Dynamics of Bose-Einstein condensates near a
Feshbach resonance in 85 Rb
byNeil Ryan Claussen
B.S., Wake Forest University, 1996

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
2003

This thesis entitled:
Dynamics of Bose-Einstein condensates near a Feshbach resonance in 85 Rb
written by Neil Ryan Claussen
has been approved for the Department of Physics

Carl Wieman

Eric Cornell

Date

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

iii
Claussen, Neil Ryan (Ph.D., Physics)
Dynamics of Bose-Einstein condensates near a Feshbach resonance in 85 Rb
Thesis directed by Distinguished Professor Carl Wieman

I describe experiments with Bose-Einstein condensates (BEC) of 85 Rb near a
Feshbach resonance. The Feshbach resonance affects the collisions between atoms, allowing us to change the self-interaction energy of the BEC by applying a magnetic field. We
used the Feshbach resonance to study the startling details of the BEC collapse after weabruptly switched the self-interaction from repulsive to attractive. This change caused asupernova-like implosion/explosion cycle, leaving behind a small remnant BEC. We also
applied short magnetic field pulses toward the resonance to investigate the possibility ofcollisional coupling between atoms and diatomic molecules. We later created an atomic
and molecular BEC superposition state and probed this state’s coherence by applying
two rapid magnetic field pulses. Careful measurements of the atom-molecule oscillation
frequency allowed us to dramatically improve our determination of the Feshbach resonance parameters. All of these experiments have stimulated new theoretical work andhave improved our understanding of the physics of resonant BEC.

Dedication

This work is dedicated to my wife, Lisa, and to the rest of my family: Susie, Pete,
Eric, Grant, and Craig.

v

Acknowledgements

My time working at JILA has been greatly enriched by a number of people. First,
I want to acknowledge my advisor, Carl Wieman. Carl has taught me a great deal about
experimental physics and how to attack the many complex, real-world problems that
occur in physics research. Carl’s experimental skills and intuition are really amazing.
Also, Carl nearly always has an optimistic viewpoint and his enthusiasm for exploring
new areas is infectious. Another professor who has provided much advice and feedback is Eric Cornell. Eric was very involved in our experiments with 85 Rb from the
beginning. He took an active interest in our results, even when they seemed bizarre
and hard to understand, as in our initial studies of BEC collapse. Eric often startles
you with great ideas and insights into the BEC physics. I am impressed that Carl andEric have continued to pursue their research goals so diligently despite the many othercommitments they now possess after receiving the 2002 Nobel prize.
There are a number of other professors I would like to thank. On the experimentalside of things, Debbie Jin has been a wonderful source for ideas and suggestions. Debbie
has a cheerful attitude and she has a knack for asking questions that expose the holes
in your thinking. Chris Greene has greatly improved my knowledge of physics. I reallyenjoyed his Math Physics course, which gave me my first and only taste of the fascinating
aspects of group theory. I also appreciate Chris’ patience with my seemingly endlessquestions about the scattering theory and Feshbach resonances. Murray Holland has
shown similar patience with me as we discussed the intricacies of the pairing theory

vi
he developed (with Servaas Kokkelmans) to explain our experimental data. Outside ofwork, I had a lot of fun as a soccer teammate of Murray’s in the Boulder city recreational
league, where Murray proved a very enthusiastic and agile striker. Another person I
want to thank is John Bohn. In the times I saw him giving talks and in class, I reallyenjoyed John’s sense of humor. I also had fun working with John on the written QM
comps exam, even though no one decided to take our test!
The graduate students at JILA are a fantastic bunch of people. I especiallyenjoyed working “in the trenches” with Jake Roberts and Sarah Thompson. Jake taught
me a lot about good experimental techniques — especially how to think about error
bars. He was an excellent teacher in the lab and from what I have heard, also as a TA
in the classroom. In addition, Jake has been a good friend of mine and we shared many
trials and tribulations during the course of the first experiments with 85 Rb. I hope that
he will enjoy CSU and Rams football. Sarah Thompson joined our experiment shortly
after Jake graduated. I’ve really enjoyed working with her for the past few years. She
has a very strong aptitude for experimental physics and communicating with others. I
am confident that she will continue to do great research in the future (hopefully the labcomputers will stop crashing!) I have also learned a lot about exotic pets from Sarah.
Although I have not worked directly with them, there are many other grad students that I should acknowledge. Mike Matthews and Jason Ensher were fun to talkwith and they were always full of laughs. Kristan Corwin was a good friend and a very
enjoyable person to work with. Kurt Miller has always been ready to lend an ear to
my comments and complaints. I appreciate his friendship and advice. I had a greattime visiting Corsica for a BEC summer school with Heather Lewandowski and PaulHaljan. I think the tastes of the infamous “charcuterie du Corse” and fish soup will
always stick with me, as will the sea-kayaking expedition. Brian DeMarco and Dirk
Mueller were classmates of mine at JILA. I was always inspired by how well Brianspoke about physics and by his enthusiasm and skill for research. Dirk was a great

vii
guy with a strong determination to succeed. He always seemed cheerful, no matter
what the circumstances (except maybe at the end of our hike up Long’s Peak). Brian,
Dirk, and I had a tradition that I miss now — we would occasionally go out for a beer(preferably Guinness stout) and talk about practically everything under the sun. I havereally enjoyed getting to know Scott Papp and Josh Zirbel after they took up residence
across the hall on the new 85 Rb experiment. I am excited to see what interesting results
will come out of their 2nd generation BEC experiment. I also enjoyed all of the rockclimbing expeditions with Josh and the planning for the ascent of Devil’s Tower (Ohyeah!) Other grad students that I was privileged to know while at JILA are Chris Myatt, Rich Ghrist, Kurt Vogel, Ian Coddington, David Harber, Peter Schwindt, Ying-ju
Wang, Jonathan Goldwin, Josh Dunn, and Cindy Regal.
The post-docs that I worked closely with at JILA were Simon Cornish and Elizabeth Donley. Simon was a very careful experimentalist who devised several ingenious
tools to make our lives easier in the lab. He was fun to work with and he advanced theprogress of the experiment immensely. I also had fun skiing with Simon and his wife.
Elizabeth was a real inspiration for me. She provided a lot of energy to explore newresearch directions after Jake graduated and Simon returned to Britain. I was privileged
to work with Liz and also to rock climb with her on several occasions. I acknowledge another post-doc, Servaas Kokkelmans, who help me greatly during the process of studying
the weakly bound state of the Feshbach resonance. Servaas performed very intricate
calculations to allow us to use our measurements of the molecular binding energy toconstrain the Feshbach resonance parameters. He worked with Murray Holland to develop a theoretical model for our atom-molecule oscillation data. Servaas also cheerfully
answered a huge number of my questions about the Feshbach resonance and scattering
theory. I am very grateful for his friendship and help. Another post-doc that I would
like to thank is Brian Anderson. I met Brian at DAMOP in 1998 and was thrilled to
discover that he could image around 100 atoms in a BEC (we wanted to do similar

viiithings with 85 Rb). After Brian came to JILA, he helped us out with some clever image
processing tools and lots of helpful questions about our data. Brian was also a masterhiker who helped me broaden my horizons for exploring the mountains around Boulder.
Recently, I met Eleanor Hodby after she joined our experiment. Although I have not
had the opportunity to work with her very much, I am very glad for the chance to know
her. I can see that Eleanor is quite talented and will make many exciting discoveries
with 85 Rb. I should also acknowledge several other post-docs, including Eric Burt, whotaught me to optimize the experiment “from soup to nuts”; Stefan Dürr, for his great
physics knowledge and help with interpreting our results; David Hall, Simon Kuppens,
Dwight Whittaker, and Peter Engels.
There were a large group of undergraduate students that I worked with during
my time at JILA. Whether they were summer REU students or regular CU students,
the undergrads performed a great deal of hard work for us. They built useful electronic
circuits, analyzed data, and worked with diode lasers. I would like to acknowledge thepeople who worked in our lab: Carter Hand, Ryan Epstein, Jeff Collins, Nick Cizek,
John Obrecht, Margaret Harris, and Alex Kane. I would also like to thank Estelle
Andrews for her friendship and help with various projects.
It is hard to say enough about the JILA staff members. The fact that so many
talented and bright people have chosen to work at JILA is a very large reason for the
successful research that goes on here. I am thankful for the electronics shop staff,
particularly Terry Brown, James Fung-a-Fat, Mike Whitmore, and David Tegart. The
computing staff — in particular, Joel Frahm, Peter Ruprecht, and Jim McKown — havebeen indispensable. In addition, I want to thank several members of the instrument
shop for their help: Blaine Horner, Hans Green, Seth Wieman, Hans Rohner, and Todd
Asnicar.
I acknowledge my excellent undergraduate advisor, Keith Bonin, for encouraging
me to pursue experimental physics. Keith was a wonderful person to work for. He always

ix
had time to chat in his office about physics and many other interesting things. I reallyenjoyed working in the lab at Wake Forest — learning about diode lasers, atomic clusters, and basic machining. Keith found my passion for turkey hunting very entertaining
and he actually made an effort to understand the basic principles. I really appreciate
the advice Keith gave me about where to look for good experimental programs in AMO
physics.
I also thank my good friends, the Swartz family. I have really enjoyed getting toknow them during the past couple of years. Their children are truly wonderful and fun
to hang out with. My parents and brothers have been very loving and supportive of myefforts in graduate school. I have been blessed to have such a close relationship withthem, despite the fact that we live far apart. Finally, I want to thank my wife, Lisa
Courtney. Lisa has supported me throughout my time here. She has been remarkably
understanding of the long hours and occasional frustration that I felt. She encouraged
me to do my best.

xContents

Chapter
1 Introduction
1.1

1

Breakdown of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

The 155 G 85 Rb Feshbach resonance (Chapter 2) . . . . . . . . .

3

1.1.2

The 85 Rb experimental apparatus: new tools and upgrades (Chapter 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.3

Dynamics of collapsing and exploding 85 Rb Bose-Einstein condensates (Chapter 4) . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.4

1.1.7

4

Atom–molecule coherence in a Bose-Einstein condensate (Chapter 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.6

4

Microscopic dynamics in a strongly-interacting Bose-Einstein condensate (Chapter 5) . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.5

3

5

Very high precision bound state spectroscopy near a 85 Rb Feshbachresonance (Chapter 7) . . . . . . . . . . . . . . . . . . . . . . . .

5Prior publication of our work . . . . . . . . . . . . . . . . . . . .

6

2 The 155 G 85 Rb Feshbach resonance7

2.1

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

7

2.2

Atomic interactions in a BEC . . . . . . . . . . . . . . . . . . . . . . . .

7

2.3

Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

xi
2.3.1

Simple picture of the resonance . . . . . . . . . . . . . . . . . . .

8

2.3.2

Real 85 Rb Feshbach resonance . . . . . . . . . . . . . . . . . . .

11

2.3.3

Variable scattering length . . . . . . . . . . . . . . . . . . . . . .

12

2.3.4

Bound state properties for the 85 Rb Feshbach resonance . . . . .

13

2.3.5

Physical relevance of weakly bound state17. . . . . . . . . . . . .

3 The 85 Rb experimental apparatus: new tools and upgrades

18

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.2

New collection MOT laser . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.2.1

General discussion of collection MOT . . . . . . . . . . . . . . .

21

3.2.2

High power MOPA system . . . . . . . . . . . . . . . . . . . . .

22

3.2.3

Transfer of atoms between MOTs . . . . . . . . . . . . . . . . . .

23

Improved magnetic field measurement technique . . . . . . . . . . . . .

27

3.3.1

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

27

3.3.2

Rf apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.3.3

Lineshape measurements . . . . . . . . . . . . . . . . . . . . . . .

29

3.3.4

Obtaining the B-field from lineshape data . . . . . . . . . . . . .

36

3.3.5

Outlook for rf lineshape measurements . . . . . . . . . . . . . . .

40

New auxiliary coil for rapid B-field pulses . . . . . . . . . . . . . . . . .

41

3.4.1

Motivation for auxiliary coil . . . . . . . . . . . . . . . . . . . . .

41

3.4.2

Auxiliary coil design . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.4.3

Capacitor bank for auxiliary coil current . . . . . . . . . . . . . .

42

3.4.4

Servo electronics and transistor . . . . . . . . . . . . . . . . . . .

44

3.4.5

Auxiliary coil interferes with magnetic trap servo . . . . . . . . .

46

3.4.6

Calibrating the total magnetic field . . . . . . . . . . . . . . . . .

47

Improved BEC diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.5.1

53

3.3

3.4

3.5

Absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . .

xii
3.5.2

Motivation for expanding the BEC . . . . . . . . . . . . . . . . .

54

3.5.3

Increasing repulsion to expand the wavefunction . . . . . . . . .

55

3.5.4

Low temperature calorimetry: shrinking the BEC . . . . . . . . .

59

3.5.5

Making bigger and better condensates . . . . . . . . . . . . . . .

62

4 Dynamics of collapsing and exploding 85 Rb Bose-Einstein condensates

63

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

63

4.1.1

Stability condition for BEC with a<0 . . . . . . . . . . . . . . .

64

4.1.2

Overview of collapse dynamics . . . . . . . . . . . . . . . . . . .

66

4.2

Preparing and manipulating the condensate . . . . . . . . . . . . . . . .

67

4.3

BEC number loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

4.3.1

BEC number versus tevolve. . . . . . . . . . . . . . . . . . . . .

69

4.3.2

Variations in tcollapse . . . . . . . . . . . . . . . . . . . . . . . . .

72

Energetic burst of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.4.1

Nburst versus tevolve . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.4.2

hEburst i versus acollapse . . . . . . . . . . . . . . . . . . . . . . . .

78

Remnant BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.5.1

Nrem exceeds Ncrit . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.5.2

Missing atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

4.5.3

Strange density patterns: jets . . . . . . . . . . . . . . . . . . . .

87

4.6

Collapse at B=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

4.7

Discussion of theoretical models . . . . . . . . . . . . . . . . . . . . . . .

92

4.7.1

Overview of observations . . . . . . . . . . . . . . . . . . . . . .

92

4.7.2

Mean-field models for collapse . . . . . . . . . . . . . . . . . . . .

93

4.7.3

Beyond mean-field theory . . . . . . . . . . . . . . . . . . . . . .

95

4.1

4.4

4.5

5 Microscopic dynamics in a strongly-interacting Bose-Einstein condensate
5.1

Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . .

96
96

xiii
5.2

5.3

5.4

5.5

Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.2.1

Producing short B-field pulses . . . . . . . . . . . . . . . . . . . .

98

5.2.2

Counting the BEC atoms . . . . . . . . . . . . . . . . . . . . . . 100

5.2.3

Determining burst properties . . . . . . . . . . . . . . . . . . . . 101

BEC number loss from short B-field pulses . . . . . . . . . . . . . . . . . 103
5.3.1

BEC number loss versus pulse amplitude . . . . . . . . . . . . . 104

5.3.2

BEC loss versus pulse shape . . . . . . . . . . . . . . . . . . . . . 106

5.3.3

Unconventional loss: Rabi-like oscillations . . . . . . . . . . . . . 108

Burst atoms created by short pulses . . . . . . . . . . . . . . . . . . . . 110
5.4.1

Overview of burst characteristics . . . . . . . . . . . . . . . . . . 110

5.4.2

Burst dependence on pulse shape . . . . . . . . . . . . . . . . . . 113

5.4.3

Burst dependence on pulse amplitude . . . . . . . . . . . . . . . 116

5.4.4

Search for secondary collision mechanism . . . . . . . . . . . . . 118

Modeling BEC loss and burst production . . . . . . . . . . . . . . . . . 119
5.5.1

Toy Landau-Zener model for BEC loss . . . . . . . . . . . . . . . 121

5.5.2

Modifying the LZ model for triangular B-field pulses . . . . . . . 124

5.5.3

A better avoided crossing model . . . . . . . . . . . . . . . . . . 128

5.5.4

Modeling the resonance physics with quantum field theory . . . . 131

6 Atom–molecule coherence in a Bose-Einstein condensate

134

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2

Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3

Observation of population oscillations . . . . . . . . . . . . . . . . . . . 139
6.3.1

BEC and burst number oscillations versus tevolve . . . . . . . . . 139

6.3.2

Nmissing oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.3.3

Oscillation frequency versus Bevolve . . . . . . . . . . . . . . . . . 143

6.3.4

Interpreting the oscillations . . . . . . . . . . . . . . . . . . . . . 145

xiv
6.3.5
6.4

6.5

Mechanisms for missing atoms . . . . . . . . . . . . . . . . . . . 147

Controlling the oscillation characteristics . . . . . . . . . . . . . . . . . . 149
6.4.1

Amplitude of BEC number oscillations . . . . . . . . . . . . . . . 149

6.4.2

Strong density dependence of Nmissing . . . . . . . . . . . . . . . 150

6.4.3

Damping of Nrem oscillations . . . . . . . . . . . . . . . . . . . . 154

6.4.4

Relative phase between Nrem and Nburst oscillations . . . . . . . 156

6.4.5

Burst energy oscillations . . . . . . . . . . . . . . . . . . . . . . . 161

Discussion of theoretical models . . . . . . . . . . . . . . . . . . . . . . . 163
6.5.1

Kokkelmans and Holland model . . . . . . . . . . . . . . . . . . . 164

6.5.2

Kohler, Gasenzer, and Burnett model . . . . . . . . . . . . . . . 166

6.5.3

Overview of experiment and theory . . . . . . . . . . . . . . . . . 169

7 Very high precision bound state spectroscopy near a 85 Rb Feshbach resonance 170
7.1

7.2

7.3

Atom-molecule coherence as a novel Feshbach resonance probe . . . . . 170
7.1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.1.2

Sensitivity of binding energy to Feshbach resonance parameters . 171

7.1.3

Advantages of the present method over previous ones . . . . . . 174

7.1.4

Improved determination of Bzero . . . . . . . . . . . . . . . . . . 174

Measuring the B-field dependence of oscillation frequency . . . . . . . . 175
7.2.1

Production of BEC and measurement of total number . . . . . . 175

7.2.2

Atom-molecule oscillations

7.2.3

Limits to precision of frequency measurement . . . . . . . . . . . 177

7.2.4

Magnetic field determination . . . . . . . . . . . . . . . . . . . . 178. . . . . . . . . . . . . . . . . . . . . 175

Coupled-channels analysis of data . . . . . . . . . . . . . . . . . . . . . . 180
7.3.1

Coupled-channels theory . . . . . . . . . . . . . . . . . . . . . . . 180

7.3.2

Fitting the CC model to the data . . . . . . . . . . . . . . . . . . 181

7.3.3

Extracting Feshbach resonance parameters from best fit . . . . . 182

xv
7.3.4
7.4

7.5

Revised stability prediction for condensates with a <0 . . . . . . 185

Search for collective effects: mean field shifts . . . . . . . . . . . . . . . 186
7.4.1

Inadequacy of 2-body scattering theory . . . . . . . . . . . . . . 186

7.4.2

Statistical method for determining mean-field shifts . . . . . . . 187

Low frequency oscillations: a simple model. . . . . . . . . . . . . . . . 188

7.5.1

Damping of oscillations: experiment and theory . . . . . . . . . . 188

7.5.2

Comparison of dephasing model to data . . . . . . . . . . . . . . 191

7.5.3

Additional damping from atom loss

7.5.4

Magnitude of mean-field shifts . . . . . . . . . . . . . . . . . . . 194

7.5.5

Limitations of current method and possible future experiments . 195

Bibliography. . . . . . . . . . . . . . . . 191

197

xvi

TablesTable
2.1

Closed channels involved in the 85 Rb Feshbach resonance . . . . . . . .

6.1

Comparison of experimental observables and trends with two theoretical

11

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.1

Measured values of magnetic field and oscillation frequency . . . . . . . 182

7.2

Sensitivity of the determined interaction parameters vDS , vDT and C6 tofractional uncertainties in C8 , C10 , φE
T and J

7.3

. . . . . . . . . . . . . . . 185

Total χ2 and Bpeak as a function of the number of included data points
in the coupled-channels fit . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7.4

Measured values of magnetic field and corresponding mean-field shifts tothe molecular binding energy . . . . . . . . . . . . . . . . . . . . . . . . 188

xvii

FiguresFigure2.1

A simplified representation of the interatomic potentials involved in a
Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

Magnetic field dependence of scattering length near the Feshbach resonance 14

2.3

Schematic representation of the Feshbach resonance in 85 Rb . . . . . . .

16

3.1

Schematic of MOPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.2

Schematic of MOPA beam-shaping optics . . . . . . . . . . . . . . . . .

25

3.3

Schematic of apparatus used to control the rf radiation . . . . . . . . . .

30

3.4

Comparison of lineshape data to theory for a sample of BEC atoms . . .

32

3.5

Rabi-flopping in a BEC . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.6

Lineshape data and theory for a thermal cloud . . . . . . . . . . . . . .

35

3.7

B-field shifts due to finite rf pulse power . . . . . . . . . . . . . . . . . .

38

3.8

Comparison of two methods to measure rf power shift . . . . . . . . . .

39

3.9

Schematic of the main magnetic coils . . . . . . . . . . . . . . . . . . . .

43

3.10 Circuit schematic for the auxiliary coil current control . . . . . . . . . .

45

3.11 Schematic diagram of the sample-and-hold circuit . . . . . . . . . . . . .

48

3.12 Compensating for induced currents . . . . . . . . . . . . . . . . . . . . .

51

3.13 Dependence of induced B-field decay rate on auxiliary coil pulse height .

52

3.14 Increasing the repulsion to expand the BEC . . . . . . . . . . . . . . . .

56

xviii3.15 Absorption image of an expanded condensate . . . . . . . . . . . . . . .

58

3.16 Shrinking the BEC to extract the thermal fraction . . . . . . . . . . . .

61

4.1

An example of experimental control of the scattering length . . . . . . .

68

4.2

The condensate number versus time after a jump to a negative scattering
length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.3

The collapse time versus acollapse for 6000 atom condensates . . . . . . .

73

4.4

A burst focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.5

Growth of burst number with tevolve . . . . . . . . . . . . . . . . . . . .

79

4.6

Average burst atom energy versus scattering length . . . . . . . . . . . .

80

4.7

Remnant BEC number versus initial BEC number . . . . . . . . . . . .

83

4.8

Dependence of remnant number on initial BEC number . . . . . . . . .

85

4.9

Jet images for a series of tevolve values . . . . . . . . . . . . . . . . . . .

89

4.10 Quantitative jet measurements . . . . . . . . . . . . . . . . . . . . . . .

90

5.1

Magnetic field vs. time for a typical auxiliary coil pulse

99

5.2

Determination of burst properties from an absorption image . . . . . . . 102

5.3

Ramp time dependence of BEC number for different triangular pulse. . . . . . . . .

amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.4

Dependence of remnant BEC number on pulse ramp time for various holdtimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5

Fraction of BEC remaining versus pulse hold time . . . . . . . . . . . . 109

5.6

Rabi-like oscillation in the number of BEC atoms . . . . . . . . . . . . . 111

5.7

Dependence of burst characteristics on pulse length . . . . . . . . . . . . 115

5.8

Dependence of burst properties on B-field pulse amplitude . . . . . . . . 117

5.9

Schematic of the B-field pulse to test the secondary collision mechanism

120

5.10 Toy model for 85 Rb Feshbach resonance . . . . . . . . . . . . . . . . . . 123

xix
5.11 Comparison of the Landau-Zener model predictions for single- and doublepassages through an avoided crossing . . . . . . . . . . . . . . . . . . . . 125
5.12 Toy model predictions for finite magnetic field pulses . . . . . . . . . . . 127
5.13 Energy eigenvalues versus B-field for an interacting atom pair near the
Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.1

Magnetic field pulse shape . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2

An absorption image taken after the fast magnetic-field pulse sequenceand the mean-field expansion . . . . . . . . . . . . . . . . . . . . . . . . 138

6.3

Coherent oscillations in BEC number and burst number . . . . . . . . . 141

6.4

Population oscillations in 3 different components . . . . . . . . . . . . . 142

6.5

Oscillation frequency versus magnetic field . . . . . . . . . . . . . . . . . 144

6.6

Number versus tevolve for high density condensates . . . . . . . . . . . . 152

6.7

Fraction of missing atoms as a function of the initial BEC density . . . 153

6.8

Damping of BEC number oscillations . . . . . . . . . . . . . . . . . . . . 155

6.9

Enhanced Nmissing oscillations . . . . . . . . . . . . . . . . . . . . . . . 158

6.10 Relative oscillation phase versus final ramp time of 2nd B-field pulse . . 159
6.11 Average axial burst energy versus tevolve . . . . . . . . . . . . . . . . . . 162
7.1

Sensitivity of binding energy to Feshbach resonance parameters . . . . . 173

7.2

BEC number versus pulse spacing tevolve . . . . . . . . . . . . . . . . . . 176

7.3

Phase noise in BEC number oscillations . . . . . . . . . . . . . . . . . . 179

7.4

Molecular binding energy versus magnetic field . . . . . . . . . . . . . . 184

7.5

Low frequency BEC number oscillations . . . . . . . . . . . . . . . . . . 189

7.6

Theoretical model for density-dependent dephasing . . . . . . . . . . . . 192

7.7

Comparison of data and theory for damping rates versus B-field . . . . . 193

Chapter 1
Introduction

1.1

Breakdown of the thesis
This thesis describes experiments with a novel state of matter that only exists

at very low temperature, known as a Bose-Einstein condensate or BEC. The different
atoms in a BEC are so cold and slow-moving that they cannot be distinguished fromone another. Using Heisenberg’s uncertainty principle, one finds that in a BEC, theuncertainty in the position of each atom is comparable to the average spacing between
the atoms. Thus, it is almost meaningless to talk about individual atoms anymore —the BEC is a kind of “super-atom”. The task of creating a condensate is very difficultand depends on a number of fairly exotic experimental techniques that were developed
quite recently (in the past 10 to 15 years). To make a BEC, one usually starts with aroom temperature sample of alkali atoms, such as lithium, sodium, rubidium, or cesium.
These atoms are confined in a vacuum chamber and then cooled to a few 100 µK using
laser-cooling and trapping techniques. Although the laser cooling can dramaticallyreduce the temperature of the atoms, this type of cooling is not sufficient for creating
a condensate. To achieve the conditions for BEC, the atoms must be further cooled(to 50-100 nK) using evaporative cooling — a method for selectively removing only thehottest atoms from the sample.
After forming a BEC, one may study a wealth of different quantum-mechanicalphenomena. One of the most interesting properties of an alkali atom BEC is the presence

2of inter-atomic interactions in the condensate, which are mediated by elastic collisions
between the atoms. Due to a fortuitous internal atomic structure, the collisions of 85 Rb
atoms can be strongly affected by applying an external magnetic field. This magnetic
field dependence is known as a Feshbach resonance. The existence of the Feshbachresonance allows one to change the BEC interaction in a highly controlled fashion. We
took full advantage of the tunable interaction to study a number of interesting behaviorsof 85 Rb BEC. These experiments are discussed at length in the present thesis.
Before conducting the experiments described in this thesis, I worked with Jake
Roberts on the first experiments with 85 Rb Bose-Einstein condensates [1]. Initially, wedeveloped techniques to cool and magnetically trap the rubidium atoms. We used cold
samples of atoms to locate the Feshbach resonance by studying the effects of a variablemagnetic field on the elastic scattering of the trapped atoms. The discovery of the 85 Rb
Feshbach resonance allowed us to improve the efficiency of evaporative cooling so that wewere eventually able to create condensates. This achievement marked the first time thatthe 85 Rb isotope was cooled to quantum degeneracy. After forming condensates in 1999,
we utilized the Feshbach resonance to control the atomic interactions in the BEC. This
control allowed us to study a number of fascinating effects, including an adjustable sizeand shape for the quantum-mechanical wavefunction of the BEC. By changing the signof the interaction from repulsive to attractive, we induced an implosion or “collapse” ofthe condensate.
The collapse experiments mark the point in time when Jake’s thesis ends and mythesis begins. The present thesis begins with a detailed description of the dynamics ofcollapse in a BEC with attractive interactions. We also describe a series of new experiments involving time-varying magnetic fields applied to the BEC near the Feshbachresonance. By improving our magnetic field control apparatus, we were able to probe
the response of the condensate to B-field changes on time scales as short as 10 microseconds. Surprisingly, the rapid magnetic field changes caused significant number loss from3
the BEC, which implied the presence of microscopic dynamics occurring between the
BEC atoms. We later showed that the mysterious BEC loss resulted from non-adiabatic
transitions to another state — a diatomic molecular state that was nearly degeneratewith the initial BEC state due to the Feshbach resonance. We observed evidence foratom-molecule coherence in the BEC and we used this coherence to precisely determine
many detailed properties of the Feshbach resonance and interatomic interactions.
In the rest of the Chapter, we outline each of the subsequent parts of the thesis.
1.1.1

The 155 G 85 Rb Feshbach resonance (Chapter 2)

This Chapter provides a brief description of the Feshbach resonance that was soimportant for our experiments. We motivate the importance of the Feshbach resonanceby showing how it affects the interactions in a BEC. Some discussion of the two-bodyscattering physics is given, and we show how the elastic scattering length varies as afunction of magnetic field near the resonance. We also explain the most important
properties of the weakly bound molecular state that causes the Feshbach resonance.
The magnetic field dependence of the both the energy and spatial size of the molecularstate can be adjusted via the applied B-field.
1.1.2

The 85 Rb experimental apparatus: new tools and upgrades (Chapter 3)

Here we explain new techniques and additions to the experiment that allowed us
to collect the data described in Chapters 4-7. The first part of the Chapter describes
a high-power laser upgrade to the experiment. We used the new laser to improve theprocess of collecting rubidium atoms from a room temperature vapor. We also devised
a greatly improved method to measure the magnetic field on short time scales using
short pulses of rf radiation. The new B-field measurement technique was essential forcharacterizing the magnetic field produced by another addition to the experiment — an

4auxiliary magnetic field coil. We built the auxiliary coil to apply rapid B-field changes tothe BEC. By varying the current through the auxiliary coil, we could precisely control
the magnetic field on short time scales. The final part of the Chapter discusses our
improved methods for measuring the most important properties of the condensed atoms— their number and temperature. We used the variable interaction strength afforded by
the Feshbach resonance to change the BEC size and shape, which enhanced our ability
to determine the BEC characteristics.
1.1.3

Dynamics of collapsing and exploding 85 Rb Bose-Einstein condensates (Chapter 4)

In this Chapter, we describe the startling response of a BEC to a change in thecondensate self-interaction from repulsive to attractive. The sudden change caused avariety of unusual effects in the condensate. After a short delay that followed the change
in the interaction, the BEC abruptly began to lose atoms. The atoms disappeared fromthe condensate over a time scale of a few milliseconds. During the same time period, an
explosion of hot atoms was emitted from the collapsing condensate. We also observed
strange features in the BEC density distribution. After the completion of the number
loss, part of the initial condensate remained visible in a highly excited state of motion.
We discuss the detailed behavior of the BEC collapse dynamics and summarize theseresults. At the end of the Chapter, the prominent features of the data are compared totheory.
1.1.4

Microscopic dynamics in a strongly-interacting Bose-Einstein condensate (Chapter 5)

Here we discuss condensate number dynamics resulting from rapid changes tothe magnetic field. Using the auxiliary coil, we applied a short B-field pulse toward
the Feshbach resonance to the condensate. The condensate responded to the pulse

5
by losing a large fraction of its initial number. Some of the atoms that left the BEC
were visible in an explosion that was quite similar to the explosion seen in the collapse
experiments of Chapter 4. We studied the dependences of both the BEC loss and the
explosion characteristics on the exact shape and size of the magnetic field pulse. The
time dependence of the loss suggested the presence of non-adiabatic transitions of thecondensate atoms to the molecular state associated with the Feshbach resonance. We
give some justification for this claim using a simple avoided crossing model for the
Feshbach resonance.
1.1.5

Atom–molecule coherence in a Bose-Einstein condensate (Chapter 6)

This Chapter is an extension of the work in Chapter 5. Rather than using asingle pulse, here we applied two magnetic field pulses to the condensate. The pulses
approached the Feshbach resonance and were separated in time by a variable delay. As afunction of the delay time between pulses, we observed oscillations in the number of BEC
atoms. The oscillation frequency matched the binding energy for the molecular stateof the Feshbach resonance. We also observed oscillations in the population of atoms
in the explosion that resulted from the B-field pulses. These atom-molecule coherence
data inspired a flurry of theoretical activity. We present a comparison of the data andtheory at the end of the Chapter.
1.1.6

Very high precision bound state spectroscopy near a 85 Rb Feshbachresonance (Chapter 7)

The atom-molecule oscillations observed in Chapter 6 allowed us to dramatically
improve our knowledge of the Feshbach resonance. We made very precise measurements of the oscillation frequency corresponding to the molecular binding energy. The
frequency measurements were combined with precise measurements of the magnetic field6
using a new experimental technique from Chapter 3. We fit the frequency and magnetic
field data to a theoretical model of the scattering physics. The best fit to the data
allowed us to extract greatly improved values for the Feshbach resonance properties,
including its position and width in magnetic field. We improved the precision of theposition and width determinations by more than a factor of 20. In addition, we used
our data to obtain new values for several important parameters of the interatomic potentials for 85 Rb. Our results for the potential parameters had comparable precision to
those of other recent high-precision experiments in rubidium. The agreement between
our results and the results of the other analysis was very good.

1.1.7

Prior publication of our work

Most of the experiments described in this thesis were previously published inscientific journals. The experiments of Chapter 4 appeared in Ref. [2], while Chapter 5
was published in Ref. [3]. The work in Chapter 6 appeared in Ref. [4]. The observationsof Chapter 7 were submitted but not yet published at the time of writing the thesis.

Chapter 2
The 155 G 85 Rb Feshbach resonance

2.1

Introduction
This Chapter provides a brief overview of a Feshbach resonance in the collisions of

85 Rb atoms. In addition to allowing the creation of 85 Rb Bose-Einstein condensates, theFeshbach resonance also made possible a variety of experiments involving changes to theself-interaction of the BEC. The effects of this variable self-interaction were numerous
and they form the subject matter for the rest of this thesis.

2.2

Atomic interactions in a BEC
One of the most interesting properties of Bose-Einstein condensates in diluteatomic gases is the existence of interactions between the constituent atoms. Although
the BEC is typically 5 orders of magnitude less dense than air, the interatomic interactions strongly affect a number of the properties of the condensate. The interactions in a
BEC have been successfully described by mean-field theory [5], in which each BEC atom
moves in an effective mean-field due to the other atoms. The mean-field interaction ismediated by the elastic collisions between atoms.
The well-known Gross-Pitaevskii (GP) equation can be used to describe a condensate in the limit of zero temperature and neglecting all correlations between the8atoms. The GP equation for a trapped BEC has the form
· 2 2
¸
∂
~ ∇
2
i~ Φ(~r, t) = −+ Vext (~r) + g|Φ(~r, t)| Φ(~r, t),
∂t
2m

(2.1)

where Φ is the BEC order parameter, Vext is the trap potential energy, and the coeffi-~ a characterizes the pair-wise interatomic interactions in the BEC throughcient g = 4πm2

the s-wave elastic scattering length, a. The solution to the GP equation (2.1) is thecondensate order parameter, Φ, which for most purposes can be regarded as the macroscopic wavefunction of the BEC atoms. The GP equation has the form of a nonlinear
Schrodinger equation, where the nonlinear term arises from the mean-field interactionof one atom with all of the others. The mean-field interaction term, often called the selfinteraction energy of the condensate, depends on the density of atoms and the scattering
length. Thus, the GP equation predicts that the sign and magnitude of a determine thestrength of the self-interaction and whether this interaction is attractive or repulsive.
For controlling the BEC self-interaction, it would clearly be desirable to findsome method to change the scattering length. Almost one decade ago, it was suggested
that the scattering length could be influenced using an external magnetic field [6].
The magnetic field would allow one to shift the energy of a molecular bound state tonear-degeneracy with the energy of a colliding pair of atoms, thereby altering the elastic
scattering properties. Such an effect is called a Feshbach resonance and was first studiedin nuclear scattering. The physics of Feshbach resonances will be discussed in the restof this Chapter.

