of S(k) doneon He II in Fig. 3.7 [81]. This peak in S(k) can be understood through the aforementioned Feynman
relation. The static structure factor, then, presents the experimenter with another measurable
6

We should note that the above results are in the zero temperature limit. For ﬁnite temperatures S(k) is greaterthan unity for kξ < 1 [33]. This has yet to be experimentally observed, however. With the current techniques ofmeasuring the Bragg response, the contributions from thermal excitations cancels out [70].

68
observable of a roton excitation.
In light of the He II peak in S(k), the structure factor for an atomic gas BEC at large
na3 has been calculated using Monte-Carlo techniques [82]. These calculations also revealed acorresponding peak in S(k), which is analogous to that of a roton. The results from Steinhauer et
al. show that for an na3 = 0.011 (which corresponds to an LHY correction to the energy densityof 64%) a maximal value of S(k) = 1.08 is found. It should be noted that this, however, is a
non-universal result, as the peak in S(k) is sensitive to the inter-atomic potential at lengthscalesof 100a0 , which corresponds to the eﬀective range of the interaction. We mention this, however, tofurther illustrate the importance of measurements of S(k) when considering pursuit of the elusive
roton.
In light of these observations, we reexamine our 2008 Bragg experiments in Fig. 3.8. Here, weplot S(k) versus the interaction strength. The data has been scaled so S(k) = 1 at small values ofa. The trend towards smaller S(k) as a increases is due to the excitations approaching the phonon
regime, as kξ also gets correspondingly smaller at larger a. This eﬀect is reﬂected in the theory
line shown in red. Ultimately this data appears too noisy to measure the small peak in S(k) thathas been predicted. These experiments would have to be reworked if an accurate measure of S(k)is the goal.

3.4

Challenges in resolving ω0
While Bragg spectroscopy serves as a powerful probe of strongly interacting BECs, accurately

measuring ω0 is not without its own hurdles. Here I lay out some of the challenges in determining
ω0 from the net Bragg response; these are important to keep in mind for future experiments.

3.4.1

Contributions to the Bragg width, σFor the goal of resolving ω0 spectroscopically, it is always the case that, with all things being
equal, the smaller the spectral width (σ) of a feature, the better one can measure ω0 . To give asense of scale, we consider an experiment aimed at measuring the LHY correction at a modest value

69

1.0

S(k)

0.8

0.6

0.4

0.2

0.0
0

n

pk

~ 1.2 x 10

200

400

14

cm

-3

600

800

1000

scattering length (a )0

Figure 3.8: Static structure factor measured in the 2008 Bragg experiments (black circles) [54].
We plot S(k) on the vertical axis and the scattering length on the horizontal axis, given in unitsof Bohr radii. The experimental data has been scaled to give the predicted result in the wellunderstood case of a weakly interacting BEC, at 300 a0 . The downward trend seen here is indicativeof the experiments accessing smaller values of kξ as the interaction energy is increased (henceξ = (8 πna)−1/2 is decreased) at ﬁxed k = 16 µm−1 . A simple theoretical prediction from Eq. 3.21
illustrating this eﬀect is seen in the red line.

70
of na3 = 6 × 10−4 . For a typical density in our experiment of 5 × 1012 cm−3 , the LHY correction
would be 15% of 2π × 350 Hz worth of interaction energy, or 50 Hz! This illustrates the importanceof taking every possible step towards minimizing σ so that one can best measure ω0 . For thisreason, we list the factors that contribute to the width of Bragg spectra so that the experimenter
can attack these problems head-on. We list all of the factors that we know to contribute to thewidth of the Bragg spectra in terms of rms gaussian widths, so that we may take all of these factors
and add them in quadrature to predict σ.

3.4.1.1

Inhomogeneous density

Provided the density of our gas was homogeneous, the excitations inﬁnitely long lived, withno extraneous mechanisms for broadening, our Bragg lineshapes would be delta functions. Ourcondensates, however, are trapped, with an inhomogeneous density that is peaked at the center of
the trap. In Bragg spectroscopy, diﬀerent parts of the condensate will have diﬀerent interactionenergies, and hence diﬀerent, local ω0 .
We therefore use a local density approximation to account for the density distribution of ourcondensate. Because the density distribution is asymmetric about the average density, the resultant
spectrum is correspondingly asymmetric about the average ω0 . We calculate the rms width of this
distribution in the single-particle (large kξ) regime and get [83]√σInhom S-P =8
gnpk /h̄147(3.23)

where npk denotes the peak density of the condensate. In the phonon regime (small kξ), thiscontribution to the width is approximated as [80]σInhom Ph ≃ 0.3 c̄k
32 √where c̄ = 15π
gnpk /m is the average speed of sound in the condensate.

(3.24)

71
3.4.1.2

Fourier broadening

In a perfect world, we would probe the condensate perturbatively, with pulses that wereinﬁnitely long and inﬁnitely weak. In reality this is not feasible for our trapped gas, as the duration of the Bragg pulse must be short compared to the trap period in order to have well-deﬁned
momentum states for the entirety of the pulse duration. In addition, for experiments where thedensity is changing (e.g. due to expansion of the cloud or due to collective density oscillations of
the BEC) one would also like to keep the pulse length short compared to the timescale for densitychange, so that the interaction energy, gn, remains relatively constant.
We deﬁne Fourier broadening as the energy uncertainty associated with the ﬁnite durationof a Bragg pulse, τpulse , which, more often that not, is the dominant contributor to the spectral
width of Bragg lineshapes. Historically, we have used square pulses to turn on and oﬀ our Bragg
light; this gives rise to the well-known sinc function in the frequency domain. Because the widthof our Bragg lineshape will be a convolution of all the broadening mechanisms, we ﬁnd it useful toapproximate the sinc function with a gaussian so that we may add the widths in quadrature. It
has been shown that the width of this gaussian due to a square pulse of duration τpulse is [49]σFourier = 2π ×3.4.1.3

0.36.
τpulse(3.25)

Doppler broadening

As mentioned earlier, the uncertainty principle tells us that the ﬁnite size of our condensate comes at the cost of an associated broadening of the BEC momentum distribution, ∆p =√21/8h̄/R, where R is the size of the BEC in the direction of ⃗k. This momentum broadening gives

rise to an associated Doppler broadening of the Bragg lineshape, which is most easily understoodin the moving grating picture. The resultant contribution to the Bragg lineshape is then [84]√σDoppler =

21 h̄
k
8 mR

(3.26)

72
3.4.1.4

Breathes

Just as ∆p gives rise to Doppler broadening, so will other contributions to the characteristic
width of the BEC momentum distribution. This is particularly important for studies of BEC at
diﬀerent na3 , as breathing modes are often induced after ramping the scattering length (as discussed
in Ch. 2), and this can broaden the BEC momentum distribution. For a breathe mode with anrms velocity averaged over the cloud of ⃗vbreathe , the resultant contribution to the Bragg lineshapewidth isσbreathe = ⃗vbreathe · ⃗k

3.4.1.5(3.27)

Collisional broadening

Scattering of excited atoms will also contribute to broadening of Bragg spectra. A collision
can be thought of as a decay mechanism that eﬀectively sets a lifetime on the |⃗k⟩ state, and thislifetime in turn results in a lorentzian lineshape. Again, because we typically work in a regime
where this is only one factor contributing to the overall width, we make a gaussian approximation
to the lorentzian so that we may later add all of the widths in quadrature. The width due toscattering of high momentum excitations in the single-particle regime is given byσcollision = 0.48 × 8πna2

3.4.1.6h̄km(3.28)

Momentum uncertainty from the beam waist

In general, Bragg spectroscopy experiments are set up with collimated beams. As we will
see in the next chapter, however, there are occasions where it is beneﬁcial to focus the beam down
to a small waist. Because of the tight focus of the beam, the k-vector of the light is spread out
and this momentum uncertainty contributes a width of its own. Bragg spectroscopy with a tightly
focussed beam is referred to as Kapitza-Dirac scattering, and can be used to scatter into multiple
momentum states [85, 86]. For a Bragg beam waist of wlaser , we can describe this broadening as

73

σK-D =

σK-D =

3.4.2

4
√

dω
wlaser dk

wlaser 2m

2(3.29)h̄2 k 2 /2m + gn√h̄2 k 2 /2m + 2gn(3.30)σ versus ω0

Aside from the obvious beneﬁts of minimizing σ for accurate measurements of ω0 , there isanother, more subtle motivation that is particularly relevant when measuring a net response, suchas the total momentum transfer. We have found that the characteristic, antisymmetric shape of
our Bragg spectra (positive peak at +ω0 and negative peak at −ω0 ) requires us to keep the ratioof σ/ω0 small. In Fig. 3.9(a), we show this eﬀect by plotting simulated Bragg spectra of diﬀering
widths, keeping the amplitudes and resonance frequencies constant. The relevant parameter whencomparing these spectra is σ/ω0 , the ratio of the width to the absolute frequency of the Bragg
resonance. Shown are σ/ω0 = 0.2, 0.5, 2.0 in green, blue and red, respectively. The problem, which
will manifest itself as a ﬁtting systematic, is that at relatively large widths (e.g. σ/ω0 = 0.5 or2.0) the two gaussian lineshapes are no longer independent, as one can see in Fig. 3.9(a). In thisregime, the competing ±h̄⃗k lineshapes have “eaten” into each other, resulting in ﬁt parametersthat are highly coupled. One can understand the ﬁtting routine’s struggle in determining ω0 atlarge σ/ω0 by comparing the dashed black line to the red line in Fig. 3.9(a). The dashed black line
here corresponds to an ω0 that is twice that of the red spectrum, yet the two look nearly identical.
To determine the necessary criteria for a trustworthy measurement of ω0 , we simulated Bragglineshape data and performed least-squares ﬁts to determine ω0 , just as we would in an actual experiment. The results of our analysis are shown in Fig. 3.9(b). Here, we ﬁt to the same antisymmetric
′ 2

′2

′ 2

′2

gaussian function of the form A′ (e−(ω−ω0 ) /2σ − e−(ω+ω0 ) /2σ ). We plot the accuracy of the ﬁt(ω0′ /ω0 ) versus the relative ﬁtted widths of the lineshapes σ ′ /ω0′ . From these simulations, we seethat once σ ′ /ω0′ > 0.5, signiﬁcant error is introduced to the measurement of ω0 . Especially disconcerting is that this error is not reﬂected in the error bars returned by our ﬁtting routine. From(a)momentum transfer (arb)

74

1.0
0

=1, A =1
/

0.5

/
/

0

0

0

= 0.2
= 0.5
= 2.0

0.0

-0.5
0

=2, A = 0.6
/

-1.0

-4

-2

0

0

= 0.8

2

4

0

2
66 points of "data"
SNR = 3fitted position (

0

' /

(b)

)

(arb)

1.5

SNR = 5
SNR = 10

1

0.5
0.0

0.2

0.4

0.6

fitted width ( '/

0.8

1.0

' )
0

Figure 3.9: Diﬃculty in measuring ω0 at large σ/ω0 . (a) Bragg lineshapes illustrating the eﬀects ofa large σ/ω0 . Green, blue and red lineshapes all correspond to the same amplitude and resonanceω0 , but have diﬀerent widths σ (see the legend). The dashed black line, on the other hand, is alineshape with a factor of 2 larger ω0 than the red lineshape it resembles. This shows how theparameters of the lineshapes, A, ω0 and σ are now strongly coupled, making ﬁts of lineshapes withlarge σ/ω0 untrustworthy. (b) Results of ﬁts to artiﬁcial data generated at various values of σ/ω0 .
On the vertical axis is the ratio of the ﬁtted Bragg resonance (ω0′ ) versus the actual Bragg resonanceused to generate the data (ω0 ). On the horizontal axis is the ﬁtted width, σ ′ normalized to ω0′ .
The dashed line indicates a good ﬁt, where ω0′ = ω0 . Diﬀerent colored points represent diﬀerent
amounts of noise added into the artiﬁcially generated data. These “fake” data sets consisted of
66 data points. We add in artiﬁcial noise by replacing A with random numbers generated froma normal distribution. The signal-to-noise ratio (SNR) described in the legend is deﬁned as theratio of the peak to the variance of the distribution used to generate A. In other words SNR =A/σvariance . From this analysis we conclude that to accurately determine ω0 , one requires thatσ ′ < 0.5 × ω0′ .

