Experiments with a two-species Bose-Einsteincondensate utilizing widely tunable interparticle
interactions
by
Scott B. Papp
B.A. Physics, University of Colorado, 2001

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
2007

This thesis entitled:
Experiments with a two-species Bose-Einstein condensate utilizing widely tunableinterparticle interactions
written by Scott B. Papp
has been approved for the Department of Physics

Dr. Carl E. Wieman

Dr. Deborah S. Jin

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
Papp, Scott B. (Ph.D., Physics)
Experiments with a two-species Bose-Einstein condensate utilizing widely tunable interparticle interactions
Thesis directed by Distinguished Professor Dr. Carl E. Wieman
This thesis presents experiments with a quantum-degenerate gas of bosonic atoms
where interparticle interactions can be precisely controlled using a magnetic-field tunable Feshbach resonance. Using our new apparatus, we have the capability to study
either single species Bose-Einstein condensates of 85 Rb or 87 Rb, or a mixture of both
species. With such a mixture we have created and studied the first ultracold heteronuclear Feshbach molecules. Further work with two-component gases includes the creationof a dual-species BEC in which the miscibility of the two quantum fluids can be controlled. Finally, with a single species 85 Rb condensate we have performed the firstdetailed measurements of the excitation spectrum of a BEC as a function of interaction
strength. With strong interparticle interactions, we have observed that the excitationenergy is smaller than that predicted by mean-field theory.

Dedication

This work is dedicated to my wife Cindy and to my family Ann Rae, Larry, and Jeff.

v

Acknowledgements

My long tenure as both an undergraduate and graduate student at JILA has been
greatly enriched by the help and support of many people. Here I would like to take theopportunity to thank the people who have in no small way made the work in this thesis
possible.
First and foremost, I thank Carl Wieman for providing me with an exceptional
graduate education. Carl is a brilliant experimental physicist who is enthusiastic about
even the most daunting challenges. Even with all of his other responsibilities, Carl
has always been very generous with his time and thoughtful advice. I have profoundly
benefited from my association with him. I can only hope that during the past several
years I have been able to absorb some of his intuition for interesting physics.
As is always the case in the JILA ultracold atom groups, Eric Cornell and Debbie
Jin played an important role in my thesis work. Debbie taught me how to work ina lab and how to understand and solve experimental problems. I gratefully thankDebbie for her guidance during my undergraduate days and throughout my graduate
studies. Recently, Eric took over leadership of our 85 Rb BEC efforts and it has been agreat pleasure working directly with him. I have benefited from his innovative ideas forresearch directions.
I had the privilege of working directly with several talented graduate students.
Josh Zirbel has been a really good friend and colleague to me throughout my graduate
career. Josh has a keen aptitude for physics and a good intuition for how to make things
work in the lab. In the summer of 2001, we were the founding members of the “new”
85 Rb BEC project. Josh deserves my thanks for all the hard work he put in helping todesign and construct the current version of the apparatus. Since those early days, Josh
has gone on to build an even better apparatus to study K–Rb Feshbach molecules. My
successors, Juan Pino and Rob Wild, both contributed significantly to the work in thisthesis. Juan played an important part in both the construction and optimization of ourapparatus. He has also been a good friend over the past few years. Rob only recently

vijoined our project, but with a few years of experimental BEC experience already under
his belt, he immediately made significant contributions. His insights and pleasant nature
were greatly appreciated. I look forward to hearing about all the wonderful experiments
Juan and Rob perform with 85 Rb BECs.
All of the many (too many to list) current and former students and post-docsof the Wieman-Cornell-Jin supergroup deserve thanks. I would like to acknowledge
the members of the original 85 Rb BEC project, graduate students: Jake Roberts, Neil
Claussen, and Sarah Thompson and post-docs: Simon Cornish, Liz Donley, and Eleanor
Hodby. Their work was the motivation for this thesis. Kurt Miller and Stephan Dürr
worked on the other side of the curtain in our lab and they were a helpful source ofadvice. Wieman group visitors Andrew Wilson and Christine Steenkamp also provided
useful advice and I thank them for their contributions. From my undergraduate daysin the Jin Group I thank Brian DeMarco and Jon Goldwin.
I would like to acknowledge many useful interactions with the theoretical community at JILA. In particular, John Bohn has patiently helped me to understand various
aspects of scattering theory and Feshbach resonances. More recently Shai Ronen has
been instrumental in helping us to understand our Bragg spectroscopy data. I also
enjoyed interacting with Murray Holland, Chris Greene, and Servaas Kokkelmanns.
I worked with a number of undergraduate students during my time in the Wieman
group. Some students were participating in the REU summer program at CU or werevisiting from other universities, while others were CU students. Each of them built
valuable pieces of the apparatus. I would like to acknowledge the people that worked
in our lab: Jason Smith, Tyler Powell, Christoph Kohstall, Jenelyn Amos, Jeff Giffin,
and Joan Dreiling.
For good reason JILA is renown for its scientific support staff. The efforts of
the JILA electronics and machining shops have been invaluable for the construction of
our apparatus. In particular, I would like to thank Terry Brown, Mike Whitmore, Carl
Sauer, Blaine Horner, Hans Green, Tracy Keep, Todd Asnicar, and Tom Foote for building critical parts for our apparatus. All the hardworking members of the supply officealso deserve thanks for efficiently purchasing the equipment used in the experiments. I
also want to thank Krista Beck for lots of assistance to me personally.
Finally, I would like to acknowledge my wonderful wife Cindy Regal. She is a
gifted scientist from whom I have learned a great deal over the last several years. It
is impossible to describe how important she is in my life. She is my best friend anda constant source of joy and support. The many times she cheered me up after adiscouraging day in the lab kept me on track to complete this work.

Contents

Chapter
1 Introduction

1

1.1

An overview of my work . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Ultracold Bose gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.1

Thermodynamics of a non-interacting Bose gas . . . . . . . . . .

4

1.3.2

An interacting Bose-Einstein condensate . . . . . . . . . . . . . .

6

1.3.3

Elastic and inelastic atomic collisions . . . . . . . . . . . . . . . .

7Contents of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4

2 The Apparatus

10

2.1

Chapter overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.3

Two-species MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.4

Quadrupole trap loading procedure . . . . . . . . . . . . . . . . . . . . .

21

2.4.1

Compressed MOT and polarization-gradient cooling . . . . . . .

23

2.4.2

Optical pumping and spin-state filtering . . . . . . . . . . . . . .

24

Magnetic trap transfer to the science cell . . . . . . . . . . . . . . . . . .

29

2.5.1

Multi-coil transfer . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.5.2

Moving coil transfer and science cell optical pumping . . . . . . .

34

2.5

CONTENTS
2.5.3

viii
Concluding thoughts on magnetic transfer . . . . . . . . . . . . .

37

Ioffe-Pritchard trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.6.1

Driving radiofrequency transitions in the science cell . . . . . . .

45

2.7

Optical dipole trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.8

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

50

2.9

Chapter conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

2.6

3 Components of the apparatus
3.1

53

Diode laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.1.1

Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.1.2

Widely tunable laser frequency stabilization . . . . . . . . . . . .

56

3.1.3

Stable external cavity diode laser design . . . . . . . . . . . . . .

61

3.2

Computerized control system . . . . . . . . . . . . . . . . . . . . . . . .

64

3.3

Magnetic-field coil current control electronics . . . . . . . . . . . . . . .

66

4 Evaporative cooling of 85 Rb to BEC

73

4.1

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

73

4.2

Simulations of 85 Rb evaporative cooling . . . . . . . . . . . . . . . . . .

75

4.2.1

Simple model of evaporative cooling . . . . . . . . . . . . . . . .

75

4.2.2

Analytic model of sympathetic cooling . . . . . . . . . . . . . . .

80

4.2.3

Motivation for sympathetic cooling based on elastic collision crosssections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.3

RF for evaporation in the Ioffe-Pritchard trap . . . . . . . . . . . . . . .

83

4.4

Evaporation of 87 Rb to BEC . . . . . . . . . . . . . . . . . . . . . . . .

85

4.4.1

Forced rf-evaporation in the magnetic trap . . . . . . . . . . . .

86

4.4.2

Cooling 87 Rb in the optical trap . . . . . . . . . . . . . . . . . .

87

Sympathetic cooling of 85 Rb in the magnetic trap . . . . . . . . . . . . .

94

Effects of 87 Rb heat capacity on sympathetic cooling . . . . . . .

96

4.5

4.5.1

CONTENTS

4.6

4.7

ix

4.5.2

Effects of inelastic collisions on 85 Rb evaporation . . . . . . . . .

4.5.3

Toward 85 Rb BEC in the magnetic trap . . . . . . . . . . . . . . 103

Simultaneous evaporation of 85 Rb and 87 Rb

98. . . . . . . . . . . . . . . 108

4.6.1

Evaporation in the tight optical trap . . . . . . . . . . . . . . . . 111

4.6.2

Creation of 85 Rb BEC in the weak optical trap . . . . . . . . . . 113

Chapter conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 Two-species BEC

126

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2

Spatial separation measurements . . . . . . . . . . . . . . . . . . . . . . 127

5.3

Tuning the spatial separation with 85 Rb interactions . . . . . . . . . . . 130

5.4

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6 Heteronuclear Feshbach molecules

134

6.1

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

6.2

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3

Discovery of heteronuclear Feshbach resonances . . . . . . . . . . . . . . 135

6.4

Creating Feshbach molecules with magnetic-field sweeps . . . . . . . . . 136

6.5

Determination of the Feshbach resonance locations . . . . . . . . . . . . 136

6.6

Heteronuclear molecule creation efficiency . . . . . . . . . . . . . . . . . 141
6.6.1

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.6.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6.7

Feshbach molecule binding energy

6.8

Conclusion. . . . . . . . . . . . . . . . . . . . . 144. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7 Bragg spectroscopy

149

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2

The excitation spectrum of a BEC . . . . . . . . . . . . . . . . . . . . . 150

CONTENTSx

7.3

Bragg spectroscopy of a BEC . . . . . . . . . . . . . . . . . . . . . . . . 151

7.4

Obtaining Bragg spectra with strongly interactions . . . . . . . . . . . . 155

7.5

Momentum signal for a weakly interacting gas . . . . . . . . . . . . . . . 158

7.6

Bragg lineshape function for a trapped BEC . . . . . . . . . . . . . . . . 159
7.6.1

Line shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.6.2

Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.6.3

Two-body scattering physics for k a ∼ 1 . . . . . . . . . . . . . . 171

7.7

Simulations of the BEC density . . . . . . . . . . . . . . . . . . . . . . . 172

7.8

Bragg spectra of a strongly interacting 85 Rb BEC . . . . . . . . . . . . . 175

7.9

Systematic effects on the Bragg line shift. . . . . . . . . . . . . . . . . 181

7.9.1

Measurements of BEC fraction from absorption images. . . . . 181

7.9.2

Bragg spectra after ramping to large a . . . . . . . . . . . . . . . 183

7.9.3

The temperature dependence of the Bragg line shift . . . . . . . 185

7.10 Bragg spectra at lower density . . . . . . . . . . . . . . . . . . . . . . . 187
7.11 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8 Conclusion

190

Bibliography

191

Tables

Table
2.1

4.1

4.2

4.3

4.4

4.5

Physical parameters of the coils used for the multi-coil transfer to thescience cell. The radius and spacings listed here are for the average turnof the coil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Various relevant parameters that are used in our simulations of evaporative cooling are collected here. For each parameter the relevant magnetic
field or range of fields is indicated. . . . . . . . . . . . . . . . . . . . . . 80
Evaporation parameters optimized for 87 Rb in the magnetic trap. The
frequency of the final stage sets the final cut which changes slightly andis near 1.25 MHz. The power referenced here is what is programmed
into the synthesizer and does not include the gain of the amplifier or loss
through various rf components. . . . . . . . . . . . . . . . . . . . . . . . 85
Evaporation parameters used for the tight optical trap. There are fourstages in the tight trap evaporation each with an initial power, final power
and ramp rate. Powers here are listed as the control voltage entered into
the control computer. The calibration is roughly 1.2 W / 1.7 V. Listed
in the last column is the calculated radial trap frequency at the end of
the stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Evaporation parameters optimized for 87 Rb in the weak magnetic trap.
The parameters are optimized by searching for the largest 87 Rb phasespace density. The power referenced here is what is programmed into thesynthesizer and does not include the gain of the amplifier or loss throughvarious rf components (see Section 4.3). . . . . . . . . . . . . . . . . . . 106
Evaporation parameters used for the weak optical trap. There are six
stages in the weak trap evaporation each with an initial power, final power
and ramp rate. Powers here are listed as the control voltage entered into
the control computer, the calibration is roughly 1.2 W / 1.7 V. The
parameter tend is the end time of each stage measured from the startof evaporation. Listed in the last two columns are the calculated trap
frequencies at the end of each stage. . . . . . . . . . . . . . . . . . . . . 120

FIGURES

xii

FiguresFigure
2.1
2.2

2.3

2.4

2.5

2.6

2.7

A photograph of the apparatus during vacuum assembly and a current
photograph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of the vacuum system used in this thesis. Atoms are laser
cooled and trapped in the collection MOT chamber and evaporative cooling is performed in the science cell. The chambers are connected by along thin transfer tube. . . . . . . . . . . . . . . . . . . . . . . . . . . .
A plot of partial pressure as a function of mass showing the residual
gases in our vacuum system without a titanium sublimation pump. The
system was sealed and the ion pumps were left off for one hour for thismeasurement. The red dashed line indicates the sensitivity limit of the
RGA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic diagram of the MOT region including the anti-Helmholtz MOT
coils, the three shim coils, and the lens used to collect MOT fluorescence
onto the photodiode. Not shown: quarter-wave plates just before each
MOT beam enters the vacuum system and the up/down direction MOTbeams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optical arrangement used to combine MOT light with the same polarization for the two species. After the final lens the two beams are collinearwith an offset less than 1 mm. . . . . . . . . . . . . . . . . . . . . . . .
Number of atoms in the two-species MOT as a function of time. At t=0
the trapping and repump light for 85 Rb is applied and that MOT begins
to fill. At the same time the trapping light for 87 Rb is switched on. At
t=15 sec the repump light for 87 Rb is switched on so that atoms areloaded into that MOT. At t=37 sec the 87 Rb repump light is switched
off and the atoms in the MOT are rapidly lost. After the 87 Rb MOT
is switched off, the 85 Rb MOT refills to its initial number. Number loss
occurs in the 85 Rb MOT due to light-assisted collisions as described in
the main text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured density in the 87 Rb compressed MOT as a function of the
CMOT laser detuning. We set the detuning to -35 MHz which gives thebest performance. Here the detuning is measured with respect to the
87 Rb cycling transition. . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

14

17

18

19

20

23

FIGURES

xiii

Level diagram for 87 Rb which shows the optical pumping scheme used tospin polarize the gas into the stretched state. The repump light (green
dashed line) depopulates the lower hyperfine state quickly while the optical pumping light (red solid line) leaves atoms in the dark F = 2, mF = 2
state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.9 Level diagram for 85 Rb which shows the optical pumping scheme used tospin-polarize the gas into the stretched state. The repump light (green
dashed line) depopulates the lower hyperfine state quickly while the optical pumping light (red solid line) leaves atoms in the dark F = 3, mF = 3
state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.10 Fraction of the 87 Rb MOT that is optically pumped into the stretchedstate as a function of the MOT fill. When there are a large number ofatoms in the MOT the optical depth of the gas is large and the efficiency
of optical pumping is limited by re-absorption of spontaneously emitted
photons. When we decrease the optical depth of the MOT by lowering
the number of atoms, the optical pumping efficiency rises due to the
reduction in photon reabsorption. . . . . . . . . . . . . . . . . . . . . . . 27
2.11 Fraction of the 87 Rb MOT loaded into the QT as a function of the QT
current. At the smallest currents the trap is too weak to hold any atoms.
Based on the calculated magnetic fields of the QT, a current of 42 A is
required to trap atoms in the stretched state. Twice as much current isrequired for the other states that have a magnetic moment of µ = 1/2 µB .
If we hold the trap current at about 60 A all the atoms in unwanted spinstates slowly fall out of the trap. . . . . . . . . . . . . . . . . . . . . . . 28
2.12 A schematic of the transfer tube region and the magnetic-field coils forthe multi-coil transfer. The length from the center of the MOT chamberto the tube corner is 8 in. The T coils are mounted on a motorizedtranslation stage to move the coils in and out of the page. . . . . . . . . 30
2.13 Optimized currents for the mulit-coil transfer as a function of location
along the axis of the transfer tube. At peak current each coil pair provides
a field gradient of approximately 100 G/cm in the direction against gravity. 32
2.14 (a) The calculated position of the atoms as a function of time. (b) The
velocity versus time. The velocity starts from zero so that the beginningof the transfer is smooth. Likewise, near the corner the velocity is slowly
reduced to zero. Number loss and heating are observed if the atoms are
violently accelerated to the 0.3 m/s top speed of the transfer. If the startof the transfer is too slow, atoms are lost due to collisions with the Rbvapor in the collection chamber. . . . . . . . . . . . . . . . . . . . . . . 33
2.15 Level diagram for 87 Rb that shows the optical pumping scheme used tospin-polarize the gas into the F = 1, mF = −1 state. The atoms all startin the F = 2, mF = 2 state. The de-pumping light (green dashed line)
depopulates the upper hyperfine state quickly while the optical pumping
light (red solid line) leaves atoms in the dark F = 1, mF = −1 state. . . 35
2.8

FIGURES
2.16 Top-view schematic showing the science cell portion of the vacuum chamber. The positions of the IP trap coils are shown. Along the east/west
direction of the science cell, optical pumping light and the optical trap
beam are combined on a dichroic beamsplitter. Probe light for both
species resonant with the cycling transition is aligned down the transfer
tube, through the atoms in the science cell, and onto the CCD camera
for absorption imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.17 Schematic drawing of a Ioffe-Pritchard trap. Radial confinement is supplied by the magnetic fields of the four parallel wires (purple) and axial
confinement comes from the pinch coils (yellow) along the axial direction.
In order to get stronger confinement along the radial direction, the bias
coils (orange) are set to oppose the field from the pinch coils. Since thefield of the bias coils is uniform, the axial curvature provided by the pinch
coils is not modified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.18 An exploded view of the IP trap coils and the phenolic holder pieces. The
central phenolic holder was shaped so that all the coils fit snugly against
it. The coils are held in compression against the central holder by thephenolic clamp pieces shown. The bottom attachment piece connects thecentral holder to the optical table mount. . . . . . . . . . . . . . . . . .
2.19 A machining drawing of the IP trap setup and a recent photograph of
the IP trap. The six trap coils are rigidly attached to a main holder
piece made of phenolic. The Ioffe bars and the bias coils are visible fromthe outside of the holder. The pinch coils are not visible. The IP trap
is mounted on a sliding dovetail mount so that the trap can be easily
positioned near the science cell. Below the dovetail is a large aluminum
block that mounts the trap to the optics table. . . . . . . . . . . . . . .
2.20 Magnetic fields of the IP trap as a function of position. The data was
acquired with a calibrated gaussmeter. A stable current of 50 A was used
to generate the fields. These measurements are in excellent agreementwith the prediction of our model. . . . . . . . . . . . . . . . . . . . . . .
2.21 The optical setup used to create a 46 micron waist trap. Both lenses are
made by Melles Griot and have good AR coatings near 1030 nm. Lightfor the trap comes over an optical fiber from the Versadisk laser that ismounted on a second optical table. To focus the trap light on the atoms
we translate the position of the fiber output facet with respect to thefirst lens assembly. Not shown are the mirror for trap alignment and thedichroic beamsplitter. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1

Schematic diagram of the laser system for both 85 Rb and 87 Rb. Thelines indicate the source of the frequency stabilization signal for eachlaser. The 85 Rb master is locked via saturated absorption to a Rb line.
The other laser frequencies are stabilized using the offset lock technique.
There is an injection locked laser to amplify the power for the two MOTs.
There is also an injection locked laser for the 87 Rb repump light in order
to increase the available power. . . . . . . . . . . . . . . . . . . . . . . .

xiv

36

38

42

43

44

48

55

FIGURES
3.2

3.3

3.4

3.5

3.6

3.7

Schematic of laser system layout to generate the probing and opticalpumping light for each species. AOMs (shown as green squares) are usedin each beam path to generate short pulses and to control the intensity ofthe beams. There is also a mechanical shutter (not shown) in each beam
path to block light that leaks through the AOMs. A large number ofmirrors, which are not shown in the schematic, are required to get all the
beam paths aligned. The long term stability of the setup is good sincemost of the light is fiber coupled. The beamsplitter cubes are labeledeither BS for a non-polarizing model or PBS for a polarizing one. . . . .
Schematic diagram of the offset locking system used in our experiments.
Light from two separate lasers is superimposed onto a fast photodetector.
The radiofrequency beatnote signal is amplified and then downconverted
to approximately 50 MHz using a mixer. The phase of the downconverted
signal is compared to a stable 50 MHz source by a digital phase-frequency
detector (Analog Devices AD9901) from which an error signal is derived.
A servo uses the error signal to stabilize the frequency of the slave laser
by feeding back to the current and PZT of the slave laser. . . . . . . . .
The red curve (top) is a typical output voltage signal from the digitalphase-frequency detector as a function of beatnote frequency between
the slave laser and master laser. The frequency of the reference laser
was stabilized for this data and the frequency of the unknown was swept
over a wide range. For comparison the black curve shows the Doppler
broadened Rb signal that was simultaneously acquired. With respect tothe lower trace, a larger absorption is up in this figure. . . . . . . . . . .
Photograph of the stable laser setup. The red arrow indicates the pathof the laser light. The diode mount, the grating mirror mount, and thecavity block are indicated in the picture. The main cavity block is madefrom a solid piece of aluminum. In this design the cavity block is extremely thick to provide rigidity. The main cavity block sits within ahermetically sealed box to help keep water from condensing on the temperature controlled laser parts. The entire assembly rests on a hollowedout block of aluminum. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured laser frequency noise as a function of frequency. The frequency
noise is determined by monitoring fluctuations in an atomic absorption
signal. The red trace (top) shows the results for a New Focus Vortex laser
and the black trace (bottom) is the results for the stable laser design. The
stable laser has approximately ten times lower frequency noise and thenoise occurs predominantly at low frequency where our servo is mosteffective. The large feature at 5 kHz in the red trace is a PZT resonancein the Vortex laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of the computerized control system of the experiment. The
DIO-128 generates TTL signals on 64 different lines. Most of the lines
go directly to equipment in the apparatus. The final 16 digital lines areused to create 16 analog voltage with our homemade DAC system. The
DIO-128 also controls the timing of GPIB command execution and it
triggers the PCI-6733 board. . . . . . . . . . . . . . . . . . . . . . . . .

xv

57

59

60

62

63

65

FIGURES
A schematic of the high current circuit for the bias coils. Current flows
from a 15 V, 440 A power supply through the coils, a set of four parallel
FETs, and a current probe. The return of the power supply is grounded
to the AC wall ground. Note that the actual FET arrangement in oursystem is different. An H-bridge configuration is used so that the current
in the bias coils can be reversed via a digital signal from the computer.
It is important to be able to reverse the current of the bias coils so thata wide range of magnetic fields can be achieved in the IP trap. A simple
digital switching circuit routes signals from the servo electronics to the
FET gates based on the desired direction of current flow. . . . . . . . .
3.9 Simple schematic of the bias coil servo. The complete set of diagrams
is available from the JILA electronics shop under file WC063A1. The
dashed box indicates the grounded box; all external inputs entering the
grounded box are buffered to avoid ground loops. In this simple schematic
only the current probe signal and the reference voltage signals are shown.
The servo works by comparing the current probe voltage to the totalreference voltage. The output of the servo is sent to the FET gates. . .
3.10 A schematic of the feedback network (loop filter) response of the servo asa function of frequency. The calculated corner frequencies of the bias coil
servo are shown for reference. This loop filter provides excellent longterm stability of the current and it guarantees the currents accurately
tracks rapid changes in the reference signal without overshoots. . . . . .
3.11 Measured open loop response (a) and phase shift (b) of the bias coil and
FET system obtained with a network analyzer. The network analyzer
driving signal was attached directly to the FET gate and the response
was determined from the current probe signal. A slow integrator wasused to keep a stable current running through the system. Beyond 200
Hz the data displays the response of the coil and FET system. Since anintegrator requires a phase shift of less than 90◦ , our measure response
indicates that the servo loop filter must be switched to proportional gain
at a frequency of 1 kHz or less. Based on the measured phase shift wecan accurately predict the required servo electronics parameters. . . . .

xvi

3.8

4.1

4.2

4.3

Number of particles lost and energy lost in a gas as a function of thetruncation parameter η. Forced evaporation typically utilizes η ≈ 4 − 6
so that only a small fraction of the gas is lost in each cooling iteration.
Results of our single species 87 Rb evaporative cooling simulation. The
number of particles in the 87 Rb gas (left) and the temperature of the
87 Rb gas (right) is shown as a function time. For this simulation we used
K3 = 4 × 10−29 cm6 /s, and the heating rate was 10 nK/s. . . . . . . .
The elastic collision cross section for 85 Rb, 87 Rb, and collisions between
the two species as a function of the collision energy. The total crosssection for 87 Rb and interspecies collisions is roughly flat over a very
wide range of temperature. In contrast, near 375 µK the 85 Rb s-wave
cross section suffers a Ramsauer-Townsend type minimum. This figure
was reproduced from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . .

68

69

70

72

77

79

82

FIGURES
Energy level diagram for evaporative cooling as a function of the distancey from the center of our magnetic trap. The plot shows the level diagramof the 85 Rb |f = 2, mf = −2i and 87 Rb |f = 1, mf = −1i states. We
can selectively evaporate only 87 Rb atoms since the g-factors of the two
species are different. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 (a) Single-species evaporation trajectory of 87 Rb plotted as Log-Log. The
points are the measured atom number at each temperature, the reddashed line is a linear fit to the data to extract αevap = 1.60(5), andthe black solid line is the result of our simulation. (b) The points are
the measured phase space density for a classical gas extracted from thenumber and temperature of the gas. The black line is the calculated
peak phase space density from our simulation. These data represent thecurrent best performance of the 87 Rb evaporation. . . . . . . . . . . . .
4.6 Measured 87 Rb number after evaporation to a fixed rf-cut as a functionof the number of atoms loaded into the MOT. The evaporation is run
with the same rf ramps for each value of NM OT . Roughly 25% of theatoms from the MOT are delivered to the magnetic trap for evaporation.
4.7 Evaporation trajectory of 87 Rb with two different amplitudes of magnetictrap current noise. Here the noise amplitude is the total integrated AC
signal from 10 Hz to 100 kHz. During normal operation the current noiseis 6 ppm on the IP trap coils. When the amplitude of noise on the trap
current was decreased, thelowest achievable temperature also dropped
by a similar amount. This data is an example of poor evaporation due
totechnical limitations in the apparatus. The noise here is limited by theinherent current instability of the commercial power supply we used at
the time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Measured 87 Rb number after evaporation in the optical trap as a functionof the final cut temperature in the magnetic trap. The optical trap hasa finite depth so the best loading will occur at lower temperatures. But
evaporation in the optical trap evaporation is likely a little more efficientthan the magnetic trap so loading the coldest atoms does not produce
the best result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 87 Rb evaporation trajectory in the tight optical trap demonstrating efficient evaporation. This data represents the baseline performance in the
optical trap and is useful for comparing the simultaneous evaporation
efficiency of 85 Rb and 87 Rb. . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Standard three absorption images of 87 Rb BEC formation in the tightoptical trap. Since the axial trap frequency is very low, the image represents the spatial distribution in that direction and anisotropic expansion
is not observed. The first signature of condensation in the optical trap isthe appearance of a bi-modal density distribution. . . . . . . . . . . . .

xvii

4.4

84

88

89

90

91

93

95

FIGURES
4.11 Evaporation trajectories for sympathetic cooling of 85 Rb (black points)
via collisions with 87 Rb (red open triangles). For comparison the data
points with filled green circles are the single-species 87 Rb evaporation
trajectory. Sympathetic cooling is successful since the observed 85 Rb
number does not change while the temperature drops more than a factorof 100. Below 10 µK the number of each species is roughly the sameand cooling of 87 Rb is very inefficient. The inefficiency is characterized
by the large fraction of the 87 Rb gas evaporated in order to lower thetemperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Evaporation trajectories of 85 Rb (black filled points) and 87 Rb (red open
triangles). The solid lines are the result of our simple model for evaporation. The lines are in good qualitative agreement with the data. The
efficiency of 87 Rb cooling remains high throughout the entire evaporation indicating that the heat-capacity limit was not reached. The coldest
87 Rb point is below the BEC transition temperature. Frequent inelastic
collisions cause poor cooling efficiency in the 85 Rb gas. . . . . . . . . .
4.13 Measured average density as a function of time after the completion of
evaporation to a final temperature of (a) 3.5 µK and (b) 1.0 µK. The filled
circles are our data points and the lines are fits to the data. The red solidline is a double exponential function with a fast decay time of 11 sec in(a) and a fast decay time of 2.3 sec in (b). In both (a) and (b) the second
time constant was fixed to 300 sec according to the measured background
lifetime. The dashed line (black) and dotted line (blue) are fits to twoand three-body inelastic collision rates, respectively. We extract a valueof K2 ≈ 5 × 10−13 cm3 /s from the fit. The three-body fit does not agreewith our data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.14 Our most efficient evaporation trajectory during sympathetic cooling in
the magnetic trap. The filled black circles are 85 Rb, the open red triangles
are 87 Rb while cooling 85 Rb. The 87 Rb single species trajectory (green)is shown for comparison. Below approximately 5 µK 85 Rb atoms are
rapidly lost due to inelastic collisions as indicated by the results of oursimulation. The experimental conditions of number and temperatureshown here represent an excellent starting point for cooling in an optical
trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 Measured temperature following a 10 sec hold in the tight trap as afunction of rf-shield frequency. The red line indicates the expected temperature of the gas due to the known heating rate. The black line is theinitial temperature just after evaporation stops. When the rf-shield is
too low a fraction of the gas is evaporated. No number loss or heating
occurs for a shield frequency of more than 3.5 MHz. . . . . . . . . . . .
4.16 Schematic of the two rf shields used to control the heating rate in themagnetic trap. The solid red and blue curves represent the trapping
potential for 87 Rb and 85 Rb and the fill is the surface of the trapped
cloud. The 87 Rb gas is evaporated by rf1 and rf2 is the tunable shield for
85 Rb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xviii

97

99

101

102

104

105

FIGURES
4.17 A compilation of three evaporation trajectories in the weak magnetic
trap. (1) The filled data points are a continuation of the cooling from
Figure 4.14 after decompression to the weak trap. (2) To reach even lower
temperature (open data points) we optimized the initial number of 85 Rb
atoms at the start of cooling. (3) The larger solid points represent ourbest results in the magnetic trap. . . . . . . . . . . . . . . . . . . . . .
4.18 Dependence of the 85 Rb and 87 Rb atom number near the end of evaporation at 600 nK in the weak trap as a function of the number loaded into
the 85 Rb MOT. The MOT number is changed by varying duration of the
MOT fill time. Due to inelastic collisions, the final 85 Rb number is notvery sensitive to the 85 Rb MOT number. In contrast the 87 Rb number
is very sensitive to the 85 Rb MOT. . . . . . . . . . . . . . . . . . . . . .
4.19 A comparison of the 85 Rb population decay as a function of time withand without 87 Rb present. Since the decay rate is roughly the same forthe two cases we conclude that interspecies inelastic loss is negligible. .
4.20 Number of 85 Rb atoms after evaporation to 110 nK as a function ofmagnetic field. The peak of the 85 Rb Feshbach resonance is 155 G whereinelastic losses are expected to be severe. Near 166 G both elastic andinelastic collisions are suppressed and we expect the best evaporation
performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.21 (a) Evaporation trajectory in the tight optical trap demonstrating ourbest results. (b) Peak phase space density at the center of the trap
calculated from the number and temperature during evaporation. The
85 Rb cooling saturates at a phase space density of 0.1, which is less than
the requirement for BEC. Note that the circled points are for condensates
and the phase space density is not known very accurately. . . . . . . .
4.22 The number (black, left side) and temperature (blue, right side) as afunction of magnetic field during single-species 85 Rb evaporative cooling.
The elastic cross section is minimized near 165.7 G and evaporation isinefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.23 Our first evaporation trajectory that makes a single-species 85 Rb BEC.
The key to efficient cooling was very weak confinement to reduce losses.
Near the end of evaporation the 87 Rb number fluctuates wildly shot-toshot causing large instability in the 85 Rb condensate shape and density.
This makes quantitative measurements such as Bragg spectroscopy withthe 85 Rb BEC impossible. . . . . . . . . . . . . . . . . . . . . . . . . .
4.24 Dependence of the 85 Rb and 87 Rb atom number at 900 nK in the weak
optical trap as a function of the number loaded into the 85 Rb MOT.
The MOT number is changed by varying duration of the MOT fill time.
These data are very similar to what we observe in the magnetic trap (forexample see Figure 4.18). . . . . . . . . . . . . . . . . . . . . . . . . . .
4.25 (a) Log plot of the optical trap power as a function of time during evaporative cooling to make 85 Rb BEC. (b) The cooling rate during evaporation
which is related to the time derivative of (a). The increase in cooling rate
at the end of evaporation is used to suppress the final 87 Rb atom numberin the trap. When the 87 Rb number is suppressed a pure 85 Rb condensate
can be repeatedly created over consecutive runs of the apparatus. . . .

xix

107

109

109

112

114

115

117

119

121

FIGURES

xx

4.26 The optimized number of 85 Rb atoms cooled in the optical trap as afunction of the scattering length. Each point represents a separate optimization of the evaporation ramps for a different value of a. The gas
was cooled to a temperature of 200 nK. While performance may not appear to vary significantly the temperature must be lowered by anotherfactor of five so that a small advantage here is more important at lower
temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.27 Evaporation trajectory in the weak optical trap to create a pure BEC
of 85 Rb with a large number of atoms. In these data the temperature isalways determined from the 87 Rb gas since it does not form a condensate.
The final 85 Rb number is small enough that if a small fluctuation in thenumber ratio at the start of cooling occurs it is unlikely to lead to 87 Rb
BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.1

5.2

5.3

5.4

6.1

6.2

Absorption images of 85 Rb (left column) and 87 Rb (right column) condensates demonstrating the immiscibility of the quantum gases. For clarity
the intensity of each 87 Rb image was scaled by a factor of 0.5. The sizeof each image is 330 µm × 150 µm. The number of 85 Rb particles varies
from approximately 12,000 in (a) to 22,000 in (e); the 87 Rb number varies
from 140,000 in (a) to 3,000 in (e). Since the trap frequency in the elongated axis is low this direction of the image is interpreted as the spatial
density of the gas. For comparison (f) shows single-species condensates. 128(a) A schematic drawing of the optical trap laser beam and the magneticfield coils. The relative position of the minimum of the trapping potential
for each species can be adjusted by changing the position of the opticaltrap focus with respect to the center of the bias field coils. Here δ indicatesthe relative difference between the focus of the optical trap beam and thecenter of the bias field coils. Absorption images of immiscible 85 Rb (b)and 87 Rb (c) condensates demonstrating a symmetric density pattern in
the radial direction after recompression of the optical trap. . . . . . . . 129
Absorption images (a-d) and corresponding axial cross sections at two different 85 Rb scattering lengths with significantly different spatial separation. The number of 85 Rb (87 Rb) atoms is approximately 40,000 (90,000). 131
Measured axial position of the 85 Rb and 87 Rb gases as a function of theparameter ∆0 . Here we have varied ∆0 by tuning the 85 Rb scattering
length from 50 a0 to 900 a0 . The uncertainty in the position reflects the
shot-to-shot reproducibility of our measurements. . . . . . . . . . . . . 132
Investigation of Feshbach resonances in the 87 Rb and 85 Rb system via
the enhanced inelastic number loss at the resonances. The asymmetricnature of the left peak is characteristic of higher partial-wave Feshbachresonances. The right peak corresponds to a heteronuclear s-wave resonance. The temperature of the two species gas was 1000 nK for thisdata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Doublet structure of a heteronuclear p-wave Feshbach resonance observed
via inelastic losses at the Feshbach resonance. The temperature of thetwo species gas was 100 nK for this data. . . . . . . . . . . . . . . . . . 138

FIGURES
6.3

6.4

6.5

6.6

6.7

6.8

Absorption-image axial cross sections of the 85 Rb gas demonstrating reversible molecule creation. The measured two-dimensional optical density(OD) was summed in the remaining radial direction of the absorption image. (a) Prior to sweeping the magnetic field the atom number is 26,500.
(b) After sweeping through the resonance 53% of the gas is converted tomolecules. (c) By reversing the molecule creation process approximately
85% of the initial atom number is observed to reappear. Note that mostof the 85 Rb loss occurs in the center of the gas where the 87 Rb BEC
density is largest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Molecule conversion efficiency as a function of sweep rate. The black
points show the rate required for conversion at the 265 G resonance, while
the red points show the required rate at the wider 372 G resonance. Bothdatasets were taken starting with similar atom number and temperature.
This data has not been corrected for the effects of atom loss not due tomolecule formation, therefore the number of molecules created in thesedata is less than the total atom loss. . . . . . . . . . . . . . . . . . . . .
Atom loss after magnetic-field sweeps through a Feshbach resonance. The
number of 85 Rb atoms is a function of the final magnetic field during thesweeps near the (a) 265 G and (b) 372 G Feshbach resonances. Initiallythere are 2.1 − 2.3 × 105 87 Rb atoms at T/Tc between 0.82 and 0.84. The
data are fitted to an error function to extract the center position andwidth. The resulting positions of the two transitions are 265.44 ± 0.15 G
and 372.4 ± 1.3 G with the uncertainty given by the fitted RMS width.
Heteronuclear molecule conversion efficiency at the 265 G Feshbach resonance as a function of 87 Rb T/Tc . At our largest conversion efficiency
the 85 Rb gas has T/Tc = 2.6. The solid line shows a simulation basedon our conversion model, and the dashed lines represent the uncertainty.
The conversion drops below T/Tc = 1 since the BEC is spatially smaller
than the thermal 85 Rb gas and the conversion process depends on theproximity of two atoms in phase space. Some of the data for this figure has been averaged together. The error bars represent the weighted
average of the points. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Resonant frequency of atom loss as a function of magnetic field. The
solid line is a fit to the data based on the universal binding energy of swave Feshbach molecules and the dashed lines represent the uncertainty
in the Feshbach resonance width. (Inset) An atom-loss spectrum at 266.5
G as a function of the modulation frequency. The modulation converts
roughly 50% of the gas to molecules. The loss is centered at 21.7 kHzwith a width of 0.6 ± 0.2 kHz. The solid line is a gaussian fit to the data.
We report the uncertainty in the binding energy as the width of the loss
spectrum because we lack a detailed understanding of the lineshape. . .
Preliminary measurements of the binding energy of p-wave heteronuclear
Feshbach molecules. Each binding energy measurement represents theresonant frequency of atom loss. The solid lines are a linear fit to the datawith slope 58(4) kHz/G. The splitting between the two ml resonances isapproximately 0.5 G. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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139

140

142

143

145

147

FIGURES
7.1

7.2

7.3

7.4

7.5

7.6

Bogoliubov excitation spectrum (Eqn. 7.1) for a condensate (black curve)
and the free particle energy (red curve) as a function of momentum. The
black curve has n = 1014 cm−3 and a scattering length of 150 a0 . The
green vertical line represents the inverse healing length of the condensate
and the crossover to free-particle excitations. . . . . . . . . . . . . . . .
Simplified schematic of the science cell area including the Bragg laser
beams, the science cell optical pumping light, and the optical trapping
beam. The Bragg beams (shown as a dashed green line) are deliveredby two optical fibers. The Bragg light is overlapped with the opticaltrapping beam to assure that Bragg excitations are along the axis of
the condensate. The optical pumping and Bragg light are misaligned
by a few degrees so that the beams do not need to be combined with abeamsplitter. The Bragg beam diameter is a few mm at the center of thescience cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A comparison of different Bragg spectroscopy detection techniques. (a)The number of Bragg scattered atoms is directly counted after the atoms
exit the BEC. The scattering length was set to 150 a0 during the Bragg
pulse. For this technique to be successful collisions between the Bragg
scattered atoms and the BEC must be minimized. Immediately following the Bragg pulse the scattering length is tuned close to zero. (b) The
scattering length was not tuned toward zero after the Bragg pulse. Many
collisions occurred between the Bragg scattered atoms and the condensate, and our signal is diminished. (c) Our method for Bragg spectroscopy
in which all the Bragg scattered atoms collide many times with the BEC.
The momentum is deposited into the condensate which generates a COM
oscillation. The amplitude of this oscillations is our signal. The center
position of the condensate in (c) is clearly offset from the condensate on
the left hand side of (a) and (b). . . . . . . . . . . . . . . . . . . . . . .
A center-of-mass condensate oscillation along the axial direction of the
optical trap induced by two-photon Bragg transitions. The amplitudeof this motion is our signal for Bragg spectroscopy. Bragg spectra are
obtained by monitoring the condensate position at 230 ms. . . . . . . .
A typical Bragg spectrum acquired at a scattering length of 900 a0 , withn a3 = 0.01. The Bragg pulse duration was 0.087 ms. We extract thecenter position and width of the Bragg spectrum by fitting the data to thedifference of two gaussians. Since our data is anti-symmetric about zero,
the gaussian fit is performed with only a single value of the amplitude,
the center position, and the width. . . . . . . . . . . . . . . . . . . . .
(a) Fraction of the condensate excited as a function of Bragg pulse length.
The Bragg laser powers were fixed at 35 µW and 2.5 µW and the beam
1/e2 radii are approximately 2.3 mm and 1.3 mm, respectively. The data
points at 1 ms are noisy since the linewidth of the Bragg resonance is sonarrow. (b) Fraction of the condensate excited as a function of Bragg
laser power. The pulse length was fixed at 0.5 ms and one of the laser
beams had a power of 35 µW while the other power was varied. The solidlines are the prediction for the fraction of the condensate excited from
Eqn. 7.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxii

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154

156

157

158

160

FIGURES
7.7

xxiii

The solid black line is the calculated Bragg lineshape plotted as a functionof the frequency ω/2π. The lineshape is asymmetric due to the ThomasFermi density. The vertical dotted line indicates the mean excitationenergy of the condensate. The red dashed line shows a gaussian fit to theBragg lineshape which accurately determines the mean excitation energy. 162
7.8 Excitation energy as a function of the wave vector k. The black curveis the Bogoliubov spectrum for weak interactions. The red and blue
curves are the respectively low-k and high-k limits of the excitation energy
including the n a3 correction. The scattering length is 900 a0 and thedensity is 1014 cm−3 . The red and blue curves were arbitrarily broken at
9 µm−1 . For each theory line shown here we assume that k a = 0. . . . 164
7.9 (a) Theoretical Bragg line shift of a homogeneous condensate as a functionof scattering length. The curves, in order of smallest to largest shift, correspond to the following theories: Hartree-Fock-Bogoliubov (solid black),
Beliaev first approximation (dashed red), Bogoliubov (dash-dot blue),and Beliaev second approximation (dash-dot-dot magenta). Calculations
were performed with a mean density of 8.5 × 1013 cm−3 and an excitation
momentum of 2 h̄ kL . For reference the top axis shows the size of the
LHY correction to the condensate chemical potential. . . . . . . . . . . 166
7.10 A comparison of the lineshape due to the inhomogeneous density of thecondensate (black dotted line) and the sinc-function lineshape of the
Bragg pulse (green dotted line). The convolution of these two functionis the blue solid line. We extract the total width from a gaussian fit (reddashed line) to the convolution. . . . . . . . . . . . . . . . . . . . . . . 168
7.11 Gaussian width from a fit to the convolution of the three lineshape functions versus
the lorentzian width γ. The data are fit to the functionpa γ + bγ 2 + w2 to extract the parameters a and b. This fitting function resembles that which is used to understand the total width of a
Voigt profile [2]. A Voigt profile is the convolution of a gaussian and alorentzian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.12 Prediction of the Bragg resonance width as a function of scattering length.
The dotted lines correspond to different contributions to the total RMS
width including the lorentzian FWHM width due to condensate-excitation
collisions (blue dashed line), the gaussian RMS widths due to the inhomogeneous density (red dotted line), and the pulse duration (green dashdotted line). The solid black line is the prediction for the total widthof the Bragg resonance based on a convolution of the profiles for eachcontribution. To get an idea of the size of beyond mean-field effects on
the width, the black open circles are the predicted total width including
the LHY chemical potential in the inhomogeneous density contribution.
Calculations were performed with a mean density of 8.5 × 1013 cm−3 andan excitation momentum of 2 h̄ kL . . . . . . . . . . . . . . . . . . . . . 170

FIGURES

xxiv

7.13 (Top) The elastic collision cross section for quasiparticle-condensate collisions with wave vector kcm . (Bottom) The k-dependent scattering length
derived from the real part of the scattering amplitude with wave vector
kcm . Several quantities are plotted on each graph. The red line is theunitarity limit, the blue line is σ = 8πa2 , the black line is from Ref. [3],and the green dotted line is our analytic results given by Eqn. 7.22 andEqn. 7.21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.14 Measured Bragg line shift (left) and width (right) as a function of time.
A large density oscillation was excited in the condensate, and we detectedthat oscillation with Bragg spectroscopy. The solid lines are a theoretical prediction based on the model of Ref. [4] for the time-dependent
condensate density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.15 (a) Scattering length sequence to increase the condensate density. Knowledge of the scattering length is derived from measurements of the magneticfield and a previous experimental calibration of the 85 Rb Feshbach resonance [5]. The Bragg pulse is initiated at 0 ms immediately following
a rapid (0.12 ms) increase to aBragg . Not shown is the Bragg pulse timing, the ramp to 890 a0 to excite a condensate center-of-mass oscillation,and the scattering length ramps used during absorption imaging. (b)Expanded view of the scattering length ramps near 0 ms. The radial sizeof the BEC is lowered by setting the scattering length to aevolve . (c, d)Calculated time-dependent density during the scattering length sequencebased on Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.16 Measurements of the condensate axial size (left) and radial size (right)
as a function of time during the compression sequence. The solid curves
are the result of the GP-equation modeling of the condensate size. The
dashed curves indicate the expected sizes if the density were 30% higher
or lower. See the main text for more details. . . . . . . . . . . . . . . . 177
7.17 Typical Bragg spectra at a scattering length of 100 a0 (blue crosses), 600
a0 (red circles), and 900 a0 (black squares). Lines are fits of the data as
described in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

FIGURES
7.18 (a) Bragg line shift and (b) width extracted from a gaussian fit to theBragg resonance as a function of scattering length. The points are ourobservations and the error bars represent standard deviations of the fit.
The mean density during the measurements ranges from 6 × 1013 cm−3
to 8.5 × 1013 cm−3 due to three-body recombination during the Bragg
pulse. In (a) the theory points, in order of smallest to largest shift, correspond to the following theories calculated for our trapped gas using a
local density approximation: Hartree-Fock-Bogoliubov with n a3 and k afinite but k ξ À 1 (black downward triangles), Beliaev first approximation with k a finite, n a3 = 0 and k ξ any value (red boxes), Bogoliubov
with n a3 = 0 and k a = 0, but k ξ any value (blue upward triangles),and Beliaev second approximation with n a3 finite, k a = 0, and k ξ any
value (magenta open circles). For reference the top axis shows the sizeof the LHY correction to the condensate chemical potential. In (b) thecrosses represent a theoretical prediction for the resonance width under
the conditions of our experiments. The open squares represent the predicted width when the LHY chemical potential is used to calculate theinhomogeneous density contribution. . . . . . . . . . . . . . . . . . . .
7.19 (Left) Measured atom number and (Right) measured condensate fraction
after the condensate density compression and scattering length ramp for
the Bragg pulse as a function of a. The solid line is the predicted atom
number loss based on the measured three-body recombination rate of
85 Rb in Ref. [6]. The measured atom number loss is used to estimate thedensity during the Bragg pulse for the experiments shown in Fig. 7.18.
7.20 Measured Bragg linewidth as a function of time after the scattering length
was increased to 900 a0 (left) and 600 a0 (right). The blue curves are a
prediction based on our model for the Bragg width, and the red curves
add some condensate heating to the prediction. . . . . . . . . . . . . .
7.21 Measured Bragg line shift as a function of the condensate fraction. We
the fraction is zero the temperature of the gas is slightly below the transition temperature. Our data indicate a significant increase in the Bragglineshift as the condensate fraction is increased. The curve represents theprediction of our model where the lineshift at 100% condensate fractionis 18.3 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxv

180

182

184

186

FIGURES

xxvi

7.22 (a) Bragg line shift as a function of scattering length with an averagedensity of approximately 2.0 × 1013 cm−3 . The theory lines, in order ofsmallest to largest shift, correspond to the following theories calculated
for our trapped gas using a local density approximation: Hartree-FockBogoliubov with n a3 and k a finite but k ξ À 1 (black solid line), Beliaev
first approximation with k a finite, n a3 = 0 and k ξ any value (red dashedline), Bogoliubov with n a3 = 0 and k a = 0, but k ξ any value (bluedotted line), and Beliaev second approximation with n a3 finite, k a = 0,
and k ξ any value (magenta dash-dotted line). (b) Bragg width as afunction of scattering length at a density of 2.0 × 1013 cm−3 . The lines
show the contributions of the pulse duration (green dashed line), theinhomogeneous density of the condensate (red dashed line), and the width
due to collisions during Bragg spectroscopy (blue dshed line). The blacksolid line is a combination of three contributions following Eqn 7.16. . . 188

Chapter 1
Introduction

1.1

An overview of my work

My thesis work began during an exciting time in the Wieman group. Aftera few years of optimization, the first Bose-Einstein condensate (BEC) of 85 Rb was
created in 2000, and right away their work was rewarded with the ability to study acondensate with tunable interactions. Atom-atom interactions in a ultracold gas of
85 Rb can be tuned using the experimentally convenient Feshbach resonance near 155 G.
Initial studies included observations of the equilibrium condensate shape and density
over a wide range of interaction strength [7] and a quantitative time-resolved analysisof the condensate collapse process for negative interactions [8, 9]. When I started in
the group during the summer of 2001, the focus was on understanding the role of thebound molecular state associated with the 85 Rb Feshbach resonance. This focus led tothe dramatic observation of coherent atom-molecule oscillations [10]. However, by 2001
the 85 Rb BEC apparatus was already old and it would need to be significantly upgradedto incorporate modern technologies like Bragg spectroscopy, optical trapping, etc. My
thesis work started by designing and constructing a new apparatus. The goal of the
new apparatus was to efficiently produce a quantum degenerate gas of 85 Rb for use inexperiments studying the strongly interacting regime.
We had several design goals for the new 85 Rb BEC apparatus so that experiments
could be performed in interesting new regimes. By 2001 the route to 85 Rb BEC via
direct evaporative cooling had been optimized, but the overall efficiency of the cooling
was low. Since the efficiency was low, the apparatus had to be well adjusted in order forcondensates to be made. Therefore our goal was to robustly create 85 Rb condensateswith a larger atom number. Since direct evaporation was difficult, we decided to pursue
a new approach in which sympathetic cooling with 87 Rb would be used.1 To perform
experiments in new regimes, we would also need to include more technology than was
feasible with the old apparatus. Techniques like optical trapping and Bragg spectroscopy
are a main focus of the work presented in this thesis, but they were not available in theold apparatus where optical access was limited.
Our motivation for sympathetic cooling is that, in comparison to 85 Rb, evaporative cooling of 87 Rb is easy. By starting evaporation with a large number of 87 Rb
atoms and a tiny 85 Rb number, we evaporate the 87 Rb gas while the 85 Rb gas cools
due to thermal contact. This technique works well for lowering the temperature down1

By 2001 another group had attempted sympathetic cooling of 85 Rb with 87 Rb, but they wereneither able to create a 85 Rb condensate nor access Feshbach resonances [11].