2.3

Feshbach resonance

2.3.1

Simple picture of the resonanceIn a simplified picture, a Feshbach resonance occurs when the energy of a boundstate of the interatomic potential is equal to the kinetic energy of a colliding pair ofatoms. Assuming a finite kinetic energy, such a degeneracy can occur only when the9bound state exists in a potential that has a higher threshold energy than that of thecolliding atom pair. This condition can be satisfied in ultracold gases of alkali atoms, due
to the low collision energy of the atoms and the existence of atomic hyperfine structure.
Since the different hyperfine states generally possess different spin configurations andmagnetic moments, one can sometimes tune the bound state energy into resonance withthe colliding atom energy via the different Zeeman shifts in an external magnetic field.
Assuming that both colliding atoms are in the lower hyperfine state, it may happen thatan interatomic potential associated with the upper hyperfine state supports a boundstate nearby in energy, as shown schematically in Figure 2.1.
The schematic curves in Figure 2.1 show the potential energy of the atoms asa function of their internuclear separation, R. Neglecting the Zeeman interaction, thelarge R asymptotic energy is determined entirely by the sum of the hyperfine energies forthe colliding atoms. Each asymptote is called a collision channel and different channels
are labeled by the hyperfine quantum numbers (F,mF ) for the atom pair.
If two atoms begin an elastic collision in the lower channel with kinetic energy
much smaller than the hyperfine splitting, ∆EHF , the atoms cannot exit the collisionin the upper channel because of energy conservation. Thus, the upper channel is energetically “closed”, while the lower channel is “open”. In the case of 85 Rb experiments,
the hyperfine splitting of 140 mK greatly exceeds the kinetic energy of the degenerate
gas sample, which is <10 nK. This means that the Feshbach resonance will be “wellresolved” because the energy spread of the BEC atoms is far smaller than the energy
difference between the two hyperfine states.
In addition to low collision energy and hyperfine structure, another requirement
for the existence of a Feshbach resonance is a coupling between the open and closed
channels. The coupling is provided by the Coulomb (or exchange) interaction, which
couples together different hyperfine states at short internuclear distance [7]. As theatoms move together during a collision, the strong electrostatic interaction between the10EnergyClosed channel∆EHF
Open channel

Internuclear Distance
Figure 2.1: A simplified representation of the interatomic potentials involved in a
Feshbach resonance. The solid lines represent the potential energy versus internuclear
separation for the colliding atoms. The dashed lines show the threshold or asymptotic
energies of the potential curves, which are separated by the hyperfine splitting, ∆EHF .
A bound state of the upper potential, shown by the short horizontal line, is nearly degenerate with the threshold energy of the lower potential. The definition of energetically
open and closed channels is given in the text.

11
nuclei and the electron clouds overwhelms the relatively weak hyperfine interaction,
allowing a spin flip to occur. Due to the symmetry of the binary system, the total
spin projection quantum number mF (tot) =mF (1) +mF (2) is conserved, but the total spin
Ftot =F1 +F2 is not. The symmetry requirement determines whether a given closed
channel can couple to the open channel.

2.3.2

Real 85 Rb Feshbach resonanceIt turns out that the idealized picture of the Feshbach resonance caused by theinteraction of a single closed channel with the open channel is too simplistic. Becauseof the complicated hyperfine structure of alkali atoms, any real resonance must involve
coupling of the open channel to a number of different closed channels. As discussed in
section 2.3.1, the symmetry of the Hamiltonian describing the colliding atoms dictatesthat only those channels with the same total spin projection as the open channel canbe coupled. For the 155 G 85 Rb Feshbach resonance, the open channel is |F1 , mF (1) i +|F2 , mF (2) i = |2, −2i + |2, −2i. The symmetry requirement that mF (tot) = −4 restricts
the number of closed channels to four. We list the relevant 85 Rb closed channels inorder of increasing threshold energy in Table 2.1.
Table 2.1: Closed channels involved in the 85 Rb Feshbach resonance. Each channel isa sum of two different hyperfine states, as indicated by the quantum numbers in thesecond column. The threshold energies and magnetic moments of the closed channels
relative to the open channel are calculated using the Breit-Rabi equation at a magnetic
field of 175 G. Note that ∆EHF = 3.036 GHz.
Label
(a)
(b)
(c)
(d)

Closed channel|2, −1i + |3, −3i|2, −2i + |3, −2i|3, −1i + |3, −3i|3, −2i + |3, −2i

Energy/∆EHF at B=175 G
0.840
0.900
1.799
1.800

Magnetic moment (MHz/G)
−2.72
−1.57
−3.22
−3.14

All of these closed channels can potentially interact with the open channel. There-12
fore the simplified picture of a Feshbach resonance given in Figure 2.1 is inadequate. In
fact, to describe the resonance when the magnetic field approaches the resonance value,
one must include the effects of multi-channel coupling [7, 8, 9]. In a coupled-channels
approach, the wavefunction of the colliding atom pair is expanded as a sum of the open
channel state and the four closed channel states. The expansion coefficients are dependent on collision energy, magnetic field, and the internuclear separation of the colliding
atoms. When one substitutes the expanded wavefunction into the Schrodinger equation
containing an approximate Hamiltonian for the interatomic interactions, one obtains a
set of coupled equations that describe the elastic collision. By solving these equations,
one can accurately determine the most important property of the elastic collision – thescattering length, a. The existence of the Feshbach resonance profoundly affects thescattering length by introducing a magnetic field dependence to a.

2.3.3

Variable scattering length

A well-known feature of low energy scattering from an attractive potential well isresonance scattering [10, 11]. Resonance scattering occurs when a bound state of thepotential is very close to the collision energy of the atoms. The presence of the boundstate near zero energy profoundly affects the scattering physics. This is because thecolliding atoms can make a transition to the bound state and dwell there briefly before
moving apart again after the collision. The biggest effect on the scattering occurs when
the two levels have exactly the same energy, which causes the elastic cross section andscattering length to reach infinite values. In a Feshbach resonance, one can adjust theenergy of a bound state relative to the collision energy by tuning the magnetic field.
Even though the bound state exists in a different interatomic potential from that of
the colliding atoms, the variable bound state energy can still dramatically alter the
scattering length.
From the coupled-channels scattering theory of Feshbach resonances, one can13derive an approximate analytic expression for the variation of the scattering lengthwith magnetic field [7]. In terms of the Feshbach resonance parameters, the scattering
length is [7, 12]µa = abg 1 −∆B − Bpeak

¶
,

(2.2)

where Bpeak is the resonance position and is defined to be the magnetic field where themagnitude of a becomes infinite, abg is the background scattering length, and ∆ is theresonance width in magnetic field. Although equation (2.2) is only approximate, wehave found that the analytic expression works very well. In fact, the scattering lengthfrom equation (2.2) agrees with a calculated from a full coupled-channels theory tobetter than 1% percent from 155 G to 250 G [13]. Figure 2.2 displays the scattering
length predictions from equation (2.2).

2.3.4

Bound state properties for the 85 Rb Feshbach resonanceVery near the 85 Rb Feshbach resonance on the high B-field side, the scattering
length is large and positive. From zero-energy scattering theory [10, 11], we knowthat when a is much bigger than the effective range of the attractive potential well(a >> Reff ∼25 a0 ), there must exist a weakly bound state just below the thresholdenergy. The binding energy of the weakly bound state bears a simple relationship to
the scattering length:²bind = −~2 /(ma2 ),(2.3)

where m is the atomic mass. The arguments leading to this result are very general,so the equation must be valid as long as a is big. It is therefore not necessary to deal
with a complicated coupled-channels calculation of ²bind in the large scattering length
regime, provided that the parameters of the Feshbach resonance are well known and
one can use equation (2.2) to determine the scattering length.
However, to obtain the full magnetic field dependence of the molecular binding

14

2000

2000

scattering length (a0)

1000
0

1000

-1000
-2000
0

50

100

150

200

250

0

-1000

-2000
135

145

155

165

175

B-field (G)Figure 2.2: Magnetic field dependence of scattering length near the Feshbach resonance.
The solid line is the scattering length calculated from equation (2.2) with the current
best values for the resonance parameters (see Chapter 7). For the resonance position
and width, we have Bpeak =155.041 G and ∆=10.709 G, respectively. The value ofthe background scattering length is -443 a0 . The horizontal dashed line indicates zeroscattering length, which occurs at Bzero =165.75 G. In the inset, we show the scattering
length variation over a larger range of B-field.

15
energy, a coupled-channels calculation is needed [9]. When the interactions between
all of the channels are accounted for, one obtains a rather complicated magnetic fielddependence in ²bind . The coupling matrix element between the open channel and eachclosed channel varies strongly with B-field, which leads to a nonlinear magnetic fielddependence of the binding energy. Figure 2.3 shows this dependence along with thescattering length variation for magnetic fields above the 85 Rb Feshbach resonance.
In addition to the binding energy, another important property of the molecularstate is the spatial size of the wavefunction. Zero-energy scattering theory provides auseful prediction for the shape of the molecular state wavefunction. Outside the effective
range of the potential, the wavefunction for the molecular state can be shown to havethe form [10]φ(r) ∼

exp (−r/a)
,
r

(2.4)

where a >> Reff is the large, positive scattering length and Reff is the range of thepotential. Equation (2.4) shows that the decay of the bound state occurs over a lengthscale equal to the scattering length, which can easily exceed the potential range by
several orders of magnitude. This behavior is possible because of the proximity of thebound state to zero energy. The elongated spatial size of the molecule has some bizarre
consequences. For instance, when the scattering length is very large, the atoms in the
weakly bound molecule spend the majority of their time at internuclear separations thatare best described by the hyperfine basis. The molecular state is far too large to belabeled according to the conventional singlet/triplet molecular quantum numbers.
It is worth emphasizing that the characteristics of the weakly bound molecularstate are completely different from those of the single closed channel state of conventional
Feshbach theory. For instance, the weakly bound state extends far past the short-range
part of the interatomic potential. The size of the weakly bound state is basically givenby the scattering length when a >> 0. In contrast, the closed channel bound state

16

a →∞

a
(a)

B
2-atom threshold(BEC atoms)

E
(b)

εbind
molecular state

Bpeak=155 G

166 G

B

Figure 2.3: Schematic representation of the Feshbach resonance in 85 Rb. (a) Variationof scattering length for B>Bpeak . The two vertical dotted lines indicate the range ofmagnetic field where a > 0. (b) Dependence of atomic and molecular energies on Bfield. The dashed line shows the colliding atom threshold energy, while the solid curve isthe molecular state energy, which intersects the atomic threshold at Bpeak . For magnetic
fields significantly above resonance, the molecular energy decreases linearly with B-field
at a rate that is very similar to the magnetic moments of closed channels (c) and (d),as listed in Table 2.1.

17
is confined to the part of the interatomic potential where molecular forces dominate,
which is roughly 25 a0 in extent. Another property of the weakly bound state that sets
it apart from the closed channel state is the magnetic field dependence. While the closed
channel state exhibits a linear variation of energy versus B-field, the weakly bound state
shifts in a highly nonlinear fashion (see Figure 2.3).

2.3.5

Physical relevance of weakly bound state

Prior to the publication of our recent measurements of the molecular state binding
energy (see [4] and also Chapters 6-7), the existing theoretical descriptions of alkali atom
Feshbach resonances [14, 15] were mostly based on the simple picture of a single closed
channel coupled to the open channel (see section 2.3.1). In the course of our studiesof 85 Rb condensates, we found that the simple picture was totally incorrect — it failed
to describe the atom-molecule oscillation data in Chapter 6. Our measurements clearly
demonstrated the physical relevance of the weakly bound molecular state. Several new
theoretical models for the 85 Rb Feshbach resonance are currently being developed [9,16, 17, 18] to describe the data.

Chapter 3
The 85 Rb experimental apparatus: new tools and upgrades

3.1

Overview
This chapter explains new features added to the original 85 Rb experimental ap-

paratus. The vast majority of the apparatus and techniques that we used to produce
85 Rb Bose-Einstein condensates were already carefully documented by Jake Roberts [1].

I refer the reader to Ref. [1] for a more comprehensive picture of the 85 Rb machine andthe actual BEC production process. In the first part of the chapter, I give a general
overview of our recipe for BEC. The last part of the chapter contains much more detailed
descriptions of several new experimental techniques and additions to the apparatus thatwere used to conduct the experiments described in Chapters 4-7. The detailed descriptions begin with an explanation of general physical principles and conclude with thepractical details.
Our procedure for BEC production depends on two cooling techniques — laser
cooling and evaporative cooling — to lower the temperature of a confined gas of atoms
from room temperature (300 K) to a few billionths of a degree (∼10 nK). We accomplish
this cooling without any conventional cryogenic methods; the only cold part of thesystem is the trapped atom cloud itself. This is possible because the rubidium atoms
are trapped in a “thermos bottle” of confining optical and magnetic fields that arehoused inside an ultra-low pressure vacuum system. When the 85 Rb atom cloud cools
to 10 nK, the quantum behavior of the atoms in the gas becomes apparent and they all19pile up in the ground state of the trap, forming a Bose-Einstein condensate. The BEC
is remarkable — it is described by a macroscopically-sized quantum wavefunction that
we can observe directly with a suitable microscope.
In our apparatus, we use a pair of magneto-optical traps (MOTs) to laser-cooland trap a few hundred million 85 Rb atoms. One MOT, called the “collection” MOT,
has the important task of cooling the atoms down to roughly 100 µK from a room
temperature vapor. We transfer these collected cold atoms to another (“science”) MOT
by pushing them through a narrow tube that connects the two MOT vacuum chambers.
The chamber that encloses the science MOT is only weakly coupled to the collection
MOT chamber containing the rubidium vapor, so the science chamber has roughly 100×
lower pressure (∼10−11 torr). This low pressure is crucial for the cooling efficiency ofthe next experimental stage — evaporative cooling.
Before starting the evaporative cooling process, we first illuminate the atoms with
an optical pumping laser to force them into a particular internal hyperfine state, thenwe load the atoms into a purely magnetic trap. The magnetic trap confines the atoms
spatially and has a much longer storage time than the MOT due to the absence of laser
light, which continually excites the atoms and leads to trap loss. However, the magnetic
trap has the disadvantage of being much more shallow than the MOT and lacking thecapability to cool the atoms (because the magnetic trapping potential is conservative,
there is no dissipation and this trap has no intrinsic cooling ability). In our system,
we typically load the magnetic trap with 6 × 108 85 Rb atoms in the |F, mF i = |2, −2i
hyperfine state at a temperature of 40 µK.
We evaporatively cool the sample by applying a radio-frequency (rf) radiation
field that selectively removes the hottest atoms from the magnetic trap. The removal
is accomplished by inducing ∆mF =+1 transitions to an untrapped spin state. Aftermaking such a transition, an atom falls away from the trapped cloud, thereby removing
energy from the sample. The most important feature of evaporative cooling in the20magnetic trap is the energy selectivity of the rf transitions. Due to the Zeeman energy
shift of the trapped atoms, the most energetic atoms spend the greatest time in higher
magnetic field regions of the trap, where the rf transition frequency is largest. Bylowering the rf radiation frequency in a time-dependent fashion, one can gently lower
the temperature of the trapped atom cloud. The optimal procedure for evaporative
cooling is quite complex and we devoted a great deal of time to studying this process.
If the cooling process can be made efficient enough so that the number of atoms
removed per unit time is relatively low and the energy removed per time is relatively
high, then the phase space density (PSD) of the atom cloud will increase as evaporation
continues. Eventually the PSD approaches unity, which means that the average interparticle separation in the gas becomes equal to the deBroglie wavelength of an atom.
Because 85 Rb atoms are bosons, their position space wavefunctions overlap with one
another and the system undergoes a transition to a new “superatom” state, the BoseEinstein condensate. In our system, we typically see a BEC begin to form at ∼15 nK.
We continue to evaporate the cloud to further lower the temperature until we obtaincondensates of 20,000 atoms at T∼5 nK. Thus, the evaporative cooling process leads to
PSD∼1 after reducing N by four orders of magnitude and reducing T by ten orders ofmagnitude.
The resulting 85 Rb BEC is a fascinating object. It is a dilute, weakly-interacting,ultracold gas of atoms whose behavior is entirely quantum-mechanical. The interactions
between atoms in the condensate are mediated by low energy collisions and can be fairly
simply described in terms of the s-wave scattering length, a. Since the scattering length
can be varied near a Feshbach resonance (see Chapter 2) by changing the magnitudeof the magnetic field, our 85 Rb BEC in a magnetic trap offers a wealth of possibilities
for studying the atom-atom interactions. In fact, the formation of the condensate at
B=162.3 G (a=210 a0 ) is only possible because of the proximity of the B-field to the
Feshbach resonance at 155 G. At magnetic fields more than 11 G away from resonance,

21
the scattering length is large and negative, so the BEC self-interaction is attractive andonly ∼80 atoms can exist in the BEC ground state (attractive interactions are discussed
in Chapter 4).
After creating a stable 85 Rb BEC, we generally change the magnetic field andstudy the effect on the condensate. Depending on the details of the B-field change— its time-dependence and magnitude, one can observe very complex BEC dynamics,
including implosions and explosions of the wavefunction, collision-induced losses, andcoherent transfer between a single atom state and a diatomic molecular state. Ourstudies of these phenomena comprise the subject matter for Chapters 4-7 of this thesis.

3.2

New collection MOT laser

3.2.1

General discussion of collection MOT

Perhaps the most important step on the road to BEC is laser cooling and trappingof the atoms in a magneto-optical trap (MOT). We utilize a double MOT system [19]consisting of a collection MOT for gathering the atoms from a room temperature vapor
and a science MOT for further cooling and transfer of the atoms into a magnetic trap.
The double MOT apparatus allows us to collect and transfer multiple bunches of atoms
from the collection chamber to the science chamber. After a capturing a sufficientnumber of atoms in the science MOT, we load them into the magnetic trap and proceed
to evaporative cooling.
The collection MOT has the important task of cooling a fraction of the atoms in
the room temperature vapor to roughly 100 µK. The fraction that is cooled is determined [20] by the velocity distribution in the gas and the capture velocity of the MOT,
vc , which is the maximum velocity that an atom can have and still be slowed enoughto become trapped in the MOT. When an atom with larger velocity than vc enters thetrapping region, it slows down but does not remain in the MOT volume. The number

22of atoms in the MOT depends strongly on vc : NMOT ∝ (vc /vtherm )4 , where vtherm is theaverage velocity of the atoms in the vapor. The capture velocity increases with increasing laser intensity in the MOT beams because vc depends on the photon scattering ratefrom the laser beam(s) that oppose the atom’s motion. It is therefore desirable to use
a relatively high power laser to maximize both loading rate and the number of atoms
trapped in the MOT.

3.2.2

High power MOPA system

In our early work with 85 Rb, we used a Hitachi diode laser to supply the trapping
light for the collection MOT. After stabilization with grating feedback, this laser produced roughly 20 mW of laser power, which was then divided into three retro-reflectedbeams. More recently we switched to a much higher power laser system, called a masteroscillator-power-amplifier (MOPA). The MOPA is capable of producing a power of several hundred mW and allows us to significantly increase the number of atoms in thecollection MOT.
The MOPA consists of a frequency-stabilized master laser that sends its light into
the power amplifier. The amplifier is a single-pass semiconductor device with a tapered
shape to maximize the gain while reducing the power density inside the medium. The
PA has high gain and very low reflectivity (<0.1%) on the input/output facets. The
power amplifier thus has a wide (∼10 nm) gain spectrum limited only by the gain
bandwidth of the semiconductor material. This makes the PA flexible enough for avariety of experiments with different alkali atoms. In addition, one can send multiple
frequencies of laser light into a single PA — either by combining two laser beams orby simply modulating the MO laser. Modulation is a particularly attractive approachthat allows one to simultaneously trap two different isotopes in overlapping MOTs (weoriginally designed our MOPA to trap 85 Rb and 87 Rb). Another possibility is to generatetrapping light and repump light from the same MOPA.

23
One of the nicest features of the MOPA is its adjustable gain. By adjusting thecurrent flowing through the PA one can vary the gain of the amplifier. The maximum
gain for our PA is ∼100 for a 1.8 A current and an input power from the MO laser of
5 mW. We typically operate at much lower PA gain because we do not need such highpower levels for the collection MOT and also to lengthen the lifetime of the PA.
Our MOPA system is based on Brian DeMarco’s design [21], which was widely
copied at JILA until the manufacturer (SDL) abruptly ceased to make the amplifiers.
However, a similar PA chip is now available from TUI optics that can be used as areplacement for the SDL chip. Figure 3.1 displays the basic optical setup for our MOPA
and Figure 3.2 shows the beam-shaping optics needed to correct the large astigmatismand ellipticity of the output from the PA. The PA output is ultimately sent througha fiber to spatially filter the beam and transport the light from one optical table toanother.

3.2.3

Transfer of atoms between MOTs

After collecting 1010 atoms in the collection MOT, we push a small fraction ofthem down the transfer tube to the science MOT. This transfer is accomplished byturning off the collection MOT beams with a mechanical shutter while leaving on a fixedpower push beam that is directed along the transfer tube axis. Because the transfer
velocity of 12 m/s is less than the capture velocity of the science MOT, all of theatoms that successfully travel down the tube without sticking to the stainless steel walls
are trapped again by the second MOT. The sticking probability is reduced by a setof permanent guide magnets arranged in a hexapole configuration around the transfer
tube.
Defining the efficiency of the transfer as the fraction of the initial collection MOT
number that ends up in the science MOT, we find that our efficiency is quite low (0.1
to 0.2 %). Thus, to fill the lower MOT to our typical value N=6 × 108 requires 25 to 50

24

feedback
MO
diode
laser

OutputRb cell( DAVLL )60 dB
isolation
PA
Wedge

MO
Close - up

PA
Close - up

grating

lensPA

output8 mW

500 mW

lens
diode

lens

5 mW

Figure 3.1: Schematic of MOPA. At top is the overall design of the MOPA, while thebottom part of the figure shows more of the details of the master oscillator and thepower amplifier. We stabilize the MO using a DAVLL lock [22] and a rubidium cell.
The output beam of the PA is shaped by the optical system shown in Figure 3.2.

25

18 cm

32 cm

PA
25 cm
(cyl)

-10 cm
(cyl)

30 cm
(sph)

6 cm
(sph) isolator λ /2

36 cm

to
MOT

fiber
coupler

λ /2

Figure 3.2: Schematic of MOPA beam-shaping optics. The first two lenses are cylindricaland only affect the vertical axis of the PA beam. After the second lens, the PA beam isroughly collimated with a square shape (∼1.5 cm on a side). The second pair of lenses
form a telescope to shrink the beam until it will pass through the isolator aperture.
Finally the beam travels through an optical fiber with 60% coupling efficiency before
traveling to the collection MOT chamber. The collection MOT beams are 1” in diameterand each of the three retroreflected beams contains ∼20 mW for a MOPA current of
0.875 A. An additional pushing beam (with 5 to 10 mW power) is split from the main
MOPA beam and is used to transfer atoms to the science MOT.

26
pushes. Since we actually do not allow the collection MOT to completely refill between
pushes, the real number of transfers can be somewhat higher. In fact, a better figureof merit for the atom transfer is the total number transferred per unit time. For ourtypical repetition rate of 0.3 s and loading time of 20 to 40 s, we obtain a total transfer
rate of 1 to 3 × 107 atoms/s.
I speculate that the primary bottleneck in the transfer process is going from thecollection MOT to the transfer tube itself. The push beam that accelerates the atoms
toward the science MOT causes a large amount of transverse heating of the atoms before
they enter the tube. Only a small fraction of the collection MOT atoms near the centerof the MOT may have a significant probability of successfully entering the tube.
This picture of the transfer process as being limited by the poor efficiency ofpushing the collection MOT atoms into the transfer tube is supported by experimental
data. We made measurements of the single load size as a function of power in thescience MOT, for a fixed number of atoms in the collection MOT. The power variation
was accomplished by picking off some of the MOPA laser light to supply the science
MOT. Our measurements showed that we could reduce the number of atoms captured
in the science MOT by turning down the beam power until the capture velocity dropped
below 12 m/s. However, as long as we kept the power above this critical value, we could
not further increase the number captured in the science MOT by increasing the power.
The single-load number of atoms saturated as the science MOT power increased, despite
the fact that the overall transfer efficiency from collection MOT to science MOT wasvery low (∼0.2 %). This implies that we can capture all of the atoms that enter thetransfer tube in the science MOT, but the number entering the tube must be far below
the number in the collection MOT.
We have also observed that the number of atoms transferred to the science MOT
in a single load is directly proportional to the number in the collection MOT. Using
the MOPA laser for the collection MOT allowed us to increase the single load size by a

27
factor of 2 or more, depending on the laser power. In addition, we took advantage of thehigher collection efficiency of the MOPA-powered MOT by decreasing the temperatureof the Rb ampoule by several degrees yet maintaining the same MOT fill rate (1/e timeof 2 s) and maximum build (Nmax =1010 ). This allows for a lower overall rubidium
pressure in the vacuum system, which improves the magnetic trap lifetime.

3.3

Improved magnetic field measurement technique3.3.1

Introduction

Studies of the 85 Rb Feshbach resonance depend critically on knowledge of themagnetic field strength. To measure the B-field, we developed a simple, precise methodthat relies on the Zeeman effect in a trapped cloud of atoms. In our technique, which isrelated to early NMR experiments [23], we induce spin-flip transitions in the atoms witha short (∼10 µs) pulse of radio frequency (rf) radiation. The rf pulse coherently drives
atoms from the original |F, mF i = |2, −2i hyperfine state to the |2, −1i state, which is
weakly confined in the B-field range of interest (∼160 G). Due to the weak confinementof the |2, −1i state, atoms that undergo a ∆mF =+1 transition find themselves far fromtheir equilibrium position in the magnetic trap. These atoms experience a force thatpushes them out of the original cloud of |2, −2i atoms. Thus, spin-flip transitions lead tothe loss of atoms from the sample. The loss is frequency dependent; the rf frequency thatmaximizes the loss corresponds to the energy splitting between the Zeeman states. Afterthe energy splitting is inferred from the rf frequency, we use the known hyperfine andZeeman Hamiltonians in 85 Rb to extract the magnetic field. This B-field determination
can be quite precise (± 10 mG) due to the inherently narrow (Fourier transform-limited)
linewidth of the rf pulse transition lineshape.
The short rf pulse method described here has several important advantages over
our earlier method for magnetic field measurement by rf “carving” [1]. In the carving

28
technique, the frequency of the rf “knife” ramps down in ∼10 s from a chosen startvalue to a stop value. By varying the stop value and also the direction of the rf ramp,
one can find the frequency at the center of the cloud and the corresponding magnetic
field. Although the precision of such a B-field measurement can be even better than the
precision yielded by the short rf pulse technique, the rf carving method has limited utility
because of its long integration time. The rf carving averages the B-field during a 10 s
window; in contrast, the rf pulse method averages the magnetic field for a time period
equal to the pulse length, which can easily be as short as 10 µs. Clearly, measuring Bfield with an rf pulse is well-suited to studies of the response of a BEC to rapid B-fieldchanges — as discussed extensively throughout this thesis.
An additional advantage of the rf pulse method is its simple theoretical description
using two-level quantum theory [24]. Although the 85 Rb atom is definitely not a spin-1/2
system, the simple two-level theory is still quite valid because of the typical magnetic
field strengths used. At relatively strong magnetic fields, the Zeeman shift for a given
spin state is not linear with B. Also, the splitting between a pair of adjacent spin states
is not degenerate with the splitting of the next closest pair. For example, at B=162 G,
the mF =-2 → mF =-1 transition frequency is 81.1 MHz while the mF =-1 → mF =0
transition frequency is 76.8 MHz. Hence, if the rf radiation is in resonance with the (-2→ -1) transition, the radiation is blue-detuned from the next closest transition (-1 → 0)
by 4.3 MHz. This means that the short rf pulse cannot couple together more than one
pair of spin states, provided that the spectral width of the pulse is sufficiently narrow.
The spectral width for a typical 10 µs rf pulse is 89 kHz FWHM (full-width at halfmaximum of Fourier transform), easily satisfying the conditions for the applicability of
two-level theory. It is important to note that the different spin states are stable against
spontaneous decay, so the intrinsic energy width of a given state is less than 1 Hz.

29
3.3.2

Rf apparatus

To utilize our new rf pulse method, we needed precise and repeatable timing
control of the rf radiation. The apparatus for controlling rf and delivering it to theatoms is shown schematically in Figure 3.3. Continuous wave rf radiation is createdby an HP8656B frequency synthesizer. The synthesizer output frequency and power iscontrolled via a GPIB computer interface.
The radiation was truncated into a short pulse with a Mini-Circuits rf switch(SPDT), model ZASWA-2-50DR. The switch was controlled by a variable pulse generator (Wavetek model 801) that could be externally triggered via a TTL signal fromthe computer TTL board. Finally, the rf pulse was amplified and applied to the atoms
through a coil situated very near the magnetic trap.
Using our apparatus, we developed a simple procedure to drive rf transitions in atrapped cloud of cold atoms. First we load the atom cloud into the magnetic trap andevaporatively cool it to a given temperature (T≤60 nK). Next we apply a short pulseof rf radiation from the rf coil. After the pulse, the trap remains on for 10-20 ms toallow the spin-flipped atoms to fall away, then the trap is switched off (B→0) and theremaining |2, −2i atoms are imaged 12.8 ms later. We count the number of atoms in
the absorption image, then repeat the procedure multiple times with different rf pulse
frequencies to obtain a transition lineshape, as shown in Figure 3.4.

3.3.3

Lineshape measurements

The theory describing the response of the atom cloud to an rf pulse was derived
in the early studies of magnetic dipole transitions in molecular beams [24]. When a twolevel atom is subjected to a pulsed oscillatory coupling field, the probability of making
a transition from the initial state (a) to the final state (b) is given byΩ2
Pa→b = 2 0 2 sin2 (δ + Ω0p
δ 2 + Ω20 τ
),
2

(3.1)

30

from
computer(GPIB)

fromcomputer
(DIO)
pulse
generator
TTL

HP8656B
synthesizer

10 W
amplifierrf switch

to
rf
coil

Figure 3.3: Schematic of apparatus used to control the rf radiation. A GPIB-controlledfrequency synthesizer produces the rf at a fixed frequency and power. To truncate
the rf wave into a pulse, we use a pulse generator and rf switch. Due to the extreme
sensitivity of the atoms to rf fields, we chose a switch with very high isolation (-90 dB)
and good switching characteristics (10%-90% rise/fall time=5 ns). From the measured
rf transition linewidths (see section 3.3.3), we conclude that the spectral purity of thefrequency synthesizer and the switching behavior of the rf switch are nearly ideal.

31
where τ is the pulse length, δ = (ωrf − ω0 ) is the detuning of the rf pulse from theresonance frequency, ω0 , and Ω0 is the Rabi-flopping frequency at zero detuning. The
Rabi frequency is proportional to the atom’s magnetic moment and to the amplitude of
the oscillating magnetic field: Ω0 = µB/(2~). In a typical lineshape measurement, τ is
fixed and we tune δ until the transition probability reaches a maximum at δ=0. This
detuning causes the maximum loss from the atom cloud in the experiment. To ensurethat Pa→b (δ = 0) is reasonably large (roughly 0.5), we adjust the rf power and pulse
length to achieve Ω0 τ ∼ π/2.
The predictions of equation (3.1) agree quite well with experimental data. Figure 3.4 shows lineshape data and theory for a particular choice of magnetic field, rf
power, and (fixed) rf pulse length. We also varied the pulse length with fixed rf frequency to map out coherent Rabi oscillations (see Figure 3.5).
For most of our magnetic field measurements, we found that using cold thermal
clouds was more convenient than using a condensate. Since a BEC can undergo huge
number losses when the magnetic field is changed in the vicinity of the Feshbach resonance (see chapter 5), a BEC cloud is not suitable for measuring the field during ourfast B-field pulse sequence. With a BEC cloud, it is difficult to separate the Feshbachresonance-induced number loss from the rf pulse-induced number loss. We therefore
measure the magnetic field with cold, low-density thermal clouds. The low density prevents density-dependent losses from occurring during our rapid B-field changes, while
the cold temperatures allow for fairly narrow rf transition linewidths. Thermal clouds
have the additional advantage of being simpler and faster to produce than BECs. The
cycle time for producing ∼60 nK, low density thermal clouds is roughly 40% shorter
than the time to make a condensate.
There is one disadvantage to using thermal clouds, however. Since thermal clouds
are hotter than condensates, they experience a larger magnetic field variation. The Bfield variation arises from gravitational “sag” in the magnetic trap. Due to our rather

32

1.0
fraction remaining

0.9
0.8
0.7
0.6
0.5
0.4
0.3
81.0

81.1

81.2

81.3

81.4

rf frequency (MHz)Figure 3.4: Comparison of lineshape data to theory for a sample of BEC atoms. The
points are the fraction of atoms remaining after the rf pulse. The line shows the quantity(1-Pa→b ) computed from equation (3.1), with τ =10 µs and Ω0 = 2π × 28 kHz (the RF
synthesizer power was +5 dBm). These conditions correspond to a Ω0 τ =π/1.9 pulse.
We used a BEC cloud at T∼3 nK for the lineshape, so there is no indication of any
linewidth broadening beyond the Fourier transform-limited theory (FWHM=89 kHz).

33

18000
16000
BEC Number

14000
12000
10000
8000
6000
4000
2000
0

10

20

30

40

50

60

Pulse length (µs)
Figure 3.5: Rabi-flopping in a BEC. The points are the number of BEC atoms remaining
after the RF pulse, while the line is given by equation (3.1) with Ω0 = 2π × 28 kHz
and δ = 2π × 23 kHz. The data set is somewhat undersampled, but it seems consistent
with the theory at short times. However, at longer times, the Rabi flopping seems todamp out. This could be related to the large negative scattering length of the |2, −1istate, which does not have a nearby Feshbach resonance like the initial |2, −2i state. In
addition, gravity should lead to decoherence because the |2, −1i state has a factor of 3
weaker magnetic moment than the initial state.

34
weak trapping potential, the atoms sag downward, away from the position of the Bfield minimum, until the force from the (increasing) magnetic field gradient becomesstrong enough to counteract gravity. In equilibrium, this leads to an approximately
linear gradient of 26 G/cm along the vertical direction. Thus, the spread in B-field
across the cloud is simply determined by the cloud’s vertical width, σr , which depends√
on temperature as σr ∝ T .
The magnetic field width of the cloud translates directly into a spread in rf transition frequency, which leads to inhomogeneous broadening of the rf lineshape. The
broadening is very simple to model as long as the rf pulse length is much less than a
trap period because the atoms do not move during the pulse. To first order, each atom
experiences a constant magnetic field with a magnitude that depends on the positionof the atom when the pulse begins. One can calculate the correct transition probability
by averaging over the distribution of transition frequencies along the vertical directionof the thermal cloud. The average probability is
1hPa→b i =
N

Z ∞
−∞

Pa→b (ω0 , ωrf )N(2π)3/2 σr αδ

exp [−

1 (ω0 − ωrf )2]dω0 ,
2 σr2 α2 (2πδ)2(3.2)

where Pa→b (ω0 , ωrf ) is the single atom probability from equation (3.1), ω0 is the angular transition frequency at a given position, ωrf is the angular rf frequency, N is thenumber of atoms in the thermal cloud, σr is the rms radial width of the gaussian density
distribution, α=2600 G/m is the magnitude of the B-field gradient, and δ is the slopeof transition frequency versus B-field, which is 530 kHz/G near 160 G.
Typically, we observed thermal cloud linewidths (FWHM) of ∼100 kHz with
10 µs pulses and T∼60 nK. To look for effects of inhomogeneous “thermal broadening”
in the lineshapes, we increased the pulse length. For example, a pair of thermal cloudlineshapes measured with different rf pulse lengths are shown in Figure 3.6. These data
show a significant amount of inhomogeneous broadening from the finite temperature of
the clouds.

35

1.1

fraction remaining

1.0
0.9
0.8

(a)

0.7
0.6
80.3

80.4

80.5

80.6

80.7

80.8

80.9

81.0

rf frequency (MHz)1.05

fraction remaining

1.00
0.95
0.90

(b)

0.85
0.80
0.75
77.8

77.9

78.0

78.1

78.2

78.3

78.4

rf frequency (MHz)Figure 3.6: Lineshape data and theory for a thermal cloud. The points are the fractionof atoms remaining after the rf pulse and the lines are predictions from equation 3.2.
The average thermal cloud density is 1011 cm−3 . (a) Data and theory for rf pulse
length τ =10 µs, temperature T=50 nK, and Rabi frequency Ω0 =2π × 20 kHz. Here theFourier transform limited linewidth is 89 kHz; however, thermal broadening increases
the FWHM above the transform limit by 30%. (b) In this case, we have τ =50 µs,
temperature T=60 nK, and Rabi frequency Ω0 =2π × 6 kHz. Although the transform
limit decreases to 18 kHz, thermal broadening causes the linewidth from equation (3.2)
to be 82 kHz. The lineshapes in (a) and (b) were measured at slightly different magnetic
fields, so the center frequencies for the lineshapes are different.

36
The total width of a thermal cloud rf lineshape depends primarily on the pulse
length (through the Fourier transform limit) and inhomogeneous broadening from spatial magnetic field differences across the cloud. Another potential source of line-broadening
is time variations of the B-field during the rf pulse. We sometimes observed broadeningthat was larger than expected from the transform limit and finite temperature alone.
In fact, one can use the observed linewidth to estimate the magnitude of B-field noise
during a rf pulse. The noise can only be estimated in this way if it causes significant
broadening beyond that due to the transform limit and finite temperature of the atom
sample.
As an aside, it is worth mentioning that for the majority of our lineshape measurements, we fit the data to a Lorentzian function. Although this simple function does
not capture the details of the full lineshape described by equation (3.2), the Lorentzian
fit provides a good estimate of the linewidth and center frequency. In general, the data
were not sufficiently precise to distinguish differences between the chosen fit function
and the theoretical form discussed here.