75
the simulations, we can also see that the general trend is a systematic increase in ω0′ . From theseresults we conclude that an accurate measure of ω0 requires that the measured width be less thanhalf the measured resonance position, i.e. σ ′ /ω0′ < 0.5.
3.4.3

Detection

While we have presented a number of hurdles to overcome in the measurement of a Bragg
signal, another primary concern is the detection of a Bragg signal at large na3 . As mentioned, oneof the goals of this thesis is to use Bragg spectroscopy to probe smaller values of kξ while accessing
higher values of na3 . The most straightforward way to do this would be to change the geometryof our Bragg beams to access smaller values of k.7 This is challenging on two fronts. First is thetricky business of measuring low momentum excitations. The groups that have successfully done
this employed a great deal of image processing where multiple absorption images are ﬁrst averaged
together to minimize noise. They then performed surface ﬁts on the images with three independent
gaussian peaks, corresponding to the original condensate momentum (h̄k = 0) and the peaks at±h̄k momentum. In addition, the condensates used in these experiments are much larger in number(107 for the experiments in Ref. [70], 105 for the Ref. [27]) than the 4 × 104 we typically work withfor 85 Rb.
The second complication arises from the small mean free path, given by (8πna2 )−1 inducedby our jump to large a. We found in our ToF images at large na3 that the cloud we expected
at momentum h̄k had been largely extinguished (see Fig. 3.10). To circumvent this diﬃculty weborrowed a trick from a colleague, Cindy Regal. After the Bragg pulse, we made the scatteringcross-section so large that excitations never had a chance to escape the BEC, but instead depositedtheir momentum to the center-of-mass (COM) momentum of the gas as a whole. Measuring thechange in the COM momentum then gives us the net eﬀect of the Bragg excitations [49].
Practically speaking a marriage of these two techniques is irreconcilable. The small momen7

We note that in Fig. 3.3 the purple line corresponding to the 2008 experiments was done in a much diﬀerent
trap, and hence density, than we currently have, as discussed in Ch. 2. If we were to perform Bragg spectroscopy
at k = 16 µm−1 (θ = 1800 ) with these new condensates, we would experimentally explore a line that began at
na3 = 10−5 , kξ ∼ 10.

76

0

momentumk

Figure 3.10: Eﬀects of large a on ToF images used in Bragg analysis. Due to the much shortermean free path at large a, the excitations do not cleanly escape from the condensate during the
expansion. This contrasts the behavior at small a shown in Fig. 3.1(a). The scattering of excitationin ToF complicates measuring the Bragg response, as described in the text.

tum we are looking to excite via the Bragg process would shift the COM momentum only a small
amount, and our signal-to-noise in the previous experiments was already marginal for the largest
values of na3 . What is needed is a diﬀerent approach to the problem of measuring phonons in astrongly interacting condensate. Our development of one such approach is presented in the next
chapter.

Chapter 4

Photon counting for Bragg spectroscopy

In the last chapter we presented the reasons why Bragg spectroscopy is a useful probe in the
study of interacting condensates, and we presented our results of Bragg spectroscopy measurementsin the strongly interacting regime. However, all of this work at large na3 has, to date, been
performed at relatively high momentum transfers (k = 16 µm−1 ), where time-of-ﬂight imaging
of Bragg excitations in a system with strong interactions was viable. To fully understand thedispersion relation of our system, one would like to probe low-momentum excitations as well.
Typical measurement schemes for Bragg excitations would have to be reworked at large na3 andlow-momentum transfer, for reasons put forth in the previous chapter. With this thought in mind
we embarked on a new type of measurement technique for Bragg spectroscopy: photon counting.
We present here our eﬀorts and results towards that end, which can also be found in Ref.
[87]. This section is meant to not only chronicle our system, but also to present the reader with thenecessary knowledge to implement a similar system on their own. I believe that photon-countingfor Bragg spectroscopy could be quite fruitful not only for measuring the dispersion of a stronglyinteracting condensate, but also as a novel probe of cold atomic gases. A few possible experiments
are mentioned at the conclusion of this chapter.

4.1

Motivation
Any experiment designed to probe excitations requires the same essential components: a way

to excite the sample and a way to measure the response. To drive the excitations one applies a

78
ﬁeld to perturb the typically small (∼ 105 ) number of atoms, and obtains spectra by measuring
the response of the ultracold gas as a function of the driving ﬁeld frequency. In ultracold atom
experiments, this response has almost always been measured in essentially the same way: after the
perturbation, the atom cloud is imaged and the response is observed as some change in the density
distribution of the imaged cloud.
There exists, however, in any excitation spectroscopy, an alternative way to measure theresponse of the quantum gas to a driving ﬁeld. Just as the quantum gas has responded to the probe
ﬁeld, the probe ﬁeld must have an equivalent response to the quantum gas. In this chapter, we
present a technique to measure this complementary information in Bragg spectroscopy by measuring
the change in the number of photons in one of the laser ﬁelds used to drive the Bragg excitations.
While in this work we use Bragg spectroscopy, this technique may more generally be applicableto other excitation spectroscopies as well. For ultracold atoms the challenge, of course, is to haveadequate signal-to-noise in measuring photon number to detect the Bragg response. For smallish
atom samples, this will typically require a photon shot-noise limited measurement. To that end,
we detail the experimental setup and the techniques of a heterodyne based detection scheme used
to realize shot-noise limited photon counting. We also explore the advantages and limitations of
this technique.
Regardless of the method used to measure a Bragg response, adequate signal-to-noise will
always be a prerequisite. The signal is set by the number of Bragg excitations, which is typicallyless than 10% of the total atoms in the sample in order to probe the linear response. For our 85 Rb
condensates, that would correspond to some 4,000 excitations. If we allow for 100,000 photons√
in the weak beam, this corresponds to a signal-to-noise ratio (SNR) of 4, 000/ 2 × 105 = 9 onresonance. This assumes a quantum eﬃciency of 1, as well as shot-noise limited detection. With
these considerations in mind, it becomes clear that for a successful measurement of a Bragg responseusing a photon counting technique, only a shot-noise limited measurement on the weak beam
photons will suﬃce.
One well established tool for making shot-noise limited measurements of light is to use de-79
tectors in their discrete, or Geiger, mode of operation. Both avalanche photodiodes and photo
multiplier tubes can be used for such measurements. The principal behind this technique is totime resolve the individual photons so that one measures them digitally. The advantage of such ameasurement is that the photon signal itself is digitized, only has to be above some threshold toregister, and hence the noise is of little consequence. However, in considering this option for our
purposes, we found that avalanche photodetectors suﬀer from pulse-pile-up for the powers we use,
while photo multiplier tubes suﬀer from quantum eﬃciencies in the IR, only 5% or so, which is too
low to be useful.
Let us consider the alternative of using detectors in their linear mode of detection. Silicon
photodetectors have quantum eﬃciencies as high as 90% and capacitances on the order of 3 pF.
However, for the pulse lengths we seek (∼ 1 ms) the Johnson noise associated with the necessary
feedback resistor corresponds to the shot noise of a 1 µW beam at 780 nm and therefore makes
direct shot-noise-limited measurements impossible for the powers used in these experiments. Our
solution to these challenges is one as old as the AM radio: heterodyning.

4.2

Heterodyne detection
To further motivate heterodyne detection, we consider the following scheme. We direct the

weak Bragg beam onto a photodiode, and after some stages of ampliﬁcation directly measure a
voltage, Sdirect proportional to the number of photons per second incident on the detector. To
illustrate the role of shot-noise, we consider our signal to be proportional to the number of photons,
Nweak , that hit the detector in some time τ . In terms of an average voltage, the signal is Sdirect =ReηNweak /τ , where η is the quantum eﬃciency of the detector, e is the charge of an electron, andR is the transimpedance gain of the ampliﬁer.
The signal-to-noise ratio, SNRdirect , on this direct weak beam detection is given by
SNRdirect =ReηNweak /τ
√δS2 + δe2

(4.1)√
where the shot-noise associated with the photocurrent is δS = (Re/τ ) ηNweak . Other sources of

80
noise (dark noise from the detector, Johnson noise introduced in the ampliﬁcation stages, noiseon background light incident on the detector, etc.) will be referred to as electronic noise, and are
represented by δe . Shot-noise limited detection is deﬁned as the regime where δe is an insigniﬁcant
contribution to the total noise (δe2 ≪ δS2 ). In this case,
SNRshot-noise =√ηNweak .

(4.2)

Shot-noise limited detection is diﬃcult to achieve in direct detection, where for typical experimental√
values of τ = 1 ms, Nweak = 105 , η = 85 %, we might expect δe = 500 × δS for the 0.001 pA/ Hz
of a low-noise transimpedance preampliﬁer.
We use a heterodyne scheme to overcome these diﬃculties. The idea of heterodyne detection
is to amplify the signal optically before detection, so that electronic noise is of no consequence. We
do this by measuring the beat of the weak beam against another, more intense beam, which wewill refer to as the local oscillator (LO). Because the beat signal, Sbeat , goes as the square root of
the product of the intensities of the weak and the LO beam, we are left with a signal that is muchstronger than that of the original, weak beam signal.
For the purposes of illustrating the heterodyne technique, we describe the weak and LO ﬁeld
as plane waves
Eweak = Eweak cos (kz − ωweak t)x̂

(4.3)

ELO = ELO cos (k ′ z − ωLO t + ϕ)x̂

(4.4)

where we have included a relative phase, ϕ. When we spatially overlap the laser ﬁelds by matching
the modes of the beams to make k ′ as close as possible to k, the resulting beat ﬁeld isEbeat = Eweak + ELO(4.5)

Of course, what we measure are not the ﬁelds themselves, but rather the total power at the photodetector. After time-averaging over optical frequencies (represented by the angled brackets below),
we write the optical power of the overlapped beams as
Pbeat = cϵ0 ⟨|Ebeat |2 ⟩ = cϵ0 (

2
E2
Eweak
+ LO + CEweak ELO cos (∆ωt + ϕ))
2
2

(4.6)

81
where ∆ω = ωLO − ωweak , and for simplicity, we have assumed z = 0. The contrast, C, is a number
between zero and one that describes the quality of the mode-matching between the two beams.
The signal of interest to us is the amplitude of the heterodyne signal at frequency ∆ω. To
measure this signal, we shine both beams onto a photodiode and then send the resultant photocurrent to a double-balanced mixer. The mixer is driven by an rf source at frequency ∆ω, thereby
mixing down the heterodyne signal to DC at the mixer output. We then use a low-pass ﬁlter on
the output, leaving only frequencies well below ∆ω.
After ﬁltering and ampliﬁcation our signal in volts is
Sbeat =√
Reη
Pbeat = (2CReη/τ ) NLO Nweak cos ϕ
hc/λ

(4.7)

where the transimpedance gain, R, includes the gain from the mixer. As one can see, our signal
goes as cos ϕ, the phase diﬀerence between the two Bragg beams. Later we will show how wemeasure the other quadrature of this signal, which goes as sin ϕ. For now, however, we will assume
that ϕ = 0 for simplicity, and we have a SNR of
SNRbeat =√2CReη NLO Nweak /τ
√δS2 + δe2

(4.8)

The advantage of heterodyne is that one may arbitrarily increases the intensity of the LO
light, so that the shot noise from the LO dominates the electronic noise, as well as the shot-noisefrom the weak beam itself. The total noise is then√√δS2 + δe2 ≃ δS = (Re/τ ) ηNLO

(4.9)

and we are left with√

SNRbeat = 2C ηNweak(4.10)Of interest to us, of course is the weak signal, rather than that of the beat. The corresponding
SNR of Nweak is then√1
SNRweak = SNRbeat = C ηNweak2(4.11)