CHAPTER 1. INTRODUCTION2

to 1 µK. To cool all the way to BEC, however, we switch from sympathetic coolingto simultaneous cooling of 85 Rb and 87 Rb. Especially in these last stages of evaporation we must be very careful to minimize 85 Rb density-dependent inelastic collisions[7, 12, 6]. In particular, we work with very weak trapping potentials to maintain a low
density. We also take full advantage of the 85 Rb Feshbach resonance to optimize theelastic and inelastic collisions rates of 85 Rb. After spending a lot of time improving theevaporation efficiency, we were pleased to find that we could make 85 Rb condensateswith up to 80,000 atoms, which is a factor of five improvement over the past results.
Even better was the fact that condensate production was robust. Condensates could
still be produced even if the apparatus was not perfectly optimized. Our system now
has the unique ability to produce single-species condensates of 85 Rb and 87 Rb, as well
as a two-species BEC.
Using both isotopes of Rb has advantages beyond an improvement of 85 Rb coolingefficiency. Experimentally accessible and interesting interspecies Feshbach resonances
between the two isotopes were predicted to exist [13], but none had been observed
experimentally. Once we were able to cool a gas of 85 Rb and 87 Rb to microKelvintemperatures in an optical dipole trap, we set out to find the predicted resonances. We
discovered a total of four new Feshbach resonances, two of which were s-wave and two of
which were p-wave. These Feshbach resonances not only allowed us to tune interactions
between the two species, but also to create and study the first heteronuclear Feshbach
molecules. Ultracold heteronuclear molecules in low-lying vibrational states are particularly interesting since they are predicted to exhibit a permanent dipole moment due tothe unequal distribution of electrons. Although no significant permanent dipole moment
is expected to exist in a 85 Rb–87 Rb molecule, our work demonstrates a first step toward
the efficient production of ground-state ultracold heteronuclear molecules. In the future
it may be possible to lower the vibrational state of our Feshbach molecule via, for example, stimulated two-photon Raman transitions. The molecule creation process allowed
us to determine the location and width of Feshbach resonances in the two-species system; this information will be required for future studies of the 85 Rb–87 Rb system witha tunable interspecies interaction.
Prior to our work, the production of a two-species condensate with 85 Rb and 87 Rb
had never been accomplished. Since the early days of dilute gas BEC, there has been
great experimental and theoretical interest in multi-component condensates. While oursystem is not the first two-component BEC, it is unique due to our ability to control
interactions. We are able to study the miscibility of two interacting quantum fluids asa function of the 85 Rb scattering length. The controllable spatial separation of the twocomponents is observed due to interparticle interactions. Surprisingly, the immiscible
condensates are observed to form multiple spatially-separated cloudlets.
As mentioned earlier, the main goal of our new 85 Rb BEC machine was to be able
to make large, stable 85 Rb condensates in a system with plenty of optical access thatwould allow more sophisticated probes of a strongly interacting BEC. The final projectof my thesis, in which we measured the excitation spectrum of a strongly interacting
85 Rb BEC, represents the realization of this goal. Bragg spectroscopy was used torecord the excitation spectrum over a very wide range of scattering length. For a
weakly interacting BEC, the excitation spectrum is given by the famous Bogoliubov
formula. The Bogoliubov spectrum is valid only in the limit that the condensate isdilute (n a3 ¿ 1) and in the limit in which two-body scattering is independent ofCHAPTER 1. INTRODUCTION

3

momentum (k a ¿ 1). With our apparatus we can achieve experimental conditions ofdensity and scattering length under which neither of these conditions are met.

1.2

Historical overview

Lowering the temperature of a sample into a regime in which new physical effects
can be observed has long been a goal of experimental physics. A great breakthroughoccurred in 1911 with the discovery of superconductivity in a low-temperature sample ofmetallic mercury [14]. By monitoring the resistance of mercury as the temperature waslowered below 4.2 K, a dramatic transition to the superconducting state was observed.
Years later, in 1938, careful studies of liquified 4 He at a temperature below 2.17 K
revealed the appearance of viscosity-less fluid flow, the characteristic of a superfluid[15, 16]. The peculiar properties of this superfluid Bose liquid were discovered in thefollowing decades. In particular, the elementary excitation spectrum of superfluid 4 He
displays complicated behavior. The excitation spectrum refers to the energy required to
add a single particle at a given momentum to the system.2 Interparticle interactions
tend to modify the excitation spectrum. In fact, the low momentum excitations ofsuperfluid 4 He behave like phonons, and at larger momenta a so-called “roton-minimum”
appears, at which the excitation energy has a local minimum. While the form of theexcitation spectrum can be predicted from physical principles [17, 18], the physicalinterpretation of a roton is complicated [19].
In both of the examples given above, the temperature of the samples was lowered
into a regime in which quantum-mechanical effects are important. Following these spectacular discoveries a strong experimental focus was to achieve ever lower temperaturesin order to expose quantum-mechanical effects in a range of new systems. Of course,
the effects of quantum mechanics are observed outside the regime of low-temperaturephysics. The electrons in a room-temperature metal behave quantum mechanically since
the Fermi temperature of the system is so high. Likewise, a neutron star is a massive
quantum degenerate gas due to the extreme density.
Jumping forward in time, in the 1990’s a new superfluid system was realized
in a dilute atomic vapor. The landmark experimental observation of dilute gas BoseEinstein condensation (BEC) has opened the door to a wide range of new studies [20,21, 22, 23]. If a low enough temperature is reached, then a phase transition occurs inwhich the ground state of the system is macroscopically occupied. There are several
advantages to working with a dilute atomic vapor in comparison to a dense liquid like
4 He. Strong interactions in a 4 He superfluid severely deplete the number of particles in
the lowest quantum state, while weak atom-atom interactions in atomic gas BEC resultin a large ground-state occupation. The properties of a weakly interacting BEC may
be accurately described by a mean-field theory. Finally, interactions in a quantum gas
may be controlled via magnetic-field tunable Feshbach resonances. Some of the firstwork with condensates produced dramatic evidence for coherence via the interferenceof two condensates [24] and for superfluidity via the observation of quantized vortices[25]. More recently atomic BECs have been studied in trapping potentials where thedimensionality can be changed. This led to the observation of the Mott-Insulator phase2

For example, in a gas described by Maxwell-Boltzmann statistics the excitation energy of thecondensate is p2 /2 m, where p is the momentum and m is the mass.

CHAPTER 1. INTRODUCTION

4transition in a 3D optical lattice potential [26] and the observation of a Tonks-Girardeau
gas in a 1D optical lattice [27].
The experimental techniques used to create the first dilute gas BEC have also
been applied to cool a Fermi gas of atoms to a low enough temperature that the effectsof quantum mechanics are visible. Since fermions obey the Pauli Exclusion principle, at
low temperature the particles stack up with one particle per quantum state. Without themacroscopic occupation of a single quantum state, ultracold fermions do not experiencea phase transition, but the effects of Fermi statistics can be observed in the behavior of
the energy in the gas as the temperature is lowered below the Fermi temperature [28, 29].
A degenerate Fermi gas has much more energy than a similar gas of Maxwell-Boltzmann
particles at ultralow temperature [28].
A key theme of recent work in the field of ultracold quantum gases has been
the exploration of the strongly interacting regime. There are two methods by whichstrong interactions can be achieved: by increasing the confinement or by using a Feshbach resonance to increase the interaction strength. Here we focus on experimentswith tunable scattering length that are enabled by a Feshbach resonance. Much of theoutstanding work during the last few years utilized Feshbach resonances in a Fermi gasof atoms. The culmination of these experiments with fermions was the realization of the
BCS-BEC crossover in which pairing occurs between fermions in the gas [30, 31, 32].
On the side of the Feshbach resonance with attractive interactions the paired fermions
resemble Cooper pairs. Whereas on the side of the Feshbach resonance with repulsive
interactions, the paired fermions resemble Bose-condensed bound molecules [33, 34, 35].
A strongly interacting Fermi gas represents a single low-temperature system in which
the quantum mechanical effects reminiscent of the physics of superconductivity and thephysics of a Bose superfluid are observed.
In general, it is more difficult to study a Bose gas near a Feshbach resonance due
to large losses associated with inelastic three-body recombination. Therefore studiesof the strongly interacting regime with fermions dominated the field, since the rate ofthree-body processes is smaller [36, 37, 38, 39]. In spite of the strong losses, importantresults were also obtained with Bose gases. Coherent atom-molecule oscillations wereobserved in 85 Rb that helped elucidate the relationship between atom and molecule
quantum states [10]. Later the efficient production of ultracold Feshbach molecules froman ultracold bosonic gas was demonstrated using a Feshbach resonance [40, 41, 42, 43].
The goal of my thesis is to continue this line of studies to develop a more detailed
understanding of a strongly interacting Bose gas. An important long-term goal is to the
explore the degree to which a strongly interacting quantum gas of bosons resembles aquantum liquid like superfluid 4 He.

1.3

Ultracold Bose gases

1.3.1

Thermodynamics of a non-interacting Bose gas

In this section we will briefly describe the thermodynamics of a trapped noninteracting Bose gas. The result of this analysis quantifies the temperature that we
strive towards throughout this thesis to create a 85 Rb BEC. Based on a formulation using
the Grand Canonical Ensemble we are able to calculate the transition temperature forthe formation of a BEC, the phase-space density of the Bose gas, and the occupation

CHAPTER 1. INTRODUCTION

5of the ground state.3 This subject has been covered in detail in Refs. [44, 23, 45]
and therefore only the most important results that are used throughout this thesis arerecounted here.
We start with the Hamiltonian of a single particle in a harmonic trapping potential
which is given by1
p2
2+ m ωρ2 r0
(1.1)
H=
2m 2
where p is the magnitude of the total momentum, r0 2 = x2 + y 2 + Z 2 is the (re-scaled)
position coordinate, and m is the atomic mass. Here we define Z = λ z and λ = ωz /ωρwhere ωρ (ωz ) are the radial (axial) trap frequencies. In a harmonic potential the densityof states is well approximated by [44]
ρ(²) =²2.
2λ(h̄ ωρ )3(1.2)

The statistics of a Bose gas are given by the Bose-Einstein distribution, which determines
the occupation number of each quantum state n (with energy levels ²n ) in the system
and it is given by
1(1.3)
f (²n ) =
1
(²
−µ)
−1
e kB T n
where T is the temperature of the gas, µ is the chemical potential, and kB is Boltzmann’sconstant. In the following we assume that the energy levels can be treated as continuous.
The chemical potential is determined by normalizing the distribution to the total numberof particles, to find the chemical potential the following equation is solved implicitly for
µ
Z
N=

∞

0

ρ(²) f (²)d².

(1.4)

For a non-interacting Bose gas the chemical potential reaches zero when the temperatureis equal to the transition temperature. Therefore the transition temperature Tc of the
non-interacting condensate is given byµkB Tc = h̄ ωho

N
ζ(3)

¶1/3

(1.5)

where ζ(3) = 1.202 is the Riemann zeta function and ωho = (ωρ2 ωz )1/3 . In our experiments the transition temperature for 87 Rb is typically about 500 nK while the transition
temperature for 85 Rb is approximately 100 nK. The difference lies in the efficiency of
evaporative cooling, since many more 87 Rb atoms can be cooled. The fraction of the total number of atoms in the gas that are in the ground state is related to the temperatureand the transition temperature by
N0
=1−
N

µ

T
Tc

¶3

(1.6)

where N0 is the number in the ground state and N is the total number. The peak phasespace density D, which is the product of the density at the center of the trap and the
3

In Chapter 4 we determine the density distribution of the trapped gas, from which the averagedensity and mean-relative speed may be calculated.

CHAPTER 1. INTRODUCTION

6
2

thermal de Broglie wavelength (λT = ( m2 πkBh̄ T )1/2 ), is given by
D = λ3T × nBE (0)(1.7)

where the density of a harmonically trapped Bose gas (nBE ) is given byµnBE (r0 ) =−1
1( 1 m ω2 r0 2 −µ)g3/2 e kB T 2 ρ
3
λT

¶

(1.8)

where g3/2 is the commonly tabulated Bose-Einstein function defined in Ref. [45].
1.3.2

An interacting Bose-Einstein condensate

In the absence of interactions the quantum-mechanical wave function of a fully
condensed gas is given by the product of the single-particle wave functions [44]Ψ(r~1 , r~2 , ..., r~N ) =

N
Y

φ(~
ri )

(1.9)

i=1

where N is the total number of particles in the gas and φ(~ri ) are the single-particle wave
functions which are normalized to one. Interactions require modifications to the total
wave function. The total Hamiltonian of the system including interactions is given by
H=

N
X
i=1

Ã

!

X
p2i
+ Vext (~ri ) + g
δ(~ri − ~rj )2mi<j(1.10)

where atom-atom interactions are specified by the contact interaction g δ(~ri − ~rj ) in2
which g = 4 πmh̄ a [44]. By minimizing the total energy of the system one finds the wellknown Gross-Pitaevskii (GP) equation for the ground state of the trapped condensate−h̄2 2∇ ψ(~r) + Vext (~r) ψ(~r) + g |ψ(~r)|2 ψ(~r) = µ ψ(~r)
2m(1.11)

where ψ(~r) is the wave function of the condensed state, |ψ(~r)|2 is the density of thegas, and Vext is the external confining potential. An important simplification of theGP equation can be made if the kinetic energy term is neglected. This simplification is
known as the Thomas-Fermi (TF) approximation and its use is justified only when
N

a
À1
aho

(1.12)q

where aho = m h̄ωho . In this limit the condensate size is large due to repulsive atomatom interactions, therefore the kinetic energy is small. The density of the trapped
condensate in the TF limit is given byn(~r) =

1(µ − Vext (~r)) .
g(1.13)
2/5 h̄ ω

In the TF limit the condensate chemical potential is µ = 15

2

³
ho

Na
aho

´2/5

[23].

CHAPTER 1. INTRODUCTION

7

A major goal of the work presented in this thesis is to reach an experimental
regime where the GP equation no longer accurately describes the properties of a condensate. The GP equation is valid in the dilute-gas regime in which the interparticle
spacing in the gas is larger than the scattering length, i.e. n |a3 | ¿ 1. The availability of magnetic-field Feshbach resonances in an atomic system allows access to strong
atom-atom interactions. When the scattering length is large, corrections to the GP
theory are expected to be important. These correction were initially studied during the1950s for a homogeneous gas. For ³example, the first-ordercorrection to the condensate´p3 /π . This was first calculated by Lee,chemical potential is µLHY = g n 1 + 32na3
Huang, and Yang (LHY) [46, 47]. Strong interparticle interactions also tends to reduce
the probability of an atom being in the ground state at zero temperature,
the amount√8
3 where Nof this quantum depletion of the condensate is given by NNout = 3 √
n
a
out
π
is the total number of depleted atoms. Since quantum depletion is neglected in the GP
equation, the size of the depletion is a good indicator of the validity of the GP theory
[23].
1.3.3

Elastic and inelastic atomic collisions

A central theme in this thesis will be the differentiation between elastic andinelastic atomic collisions. Elastic collisions between atoms in the context of ultracold
gas experiments can be considered good collisions. These collisions rethermalize thegas but do not change the internal state of the atoms. On the other hand a constantconcern in ultracold gas experiments are inelastic collisions, often referred to in thisthesis as bad collisions. In these collisions the products involve particles in different
internal states, often of lower energy. These collisions lead to loss from the experiment,
as the different internal states are often no longer detectable in our experiments. They
also lead to detrimental heating because the difference in energy between initial atoms
and the products of the collision must be released as kinetic energy. Inelastic collisions
that are discussed in this thesis involve either two atoms (two-body collisions) or threeatoms (three-body collisions). An excellent review of inelastic collisions in ultracoldgases is found in Ref. [1].
To describe an elastic collision between a pair of atoms in an ultracold gas, thewave function is found by solving the time-independent Schrödinger equation. Scattering theory is described in great detail elsewhere [1, 6, 48, 49, 44, 50], here we give a few
details that are used later in this thesis. The wave function Ψ for the relative motionof the two atoms is given by [48]
Ψ = ei k z + f (θ)

ei k rr(1.14)

where the first term represents the incoming wave function, which is a plane wave withrelative motion along the z-axis. The second term represents the scattered wave function, which is a spherical wave with wave vector k. The factor f (θ) is the scattering
amplitude where the angle θ describes the difference of the relative momentum wave
vector before and after two-body scattering occurs. Important quantities that are used
to characterize elastic collisions in our system are calculated fromthe scattering ampliR1
tude, for example the elastic collision cross section is σ = 2 π −1d(cos(θ))|f (θ)|2 [44].

CHAPTER 1. INTRODUCTION
The scattering amplitude isf=

81
k cot(δ) − i k(1.15)

where δ is the scattering phase shift. This phase shift of the scattering wave function is
calculated based on the interaction potential of the two colliding atoms [1]. The phaseshift may also be determined from the effective range approximation; see Section 7.6.3
for more details. Note that in the limit of very low energy, the scattering amplitude isconstant and equal to −a, then the wave function is [44]aΨ=1− .
r(1.16)

Therefore the scattering length describes the asymptote of the wave function on ther-axis.
Another central theme is our ability to tune the interactions in our gas with a
Feshbach resonance. At a Feshbach resonance simply by tuning a magnetic field B
we can, through the differential Zeeman tuning, bring a bound state near the atomic
scattering threshold. This leads to a divergence in the scattering properties of thesystem according to the following equationµa(B) = abg∆
1−
B − B0¶(1.17)

Here abg is the background (non-resonant) scattering length, B0 is the magnetic fieldposition of the Feshbach resonance, and ∆ is the magnetic-field width of the Feshbachresonance, defined as the distance in magnetic field between B0 and the magnetic field at
which a = 0. Readers interested to know more about the physics of Feshbach resonancecan consult a now very large literature on the effect [51, 52, 53, 54, 6, 32, 55, 56, 57].
It is important to recognize that near a Feshbach resonance inelastic collisions can beenhanced just as much as elastic collisions. For a discussion of the relationship between
elastic and inelastic collisions near the 85 Rb Feshbach resonance see Ref. [12, 6].

1.4

Contents of this thesis

In the first half of this thesis I will concentrate on the apparatus and on theevaporative cooling sequence that we use to create condensates. We have the uniqueability to produce a BEC with 85 Rb, 87 Rb, or a mixture of both species. The realizationof this system was the culmination of several years work on building and optimizing ourapparatus. The success of our apparatus is indicated by the significant improvement in
the efficiency of 85 Rb condensate production that we achieved. For the benefit of future
students using this apparatus, or a similar BEC machine at JILA, I have tried to makethe description of the apparatus and cooling procedures as detailed as possible.
The main scientific results of this thesis are presented in Chapters five, six, andseven. Chapter 5 contains the results of experiments with a dual-species BEC of 85 Rb
and 87 Rb. The miscibility of this system of two interacting quantum fluids was studied as a function of the 85 Rb scattering length. Chapter 6 describes the creation of
the first heteronuclear Feshbach molecules. We studied two different methods to create
the molecules, and using signatures of the molecule creation we precisely measured theposition of two Feshbach resonances in our two-species system. Chapter 7 describes

CHAPTER 1. INTRODUCTION

9

our measurements of the excitation spectrum of a strongly interacting BEC. In particular, we have observed beyond mean-field corrections to the excitation energy of acondensate.

Chapter 2
The Apparatus

2.1

Chapter overview

This chapter describes the new apparatus that was used to obtain all the results detailed in the rest of this thesis. The overall goal for the new apparatus was toproduce 85 Rb Bose-Einstein condensates in order to study a strongly interacting BEC.
The motivation was the ground-breaking work done in the Wieman group using theapparatus detailed in Ref. [6, 58]. Despite the spectacular scientific results [7, 8, 9, 10],the original 85 Rb BEC machine had several shortcomings. Primarily it was quite old,
built in 1994-1995 [59], and therefore not designed to incorporate modern technologies
used in BEC experiments like Bragg spectroscopy, optical trapping, etc. Furthermore
the original 85 Rb BEC machine relied on the difficult task of direct 85 Rb evaporative
cooling. That task requires a very high quality vacuum, careful attention to nearly all
experimental parameters, and precise optimization of the rf-evaporation trajectory. In
this thesis, I will present an alternative approach that significantly relaxes the experimental requirements to produce 85 Rb BEC while also improving the efficiency of 85 Rb
condensate production. We use sympathetic cooling with 87 Rb to reduce the difficulties
associated with 85 Rb evaporative cooling. The evaporative cooling procedure we use isdiscussed in Chapter 4. Our apparatus has been specifically designed for this approachand the details of the apparatus are described in this chapter.
In the vast majority of quantum gas experiments two very different techniques
are required to achieve ultralow temperatures: laser cooling and magnetic trappingwith evaporative cooling. We use a standard three-dimensional, magneto-optical trap
(MOT) to collect and cool room temperature Rb atoms from a background vapor.
Unfortunately the lowest attainable temperatures in the MOT are not sufficient tocreate a BEC. Therefore the atoms are further cooled to ultralow temperature using
forced evaporation in a magnetic trap. The primary obstacle in operating a MOT andevaporative cooling in the same vacuum system is the disparate vacuum conditions
required. To efficiently collect atoms into the MOT, a relatively large vapor pressureof Rb is required, but evaporative cooling requires very low vacuum pressure. A now
standard solution is to use physically separated MOT and evaporative-cooling chambers.
The two chambers are connected by a long, thin tube through which laser-cooled atoms
from the MOT chamber can be transported to the science chamber. The thin tube
isolates Rb vapor in the MOT chamber from the evaporation chamber, however in theoriginal 85 Rb apparatus it was determined that over several years a detrimental Rbvapor can build up in the evaporation chamber. Our apparatus is designed to overcome
this problem by removing line-of-sight between the two chambers with a curved transfer

CHAPTER 2. THE APPARATUS

11

tube. Rb vapor that diffuses into the transfer tube sticks to the walls before reaching
the evaporation chamber. A significant vapor pressure would have to develop in thetransfer tube for much Rb to leak into the evaporation chamber. We prevent alkalibuildup in the transfer tube by heating the tube to drive the vapor back towards the
MOT chamber.
Essentially two techniques exist (with many variants) to transport cold atoms
between the two chambers. The double MOT system where atoms are pushed by aresonant laser beam from a collection MOT through the transfer tube, then the atoms
are caught in a second MOT [60]. The other technique is moveable magnetic traptransport, in which the atom trapping potential is translated to move the atoms [61, 62].
Our requirement of removing line-of-sight between the two chambers complicates theimplementation of either technique. A significant portion of the early work in this thesis
focused on testing new transport implementations that incorporate a curved transfer
tube.
Once cold atoms can be delivered from the collection chamber to the science chamber evaporative cooling can begin. In a typical ultracold gas experiment the trapping
potential is tightly compressed for evaporative cooling in order to increase gas densityand enhance the collision rate. With 85 Rb this route is blocked since the elastic collisioncross section is significantly lower at high temperature, and inelastic density-dependent
loss rates are significantly larger [6]. Direct evaporation of 85 Rb is possible in a very
weak trap, but the efficiency of the cooling is poor and long evaporation times are required. To improve efficiency of 85 Rb production, we implemented sympathetic coolingwith 87 Rb. This required the added complexity of a dual-species MOT (described in
Section 2.3) and simultaneous transfer of the two species, but it allowed significantly
faster, more efficient, and more reliable production of 85 Rb BEC.
Evaporative cooling of the dual-species gas begins in a Ioffe-Pritchard-style magnetic trap. The design and construction of this magnetic trap is focused on satisfying
two competing criteria. Evaporation requires an extremely stable and low noise trapping potential, while the science goals of our apparatus require the ability to quickly
and precisely change the magnetic field value at the center of the trap. In Section 2.6
the magnetic trap used to obtain all the results of this thesis is described.
Due to several inherent limitations, sympathetic cooling in the magnetic trap is
not able to produce a 85 Rb BEC. Finally, the solution is to use an optical trap andperform simultaneous cooling of 85 Rb and 87 Rb. The optical trapping setups that wereused for the work in this thesis are described in Section 2.7. Our setups provide a very
stable potential with low heating rate which is ideal for evaporation. With an opticaltrap, an arbitrary magnetic field can be applied to the atoms without significantly
altering the confining potential. During evaporation the magnetic field is tuned near
the 85 Rb Feshbach resonance in order to control both the elastic and inelastic collision
rates in the 85 Rb gas.
Most of the data taken with our apparatus is obtained from absorption images.
Section 2.8 gives a brief overview of our imaging setup. The procedure that we use forabsorption imaging is now quite standard, therefore only a brief review of our work isprovided and references are provided for further information.

CHAPTER 2. THE APPARATUS

Before12

After

Figure 2.1: A photograph of the apparatus during vacuum assembly and a current
photograph.

CHAPTER 2. THE APPARATUS

2.2

13

Vacuum system

Our vacuum system is constructed from ultrahigh vacuum (UHV) compatible
components such as stainless steel and pyrex glass; joints are sealed with welds or with
ConFlat-type copper compression fittings. A schematic of the vacuum system is shownin Figure 2.2. The system consists of two chambers separated by a long, thin tube. The
collection MOT is a ten-way stainless-steel cross. Five ports have 2.75 inch Conflat (CF)fittings to which viewports with 1.5 inch clear aperture were attached; the MOT beams
travel through these ports. Another 2.75 inch CF port is connected to a 2.75 inch tee inorder to mount an ion pump and provide optical access for the remaining MOT beam.
The remaining four ports are 1.33 inch mini-CF flanges; two have mini-CF viewports
attached and allow for monitoring of the MOT fluorescence, one has a mini-CF viewport
and a connection to the ampoule valve, the last port connects to the transfer tube. The
collection chamber is pumped by a 20 L/s ion pump made by Physical Electronics (nowGamma Vacuum), the pumping element is a combination of tantalum and titanium tohave a good pumping speed for noble gases and other gas loads. The collection MOT
ion pump has been disabled for the last several years since the vacuum pressure in thecollection chamber is dominated by Rb. In fact, the collection MOT ion pump is likely
saturated with Rb vapor. It would have to be cleaned out before it could be used topump on the collection chamber again.
The source of Rb for the collection MOT is an ampoule. A solid mass of Rb
is enclosed in a glass ampoule tube sealed under nitrogen. The ampoule was placed
inside a stainless-steel tube that was attached to the rest of the vacuum system. The
ampoule was carefully broken under vacuum by pinching the steel tube prior to thefirst vacuum bakeout. An inexpensive valve was used to separate the Rb ampoule fromrest of the vacuum system. We control the Rb vapor pressure in the collection MOT
chamber by heating the steel ampoule tube. We choose to use a Rb ampoule instead ofalkali dispensers (“getters“) since the ampoule is guaranteed to produce Rb vapor anda significant amount of vapor is initially required to coat the steel walls of the collection
MOT chamber.1 Getters work very well in an all-glass chamber but in our experience
their performance is marginal in a big stainless chamber like ours.
The science chamber is located at the other end of the transfer tube. The sciencecell is a commercially available glass fluorometer cell sold by Starna Cells, Inc. It is
made of high optical quality pyrex glass and it is rectangular with dimensions: 10 mm
tall, 20 mm wide, and 45 mm long. The cell is attached to a 1 inch diameter pyrex tube
and then a 2.75 inch CF flange with a glass-to-metal seal. The science cell is quite small,basically only large enough so that a hot gas (800 µK) tightly confined in a magnetic
trap fits inside. The small glass cell provides several advantages: magnetic trapping
coils can be located close to the atoms, the optical quality is good across the entire cell,and induced currents from changing magnetic and rf fields are reduced.
On the science cell side of the transfer tube there are two vacuum pumps and an
ultrahigh vacuum valve. The valve (made by Granville-Phillips, Model 203) is used toconnect a turbopump for the initial pump down of the vacuum system from atmosphericpressure. There is another 20 L/s ion pump on the science cell side of the vacuum
system which is identical to the collection MOT pump. Since there is little Rb vapor
on science cell side of the system, this pump runs continuously to remove gas from the1

Alkali dispensers are sold by SAES Getters Inc.

CHAPTER 2. THE APPARATUS

14

Science Cell

UHV bakeout
valve

20 L/stitanium sublimationpump

Collection MOT

0
2
s
L/

to ampoule valve
and Rb source

Figure 2.2: Schematic of the vacuum system used in this thesis. Atoms are laser cooled
and trapped in the collection MOT chamber and evaporative cooling is performed in
the science cell. The chambers are connected by a long thin transfer tube.

CHAPTER 2. THE APPARATUS

15

entire vacuum system. There is also a titanium-sublimation pump (TSP) on the sciencecell side of the system which is used to remove residual gas that diffuses out of thestainless-steel vacuum parts.
The transfer tube which connects the collection MOT chamber to the science cellis constructed from a long, thin stainless steel tube. It is composed of two segments
welded at either end. The segments are connected in the middle with a mini-CF connection. The segment at the collection chamber end is about 4 inches long with an
inner diameter of 9 mm and the segment at the science cell is 15 inches long with an
11 mm inner diameter. The conductance C in L/s of a long thin tube is C = 12D3 /L
where the diameter and length are in centimeters [63, 64]. The total conductance of the
tube is 0.3 L/s for air, not including the effects of the 90o bend which should lower the
conductance by another factor of three for air [64].
The bend in the transfer tube has been successful at limiting Rb vapor from
polluting the vacuum of the science cell. Since the first pump down of the vacuum
system we have had consistently excellent vacuum conditions in the science cell. The
only vacuum problems that we experienced were due to buildup of Rb in the collection
chamber side of the transfer tube. The magnetic-trap transfer efficiency is an excellent
indicator of the transfer tube pressure; we use it to diagnose the vacuum pressurein the transfer tube. We installed several heaters on this section of the tube; theheaters are constantly run to keep the tube temperature about 50-100◦ C above ambient.
The heaters are designed to maintain a thermal gradient along the transfer tube. The
collection chamber end is kept the coldest so that any vapor buildup on the tube walls
is driven back towards the MOT area. We determined experimentally that the thermal
gradient along the tube is important to maintain the best conditions.
Our vacuum system was designed with the minimum number of CF seals to makeassembly of the system easier and to reduce the potential for leaks. Wherever possible we
designed custom parts that were welded together instead of utilizing multiple commercial
pieces. The welds in a custom part are typically easy to leak check after construction
and are made reliably by the skilled machinists at JILA. In contrast, a large systemwith many CF seals can be awkward to leak check. Welded seals are also reliable
to much higher temperature than a CF joint so that baking restrictions are relaxed.
Unfortunately welded seals are sometimes magnetic, but in our experience it is more
likely to find screws and nuts that are magnetic since they are made from lower quality
steel.
Once the vacuum system was assembled and fully leak checked, we pumped thesystem down using a turbopump backed by a scroll pump. These pumps were connected
through the UHV valve. We degassed the TSP by running each filament at 30 A for
20 minutes and then 40 A for 3 minutes. The vacuum system including the ion pumps
was baked out for several days to reduce the outgassing rate. The stainless steel parts
were wrapped in fiberglass heater tape, and aluminum foil was used for insulation. We
constructed a fire-brick oven around the science cell, since heater tape could not be
wrapped directly on the glass. The collection MOT chamber and ion pump were heated
to 170 ◦ C, but the ampoule temperature was kept below 90 ◦ C. The transfer tube washeated to 225 ◦ C and the science cell area was heated to 260 ◦ C. The ion pump magnets
were removed for the bake. We monitored the pressure at the turbopump during thebake, and a minimum pressure was reached after four days. At this point we closed
the UHV valve and cooled the ion pumps so that they could be activated. The baking

CHAPTER 2. THE APPARATUS

16

continued into the ion pumps for another two days before we slowly cooled the system
down. The TSP was fired at 47 A for 3 minutes and then 48 A for 1.5 minutes after
the vacuum system was brought to room temperature.
There is some controversy about the effectiveness of titanium sublimation pumps.
Some BEC machines in JILA (see Ref. [6]) have seen little reduction in the pressure
after firing the titanium sublimation pump. Therefore we initially constructed oursystem without a titanium sublimation pump expecting that the ion pumps would be
able to lower the vacuum pressure sufficiently. The most direct measurement of vacuum
pressure in our system is the lifetime of magnetically trapped atoms. The population of
atoms in the trap decays exponentially with a time constant that is related to vacuum
pressure. Without a titanium sublimation pump, we measured a magnetic trap lifetimeof 30 seconds. We performed a careful analysis of residual gas in the vacuum systemusing an SRS residual-gas analyzer; the results are shown in Figure 2.3. We foundthat the dominant contributions were from chemical species that typically desorbe anddiffuse from stainless steel. We found no evidence of vacuum leaks to atmosphere. We
installed a titanium sublimation pump to remove this residual gas, and we observed
magnetic trap lifetimes greater than 500 seconds.

2.3

Two-species MOTThis section describes the two-species MOT used to collect and cool 85 Rb and87 Rb from a room temperature vapor. The precise functioning of a MOT has been
described elsewhere in detail [65, 6, 66] and will not be repeated here. A schematic of
the MOT region is shown in Figure 2.4. Our MOT configuration has three orthogonal,
retroreflected laser beams with σ + -σ − polarization; the laser beams are expanded tocompletely fill the vacuum system viewports.2 A pair of anti-Helmholtz coils called the“MOT-coils” provide the magnetic-field gradient to establish the position of the MOT.
These coils also provide the first quadrupole trap for our magnetic transfer sequence,and they will be discussed in more detail later. To load atoms into the MOT, 30 A ispassed through the MOT coils to create a field gradient of 10 G/cm. The current in
the MOT coils is controlled with an electronic servo, for details see Section 3.3. The
reference voltage for the servo is supplied by an analog voltage from the computer.
Three other utility coils are used to shim the MOT position, they are referred to by
the direction of their axes “up/down”, “east/west”, and “north/south”. The shim coils
are made from 22 AWG magnet wire with about 100-200 turns per coil. The current
through the coils is controlled with an analog voltage from the computer.
For both 85 Rb and 87 Rb trapping light for the MOT is derived from injection
locked diode lasers (see Chapter 3 for more specific details) and delivered to the MOT
region of the optical table with polarization maintaining optical fibers. The fibers provide very stable long-term pointing of the MOT trapping beams; in particular when
the injection locked diode is replaced, only the input to the fiber need be realigned.
In fact, since installing the fibers in 2004, the two-species MOT has not required serious realignment. Simultaneously collecting 85 Rb and 87 Rb into the MOT requires the
combination of different laser beams at nearly the same frequency and with the same
2

A MOT with a large number of atoms has a significant optical depth leaving a low-intensity shadow
in the retroreflected beams. This typically results in a non-uniform MOT cloud shape which is not ideal
but can be transiently corrected by reducing the optical thickness of the MOT.

CHAPTER 2. THE APPARATUS

17

1E-6

H2

P (torr)

1E-7

1E-8

Ar

CO

CH4
C

1E-9

1E-10

1E-11

1E-12
0

10

20

30

40

50

amu

Figure 2.3: A plot of partial pressure as a function of mass showing the residual gases
in our vacuum system without a titanium sublimation pump. The system was sealed
and the ion pumps were left off for one hour for this measurement. The red dashed line
indicates the sensitivity limit of the RGA.

CHAPTER 2. THE APPARATUS

18

Collection MOT

6.0 inN/S shim

U/D shim

E/W shim

MOT coils

retroreflected
MOT beam
l/4
MOT
beams

MOT
optical pumping
lens
MOT
photodiodeto op-amp

Figure 2.4: Schematic diagram of the MOT region including the anti-Helmholtz MOT
coils, the three shim coils, and the lens used to collect MOT fluorescence onto thephotodiode. Not shown: quarter-wave plates just before each MOT beam enters thevacuum system and the up/down direction MOT beams.

CHAPTER 2. THE APPARATUS

19

polarization. This is trivial with a non-polarizing beamsplitter as long as half the powerin each beam can be sacrificed, but for our MOTs this is unacceptable. Approximately
65 mW of 87 Rb trapping light and 12 mW of 85 Rb trapping light is available after the
optical fibers. We devised an optical system that combines the beams with the samepolarization and no loss of power, but with a slight misalignment. The optical setup we
use to combine the 85 Rb and 87 Rb MOT beams is shown in Fig. 2.5. The two beams
are tightly focused with a 3 cm focal length achromatic lens very close to a dielectric
mirror with a finely polished edge. The 87 Rb light passes by the edge of the mirror
while the 85 Rb light is steered to reflect off the polished mirror into the same direction
as the 87 Rb light. Both laser beams, now collinear and nearly completely overlapped,
are expanded and collimated to the size of the MOT chamber viewports.
polished mirror

87

Rb
fiber

lens

to MOT

85

Rb
fiber

lens

Figure 2.5: Optical arrangement used to combine MOT light with the same polarization
for the two species. After the final lens the two beams are collinear with an offset lessthan 1 mm.
Repump light for both species is derived from diode lasers (see Chapter 3) anddelivered to the MOT region. The 87 Rb repump light comes from an injection-locked
diode laser, but it is not fiber coupled. The 85 Rb repump light comes directly from anexternal cavity diode laser; the light is fiber coupled to enhance the pointing stability.
Approximately 15 mW of 87 Rb repump light and 4 mW of 85 Rb repump light is available
for the MOT; this amount of power is more than enough to collect a large number ofatoms into each MOT.
The trapping and repump beams for both species are combined using a polarizing
beamsplitter cube since they do not require the same polarization. After combination
the MOT light is split into three beams using a pair of polarizing beamsplitter cubes;
each beam is aligned through a quarter-wave plate to produce circular polarization,then the beam is aligned into the MOT chamber with a pair of dielectric mirrors. A
significant advantage of the stainless steel vacuum chamber is the ease of alignment of
the all the MOT beams. Simply by verifying that the beams travel straight throughthe chamber, angular alignment of better than one degree is assured. The retroreflectedbeams are initially aligned to overlap the incoming beam as well as possible. With
atoms collected into the MOT, the retroreflected beams are adjusted slightly to give themost compact and symmetric shaped cloud.
The absolute number of atoms trapped in the MOT can easily be determinedwithin a factor of three by measuring the fluorescence emitted by the atoms. Fluores-CHAPTER 2. THE APPARATUS

20

Light
on

87

Rb
85

Rb
time

4

85
87

Rb MOT fill + re-fill
Rb MOT fill

0.8
0.6
0.4

9

N (x 10 )

3

0.2

2

0.0
30

35

40

45

50

1

0
0

10

20

30

40

50

60

time (sec)Figure 2.6: Number of atoms in the two-species MOT as a function of time. At t=0
the trapping and repump light for 85 Rb is applied and that MOT begins to fill. At thesame time the trapping light for 87 Rb is switched on. At t=15 sec the repump light for
87 Rb is switched on so that atoms are loaded into that MOT. At t=37 sec the 87 Rb
repump light is switched off and the atoms in the MOT are rapidly lost. After the 87 Rb
MOT is switched off, the 85 Rb MOT refills to its initial number. Number loss occurs in
the 85 Rb MOT due to light-assisted collisions as described in the main text.

CHAPTER 2. THE APPARATUS

21

cence from the MOT is collected onto a calibrated photodiode. The number of atoms
is determined based on the expected scattering rate of a single atom given by [66]




1
1

Γ= γ 
2
2)
1 + s0 + ( γ √2(δ)
1+s

(2.1)

0

where γ is the natural linewidth of the atomic transition, s0 = I/Isat is the saturation
parameter, and δ is the laser detuning from atomic resonance. The absolute numberof atoms in the MOT as measured by the fluorescence depends on the MOT detuning,
the MOT laser power, and the collection efficiency of fluorescence onto the photodiode.
The absolute number also depends on the details of the atomic level structure, see
Ref. [65, 67] for details. The MOT can be viewed through various vacuum ports but
calibration of the solid angle is difficult since the inside of the vacuum system behaves
as a light funnel. While loading atoms from the vapor in the collection chamber, the87 Rb (85 Rb ) MOT detuning is set to -21 MHz (-13 MHz) and both repump lasers are on
resonance. We are typically able to collect approximately 3×109 87 Rb atoms and a little
more than 108 85 Rb atoms based on the fluorescence signal. The fluorescence signal from
the MOT is used to monitor the atom number during operation of the apparatus. For
reproducible behavior of evaporative cooling in the science cell, we would like the MOT
number to be stable to much better than 1%. However fluctuations in the parametersthat determine the MOT number via fluorescence tend to limit the stability of MOT
fluorescence to about 1%, meaning that we cannot measure the relative number better
than this. To significantly improve our ability to measure the MOT number, we would
need to stabilize the laser frequencies to better than 0.1 MHz and the MOT laser power
to better than 1%.
Figure 2.6 shows the measured number in the two species MOT as a functionof time during loading. For this experiment the 85 Rb MOT was loaded before the
87 Rb MOT was initiated. The presence of 87 Rb atoms tends to cause number loss inthe 85 Rb MOT due to light assisted collisions between excited-state and ground-state
atoms [68, 69, 70]. A pair of 85 Rb –87 Rb atoms can absorb a photon with energyh ν1 from the MOT light during a collision and interact via an attractive excited state
potential. The pair emits a photon with lower energy h ν2 during the collision, due tothe attractive potential. The energy difference between the two photons goes into thekinetic energy of the pair. If this kinetic energy is larger than the depth of the MOT,
then the atoms are lost. This type of loss is evident in Figure 2.6, and we see a re-fillingof the 85 Rb MOT after the trap for 87 Rb is switched off. We typically load a muchlarger number of 87 Rb atoms into the MOT and therefore light assisted losses are more
severe in the 85 Rb MOT.

2.4

Quadrupole trap loading procedure

The two-species MOT collects atoms and cools them to approximately 100 µK,
but this is not cold enough to make a BEC. Atoms from the MOT must be transferredto the science cell for evaporative cooling. In this section we describe the first stepin the transfer process, atoms from the MOT are loaded into a quadrupole magnetic
trap. Our goal is to load as many of the MOT atoms into the quadrupole trap (QT) as
possible without raising the temperature significantly. If we simultaneously turned off

CHAPTER 2. THE APPARATUS

22

the MOT and switched on the QT, only a small fraction of the atoms would be loaded
since the experimental conditions for collecting atoms into the MOT are different fromthe optimum conditions for loading the QT. Instead we use a multistep procedure toload atoms from the MOT into the QT. First, the shape and position of the MOT cloudare optimized for QT loading by initiating a compressed MOT (CMOT) stage [71]. The
CMOT increases the density and tends to shape the cloud into a smooth ball. Next we
use polarization-gradient (PG) cooling with the MOT light to remove as much kinetic
energy from the atoms as possible. Following the completion of PG cooling, the MOT
light is quickly switched off, and the atoms begin to expand and fall under gravity. We
then quickly prepare all the atoms from the MOT into the correct spin state for QT
loading. Finally the QT is rapidly switched on to catch the atoms.
Both species must be loaded into the QT from the MOT with reasonable efficiency.
Although we can collect many more 85 Rb atoms into the MOT than are needed, we stillmust load 85 Rb efficiently or else stability is poor and a different number is loaded
each run. Some experimental parameters can be set independently for the two species,
but many parameters like the magnetic fields, laser polarizations, and duration of theloading procedure must be the same. The apparatus is first optimized to load 87 Rbfrom the MOT into the QT as well as possible. The remaining independent parameters
for the 85 Rb MOT are then optimized.
The quadrupole trap potential is created with the same anti-Helmholtz coil pairused to produce the MOT field gradient. Each MOT coil is wound with 14 turns ofsquare-shaped hollow copper tubing. The copper tubing is about 0.16 in. per side and it
is insulated with a layer of kapton adhered to the copper with silicon. The copper tubing
is hollow so that water can flow through the center of the conductor to remove the heatgenerated when the coil is energized with up to 300 A of current. In cross section there
are three turns along the axis of the coil and five turns along the radius. The final turnof the coil is missing to make a total of 14 turns. The radius of the average MOT coilturn is 1.88 in, and the distance to the atoms is 2.0 in. The MOT coils provide a uniform
magnetic field gradient of 0.37 cmGA . The current running through the coil is controlledusing our standard servo system discussed in Section 3.3. The reference signal for the
MOT coil servo system is provided by an analog voltage from the control computer.
We use two diagnostic tools, MOT recapture and absorption imaging (see Section2.8), to study the QT loading efficiency and the temperature of the gas in the QT.
We perform MOT recapture during any stage of the QT loading procedure simply byswitching all the MOT parameters to the values used during normal MOT operation.
We record the MOT fluorescence immediately after the parameters are reset to exclude
atoms loaded from the vapor. Of the two tools, MOT recapture is the most usefulsince it provides a direct measure of the fraction of the MOT loaded into the QT and
it only requires a simple photodiode to record MOT fluorescence. However with MOT
recapture it is difficult to get an accurate measure of temperature, when this informationis required we resort to absorption imaging. There is an optical path available in thecollection MOT for absorption imaging but the CCD camera must be moved from its
normal location near the science cell.