3.3.4

Obtaining the B-field from lineshape data

Once the center frequency for the rf lineshape is measured, we calculate themagnetic field with the Breit-Rabi equation [24]. The form of this equation for 85 Rb isdescribed in Ref. [1] and will not be repeated here. Since the Breit-Rabi equation is anexact analytical result for the B-field dependence of the energies of the various hyperfine
states, we simply insert the measured energy difference between two states (the center
frequency from the rf lineshape) and then invert the equation to obtain the magnetic
field. The B-field determination has a precision limited only by the precision of the rf
frequency measurement.
In principle, the accuracy of this B-field determination should also be very good.
However, the measurement is distorted by one significant systematic effect from finite rf

37
power. The rf power systematic is a shift to the magnetic field due to interference of therf radiation with the magnetic field control circuitry. The current sensor for the auxiliary
coil current servo (described in section “aux-coil hall probes”) detects the rf radiationand filters it, leading to a change in the current and a “DC” shift to the field. Examples
of rf power shifts to the Hall probe current sensors are displayed in Figure 3.7. We usedthe Hall probe signals, which were previously calibrated with respect to magnetic field(see section 3.4.6), to correct for the rf power shift to the B-field.
We have an additional, independent way to determine the size of the rf power
shift. In this method, we vary the power of the rf pulse and measure a lineshape foreach value of power. The rf power shift causes the center frequency of the lineshape todecrease linearly with rf power (see Figure 3.8). By extrapolating to zero rf power, we
obtain another estimate of the true rf transition frequency (B-field).
The rf power shift discussed here is only present due to interference of the rfradiation with the auxiliary coil current servo. There is no noticeable effect of a short
rf pulse on the magnetic field produced by the baseball and bias coil currents, which
provide the majority of the B-field experienced by the atoms.1 Thus, in experiments
where we change the magnetic field slowly, without using the auxiliary coil, there is norf power shift to the measured B-field.
Because 85 Rb is not a two-level atom, the rf radiation causes another (small)
power shift to the measured rf transition frequency. The shift arises because the rfradiation — tuned near the |2, −2i → |2, −1i resonance — simultaneously couples the|2, −1i and |2, 0i states. This latter coupling leads to a small AC Stark shift of the
mF =-1 state. The Stark shift is equal to Ω20 /|4δ| '0.3 kHz, where δ=4.3MHz is therelevant detuning at 162 G (see section 3.3.1). Of course, the calculated shift is totally1

The absence of an effect is due to two things: the servo that controls the current for the trapping
coils (baseball and bias coils) has much lower bandwidth than the auxiliary coil servo, so it is not able
to respond to the rf pulse during the time of the pulse. Also, the trapping coils have a larger inductancethan the auxiliary coil and thus they offer a larger impedance to high frequency current changes.

38

relative B-field (G)

-0.8

-0.9

-1.0

-1.1

-1.2
45

50

55

60

65

70

time (µs)
Figure 3.7: B-field shifts due to finite rf pulse power. Solid lines are the B-fields versus
time calculated from the calibrated auxiliary coil current sensor. Vertical dotted lines
show the 11 µs duration of an applied rf pulse. Each solid line corresponds to a different
rf pulse power. From top to bottom, the rf power from the frequency synthesizer is-120 dBm, -1 dBm, 2 dBm, and 5 dBm. The current sensor output signals are filteredusing a 2 µs averaging window, so there is no indication of oscillations at the rf frequencyof ∼80 MHz. To estimate the B-field shift, we calculate the average difference between
the line with -120 dBm and the line with a given rf power, within the time window
for the rf pulse. We estimate the uncertainty on the B-field shift from the standard
deviation of the average field difference.

39

B-field shift (G)

0.00
-0.05
-0.10
-0.15
-0.20
-0.25
0

1

2

3

rf synthesizer power (mW)
Figure 3.8: Comparison of two methods to measure rf power shift. Black points witherror bars are B-field shifts as measured by the auxiliary coil current sensor as a functionof rf pulse power (see caption to Figure 3.7). White points with error bars are B-field
shifts determined directly from rf lineshape data. An arbitrary offset was added to thelineshape data to make the point at ∼0.8 mW overlap the corresponding current sensor
measurement. The solid line is a weighted fit to the lineshape data, showing that theextrapolation to zero rf power agrees well with the current sensor data. The slope fromthe fit is -49(4) mG/mW.

40
negligible compared to a typical rf linewidth of 100 kHz, so we can safely ignore it.
Ignoring the shift is reasonable as long as the magnetic field is large enough to breakthe degeneracy between adjacent ∆m=+1 spin-flip transitions.

3.3.5

Outlook for rf lineshape measurements

The present technique for measuring the magnetic field made possible the precise
characterization of the 85 Rb Feshbach resonance discussed in chapter 7. This method
was also essential for understanding the time-variations of the B-field produced by theauxiliary coil (see section 3.4). There are at least two other possible experimental
applications for the rf lineshape technique, although we have not fully explored them.
First, one can use the rf lineshapes to determine the nature of magnetic field noisein the experiment. By increasing the length of the rf pulse (and thereby decreasing thelinewidth due to the Fourier transform limit), one can look for a linewidth “floor”
corresponding to time variations in the B-field. The magnitude of these field variations
could be explored by careful study of the lineshapes.
Second, the rf lineshape technique might be used to measure the average meanfield energy of a BEC. After subjecting a condensate to a rapid change in magnetic field(and scattering length), one could then measure an rf lineshape and look for a shift or
broadening of the lineshape from the BEC self-interaction. Perhaps it would be possible to measure a beyond mean-field energy shift in this way [1]. Unfortunately, sucha measurement would be difficult because condensate atoms experience non-adiabatic
transitions to a molecular state when the B-field is changed quickly near the Feshbachresonance, as noted in chapter 5. Since the BEC loss occurs on a time scale (∼10 µs)
that is comparable to the minimum rf pulse length needed for a precise frequency measurement, interpetation of the rf lineshape would likely be very complicated. However,a compromise might be found in which the B-field change was slow enough to avoid
non-adiabatic transitions but fast enough to prevent changes to the condensate density

41(see section 3.5.3).

3.4

New auxiliary coil for rapid B-field pulses

3.4.1

Motivation for auxiliary coil

To study the response of a 85 Rb BEC to rapid magnetic field changes near the
Feshbach resonance, we developed a specialized “auxiliary” magnetic field coil. This
coil and its accompanying electronics were designed to provide controlled B-field pulses
toward the resonance, allowing us to explore the resulting BEC dynamics over a wide
range of time scales. In the following sections, the details of the auxiliary coil designand characterization are given.
Before building the auxiliary coil, we made some preliminary attempts to pulse
the magnetic field quickly by rapidly changing the reference voltage used in the current
servo for our main “baseball” (BB) trap coil. This is the same method we used to study
condensate collapse dynamics, as described in Chapter 4. By using short magnetic
field pulses, we hoped to avoid the issues of condensate heating and 3-body collisional
losses that we previously observed with slower B-field ramps [1]. However, the BB coilramping scheme had limited utility. Due to the 8 kHz bandwidth of the BB current
control servo, a 10 G B-field pulse toward the Feshbach resonance from the initial fieldof 166 G required a total time of ∼250 µs. Surprisingly, we observed large BEC number
loss under these conditions, with most of the loss occurring during the ramp itself. Thisevidence for rapid condensate number variations prompted us to explore even shortertime scales, so we developed a fast, high current control circuit that drives a separate
auxiliary coil.
We decided there were at least two good reasons to start from scratch and build
a separate coil rather than modifying the existing BB trap/servo system. Most importantly, the current control for the trap coils was previously optimized to give good42
stability during rf evaporation and BEC production [1]. We wished to avoid the possibility of messing up one of the best parts of the apparatus and/or tying up the experiment
during the extensive debugging process that would follow a change to the trap. In addition, we realized that we could carefully design the auxiliary coil to meet the specialized
requirements for rapid B-field ramps, including low coil inductance and high bandwidth
for the servo.

3.4.2

Auxiliary coil design

We designed the auxiliary coil to give a relatively large magnetic field at theposition of the trapped atom cloud and to have a low self-inductance, L. Since theinductance for a solenoid is proportional to the area and the number of turns squared,
we constructed the auxiliary coil with only 3 turns of square magnet wire with a coildiameter barely big enough to fit over the glass science cell. The coil is placed as close
as possible to the trapped atoms and is oriented coaxially with the BB and bias trap
coils, so that the B-fields in the z-direction add linearly. The position of the auxiliary
coil with respect to the trapped atoms and the other major magnet coils is shown in
Figure 3.9. For this geometry, the self-inductance of the auxiliary coil is ∼2 µH andthe coil produces a magnetic field at the trap center of 10G when driven by a currentof 167A.

3.4.3

Capacitor bank for auxiliary coil current

To switch such large currents rapidly, we built a circuit analogous to a short-pulse
flashlamp driver [25]. We first charge a 170µF capacitor bank to 580V, then discharge itthrough the auxiliary coil at a rate determined by a transistor. The large capacitor bankat high voltage allows for large amounts of stored charge and also a large current ramp
8rate, dI/dt. From Faraday’s law, V = L dI
dt , the maximum ramp rate is 2.9 × 10 A/s! In

practice however, the transistor resistance limits the actual value of dI/dt to the more

43bias coil
BB coilbias coil
auxiliary coilFigure 3.9: Schematic of the main magnetic coils. The shaded grey tube represents theglass cell and the black dot shows the position of the trapped atom cloud. The bias
coils have average radii of 4.5 cm and are separated by 4 cm. The BB coil has a radiusof 3.2 cm. The auxiliary coil barely fits over the cell with its 1.6 cm radius, and it islocated 4 cm from the atoms.

44
modest value of 2.6 × 107 A/s.
In the course of testing the discharge rates of various capacitors, we learned that
all capacitors are not created equal. In fact, different types of capacitors with the samevalues for C do not necessarily discharge at the same rates. For rapid discharge, i.e.,
ideal capacitor performance, one needs a capacitor with low effective series resistance(ESR). The ESR of a capacitor depends strongly on its innards — the materials used tostore the charge. For example, electrolytic capacitors have very large ESR compared tometal film or oil-filled capacitors. To form our capacitor bank, we dug up several ancient,toxic, PCB-filled capacitors from the JILA electronics shop and also bought some newpaper/polyester film capacitors from CSI Technologies ($100 per 50 µF capacitor). The
capacitor bank voltage is limited by the finite capacitor working voltage of 600 V, whichleads to a total stored charge Q=CV=0.1 Coulombs.

3.4.4

Servo electronics and transistor

To regulate the discharge rate of the capacitor bank, we use a negative-feedback
current servo and a transistor. The servo compares the output of a fast Hall-effectcurrent sensor (Hall probe model CLN-300) with a reference voltage and sends thedifference signal to the transistor, as displayed schematically in Figure 3.10. The Hall
probe used in this circuit is identical to those used for sensing the BB and bias coilcurrents [1]. These current sensors have excellent dynamic range and linearity, as well
as a large bandwidth (150 kHz specification). We verified that the Hall probe waslinear up to dI/dt values of 4 × 107 A/s by using a silicon-controlled rectifier (SCR) to
discharge the capacitor bank very quickly.
The auxiliary coil current servo is quite simple — a single op-amp compares
the hall probe signal to the reference voltage, producing an output proportional to thedifference voltage. We chose the feedback loop using a variable RC click-box, also known
as “Terry’s Magic Box”, to optimize the transient response of the servo to an input step

45

+600V
+V

torroid
chokes
1k

2uH

DIODE
170uF
IGBT
CM200HA-24H

CLN-300
hall probe

69k
Function Gen.

10k

OPA627

33

SRS
10k
50

Figure 3.10: Circuit schematic for the auxiliary coil current control. The auxiliary coilis shown as the 2 µH inductor and the capacitor bank is the 170 µF capacitor. An
important part of the circuit is the SRS function generator (model DS345), which wasprogrammed via a GPIB interface with our Labview computer. The function generator
allowed us to apply very precisely-tailored magnetic field pulses to the BEC atoms. The
torroid chokes between the capacitor bank and the auxiliary coil prevent “parasitic”
oscillations, which are created by miniature LC oscillators from the finite inductanceand capacitance of the various circuit elements.

46
function pulse on the reference voltage. The best behavior was obtained with a simple
proportional gain of ∼7. Terry’s box allowed us to sidestep the calculation of thecomplicated frequency response of various circuit elements, including the Hall probe
and transistor. We empirically determined the best feedback loop by examining theservo behavior for various R, C, and RC combinations.
The transistor used to control the capacitor discharge is an insulated-gate-bipolartransistor (IGBT): Powerex model CM200HA-24H. This device has the useful property
of a very large maximum voltage rating for the collector-emitter voltage (1200 V), which
far exceeds the capacitor bank working voltage of 600 V. The IGBT has a maximum
average power dissipation of 1.5 kW, which is generally not exceeded during our typicalfew hundred Amp current pulses. To ensure that we do not kill the transistor during
a given current pulse, we use the minimum capacitor bank voltage necessary to supply
the pulse. In this way we can minimize the voltage drop across the transistor (andtherefore the power dissipation) during the discharge process. The most important
advantage of the IGBT is that it can be used alone; there is no need to put several suchdevices in parallel. We found that using multiple transistors in parallel caused massive
oscillations in the current due to competition between the devices. One disadvantage of
the IGBT relative to a more common MOSFET is the higher ON resistance. The ON
resistance of the transistor limits the maximum discharge rate of the capacitor bank— the auxiliary coil circuit can be modeled as a LCR damped oscillator, where the Rvalue of the transistor determines the damping rate. Although the IGBT resistance ishigher than a MOSFET, the IGBT is still preferable due to its better voltage and power
ratings.

3.4.5

Auxiliary coil interferes with magnetic trap servo

In our initial attempts to pulse the magnetic field with the auxiliary coil circuit,
we discovered that the auxiliary coil magnetic field change induced a corresponding

47
magnetic field change in the BB and bias coils. Mutual inductance between the various
coils causes this effect, which forces the BB and bias servos to oscillate for several milliseconds. To avoid this undesirable interference of the auxiliary coil with the magnetic
trap servos, we devised a scheme to briefly decouple the BB and bias servos from theirrespective FETs during the auxiliary coil pulse. We built a sample-and-hold circuit for
the BB and bias coil servos, as shown in Figure 3.11. The operation of the circuit is
as follows. Just before the beginning of the auxiliary coil pulse, we send a HOLD TTL
signal to the LF398 sample-and-hold chip, instructing it to start holding the voltage
constant. Soon afterward, we send a SWITCH TTL signal to the DG201 switches,causing one switch to open and one to close. Now the servo is decoupled from the FET
gate and the sample-and-hold controls the FET. After the auxiliary coil pulse ends, we
switch back to servo control and allow the LF398 to start sampling the input voltage.
Of course, the sample-and-hold circuit cannot prevent the auxiliary coil from inducing
currents in the other coils, but the circuit does remove the unwanted transient responseto the induced currents.

3.4.6

Calibrating the total magnetic field

Before attempting any experiments to study condensate loss, we had to calibrate
the auxiliary coil. We made a simple estimate of the magnetic field experienced by thetrapped atoms for a given auxiliary coil current using the solenoid equation:
B(z) =¤−3/2µ0 N I £1 + z 2 /R2
,
2R

(3.3)

where B(z) is the z-component of the magnetic field evaluated on the axis of the solenoid,µ0 =4π × 10−7 N/A2 is the free-space permeability, N is the number of turns in thesolenoid, I is the current, and R is the radius. Using the measured coil radius andposition (see Figure 3.9), the solenoid equation predicts a B-field to current ratio of-0.06 G/A, or -3.57 G/VHall , where VHall is the Hall voltage — the voltage produced

48

SWITCH TTLInverter
1k

DG201
TTL switch

+
V_OUT
-

1k
V_IN

+
DG201
TTL switch
-

(to
FET
gate)

LF398
sample & hold(from
Current
Servo)

200k

3.3nF
HOLD TTL

Figure 3.11: Schematic diagram of the sample-and-hold circuit. The circuit is shownin the HOLD configuration, so the trap servo is effectively disconnected from the FET.
Both the BB and bias coil current servos have a sample-and-hold circuit like this one.

49
by the Hall current flowing through a resistor RHall =33.5 Ω. The fractional uncertainty
in this calculation is roughly 20% due to uncertainty in measured values of R and z.
We tested this prediction with two different experiments. In the first experiment,
we connected the auxiliary coil to a DC power supply and used a fixed 16 A current. We
then loaded atoms into the magnetic trap and removed them with rf radiation from therf coil. The rf frequency was smoothly ramped downward to cut away more and more
atoms until none remained. We recorded the final rf frequency required to remove allof the atoms as a function of the auxiliary coil power supply current and converted therf transition frequency to magnetic field using the Breit-Rabi equation. This proceduregave an auxiliary coil calibration of -3.696(2) G/VHall , which agrees quite well with therough estimate given above.
We obtained a second calibration of the auxiliary coil using short rf pulse spectroscopy (discussed in section 3.3). Two 10 µs rf radiation pulses were applied at
different time delays during the auxiliary coil pulse. The short rf pulse calibration was-2.26 G/VHall , which is only 60% of the “DC” calibration discussed previously. The
reason for the discrepancy is the difference in induced currents for the two different
calibration experiments. With a steady current flowing through the auxiliary coil, there
are no induced currents in other coils or pieces of metal near the atom cloud. However,when we apply a short, high current pulse to the auxiliary coil, there is a tremendous
change in magnetic flux through the other coils, leading to significant induced magnetic
fields that oppose the auxiliary coil B-field. The presence of the induced currents limited
the maximum ramp speed of the total field to ∼1 G/µs.
To compensate for the effects of induced currents, we had to first measure theirtime-dependence. As displayed in Figure 3.12, we used short rf pulses to map out B(t)for a trapezoidal auxiliary coil current pulse. In this experiment, the current remains
constant during the hold time, thold . Although the auxiliary coil magnetic field, Baux ,must therefore be constant during thold , we observe that the total field decreases with

50
time, which means that the induced B-field is decreasing in magnitude (Lenz’s law
predicts that the induced B-field is opposite in sign to that of the auxiliary coil). Clearly,
in the limit that thold → ∞, the induced fields should decay to zero and the auxiliary
coil calibration must return to its DC value of -3.696 G/Vhall .
By tailoring the reference voltage waveform, we changed the time-dependence ofBaux until the total B-field was approximately constant during thold . In most cases therequired correction is a linear “droop” of the auxiliary coil current, as in Figure 3.12.
We adopted an iterative procedure and found this linear slope empirically because adetailed model of the coupled coil system would be very complex. Figure 3.13 shows
the measured dependence of the droop rate of the induced magnetic field on the auxiliary
coil pulse height. This plot proved very useful for many of our BEC experiments, wherea constant evolution B-field was important.

3.5

Improved BEC diagnostics
To conduct the experiments described in this thesis, we developed new methods

to measure the number and temperature of the condensed sample. Accurate knowledgeof these properties (especially the number of atoms) is crucial for studying the dynamics of the BEC when subject to changes in the magnetic field and scattering length.
In our number and temperature measurements, we made extensive use of the Feshbachresonance to change the ground state size of the BEC wavefunction in a very highlycontrolled fashion. Our manipulation of the wavefunction via the tunable scattering
length in the 85 Rb condensate can be viewed as an example of quantum engineering,where the detailed quantum-mechanical characteristics of a system may be changed atwill. Basically, we can use the well-understood mean-field interaction that depends on
a to either expand or shrink the BEC wavefunction, allowing us to count the condensate number or measure its temperature. We then use these techniques to study thecondensate dynamics in regimes that are not well-understood, such as the collapse of an51

2

relative B-field (G)

0
-2
-4
-6
-8
-10
0

100

200

300

400

500

600

time (µs)
Figure 3.12: Compensating for induced currents. The black line shows the auxiliary
coil B-field calculated from the Hall probe with the calibration -2.26 G/V. In this case,the auxiliary coil current was roughly constant during the hold time. The Hall probe
voltage was filtered with a 2 µs averaging window. Black points with error bars aremeasurements of the total magnetic field experienced by the atoms from rf pulse spectroscopy (pulse length=22 µs) corresponding to the conditions of the black line. The
decrease in total B-field is related to the droop in the induced magnetic field. The greyline shows the auxiliary coil magnetic field when a linear correction is applied to theservo reference voltage. The white points are the rf pulse B-field data corresponding
to the grey line. By precisely adjusting the auxiliary coil current, we made the totalmagnetic field constant to within 30 mG for over 400 µs.

52

observed droop rate (mG/µs)

2
0
-2
-4
-6
-8
0

1
2
3
auxiliary coil hall voltage (V)4

Figure 3.13: Dependence of induced B-field decay rate on auxiliary coil pulse height.
The black points with error bars are obtained by fitting a straight line to a seriesof consecutive magnetic field measurements, as in Figure 3.12. The induced B-field is
calculated to be the difference between the total magnetic field (measured by the rf pulse
spectroscopy) and the auxiliary coil magnetic field, Baux (t)=-2.26 G/VHall × VHall (t).
The pulse height of the auxiliary coil current pulse, which is used on the x-axis, is definedas the initial value of the Hall probe voltage after the completion of the first rapid B-fieldramp(s). We used this plot as a look-up table for adjusting the auxiliary coil droop rate
to make the total magnetic field constant for a particular value of magnetic field, Bhold .

53initially stable BEC (Chapter 4) or the non-adiabatic mixing of atomic and molecularstates (Chapters 5 and 6).

3.5.1

Absorption imaging

All of our measurements of BEC properties depend on absorption imaging of theatom cloud. We illuminate the cloud with an on-resonant laser beam and then image
the shadow onto a CCD camera. The “darkness” of the shadow gives information about
the number of atoms casting the shadow. To obtain the most sensitive and accurate
measurements of number, we find that the magnetic trap must be turned off and thecondensate spatial size must be significantly larger than the 7 µm resolution limit of the
optical system. Turning off the magnetic trap is important to maximize the number ofphotons scattered by each atom, which improves the signal-to-noise ratio of the shadow
images.
The detection of the BEC number by absorption imaging is necessarily a destructive measurement because of the photon scattering and the trap turn-off. The BEC
atoms are dramatically heated by the recoil momenta of the scattered photons, effectively destroying the ultracold condensate. Even if the photon heating could be reducedby turning down the laser intensity, turning off the magnetic trap would also destroy
the condensate because when B→0, the scattering length changes sign and magnitude(a → -470 a0 ). This leads to a strong attractive self-interaction in the BEC, whichshould cause the condensate to self-destruct by collapsing (see Chapter 4).
To count the number of atoms in the BEC, we use the well-known relationshipbetween the amount of absorption (the optical depth) and the number [1]. We typically
fit the shadow image to a Gaussian function and then calculate the number of atoms
from the fit parameters. Although the Gaussian function does not always fit the images perfectly, it provides a good approximation for most cases. A more robust way toestimate the number of atoms is to calculate the total optical depth by summing all of

54
the pixels in the shadow image. The number of atoms is directly proportional to this
optical depth summation. One advantage of the pixel sum method is its insensitivity
to the density distribution of the atoms that determines the precise spatial shape ofthe shadow image. Even non-standard density profiles or multi-component spatial distributions of atoms can be accurately counted using the pixel sum. We also developed
more specialized fitting routines that combine Gaussian fitting and pixel sums for the
dynamics experiments described in Chapters 4,5, and 6.

3.5.2

Motivation for expanding the BEC

There are a number of systematic effects that plague the experimenter attempting
to measure the number of atoms in his 85 Rb BEC. The student cannot simply use the“standard” approach of turning off the magnetic trap, waiting for the condensate toexpand for a few milliseconds, then finally taking a picture. In fact, there are a numberof unusual features of the 85 Rb BEC that render the standard procedure given abovequite useless. Some of the unusual features of the 85 Rb BEC system are the small spatial
size of the BEC (comparable to the finite resolution limit of the optical system), therelatively low kinetic energy of the BEC owing to the weak magnetic trap confinement,
the large density-dependent losses experienced by the condensate as the B-field crosses
the Feshbach resonance [1, 26], and the large negative scattering length of 85 Rb at zeromagnetic field (a=-470 a0 ).
Fortunately, one can avoid all of these potential problems by expanding the BEC
while it remains in the magnetic trap. Expanding the wavefunction above the resolution
limit removes any systematic effects due to imaging a small object. Also, by expanding
the condensate in the magnetic trap, we can dramatically reduce the density before
crossing the Feshbach resonance at the trap turn-off. Finally, we have found that a lowdensity, rapidly expanding BEC does not collapse at B=0, despite the large negative
scattering length.

55
3.5.3

Increasing repulsion to expand the wavefunction

We devised a simple method to expand the BEC wavefunction that utilizes thetunable scattering length in 85 Rb. We ramp the B-field to quickly increase the scattering
length and excite a large amplitude collective oscillation of the BEC. In this case, theenergy imparted to the condensate comes directly from the changing magnetic field,
which is provided by a changing current in a Hewlett-Packard power supply.2

Afterincreasing the mean-field repulsion, the BEC does not expand indefinitely in the trap;
the confining potential provides a restoring force that causes large amplitude breathingmode oscillations. The period for the radial oscillation is roughly 1/(2 νr ) = 29 ms.
We therefore turn off the magnetic trap well before the condensate reaches the outerturning point of its radial oscillation, as shown in Figure 3.14. We set the timing of the
expansion ramp to maximize the outward expansion velocity of the condensate at the
moment that the trap turns off. This allows the condensate to continue to expand quasiballistically at B=0 (a=-470 a0 ), despite the strong attractive mean-field interaction.
For the conditions of Figure 3.14, the expansion velocities of the BEC radial and axial
widths were measured to be ∼1 mm/s, corresponding to an average kinetic energy per
atom of 5 nK.
As the BEC expands, the volume of the atom cloud increases and takes on aroughly spherical shape. The density of the BEC decreases, so that the probe beam
used in absorption imaging becomes less heavily absorbed. The decrease in absorption
occurs because a given laser photon has a lower probability of encountering an atom
as it passes through the BEC. We typically expand our N=16000 atom condensates
until the on-resonant optical depth of the cloud decreases to approximately unity. This
corresponds to a volume increase of the BEC of a factor of 1600, assuming that the2
The change in energy is really very small: for an average excitation energy of 5 nK per atom,
the total energy of an expanding 16000 atom BEC is ∼80 µK or 10−8 eV. As a side note, it is quite
interesting to think about the way that the energy is transferred from power supply to atoms — thetransfer is ultimately mediated by atomic collisions!

56

2000

scattering length (a0)

1500

(a)

1000
500
0
-500
-1000
-5

0

5

10

15

20

25

Time (ms)32
radial BEC width (µm)

28
24
20
16
12
8

(b)

4
0
-5

0

5

10

15

20

25

time (ms)Figure 3.14: Increasing the repulsion to expand the BEC. (a) The scattering length
versus time is shown as the thin solid line. The magnetic trap is turned off near 11 ms,
when the scattering length changes sign. The thick black line represents the probe laser
pulse for the absorption image, for a particular choice of expansion time. (b) Here wecompare the observed expansion of the BEC to the prediction of the PG model. Black
points show the measured rms radial width of the BEC versus time, as determined from a
Gaussian fit. The white points connected by a line show the PG model prediction for the
BEC rms width during the scattering length ramp shown in (a). The small disagreement
between the data and theory is likely due to a systematic underestimate of the width
in the PG model due to the Gaussian approximation for the BEC wavefunction. We
estimate this effect to be of order 15% when the scattering length is large enough thatthe BEC is in the Thomas-Fermi limit.

57
BEC starts near zero scattering length. To emphasize the effect of the expansion on theabsorption imaging, we show an actual image of an expanded condensate in Figure 3.15.
For a simple intuitive explanation of the BEC expansion resulting from the change
in scattering length, one can use a mean-field picture. The equilibrium BEC radius along
a given direction is determined by a balance of forces consisting of outward quantum
pressure due to the curvature of the confined wavefunction, a repulsive mean-field force
that is proportional to the gradient of the atom density, and the confining force of theharmonic trap. When we quickly jump the scattering length to a larger value, theBEC wavefunction retains its shape immediately after the jump, but now the repulsive
mean-field force is much larger. Although the mean-field force increases due to thejump, the confining force from the trap potential remains constant. Thus, the repulsive
and attractive forces no longer balance one another and the BEC will begin to expandin size. Expansion continues until the trap force dominates the mean-field repulsion andcauses the BEC to contract. As a function of time, a periodic oscillation occurs in the
BEC radius, which is called a breathing mode oscillation [5].
To obtain a quantitative understanding of the BEC expansion, we use a model
developed by Pérez-Garcı́a et al. [27], hereafter referred to as the “PG model”. Thismodel involves an analytic variational solution to the nonlinear Gross-Pitaevskii (GP)equation using a Gaussian trial wavefunction. When adapted to our cylindrical trapping geometry, the PG model yields a pair of coupled nonlinear ordinary differential
equations for the condensate widths in the radial and axial directions. We numericallysolve these equations in Maple to predict the BEC expansion (or contraction) for an
arbitrary scattering length ramp. A comparison of the results of the PG model withthe observed BEC expansion rate is given in Figure 3.14. In addition to modellingcondensate excitations, we also utilize the PG model for predicting the equilibrium sizeand density of the condensate for different values of the scattering length.
It is important to note that the PG model is limited to the conditions of appli-

58

280 µ mFigure 3.15: Absorption image of an expanded condensate. Here we used the scattering
length ramp in part (a) of Figure 3.14 to expand the BEC. The false color image
indicates the optical depth as a function of position in the expanded condensate. The
white ellipse represents the equilibrium size of a 16000 atom condensate after formation
at 162.3 G (210 a0 ), as calculated with the PG model (see text). The principal axes ofthe ellipse are equal to 4 × σi , where σi is the Gaussian rms width along a particular
direction. Comparing the BEC to the ellipse, we see how the signal-to-noise ratio of thenumber measurement can be improved by expansion because of the increased numberof pixels in the absorption image. The area of the BEC image is 22 times larger thanthat of the ellipse.

59cability of mean-field theory because the model is based on the GP equation. Anotherlimitation of the model is the assumption of a Gaussian wavefunction for the condensate, which leads to a systematic error in the predicted BEC radii for the strongly
repulsive interaction limit (Thomas-Fermi limit). Nevertheless, the PG model provides
a useful and accessible analytic theory that can interpolate reasonably well between
the non-interacting and strongly interacting limits. One can even model the attractive
interaction regime for the condensate fairly well, as long as the BEC is not on the vergeof collapse.

3.5.4

Low temperature calorimetry: shrinking the BEC

Since evaporative cooling to form Bose-Einstein condensates cannot remove all of
the entropy from the atom cloud, there will always be a significant number of “thermal”
atoms that do not occupy the ground state. The ratio of the number of thermal atoms,
Ntherm , to the total number of atoms, Ntotal = N0 + Ntherm , provides a measure of
the temperature of the sample. Thus, we need to determine both Ntherm and Ntotal ina given experiment. Unfortunately, the mean-field expansion described in the previous
section only provides a method to accurately measure Ntotal ; the expansion method does
not distinguish between BEC atoms and thermal atoms. In fact, the thermal cloud isprobably not static during the mean-field expansion due to coupling between thermal
atoms and the expanding BEC. We observe that the thermal cloud and the condensate
have significant spatial overlap in the absorption images of expanded BECs.
To separate the two components of our BEC samples and accurately measure
Ntherm , we again make use of the tunable self-interaction of the condensate. In this case,we shrink the BEC wavefunction by slowly decreasing the scattering length toward zero.
As a approaches zero, the condensate wavefunction shrinks inward (while the thermal
cloud remains unaffected) until the BEC radii match the harmonic oscillator lengths of√the trap (aho, radial =2.6 µm, aho, axial =4.2 µm). After dividing these widths by 2 to

60
account for the difference between the widths of the BEC wavefunction and the observed
BEC density profile, one finds that the BEC widths are 5 to 8 times smaller than thecorresponding radii of a 10 nK thermal cloud. We can therefore spatially resolve the
BEC from the surrounding thermal cloud “halo”. Our low-temperature calorimetry
procedure and the data from an example absorption image are given in Figure 3.16.
There is a small gloss in the above description of the Ntherm determination.
Shortly after the completion of the adiabatic decrease in the scattering length to shrink
the BEC, we also impart a small outward velocity to the condensate before we turnoff the magnetic trap. Although this seems counter-productive for the goal of spatiallyseparating the two components, it turns out that a small amount of BEC expansion isvery useful for two reasons. First, the expansion velocity decreases the density before
we cross the Feshbach resonance and this reduces density-dependent losses and burst
production (see Chapter 5). Second, the outward velocity of the BEC prevents the condensate from collapsing at B=0, which also produces a burst of hot atoms (Chapter 4).
Whether the burst atoms are generated during the magnetic trap turn-off or during thebrief delay time at B=0, the presence of the burst atoms can potentially distort the thedetermination of Ntherm .
In practice, we determined the parameters of the expansion ramp by finding thebest compromise between minimum spatial size of the BEC and minimum number ofburst atoms created. The best expansion ramp consisted of a 1 ms linear B-field ramp
to 161.5 G (a '290 a0 ) followed by a 2.4 ms hold. The trap was then turned off andan absorption image was taken 1.6 ms later. This ramp was used to obtain the image
shown in part (b) of Figure 3.16 (to avoid confusion and because of its short duration,
the ramp was not shown in part (a)). Our sensitivity limit to thermal atoms with thecondensate shrinking technique is very low — roughly 500 thermal atoms.

61

scattering length (a0)

400
200
0

(a)

-200
-400
-600
0

200

400

600

800

1000

Time (ms)
3.5
3.0

optical depth

2.5
2.0

(b)

1.5
1.0
0.5
0.0
-0.5
0

10

20

30

40

50

60

70

80

horizontal pixel

Figure 3.16: Shrinking the BEC to extract the thermal fraction. (a) Scattering length
versus time and probe laser pulse. The scattering length is adiabatically (∼0.5 s) ramped
down to a value near zero, then the magnetic trap is turned off. We take an image after
a short delay of 1.6 ms to allow induced currents to die away. The slow decrease in acauses the BEC to contract as the mean-field repulsion vanishes. (b) Absorption image
axial cross-section for a BEC plus thermal cloud after the scattering ramp shown in(a). Black points are the measured optical depth versus pixel position on the camera.
One pixel length equals 2.54 microns. The dashed line and solid line are simultaneous
Gaussian fits to the condensate and thermal cloud, respectively. From the thermal cloud
fit, we find Ntherm ' 5000 atoms. The condensate fit is subject to systematic effects(resolution limit and optical depth ceiling), so we do not use this to determine N0 .
When we combine the accurate Ntherm measurement with a subsequent measurementof Ntotal ' 16000 as in section 3.5.3, the BEC temperature is quite well-known.

62
3.5.5

Making bigger and better condensates

As a result of our clever schemes to manipulate the BEC wavefunction, we havebeen able to produce larger condensates with smaller thermal fraction than we could
previously [1]. The ability to accurately measure both Ntotal and Ntherm for each cloudof atoms enabled us to re-optimize the final stages of evaporative cooling. We adjusted
the trajectory of the rf frequency versus time to minimize the thermal number while
maximizing the total number in the condensed sample. As a result, in our day-to-day
experiments we typically work with Ntotal =16,000 and Ntherm ≤500, which corresponds
to a condensed fraction of 97 %. This represents a dramatic improvement over ourearliest BEC experiments in August 1999, when the optimum value for the condensed
fraction was roughly 10 %. The very best condensates created in our experiment had
Ntotal =30,000 with fewer than 500 thermal atoms. As we have not been able to reproducibly obtain such low temperature condensates, we suspect there may have been aparticularly advantageous planetary alignment at that time.

Chapter 4
Dynamics of collapsing and exploding 85 Rb Bose-Einstein condensates

4.1

Introduction
This Chapter presents a description of our experiments with 85 Rb BEC in theattractive self-interaction regime, where the condensate may become unstable and undergo a “collapse” if the attraction is sufficiently strong. The first part of the Chapter
provides some background information about BEC collapse physics, including our earlyexperiments to study them. We move on to a detailed description of the response of
the BEC when the sign of the scattering length is quickly switched from positive tonegative. At the end of the Chapter, we compare our data to recent theoretical models
for the collapse dynamics.
The 85 Rb BEC system is an ideal candidate for studying the collapse instability
because of the condensate’s tunable interactions. By ramping the magnetic field, we
can easily change the sign and magnitude of the scattering length (and interaction
strength). As discussed in Chapter 2, the BEC self-interaction is repulsive when the
scattering length is positive, and attractive when a is negative. The existence of the
85 Rb Feshbach resonance, which leads to a magnetic-field-dependent scattering length,

allowed us to gain tremendous insight into the collapse physics.
Before our experiments in 85 Rb, condensates with attractive interactions had received only limited study in a 7 Li BEC system with fixed negative scattering length.
The early 7 Li BEC studies [28] showed evidence for an instability, but the data did

64
not allow for stringent tests of theory. More recent experiments in 7 Li have focusedon the collapse process in a high temperature condensate, where the BEC was continually replenished by a cold thermal cloud, leading to a sequence of growth and collapse
cycles [29, 30]. In contrast, our 85 Rb BEC experiments were conducted with nearly
thermal-atom-free condensates, and we took full advantage of our experimental controlof the scattering length to extract more information about the collapse process.