82
which, for perfect contrast (C = 1), realizes our stated goal for a shot-noise limited measurement
as in Eq. 4.2. These principles of optical heterodyning are well established, and we refer the reader
to Refs. [88, 89, 90] for a more thorough discussion.
Keep in mind that to this point the SNR we have been considering is that of the weak beam.
For the actual Bragg experiments our signal is not the weak beam, but NBragg , the number of Bragg
excitations. These Bragg excitations are subtraction measurements, measured with respect to a
background number of weak beam photons. We can write the associated signal-to-noise ratio as√
C η NBragg
1 NBragg
SNRBragg = √SNRweak = √ √.
2 Nweak
2 Nweak(4.12)

From Eq. 4.12 one can see that to maximize the SNR of a Bragg experiment one would like toboth maximize the number of Bragg photons (again, typically ﬁxed by the overall condensate
number) while minimizing the number of weak beam photons. The tradeoﬀs of such requirementsfor optimum signal-to-noise ratios will be discussed in following sections.
The measured LO shot noise also serves to calibrate the overall gain of our system, R.
Provided one knows η (which is readily available from the photodiode’s datasheet) and has acalibration for NLO (which is straightforward for the relatively high power of the LO beam), one
can rewrite Eq. 4.9 as(τ /e)
R = δS √ηNLO

4.3(4.13)

Layout and construction
While the principle behind shot-noise limited heterodyne detection is straightforward, theimplementation of such a technique has a number of subtleties which we lay out here. We illustrate
both the optical and the rf design in Fig. 4.1. The two Bragg beams, which we label “weak” and“strong”, intersect at the location of the BEC. The third, much more intense LO beam functions
only to amplify the weak beam for our heterodyne detection, and avoids the atoms altogether.
All three beams are derived from the same laser, and are oﬀset in frequency after passing through83
acousto-optical modulators (AOM). The rf sources driving the AOMs are phase-locked to each otherin order to ensure coherence between the Bragg beams. The weak and LO beams are combined ona beamsplitter that provides good spatial overlap of the two beams, but causes some attenuationof the weak beam. This, along with other sources of attenuation (e.g. scattered weak beam lightfrom the cell wall, imperfect reﬂections oﬀ of mirrors, etc.) can be included in Eq. 4.11, simply
by replacing η with a lower, eﬀective quantum eﬃciency for the detection system. For the best
signal-to-noise ratio we would like to measure as much of the weak beam as possible. This must
be balanced, however, with our need for large enough LO power to overwhelm the electronic noise.
Typically, we work with 90% reﬂection of the weak beam and 10% transmission of the LO beam,
which corresponds to 250 µW of LO power.
In the following sections, we discuss how we maximize our SNR for the photon-counting
technique. We begin the discussion with the considerations of the optical design, and then moveon to the rf considerations.

4.3.1

Mezzanine level

A primary challenge for shot-noise limited detection with a heterodyne interferometer is tominimize any jitter in the relative phase (ϕ) between the two beams. As we will see later, this phasejitter introduces noise that prevents us from achieving shot-noise limited detection for long Bragg
pulses. We ﬁnd that the biggest source of instability in the relative phase is due to the opticalﬁbers. These ﬁbers are sensitive to mechanical vibrations and acoustical noise. Therefore, we use
relatively short ﬁbers, 1 m in length, to minimize phase jitter between the two beams, while still
providing the necessary ﬁltering of the spatial mode. In order to have such short ﬁbers, we built
the Bragg optics shown in Fig. 4.1 on a small breadboard erected above our larger optical table.
We lovingly refer to this as the mezzanine level, which can be seen in Fig. 4.2. Also shown in theﬁgure are the various other steps we have taken to reduce the coupling of mechanical vibrations tothe ﬁbers: these steps are described in the caption. We found that minimizing the physical contact
between the ﬁbers and other elements, such as posts, proved crucial in achieving shot-noise limited

weak

LO

PM

I Q

to oscilloscope

“LO”

RF

mixer

photodiodebeamsplitter

Servo

BEC

Figure 4.1: Diagram of the heterodyne setup for Bragg scattering via photon counting. Three beams (labeled weak, LO and strong) are
derived from the same laser source and individually ﬁber coupled. The optics for the LO telescope are chosen to give the same spatial
mode as that of the weak beam. After being combined on a beamsplitter, the weak and LO beams illuminate a photodiode and the beatsignal is sent to a demodulating mixer. The two quadrature outputs of the demodulator, I and Q, are sent to an oscilloscope for themeasurement. Also illustrated in the ﬁgure is a servo loop that functions to minimize the phase ﬂuctuations between I and Q. The servo
minimizes the demodulator output, I, by feeding back to a phase modulation port of the synthesizer that drives the LO port of the mixer.
This synthesizer is phase-locked to the synthesizer driving the weak beam AOM. Not shown in the schematic is the synthesizer driving
the strong beam AOM, which is also referenced to the weak AOM synthesizer.

AOM

AOM

strong84

85
measurements on longer timescales.

4.3.2

Contrast

A principal challenge in maximizing the signal-to-noise ratio of the heterodyne detection isachieving a contrast, C, as close to unity as possible. In Eq. 4.11, we see that the signal-to-noise
ratio scales linearly with the contrast. Therefore, we would like the best possible mode-matching
between the two beams, both spatially as well as in terms of polarization. In light of these two
requirements, we launch the two beams into identical, polarization-maintaining ﬁbers (shown in
Fig. 4.1 and Fig. 4.2). The weak and the LO beams subsequently pass through diﬀerent setsof lenses, which serve to match the focussed size of the weak beam to that of the BEC, as wellas to match the spatial mode of the LO beam to that of the weak when they recombine on the
beamsplitter. To avoid interactions between the intense LO light and the atoms, the LO beam
has its own optical path that avoids the vacuum chamber and the atoms. In addition, we have an
optical isolator (not shown in ﬁgure) in between the beamsplitter and the photodiode to extinguish
LO light backreﬂections oﬀ of the photodiode that interact with the atoms.
To measure and maximize the contrast we replace the light launched into the weak beamﬁber with light at the same frequency as the LO. With this homodyne interferometer, one caneasily assess the quality of the contrast by eye, by looking at the pattern of the interference fringes,
as shown in Fig. 4.3(a). Our procedure is to ﬁrst align the beam such that the fringes are a circularpattern of rings, as seen in Fig. 4.3(b). We then optimize the collimation of the LO beam (achieved
by manipulating the relative distance between two lenses in the LO telescope) by minimizing thetotal number of rings, and are left with an interference pattern like that shown in Fig. 4.3(c). We
found this to be an essential step in getting good contrast.
Once the beams are closely mode matched by eye, we perform the ﬁnal steps of alignment bydirectly monitoring the DC output of the photodetector. We found it useful to modulate the phaseof one of the beams, enough to wrap around 2π, by placing a glass slide in one of the beam paths.
This slide was angled so that its surface made an acute angle with the beam. We mounted this86

Figure 4.2: A photograph of our newly remodeled optical table, showing what we refer to as the“mezzanine level”. We found this additional breadboard necessary so that we could work withshort, 1 m ﬁbers. Several steps were taken to reduce phase jitter between the two beams. Large,
1/2 inch posts are used for all of the optics on the mezzanine. The weak and LO ﬁbers are bothwrapped in foam to minimize acoustical coupling to the ﬁbers through air. We also minimize
the number of times these ﬁbers physically connect to the table or other posts. Two places areunavoidable: the input and output ﬁber couplers. We managed to only have the ﬁbers touch one
more post in the system, and here we have heavily wrapped the contact point with foam, as shownin the picture. Finally, the mezzanine itself is supported by 4 posts that are ﬁlled with lead shot
to further dampen vibrations.

87
slide on a oscillating motor that would rotate the slide roughly 20◦ about its described equilibrium
position. This phase wrapping gives us both fully constructive (Pconst ) and fully destructive (Pdest )interference. Using the measurements of Pconst and Pdest we are able to quantify the contrast by
analyzing Eq. 4.6. For the case of homodyne, ∆ω = 0 and we can take Eq. 4.6 and write fully
constructive interference as
2
Eweak
E2
+ LO + CEweak ELO )22(4.14)

2
Eweak
E2+ LO − CEweak ELO ).
2
2(4.15)

Pconst = cϵ0 (and fully destructive interference as
Pdest = cϵ0 (If we then evaluate

Pconst − Pdest=C
Pconst + Pdest(

2Eweak ELO
2
2
Eweak
+ ELO

)

(4.16)

Which, if one is careful to balance the powers of the weak and the LO beam so that they are equal,
reduces to
C=

4.3.3

Pconst − Pdest
Pconst + Pdest(4.17)

Strong beam

A stated goal is to minimize the total number of photons in the weak beam. If one hopes
to keep the excitation rate (and hence the signal) constant for very low weak beam intensities, the1 Ω2intensity of the strong beam must be correspondingly increased (recall Eq. 3.13 Ω2-ph = Ω2∆).

Note that this large mismatch between the two Bragg beam intensities is quite diﬀerent from theusual situation when using atom-response detection, where there is no reason not to have equal
intensities in the two beams.
Single photon processes, in particular oﬀ-resonant scattering of the stronger beam, limits the
maximum permissible intensity in the strong beam, and this is one limit on the minimum photon
number in the weak beam. We write the oﬀ-resonant scattering rate as [77]
Γoﬀ-resonant =(I/Isat )(Γ/2)
1 + I/Isat + 4(∆/Γ)2(4.18)

88

(a)

(b)

(c)

Figure 4.3: Pictures of homodyne interference fringes used for coarsely tweaking up the contrast.
Both beams here at roughly equal powers. A −5 cm focal length lens was used to better see thefringes, which are incident on a white index card and seen through an IR viewﬁnder. (a) Densely
packed parallel fringes indicating a large misalignment between the two beams. From these initialinterference patterns one walks one of the beams so that the fringes become less dense and begin
to look more circular, as in (b). Once the fringes are centered and circular all that is left to do is
to manipulate the collimation of one of the beams, again trying to reduce the density of the rings
until one is left with something like (c). From here one can use the DC output of the heterodyne
photodiode to maximize the contrast, as described in the text.

89
where I is the intensity of the light, Isat is the saturation intensity, deﬁned in Ch. 3, Γ is the naturallinewidth of the electronic transition, and ∆ the detuning of the light from the electronic transition.
In the limit of large detuning, which is typical for Bragg experiments, the oﬀ-resonant scattering
can be approximated as
Γoﬀ-resonant ≈ΓI/Isat8(∆/Γ)2(4.19)

One can ask if there is any beneﬁt to working at a larger detuning, as Γoﬀ-resonant goes as1/∆2 . However, the Bragg excitation rate (in the low excitation regime) go as Ω22-ph and has thesame 1/∆2 dependence on the detuning. Hence, whatever is won in terms of a reduced singlephoton scattering rate at larger ∆ is subsequently lost in the signal. In other words, for a ﬁxed
weak beam intensity and Bragg excitation rate, we will always suﬀer the same amount of strong
beam scatter, in the limit of large detuning.
As the eﬀects of strong beam scatter are independent of ∆, we choose to work with a smaller
detuning of 2π ×2.3 GHz compared to our previous Bragg experiments [54]. We do this to minimize
the amount of rf that is sent to the strong beam AOM, as it is picked up on our photodiode, writing
extra noise to our weak beam measurements. Working at smaller detunings allows us to accesshigher Ω22-ph for the same amount of rf power sent to the AOM.
4.3.4

LO beam

The LO light provides optical gain for the weak beam. Because this LO power is used toultimately determine the number of photons we detect in the weak beam (see Eq. 4.7) we ﬁnd
it useful to servo the power of the LO beam to keep it from drifting shot-to-shot. We do this bymonitoring the DC output of the photodiode used for the heterodyne detection and actively feedingback to a AOM that controls all of the Bragg light power before it is split oﬀ for the strong, weak,and LO beams. It was hoped that the reduced intensity noise might make our photon-counting
quieter on a per-shot basis, but this was not the case. This would be understandable if the LO
power drift were caused by something that only aﬀected the LO beam path and not the weak beam,
as the servo would be quieting the LO beam at the expense of making the weak beam noisier. This

90
further supports our belief that the ultimate limit on the photon-counting noise performance is notintensity noise of the laser beams but phase jitter.
In spite of putting much care into diverting the LO beam from the science chamber (ensuringthat it does not interact with the condensate), we still saw an eﬀect of the LO light on the BEC.
We tracked this down to light that had been reﬂected oﬀ the face of the photodiode and madeit back to the atoms via the weak beam path. We subsequently installed an optical isolator justbefore the photodiode and ﬁxed this issue.