CHAPTER 2. THE APPARATUS
2.4.1

23

Compressed MOT and polarization-gradient cooling10-3

average density (10 cm )The density of atoms in a MOT is limited by number losses due to several processes like light-assisted collisions (see Refs. [72, 66]) and by effective repulsive atomatom interactions through re-scattered light. The emitted photon from a trapped atom
may be re-absorbed by another atom in the MOT before the photon leaves the trapping volume. Averaging this process over many absorption-emission events causes aneffective repulsive interaction between atoms in the MOT that lowers the density.

3.0

2.5

2.0

1.5

1.0

20

40

60

80

100

120

- CMOT detuning (MHz)Figure 2.7: Measured density in the 87 Rb compressed MOT as a function of the CMOT
laser detuning. We set the detuning to -35 MHz which gives the best performance. Herethe detuning is measured with respect to the 87 Rb cycling transition.
One way to overcome this density limitation of the MOT is to quickly lower therate of photon scattering by increasing the detuning of the trapping light and lowering
the repump power. In this case, both the rate of light-assisted collisions and the rateof MOT light re-absorption are reduced and the density of the MOT is transiently
increased. To realize a CMOT for 87 Rb (85 Rb) in our system, we change the trapping
laser frequency detuning from -21 MHz (-13 MHz) to -35 MHz (-20 MHz) and we
decrease the intensity of the repump laser from 4.8 mW/cm2 (1.3 mW/cm2 ) to 0.3
mW/cm2 (0.15 mW/cm2 ). We also change the MOT shim coil currents to new values
in order to position the CMOT at the best location for QT loading. These conditions
are maintained for approximately 40 ms until the maximum CMOT density is reached.
Figure 2.7 shows the 87 Rb CMOT density as a function of the laser detuning. The best
result is obtained near a -35 MHz detuning.
Following the CMOT we use polarization-gradient (PG) cooling to quickly lower
the temperature of the gas. To initiate PG cooling the MOT magnetic-field gradientis switched off, the MOT shim coil currents are changed, and the laser detuning for
87 Rb (85 Rb) is tuned 103 MHz (60 MHz) red of the cycling transition. The shim coilsettings during PG cooling are optimized by watching the real-time expansion of the
MOT on a TV monitor after the PG cooling is initiated. The shim coils are tuned until
the expansion is very slow and uniform as the cloud falls under gravity; we particularly

CHAPTER 2. THE APPARATUS

24

avoid any rapid center-of-mass jumps in the cloud position. The temperature after
40 ms in the CMOT was measured to be 150 µK, and after 3 ms of PG cooling thetemperature was lowered to 50 µK. During the PG cooling the density drops by about
15%. Using MOT recapture we determined that only 5% of the atoms initially loaded
into the MOT are lost during the CMOT and PG cooling stages.
Following the PG-cooling stage, the MOT light for each species is switched off
quickly with a mechanical shutter made by Vincent Associates. The shutter is placedat the focus of the laser so that the switch off time is less than 0.1 ms. At this pointin the loading procedure we have prepared a cold, dense two-species gas that is nearly
ready to be loaded into the quadrupole trap. The only remaining step is to opticallypump the atoms into the desired spin states before the QT is switched on.
2.4.2

Optical pumping and spin-state filtering

The CMOT and PG-cooling stages leave the atoms in a somewhat random mixture of all possible spin states. For magnetic trapping, all the atoms need to be spinpolarized into a low-field-seeking state, and for transfer to the science cell we needboth species prepared into the stretched states with F = 2, mF = 2 for 87 Rb andF = 3, mF = 3 for 85 Rb where the magnetic moment is maximum. Figure 2.8 shows alevel diagram for 87 Rb and the two laser frequencies required for optical pumping. Any
87 Rb atoms initially in the F = 1 lower hyperfine state are quickly repumped to the
F = 2 state. Once there the atoms are optically pumped to the 87 Rb F = 2, mF = 2
state which is a dark state due to polarization. The optical pumping scheme for 85 Rb
is qualitatively identical, a diagram for 85 Rb is shown in Fig 2.9.
F’=3
F’=2

mF = -3

-2

-1

0

1

2

mF = -2

-1

0

1

2

F’=1

3

mF = -1

opticalpumping
repump

F=2
F=1

mF = -2-1

0

1

mF = -1

0

1

2

Figure 2.8: Level diagram for 87 Rb which shows the optical pumping scheme used tospin polarize the gas into the stretched state. The repump light (green dashed line)
depopulates the lower hyperfine state quickly while the optical pumping light (red solidline) leaves atoms in the dark F = 2, mF = 2 state.

A total of four optical pumping and repump laser beams for the two species enterthe vacuum system along the north/south axis of the collection MOT chamber, as shownin Figure 2.4. The optical pumping light for 87 Rb (85 Rb) is tuned within a few MHz toCHAPTER 2. THE APPARATUS

F’=4
F’=3

mF = -4

25

-3

-2

-1

0

1

2

3

mF = -3

-2

-1

0

1

2

3

mF = -2

-1

0

1

2

F’=2
opticalpumping

F=3
F=2

mF = -3

4

repump

-2

-1

0

1

2

mF = -2

-1

0

1

2

3

Figure 2.9: Level diagram for 85 Rb which shows the optical pumping scheme used tospin-polarize the gas into the stretched state. The repump light (green dashed line)
depopulates the lower hyperfine state quickly while the optical pumping light (red solidline) leaves atoms in the dark F = 3, mF = 3 state.

CHAPTER 2. THE APPARATUS

26

the F = 2 → F 0 = 2 (F = 3 → F 0 = 3) transition and the repump light for 87 Rb (85 Rb)
is tuned on resonance with the F = 1 → F 0 = 2 (F = 2 → F 0 = 2) transition. Each laser
beam is shuttered by an AOM so that short pulses can be created. Before the four laser
beams enter the collection chamber, the light reflects off a polarizing beamsplitter cube
which creates a clean linear polarization and then the light passes through a quarterwave plate to create circular polarization. Both the repump and optical pumping beams
are polarized to drive σ + transitions. The light is aligned parallel to the axis of the N/S
MOT shim coil which provides a 2 G magnetic field that defines a quantization axis.
There are a number of parameters to optimize for the optical pumping including
the laser frequencies, the polarization of the light, the intensity of the light, and theduration of the optical pumping stage. Since the density of the gas drops during theoptical pumping stage, we set the total duration of optical pumping to be a little lessthan 1 ms. We experimentally optimized the remaining parameters by measuring theloading efficiency of the QT with MOT recapture. The polarization is clearly optimized
when set to σ + , and the best efficiency was obtained with the lasers tuned onto resonance. We were careful to keep the laser intensities as low as possible to avoid heating
the sample with unnecessary photon scattering.
A reasonable question is: why does optical pumping work at all for a quadrupole
trap? In the QT the magnetic-field vectors near the center of the potential do not
all point in the same direction. Along the axis of the anti-Helmholtz coils the field
vectors point inward, and along the radial direction of the coils the vectors point radiallyoutward. Therefore if we perfectly spin-polarize the sample and load at the center of
the trap, we should expect only about a third of the atoms to be loaded. The othertwo-thirds of the sample should be ejected from the trap since the spins are anti-alignedwith the magnetic-field vector. We are definitely able to load much more than a thirdof the available atoms into the trap. There are likely two reasons for this unexpectedly
high efficiency. The QT switch on is not instantaneous and there is enough time forthe spins to rotate with the field vector. Also the experimentally determined optimum
CMOT location likely includes a slight offset from the true trap center which tends toincrease the efficiency of loading.
Our current best optical pumping efficiency is about 70% for 87 Rb and slightly
less than that for the 85 Rb atoms. We studied the performance of the 87 Rb opticalpumping stage and found that the high optical depth of the sample was the limiting
factor. Re-absorption of a spontaneously emitted photon has the opposite effect of
optical pumping. Keeping all parameters of the apparatus fixed and simply lowering
the number of atoms loaded into the MOT, we observe greater than 90% of the available
atoms are loaded into the QT. We are further able to directly check that the degree ofspin polarization is higher for a sample with low optical depth. We directly measured
the fraction of the atoms in the QT that were properly spin polarized into the 87 Rb
F = 2, mF = 2 state; this data is shown in Figure 2.10. If the MOT is fully loaded then
only 70% of the atoms are successfully optically pumped. But that percentage increases
toward 100% if a smaller number is loaded into the MOT.
The fact that the optical pumping stage in our system does not work perfectlyis unfortunate. Some of the atoms are left in magnetically trappable spin states otherthan the stretched state. The fraction of the sample in the wrong state is usually about
ten percent. This would hardly be a problem except for the presence of 85 Rb. Since
the 85 Rb number in the MOT is so small, the population of 87 Rb in bad spin states

27

100

90

87

percentage of Rb atoms
pumped into |2, 2>

CHAPTER 2. THE APPARATUS

80

70
0

20

40

60

80

100

percentage of MOT filled
Figure 2.10: Fraction of the 87 Rb MOT that is optically pumped into the stretched state
as a function of the MOT fill. When there are a large number of atoms in the MOT
the optical depth of the gas is large and the efficiency of optical pumping is limited byre-absorption of spontaneously emitted photons. When we decrease the optical depthof the MOT by lowering the number of atoms, the optical pumping efficiency rises dueto the reduction in photon reabsorption.

CHAPTER 2. THE APPARATUS

28

can actually be greater the total 85 Rb number. The rate of spin exchange collisions
and inelastic collisions between 85 Rb and the bad 87 Rb spins is large. This leads to a
rapid depletion of the 85 Rb population once the gases are loaded into the QT. We haveobserved an unacceptable factor of ten loss of 85 Rb number during the transfer to thescience cell.
We devised a method to selectively remove only the unwanted spin states fromthe QT. The magnetic-field gradient of the QT must be large enough to support an
atom against gravity. The required gradient β to balance gravity is given by β = mµg
where µ is the magnetic moment of the atom and m g is the force due to gravity. We
can filter out the 87 Rb atoms in the bad spin states by holding the sample in a QT
with a field gradient only large enough to hold the stretched state. Since the massesof 85 Rb and 87 Rb are nearly the same and the magnetic moments are identical, thestretched state of 85 Rb is also trapped. Figure 2.11 shows the results of an experiment
designed to test the QT as a spin state filter. We loaded the QT with 87 Rb atoms from
the MOT with a large enough QT current so that any magnetically trappable state isheld. The QT current was then slowly decreased to different values, and the numberof atoms remaining in the trap was recorded with MOT recapture. In the figure there
are two distinct levels as the QT current is decreased. The highest level corresponds totrapping all 87 Rb spin states, and the slightly lower level corresponds to trapping only
the stretched state. Below a certain current no atoms are held in the trap.

1.0
0.8|2, 2> only

0.6
0.4
0.2

87

Rb MOT to QTtransfer efficiency

all states

0.0
020 40 60 80 100 120 140 160

MOT coil current (A)Figure 2.11: Fraction of the 87 Rb MOT loaded into the QT as a function of the QT
current. At the smallest currents the trap is too weak to hold any atoms. Based on
the calculated magnetic fields of the QT, a current of 42 A is required to trap atoms
in the stretched state. Twice as much current is required for the other states that havea magnetic moment of µ = 1/2 µB . If we hold the trap current at about 60 A all theatoms in unwanted spin states slowly fall out of the trap.

We use the QT as a spin state filter during normal operation of the apparatus.
Immediately after the optical pumping stage terminates, the quadrupole trap is rapidly
switched on with a current designed to only hold the stretched spin states of 85 Rb andCHAPTER 2. THE APPARATUS

29

87 Rb. This current is maintained for 250 ms to allow all the bad spin states to fall outof the trap. Then the current is increased in 150 ms to provide a field gradient of 100
G/cm. At this point the atoms in the QT are ready for transfer to the science cell. The
QT loading efficiency is approximately 70% (65%) for 87 Rb (85 Rb) with a total 87 Rb(85 Rb) number of approximately 109 (4×106 ). The two gases are in thermal equilibriumwith a temperature in the range 180-200 µK in the fully compressed QT. We estimate
the temperature is about 70 µK when the atoms are loaded into the weak QT.

2.5

Magnetic trap transfer to the science cell

After loading the two-species spin-polarized atomic sample into the quadrupole
trap, the transfer to the science cell begins immediately to minimize number loss due
to collisions with background gas. To transfer the sample between the two vacuum
chambers, we translate the center of the quadrupole trapping potential. Atoms in thetrap will precisely follow the translating potential and the temperature of the gas will
not rise as long as the trap center does not move too quickly. The trapping potential
is translated in two stages: along the transfer tube and around the 90◦ bend. In thefirst stage the currents running through a series of spatially overlapped anti-Helmholtz
coil pairs are varied as a function of time. The currents are optimized so that thetrapped atoms are smoothly passed between each pair of the overlapped coils from thecenter of the MOT chamber to the transfer tube corner. The final pair of overlapped
coils is centered exactly at the corner; this coil pair is mounted on a sliding, motorizedtranslation stage, and the motion of the stage is controlled via computer. The final coilpair and the trapped atoms are physically translated from the corner to the science cell.
2.5.1

Multi-coil transfer

The first stage of our magnetic transfer, which we refer to as the multi-coil transfer(MCT), is described here. This method was first demonstrated by the Munich group,and their implementation is described in Ref. [61]. An undergraduate student in ourlab also helped with the MCT, see Ref. [73]. The partially overlapped anti-Helmholtz
coil pairs are aligned in a row along the transfer tube. The combined magnetic fieldsof the coils create a quadrupole-like trapping potential. Due to the symmetry of oursetup, the center of the trap always lies somewhere along the axis of the transfer tube.
The remaining degrees of freedom are the trap center position along the tube and thethree magnetic field gradients near the trap center. Only two of the field gradients are
truly independent since ∇ · B = 0 must be satisfied. We only energize three of the coilpairs, out of a total of six coils pairs, at a single time; these three currents allow us topick the trap center position, the field gradient against gravity, and the field gradientperpendicular to the axis of the tube.
Figure 2.12 shows a side-view diagram of the collection MOT chamber, the transfer tube, and the six coil pairs involved in the MCT. As shown in the figure our coordinate system is aligned so that the x-direction points along the axis of the transfer
tube and the y-direction points against gravity. The location and other properties of
the coils are listed in Table 2.1. Atoms from the MOT are loaded into the QT provided
by the MOT coils; see Section 2.4 for details. The coil pairs labeled a, b, c, d, and T
are sequentially energized to transfer the trapped atoms to the corner. Coil pairs a-d

CHAPTER 2. THE APPARATUS

30

y

x

z
a

b

d

T coilsc

MOT coils

4"
Figure 2.12: A schematic of the transfer tube region and the magnetic-field coils for themulti-coil transfer. The length from the center of the MOT chamber to the tube corneris 8 in. The T coils are mounted on a motorized translation stage to move the coils inand out of the page.

CHAPTER 2. THE APPARATUS

31

Table 2.1: Physical parameters of the coils used for the multi-coil transfer to the sciencecell. The radius and spacings listed here are for the average turn of the coil.
Coil
MOT
a
b
c
d
T

radius (in)1.875
1.31251.31251.31251.31251.75

spacing (in)
4
3
2
3
2
4.5

turns
14
64
64
64
64
19

x position (in)
0
2.325
3.95
5.575
7.2
8

(shown in red) are composed of 64 turns of solid 18 AWG magnet wire.3 These coils
are air cooled since they are only energized for a few hundred milliseconds during thetransfer. The fact that water cooling is not required represents a significant advantage
of our system. To simplify alignment of the coils with respect to the tube, the coils are
all glued onto an aluminum plate. The plate is referenced to the transfer tube for easyalignment and it is rigidly mounted to the optical table. The construction of the T-coilpair is very similar to the MOT coils.
The currents running through all the coils for the MCT are controlled by a simpler
version of the servo electronics described in Section 3.3. To minimize heating of the gas
during the MCT, the current noise must be kept below about 0.1%, which is easily
obtainable. There is a servo setup for each coil and the reference voltages come fromthe control computer. To minimize the number of analog voltages required for the MCT,
we use three digital lines from the computer boards to multiplex three analog signals.
The digital lines program electronic that route the analog signals to the proper coil.
Coils pairs a-d are powered by an HP power supply with an output current up to 50 A.
The MOT coil pair and the T coil pair are powered by the main Agilent power supply
which can output up to 440 A.
We numerically optimized the time-dependent currents that flow through all sixof the MCT coils. The objective is to find current settings that smoothly translate
the combined trapping potential from the center of the MOT chamber to the transfer
tube corner. The three-dimensional magnetic field of each coil pair is modeled using
Eqn. 2.5, which is discussed later in this chapter. The field components of each coilpair are summed together to find the total quadrupole trapping potential. There areseveral methods that can be used to optimize the coil currents for the transfer. We
found a particularly simple and efficient approximate solution to the problem which is
described here. For simplicity we consider only the first three of the MCT coil pairs. Our
optimization procedure is run at every location along the axis of the tube at intervalsof 0.005 in. Since there are three currents, we can constrain three parameters of thetrapping potential. We choose to constrain the sum of the x-direction magnetic-fieldcomponents of the coils, the sum of the z-direction magnetic-field gradients of the coils,
and the sum of the y-direction magnetic-field gradients. The x-direction magnetic-field3

We ordered these custom-designed coils from Custom Coil Inc. The price per coil was only about$20. The construction quality of the coils is excellent. Each coil is securely glued with high-quality
epoxy that can withstand a temperature close to 200◦ C

CHAPTER 2. THE APPARATUS

32

components and the two field gradients need only be calculated at the center of the
tube for a given x location. Essentially we have the following three linear equations in
the three unknown currents
Bm,x (x) Im (x) + Ba,x (x) Ia (x) + Bb,x (x) Ib (x) = 0(2.2)

βm,z (x) Im (x) + βa,z (x) Ia (x) + βb,z (x) Ib (x) = βz

(2.3)

βm,y (x) Im (x) + βa,y (x) Ia (x) + βb,y Ib (x) = βy(2.4)

where Im (x), Ia (x), and Ib (x) are the currents in the coils as a function of x, respectively. The calculated x-direction magnetic field per amp is labeled Bm,x (x), Ba,x (x),and Bb,x (x) for the MOT, a, and b coils. The calculated y-direction and z-direction
magnetic-field gradients per amp of current in the MOT, a, and b coils are labeled by,for example, βm,z (x). The constants βy and βz fix the strength at the center of the composite trapping potential provided by the three coils. These gradients are set to be 58
G/cm and 100 G/cm, respectively, for the transfer. By solving this simple linear systemof equations for the currents Im , Ia , and Ib at every point along the transfer tube, weare guaranteed that the field gradients of the trapping potential will be nearly constantthroughout the MCT. We verify that the approximate solution works by calculating thefull trapping potential with Im , Ia , and Ib .

20
300

200
150
100

b current (A)250

a current (A)MOT current (A)

30

20

10

15

10

5

50
0
0

2

4

6

0
0

8

2

4

6

0
0

8

2

x (in)x (in)

4

6

8

6

8

x (in)

35
250

20
15
10

T current (A)1525

d current (A)c current (A)

30

10

5

150
100
50

5
0

200

0

2

4

x (in)

6

8

0
0

2

4

x (in)

6

8

0
0

2

4

x (in)

Figure 2.13: Optimized currents for the mulit-coil transfer as a function of location along
the axis of the transfer tube. At peak current each coil pair provides a field gradient ofapproximately 100 G/cm in the direction against gravity.

The optimized currents for the MCT transfer are shown in Figure 2.13 as afunction of location along the tube. The atoms start at the center of the MOT chamberwhere only the MOT coil is energized. To move the potential toward the corner, theCHAPTER 2. THE APPARATUS

33

MOT-coil current is slowly lowered while the a-coil current is ramped up.4 This generaltrend continues until only the T coil is energized and the atoms are trapped at the corner.
The transition at which one coil is switched off and a new coil is switched on occurs “onthe-fly” as the atoms are being transferred, i.e the motion of the atoms is not stopped
when a switch-over between coils is necessary. For example, at x = 2.75 in. the MOT
coil current is zero and current begins to flow in the c coil. The physical placement anddegree of overlap between the coil pairs is critical in making these transitions smooth.

0.3

(a)
velocity (m/s)

position (m)

0.2

0.1

0.2

0.1

(b)
0.0
0.00

0.25

0.50

time (sec)

0.75

0.0
0.00

0.25

0.50

0.75

time (sec)Figure 2.14: (a) The calculated position of the atoms as a function of time. (b) The
velocity versus time. The velocity starts from zero so that the beginning of the transfer
is smooth. Likewise, near the corner the velocity is slowly reduced to zero. Number loss
and heating are observed if the atoms are violently accelerated to the 0.3 m/s top speedof the transfer. If the start of the transfer is too slow, atoms are lost due to collisions
with the Rb vapor in the collection chamber.

Using our optimization procedure we found the required currents as a functionof x along the transfer tube. Based on this information, we can generate any desired
time-dependent trajectory x(t) for the atoms to follow. We picked the trajectory thatis shown in Figure 2.14. The acceleration out of the MOT chamber was experimentally
optimized, as well as the maximum speed of the transfer. Since there are no mechanicalpieces in the MCT, the maximum speed is limited to the rate at which the currents
can be ramped. Our electronic control system for the MCT coil currents is capable ofsmoothly moving the trapping potential from the center of the MOT chamber to thetransfer tube corner in approximately 10 ms.
The multi-coil transfer works well. We are able to transport at least 75% of theatoms loaded into the QT to the corner. We test the transfer efficiency by moving atoms
to the corner and bringing them back to the MOT for MOT recapture. The limitation in
the efficiency is due to the short vacuum lifetime in the collection MOT chamber. One
useful diagnostic of the MCT is to vary the distance that the atoms are transferred into
the tube, and then bring them back to the MOT. It allows us to pinpoint the specific4

For the first bit of the MCT we actually only use two coils due to the details of our system.

CHAPTER 2. THE APPARATUS

34

location of any number losses. In particular, this diagnostic helped us to characterize
and understand the optimized current ramps. The MCT setup is quite reliable androbust. Since ironing out several problems during the initial testing of the setup, we
have observed good performance and the setup has required little maintenance. As longas the computer control sends out the proper signals to the MCT electronics, we can
expect that the transfer will work.
2.5.2

Moving coil transfer and science cell optical pumping

Once the atoms reach the transfer tube corner, the second stage of magnetictransfer can begin. Atoms are transferred to the science cell using the quadrupole trap
provided by the T coils. The coils are physically moved with a motorized translation
stage. As long as the stage does not move too fast the atoms remain trapped and they
are delivered to the science cell. This method of magnetic transfer, via a physicallymoving coil pair, was developed by the Cornell group at JILA; For further information
see Refs. [62, 74].
We use a motorized translation stage made by Parker-Daedal Automation. The
moving platform on the translation stage is driven by a precision ballscrew and it isguided by a pair of steel rails. The ballscrew is rotated by a small motor and aninternal rotary encoder precisely determines the location of the moving platform. Thereis an external controller that powers the motor and processes the signal from the rotary
encoder. The controller also accepts external TTL signals so that the motion of theplatform can be precisely timed. The hardware controller communicates with one of
the experiment control computers to program the motion of the platform.
We mounted an aluminum structure on top of the moving platform that holds
the T coils. The structure was designed to take up as little space around the sciencecell as possible, making it easier to align light into the science cell for optical trapping
and Bragg spectroscopy. To accomplish this goal, the T coils are suspended well off theaxis of the motorized translation stage. The structure is rigid enough to create a stable
magnetic trap for the atoms.
The transfer efficiency from the transfer tube corner to the science cell is very
good. We estimate that the transfer efficiency must be greater than 75% based on themeasured atom numbers in the MOT and the science cell. There also appears to be
little or no heating that occurs during this stage of the transfer. It takes about threeseconds to transfer the atoms a distance of 19 in from the corner to the science cell.
The translation stage delivers the atoms to the exact center of the Ioffe-Pritchard trap.
The atoms are released from the T coils by switching off the current in less than 0.1 ms.
Before the IP trap currents can be switched on, the atoms must be spin polarized
into the desired spin state for evaporation. During the transfer we want the 85 Rb and87 Rb atoms in the stretched states to take advantage of the larger magnetic moment.
For evaporation in the IP trap we want the 87 Rb (85 Rb) atoms in the F = 1, mF = −1
(F = 2, mF = −2) state. As will be seen in later chapters, these are the states thathave magnetic-field Feshbach resonances. To prepare the atoms into the desired spinstates we perform another iteration of optical pumping similar to the one described in
Section 2.4.2.
Figure 2.15 shows a level diagram for 87 Rb and the two laser frequencies requiredfor optical pumping. During optical pumping in the science cell, any 87 Rb atoms initially

CHAPTER 2. THE APPARATUS

35

in the F = 2 upper hyperfine state are quickly de-pumped to the F = 1 state. On
average only two photons must be absorbed on the de-pumping transition for an atom
to get to the lower hyperfine state. Once in the lower hyperfine state, the atoms are
optically pumped until they reach the 87 Rb F = 1, mF = −1 state which is dark to all
the applied light due to the polarization of the light. The optical pumping scheme for
85 Rb is qualitatively identical.
F’=3
F’=2

mF = -3

-2

-1

0

1

2

mF = -2

-1

0

1

2

F’=1

3

mF = -1

de-pumping

F=2
F=1
87

mF = -2-1

0

1

mF = -1

0

1

2

opticalpumping

Rb

Figure 2.15: Level diagram for 87 Rb that shows the optical pumping scheme used tospin-polarize the gas into the F = 1, mF = −1 state. The atoms all start in the
F = 2, mF = 2 state. The de-pumping light (green dashed line) depopulates the upper
hyperfine state quickly while the optical pumping light (red solid line) leaves atoms inthe dark F = 1, mF = −1 state.

A total of four optical pumping and de-pumping laser beams for the two species
enter the vacuum system along the east/west axis of the science cell, as shown in Figure7.2. The optical pumping light for 87 Rb (85 Rb) is tuned within a few MHz to the
F = 1 → F 0 = 1 (F = 2 → F 0 = 2) transition and the de-pumping light for 87 Rb(85 Rb) is tuned on resonance with the F = 2 → F 0 = 2 (F = 3 → F 0 = 3) transition.
Each laser beam is shuttered by an AOM so that short pulses can be created and theintensity can be varied with a computer-controlled voltage. Before the four laser beams
enter the collection chamber, the polarization is set to be circular by a quarter-wave
plate, and the four beams reflect off a dichroic beamsplitter plate that combines the
optical pumping beams with the optical trapping beam. The light is aligned parallel tothe axis of the E/W science cell shim coil which provides about a 2 G magnetic field toprovide a quantization axis for the atoms.
The absolute efficiency of the science cell optical pumping stage is somewhat
difficult to measure. Since there is no MOT at the science cell we cannot perform MOT
recapture. Moreover, absorption imaging is inaccurate for the spatially large clouds at
this point in the experiment. We estimate that the optical pumping efficiency is better
than 50% based on a comparison of absorption images of the gas in the quadrupole
trap and in the IP trap. To optimize this stage of optical pumping we measure theCHAPTER 2. THE APPARATUS

36

to CCDdichroic BS
opticaltrap

IP trap
optical pumping

20 L/sprobe beam

Figure 2.16: Top-view schematic showing the science cell portion of the vacuum chamber. The positions of the IP trap coils are shown. Along the east/west direction of
the science cell, optical pumping light and the optical trap beam are combined on adichroic beamsplitter. Probe light for both species resonant with the cycling transitionis aligned down the transfer tube, through the atoms in the science cell, and onto the
CCD camera for absorption imaging.

CHAPTER 2. THE APPARATUS

37

number of atoms after some evaporation in the magnetic trap where accurate number
determination is possible. The parameters of optical pumping (e.g. laser power, laser
detuning, etc.) are systematically varied to produce the largest number of atoms after
evaporation.
2.5.3

Concluding thoughts on magnetic transfer

We are in a unique position to evaluate the relative advantages of the two magnetictrap transfer techniques, the multi-coil transfer (MCT) method and the motorized translation stage (track) method. Our decision to use both methods was based on our desire
to have a bend in the transfer tube.5 Both methods can transfer atoms with about thesame efficiency. Both methods take up about the same amount of valuable space in thescience cell area. Based on our experience, the motorized track system is significantly
faster to implement.

2.6

Ioffe-Pritchard trap

The Ioffe-Pritchard (IP) trap is the most critical single piece of equipment in theapparatus. We use the IP trap to create a harmonic potential suitable for evaporative
cooling of the 87 Rb and 85 Rb gases. A pair of Helmholtz coils that are a part of the
IP trap are used to provide the magnetic bias field for all of our Feshbach resonancework. A good deal of time went into the design of our magnetic trap; the design
was guided by several criteria related to producing 85 Rb condensates, and performing
experiments near a Feshbach resonance. We require excellent optical access to thecondensate from multiple different angles for imaging and to perform measurements like
Bragg spectroscopy. The size and placement of the trapping coils must be optimized
to provide the necessary fields without taking up too much space. Likewise the holder
pieces that mount the coils to the optical table must be rigid, but not take up too muchspace. For evaporative cooling we require that the magnetic fields be very stable. In
contrast, for many experiments we need the ability to quickly change the magnetic field.
This section describes our trap design, which satisfies all of these requirements.
The term Ioffe-Pritchard trap refers to a particular arrangement of coils that produce a harmonic trapping potential as compared to the linear potential of a quadrupole
trap. A typical arrangement is shown in Fig. 2.17 where a set of four current carrying
wires (called Ioffe bars) placed along the z-direction, provide a strong radially-oriented
magnetic-field gradient. A pair of coils (pinch coils) with axes placed parallel to thez-direction are optimized to produce a field curvature, but they also create a large magnetic field near the center of the trap. A second pair of Helmholtz coils (bias coils) are
optimized to produce a uniform field that opposes that of the pinch coils and eliminates
the large field at the trap center. With this arrangement tight confinement is achieved
along the radial direction of the trap, while relatively weak confinement exists along theaxis of the trap. The trapping potential is harmonic and is characterized by a radial
and an axial frequency.
5
We could have used the MCT to transport atoms all the way to the science cell, but that would
require a lot of coils near the science cell. Those coils would take up a lot of space around the science
cell, limiting the optical access and add mutual inductance to the other coils which slows the rate at
which the magnetic field can be changed. Coils mounted on the track can be moved away from thescience cell area after the atoms are dropped off, therefore the inductance of the coils is irrelevant.

CHAPTER 2. THE APPARATUS

38

y
z
x

Ioffe bars
pinch coils
bias coils
Figure 2.17: Schematic drawing of a Ioffe-Pritchard trap. Radial confinement is supplied by the magnetic fields of the four parallel wires (purple) and axial confinement
comes from the pinch coils (yellow) along the axial direction. In order to get stronger
confinement along the radial direction, the bias coils (orange) are set to oppose the fieldfrom the pinch coils. Since the field of the bias coils is uniform, the axial curvatureprovided by the pinch coils is not modified.

CHAPTER 2. THE APPARATUS

39

There are a number of different coil arrangements that can create a harmonic
potential. At JILA there are examples of a “TOP” trap [75], a “baseball-style” trap[59], a “cloverleaf” trap [76], and a couple of different IP-style trap designs. We chosea design which closely mimics the diagram shown in Figure 2.17, since the coils are
relatively easy to fabricate and the physical size of the complete trap can be small.
The design of the IP trap is broken into two tasks. First we determine what coilshapes, sizes, and placement are required to give the desired magnetic fields. The main
factor in designing the coils is a desire to keep the number of turns N small, since thecoil self-inductance scales as N 2 . This decision means that the coils have to be close to
the atoms to create the desired field with a reasonable amount of current. After deciding
on the coils, we have to figure out how to hold all the pieces together in a rigid mannerwithout sacrificing too much optical access. The coil holder pieces must be electricallyinsulating so that induced fields due to eddy currents are avoided. Since a drift in thecoil positions will significantly change the magnetic field, the holder pieces must have asmall coefficient of thermal of expansion.
We modeled the magnetic fields of our IP trap using standard electromagnetic
theory. Here we review the three dimensional magnetic-field formulae necessary for themodel. The fields of a coil placed at the Cartesian coordinates (x0 , y 0 , z 0 ) with radius
R are given by [77]Bx (x, y, z) =z − z0x − x0µ0 Ip2 π ρ (R + ρ)2 + (z − z 0 )2 ρ

(2.5)

!

Ã

R2 + ρ2 + (z − z 0 )2
EE (ks )× −EK (ks ) +(R − ρ)2 + (z − z 0 )2
By (x, y, z) =Ãµ0 Iz − z0y − y0p
2 π ρ (R + ρ)2 + (z − z 0 )2 ρ× −EK (ks ) +R2 + ρ2 + (z − z 0 )2
EE (ks )(R − ρ)2 + (z − z 0 )2Bz (x, y, z) =
Ã

2π

(2.6)

!

µ0 I
(R + ρ)2 + (z − z 0 )2

(2.7)

p

!

R2 − ρ2 − (z − z 0 )2
× EK (ks ) +EE (ks )(R − ρ)2 + (z − z 0 )2q

ρ(x, y) =(x − x0 )2 + (y − y 0 )2

(2.8)

4Rρ
(2.9)
(R + ρ)2 + (z − z 0 )2
where EK and EE are commonly tabulated elliptic integrals, I is the current, andµ0 = 4π × 107 N/A2 . To accurately model the field gradient produced by the Ioffe bars
we note that the field at the Cartesian coordinates (x, y, z) due to a finite length L pieceof wire located at (x0 , y 0 ) carrying a current I is given byks (x, y, z) =Bx = −µ0 I (y − y 0 )4π((x − x0 )2 + (y − y 0 )2 )

¶

µ

L+2z
L−2z
+√
)
4((x−x0 )2 +(y−y 0 )2 +(L−2z)2 )4((x−x0 )2 +(y−y 0 )2 )+(L+2z)2 )× √(2.10)

CHAPTER 2. THE APPARATUS

40
By =

µ

µ0 I (x − x0 )4π((x − x0 )2 + (y − y 0 )2 )(2.11)

¶

L−2z
L+2z
+√
)
4((x−x0 )2 +(y−y 0 )2 +(L−2z)2 )4((x−x0 )2 +(y−y 0 )2 )+(L+2z)2 )× √.

To find the total magnetic field of the trapping coils weq
sum together the three components of each coil and calculate the magnitude Btot = Bx2 + By2 + Bz2 . The trapping
frequencies
are related to the second derivative of the total field and they are given byr
ωi =µ d2 Bm dx2i , where the i subscript indicates the different Cartesian directions, µ is themagnetic moment of the atoms, and m is the mass. Since the above equations are valid
anywhere in space, they can be used to estimate the effects of anharmonic terms in thetrapping potential.
Our IP trap is composed of three coil pairs. For each coil we specify the parametersof the “average” turn. In our calculation of the fields, we assume that each turn islocated at the average location. We verified that this approximation is accurate at the
few percent level for our system by comparing to a calculation of the fields that includes
the individual turns. The bias coils are in approximately Helmholtz configuration, in
which the coil radius is equal to the coil pair separation. The bias coils have a radius of
1.5 in., a separation of 1.875 in., and are composed of 8 turns per coil. The measured
resistance of the bias coils is 7.23(1) mΩ. The pinch coils are designed to provide a large
field curvature so their radius is small compared to the pair separation. The pinch coils
have a radius of 0.4375 in., a separation of 1.87 in., and are composed of 6 turns percoil. The measured resistance of the pinch coils is 2.119(4) mΩ. The Ioffe bars are 4
in long and each bar has seven conductors. The Ioffe bars are placed along the radial
direction with coordinates x0 = 0.625 in. and y 0 = 0.656 in; the zero of the coordinate
system is at the center of the bias coils. The measured resistance of the Ioffe bars is
7.61(2) mΩ.
With the above coil dimensions the magnetic fields of our trap can be calculated.
For the pinch and bias coils we characterize the coils by the field generated per amp
of current and the field curvature per amp. The model predicts that the bias coil field
is 1.61 G/A with a curvature of 0.097 G/cm2 /A. The pinch coil supplies 0.515 G/A
and a curvature of 0.695 G/cm2 /A. The Ioffe bars are characterized by the radial field
gradient, which is 1.04 G/cm. During evaporation we pass about 312 A through the
Ioffe bars and pinch coils, and about 90 A flows through the bias coils. The model
indicates that these currents provide a radial (axial) trap frequency of about 210 Hz(13 Hz) for 87 Rb at a bias field of about 1.6 G. Tighter confinement is possible with asmaller bias field at the center of the trap.
Each of the six IP trap coils are wound from square, hollow copper tubing thatis about 0.16 in. per side and is insulated with a layer of kapton. The resistance ofthe tubing is 1.73 mΩ/m. This is the same tubing that is used for the MOT and T
coils. The hollow core of the tubing allows us to water cool the coils.6 7
The IP
6

It is important to interlock the high current power supply against flow of the cooling water. We
use a TTL based flow switch and the interlock feature of the power supply.
7
We have a simple model for the water flow rate required to cool the coils. The flow rate is ρw VPcoil
cw ∆T
where Pcoil is the power dissipated in the coil, ρw is the density of water, cw = 4187J/kg K is the heat
capacity of the water, V is the volume of water in the coil, and ∆T is the temperature increase of thecooling water between the entrance and exit of the coils. We observed excellent agreement between thismodel and the observed temperature increase of the water.

CHAPTER 2. THE APPARATUS

41

coils were wound around custom-machined forms made of aluminum and plastic. The
gaps between different turns of a coil were filled with two-ton epoxy to keep the coilfrom unraveling when it is removed from the form. For simplicity, the upper and lower
pairs of Ioffe bars are actually wound as racetrack-shaped coils. The section of the theseracetrack coils near the center of the trap is straight.
With all the parameters of the coils picked out, we focus on the design of the
IP trap mounting structure. All the coil holder pieces are made from phenolic which
is basically layers of cotton cloth held together with epoxy and compressed to form arigid, machinable material. Phenolic is an excellent electrical insulator and it has areasonably small coefficient of thermal expansion. A machine drawing of the coil holder
pieces is shown in Figure 2.18. All the coils are mounted onto a central phenolic form.
We started with a solid block of phenolic and removed material in the shape of thecoils until they fit snugly. The coils are held in compression against the central form byvarious phenolic clamp pieces. We use titanium screws to attach the clamps; titanium
is used since it is non-magnetic. Any eddy currents generated in these screws should besmall. The central phenolic form is supported from below by the attachment piece asshown in the figure. This piece is bolted to a solid aluminum structure which in turnis attached to the optical table. Figure 2.19 shows a machine drawing of the complete
IP trap setup along with the aluminum structure. The IP trap coils and holder piece
are carefully positioned near the science cell with a sliding dovetail mount. The overall
clearance between the science cell and the IP trap setup is only about 2 mm. The
dovetail structure assures that the trap slides into the correct position around the cell.
We tested the IP trap by directly measuring its magnetic fields, the results areshown in Figure 2.20. A FW Bell calibrated gaussmeter was used to measure the field.
We recorded the field as a function of position along the axial and radial directions of the
trap. We extracted from the data a radial magnetic-field gradient of 99.0(1) G/cm andan axial curvature of 34.0(5) G/cm2 ; these measurements agree with our calculations
to within 5% and 2% respectively. We measured the magnetic field at the center of thebias coils and found it to be 1.618 G/A, in excellent agreement with our calculations.
We measure the trapping frequencies of the IP trap by monitoring the motion ofcold, trapped atoms. To measure the radial trap frequency, we transiently change thecenter position of the trap by lowering the bias coil current for about 5 ms. Then, as afunction of time, we record the center-of-mass position of the gas. We extract the trap
frequency with a fit to the oscillating position. In the axial direction of the trap wetransiently change the position by applying a field from the E/W science cell shim coil.
We have also characterized the stability of the IP trap magnetic fields over the
past several years. On a many-week timescale or shorter we have measured that thetrap fields are stable to a little better than 10 mG. Over many months we sometimes
observe drift in the field by about 50 mG. We have not characterized this drift very
carefully, but it seems to be correlated to seasonal changes in the water temperature.
The excellent long terms stability of the IP trap is due to the mechanical design of the
setup as well as active stabilization of the coil currents. In order to realize these levelsof stability, we put a great deal of effort into the electronics used to control the current
in the IP trap coils. Section 3.3 is devoted to those details.

CHAPTER 2. THE APPARATUS

42

Ioffe bar clamp

bias coil clamp

bias coilpinch coil

Ioffe barphenolic
central holder
phenolic
attachment piece

Figure 2.18: An exploded view of the IP trap coils and the phenolic holder pieces. The
central phenolic holder was shaped so that all the coils fit snugly against it. The coils
are held in compression against the central holder by the phenolic clamp pieces shown.
The bottom attachment piece connects the central holder to the optical table mount.

CHAPTER 2. THE APPARATUS

43

aluminumsliding dovetail mountup/down, yeast/west, zbias coil
Ioffe barglass cell

north/south, x

Figure 2.19: A machining drawing of the IP trap setup and a recent photograph of the
IP trap. The six trap coils are rigidly attached to a main holder piece made of phenolic.
The Ioffe bars and the bias coils are visible from the outside of the holder. The pinch
coils are not visible. The IP trap is mounted on a sliding dovetail mount so that the trap
can be easily positioned near the science cell. Below the dovetail is a large aluminum
block that mounts the trap to the optics table.

CHAPTER 2. THE APPARATUS

4440

By (G)

20
0
-20
-40

0.990(1) G/cm/A
-60
0

2

4

6

8

10

y, against gravity (mm)32

20.680(10) G/cm /A

Bz (G)

30
28
26
24
-10

-5

0

5

10

z, along bias field (mm)Figure 2.20: Magnetic fields of the IP trap as a function of position. The data was
acquired with a calibrated gaussmeter. A stable current of 50 A was used to generate
the fields. These measurements are in excellent agreement with the prediction of ourmodel.

CHAPTER 2. THE APPARATUS
2.6.1

45

Driving radiofrequency transitions in the science cell

There are a few other small coils in the vicinity of the IP trap that have not been
discussed yet. These coils are used to drive radiofrequency atomic transitions in the gases
confined to both the magnetic trap and the optical trap. Driving rf transitions between
internal Zeeman states of 85 Rb and 87 Rb is critical for both evaporative cooling (seeChapter 4) as well as precise calibrations of the magnetic-fields used in our experiments.
This section describes how we deliver rf to the trapped atoms and our procedure forcalibrating the magnetic-field. The key to delivering a large enough radiofrequency field
to the trapped atoms is the impedance match between the broadcasting coil and the 50Ω transmission line that connects the rf coil to a high power amplifier.
To calibrate the magnetic fields that are applied to the trapped atoms we excite
Zeeman transitions. Here we only describe what was done in the optical trap since all thedata presented in later chapters was obtained in an optical trap. Following evaporation
to a temperature less than 1 µK, a short pulse (5 µs - 100 µs) of rf is applied to theinitially spin-polarized gas in order to excite Zeeman transitions. We then release theatoms from the optical trap and let the gas expand. During the expansion we apply amagnetic-field gradient of approximately 30 G/cm with a coil mounted directly below
the IP trap; the axis of the coil is oriented vertically. This field gradient spatiallyseparates the expanding gas into clouds corresponding to the different Zeeman states.
To calibrate the magnetic field, we measure the fraction of the gas that makes an rf
transition as a function of the drive frequency. The frequency that excites the largest
fraction of the gas corresponds to the magnetic field at the center of the trap.
We use rf (Agilent E4400B and E4420B) synthesizers to generate the signals that
we need. The output of the synthesizers is amplified (ZHL-1-2 made by Mini-Circuits)
with 5 W total output, gain of 40 dB, and a nominal 50 Ω impedance. Since we operate
the magnetic-trap bias field in the range 1 G to 500 G, the frequencies that are needed
range from 1 MHz to 350 MHz. The rf coil used in the experiment is two turns of 18
AWG magnetic wire with a diameter slightly larger than 1 inch. The coil is placed along
the axis of the transfer tube which is perpendicular to the bias magnetic field and itsits within approximately 2 cm from the atoms. We optimized the impedance match
between the rf coil and the 50 Ω output impedance of the amplifier over this frequency
range. Coaxial cable is used to bring the rf close to the magnetic trap but for the last
4 inches the leads are untwisted magnet wire. Note that several standard rf cable types
are magnetic (RG-174) and should not be placed close to the magnetic trap. In orderto drive rf transitions between the magnetic sub-levels only the magnetic field of theradiated power is relevant, and so the axis of the rf coil is placed perpendicular to themagnetic-field direction of the trap.
The rf coil in our experiment is matched to the transmission line using a single
1/2 Watt 50 Ω resistor. This provides the most broadband impedance match possible
and is only limited by the parasitic inductance of the resistor and the inductance ofthe coil. The amplitude of the rf signal that is delivered to the atoms is adequate forour requirements. For example, it takes approximately 5 µs to drive a “π-pulse”, whichtransfers 100% of the atoms, in a magnetic field of 165 G.
Since the rf coil is just a loop of wire it basically looks like a short to the amplifier.
The rf coil could also be matched to 50 Ω by using a lumped LC network. Following theuseful analysis of Pozar [78] the rf matching network is easily designed. Construction of

CHAPTER 2. THE APPARATUS

46

the rf coil and the matching network is critical but straightforward. Stray capacitanceand inductance is avoided by keeping lead lengths as short as possible. In the ideal casethe rf from the amplifier is carried through coax right up to the rf coil. The matching
network is soldered This impedance match is resonant at a frequency which is easily
tuned in-situ with either a variable inductor or capacitor. A distinct feature of thistechnique is that the impedance match is performed with reactive components withlittle resistance; therefore the current flowing in the coil is large and a large oscillating
magnetic field is produced.
We have also developed the ability to efficiently impedance match coils into the
GHz frequency range. In this case we construct the coil directly on a small piece ofhigh quality microwave circuit board (free samples are available from Rogers Corporation). The circuit board has spatially uniform dielectric constant that does not varysignificantly with frequency. A track in the shape of a coil can be easily patterned onto
the board either with a circuit board routing machine or the tracks can be cut by handwith a razor blade. The coil shaped track is matched to 50 Ω using a technique calledstub tuning; see Ref. [78] for details. The JILA electronics shop can make and stubtune coils over a very wide range of microwave frequencies. In particular, coils havebeen made that are centered at 3 GHz and 7 GHz; these coils have a useable range ofat least a few hundred MHz. Using a microwave amplifier with an output power of afew Watts, Rabi frequencies of 2 π × 50 kHz have been attained.