4.1.1

Stability condition for BEC with a<0

A stable, harmonically trapped condensate with negative scattering length has amaximum or critical allowed number of atoms, Ncrit . When the number in the condensate exceeds the critical number, NBEC >Ncrit , the attractive self-interaction dominates
the repulsive zero-point energy of the confined condensate wavefunction. The stability
condition for the BEC is defined as
Ncrit |a|= kcollapse ,
aho

(4.1)

p
~/(mω) is the mean harmonic oscillator length of the trap potentialp(with mean frequency ω = 3 ωr2 ωz ) and kcollapse =0.55 is a constant [31]. One canwhere aho =also view equation (4.1) as defining a critical value for the magnitude of the negative
scattering length, acrit , if the number of BEC atoms is assumed to be fixed. The stability
condition expressed in equation (4.1) was determined by studying the ground state of
the GP equation as a function of the product N |a| [31, 32]. When this product exceeds
the stability condition, then no stable solution can be found for the ground state. Thus,
one expects an initially stable BEC that experiences a change in scattering length suchthat (ainit > 0) → (af inal < acrit ) to become unstable and collapse in some fashion.
In our early experiments [33], we found that an initially stable BEC undergoes adramatic (∼ 50%) number loss some time after the scattering length is decreased below a
critical value. Typically, we label this number loss as the BEC “collapse”. By measuring

65
the critical number of atoms for collapse at several different values of negative scattering
length, we verified the functional form of equation (4.1). We also determined the valueof kcollapse , which was significantly lower than the prediction of existing theory [32].
However, our more recent high-precision measurement of the Feshbach resonance parameters (see Chapter 7) has allowed us to revise our previous estimate of kcollapse . The
new value is completely consistent with theory [31].
The collapse of the condensates was an all-or-nothing process. We never observed
partial collapses; a given BEC would either collapse at negative a or remain stable,
depending on whether or not the number exceeded Ncrit . We saw no evidence fora macroscopic quantum tunneling phenomenon in the collapse process, as some theory
work had suggested(see Ref. [34] and references therein). In fact, collapse process turned
out to be so reproducible that we could use it to precisely measure the magnetic field.
We accomplished this by first measuring Ncrit and acrit and then using our knowledgeof the variable scattering length due to the Feshbach resonance.
On a daily basis, we found the initial magnetic field using a fairly simple procedure. We first prepared a stable BEC at some initial value of B-field that corresponded
to a positive scattering length. We then ramped the B-field to change the sign and magnitude of the scattering length to a negative value near acrit . We observed the resulting
BEC number and noted which final magnetic fields (final scattering lengths) caused
large BEC number loss. After determining Ncrit and acrit in this way, we found theabsolute magnetic field using equation (4.1) in combination with the known dependenceof scattering length on magnetic field. We extrapolated from Bcrit to Bzero , the field
where the scattering length vanishes (corresponding to Ncrit = ∞). The value of Bzero
provided us with a useful calibration point.
Utilizing the BEC collapse to measure the magnetic field proved to be a rapid
and precise technique. We repeated the experimental cycle with different B-field ramps
until Ncrit and acrit were known. In a period of about one-half hour, we could determine

66
the initial magnetic field to a ∼5 mG precision. Precise knowledge of Binit and ainit
was quite useful for a variety of other experiments described in this thesis.

4.1.2

Overview of collapse dynamics

Although our early BEC experiments in the attractive self-interaction regime
verified the stability condition and showed that condensate collapse involves number
loss, our data actually raised more questions than they answered. For instance, we
could not initially determine the fate of the atoms that left the BEC. We had otherunanswered questions regarding the details of the number loss: how quickly were theatoms going away and what was setting the timescale of the collapse? Furthermore,
what exactly happened to the BEC wavefunction during the collapse process?
At the time of our experiments, existing theories of BEC collapse [35, 36, 37]predicted that an unstable BEC would implode on itself. The contraction of the BEC
would presumably take place on the time scale of a trap oscillation, and the density
would sharply increase after Trad /4 ' 14 ms, where Trad is the radial trap period. As
a result of the density growth, 3-body recombination losses were expected to cause a
rapid depletion of the BEC, preventing the formation of a true singularity in the densityof the atom cloud.
To explore the dynamics of the collapse process and check the validity of the
proposed theoretical pictures, we took full advantage of the variable scattering length of
85 Rb condensates. We changed the sign of a and observed the time-dependent responseof the BEC. We were quite surprised by the complexity of the BEC collapse. Various
phenomena were observed — an initial gradual contraction was followed by a supernovalike explosion of hot atoms. At the end, a small remnant condensate was left behind in an
excited oscillatory state, and much of the initial condensate had completely disappeared
from view. In the rest of this chapter, we detail the prominent features of the collapse
process as well as the methods used to study them.

67

4.2

Preparing and manipulating the condensate
The first stage in our experiment was to form a stable 85 Rb condensate[26]. A

standard double magneto-optical trap (MOT) system[19] was used to collect a cold
sample of 85 Rb atoms in a low-pressure chamber. Once sufficient atoms had accumulated
in the low-pressure science MOT, the atoms were loaded into a cylindrically symmetriccigar-shaped magnetic trap with frequencies νradial = 17.5 Hz and νaxial = 6.8 Hz.
Radio-frequency evaporation was then used to cool the sample to ∼3 nK to form purecondensates containing >90% of the sample atoms. The final stages of evaporation were
performed at 162.3 G where the scattering length is positive and stable condensates of
up to 20,000 atoms could be created. After evaporative cooling, the magnetic field
was ramped adiabatically to 166 G, where the scattering length was near zero. This
provided a well-defined initial condition — the BEC would shrink to approach the sizeand shape of the harmonic oscillator ground state (see section 3.5.4). We note that in a
few cases, the initial scattering length was chosen to be significantly different from zero;
this allowed us to vary the initial density of the condensates over a wide range.
We developed various different magnetic field ramps into useful tools for studying
the BEC collapse dynamics. The B-field control allowed us to adjust the mean-fieldinteractions within the BEC to a variety of values on time scales as short as 0.1 ms. The
obvious manipulation was to jump to some value of a < acrit to trigger a collapse, but
the tunability of a also greatly aided in imaging the sample. Usually the condensate size
was below the resolution limit of our imaging system (7µm FWHM). However, we could
ramp the scattering length to large positive values and use the repulsive interatomic
interactions to expand the BEC before imaging, thus obtaining information on the preexpansion condensate shape and number. A typical a(t) sequence is shown in Figure 4.1.
We have used a variety of such sequences to explore many aspects of the collapse andenhance the visibility of particular components of the sample.

68

2000

scattering length (a0)expand

1500
1000
500
tevolve0

init
quenchcollapse

-500
0

5

10

15

20

time (ms)Figure 4.1: An example of experimental control of the scattering length. The thinsolid line is the time dependent scattering length while the thick vertical line indicates
the timing of the probe laser pulse for absorption imaging. We used a variety of a(t)sequences similar to what is shown here to study the dynamics of the BEC collapse. To
determine a(t), we first calculated the magnetic field using a calibrated current sensor
on the magnetic trap coil. Then we converted the B-field into scattering length using
equation (2.2), the Feshbach theory expression for a(B).

69
In Figure 4.1, the scattering length is jumped at t = 0 in 0.1 ms from ainit ' 0
to acollapse < 0, where the BEC evolves for a time tevolve . We carefully control the
B-field so that magnetic field noise translates into fluctuations in acollapse on the orderof ∼ 0.1 a0 in magnitude. After the time tevolve , the BEC collapse process is theninterrupted with a jump to aquench , and the field is ramped in 5 ms to a large positive
scattering length which makes the BEC expand. After 7.5 ms of additional expansion,
the trap is turned off in 0.1 ms and 1.8 ms later the density distribution is probed using
destructive absorption imaging with a 40 µs laser pulse (indicated by the vertical bar in
Figure 4.1). In some instances, we increased the expansion time after the trap turn-off
to further expand the BEC wavefunction.
As discussed in section 3.5.3, the increase in a from acollapse to aexpand is far too
rapid to allow for the BEC to expand adiabatically. On the contrary, the smaller the
BEC before expansion, the larger the cloud at the moment of imaging. Thus we canreadily infer the relative size of the BEC wavefunction just prior to the jump to aquench .
After the mean-field expansion, the density of the expanded BEC is so low that the
rapid transit of the Feshbach resonance [38] during the trap turn-off and the subsequent
time spent at B = 0 (a = −470 a0 ) both have a negligible effect.

4.3

BEC number loss

4.3.1

BEC number versus tevolve

When the scattering length is jumped to a value acollapse < acrit , the BEC kinetic
energy no longer provides a sufficient barrier against collapse. After making such ascattering length jump, we observed a delayed and abrupt onset of number loss fromthe condensate. As a function of the time spent at a < 0, the BEC number remained
steady for a while, then dropped exponentially with a time constant τdecay . This trend
is illustrated by Figure 4.2, which is a plot of the condensate number NBEC versus the70evolution time, tevolve at a negative scattering length. Here the observed delay time
before the onset of number loss is defined as the collapse time, tcollapse . The exponential
number decay finally leveled off at a value Nrem , which we define as the remnant BEC
number. Nrem was typically ' 40% of the initial number, Ninit .
Although NBEC was constant for a short time after the jump to acollapse , we didobserve some slight changes to the expanded condensate radius during tevolve . Theseradius changes indicated that the BEC was contracting during the evolution time. Since
the pre-expansion BEC was smaller than our resolution limit, we could not observe thecontraction directly; however, we could infer the extent of the contraction by observing
the degree of mean-field expansion. In fact, the BEC expanded more and more during
the expansion ramp of Figure 4.1 as it contracted more and more during tevolve . We
actually observed that the post-expansion condensate widths changed very little withtime tevolve before tcollapse . For the data in Figure 4.2, the expanded radial BEC width
increased by only 10% as tevolve increased from 0 ms to 3.5 ms. The axial BEC width
showed even less of a change.
From this we infer that the cloud of condensate atoms did not contract dramatically before loss began. To obtain a quantitative estimate for the amount of contraction,
we use the equations in Ref. [27]. The PG model results provide a fairly good match tothe observed expanded BEC shape over a moderate range of acollapse and time before
collapse. We can therefore have some confidence in using the equations of Ref. [27] toestimate the density before collapse. Surprisingly, we find that the predicted contraction corresponds to only a 50% increase in the average density (to 2.7 × 1013 cm−3 )before the beginning of the number loss. Using the inelastic collision rates for 85 Rb that
we measured in Ref. [39], this density predicts a loss time that is far longer than what
we observe (τinelastic ∼2.5 s) and does not have the observed sudden onset. Thus, oursimple estimates of the BEC contraction and inelastic loss do not jibe with the observed
behavior in Figure 4.2. The disagreement implies that the conventional simple picture

71

collapse

BEC number

16000

Ninit

12000
8000
Nrem

4000
0

2

4

6

8 10 12 14 16 18 20 22

tevolve (ms)Figure 4.2: The condensate number versus time after a jump to a negative scattering
length. Points are the number of atoms remaining in the BEC versus tevolve at acollapse =−36 a0 . The initial scattering length before the jump was ainit = +8 a0 and the initial
number of condensate atoms was Ninit =16,000. These conditions produced an estimatedaverage initial density of 1.8×1013 /cm3 . We observed a delayed and abrupt onset of loss
from the BEC number. The solid line is a fit to the data using a delayed exponentialfunction with a best fit value of tcollapse ' 3.7(5) ms for the delay. The number of atoms
remaining in the condensate after the end of the decay process is Nrem ∼ 5000.

72
of the collapse as a global implosion of the entire BEC wavefunction is incorrect. We
will provide additional discussion of this point in section 4.5.3.
For the data in Figure 4.2 and most other data presented in this Chapter, we
jumped to aquench = 0 in 0.1 ms after a time tevolve at acollapse . We believe thatthe number loss stopped immediately after the jump. This interpretation is based on
the surprising observation that the quantitative details of curves such as that shownin Figure 4.2 did not depend on whether the collapse was terminated by a jump toaquench = 0 or aquench = 250 a0 . Presumably, if there were a significant amount of loss
occurring after the “quenching” ramp, then quantities such as Nrem and tcollapse should
depend on the precise value of aquench ; however, this was not the case.
4.3.2

Variations in tcollapse

We have measured number decay curves like that in Figure 4.2 for many different
values of the negative scattering length. The measured collapse time versus |acollapse|
for fixed initial density is presented in Figure 4.3. As we increased |acollapse | from 7 a0
to 65 a0 , the collapse time decreased by more than one order of magnitude. The data
display a nearly perfect inverse dependence on the magnitude of the scattering length,
as shown by the solid line.
We also studied the dependence of tcollapse on the initial BEC density. By reducing
Ninit , we decreased the initial density from hninit i = 2.8 × 1013 cm−3 to hninit i =1.1 × 1013 cm−3 (a factor of ∼ 2.5). This change caused the collapse time to lengthen
by a factor of 2.4(8) at acollapse =-8 a0 . We verified that tcollapse was truly proportional
to density and not only to Ninit by reducing the density through a volume change tothe initial condensate. Holding NBEC fixed at roughly 16,000 atoms, we decreased theinitial density by a factor of 3.5 (with a corresponding increase in volume) by setting
ainit = +105 a0 instead of our more typical value, ainit = +8 a0 . At acollapse = -15 a0 ,the volume-induced density change caused tcollapse to grow by roughly a factor of 3.

73

14
12

tcollapse (ms)

10
8
6
4
2
0
0

20

40

60

|acollapse| / a0
Figure 4.3: The collapse time versus acollapse for 6000 atom condensates. Points witherror bars are the measured values of tcollapse . The solid line is a weighted fit to thefunction: t = Q/acollapse , with Q=57(6) ms a0 . The vertical dashed line indicates acritfor N0 = 6, 000. The data were acquired with ainit = aquench = 0(2) a0 , so that theinitial BEC density was hninit i = 3(1) × 1013 cm−3 . The average BEC densities given
here were computed using the PG model (section 3.5.3) to determine the volume of
the BEC wavefunction for a given value of the initial scattering length. We found that
the PG model accurately predicts the mean density over the entire range of repulsive
interactions, from the Thomas-Fermi limit of strong interactions to the non-interacting
limit.

74
The observed dependences of the collapse time on scattering length and density
indicate that there is some critical condition that must be reached before the onset of
number loss. The time to reach the critical condition becomes smaller as we increasehninit i and |acollapse |. All of our evidence supports the idea that tcollapse is inversely
proportional to the magnitude of the attractive mean-field interaction: Emf ∝ hnia.
This dependence seems reasonable since the rate of implosion of the BEC should dependon Emf .
In contrast to tcollapse , the atom loss time constant τdecay depended only weakly onacollapse and hninit i. For the range of acollapse shown in Figure 4.3, τdecay did not dependon acollapse or hninit i by more than the ∼20% experimental noise in the τdecay determination. On average, τdecay was 2.8(1) ms. For the very negative value of acollapse ' −290 a0 ,however, τdecay did decrease to 1.8 ms for Ninit = 6, 000 and 1.2 ms for Ninit = 15, 000.

4.4Energetic burst of atoms
Thus far we have not explained what happens to the atoms after they exit thecondensate during a collapse. As displayed in Figure 4.2, a large fraction of the atoms
leave the BEC during the exponential decay. There are at least two components tothe expelled atoms. One component (known as the “missing atoms”) is not detectedin the absorption images. The other component emerges as a burst of detectable, spinpolarized atoms with energies much greater than the initial condensate’s energy but
much less than the magnetic trap depth. The bizarre properties of the burst and its
surprising formation during the BEC collapse prompted Carl Wieman and Eric Cornell
to name it a “Bosenova”, in analogy with a supernova. In our attempts to understandthe origins of the burst, we have studied the details of the burst atom distribution undera wide range of conditions. The average energy and number of the burst atoms depend
in a complex manner on acollapse and Ninit . Although the details are complicated, we
present them here because they provide a stringent test of collapse theories.

75
The angular kinetic energy distribution with which the burst atoms are expelledfrom the condensate can most accurately be measured by observing the harmonic oscillations of the atoms in the trap, as illustrated in Figure 4.4(a). For example, one-half ofa radial period after the expulsion (Trad /2), all burst atoms return to their initial radial
positions. At times significantly before or after this radial “focus”, the burst cloud is
too dilute to be observed. Fortunately, at the radial focus, the oscillation trajectories
along the axial trap axis are near their outer turning points. Therefore the axial energy
can be found from the length of the stripe of atoms along the axial direction. The radial
energy can be found with an identical procedure in an axial focus.
One can also define the “sharpness” of the burst focus as the minimum width
in the narrow dimension. The harmonicity of the magnetic trap potential ensures thatthe sharpness of the burst focus should only depend on the duration of time that burst
atoms are expelled from the BEC (the length of the explosion) and the initial spatial
distribution of the atoms before the explosion. In principle we can therefore determine
the formation time for the burst atoms by measuring the narrow width of the burst focus.
However, this technique only allows us to set an upper limit on the burst creation time
because of significant broadening of the images from our finite optical resolution limit.
The upper limit from the burst focus sharpness is consistent with a direct measurementof the burst generation time (∼1.2 ms) in section 4.4.1.
Figure 4.4(b) shows an image of a radial focus. The size scales for the burst focus
and the remnant were always well separated since the latter was not expanded when we
imaged the burst. Figure 4.4(c) shows axial cross sections of the image in part (b) andthe fit cross sections for the burst and thermal clouds. The burst spatial distribution
was well fit by a 2-D Gaussian function whose width in the long dimension corresponded
to the average energy of the burst atoms.
Since the burst atom cloud extended far out into the edges of the absorption
images and the optical depth was fairly low in the outer wings of the atom distribution,

76

t = tburst + Trad /2

t = tburst
r

(a)
z

remnant(b)
burst focus

averaged optical depth

0.4

(c)

0.3

0.2

0.1

0.0
0

50

100

150

200

250

300

position (µm)

Figure 4.4: A burst focus. (a). Conceptual illustration of a radial burst focus. (b) An
image of a radial burst focus taken 33.5 ms after a jump from ainit = 0 to -36 a0 for
N0 = 15, 000. Here Trad /2 = 28.6 ms, which indicates that the burst occurred 4.9(5) ms
after the jump. The average axial energy for this burst was hEburst,axial i = 62 nK. The
image dimensions are 60 × 310 µm. (c) Radially averaged cross section of the imagein (b) along with a Gaussian fit to the burst spatial distribution. The central 100 µm
were excluded from the fit to avoid fitting distortions from the condensate remnant
and the thermal cloud. The rms widths of the BEC remnant and thermal cloud wereσ = 9µm and σ = 17µm, respectively. A small cloud of thermal atoms is present in the
pre-collapse sample due to the finite temperature; these atoms appear to be unaffectedby the collapse. The dashed line indicates the fit to this initial thermal component.
There is a clear offset between the centers of the burst and the remnant. This offset
varies from shot to shot by an amount comparable to the offset shown.

77
we utilized an image-processing tool to reduce background noise and improve the image
signal-to-noise ratio. Much of the “noise” in the absorption images is actually due tointerference fringes from the coherent probe laser beam. We could therefore improve
the fitting procedure by removing the periodic stripes from the images. The removal
was done by calculating the spatial Fourier transform of the optical depth data, thenremoving the probe laser stripes from the image, and finally transforming back to real
space. In this way, we could obtain reliable measurements of the average burst atomenergy along the radial and axial directions.
The average energy was generally different for the two trap directions. The burst
energy fluctuated from shot to shot by up to a factor of 2 for a given acollapse . Thisvariation is far larger than the measurement uncertainty or the variation in initial number (both ∼ 10%), and its source is unknown. The burst energy and the dependencesof hEburst i on Ninit and acollapse will be described in section 4.4.2.
4.4.1

Nburst versus tevolve

One of the most interesting aspects of the burst behavior was the increase inNburst during the collapse process. When we interrupted the BEC collapse with a jumpback to aquench = 0 (see section 4.3.1), we also interrupted the growth of the burst.
The “interrupted” burst atoms still refocused after sitting at a = 0 for the requisite
one-half of a trap period. We found the energy of the atoms in the interrupted bursts
to be the same, but the number of atoms was smaller than in the uninterrupted case.
By changing the time at which the collapse was interrupted we could measure the timedependence of the creation of burst atoms.
The increase in the burst number with tevolve is illustrated in Figure 4.5. The
number of burst atoms Nburst grows exponentially with a time constant of 1.2 ms,
starting at tevolve =3.5 ms and reaching an asymptotic final number of ∼2500 atoms for
all times ≥7 ms. Nburst also levels off at roughly the same time as the remnant BEC

78
number displayed in part (b) of the figure. Despite the observed large variations in
the burst number, the average energy of the burst atoms (∼ 85 nK) does not change
significantly over the range of tevolve shown in Figure 4.5.
The interrupted burst data clearly show that the growth of the burst with time atnegative scattering length mirrors the decay of atoms from the condensate — the timedependences for these two processes are quite similar. However, by comparing parts(a) and (b) of Figure 4.5, one can see that the number of burst atoms is far smaller
than the number of atoms leaving the BEC — there are at least 11000 − 2500 = 8500
missing atoms. At present, the fate of the missing atoms has not been determined byexperiment. The prevailing hypothesis is that the missing atoms are due to 3-bodyrecombination into molecules, where the molecular binding energy is converted into
kinetic energy that is shared by the collision partners. We discuss the scattering length
dependence of the missing fraction of atoms in section 4.5.2.
Somewhat surprisingly, the number of atoms in the burst did not depend on
the scattering length, acollapse . Although Nburst varied randomly by ∼20% as |acollapse |increased from nearly 0 a0 to 350 a0 , on average the fraction of atoms going into the burst
remained fixed at roughly 20% of the initial BEC number. Larger initial condensates
produced larger bursts in direct proportion, so that the burst fraction remained constantas a function of Ninit .
4.4.2

hEburst i versus acollapse

We carefully examined the scattering length dependence of the burst energy as
one part of our collapse dynamics experiment. Although the burst energies were seen
to vary from shot to shot, the average value for the energy was well defined and showedlarge-scale trends that dominated the shot-to-shot variations. We show the axial andradial burst energies versus acollapse in Figure 4.6(a) and 4.6(b) for Ninit = 6, 000 and
15,000, respectively.

79

number of burst atoms

3000
2500
2000
1500
1000

(a)

500
0
0

2

4

6

8

10

12

tevolve ( ms )20000

BEC number

(b)
15000
10000
5000
0
0

2

4

6

8

10

12

tevolve ( ms )Figure 4.5: Growth of burst number with tevolve . The data were collected under thesame conditions as for Figure 4.2. (a) The points are Nburst versus evolution time atacollapse =-36 a0 . The solid line shows the best fit of the data to a rising exponentialfunction with time constant τ =1.2(3) ms. The dashed vertical line indicates the best fit
value for tcollapse from a fit to the BEC loss data in part (b). Figure (b) displays the
BEC number decay for comparison to the burst growth data. The burst growth begins
and ends at about the same time as the condensate loss. The time constants for BEC
loss and burst growth are also comparable: τBEC =2.7(3) and τburst =1.2(3) ms.

80

average kinetic energy (nK)

200

(a)
150

100

50

0
0

100

200

300

400

300

400

|acollapse|/a0

average kinetic energy (nK)

500

(b)
400
300
200
100
0
0

100

200
|acollapse|/a0

Figure 4.6: Average burst atom energy versus scattering length. (a) and (b) show theaverage axial and radial burst energies versus |acollapse | for Ninit ' 6, 000 and Ninit '15, 000, respectively. Filled points are the radial burst energies; open points are the axial
energies. The average energy is calculated from the rms width, σ, of the 2-D Gaussian
fit to the data as shown in Figure 4.4. For a given dimension, the average energy is
hEburst, i i = mωi2 σ 2 /2, where ωi is the trap frequency in the ith dimension. On average,ten burst focuses were measured for each trap direction at each value of acollapse studied.
The vertical and horizontal error bars indicate the standard error of the measurements
and the uncertainty in acollapse arising from the magnetic-field calibration, respectively.
For some of the points, the uncertainties are comparable to the symbol size.

81
The most striking feature of the data is the strong dependence of the averageenergy on the number of condensate atoms. Comparing (a) and (b), one sees that for
fixed |acollapse | ' 30 a0 , the radial and axial burst energies increase by a factor of ∼3as the initial condensate number increases by 2.5×. In fact, the energies were higher
for the larger Ninit condensates over the full range of acollapse studied. An additional
observation is the overall trend of increasing energy with increasing |acollapse | in the low
Ninit data (Figure 4.6(a)). However, the high Ninit data in part (b) display much more
complex behavior that is not monotonic with acollapse .
We also note that over much of the scattering length range shown in Figure 4.6
there is a significant anisotropy in the burst energy. The radial energy is always greaterthan or equal to the axial energy. We discovered that the burst energy anisotropy
depended on Ninit , acollapse , and ainit . The anisotropy showed a peaked behavior for15000 atom condensates, with maximum near |acollapse | ∼100 a0 . For values of |acollapse |only slightly larger than |acrit |, the burst was isotropic for both Ninit ' 6, 000 and
Ninit ' 15, 000. At larger values of |acollapse |, condensates with larger number gave rise
to stronger anisotropies.
We cannot rule out a possible dependence of the burst energy anisotropy on
the initial aspect ratio of the cloud, λ. We define λ as the ratio of the axial andradial rms widths of the BEC: λ = σz /σr . For the data with Ninit = 6000, the initial
scattering length was ainit ' 0 a0 , leading to λ = 1.6. In contrast, the high number
data with Ninit = 15,000 had ainit ' +8 a0 and λ = 2. When instead we started atainit = +100 a0 (Ninit =15,000), the BEC was initially more anisotropic (λ = 2.4), but
the burst became more isotropic, with hEax i going up by ∼40% and hErad i droppingby ∼60% at acollapse = −100 a0 . These various results for the burst energy and theenergy anisotropy are clearly very complicated. It is impossible to say how much of theobserved differences between the low number and high number data in Figure 4.6 aredue to the change in Ninit because we also changed the initial scattering length between

82
these two data sets. Nevertheless, the measured burst energy dependences can be used
to test the validity of several mean-field models that were recently proposed to explain
our results (see section 4.7).

4.5

Remnant BEC

4.5.1

Nrem exceeds Ncrit

After each collapse, a “remnant” condensate containing a fraction of the atoms
survived with nearly constant number for more than one second. The remnant BEC
oscillated in a highly excited collective state with the two lowest breathing modes atν ' 2νaxial and ν ' 2νradial being predominantly excited. The measured frequencies
were ν = 13.6(6) Hz and ν = 33.4(3) Hz. To find these oscillation frequencies, wemeasured the widths of the condensate as a function of the time spent at acollapse . As
usual, we utilized a mean-field expansion of the BEC to measure the widths. Whilethe expansion leads to large increases in the widths, it does not distort their periodic
modulation in time.
The number of atoms in the remnant depended on acollapse and Ninit , and ingeneral was not limited by the critical number, Ncrit . Although the stability conditionin equation (4.1) certainly determined whether or not the condensates experienced acollapse event [33], the number of atoms left behind was often significantly larger thanNcrit . This behavior is shown in Figure 4.7, where we plot the remnant number as afunction of scattering length.
In Figure 4.7, the 6000 atom data do mostly lie on or below the line showing
the value of the critical number. In fact, for Ninit = 6, 000 and |acollapse | < 10 a0 ,more atoms were lost than the number required to lower Nrem below Ncrit . Since
we did not take the large Ninit data until much later, we initially believed that Ncrit
truly did constrain Nremnant . However, the agreement observed in the Ninit =6000 data

83

7000

Ninit=6000

remnant number

6000

Ninit=15000

Ncrit

5000
4000
3000
2000
1000
0
10

100

| collapse |/ 0
Figure 4.7: Remnant BEC number versus initial BEC number. The dashed line isthe critical number of BEC atoms as calculated from the stability equation (4.1) withkcollapse =0.55 [31]. The black points and white points are the values of Nrem for different
initial number, as explained in the legend. Note that the Ninit =6000 remnant data
(black points) correspond to the data we used for determining the collapse time as afunction of scattering length in Figure 4.3.

84
turned out to be entirely coincidental; when we increased the initial number to ∼16,000
atoms, the remnant number always exceeded Ncrit for the entire range of |acollapse | that
we investigated. The fraction of initial number of atoms that went into the remnant
decreased with |acollapse |, and was ∼40% for |acollapse | < 10 a0 and ∼50% for |acollapse | >100 a0 .
We found that a fixed fraction of Ninit went into the BEC remnant independentof the value of Ninit , so that smaller condensates often ended up with Nrem < Ncr , butlarger condensates rarely did. After discovering this surprising result, we studied theremnant dependence on initial number in the condensate. Figure 4.8 demonstrates the
direct proportionality between Nrem and Ninit for fixed scattering length. Increasing
the value of |acollapse | merely shifts the proportionality constant downward — the linear
dependence remains.
The observation that the remnant number routinely exceeded the critical number is surprising, but not impossible to explain. One reason that we might expect thisbehavior is the highly excited state of the remnant. Due to the excitations and large
number losses, it is difficult to know the proper temperature of the cloud. It is possiblethat the remnant is not actually a condensate, although the observed mean-field expansion of the atom cloud indicates that the phase space density is comparable to unity.
Another possible explanation for Nrem exceeding Ncrit is an inherent stabilizing natureof the excitations. Pattanayak et al. [40] have shown that a BEC with a particular typeof collective excitation known as a quadrupolar shape oscillation can impart enoughangular momentum the BEC to provide a centrifugal barrier to collapse. The excitation can stabilize a condensate by creating an additional quantum pressure beyond thatprovided by the confining trap potential. However, Ref. [40] also predicts that radial
breathing mode excitations should destabilize a BEC with attractive interactions. Since
we have observed breathing mode oscillations in the condensate widths, the Pattanayak
et al. mechanism for stabilization is unlikely to explain our results. It would be inter-85-25 a0-36 a06000
remnant number-300 a0
Ncrit for -25 a0

4000

2000

0
0

3000

6000

9000

12000

15000

initial BEC number

Figure 4.8: Dependence of remnant number on initial BEC number. The points are themeasured values of Nrem versus Ninit . The legend lists the various different values of|acollapse | that were used. The dashed line shows the critical number for the smallest
relevant value of |acollapse |=25 a0 . Most of the data clearly lie above this stability line.
For the data with |acollapse |=25 a0 , we found no discernible dependence of Nrem on
initial scattering length when ainit was varied between −1 a0 and +6 a0 .

86esting to return to the issue of the BEC excitations and the apparent stabilization of
the remnant in the future. One observable that might give useful information, but was
not yet studied carefully is the excitation amplitude of the remnant cloud. Perhaps we
could determine whether the oscillation amplitude is bigger for the collapse events inwhich Nrem greatly exceeds Ncrit .
4.5.2

Missing atoms

In the collapse process, a large fraction of the initial BEC disappears. The missing
atoms do not show up in the absorption images of the post-collapse BEC. Although
we cannot yet say what happens to these expelled atoms, we are able to discuss thedependence of the missing atoms on scattering length and initial number because weknow the corresponding trends for the burst and remnant numbers. Since Nburst was
independent of |acollapse | but Nrem decreased with |acollapse |, the number of missing
atoms grew larger with |acollapse |.
Interestingly, the absolute number of missing atoms also increased with Ninit .
The increase was proportional to the initial number so that the missing fraction dependence on |acollapse | was identical for both Ninit = 6, 000 and Ninit = 15, 000. For|acollapse | < 10 a0 , we measured a missing fraction of ∼40%, while the fraction was∼70% for |acollapse | ≥ 100 a0 . The missing atoms were presumably either expelled fromthe condensate at such high energies that we could not detect them (> 1 µK), or they
were transferred to untrapped atomic states or undetectable molecular states.
We made an attempt to test the hot atom hypothesis for the missing atoms byturning off the magnetic trap immediately after a collapse and then recapturing theatoms in the MOT. Because the MOT trap depth of order ∼1 K is much deeper thanthat of the magnetic trap (∼100 µK), the recapture process should collect atoms withmuch higher energy than we could ever observe with absorption imaging. An additional
advantage of the MOT is its large trap volume, which greatly exceeds the magnetic

87
trap volume. However, recapture in a MOT and subsequent fluorescence detection
suffers from two major defects that we could not surmount. The first defect is the poorsensitivity of fluorescence detection versus absorption imaging. For small atom samples,
there simply is not much light scattered by the atoms for detection. The sensitivity
problem can be solved by careful design of a high-gain photodiode circuit. However,there is another problem with the MOT recapture, namely, the large background signal
due to hot “Oort cloud” atoms in the magnetic trap. The Oort cloud is thought toform during the evaporative cooling process and to interact only very weakly (if at all)with the BEC [41]. In fluorescence detection, the Oort cloud atom signal swamps the
BEC signal due to a difference in the numbers of atoms of many orders of magnitude.
As a result, we could not make a quantitative search for the missing atoms with themagneto-optical trap.

4.5.3

Strange density patterns: jets

Under certain experimental conditions, we observed streams of atoms with highly
anisotropic velocity distributions emerging from the collapsing condensates. Atoms in
the “jets” possess distinctly different properties from the burst atoms. First, the jets
have much lower kinetic energy than the burst (a few nK compared to ∼100 nK). The
average velocity of the jet atoms is nearly purely radial, so that the energy anisotropy
is at least 10. In addition, we have discovered that the jets appear only when the
BEC collapse process is interrupted (i.e., by jumping to aquench = 0) during the periodof BEC number loss. When the collapsing condensate was allowed to complete thecollapse process so that NBEC →Nrem , no jets were emitted from the BEC. In contrast
to this behavior, the burst atom explosion always seemed spontaneous; bursts would
form whether or not we quenched the collapse by quickly ramping to zero scattering
length.
Examples of the strange jet density patterns we have observed are shown in88
Figure 4.9 for different values of tevolve and the same conditions as Figure 4.2. We
observed that the jet size and shape varied from image to image even when all conditions
were unchanged, and as many as three jets were occasionally emitted from the collapseof a single condensate. Surprisingly, the jets were not always symmetric about thecondensate axis. We even observed that some jets were emitted at a slight angle withrespect to the radial direction.
We hypothesize that the jets are manifestations of local “spikes” in the condensate
density that grow while the BEC is imploding and then expand when the balance offorces is changed by quenching the collapse. If this interpretation is correct, we canestimate the size of the density spikes using the uncertainty principle. After a jump toaquench = 0, the kinetic energy of the atoms in the resulting jet must be equal to theconfinement energy that the spike had prior to quenching the collapse. If we assume
a Gaussian density spike in the BEC wavefunction with a width σ, then the averagekinetic energy of the jet atoms is related to the width of the spike by~2
1
hT i = mhvi2 =
.
2
4mσ 2

(4.2)

In the context of the spike picture, the observed anisotropy of the jets indicatesthat the spikes from which they originated were also highly anisotropic, being narrower
in the radial direction. From the calculated widths (using equation (4.2)) and themeasured number of atoms in the jets, we can estimate the density in the spikes. We
present plots of the number of jet atoms and the inferred spike density versus tevolve in
Figure 4.10. The jet data exhibited variability in energy and number significantly larger
than the ∼ 10% measurement noise.
The jets expanded with velocity v ' 1 mm/s, which corresponds to a kinetic energy of ∼5 nK and a radial pre-quench Gaussian rms width of ∼ 0.5 µm. Since the axial
size of the jets was below our optical resolution limit, we could not measure the axial

89

1

0.5

(a)

(b)

(c)

(d)

(e)

(f)

0

Figure 4.9: Jet images for a series of tevolve values. The absorption images were takenwith the conditions of Figure 4.2. The evolution times were 2, 3, 4, 6, 8, and 10 ms (from(a) to (f )). Each image has dimensions 150 × 255 µm. The shaded bar indicates the
optical depth scale. For these data, we applied an expansion ramp to aexpand = +250 a0 ,so the jets are longer than for the quantitative measurements explained in the text. The
jets were longest (i.e., most energetic) and contained the most atoms at values of tevolvefor which the slope of the number loss curve (Figure 4.2) was greatest. A tiny jet isbarely visible for tevolve ' 2 ms (image (a)), which is 1.7 ms before tcollapse . The centralcomponent of the images is the remnant BEC, which clearly decreases with time. The
time between the application of aquench and the acquisition of the images was fixed at
5.2 ms.