4.3.5

Weak beam

For the best signal-to-noise, provided a shot-noise limited measurement is available, we needto minimize the total number of photons in the weak beam, as seen in Eq. 4.12. One considerationin the weak beam photon number is the spatial proﬁle of the weak beam with respect to thecondensate. A beam whose cross section is much larger than that of the condensate, for example,
would have photons that could never interact with the condensate, and these photons would only
add to the shot-noise of the measurement and not to the signal. We tailor the spatial proﬁle of
the weak beam by focussing the weak beam onto the atoms, attempting to match the transverse
spatial proﬁle of the weak beam to that of the condensate in order to minimize the number of extra
photons that will never contribute to the signal.
We can model this eﬀect by taking the overlap integral between the BEC density proﬁle andthe gaussian proﬁle of the weak beam to calculate the number of Bragg excitations expected onresonance. The overlap integral is calculated as∫ ∞ ∫ ∞ ∫ 2π
0

−∞ 0

S(k)

where

−1
2
π Ω2-ph (ρ, z)τ σBragg√
nBEC (ρ, z)ρ dρ dz dϕ22π(Ω2-ph (ρ, z) =

Γ2
4∆


) √
Istrong (ρ, z)Iweak (ρ, z)


′
Isat

(4.20)(4.21)′ the saturation intensity scaled by thewhere Eq. 4.21 is the same as that given by Eq. 3.19, with Isat

Clebsch-Gordan coeﬃcients. The intensity of the weak beam as a function of ρ and z is given by

91
Iweak (ρ, z) (similarly for the strong beam). The BEC density proﬁle is given by nBEC (ρ, z), which
we calculate in the Thomas-Fermi limit, as in Ref. [55], S(k) the static structure factor calculated
from Eq. 3.21 and σBragg is the rms width of the Bragg lineshape, which for these calculations weassume to be dominated by Fourier broadening, as described by Eq. 3.25.
We ﬁnd that for our 85 Rb condensates with a Thomas-Fermi radius of 13.5 µm, a 15 µm
beam waist matches the spatial proﬁle of the BEC quite well. In this thesis, we deﬁne the beam
waist as the radius at which the intensity has dropped oﬀ to 1/e2 of its peak value. From thismodel we calculate that for a 1 ms long Bragg pulse, with the strong and weak beams intersecting
at a 10.5◦ angle (corresponding to S(k) = 0.8) and 105 photons in the weak beam, we will excite
7 × 103 atoms out of 4 × 104 BEC atoms, when on resonance. This corresponds to a SNR of 10,
with only 3% of the condensate scattered due to the strong beam, provided we are measuring 90%of the weak beam light.
Because a 15 µm beam waist is diﬃcult to measure directly on a CCD camera due to theﬁnite size of the pixels, we determine the beam waist with a simple knife-edge measurement, asshown in Fig. 4.4. This measurement is done by measuring the beam power on a photodiode as aPower (arb.)

1.5

1.0

0.5

0.0
6.76

6.78

6.80

6.82

6.84

knife edge position (mm)Figure 4.4: Typical knife-edge measurement to determine the weak beam waist. Plotted on thevertical axis is the light power incident on a photodiode versus the position of a razor blade (horizontal axis) as it is moved into the beam. The ﬁtting method to determine the weak beam waist
is described in the text. The data shown here corresponds to a beam waist of 16.0(3) µm.

92function of the position of a razor blade which is perpendicular to the beam and is drawn acrossthe beam. We record the fraction of light that makes it to the detector as a function of the positionof the razor blade, which is mounted on a translation stage. From this, we determine the beam
waist by ﬁtting the data shown in Fig. 4.4 to a functional form laid out by de Araújo et al. [91]P (x) =

P0a
1 + e 1 s+a3 s3(4.22)

where P0 is the power in the beam and s = 2(x − x0 )/wweak is described by the beam waist
wweak , the position of the razor blade x, and the position of the center of the weak beam x0 . The
coeﬃcients a1 = 1.597106847 and a3 = 7.0924013 × 10−2 have been determined by de Araújo et al.
In minimizing the number of weak beam photons, it is possible to approach a regime where thenumber of Bragg photons scattered is a signiﬁcant fraction of the weak beam photons themselves.
In this regime, the transverse spatial proﬁle of the weak beam intensity along the ⃗k direction has
been modiﬁed due to the scatter of Bragg photons out of (or into) the weak beam, which can, ineﬀect, burn a “hole” in the probe. We therefore keep the total number of photons in the regionof overlap between the weak beam and the BEC cross-section large compared to the number of
excitations.

4.3.5.1

Weak beam alignment

If aligning a 50 µm optical trap onto a relatively hot gas of atoms is akin to pulling teeth,
then aligning a 15 µm beam onto a Bose condensed gas would be something more along the linesof pulling tonsils.1

In order to control the position of the weak beam at the atoms in a precise

way, we tilt a plate of glass that is mounted inside a telescope for the weak beam, as shown in Fig.
4.5. This design has the advantage of giving ﬁne control of the weak beam position (50 µm at theatoms for every mm of micrometer displacement on the glass plate’s mount) while also having no
measurable hysteresis.
In Fig. 4.6 we show typical data used for alignment of the weak beam light to the trapped
condensates. The alignment procedure is to essentially blast the atoms out of the trap with weak1

For a good reference on other “tonsil pulling endeavors” I refer the reader to Dr. Stutz’ thesis [92]

glass
plate

lens

93
lens

Figure 4.5: Glass plate placed in between the two lenses of a telescope for the weak beam. We
precisely align the weak beam onto the BEC by changing the angle of the plate with respect to theweak beam. Because the weak beam will pass through a focussing lens just before the atoms (seeFig. 4.1), the beam angle with respect to the axis of the focussing lens must be changed in orderto manipulate the weak beam position at the focus. For this reason we cannot place the glass plateoutside the telescope, as its eﬀect on a collimated beam would be to only displace it with respect
to the focussing lens axis.

beam light that we shift to be on resonance as we vary the position of the weak beam with respect
to the atoms. We then ﬁt the log of the number of atoms remaining to an inverted gaussian, thisis shown in red in the ﬁgure. We take the center of the gaussian lineshape as the best alignment.
Typically, we use very pure 85 Rb condensates for this procedure, with the rf 85 Rb evaporation knife

4.2

log(N)

4.0

3.8

3.6
120

140

160

180

vertical weak beam position ( m)Figure 4.6: Typical number loss data used to align the weak beam onto the condensate. On thevertical axis, we plot the log of the number of BEC atoms, and on the horizontal axis we plot theposition of the weak beam near the condensate. We should mention that the rms width of thisfeature does not predict the correct weak beam waist (wweak = 2σRMS ), but instead predicts a
waist that is roughly 3 times larger than measured with the knife-edge for this data. This could be
attributed to saturation eﬀects of the high OD condensate on the 10 µs resonant blast light.

94
on for both the 10 µs blast pulse as well during 500 ms hold time in the trap after the blast. This
500 ms hold is meant to enhance the eﬀect of the light on the measured loss, allowing for more
rethermalization and giving the rf knife a chance to remove more atoms from the heated cloud.

4.3.6

RF

We also take steps to minimize the electronic noise (δe ) that the LO’s shot-noise must overcome. The relevant noise for the heterodyne setup is at the frequency of the beat note, ∆ω.
Our photodiode circuit is designed to minimize the noise at this frequency (2π × 70 MHz) by
way of a standard “tank” circuit (an inductor that gives signals at ∆ω a high impedance path toground), shown in Fig. 4.7. We are able to reduce the receiver dark noise (dark noise from both√the photodiode as well as the ampliﬁer) to 2 pA/ Hz, which is the limit of today’s state of the art
transimpedance ampliﬁers.2

The subsequent stages of ampliﬁcation are chosen so that the noise

they add is small compared to this noise.
The voltage from the photodetector then goes to an rf mixer. As shown in Fig. 4.1, the mixer
is a four-terminal device, called an I,Q-Demodulator; this is essentially two mixers in one, with the
rf input split between the two. Port I is the same as the IF output of a standard mixer, while the
Q port is the output of a second mixer whose “LO” drive has a phase oﬀset 90 degrees with respect
to the phase of the I port’s “LO” drive. In our application, the outputs of the demodulator havethe form√

SI = 2CReηNLO Nweak
cos ϕ
τ
τ

(4.23)NLO Nweak
sin ϕ
τ
τ

(4.24)√

SQ = 2CReη

By summing the squares of SI and SQ , we are able to measure the amplitude of the beat signal,
regardless of the relative phase ϕ. For a known LO beam power, ∝ NLO /τ , our signal is directly
proportional to the rate of weak beam photons, ṅ = Nweak /τ . We measure SI and SQ with adigitizing oscilloscope and perform the sum in subsequent software analysis. We now have the2

For our photodetector, we use a Philips SA5211 Transimpedance ampliﬁer. These transimpedance ampliﬁers
are commonly used in the telecom industry to do exactly this: amplify a photocurrent at high BW with low-noise.

95

+V

CPD

L

Figure 4.7: Tank circuit to minimizing the electronic noise of the photodiode circuit. The motivation
is to reduce the voltage noise’s (vn ) contribution to the electronic noise δe , leaving the current noise(in ) as the dominant electronic noise source. The principle behind the tank circuit is that in choosing
an appropriate inductor value, L, for a given
internal capacitance of the photodiode (CPD ), one can√have a resonant LC circuit at ωres = 1/ LCPD . The impedance of a parallel (both ground and +Voﬀer zero impedance to an AC signal) LC circuit is given by Z = −jωL/(ω 2 LCPD − 1), which, atω = ωres is inﬁnity. In reality there are stray resistances that keep you from inﬁnite impedance, by
broadening the Q of the circuit. However, this high impedance Z(ωres ) does limit the contributionof the voltage noise, as δe = in + vn /Z(ωres ). We have veriﬁed that the electronic noise of ourreceiver is at the current noise limit for our transimpedance ampliﬁer.

96
following expression for the total measured photon number in the weak beam
2
Nweak = (SI2 + SQ)(

τ
)2 /NLO
2CReη(4.25)

While, in principle, we now have a signal that is insensitive to the relative phase ϕ between
the weak beam and the LO beam, the demodulator is imperfect, with non-linearities and voltage
oﬀsets in the I and Q outputs as well as deviations from the 90 degree phase oﬀset applied to
the LO drive. We therefore servo the phase of the rf driving the LO port of the demodulator, as
illustrated in Fig. 4.1. The servo minimizes SI by actively feeding back to the phase-modulation
input of the demodulator’s synthesizer (which is phase-locked to the other synthesizers used in the
Bragg setup). This makes the quadrature sum less sensitive to demodulation imperfections.