2.7

Optical dipole trap

It is in the optical dipole trap that the final stages of the experiment occur.
As discussed previously in this thesis, the main advantage of the optical trap for our
experiments is its ability to hold the two species in the same spatial location independentof the spin states. Another important feature is that an arbitrary magnetic field can beapplied; we use this feature to access various Feshbach resonances. Optical traps havebecome standard technology in BEC experiments so we will not describe the conceptof an optical trap in detail. For a useful review article on the concept see the review in
Ref. [79]. Three useful JILA theses by Kristen Corwin [80], Kurt Miller [81], and Cindy
Regal [32] discuss optical traps in great detail.
The trap we use is a far red-detuned optical dipole trap formed at the focus of asingle laser beam at 1030 nm. The trapping potential is related to the gaussian shapeof the laser beam and is given by [79]Udip (r, z) =2PL2r2
3πc2 Γa×exp(−)
3
πw2 (z)w2 (z)
2 ω0 ∆(2.12)

where ω0 is the atomic transition frequency, Γa is the linewidth of the atomic transition,∆ is the detuning of the optical trap from the atomic resonance, and the waist w(z) of
the laser along the axis of the beam is given bysw(z) = w0 1 +

µ

z
zR

¶2

(2.13)

with w0 equal to the waist at the focus and zR is the Rayleigh range. These formulae
are relevant in the case that the optical trap laser detuning from the atomic resonance isCHAPTER 2. THE APPARATUS

47

large compared to the fine structure splitting, the polarization of the trap laser is linear,and the rotating wave approximation (|∆| ¿ ω0 ) can be made. In our experiments we
use a waist w0 of either 23 or 46 microns. This waist size is important to the evaporation
process as discussed in Chapter 4, since a trap with larger waist will hold a gas at lower
density. One must also consider the role of gravity; the main effect of gravity on an
optical dipole trap potential is to lower the trap depth, but the radial trap frequency
in the direction against gravity is also modified. These effects become important in thevery weakest traps during the final stages of evaporative cooling.
For efficient evaporation in the optical trap, we require both a very low heating
rate as well as a smooth potential that maintains the expected depth at the end of theevaporation. To maintain a low heating rate, both good intensity stability and good
beam pointing stability must be achieved. For our optimized setup described below, we
realize a heating rate of 1 nK/sec, which is even better than we achieve in our magnetic
trap.
The optical trap laser is installed on a second optical table in a room next tothe main experiment. It is nice to have the higher power laser away from the main
apparatus for safety reasons, and since we do not have space on the main table. We
typically run the optical trap laser with an output power of about 5 W. The light is sent
through an AOM that is used for intensity stabilization and then the light is aligned
into a polarization maintaining (PM) optical fiber. This fiber is made by Oz Optics,
and it is the same as the other PM fibers in our laboratory except it has a stainlesssteel protective layer to prevent damage to the fiber and to contain the light if the fiber
breaks. The fiber carries the light over to the main optical table where it is collimatedand then focused into the science cell. The optimized optics to collimate and focus
the light from the fiber are discussed below. After exiting the science cell, the light iscollected and a small fraction is sent to a calibrated photodiode. The signal from thisphotodiode is used to feedback to the AOM to stabilize the intensity of the trap. The
reference signal for the servo system comes from the computer, so that the intensity ofthe optical trap can easily be controlled.
Laser light for the optical trap is derived from a diode-pumped solid state Yb:YAG
Versadisk laser sold by ELS. Currently this laser is adequate for our needs but it has
a long and troubled history in our lab. As specified on paper the Versadisk system isan excellent laser, however in our experience few of the specifications are met in reality.
We have measured that neither the output power stability or the frequency stability of
the laser are as good as they should be. There appears to be significant frequency noisein the 10’s of Hz range that gives the laser a linewidth of about 40 MHz. The sourceof this noise seems to be related to water cooling of the cavity and the gain medium.
It may be possible to remove this noise by stabilization of the frequency to an external
cavity. We should point out that on a millisecond timescale the linewidth of the laser
is 100 kHz. The pointing stability of the Versadisk laser appears to be adequate since
the coupling efficiency into our PM fiber remains high for many weeks at a time.
We have constructed a diffraction-limited optical system to take the optical trap
light from the polarization maintaining fiber and focus it onto trapped atoms in thescience cell. A schematic drawing of the optics used in our experiments to make a 46
micron waist is shown in Figure 2.21.8 The first two optics collimate the light exiting
8

The optics for our 23 micron waist optical trap include the following elements to collimate theCHAPTER 2. THE APPARATUS

48

PM fiber

GLC 003
f = 14 mm

LAO 189f = 200 mm

Figure 2.21: The optical setup used to create a 46 micron waist trap. Both lenses are
made by Melles Griot and have good AR coatings near 1030 nm. Light for the trap
comes over an optical fiber from the Versadisk laser that is mounted on a second optical
table. To focus the trap light on the atoms we translate the position of the fiber output
facet with respect to the first lens assembly. Not shown are the mirror for trap alignment
and the dichroic beamsplitter.

CHAPTER 2. THE APPARATUS

49

the fiber, and the last lens focuses the light. Not shown in the figure are a mirror tosteer the beam and a dichroic beamsplitter plate to combine the optical trap light withlight at 780 nm. These optics are between the collimating lenses and the focusing lens.
The fiber output is mounted on a translation stage to move the location of the focus.
For beam-pointing stability, we specially machined the holders for all the optics andmounted the entire setup on a stable, raised platform. The Thorlabs cage assemblies
are useful for mounting the optics; we machined custom lens holders from blank cage
plates. We tested the optics using a beam profiling CCD camera to make sure the focus
size is what we expect and to make sure that astigmatism is minimized.9 Astigmatism
is induced by off-center and tilted optics, so we built a special mounting jig so thatlenses could be properly mounted in the Thorlabs cage assemblies.
The most important parameter of the optical trap is the focused waist size, whichfis given by w0 = πλ w
, where f is the focal length of the focusing lens and w is thesize of the laser beam at the lens. We change the optical trap waist by using different
collimation lens systems to give a different beam size entering the focusing lens. Melles
Griot sells a variety of high quality, diffraction-limited lenses suitable for collimating
the 1030 nm light that exits the PM fiber. The GLC series lenses are convenient sincedifferent focal lengths come in the same package size so that switching between different
lenses, and different waists, is easy. Matched pairs of achromats and meniscus lenses
like the one shown in Figure 2.21 are also available.
We align the optical trap onto the position of the Ioffe-Pritchard trap by translating the focus in three dimensions. To enhance the stability of the optical trap setup
we have only one knob for each direction. It can be somewhat difficult to align the two
traps from scratch, but once atoms can be held in the optical trap the three knobs are
independently tuned to maximize the initial number in the trap.10
In the most recent experiments with 85 Rb condensates, the 46 micron waist opticaltrap is used. It is important to note that with such a large waist the confinement along
the axial direction of the trap is very weak when the trap power is low. With thecurrent apparatus, the axial confinement is enhanced by the curvature of the magneticfield produced by the bias coils. The harmonic trapping frequencies of the magnetic-fieldpotential and the optical trap add in quadrature so that the magnetic-field curvatureis the dominant effect. Since the magnetic moments of 85 Rb and 87 Rb are different,
the two species have slightly different axial trap frequencies. Near the 85 Rb Feshbachresonance at 155 G the axial frequency of 85 Rb was measured to be 2 π × 2.9 Hz, whichis in good agreement our expectations.
light from the optical fiber: a plate of BK7 glass with 12 mm thickness, a Melles Griot LAM-059/077
meniscus lens, and a Melles Griot LA0-059/077 achromat lens. Otherwise the 23 micron and 46 micron
optical trapping setups are identical.
9
The WinCamD CCD camera made by www.dataray.com was used for testing. It has small 4.7
micron pixels for excellent spatial resolution and good dynamic range. The software for the WinCamD
is useful since it performs online gaussian fits to extract the beam waist.
10
We have only had to align the optical trap from scratch once even though we have changed theoptics multiple times. Before changing the optics, we image the trap laser beam after it exits the sciencecell onto the beam profiling CCD camera and mark the pixel coordinates. We then switch the optics
and position the focus of the new trap onto the CCD camera pixel mark.

CHAPTER 2. THE APPARATUS

2.8

50

Absorption imaging

Nearly all of the data in this thesis comes from absorption images of gases expanded either from the magnetic trap or the optical trap. Absorption imaging is such astandard technique in the ultracold gas community that it will only be discussed briefly
here. There is an excellent discussion of absorption imaging in Jake Robert’s thesis[6] and in the review article by the Ketterle group [82]. For reference we present theformulae for the number and temperature of an expanded Maxwell-Boltzmann gas given
by
Ã
!
2π
∆2
N=
pkOD σx σy 1 + 4 2(2.14)σ0ΓT =(2πfr )2 σr2mkb (1 + (2πfr )2 t2exp )2(2.15)

where σ0 = 3 × 780nm/2π, pkOD is the optical depth at the center of the cloud, σi are
the cloud widths, texp is the expansion time, ∆ is the detuning from atomic resonance,and Γ is the linewidth of the atomic transition, and fr is the radial trapping frequency.
For a classical gas described by Maxwell-Boltzmann statistics, we fit the absorption
image to a gaussian. We neglect any corrections to the atom number due to saturationof the probe transition, because we keep the intensity low.
Since we take up to several hundred absorption images per day we have developed
automated image processing software. The software is written in Visual Basic 6 andcommunicates directly with the camera controller hardware. It automatically sets all
the parameters of the camera and receives the raw data when an image is acquired.
The data are automatically saved and organized. The absorption images can be fit via
a reduced χ2 algorithm to a variety of two-dimensional trial functions like a gaussian,double gaussian, or an inverted parabola. The software is also able to acquire images
using the kinetics mode of operation of the camera.
We use a CCD camera from Roper Scientific (Model 1024B) with 1024×1024
square pixels each about 13 microns per side. The CCD is back-illuminated for thebest quantum efficiency. This camera is completely adequate for our needs but two
things might be improved. The CCD array suffers from a larger than normal amount of
etaloning compared to other models. Etaloning comes from reflections between the front
and back surfaces of the CCD array. The particular CCD model used in our camera
has particularly severe etaloning in the near-IR. The etalon fringes that we observe arequite stable and therefore they are subtracted from the absorption images. Also, thecharge shift rate from row-to-row could be faster so that images could be acquired more
rapidly. The readout noise of the camera appears to meet the specification and is low
enough to not degrade the signal-to-noise of our images.
We acquire absorption images along the north/south direction of the science cellwith an optimized two lens imaging system. The first lens is an achromat (Melles Griot
model LAI-009) optimized specifically for near-IR wavelengths. It has a focal length of
8 cm and is located about one focal length from the atoms. The second lens (LAI-015)
is also optimized for the near-IR and has a focal length of 18 cm. This lens is placed
about a focal length from the camera so that the magnification of the imaging system isabout 2.25. In off-line tests we verified that the imaging system spatial resolution was
diffraction limited by taking pictures of a calibrated test target. The size of a camera

CHAPTER 2. THE APPARATUS

51

pixel is approximately 5.8 microns and that is about the resolution limit of the first
optic. It would be helpful for several of our experiments to increase the magnification
so that the image is spread over a larger number of pixels, but we would likely haveto also lower the resolution limit of the imaging system.11
The imaging system isfocused by translating the first lens and imaging a small sized gas. The magnification
is calibrated by observing the acceleration of atoms released from either the magnetic
trap or the optical trap.
We acquire an absorption image of the gas after releasing the atoms from either
the magnetic trap or the optical trap. The gas is allowed to expand for at least a few
ms, and typically 20 ms, to lower the optical depth and increase the size. Since both
85 Rb and 87 Rb are trapped in the lower hyperfine state, we apply a short and intense
pulse of repump light just before the probe to pump the atoms into the upper state.
The probe beam is circularly polarized to drive either σ + or σ − transitions, so that theatoms are optically pumped into a stretched state. We apply a 2 G bias field with a coilplaced along the imaging axis. This field is switched on immediately after the atoms
are released from the trap. To obtain the best signal to noise in the absorption images
we set the probe beam intensity to about 0.15 times the saturation intensity and theduration of the probe pulse is 50 µs.
Since a single measurement of a condensate with absorption imaging is destructive, experiments are performed with multiple runs of the apparatus. Ideally eachidentical run would yield the same number of atoms and temperature, but fluctuationsin the apparatus cause noise. Even if we could produce exactly the same number ofatoms at the end of each run, some noise is generated during absorption imaging. The
biggest source of the number noise is laser frequency fluctuations. We spent a good dealof time optimizing our laser frequency stabilization electronics to lower number noisefrom the imaging. The shot-to-shot imaging noise is usually about 5% for 87 Rb andslightly higher for 85 Rb, see Chapter 3 for details on the laser system.
Our best diagnostic test on the accuracy of absorption imaging is a comparisonof the calculated BEC transition temperature with experimental observations. In theabsorption images it is straightforward to observe when the transition temperature isreached, a strong OD peak in the center of the cloud appears. We can observe a
condensate fraction of only a few percent and extract the temperature of the gas preciselyfrom the thermal component. Based on a knowledge of the trapping frequencies andthe number of particles, we can calculate the transition temperature. To estimate theaccuracy of the absorption images, we calculate the number of atoms, Ncalc , basedon the measured BEC transition temperature. We compare Ncalc with the number ofatoms, Nabs determined from the total absorption of the gas. When the imaging system
is reasonably tweaked-up, the accuracy is better than 10%.

2.9

Chapter conclusion

This concludes the chapter on our apparatus. Throughout the remainder of thethesis we will refer back to this chapter concerning some of the experimental details of
our measurements. In particular, this chapter is related to the work that is presented in
11

It might be possible to improve the resolution of our imaging system without inducing aberrations,by using a meniscus lens to lower the f-number of the first lens.

CHAPTER 2. THE APPARATUS

52

the chapters that cover experimental components of the apparatus and the chapter that
describes the evaporative cooling that is performed in the magnetic and optical traps.

Chapter 3
Components of the apparatus

Ultracold gas experiments do not require any state-of-the-art technology, butrather combine an impressive array of advanced techniques that must all work together
seamlessly. In this chapter various components of the apparatus are described including
the diode laser system, the computer control system, and the magnetic trap current
stabilization electronics. The role of this chapter is to give a brief technical overview
of these different systems since they are not documented elsewhere. The main focus
here is on technology that we worked on which might be useful for the other quantum
gas experiments at JILA. In particular, we will discuss a new external cavity diodelaser design with low sensitivity to optical table vibrations, a simplified offset frequency
stabilization setup, and our low-noise low-drift electronics for current stabilization.1

3.1

Diode laser system

Currently ten separate laser systems are required to operate the experiment. Nineof those are low power diode lasers; the tenth is a high power solid-state laser for opticaltrapping. The optical dipole trap laser has already been described in Section 2.7. We
require a large number of diode lasers for laser trapping and cooling of both 85 Rb
and 87 Rb, and to probe the quantum gases after evaporative cooling. Keeping all thelasers in the apparatus stabilized to the desired frequencies is challenging. A further
complication is that the frequency of each laser must be tunable over a wide range. We
use a frequency offset stabilization technique (“offset lock”) to control the frequency of
our lasers. The offset lock allows us to precisely tune the laser frequency over several
hundred MHz and to stabilize the frequency to better than 1 MHz. We will also describe
a mechanically stable diode laser setup that we developed. This “stable laser” system
is significantly less sensitive than previous designs to mechanical vibrations present in
our laboratory. The stable laser design is currently in use throughout the ultracold gas
laboratories at JILA, with at least ten setups in use today.
3.1.1

Laser system

A diagram of the entire laser system is shown in Figure 3.1. There are five external
cavity diode lasers (ECDLs), each laser is in the Littrow configuration. We refer to thelasers in our apparatus by their function; for example the ECDL that is closest to the1
Detailed mechanical diagrams and electrical circuit diagrams for much of the work described inthis chapter are available from the JILA shops.

CHAPTER 3. COMPONENTS OF THE APPARATUS
87 Rb F54

= 2 → F 0 transitions is called the 87 Rb trap laser. The fifth ECDL is called

the 85 Rb master, and it is discussed below in more detail. The frequency of the ECDLs
is tuned by changing the length of the cavity. The frequency is stabilized via feedback
to a piezoelectric transducer (PZT) placed on the diffraction grating, and it is further
stabilized by feedback to the diode laser current. The source of the stabilization signal
either comes from saturated absorption (only for the 85 Rb master laser), or from ouroffset locking scheme that is described below. There are three injection locked diodelasers. The frequency of these lasers is stabilized with a low power seeding laser beamfrom one of the ECDLs. The arrows in the figure indicate the source of the stabilization
signal for each laser.
The most important laser in our system is the 85 Rb master. It is the only laser
stabilized directly to a Rb saturated absorption line.2 All the other lasers in the system
are stabilized with respect to the frequency of the 85 Rb master. Since the frequency
stability of this laser is critical, its design is of the “stable laser” variety described in
Section 3.1.3. The laser is locked to a derivative of the saturated absorption spectrum.
To generate the derivative signal, we modulate the current of the master laser diode at
300 kHz, and we detect the modulation via a lock-in amplifier. The gain bandwidthof both the photodetector that acquires the modulation signal and the lock-in detector
were carefully optimized to increase the size of the derivative signal. We estimate that
the RMS frequency noise on the laser is approximately 0.3 MHz. The frequency noise ismeasured by monitoring fluctuations in the atomic absorption when the laser frequency
is tuned onto the side of a saturated absorption line. The frequency noise is limited by
the shot noise in the saturated absorption spectrometer photodiode.3
In our apparatus there is an ECDL for both the trapping and repump transitions
for each species. These lasers give us flexibility in the frequencies of the trapping
and repump light. The frequencies of these ECDLs are stabilized via an offset lock(see Section 3.1.2) with respect to the 85 Rb master laser. Typically we can tune thefrequencies of these lasers over all the hyperfine lines in a particular saturated absorption
manifold. The ECDLs do not provide enough laser power to collect a large number ofatoms into the MOTs. To increase the available MOT laser power we use injection
locked diode lasers. A small amount of light from the 87 Rb trap laser, the 85 Rb trap
laser, and the 87 Rb repump laser is used to stabilize the associated injection locked
lasers. The injection locking light for each ECDL is aligned into the laser to be lockedwith an optical isolator that has a “feedback” port. In the injection locked laser setups
we use either a 120 mW Sharp laser diode or a 80 mW Sanyo diode. The performanceof the Sharp diode is not as good as the Sanyo diode. Therefore the Sharp diode is
only used in the case that the largest available optical power is required. We monitor
the frequencies of the injection locked lasers using a confocal scanning Fabry-Perot (FP)interferometer. We have constructed a low-cost confocal FP cavity using a pair of curved
mirrors and Thorlabs cage components. One of the mirrors is placed on a PZT to scan
the length of the cavity over a free spectral range. The cost of this setup is significantly
less than a commercial FP cavity and the performance is comparable.
Most of the light used for laser cooling and trapping in the MOTs is delivered2

The 85 Rb master is locked to the peak of the F = 3 → F = 20 /F = 40 crossover transition
A significant improvement to the frequency stability of this laser would be attained if an electrooptic modulator were used to increase the amplitude of the frequency modulation sidebands in thelocking setup.
3

CHAPTER 3. COMPONENTS OF THE APPARATUS

87

Rb

85
15 mW

5 mW
(MOT, probe,
optical pump)

87
Rb
trap

injectionlockeddiode87
Rbrepumpinjectionlockeddiode85
RbmasterECDL

3 GHz
offset lock

(hyperfine repump,
optical pumping)

5 mW

Rb

85
Rb
trap

0.4 GHz
offset lock

1.1 GHz
offset lock

5.4 GHz
offset lockpolarization
maintaining
fiber

55

85
Rb
repumppolarization
maintaining
fiber

15 mW

70 mW
to MOT

15 mW
to MOT
injectionlockeddiodepolarization
maintaining
fiber

40 mW

5 mW

Figure 3.1: Schematic diagram of the laser system for both 85 Rb and 87 Rb. The lines
indicate the source of the frequency stabilization signal for each laser. The 85 Rb masteris locked via saturated absorption to a Rb line. The other laser frequencies are stabilizedusing the offset lock technique. There is an injection locked laser to amplify the power
for the two MOTs. There is also an injection locked laser for the 87 Rb repump light inorder to increase the available power.

CHAPTER 3. COMPONENTS OF THE APPARATUS

56

with polarization maintaining optical fibers in order for the MOT beams to have good
pointing stability. Only the 87 Rb repump light is not fiber coupled. We typically achieve
75% fiber coupling efficiency using optical fibers in which the distance from the fiber
facet to the coupling lens can be optimized. We have up to 70 mW available for the
87 Rb MOT. This is plenty of laser power to collect more than 109 atoms.
As we described in Chapter 2, a number of different laser beams are required forabsorption imaging and optical pumping in our experiments. We send a total of four
beams into the north/south (N/S) axis of the collection MOT chamber, the four beams
consist of repump and trap light for both species. We also send the same set of four
laser beams into the east/west (E/W) port of the science cell. For absorption imaging
we send trapping light for the two species along the transfer tube into the N/S directionof the science cell. Figure 3.2 shows the source of all these laser beams for imaging andoptical pumping. In each case the light is passed through an 80 MHz AOM, the functionof which is to generate short pulses and to control the optical power in the beams. The
frequency of the laser beams for probing and optical pumping is set exclusively by theoffset locking setups, we never tune the AOM drive frequency.
There is one more ECDL that has not yet been discussed. This laser is of the“stable” design variety and it is used for Bragg spectroscopy. The laser is locked tothe peak of a saturated absorption feature in the 87 Rb trapping manifold. Therefore
the laser is approximately 4.2 GHz detuned from the 85 Rb atomic resonance. This
detuning is one of the parameters that determines the two-photon Rabi frequency for
Bragg spectroscopy. Further details of this laser are given in the Bragg spectroscopy
chapter.
3.1.2

Widely tunable laser frequency stabilization

Precise absolute frequency control of laser light is critical for many of the experiments described in this thesis. The natural linewidth of Rb is approximately 6 MHz.
Therefore the laser frequency must be controlled to less than 1 MHz in a bandwidth of
100 kHz. The laser frequency must also be continuously tunable in order to reach thedifferent atomic transitions in 85 Rb and 87 Rb, which are separated by a few hundred
MHz. Therefore direct stabilization to a single atomic line is impractical. A large bodyof literature exists detailing numerous methods to stabilize and tune the frequency ofa laser [83]. For an excellent review of different stabilization techniques see Ref. [84].
Several techniques exist which are low-cost and are simple to implement, see for example
Refs. [85, 86]. Recent start-of-the-art results have been described in Refs. [87, 88, 89].
One technique used extensively at JILA is the DAVLL [90]. The DAVLL provides awide tuning range since the signal is derived from the Doppler broadened absorption
features of Rb. There are a few disadvantages to the DAVLL including a large temperature dependence of the locking signal and a lack of absolute frequency calibration.
In this section we describe the offset locking scheme that we use to precisely stabilize
and tune the frequency of our lasers. Our scheme has a frequency tuning range that islarger than that of the DAVLL, a negligible temperature dependence, and an absolutecalibration of the laser frequency is readily available. Offset frequency stabilization ofa slave laser with respect to a master laser is a well-known technique; see Ref. [83]and citations therein. The offset locking scheme described here uses cheap and easily
accessible components and it provides all the flexibility and performance required forCHAPTER 3. COMPONENTS OF THE APPARATUS

to MOT

57

fiber+80 MHZ

85

Rb
F=3
ECDL

injectionlocked
diode

PBS
fiber
BS

+80 MHZ

to science cell
N/S

87

Rb
F=2
ECDL

to MOTBS+80 MHZ
87

Rb
F=1
ECDL

injectionlockeddiodeto science cell
E/W
BS
fiber
+80 MHZ

85

Rb
F=2
ECDL

to collection MOT
chamber N/S

Figure 3.2: Schematic of laser system layout to generate the probing and optical pumping light for each species. AOMs (shown as green squares) are used in each beam pathto generate short pulses and to control the intensity of the beams. There is also a mechanical shutter (not shown) in each beam path to block light that leaks through theAOMs. A large number of mirrors, which are not shown in the schematic, are required
to get all the beam paths aligned. The long term stability of the setup is good sincemost of the light is fiber coupled. The beamsplitter cubes are labeled either BS for a
non-polarizing model or PBS for a polarizing one.

CHAPTER 3. COMPONENTS OF THE APPARATUS

58

our quantum gas experiments.
The scheme is based on determining the frequency of a slave laser by measuring
the heterodyne beat between that laser and a calibrated master laser. The phase of thebeatnote frequency is compared to a stable rf source by an electronic phase detector to
produce an error signal. Using standard JILA electronics the error signal is filtered andamplified and then used as feedback for the slave laser.4 In principle this system iscapable of stabilizing the difference between the slave laser frequency and the master
laser frequency to approximately the level of the rf source. The absolute frequency
stability of the slave laser is then given by the stability of the master. To meet ourrequirements all that is needed is a cheap voltage controlled oscillator (VCO). Tuningof the unknown laser frequency is accomplished simply by changing the rf from the
VCO.
A schematic diagram of our stabilization scheme is shown in Fig. 3.3. There
are four separate realizations of this scheme with offset frequencies of approximately
400 MHz, 1.2 GHz, 3 GHz, and 5.4 GHz respectively. These provide the four different
laser frequencies needed for our experiments. The heterodyne beatnote between themaster and slave lasers is measured with a fast photodiode. We use a digital phasefrequency detector made by Analog Devices (AD9901).5
This chip has two digital
inputs into which a reference rf source and the rf beatnote are applied and two digital
outputs that report the phase difference. There are two automatically selected modesof operation of the chip. When the two rf input signals are nearly in phase the outputof the chip is a voltage which is linear in phase. When the two lasers are exactly inphase the output of the chip is zero. When the two rf inputs are far out of phase,
therefore not at the same frequency, the output of the chip is a constant positive or
negative output of approximately 1 V. The sign of the constant signal depends on therelative frequency difference of the two lasers. The AD9901 can only accept rf input
frequencies of about 100 MHz. Therefore higher beatnote frequencies must be reduced.
Up to a frequency of 3 GHz commercial VCOs are readily available (Mini-circuits) andare used with a double-balanced mixer to reduce the beatnote frequency into the proper
range for the AD9901. VCOs above 3 GHz are more difficult to obtain (although they
have recently become available from Hittite Microwave) so an ultrafast digital prescaler(Hittite Microwave) is used to downconvert the beatnote from 5.5 GHz to 690 MHz.6
The signal from the phase detector is the error signal that the servo electronics uses tocontrol the frequency of the ECDL.
A typical error signal produced by the AD9901 phase detector is shown in Fig.
3.4 (top) as a function of the beatnote frequency between the two lasers. For comparison
the Doppler broadened absorbtion signal is also shown by the black line (bottom) in the
figure. The slave laser is locked to one of the sharp transitions where the output signal
crosses zero. The sharp transition occurs when the phase of the rf reference signal
is the same as the beatnote phase after the down-conversion. The frequency of theslave laser is tuned by changing the down-conversion frequency. Effectively this changes
the frequency at which the sharp transition occurs. The two sharp transitions eachcorrespond to a different sign of the beatnote signal. The width of a sharp transition is4

Most of the components in our system are available from the JILA electronics shop.
This phase detector chip is used in the JILA circuit WC061A2.
6
A circuit board from the electronics shop exists for these Hittite prescaler chips. The board number
is WC065B1.
5

CHAPTER 3. COMPONENTS OF THE APPARATUS

59

lock to Rb
line

masterlaserbeat note
nbeat = nref - ntune
0.5 mWnbeat - RFlens

0.5 mW

mixer

amplifiedphotodiode

slave
laser

RF

servo forPZT andcurrent

large BW

digitalphase-frequency
detectorlarge capture
range

Figure 3.3: Schematic diagram of the offset locking system used in our experiments.
Light from two separate lasers is superimposed onto a fast photodetector. The radiofrequency beatnote signal is amplified and then downconverted to approximately 50 MHz
using a mixer. The phase of the downconverted signal is compared to a stable 50 MHz
source by a digital phase-frequency detector (Analog Devices AD9901) from which anerror signal is derived. A servo uses the error signal to stabilize the frequency of theslave laser by feeding back to the current and PZT of the slave laser.

Error Signal (V)CHAPTER 3. COMPONENTS OF THE APPARATUS

60

2
0
-2
-4
-0.5

0.0

0.5

1.0

1.5

2.0

beat frequency (GHz)
Figure 3.4: The red curve (top) is a typical output voltage signal from the digitalphase-frequency detector as a function of beatnote frequency between the slave laser
and master laser. The frequency of the reference laser was stabilized for this data andthe frequency of the unknown was swept over a wide range. For comparison the black
curve shows the Doppler broadened Rb signal that was simultaneously acquired. With
respect to the lower trace, a larger absorption is up in this figure.

CHAPTER 3. COMPONENTS OF THE APPARATUS

61

only about 1 MHz. This is significantly more narrow than any available Rb absorption
line. The capture range of the offset lock is at least several hundred MHz. In orderfor the slave laser to lose lock, the frequency of the slave laser must deviate from themaster laser by a significant amount comparable to the capture range of the system. A
slave laser stabilized using our offset locking setup can routinely remain locked for an
entire day. A typical level of vibration on the optical table, for example the movementof the motorized translation stage during transport from the vacuum system corner tothe science cell, does not cause the laser to lose lock. The most common reason for aloss of lock is a diode laser mode-hop.
3.1.3

Stable external cavity diode laser design

We developed a new mechanical design for a stable external cavity diode laser(ECDL) using the Littrow configuration.7 For an excellent review of diode lasers andapplications to ultracold gas experiments see the review article in Ref. [84]. Like alllasers, an ECDL is extremely sensitive to mechanical vibrations of the cavity since thefrequency is set by the length of the cavity. The goal of our “stable laser” design is to
reduce the sensitivity of the laser frequency to the mechanical vibrations present in thelaboratory. Figure 3.5 shows a photograph of the design with labels indicating someof the components. Our design builds on previous work at JILA by including a few
new features. The most important feature is the monolithic design of the mechanicalpieces that hold the laser diode and the diffraction grating. This reduces the effectsof laboratory vibrations on the laser frequency. Other measures described below weretaken to further reduce vibrations. Another feature of our design is that it can be
rapidly duplicated by the JILA shop.
All the components of the laser are made from standard 6061 aluminum. The
cavity is formed around a solid metal block (main cavity block) to which the diffraction
grating and the laser diode holder are mounted. The grating sits on a mirror mountthat is incorporated into the main cavity block for stability. Since the main cavity
block is rigid, it is difficult for the cavity length to change quickly. Therefore, the main
components of noise on the laser frequency are at low frequency where our servo is mosteffective. Below the cavity block is a large heat sink that also raises the block off thesurface of the optical table. The cavity block is surrounded by a commercial Al box.
A large fraction of the heat sink was hollowed out so that lead shot could be added
to further reduce vibrations. The entire laser assembly is vibration isolated from theoptical table with rubber o-rings.
A feature of the stable laser design is that all the pieces are machined on a CNC
milling machine by the JILA shop. This helps to guarantee that each new system will
have the same characteristics. This also allows several laser systems to be made all at
the same time. It takes about two weeks of shop time to build and assemble ten entire
laser systems. To date at least ten of these setups are in use at JILA.
We have tested the performance of the stable laser setup. We characterize thefrequency noise of the laser by monitoring fluctuations of atomic absorption when thelaser frequency is tuned near resonance. Figure 3.6 shows the measured spectrum of7

A second stable laser design currently exists that is much more compact. It is a little smaller thana New Focus Vortex laser head. Only two setups with this compact design have been tested, but theperformance is similar to what is described here. See the JILA machine shop for more details.

CHAPTER 3. COMPONENTS OF THE APPARATUS

diode mount62

cavity block

grating mountFigure 3.5: Photograph of the stable laser setup. The red arrow indicates the path ofthe laser light. The diode mount, the grating mirror mount, and the cavity block areindicated in the picture. The main cavity block is made from a solid piece of aluminum.
In this design the cavity block is extremely thick to provide rigidity. The main cavity
block sits within a hermetically sealed box to help keep water from condensing on thetemperature controlled laser parts. The entire assembly rests on a hollowed out blockof aluminum.

frequency noise (MHz)CHAPTER 3. COMPONENTS OF THE APPARATUS

631
Vortex laser

0.1

stable
laser

0.01

0.1

1

10

frequency (kHz)
Figure 3.6: Measured laser frequency noise as a function of frequency. The frequency
noise is determined by monitoring fluctuations in an atomic absorption signal. The red
trace (top) shows the results for a New Focus Vortex laser and the black trace (bottom)
is the results for the stable laser design. The stable laser has approximately ten times
lower frequency noise and the noise occurs predominantly at low frequency where our
servo is most effective. The large feature at 5 kHz in the red trace is a PZT resonancein the Vortex laser.

CHAPTER 3. COMPONENTS OF THE APPARATUS

64

laser frequency noise for the stable laser design and for a New Focus Vortex laser.
The frequency noise of the stable laser is significantly less than that of the Vortex,
particularly at 1 kHz and above. The un-stabilized RMS noise of the stable design is
0.2 MHz measured in a bandwidth of 100 kHz. The un-stabilized RMS frequency noise ofa Vortex laser is 1-2 MHz. Other homemade laser designs used in our lab typically have
1 MHz of frequency noise. The un-stabilized frequency noise of the other homemadelasers is dominated by mechanical vibrations. The fact that the frequency noise of
the stable laser is a factor of five smaller is due to the more rigid design. To further
characterize the frequency noise, two identical stable laser setups were constructed in
the same laboratory by the Jin group. The outputs of the lasers were heterodyned
together and a linewidth of less than 100 kHz was measured on a few ms timescale. We
have also characterized the frequency drift of the stable laser design over several hours
and found it to be approximately 50 MHz/hour. This level of drift is comparable toother homemade lasers in our apparatus.

3.2

Computerized control system

We built a computerized system to control the various pieces of equipment thatmake up the apparatus. The computer system is capable of controlling the apparatus
with precise timing (at the few microsecond level) and high resolution (typically at the 1
part in 10,000 level or better). Our goal was to produce a computer code that is highly
flexible and reliable. The ability to quickly change the parameters of an experimentor to switch to an entirely new set of timings is extremely important. The current
computer control apparatus is based on a combination of commercial output boards
and homemade electronics. All of the software to program these boards is written in
the graphical language of LabView. A schematic of the computer control system isshown in Figure 3.7. In this section we briefly describe the various elements of thecomputer system.
The most important hardware board is the DIO-128 by Viewpoint Systems which
provides 64 programmable digital output lines. The central feature of the DIO-128 isa digital signal processor which changes the output level of the digital lines only whenrequired. Complex digital patterns can be generated with precise time resolution (1µs) without the need to specify the output state of every digital line at every clock
cycle. The DIO-128 has an onboard clock that seems to be stable at the few ppm levelbut operates at 1.000490 MHz. Since this offset from 1 MHz is well within the timing
resolution of the board, it is corrected in software. Each output line from the DIO-128 isbuffered and opto-coupled in an attempt to shield the experiment from electrical noisein the computer and to protect to DIO-128 from accidental short-circuits.
Seventeen digital lines from the DIO-128 are multiplexed to control a set of 16
independent digital-to-analog converter chips (AD767) with 12-bit resolution (5 mV
resolution). Twelve of the sixteen lines provide data for the AD767 DAC chips. Anotherfour digital lines are encoded with the address of a single DAC chip. Based on theaddress lines a signal from the seventeenth digital line is routed to update the DAC.
Only three clock cycles are required to update any single DAC out of the sixteen; one
cycle to load the data and address lines and two cycles to arm and trigger the chip. In
practice 160 µs are required to change the analog output value of all the DACs. This
homemade DAC system has proved to be very useful. Nearly all the laser frequencies

CHAPTER 3. COMPONENTS OF THE APPARATUS

DIO-128

GPIB board
1-48

freq. synths, etc..

65

PCI-6733 DAC sys.,

49-64

16 bits, 0.025 ms
resolution

TTL lines

DACs for bias coils,
optical trap, etc..

Homemade DAC sys.,
12-bit, 2 ms settle time

1-16

DACs for laser freqs,
laser intensity, etc..

Figure 3.7: Schematic of the computerized control system of the experiment. The
DIO-128 generates TTL signals on 64 different lines. Most of the lines go directly toequipment in the apparatus. The final 16 digital lines are used to create 16 analog
voltage with our homemade DAC system. The DIO-128 also controls the timing of
GPIB command execution and it triggers the PCI-6733 board.

CHAPTER 3. COMPONENTS OF THE APPARATUS

66

and the laser powers as well as many of the magnetic coil currents are controlled with
voltages from this system. The system is also quite flexible since the AD767 chips canbe replaced with a different part number if better specifications or different features are
required.
We also use a National Instruments PCI-6733 (16-bits, 0.3 mV resolution, 8 lines)
analog output board when computer controlled voltage ramps of long duration or more
precise resolution are required. For example the multi-coil transfer, optical-trap evaporation, and B-field ramps near the Feshbach resonance are controlled with this board.
The 6733 board utilizes an on-board clock (40 kHz update rate) for precise timing andis always triggered to start from the DIO-128 so that the two boards remain synchronized at the 10−5 level. Output lines from the 6733 board are buffered by op-amps forprotection against short circuit. A significant disadvantage of the 6733 board is thatdata is required to be output at each clock cycle.
Several instruments in the experiment are controlled via a GPIB board in thecomputer. These include rf synthesizers for the magnetic trap evaporation and Bragg
spectroscopy and arbitrary waveform generators that produce rf sweeps. The timingof GPIB signals is synchronized to the DIO-128 clock with a few ten’s of millisecond
precision via software written in LabView. This level of time resolution is adequate
since it can take several hundred milliseconds for the GPIB instruments to execute acommand.

3.3

Magnetic-field coil current control electronics

Precise control of magnetic fields at the few milliGauss level is required for theexperiments described in this thesis. For evaporative cooling in the Ioffe-Pritchard (IP)trap, the magnetic-field at the center of the trap must be constant throughout the cooling. Noise on the currents that produce the magnetic field leads to inefficient evaporation
since the noise reduces the energy selectivity of the RF knife. In addition, for our Feshbach resonance experiments we need the ability to create complicated magnetic-field
waveforms in which the field is rapidly changed. This section describes the electronic
feedback system which satisfies both of these requirements, see Refs. [91, 6] for further
information on magnet coil servos. The electronics are optimized to produce long-term
stability as well as the ability to quickly change the magnetic fields. Here we describe
the electronics for only the IP trap bias coils; this particular system is representative ofthe techniques used to control other magnetic-field coils used in the apparatus. A large
current is passed through the bias coils to create up to 700 G and the amount of current
can be controlled with a field-effect transistor (FET). The servo electronics feedback to
the FET based on a real-time measurement of the current.
Figure 3.8 shows a schematic drawing of the bias-coil high-current circuit. Current
flows from the Agilent power supply, through the bias coils, into a set of four field-effecttransistors (FETs) connected in parallel, through a precision current probe, and thenthe current is returned to the power supply. The current probe generates a voltage
signal which is proportional to the current flowing in the bias coils.8 The 20 Ω resistor
8

We initially used a Hall effect current probe from FW Bell. Unfortunately the response of this typeof probe depends on temperature at the 0.01% level, in our system this large temperature coefficientis unworkable. We switched to current probe made by Danfysik with a temperature coefficient that is1000 times smaller. The temperature response of the Danfysik probes is so much lower since they are

CHAPTER 3. COMPONENTS OF THE APPARATUS

67

for the currents probe has a very low temperature coefficient of resistance of 2 ppm/◦
C. Any drift in the value of this resistor would cause a drift in the coil current. We usehigh power FETs from Advanced Power Technology. The FETs have a capacitance ofabout 10 nF between the gate and the source and between the drain and the source.9
It is important to have a small FET capacitance since this capacitance in series with thetrap coils’ inductance forms a resonant circuit. The presence of this resonance limits theuseful bandwidth of our servo system. The voltage signal at the FET gate connection
comes from the servo electronics, which are described below. The return of the Agilent
power supply is grounded to a connection from the regular room AC ground in only asingle location. It is critical that ground be attached only once for the entire system toavoid ground loops.
To stabilize the current flowing through the bias coils, an electronic servo monitors the difference between current probe signal and a reference signal. Based on the
difference of these signals, the servo regulates the current flowing through the bias coilswith feedback to the FET gate. The goal of the servo is to make the difference between
the reference signal and the current probe signal zero. A simple schematic of our servo
electronics is shown in Figure 3.9. The servo electronics reside in a grounded box thathas many inputs and outputs. All signals that enter the servo box are buffered by an
instrumentation amplifier to avoid ground loops.10 For simplicity we will assume thatthe servo has two inputs and a single output that is connected to the FET gates. The
input signals come from the current probe and from the computer control so that thereference voltage can be adjusted. For maximum flexibility, the total reference voltage
is the sum of a stable DAC signal which is generated inside the servo box and several
signals from the control computer. All these signals are combined with a single summing amplifier.11 The servo works by comparing the current probe voltage to the totalreference voltage. The difference of these two signals is amplified and used as feedback
at the FET gate to control the amount of current flowing through the bias coils. During normal operation of the servo the sum of the current probe voltage and the totalreference voltage is zero.
The highest current stability is attained when the gain of the servo is as large as
possible. A large gain allows the observation of a small voltage difference between thereference signal and the current probe signal. Of course the servo gain is limited by
the delay in the response of the current to a change in the servo output. If the delay
time is too long the servo will provide positive feedback instead of negative feedback at
based on a fluxgate whereas the FW Bell probes are based upon a Hall effect sensor.
9Note that the actual FET arrangement in our system is different. An H-bridge configuration isused so that the current in the bias coils can be reversed via a digital signal from the computer. It is
important to be able to reverse the current of the bias coils so that a wide range of magnetic fields canbe achieved in the IP trap. A simple digital switching circuit routes signals from the servo electronics
to the FET gates based on the desired direction of current flow.
10
Commercial instrumentation amplifiers (IAs) have several nice features including a high input
impedance, low temperature drift, and excellent common-mode rejection. In our experiments we havefound it necessary to improve the IA common-mode rejection for MHz frequencies, see the Analog
Devices Application Note AN-671 for details. Commercial IAs do have a downside which is large noisefor a gain of only one.
11
For the best long-term stability and low noise, a large fraction of the reference voltage comes fromthe stable DAC. The computer signals are much smaller and therefore do not need to be as stable. For
the best immunity to rf signals in the lab we found it necessary to low-pass filter all the op-amps in the
servo at about 100 kHz.

CHAPTER 3. COMPONENTS OF THE APPARATUS

68+15 V, 440 A
Agilent
supplybias coilhigh power FETs
x4

from
servo
30 W

Rsense = 20 W
groundFigure 3.8: A schematic of the high current circuit for the bias coils. Current flows froma 15 V, 440 A power supply through the coils, a set of four parallel FETs, and a current
probe. The return of the power supply is grounded to the AC wall ground. Note thatthe actual FET arrangement in our system is different. An H-bridge configuration isused so that the current in the bias coils can be reversed via a digital signal from thecomputer. It is important to be able to reverse the current of the bias coils so that awide range of magnetic fields can be achieved in the IP trap. A simple digital switchingcircuit routes signals from the servo electronics to the FET gates based on the desired
direction of current flow.

CHAPTER 3. COMPONENTS OF THE APPARATUS

69

feedback

current
probe

+1
servo
op-ampinput #1

+1

input #2

+1

stableDAC+

to FETgate

referencevoltagemetal
enclosure

Figure 3.9: Simple schematic of the bias coil servo. The complete set of diagrams isavailable from the JILA electronics shop under file WC063A1. The dashed box indicatesthe grounded box; all external inputs entering the grounded box are buffered to avoid
ground loops. In this simple schematic only the current probe signal and the reference
voltage signals are shown. The servo works by comparing the current probe voltage tothe total reference voltage. The output of the servo is sent to the FET gates.

CHAPTER 3. COMPONENTS OF THE APPARATUS

70

servo gain

some frequency, and the bias coil current will oscillate. Therefore we reduce the gainof the servo to prevent positive feedback. Figure 3.10 shows a plot of the servo loop
filter response as a function of frequency. In order to realize the best DC stability of
the current we use an integrator. Since the response of the coil and FET system is not
perfect at every frequency (see below for details) we must switch to proportional gain
at a frequency of about 0.25 kHz. At even larger frequencies we use a differentiator toextend the useable frequency range of the servo. The differentiator turns on at about
11 kHz. At frequencies beyond about 90 kHz our servo cannot possibly control thecurrent effectively because of the delay between a change in the servo output and thecorresponding change in current. Therefore we low-pass filter the loop filter responseat 90 kHz. This complicated loop filter arrangement is not necessary for good DC
current stability. For good stability only the integrator and first proportional section
are needed. The differentiator and second proportional sections of the loop filter arenecessary to provide good dynamic current stability. For example, the coil current
should accurately follow a rapid change in the reference voltage signal without any
overshoot or oscillation. The high frequency sections of the loop filter response are
critical in eliminating unwanted overshoots and oscillations after a rapid change in thereference voltage.

integrator
differentiator

proportional
low-pass

proportional

1

0.25

11

45

90

freq (kHz)

Figure 3.10: A schematic of the feedback network (loop filter) response of the servo as afunction of frequency. The calculated corner frequencies of the bias coil servo are shown
for reference. This loop filter provides excellent long-term stability of the current and it
guarantees the currents accurately tracks rapid changes in the reference signal without
overshoots.

The bias coil loop filter parameters (i.e. the frequency at which we switch froman integrator to proportional, etc.) were primarily determined based on measurementsof the response of coil and FET system.12 The parameters of the servo are shownin Figure 3.10. We directly measured the transfer function of the bias coils and FET
12

A small amount of trial and error was used to set the highest frequency parameters in the loop filter
since accurate measurements of the bias coil and FET system response are difficult at high frequency.

CHAPTER 3. COMPONENTS OF THE APPARATUS

71

circuit using a network analyzer. The measured response and phase shift of the circuit
is shown in Figure 3.11 as a function of frequency. The network analyzer output signal
was directly attached to the FET gate and the current probe signal was fed into the
input of the network analyzer. To obtain this data we use a servo to control the DC
current flowing in the circuit. The servo gain bandwidth was limited to approximately
200 Hz. We choose 200 Hz since it is much lower than the expected bandwidth ofthe coils and FET system. Based on our measurements we can accurately predict the
optimum parameters of the servo loop filter. For example the phase shift of the coils
and FET system is around 90 degrees at 1 kHz, this means that with an integrator
the current will oscillate at 1 kHz. Therefore the gain of the servo would have to be
sufficiently low (i.e a closed loop gain of less than one is required) at 1 kHz to eliminate
the oscillation.
We characterize the success of the servo electronics by both the short-term (atimescale of milliseconds) and long-term (a timescale of seconds or more) DC stability
attained and the rate at which we can change the coil currents and thus the magnetic
field. Based on measurements of the field using rf transitions between Zeeman levels wehave determined that the short-term (100 µs) magnetic-field noise is only a few mG.
Over a timescale of weeks the magnetic field appears to be stable to better than 10 mG.
For longer times it appears that seasonal variations do effect the magnetic field at the
50 mG level. However the source of this drift has not been carefully analyzed. Finally,
with our servo system we are able to change the magnetic-field at a maximum rate of 6
µs/G without any overshoot or any substantial field oscillations following the change.

CHAPTER 3. COMPONENTS OF THE APPARATUS

0

0

-20

phase (deg)response (dB)

72

-40

-90

-180

-60

(a)
-80

(b)
1k

10k

frequency (Hz)

100k

-270

1k

10k

100k

frequency (Hz)Figure 3.11: Measured open loop response (a) and phase shift (b) of the bias coil and
FET system obtained with a network analyzer. The network analyzer driving signal was
attached directly to the FET gate and the response was determined from the current
probe signal. A slow integrator was used to keep a stable current running through thesystem. Beyond 200 Hz the data displays the response of the coil and FET system. Since
an integrator requires a phase shift of less than 90◦ , our measure response indicates thatthe servo loop filter must be switched to proportional gain at a frequency of 1 kHz or
less. Based on the measured phase shift we can accurately predict the required servo
electronics parameters.