90

2000

(a)
number in jet

1500

1000

500

0
0

2

4

6

8

10

12

2

4

6

8

10

12

density in spike (1014 cm-3)

2.0

(b)
1.5

1.0

0.5

0.0
0

tevolve (ms)Figure 4.10: Quantitative jet measurements. (a) The number of atoms in the jets versus
tevolve for the conditions of Figure 4.2. (b) The spike density inferred from the kinetic
energies of the jets. The vertical bars indicate the full range of shot-to-shot variability,
which was rather high. For the analysis, we assumed the jets were disk-shaped since
the magnetic trap has cylindrical symmetry. The images were taken perpendicular to
the axial trap axis, corresponding to an edge-on view of the disks.

91
expansion rate. We therefore assumed an axial width equal to the harmonic oscillator
length for estimating the spike density. The atom density in the spikes decreased forlarger values of |acollapse |, and was half as large for acollapse = −170 a0 as for −36 a0 .
The observation of jets in the collapsing condensates suggests that the collapse
process involves a clumping of atoms into bunches. The implosion of the BEC probably
cannot be simply characterized as an overall contraction with a single density peakat the center of the cloud [37]. In fact, we have evidence from mean-field expansionof the collapsing condensates (see section 4.3.1) that the overall density contraction isrelatively small — with an average density increase on the order of 50%. So a moreplausible picture of the collapse is one with internal structure, as suggested by more
recent theoretical models. A discussion of the recent theory is given in section 4.7.

4.6

Collapse at B=0
In addition to the measurements described in this Chapter, it is potentially very

interesting to examine the collapse of an 85 Rb BEC after turning off the magnetic trap.
When the trap potential is switched off, we have the novel situation that the scattering
length is large and negative (∼-470 a0 at B=0). We have not extensively studied thecollapse dynamics at B=0 because most of the special techniques we have developed,
such as mean-field expansion of the BEC and focusing the burst atoms, do not work atzero field.
However, when we quickly turn off the magnetic trap after forming a high density
condensate, we do observe a burst of energetic atoms emerging from the BEC. At
the same time, the number of condensate atoms decreases rapidly. Unfortunately, it issomewhat difficult to determine how much of the burst and number loss is due to collapse
physics and how much is due to ramping the B-field across the Feshbach resonance (seeChapter 5). Nevertheless, the BEC collapse at B=0, and also at magnetic fields slightly
below the Feshbach resonance, may provide an interesting avenue for future research.

92

4.7

Discussion of theoretical models

4.7.1

Overview of observations

Collapsing 85 Rb condensates display very dramatic and complex behavior. At
the time that the data were collected, most of the observed details of the collapse could
not be easily explained in terms of existing theoretical models for the collapse [36,37, 42, 43, 44]. These theories were developed to describe 7 Li experiments, where
the scattering length was fixed and the BEC interactions with a large thermal cloud
proved very important. The peculiar and complex collapse behavior seen in our 85 Rb
experiment prompted a number of theorists [45, 46, 47, 48] to adapt their models to ourexperimental conditions. To facilitate comparison of the prominent features of our datawith the predictions of the theories, we can summarize our main observations.
(1) There is a delayed and abrupt onset of number loss from the condensate aftera → acollapse .
(2) The delay time before the loss depends inversely on both the initial density and|acollapse |.
(3) The loss decay constant τdecay is independent of both Ninit and |acollapse | for|acollapse | < 100 a0 , and only weakly depends on these quantities for larger|acollapse |.
(4) Part of the BEC number loss is due to formation of an energetic burst of atoms.
The average burst energy and energy anisotropy exhibit dramatic variations
with initial condensate number and |acollapse |.
(5) The number of cold remnant BEC atoms surviving the violent collapse varies
between much less and much more than Ncrit , depending on Ninit and acollapse .

93
4.7.2

Mean-field models for collapse

All of the papers that model the collapse dynamics experiments described inthis Chapter [46, 47, 48] use a mean-field approach that includes density-dependent
decay from the BEC. To account for 3-body recombination losses and also describe
the attractive self-interaction in the BEC, the authors of Refs. [46, 47, 48] integrate a
modified Gross-Pitaevskii (GP) equation that possesses an imaginary loss term [37].
Santos and Shlyapnikov [46] numerically integrate the modified GP equation for
our anisotropic magnetic trap potential, using a particular value for the 3-body recombination loss coefficient, L3 . The authors choose L3 to be within the limits set by ourprevious experiments [39], with a typical choice being L3 = 2 × 10−26 cm6 /s. The modelof Ref. [46] shows that 3-body recombination losses set an upper to the contraction of
the collapsing BEC. The density should grow until the loss “burns” a small numberof atoms in the dense central region, then the region is quickly refilled by surrounding
atoms. Santos and Shlyapnikov observe a series of such intermittent collapses of the
BEC, with each collapse destroying only a small number of the condensate atoms. As
the number decreases due to the losses, the balance of forces controlling the condensate wavefunction changes. The attractive mean-field interaction becomes weaker andis overcome by a repulsive force from the tight compression of the BEC wavefunction.
Thus, each intermittent collapse has a compression phase followed by a rapid expansion
or explosion of BEC atoms.
The authors of Ref. [46] therefore identify the burst atoms observed in the experiment as excited condensate atoms produced during the intermittent collapses. Obviously
then, the missing atoms are the atoms that are burned off by 3-body recombination.
Santos and Shlyapnikov make quantitative predictions for several experimental quantities, including tcollapse , τdecay , and hEburst i in the axial and radial directions. The model
reproduces several of the prominent features of the experimental data, including the

94
BEC loss characteristics (features (1)-(3) above) and the observed fact (5) that Nrem isoften much greater than Ncrit . There is no explanation given for Nrem > Ncrit in themodel — the authors simply state that this is “expected”!
Santos and Shlyapnikov assert that the energetically expanding BEC atoms are
the burst atoms. The calculated burst energies are predicted to scale with the ratio
a2collapse /L3 . By allowing L3 to vary with scattering length from ∼ 2 × 10−27 cm6 /s ata ' −30 a0 to ∼ 2 × 10−25 cm6 /s at a ' −250 a0 , Santos and Shlyapnikov obtain good
agreement with the experimental data for hEburst i with Ninit =6000 atoms. However,
the agreement with the larger Ninit =15,000 data is quite poor. Overall then, Santos andShlyapnikov cannot model the full experimental burst behavior (feature (4) above), evenwhen they allow the loss rate coefficient to vary by a factor of 100. While Ref. [46] has
intuitive appeal and does qualitatively explain all of the observations, it is impossible tofully evaluate the validity of the theory without better knowledge of the 3-body decay
rates in the negative scattering length regime.
Other theoretical models have been developed to explain the collapse dynamics [47, 48]. Most notably, Saito and Ueda [47] have proposed a mean-field model that isnearly identical to that of Ref. [46], with the only quantitative difference being a smaller
choice for the 3-body decay rate coefficient (L3 = 2 × 10−28 cm6 /s). Saito and Ueda also
predict details of the BEC loss that agree with experiment and they observe a burst ofenergetic atoms. However, the authors make no attempt to model the burst energy because of the unknown L3 dependence on scattering length. One unique prediction from
Saito and Ueda regards the jets. The authors explain that the jets do not result fromthe released confinement energy of density spikes but instead are due to interference ofmatter waves emanating from different density clumps in the imploding BEC.

95
4.7.3

Beyond mean-field theory

Although the published theoretical models for the 85 Rb BEC collapse behavior
are all based on conventional BEC mean-field theory, it is possible that the physics
might require more complicated theories that go beyond the level of the GP equation.
Murray Holland has expressed the opinion that the 85 Rb collapse is special because of
the Feshbach resonance. He suggests that a quantum field theory that includes higherorder correlations between the atoms is necessary to fully explain our observations. In
addition, T. Kohler and K. Burnett have also been pursuing a beyond-GP-equation
approach to explain the burst atoms we observed during the collapse process (Ref. [46]contains a note about this). It will be interesting to see whether a full theory of resonant
BEC can be developed to explain both the collapse experiments and the atom-molecule
coherence experiments in Chapter 6.
On a related note, Jake Roberts has proposed an interesting experiment to study
the collapse by changing the scattering length in a different way. We could use an
RF π pulse to change the spin state of the atoms from |2, −2i to |2, −1i. This state
has a large negative scattering length and no nearby Feshbach resonance. Thus, bystudying the collapse dynamics (especially the possible presence of burst atoms), one
could potentially separate the physics of the collapse at fixed scattering length from thephysics of resonant BEC.

Chapter 5
Microscopic dynamics in a strongly-interacting Bose-Einsteincondensate

5.1

Introduction and motivation
This chapter describes experiments that explore the response of an 85 Rb BEC to

rapid magnetic field variations. We begin by giving a motivation for the experiments.
Next comes a detailed explanation of the experimental methods and data. Finally, we
review some of the relevant theory for modeling the experiments.
In contrast to the negative scattering length experiments of Chapter 4, here we
discuss BEC behavior in the positive a region near the 85 Rb Feshbach resonance, where
the condensate self-interaction is large and repulsive. We used rapid magnetic field
variations to probe the strongly interacting regime in the condensate. We changed
the magnetic field to approach the Feshbach resonance from above, which causes the
positive scattering length to increase. As na3 becomes comparable to or larger thanone, the customary mean-field description of the dilute gas BEC, which assumes nocorrelations between the atoms, breaks down. Correlations between the atoms become
increasingly important and a BEC with sufficiently large na3 will eventually reach ahighly correlated state such as that found in a liquid. We are interested in the natureof this transition to a highly correlated state and the time scale for the formation of thecondensate correlations. In a strongly-correlated BEC with large scattering length, itshould be possible to discover effects that go beyond the standard mean-field theoretical

97
description.
In addition to such interesting physics, our time dependent experiments also allow
us to investigate the possibility of collisional coupling between pairs of free atoms andbound molecules. Since the atomic and molecular BEC states are degenerate at the
Feshbach resonance, a number of authors [14, 49, 50, 15, 51, 52, 53] have predicted that
atom-molecule transitions can occur during a B-field ramp. These theories were mainly
inspired by the Na BEC experiments at MIT [12, 38], in which very large number losses
to the condensate were seen when the magnetic field was ramped through a Feshbachresonance. However, these observations always revealed that the more time spent near
the Feshbach resonance, the greater the number loss. This time dependence is consistent
with a basic mean-field description of the BEC that includes a density-dependent loss
process, such as three-body recombination into molecules [54, 41]. Thus, the Na BEC
experiments could neither prove nor disprove the hypothesis that coherent molecule
creation was the cause of the observed BEC number loss.
We also observed BEC number loss due to ramping the B-field across the resonance. In fact, after producing 85 Rb condensates in 1999, our most important task was
to determine the effect on the BEC of crossing the Feshbach resonance. Except in special cases, we always turn off the magnetic trap before imaging the BEC, which requiresthat we ramp the magnetic field quickly from ∼160 G to 0 G. Any perturbations to thecondensate caused by crossing the Feshbach resonance at 155 G could potentially distort
our measurements of BEC dynamics in the trap. In our early tests, we saw number loss
from the BEC of up to 75% when we crossed the Feshbach resonance. The number loss
became steadily smaller as we ramped more quickly to zero B-field, until the fastest
ramp we could produce (∼1 G/µs) caused N0 to drop by less than 10% [26]. In later
experiments, we found that the BEC number loss was density-dependent [1]; therefore,
we could completely eliminate the loss by expanding the condensates in the magnetic
trap before crossing the resonance (see section 3.5.2). These measurements stimulated

98
our curiosity. We wanted to discover how much of the number loss was due to ordinary
inelastic collisions, which become increasingly frequent as one approaches the Feshbachresonance [39], and also whether any of the loss might be due to the coherent formationof molecules suggested by Refs. [14, 49, 50, 15, 51, 52, 53].

5.2

Experimental methods

5.2.1

Producing short B-field pulses

To understand time dependence of loss in our 85 Rb BEC, we examined the response of the condensate when we briefly approached the Feshbach resonance by applying a short magnetic field pulse (much shorter than the period for collective excitations).
We initially generated these B-field pulses using a time-dependent current in the baseball coil. From the outset, it was clear that the pulse had a dramatic effect on the BEC— depending on how closely we approached the resonance, we saw large number loss
from the condensate and also a relatively hot (∼150 nK) burst of atoms (this burst
was similar in character to that produced during the collapse dynamics experiments in
Chapter 4). The most obvious feature of the BEC response to the magnetic field pulse
was a loss of atoms that increased when we made our pulses shorter, until the loss became very large for very short pulses. Since the BEC number loss occurred very quickly(pulse duration < 100 µs), we had to improve our B-field control to make very shortpulses. We designed an auxiliary B-field coil (details given in Chapter 3) for generating
precisely tailored pulses toward the Feshbach resonance, as shown in Figure 5.1.
Our ability to carefully control the B-field allows us to probe the effects of microscopic physics in the BEC. By applying a short B-field pulse, we obtain a remarkableseparation of the time scales for microscopic versus macroscopic changes to the condensate. For instance, we can change the magnetic field by 10 G in a time of 10 µs,
which is a factor of 103 shorter than the oscillation period of the atoms in the trap.

99

-70
0

164

Hold Time

160

160
520

Ramp
Time

1900

156

Feshbach Res.
152≈∞

scattering length (a0)magnetic field (G)

168

-2150

0

50

100
time (µs)

150

Figure 5.1: Magnetic field vs. time for a typical auxiliary coil pulse. The solid line is thefield calculated from measurements with calibrated Hall-effect current sensors. On theright vertical axis, the corresponding variation in scattering length is shown. The open
circles are independent measurements of the magnetic field obtained by determining theresonant frequency for a 10 µs RF radiation pulse that drives atoms to the mF =–1 spinstate (error bars are smaller than the points). The dashed line shows the position of the
Feshbach resonance, where the scattering length becomes infinite. The field variationon the peak of the pulse was typically ∆B . 0.1 G.

100This means there is negligible motion of the atoms during a short pulse – the BEC
wavefunction remains static for the pulse duration. In fact, the brevity of the pulse
prevents any BEC expansion despite the fact that the repulsive self-interaction energyof the condensate typically increases during the pulse. We have verified this assertion
by modeling the BEC density with the PG model (see section 3.5.3) for the pulses usedin the experiment.

5.2.2

Counting the BEC atoms

To study BEC number loss we first form a 85 Rb condensate, following the procedure given in Ref. [26]. A sample of 85 Rb atoms in the F=2 mF =–2 state is evaporatively
cooled in a cylindrically symmetric magnetic trap (νradial = 17.5 Hz , νaxial = 6.8 Hz).
The magnetic field at completion of evaporation is 162.3 G, corresponding to a scattering length of 210 a0 . Typically, the cooled sample has N0 =16,500 BEC atoms andfewer than 1000 noncondensed atoms. The magnetic field is then ramped adiabatically(in 800 ms) to ∼166 G where the scattering length is positive but nearly zero [33], andthe BEC assumes the shape of the harmonic trap ground state.
We next apply a short magnetic field pulse (duration < 1 ms) so that the fieldbriefly approaches a value moderately close to the Feshbach resonance at ∼155 G, as
displayed in Figure 5.1. We use destructive absorption imaging to look at the numberof atoms remaining in the condensate. This experiment is repeated with a variety ofdifferently shaped magnetic field pulses.
We find that the magnetic trap must be turned off and the condensate spatial
size must be significantly larger than our resolution limit to obtain the most sensitive
and accurate measurements of number (see section 3.5.2). To expand the BEC after
the short pulse, we ramp in 5 ms to ∼157 G (a=2000 a0 ), and then hold at that Bfield for 7.6 ms. The mean-field repulsion during the ramp and hold times decreases
the density by about a factor of 30 before the trap is turned off. The density decrease

101
avoids density-dependent losses that we have observed during the trap turn-off (seesection 5.1). After turning off the trap, we allow the BEC to continue to expand for12.6 ms at B=0 before taking an absorption image. From the absorption image of the
expanded BEC we calculate the number of atoms remaining after a given auxiliary coilpulse.

5.2.3

Determining burst properties

The burst atoms generated by the B-field pulse have a very different energy
distribution from the condensate, so we employ a different procedure to determine theirproperties. The best way to find the total number, Nburst , and average energy, hEburst i,of the burst atoms is to image them at a focus of their simple harmonic motion (seesection 4.4). To focus the burst atoms onto the axial axis of the trap, we allow themto oscillate in the harmonic potential for 20.4 ms, which is more than one-half of aradial period (Trad /2=29 ms). We obtain this delay without interfering with the BEC
expansion ramp by adding a 7.8 ms delay between the end of the short B-field pulse andthe start of the BEC expansion ramp. After 20.4 ms in the trapping potential, all of
the burst atoms have an inward radial velocity. We then turn off the trap and the burst
atom spatial distribution continues to contract ballistically until reaching a minimum
radial size or focus 12.6 ms later.
A typical absorption image of the burst atom distribution superimposed on the
BEC remnant is shown in Figure 5.2. This image was taken after preparing the sample
of atoms according to the procedure given above. We use a 2-D Gaussian to fit the burst
atom image, but we exclude the central region containing the condensate remnant fromthe fit. The excluded Gaussian fit easily separates the burst from the BEC remnant
because of the large difference in the spatial extent of the two parts of the image.
The Gaussian fit parameters for a particular absorption image allow us to extract
both Nburst and the value of hEburst i for either the axial or radial direction. The focused

102

original image

center excluded

Gaussian fit

500 µ moptical depth x 1000

600

500

400

300

200

100

0
0

20

40

60

80

100

120

140

160

180

pixel position
Figure 5.2: Determination of burst properties from an absorption image. At top we show
a false-color absorption image taken after applying a short B-field pulse to the BEC,
then expanding the condensate and focusing the burst as described in the text. Here
we used a B-field pulse from Binit =165.5 G to Bf inal =157.1 G with ramp times of 13 µs
and thold =30 µs. The images show the measured optical depth times 1000 versus spatial
position on the camera pixel array (see colorbar). With the three images, we show how
the central region of the image containing the BEC is excluded from the 2-D Gaussian
fit. The excluded region size is chosen to be large enough so that the fit parameters are
independent of the box size. At the bottom is an axial cross-section of the data and fit.
Note the dramatic difference between the two spatial distributions, owing to the large
difference in their mean energies (hEburst i = 150 nK ' 50 × hEremnant i).

103burst looks like an elongated cigar with an aspect ratio of roughly 20. In a fit to theburst focus, the long dimension of the Gaussian (usually along the axial direction) gives
an estimate of the rms energy of the burst atom distribution. The short dimension of
the Gaussian is very narrow; we find that the observed size of the burst is limited by thefinite resolution of our imaging optics. In principle, with perfect imaging optics and atruly harmonic potential, we could determine the burst formation time from the extent
of the narrow focused dimension.
We can use the observed signal-to-noise (S/N) ratio in images such as Figure 5.2
to estimate our sensitivity to higher energy atoms in the burst. From the typical imaging
S/N we believe there are no burst atoms with energy between 300 nK and 1 µK. Of
course, it would be harder to see such atoms if there were two distinct components tothe burst energy distribution, but this seems unlikely given the observed smooth spatial
distribution that is fit very well by a Gaussian function.
To determine the BEC number from an image such as Figure 5.2, we first perform
an excluded Gaussian fit to the burst atoms, then we subtract the Gaussian fit fromthe absorption image. The remaining image pixels due to the BEC remnant are thensummed to find the total optical depth of the remnant. We compute the number of
BEC atoms from the optical depth sum.

5.3

BEC number loss from short B-field pulses
In this section, we simplify the description of the BEC response to the magnetic

field pulses by concentrating on the most obvious effect — the time dependent number
loss from the condensate. We also observed that a significant fraction of the initial
BEC number was converted into burst atoms by the short B-field pulses. However, theburst atoms did not account for all of the BEC number loss — there were also someundetected or missing atoms as in Chapter 4. Discussion of the burst atoms will be
postponed until section 5.4.

104
5.3.1

BEC number loss versus pulse amplitude

One of our first experiments was to explore the amount of BEC loss as a functionof the approach distance to the Feshbach resonance. We used triangular magnetic
field pulses with variable amplitude and ramp time. The amplitude was varied to
examine fields from 158.0 G (a=1200 a0 ) to 156.0 G (a=4500 a0 ). Here we list thecorresponding scattering lengths that we have observed by slowly adjusting the magnetic
field, as in Ref. [26]. The scattering length was calculated from the equation: a(B) =abg (1−∆/(B −Bpeak )) with the Feshbach resonance parameters as given in section 7.3.3.
For the range of magnetic fields examined here, the maximum value of the dilutenessparameter for the BEC varied from na3 = 0.01 for a=1200 a0 to na3 = 0.7 for a=4500 a0 .
The resulting BEC loss data are displayed in Figure 5.3. Here the dominant trend
is an increase in the number loss as the pulses approach closer to resonance. For a fixed
pulse amplitude, the BEC remnant number, Nrem , is largest at the shortest ramp time
and then decreases with ramp time until it reaches a minimum. Then longer ramp times
cause Nrem to increase over a time scale of tens of microseconds. The minimum in Nrem
versus ramp time corresponds to a maximum in the number of lost atoms. The data in
Figure 5.3 demonstrate that the ramp time that induces maximal loss becomes longer
as the pulses approach the Feshbach resonance. In addition, the width of the minimum
is largest for the pulses that come closest to resonance.
The triangular B-field pulses produce a very interesting and non-intuitive timedependent number loss. Under some conditions, we see that the loss increases for lesstime spent near resonance. Of course, the loss is not instantaneous; as the pulse lengths
approach zero time, the loss time dependence reverses and the loss also goes to zero. Hereit is useful to recall that conventional condensate loss is characterized by a rate constantfor a density-dependent decay process, and thus the loss increases monotonically withtime. In contrast, we measure an enhanced loss when the ramp time decreases, which

105

BEC number (x10-3)

16
14
12
10
8
6






42
     !  

0
0

10

20

30

40

50

60

70

scaled ramp time ( µs )Figure 5.3: Ramp time dependence of BEC number for different triangular pulse amplitudes. The legend gives the magnetic field (and scattering length) at the peakof the pulse, which may be compared the Feshbach resonance B-field of 155 G.
The BEC number and density at the beginning of the pulse were N0 =16,600 and
hni0 = 1.9 × 1013 cm−3 . The initial scattering length was ainit =7 a0 . The scaled
ramp time on the x-axis is equal to the actual pulse ramp time divided by four, showing the time required to ramp from 75% to 100% of the pulse amplitude. This scalingreflects the observed fact that most of the loss occurred at fields closest to the Feshbachresonance. The pulse hold time was fixed at 1µs for these data. Data points are larger
than the error bars.

106
suggests the presence of microscopic BEC physics such as non-adiabatic transitions toanother state.

5.3.2

BEC loss versus pulse shape

We also measured how the BEC number loss depends on the ramp time of themagnetic field pulse for a variety of different pulse hold times. As shown in Figure 5.4,
we varied the ramp time from 12.5 µs to 250 µs and changed the hold time at the pulse
peak from 1 µs to 100 µs. For hold times thold ≤ 15 µs, there is an initial decrease in
Nrem as the ramp time increases. Then the slope changes and fewer atoms are lost for
longer ramp times. Thus, the short hold time data show a distinct minimum in Nremvs. ramp time, which shifts toward shorter ramp times as the hold time is increased. Allof the hold time data display an upward slope in Nrem versus ramp time over some rangeof times, but the range is largest for the 100 µs hold time. This increase in remnant
number for longer ramp time provides clear evidence that the loss is not conventional
inelastic decay that is characterized by a rate constant.
Of course, the above interpretation of our data would be modified if the densityof the BEC were changing due to the rapid increase in the mean-field interaction, but
the characteristic time for such readjustments in cloud shape is far longer (of order
1/(2νradial )=29 ms) than the time scales for the B-field pulses considered here. For
example, using the analytic PG model (section 3.5.3), we calculate that for a 250 µs
ramp to Bf inal =156.0 G (4500 a0 ), the change in mean-field energy causes the BEC
density to decrease by only 1% from its initial value. Thus, the observed dependenceson ramp time must reflect microscopic physics in the BEC and not any macroscopicchanges in the shape of the condensate.

107

BEC number (x10 -3)

14
12
10
8
6


4
2
0
0

10

20

30

40

50

µ
µ
 
  µ
 µµ

60

70

scaled ramp time ( µs )Figure 5.4: Dependence of remnant BEC number on pulse ramp time for various holdtimes (see legend) with N0 =16,500 (hni0 = 1.9 × 1013 cm−3 ). For the majority of thedata points, the symbol is larger than the statistical error bar (not shown). The lines
are spline fits to guide the eye. The ramp times were multiplied by a factor of 1/4 as in
Figure 5.3. The magnetic field during the hold time was Bhold =156.7 G (2400 a0 ).

108
5.3.3

Unconventional loss: Rabi-like oscillations

An alternative way to demonstrate the unconventional time dependence of the
BEC number loss is by measuring the remnant BEC number as a function of hold time
for the magnetic field pulse. For example, we can take the data from Figure 5.4 andplot Nrem versus thold for a B-field pulse with fixed ramp time. Using a linear ramp of
12.5 µs duration, we observed BEC number loss of 10-20% when the hold time was set
to zero (corresponding to a triangular pulse). The number of BEC atoms then showeda smooth exponential drop as the hold time increased (see Figure 5.5).
At first glance, the exponential time dependence of the data in Figure 5.5 seems
to support the hypothesis that the number loss came from density-dependent decay, as
in Refs. [54, 41]. However, we showed that this was not true because the decay rate
was insensitive to the initial BEC density. The BEC density before the short magnetic
field pulse can be varied over a wide range by slowly changing the scattering length [26].
Surprisingly, when we used this method to reduce the initial BEC density by more thana factor of 2, the time constant for number decay was nearly unchanged. In addition,
the low[high] density data exhibited a decay time constant that was over 2[1] orders ofmagnitude shorter than predicted by our previous inelastic loss measurements with cold
thermal clouds [39].
To further explore the BEC number loss with hold time, we also varied the magnetic field at the peak of the pulse, Bhold . At Bhold =157.1 G, we observed a striking
damped oscillation in Nrem versus thold . As displayed in Figure 5.6, the initial number
loss actually reverses sign near thold =20 µs! Clearly this behavior cannot be explainedin terms of a monotonic density-dependent number loss. To the contrary, the oscillation
data suggest that the BEC number loss may involve coherent transitions to anotherstate that are analogous to Rabi-flopping oscillations. However, the actual physicalprocess appears to be more complex — there is a strong damping to the Nrem oscilla-

109

fraction remaining

1.0
0.8
0.6
0.4
0.2
0.0
0

20

40

60

80

100 120

hold time ( µs )Figure 5.5: Fraction of BEC remaining versus pulse hold time. The pulse ramp times
were 13 µs and the magnetic field during the hold was Bhold =156.7 G (2420 a0 ). Number
decay was measured for two different initial densities: hni = 1.9 × 1013 cm−3 (•) andhni = 0.7 × 1013 cm−3 (◦). Fitting the data to exponential functions (solid lines) gave
time constants of 13.2(4) µs and 15.4(14) µs. Thus, reducing the density by a factor of
2.6 caused an increase of only 17(11)% in the decay time constant.

110
tions and there is an overall irreversible number loss from the BEC. From a damped
sinusoidal fit to the data, we find an oscillation frequency of roughly 25 kHz, which issimilar to the binding energy of the weakly bound molecular state associated with the
Feshbach resonance. Attempting to extract the frequency precisely from the data isquite difficult, however, because of the rapid damping of the oscillations and the overall
exponential decay of the BEC number.
Experimentally, the Rabi-like oscillations were quite difficult to observe and could
only be seen with Bhold centered in a ∼0.1 G wide window of magnetic field around157.1 G. We believe this narrow window results from a trade-off between being closeenough to the resonance to have strong coupling with the molecular state and far enoughfrom the resonance to have an oscillation period that is short compared to the overall
BEC number decay time. We have discovered a way to avoid these limitations with an
improved method for inducing atom-molecule oscillations. As described in Chapter 6,
we use two magnetic field pulses separated in time to obtain Ramsey-like oscillations [24]in the BEC number.

5.4

Burst atoms created by short pulses
In our short pulse experiments, we always observed the formation of a relatively

hot burst of atoms accompanying the disappearance of the BEC atoms. This section
begins with an overview of the burst characteristics and then describes a more detailed
experimental study of the burst.

5.4.1

Overview of burst characteristics

When we first realized that it was possible to generate an explosion or burstof atoms using a short magnetic field pulse toward the Feshbach resonance, we wereshocked. The existence of this burst was very surprising because of its extremely rapid
formation time of less than 20 µs. For comparison, the qualitatively similar burst

111

14000
13000

BEC number

12000
11000
10000
9000
8000
7000
6000
0

20

40

60

80

100

120

140

hold time (µs)
Figure 5.6: Rabi-like oscillation in the number of BEC atoms. Black points with error
bars are the measured BEC number as a function of hold time for a trapezoidal B-field
pulse. The ramp times for the pulse were fixed at 13 µs and the pulse decreased the
B-field from the initial value of ∼165.5 G to the value Bhold =157.12 G (1840 a0 ). The
black line is a fit to the data using a damped oscillation superimposed on an exponential
decay, with the functional form N (t) = N0 − γ exp (−t/τ ) + A exp (−βt) sin (ωe t + φ).
For this fit, the BEC number decay time τ =110 µs is somewhat longer than the damping
time for the oscillation, 1/β=32 µs and the oscillation period 2π/ωe =40 µs.

112
of atoms observed in the BEC collapse experiments (Chapter 4) was generated in atime of several milliseconds.1 The BEC collapse process seemed to create burst atoms
spontaneously during the slow implosion of the condensate at fixed magnetic field. In
contrast to that scenario, we saw that short B-field pulses allowed us to generate an
atom burst from a stable BEC and to influence the burst during its formation. In
fact, we observed that the short-pulse burst characteristics were quite sensitive to themagnetic field pulse shape and amplitude, as discussed in section 5.4.2. Note that for
the remainder of the Chapter, the short-pulse burst will simply be referred to as “theburst” or “the burst atoms”.
Because the burst formation time is much less than the harmonic oscillation
period of the magnetic trap, Tr = 2π/ωr =57 ms, the trap potential focuses the burst
atoms onto a symmetry axis after a time t=Tr /2. The cylindrical anisotropy of the trap
potential leads to radial and axial burst focuses at different delay times in the magnetic
trap. Depending on whether the atoms form an axial or radial focus, we measure adifferent average burst atom energy, with the radial energy being roughly 2× the axial
energy. Presumably this energy anisotropy results from the initial spatial anisotropy of
the BEC before application of the short B-field pulse. We could study this by varying
the initial condensate aspect ratio using the tunable scattering length, but we have nottried this yet.
It is interesting to note that the energy anisotropy of the burst atom distribution
can persist indefinitely in the magnetic trap due to the absence of thermalizing collisions.
The density of the burst atoms remains very low throughout their motion in the trap.
Simply put, the burst atoms do not form an equilibrium distribution! We may therefore
be able to use the observed energy anisotropy to glean information about the history of
the burst, i.e., the mechanism for the burst generation.
1

Despite the fact that it was created under very different conditions, the short-pulse burst atom
distribution had comparable number (∼3000) and average energy (∼150 nK) to the collapse burst.

113
In addition to their energy, another important characteristic of the burst atoms istheir spin-polarization. From the observed occurrence times and spatial locations of the
burst focuses, we can infer that the oscillation frequency of the burst atoms corresponds
to the |F, mF i = |2, −2i state, which is the same as that of the BEC atoms. We looked
for any additional burst atoms in other spin states with comparable energies to those of
the |2, −2i atom burst. This was done by reducing the total time delay before taking anabsorption image to the rather short value of 2.1 ms. Since our absorption imaging issensitive to all spin states of 85 Rb [1], the short delay time should allow us to count burst
atoms not in the |2, −2i state before they could expand appreciably. We saw no evidence
for any such “extra” burst atoms in other spin states — our experimental sensitivity
was ∼500 atoms, or 8% of the |2, −2i burst number. This observation indicates thatthe burst generation mechanism does not involve Zeeman spin flip transitions.
We have found that the rapid magnetic field variations only create burst atoms
from Bose-Einstein condensates and not from cold thermal clouds. We searched forburst atoms after subjecting a 40 nK thermal cloud with density < n >= 7 × 1011 cm−3
to a standard B-field pulse, but we did not observe any trace of a burst. The absenceof the burst may have been due to a strong density-dependence in the burst formation
process, or perhaps the burst cannot be formed from a thermal cloud at all. For instance,
the observed burst formation from a BEC might depend on the fact that the atoms are
initially all in the same ground state of the harmonic trap instead of being distributed
over many trap states, as in a thermal cloud. Since we have not repeated the experimentwith condensates that have similar density to the thermal cloud, we cannot presently
distinguish between the two possibilities.

5.4.2

Burst dependence on pulse shape

As a complement to the experiments on BEC number loss described in Section 5.3,
we carefully examined the dependence of the burst properties on magnetic field pulse

114
shape. We first studied the burst number and energy as a function of the B-field pulse
length, as shown in Figure 5.7. Here we see Nburst smoothly grows with increasing thold
before saturating at roughly 4000 atoms. During the same time period, the average
axial energy of the burst atoms shows a significant decrease from 200 nK to 100 nK.
We did not measure the radial energy of the burst atoms for these conditions, but weassume that this energy obeys a similar time dependence.
One possible mechanism for the burst energy that was initially quite appealing
to us is the sudden increase in mean-field energy due to the change in scattering length
during the fast B-field pulse. The idea is that the BEC suddenly finds itself far fromequilibrium due to the dramatically increased repulsive interactions. After completion of
the first magnetic field ramp, the BEC should therefore immediately start to expand as
the increased mean-field energy (potential energy) converts into kinetic energy. Ramping
back to the initial B-field before the expansion has proceeded very far should halt theprocess by removing the excess potential energy.
To investigate the validity of the mean-field impulse hypothesis, we again utilized
the PG model. With the magnetic field pulse of Figure 5.7 as input to the model, we determined the time dependent scattering length and integrated the PG model equations.
The prediction of the PG model for the amount of mean-field energy that is converted
into kinetic energy is shown in the inset to part (b) of Figure 5.7. Not only is themagnitude of the predicted impulse energy far too small, but it also shows the opposite
time dependence to that observed in the data. There is simply not enough time for theincrease in mean-field potential energy to be converted into significant kinetic energy.
Although the changing mean-field energy of the BEC clearly cannot explain thehold time dependence of burst energy, the energy for burst formation certainly must
be related to the time-varying magnetic field. For the data in Figure 5.7, the mostenergetic burst atoms are produced by the shortest values of thold , which correspondto the B-field pulses that are closest in shape to a spike or triangle pulse. Perhaps

115

5000

burst number

4000

3000

2000

(a)

1000
0

20

40

60

80

100

120

140

hold time (µs)

axial burst energy (nK)

300

1000
100

250

10
1

200

0.1
0.01

150

0 20 40 60 80 100 120 140

100
50

(b)

0
0

20

40

60

80

100

120

140

hold time (µs)

Figure 5.7: Dependence of burst characteristics on pulse length. (a) Burst number
versus pulse length. Black points with error bars are the measured number of burst
atoms versus hold time for a trapezoidal pulse. In this pulse, the magnetic field wasramped in 13µs from ∼165.5 G to Bhold =157.12 G (1840 a0 ). After the completion of
the hold time, the B-field was ramped back to roughly 165 G. (b) Burst energy versus
pulse length. Black points with error bars are the measured average burst atom energy
in the axial direction. The inset shows the same data plotted on a log scale for they-axis. The black line is the PG model prediction for the mean-field impulse energy
imparted to the BEC.

116
there is some relationship between the Fourier transform of the pulse and the energy
imparted to the burst atoms. Murray Holland is conducting simulations to investigate
this idea for the burst energy dependence on pulse shape [55] using a quantum field
theory approach [56]. Some discussion of this theory will be given in section 5.5.4.

5.4.3

Burst dependence on pulse amplitude

The characteristics of the burst atoms produced by the magnetic field pulses also
displayed a strong dependence on the pulse amplitude. Figure 5.8 gives the measured
variation of both Nburst and average radial burst energy versus pulse height. In thisfigure, we combine data from two slightly different magnetic pulses. The black points
were taken with a ramp time of 100 µs and hold time of 1 ms, while the white points
had tramp =50 µs and thold =200 µs. We believe it is reasonable to plot the B-fielddependent data for these two different pulse shapes on same graph because both pulses
were sufficiently long so that the burst should have reached “steady state” values forthe number and energy, as observed in Figure 5.7.
The burst number shows very little magnetic field dependence over most of therange given in part (a) of Figure 5.8. From ∼155 G to 159 G, the number of atoms inthe burst remains fairly constant. However, Nburst decreases very sharply as Bhold rises
above ∼159 G. For pulses with values of the hold B-field greater than 160 G, we did
not observe any burst atoms.
In contrast to the saturating dependence of the burst number with B-field, theaverage radial energy grows in a linear fashion with increasing pulse amplitude (decreasing Bhold ). This behavior continues all the way down to magnetic fields very near the
Feshbach resonance at 155 G. It would be interesting to see whether the linear trend
continues past 155 G, because this would imply that the burst formation has relatively
little to do with the Feshbach resonance. We note that the average axial burst energy
depends on magnetic field in a similar fashion to the radial energy, but the axial energy

117

6000

burst number

5000
4000
3000
2000
1000
0
155

(a)
156

157

158

159

160

161

159

160

161

B-field (G)radial burst energy (nK)

300
250
200
150
100
50
0
155

(b)
156

157

158
B-field (G)Figure 5.8: Dependence of burst properties on B-field pulse amplitude. (a) Number ofatoms in the burst versus Bhold for an initial field Binit ∼166 G. Black points with error
bars are the number of atoms measured in a radial burst focus after a trapezoidal pulse
with tramp =100 µs and thold =1 ms. White points show the burst number generated bya slightly different B-field pulse shape with tramp =50 µs and thold =200 µs. (b) Averageradial burst energy as a function of pulse amplitude, for the same B-field pulses as inpart (a).