4.4

Calibration and performance
As we mentioned previously, the overall calibration of the transimpedance gain of our system

is calculated from Eq. 4.13. To do this, we measure the voltage noise δs , which is just the standard
deviation of the voltage from one of the quadratures, at diﬀerent LO powers. The LO powers aremeasured using a power meter. We plot the results of these measurements in Fig. 4.8. In this ﬁgure,
we have calculated what we would expect for the LO photon shot noise (in W of optical power)√√using (hc/λ)(1/τ ) NLO = PLO hc/τ λ for the particular pulse length τ , and LO power PLO . The
slope, β of the linear ﬁt to δs versus√

PLO hc/τ λ is β =√η(eλ/hc)R, with units of V/W directly

S 2 +S−Q2

I
giving the experimenter a useful conversion (Pweak = ηC
2 (2β)2 ) as well as the transimpedance gainof our system. In measuring δs , one must be careful that the noise is completely dominated by theshot noise of the LO photons. We therefore disregard noise data that is too close to the noise ﬂoorof the receiver (which is easily measured by turning oﬀ the LO light). We also disregard noise datataken at too high a bandwidth, where an inline low-pass AC ﬁlter would suppress ﬂuctuations of
the voltage above a certain bandwidth, thereby smoothing over the noise.
In Fig. 4.9, we present data illustrating the shot-noise limited performance of our heterodyne
detection. We measure noise by taking the standard deviation σ in the measured weak beam photon

measured noise (mV)

97

1

0.1

1

10

expected LO noise (nW)

Figure 4.8: Typical calibration data used to determine R, the overall gain of our heterodyne system.
On the vertical axis is the electronic noise, which we know to be dominated by the photocurrent
shot-noise associated with the LO light. On the horizontal axis we plot the expected LO photon
shot noise in terms of optical power (nW). The slope of this data, β has units of V/W, conveniently√
giving one a useful calibration for the system. In addition, one can determine R = hc/λe η β fromthis slope. For this calibration we varied the LO power between 100 µW and 720 µW, with τranging from 5 µs to 100 µs to give us a slope of 97,200(400) V/W, which corresponds to a gain of
R = 165.6(7) kΩ.

98
number for M consecutive pulses of equal length,
M
−1
∑

1
σ=√
2 i=1

√(

Nweaki+1 − Nweaki
M −1

)2

(4.26)

where Nweaki is the measured number of photons in the weak beam for pulse i. We plot thenoise on our measurements of weak beam photon number, normalized to the expected shot-noise√(δshot = ηNweak ) for that particular pulse duration and laser power, as a function of the durationof a ﬁxed intensity pulse. In the inset of the ﬁgure, we plot the same noise measurement for the casewhere the weak beam intensity is varied to keep the total number of photons ﬁxed at a constant105 .
For a large range of pulse durations, the measurement is at, or within a factor of two of,the shot-noise limit. For pulses shorter than 1 µs the noise is artiﬁcially low due to an inline,
low-pass ﬁlter. The increased noise at longer timescales sets an upper limit to the time available
for our Bragg measurements and is probably caused by residual phase drift in our system. Servoing
the laser power provided no signiﬁcant improvement in the noise performance of our heterodyne
detection.

4.5

Bragg experiments with photon counting
In Fig. 4.10, we show an example of a measured Bragg lineshape for a weakly interacting

85 Rb BEC. This data both demonstrates the viability of the photon-counting technique, as wellas tests it against the benchmark of Bragg spectroscopy measurements: ToF imaging. On thehorizontal axis, we have the frequency diﬀerence between the two Bragg beams, which sets theenergy of the Bragg excitation. On the vertical axis, we have the Bragg signal, which is the numberof excitations due to the Bragg process. We deﬁne this signal such that it can be either positiveor negative, reﬂecting the direction of the momentum transfer. The number of excitations due tothe Bragg pulse are counted in two diﬀerent ways, and one can see that the photon counting andthe time-of-ﬂight imaging signals agree well with each other. The two sets of data were acquired
simultaneously, with each cycle of the experiment providing both a photon-counting and an atom-

99

3

5

noise (

noise

10

/

shot

)

constant 10

photons

2

1
100µs

1ms

10ms

pulse length

11000 pW
100 pW
10 pW
2 pW

100ns 1µs 10µs 100µs 1ms 10ms 100ms
pulse length

Figure 4.9: Noise performance of heterodyne detection for a LO power of 250 µW. The vertical scale
is normalized to the shot-noise expected for the relevant pulse length and laser power detected. The
legend shows the diﬀerent weak beam powers used. At long time scales, our heterodyne scheme is
no longer shot-noise limited, presumably due to the phase ﬂuctuations between the LO and weak
beams. The inset has the same normalized vertical scale, and shows data at diﬀerent weak beam
intensities that correspond to a ﬁxed 100,000 photons. The inset is more relevant experimentally, as
we typically try to keep the same ratio of Bragg excitations (ﬁxed by the condensate atom number)to Nweak constant.

100
imaging data point, demonstrating the complementary nature of the two techniques. The lines inFig. 4.10 are individual ﬁts of the Bragg spectrum to two Gaussian functions. These ﬁts can beused to extract a center frequency and an rms line width.
A potentially useful feature of a photon-counting measurement is the ability to measure thedynamics of Bragg excitations during a single laser pulse. We demonstrate this capability in Fig.
4.11, where we plot the number of excitations, Nexc , as a function of time, τ . The data were takenusing a condensate of 400,000 87 Rb atoms, with the Bragg detuning set to be on resonance with themeasured Bragg transition at ω = 2π ×250 Hz for a momentum transfer given by k = 1.5 µm−1 . We
expect Nexc to go as τ 2 , however an interesting feature illustrated in Fig. 4.11 is the suppression of
signal for pulses short compared to the inverse Bragg resonance, τ < ω −1 = 0.3 ms. For these shortpulses, the associated energy uncertainty makes it impossible to resolve a +k excitation from a −kexcitation. Photon emission from the one process cancels photon absorption from the other process.
In our experiments, we did not observe Rabi ﬂopping in the time-dependent data, which may be
due to dephasing. For the measurements here, the weak beam proﬁle used here was much smaller (7µm 1/e2 waist) than that of the BEC (22 µm Thomas-Fermi radius), which complicates the Bragg
response. We present this data, however, to illustrate a promising feature: the ability to resolve
excitations in time within a single Bragg pulse. This feature is unavailable to the experimenterwith traditional TOF imaging.
We have also observed asymmetries in the strength of the positive and negative frequency
features of the Bragg lineshapes. We attribute this to the relatively small number of Nweak photons necessitated by the photon counting approach. A typical asymmetric lineshape is shown in
Fig. 4.12. This eﬀect was observed both using photon counting as well as traditional time-of-ﬂight
imaging to measure the Bragg excitation. We think that these asymmetries are related to the propagation eﬀects (the aforementioned “hole burning”) inherent when the number of Bragg photons
is a signiﬁcant fraction of Nweak , although we have not been able to experimentally conﬁrm this.
The shape of the asymmetries, with the larger peak on the side where the weak Bragg beam isenhanced by Bragg photons, does support the idea of propagation eﬀects being the culprit.

101

4

1x10

photons
atoms

3

N

exc

5x10

0
3

-5x10

4

-1x10

-20

-16

-12

12

16

20

Bragg frequency (kHz)
Figure 4.10: Bragg spectrum of a weakly interacting 85 Rb BEC of 5.5 × 104 atoms at k = 16 µm−1 ,measured in two diﬀerent ways. The horizontal axis shows the frequency diﬀerence between the
Bragg beams. The vertical scale is the number of excitations measured using either traditional
absorption imaging of ejected atoms (hollow circles), or the photon counting technique presented
here (black triangles). Error bars on black points represent the shot-noise in the photon counting
measurements. The photon counting measurements used three pulses of equal length. The ﬁrst
and the third pulses used only a single weak beam to make an average background measurement,with no Bragg excitation. During the second pulse, both Bragg beams illuminate the condensate toinduce Bragg scattering and we subtract the averaged background to count the number of photons
gained or lost in the weak beam due to the Bragg excitation.

102

15k

(a)

N

exc

10k

5k

0
0

1

2

3

(ms)

30k

(b)

N

exc

20k

10k

0
0

5

10
(ms)

Figure 4.11: Integrated Bragg excitations measured with photon counting, as a function of time.
Diﬀerent colors correspondto the diﬀerent drive amplitudes. The number of Bragg excitations is∫given by Nexc = 0τ (ṅ(t) − ṅavg )dt, where ṅavg is the average rate of weak beam photons measured
when no strong light is present. The Bragg beams are turned on at 0 ms, beginning the Bragg
excitation process. Experiments are performed on resonance, for a BEC of 400,000 87 Rb atoms.
The two plots (a) and (b) correspond to the same data, plotted on diﬀerent scales.

103

4.6

Outlook
We have demonstrated photon counting as a viable technique for Bragg spectroscopy inultracold atoms. By measuring the response of the driving ﬁeld to the atoms, one has a measurement
independent of the traditional time-of-ﬂight imaging of the atom cloud. Our heterodyne scheme
achieves shot-noise limited detection and allows us to measure the number of photons added to, or
depleted from, one of the Bragg beams.
Photon counting for Bragg spectroscopy is not without its own unique complications, however.
Because the weak Bragg beam proﬁle is roughly matched to the transverse proﬁle of the condensate,
careful alignment must be maintained for the photon counting approach. Also, this method isinherently a measure of the net eﬀect on the probe. This is problematic for small momentum
transfers, where the positive and negative frequency features of the lineshapes seen in Fig. 4.10
begin to merge for Bragg frequencies near zero. This complicates the analysis as described in theprevious chapter. Ultimately, for this approach to be successful, I believe that investigations ofphoton counting would have to be at slightly larger momentum transfers where these eﬀects areless important. In this regime, the technique could be investigated and characterized, and later
extended to small k.
Still, this technique could ﬁnd more success in systems whose atom number is large comparedto the 40,000 we employ in our 85 Rb BECs. Furthermore, measurements of the scattered Bragg
photons could open the door to new investigations of the temperature dependent structure factor,
as proposed by Stamper-Kurn et al. [70]. This technique of measuring the probe in order todetect atom-light interactions could be applied to other types of ultracold atom spectroscopy aswell [93], and it seems that the two technologies, photon counting and time-of-ﬂight imaging, marrynicely.

104
4

2x10

(a)

4

Nexc

1x10

0
4

-1x10

4

-2x10
-40

-20

0

20

40

Bragg frequency (kHz)

(b)
y

kweak
z

Iweak

z
G

z
Figure 4.12: (a) Example of an asymmetric lineshape. Here the excitations were measured using thephoton counting technique, but traditional time-of-ﬂight imaging of Bragg excitations gave results
consistent with the photon counting data. The peak corresponding to “extra” weak beam photons(here the one at -18 kHz) is larger in amplitude than the peak corresponding to depletion of the
weak beam. For this reason, we speculate that this could be due to depletion of the weak beam.
In (b) we illustrate the concept of depletion. For the case where the Bragg photons are large innumber compared to the weak beam photons we see the weak beam being attenuated as it passes
through the condensate. This, in turn, lowers the Bragg response of the weak beam, resulting inthe smaller amplitude peak for the Bragg resonance corresponding to absorption of weak beam
photons during the Bragg process. Attempts to quantify this eﬀect, however, have so far been
unsuccessful, as strong beam scatter makes studies in these regimes diﬃcult.

Chapter 5

Contact spectroscopy

In this chapter I present the concept of the contact interaction and its usefulness in probing
strongly interacting BEC. We will see that the contact connects the microscopic physics of shortrange, two-body interactions to macroscopic, thermodynamic quantities in the system. Using rf wehave made preliminary measurements verifying the existence of the contact interaction in a BEC.
We observe both a dependance on a and see evidence of the high-momentum states that we areprobing, in line with the predictions for a contact in a BEC.
The technique of contact spectroscopy looks promising in measuring the LHY correction to
the energy of a BEC, something which to date has only been accomplished with extremely weakly
bound dimers made of fermionic atoms [94]. Remarkably, this measurement matches the LHY
prediction, even though the composite bosons in these systems are clearly not the hard spheres orpoint-like bosons described by the LHY theory. These intriguing results from strongly interacting
fermionic atom gases highlight the need for a measurement of the LHY correction in a dilute gasof bosonic atoms, where one can directly test the LHY theory.