Chapter 4
Evaporative cooling of 85 Rb to BEC

4.1

Introduction

The main goal of our apparatus is to create a 85 Rb BEC near the Feshbachresonance at 155 G. Using a Feshbach resonance the two-body scattering length may
be tuned to access new physical regimes in a quantum degenerate gas. In particular wewould like to study the properties of a condensate in the strongly interacting regime
where beyond mean-field effects are expected to be important. In the previous two
chapters we described the process of collecting atoms into a MOT, transferring the atoms
to the science cell, and loading the atoms into the IP trap. This chapter describes our
new approach to evaporative cooling of 85 Rb, in which the 85 Rb gas is sympatheticallycooled via thermal contact with a 87 Rb gas in both a magnetic trap and an optical dipole
trap. The results presented here are different from previous work in which a 85 Rb gas
was cooled to BEC using direct evaporation [7, 6] and 85 Rb was sympathetically cooled
to 1 µK via collisions with 87 Rb [11]. Our evaporative cooling approach represents asignificant improvement over this previous work since we can produce larger number
85 Rb condensates than ever before, and since production of 85 Rb condensates is robustin the sense that condensates can be produced even if the apparatus is not perfectly
optimized.
The intent of evaporative cooling is to lower the temperature of a trapped gas
while simultaneously increasing the phase-space density of the gas. The requirements forthe formation of a condensate were discussed in Chapter 1. The efficiency of evaporative
cooling is determined by the ratio of the elastic collision rate to the rate of inelastic
collisions that cause number loss and heating. Both the elastic and inelastic collision
rates for 85 Rb have been carefully studied and are documented in Refs. [8, 92, 12].
Since the collision properties of 85 Rb are quite poor, efficient evaporation is extremely
difficult to implement. In comparison, 87 Rb suffers little loss or heating due to inelastic
collisions and this represents a key feature of our system. We verified the ability of
our apparatus to perform evaporative cooling (i.e., stability of the trapping potential,
background gas in the science cell, etc.) with 87 Rb since its collision properties arenearly ideal. The performance with regard to 87 Rb is excellent and condensates witha few million atoms can be produced. Of course our goal is to make 85 Rb condensates
and the achievement of that goal is the main focus here.
There are two central roadblocks to efficient evaporative cooling of a 85 Rb gas: azero in the s-wave elastic collision cross section near 0.4 mK and large two- and threebody inelastic collisions that lead to atom loss and heating. At high temperature theelastic collision cross section of 85 Rb is suppressed causing evaporation to be inefficientCHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

74(see Section 4.2.3 for more information). As the temperature is lowered the elastic crosssection sharply increases and evaporation can proceed more rapidly. However in efficientevaporation lowering the temperature of the gas will also increase the density. Since therate constants of density-dependent inelastic collisions in 85 Rb are large, an increased
density leads to more rapid number loss from the gas, which causes evaporation to beinefficient. These two sources of inefficiency in the cooling process are the roadblocks
that must be avoided. At the beginning of evaporation we use sympathetic cooling with
87 Rb which circumvents the first roadblock (small collision cross section). Once inelastic
number loss becomes significant during the cooling, we load the atoms (85 Rb and 87 Rb)
into an optical trap in which we are able to apply a magnetic field to reach the 85 Rb
Feshbach resonance. Using the Feshbach resonance elastic and inelastic collision rates
in the 85 Rb gas can be controlled; in particular we tune the magnetic field to minimize
inelastic collisions and this capability allows us to circumvent the second roadblock. In
the optical trap we simultaneously evaporate the two gases; the 85 Rb gas rethermalizes
primarily through collisions with the 87 Rb gas. The combination of the control affordedby the Feshbach resonance and the presence of 87 Rb for sympathetic cooling represents
a significant advantage for cooling 85 Rb in our system.
The optimized evaporation procedure we currently utilize to cool a gas of 85 Rb toultralow temperature can be summarized by the following steps. The evaporative coolingprocess begins with sympathetic cooling in a magnetic trap. We load approximately
5 × 108 87 Rb atoms and 4 × 106 85 Rb atoms into the IP trap. We directly evaporate the
87 Rb gas to a temperature of 8 µK; the 85 Rb gas is cooled due to thermal contact. Thisevaporation in the IP trap takes 44 seconds and approximately 3 × 106 atoms of eachspecies remain in the IP trap. To further cool the mixed species gas we load into an
optical trap. Both species are simultaneously cooled by lowering the depth of the opticaltrapping potential. The duration of the optical trap evaporation is 11.5 seconds. During
the first few seconds of the optical trap evaporation the magnetic field at the center of
the trap is tuned to 168.8 G. At this field 85 Rb inelastic collisions are minimized. Inorder to create a BEC of only 85 Rb, prior to the last two seconds of the cooling, we ramp
the magnetic field to 163 G and increase the speed of evaporation. With these settings
87 Rb the efficiency of 87 Rb cooling is low and a significant fraction of the gas is lostfrom the trap. At the end of evaporation we can produce 85 Rb condensate with up to
80,000 atoms and a condensate fraction of greater than 80%. A residual non-condensed
gas of 87 Rb with approximately 10,000 atoms remains in the trap.
While our cooling procedure currently works well and can be summarized in
the short paragraph above, arriving at this final, complex procedure required a longseries of experiments and studies. In our first work towards cooling 85 Rb we worked to
optimize sympathetic cooling in our IP magnetic trap. In this trap we carried out many
experiments that, while they never allowed us to fully reach 85 Rb BEC, were essential
to identifying the techniques we use to circumvent the roadblocks to cooling 85 Rb. In
the end these careful studies led us toward our current use of an optical trap for the
final cooling. Readers interested in an understanding of the intricacies of 85 Rb coolingvia 87 Rb should follow our historical discussion of our cooling work presented in theentire chapter, while readers mostly interested in our current experimental procedure
should skip towards the end of the chapter where the optical trap cooling is described.
This chapter is organized into several sections. We begin with a discussion of amodel for the evaporation process that we use to understand our cooling results. The

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

75

results of the simulations motivate our route to 85 Rb BEC using sympathetic cooling.
Next evaporation of 87 Rb to BEC in both the magnetic and optical traps is discussed.
This information provides a baseline for the best performance we could hope to observein 85 Rb. Unfortunately this just sets our hopes far too high. Then the results andlimitations of sympathetic cooling of 85 Rb with 87 Rb in the magnetic trap are discussed.
Finally evaporative cooling of both 85 Rb and 87 Rb in the optical trap is discussed
including cooling into the quantum-degenerate regime. Efficiently producing a 85 Rb
BEC in the optical trap required a very weakly confining potential to maintain low
density as well as detailed control of the 85 Rb collision rates during the evaporation.

4.2

Simulations of 85 Rb evaporative cooling

The concept of evaporative cooling in a trapped gas is straightforward but very
powerful [93, 94, 95, 96, 97, 98, 99]. To lower the temperature of an equilibrium atomicgas, a small fraction of the most energetic atoms are selectively removed. The remaining
non-equilibrium gas has a smaller total energy and elastic collisions tend to rethermalize
the gas to a lower temperature. By repeating the process many times the temperature
can be lowered by many orders of magnitude. Evaporative cooling can be a very efficient
process. In an extreme case a large fraction of the energy in a gas can be contained ina small fraction of the atoms. By removing just those energetic atoms, the total energy
in the gas drops significantly but the total particle number is nearly unchanged. This
extreme situation can only occur, however, if very low loss and heating of the gas aremaintained.
We have performed a simulation of the evaporation process based upon a simple
model. These simulations serve as a guide in understanding the most efficient routeto BEC with 85 Rb. The model includes the removal of energetic atoms from trap andthe rethermalization of the gas to lower temperature. It is critical to include sources ofatom loss and heating in the model. During the rethermalization process atom loss andheating can occur due to inelastic density-dependent collisions as well as collisions withthe background vapor in the science cell. The best route to BEC must balance the atom
loss due to inelastic collisions against a more efficient slow evaporation. Unfortunately
our initial attempts at cooling 85 Rb were an abject failure for a variety of reasons.
Comparison of the experimental data to our simulations allowed us to pinpoint the
failure and devise an improved strategy. In this section our simple model is developed
based on calculations of the collision rate in the gas including the energy dependenceof the elastic cross section. Truncated distributions of the energy in a gas are used tomodel energy removal. Based on this information we lay out our strategy for evaporative
cooling.
4.2.1

Simple model of evaporative cooling

There is a significant body of theoretical work on evaporative cooling (for example
see [96]) including non-trivial effects due to the quantum statistics of the particles [100].
This is particularly evident in evaporation of a Fermi gas into the quantum-degenerate
regime, where suppression of collisions due to the quantum statistics of the particles
changes the efficiency of evaporative cooling [101]. Our simulations draw from Refs.
[96, 6, 102] and always assume that the gas behaves classically since for our Bose gas

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

76changes to thermodynamic quantities like heat capacity due to quantum statistics should
be small compared to the effects of inelastic losses. The elastic collision rate per atom in
the gas sets the timescale for rethermalization and it is given by the following expression
Γ = 2hni σ hvrel i(4.1)

where hni is the average density, hvrel i is the mean relative speed, and σ is the elastic
cross section (see Section 4.2.3). The factor of 2 represents an increase in the collision
rate of identical non-condensed bosons due to atom-atom correlations. We will calculate
the average density and mean relative speed from the gas density distribution. We
will consider the elastic cross section later in this chapter (see Section 4.2.3). For acylindrically symmetric harmonic trap with radial (axial) frequency ωρ (ωz ) the potential
is1
U = m(ωρ2 ρ2 + ωz2 z 2 )(4.2)
2
and the distribution function is given byn(~r, p~) = npk e− k 1T H(~
r,~
p)
b

2

= npk e− k 1T ( 2pm +U (~
r))
b

(4.3)

whereR npk is the density at the center of the trap and the normalization is given by
N = n(~r, p~) d3~r d3 p~. It is useful to re-scale the coordinate system with Z = λ z where
λ = ωz /ωρ to obtain a spherically symmetric potential U = 21 mωρ2 (ρ2 + Z 2 ) = 12 m ωρ2 r0 2
with r0 2 = ρ2 + Z 2 . For a Bose gas with temperature above the transition point, themomentum integral is easily calculated to obtain the spatial density distribution of thegaswhere npk = N ωρ2 ωzdensity is³

m
2πkb T´3/2n(~r) = npk e

2 r0
−m ωρ
2 kb T2(4.4)

is the density at the center of the trap. The average

Z

1
N
hni =
n(~r)d3~r = ωρ2 ωzN
8and the mean relative speed is [6]

µ

m
πkb T

¶3/2

√
= npk / 8

(4.5)

s

hvrel i =16 kb T
.
πm

(4.6)

For our simulations of evaporative cooling it is useful to determine the occupancy
of a particular quantum state of the system with energy ²n . Each quantum state islabeled by the quantum number n, and the occupancy is−1f (²n ) = e kb T(²n −µ)(4.7)

where we have assumed a Maxwell-Boltzmann (MB) gas with a chemical potential µ.
For the remainder of this discussion we will assume the energy takes on a continuous
range of values.1 The density of quantum states in the three-dimensional harmonic
potential is²2ρ(²) =.
(4.8)
2λ(h̄ωρ )3
1

For a MB gas this is a reasonable assumption since the temperature of the gas is larger than thespacing of a quantum states with kB T À h̄ ωρ .

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

77

For a classical (MB) gas the chemical potential is easily determined by normalizing theenergy distribution to the number of total number of trapped particlesµZ

ρ(²)f (²)d² → eµ/kb T = λ N

N=

h̄ ωρ
kb T

¶3

.

(4.9)

In our simulations we assume that the evaporation process truncates the distributionof energies in the gas at some maximal value η. It is convenient to scale the truncationenergy to the temperature of the gas as η = ²kcut. We can find the number of particlesbT
lost and the total energy lost in the gas by truncating the distribution at η by performing
the following integrals
Nlost =Z ∞Z ∞η

ρ(²)f (²)d² =N −η(e (2 + 2η + η 2 ))2

(4.10)

N kb T −η
(4.11)(e (6 + 6η + 3η 2 + η 3 )).
2η
In Fig. 4.1 these quantities are plotted as a function of η. In our simulation the valueof η is set to 5.5 so that only a small fraction of the atoms and energy in the gas isremoved. In our calculations we assume that the time scale for removal of atoms fromthe trap is very fast compared to the rethermalization rate of the gas.
Elost =² ρ(²)f (²)d² =

1.00

T)

1.00

/ (3 N k

/ N
0.50

E

0.25

0.00
0

0.50

lost

N

lost

0.75

B

0.75

2

4

6

8

10

0.25

0.00
0

2

4

6

8

10

Figure 4.1: Number of particles lost and energy lost in a gas as a function of thetruncation parameter η. Forced evaporation typically utilizes η ≈ 4 − 6 so that only asmall fraction of the gas is lost in each cooling iteration.
To simulate the forced evaporation process in our experiments, we break theprocess into a large number of iterations. At each iteration a small fraction of theatoms in the gas are removed and therefore a small fraction of the total energy is lost.
We then estimate the rethermalization time based on the collision rate in the gas. Fora specific value of η and initial temperature T and number N we calculate the numberof atoms (Nlost ) and energy (Elost ) removed as well as the time required to repopulate
the high-energy trap states. This required time represents the relevant rethermalization
time for evaporation and is approximated by [96]1
tevap

= Γ × b exp −η.

(4.12)

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

78

where we have added a scale factor b that is described below. Our model significantly
underestimates the time required for evaporation as compared to the observed timein our experiments. The model predicts that efficient 87 Rb evaporation can occur ten
times faster than we observe in our experiments. We account for this underestimation bychanging the scale factor b ≈ 0.1 to set the total duration of many iterations to roughly
coincide with our experimental observations. The likely cause of the underestimation isthat full thermodynamic equilibrium throughout the gas is required for efficient evaporation. For the duration of this rethermalization time, loss of atoms and heating canoccur due to density-dependent inelastic collisions and collisions with background gas.
A value for the heating rate, Rheat , is obtained from experimental data since we lackan adequate model of this process. We simulate the number loss using the following
differential equationdN
N= −K2 hniN − K3 hn2 iN −(4.13)dtτbg
where the terms with K2 and K3 represent two and three-body loss and τbg is the
exponential lifetime due to background gas. For a non-condensed gas of bosons in aharmonic potential we find8
hn2 i = √ hni2
(4.14)
27
0
For now we consider single species 87 Rb cooling. The temperature T 0 and number N87
after a single iteration of the evaporation simulation is approximated by

3N kB T − Elost+ Rheat tevap
3N 0 kB(4.15)

0N87= N − Nlost − K3 hn2 i N tevap(4.16)

T0 =where we have neglected two body losses since K2 for 87 Rb is small. The next iterationof the simulation begins with these values and a new rethermalization time is calculated.
The simulation runs until no atoms remain in the gas or a minimum temperature has
been reached.
We have performed simulations using our model under a variety of different conditions, and we find qualitatively reasonable behavior. Figure 4.2 shows the result of
our simulation for single species 87 Rb evaporation under the typical conditions of ourexperiments. The radial (axial) trapping frequency is 2π × 215 Hz (2π × 12.5 Hz). In
the absence of loss and heating, as η is increased the number of atoms lost is lower
and the duration of evaporation is expected to be longer. Also increasing the elastic
cross section or the trap frequencies shortens the duration of the cooling because thesechanges increase the elastic collision rate. As expected, starting the evaporation witha larger number of atoms or a lower temperature tends to increase the final number.
When loss and heating are included we also observe reasonable behavior. In particular
as the value of K3 increases, the atom number remaining after evaporation decreases.
So far our model is limited to single-species evaporation but including a second
species is straightforward. In our simulations the magnetically trapped 85 Rb gas will
always be sympathetically cooled by the 87 Rb gas. In order for sympathetic coolingto be effective there must be good thermal contact between the two species; i.e the85 Rb and 87 Rb gases must be spatially overlapped so that collisions occur between thetwo species, and the elastic collision cross section must be sufficiently large. In ourCHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

9

1000

Temperature (µK)

108Number

10

7

10

6

10

5

10

0

79

10

20

30

40

time (sec)

50

60

100

10

1

0.1
0

10

20

30

40

50

60

time (sec)Figure 4.2: Results of our single species 87 Rb evaporative cooling simulation. The
number of particles in the 87 Rb gas (left) and the temperature of the 87 Rb gas (right)
is shown as a function time. For this simulation we used K3 = 4 × 10−29 cm6 /s, andthe heating rate was 10 nK/s.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

80

Table 4.1: Various relevant parameters that are used in our simulations of evaporative
cooling are collected here. For each parameter the relevant magnetic field or range offields is indicated.
Parameter
K3 87 Rb
K2 87 Rb
K3 85 Rb
K2 85 Rb

B (G)
all
all
2
2

Value
4 × 10−29≈ 10−18
5 − 10 × 10−25
3 × 10−14

units
cm6 /scm3 /scm6 /scm3 /sReference
[104]
[13]
[6]
[6]

simulation we assume good thermal contact since the elastic cross section is large (theinterspecies scattering length is 213 a0 [103, 11]) and our IP trap assures spatial overlap
between the two gases. To include sympathetic cooling in our model, we modify thetemperature T 0 and number N 0 after a single iteration of the evaporation simulation tobe3(N85 + N87 )kB T − ElostT0 =+ Rheat tevap(4.17)
0 + N 0 )k
3(N85
87 B
0
N87
= N87 − Nlost,87 − K3,87 hn287 iN tevap(4.18)

0
N85= N85 − K3,85 hn285 iN tevap − K2,85 hn85 iN tevap .

(4.19)

Here we have included separate rates of inelastic atom loss for each species. Loss frominelastic interspecies collisions is ignored since the rate constants are known to be small[13, 1]. The actual rate constant values used in the model are listed in Table 4.1.
These values were derived from previous work in the Wieman group (see the table forreferences). In our model, the time required for a single iteration does not change withthe addition of 85 Rb. This is because we have assumed good thermal contact between
the two species so that thermal equilibrium is maintained, therefore the timescale forrethermalization is the same for both species. Energy is only directly removed fromthe 87 Rb gas. Sympathetic cooling works best in the limit that the number of 87 Rb
atoms is much larger than the number of 85 Rb. When the number in each species isroughly the same the 87 Rb cooling becomes inefficient. The inefficiency is characterized
by the large fraction of the 87 Rb gas that has to be evaporated in order to lower thetemperature of the two-species gas by an appreciable amount. When the cooling isinefficient, the number of 87 Rb atoms is rapidly depleted and no further reduction intemperature occurs. The specific results of a sympathetic cooling simulation based on
this model will be presented later in conjunction with experimental data (see Fig. 4.11).
4.2.2

Analytic model of sympathetic cooling

To conclude the discussion of evaporation models, we briefly mention an analytic
model of the sympathetic cooling process that is based on an energy budget. As described in Ref. [102], the model takes into account the finite heat capacity of the gas thatis used as the coolant during sympathetic cooling. This model is useful for understanding the limitations of sympathetic cooling when the number of particles in the two gases
are nearly equal. The key result of Ref. [102] is a prediction for the temperature of theCHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

81

two-species gas as a function of particle number of each species during the cooling. Thisprediction will be compared to our experimental observations of sympathetic coolingpresented in Section 4.5.1. In a two-species gas with particle numbers N1 and N2 thetotal energy is E = 3(N1 + N2 )kB T and the change in total energy for a single step of
evaporation is dE = dN1 (η kb T ).2 After rethermalization the temperature of the gas
may be found from E + dE = 3(N1 + N2 )kb (T + dT ). We define α = η/3 − 1, and then
the temperature as a function of particle numbers N1 and N2 during the evaporation is
Ã

T =T

i

N2
N1i

!α

µ

×

¶

α
N1
+1 .
N2

(4.20)

Here T i is the initial temperature of the two gases prior to evaporative cooling andN1i is the initial number of that species. During sympathetic cooling the number N1
decreases as atoms from that gas are evaporated, the number N2 remains constant, andthe temperature of the two-species gas drops. When theµ number
N1 reaches zero, the¶cooling ceases and the final temperature is Tmin = T i ×
4.2.3

N2
N1i

α

.

Motivation for sympathetic cooling based on elastic collision crosssections

With the theoretical results obtained in the previous section in mind, in particular the scaling of evaporation rate with collision rate, several conclusions about the
expected performance in our system can be drawn. The traditional route to efficiently
producing alkali-atom BEC involves increasing the trap frequencies as much as possible
since the elastic collision rate scales as wρ2 . The temperature of the gas also increases³ ω0 ´2/3

so an overall increase in collision
with increasing ωρ but only fractionally as ωρρrate is still obtained. With a larger elastic collision rate the evaporation process is fastercompared to the loss caused by collisions with background gas in the vacuum system,and therefore evaporation can be quite efficient. In fact a highly desirable “runaway”
process can occur where evaporative cooling actually increases the elastic collision rate
despite significant loss of atom number [99, 105, 97]. In this condition as the gas is cooled
to lower temperature the collision rate rises and the efficiency of the process increases.
Runaway evaporative cooling is quite easily achieved for many different species but thisroute assumes no negative side-effects of the increased density or initial temperature.
Unfortunately achieving runaway evaporation with 85 Rb is difficult since increasing thetrap frequencies actually leads to a decrease in the elastic cross section and an overallloss of collision rate. Large inelastic collision rates for 85 Rb further preclude the increaseof trap frequencies.
The elastic collision cross section is calculated from the energy-dependent scattering amplitude in a partial-wave expansion [49, 48]
fl =1
k cot(δl ) − ik(4.21)

where√ l is the angular momentum of the collision, k is related to collision energy bycoll
k = 2µE, µ is the reduced mass, and δl is the two-body scattering phase shift. Theh̄
2

In this model N1 and N2 would correspond to the number of 87 Rb and 85 Rb atoms respectively.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

82

J. Burke, JILA Thesis

Figure 4.3: The elastic collision cross section for 85 Rb, 87 Rb, and collisions between
the two species as a function of the collision energy. The total cross section for 87 Rb
and interspecies collisions is roughly flat over a very wide range of temperature. In
contrast, near 375 µK the 85 Rb s-wave cross section suffers a Ramsauer-Townsend typeminimum. This figure was reproduced from Ref. [1].

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

83

term with −ik represents the unitarity limit since kfl cannot exceed 1. At unitarity theelastic cross section is not larger than 4k2π . The cross-section is then [49, 48]
σtot =

4π X(2l + 1) sin2 (δl ).
k 2 l=0(4.22)

At low temperature only s-wave scattering is relevant and the total cross section is wellapproximated by σ = 4 πa2 . In Fig. 4.3 a plot of the elastic cross section as a functionof collision energy from Jim Burke’s PhD thesis is shown to illustrate several advantages
and challenges in our system (see Refs. [13, 1]). The elastic cross section for both 87 Rb
and interspecies collisions is constant and large over the range in temperature relevant
for our experiments (0-1 mK). In contrast the 85 Rb cross section, the solid line is thetotal cross section for all partial waves and the dashed line is for s-wave scattering
only, is small over much of that range due to the presence of a Ramsauer-Townsend
type minimum near 0.4 mK. To understand the origin of this suppression in the crosssection, consider only l = 0 scattering for which the cross section is 4k2π sin δ0 2 . When thephase shift approaches π the cross section is minimized. If the higher partial waves aresuppressed by the low temperatures attained in our experiments, then the total crosssection is also small. This represents a significant challenge in cooling 85 Rb into thequantum regime and has been the subject of discussion in Refs [7, 6].
There are two strategies for overcoming the small elastic cross section of 85 Rb. In
the first strategy direct evaporation is performed with reasonable efficiency as long as thecooling is initiated from a low temperature. In the initial Wieman group experimentswith 85 Rb great care was taken to load a very cold gas into the magnetic trap andcompression of the trap was avoided. This resulted in a low but manageable initialcollision rate so that evaporation could proceed slowly. Another strategy indicated by
Fig. 4.3 and first proposed in Ref. [13] is to utilize the large elastic cross section between
85 Rb and 87 Rb. The 87 Rb gas is then cooled by direct evaporation while the 85 Rb iscooled through thermal contact. Sympathetic cooling is the strategy that we use, andtherefore in principle we only need to worry about cooling 87 Rb efficiently.

4.3

RF for evaporation in the Ioffe-Pritchard trap

In the magnetic trap the most energetic atoms are selected by driving radiofrequency (rf) transitions between different magnetic sub-levels. Figure 4.4 shows aschematic diagram of the rf transitions leading to evaporative cooling in 87 Rb. The
evaporation frequency is set to νrf = ν0 + η kb T /h where η sets the fraction of the gasthat is removed and ν0 represents the bottom of the trap given by ν0 = gF mF µB B0 /hwhere gF depends on the hyperfine state of the atom and B0 is the field at the centerof the trap. Since the g-factor of 85 Rb is smaller than that of 87 Rb, the frequency toremove atoms at a given energy is higher. Therefore if the two gases are in thermal
equilibrium only 87 Rb atoms will be evaporated.
An Agilent E4400B rf synthesizer is used to generate the rf for evaporative cooling.
This signal is sent to an amplifier (ZHL-1-2 made by Mini-Circuits) with 5 W total
output, gain of 40 dB, and a nominal 50 Ω impedance. The frequency and output
power of the synthesizer are set by the control computer over the GPIB interface. Since
the timing precision of the GPIB interface is poor the rf is turned on and off withmicrosecond precision using an rf switch. The amplified signal is broadcast to theCHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

384

mf = -1
87

Rb
85
Rb

energy (mK)2

mf = -1
mf = -1

νrf

1
0

mf = 0-1
mf = 1-2-3

mf = 1

-0.30

-0.15

0.00

0.15

0.30

mf = 2y (mm)
Figure 4.4: Energy level diagram for evaporative cooling as a function of the distancey from the center of our magnetic trap. The plot shows the level diagram of the 85 Rb|f = 2, mf = −2i and 87 Rb |f = 1, mf = −1i states. We can selectively evaporate only
87 Rb atoms since the g-factors of the two species are different.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

85

atoms with a simple coil that is placed as close to the atoms as possible. The coil isapproximately 2 cm from the atoms and it has a radius slightly larger than 1 inch. See
Section 2.6.1 for more details on our rf setup.
It might seem advantageous to continuously sweep the rf source to obtain the mostefficient evaporation, but this is both technically very challenging and in practice unnecessary since the synthesizer frequency can be changed on a timescale roughly equal to
the rethermalization time in the gas. In our experiments the rf is swept discontinuouslywith a frequency change every 50-100 ms.
The evaporation frequency sweep is broken into several stages in which the parameters of the sweep are separately optimized. Typically we choose the start and stopfrequencies to break the evaporation into stages in which the temperature of the gas islowered by a factor of 2-4. The rf power and time constant τstage of each stage are varied
over a wide range to optimize the atom number after the cooling. If the evaporation isreasonably efficient the temperature is determined by the rf, and the temperature will
not change for different values of the rf power and time constant. In general, the rf
power and time constant for evaporation are only sensitive at approximately the 30%
level and tend to behave smoothly. The parameters for the magnetic trap evaporation
currently used in the experiment are shown in Table 4.2.
Table 4.2: Evaporation parameters optimized for 87 Rb in the magnetic trap. The
frequency of the final stage sets the final cut which changes slightly and is near 1.25
MHz. The power referenced here is what is programmed into the synthesizer and does
not include the gain of the amplifier or loss through various rf components.
Stage
1
2
3
4
5

fstart (MHz)
50
20
7
3
1.8

fstop (MHz)
20
7
3
1.8
x

τstage (sec)
20
15
6
3
6

fof f set (MHz)
1.4
1.4
1.4
1.3
1.05

rf power (dBm)
-18
-21
-24
-24
-24

The pattern of the sweep is an exponential function given by
f (t) = (fstart − fstop ) exp−(t − tstart )+ fof f set
τstage(4.23)

with start, stop, and offset frequencies. The offset frequency can be used to vary the
speed of evaporation. If the offset is far below the final frequency of a stage, then thetime it takes to reach the stage end frequency is shorter. To maintain the best possible
timing accuracy the frequency is calculated by the control computer “on-the-fly” during
the evaporation using the internal Windows clock. This suppresses the accumulation of
unknown timing lags during the evaporation.

4.4

Evaporation of 87 Rb to BEC

This section describes our 87 Rb single species evaporative cooling procedure inboth the magnetic trap and the optical trap. The focus of this section is to document

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

86

the excellent performance of evaporative cooling that we observe with the 87 Rb gas.
This information is important for two reasons. First, it allows us to verify and monitor
the proper functioning of our apparatus.3 With a baseline for 87 Rb performance established, when poor evaporation of 85 Rb is observed, we can eliminate many possible
technical malfunctions of the apparatus as the cause of the poor 85 Rb cooling performance. With this knowledge we can focus our efforts on improving the 85 Rb coolingby minimizing inelastic collisions. Secondly, the information in this section provides auseful comparison to our observations of the efficiency of sympathetic cooling. During
sympathetic cooling, we monitor the number of 87 Rb atoms that remain in the magnetic trap in order to understand the heat load due to the 85 Rb gas. If the number of
85 Rb atoms in too large, a large fraction of the 87 Rb gas must be evaporated and a low
enough temperature after cooling may not be reached. We compare the final numberof 87 Rb atoms after the cooling with and without 85 Rb to understand whether a large
number of 85 Rb atoms may be cooled.
4.4.1

Forced rf-evaporation in the magnetic trap

Prior to evaporative cooling in the magnetic-trap, atoms are delivered to thescience cell from the MOT located in the collection chamber. Typically the number of
87 Rb atoms delivered to the science cell is in the range of 5−10×108 and the temperature
is 220 µK. The atoms are optically pumped into the |f = 1, mf = −1i state and loaded
into the Ioffe-Pritchard magnetic trap. The magnetic trapping frequencies are slowly
increased over 3 seconds to ωρ = 2π × 215 Hz and ωz = 2π × 12.5 Hz just before
evaporative cooling begins. This adiabatic compression raises the temperature of the0³ ω0 ´2/3

so that cooling begins from approximately 700 µK. It
gas according to TT = ωρρis also the case that atoms are lost due to contact with the science cell walls during
the compression. We made efforts to minimize this loss such as increasing the physicalsize of the glass cell along the weak direction of the trap. But the residual loss of theseenergetic atoms does not significantly hinder the performance of evaporation.
To characterize the efficiency of the cooling, we measure the number of atoms
and temperature of the gas at different values of the rf-cut; we refer to this data as anevaporation trajectory. The term rf-cut (or simply cut) describes the final frequency
used to cool the atoms. If the evaporation is working reasonably well then the cut will be
proportional to the temperature of the gas. An important indicator of bad evaporation
is a large mismatch between the cut and temperature. Figure 4.5 shows a trajectorythat represents nearly our best evaporation results. We characterize the efficiency ofthe cooling by the slope of the data points in Fig. 4.5 (a), this slope is defined as
αevap =d(ln T )d(ln N )(4.24)and the data were fit to a line to extract α = 1.60(5). For every factor of ten loss inatom number the temperature of the gas drops by a factor of fifty. This result is ingood agreement with other examples of excellent evaporation performance such as that3

For example, good evaporation performance of 87 Rb indicates that the magnetic-trap currents arestable. Good performance is also an indication that the transfer from the MOT to the science cell isfunctioning properly.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

87

described in Ref. [106]. In Fig. 4.5 (b) the measured peak phase space density calculated
based on a Maxwell-Boltzmann distributions is shown as a function of temperature. The
BEC transition is reached when the peak phase space density reaches ζ(1.5) ≈ 2.612.
The solid black lines in Fig. 4.5 (a) and (b) are the result of our previously discussed
simulation with η = 5.5 and are in reasonable agreement with the data. The total
duration of the simulated evaporation was set to be equal to the 50 seconds used in theexperiment. A three-body loss rate with K3 = 4 × 10−29 cm6 /s and heating rate of 20
nK/s were used in the simulation. Two-body losses are expected to be small and theagreement between our data and the simulations confirms this is true. Since there isgood agreement with the simulation we can safely conclude no other factors significantly
contribute to the evaporation efficiency.
The number of 87 Rb atoms cooled to microKelvin temperatures depends on theinitial number at the beginning of evaporation. In Fig. 4.6 we show a plot of thenumber at a fixed rf-cut in which we varied the initial number by changing the MOT
fill. The temperature of the gas after evaporation is constant over the data except
for the two points with lowest number. For these points that evaporation was very
inefficient. The data in the figure suggests that we could increase the performance ofthe experiment by loading more atoms in the MOT or delivering a greater fraction of
the MOT to the science cell. Increasing the MOT number might be accomplished byadding more trapping laser power or delivering more number to the science cell, which
requires improving the transfer tube vacuum or the optical pumping. Given the current
performance of the experiment, none of these improvements seem necessary.
The results shown in Figure 4.5 reflect the baseline performance of our apparatus
under typical experimental conditions. On a daily basis we measure the number of 87 Rb
atoms at an rf cut of 2.5 MHz. Under these conditions the temperature is 8–9 µK and thenumber of particles is 21 million. When the apparatus is not functioning properly a dropin the number of atoms at this cut is observed. It is of course important to understandthe causes of poor evaporation performance. In particular, it is important to understandthe difference between poor evaporation that is caused by the unfavorable collision
properties of the gas and poor evaporation which is caused by technical malfunctioningof our apparatus. A good example of a technical malfunction is noise on the magnetictrap currents, which leads to an effective broadening of the rf knife for evaporation
since the magnetic field at the center of the trap is rapidly changing. A broad rf knife
indiscriminately removes atoms from the trap over a wide range of energy. As thetemperature of the gas drops the rf knife does not have adequate energy selectivity for
efficient evaporation. This results in a reduced efficiency of the cooling and a reduction in
the final number of atoms. Figure 4.7 shows a comparison of two evaporation trajectories
for different values of the magnetic-trap current noise amplitude. When the rf-knife is
too broad (large noise amplitude) a dramatic loss of number occurs. As shown in the
figure, when we decrease the amplitude of the noise by a factor of ten, we are able tocool the gas to a temperature approximately ten times lower.
4.4.2

Cooling 87 Rb in the optical trap

This section describes the single-species evaporation of the 87 Rb gas in an optical
trap. It represents the baseline performance of our apparatus with respect to 87 Rb
cooling efficiency. The work described in the later chapters of this thesis was performed

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

88

9

10

(a)

Number

8

10

7

10

αevap = 1.60(5)
6

10

1

10

100

peak phase space density

Temperature (µK)
0

10

(b)
-2

10

-4

10

-6

10

-8

10

1

10

100

Temperature (µK)

Figure 4.5: (a) Single-species evaporation trajectory of 87 Rb plotted as Log-Log. The
points are the measured atom number at each temperature, the red dashed line is alinear fit to the data to extract αevap = 1.60(5), and the black solid line is the result of
our simulation. (b) The points are the measured phase space density for a classical gas
extracted from the number and temperature of the gas. The black line is the calculated
peak phase space density from our simulation. These data represent the current best
performance of the 87 Rb evaporation.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

89

6

N after evap (10 )

6

4

2

0
0

1

2

N

(10 )

3

9

MOT

Figure 4.6: Measured 87 Rb number after evaporation to a fixed rf-cut as a function of
the number of atoms loaded into the MOT. The evaporation is run with the same rf
ramps for each value of NM OT . Roughly 25% of the atoms from the MOT are delivered
to the magnetic trap for evaporation.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

90

8

10

42 ppm noise
420 ppm noise
7

Number

10

6

10

5

10

4

10
0.1

1

10

100

Temperature (µK)
Figure 4.7: Evaporation trajectory of 87 Rb with two different amplitudes of magnetictrap current noise. Here the noise amplitude is the total integrated AC signal from
10 Hz to 100 kHz. During normal operation the current noise is 6 ppm on the IP
trap coils. When the amplitude of noise on the trap current was decreased, thelowest
achievable temperature also dropped by a similar amount. This data is an example ofpoor evaporation due totechnical limitations in the apparatus. The noise here is limitedby the inherent current instability of the commercial power supply we used at the time.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

91

with two different optical trapping setups: the “tight” trap with a 23 micron waist andthe “weak” trap with a 46 micron waist. We attain a very similar efficiency for single
species 87 Rb evaporation in both traps. To avoid confusion, in this section we only
discuss evaporation in the tight trap. To begin single-species 87 Rb cooling in the tightoptical trap, we pre-cool a gas in the magnetic trap and then we load the atoms into an
optical dipole trap for further cooling. We perform evaporative cooling in the opticaltrap by slowly lowering the depth of the potential to let the most energetic atoms spillout. Then the remaining atoms in the gas rethermalize to lower temperature just as
in the magnetic trap. After several seconds of optical trap evaporation we are able tocreate a condensate with more than a million atoms.
After evaporation in the magnetic trap to 3-30 µK, we load the optical trap as
described in Section 2.7. The final temperature in the magnetic trap is chosen to maximize the number of atoms remaining in the optical trap after evaporation to 1 µK.
Figure 4.8 shows the number of atoms remaining in the optical trap after evaporation
to 1 µK as a function of the final temperature in the magnetic trap. The rf evaporation
was stopped at different frequencies in the range 1.8 - 5 MHz, and then the optical trap
evaporation sequence was performed. Loading the optical trap with a higher temperature gas is inefficient since only a fraction of the atoms fit into the finite initial depth of
the trap. Starting the optical trap evaporation with a colder gas is slightly less efficient,since more atoms are consumed by the magnetic trap cooling than are consumed by the
optical trap evaporation. Optimum performance is obtained for a final magnetic trap
temperature of approximately 10 µK.

5

N

87

6

(10 )

4
3
2
1
0
0

10

20

30

MT final cut ( K)

Figure 4.8: Measured 87 Rb number after evaporation in the optical trap as a functionof the final cut temperature in the magnetic trap. The optical trap has a finite depth so
the best loading will occur at lower temperatures. But evaporation in the optical trap
evaporation is likely a little more efficient than the magnetic trap so loading the coldest
atoms does not produce the best result.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

92

Table 4.3: Evaporation parameters used for the tight optical trap. There are fourstages in the tight trap evaporation each with an initial power, final power and ramp
rate. Powers here are listed as the control voltage entered into the control computer.
The calibration is roughly 1.2 W / 1.7 V. Listed in the last column is the calculated
radial trap frequency at the end of the stage.
Stage
1
2
3
4

Pstart (V)
1.7
0.3
0.1
0.03

Pstop (V)
0.3
0.1
0.03
0.02-0.018

rate (V/sec)
5
0.2
0.01
0.004

νradial (Hz)
1900
780
451
∼ 200

νaxial (Hz)
18
8
4
∼2

To evaporate in the optical trap we linearly decrease the power as a function oftime.4 The only input parameters are the initial power, final power, and the ramp rate.
The power of the optical trap is continuously monitored with an amplified photodiode.
The photodiode signal is used as feedback to an acousto-optic modulator in order tostabilize the trapping power. An analog voltage from the experiment-control computer
supplies the reference voltage for the servo electronics that stabilize the trapping power.
The optical trap evaporation is broken into several stages each with a different ramp
rate. The start and stop powers are chosen to give a change in temperature of approximately a factor of two. The ramp rate is optimized to give the largest number of atoms
remaining after evaporation as long as the target temperature is achieved. In Table 4.3
we have listed the parameters for each stage of evaporation in the tight trap. The entire
evaporation takes nearly 11 seconds. The optical trap evaporation is straightforward to
optimize since there is only one parameter per stage to set, the ramp rate. The ramp
rate is varied over a wide range to find the largest number of atoms after evaporation.
Since we are searching for the largest number at a fixed temperature and a fixed opticalpower, the ramp rate that optimizes the number also corresponds to the highest phasespace density.
We have recorded an evaporation trajectory of 87 Rb in the tight optical trap; thedata is shown in Figure 4.9. The optical trap power is lowered from 1.2 W to 0.035 W
in 11 seconds. The first two stages of the cooling are fast and take only 1 sec while thelast two stages take much longer. The BEC transition temperature is reached with atleast two million atoms, and we are routinely able to produce condensates with a large
atom number. Since the gas cools by adiabatic decompression as well as evaporation,
extracting an efficiency from the data is complicated. The adiabatic decompression
also complicates a comparison of the data and our evaporation model, since our model
does not account for the decompression. Our data indicates that the evaporation in
the optical trap is efficient since the final number of atoms after the formation of acondensate is comparable or slightly better than what we observe in the magnetic trap.
In Figure 4.10 three absorption images of 87 Rb after 20 ms of expansion from theoptical trap are shown. The images were acquired at different temperatures near the4

We have verified that exponential ramps do not improve the performance of 87 Rb evaporation in
the optical trap.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

1.2 W

7

Number10

93

35 mW , T/T

c

= 1

6

10
0.1

1

10

Temperature ( K)

Figure 4.9: 87 Rb evaporation trajectory in the tight optical trap demonstrating efficientevaporation. This data represents the baseline performance in the optical trap and isuseful for comparing the simultaneous evaporation efficiency of 85 Rb and 87 Rb.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

94

BEC transition point. The last image is a good representation of the appearance of a
87 Rb BEC after expansion from the optical trap. The vertical direction of the images
correspond to the radial direction of the gas. In this direction the images represent the
momentum distribution of the trapped gas. Since confinement along the axis of the gas isweak, the images in this direction are to be interpreted as the spatial distribution of theatoms in the trap. Further, given the typical expansion times used in our experiments(3-100 ms) and the weak axial confinement, we do not observe anisotropic expansion of
the Bose condensed gas. However, the appearance of a condensate is readily observed
through the appearance of a bi-modal distribution in both the axial and radial directionsof the absorption images when the temperature falls below Tc .

4.5

Sympathetic cooling of 85 Rb in the magnetic trap

The remainder of this chapter describes our work cooling a gas of 85 Rb atoms into
the quantum degenerate regime. In this section we discuss sympathetic cooling of 85 Rb
in the magnetic trap using a 87 Rb gas as the coolant. In particular we will describe
our successful sympathetic cooling results in which a temperature of 5 µK is achieved.
The cooling is judged to be successful since no 85 Rb atoms are lost from the trap as thetemperature is lowered from an initial value of nearly 1 mK. The achievement of thislow temperature represents a solution to the first roadblock on the path to 85 Rb BEC,
namely the zero in the elastic cross section at 0.4 mK.
However, a temperature of 5 µK is not cold enough to form a BEC. Therefore
we have explored the limitations of sympathetic cooling in the magnetic trap in orderto understand the coldest attainable temperatures in our system. One limitation is thefinite heat capacity of the 87 Rb gas; in Section 4.5.1 we show that sympathetic cooling
becomes inefficient when the number of 85 Rb atoms is nearly equal to the number of
87 Rb atoms. This observation is consistent with the predictions of the evaporation
models previously discussed. Another limitation is 85 Rb inelastic collisions; in Section
4.5.2 we directly demonstrate the effect of inelastic collisions on the cooling efficiency.
Below a temperature of 5 µK in the magnetic trap, sympathetic cooling is difficult since
85 Rb inelastic collisions occur frequently and they lead to significant 85 Rb number loss.
This loss of number limits the phase-space density of the gas to a value less than that
required for BEC. In Section 4.5.3 we investigate the possibility of improved 85 Rb coolingefficiency by lowering the density of the two-species gas in the magnetic-trap. With alower density the frequency of 85 Rb inelastic collisions decreases, therefore 85 Rb number
losses are also decreased and a larger phase-space density is achievable. However, thisphase space density in our magnetic trap experiments doesn’t quite reach the values
required for BEC. Our experiments in which we do achieve BEC by doing the final
cooling in an optical trap will be described in subsequent Section 4.6.
The general concept of sympathetic cooling is rather simple. A hot substancewith small heat capacity is placed in thermal contact with another substance at colder
temperature and larger heat capacity. Sympathetic cooling in ultracold neutral atomgases was first demonstrated by the Wieman group with two different hyperfine statesof 87 Rb in Ref. [106]. Since then the technique has been used with a variety of different
species, in particular Bose-Fermi mixtures (6 Li and 7 Li [29, 107], 6 Li and Na [108], K
and Rb [109, 110, 111]). Sympathetic cooling of a mixture of 85 Rb and 87 Rb was firstreported in Ref. [11], but 85 Rb BEC could not be reached due to severe inelastic number

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

95

21 mW

17 mW

13 mW

87

Rb BEC

Figure 4.10: Standard three absorption images of 87 Rb BEC formation in the tightoptical trap. Since the axial trap frequency is very low, the image represents the spatial distribution in that direction and anisotropic expansion is not observed. The firstsignature of condensation in the optical trap is the appearance of a bi-modal density
distribution.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

96

loss and the negative background scattering length. A significant amount of theoreticalwork has also been reported including predictions for our system in Ref. [13, 112].
There are two critical prerequisites for sympathetic cooling. First we must spinpolarize the gases into the 85 Rb |f = 2, mf = −2i and 87 Rb |f = 1, mf = −1i statessince loss due to spin-exchange collisions is large. As described earlier we spin polarize
the gases quite well before loading into the magnetic trap for evaporation, and hereafter
we ignore this type of loss. Another critical issue in the magnetic trap is the differencein magnetic moments of the two states. At low field the magnetic moment in the 85 Rb|f = 2, mf = −2i state is 2/3µB while the moment in the 87 Rb |f = 1, mf = −1i stateis 1/2µB . In the tight trap initially used for evaporation, the 85 Rb radial (axial) trap
frequency is 2π × 250 Hz (2π × 14 Hz). In a magnetic trap with harmonic confinement
gravity tends to “sag” the central position of the trapped gases by an amount equal tog/ωρ2 . Since the trap frequencies of each species are different, there is an offset in thecenter position of each gas given by the differenceÃ∆y = g

1
2
ωρ,87

−

1
2
ωρ,85

!

.

(4.25)In the magnetic trap this difference is only a few µm. We expect good thermal contact
between the two species until the temperature is lowered to approximately 100 nK. At
this temperature the difference between the central locations of two species is on orderof the trapped size of the two gases.
4.5.1

Effects of 87 Rb heat capacity on sympathetic coolingSympathetic cooling is only efficient as long as the 85 Rb number is smaller than
the 87 Rb number. Once the number of particles is roughly equal the coolant gas does nothave sufficient heat capacity and rapid loss of the coolant occurs. This effect sets eithera maximum number of 85 Rb atoms that can be cooled, a minimum number of 87 Rb
atoms required, or a minimum achievable final temperature of the two gases. In thissection we demonstrate the limitation of sympathetic cooling due to the heat capacityof the 87 Rb gas.
We begin sympathetic cooling by loading 85 Rb and 87 Rb into the magnetic trap.
To evaporate, the same radio-frequency ramps that were optimized for single-species
87 Rb cooling are applied to the mixed species gas. We optimized the rf ramps specificallyfor sympathetic cooling, but we only found a slight improvement in the final number ofatoms. In Fig. 4.11 several evaporation trajectories for sympathetic cooling are shown.
The dashed lines in the figure show the BEC transition temperature for the 85 Rb (black)and 87 Rb (green) gases. Therefore the line represent our goal for evaporative cooling.
This data demonstrates successful sympathetic cooling of the 85 Rb gas to 5 µK, since noatoms are lost while the temperature is decreased by more than two orders of magnitude.
While we could not cool into the quantum-degenerate regime, this data was the first
demonstration of sympathetic cooling in these spin states and represented a significant
milestone in our efforts to reach 85 Rb BEC. Finally we note that the temperatures
reached with sympathetic cooling are sufficient to provide a large elastic cross section
for direct 85 Rb evaporation, lifting the first roadblock to 85 Rb BEC.
The solid lines in Fig. 4.11 are a fit to the data using the analytic model of
sympathetic cooling from Ref. [102] that is described in Section 4.2.2. In each case theCHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

97

8

10

Number

87

Rb

7

10

87

85

Rb cooling

6

10

Rb

85

Rb

5

10
0.1

1

10

100

1000

Temperature ( K)

Figure 4.11: Evaporation trajectories for sympathetic cooling of 85 Rb (black points) via
collisions with 87 Rb (red open triangles). For comparison the data points with filledgreen circles are the single-species 87 Rb evaporation trajectory. Sympathetic coolingis successful since the observed 85 Rb number does not change while the temperaturedrops more than a factor of 100. Below 10 µK the number of each species is roughly
the same and cooling of 87 Rb is very inefficient. The inefficiency is characterized by thelarge fraction of the 87 Rb gas evaporated in order to lower the temperature.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

98

only fitting parameter α is related to the overall efficiency of evaporation. We first fit
the single-species 87 Rb data to extract a value of α = 1.6 which is consistent with theresults discussed previously (Sec. 4.4.1).5 Rapid loss of 87 Rb occurs while cooling 85 Rb
when the number of atoms in each gas is roughly equal. We fit the 87 Rb data to themodel with N85 = 106 and extracted α = 1.3 which indicates less efficient cooling. The
qualitative agreement between the data and the analytic model indicates that no othereffects play a significant role in the rapid loss of 87 Rb number.
4.5.2

Effects of inelastic collisions on 85 Rb evaporation

In this section we consider the effect of inelastic collisions on the efficiency of
To demonstrate the effects of these bad collisions, we have obtained anevaporation trajectory during sympathetic cooling in which the number of 85 Rb atoms
is always significantly less than the number of 87 Rb. Under these conditions (Fig. 4.12)
sympathetic cooling is not limited by the heat capacity of the 87 Rb gas, and therefore
the temperature of the two-species gas can be lowered much further than what had
previously been possible.6 In fact a temperature significantly below 1 µK is attained,
but at these temperatures a rapid loss of 85 Rb atoms from the trap is observed. The
focus of this section is to demonstrate that this behavior is consistent with inelastic
collisions.
The data shown in Fig. 4.12 demonstrate efficient sympathetic cooling, with noloss in the 85 Rb number, until a temperature of approximately 3 µK. Below this temperature rapid 85 Rb number loss occurs. Also shown in Fig. 4.12 is the measured peakphase-space density of the two gases. This quantity is calculated based on the measured
number and temperature of the gas along with a knowledge of the trap frequencies. Our
goal is to increase the peak phase-space density of the 85 Rb gas to 2.612 but the largest
phase-space density achieved here is still a factor of 100 too small. The data covers
a very wide range of temperature. We purposely continued to cool the 87 Rb gas until
BEC was reached, in spite of the fact that only a few thousand 85 Rb atoms remain. We
are able to continue cooling 87 Rb since the number in the two species is never equal.
For the lowest temperatures it is likely the case that the gases were no longer in perfectthermal contact due to gravitational sag. Therefore the two species are not in thermal
equilibrium. For reference, we note that at a temperature of 5 µK the average densityof the 85 Rb gas is 7 × 1011 cm−3 .
In order to understand the the poor 85 Rb evaporation performance, we havecompared the measured evaporation trajectory to the predictions of the simple model
described in Section 4.2.1. Number loss and heating occur during the time required
for evaporative cooling, and as the density increases this loss worsens and eventually
the gas cannot be cooled fast enough. The solid lines in Fig. 4.12 are the results of a
85 Rb cooling.