118
is consistently about a factor of 2 smaller in magnitude (the axial energy data wereomitted to improve the clarity of Figure 5.8).

5.4.4

Search for secondary collision mechanism

In an attempt to explain some of the observed burst properties, we looked forevidence that the burst atoms might be generated by secondary collisions with “superhot” (∼0.1 to 1 mK) atoms. Beijerinck and co-workers [57, 58] have proposed that theenergetic decay products of 2- and 3-body collisions might cause considerable heatingof a BEC if these super-hot atoms collide with the near-stationary BEC atoms while
exiting the cloud. The density-dependent inelastic decay rates are highest in the dense
central region of a condensate, so the decay products usually must travel through asignificant region of space containing many BEC atoms before escaping. If the crosssection for collisions with BEC atoms is high enough, there should be many secondary
collisions during this time. The authors in Ref. [58] also suggest the possibility that thesecondary collision products may also collide with the BEC atoms, leading to a sort ofpositive-feedback or avalanche effect.
Originally, these ideas seemed somewhat plausible to us. After the fast B-field
pulses, we had observed missing atoms, burst atoms, and the BEC remnant. We suspected that the missing atoms might correspond to the super-hot inelastic decay products while the burst atoms could be created in secondary collisions. We believed that weshould be able to test this possible burst generation mechanism by varying the elastic
cross-section of the BEC atoms after producing a bunch of super-hot atoms.
To test the secondary collision hypothesis, we devised a special B-field pulse
consisting of an initial brief triangular spike followed by a long plateau with an adjustableheight (see Figure 5.9). The idea was to generate the hot 3-body recombination products
during the brief spike toward the Feshbach resonance, then to alter the s-wave elastic
cross-section, σs = 8πa2 , by changing the level of the plateau. If the burst atoms

119
were coming from secondary collisions, then we should be able to strongly influence thenumber and energy of the burst by varying the plateau scattering length.
Unfortunately, the BEC number loss and burst data resulting from the spike/plateau
pulse did not allow for a conclusive test of the hypothesized burst generation mechanism. For instance, while Nburst did grow as the plateau scattering length was increased,
the average burst energy was reduced. In addition, the total number of missing atoms
increased with increasing plateau scattering length. This means that we cannot obtain aconsistent secondary collision picture because it is impossible to say that the super-hot
atoms were only generated during the initial brief spike part of the pulse. Also, theobserved values for the burst energy hEburst i ' 100 to 200 nK seem far too small for thepredictions of the secondary collision model. If the posited super-hot atoms really haveenergies of 100 to 1000 µK, then it is difficult to understand how they could impart sucha tiny fraction of their energy to the burst atoms. Due to the difficulties with the secondary collision mechanism, we decided to abandon these studies. While there may besome 3-body recombination and subsequent secondary collisions occurring in our shortB-field pulse experiments, we believe that most of the observations can be explained interms of non-adiabatic transitions to another state, as described in section 5.5.

5.5

Modeling BEC loss and burst production
The observed time dependence of the BEC number loss from magnetic field pulses

suggests the presence of non-adiabatic transitions to another state. In our case, the obvious candidate for this other state is the weakly-bound molecular state that causes the
Feshbach resonance. At B-fields larger than the resonant value of 155 G, the molecularbound state lies just below the zero-energy threshold for two colliding BEC atoms. As
one decreases the magnetic field to values below 155 G, the molecular state rises above
threshold and becomes a quasi-bound state with a finite lifetime due to coupling with acontinuum of scattering states with finite collision energy. These considerations suggest

120

t=250 µs

t=0
166 G

161 G

variable
Bhold

157 G

Figure 5.9: Schematic of the B-field pulse for testing the secondary collision mechanism.
At top, we show the time-dependent magnetic field during the pulse. The total lengthof the spike part of the pulse is 50 µs and the plateau length is typically 200 µs. The
lower part of the figure displays a simplified picture of the secondary collision mechanism
proposed in the text. The ellipses show the condensate spatial extent. In the condensate
at left, the explosion represents hot products of 3-body recombination. The next BEC
picture shows the expansion of these decay products into the BEC volume during the
B-field pulse plateau.

121
the possibility of modeling the Feshbach resonance as an avoided crossing of the atomic
and molecular states, as in the well-known Landau-Zener (LZ) model. In this section,
we discuss a simple “toy” model for the Feshbach resonance to gain some qualitative understanding of our experimental data. To improve upon the toy model, we also consider
a more elaborate curve-crossing model from Josh Dunn and Chris Greene that includes
many trap states. Finally, we give a brief review of published atom-molecule coupling
theory as it applies to our data, including theory work from JILA (Servaas Kokkelmans
and Murray Holland) and Oxford (Kohler, Gasenzer, and Burnett).

5.5.1

Toy Landau-Zener model for BEC loss

In an attempt to understand the BEC number loss in terms of non-adiabatic
transitions, we developed a simple toy model for the Feshbach resonance. Although themodel is totally useless for quantitative comparison with experiment, it does provide
some insight into the time dependence of the BEC loss for various magnetic field pulses.
By investigating the limitations of the toy model, one can learn more about the real
physics of the Feshbach resonance. For instance, one major flaw of the toy model is
its neglect of the burst atoms. In the toy model avoided crossing (see Figure 5.10),
there are only two states – the initial colliding atom state and the molecular state.
These states are coupled by an off-diagonal matrix element in the Hamiltonian so thattransitions between the atomic and molecular states are possible. Using this model,
one might postulate that the BEC number loss observed in the experiment is due tothe formation of molecules (the “missing atoms”), but then how do the burst atoms
fit into the picture? In fact, the burst atoms do not fit into a two-level picture at all!
This point caused us considerable confusion when we first observed the burst atoms andwere unable to identify them with a particular state of the 2-level toy model. The burst
atoms can only be explained in terms of excited states with higher energy than the BEC
ground state. The excited states may form a series of discrete levels of the harmonic

122
trap or a continuum of states if the trap is neglected.
Before calculating transition probabilities with the toy model from a pulsed magnetic field, it is illuminating to review the standard Landau-Zener theory. Let us assume
that the initial state of the system is a stable BEC of atoms at a magnetic field above
the Feshbach resonance. This initial condition corresponds to a probability of unity of
being in the colliding atom state of the toy model (the upper dashed line to the rightof the avoided crossing in Figure 5.10). For a single linear ramp through the avoided
crossing, the probability of making a non-adiabatic transition from one adiabatic ordressed state to the other is [59, 60]
p = exp (−2π

ω02
),
4ω̇

(5.1)

where ω0 is the coupling matrix element divided by ~ and ω̇ is the time rate of changeof the detuning between the uncoupled atomic and molecular states. The single-passage
transfer probability becomes smaller as the ramp rate decreases, so that an infinitely
slow ramp produces zero transition probability. In the toy model, this infinitely slow,
adiabatic ramp causes the atomic state to smoothly change into the molecular state
while gaining potential energy from the magnetic field. In contrast, a very fast ramp
through the crossing leads to a large non-adiabatic transition probability p'1, so thestate of the system during the ramp is best described by the uncoupled atomic state,
shown as the horizontal solid line in Figure 5.10.
For our pulsed magnetic field experiments, a much more relevant case to consider
is that of a double passage through the avoided crossing, in which the field is ramped
linearly from far above resonance to far below resonance, then back again. In the limitthat the initial, intermediate, and final detunings are infinitely far away from resonance,
the probability of making a non-adiabatic transition from the upper dressed state to
the lower dressed state is
Pdouble = 2p(1 − p),

(5.2)

123



εbind

2-atomstate (BEC)

molecular
state
Bpeak = 155 G

 

Figure 5.10: Toy model for 85 Rb Feshbach resonance. Here the resonance is represented
as a simple avoided crossing of two states: the zero-energy state of two colliding BEC
atoms and the molecular state (solid lines). The different magnetic moments of the twostates cause them to cross at 155 G, but this degeneracy is broken by a weak coupling
between the states. The adiabatic or dressed state eigenvalues of the total Hamiltonian
are shown by the dashed lines.

124
where p is the single-pass transition probability from equation (5.1). The structure ofthis equation leads to a very different time dependence of the transition probability. For
fast ramps (short magnetic field pulses), the single-pass probability is near unity, so thatPdouble is almost zero. The state of the system during the entire pulse is approximatelythat of the uncoupled or diabatic atom-atom state. The opposite condition of very
slow ramps (long magnetic field pulses) causes p→0, which again leads to a vanishingly
small value for Pdouble . In this case, the atoms “ride along” the adiabatic energy level
during the pulse before coming back to where they started at the beginning. The only
way to get a significant double-pass transition probability out of the initial state is touse intermediate ramp speeds such that p has an intermediate value between 0 and
1. In fact, the largest transition probability is obtained for p=1/2. The different timedependences for single- and double-passages through resonance are shown graphicallyin Figure 5.11 for a particular choice of the avoided crossing parameters.

5.5.2

Modifying the LZ model for triangular B-field pulses

After discussing the standard LZ model predictions in the context of the toy modelfor the Feshbach resonance, we are now in a position to explore the effect of finite B-field
pulses. In the actual experiment, we mostly use magnetic field pulses that approach,but do not cross the Feshbach resonance B-field of 155 G. To predict the effects ofsuch a pulse in the toy model, it is necessary to move beyond the simple LZ analyticalexpressions presented above. To do this, we use the concise formulation of a standard
avoided crossing from Ref. [61]. We numerically integrate the coupled Schrodinger
equations for the two-level system and calculate the non-adiabatic transition probability.
The numerical integration allows us to simulate various differently-shaped B-field pulses
subject to the initial condition that all of the probability resides in the atomic statecorresponding to a pair of BEC atoms.
We observe that the behavior when the B-field approaches and then backs away

125

1.0

probability
Probability

0.8
0.6
0.4
0.2
0.0
0

100

200

300

400

500

600

Ramp
ramp time
time (µs)(µs)
Figure 5.11: Comparison of the Landau-Zener model predictions for single- and doublepassages through an avoided crossing. The dashed line is the non-adiabatic transition
probability, p, as a function of ramp time for a single-passage through resonance. In
the context of the toy model, the dashed line gives the probability of remaining in theatomic state after the ramp. The solid line is the quantity 1-2p(1-p), which corresponds
to the probability of remaining in the atomic state after sweeping over resonance andcoming back. Although both predictions show a decrease in probability as the ramp
time increases from zero, the minimum in the double-passage probability versus ramp
time leads to a strikingly different time dependence for longer, slower ramps. For both
the dashed and solid line predictions, the B-field is swept so that the detuning changesby |∆ω| = 4.3 × 108 s−1 during the ramp time and the coupling strength is fixed at
ω0 = 1.4 × 106 s−1 .

126from the crossing point with a triangular pulse is qualitatively similar to the morefamiliar double-passage case discussed previously.2

As shown in Figure 5.12, thefinite B-field pulses still produce a minimum in the probability remaining versus pulse
ramp time. However, the depth of the probability minimum depends strongly on theapproach distance to resonance and is generally less than for the infinite ramp case.
This behavior can be understood in terms of the magnetic field width of the avoided
crossing, defined as the ratio of the coupling strength to the slope of detuning with
B-field: ω0 /(dω/dB). The transition probability goes up as the pulses penetrate moredeeply into this strong coupling region. Another notable difference between the finite
B-field pulse result and the infinite double-passage result is the ramp time at which
the minimum occurs (∆tmin ) . For finite pulses, the probability minimum occurs at a
ramp time that is proportional to the inverse of the squared dressed state splitting at
closest approach. The dressed state splitting decreases as one approaches resonance,
so that ∆tmin becomes larger. It is very interesting to observe these trends in the toymodel because the same qualitative features are observed in the experimental data fortriangular B-field pulses (see Figure 5.3). In those data, the number remaining in thecondensate after a triangular pulse toward resonance has a clear minimum as a functionof ramp time. The depth and position of the minimum vary with approach distance toresonance, as in the toy model.
Given the qualitative agreement between the toy model predictions and the triangle pulse experimental data, one might hope to make a quantitative comparison oreven try to fit to the data. In principle, this would allow a determination of the appropriate Feshbach resonance coupling strength and relative magnetic moment between
the atomic and molecular states. However, all of our attempts to fit the toy model to2

Even when the B-field pulse does cross resonance, there is one conspicuous difference between the
numerical integration and the LZ prediction – the integration yields large amplitude, high frequency
probability oscillations, called Stückelberg oscillations, as a function of the ramp time. The LZ model
expression 2p(1-p) is a result of phase-averaging over these rapid oscillations. Stückelberg oscillations
were never seen in the experiment.

127

1.05

probability

1.00
0.95
0.90
0.85
0.80
0.75
0

50

100

150

200

250

300

ramp time (µs)
Figure 5.12: Toy model predictions for finite magnetic field pulses. The probability ofremaining in the initial atomic state is shown versus ramp time for a triangular pulse.
The conditions are Bcross =155 G, Binit =165.6 G, and Bpulse =158 G (white points) and
156 G (black points). The coupling strength and relative magnetic moment between
the atomic and molecular states are (arbitrarily) set to be 11 kHz and 7.6 kHz/G,
respectively. We are unable to obtain quantitative agreement with the experimental
data of Figure 5.3 even when we allow these two parameters to float freely.

128
the data have failed. We find that the toy model does a rather bad job at reproducing
quantitative features of the data, even when we allow the coupling strength and relative
magnetic moment between the two states to be arbitrary parameters in the numericalsimulation. In addition, the production of the burst atoms in the experiment cannot
be explained by the two-level toy model. It is clear that a more sophisticated avoided
crossing theory is needed to accurately model the real Feshbach resonance.

5.5.3

A better avoided crossing model

Josh Dunn, Chris Greene, and coworkers developed a fairly simple theory of the
85 Rb Feshbach resonance [62, 63] (see also Ref. [64] for related picture) that has several

advantages over the toy model discussed above. They model the Feshbach resonanceas a series of avoided crossings between the resonance state (molecular state) and theeigenstates of the harmonic trapping potential. In the context of the Dunn/Greenemodel, one can see how a B-field pulse toward resonance could generate burst atoms.
Because the molecular state is coupled to many trap states, the molecular state canserve as an intermediate state for transitions from the lowest trap level (BEC) to theexcited levels. This provides a potential mechanism for the production of burst atoms in
the experiment. While this model has some appealing features, we have not successfully
applied it to the Feshbach resonance data discussed in Section 5.3 and Section 5.4. Here
we provide a brief discussion of the model and its limitations.
The Dunn/Greene theory is based on the exact quantum mechanical descriptionof a pair of harmonically-trapped atoms that interact via a short range (δ function)
potential characterized by the scattering length, a [65]. While the δ function potential
is an approximation to the real one, this approximation is justified when the atoms
collide at low energy so that their deBroglie wavelength greatly exceeds the range of thephysical potential. This means that the scattering length is sufficient to describe theinteraction between pairs of condensate atoms. For the atom pair, the total Hamiltonian

129
can be broken into two parts — one for the center-of-mass motion and one for the relative
motion. Since the relative motion Hamiltonian describes the atomic collisions, this partcontains the interaction potential that depends on the scattering length. The eigenvaluesof the relative motion Hamiltonian are simply the standard harmonic oscillator levels,
but shifted according to the sign and magnitude of a. At positive scattering length or
repulsive atom-atom interaction the energy levels are shifted upward. The opposite istrue for negative a. Because the Feshbach resonance causes a large variation in a withmagnetic field, the eigenvalues of the atom pair show a pronounced B-field dependence,as shown by the solid lines in Figure 5.13.
The actual calculation of the energy eigenvalues of the interacting atom pair
involves solving a transcendental equation containing the energy and B-field-dependent
scattering length. One can solve the equation iteratively or graphically, but there isno analytic expression for the eigenvalues as a function of magnetic field. Since this israther cumbersome for predicting transition probabilities when the B-field is ramped
near resonance, the authors of Ref. [62] create an approximate model Hamiltonian inwhich the molecular state undergoes repeated avoided crossings with all of the harmonic
trap states. The uncoupled molecular state energy is assumed to vary linearly with
B-field and the coupling matrix elements to the excited trap states obey the Wigner√
threshold law [52] (ω0 ∝ E). By adjusting the parameters of the model Hamiltonianand diagonalizing it, one can obtain very good agreement with the exact eigenvalues
over a small range of magnetic field close to the Feshbach resonance (see part (a) of
Figure 5.13). However, the agreement becomes quite terrible as the B-field increases
by a few tenths of a Gauss above the resonance. The disagreement clearly shows thatthe curvature in the molecular bound state energy is not due to a simple coupling withthe trap states! In fact, the binding energy curvature is very strong even outside theB-field region where the trap states mix strongly with the molecular state. In the exact
solution to the interacting atom pair, the binding energy curvature is determined by

130energy (oscillator units)

10

5

0

-5

(a)

-10
154.5

155.0

155.5

156.0

B-field (G)energy (oscillator units)

0

(b)

-200
-400
-600
-800
-1000
154.5

155.0

155.5

156.0

156.5

157.0

B-field (G)Figure 5.13: Energy eigenvalues versus B-field for an interacting atom pair near the
Feshbach resonance. (a) Solid lines are the exact eigenvalues calculated by Josh Dunn.
Black points are the eigenvalues of an approximate model Hamiltonian in which the
diabatic bound state crosses zero energy at 155.3 G with a slope of -244,108 ~ω/G.
The coupling matrix elements are equal to (113 ~ω) n1/4 , where n ≥ 1 is the quantum
number of a given excited state. The agreement between the model Hamiltonian andthe exact solution is quite good near resonance. (b) Same as in (a), but with a larger
energy and magnetic field range. Here the molecular bound state (solid line) lies well
below the excited trap states, which are visible as a thick bar of black at the top. White
points are the bound state energies from a coupled-channels calculation by Servaas
Kokkelmans. These calculations agree quite well with the eigenvalues from the atompair model. However, the model Hamiltonian eigenvalues (black points) disagree verystrongly with both the atom-pair model and the CC calculation over this B-field range.

131
the variation of scattering length with B-field.
Because of the failure of the model Hamiltonian to reproduce the molecular binding energy curve over the relevant magnetic field range, we have not attempted to model
the effects of the experimental B-field pulses with the approximate Hamiltonian of theDunn/Greene model. In the experiment, the pulses typically jump from ∼166 G to∼156 G. This magnetic field range corresponds to the region where the model Hamiltonian from Ref. [62] fails to accurately describe the energy difference between the atomic
and molecular states. Another potential obstacle to modeling the number loss and burst
formation is the large number of trap states that we would need to include. Since theburst energy is of order 100 nK or 2 kHz, we expect that the number of trap statesrequired would be roughly Eburst /~ω = 2 kHz/12.7 Hz ' 160. However, calculatingtransition probabilities to a large number of trap states might not be necessary; one
alternative is to reduce the number of states by changing the trap frequency used in
the model (and adjusting the coupling matrix elements accordingly). We believe thiswould be justified because the exact structure of the harmonic trap potential should
not affect the details of the non-adiabatic transitions, which occur on a time scale far
shorter than an actual trap oscillation period. In the future, it may be possible to devise
an improved model Hamiltonian that would allow one to predict the dependences of theexperimental observables on the magnetic field pulses. Another possibility would be tofind a way to integrate the Schrodinger equation using the exact eigenvalues from thetranscendental equation in the Dunn/Greene model. Whether or not the BEC number
loss and burst formation can be understood quantitatively in terms of simple 2-bodyinteractions remains to be seen.

5.5.4

Modeling the resonance physics with quantum field theory

Servaas Kokkelmans and Murray Holland have developed a quantum field theory
to describe the physics of near-resonant 85 Rb BEC [56, 9]. This theory has been highly

132
successful in describing the atom-molecule coherent oscillation experiments of Chapter 6,
so it is reasonable to expect good agreement with the single pulse observations of thepresent Chapter. However, a detailed comparison between theory and single pulse data
has not yet been made.
The Kokkelmans/Holland theory moves far beyond the level of the two-body scattering models discussed in the previous sections. In this theory, the BEC is described bya mean-field theory that includes pair correlations between atoms. Although all correlations between atoms are usually neglected in the standard Gross-Pitaevskii equation,
the authors of Ref. [56] argue that the pair correlations are essential for understanding
the BEC physics near a Feshbach resonance. The resonant two-body scattering physics
is accurately contained in the effective field theory, where atomic and molecular field
operators describe the open and closed channel states of the Feshbach resonance. The
closed channel is coupled to a continuum of scattering states above the open channel
threshold.
The effect of rapid B-field variations on the BEC can be determined by numerically solving the equations in Ref. [56]. As in the experiment, the model shows thata magnetic field pulse leads to number loss from the BEC. Most of the loss is due toformation of relatively hot noncondensate atoms (identified by Kokkelmans and Hollandas the burst atoms). These noncondensate atoms are very strongly correlated because
they are created as pairs of atoms with equal and opposite momenta (±p). Each pairof burst atoms can be viewed as a dissociated molecule. The model also predicts thecreation of a fairly small number of bound molecules, which are assumed to be undetected or missing in the experiment. Focusing on the burst, the Kokkelmans/Holland
theory predicts values for both the number and energy distribution of these atoms. The
predictions for Nburst and Eburst agree quite well with the double magnetic field pulse
experiments in Chapter 6 and we expect that the model will also accurately describe
the single pulse experiments [55].

133
While the mean-field theory from Kokkelmans and Holland is conceptually much
more complex than the avoided crossing models described previously, the field theory has
the important advantage that it has successfully described the atom-molecule coupling inour double magnetic field pulse experiments. We will therefore provide more discussionof the Kokkelmans/Holland theory and its relationship to the experimental data in thenext chapter.
A different quantum field theory for our BEC experiments was recently proposedby the Oxford theory group [16]. This theory shares some similar features with the
Kokkelmans/Holland work, but the authors use a different method to incorporate theunderlying two-body scattering physics into the many-body Hamiltonian. In addition,
Kohler et al. examine the effect of a harmonic trapping potential on the physics of theatom-molecule coupling. Like the Kokkelmans/Holland theory, the theory of Ref. [16]yields a number of predictions that agree with experimental observables. More discussion of the Kohler et al. model will be postponed until Chapter 6.

Chapter 6
Atom–molecule coherence in a Bose-Einstein condensate

6.1

Introduction
The strange time dependence of BEC number loss described in Chapter 5 strongly

suggested the presence of non-adiabatic transitions to an undetected molecular state.
To further explore this loss process, we utilized a pair of carefully controlled magnetic
field pulses, separated in time, in the vicinity of the Feshbach resonance.
The original motivation for the double B-field pulse sequence was to measure the
BEC recovery time after a single pulse. We hypothesized that the first pulse would
cause a large amount of 3-body recombination-induced number loss to the condensate.
The abrupt increase in recombination should produce a multitude of tiny holes or voidsin the BEC wavefunction. The sudden creation of voids would lead to condensate
excitations because of the localized changes in the mean-field interaction potential [66].
One might ask the question of how long does it take for these voids in the BEC to refill
as the surrounding atoms move in. Presumably the condensate recovery time could
be determined by applying a second, identical B-field pulse shortly after the first one.
One could increase the delay time between pulses until the 2nd pulse produced thesame BEC loss as the first pulse; this delay time would define the recovery time andit should give information about the microscopic physics occurring inside the BEC. By
measuring the BEC response on such short time scales, one might hope to find evidence
for a breakdown in the predictions of mean-field theory.

135
However, the recovery time picture proved too simplistic. The main defect of thishypothesis lies in its assumption that the BEC number loss is incoherent; in contrast, we
actually discovered that a significant part of the loss is due to coherent transfer to another state. We measured coherent oscillations in the BEC atom number as the spacing
between the two magnetic field pulses was varied. The observation of coherent oscillations provides us with definitive evidence for coupling between the atomic and molecularstates associated with the 85 Rb Feshbach resonance. These experiments demonstrate
a remarkable quantum superposition between two distinct chemical species: free atoms
and weakly bound molecules.
Although the single B-field pulse data from Chapter 5 also suggest that there
is nonadiabatic mixing of the atomic and molecular states, these data were not as
convincing as the results described here. The experiments discussed in this Chapter have
led to a much more accurate theoretical picture of Feshbach resonances; in particular,
the physical importance of the shallow s-wave bound state has become clear. As a resultof our observations, the phenomenon of atom-molecule coupling and the role of the swave bound state in the dynamics of the near-resonant 85 Rb BEC have been exploredrecently by several theoretical groups [56, 16, 67, 68]. A discussion of the present
agreement between experiment and theory will be given at the end of the Chapter.

6.2

Experimental methods
To study atom-molecule coherence, we first created 85 Rb condensates typicallycontaining 16,500 atoms, with fewer than 1,000 non-condensed thermal atoms. The
initial number, Ninit , fluctuated from shot to shot by ∼500 atoms (∼ 3% number noise).
After producing a condensate at a B-field of 162.3 G, we ramped the magnetic field
adiabatically to 165.5 G, corresponding to an initial scattering length ainit ' 7 a0 . The
spatial distribution of the atoms was Gaussian with a typical average atom density ofhninit i = 1.8 × 1013 cm−3 . For some experiments where a lower initial density was

136
desired, the adiabatic ramp to 165.5 G was omitted so that the mean field repulsion at
162.3 G (ainit ' 200 a0 ) reduced the average initial density to hninit i = 3.5 × 1012 cm−3 .
After preparing the condensate, we applied a selected fast magnetic field pulse
sequence by sending an appropriate time-dependent current through the auxiliary magnetic field coil. A representative set of double B-field pulses is shown in Figure 6.1.
The sequence is composed of two nearly identical short trapezoidal pulses separated bya region of constant magnetic field (Bevolve ). In our experiments, we varied the evolution time between pulses, tevolve , from 0 to 400 µs, while Bevolve ranged from 162 G to
156 G. We determined the value of Bevolve between the two pulses using the short rf
pulse technique described in section 3.3.3. We also found it useful to alter the initial
magnetic field, Binit , to change the initial BEC density and distance from the Feshbachresonance. Usually, Binit was 162.3G or 165.5 G. We chose the B-field pulse amplitude
so that the pulses always approached to within ∼2 G of the Feshbach resonance.
Upon completion of the fast-pulse sequence in Figure 6.1, we ramped the magnetic
field from 165.5 G to ∼157 G in 5 ms and held the B-field constant for an additional 7 ms
to allow the repulsive mean-field energy to expand the condensate (see section 3.5.3).
Then we turned off the magnetic trap and used destructive absorption imaging 12.8 ms
later to observe the condensate and burst atoms. Our detection scheme was not sensitive
to atoms with kinetic energies larger than ∼ 2 µK nor to atoms in off-resonant molecularstates.
As we observed for single pulses toward the Feshbach resonance, there were two
distinct components of atoms visible in the absorption images and a third missing component that we could not detect. One of the observed components was a cold remnant
BEC which was not noticeably heated or excited by the fast-pulse sequence (< 1 nK
kinetic energy was imparted by the pulse sequence), while the other component was a
relatively hot (∼200 nK) burst of atoms that remained magnetically trapped during the
BEC expansion time. Using the PG model to simulate the mean-field expansion that

137

166
164

B (G)

162
tevolve

160
Bevolve

158
156

Pulse #1

Pulse #2

154
0

20

40

60

80 100 120 140
t (µs)

Figure 6.1: Magnetic field pulse shape. B-fields shown for pulses #1 and #2 correspondto scattering lengths of ∼3300 a0 , and the free precession field, Bevolve , corresponds toa scattering length of ∼600 a0 . The dashed line indicates the position of the Feshbachresonance, where the scattering length diverges to infinity. In the text, we refer to thefree precession time as tevolve . The average ramp time for most of the pulses was 14 µs.

138

Figure 6.2: An absorption image taken after the fast magnetic-field pulse sequenceand the mean-field expansion. The horizontal bar indicates the optical density. The
horizontal and vertical directions coincide with the axial and radial axes of the trap,
respectively. The dimensions of the image are 366 × 52 µm. The BEC remnant is
the roughly spherical cloud at the center, while the burst atoms are focused into athin line along the axial direction. There is a dramatic difference between the two
spatial distributions, owing to the large difference in the mean energies of the burst andexpanding BEC remnant (hEburst i ' 50 × hErem i).

139
we applied to the BEC remnant to measure its number, Nrem , we found that we should
impart ≤ 3 nK expansion energy to the remnant before imaging. This estimate agrees
well with the expansion velocity that we measured after the trap turn-off. Thus theremnant BEC was more than 60× colder than the burst. A typical absorption imageof the atoms remaining after the double B-field pulses and time-of-flight expansion isshown in Figure 6.2.
To find the number of atoms in the remnant BEC and the number of burst atoms,
we allowed the magnetic trap to focus the burst cloud before imaging. We fit the focused
burst (which had a much larger spatial extent than the remnant) with a two-dimensional
Gaussian surface, excluding the central region of the image that contained the remnant.
This fit yielded the number of burst atoms, Nburst . Subtracting the fit from the image
and performing a pixel-by-pixel sum of the central region of the image gave the remnant
number, Nrem .

6.3

Observation of population oscillations

6.3.1

BEC and burst number oscillations versus tevolve

We observed large-amplitude oscillations in the number of BEC and burst atoms
as we varied the delay time between magnetic field pulses. Representative number
oscillation data for the condensate and burst atoms are displayed in Figure 6.3. The
number oscillations were only visible after two B-field pulses; in fact, with only pulse #1
and the subsequent constant B-field but with no pulse #2, Nrem showed no variation
except for a slow decay. The gradual decay rate of a few atoms per microsecond isconsistent with the loss rate expected for a single pulse to the evolution magnetic field,
as discussed in Chapter 5. Although we studied the oscillations under a wide variety
of different conditions, we never observed the double-pulse value of Nrem to be larger
than the single-pulse value, implying that each B-field pulse causes some incoherent or140
irreversible loss to the BEC.

6.3.2

Nmissing oscillations

The BEC and burst atom oscillations in Figure 6.3 have very similar amplitudes
and frequencies and they appear almost completely out of phase. Nburst grows larger
when Nrem decreases, and vice versa. However, the measured phase shift between thetwo oscillations is actually 154(4)◦ , which is significantly different from 180◦ . Thus, the
BEC and burst number oscillations do not exactly cancel out — if one calculates thetotal number of atoms, Ntot =Nrem +Nburst , this quantity also displays clear oscillations
versus evolution time. As shown in Figure 6.4, the amplitude of the Ntot oscillations is
significantly smaller than that of the other two number components.
Figure 6.4 clearly demonstrates that the time-averaged total number of atoms
after the two B-field pulses does not equal the initial number of BEC atoms. There are
a number of missing atoms, Nmissing , and the measured oscillations in Ntot correspondto oscillations in Nmissing . We therefore have evidence that some of the BEC atoms are
oscillating into and out of a different state that is not detectable by the experiment.
This phenomenon of oscillations to a state that is invisible to the detector may becompared to the well-known neutrino oscillations in astrophysics. We assume that theundetected state in our BEC experiment is the weakly bound molecular state of the
Feshbach resonance.
As a side note, the data in Figure 6.4 were measured with fairly low-density
condensates to minimize the number of missing atoms that do not oscillate and thereby
allow for more atoms in the remnant BEC.1 This improved the visibility of the remnant
and the signal-to-noise ratio for the absorption images. To produce condensates withlow density for these measurements, we altered the magnetic pulses somewhat from1

We observed that the B-field pulses always produce some missing atoms whose number does notoscillate with tevolve . These atoms may or may not be converted into the weakly bound molecular state.
In any case, we found that reducing the BEC density decreased the number of non-oscillating missing
atoms. For further discussion of this point, see section 6.4.2.

141

14000

BEC number

13000
12000
11000
10000
9000

(a)

8000
10

15

20

25

30

35

40

30

35

40

tevolve (µs)
7000

burst number

6000
5000
4000
3000
2000

(b)

1000
10

15

20

25
tevolve (µs)

Figure 6.3: Coherent oscillations in BEC number and burst number. (a) Black points
are the remnant BEC number, Nrem , versus evolution time, tevolve , between the twomagnetic field pulses. The solid line is a fit to the data using a damped sine wave witha frequency of 196(1) kHz and a damping rate of 11(4) × 103 s−1 . The open circles
near Nrem = 13000 indicate the number remaining after only pulse #1 and tevolve at
159.84 G. (b) Black points are the number of burst atoms versus tevolve . The fit (solidline) yields an oscillation frequency and damping rate that are consistent with thosefrom the fit to the BEC number in part (a). The clear circles are the burst number
measured after a single pulse plus a plateau at Bevolve . For the data in (a) and (b), theinitial BEC density was hninit i = 3.5 × 1012 cm−3 and Binit =162.25(5) G. The evolution
B-field was Bevolve =159.84(2) G.

142

number of atoms

16000
12000
8000
4000
0
10

15

20

25

30

35

40

tevolve (µs)
Figure 6.4: Number versus tevolve for low density condensates (hninit i = 3.5×1012 cm−3 ).
From bottom to top, the data are Nburst (open circles), Nrem (filled circles), and the total
number of observed atoms (gray squares). Each data set is fit by a damped sine wave asshown by the lines. Here the fitting function is y = y0 +A exp (−t/τdecay ) sin (2πν0 t + φ),
where ν0 is the frequency and τdecay is the decay time constant. The initial BEC number,
Ninit = 17,100, is indicated by the flat dashed line. The fit to the BEC remnant data
gives an oscillation frequency of 196(1) kHz and τdecay = 91(33) µs (the longer time
data used to determine τdecay are not shown).

143
the shapes shown in Figure 6.1. We set the initial magnetic field before the fast-pulse
sequence to 162.25(5) G and reduced the amplitudes for pulses #1 and #2 to ∼7 G.
The evolution B-field was Bevolve = 159.84(2) G. Under these conditions, the initial
BEC density was relatively low: hninit i = 3.5 × 1012 cm−3 .
6.3.3

Oscillation frequency versus Bevolve

Very strong evidence for the importance of the weakly bound molecular state tothe observed BEC dynamics can be obtained from the magnetic field dependence ofthe number oscillation frequency. We have discovered that the oscillation frequency(ν0 ) for the BEC, burst, and missing atom populations matches the binding energy ofthe molecular state: hν0 =²bind . We explored the B-field dependence of the oscillation
frequency by applying different values of Bevolve during the evolution time between themagnetic field pulses. As shown in Figure 6.5, we mapped out ν0 versus B-field overa ∼5 G range above the Feshbach resonance field of 155 G. The oscillation frequency
varied by almost two orders of magnitude as Bevolve increased from 156.6 G to 162 G.
Along with the measured frequencies and magnetic fields, Figure 6.5 also displays
theoretical predictions for the molecular bound state energy relative to the atomic state.
One prediction, shown by the dotted line, comes from basic zero-energy scattering theory. In the regime where the scattering length is positive and much larger than theradius of the interatomic potential well, the bound state energy for an arbitrary attractive potential can be approximated by ² = −~2 /ma2 [11], where ~ is Plank’s constantdivided by 2π, m is the atomic mass, and a is the scattering length. The same equation
relates the bound state energy to the effective scattering length, which is calculated fromthe Feshbach resonance parameters through the relation a = abg × (1 − B−B∆peak ), whereabg is the background scattering length, ∆ is the width of the Feshbach resonance, andBpeak is the resonant magnetic field. The quantity |²|/h is plotted with no adjustable parameters in Figure 6.5. The measured oscillation frequencies are in excellent agreement

144

1400

150

frequency (kHz)

1200
1000
800
600

100
50
0
157

400

158

159

200
0
157

158

159
160
Bevolve (G)

161

162

Figure 6.5: Oscillation frequency versus magnetic field. The points are the measured
frequencies. The solid line is the energy difference between the atom–atom threshold
and the bound molecular state found by S. J. J. M. F. Kokkelmans with a coupledchannels scattering calculation. The dotted line is a plot of |²| = ~2 /ma2 . The inset
is an expanded view of the lower-frequency data. The magnetic-field measurements forthe points with the smallest horizontal error bars were performed on the same days as
the corresponding frequency measurements. The error bars for the points with larger
field uncertainties were inflated by 100 mG to account for estimated day-to-day field
drifts.