5.1

Making contact with an interacting BEC
Since the ﬁrst chapter of this thesis, we have argued that the two-body interactions describedby the scattering length a dominate the microscopic and macroscopic behavior that has been
observed in BEC. In this chapter, we present the contact, a quantity that directly relates the twobody physics of s-wave scattering to bulk thermodynamic quantities. This is a universal result that106
works in the context of Fermi gases as well, as it is only concerned with two-body physics.
One such way that the two-body interaction manifests itself in other physical characteristicsof the interacting gas is through the momentum distribution, n(k). For a certain range of momentum, all gases with two-body interactions will exhibit the same characteristic 1/k 4 momentum
distribution (this can be seen by taking the Fourier transform of the two-body scattering wavefunction). The contact quantiﬁes the strength of this 1/k 4 momentum tail for the particular gas inquestion, given bylim

1/r0 > k > n1/3

n(k) = C/k 4(5.1)

where C is the contact (also referred to as the contact interaction), with units of length−1 . Notethat the contact is an extensive parameter of the system, and for this reason we will often times
normalize the contact to the total number of particles in the gas and write the contact per particle
C/N .

rY

r

r0 a2 a1

Y1

n

-1/3

Y2

Figure 5.1: Two scattering wavefunctions (the Fourier transforms of which give n(k)) at diﬀerenta, illustrating the limits of the contact interaction. For r < r0 , details of the interatomic potential
cannot be ignored, setting a lower (upper) limit on r (k). For r ∼ n−1/3 the two wavefunctions arequite similar, as far as the Fourier transform is concerned, and can hence also be disregarded when
considering the projection to the 1/k 4 tail, setting an upper (lower) limit on r (k).

We can understand the limits in which Eq. refeq:5n2body is valid, 1/r0 and n1/3 , by considering the associated lengthscales. In Fig. 5.1 we present two scattering wavefunctions, each with
its own scattering length, a1 , a2 . The lower limit of Eq. 5.1 (in terms of lengthscales) is set by the

107eﬀective range of the interaction, r0 . This is the lengthscale where the wavefunction can no longer
be described by one simple scattering parameter a, and the details of the interatomic potential must
be taken into account. For 85 Rb this is roughly 80 a0 [95]. Because these short-range potentials
are particular to each species, one does not expect a universal result (such as the 1/k 4 tail) in thisregime. We therefore ignore the curvature in the wavefunction for r < r0 when we take the Fouriertransform. Hence an n(k) ∼ 1/k 4 must only be only valid for k < r0−1 .
The other limit is set by the interparticle spacing n−1/3 . Notice in Fig. 5.1 that at suchlarge distances there is only a subtle diﬀerence, essentially a phase shift, between ψ1 and ψ2 .
When taking the Fourier transform of this wavefunction, this subtlety is washed away and the1/k 4 tail is insensitive to changes in a at these large lengthscales (small k scales). In reality, theactual wavefunction (as opposed to the simpliﬁed picture of the scattering wavefunction) will be
modiﬁed by the presence of other particles at distances n−1/3 , so that n(k) ∼ 1/k 4 is unphysicalfor k < n−1/3 .
What is left is a parameter a that characterizes a universal feature seen in all gases, the1/k 4 tail. It would turn out that a will characterize a great number of macroscopic parameters,
all of which can be related through C [96, 97, 98]. While the universal relations connecting C
to these macroscopic parameters have been developed and tested in the context of Fermi gases[99, 100, 101, 102, 103, 104, 105], we know that the contact interaction is two-body. Becauseof this, the derivation of these universal relations also does not depend on the particle statistics
and can therefore be generalized to our 85 Rb Bose gases as well [106]. We use these ideas as ameasurement tool of the LHY correction.
One of these universal relations connects the contact of a gas to its energy through what iscalled the adiabatic sweep theorem [97]
h̄2
dE
=
CBEC
da
8πma2

(5.2)

where E is the total energy of the gas. We write CBEC here to alert the reader that this universal
relation diﬀers by a factor of two from the one considered by Tan [97] due to our Bose gas being

108
completely spin-polarized [107]. Combining this, with the LHY result (Eq. 1.3), yields the predictedcontact for a condensate
(

)

CBEC
5 128 √ 3√
= 16π 2 na2 1 +na + . . .
NBEC
2 15 π(5.3)

where NBEC is the condensate atom number. Notice that the LHY term is a 5/2 bigger correction
for the contact interaction compared to the energy density (Eq. 1.3). This is because the adiabatic
sweep theorem (to which the contact is proportional) does not address the energy directly, butinstead tells us how the energy directly changes with a. This important distinction makes thecontact a powerful tool for probing beyond-mean ﬁeld physics, allowing us to probe large corrections
to the contact at smaller values of na3 than would be necessary if resolving the energy directly.
This is an important consideration since we must always ﬁght timescales associated with three-bodyrecombination.

5.2

How rf spectroscopy connects
Spectroscopy in the radio domain has been a highly successful probe in the cold atom com-munity, and has helped to unwrap the physics of the Feshbach resonance upon which this thesis
is built [46]. For a more detailed discussion of rf spectroscopy and its usefulness I refer the reader
to Dr. Stewart’s thesis [108]. One might hope that it would again be a useful frequency domain
for spectroscopy of strongly interacting condensates, falling somewhere between the success of the
optical domain (the aforementioned Bragg spectroscopy), and the failures inherent in the spectroscopy of breathing modes (being too slow with respect to three-body loss). The problem withrf spectroscopy as a direct probe of interacting condensates, however, is due to the nature of theenergy splitting between the states. With rf we probe a transition between diﬀerent magnetic
Zeeman sublevels. If we intend to use rf to directly resolve the interaction energies of a BEC (afew 100 of Hz) we must then be comfortably above the noise associated with this energy splitting:magnetic ﬁeld noise. In our system, this noise is on the order of 5 mG or 30 ppm of our total ﬁeld.
In terms of the actual energy splitting, this corresponds to 4 kHz, which is 10 times larger than the

109energy we are trying to resolve. If our goal is more bold, say resolving an LHY correction of 30%,
the prospects of directly probing the energy with rf spectroscopy is even more bleak.
What the contact, and Tan’s relations, allows for is a way to use rf to probe the energy,
or more accurately dE/da, with rf that is detuned from the splitting associated with the Zeeman
states. An extension of Tan’s universal relations asserts that the 1/k 4 tail gives rise to a tail in therf lineshape that goes as 1/ω 3/2 for large detunings [109, 110], which was later veriﬁed [105]. Being
able to work at such large detunings, where one is tasked with measuring the strength of our signal(rather than the resonance of our signal), makes our magnetic ﬁeld noise tolerable.

5.2.1

Final-state eﬀects

In our experiments, the rf transfers a small fraction of the spin-polarized atoms from the|F = 2, mF = −2⟩ state into the |F = 2, mF = −1⟩ state, which we refer to as the ﬁnal state. We
modify Braaten’s result [111] for a gas with two spin states to take into account the fact that ourgas is spin polarized and writeη τ Ω2 α(a)lim S(ω) =ω→∞
8π ω 3/2 β(ω)
where

√

h̄
C,
m

(5.4)

∫ ∞

π
S(ω)dω = η τ Ω2 N.
2
−∞

(5.5)

Here τ is the rf pulse duration, Ω is the Rabi frequency, and N is the total number of atoms in oursystem. The imaging eﬃciency, η, will be discussed later in more detail. Despite the fact that Eq.
5.4 and Eq. 5.5 diﬀer from Braaten’s result by a factor of two, the ratio of the two formulas is thesame in both cases. This is important because we will later be using Eq. 5.5 to normalize Eq. 5.4.
In Eq. 5.4, the ﬁnal-state corrections are encompassed in α(a) = (a′ /a − 1)2 , and β(ω) = 1+h̄ω/E ′ ,where E ′ = h̄2 /ma′2 . These ﬁnal state corrections are due to interactions between atoms in the ﬁnal
spin state and atoms in the initial spin state, and are characterized by an interspecies scattering
length a′ = −565a0 for our system [112], a0 being the Bohr radius.

110

5.3

Contact spectroscopy experiments
Our experiments probe 4 × 104 Bose-condensed 85 Rb atoms in a gas with a 60% condensate

fraction. During the ﬁnal stages of evaporation, the ﬁeld is set to give a scattering length of 130 a0 ,and the calculated averaged number density is 5 × 1012 cm−3 . After evaporation, we ramp the bias
ﬁeld in order to change a on a timescale that is fast compared to the trap period, but adiabatic withrespect to h̄/ma2 (ȧ/a < 0.01h̄/ma2 , ȧ being the time derivative of a). To minimize magnetic-ﬁeld
gradients to less than 5 G/cm, we leave the atoms trapped but change the center position of thevertical conﬁning potential.
An example of rf contact spectroscopy is shown in Fig. 5.2(a). We probe the condensate
roughly 1 ms after the magnetic-ﬁeld ramps using an rf pulse to drive the |F = 2, mF = −2⟩ to|F = 2, mF = −1⟩ transition, and our signal S ′ (ω) = S/ π2 τ Ω2 N (black circles) is the number ofatoms transferred to the ﬁnal spin state (S), normalized for the diﬀerent pulse lengths and powers.
The rf pulses are amplitude modulated to a Gaussian envelope and we selectively image ﬁnal state
atoms by ARPing them to an imaging, |3, 0⟩ state, where our probe laser is far detuned from theatomic transition for atoms in the BEC. We ﬁt S ′ (ω) to a Gaussian lineshape, also shown (blackline in Fig. 5.2(a)). The center of this ﬁt is taken to be the bare Zeeman splitting (ω0 ). For small
detunings, we use short pulses, with an rms width of 25 µs. At large detunings, we increase theweak signal by using pulses with a 200 µs rms width and 100 times more rf power. We outcouple
at most 10% of the condensate and check that the signal depends linearly on τ Ω2 . We correct thesignal for a small measured nonlinearity due to saturation eﬀects, at the level of 10% for the tailand 7% for the main lineshape. For large negative detunings, we observe the expected 1/ω 3/2 tail(see inset of Fig. 5.2(a), with the Fourier-limited lineshape in black for reference). Because the rf
drives stimulated emission to a lower energy ﬁnal state, the interaction-induced 1/ω 3/2 tail is on
the low frequency side of the rf resonance. At large positive detuning (see Fig. 5.3), we ﬁnd thatthe signal is consistent with zero.
Shown in Fig. 5.2(b) is the spatial size of the outcoupled atom cloud in time-of-ﬂight ex-

111

-8

-5

6x10

1.2x10(a)-6-8

9.0x10S' (s)

3x10

-6

0
80.03

6.0x1080.06-6expansion energy / h (MHz)

3.0x10

0.0
0.04

(b)

0.02

0.00
80.02

80.07

(

+

)/2

0

80.12

(MHz)

Figure 5.2: Example
of rf contact spectroscopy. (a) Normalized number of outcoupled atoms,∫∞′
S (ω) = S(ω)/ −∞ S(ω)dω. We ﬁt to a Gaussian lineshape (black line) and take the center to be
the single atom resonance ω0 . In the inset, we show the 1/ω 3/2 tail at large negative detuning,where the signal is much larger than would be expected from the Gaussian ﬁt (black line). The
red dashed line is a ﬁt of Eq. 5 to the 1/ω 3/2 tail. (b) Kinetic energy (K.E.) of outcoupled atom
cloud, in units of MHz, measured in expansion. We calculate the energy√ from the width of the3
2
outcoupled atom cloud (σ), assuming K.E. = 2 mvkick /h, where vkick = σ 2 − σ02 /T . Here σ0 is
the inherent width of the expanded cloud, measured on resonance, and T is the time between therf pulse and the absorption image (4.5 ms). We use T instead of the expansion time because our rf
couples the atoms into a less trapped state, and the subsequent ARP puts them in an untrappedstate, expanding freely before we have turned oﬀ the magnetic conﬁnement. In black we plot a
prediction for the expansion energy divided by h, − 12 (ω − ω0 )/2π. We predict that the expansion
energy should goes as 12 h̄ω, since the extra energy from the rf photon is split between two particles,
owing to the pair-wise contact interaction. Our data shows good agreement with this prediction.