5

We point out that this agreement is interesting since the two sets of data were taken under very
different conditions in the experiment and separated in time by more than a year. The data in Sec.
4.4.1 has nearly an order of magnitude more atoms yet the efficiency of evaporation is the same.
6
Note that the number of 85 Rb and 87 Rb atoms at the beginning of evaporation is significantly
higher in the data shown in this section. Following our initial demonstration of sympathetic cooling,
we increased the efficiency of the transfer to the science cell allowing a larger initial number for cooling.
We were also able to increase the efficiency of optical pumping from the MOT into the quadrupole
trap. This increase in atom number was carried throughout the cooling to produce a larger final atom
number.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

2

8

10

10

87

Rb

Number

6

10

85

Rb

5

87

0

Rb

10

pk PSD

7

10

10

-2

10

85

Rb

-4

10

-6

4

10

10

-8

3

10

99

0.1

1

10

T ( K)

100

10

0.1

1

10

100

T ( K)

Figure 4.12: Evaporation trajectories of 85 Rb (black filled points) and 87 Rb (red open
triangles). The solid lines are the result of our simple model for evaporation. The lines
are in good qualitative agreement with the data. The efficiency of 87 Rb cooling remains
high throughout the entire evaporation indicating that the heat-capacity limit was notreached. The coldest 87 Rb point is below the BEC transition temperature. Frequentinelastic collisions cause poor cooling efficiency in the 85 Rb gas.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

100

simulation based on our model. The predictions of the model are in good qualitative
agreement with the data. Simulations were performed using the values for inelastic
loss coefficients (K2 and K3 ) shown in Table 4.1. The model roughly predicts thetemperature at which loss of 85 Rb begins and it predicts how many atoms will be lost
during the evaporation. Furthermore the model predictions of the phase-space density
during sympathetic cooling are also in qualitative agreement with our observations. Our
results strongly indicate that 85 Rb condensate formation is highly unlikely under these
conditions. Based on the agreement between the simulation and our data we are able
to conclude that 85 Rb inelastic collisions are the primary limitation of the sympathetic
cooling.
We have performed experiments to directly characterize the 85 Rb inelastic collisions in our system by measuring the 85 Rb population decay as a function of time.
From this data we can extract an estimate for the two-body inelastic rate constants andcompare this with the values derived from previous experiments and used in our simulations (Table 4.1). Lifetimes of the gas were acquired at two different temperaturesnear the end of evaporation, one at a temperature just as the evaporation begins to failand the other well into the poor cooling regime. The evaporation is stopped either at
3.5 µK or 1 µK, and the number of atoms and temperature of the gas is monitored asa function of time. We calculate the average density of the gas using the trapping potential parameters. The results for 1 µK and 3.5 µK are shown in Fig. 4.13 in a log-log
plot as a function of time. In each plot the data were fit to a exponential function (redline) with two time constants. The second time constant was fixed to the previously
measured background lifetime in the trap of 300 s. At 1 µK the first time constant was
only 2.2 s, a factor of five shorter than at 3.5 µK. The data in Fig. 4.13 have also been
fit to extract the values of the inelastic rate constants K2 and K3 ; the rates that we
extracted are given in the figure. The atom loss in the magnetic trap is primarily due
to two-body collisions with a measured rate constant of K2 ≈ 5 × 10−13 cm3 /s. Thereis reasonable agreement between this measured rate and previous work [6]. Our data
does not allow us to determine the rate of three-body collisions. From the measured
temperature as a function of time we also extract the heating rate. It was 70 nK/s forboth data sets. We do not have a detailed understanding of this heating rate; in thefollowing section we will discuss experimentally exploring its origin.
For future reference we present an evaporation trajectory (Fig. 4.14) during
sympathetic cooling in the magnetic trap that demonstrates the best evaporation results
obtained to date with our apparatus. We are able to cool several million 85 Rb atoms
without loss to a temperature of several microKelvin. At this temperature nearly 10
million 87 Rb atoms remain in the trap. These conditions are the starting point for
further cooling in an optical dipole trap in which 85 Rb condensates are formed. As
discussed above, the heat-capacity limit is never reached in this data indicating thateven more atoms could be cooled. We have measured the heating rate of the two-species
gas after stopping the evaporation at a temperature of 9 µK. We observe a single-species
87 Rb heating rate of 80 nK/s. When 85 Rb is also present in the trap the rate soars to 400
nK/s. The two-species heating rates are significantly larger than the 70 nK/s reported
above and the likely cause is the increase in overall 85 Rb number.

-3average density (cm )CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

101

Tinitial = 3.5 µK

(a)
1E11

τ = 11(2) sec-13

K2 = 3 x 10
1E10

3

cm /s

1

10

100

-3

average density (cm )hold time (sec)
Tinitial = 1 µK

(b)
1E11

τ = 2.3(3) sec
-13

1E10

K2 = 9 x 10
1

3

cm /s
10

100

hold time (sec)Figure 4.13: Measured average density as a function of time after the completion of
evaporation to a final temperature of (a) 3.5 µK and (b) 1.0 µK. The filled circles
are our data points and the lines are fits to the data. The red solid line is a double
exponential function with a fast decay time of 11 sec in (a) and a fast decay time of 2.3
sec in (b). In both (a) and (b) the second time constant was fixed to 300 sec according
to the measured background lifetime. The dashed line (black) and dotted line (blue)
are fits to two- and three-body inelastic collision rates, respectively. We extract a valueof K2 ≈ 5 × 10−13 cm3 /s from the fit. The three-body fit does not agree with our data.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

Number

108

87

Rb

87

Rb cooling

7

10

85

Rb

85

6

10

105

102

Rb

1

10

100

1000

Temperature ( K)

Figure 4.14: Our most efficient evaporation trajectory during sympathetic cooling in
the magnetic trap. The filled black circles are 85 Rb, the open red triangles are 87 Rb
while cooling 85 Rb. The 87 Rb single species trajectory (green) is shown for comparison.
Below approximately 5 µK 85 Rb atoms are rapidly lost due to inelastic collisions as
indicated by the results of our simulation. The experimental conditions of number andtemperature shown here represent an excellent starting point for cooling in an opticaltrap.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC
4.5.3

103

Toward 85 Rb BEC in the magnetic trap

In the previous section we established inelastic collisions as the limitation tocooling 85 Rb in the magnetic trap. In this section we will focus on work we did tolower the inelastic collision rate in our sympathetic cooling experiments. Specifically wehave explored evaporative cooling in a magnetic trap with a significantly lower radialtrapping frequency. To avoid confusion, hereafter we refer to the magnetic trap with
reduced radial frequency as the “weak” trap. The magnetic trap with higher frequency,that has been used in all previous experiments, is now called the “tight” trap. The
density of the 85 Rb gas is reduced in the weak trap; therefore the rate of inelastic
collisions is also reduced. Under these conditions a low heating rate in the trapped
gases is critical, so we added a second radio-frequency source to remove energetic 85 Rb
atoms generated during inelastic collisions. We first discuss our experimental setup tocontrol the magnetic trap heating rate, and then we present an improved evaporation
trajectory in the weak magnetic trap. With a lower 85 Rb density, we are able to improvethe 85 Rb phase-space density at the end of evaporation.
4.5.3.1

RF shielding to lower the magnetic trap heating rate

We characterized the heating rate in the tight magnetic trap by applying an rfshield to a 87 Rb gas. A radio frequency is applied just above the temperature of thegas in much the same way as evaporation. Heating can either occur internally in thegas or as a result of elastic collisions with a low density, high temperature secondarygas sometimes called an “Oort cloud” [59, 113]. The source of this secondary gas is
typically from incomplete evaporation, in which a small fraction of the atoms removed
during the evaporation do not leave the trap. An Oort cloud may also be created fromthe remnants of inelastic collisions. The Oort cloud is only in weak thermal contact
with the much colder gas at the center of the trap. Infrequently an energetic atom fromthe Oort cloud collides with the cold gas depositing a lot of energy. In the tight trap wetest the shield by holding the 87 Rb gas immediately after evaporation stops. During thisholding time the gas heats up at a known rate. We characterize the effect of different
shield frequencies by measuring the temperature after the hold; the data are shown in
Figure 4.15. As long as the shield frequency is larger than 3.5 MHz, no heating occurs
during the hold and no number loss is observed. The fact that similar shielding of thegas is observed for a wide range of frequencies is a clear indication that the heating
rate is mostly due to the external Oort cloud. This is because the temperature of theOort cloud is much larger than the temperature of the gas at the center of the trap.
Using the rf shield we were able to reduce the 87 Rb heating rate in the tight trap toapproximately 30 nK/s from 80 nK/s and the heating rate of the two-species gas toapproximately 100 nK/s from 400 nK/s.
By applying an rf-shield to the atoms in the weak trap, the heating rate of the gas
is even smaller. We cool a two-species gas in the tight magnetic trap to a temperatureof 8 µK with nearly 107 87 Rb atoms and 2 × 106 85 Rb atoms. We reduce the density
in the magnetic trap by slowly lowering the current in the bias coil over a few seconds
until the radial trap frequency for 87 Rb (85 Rb) is 2 π × 45 Hz (2 π × 53 Hz) while theaxial frequencies remain unchanged. The bias field at the center of the trap is set to31.6 G. The trap frequencies are chosen to give a low density, but still maintain thermal
overlap between the two species. We verified that the phase space density of the gas

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

104

10

T ( K)

expected T

9initial T

8

7
2

3

4

5

6

7

8

rf shield freq. (MHz)Figure 4.15: Measured temperature following a 10 sec hold in the tight trap as a functionof rf-shield frequency. The red line indicates the expected temperature of the gas due tothe known heating rate. The black line is the initial temperature just after evaporation
stops. When the rf-shield is too low a fraction of the gas is evaporated. No number loss
or heating occurs for a shield frequency of more than 3.5 MHz.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

105

energy = m B

does not change during the trap decompression, meaning that the ramp is adiabatic.
Using the rf shield we were able to reduce the 87 Rb heating rate in the weak trap toapproximately 3 nK/s from 35 nK/s.
Since the spin states of 85 Rb and 87 Rb that we use have different magnetic moments, in the weak magnetic trap it is possible to place an rf shield in-between the two
species. Figure 4.16 shows a diagram of the trapping potential for the two species andthe location of the two rf shields. The rf transition between the 85 Rb |2, −2i → |2, −1i
is sufficiently different from the 87 Rb |1, −1i → |1, 0i transition that each gas can beaccessed separately. The field rf1 is used to evaporate the 87 Rb gas, and it is used asa shield after the cooling. We place the field rf2 just above the 85 Rb gas but far below
the bottom of the 87 Rb trap. The only use of this shield is to protect the 87 Rb gas
against energetic atoms generated inside the 85 Rb cloud. Without the second shield,
85 Rb atoms are trapped with energy up to 0.4 mK before possible removal by the 87 Rb
rf knife. If the shield were not present then the 87 Rb cloud absorbs a lot of energy
from this Oort-like 85 Rb gas. Using both the rf1 and rf2 shields the heating rate of thetwo-species gas is reduced to approximately 15 nK/s from the original 100 nK/s. This
heating rate is a factor of 27 better than the rate observed in the tight trap, see Figure
4.14.

87

Rb
rf1 = 25 MHZ

85

Rb

rf2 = 18 MHZ

xFigure 4.16: Schematic of the two rf shields used to control the heating rate in themagnetic trap. The solid red and blue curves represent the trapping potential for 87 Rb
and 85 Rb and the fill is the surface of the trapped cloud. The 87 Rb gas is evaporated
by rf1 and rf2 is the tunable shield for 85 Rb.

4.5.3.2

Evaporation in the weak magnetic trap

Here we describe the results of sympathetic cooling of the 85 Rb gas in the weakmagnetic trap. In particular we identified two critical parameters of our system thatprimarily determine the efficiency of the cooling. The first critical parameter of oursystem is the relative number of atoms of the two species that are loaded into theCHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

106

magnetic trap for evaporation. The second critical parameter is the rf shield (rf2 ) thatis placed above the the 85 Rb gas but below the 87 Rb gas. This shield is particularly
effective at protecting the 87 Rb gas from energetic 85 Rb atoms generated during inelastic
collisions. We optimized these two parameters in order to achieve the best evaporation
results.
Evaporation of 87 Rb in the weak trap is very similar to all the discussion in theprevious sections of this chapter. We modified the radio-frequency sweeps to account forthe different magnetic field at the center of the trap (see Table 4.4). Since the collision
rate in the gas is lower the optimum duration of the sweeps is approximately five times
higher than in the tight trap. We verified that the efficiency of cooling 87 Rb is still
good. For every factor of two reduction in number the temperature drops by a factorof 3.4. The BEC transition temperature is reached with 3 × 106 atoms.
Table 4.4: Evaporation parameters optimized for 87 Rb in the weak magnetic trap. The
parameters are optimized by searching for the largest 87 Rb phase space density. The
power referenced here is what is programmed into the synthesizer and does not include
the gain of the amplifier or loss through various rf components (see Section 4.3).
Stage
1
2

fstart (MHz)26.5
25

fstop (MHz)25
24.73-24.8

τstage (sec)20
15

fbase (MHz)
24.7
24.7

rf power (dBm)-27-27

The results of sympathetic cooling in the weak magnetic trap are shown in Fig.
4.17. The figure is a compilation of three different data sets that demonstrate the two
critical parameters of our system. The first set of data at the highest temperatures(filled circles and triangles) is an evaporation trajectory immediately following the decompression into the weak trap. The 85 Rb gas is cooled below 1 µK, but inelastic
collisions are constantly removing atoms from the trap. We do not use the rf2 shield yet
so the heating rate is still large. Therefore much of the 87 Rb gas is evaporated. By 500
nK nearly all the 87 Rb number has been evaporated in order to reach that temperature,and cooling cannot continue.
Since we are constantly losing 85 Rb atoms to inelastic collisions there is no benefit
to starting the evaporation (i.e the initial cooling in the tight trap) with a large numberof 85 Rb atoms. To maximize the number of 85 Rb atoms at the end of evaporation wevary the number of that species loaded into the MOT. We have directly measured thefinal number of each species after evaporation (Fig. 4.18) to 600 nK as a function of
the initial 85 Rb number. In this experiment the initial number of 87 Rb atoms was fixed.
Interestingly the final 85 Rb number is unchanged as the 85 Rb MOT number varies by
a factor of two. In contrast the 87 Rb number dramatically increases if there are fewer
85 Rb atoms to cool in the magnetic trap. There is a significant advantage to starting theevaporation with fewer 85 Rb atoms. The limitation to the trajectory with filled points
in Fig. 4.17 is the large fraction of the 87 Rb cloud that is evaporated to reach only 500
nK. By starting the cooling with fewer 85 Rb atoms a lower temperature is reached. The
trajectory with open data points in the Fig. 4.17 has an optimized initial 85 Rb MOT
load allowing that gas to be cooled significantly closer to BEC without evaporating away
the entire 87 Rb cloud.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

7

Number

10

107

BEC transition point

1. Evaporation in the weak trap
87
Rb
85
Rb
---------------------------------85
2. Optimized Rb MOT number
87
Rb
85
Rb
---------------------------------3. Best Magnetic Trap Evaporation:
85
Optimized Rb MOT number and rf2 shield

6

10

5

10

87
85

Rb
Rb

4

10

0.1

1

Temperature (µK)
Figure 4.17: A compilation of three evaporation trajectories in the weak magnetic trap.
(1) The filled data points are a continuation of the cooling from Figure 4.14 after
decompression to the weak trap. (2) To reach even lower temperature (open data
points) we optimized the initial number of 85 Rb atoms at the start of cooling. (3) The
larger solid points represent our best results in the magnetic trap.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

108

Based on the data in Fig. 4.18, we developed a physical picture to understand theresults of evaporative cooling in our system. In order to create a 85 Rb BEC the peakphase-space density of the gas much reach 2.612. Therefore our goal is to evaporate
the largest possible number of 85 Rb atoms to the corresponding temperature for BEC.
Since we can cool 87 Rb with high efficiency, the final temperature achievable is only
set by the heat load of 85 Rb. Here the heat load is a combination of the atom number
plus any heating due to 85 Rb inelastic collisions. Therefore a colder temperature can beobtained by starting the evaporation with slightly less 85 Rb. Of course this assumes goodthermal contact so that the two gases are in thermal equilibrium. The final number of
85 Rb atoms will depend on many factors including trap strength, the 87 Rb cooling rate,
and 85 Rb inelastic rate constants but the key point is that sufficiently low temperature
for condensation may be obtained.
In order to further improve the evaporation we used the rf2 field to shield the87 Rb gas from energetic 85 Rb atoms generated during inelastic collisions. This is thesecond critical parameter of our system. In conjunction with an optimized 85 Rb MOT
number, the rf2 shield improves the phase-space density after evaporation. The resultswith the rf shield are shown in Fig. 4.17 as the filled square and diamond points. The
large diamond-shaped points with 30,000 85 Rb atoms near 100 nK indicate the highest
peak phase-space density (0.1) of 85 Rb achieved in the magnetic trap. This phase spacedensity is a factor of 500 improvement over our best tight magnetic trap result. The
transition temperature for 87 Rb is reached, but at this temperature we believe thermal
contact between the two species is no longer adequate to establish thermal equilibrium
due to gravitational sag. At the lowest rf-cuts we perform we still observe 87 Rb cooling
but the 85 Rb temperature has stopped changing. The temperature at which coolingstops is roughly consistent with the known sag.
To be certain that only 85 Rb inelastic collisions are responsible for the poor evaporation, we have measured the lifetime of the 85 Rb gas with and without a 87 Rb gas
present at a temperature of 0.9 µK. The results of this experiment are shown on a log-log
plot in Figure 4.19. The initial number of atoms was slightly different since some 85 Rb
loss occurred during the removal of 87 Rb. From the plot it is clear that the decay rateof the two data sets is the same, meaning that 85 Rb inelastic collisions are primarily thesource of the observed atom loss from the trap.
In conclusion, we have sympathetically cooled a gas of 30,000 85 Rb atoms toa peak phase-space density of 0.1 in our weak magnetic trap. The limitation of thesympathetic cooling is poor thermal contact with the 87 Rb cloud. Further improvements
to the sympathetic cooling might be possible if thermal contact of the two species could
be achieved. One possibility might be to use an optical dipole laser beam in conjunction
with the weak magnetic trap in order reduce the relative gravitational sag of the two
clouds. If such an arrangement could be implemented, in might be possible to tune themagnetic field at the center of the trap to the 85 Rb Feshbach resonance at 155 G. There
the 85 Rb inelastic collision rate can be tuned in order to optimize the conditions for
sympathetic cooling further.

4.6

Simultaneous evaporation of 85 Rb and 87 Rb

This section describes the simultaneous evaporation of 85 Rb and 87 Rb in an optical dipole trap to create a 85 Rb BEC. The evaporation of both species is performed

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

109

6

Number (10 )

1.6

87

Rb

1.2

0.8

0.4

0.0
0.0

85

Rb

0.1

fraction

0.2

0.3

85

Rb MOT fill

Figure 4.18: Dependence of the 85 Rb and 87 Rb atom number near the end of evaporation
at 600 nK in the weak trap as a function of the number loaded into the 85 Rb MOT. The
MOT number is changed by varying duration of the MOT fill time. Due to inelastic
collisions, the final 85 Rb number is not very sensitive to the 85 Rb MOT number. In
contrast the 87 Rb number is very sensitive to the 85 Rb MOT.

8
7
5

5

N (10 )

6

4
3
with 87
no 87

2

1

10

hold time (sec)Figure 4.19: A comparison of the 85 Rb population decay as a function of time with andwithout 87 Rb present. Since the decay rate is roughly the same for the two cases weconclude that interspecies inelastic loss is negligible.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

110

simultaneously since the depth of the trapping potential for the two species is nearly
the same and it is not independently tunable. The work described in the later chaptersof this thesis was performed with two different optical trapping setups: the “tight” trapwith a 23 micron waist and the “weak” trap with a 46 micron waist. We note that theevaporation efficiency of 85 Rb is very different in the tight and weak traps. The differences will be described in the later sections of this chapter. To put these sections into
context, we point out that the heteronuclear molecule work (Chapter 6) was performed
with the tight trap in which cold enough gases could be obtained to make Feshbach
molecules with 85 Rb and 87 Rb. The dual BEC work (Chapter 5) and the Bragg spectroscopy work (Chapter 7) were performed in the weak trap where evaporative coolingis more efficient and 85 Rb condensates can be produced.
After understanding the difficulties inherent in producing a 85 Rb condensate in
the magnetic trap via sympathetic cooling, we switched to evaporation in an opticaltrap which has several advantages. The optical trap holds each species in the same
location without significant relative gravitational sag of the two species. The depth ofthe optical trap during evaporation is held only slightly above the maximum energy ofthe gas. Therefore any hot atoms produced via density-dependent loss are naturally
lost and the heating rate in the gas is significantly reduced. Another advantage isthat simultaneous cooling in the optical trap is straightforward since the depth of thetrap is nearly the same for both species. Finally, the biggest advantage is that any
arbitrary magnetic field may be applied to the atoms while in the optical trap withoutchanging the trapping potential significantly. This allows us to tune near the 85 Rb
Feshbach resonance where we can suppress inelastic collisions and we can control theelastic collision cross section.
Our goal is to overcome the limits of evaporation that were encountered in themagnetic trap. To evaporate we simply lower the depth of the optical trapping potential
on a timescale set by the rethermalization rate of the two-species gas. Since both
species are cooled simultaneously the heat capacity of 87 Rb is no longer a limitation.
As the depth is lowered by reducing the optical power, the trap frequencies are also
reduced. The optical trap evaporation simultaneously lowers the temperature of thegas by removing the most energetic atoms and by decompression of the trap. A simple
model was presented in Ref. [114] which includes both of these effects and agrees
reasonably well with experiments. From our experience cooling in the magnetic trap,
we can qualitatively understand the results in the optical trap without comparison toan evaporation model.
During our initial work in the tight optical trap (Section 4.6.1), our cooling routine resembles the routine that is used in the magnetic trap. Apart from the trapping
potential there are only two fundamental changes. First, since both species are simultaneously evaporated, we begin the cooling with roughly equal atom numbers. Second,the 85 Rb Feshbach resonance is used to minimize inelastic collisions as much as possible.
With that setting the 85 Rb elastic collision cross section is small. Therefore during thecooling, the 85 Rb gas primarily rethermalizes via collisions with the 87 Rb gas. Thisis feasible since the cross section for interspecies elastic collisions is large.7 In fact,assuming that the number of particles in each gas is the same and the trapping potential
is identical for both species, the timescale for rethermalization of the two-species gas is
7

The interspecies scattering length is approximately 213 a0 . The rate of inelastic 85 Rb -87 Rb
collisions is expected to be small and we have verified this fact experimentally (see below).

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

111

limited by the rate of 87 Rb collisions. During the simultaneous evaporation each trapped
85 Rb atom should behave very much like a 87 Rb atom since we purposely suppressed
85 Rb collisions. Therefore the cooling routine during the simultaneous evaporation in
the tight optical trap is optimized for the 87 Rb gas.
Unfortunately the 85 Rb evaporation in the tight optical trap is not efficient enoughto produce BEC. But from our experience with the tight trap, we were able a gain
insights into the best procedures for cooling 85 Rb. In particular, we learned that 85 Rb
inelastic collisions still preclude the formation of a 85 Rb BEC. We have applied thisknowledge to cooling in a weaker optical trap (Section 4.6.2) where the density of thegases can be lower. In this weak trap we are able to cool the 85 Rb gas with a highenough efficiency to create a condensate for the first time. Soon after making our first
85 Rb BEC, we discovered that the shape and atom number of the condensed 85 Rb gas
depends sensitively on the co-existence of a 87 Rb condensate in the optical trap. Thisis because the two quantum fluids are immiscible (they spatially separate) under the
typical conditions of our experiments; see Chapter 5 for more details. Fluctuationsin the shape of the 85 Rb condensate cause a significant experimental uncertainty indensity of the gas. In order to perform meaningful experiments with a single-species
85 Rb condensate, we modified the cooling routine in the weak optical trap to suppress
the formation of a 87 Rb BEC. We now use an improved evaporation routine that not
only produces a 85 Rb BEC, but the new routine also suppresses the 87 Rb atom number
by making the evaporation inefficient for that species.
4.6.1

Evaporation in the tight optical trap

Simultaneous cooling of 85 Rb and 87 Rb in the tight optical trap is performed inmuch the same way as the single-species 87 Rb optical trap evaporation described in
Section 4.4.2. After evaporative cooling in the tight magnetic trap to approximately
9 µK the atoms are loaded into the optical trap following the procedure outlined in
Section 2.7. The optical trap power is decreased in several stages using the ramps
listed in Table 4.3. After the optical trap is loaded we ramp the bias-coil current in
100 ms to provide a magnetic-field near 168.8 G, where a ≈ −100 a0 . The opticaltrap evaporation is initiated during this ramp of the magnetic field. The efficiency of
the 85 Rb evaporation depends strongly on magnetic field as expected. To optimize theevaporation, the total number of 85 Rb atoms after cooling is measured as a function of
the magnetic field; the results are shown in Figure 4.20. For this data we cooled until
a 87 Rb condensate with T /Tc = 0.75 was formed. The temperature of the 85 Rb gas
was 110 nK and T /Tc ∼ 2.5. The best results are obtained near 168.8 G where the85 Rb scattering length is a ≈ −90 a , suggesting that the minimum of inelastic losses
0
does not coincide with the exact point that a = 0.8 Anywhere near the peak of the
Feshbach resonance and above 185 G the entire 85 Rb gas is lost before 110 nK is reached.
Unfortunately we were not able to form a 85 Rb condensate in this trap even though we
could access the Feshbach resonance.
We characterized the optical trap cooling by recording an evaporation trajectory;
8

In fact we have repeatedly determined that simultaneous cooling with 87 Rb works best near 168.8
G. Later in this section we present evidence that the elastic cross section is minimized at 165.7 G as
expected. This effect has been previously observed in the measurements of 85 Rb inelastic loss presented
in Ref. [6].

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

112

85

4

Rb Number (10 )

5
4
3
2
1
0
160

170

180

190

200

B-field (gauss)

Figure 4.20: Number of 85 Rb atoms after evaporation to 110 nK as a function of magnetic field. The peak of the 85 Rb Feshbach resonance is 155 G where inelastic losses are
expected to be severe. Near 166 G both elastic and inelastic collisions are suppressed
and we expect the best evaporation performance.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

113

the data are shown in Figure 4.21 (a). The calculated phase space density of each gas isshown in Fig. 4.21 (b). Following the same color coding used previously the black points
are 85 Rb, the red open triangles are 87 Rb while cooling 85 Rb, and the green points are
single-species 87 Rb cooling shown for comparison. There are several interesting aspects
to the data. Using the single-species 87 Rb trajectory as a guide, at the beginning of
evaporation we observe good cooling efficiency for both 85 Rb and 87 Rb, and it continues
well below 1 µK. Near 300 nK a rapid decline in the 85 Rb number begins and the phasespace density does not exceed a value of 0.1. As the cooling begins to fail, the averagedensity is approximately 1012 cm−3 and the poor evaporation performance is consistent
with our observations in the magnetic trap where inelastic loss was the limitation. The
measured lifetime of the 85 Rb gas at 300 nK is less than the time required to cool, aclear indication that the efficiency will be poor. At this temperature we systematicallyvaried both the evaporation rate and the magnetic field in search of better performance,
but we were not able to find a more favorable combination. Since the gas is still far
from the BEC transition temperature, condensate formation and subsequent collapse
due to the negative scattering length is not the problem.
Since the optical trap evaporation starts at such a low temperature, in principle
we have the ability to directly cool 85 Rb without any 87 Rb present. As a last-ditch
effort to produce a 85 Rb BEC in the tight optical trap we attempted single-species 85 Rb
evaporation. This experiment is also a nice check on our understanding of number loss
during evaporation, as we can test that interspecies inelastic collisions are unimportant.
The cooling was optimized by varying both the ramp rates and the magnetic fieldindependently. The first step was to locate a favorable range of magnetic field to performthe cooling. Since there is no 87 Rb gas to rely on, the 85 Rb scattering length was
set to provide a large elastic cross section. In Figure 4.22 the measured number andtemperature after running the evaporation to a fixed cut are shown as a function ofmagnetic field. For each point we used the same cooling ramps so that the data roughly
indicate the difference in collision properties. Near 166 G the temperature of the gas ishigh since a low elastic cross section inhibits cooling. Another consequence of the higher
temperature at 166 G is that fewer atoms fit in the trap. There is a small range on eitherside of 166 G where the evaporation works reasonably well, but closer to the peak of the
Feshbach resonance and above 170 G the cooling is less efficient. This data is a coarse
measurement of the elastic collision rate. The points with largest temperature give an
estimate of 165.7 G for the magnetic field where a = 0, which is in good agreement withthe precise determination of 165.75 G found in Ref. [5].
After optimization of the entire single-species evaporation trajectory, we foundthat our largest peak phase space density was a factor of two smaller than that achieved
using simultaneous cooling of 85 Rb and 87 Rb. Even though the cooling is not quite
as efficient, we were encouraged that a further reduction in the optical trap strength
might finally provide the conditions to make BEC. We also confirmed that interspecies
inelastic collisions do not hinder the efficiency of evaporation.
4.6.2

Creation of 85 Rb BEC in the weak optical trap

In this section we describe evaporative cooling in the weak optical trap, in which
we are able to produce either a single species 85 Rb or 87 Rb condensate or a dual BEC
of both species. To control what species remain in the optical trap after evaporation,

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

(a)

114

87

Rb only

6

Number

10

5

10

85

Rb

87

Rb cooling
85
Rb

4

10
0.01

0.1

1

10

peak phase space density

Temperature (µK)
100

(b)

87

Rb cooling
Rb

10

85

1
0.1

87

Rb only

85

Rb

0.01
1E-3
0.01

0.1

1

10

Temperature (µK)

Figure 4.21: (a) Evaporation trajectory in the tight optical trap demonstrating our best
results. (b) Peak phase space density at the center of the trap calculated from thenumber and temperature during evaporation. The 85 Rb cooling saturates at a phasespace density of 0.1, which is less than the requirement for BEC. Note that the circled
points are for condensates and the phase space density is not known very accurately.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

0.7

5

0.6
1.0
0.5
0.5
0.4

165

170

175

Temperature (µK)

Number (10 )

1.5

0.0
160

115

0.3

magnetic field (G)Figure 4.22: The number (black, left side) and temperature (blue, right side) as afunction of magnetic field during single-species 85 Rb evaporative cooling. The elastic
cross section is minimized near 165.7 G and evaporation is inefficient.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

116

we vary the relative number of each species in the MOT. The evaporation in the weak
optical trap is quite successful. Condensates with up to 80,000 85 Rb atoms are routinely
created under the best conditions. This represents a significant improvement over theprevious record of 17,000 [6, 58]. The production of 85 Rb BEC is robust; for example we
can tolerate significant loss of number in either species’ MOT and still make condensates.
The production of condensates is also stable, with atom number reproducibility of 10%.
We note that this level of stability is inferior to the previous Wieman group 85 Rb
apparatus [6, 58].
In order to achieve this 85 Rb evaporation performance, two critical changes havebeen made to our system. First, we lowered the confinement of the trapped gas byincreasing the waist of the optical trapping beam by a factor of two. This was done tomaintain a low density during evaporation. In this new optical trap a 85 Rb condensate
can be produced by simultaneously cooling the two-species gas. During the coolingthe magnetic field was tuned to 168.8 G where 85 Rb collisions are minimized and theoptical trap power ramps were optimized for the best 87 Rb evaporation. The presenceof a 87 Rb condensate in the optical trap distorts the 85 Rb condensate so much that the
85 Rb condensate is not usable for some of our experiments (i.e. Bragg spectroscopy).
We can minimize the number of 87 Rb atoms at the end of evaporation by startingwith the appropriate relative numbers of 85 Rb and 87 Rb in the MOT. However, as
we have already discussed the 87 Rb number after the cooling is extremely sensitive tothe relative number in the MOT. Although the shot-to-shot stability of the relative
MOT number in our system is less than 1%. This is not stable enough to guarantee
a 87 Rb condensate does not form. Therefore, the second critical change to our system
was a new evaporation routine that suppresses 87 Rb atom number near the end of thecooling. In the new routine the 87 Rb cooling efficiency at the end of evaporation is low
and severe 87 Rb number loss occurs. This has the effect of dramatically reducing 85 Rb
condensate fluctuations. Only the combination of these two critical changes has allowed
us to robustly produce 85 Rb condensates with larger atom number.
4.6.2.1

A weak optical trap

We have experimentally verified the performance of the weak optical trap setup
by performing single species 87 Rb evaporative cooling. Evaporation of 87 Rb in the weak
trap is very similar to what is described in the previous section, except the optical powerramps were optimized to slightly different values. We can produce single-species 87 Rb
condensates with two million atoms, but since the trap is so weak the duration of thecooling is close to 30 seconds. Evaporation can still be efficient despite the long timerequired since the heating rate in the weak trap is very low. The measured heating
rate in our trap is 1 nK/sec. Inelastic number losses are also minimized due to the low
density. The average density of a gas with 105 atoms at the transition temperature is1012 cm−3 . For comparison, in the tight magnetic trap with a radial (axial) frequencyof 215 Hz the density is approximately five times larger.
We have performed simultaneous cooling of 85 Rb and 87 Rb in the weak opticaltrap using optical trap power ramps that were optimized for single-species 87 Rb cooling;
the ramps are documented in Section 4.4.2. By taking advantage of the low densities
offered in the weak optical trap and by tuning the magnetic field to 168.8 G where 85 Rb
inelastic collisions are minimized, we were able to observe our first 85 Rb condensates. In

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

117

6

10

5

N

10

85

Rb

4

10

87

Rb

3

10

10

100

Temperature (nK)

Figure 4.23: Our first evaporation trajectory that makes a single-species 85 Rb BEC.
The key to efficient cooling was very weak confinement to reduce losses. Near the endof evaporation the 87 Rb number fluctuates wildly shot-to-shot causing large instability
in the 85 Rb condensate shape and density. This makes quantitative measurements suchas Bragg spectroscopy with the 85 Rb BEC impossible.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

118

our experiments, just prior to the formation of a condensate we ramp the magnetic field
to 164.4 G so that the scattering length was 60 a0 . With a positive scattering length,
interactions in the condensate are repulsive and collapse does not occur. In order toacquire an absorption image of the condensate we expand from the optical trap. The
magnetic field remains on for the first 10 ms of the expansion to keep the condensate
from collapsing due to the negative scattering length when the magnetic-field is switchedoff.
For many experiments we desire a BEC of only 85 Rb atoms. The presence ofeven a small 87 Rb condensate distorts the shape and atom number in the 85 Rb gas so
dramatically that measurements cannot be performed. We have studied the evaporationin the weak optical trap (Fig. 4.23) to understand the optimal conditions with which to
make pure 85 Rb condensates. Until the the vary end of evaporation when the magnetic
field is ramped to 164.4 G, rethermalization of the 85 Rb gas occurs primarily due tocollisions with 87 Rb. Therefore the evaporation routine used to obtain the trajectory
shown in Fig. 4.23 is optimized to obtain the best results for 87 Rb. The trajectory in the
figure demonstrates the creation of a pure 85 Rb condensate. To create this quantum gas
we optimized the relative number of atoms in the MOT so that the 87 Rb gas was mostly
evaporated away. In order to reliably create a 85 Rb condensate in this way the MOT
number must be stable to better than 0.5% on every run of the experiment. When ourapparatus is well optimized the stability reaches that level, but obtaining a significant
amount of data under this stability requirement is challenging. The observed stabilityof the MOT number is typically only slightly better than 1%.
MOT number stability of 0.5% is required because number of 87 Rb atoms at theend of the cooling is very sensitive to the relative number of 85 Rb and 87 Rb in the MOT.
The sensitivity results from that fact that the 87 Rb gas acts as a coolant throughout
the evaporation; more 87 Rb atoms are consumed during evaporation when the 85 Rb
atom number in the MOT is increased. Figure 4.24 shows the steep dependence of
the 87 Rb number after cooling on the 85 Rb MOT number. To inhibit the formationof a 87 Rb condensate, we choose the 85 Rb MOT number to suppress the final 87 Rb
number. However, the 85 Rb MOT number cannot be too large or else a low enoughfinal temperature is not reached. In order to produce single-species 85 Rb condensatesusing the routine described in this section, shot-to-shot fluctuations in the MOT number
must be smaller than 0.5%. In the next section we describe a new evaporation routinethat relaxes the MOT number stability requirements to reliably make 85 Rb condensates.
4.6.2.2

Achieving reproducible evaporation performance

We have developed a new evaporation routine that suppresses the formation of a
87 Rb BEC. The 85 Rb Feshbach resonance is used to make the optimal rates of the two-

species evaporation significantly different. By evaporating at a rate that is unfavorable
for the 87 Rb gas, we suppress the final number of that species. First we present aphysical picture of the cooling procedure. Later on in this section the experimental
details of the cooling routine are documented and our most efficient 85 Rb evaporation
results are presented.
The timings of the evaporation optical power ramps described in the previous
section have been modified in order to make 87 Rb cooling inefficient so that a large
fraction of that gas is lost. We simultaneously cool a gas of 85 Rb and 87 Rb with roughlyCHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

119

3

Rb

6

Number (10 )

87

2

1
85

Rb

0
0.0

0.2

0.4

0.6

85

fraction of Rb MOT filled
Figure 4.24: Dependence of the 85 Rb and 87 Rb atom number at 900 nK in the weak
optical trap as a function of the number loaded into the 85 Rb MOT. The MOT number
is changed by varying duration of the MOT fill time. These data are very similar towhat we observe in the magnetic trap (for example see Figure 4.18).

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

120

Table 4.5: Evaporation parameters used for the weak optical trap. There are six stages
in the weak trap evaporation each with an initial power, final power and ramp rate.
Powers here are listed as the control voltage entered into the control computer, thecalibration is roughly 1.2 W / 1.7 V. The parameter tend is the end time of each stage
measured from the start of evaporation. Listed in the last two columns are the calculated
trap frequencies at the end of each stage.
Stage
1
2
3
4
5
6

Pstart (V)
1.7
0.7
0.35
0.2
0.15
0.11

Pstop (V)
0.7
0.35
0.2
0.15
0.11
0.100 - 0.107

rate (V/sec)
1
0.1
0.06
0.014285
0.025
0.014285

tend (sec)
1
4.5
7
10.5
12.1
12.52

νradial (Hz)
1900
780
451
∼ 200
∼ 150
∼ 100

νaxial (Hz)
3
3
3
∼ 2.9
∼ 2.9
∼ 2.9

equal number to 250 nK. At this temperature, the magnetic field is ramped from 168.8
G to 163.05 G which changes the scattering length from approximately -85 a0 to 150
a0 . At the lower magnetic field the 85 Rb elastic collision cross section is approximately
a factor of three larger. Therefore the timescale for rethermalization of the 85 Rb gas isshorter. Slightly after the field ramp, we speed up the evaporation rate in the opticaltrap to a value that is optimized with respect to the efficiency of 85 Rb cooling. Since
the 87 Rb elastic cross section has the same value at 169 G and 163 G, that gas does
not rethermalize fast enough and a significant loss of atoms from the trap occurs. Thispreferential loss of 87 Rb atoms suppresses the formation of a 87 Rb condensate, and thefinal number of 85 Rb atoms is larger than the final of 87 Rb atoms by a factor of 5-10.
In fact, using our new routine the final 85 Rb number can be as large as 80,000.
The suppression of final 87 Rb number moderates our sensitivity to the relative
number of 85 Rb and 87 Rb atoms in the MOT. This fact represents an important featureof our new routine. With this new evaporation routine the required stability of the
MOT number is 4%, which is much better than the previous requirement of 0.5%. The
4% requirement is well within the capabilities of our system.
Table 4.5 lists the parameters of the weak optical trap power ramps that are usedin our new cooling routine. The evaporation is broken into a series of consecutive stages;
each stage lowers the temperature of the gas by approximately a factor of two. In eachstage the power decreases linearly as a function of time. Unlike all the other sets of
optical trap power ramps described in this chapter, this one is not optimized for thebest 87 Rb performance. A plot of the ramps is shown in Figure 4.5 (a) and (b) shows
the time derivative of the ramps, which is related to the cooling rate of the gas. The
increase in cooling rate occurs at 10.5 seconds when the temperature of the gas in the
optical trap is 250 nK. The first three stages of evaporation are similar to what wasused in the tight optical trap except the cooling rate is slower. The magnetic-field is
set to 168.8 G so that the 85 Rb inelastic losses are minimized. Before the end of the
fourth stage, we ramp the scattering length to 150 a0 in 500 ms by changing the field
to 163.05 G. The fifth (sixth) stage of evaporation is then performed a factor of five

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

121

1

(a)
ramp rate (V/s)

optical trap cut (V)(three) faster than optimum for 87 Rb. The parameters of the evaporation ramps listed
in Table 4.5 are optimized to produce the largest number of 85 Rb atoms at the end of
the cooling. To optimize the evaporation ramp rates, we varied the rates and recorded
the number of 85 Rb atoms after the cooling. The total time of the two stages is about
1.5 seconds. The measured lifetime of a 85 Rb condensate is about 3 seconds, which isa little bit longer than the formation time. After the cooling ends, we linearly increase
the depth of the optical trap from the final cut to 1 µK in a fixed time of 50 ms.
The value of the magnetic-field during the evaporation has also been optimized.
Of course the optimum value of this field depends on the temperature of the gas. During
the first few weak optical trap stages, the field must be set to 168.8 G to minimize
inelastic collisions. We have systematically optimized the evaporation ramps during
the first few stages of the cooling for different values of the 85 Rb scattering length;
the data are shown in Fig. 4.26. As we had demonstrated previously (Fig. 4.20),
the best performance occurs when the scattering length is set near -85 a0 . During thefinal few stages of evaporation, the magnetic field is tuned to 163.05 G to produce
85 Rb condensates with the largest atom number. We studied the range of magnetic
fields suitable for the final evaporation stages and we determined that the field must bewithin ±0.5 G of the optimum value.

1

0.1
0

2

4

6

8

time (sec)

10

12

(b)

0.1

0.01
0

2

4

6

8

10

12

time (sec)Figure 4.25: (a) Log plot of the optical trap power as a function of time during evaporative cooling to make 85 Rb BEC. (b) The cooling rate during evaporation which is related
to the time derivative of (a). The increase in cooling rate at the end of evaporation isused to suppress the final 87 Rb atom number in the trap. When the 87 Rb number issuppressed a pure 85 Rb condensate can be repeatedly created over consecutive runs of
the apparatus.
We have developed an understanding of how the final 87 Rb atom number depends
on the relative number of 85 Rb and 87 Rb atoms in the MOT. In particular, a procedure
has been developed to lower the final 87 Rb number sensitivity to the relative MOT
number. For each run of the apparatus the loading time of the 87 Rb MOT is the same.
The optimum value of the relative atom number in the two species MOT is determinedby varying the 85 Rb MOT loading time and observing the number of atoms in eachspecies after the cooling. Our goal is to produce a 85 Rb BEC and a small non-condensed

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

122

gas of 87 Rb. The minimum value of the 85 Rb MOT loading time is set by the initial
85 Rb number required to suppress the final 87 Rb atom number. In contrast, the 85 Rb
MOT number cannot be too large or else a low enough final temperature will not bereached. By studying the evaporation performance at the very end of the cooling, wefound a dependence between the lowest optical trap power of the evaporation (the finalcut) and the final 87 Rb sensitivity to the relative MOT number. Our measurements
showed that the 87 Rb final number suppression can be enhanced (i.e less 87 Rb atoms)
by the combination of an increased 85 Rb MOT number and a lowered final cut. We
note that the number of atoms in the 85 Rb condensate tends to be smaller.

45

Number (10 )

5

3
2
1
0

-100

0

100

200

300

scattering length (a0)Figure 4.26: The optimized number of 85 Rb atoms cooled in the optical trap as afunction of the scattering length. Each point represents a separate optimization of theevaporation ramps for a different value of a. The gas was cooled to a temperature of
200 nK. While performance may not appear to vary significantly the temperature must
be lowered by another factor of five so that a small advantage here is more important
at lower temperature.

Figure 4.27 shows an evaporation trajectory typical of our current experimental
setup. To obtain these data the evaporation was performed using the optical trap powerramps of Table 4.5. For these data the initial number ratio in the two-species MOT was
set to reproducibly create a 85 Rb condensate with 60,000 atoms and a non-condensed
87 Rb gas with approximately 10,000 atoms. In the weak optical trap we actually begin
with a slightly larger number of 85 Rb than 87 Rb; this helps suppress the 87 Rb number
after cooling. At the start of evaporation the efficiency for both species is roughly thesame and equal numbers from each gas are evaporated. If this trend were to continue,
we would end up with a dual BEC. But as the temperature of the gas reaches 250
nK, which is equivalent to T /Tc ≈ 1.5 for 85 Rb, the cooling rate is increased and thesteep loss of 87 Rb begins. Since the 87 Rb number is low at the end of the cooling,small fluctuations are much less likely to cause random 87 Rb condensate formation.
The results here with a suppressed final 87 Rb number should be compared against theCHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

123

results shown in Fig. 4.23 where the 87 Rb fluctuated wildly at the end of the cooling.
By varying the relative number in the two-species MOT, we have measured that thestability of the MOT number must be better than 4%, i.e fluctuations larger than 4%
cause random shot-to-shot condensate formation.
We lack a detailed theoretical model to completely understand the cooling process
at the end of the evaporation routine. It is likely that the different gravitational sagof the two species plays a role in the final stages of evaporation. Since 87 Rb is slightly
heavier it tends to spend more time near the evaporation surface at the lower lip of
the optical trap potential in the direction parallel to gravity. There may also be a nonnegligible mean-field repulsion between the two species which leads to a preferential
loss of 87 Rb from the optical trap. Given the scattering lengths in our system a twospecies superfluid is immiscible and tends to spatially separate (see Chapter 5). Spatial
separation of the two species might explain some suppression of the final 87 Rb number
but, we have not been able to systematically study this possibility since reproducibility
in the experiment is poor.
It may be possible to further improve the production of 85 Rb condensates via
evaporation in our system. Possible improvements include an increase in the 85 Rb final
number and an enhanced suppression of the 87 Rb number. Condensates of 87 Rb withtwo million atoms can be made with our system, but to date we have been limited
to 85 Rb condensate with at most 80,000 atoms due to inelastic collisions. Performing
the optical trap evaporation in a still weaker trap might increase the final number ofatoms in the 85 Rb condensate.9 It may be possible to further suppress the final 87 Rb
number by utilizing either resonant light or resonant radio-frequencies. Since the atoms
are primarily confined in the axial direction of the weak optical trap by the curvatureof the ∼ 160 G magnetic field, only magnetically trapable spin states are confined. By
driving rf transitions 87 Rb atoms may be removed from the trap. One complication ofthis technique is that the atoms exit the trap in only the axial direction. Therefore they
can collide with trapped atoms and heat up the trapped gas.10

4.7

Chapter conclusion

In this chapter we described the evaporative cooling routine that is used to produce a 85 Rb BEC. Our system is also capable of robustly producing a 87 Rb condensate
as well as a dual BEC of the two species. Evaporation begins in a magnetic trap where
sympathetic cooling of 85 Rb is performed using the 87 Rb gas as a coolant. We havethoroughly investigated the limitations of 85 Rb sympathetic cooling in the magnetic
trap. The limitations include the finite heat capacity of the 85 Rb gas and inelastic collisions between 85 Rb atoms. In the magnetic trap these 85 Rb inelastic collisions haveso far limited the maximum obtainable phase space density to a value of 0.1, a factorof 25 smaller than that required for BEC. In the magnetic trap it might be possible
9
We tried an optical trap with an 83 µm waist, but single species evaporation of 87 Rb was notefficient since only a fraction of the atoms from the magnetic trap could be transferred into this weak
optical trap.
10
We have observed a slight decrease in the final 87 Rb number by using a fixed frequency resonantwith atoms near the bottom of the trap. The time during evaporation at which the rf drive is applied is
critical; the 87 Rb atoms cannot be removed prematurely or else a low enough final temperature is notreached. The optimum time was determined to be at the end of stage 5. It is likely that sweeping thefrequency from the edge of the gas inward toward the center may provide better results.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

124

Number

106
87

105

104
10

Rb

85

Rb

100

1000

Temperature (nK)

Figure 4.27: Evaporation trajectory in the weak optical trap to create a pure BEC of
85 Rb with a large number of atoms. In these data the temperature is always determinedfrom the 87 Rb gas since it does not form a condensate. The final 85 Rb number is smallenough that if a small fluctuation in the number ratio at the start of cooling occurs it
is unlikely to lead to 87 Rb BEC.