145
with this simple model over the range of magnetic fields where the model is expected
to be valid.
The solid line in Figure 6.5 is a prediction of the binding energy from a coupled–channels (CC) scattering calculation by Servaas Kokkelmans. This calculation is basedon the results of a recent determination of the rubidium collisional interactions fromseveral high-precision data for 85 Rb and 87 Rb [69]. We see that the coupled-channels
predictions for the binding energy are in excellent agreement with the oscillation frequency data over the entire range of magnetic field. It is important to emphasize thatboth the simple expression and the more sophisticated CC calculation provide predictions for the binding energy; they are not fits to the oscillation frequency data. The
magnetic field dependence of ν0 is discussed in greater detail in Chapter 7, where we use
precision measurements of the frequency and B-field to tightly constrain the parametersof the Feshbach resonance.

6.3.4

Interpreting the oscillations

The fact that the population oscillations occur at exactly the frequency corresponding to the molecular binding energy clearly indicates that we are creating a coherent superposition of atoms and molecules with the sudden magnetic-field pulses.
We suggest the following interpretation of our observations. Each magnetic field pulse
provides a sufficiently rapid perturbation to result in coherent nonadiabatic transitions
between the atomic and molecular states. The first B-field pulse produces an atommolecule superposition state and the relative phase between the two states in the superposition evolves according to their relative energy difference during tevolve . This energy
difference is determined by the B-field-dependent binding energy at Bevolve . Becauseof the relative phase evolution, the final transition probability from the atomic state to
the molecular state after the second pulse depends on tevolve . The observed population
oscillations are due to a quantum-mechanical interference between the transition am-

146plitudes for pulse #1 and pulse #2. Another way to understand the oscillations is toconsider that the 2nd pulse transfers some population from the molecular part of thesuperposition to the atomic state. This “new” probability amplitude in the atomic state
can interfere with the amplitude that remained in the atomic state during tevolve . The
observed oscillation behavior is somewhat analogous to that seen in Ramsey’s methodof separated oscillatory fields [24], in which two microwave pulses, separated in time,
drive transitions in a two-level atom.
Although our experiment possesses some similarities to the Ramsey 2-pulse experiment, there are also many differences. It is illustrative to explore these differences.
For instance, we observe three distinct components of atoms oscillating as a function ofevolution time, each with a different oscillation phase and amplitude. In contrast, theconventional Ramsey oscillations occur in a two-level system where the populations of
the two states are always exactly out of phase with one another. An additional difference between our experiment and the Ramsey case is the precise nature of the coupling
pulses. In the Ramsey experiment, the two states are coupled by a pair of rf or microwave radiation pulses that are nearly resonant with the transition frequency between
the two states. Each radiation pulse has a duration that is much longer than the periodof the rf or microwave radiation. Our coupling pulses, on the other hand, involve nonperiodic time-variations of a magnetic field. We achieve coupling between the atomic
and molecular states by briefly ramping down the relative energy difference between
the states toward zero, as in a Landau-Zener experiment. In fact, our experiment isprobably best pictured as a combination of Landau-Zener magnetic field coupling pulses
separated by a Ramsey-like relative phase evolution between the atomic and molecularstates.
The experimentally observed population oscillations in the BEC number indicatethat we have produced a coherent superposition of atomic and molecular Bose-Einsteincondensates. We propose a simple argument to justify our claim for the existence of

147
the molecular BEC (mBEC). First, we know from the phase coherence property ofcondensates that the initial atomic BEC possesses a common phase across the sample
of atoms. After the double B-field pulses, we measure large contrast oscillations in
the number of BEC atoms, which implies that there is a large fraction of the atomic
BEC participating in the population oscillations. If a large fraction of the atomic BEC
participates in the oscillations, then the atom/molecule superposition must exist at
significantly different spatial positions within the original atomic BEC volume. Thus,
we produce molecules that are distributed in space and are phase coherent — these are
two important properties that identify a BEC! The creation of a molecular BEC is aremarkable achievement and should provide an abundance of avenues for future study.

6.3.5

Mechanisms for missing atoms

If our interpretation of the BEC number oscillations is correct, then one mightask how it is that we distinguish between the atomic and molecular states. In any
interference experiment, there must be some way to discriminate between the two statesthat are interfering with one another. Since we find that some atoms are missing fromthe absorption images, we evidently do have a discriminating mechanism that is “builtin” to the experiment.
In our experiment, missing atoms could be created by at least two different mechanisms involving the molecular state. One possibility is that the molecules formed by
the double B-field pulses are destroyed when we turn off the magnetic trap and cross the
Feshbach resonance. A related idea is presented in Ref. [15]. By turning off the trap,
we cause the molecular state to become unbound (the molecular state lies far above thecolliding atoms threshold energy for B=0). The molecules could therefore dissociate
into sufficiently high energy atoms during the trap turn-off (> 1 µK) that we would notdetect them in the absorption imaging.
Another mechanism for the generation of missing atoms is the possibility of in-148
elastic molecule-atom or molecule-molecule collisions during the ∼30 ms time between
the double pulse sequence and the magnetic trap turn-off. In principle the molecules
should be confined in the magnetic trap because their magnetic moment is very similar to that of the |2, −2i atomic state for magnetic fields near the Feshbach resonance.
During the time that a given molecule resides in the magnetic trap, if the molecule
collides with an atom or another molecule, it may drop to a lower energy vibrational
level, gaining substantial kinetic energy in the process. In this case, the deeply boundmolecule would not experience a Feshbach resonant coupling to the atomic state and themolecule would remain a molecule after the magnetic trap turnoff. However, we would
not see the molecule because of its large kinetic energy and low probability of scattering
photons from the probe laser. It is important to note that the cross-sections and rates
for collisions between highly-vibrationally-excited molecules and ultracold atoms havenot been calculated for the heavy alkali atoms [51, 70], although some results have beenobtained for He-H2 collisions [71]. Measuring such a cross-section for Rb2 would be amajor experimental breakthrough.
To investigate the mechanism for missing atom generation, we attempted to measure the number oscillations by imaging the BEC in the magnetic trap. The goal was
to look at the BEC immediately after the fast pulse sequence or even during tevolve
between the pulses. In theory, the presence or absence of oscillations in the data would
then help us to identify how the missing atoms are removed from the sample under ournormal operating conditions.
Unfortunately, our attempts to extract useful information from the in-trap imaging of the condensates proved inconclusive. Due to a number of systematic effects related
to the small size of the unexpanded BEC, the large Zeeman shift of the trapped atoms,
and optical pumping effects that reduce the number of photons scattered by each atom,
the absorption images were difficult to interpret. For example, one consequence of notallowing the BEC and burst atoms to expand is the inability to distinguish between the

149
BEC and burst populations in the images. When we tried to circumvent this problem
by expanding the BEC in the trap and then imaging the atoms, the data indicated thatthe missing atoms were still missing before the trap turn-off. However, this result should
be considered preliminary and needs to be confirmed with more data. Although we may
return to this experiment in the future, at present we have not definitively determined
the mechanism for the missing atoms.

6.4

Controlling the oscillation characteristics
After making some preliminary measurements of the atom-molecule population

oscillations, we made a detailed study of the oscillation characteristics. To learn more
about the physics of this unique system, we explored the dependence of the oscillations
on the initial BEC density as well as the particular shapes of the two magnetic field
pulses. The behavior of the atom-molecule oscillations under a variety of different
conditions is described in this section.

6.4.1

Amplitude of BEC number oscillations

The first property of the number oscillations that we studied was their amplitude.
We desired to increase the amplitude as much as possible, both to increase the numberof molecules created and also to improve the signal-to-noise ratio for determining otherproperties, such as the frequency and phase.
Keeping the pulse lengths fixed at 15 µs, we varied the double pulse amplitudes
to change the approach distance to the Feshbach resonance. For pulses to ∼156.3 G,
we measured a BEC number oscillation amplitude of roughly 1700 atoms. Increasing
the pulse amplitudes to reach ∼155 G caused no change to the oscillation amplitude.
The lack of a change is not surprising when one considers that the molecular binding
energy has very little B-field dependence from ∼156.3 G to ∼155 G (see Figure 6.5) eventhough the scattering length increases from ∼3000 a0 to more than 20,000 a0 . Since

150
the magnetic field dependence of ²bind is nearly quadratic with B-field to the right of
the Feshbach resonance, we expect that the population oscillations should dramaticallydecrease in amplitude for pulses that only approach to within 3 or 4 G from resonance.
We did not attempt to study this expected decrease in oscillation amplitude. We feltthat the single pulse data on BEC number loss versus pulse amplitude (see section 5.3)
should be adequate for predicting the trend.
Based on the single pulse data for BEC loss, we suspected that varying the pulse
lengths of the double pulses could substantially change the amplitude of the population
oscillations. Although this proved to be true, nevertheless our original choice of 15 µs forthe pulse lengths turned out to produce the largest number oscillations. We studied thepulse length dependence of the oscillation amplitude as the length was increased from
5 µs to 50 µs. The shortest pulses produced very little loss from the condensate, so theoscillation amplitude was small. In contrast, the longest pulses caused huge losses fromthe condensate (more than 50%), but the oscillation amplitude actually decreased. We
found that simply increasing the overall number of missing atoms does not necessarily
increase the missing atom oscillation amplitude, implying that there may be two parts
to Nmissing . Some of the loss is coherent and leads to number oscillations while anotherpart of the loss is incoherent. The incoherent number loss grows very rapidly withincreasing magnetic pulse length.

6.4.2

Strong density dependence of Nmissing

In addition to the dependence of Nmissing on pulse length for the double pulses,
we also observed a strong dependence on the initial condensate density. When the
BEC density was increased from hninit i = 3.5 × 1012 cm−3 by more than one orderof magnitude, the missing fraction increased from ∼8% to ∼40% of the initial BEC
number. In addition, the number of burst atoms increased dramatically. As a result,the average number of BEC remnant atoms was less than the average number of burst

151
atoms.
We show the three populations of atoms versus tevolve for high initial BEC density
in Figure 6.6. While the burst and condensate numbers apparently still oscillate at thesame frequency for these data, the phases and amplitudes of the oscillations in Nrem
and Nburst are such that the total number oscillations are not visible. Under these highdensity conditions, the number of missing atoms is huge — roughly 10,000 atoms! HereNmissing exceeds the average value of Nrem , which is about 7000.
We took data similar to those shown in Figure 6.6 for several values of the BEC
density and studied the resulting number of missing atoms. To isolate the density
dependence of Nmissing , we used fixed values of Bevolve and pulse length. We varied theinitial density by two different techniques. In the first method, we changed the numberof atoms in the BEC by holding the BEC in the magnetic trap for 5 to 10 s after
completion of evaporative cooling. Density-dependent losses then reduced the numberof BEC atoms without appreciably heating the condensate [1]. Our second method
for changing the density was by adiabatically varying the initial magnetic field before
the set of double pulses, thereby altering the repulsive self-interaction strength in thecondensate. This caused the volume occupied by the BEC ground state wavefunction
to vary with the magnetic field. In the latter method, we also adjusted the amplitudeof the double pulses to ensure that they maintained the same approach distance to the
Feshbach resonance.
The dependence of Nmissing on BEC density is illustrated in Figure 6.7. Overthe range in the plot, the missing fraction rises from nearly zero to 0.4. For somereason, the smooth growth of the missing fraction with density saturates above hninit i '2 × 1013 cm−3 . Further investigation of the missing atoms is needed to understand thistrend as well as the origin of Nmissing .

152

30000

number of atoms

25000
20000
15000
10000
5000
0
10

20

30

40

50

tevolve (µs)

Figure 6.6: Number versus tevolve for high density condensates (hninit i = 4.4 ×
1013 cm−3 ). As in Figure 6.4, we show Nrem (filled circles), Nburst (open circles), andthe total number of observed atoms (gray squares). The lines are damped sinusoidal
fits to the different populations of atoms. The initial BEC number, Ninit = 27,800,
is indicated by the flat dashed line. The fit parameters for the remnant oscillationsare ν0 = 133(1) kHz and τdecay = 38(8) µs. We added an additional term to the fit
function to account for an overall linear loss rate to the number. The fit gave a loss
rate of -19(6) atoms/µs. The evolution B-field was Bevolve = 159.35(5) G. Because theinitial magnetic field before the fast-pulse sequence was 165.71(15) G, we used pulse
amplitudes of ∼10 G to closely approach the Feshbach resonance.

153

0.5

missing fraction

0.4
0.3
0.2
0.1
0.0
0

1e+13

2e+13

3e+13

4e+13

5e+13

initial BEC density (cm-3)

Figure 6.7: Fraction of missing atoms as a function of the initial BEC density. The
missing fraction is defined as 1-hNtotal i/hNinit i, where hNtotal i is the average sum of the
burst atoms and the BEC remnant atoms and hNinit i is the average number of BEC
atoms before the double pulse sequence. We can determine the density by measuring
the initial magnetic field (scattering length) and calculating the volume of the BEC
using the PG model. After finding the volume, we use the measured value of hNinit i to
calculate the average density. The typical uncertainty in this density determination is∼20% due to uncertainty in the measured number of atoms and the initial scattering
length used in the PG model.

154
6.4.3

Damping of Nrem oscillations

Damping of the number oscillations can be seen in most of the data that wecollected. For values of Bevolve between 157 G and ∼161 G, the number oscillations areunderdamped, so that the damping time is much longer than the oscillation period. To
obtain a high precision measurement of τdecay it is therefore necessary to measure thenumber oscillations over many cycles. One obvious method to obtain the damping time
without requiring too many points is by taking closely-spaced data in windows that are
separated by longer time intervals, ∆t >> 1/ν0 . Then one can fit the entire set to adamped sine wave that interpolates in the intervals between the data windows.
Unfortunately, we found this method did not work well in practice because of thepresence of phase noise in the data. We observed shot-to-shot noise in the oscillation
phase for different repetitions of the experimental cycle. The noise was bigger for longer
tevolve , indicating that the source of the noise was shifts to the oscillation frequency.
The frequency noise was likely due to shot-to-shot magnetic field noise. We provide
further discussion of this point in section 7.2.3.
To determine the damping rate despite the presence of phase noise, we adopted
another technique for data measurement. We measured Nrem versus tevolve in several
windows having a length of one or two oscillation periods. The spacing between windows
was chosen to be comparable to our estimated 1/e time for the damping. We analyzed
the data by computing the maximum variation in remnant BEC number within eachtime window. We plotted ∆Nmax versus tevolve and fit the data to a decaying exponentialfunction, as shown in Figure 6.8.
We discovered that the damping rate was quite insensitive to both initial BEC
density and magnetic field in the B-field range from 157 G to ∼161 G. Recently, we havefound that the rate (1/τdecay ) actually increases rapidly as the magnetic field increases
above ∼161 G, which is possibly related to the rapid variation in binding energy with

155

11000

(a)

BEC number

10500
10000
9500
9000
8500
8000
0

50

100

150

200

250

300

350

400

tevolve (µs)

∆Nmax in 14 µs window

3000
2500

(b)

2000
1500
1000
500
0
0

100

200

300

400

tevolve (µs)

Figure 6.8: Damping of BEC number oscillations. (a) BEC remnant number (Nrem )versus tevolve . The data are grouped into 4 windows with each window consisting of
2 oscillation periods. (b) Peak-to-peak variation in remnant number, ∆Nmax , versus
tevolve (black points). ∆Nmax is calculated for a single oscillation period, which isroughly 14 µs in length. The line is an exponential decay fit to the data, with thebaseline fixed at the average of the four longest time data: Nbase =640 atoms. The
damping time from the fit is 82(14) µs (damping rate γ = 12(2)×103 s−1 ). The baselineof 640 atoms is quite reasonable considering the peak-to-peak number noise observed
during the measurement. Note that the peaks of the oscillations in part (a) decrease
more rapidly than the troughs due to an overall loss of BEC atoms during tevolve . We
find that the loss is linear with time and occurs at a rate of -2.2(2) atoms/µs.

156
magnetic field. Both the B-field dependence and possible mechanisms for the damping
are presently under investigation and will be the subject of a future publication by ourgroup.
We also looked for a temperature dependence of both the damping and the oscillation frequency at ν0 ∼ 200 kHz and did not see any. The high-temperature data weremuch noisier than the data for pure condensates, due to unexplained enhanced noisein the number of thermal atoms after the magnetic field pulses, but when the initial
thermal fraction was increased from <5% to 30%, the data still displayed oscillations
with frequency, amplitude, and damping consistent with what was observed with low
temperature data. Since we changed Ntherm by more than a factor of 6 and saw noeffect, any dependence of the damping rate on temperature must be very weak.

6.4.4

Relative phase between Nrem and Nburst oscillations

A particularly intriguing aspect of our data is the observation of three distinct
oscillations in the various number components visible from the absorption images. Because the relative phase between the BEC remnant and the burst oscillations does notequal 180◦ , there is a third oscillation visible in Nmissing (or Ntotal ). We believe that the
Nmissing oscillation amplitude may correspond to the number of molecules produced by
the double pulse sequence.
For reasons that will be elucidated later in this section, we suspected that the
number of molecules might be increased by ramping the B-field more slowly at theend of magnetic field pulse #2. We therefore examined the amplitude of the Nmissing
oscillations as a function of the final ramp time of the second B-field pulse. As aresult of changing this ramp time, there were variations in the relative phase between
the BEC remnant and burst number oscillations, ∆φ = φburst − φBEC . The relative
phase variations caused the Nmissing oscillation amplitude to change significantly. For
example, increasing the final ramp time of pulse #2 from our default value of 11 µs to

157
160 µs caused the missing number oscillation amplitude to increase from 1000 atoms tonearly 2800 atoms. The number oscillations for the 160 µs final ramp time are displayedin Figure 6.9.
By comparing different data sets such as those shown in Figure 6.9, we observedthat the relative phase shift , ∆φ, decreased as the final ramp time for pulse #2 became
longer. The phase shift behavior was entirely due to a phase shift in the burst number
oscillations; the BEC remnant phase remained fixed. As the final ramp time increased,
the burst oscillation phase shifted in such a way that the lowest curve in Figure 6.9 moved
smoothly to the right. We show the quantitative details of this trend in Figure 6.10.
There ∆φ is seen to decrease from ∼150◦ to ∼80◦ as the ramp time goes from 10 µs
to 330 µs. After an initially rapid decline, the relative phase eventually levels out forslower and slower ramps.
After demonstrating that the relative phase shift could take on values very different from 180◦ , we scratched our heads and wondered about the origins of the shift.
Since we had generally observed that fewer atoms in the BEC correspond to more atoms
in the burst, we speculated that the phase shift between the Nrem and Nburst oscillations might originate in a time delay between the disappearance of the BEC and thecreation of the burst atoms. To determine whether ∆φ was a true phase shift or was
simply a shift caused by a fixed time delay, we measured ∆φ at two different values ofthe oscillation frequency. While holding the pulse #2 final ramp time fixed at 160 µs,
we changed Bevolve to alter the oscillation frequency. Decreasing ν0 from ∼200 kHz to
56 kHz caused the relative phase shift to change from 68(7)◦ to 88(9)◦ . The relative
phase shift therefore increased by a factor of 1.3(2). However, if the phase shift were
due to a fixed time delay, then decreasing the frequency by a factor of 4 should cause∆φ to drop by a factor of 4 also (so we would expect 68◦ /4 = 17◦ in this case). Thus,
the time delay idea is clearly not correct.
Nevertheless, it remains true that the second magnetic pulse shape affects the158number of atoms

16000

12000

8000

4000

0
2

4

6

8

10

12

14

16

tevolve (µs)

Figure 6.9: Enhanced Nmissing oscillations. Number versus tevolve for a relatively longpulse #2 final ramp time of 160 µs. From bottom to top, the data are Nburst (opencircles), Nrem (filled circles), and the total number of observed atoms (gray squares).
The different data sets are fit by damped sine waves as shown by the lines. The relative
phase between burst and remnant oscillations is φburst − φBEC =68(7)◦ , which is morethan 2× smaller than the 147(8)◦ phase difference for our typical 11 µs final ramp time(see Figure 6.4 and related discussion in the text).

159

180
phase difference (deg)

160
140
120
100
80
60
40
0

50

100

150

200

250

300

350

2nd pulse final ramp time (µs)

Figure 6.10: Relative oscillation phase versus final ramp time of 2nd B-field pulse. Black
points with error bars are the phase difference φburst − φBEC . To calculate the phasedifference, we first fit each population oscillation (remnant BEC atoms, burst atoms, andtotal number of atoms) to a sinusoidal function. From these three fits, we then computethe weighted average oscillation frequency. We next fit each component of atoms again,
constraining the oscillation frequency for the fit to be equal to the weighted averagevalue. This procedure allows us to determine the phase shift between the burst andremnant BEC number oscillations with improved precision.

160
ratio of BEC atoms to burst atoms at fixed evolution time. For instance, if we choose
tevolve such that Nrem is at a minimum or trough of the oscillation, then lengthening
the 2nd pulse final ramp time causes the number of burst atoms to decrease and themissing number to increase. Another way to see this is to compare the phases of theoscillations in Nrem and Nmissing . As the final ramp time becomes longer, the remnant
oscillations become more and more out of phase with the missing number oscillations.
Concurrently, the amplitude of the Nmissing oscillation grows larger.
These observations suggest that we preserve more molecules (more missing atoms)
after pulse #2 by slowing down the final ramp. We believe that increasing the 2nd
pulse final ramp time enhances the probability that the molecules formed during thecoupling pulse survive the ramp back toward higher B-field (∼166 G). The justification
for this claim is based on the arguments of Chris Greene and Murray Holland. Since
the radial wavefunction of the weakly bound molecular state has a size characterized
by the scattering length (see Chapter 2), a longer pulse should allow a more gradual or
adiabatic decrease in the size of the wavefunction toward smaller and smaller radius. In
contrast, the shorter ramp times are more sudden and they lead to a higher probability
that the molecule will simply dissociate into a pair of free atoms during the ramp.
It is possible that the hugely extended molecules that are created near the Feshbachresonance are very fragile and they dissociate when the B-field jumps quickly back tosmall a. If this interpretation is correct, then the data imply that the dissociation of
the molecular state can be influenced by the magnetic field ramp rate — with slower
ramps leading to less dissociation.
One must take some care when talking in this way about the state of the molecule
at a particular B-field (scattering length) because of the time variation of the field.
During the double magnetic field pulses, the state of the system is far from equilibrium.
We only observe the populations of the BEC remnant and burst atoms long after theatom-molecule coupling dynamics have ended.

161
While the “fragile molecule” hypothesis discussed above appears to have somevalidity, it is far too simple to describe all of the effects of changing the ramp time ofpulse #2. In addition to the relative oscillation phase shifts and the oscillation amplitude
of Nmissing , we observed that several other observables depended on the 2nd pulse final
ramp time. For example, the oscillation amplitude of Nrem and Nburst exhibited a slightdecrease as the final ramp lengthened. There was also a decline in the average energy ofthe burst atoms for slower ramps. These trends have not been explained at the present
time.

6.4.5

Burst energy oscillations

The final feature of the atom-molecule coherence experiments that we describe inthis Chapter involves the energy of the burst atoms. We measured a periodic modulationof the average energy of the burst atoms as a function of tevolve , as shown in Figure 6.11.
The data in the Figure are the average energies corresponding to the burst atom number
data that was plotted previously in Figure 6.6. For these data, number of burst atoms
was the highest we ever observed, so that Nburst >Nrem . The large number of burst
atoms in the absorption images facilitated the determination of the average energy ofthe distribution along the axial direction. As described in section 5.2.3, we fit each burst
atom distribution to a 2-D Gaussian function and used the width of the Gaussian in thelong dimension to calculate the average energy. In Figure 6.11, the observed variationin energy of ∼50 nK is equivalent to a modulation of ∼17 µm in the rms width of the
focused burst atom cloud. The energy oscillations appear to damp out fairly rapidly
until they become lost in the noise after ∼35 µs. Our measurements show that boththe number of burst atoms and their energy distribution change with tevolve .

162

300
280
<Eburst> (nK)

260
240
220
200
180
160
5

10

15

20

25

30

35

40

45

tevolve (µs)

Figure 6.11: Average axial burst energy versus tevolve . Black points are the average
axial burst energy at different values of the evolution time between the double pulses.
The line is a damped sine wave fit to the data for tevolve ≤30 µs. The fit yields theparameters: 222(2) nK average energy, 83(15) nK oscillation amplitude, 133(3) kHz
oscillation frequency, and 70(16)× 103 s−1 damping rate. The oscillations in the averageenergy occur at the same frequency and phase as the burst number oscillations (seeFigure 6.6), while the damping of the energy oscillations is roughly 3× faster than in
the burst number oscillations.

163

6.5

Discussion of theoretical models
The observations of atom-molecule coherence described in this Chapter raise anumber of outstanding questions. First, do the magnetic field pulses really produce
molecules and if so, do the molecules form a molecular BEC? As discussed in section 6.3.4, the fact that we see an oscillation involving a large fraction of the atomic
BEC strongly supports the existence of a molecular BEC. There are many remaining
questions, however. For instance, exactly how many molecules can we produce and how
can we increase this number by optimizing the shape of the B-field pulses? Anotheroutstanding issue is the precise relationship between the burst atoms and the molecular state. Finally, what happens to the molecules after they are created? Are there
collisions and/or mean-field interactions with the surrounding atoms [72]?
For a definitive experimental answer to these questions, we need to devise amethod to directly detect the molecules and their spatial distribution in the magnetic
trap. New experiments are currently being developed at JILA that should allow molecule
detection by photo-ionization and subsequent detection of the molecular ions. However,many useful insights and possible answers to the questions raised by our data can beobtained from recent theoretical work. A number of theorists [9, 16, 67, 73, 68], inspiredby our oscillation data, have studied the 85 Rb BEC system. Some of the theorists havedeveloped models that capture most of the prominent experimental features, especiallythe existence of the different oscillating components of atoms that we observe. We
describe the successes of some of the relevant theories in this section. We also point out
areas that remain unexplored theoretically. At the end of the section, we use a table topresent a concise comparison of the theories to the relevant experimental observables.

164
6.5.1

Kokkelmans and Holland model

Kokkelmans and Holland’s theoretical description of the atom-molecule coupling
experiments [9] was briefly described in Chapter 5. The authors emphasize that it is very
important to properly incorporate the two-body scattering physics into their effective
quantum field theory. The Feshbach resonance in the two-body atomic scattering can be
described most accurately by a coupled-channels calculation, which predicts the B-fielddependent scattering length, the binding energy of the molecular state, and the boundstate wavefunction (see Chapter 2). Kokkelmans and Holland show how to include
the complex scattering physics into a simple effective Hamiltonian used in mean-field
theory [56]. This effective Hamiltonian can then be used to predict the response of anear-resonant condensate to variations in magnetic field.
Kokkelmans and Holland solve a set of time-dependent equations for their effective
Hamiltonian, which depends on magnetic field. Using the experimental B-field pulses,
they integrate the equations for a homogeneous BEC. To model the inhomogeneousdensity distribution of the real condensates, Kokkelmans and Holland conduct theircalculations for a range of densities, finally performing a Gaussian weighted averageof their results. This corresponds to a local density approximation for the BEC. The
authors assert that they can safely ignore any other physics due to the trapping potential
because of the short time duration of the magnetic field pulses (much shorter than themagnetic trap oscillation period).
The theory successfully predicts a number of details of the experiment. Kokkelmans and Holland identify the condensate remnant as the atomic BEC field in their
theory, while the burst atoms of the experiment correspond to noncondensate atoms
that are created in strongly correlated pairs. The missing atoms are identified as amolecular BEC. Kokkelmans and Holland suggest that the reason the molecules turninto missing atoms in the experiment is because of inelastic collisions with BEC atoms

165
before the magnetic trap turn-off.
For a double pulse sequence beginning at Binit =162 G (low initial BEC density),
the Kokkelmans/Holland theory predicts striking oscillations in the number of BEC
and burst atoms, with an oscillation frequency that matches the binding energy of themolecular state. In the theoretical results, there is a phase shift between the remnant
and burst oscillations that agrees with the experiment. In addition to these predictions,
Kokkelmans and Holland find that the average energy of the burst atoms is similar tothat observed experimentally.
There are only two important aspects of our experimental data that are not described by this theory. First, the theory does not contain any mechanism to cause
the large numbers of non-oscillating missing atoms that we observe. We found in section 6.4.2 that increasing the BEC density caused the average value of Nmissing toincrease dramatically. These missing atoms may be removed from the BEC by 3body recombination, which is not contained in the Kokkelmans/Holland model. In
the recombination process, two atoms spontaneously form a molecule during a collision
with a third atoms. An alternative explanation to the recombination idea is that the
non-oscillating missing atoms are in fact coherently formed molecules, but the Kokkelmans/Holland theory fails to predict their presence. An additional feature of the experiment that is not explained by the theory is the strong dependence of the relative phase,∆φ, between burst and BEC remnant oscillations, on the final ramp time of the second
B-field pulse. While Kokkelmans and Holland did observe a such a dependence, it wasmuch weaker than in the data. Since we have interpreted the relative phase variationin terms of making more molecules, it would be interesting to further study the trend
in the context of the Kokkelmans and Holland theory.
Overall, the success of this theory is remarkable. The Kokkelmans/Holland theory
provides an explanation of the burst atoms — they are noncondensate atoms thatdevelop strong pair correlations due to the B-field ramps near the Feshbach resonance.

166
According to the model, the burst atoms significantly outnumber the molecules due tothe large spatial overlap of the weakly bound molecular state with the pairing field that
describes the two-body correlations. For the B-field pulse and initial BEC density used
in Ref. [9], the molecular fraction is predicted to be much smaller than the burst — only
∼ 2% compared to ∼ 25%. In addition, the authors identify the oscillating populationof missing atoms as a molecular BEC.

6.5.2

Kohler, Gasenzer, and Burnett model

Another theoretical paper describing our 85 Rb BEC experiments was recently
produced by the theory group at Oxford [16]. In this work, Kohler, Gasenzer, and Burnett utilize a generalization of the mean-field theory leading to the well-known GrossPitaevskii equation. The theory goes beyond the GP equation by including first andsecond-order correlations between the atoms. This leads to a nonlinear Schrodinger
equation for the atomic mean field (atomic BEC). The Schrodinger equation describes
the relative motion of two colliding atoms through a time evolution operator. The operator and its related coupling function capture the full time-dependence of the magnetic
field or scattering length in the experimental B-field pulses. The authors of Ref. [16]assert that their microscopic dynamics approach avoids any assumptions about the typeof states involved in the system of colliding BEC atoms. They determine the state of theatoms after a double pulse sequence by projecting onto a given set of basis states, but
the results are not dependent on the choice of the states. In addition, Kohler, Gasenzer,
and Burnett examine the effect of including the magnetic trap potential. They compare
their results for both a homogeneous gas and a trapped gas to learn more about thephysics of the atom-molecule coupling.
Like the Kokkelmans/Holland effective field theory, the model of Ref. [16] predicts
oscillations in the number of BEC atoms as a function of tevolve between a pair ofmagnetic field pulses toward the Feshbach resonance. Kohler, Gasenzer, and Burnett

167
also identify a relatively hot component of noncondensate atoms in their theory with theburst atoms observed in the experiment. A population of bound molecules exists after
the B-field pulses as well; according to the authors, these bound molecules correspond tothe missing atoms in the experiment (no consideration is given as to how the molecules
actually become missing atoms). The properties of the weakly bound molecular statenear the Feshbach resonance are explored by the authors of Ref. [16] in a useful appendix
to the paper.
For the case of a homogeneous BEC subject to a set of B-field pulses similar to
those used in the experiment, the theory predicts that the BEC and burst populations
oscillate slightly out of phase with one another, so that their sum, Ntotal , also oscillates.
The amplitude of the Ntotal oscillations is reduced relative the that of the BEC andburst numbers, as was seen in experimental data.
When Kohler, Gasenzer, and Burnett include the magnetic trap in their simulations, they observe the same qualitative behavior in the number oscillations, but there
are significant quantitative differences. The average number of burst atoms and missing
atoms is seen to depend strongly on the presence of the trap potential. For instance,
adding the trap potential causes the average burst fraction to increase from ∼15% to∼40% while the average missing fraction decreases from ∼10% to ∼5%. However, thephases and frequencies of the population oscillations remain unchanged when the magnetic trap is included in the Schrodinger equation. The authors of Ref. [16] explain thatthe dependence of the quantitative details of the oscillations on the presence of the trap
potential arises from the first B-field pulse toward the Feshbach resonance. During thispulse to 155.5 G, the magnitude of the scattering length approaches the harmonic oscillator length of the trap, which strongly affects the interactions of the atom pairs. Since
the oscillator length does not appear in the homogeneous theory, it seems reasonable
that the results should depend somewhat on the inclusion of the trap potential.
One interesting aspect of the theory is its explanation of the origin of the burst

168
atoms. The authors of Ref. [16] point out that the magnetic field pulses provide energy
for the relative motion of the noncondensate atom pairs, but there is no transfer ofmomentum. Thus, the center-of-mass motion of the burst is unaffected by the magnetic
field ramps.
In addition to providing energy for the relative motion of the burst atoms, thetime-dependent B-field can remove energy from the system so that atom pairs canbecome bound together. Kohler, Gasenzer, and Burnett also predict the formation ofa molecular BEC due to the B-field pulses. The authors state that after the pulse
sequence, the mBEC remains overlapped with the atomic BEC while the burst atoms
undergo a ballistic expansion. A density distribution for the molecular BEC is given in
the paper.
There are some results of the experiments in this Chapter that remain unexplainedby the Kohler, Gasenzer, and Burnett model. The large, density-dependent fraction of
missing atoms that do not show an oscillating population are not predicted by thismodel. In addition, there is no modeling of our observed phase shift dependence on
second B-field pulse final ramp time. Presumably this could be done and compared todata to further test the applicability of the theory; however, as in the model of Ref. [9],heavy numerical work is required to evaluate the predictions of the Kohler, Gasenzer,
and Burnett theory. It is difficult for experimentalists to apply this model to the data
to gain further intuition about the observed trends. For instance, we would like to use
one of these successful theoretical models to investigate the dependence of the Nmissing
oscillation amplitude on the second pulse final ramp time. Of course, it is not reallyfair to complain that the theory is too complicated because an accurate description of
the physics probably does require an effective field theory.

169
6.5.3

Overview of experiment and theory

To clarify the comparisons between theory and experiment described here, wesummarize them in Table 6.1. Here we list the experimental observables and trends anda brief note on the current status of agreement with theory.
Table 6.1: Comparison of experimental observables and trends with the two theoreticalmodels described in this section.
Observation

Kokkelmans/Holland

Kohler/Gasenzer/Burnett

Nrem , Nburst osc. amplitude

calculated; agrees with expt.

calculated; agrees with expt.

hNrem i, hNburst i avg. values

calculated; agrees with expt.

calculated; agrees with expt.

Nmissing oscillations

calculated; agrees with expt.

calculated; agrees with expt.

∆φ = φburst − φBEC '150◦

calculated; agrees with expt.

calculated; agrees with expt.

∆φ vs. tramp, f inal of pulse #2

calculated; does not agree

not yet calculated

νosc = ²bind /hcalculated; agrees with expt.

calculated; agrees with expt.

Damping of Nrem osc.

not yet calculated

not yet calculated

Nmissing vs. hninit i

calculated; does not agree

not yet calculated

hEburst i

calculated; agrees with expt.

calculated; agrees with expt.

hEburst i oscillations

not yet calculated

not yet calculated

In addition to the theoretical models described here, there are several other papers that study the atom-molecule coupling in 85 Rb [67, 73, 68]. In particular, thepredictions of Mackie et al. [67] bear some similarities to the observations of the experiments. However, the agreement is not particularly good. Since the Kokkelmans/Holland
and Kohler, Gasenzer, and Burnett models have been so successful in describing our
experiments, we do not include discussion of the other theories here.