112
pansion. In the region of the 1/ω 3/2 tail, the expanded cloud size increases monotonically withincreasing |ω|. This is expected as larger |ω| outcouples atoms from further out into the 1/k 4 tailin the momentum distribution. Notice that the data agrees well with the prediction (black line)
described in the ﬁgure caption.
To extract the contact from our data, we plot the 1/ω 3/2 tail of S(ω) multiplied by ω 3/2 ,as shown in Fig. 5.3. We ﬁt this to B/β(ω), where B = S(ω)β(ω)ω 3/2 , at large |ω|. For ourﬁts we only consider detunings between 2π × 30 kHz and 2π × 100 kHz, which is well outside theFourier-limited central lineshape that has an rms width of 1/25 µs= 2π × 6 kHz. The main eﬀect ofthe β(ω) ﬁnal-state correction is to suppress the tail. We illustrate this by plotting what we would
expect, given the ﬁt, had β(ω) = 1 in the blue dashed line of Fig. 5.3. There also exists a minimal
frequency dependence of β(ω) for our range of detunings.
We present preliminary results of our measurements of the contact in an interacting BEC
in Fig. 5.4. On the vertical axis, we show the contact per particle C = C/NBEC , normalized tothe average density of our cloud, with units of µm2 . We normalize the contact per particle to theaverage density to easily compare data taken at slightly diﬀerent densities, as Eq. 5.3 shows us thelinear dependence of C/NBEC on n. On the horizontal axis, we show both the scattering length√√
na3 , on theon the bottom, and, for reference, the LHY correction to the energy density, 15128
πtop. The Rabi frequency Ω is calculated by measuring the relative spin populations after rf sweeps
inducing an adiabatic rapid passage (ARP), as a function of rf power. By measuring the transferredfraction, we can then deduce Ω [77]. The error bars are underestimates of the uncertainty, reﬂecting
the error in the ﬁt to B.
We image the ﬁnal state atoms by performing a ARP with > 95% eﬃciency into the |3, 0⟩
state. Atoms are imaged with σ − light, which also optically pumps them to the |3, −3⟩ state
where the imaging light is now resonant with the |3, −3⟩ to |4′ , −4′ ⟩ cycling transition. Because theimaging light is oﬀ resonant with the initial optical-pumping transitions, and these transitions havesmaller Clebsch-Gordan factors relative to the cycling transition, the number we measure in theabsorption images of this state is systematically low. We describe this systematic with an imaging

113

3

2

3/2

(Hz

1/2

)

4

S'

1

0
-60

-40

-20

/2

0

20

40

60

(kHz)

Figure 5.3: Extracting the contact at a = 500 a0 , navg = 4.9×1012 cm−3 , where navg is the densityweighted-density of the BEC. The condensate
fraction for this data was 73%. The signal has again∫∞
been normalized so that S ′ (ω) = S(ω)/ −∞S(ω)dω, as well as multiplied by the ω 3/2 dependence.
We ﬁt this (black line) to B/β(ω) = S ′ (ω)ω 3/2 = 4π1 2 α
√
2

C
4π
N =B α√h̄ C
m N , where C = 8π√mBh̄ τ Ω2 α(a) .

Thusmh̄ , and we are able to extract the contact. We also plot the ﬁtted value of B (dashedblue line), which is what the signal would have been had β(ω) = 1, illustrating this ﬁnal-stateeﬀect.

114

LHY correction
0.0

0.06

0.17

0.3

2

C/n ( m )

2.0

1.5

( )

1.0

0.5

0.0
0

500

1000

1500

a)

scattering length (0

Figure 5.4: Preliminary BEC contact data as a function of interaction strength. Measured contact
per particle (open circles) divided by n for diﬀerent interaction strengths. The black circles show
the data after making a correction for ﬁnite temperature as described in the text. The s-wave
scattering length is given in units of a0 on the lower scale and the fractional LHY correction isshown on the upper scale. The red line shows the mean-ﬁeld prediction, using a local density
approximation, while the black line is calculated like the red, but includes the LHY correction to
the energy as well. It should be noted that it is quite possible that the apparent agreement between
the data and the theory is fortuitous, and the error bars are not indicative of the other systematics
inherent in these measurements.

115
eﬃciency η. The imaging eﬃciency is measured by comparing the number of atoms measured in
the |3, 0⟩ versus the |3, −3⟩ state in subsequent absorption images of thermal clouds of atoms. The
states are prepared by using rf pulses and ARPs and we can check them by verifying that themeasured atoms remaining in the |2, −2⟩ or |2, −1⟩ state is negligible. We later found out that thismeasured eﬃciency was highly sensitive to probe detuning, which was not constant throughout thedata presented in Fig. 5.4. As a consequence the error bars shown in Fig. 5.4 are at best a lower
limit to the actual uncertainty in the data, and we cannot accurately make an LHY measurementfrom these results.
The breathing-mode excitations discussed in Ch. 2 are modeled [56] and we ﬁnd that at thetime of the rf pulse the density has changed by less than 10%. In Fig. 3, this calculated density
decrease has been included in the results. We also consider the eﬀects of ﬁnite temperature in thecontact measurement. At a = 750 a0 , we have measured the dependence of C on the condensate
fraction, and we use this to empirically extrapolate our data (white circles) to zero temperature(black circles).
It should be noted that for the contact the next order correction is expected to be as large
as the LHY correction itself at 1200 a0 for our densities [34, 35, 36].

5.3.1

Advantages

Contact spectroscopy, oﬀers advantages compared to a direct probe of the energy of the gas.
As mentioned earlier, three-body loss limits the time available for probing a BEC with large a,and this, in turn, can limit the resolution of energy measurements. However, in measuring thecontact, the pertinent data is the amplitude of the signal far-detuned from resonance and noise on
the central resonance of the lineshape is much less of a concern. This also allows us to use a short
rf pulse if needed, since Fourier broadening of our pulse is tolerable. Finally, in Fig. 5.5, we showthat ﬁnal-state eﬀects in the case of 85 Rb serve to enhance the signal, particularly at smaller valuesof na3 . Notice how the signal goes as (a − a′ )2 , whereas the contact itself (Eq. 5.3) goes as a2 .
This will help facilitate a future measure of the LHY term, by giving an opportunity to validate

116
measurements at interaction strengths for which the LHY correction is negligible, and the physics
is well understood.

5.3.2

Challenges

Contact spectroscopy is not without challenges. The biggest challenge we have found is in
obtaining accurate measures of the BEC number. This is crucial for any contact measurement as
the contact is an extensive quantity. We had previously tried to avoid such issues by normalizing
our measurements to the total lineshape (i.e. determining τ Ω2 N ) , as given by Eq. 5.5. It turns
out there is some diﬃculty with this approach, as seen in Fig. 5.6(b), where the Rabi frequency asmeasured with an ARP is compared with a Rabi frequency measured by integrating the lineshape.
The suspected mechanism for suppressing the number of atoms outcoupled to the ﬁnal state (which
is directly proportional to τ Ω2 N ) is collapse induced by the large, negative background scattering
length of -565 a0 . This collapse would lead to large three-body loss rates, and atoms that would
never be imaged [66].
This eﬀect should only occur for measurements on resonance, and not for data at large rf
detuning. This is because the atoms outcoupled in the 1/ω 3/2 tail at large detuning have kinetic
energies (see Fig. 5.2) much larger than the attractive energy provided by a′ = −565 a0 (∼ 300
Hz).

5.4

Conclusion
In conclusion, we have presented the ﬁrst measurements of the contact in a BEC. Our resultshow the correct scaling with a, and the outcoupled atoms also reﬂect the high-k tail that we would
expect in a contact measurement. We hope this measurement technique, contact spectroscopy, willallow for future investigation of dynamics following a fast change in the interaction strength. It will
also be interesting to push these measurements into regimes where the interaction can no longer
be considered perturbative.

117

10

8

a)

6

0.5

4

(

B (arb.)

1.0

2

0.0
-500

0

500

1000

1500

a)

scattering length (

0
2000

0

Figure 5.5: Strength∫ of signal at diﬀerent a, divided by the average density, n. The data has again∞
been normalized to −∞S(ω)dω. The solid line is the prediction for mean-ﬁeld theory. The eﬀect2′of the α(a) = (a /a − 1) ﬁnal state correction is shown in the blue dashed line, and its eﬀect on
the data is clearly seen here, shifting what would be a parabola centered about 0 to one centeredabout a′ = −565a0 . This eﬀect greatly enhances our signal at small a, making future calibrationsof contact measurements at low na3 possible. This will be important for eventual measurements of
the LHY correction at high na3 .

118

2

(kHz )

32

2
1
integrated lineshape taken on a BEC
integrated lineshape taken on a BEC at half the density
integrated lineshape taken on a thermal gas
ARP's measurements made on a thermal gas
scaled rf coil power dependance on a

0
0

500

1000

1500

scattering length (a )0∫

Figure 5.6: Pitfalls of using S(ω)dω to calculate τ Ω2 N . Blue squares correspond to ARP measurements used to determine Ω2 . For the black circles, the Rabi frequency is determined by integrating
a lineshape like that shown in Fig. 5.2(a). The rf power, and therefore the Rabi frequency, isdependant on the frequency of the rf, which in turn depends on the B-ﬁeld. On the x-axis, weshow the scattering length, which is tuned by varying the B-ﬁeld. The fact that the black circles
are systematically low supports the argument that the |3, 0⟩ atoms suﬀer loss and are not fully
counted. Data taken in a similar fashion using a low density thermal gas (red circles), or BECs athalf the density (open circle) further support this argument. The black line is arbitrarily scaled
and reﬂects the response of the rf antenna as a function of B-ﬁeld.

Chapter 6

Conclusion

We began this thesis with an allegorical story of a horse chasing some mythical carrot, so it
only feels right to end it in the same way. While some of the smaller questions about the story
might have been solved by the reader (in particular, the identities of the farmers holding the stick)there exist more fundamental questions: Have we tasted the carrot? How much farther is the carrot
patch?

6.1

A small bite
While we have not been able to make a quantitative measure of the LHY correction in a BEC,

I believe we are on the doorstep, staring down upon this carrot patch. The advantages provided by
the 10 Hz spherical trap are being harnessed every day in lab, as 1 ms probes at large na3 are now
the norm, whereas before they were only a dream. Bragg spectroscopy provided our ﬁrst taste of
the carrot in the 2008 experiments we performed at large na3 . Fully motivated from the nibble, wehave been able to carve out two distinct roads for approaching our many-body physics destination.
The ﬁrst is the experimentally realized photon counting technique for Bragg spectroscopy.
This path can be a little hilly, as we always measure a net response that complicates Bragg at smallk, but steers clear entirely the problems incurred with large interactions that complicate a Bragg
signal. In addition, it oﬀers a number of tangential opportunities for the study of BEC physics.
While these might not fall in line with our strictly carrotarian diet, they might provide a satisfying
meal in their own right.

120
The second is contact spectroscopy, a new stretch of road that perhaps oﬀers the straightest
path towards an LHY measurement and our carrot patch. We have shown preliminary data verifyingthat contact measurements are feasible in our system, and look ideally suited to measure the LHY
correction. As with any new road, however, there are a spots that need to be ironed out. Primaryamong them is an accurate way to measure the number in a strongly interacting BEC, or, better
yet, a way to normalize the measurements of the rf tail. With the promise this has to oﬀer, though,it will not be long before this becomes a major highway on the way to BEC enlightenment.