CHAPTER 4. EVAPORATIVE COOLING OF 85 Rb TO BEC

125

to improve the 85 Rb sympathetic cooling efficiency if the field at the center of the trap
can be tuned to the 85 Rb Feshbach resonance. But some method must be found tocompensate for the relative gravitational sag of the two gases.
In order to create a 85 Rb condensate in our system, we pre-cool a two-species
gas in the magnetic trap. Those atoms are then loaded in an optical trap for further
cooling. In the optical trap an arbitrary magnetic field may be applied in order to control
85 Rb elastic and inelastic collisions. We are able to produce 85 Rb condensate with up
to 80,000 atoms and this represents a factor of five increase in performance over theprevious best results [6, 58]. This work opens the door to a variety of new experiments.
In particular, we now have access to a condensate in which the interactions may betuned over a very wide range. By tuning the magnetic-field at the center of the opticaltrap toward the peak of the Feshbach resonance the strongly interacting regime may bereached.

Chapter 5
Tunable spatial separation in a two-species BEC

5.1

Introduction

Two-component quantum fluids have long been known to exhibit rich physics as aconsequence of access to new parameter regimes [115]. The miscibility of two quantum
fluids represents a fundamental property of a two-component system and is partiallycontrolled by interparticle interactions. Building on the achievement of dilute-gas BoseEinstein condensation [23], mixtures of ultracold quantum gases provide a unique system
for the study of interacting quantum fluids. Moreover resonant control of two-bodyinteractions in ultracold gases is possible via magnetic-field Feshbach resonances [56]and provides a parameter which directly alters the miscibility of the dual-species BEC.
The first two-component condensate was produced with different hyperfine states
87
of Rb, and evidence for repulsive interspecies interactions was reported [106]. Further
work with different hyperfine states of 87 Rb studied the dynamics of interpenetrating
quantum fluids [116]. In spinor condensates of Na both miscible and immiscible twocomponent condensates were observed for the first time [117]. More recently a dualspecies BEC of 41 K and 87 Rb was reported in Ref. [118] where scissors-like collective
oscillations were observed as a result of off-axis collisions. Two-component quantum
gases have long been the subject of significant theoretical interest; topics of study include
the density pattern and phase separation of the components [119, 120, 121, 122, 123,124] and the spectrum of collective excitations [125, 126]. Spatial separation of two
immiscible quantum fluids in a trap is typified by a ball-and-shell structure [119] inwhich one fluid forms a low density shell around the other and it depends on the relative
strength of interspecies and single-species interactions. Other structures are expected
as well and a general classification of all possible spatial structures has been presented
in Ref. [127].
A mixture of 85 Rb and 87 Rb has long been considered a promising system forstudying the miscibility of quantum fluids and a route to dual-species BEC with 85 Rb–87 Rb was proposed in Ref. [13]. Applying the analysis of Refs. [128, 129], in which
the Thomas-Fermi limit is utilized, to the case of a 85 Rb –87 Rb mixture, we define2∆ = u85 u87 − u285−87 where ui = 4 πmh̄i ai and mi is the mass. In the case where a85−87
is always positive two regimes emerge: one with ∆ > 0 in which the two condensates
can overlap in space and one with ∆ < 0 in which overlap is restricted due to thestrong interspecies repulsion. Since the masses of 85 Rb and 87 Rb are nearly identical
the parameter ∆ can be re-scaled to depend only on the three scattering lengths in theCHAPTER 5. TWO-SPECIES BEC
system∆0 ≡∆a85 a87= 2− 1;
u285−87a85−87127(5.1)

the quantity ∆0 compares the strength of single-species and interspecies interactions.
Finally we note that gravitational sag is predicted to cause nontrivial modifications to
the expected ball-and-shell structures [128].
In this chapter I focus on experiments with a dual-species BEC of 85 Rb and87 Rb near the 85 Rb Feshbach resonance, where the 85 Rb interaction can be tuned.
Immiscibility of the two-component quantum gas was observed as a dramatic departurein the gas density distribution from that expected due to the observed symmetry of thepotential. The shape and size of the gases depended on the influence of gravitationalsag and the external trapping potential. Surprisingly we observed the robust formationof multiple, non-overlapped, single-species “cloudlets”. Furthermore we found that thedegree of spatial separation between the two species can be controlled with a tunable
85 Rb scattering length. This information will be important for further studies of thisrich dual-species system with widely tunable single-species and interspecies interactions.

5.2

Spatial separation measurements
The work in this chapter was performed after our success in creating 85 Rb and87 Rb dual-species BEC as described in Ch. 4. By precisely controlling the evaporation

rate and the number ratio of the two species at the start of evaporation, we are able tocreate in our optical trap a single species 87 Rb BEC with 3 × 106 atoms, a 85 Rb BEC
with up to 8 × 104 atoms, or a BEC of both species. Unless otherwise noted, for thiswork, after evaporation the depth of the optical trap is raised in 50 ms to approximately
1 µK with a measured 85 Rb radial (axial) trap frequency of 2π × 130 Hz (2π × 2.9 Hz).
While the 87 Rb radial frequency is nearly the same, the axial frequency that is 2π × 2.6
Hz is smaller due to a different magnetic moment. In the entirety of this work theinterspecies scattering length is +213(7) a0 [103] and the 87 Rb scattering length is +99
a0 [130].
One way in which we can demonstrate the different dual-species density patterns
that are observed is by varying the number of particles in each Rb species. Figure
6.3 shows absorption images of both 85 Rb and 87 Rb condensates immediately after
evaporation without recompression of the optical trap. The 85 Rb scattering length was
set to 81 a0 with ∆0 = −0.82. Prior to imaging, the gas is released from the optical trap
and expands for 20 ms; the magnetic field remains on for the first 10 ms of expansion
to prevent collapse of the 85 Rb gas [8, 9]. In each image a different condensate isshown since the measurement process is destructive. The elongated axial direction of
the images is interpreted as the spatial density of the gas since the trapping frequency
is small; the radial direction is interpreted as the momentum density.
When a condensate of each species is present in the trap we observe“holes” in87
the Rb image that correspond to “spikes” at approximately the same location in the
85 Rb image (Fig. 6.3 (a)-(d)). This behavior is consistent with spatial separation of
two immiscible quantum fluids since the density distributions no longer display thesymmetry of the trapping potential. However the images are not consistent with atypical ball-and-shell phase-separation structure. Surprisingly, the 85 Rb gas is observedto split into multiple separated cloudlets that appear as distinct holes in the 87 Rb gas. In

CHAPTER 5. TWO-SPECIES BEC

85

Rb

128

87

Rb

(a)
y z
(b)

(c)

(d)

(e)

(f)

Figure 5.1: Absorption images of 85 Rb (left column) and 87 Rb (right column) condensates demonstrating the immiscibility of the quantum gases. For clarity the intensityof each 87 Rb image was scaled by a factor of 0.5. The size of each image is 330 µm
× 150 µm. The number of 85 Rb particles varies from approximately 12,000 in (a) to
22,000 in (e); the 87 Rb number varies from 140,000 in (a) to 3,000 in (e). Since the trap
frequency in the elongated axis is low this direction of the image is interpreted as thespatial density of the gas. For comparison (f) shows single-species condensates.

CHAPTER 5. TWO-SPECIES BEC

129

Fig. 6.3 (a) and (b) two 85 Rb cloudlets are observed, and three cloudlets are observedin image (c). If the initial number of 85 Rb atoms at the start of evaporation is toolarge, the 87 Rb gas cannot be cooled into the quantum degenerate regime; in this case(Fig. 6.3 (e)) the 85 Rb condensate shape is unperturbed and well approximated by aThomas-Fermi model. For comparison absorption images of single species 85 Rb and87 Rb condensates are shown in Fig. 6.3 (f).

(a)

(b)

d+d

d-d

85

(c)

87

Figure 5.2: (a) A schematic drawing of the optical trap laser beam and the magneticfield coils. The relative position of the minimum of the trapping potential for eachspecies can be adjusted by changing the position of the optical trap focus with respect
to the center of the bias field coils. Here δ indicates the relative difference between thefocus of the optical trap beam and the center of the bias field coils. Absorption imagesof immiscible 85 Rb (b) and 87 Rb (c) condensates demonstrating a symmetric density
pattern in the radial direction after recompression of the optical trap.
The immiscible gases shown in Fig. 6.3 are not symmetric with respect to either
the radial or axial directions of the trap. These asymmetries are caused by gravitationalsag of the trap and can be corrected by recompression of the optical trap to increase thedepth or compensated for by careful alignment of the optical trap focus with respect tothe magnetic-field center. In Fig. 5.2 (a) and (b) absorption images of immiscible gases
were acquired after recompression of the trap; in the images symmetry in radial direction
was restored but spatial separation is still clearly observed along the axial dimension.
Similarly we have verified that offsets in the center-of-mass axial position of each species
can be controlled via alignment of the optical trap. Although the precise shape, size,and position of the immiscible gases is affected by details of the trapping potential,
the observation of spatial separation and the formation of separated 85 Rb cloudlets isrepeatable under a variety of experimental conditions. By monitoring the total numberof atoms in each gas as a function of time at a 85 Rb scattering length of 144 a0 , wefind that the lifetime of the dual species BEC is approximately 1 sec. This decay is
predominantly due to loss of 85 Rb atoms, the likely cause of which is inelastic densitydependent collisions [12]. During the decay, the dual-species BEC remains spatiallyseparated and individual cloudlets of 85 Rb do not coalesce even after many radial andCHAPTER 5. TWO-SPECIES BEC

130

a few axial trap periods.
We lack a detailed understanding of 85 Rb cloudlet formation in our experiments.
The geometry of our system is complicated by the elongated axial direction of the trap,with a trap aspect ratio near 50, and the presence of gravitational sag. In an analysisof the favorability for cloudlet formation, the energy cost of forming and maintaining
extra boundary surface area must be weighed against the cost of all particles residing inthe same cloud [122]. These questions appear to require the solution of coupled GrossPitaevskii equations for the dual-species BEC wave-functions which is beyond the scopeof this thesis [121].

5.3

Tuning the spatial separation with 85 Rb interactions

The ability to tune interactions between 85 Rb atoms in the gas represents a significant feature of our system. We have investigated the degree of spatial separation
between the two species as a function of the 85 Rb scattering length. After preparationof a dual species BEC, the magnetic field was swept from 163.1 G to a value in the rangeof 164.6 G to 158.6 G in 400 ms; the sweep covers a range in ∆0 of approximately -1 to+1. The rate of this sweep was measured to be slow with respect to the response of thegases in the radial and axial directions of the trap. Absorption images of each gas wereacquired following release from the trap as before, the results of which are shown for∆0 = −0.89 [Fig. 5.3 (a) and (b)] and ∆0 = 0.72 [Fig. 5.3 (c) and (d)]. The measured
two-dimensional optical density (OD) was summed in the remaining radial direction to
produce the cross sections shown in Fig. 5.3. As expected, spatial separation of the twogases is observed primarily when the parameter ∆0 is negative. In addition to the coexistence of each species throughout the axial direction, separated cloudlets with sharp
density spikes give way to a smooth overall density. Spatial separation was observed
to return when an initial magnetic-field sweep to ∆0 > 0 was immediately followed bya second field sweep in the opposite direction. The penetration distance in the axial
direction over which the two gases overlap is estimated to be approximately 30 µm when
∆ = −0.89. Note that this penetration was measured for an expanded cloud. All of themean-field energy is released during expansion and this effect must be accounted for in
a comparison to this measured penetration depth.
To characterize the degree of spatial separation as the parameter ∆0 is varied wehave extracted the axial center-of-mass location of each species from images similar to
those shown in Fig. 5.3. In this experiment an offset between the axial trap centers foreach species was intentionally introduced to favor the spatial separation in the center-ofmass location. The offset causes the two immiscible quantum gases to separate primarily
in the axial direction with each species preferring a well-defined side of the trap. The
results of this experiment are plotted in Fig. 5.4 as a function of ∆0 . The axial positionof each species in the trap gradually approach eachother as ∆0 is increased from -0.9
toward zero. Once ∆ > 0 is attained, the center-of-mass position of each gas depends
only weakly on the 85 Rb scattering length. This is consistent with the expectation thatthe two species can coexist throughout the axial direction of the trap. In this case, themeasured difference in the position of each gas is in good agreement with that expected
for single-species condensates. With our current imaging system, we are not able toquantify the spatial overlap of the two gases throughout the entire trapping volume,and residual gravitational sag likely induces incomplete spatial overlap.

CHAPTER 5. TWO-SPECIES BEC

131

(a)

(b)

85

87

(c)

(d)

85

87

4

summed OD (10 )

4

51 a0
3

(b)

(a)

2

1

0

-200

-100

0

100

200

position (mm)780 a04

summed OD (10 )

4

(d)

3

2

(c)

1

0

-200

-100

0

100

200

position (mm)Figure 5.3: Absorption images (a-d) and corresponding axial cross sections at two different 85 Rb scattering lengths with significantly different spatial separation. The numberof 85 Rb (87 Rb) atoms is approximately 40,000 (90,000).

axial gas position (µm)

CHAPTER 5. TWO-SPECIES BEC

132

50
87

Rb

25

0
85

Rb

-25

-50
-1.0

-0.5

0.0

0.5

1.0

∆’
Figure 5.4: Measured axial position of the 85 Rb and 87 Rb gases as a function of theparameter ∆0 . Here we have varied ∆0 by tuning the 85 Rb scattering length from 50 a0
to 900 a0 . The uncertainty in the position reflects the shot-to-shot reproducibility of
our measurements.

CHAPTER 5. TWO-SPECIES BEC
5.4

133

Conclusion

In summary, we have observed immiscibile behavior in a dual-species BEC of
Surprisingly, immiscible condensates areobserved to form multiple spatially-separated cloudlets, the exact cause of which remains
unknown.1 By varying the 85 Rb scattering length, the degree of spatial separation
was characterized. In the future studies of the collective-excitation spectrum of a twocomponent BEC may be possible [126]. Numerous Feshbach resonances are available inthe 85 Rb –87 Rb system making it possible to study a two-component condensate witha tunable interspecies interaction (see Ch. 6). In addition, the dynamics of singlespecies condensate collapse may be modified by the presence of a second condensate
component.
85 Rb and 87 Rb with tunable interactions.

1

Shai Ronen and John Bohn have performed numerical simuilations of our system using a pair ofcoupled GP equations. They find qualitative agreement between our data and the simulations.

Chapter 6
Creation of heteronuclear Feshbach molecules

6.1

Introduction

In this chapter I present our work on ultracold heteronuclear Feshbach molecules.
Starting with a 87 Rb BEC and a cold atomic gas of 85 Rb, we utilized previously unobserved interspecies Feshbach resonances to create up to 25,000 molecules. Even though
in these experiments the 85 Rb gas was non degenerate, we observed a large molecular conversion efficiency due to the presence of a quantum degenerate 87 Rb gas; thisrepresents a key feature of our two-species system. Using signatures of the molecule
creation, we precisely measured the positions of two Feshbach resonances in our two
species system to be 265.44 ± 0.15 G and 372.4 ± 1.3 G. We compared the molecule creation at these two different Feshbach resonances with different magnetic-field widths.
We also directly measured the small binding energy of the molecules through resonant
magnetic-field association.
The creation of ultracold molecules from ultracold atoms is currently a topic
of great experimental and theoretical interest [10, 131, 132, 56]. Ultracold heteronuclear molecules in low-lying vibrational states are particularly interesting since they are
predicted to exhibit a permanent dipole moment due to the unequal distribution ofelectrons. Numerous proposals for utilizing polar molecules exist, including quantum
computation [133] and the search for the electron electric dipole moment [134]. Although no significant permanent dipole moment is expected to exist in a 85 Rb–87 Rb
molecule, our work demonstrates a first step toward the efficient production of groundstate ultracold heteronuclear molecules.
Furthermore we note that the data presented in this chapter may be useful forunderstanding possible non-adiabatic corrections to the Born-Oppenheimer potentialthat describes atom-atom interactions in our system [135]. In the limit that beyond
Born-Oppenheimer corrections can be neglected, the two-body Hamiltonian which de2
scribes the interaction of two atoms can be written as H = 2 pMr + VBO (r) where Mr is
the reduced mass and VBO (r) is the interaction potential of Rb. This interaction potential does not depend on the mass of the Rb atoms in question, i.e. the same potential
describes 85 Rb–85 Rb , 87 Rb–87 Rb , and 85 Rb–87 Rb interactions. In the theoretical analysis of Refs. [136, 103, 135] it was shown that beyond Born-Oppenheimer effects did
not need to be included to accurately describe available experimental data on the Rb
potential. In this chapter we present a precise experimental determination of several
new interspecies Feshbach resonances. It would be interesting to understand whether
our experimental data can also be described without including corrections to the BornOppenheimer potentials. We note that beyond Born-Oppheimer effects have previously

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

135

been observed through precise spectroscopic studies of H2 , HD, and D2 spectral lines[137].
To date, cold heteronuclear molecules in high-lying vibrational levels have beencreated using photoassociation [138, 139, 140]. These molecules can then be pumped
toward low-lying vibrational levels by exciting bound-bound molecular transitions [132]via, for example, stimulated Raman-type transitions [133] that enhance the probabilityof populating the lowest vibrational level. The initial photoassociation step used in thisprocess is inefficient, and many final vibrational levels of the molecule are occupied. An
alternative to this initial photoassociation step is the direct conversion of two free atoms
into a molecule in the highest vibrational levels using a Feshbach resonance [10, 131,
41, 43, 141, 142, 143, 144]. High conversion efficiency using a Feshbach resonance has
been demonstrated in single-species gases via adiabatic magnetic-field sweeps across theresonance [40, 56], three-body recombination [142], resonant magnetic-field association[144], and non adiabatic magnetic-field sweeps [10, 145]. Feshbach resonances between
two different atomic species [146, 147] have previously been reported. Our work has built
upon these observations by demonstrating stable heteronuclear Feshbach molecules.

6.2

Experimental setup

The previous chapters of this thesis have described in detail the experimental
apparatus used to create an ultracold mixture of 85 Rb and 87 Rb. Here we will briefly
describe a few details of the experimental setup relevant to the results in this chapter.
The optical trap for these measurements is formed at the focus of a Yb:YAG laser beamwith a 1/ e2 radius of 23 µm. A Helmholtz pair of coils provides a magnetic field up to 700
G; the magnetic field is calibrated using rf-driven Zeeman transitions with a systematic
uncertainty of 0.01%. We perform the optical trap evaporation slightly above the 85 Rb
Feshbach resonance at 155 G where density-dependent 85 Rb– 85 Rb loss is minimized[12]. We typically produce a 87 Rb BEC with 300,000 atoms and 50% condensate fraction
and a non degenerate (T/Tc = 2.4) gas of 85 Rb with 40,000 atoms. Measurements are
performed in an optical trap with a radial trap frequency of ωr = 2 π × 200 Hz and an
aspect ratio of approximately 70. Both the optical trapping potential and the curvatureof the magnetic field contribute to the confinement of the atoms in the axial direction.
Confinement in the radial direction of the trap is primarily provided by the optical trap.
For the measurements presented in this chapter, we create a BEC of 87 Rb but
only a thermal gas of 85 Rb. There are two reasons for this choice. Away from the
Feshbach resonance at 155 G, a 85 Rb condensate is not stable against collapse due tothe attractive atom-atom interactions. If we attempt to set the magnetic field above
165.7 G or below 155 G, we measured that the 85 Rb condensate would collapse in lessthan 0.1 ms [8]. The other reason for working with a 85 Rb thermal gas is that phaseseparation can occur in a dual BEC system, this was demonstrated in the previous
chapter.

6.3

Discovery of heteronuclear Feshbach resonances

Our work on heteronuclear molecules follows from our discovery of two heteronuclear s-wave Feshbach resonances in the 85 Rb |f = 2, mf = −2i state and 87 Rb|f = 1, mf = −1i state, one near 265 G and the other near 372 G. These resonances

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

136

were predicted theoretically in Ref. [13] and were first located in our lab by detecting
the enhanced inelastic collision rate near the resonance position. For both of the resonances the binding energy of the bound molecular state increases with magnetic field
above the position of each Feshbach resonance. Figure 6.1 shows data used to locate thes-wave Feshbach resonance at 265 G. In this measurement we also discovered a p-wave(l = 1) resonance (left feature). Because the 85 Rb loss only occurs in the presence of
87 Rb, we can clearly see that the features are due to heteronuclear collisions.
We have looked more closely at the structure of the p-wave resonance (Fig. 6.2).
The doublet feature is due to the magnetic dipole-dipole interaction between the atoms[148]. This effect results in two distinct peaks characterized by their partial-wave projection onto the magnetic field axis, ml = 0 or |ml | = 1. Such a resonance could provide
a way to tune anisotropic interactions in a two-species BEC.

6.4

Creating Feshbach molecules with magnetic-field sweeps

To create heteronuclear molecules we adiabatically (450 µs/G) sweep [131] themagnetic field from low to high field through one of our s-wave Feshbach resonances.
Figure 6.3b shows an absorption-image cross section of the 85 Rb gas following a sweep
through the resonance. Prior to imaging, the gas is released from the optical trap andexpands for 3 ms; at the same time as the optical trap release is initiated, the magnetic
field is switched off within 50 µs. A significant fraction (∼ 60%) of the atoms disappear
as compared to an absorption-image cross section without sweeping the magnetic field(Fig. 6.3a). When we reverse the molecule creation process by applying a second fieldsweep in the opposite direction less than 100 µs after the first sweep ends, (Fig. 6.3c), a
large fraction of the atoms reappear [131]. The atoms that reappear represent reversible
heteronuclear molecule formation. The small fraction of atoms that do not return arelost to 85 Rb– 87 Rb inelastic collisions that remove atoms from the trap. We verified
this by sweeping across the resonance at the same rate in the opposite direction. At atypical 87 Rb density of 0.5×1014 cm−3 we observe approximately 15% 85 Rb number loss
during a sweep. By monitoring the reappearance of atoms as a function of time before
the second sweep, we find that the molecules decay with a lifetime of approximately 1
ms. This decay is due to a combination of inelastic collisions with the 87 Rb gas and a
one-body spontaneous process [149, 150].
The dependence of molecule conversion efficiency on sweep rate was measured inRef. [40] and is well characterized by a Landau-Zener model. For our two-species gas
we observe a factor of 5.3 ± 1.9 decrease in the ramp rate required to create molecules
at the 372 G Feshbach resonance as compared to the 265 G resonance (see Fig. 6.4)
for a similar number and temperature. This factor is consistent with the ratio of thepredicted widths of the two resonances, which is in the range of 6–8 [13, 151, 152].
This directly verifies the predicted inverse relationship between the Feshbach resonancewidth and the sweep rate required to create molecules.

6.5

Determination of the Feshbach resonance locations

We precisely located the Feshbach resonances with an experimental technique thatavoids the need for rapid magnetic-field sweeps. At the lower (higher) field resonance, we
sweep the magnetic field from 269 G (397 G) downward toward the resonance stopping

137

5

10

4

5x10

85

Rb Number

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

87

85

Rb + Rb
Rb only

85

4

10
250

260

270

280

290

magnetic field (gauss)
Figure 6.1: Investigation of Feshbach resonances in the 87 Rb and 85 Rb system via
the enhanced inelastic number loss at the resonances. The asymmetric nature of theleft peak is characteristic of higher partial-wave Feshbach resonances. The right peakcorresponds to a heteronuclear s-wave resonance. The temperature of the two species
gas was 1000 nK for this data.

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

138

Number (arb)

1

0.1

0.01
255

256

257

258

259

260

magnetic field (G)Figure 6.2: Doublet structure of a heteronuclear p-wave Feshbach resonance observed
via inelastic losses at the Feshbach resonance. The temperature of the two species gas
was 100 nK for this data.

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

139

summed OD

2.5

a

2.0
1.5

c

1.0

b

0.5
0.0
0.00

0.25

0.50

0.75

1.00

1.25

position (mm)Figure 6.3: Absorption-image axial cross sections of the 85 Rb gas demonstrating reversible molecule creation. The measured two-dimensional optical density (OD) was
summed in the remaining radial direction of the absorption image. (a) Prior to sweeping the magnetic field the atom number is 26,500. (b) After sweeping through theresonance 53% of the gas is converted to molecules. (c) By reversing the molecule creation process approximately 85% of the initial atom number is observed to reappear.
Note that most of the 85 Rb loss occurs in the center of the gas where the 87 Rb BEC
density is largest.

1406
τ = 310(70) µs / G

4

2

85

4

Rb Number (10 )CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULESτ = 58(16) µs / G

0

0

1

2

3

inverse ramp rate (ms / G)
Figure 6.4: Molecule conversion efficiency as a function of sweep rate. The black points
show the rate required for conversion at the 265 G resonance, while the red points show
the required rate at the wider 372 G resonance. Both datasets were taken starting withsimilar atom number and temperature. This data has not been corrected for the effects
of atom loss not due to molecule formation, therefore the number of molecules createdin these data is less than the total atom loss.

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

141

at various final values. The rate of this magnetic field sweep is fast compared to boththe timescale for molecule creation and atom loss due to inelastic collisions. The fieldis held at the final value for 0.3 ms and then returns to 269 G (397 G) at a rate of 450µs/G (70 µs/G) which is slow compared to the molecule creation rate. The magnetic
field remains here for 5 ms to ensure that any molecules made during the second sweep
decay and are lost from the optical trap [149, 150]. We then simultaneously turn off
the magnetic field and the optical trap to let the gas expand for 6 ms and measure thenumber of 85 Rb atoms remaining. In Fig. 6.5 we show the atom number remaining asa function of the final magnetic field for the two Feshbach resonances. The key to thismethod is that if the field passes through the Feshbach resonance on the first sweep,
then a fraction of the atoms will be converted into molecules by the second sweep. The
rapid onset of atom loss due to molecule creation when the magnetic field is swept below
265.44 G (372.4 G) represents crossing the peak of the Feshbach resonance.

6.6

Heteronuclear molecule creation efficiency

6.6.1

Experiment

We have investigated the adiabatic conversion efficiency from atoms to molecules.
We begin by sweeping the magnetic field upward through the 265 G Feshbach resonanceat a rate that is slow with respect to the molecule conversion rate. It is then held at
269 G for 5 ms to allow the molecules to decay and be lost from the trap. The numberof molecules formed is simply the difference in 85 Rb number before and after the sweep
with a small correction applied to account for measured inelastic atom loss during thefield sweeps. The conversion efficiency to heteronuclear molecules is shown in Fig. 6.6 asa function of 87 Rb T/Tc . We varied T/Tc by either changing the number of 87 Rb atoms
or the temperature. The 85 Rb gas had a T/Tc in the range of 2.2 to 4. The numberof 85 Rb atoms was always less than or equal to the number of 87 Rb; the 85 Rb number
ranged from 28,000 to 55,000 and the 87 Rb number ranged from 30,000 to 400,000. We
observe up to 60% conversion of the 85 Rb gas into molecules even when the 85 Rb gas isfar from quantum degeneracy. The largest conversion efficiency is observed when 85 Rb
is least quantum degenerate, indicating that the conversion efficiency primarily depends
on 87 Rb T/Tc .
6.6.2

TheoryIn a single-species gas the molecule conversion efficiency for an adiabatic fieldsweep was shown to depend only on the degree of quantum degeneracy [40, 153]. A
molecule is formed during an adiabatic sweep if two free atoms are sufficiently closein phase space so that their wavefunction can smoothly evolve to a bound molecule as
the Feshbach resonance is crossed. This model also explains the behavior we observe
here. Each particle of the less degenerate species is surrounded in phase space byparticles of the more degenerate 87 Rb gas with which molecules can be formed. We
used a Monte-Carlo simulation to model the observed conversion efficiency. Phase space
distributions of position and momenta of 85 Rb and 87 Rb are generated based on thenumber and temperature of each gas and the trap frequencies. For the 87 Rb gas at
temperatures above Tc a Maxwell-Boltzmann (MB) distribution is used. Below Tc we
use a combination of a Thomas-Fermi distribution for the BEC and a Bose-EinsteinCHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

142

(a)

85

4

Rb Number (10 )

5
4
3
2
1
0
262

264

266

268

magnetic field (G)85

4

Rb Number (10 )

5

(b)
4
3
2
1
0

360

370

380

390

magnetic field (G)Figure 6.5: Atom loss after magnetic-field sweeps through a Feshbach resonance. The
number of 85 Rb atoms is a function of the final magnetic field during the sweeps nearthe (a) 265 G and (b) 372 G Feshbach resonances. Initially there are 2.1 − 2.3 × 105
87 Rb atoms at T/T between 0.82 and 0.84. The data are fitted to an error function tocextract the center position and width. The resulting positions of the two transitions are
265.44 ± 0.15 G and 372.4 ± 1.3 G with the uncertainty given by the fitted RMS width.

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

143% 85Rb conversion

100

80

60

40

20

0
0

1

2

3

87

Rb T/Tc

Figure 6.6: Heteronuclear molecule conversion efficiency at the 265 G Feshbach resonance as a function of 87 Rb T/Tc . At our largest conversion efficiency the 85 Rb gas has
T/Tc = 2.6. The solid line shows a simulation based on our conversion model, and thedashed lines represent the uncertainty. The conversion drops below T/Tc = 1 since the
BEC is spatially smaller than the thermal 85 Rb gas and the conversion process depends
on the proximity of two atoms in phase space. Some of the data for this figure has been
averaged together. The error bars represent the weighted average of the points.

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

144

distribution for the thermal component. A MB distribution is always used for the 85 Rb
gas since the temperature of the gas does not fall below T = 2.2 Tc . For each 85 Rb
atom the simulation searches the 87 Rb gas to find a mate that is sufficiently close inphase space. After a pairing occurs, the two atoms are removed from the simulation.1
Two atoms are considered sufficiently close in phase space if either of the following
conditions is met: (1) If a BEC is present, any 85 Rb atoms inside the Thomas Fermi
radius of the condensate form a molecule, or (2) outside the BEC, an 85 Rb atom anda partner 87 Rb atom must satisfy the relation found in Ref. [40]: |δrrel m δvrel | < γ h,where δrrel is the separation of the pair, m is the atomic mass, δvrel is the relative
velocity, and γ = 0.44 ± 0.03.
The results of our simulation are shown by the solid line in Fig. 6.6. For T/Tc > 1
each gas has roughly the same spatial size in the optical trap, and therefore molecule
conversion can occur anywhere in the gas as long as the local phase space criterion ismet. At T/Tc = 1 and below, a significant fraction of the 87 Rb atoms are part of thecondensate, which is spatially small compared to the extent of the 85 Rb gas. Therefore
the conversion efficiency of atoms into Feshbach molecules drops in this regime since
the spatial overlap of the two species decreases. If the 85 Rb gas were a BEC, the two
species would have a better spatial overlap, and our simulation would predict that themolecule conversion efficiency quickly approaches 100% as the T/Tc of each gas drops
below one. There is good agreement between the experiment and simulation in boththe non degenerate and quantum degenerate regimes, suggesting that the pairing modelin Ref. [40] is also applicable to heteronuclear molecule creation.

6.7

Feshbach molecule binding energy

We have found that a useful way to spectroscopically characterize the Feshbachbound state is to create heteronuclear molecules using a small oscillating magnetic field.
The oscillating field causes two atoms to bind together and form a molecule [144, 154].
We first ramp the magnetic field from 269 G to a selected value between 266.2 and 267
G in 0.3 ms. The field modulation is then applied using the Helmholtz coil pair for 20
ms with a peak-to-peak amplitude between 0.6 and 1.0 G at a frequency up to 40 kHz.
Next the field is returned to 269 G and held there for 5 ms, allowing any molecules madeduring the modulation to decay. Finally we determine the number of atoms remaining
as described previously in this Chapter. As in Ref. [144], we observe strongly enhanced
atom loss at certain frequencies, and the loss depends on the duration and amplitude of
the modulation. A typical loss spectrum is shown in the inset of Fig. 6.7; the frequency
at which we observe maximum atom loss gives a measure of the binding energy of themolecules [155].2
1

The results of our simulation for the molecule conversion efficiency depend on the absolute numberof atoms in the two gases. The predicted molecule creation efficiency increases as the total number ofparticles in the simulations is increased. We study the behavior of the number dependence by running
our simulations over a very wide range of particle number. It takes several hours to run the simulation
at a single T /Tc for a number of particles similar to what is used in our experiments. Therefore we
determine the number dependence at only the single value T /Tc = 1.22, and we use that information toscale the simulation results when a smaller number of particles is used in the simulation. The correction
factor we use is approximately 10%.
2
The atom-loss spectrum was recorded for a 87 Rb gas with 50% condensate fraction and two noncondensed gases with temperatures of 130 and 190 nK. Over this range in temperature, the measured

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

145

100

30

% lossbinding energy (kHz)

40

20

50
0
16
20
24
28
modulation frequency (kHz)

10
0
0.0

0.5

1.0

1.5

B (G) - 265.44 G
Figure 6.7: Resonant frequency of atom loss as a function of magnetic field. The solidline is a fit to the data based on the universal binding energy of s-wave Feshbach
molecules and the dashed lines represent the uncertainty in the Feshbach resonancewidth. (Inset) An atom-loss spectrum at 266.5 G as a function of the modulation
frequency. The modulation converts roughly 50% of the gas to molecules. The loss iscentered at 21.7 kHz with a width of 0.6 ± 0.2 kHz. The solid line is a gaussian fit to thedata. We report the uncertainty in the binding energy as the width of the loss spectrum
because we lack a detailed understanding of the lineshape.

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES146

The resonant frequency of maximum atom loss is shown in Fig. 6.7 for various
magnetic fields near the Feshbach resonance. For each loss curve the amplitude usedconverted roughly half the 85 Rb gas into molecules. The solid line is a fit to the data
based on the universal form of the molecular binding energy near an s-wave Feshbachresonance [56]. In the fit the background scattering length is fixed to the value 213±7 a0[103], and the Feshbach-resonance peak position and width are varied; the best fit finds
the peak position and width to be 265.42 ± 0.08 G and 5.8 ± 0.4 G, respectively. These
results are consistent with our previously discussed determination of the Feshbachresonance peak position and with the predicted width of the Feshbach resonance in
Refs. [13, 152, 151].
Resonant association of free atoms into Feshbach molecules has recently beenused in an atomic 40 K gas to create stable p-wave molecules [156]. Two-body scattering
at p-wave Feshbach resonances has also been studied in 40 K [148, 157]. Due to thehigher angular momentum of the p-wave molecules, the binding energy scales linearly
as opposed to the quadratic dependence of the s-wave molecules. In our system, theresonant modulation technique that has already been described was extended to spectroscopically study the binding energy (Fig. 6.8) of p-wave heteronuclear 85 Rb–87 Rb
Feshbach molecules. The data presented here represents only a preliminary study of thisinteresting new system, and it might be useful as a starting point for future experiments.
The molecules are associated at the p-wave Feshbach resonance that we discoverednear 257.5 G; earlier in this chapter we investigated the doublet structure of this Feshbach resonance. To begin our experiments we ramped the magnetic field to a selectedvalue in the range of 256.7 G to 258.2 G. The field modulation is then applied using
the Helmholtz coil pair for 20 ms with a peak-to-peak amplitude between 0.1 and 0.7 G
at a frequency up to 50 kHz. Finally we determine the number of atoms remaining in
the trap with absorption imaging. As in the case of s-wave molecules, the frequency ofthe modulation at which maximum atom loss corresponds to the binding energy of themolecules. We acquired resonant modulation spectra at a few different magnetic fields
near the p-wave Feshbach resonance; the measured binding energies at these fields isshown in Fig. 6.8. We have extracted a preliminary value for the slope of the binding
energy at the p-wave Feshbach resonance, the slope is 58(4) kHz/G.

6.8

Conclusion

In this Chapter I have discussed how we created heteronuclear Feshbach moleculesfrom an ultracold gas of 85 Rb and 87 Rb. We demonstrated that molecules can be produced with two methods, magnetic-field sweeps and resonant-field modulation. The
conversion efficiency of 85 Rb into molecules can reach 60% even when that gas is notquantum degenerate. For this measured conversion efficiency the number of 87 Rb atoms
was a factor of five larger than the number of 85 Rb atoms. The presence of two species
with different quantum degeneracy provides a rich system for testing our understandingof the conversion efficiency from atoms to molecules. The heteronuclear molecules described here are ultracold and are stable for at least 1 ms. These conditions may provide
a first step toward the efficient production of ground-state heteronuclear molecules. Furthermore, the molecule creation process allows us to determine the location and width
resonant frequency shifts by only 5(2) Hz/nK.

binding energy (kHz)CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

147

40
ml = +/-1

20

ml = 0

0
-20
-40
-1.0

-0.5

0.0

0.5

1.0

B (G) - 257.79 G
Figure 6.8: Preliminary measurements of the binding energy of p-wave heteronuclear
Feshbach molecules. Each binding energy measurement represents the resonant frequency of atom loss. The solid lines are a linear fit to the data with slope 58(4) kHz/G.
The splitting between the two ml resonances is approximately 0.5 G.

CHAPTER 6. HETERONUCLEAR FESHBACH MOLECULES

148

of Feshbach resonances in the two-species system; this information will be required for
future studies of the 85 Rb–87 Rb system with a tunable interspecies interaction.

Chapter 7
Bragg spectroscopy of a strongly interacting BEC

7.1

Introduction

Many physical properties of a Bose-Einstein condensate can be described within
the framework of a mean-field theory [23]. In the mean-field theory the Hamiltonianof the system is composed of three terms, a term for the kinetic energy, a term for theenergy due to the external confining potential, and a mean-field term that describes
the energy due to interparticle interactions in the condensate. The kinetic energy of atrapped condensate is very small and is typically ignored. The mean-field term depends
on the condensate density and the s-wave scattering length a. The total energy of thecondensate and its physical properties like size and shape are determined by the trapping
potential and the strength of interactions. Mean-field theory is remarkably successful
at describing experimental observations of a BEC with weak atom-atom interactions.
However as the strength of interactions increases, the observed physical properties of acondensate are expected to deviate from the predictions of mean-field theory.
As a particular example of predicted beyond mean-field effects, consider the chemical potential of a condensate. In a mean-field theory the chemical potential is proportional to the product of density and scattering length. If the scattering length isincreased, a beyond√mean-field correction to the chemical potential is expected; thecorrection scales as n a3 . The exact form of this correction was first calculated by Lee,
Huang, and Yang (LHY) in the 1950s [47, 46, 158]. Beyond mean-field corrections also
show up in other physical properties of a condensate with strong interactions. Recently,
a beyond mean-field shift in the collective oscillation frequency of a Fermi gas on the
BEC-side of the BCS-BEC crossover was observed [159]. The shift in the oscillation
frequency is related to the LHY correction. But to date, efforts to observe the effectsof this beyond mean-field correction have not been successful in a Bose gas [6].
Measuring the excitation spectrum is an another important way of revealing theeffects of inter-particle interactions in a condensate. Here the excitation spectrum refers
to the energy required to add a particle or quasiparticle with a given momentum. In
a classical gas, which obeys Maxwell-Boltzmann statistics, the excitation spectrum isfree-particle-like for all values of momentum. A key characteristic of a superfluid isthe presence of phonon excitations at small momenta. The energy of a phonon scales
linearly with momentum and the slope is given by the sound velocity in the gas. In thecase of superfluid 4 He, measurements of the excitation spectrum not only revealed theexistence of phonons at small momentum, but the presence of a local minimum in the
spectrum at larger momentum due to rotons. The current physical understanding of
the origins of the roton minimum is complicated, see Ref. [19] for details.

CHAPTER 7. BRAGG SPECTROSCOPY

150

In this Chapter the excitations of a BEC are studied. The excitation spectrumof a condensate with no interactions scales quadratically with momentum since thequasiparticles behave as free particles. Atom-atom interactions modify the character ofthis spectrum; at low momentum the spectrum is linear due to phonon-like excitations,while at large momentum the spectrum is free-particle-like with a shift in energy due
to interactions. The excitation spectrum of a strongly interacting condensate is further
modified by beyond mean-field effects related to the correction derived by Lee, Huang,
and Yang [47, 46, 158].
As mentioned earlier in this thesis, one of the goals of our new 85 Rb BEC machine
was to be able to make large, stable 85 Rb condensates in a system with plenty of opticalaccess that would allow more sophisticated probes of a strongly interacting BEC. The
experiments described in this chapter represent the realization of this goal. By adding
a pair of counter-propagating laser beams to our science cell we are able to probe theexcitation spectrum of our 85 Rb gas through Bragg spectroscopy. Since we have accessto the 85 Rb Feshbach resonance, we can continuously tune the scattering length into thestrongly interacting regime. This allows us to probe the onset of corrections to the meanfield theory of a condensate. Our measurements of the excitation spectrum of a stronglyinteracting BEC near a Feshbach resonance have revealed some very interesting physics.
In the case of large momentum excitations, we have observed that the excitation energyof the condensate is significantly smaller than what is expected from a mean-field theory.
Our observation of a smaller than expected excitation energy is in qualitative agreementwith the free-particle regime predictions of Refs. [160, 161], which are related to the
LHY beyond mean-field corrections. However, our experiments were not performed
completely in the free-particle regime. That regime is characterized by k ξ À 1 where
k is the wave vector of the excitation and ξ is the healing length of the condensate.
Our experiments were performed with a wave vector such that k ξ ≥ 2. To understandwhat we expected for our strongly interacting condensate, we have collaborated with S.
Ronen and J. Bohn who have predicted the excitation spectrum using existing theories
that include many-body effects [160, 161].
In this chapter we describe experiments designed to probe beyond mean-field corrections to the excitation spectrum of a BEC. Bragg spectroscopy is used to record
the excitation spectrum over a wide range of scattering length. Bragg spectroscopy
has several advantages over measurements of collective excitations, including a measurement time much shorter than the radial trapping period and the ability to accessdifferent momentum regimes. To date our experimental focus has been on acquiring andunderstanding the large momentum excitations of a strongly interacting BEC. In thisregime, the excitation energy is larger than the chemical potential of the condensate.
In future experiments it may be possible to study excitations with smaller momentum,
and therefore with energy less than the chemical potential. A particularly interesting
regime to study is where the wave vector of the excitation is approximately equal to theinverse of the interparticle spacing. It is in this regime that the roton minimum of 4 He
appears.

7.2

The excitation spectrum of a BEC

The excitation spectrum of a homogeneous BEC in mean-field theory is given by
the well-known Bogoliubov formula, which states that the excitation energy (h̄ ω(k)) asCHAPTER 7. BRAGG SPECTROSCOPY

151

a function of the momentum h̄ k is given byv!
u 2 Ã 2u h̄ k 2 h̄ k 2t+ 2gnh̄ ω(k) =

2m

2m

(7.1)

where g = 4πh̄2 a/m, a is the s-wave scattering length and n is the average condensate
density. The excitations of an interacting quantum fluid differ significantly from a gas
described by Maxwell-Boltzmann statistics with free particle excitations at all momenta.
In particular at small values of k condensateq excitations are phonon-like and the energy
is approximately ²k ≈ c h̄ k with c = gmn . For large k, condensate excitations are
free-particle-like except with a shift in energy due to the interactions; the energy is
given by ²k ≈ h̄2 k 2 /(2m) + g n. The gradual crossover from the phonon√ regime to freeparticle regime is indicated by the condensate healing length ξ = 1/ 8 π n a. Figure
7.1 shows the excitation energy of a condensate and the free-particle excitation energy
as a function of momentum.
The excitation spectrum of a trapped BEC is qualitatively similar to the homogeneous case. The spectrum is calculated by averaging the Bogoliubov energy over the
inhomogeneous density of the trapped condensate. In the large momentum regime, averaging the excitation spectrum over the Bogoliubov energies again predicts a constantshift due to interactions.
So far only weak atom-atom interactions in the condensate have been considered.
The Bogoliubov excitation spectrum is valid only in the limit that the condensate isdilute (n a3 ¿ 1) and two-body scattering is independent of momentum (k a ¿ 1). With
our apparatus we can achieve experimental conditions of density and scattering length
under which neither of these conditions are met. Since we have chosen to create large
momentum excitations with Bragg spectroscopy, even at a modest scattering length of
900 a0 we have k a = 0.8. As we will see later in this chapter, we set the density of thecondensate to be large to enhance beyond mean-field effects due to n a3 physics withrespect to the momentum-dependent scattering physics.

7.3

Bragg spectroscopy of a BEC

In our experiments we use Bragg spectroscopy to precisely probe the excitation spectrum of a condensate. Bragg spectroscopy is performed by driving stimulated
two-photon transitions with two independent laser beams that are directed onto thecondensate. The interference pattern of the two laser beams creates a small-amplitude,
spatial oscillation in the BEC wave function. The energy and momentum associatedwith the wave function oscillation are given by the frequency difference and wave vector
difference between the two lasers. We can detect the effect of Bragg spectroscopy bymeasuring the number of atoms that make a two-photon transition. Since the majority of atoms in the condensate have zero momentum prior to Bragg spectroscopy, theexcitation of the BEC can be probed with good resolution.
Bragg spectroscopy of a condensate was first demonstrated at MIT in 1999.1
In their original work large momentum excitations of a Na condensate were studied,and the energy shift due to condensate interactions was directly observed [163]. They
1

The Phillips group at NIST also demonstrated two-photon Bragg transitions in a Na BEC [162].

CHAPTER 7. BRAGG SPECTROSCOPY

152

weak interactions
free particle
inverse healing length

energy (kHz)

15

10

5

0
0

5

10

15
-1

momentum (µm )Figure 7.1: Bogoliubov excitation spectrum (Eqn. 7.1) for a condensate (black curve)
and the free particle energy (red curve) as a function of momentum. The black curve has
n = 1014 cm−3 and a scattering length of 150 a0 . The green vertical line represents theinverse healing length of the condensate and the crossover to free-particle excitations.

CHAPTER 7. BRAGG SPECTROSCOPY

153

also observed that the coherence length of the BEC is larger that its size, since Bragg
spectroscopy reveals the momentum distribution of the gas. In later work they studied
excitations in the phonon regime. In particular they were able to observe a suppressionof the Bragg response for low-momentum excitations [164]. This suppression is causedby the presence of correlated atom pairs with momentum ±p. Bragg spectroscopy at
low momenta is a superposition of two events, scattering from the −p states into thecondensate and scattering from the condensate into a state with momentum p. If the
momentum change due to Bragg spectroscopy is very small, the two events occur withequal probability and the Bragg response is suppressed. These two experiments eachstudied the excitation spectrum at a single value of momentum. In 2002, the Davidson
group measured the excitation spectrum over a wide range of momentum, spanningthe phonon and free-particle regimes. Excellent agreement between their data and the
Bogoliubov spectrum was observed. Our Bragg spectroscopy work builds upon theseobservations by studying condensate excitations over a very wide range of interaction
strength. Since we have access to a Feshbach resonance, we are able to explore theexcitation spectrum in a regime of interactions where beyond mean-field corrections areimportant.
Our experiments are initiated by creating a pencil-shaped 85 Rb BEC in an optical
dipole trap. For these experiments the 46 micron waist optical trap is used. The radial
trap frequency was measured to be 2 π×129(2) Hz and the axial frequency was measured
to be 2 π × 2.9(1) Hz.2 A Helmholtz pair of coils provides a magnetic field of up to
700 G; the curvature of the field enhances confinement along the axial direction of the
optical trap.
Two-photon Bragg transitions change the energy and momentum of an atom
without changing the internal spin state. The energy and momentum imparted tothe condensate is calibrated via the frequency and wave vector difference of the twolaser beams. The momentum imparted to the BEC via Bragg scattering is given by
q = h̄ k sin(θ/2) = 2 h̄kL sin(θ/2) where kL = 7802 πnm is the magnitude of the wave
vector of the laser beams and θ is the angle between the two beams. A significant
advantage of Bragg spectroscopy is that the energy and momentum imparted to thecondensate is well calibrated. Therefore the energy and momentum of the excitation
does not need to be directly determined from the atoms that undergo Bragg transitions.
In our system the condensate is illuminated by the Bragg spectroscopy light for aduration tp . A pair of counter-propagating Bragg laser beams are oriented along the axisof the condensate. The motivation of this choice will be discussed in the next section.
Figure 7.2 shows the optics and beam paths in the science cell area. The Bragg light isdelivered to the science cell area with a pair of polarization-maintaining optical fibers.
The Bragg light is aligned along the axis of the trapped condensate onto the atoms byoverlapping the beams with the optical trap light.
The laser light for Bragg spectroscopy is derived from an external cavity diodelaser located on a different optical table from the main apparatus. The laser frequency isstabilized to the 87 Rb F = 2 → F 0 = 2/F 0 = 3 crossover line in the saturated-absorption
spectrum, which is detuned 133.5 MHz from the cycling transition. Approximately 20
mW of total light is available from the laser. For Bragg spectroscopy we require twophase stable, independently steerable laser beams with a precisely tunable frequency
2

Note that the frequency of breathing motion along the elongated direction of the condensate isapproximately 1.6 ωz [23].