Chapter 7
Very high precision bound state spectroscopy near a 85 Rb Feshbachresonance

7.1

Atom-molecule coherence as a novel Feshbach resonance probe

7.1.1

Introduction

The phenomenon of a Feshbach resonance in ultracold collisions of alkali atoms
has received much theoretical and experimental interest in recent years and has sparked
interest in the subject of resonant Bose-Einstein condensates (BEC). In addition toproviding the subject matter for this thesis, Feshbach resonances were used to control
elastic and inelastic collisions in ultracold gases [74, 75, 76, 77] and for tuning theself-interaction in BEC [12, 26, 2] by changing the magnitude of an external magnetic
field.
The magnetic field controls the self-interaction in the BEC by affecting the swave scattering length, a. Close to resonance, the scattering length varies with B-fieldaccording toµa = abg 1 −∆B − Bpeak

¶
,

(7.1)

where Bpeak is the resonance position and is defined to be the magnetic field where
the magnitude of a becomes infinite, abg is the background scattering length, and ∆ =Bzero − Bpeak is the resonance width where Bzero is the B-field where the scattering
length crosses zero. Measurements of Feshbach resonance positions and widths have

171
been used in a variety of alkali atoms to improve the determination of the interatomic
potentials. These potentials have then been used to precisely calculate a multitude ofimportant properties for trapped atomic gases [75, 78, 79, 13, 77, 69].
In 85 Rb, our recent production of an atom-molecule superposition state has allowed us to dramatically improve an existing determination of the Feshbach resonanceposition and width. The phenomenon of atom-molecule coherence was described in detail in Chapter 6. By inducing periodic oscillations in the number of condensate atoms,
we obtain a direct, high precision measurement of the molecular bound state energy.
Exploiting the resonance, we tune the molecular state very close to threshold — to ourknowledge, this is the most weakly bound state ever observed.
This chapter discusses the experimental techniques for mapping out the B-fielddependence of the molecular binding energy and how the data were fit to a theoreticalmodel based on 2-body scattering theory. As a result of the fitting, there was a substantial improvement in the determination of the Feshbach resonance parameters. The
strengths and limitations of the present method for studying the Feshbach resonance areemphasized. The final section of the chapter discusses our search for many-body effects,
which manifest themselves in a mean-field shift to the observed oscillation frequency.
7.1.2

Sensitivity of binding energy to Feshbach resonance parameters

The atom-molecule coherent oscillation frequency, ν0 , is directly related to themolecular state binding energy by ²bind =ν0 /h. Since the magnitude of this binding
energy depends strongly on the distance from the Feshbach resonance, one can use
measurements of the binding energy and corresponding magnetic field to constrain the
Feshbach resonance parameters, such as the position (Bpeak ), width (∆), and background scattering length (abg ). To understand the close relationship between ²bind andthe Feshbach resonance parameters, it is useful to consider the approximate expression²bind = −~2 /(ma2 ),

(7.2)

172
which is valid when the scattering length, a, is large and positive.1

Substituting

equation (7.1) for a versus B-field into equation (7.2) leads to an equation relating theoscillation frequency to the Feshbach resonance parameters:ν0 =(B − Bpeak )2~.
2
2πmabg (B − Bzero )2(7.3)

For magnetic field changes near resonance (B≥Bpeak ), the numerator of equation (7.3)
varies much more rapidly than the denominator and the B-field dependence is nearlyquadratic. To first approximation, a plot of binding energy versus B-field is simply aparabola centered at Bpeak and with a curvature that depends on 1/a2bg . Clearly, one
can obtain information about the Feshbach resonance parameters from the magnetic
field dependence of ²bind , as indicated schematically in Figure 7.1.
One of the best uses we found for equation (7.3) was to determine the optimal
data-gathering strategy for precisely determining a given Feshbach resonance parameter.
For example, if we hold other parameters fixed, Bpeak can be best determined by measurements of high oscillation frequencies, where the slope ∆ν0 /∆B is large. Using simple
error propagation and making reasonable assumptions for experimental uncertainties ina typical (B, ν0 ) measurement, we estimated that we should be able to measure Bpeak
to a precision of 0.047 G and abg to a precision of 4.5 a0 . Although these estimates
turned out to be fairly conservative (see section 7.3.3), they motivated us to pursue themeasurement because such precision would represent a dramatic improvement over thatobtained in earlier experiments (0.4 G for Bpeak and 21 a0 for abg [1]).
Although the present discussion of equation (7.3) provides motivation for precise measurements of the binding energy, the actual theoretical analysis used to modelthe data was considerably more complex. Servaas Kokkelmans collaborated with us tomodel the binding energy data. He utilized a highly accurate coupled-channels scattering model to fit the experimental data and extract the Feshbach resonance parameters,
1

In fact, equation (7.2) only underestimates the actual binding energy by ∼30% for a '300 a0(B'161.5 G).

173

binding energy (kHz)

800

600

Ordinary parameter values
Bpeak increased by 0.1%
abg increased by 7%

400

200

0
156

157

158

159

160

161

162

magnetic field (G)Figure 7.1: Sensitivity of binding energy to Feshbach resonance parameters. The solidline is the binding energy versus magnetic field as calculated from the approximate
equation (7.3), with our current best determinations for the Feshbach resonance parameters (section 7.3.3). The dashed line shows the effect of increasing the value of Bpeak
by 0.1% (0.15 G), while the dotted line corresponds to a reduction in |abg | by 7%. The
relative difference in sensitivity to the different parameters can be understood from a
simple Taylor expansion of equation (7.3).

174
providing strong constraints on the exact shape of the interatomic potential.

7.1.3

Advantages of the present method over previous ones

The present method for studying the Feshbach resonance through atom-molecule
oscillations offers all of the many inherent advantages of a frequency measurement, including the possibility of high measurement precision, a lack of sensitivity to errors in
the absolute atom number calibration, and a simple interpretation of the oscillation
frequency in terms of the relative energy difference between the atomic and molecularstates. When these advantages are combined with an improved method for magnetic
field calibration [3], the present technique for probing the Feshbach resonance is much
more precise than previous experiments that examined such Feshbach resonance observables as variable rethermalization rates in a trapped cloud of atoms [75, 13], enhancements of photoassociation rates [74] and inelastic loss rates near the resonance [80], andvariations of the mean-field expansion energy of a BEC [12]. In particular, the present
atom-molecule coherence experiments have exposed a systematic error in our previous
determination of the background scattering length, abg , as discussed in section 7.3.4.
Our new measurement of abg agrees with the value obtained from an analysis of several
other high precision experiments with ultracold rubidium [69].

7.1.4

Improved determination of Bzero

To complete the precise characterization of the Feshbach resonance, we also madean improved measurement of Bzero , the magnetic field where the scattering length vanishes. This experiment is very similar to our previous work [1, 13, 33], where we determined the a=0 field by measuring the critical number (Ncr ) for collapse of a BEC. For agiven magnetic field (scattering length), there is an upper limit to the number of atoms
the condensate can contain before becoming unstable. To measure this critical number,
we prepare a large (N0 ∼16000) condensate with fewer than 5% thermal fraction. The

175
condensate is formed at a B-field where the scattering length is positive (a ∼200 a0 .
We next apply an adiabatic magnetic field ramp to change the sign of the scattering
length from positive to negative, then we slowly increase the B-field until the condensate
collapses, which causes an abrupt drop in the number of atoms. From the measured
value of the collapse B-field (Bcollapse ) and the known slope of 1/Ncr versus magnetic
field [33] we can extrapolate to the magnetic field where Ncr would be infinite, thereby
determining the value of Bzero . We have improved the measurement precision of Bzero
by about a factor of 4 by improving our magnetic field calibration and using a larger
number of condensate atoms to measure the collapse. We find Bzero =165.750(13) G.

7.2

Measuring the B-field dependence of oscillation frequency

7.2.1

Production of BEC and measurement of total number

The procedure used to generate atom-molecule oscillations in 85 Rb Bose-Einsteincondensates has been described in chapter 6, so we merely outline the method here.
After creating condensates with initial number of atoms N0 '16000 at a magnetic field
B'162 G, we apply two short B-field pulses (∼40 µs duration) that approach andthen recede from the Feshbach resonance at Bpeak '155 G. The intermediate value ofmagnetic field between the pulses, Bevolve , and the time spacing between pulses, tevolve ,are variable quantities. The double pulse sequence is followed by a slow change in the
B-field to expand the BEC [2], then the trap is switched off (B→0) and destructive
absorption imaging is used to count the number of atoms remaining in the condensate.

7.2.2

Atom-molecule oscillations

As in Ref. [4], periodic oscillations in the BEC number were observed as a functionof tevolve (see Figure 7.2). We fit the BEC number oscillation to a damped harmonic

176

BEC Number
number

10000
10000

(a)

8000
8000

6000
6000

4000
4000
00

100
100

200
200

300
300

400
400

500
500

ttevolve(µs)
evolve (µs)
13000

BEC Number

(b)
11000

9000

7000
10

15

20

25

30

tevolve (µs)

Figure 7.2: BEC number versus pulse spacing tevolve . (a) Bevolve =156.840(25) G. At
this magnetic field, which is relatively close to resonance, the oscillation frequency isvery low (ν0 =9.77(12) kHz) so that the damping and atom loss significantly affectthe observed time dependence (here β=3640(750) s−1 and α=7.9(4) atoms/µs). (b)Bevolve =159.527(19) G. Farther from resonance, the time dependence of the BEC number
is dominated by the higher oscillation frequency of ν0 =157.8(17) kHz. Damping of theoscillations and atom loss are negligible in the relatively short time window used todetermine ν0 .

177
oscillator function with an additional linear loss term:
N (t) = Navg − αt + A exp (−βt) sin (ωe t + φ),(7.4)

where Navg is the average number, A is the oscillation amplitude, α and β are the nump
ber loss and damping rates, respectively, and ωe = 2π ν02 − [β/(2π)]2 . The quantityof interest here is ν0 , the natural oscillator frequency corresponding to the molecularbinding energy, ν0 =²bind /h.
We measured the oscillation frequency for values of Bevolve from 156.1 G to
161.8 G. Over this range, the frequency varies by over 2 orders of magnitude (101000 kHz), but the linear loss rate changes very little. The damping rate shows asignificant B-field dependence, increasing from β ' 2π × 0.8 kHz near 156 G to β '2π × 22 kHz near 162 G. We find no significant density dependence to the damping
at Bevolve =158.60(5) G; increasing the total atom density by a factor of 4.3(5) leads
to a damping rate increase of only a factor of 1.3(3). Atom loss from the BEC is welldescribed by a linear rate of -2 to -7 atoms/µs over the field range of interest. The rateis consistent with previous measurements of number loss due to a single B-field pulse
toward the Feshbach resonance, although we have not determined the mechanism forthis loss [3].

7.2.3

Limits to precision of frequency measurement

The oscillation frequency measurements have a typical fractional uncertainty of
1%. In principle, the precision of the oscillation frequency determination could be improved indefinitely by increasing the integration time, tevolve . However, there are several
effects that prevent us from simply increasing tevolve without limit. First, time variationin the applied magnetic field limits the number of oscillations we can observe. Since
we destroy the condensate each time we measure the number of atoms, any variation in
B-field from one shot to the next causes the B-field-dependent oscillation frequency to178shift slightly. This frequency shift manifests itself in the BEC number oscillations as aphase shift. If the phase shift between successive measurements is large enough, thenthe oscillations “wash out” and become impossible to measure. Because the phase shift
is proportional to tevolve , our sensitivity to phase noise increases for longer integration
time. Evidence for phase noise can be seen in the BEC number oscillations displayedin Figure 7.3.
Another limitation to the precision of the oscillation frequency determination is
the finite coherence time for the oscillations. The oscillation amplitude decays with time
until the amplitude becomes similar in size to the experimental noise on the number
measurement. We typically find that the effects of phase noise dominate over decoherence in limiting the precision of a given frequency measurement, except for the lowest
frequency data taken near resonance, where the oscillations damp out on the same timescale as the oscillation period (see section 7.5.1).
In addition to shot-to-shot B-field variations, it is possible to have time variationof the magnetic field during tevolve that is reproducible from shot-to-shot. For example,
if the magnetic field from the auxiliary coil is not adjusted to properly compensate
for eddy current effects (described in section 3.4.6), the B-field may change as tevolve
increases. This variation leads to an oscillation frequency variation or “chirp”. Sucha chirp significantly complicates the process of accurately determining the oscillation
frequency as a function of magnetic field.

7.2.4

Magnetic field determination

To characterize the Feshbach resonance it is necessary to know Bevolve precisely
as well as the oscillation frequency. As described in section 3.3, we measure Bevolve bytransferring atoms to an untrapped spin state by driving ∆m=+1 spin flip transitions
with an applied pulse of rf radiation (pulse length=5→25 µs). The spin flip frequency
is determined from the rf lineshape for the loss of atoms from the magnetic trap. After

179

14000

BEC number

13000
12000
11000
10000
9000
8000
10

20

30

40

50

60

70

80

tevolve (µs)

Figure 7.3: Phase noise in BEC number oscillations. Black points are the measured
number of BEC atoms versus tevolve while the black line is a fit to the short time data
tevolve ≤42 µs. The scatter in the data near 50 µs and 80 µs indicates phase noisethat is at least π radians in magnitude. From the observed phase noise and the known
dependence of frequency on magnetic field, ∆ν0 /∆B, one can estimate the minimum
amount of magnetic field noise required. The data near 50 µs imply shot-to-shot Bfield noise of 0.05 G (peak-peak). Because ∆ν0 /∆B increases rapidly as the B-field
increases above the Feshbach resonance, the higher magnetic field (and higher oscillation
frequency) data are more susceptible to phase noise.

180
measuring the rf transition frequency, we invert the Breit-Rabi equation to obtain thecorresponding B-field. To ensure that the magnetic field is sufficiently constant during
tevolve , we map out B(t) using rf pulses with lengths short compared to tevolve . Due
to interference of the rf radiation with the magnetic field control circuitry, there is asmall systematic shift of the field as a function of the rf power used. The uncertainty
for each magnetic field determination is the quadrature sum of the uncertainty due
to the lineshape measurements (10→15 mG) and the uncertainty in the extrapolation
to zero rf power(∼20 mG). Typically, the total uncertainty in the average B-field was∼25 mG and was dominated by the rf power shift. Although we were aware of the rf
power shift when we gathered the data, at that time we did not realize how the power
shift would dominate all other uncertainties in the magnetic field determination. Only
after extensive analysis did this become apparent. In the future, it should be possible toreduce the effects of rf interference in the experiment by adding shielding at appropriate
places in the auxiliary coil circuit.

7.3

Coupled-channels analysis of data

7.3.1

Coupled-channels theory

To accurately model the nonlinear magnetic field dependence of the binding energy near the Feshbach resonance, a coupled-channels (CC) scattering theory is required.
Coupled-channels scattering theory was first applied to ultracold alkali atom collisions
by Verhaar and coworkers [81, 8]. This theory offers a full quantum mechanical description of the 2-body scattering process, including the hyperfine and Zeeman Hamiltoniansof the separated atoms as well as the short-range molecular interactions and long-range
van der Waals interactions. The CC theory can be used to predict the energy and magnetic field dependence of various quantities, such as the phase shift, scattering length,
and the energy of molecular bound states. In principle, the accuracy of the CC calcula-

181
tions is limited only by the incomplete knowledge of the precise shape of the interatomic
potentials, which cannot be calculated ab initio [82].
The coupled-channels model presented here for 85 Rb was developed by Servaas
Kokkelmans, who collaborated with us to analyze the magnetic field dependence ofthe molecular binding energy [83]. Servaas’ model is based on the coupled-channels
analysis of van Kempen, Kokkelmans, Heinzen, and Verhaar (KKHV) [69], in which
several high-precision data for 85 Rb and 87 Rb were combined to perform an inter-isotope
determination of the rubidium interactions with unprecedented accuracy. The predictive
power of the KKHV analysis can be seen from Ref. [4], where the initial data on theatom-molecule coherence were already in good agreement with the predicted binding
energy of the molecular state. Another example of the accuracy of the KKHV analysis
is its agreement with more than 40 Feshbach resonances recently discovered in 87 Rb
[84].
7.3.2

Fitting the CC model to the data

The measured oscillation frequencies and magnetic fields are listed in Table 7.1
and plotted in Figure 7.4. For the purposes of the coupled-channels fitting, we choose
B-field as the independent variable and assume the magnetic field uncertainty, σB , iszero. We then increase the frequency error to include the B-field uncertainty according
to the formula [85]:rσν, tot =σν2 + (σB

∂ν 2
) ,
∂B

(7.5)

∂ν
where ∂B
is the slope of binding energy versus magnetic field and σν and σB are theactual measured uncertainties, as given in Table 7.1. For most of the data, the totalfrequency error (σν, tot ) is dominated by the second term under the square root becauseof the magnitude of σB . This B-field uncertainty comes mainly from a systematic
uncertainty related to the rf power shift (section 3.3.4). In fact, the magnetic field error
becomes even more significant for the highest oscillation frequency data, where a given

182
B-field uncertainty corresponds to a much larger frequency uncertainty due to the steep
slope of binding energy with B-field (see Figure 7.4). After determining the appropriate
uncertainties, we fit the binding energy data and the zero-crossing field Bzero to the CCmodel with a numerical nonlinear regression in Mathematica.
Table 7.1: Measured values of magnetic field and oscillation frequency.
Thedata were combined with a recent measurement of the magnetic field where a=0(Bzero =165.750(13) G). We fit the combined data set with a coupled-channels scattering
theory.
B (G)156.840
158.211158.655
159.527160.529160.887
161.771

7.3.3

σB (G)
0.025
0.025
0.010
0.019
0.021
0.025
0.017

ν (kHz)
9.77
47.4
73.6
157.8
383.1
509
1072

σν (kHz)
0.12
1.0
1.1
1.7
3.7
12
11

Extracting Feshbach resonance parameters from best fit

The authors of Ref. [69] used the best known values [13] for the resonant magnetic
field Bpeak and zero crossing Bzero . In our analysis, we ignore the relatively imprecise
value of Bpeak from Ref. [13], and instead use the measured dependence of binding
energy on magnetic field along with the new Bzero measurement given above to determine
the interaction parameters. We observe that the fitting procedure is mainly sensitive
to only three parameters: the van der Waals dispersion coefficient, C6 , and the nonintegral vibrational quantum numbers at dissociation, vDS and vDT , which determine
the position of the last bound state in the singlet and triplet potentials, respectively.
Varying the additional parameters C8 , C10 , φE
T (the first-order energy-dependence of thephase of the oscillating triplet radial wave function), and J, the strength of the exchangeinteraction, does not improve the fitting because these changes can be absorbed insmall shifts of vDS , vDT and C6 . Therefore, we take the mean values for these four183parameters [86] from the most recent determination in Ref. [84].
The best fit to Bzero and the seven highest frequency data points yields a reduced χ2 =0.30 for 5 degrees of freedom. This value of χ2 is improbably low due tothe fact that the uncertainty in the data is dominated by the systematic uncertainty
in magnetic field related to the magnitude of the rf power shift. Figure 7.4 shows theresulting theoretical binding energy as a function of the magnetic field together withthe experimental values. From the fit, we find substantially improved values for the
Feshbach resonance position Bpeak =155.041(18) G, width ∆=10.71(2) G, and background scattering length abg =-443(3) a0 . These results may be compared to previously obtained results Bpeak =154.9(4) G and ∆=11.0(4) G [13], and abg =-450(3) a0[87]. Our best interaction parameter values are C6 =4707(2) a.u., vDS =0.00918(17),and vDT =0.94659(29). Here the error bars do not include systematic errors due to theuncertainties in other interaction parameters that are not constrained by our data. To
compare our values with those of Ref. [69], we determined the sensitivity of our threeinteraction parameters to systematic shifts in the other parameters, as shown in Table 7.3.3. Using the fractional uncertainties in C8 , C10 , φE
T , and J from KKHV, wefind C6 =4707(13) a.u., vDS =0.0092(4), and vDT =0.9466(5). All of these values agreewith those given in KKHV: C6 =4703(9) a.u., vDS =0.009(1), and vDT =0.9471(2). Our
value for vDS is more precise than that of Ref.[69], while vDT and C6 are slightly lessprecise. However, if future experiments allow improvements to the other interaction
parameters, then our results will also become more precise since the systematic errors
are comparable to or larger than our statistical errors from the fit.
To understand the strong parameter constraints we obtain with our bound state
spectroscopy, it is important to consider the nonlinear dependence of the binding energy on magnetic field. The magnetic field dependence of ²bind as it approaches thecollision threshold depends sensitively on the exact shape of the long range interatomic
potentials, which are mainly characterized by the van der Waals coefficient, C6 . At

184binding energy (kHz)

1000

100

10

(a)
1
156

157

158

159

160

161

162

magnetic field (G)1200

binding energy (kHz)

1000

12
10
8

800

6
4

600
400

2
0
156.00156.25156.50156.75157.00

200
0
156

(b)
157

158

159

160

161

162

magnetic field (G)Figure 7.4: Molecular binding energy versus magnetic field, Bevolve . (a) The points
are measured values of the atom-molecule oscillation frequency, ν0 , while the solid line
represents the molecular binding energy, which we fit to the data by adjusting theparameters of a coupled-channels scattering theory. Only black points were included in
the fit; white points were excluded because they experienced a statistically significant
mean-field shift. To improve visibility, the points are larger than the error bars. (b)Same as in (a), but with a linear scale for the vertical axis. The inset shows the deviationof the lowest frequency data from the fit to the rest of the data.

185
Table 7.2: Sensitivity of the determined interaction parameters vDS , vDT and C6 tofractional uncertainties in C8 , C10 , φE
T and J. For instance, the systematic error in C6
due to a 10% uncertainty in C8 is 123 × 0.10 = 12.3 a.u.
∆vDS
∆vDT
∆C6

∆C8 /C8−1.53 × 10−4
−4.14 × 10−4
123

∆C10 /C10−6.80 × 10−5
−1.39 × 10−4
33.4

E
∆φE
T /φT
−2.59 × 10−3
2.31 × 10−3
−47.8

∆J/J
1.72 × 10−3
1.71 × 10−3
19.3

magnetic fields far from resonance, the bound state wave function is confined to shortinternuclear distance and the binding energy varies linearly with magnetic field. The
linear dependence on B-field gives relatively little information about C6 . As the Bfield approaches resonance, the detuning decreases until the bound state lies just below
threshold. Now the bound state wave function penetrates much deeper into the classically forbidden region, which causes ²bind to curve toward threshold as a function ofmagnetic field. Because the energetically forbidden region stretches out as C6 /r6 , theobserved curvature depends sensitively on the C6 coefficient. One can show [17] thatan analytical Feshbach model that includes the correct potential range and backgroundscattering processes [56] can reproduce the binding energy curve over the full range ofmagnetic field.

7.3.4

Revised stability prediction for condensates with a <0

As a result of the improved determination of the 85 Rb Feshbach resonance parameters, we find that our new value for the off-resonant or background scattering
length, abg =-442.9(23) a0 , is inconsistent with the value given in Ref. [13], where abg =380(21) a0 . The most plausible explanation we can find for disagreement is that thetheoretical expression used to relate measured rethermalization rate to cross section [1]is insufficient for the requisite level of accuracy. However, the new value for abg allows
us to revise our previous estimate for the stability condition of a BEC with negative
scattering length [33]. We use equation (7.1) to obtain the linear slope of scattering

186
length versus B-field near B=Bzero . We then find the stability coefficient for BEC collapse, kcollapse , by combining the value of ∆a/∆B=-39.86(22) a0 /G with the measured
slope of 1/Ncrit versus magnetic field [33] of 0.00128(3) (atoms G)−1 . Thus, we obtain
the revised value kcollapse =0.547(58), where the error is dominated by a 10% systematic
uncertainty in the determination of Ncrit . The present determination agrees with thetheoretical value of 0.55 [88].

7.4

Search for collective effects: mean field shifts

7.4.1

Inadequacy of 2-body scattering theory

The coupled-channels theory used in this work applies to 2-body scattering; therefore, this theory cannot account for many-body effects in the atom-molecule BEC system, such as a mean-field shift to the observed oscillation frequency [9, 68]. Any suchmean-field shift must be fractionally largest near the Feshbach resonance, where thecoupled-channels theory predicts that the binding energy approaches zero while theatom-atom scattering length increases to infinity. We searched for a mean-field shift
to the oscillation frequency when Bevolve was decreased to ∼156 G. As shown in Figure 7.4, the lowest magnetic field data display a clear frequency shift with respect tothe coupled-channels theory prediction. As Bevolve approaches resonance, the observed
shift increases to 1.7 kHz, which significantly exceeds a simple estimate for the averageatom-atom mean-field shift in the BEC: 4π~2 hnia/m ' 0.5 kHz at Bevolve =156.1 G.
We expect the mean-field shift to depend on the density of atoms (and molecules), but
we have not yet studied the density dependence. In practice, the initial BEC density
can be changed by adiabatically varying the initial scattering length before the rapid
magnetic field pulses as discussed in section 5.3.3.

187
7.4.2

Statistical method for determining mean-field shifts

Since the lowest frequency data show evidence for a mean-field shift, we exclude
these points from the (2-body) theory fit. We determine the cutoff magnetic field forthe excluded region by the following procedure. First, we fit the data set that includes
all frequency measurements satisfying ν0 ≥ 73 kHz. For these data, the expected
fractional effect of any mean-field shift is negligible; for example, the average meanfield interaction energy in the atom BEC is ≤0.1 kHz, and we expect atom-moleculeinteractions to be similar in magnitude. After fitting this minimal data set, we add thenext lower frequency/B-field measurement to the data set and repeat the theory fit.
Finally, we compare the two fits according to their minimum χ2 values. As long as theincrease in χ2 that results from addition of a new data point is reasonable (on the orderof unity), we conclude that there is no significant mean-field shift associated with thefrequency measurement.
Table 7.3: Total χ2 and Bpeak as a function of the number of included data points in the
coupled-channels fit. The abrupt jump in χ2 and Bpeak as the data set increases from
7 to 8 points indicates the presence of a mean-field frequency shift in the 8th point.
The mean-field shift magnitude increases as the added points approach closer to the
Feshbach resonance; since the shift is non-statistical, χ2 increases very rapidly as morepoints are added to the set.
Included pts.
5
6
7
8
9
10
11

Total χ2
0.42
0.77
1.3
10.0
15.4
24.4
41.7

Bpeak from fit
154.971154.994155.042154.965154.922154.885
154.847

In Table 7.3 we give the variation of total χ2 and Bpeak as a function of thenumber of points included in the fit. Both χ2 and Bpeak shift abruptly when the numberof points increases from 7 to 8 (corresponding to a decrease in Bevolve from 156.840 G

188
to 156.390 G). The observed behavior seems sensible since we expect mean-field shifts
to increase rapidly as one moves toward resonance (see Figure 7.4). Our χ2 exclusion
criterion allows us to fit the data in a manner that avoids systematic errors in the
Feshbach resonance parameters. By extrapolating the coupled-channels binding energy
prediction to lower magnetic fields, we determine the magnitude of the mean-field shifts,
which are listed in Table 7.4.
Table 7.4: Measured values of magnetic field and corresponding mean-field shifts to themolecular binding energy. The mean-field shifts were determined by subtracting theextrapolated coupled-channels binding energies from the lowest oscillation frequency
data. The shift error, σmf , is simply the error from the frequency measurement.
B (G)156.076
156.185
156.352
156.390156.840σB (G)
0.024
0.023
0.023
0.024
0.025

∆νmf (kHz)
1.69
1.26
1.07
1.00
−0.15

σmf (kHz)
0.12
0.06
0.08
0.13
0.12

7.5

Low frequency oscillations: a simple model

7.5.1

Damping of oscillations: experiment and theory

The low frequency oscillations that show evidence for a mean-field shift are quite
strongly damped, so only one or two wiggles are observed before the signal-to-noise ratio
approaches unity. In addition, the number loss rate, α ' 7 atoms/µs, causes a significant
decrease in the average number of BEC atoms during tevolve . The combination of theseeffects leads to a rather complicated time dependence of the measured BEC number, as
shown in Figure 7.5.
We use a simple model to explain part of the observed damping as due to inhomogeneous dephasing from a density dependent mean-field shift. This idea was suggested
to us by Murray Holland and Servaas Kokkelmans [9]. Assuming that the atom-molecule

189

8000

BEC number

7000
6000
5000
4000
3000
2000
1000
0

100

200

300

400

tevolve (µs)

Figure 7.5: Low frequency BEC number oscillations.
Black points correspond to Bevolve =156.390(24) G; white points were at a lower magnetic field of
Bevolve =156.076(24) G. The solid line is a fit to the black points using equation (7.4)
with ν0 =5.93(13) kHz, β=4500(1200) s−1 and α=6.1(8) atoms/µs. The dashed line fit
to the white points gives parameter values of ν0 =4.34(12) kHz, β=7600(1000) s−1 , and
α=6.4(5) atoms/µs.

190
oscillation frequency depends on the local density in the sample, we haveν(~r) = νbare + An(~r),(7.6)

where νbare is the “bare” oscillation frequency at zero density (i.e., the molecular binding
energy), A is the (positive) frequency shift per density and is determined empirically,and n(~r) is the local density in the BEC cloud. For simplicity, we assume that thedensity distribution of the condensate has an isotropic parabolic form: n(0)[1 − r2 /R2 ] r ≤ Rn(~r) =

 0
r>R

(7.7)

15N
where n(0) = 8πR
3 is the peak density, N is the total number of atoms, and R is

the radius of the parabolic condensate. Although we neglect the anisotropy of the
BEC, this is a fairly reasonable approximation since the aspect ratio is at most 2.5 in
the Thomas-Fermi limit (we approximate a football-shaped object by a sphere). The
Thomas-Fermi approximation of a parabolic density distribution is well satisfied by
the condensates since we typically prepare them at the evaporation magnetic field of
162.3 G, corresponding to a ∼200 a0 . For this scattering length, the ratio of mean fieldinteraction energy to kinetic energy in the BEC is N a/aho =55, which is far into theThomas-Fermi limit.
Using the above density distribution, we next obtain the probability distributionof the atoms in oscillation frequency space. We simply find the probability of being in agiven spherical shell of constant density (constant frequency). The probability per unit
frequency is
2πR3 (ν − νbare )P (ν) =N A ∆νmaxs1−(ν − νbare )
,
∆νmax

(7.8)

where ∆νmax = An(0) is the maximum frequency shift corresponding to the peak density
R ν +∆νmax
in the sample and P (ν) is normalized so that 0 bare
P (ν)dν = 1.
After obtaining the probability distribution in equation (7.8), we numericallycalculate the Fourier transform of P (ν) and study the oscillation amplitude versus time.

191
One can then look for the effects of density-dependent dephasing in the oscillations.

7.5.2

Comparison of dephasing model to data

In Figure 7.6, we plot the theoretical oscillation amplitude as a function of time fortwo choices of the quantities νbare and ∆νmax . We chose these values based on empiricalestimates for the mean-field shift — we estimated the average shift by extrapolating thecoupled-channels prediction for ²bind and then calculating the difference between thisprediction and the measured oscillation frequency.
The dephasing model predicts damped oscillations that are qualitatively similar
to the experimental measurements, as can be seen by comparing Figures 7.5 and 7.6.
To make a more quantitative test of the dephasing model predictions, we show themean-field shift and damping rate versus magnetic field in Figure 7.7. Here the observed oscillation frequency shifts and damping rates can be directly compared to thepredictions of the dephasing model. Figure 7.7 demonstrates that the damping rates
predicted from the dephasing model are significantly smaller than those observed in
the experimental data. There appears to be an additional B-field-independent damping
mechanism in the experiment. The most obvious additional source of damping is theobserved loss of BEC atoms during the evolution time between magnetic field pulses.
This loss has no apparent B-field dependence over the range of magnetic fields in Figure 7.7. Although the mechanism for the loss is unknown, the process must be distinct
from that of the inhomogeneous dephasing discussed here and should lead to a loss ofcoherence in the oscillations.

7.5.3

Additional damping from atom loss

To predict the expected damping rate due to the measured atom loss rate, we
assume that the only the atomic state in the atom-molecule superposition experiences
decay of population. One can show [89, 90] that if the number of atoms decreases at

192

amplitude (arbitrary units)

1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
0

100

200

300

400

time (µs)

Figure 7.6: Theoretical model for density-dependent dephasing. Points are the calculated Fourier transform of the frequency probability distribution in equation (7.8)
versus time. A phase shift of 0.8 rad was added to the Fourier transform to better
approximate the initial phase in the experiment (see Figure 7.5). The black points
correspond to a bare oscillation frequency of νbare =4.9 kHz with an average meanfield shift of ∆ν=0.99 kHz. The white points had νbare =2.7 kHz and ∆ν=1.7 kHz.
The solid and dashed lines are fits to the two different model data sets. From thesefits, we extract values for the oscillation frequency of an equivalent damped oscillator,
ν0 (see equation(7.4)), and its damping rate, β. The fits yield ν0 =5.90(1) kHz and
β=2250(120) s−1 (solid line) and ν0 =4.42(3) kHz and β=3560(220) s−1 (dashed line).
The fits allow us to compare the dephasing model directly to the data.

193

frequency shift (kHz)

2.0

(a)

1.5
1.0
0.5
0.0
-0.5
156.0

156.2

156.4

156.6

156.8

157.0

B-field (G)(damping rate)/1000 ( s-1 )

10

(b)

8
6
4
2
0
156.0

156.2

156.4

156.6

156.8

157.0

B-field (G)Figure 7.7: Comparison of data and theory for oscillation damping rates versus Bfield. (a) Mean-field shift of the oscillation frequency as a function of magnetic field.
Black points with error bars are empirically determined values of the mean-field shift,
as discussed in the text. White points are the values of the mean-field shift used in thedephasing model. (b) Damping rates versus B-field. Black points with error bars are
damping rates from fitting the BEC number oscillation data, as in Figure 7.5. White
points connected by straight lines are the predicted damping rates due to inhomogeneousdephasing from the mean-field shifts in part (a). The dotted line shows the effect ofadding a constant rate of 3500 s−1 to the dephasing model prediction.

194
an exponential rate, Na = N0 exp (−γt), the amplitude for population oscillations will
decay according to exp (−γt/2). In the magnetic field range of interest (from 156-157 G),
we observed a roughly linear loss rate from the atom BEC of -α=-7.2 atoms/µs. To
convert this linear loss rate into an effective exponential decay rate, we determine thetime interval required to decrease the total number of BEC atoms by a factor of 1/e.
The initial average number of BEC atoms at our shortest evolution times (tevolve '0)
was 6000, so the time to reach 6000/e=2210 with a linear loss rate of -7.2 atoms/µs is
527 µs. This 1/e time corresponds to an effective number decay rate of γ=1900 s−1 , so
we expect the atom-molecule oscillations to decay at the rate γ/2=800 s−1 .
As can be seen from Figure 7.7, the additional damping rate due to atom number
loss is too small to explain the discrepancy between the data and the simple dephasing
theory. In fact, to explain the observations, there must be an additional damping rate of∼3500 s−1 , which is more than 4 times larger than the expected damping from decay of
the BEC number. There may be other sources for decoherence in the system, including
a possible loss from the molecular state during tevolve . Whatever the source of theadditional damping, our data suggest that this damping is not strongly sensitive tomagnetic field. We also have evidence that the additional damping rate is insensitive
to density — at Bevolve =158.60(5) G, where the mean-field shift is negligible, we findthat increasing the BEC density by a factor of 2.5(3) leads to a damping rate increaseof only a factor of 1.24(25). Future investigation of the mean-field shifts and damping
rates should improve our understanding of the physics of atom-molecule decoherence
near the Feshbach resonance.

7.5.4

Magnitude of mean-field shifts

The magnitude of the mean-field shift shown in Figure 7.7 increases as the magnetic field approaches the Feshbach resonance. Over the 1 G range shown in this figure,the shift increases from 0 to 1.7 kHz. The shift for the lowest magnetic field data is195significantly larger than one would expect from a simple estimate of the average meanfield energy in the BEC, based on the GP equation: 4π~2 hniaevolve /m ' 0.5 kHz at
Bevolve =156.1 G. The observed shift also exceeds a recent calculation for the equilibrium mean-field shift using an effective quantum field theory [68] (<0.1 kHz). One
intriguing possibility for the discrepancy is that there is a positive mean-field shift due
to atom-molecule (or molecule-molecule) interactions. We believe that the current data
are interesting and suggestive, but we have really only touched the tip of the iceberg inthis area.
Since a mean-field shift must depend on the density of atoms and molecules, itwould be desirable to study the density dependence. Ideally, one would like to change
the density of atoms and molecules independently to identify the source of the shift(s).
At the present time, we have not measured any density dependence to the mean-field
shifts.

7.5.5

Limitations of current method and possible future experiments

As discussed above, the observed time dependence of the BEC number arises
from the interplay between the bare atom-molecule oscillation frequency and a densitydependent frequency shift. The data suggest the presence of highly interesting manybody physics, but the information we can glean from the oscillation data is limited. For
example, as we decrease the magnetic field to approach closer to the Feshbach resonance,
we observe that the oscillations rapidly damp out due to increased dephasing. Eventually the damping rate becomes comparable to the oscillation frequency and it becomes
impossible to reliably extract the underlying frequency because the system is criticallydamped. One way to avoid this limitation would be to measure the atom-molecule
oscillations as a function of position in the condensate. If we could “bin” the cloud
into different spatial regions, then we could avoid the effect of inhomogeneous dephasing [89]. However, our present method of measuring the BEC number involves a large

196
time-of-flight expansion before imaging, which effectively blurs any spatial information
about the pre-expansion condensate (see section 3.5.3).
Another practical limitation of our experiment comes from the finite amount ofcharge stored in the capacitor bank used to control the magnetic field (see section 3.4.3)
The finite amount of charge prevents us from holding the magnetic field constant for
more than ∼1 ms, so we cannot observe an atom-molecule oscillation with a period
longer than this limit.
However, we are investigating novel techniques to determine mean-field shifts
to the atom-molecule system near the Feshbach resonance. For example, it may bepossible to use rf pulse spectroscopy, as discussed in section 3.3.5, to measure the meanfield energy. Such a method allows one to probe the energy of the system on shorter
timescales and has proven useful in studies of 40 K Feshbach resonances [91].

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