6.2

Staring down on the carrot patch
I believe the future looks bright for the strongly interacting BEC machine. With the pre-liminary results contact spectroscopy has brought in, an LHY measurement looks like it might be“one week away”. My brothers-in-lab have already implemented high-ﬁeld imaging, and a hostof other technical improvements mere moments after I set foot out the door. With a few more
systematics nailed down, they should be publishing some highly sought after results in the near
future. With contact spectroscopy, there is a hope that we could also test the rates of our sweeps
of a to determine the proper criteria for adiabaticity with respect to local many-body equilibrium.
This is a current debate amongst ourselves and if we could shed light on this subject BEC dynamics
could be experimental desserts.
The photon-counting that we worked so hard for ultimately took a back seat to the contact
spectroscopy measurements that are still underway. This probe still has much to oﬀer in termsof BEC physics, however, as there is an entire jungle of experimental phase-space still left to be
explored in the many-body map (see Fig. 3.3). A few weeks before making the leap to contact
spectroscopy, we discovered that we had been using a much smaller beam waist than we hadplanned for. It would have been interesting to see what results could have come from a redesigned
weak beam at 15 µm. The ideas being tossed around at the time were of blue-detuned trap beams
to damp out slosh, moving to a higher k so that the lineshapes were cleanly separated, or probingat θ = 90◦ and θ = 180◦ , as our redesigned trap oﬀered new areas to map out the dispersion of

121
strongly interacting BEC.
Other outstanding questions could be directly answered with our photon counting technique
by using the larger 87 Rb clouds aﬀorded in our experiment. It is predicted that the temperaturedependence of S(k) could be observed by measuring the Bragg response with a photon-counting
technique [70]. In addition, with a system capable of studying the dynamics of a Bragg excitation,
fundamental questions of the coherence of these momentum excitations can be addressed, which
would have clear value to the atom-optics community. There are plenty of carrots to be had withthe 85 Rb machine, and I believe through our work, the stick holding them from our nose is thatmuch shorter.

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Appendix A

Cart switch

To provide a uniform magnetic ﬁeld in ŷ (needed to shift the center of the magnetic potentialof the 10 Hz spherical trap to compensate for sag and align to the optical trap) we modiﬁed
the circuit driving a pair of coils that is mounted on our moving track [49]. The “cart coils”
provide a magnetic quadrupole trap for the transfer of atoms from the MOT to the science cell,
where the magnetic trap resides (see Ref. [49]). To go from the anti-Helmholtz conﬁguration for
the quadrupole trap potential to the Helmholtz conﬁguration for a uniform ﬁeld in ŷ necessitateschanging the direction of the current in one of the coils. While this might sound easy enough, thesecoils have signiﬁcant inductances that we must ﬁght against in order to keep things from...well,blowing up.
The switching scheme we use to ﬂip the current in the top coil is referred to as the H-bridge,
and is shown in Fig. A.1. To switch the direction of the current, we employ the same types ofﬁeld-eﬀect transistors (FETs) that are used for servoing the current in the cart coils.1

Becausethe task of these FETs is only to switch the current direction, a mechanical relay would work ﬁne
for this application. However, for the large current (∼250 A), we would need to string a numberof solid state relays in parallel to make this work. The cost of these relays ($300/relay, with 8
relays needed) as well as our inexperience with them pushed us towards our trusted (and available)
FETs. We opted to stay away from mechanical relays (read: expensive) to avoid changing thecharacteristic capacitance and impedance of the cart coils, and, more importantly, the dynamics of1This is the APT10M07JVR FET that can be found in a number of the cold-atom groups here at JILA, and is
the same one used in our magnetic trap servo.

130
the cart servo loop. We use three FETs in parallel, for each single FET shown in Fig. A.1, and wewater-cool the FET bank in the typical way.
Complications to the switching scheme arise because of the inductance in the cart coils
themselves. We calculate the self-inductance of each coil to be on the order of 70 µH. We also candirectly measure dI
dt using a Hall-probe current sensor after quickly turning oﬀ the current. Using
V = L dI
dt and the measured voltage when the current in the coils is turned oﬀ quickly gives us an
inductance of 380 µH. This large discrepancy is most likely due to our neglecting of the resistanceof the coils, which modiﬁes the above voltage drop V = L dI
dt + IR. When installing the H-bridge
conﬁguration, we shut oﬀ the current running in series through the two coils and measured a 130 V
spike (with respect to ground) at point (a) of the bottom coil (shown in Fig. A.1) and a 60 V spike(also with respect to ground) at point (b). These large spikes in voltage assured certain death forour switching FETs, as they are only rated to take at most a transient gate-source voltage of ±40
V.
These transient voltages led us to make two modiﬁcations to the standard design. The ﬁrst
is the protection diodes seen in Fig. A.1. These diodes represent four 1N4004 diodes in parallel.
The transient voltages, now given a shunt to the power supply voltage (which is typically 6 V),
were reduced to 60 V at point (a) and 45 V at point (b).
While this helped with the problem of the FETs blowing up, our overall goal was a bit more
ambitious than that. We still needed these FETs to stay on when asked! This was especiallyproblematic, seeing as how the FETs respond to the aforementioned gate-source voltage, with oursource voltage swinging up to as much as 60 V. Seeing as how the gate voltage was being referenced
by our switching electronics, keeping the FET on (i.e. VGS > 3V ) was challenging. To that end,
we devised a scheme that works equally well for high current switching as it does for householdsmoke alarms: the ever-popular 9V battery.
The batteries depicted in Fig. A.1 are of the 9V variety and function to maintain VGS at 9
V by referencing the battery’s anode to the FET’s source. These FETs now function as switches
in the “closed” position. To keep the other pair of FETs in their oﬀ state (“open” position) we

131

+V

bottom
coil

a
30 W

30 W
top
coil

30 W

b

30 W

to cartservo

Figure A.1: Electrical schematic for H-bridge switching FETs. All of the switches shown here aretied together. As shown the current would ﬂow through the top coil from left to right. Notice theseparate grounds for the power supply and the switching electronics. Circled points are referred toin the text. The single FETs here represent three APT10M07JVR FETs in parallel. The switches
shown are actually AZ831-2C-5DSE relays which are robust to high voltages.

132
reference the gate to the electronics ground, depicted by the hollow triangle. When we have thelarge voltage spikes induced by the quadrupole trap turn oﬀ these FETs stay in their “oﬀ” state,
as VGS becomes negative, in this case. We ﬁnd that these batteries worked extremely well, despite
the fact that the voltage spike is extremely fast, rising to its peak voltage in roughly 10 ms. We
suspect that the battery is able to track the source voltage due to its negligible capacitance withthe rest of the electronics.

Appendix B

Alternative approaches to BEC in a 10 Hz trap

B.1

The 2nd law of graduate student dynamics
There are a handful of “experimental axioms” tossed around in physics laboratories. Someof these are useful in diagnosing an elusive problem (“if you can’t make things better, make things
worse”). Some more useful in working with signals (“use lock-in detection, noise is always better
at AC”). And some still for those interpreting data (“with enough ﬁt parameters I can make thedata agree too!”). Another axiom that is familiar with every graduate student in the lab is what I
call the 2nd Law of Graduate Student Dynamics. Practically speaking, it means “If it ain’t broke,
don’t ﬁx it.” This is learned through both experience (“...I touched this one knob that shouldn’t doanything and now the experiment doesn’t work...”) and through instruction from the more senior
students (“...don’t ever, EVER touch any knob on here unless you absolutely have to...”). For
better or worse this is a lab mantra, and, after doing a number of informal surveys (generally on
the way to lunch) this is something ubiquitous to nearly every graduate student that has put in timeon his or her respective machine. Much of this law borrows from the 2nd law of thermodynamics,
which tells us that entropy and disorder will always go up in a system. In the laboratory this often
means that each additional layer of control (“I made the experiment better!”) comes at the cost ofadditional phase-space for things to spiral out of control (“...and now nothing works”).
I mention this because in developing this spherical trap, we also had to develop a new evaporation scheme, throwing out a lot of what was working for us. And while our evaporation isdefensible in light of its results, knowing the 2nd Law a diﬀerent motivation is needed, at least

134
for the graduate students’ sake. In this appendix, I outline a few other ideas and paths we choseto go down, with the goal of minimizing change, and why these other avenues ultimately did not
succeed. What we found was that, for very diﬀerent reasons, none of these strategies would workto our satisfaction. It was this realization that pushed us towards developing an entirely new way
to think about evaporation and sympathetic cooling.

B.2

Crossed optical trap
Because simultaneous cooling of 85 Rb in the optical trap was successful, the simplest approach

would be to add another laser beam for trapping in order to get the spherical geometry we desired.
With this additional axial conﬁnement, of course, comes larger mean trap frequencies, which arebad for both the physics we aim to study (see Fig. 2.3), as well as being major roadblock in the
production of 85 Rb BEC [49].
In order to keep the mean trap frequency the same, we would need a reduced radial conﬁnement, and hence more gravitational sag (ysag = g/ωy2 ). This is the eventual deal-breaker of
optical trap geometries, because when the sag is on the order of the beam waist, the trap depth iscompromised and strong anharmonic components are introduced in the potential. We predicted,
however, that with substantial amounts of laser power (∼ 1 W ) at 1064 nm, and large enoughbeam waists (400 µm) a spherical trap would be possible with a potential created by two crossed
beams. With this crossed beam design, the ŷ direction would share the radial conﬁnement between
the two beams (i.e. both beams perpendicular to gravity).
In trying to implement this design, we found that with the needed laser intensities our optics
severely distorted the imaging beam, as seen in Fig. B.1. We attributed this to heating of opticalelements, causing a false signal in the OD images after the background image was taken 300 µs laterand subtracted away. Shortly after this discovery we abandoned the crossed optical trap. Althoughheating of the optical elements could perhaps be avoided with less time between exposures, thistype of trapping scheme would also require us to change out our original trap optics. Furthermore,this scheme would always be something of a compromise, having the spherical geometry, but still135
with same relatively tight conﬁnement as our previous optical trap.

Figure B.1: Absorption image taken with no atoms present after ∼ 2 W of optical trap light for the
crossed beam trap passed through optics shared with the imaging system. The contrast between
the fringes seen here goes from an optical depth of -0.05 0.1. We speculate that these fringes are dueto the heating and subsequent cooling of optical elements, distorts the beam proﬁle. Changes inthe spatial proﬁle of the imaging beam between the shadow image and the reference image (without
atoms), taken 300 ms later can cause the fringes.

B.3

BEC loading
The next approach, and perhaps the simplest one in terms of experimental changes, was toload atoms from the optical trap into the 10 Hz magnetic trap after we had created the BEC in ourconventional way (simultaneous cooling of both 85 Rb and 87 Rb atoms in an optical dipole trap).
This presented a few of its own diﬃculties, of course. First, as discussed previously, the equilibrium
size of the BEC is very sensitive to the scattering length a. In order to keep the same scattering
length throughout the loading process, we must ramp down the bias coil current in concert with
ramping up of the magnetic trap currents. This would be necessary to avoid the breathing modes
induced by a change in a, discussed previously.
The bigger issue, however, was in the gross mismatch of the shape and strength of the two
traps. We would be loading from a 135 × 135 × 3 Hz trap to one that was 10 Hz all around. Things
like residual breathes and sloshes would be easily excited and eventually we would have to devise
some way to damp these oscillations, which can be inherently diﬃcult for a superﬂuid. These136
mode-matching issues also showed up when the traps were slightly misaligned, as seen in Fig. B.2.
Here, some of the BEC has spilled over a potential “bump” halfway through the load into the 10
Hz trap and the BEC then had a two-component shape. It was at this point that we abandoned
this approach and realized that the only way a stable 85 Rb BEC was going to be happy in the 10
Hz trap was if it was born there.

Figure B.2: Image of a 85 Rb BEC after ramp down from optical conﬁnement to the weak, spherical
trap. Images such as these, with two clearly distinguishable features, are a result of misalignment
between the optical and weak magnetic traps.

B.4

Direct evaporation on 85 Rb
The last approach we tried before revamping the cooling scheme was to go back to thetechnique of the original 85 Rb experiments [48] and directly evaporate the 85 Rb gas, via forced rf
evaporation, in the 10 Hz trap. Attempts at this were unsuccessful, however, and never proved
very promising. We speculate that the extremely weak trap, with its sub-Hz collision rates, madeevaporating on few minute duty cycles diﬃcult. Considering that the original 85 Rb experiment
was producing condensates of 10,000 atoms during its prime, we abandoned this approach without
much regret, needing at least the 40,000 atoms we were getting in 2008 (and the signal-to-noise
associated therein) for the experiments we proposed.