CHAPTER 7. BRAGG SPECTROSCOPY

154

to CCDx
dichroic BS

dichroic BS

optical
trap

z
y

Bragg
fiber #1

IP trap
Bragg
fiber #2

optical pumping

Figure 7.2: Simplified schematic of the science cell area including the Bragg laser beams,
the science cell optical pumping light, and the optical trapping beam. The Bragg beams(shown as a dashed green line) are delivered by two optical fibers. The Bragg light isoverlapped with the optical trapping beam to assure that Bragg excitations are along
the axis of the condensate. The optical pumping and Bragg light are misaligned by a
few degrees so that the beams do not need to be combined with a beamsplitter. The
Bragg beam diameter is a few mm at the center of the science cell.

CHAPTER 7. BRAGG SPECTROSCOPY

155

difference. The light from the Bragg laser is split into two beams with a polarizing
beamsplitter cube. Each beam is passed through an 80 MHz AOM. One of the AOMs
is set exactly at 80 MHz and the other AOM frequency is tunable. For simplicity weuse a pair of synthesized frequency generators to create the 80 MHz signals sent to the
AOMs. In addition to controlling the relative frequency of the Bragg beams, the AOMs
are used to generate short duration pulses of the light.

7.4

Obtaining Bragg spectra with strongly interactions

This section describes the technique we use to detect the signal of Bragg spectroscopy in the strongly interacting regime. Two-photon Bragg transitions in our experiment cause atoms to be scattered from the condensate into a momentum state with avelocity of 1.2 cm/s. The total momentum of the scattered atoms is the signal for Bragg
spectroscopy. The standard method for determining the total momentum is to countthe number of atoms scattered after they exit the BEC. Since the Bragg laser beams areoriented along the axial direction in our experiments, it takes approximately 20 ms forthe scattered atoms to completely exit the BEC. During that 20 ms, collisions between
the Bragg scattered atoms and the condensate occur. Eventually enough collisions occurthat none of the Bragg scattered atoms are observed to cleanly exit the BEC. Using the
85 Rb Feshbach resonance we can minimize the collision rate in the condensate so thatfew collisions occur during the 20 ms it takes for the Bragg scattered atoms to exit the
BEC. Figure 7.3 (a) shows an absorption image acquired 40 ms after the Bragg light
was applied. For this measurement, immediately after the Bragg light is switched off
the scattering length is ramped in 0.2 ms to less than 10 a0 . The condensate appearson the left hand side of the image and the Bragg scattered atoms are on the right. We
can cleanly obtain Bragg spectra using this sequence as long as the scattering length
during the Bragg pulse is below about 300 a0 . For comparison, Figure 7.3 (b) shows anabsorption image under the same conditions as in (a), except the scattering length was
not ramped after the pulse. As is observed in the image, collisions severely compromise
the signal for Bragg spectroscopy.
Unfortunately, the successful method described above, in which the scattering
length is lowered to minimize collisions between the Bragg scattered atoms and thecondensate, only works if the scattering length during the Bragg pulse is smaller thanapproximately 300 a0 . If the scattering length is higher, collisions during the pulse are
rapid enough that our signal for Bragg spectroscopy is severely compromised. So we use
another technique to extract a signal for Bragg spectroscopy, which works equally welleven at large a. We exploit collisions between the quasiparticles and the condensate
to measure the momentum imparted to the BEC [165, 166]. Related experiments havebeen performed with a strongly interacting Fermi gas by C. Regal. For this technique
immediately after the Bragg beams are extinguished, we ramp the 85 Rb scattering
length to 900 a0 to increase the elastic collision cross section. The quasiparticles excitedby the Bragg pulse then rapidly collide with the condensate and deposit momentum
along the optical trap axis. The total momentum generated by Bragg spectroscopy
causes a center-of-mass (COM) oscillation in the condensate axial position. We detect
the momentum by monitoring the amplitude of the condensate COM oscillation as afunction of time; an example of the oscillation is shown in Figure 7.4. We fit the data
to a sine wave in order to extract the oscillation period and amplitude.

CHAPTER 7. BRAGG SPECTROSCOPY

156

BEC

(a)

(b)

Messy!

(c)Figure 7.3: A comparison of different Bragg spectroscopy detection techniques. (a) The
number of Bragg scattered atoms is directly counted after the atoms exit the BEC.
The scattering length was set to 150 a0 during the Bragg pulse. For this technique
to be successful collisions between the Bragg scattered atoms and the BEC must be
minimized. Immediately following the Bragg pulse the scattering length is tuned close
to zero. (b) The scattering length was not tuned toward zero after the Bragg pulse.
Many collisions occurred between the Bragg scattered atoms and the condensate, andour signal is diminished. (c) Our method for Bragg spectroscopy in which all the Bragg
scattered atoms collide many times with the BEC. The momentum is deposited into thecondensate which generates a COM oscillation. The amplitude of this oscillations is oursignal. The center position of the condensate in (c) is clearly offset from the condensate
on the left hand side of (a) and (b).

CHAPTER 7. BRAGG SPECTROSCOPY

157

We studied the signal-to-noise ratio of our Bragg spectroscopy signal. Since oursignal is the amplitude of condensate COM oscillations, we must detect the position of
the BEC along its axis with high precision. We optimized the probe beam intensity forour absorption imaging to produce the lowest noise per pixel in the processed absorption
image. We determined that the lowest noise is obtained when the intensity is set suchthat the fluctuations in each pixel is primarily due to shot noise. Naturally, we also
found that the axial size of the condensate is important in determining the center
position of the cloud. We acquired our Bragg spectroscopy data when the phase of theaxial breathing motion is such that the size of the cloud is smallest.

gas position ( m)

60

30

0

-30

-60

0

200

400

600

hold time (ms)Figure 7.4: A center-of-mass condensate oscillation along the axial direction of the
optical trap induced by two-photon Bragg transitions. The amplitude of this motion
is our signal for Bragg spectroscopy. Bragg spectra are obtained by monitoring thecondensate position at 230 ms.
Our technique for Bragg spectroscopy requires a stable condensate position between repeated runs of the experiment. Since the Bragg beams are aligned along theaxis of the condensate, the stability in that direction is critical. In particular, we observe
residual COM motion in the axial direction of the trap due to a slight misalignment
during the optical trap load from the magnetic trap.3 4 This “sloshing” motion appears as a Doppler shift of the Bragg resonance, and the slosh amplitude and phase is
not perfectly repeatable from run-to-run of our experiments. To solve this problem, weapplied an rf field to the atoms that was only resonant with the edges of the cloud along
its axis. The rf damps the amplitude of the sloshing motion by selectively removing themost energetic atoms from the optical trap. For the optimum removal of slosh, rf fields
were applied at frequencies near both the 85 Rb and 87 Rb resonances. The rf effectively
3

Due to the details of our system we are not able to correct this misalignment.
The COM motion does not fully decay during the evaporation since the contribution of anharmonic
terms to the axial potential is small. The optical trap evaporation primarily removes energy along theradial direction of the trap and therefore the COM motion remains throughout the cooling.
4CHAPTER 7. BRAGG SPECTROSCOPY

158

damps the sloshing motion, at the end of evaporation the position stability of the 85 Rb
condensate is better than 2.5 µm (standard deviation). Typically the size of our Bragg
signal is 60 µm.

% excitation

10

5

0

-5

-10
-60

-40

-20

0

20

40

60

frequency difference (kHz)
Figure 7.5: A typical Bragg spectrum acquired at a scattering length of 900 a0 , withn a3 = 0.01. The Bragg pulse duration was 0.087 ms. We extract the center position
and width of the Bragg spectrum by fitting the data to the difference of two gaussians.
Since our data is anti-symmetric about zero, the gaussian fit is performed with only asingle value of the amplitude, the center position, and the width.
To record the Bragg resonance of a condensate we monitor the amplitude of thecondensate oscillation as we vary the frequency difference between the two laser beams.
For each frequency difference, we measure the position after a 230 ms hold in the opticaltrap at which time the oscillation amplitude is largest. Figure 7.5 shows a typical Bragg
spectrum for a strongly interacting condensate. From the measured Bragg resonancewe extract the linewidth and line shift of the resonance by fitting the spectrum to thedifference of two gaussians. Here we define the Bragg line shift as the difference of the
2 2k.
measured center frequency and the free-particle energy h̄2 m

7.5

Momentum signal for a weakly interacting gas

Our signal for Bragg spectroscopy is the amplitude of the condensate centerof-mass oscillation along the axis of the optical trap. The on-resonance signal size
depends on the Bragg laser power, the pulse length, and the detuning from the atomic
resonance. We extract the total momentum excited from the oscillation amplitude
based on knowledge of the harmonic trapping frequency. The total momentum per
atom imparted to the condensate is
P = 2 h̄ k γ(7.2)

CHAPTER 7. BRAGG SPECTROSCOPY

159

where γ is the fraction of the condensate excited. It is useful to note that our Bragg
spectroscopy signal does not explicitly depend on the absolute number of atoms in the
BEC. Following the analysis of Refs. [167, 168] the fraction of the condensate that
undergoes a two-photon Bragg transition is given byrγ=

π Ω2R tp
2 2σ

(7.3)

where tp is the length of√the Bragg pulse, σ is the RMS width of the Bragg responseΓ2 I1 I2
lineshape, and ΩR = 4∆
Isat is the two-photon Rabi frequency and ∆ is the detuningof the Bragg laser beams from atomic resonance.5 This formula is only valid when
the fraction of the condensate excited is small, the duration of the Bragg pulse is shortcompared to the trap oscillation time but long compared to the inverse of the frequency
difference between the Bragg laser beams, and the momentum of the excitations is large(k ξ À 1).
Figure 7.6 shows the fraction of a 85 Rb condensate excited by two-photon Bragg
transitions as the Bragg laser power and pulse length are varied. The fraction excited scales quadratically with the Bragg pulse duration, and it scales linearly with the
product of the two laser beam intensities. In the figure, the solid lines represent theprediction of Eqn. 7.2 for the total momentum per atom imparted to the condensate by
Bragg spectroscopy. The measured Bragg spectra presented in this chapter were all obtained with a combination of intensity and pulse length such that the number of atoms
excited were in the range of 5% to 10%. We expect that this means the Bragg scattering
process does not dramatically alter the properties of the condensate. Furthermore, oursignal for Bragg spectroscopy, namely the amplitude of condensate COM oscillations,does not saturate until the excitation fraction is much larger than 10% (Fig. 7.6).

7.6

Bragg lineshape function for a trapped BEC

To understand the expected dependence of the line shift and linewidth of the
Bragg resonance on atom-atom interactions, we have modeled the Bragg resonancelineshape using existing theories. Here the Bragg resonance refers to the amount of
momentum imparted to the condensate as a function of the excitation energy; an example was shown in Fig. 7.5.6 To theoretically understand a lineshape such as this,we calculate the rate of Bragg scattering which determines the momentum imparted.
The Bragg scattering rate can be separated into the product of two terms; the rate isgiven by [169]2 π (h̄ ΩR )2× S(~k, E)(7.4)
ΓBragg =h̄4where ~k is the wave vector of the excitation and E is the energy of the excitation. In Eqn.

7.4 the term on the left ((h̄ ΩR )2 ) depends only on the laser beams and the single atomphoton coupling, while the right hand term (S(~k, E)) depends only on the propertiesof the condensate, such as the excitation spectrum, chemical potential, size, etc. The
dynamic structure factor S(~k, E) has been studied extensively for the case of a weakly
5
In this formula we have assumed that the excitations are free-particle-like and the static structure
factor of the condensate is unity. See the next section for details of the structure factor.
6
Experimentally the excitation energy is the frequency difference of the Bragg laser beams.

fraction of gas excited

CHAPTER 7. BRAGG SPECTROSCOPY

0.08

160

(a)

0.06
0.04
0.02
0.00
0.0

0.2

0.4

0.6

0.8

1.0

fraction of gas excited

Bragg pulse length (ms)

0.3

(b)

0.2

0.1

0.0
0

20

40

60

80

100

beam #2 power (µW)

Figure 7.6: (a) Fraction of the condensate excited as a function of Bragg pulse length.
The Bragg laser powers were fixed at 35 µW and 2.5 µW and the beam 1/e2 radii areapproximately 2.3 mm and 1.3 mm, respectively. The data points at 1 ms are noisy
since the linewidth of the Bragg resonance is so narrow. (b) Fraction of the condensate
excited as a function of Bragg laser power. The pulse length was fixed at 0.5 ms andone of the laser beams had a power of 35 µW while the other power was varied. The
solid lines are the prediction for the fraction of the condensate excited from Eqn. 7.2.

CHAPTER 7. BRAGG SPECTROSCOPY

161

interacting Bose-Einstein condensate. In particular modifications of S(~k, E) due to theinhomogeneous density of a trapped condensate have been considered [170, 169, 167].
Since the Bragg scattering rate is proportional to S(~k, E), the line shift and linewidth
can be calculated directly from S(~k, E). In this section we present the dynamic structure
factor calculated in two approximation schemes: the impulse approximation (IA) andthe local density approximation (LDA).
In the impulse approximation (IA), which can be applied when the excitation
momentum (h̄ ~k) is very large, the form of the dynamic structure factor is dominated by
Doppler broadening. Here Doppler broadening refers to the fact that Bragg scattering
is sensitive to the momenta of the trapped atoms. Due to the finite size of a trapped
condensate, there is a distribution of atomic momenta that tends to broaden the Bragg
resonance. The distribution of momenta is given by the Fourier transform of the spatial
density distribution; a small-sized BEC has a large spread in momentum. For the IAcase, S(~k, E) is given by [170]mS(~k, E)IA =h̄kZ

dpx dpy nz (px , py , pz )(7.5)

where nz (px , py , pz ) is the momentum distribution of the condensate which is integratedmalong the two directions that are perpendicular to the Bragg beams and pz = h̄k
(E −
2 2
h̄ k
Er (k)) with Er (k) = 2 m is the energy of the excitation. For fixed momentum in theIA, the width in energy of S(~k, E) scales as h̄k/Rz where Rz is the size of the condensate.
As shown in Figure 7.2, the Bragg light is aligned along the axis of the condensate, inthe direction where the BEC is larger than 0.1 mm. The contribution from the IA isvery small in our experiments with a typical value of 30 Hz, since the size Rz is typicallyon order of 0.1 mm.
In the local density approximation (LDA), the form of the dynamic structure
factor is dominated by the condensate density profile. For the LDA case, S(~k, E) is
given by [170]1
S(~k, E)LDA =h̄Z

d~r n(~r) δ(E − h̄ ω(n(~r), k))(h̄k)22 m h̄ ω(n(~r), k)(7.6)

where n(~r) is the density distribution of the BEC and h̄ ω(k) is the Bogoliubov excitationenergy (Eqn. 7.1) evaluated at the local density n(~r). Our experiments are performed
under conditions in which the condensate dynamic structure factor is described by the
LDA. This approximation is used below to calculate the excitation spectrum of a trapped
BEC with inhomogeneous density.
7.6.1

Line shift

The Bragg line shift is defined here to be the difference in the condensate excitation energy (E(k)) and the free-particle energy (Er (k)). Since we are only interested inchanges to the condensate excitation spectrum due to interactions, we subtract the freeparticle energy from E(k) to obtain the Bragg line shift.
7.6.1.1

The weakly interacting case

The center of the Bragg resonance for a trapped BEC in the weakly interacting
regime is calculated by averaging the Bogoliubov excitation spectrum (Eqn. 7.1) over

CHAPTER 7. BRAGG SPECTROSCOPY

162

the inhomogeneous density of the condensate. By applying the Thomas-Fermi approximation for the condensate density, the excitation energy for any value of k is given by[170, 168]
v
Ã
!2
u
u
h̄2 k 22 2 2t
E(k) = (c(k)) h̄ k +

(7.7)

2m

where c(k) is a complicated function of momentum and the condensate
chemical pop
32
tential. For small k in the phonon regime (k ξ ¿ 1) c(k) = 15πµ/m, and in theq

Bragg lineshape (arb)free-particle regime (k ξ À 1) c(k) = 47 µ/m. Here µ is the chemical potential of thecondensate given by µ = g n0 where n0 is the density at the center of the condensate
2
and g = 4 πmh̄ a.

1.0

0.5

0.0
15.0

15.5

16.0

16.5

ω / 2π (kHz)

Figure 7.7: The solid black line is the calculated Bragg lineshape plotted as a functionof the frequency ω/2π. The lineshape is asymmetric due to the Thomas-Fermi density.
The vertical dotted line indicates the mean excitation energy of the condensate. The
red dashed line shows a gaussian fit to the Bragg lineshape which accurately determines
the mean excitation energy.
Using the Thomas-Fermi approximation for the condensate density, the Bragglineshape is given by the dynamic structure factor
15 E 2 − Er2
S(~k, E)LDA =8 h̄ Er µ2s1−

E 2 − Er2.
2 Er µ(7.8)

The Bragg lineshape is plotted in Figure 7.7 as a function of the frequency E/h. The
center of the Bragg resonance is the mean value of this function. The Bragg lineshapeis asymmetric due to the Thomas-Fermi profile. However, it is interesting to notethat a gaussian fit to the asymmetric shape still yields the correct mean value of theCHAPTER 7. BRAGG SPECTROSCOPY

163

distribution. In the free-particle regime the mean energy is given by
4
E(k) = Er (k) + µ.
7

(7.9)

The second term, which is independent of k, is the condensate mean-field contribution
to the excitation energy. This formula for the excitation energy is valid in the limit thatthe chemical potential is small compared to the free-particle energy or that kξ À 1.
7.6.1.2

Estimates of the Bragg line shift with strong interactions

In this section we present calculations of the excitation spectrum of a stronglyinteracting homogeneous condensate using existing theories.7 Later on in this chapter
we compare the predictions of these theories to our measured Bragg spectra of stronglyinteracting 85 Rb condensates. When the atom-atom interactions in the condensate arestrong, the excitation spectrum is not accurately described by the Bogoliubov formula.
In particular, when na3 is no longer small beyond mean-field effects are expected to
be important, and when k a is no longer small two-body scattering in the condensate
is expected to be momentum dependent. Beliaev has derived two corrections to the
Bogoliubov excitation spectrum of a homogeneous BEC using perturbation theory withthe small parameter na3 [171, 160]. Note that Beliaev’s theory may be applied toexcitations with any value of (k ξ). The calculations are somewhat similar in nature
to those performed by Lee, Huang, and Yang to obtainthe first-order correction to√3
the condensate chemical potential which scales as n a . In the first correction thecondensate is assumed to be dilute with na3 = 0, but k a may take on any value andthe excitation spectrum is given by [160]
vÃ
!2
u
2
u
8
π
h̄
n <(fs (k/2)) − n g − n2 (fs (k))2
E(k) = t Er +m

(7.10)

where <(fs ) is the real part of the symmetrized s-wave scattering amplitude that is
discussed in Section 7.6.3. 8 It is useful to think of <(fs (k/2)) as a scattering lengththat depends on the momentum wave vector. The first Beliaev correction tends toreduce the excitation energy of the condensate since excitation-condensate interactions
via two-body scattering are reduced as k a is increased. This reduction is related to thefamous Ramsauer-Townsend effect in which a very large scattering phase shift associatedwith strong interactions actually suppresses the s−wave elastic cross section [48]. In thelimit that k a ¿ 1, fs (k/2) → a and the Bogoliubov formula (Eqn. 7.1) is recovered.
The Beliaev second order correction to the Bogoliubov theory considers beyond
mean-field effects by allowing na3 to take on a finite value. However in this second
approximation it was assumed that k a = 0 so that two-body scattering is independentof momentum. The excitation energy in the second approximation is given by thepoles of Eqn. (5.17) in [160]. Analytical results for the excitation energy in the second
Beliaev approximation were obtained for the limiting cases of low-momentum phonon
7

We thank Shai Ronen and John Bohn for helping us understand the theory presented here.
Shai Ronen and John Bohn have calculated that fs (k) ≈ 4 πa to a few percent for the experiments
presented in this chapter.
8

CHAPTER 7. BRAGG SPECTROSCOPY

164

20

energy (kHz)

Bogoliubov spectrum
3(n a = 0)

15

Beliaev spectrum
3(n a = 0.01)

10
low-k correction
3(n a = 0.01)

5
0
0

5

high-k correction
3(n a = 0.01)

10

15

wave vector (µm)
Figure 7.8: Excitation energy as a function of the wave vector k. The black curve is the
Bogoliubov spectrum for weak interactions. The red and blue curves are the respectively
low-k and high-k limits of the excitation energy including the n a3 correction. The
scattering length is 900 a0 and the density is 1014 cm−3 . The red and blue curves werearbitrarily broken at 9 µm−1 . For each theory line shown here we assume that k a = 0.

CHAPTER 7. BRAGG SPECTROSCOPY

165

excitations and large-momentum free-particle excitations. For small k, the excitationenergy is
r
µ
¶
gn
8 √ 3
E(k) =h̄ k 1 + √na .
(7.11)mπ
For large k excitations in the free particle regime, the energy is expected√ to be smaller
than the mean-field prediction, but the correction factor also scales as n a3 . The result
for large k isµ¶
32 √ 3√E(k) = Er (k) + g n 1 −na .
(7.12)
3 π
Equations 7.12 and 7.11 are plotted as a function of the wave vector k in Figure 7.8 fora strongly interacting 85 Rb condensate with a density of 9 × 1013 cm−3 and a scattering
length of 900 a0 . Shai Ronen and John Bohn have used the Beliaev second orderapproximation to calculate the excitation spectrum of a condensate for excitations withan arbitrary wave vector k. This allows us to connect the small-k and large-k limits.
For comparison, their calculation of the condensate excitation energy is also shown in
Fig. 7.8. Note that for the range of wave vector that is plotted, the predicted excitationenergy has not yet reached its large-k limiting value. All of the experiments described inthis chapter were performed with wave vector of 16.1 µm−1 . At this value, the Beliaev
theory with k a = 0 and finite na3 predicts that the excitation energy is larger than the
Bogoliubov prediction.
The free-particle excitation energy of a strongly interacting BEC can also be
obtained using Hartree-Fock-Bogoliubov (HFB) theory. The HFB energy is given by[172](7.13)
E(k) = Er (k) + 2 g n − µ.
The second term represents interactions between the condensate and the excitations
generated by Bragg scattering, including both direct and exchange interactions. The
third term is the chemical potential of the condensate. As shown in Eqn. 7.13, the
HFB is only valid in the limit that k a = 0 and na3 = 0. We have modified the HFB
picture to include momentum-dependent two-body scattering and beyond mean-field
effects. The energy of direct and exchange interactions between the condensate and2
the excitations is modified to be 8πh̄m n fs (k/2). We also include the first-order LHY
correction
chemical potential due to strong interactions given by³ to the√condensate´
323√µ = g n 1 + 3 π na . The energy of large-k excitations is given byµE(k) = Er (k) +4 π h̄2 n
2 <(fs (k/2)) − amµ32 √ 3
1+ √
na
3 π

¶¶

.

(7.14)In Figure 7.9 we have plotted the predicted Bragg resonance line shift from thedifferent theories presented here as a function of scattering length. Since the theories
discussed here are for a homogeneous gas, the results shown in the figure are for themean density of our trapped condensate. The top axis of the figure shows the size of the
LHY correction to the condensate chemical potential as a function of scattering length.
At 900 a0 the LHY correction is 0.6 indicating the importance of beyond mean-field
effects.

CHAPTER 7. BRAGG SPECTROSCOPY166

LHY correction
0.00

0.06

0.17

0.31

0.48

0.67

200

400

600

800

1000

line shift (kHz)

8
6
4
2
0
0

scattering length (a0)Figure 7.9: (a) Theoretical Bragg line shift of a homogeneous condensate as a functionof scattering length. The curves, in order of smallest to largest shift, correspond to thefollowing theories: Hartree-Fock-Bogoliubov (solid black), Beliaev first approximation(dashed red), Bogoliubov (dash-dot blue), and Beliaev second approximation (dash-dotdot magenta). Calculations were performed with a mean density of 8.5 × 1013 cm−3 andan excitation momentum of 2 h̄ kL . For reference the top axis shows the size of the LHY
correction to the condensate chemical potential.

CHAPTER 7. BRAGG SPECTROSCOPY
7.6.2

167

Linewidth

In order to accurately measure the Bragg resonance line shift, it is importantto quantitatively understand the width of the Bragg resonance in our experiments. In
this section we present predictions of the Bragg resonance width using existing theories.
Later on in this chapter we will compare these predictions to our experimental data.
The width of the Bragg resonance depends on several contributions, including the duration of the Bragg pulse, the distribution of momenta in the inhomogeneous condensate,and collisions during the Bragg pulse. We model the lineshape of each contribution
separately. To understand the composite lineshape that includes each of these contributions, we calculate the convolution of the contributing lineshapes. For further detailon calculating composite lineshapes see, for example, Ref. [83]. A gaussian fit to the
convolution extracts the total linewidth. Since our measured Bragg spectra are also fit
to a gaussian, we can directly compare our measurements with the model.
The lineshape associated with the duration of Bragg spectroscopy is a sinc2 function, and the lineshape due to the inhomogeneous density of the condensate is theasymmetric function shown in Figure 7.7. The convolution of these two lineshapes isshown in Figure 7.10. Since it is similar to a gaussian, we can extract the total widthwith a fit. The gaussian width σ of the convolution is
vÃ
!
Ãr
!2
u
u 0.36 2
8 µ
t
+
σ=

tp

147 h

q

(7.15)

8 µwhere tp is the duration of the Bragg pulse and 147
h is the frequency width due to theinhomogeneous density of the condensate. The linewidth of the Bragg resonance might
be increased due to beyond mean-field effects. For example the contribution due to theinhomogeneous density might be increased in the strongly interacting regime since the
LHY chemical potential is larger than the mean-field value.
Collisions between the condensate and the quasiparticle excitations created by
Bragg scattering tend to broaden the Bragg resonance. Beliaev first studied the decayof quasiparticles due to collisions in Ref. [160]. He showed that the excitation spectrum
has an imaginary component equal to the quasiparticle-condensate collision rate thatshould be interpreted as the energy width of the spectrum. We note that this imaginary
component only appears in the result of Beliaev’s second approximation where finite
n a3 is considered but the limit that k a = 0 is assumed. The lineshape associatedwith quasiparticle-condensate collisions is a lorentzian that is characterized by a full
width at half maximum (FWHM) γ = 2 π1 τ . Here τ1 is the collision rate in the gas
given by τ1 = hniσel (k)hvrel i where σel (k) is the elastic collision cross section, hni is theaverage density, and hvrel i = h̄ k/m is the mean relative speed. At low momentum, theelastic cross section for quasiparticle-condensate collisions is suppressed due to energy
and momentum conservation; this suppression has already been observed via Bragg
spectroscopy of a 87 Rb condensate [165]. Shai Ronen and John Bohn have calculated
the elastic cross section for quasiparticle-condensate collisions based on Ref. [160]. We
have used their result to predict the lorentzian FWHM.
We are able to determine the expected width of the Bragg resonance by calculating
the convolution of all three lineshapes (pulse duration, inhomogeneous density, andquasiparticle-condensate collisions) and performing a gaussian fit. Figure 7.11 shows

CHAPTER 7. BRAGG SPECTROSCOPY

168

inhomogeneous density
sinc function
convolution
Gauss rms fit to convolutionlineshape (arb.)

1.0

0.5

0.0
12

14

16

18

frequency difference (kHz)
Figure 7.10: A comparison of the lineshape due to the inhomogeneous density of thecondensate (black dotted line) and the sinc-function lineshape of the Bragg pulse (green
dotted line). The convolution of these two function is the blue solid line. We extract
the total width from a gaussian fit (red dashed line) to the convolution.

fitted gaussian width (kHz)CHAPTER 7. BRAGG SPECTROSCOPY

169

3

2

1

0
0

data
2
2 0.5
a*x + (b*x + w )

1

2

3

4

5

6

lorentzian width (kHz)

Figure 7.11: Gaussian width from a fit to the convolution of the threep lineshape functions
versus the lorentzian width γ. The data are fit to the function a γ+ bγ 2 + w2 to extract
the parameters a and b. This fitting function resembles that which is used to understandthe total width of a Voigt profile [2]. A Voigt profile is the convolution of a gaussian
and a lorentzian.

CHAPTER 7. BRAGG SPECTROSCOPY

170

line width (kHz)

6

4

2

0
0

200

400

600

800

1000

scattering length (a0)Figure 7.12: Prediction of the Bragg resonance width as a function of scattering length.
The dotted lines correspond to different contributions to the total RMS width including
the lorentzian FWHM width due to condensate-excitation collisions (blue dashed line),
the gaussian RMS widths due to the inhomogeneous density (red dotted line), andthe pulse duration (green dash dotted line). The solid black line is the prediction for
the total width of the Bragg resonance based on a convolution of the profiles for eachcontribution. To get an idea of the size of beyond mean-field effects on the width, theblack open circles are the predicted total width including the LHY chemical potential
in the inhomogeneous density contribution. Calculations were performed with a meandensity of 8.5 × 1013 cm−3 and an excitation momentum of 2 h̄ kL .

CHAPTER 7. BRAGG SPECTROSCOPY

171

the calculated gaussian width as a function of the lorentzian width. For this simulation
we fixed the Bragg pulse duration to 0.25 ms; the lorentzian width and inhomogeneousdensity width were simultaneously varied by changing the scattering length.9
The
gaussian width σ of the Bragg resonance based on our convolution modeling is given by
v
Ã
!2 Ãr
!2
u
u
0.36
8 µ
t
2
σ = aγ + bγ +
+

tp

147 h

(7.16)

where the numerical factors a = 0.287 and b = 0.089 were determined from a fit to thedata points in Figure 7.11. In general the lorentzian width γ is added to a series ofgaussian widths wi viasσ = aγ +b γ2 +

Xwi2 .

(7.17)i

In Figure 7.12 we have plotted the contributions to the Bragg resonance linewidth
as a function of scattering length. Also shown in the figure is the combined linewidth
which is obtained from a convolution of the three contributing lineshapes. To get an
idea of how large beyond mean-field effects on the resonance width might be, the black
open circles in the figure show the total width in which the inhomogeneous density
contribution includes the LHY chemical potential instead of only the mean-field chemical
potential. Since the theories discussed here are for a homogeneous gas, the results shownin the figure are for the mean density of our trapped condensate.
7.6.3

Two-body scattering physics for k a ∼ 1

In this section we review the two-body scattering physics necessary to predict theline shift and linewidth of Bragg spectroscopy in the regime with k a ∼ 1. In particular, we will calculate the dependence of the symmetrized s-wave scattering amplitude(fs (k/2)) on the momentum wave vector of a quasiparticle-condensate collision. Thisquantity depends on the zero-energy scattering length. Near the 85 Rb Feshbach resonance, the zero-energy scattering length is given by a(B) = abg (1 − B−B∆peak ) with
∆ = 10.71(2) G, Bpeak = 155.041(18) G, and abg = −443(3)a0 . A significant advantage
of our system is the detailed knowledge of the Feshbach resonance parameters whichis available from previous work in the Wieman group [5] and other groups [135]. In
experiments performed at 900 a0 the B-field was set to 158.6 G, nearly 4 G from thepeak of the Feshbach resonance.
Following the basic scattering theory in the quantum mechanics text by Sakurai[48], the scattering amplitude is calculated in a partial-wave expansion. Here we consider
only s-wave scattering and kcm refers to the relative wave vector of the collision in thecenter-of-mass frame. The scattering amplitude is given byf=1
kcm cot(δ) − i k(7.18)

which includes the unitarity limit. The scattering phase shift δ is given in the effective
range approximation as
1 1
kcm cot(δ) = − + re kcm 2(7.19)a 2
9

We verified that this analysis is the same over the range of pulse lengths used in our experiments.

CHAPTER 7. BRAGG SPECTROSCOPY

172

and the effective range parameter re was determined analytically in Refs. [173, 174] to
be
µ
¶
2
1
(7.20)
re /β6 ={1 + [1 − χe (a/β6 )]2 }
3χe (a/β6 )2with β6 = (mC6 /h̄2 )1/4 , C6 = 4707(10) a.u. for 85 Rb [5], and χe = Γ(1/4)2 /2π whereΓ is the Euler-Gamma function. The dependence of the real part of the k-dependent
scattering amplitude on the phase shift is given by<(fs (k)) = −sin(2δ).
2k(7.21)

We can also determine the dependence of the elastic collision cross section (σ(k/2)) on
the momentum wave vector. It is useful to understand how much the elastic crosssection is modified by the unitarity limit and the effective range of the scattering
potential. But here we do not consider the suppression of the elastic cross section
for low-k quasiparticle-condensate collisions.10 The elastic collision cross section for
quasiparticle-condensate collisions is given by
σ(k) =

8π
sin2 (δ)
k2(7.22)

which is also obtained from the optical theorem result Im(f ) = k σ/ 8π. The rate ofquasiparticle-condensate collisions is increased by a factor of two due to correlation
effects. Both the elastic cross section and the k-dependent scattering amplitude areplotted in Fig. 7.13 as a function zero-energy scattering length. In this analysis therelevant momentum and energy of a collision are given in the center-of-mass frame by2
cm
pcm = h̄ kcm = h̄ kL and Ecm = pm
= 370 nK.
We compared the predictions of our analytic formulae for the elastic collision crosssection (Eqn. 7.22) and the real part of the scattering amplitude (Eqn. 7.21) to theexact results from a coupled-channels calculation supplied by Chris Greene’s group [3].
The k-dependent scattering length (Eqn. 7.21) and the elastic collision cross section(Eqn. 7.22) are shown in Figure 7.13 as a function of the zero-energy scattering length.
The excellent agreement between our model and the exact calculation is an indicationthat the effective range approximation we use is accurate.

7.7

Simulations of the BEC density

A comparison of our experimental Bragg spectra to theory requires a knowledgeof the condensate density. For a static condensate the density is extracted from ameasurement of the atom number and knowledge of the harmonic trap frequencies using
the Thomas-Fermi approximation. However many of our Bragg spectra are obtainedwith non-static condensates in which the size is changing. We use a variational solutionof the Gross-Pitaevskii (GP) equation to model the density based on a knowledge ofthe scattering length during our experiments [4].
We performed a simple experiment to test our understanding of the time dependent condensate density. A large condensate density oscillation was created by quickly
10
To our knowledge, no theoretical predictions exist for the cross section of phonon-like quasiparticles
interacting with a condensate when k a may take on any value.

173

2

elastic cross section (cm )CHAPTER 7. BRAGG SPECTROSCOPY

1E-8

1E-9

1E-10

analytic model
unitarity limit
Greene code
2
8π a

1E-11
100

1000

10000

a (a0)
1200

fs(k/2) / a0

1000
800
600
400
200
0
0

analytic modelGreene code

2000 4000 6000 8000 10000

a (a0)

Figure 7.13: (Top) The elastic collision cross section for quasiparticle-condensate collisions with wave vector kcm . (Bottom) The k-dependent scattering length derived fromthe real part of the scattering amplitude with wave vector kcm . Several quantities areplotted on each graph. The red line is the unitarity limit, the blue line is σ = 8πa2 ,the black line is from Ref. [3], and the green dotted line is our analytic results given by
Eqn. 7.22 and Eqn. 7.21.

CHAPTER 7. BRAGG SPECTROSCOPY

2.0

width (kHz)line shift (kHz)

1.0

0.5

0.0

174

0

1

2

3

time (ms)

4

5

1.5

1.0

0.5

0.0

0

1

2

3

4

5

time (ms)Figure 7.14: Measured Bragg line shift (left) and width (right) as a function of time. A
large density oscillation was excited in the condensate, and we detected that oscillation
with Bragg spectroscopy. The solid lines are a theoretical prediction based on the modelof Ref. [4] for the time-dependent condensate density.

CHAPTER 7. BRAGG SPECTROSCOPY

175

ramping the scattering length from 144 a0 to 500 a0 in 0.15 ms. We acquired Bragg
spectra as a function of time immediately following the scattering length ramp. A Bragg
pulse was applied for 0.25 ms, a duration much less than the radial breathing periodof the condensate. The measured Bragg resonance line shift and width are shown in
Figure 7.14 as a function of time. We subtracted the free-particle energy Er (k) fromeach measurement of the center to obtain the Bragg line shift. In the figure the solidlines represent the theoretical prediction for the line shift (Eqn. 7.9) and the width(Eqn. 7.16) including the results from our time-dependent simulations of the density.
There is excellent agreement between the measured line shift and the theory indicating
an accurate understanding of the condensate density for some range of parameters. The
measured Bragg width is consistently 200 Hz larger than the prediction, and this discrepancy is on order of our typical line shift uncertainty based upon a gaussian fit. As
discussed previously, it is important that we understand the Bragg lineshape in orderto accurately measure the center position of the resonance and extract the Bragg line
shift.

7.8

Bragg spectra of a strongly interacting 85 Rb BEC

We have acquired Bragg spectra over a very wide range of scattering length toprobe the excitation spectrum of a strongly interacting BEC. To begin the experiment,
we create a weakly interacting 85 Rb BEC in which the scattering length is 150 a0 . The
gas has a mean density of 2.1 × 1013 cm−3 and a condensate fraction greater than 85%.
We then apply a time-dependent sequence of scattering length ramps (Fig. 7.15 (aand b)) to increase the condensate density (Fig. 7.15 (c and d)) prior to the Bragg
spectroscopy pulse. We exploit the aspect ratio νz /νρ = 0.022 of the optical trap tosemi-independently control the radial and axial size of the gas, where νρ (νz ) is theradial (axial) trapping frequency. By lowering the scattering length to 30 a0 quickly
compared to νz−1 , a condensate axial size oscillation is excited which transiently increases
the density. Along its axis the condensate size shrinks by a factor of 2.2 in 120 ms. To
further increase the density the scattering length is ramped quickly compared to νρ−1 .
First the radial size is expanded by increasing to 200 a0 in 3.7 ms, then the scattering
length is ramped to the value aevolve in 0.2 ms and held there for 1 ms. The radial size
rapidly shrinks by approximately a factor of two in the final few ms of the sequence. We
model the time-dependent condensate density (Fig. 7.15 (c and d)) using a variational
solution to the Gross-Pitaevskii (GP) equation [4].
We measured the condensate size as a function of time during the scattering length
sequence to monitor the compression; the results are shown in Figure 7.16. In these data
the final few milliseconds of the sequence were omitted to study only the axial portionof the compression. Measurements were obtained after a 20 ms ballistic expansion
from the optical trap. The curves in the figure are the result of the GP-equation
simulation (Section 7.7) including the expansion from the trap. The excellent agreement
between our data and the simulation indicates the success of the compression sequencewith respect to the axial direction of the condensate. To estimate the uncertainty incondensate density we performed simulations assuming slightly different conditions. The
results for a 30% higher or lower density are shown by the dashed lines in the figure.
We have separately verified the success of the condensate compression during the final
few milliseconds of the sequence by performing a similar experiment.

CHAPTER 7. BRAGG SPECTROSCOPY

scattering length (a0)

500

176

(a)

(b)

(c)

(d)

400
300
200
100
0

-3

density (10 cm )

10

13

8
6
4
2
0

-120

-80

-40

0 -6

-4

-2

0

2

time (ms)Figure 7.15: (a) Scattering length sequence to increase the condensate density. Knowledge of the scattering length is derived from measurements of the magnetic-field and aprevious experimental calibration of the 85 Rb Feshbach resonance [5]. The Bragg pulse
is initiated at 0 ms immediately following a rapid (0.12 ms) increase to aBragg . Not
shown is the Bragg pulse timing, the ramp to 890 a0 to excite a condensate center-ofmass oscillation, and the scattering length ramps used during absorption imaging. (b)Expanded view of the scattering length ramps near 0 ms. The radial size of the BEC
is lowered by setting the scattering length to aevolve . (c, d) Calculated time-dependent
density during the scattering length sequence based on Ref. [4].

CHAPTER 7. BRAGG SPECTROSCOPY

30

radial size (µm)

80axial size (µm)

177

60
40
20
0
-100 -500

50

100 150 200

hold time (ms)

25

20

15
-100 -500

50

100 150 200

hold time (ms)Figure 7.16: Measurements of the condensate axial size (left) and radial size (right) as afunction of time during the compression sequence. The solid curves are the result of theGP-equation modeling of the condensate size. The dashed curves indicate the expected
sizes if the density were 30% higher or lower. See the main text for more details.

CHAPTER 7. BRAGG SPECTROSCOPY

178

Following the compression of the condensate density, the scattering length isramped in 0.12 ms to a value between 50 a0 and 900 a0 . The rate of this magnetic-fieldramp does not exceed 17 µs/G. The ramp is designed to be slow compared to typical2
many body time scales as determined by h̄/Eb , where Eb = mh̄a2 is the two-body binding energy associated with the Feshbach resonance. This ensures that the condensate
remains in its many-body ground state throughout our measurements. For each value of
scattering length, the compression sequence is optimized so that the predicted averagedensity during the Bragg pulse is 8.5 × 1013 cm−3 . Immediately following this scattering
length ramp the Bragg spectroscopy pulse is initiated. Since the condensate size rapidly
expands after the scattering length is set, the Bragg pulse duration is set so that only
a 30% change in the predicted condensate density occurs during the pulse.

% momentum transfer

64
2

900 a00600 a0100 a0

-2
-4
-6
-30

-20

-10

0

10

20

30

frequency difference (kHz)
Figure 7.17: Typical Bragg spectra at a scattering length of 100 a0 (blue crosses), 600
a0 (red circles), and 900 a0 (black squares). Lines are fits of the data as described in
the text.
Figure 7.17 shows measured Bragg spectra for three values of scattering length.
As interactions are increased from zero, mean-field theory predicts a continuous increasein the line shift. A comparison of our data at 600 a0 and 900 a0 displays a decreasing
shift with stronger interactions. This reversal in the line shift trend reflects the onsetof strong interactions in the condensate. It is important to note that the characterof the condensate excitations also changes with increasing scattering length. At smalla the excitations are free-particle-like with k ξ À 1, but at 900 a0 the character ofthe excitations is intermediate of the free particle and phonon limits with k ξ = 2.
Theoretical predictions of the line shift in our system must account for this fact. As we
increase the strength of interactions, the Bragg pulse duration is shorter and we observea larger width. The large width of the resonance makes a determination of the center
less certain. The plot shown here demonstrates the typical quality of our measured
Bragg spectra in the strongly interacting regime.

CHAPTER 7. BRAGG SPECTROSCOPY179

We have studied the Bragg resonance of a 85 Rb condensate (Fig. 7.18) as afunction of scattering length from the regime of weak interactions into the stronglyinteracting regime. The measured Bragg line shift shown in Fig. 7.18 (a) exhibits a linear
dependence on the scattering length below approximately 300 a0 , which is in excellent
agreement with Bogoliubov theory. However as the scattering length is increased further,
the Bragg resonance line shift deviates significantly from the Bogoliubov prediction.
This deviation reflects the onset of beyond mean-field corrections to the excitation
spectrum. The line shift is observed to saturate near 600 a0 and then it slightly decreases
as the scattering length is raised further. Our line shift data shown in Fig. 7.18 (a)exhibits a systematic dependence on the temperature of the sample which cannot be
neglected [167]. A small number of non-condensed 85 Rb atoms in the optical trap also
respond to the Bragg pulse, i.e. momentum is transferred to the non-condensed gas
as well. This is because the Bragg resonance width of the condensate and the noncondensed atoms becomes comparable beyond 500 a0 , therefore the response of the
non-condensed atoms at the condensate is approximately the same. We varied thetemperature of the condensate to characterize this effect; a small correction is applied
to our data to represent the expected line shift (black circles in Fig. 7.18 (a)) at zerotemperature. This temperature dependence is discussed in more detail in Section 7.9.
To understand how our observations compare to the theories of the Bragg line
shift discussed earlier in this chapter, we have also plotted these theoretical predictionsin Fig. 7.18 (a). Instead of evaluating the theories at the mean density of the trapped
condensate, a local density approximation is used in which we average the excitationenergy over the density profile of the condensate. Here we assume the density of thecondensate is given by a Thomas-Fermi profile. These predictions of the line shift also
account for observed number losses during our experiments, which are approximately
consistent with the previously observed three-body recombination rate of 85 Rb [12];For more details see Section 7.9. Note that our observed Bragg resonance line shift issignificantly smaller than the Bogoliubov prediction (blue upward triangles) when the
scattering length is larger than 600 a0 . Both the first Beliaev approximation, in whichn a3 = 0 but k a and k ξ may take on any value, and the Hartree-Fock-Bogoliubov
theory, in which n a3 and k a may take on any value but k ξ À 1, predict a reduced
Bragg line shift. In particular, the HFB theory predicts the smallest line shift and it
appears to be in reasonable agreement with our data even though that theory should
only be applied to the case of free-particle excitations. Our data disagrees with theprediction of the Beliaev second approximation, in which n a3 and k ξ may take on any
value but k a = 0. A simple estimate of the size of a momentum-dependent correction
to the second approximation, given by the difference between Bogoliubov theory andthe first Beliaev approximation (red open boxes in Fig. 7.18 (a)), does not appear to
explain the discrepancy.
The measured Bragg linewidth (Fig. 7.18 (b)) increases continuously as a function of scattering length. This trend is what we expect based on the predictions shownin Fig. 7.12. The primary reason for the increase is that a shorter Bragg pulse durationmust be used at larger a, since the condensate size rapidly expands. We expect that
there are two other significant contributions to the Bragg linewidth, the inhomogeneousdensity of the condensate and collisions during the Bragg pulse. The dashed curves
in Fig. 7.18 (b) indicate the relative size of the different expected contributions to
the width. The solid curve is a combination of the three contributions based on ourCHAPTER 7. BRAGG SPECTROSCOPY180

LHY correction
0.00

line shift (kHz)

5

0.06

0.17

0.30

0.43

0.54

400

600

800

1000

(a)

4
3
2
1
0
0

200

scattering length (a0)width (kHz)

6

(b)

4

2

0
0

200

400

600

800

1000

scattering length (a0)Figure 7.18: (a) Bragg line shift and (b) width extracted from a gaussian fit to theBragg resonance as a function of scattering length. The points are our observations
and the error bars represent standard deviations of the fit. The mean density during
the measurements ranges from 6 × 1013 cm−3 to 8.5 × 1013 cm−3 due to three-bodyrecombination during the Bragg pulse. In (a) the theory points, in order of smallest
to largest shift, correspond to the following theories calculated for our trapped gas
using a local density approximation: Hartree-Fock-Bogoliubov with n a3 and k a finite
but k ξ À 1 (black downward triangles), Beliaev first approximation with k a finite,
n a3 = 0 and k ξ any value (red boxes), Bogoliubov with n a3 = 0 and k a = 0, but k ξ
any value (blue upward triangles), and Beliaev second approximation with n a3 finite,k a = 0, and k ξ any value (magenta open circles). For reference the top axis shows
the size of the LHY correction to the condensate chemical potential. In (b) the crosses
represent a theoretical prediction for the resonance width under the conditions of ourexperiments. The open squares represent the predicted width when the LHY chemical
potential is used to calculate the inhomogeneous density contribution.

CHAPTER 7. BRAGG SPECTROSCOPY

181

convolution analysis (Eqn. 7.16). At small scattering length there is reasonable theoretical agreement with our data indicating that the relevant physics is included in ourmodel. At large scattering length our observed Bragg linewidth is slightly smaller thanour expectation for the total linewidth. The discrepancy is approximately 0.5-1 kHz
and we do not currently understand its origin in detail. The open black boxes in Fig.
7.18 (b) represents a preliminary estimate for beyond mean-field effects on the total
linewidth. For this line the LHY chemical potential is used instead of the mean-fieldvalue in calculating the contribution of the condensate inhomogeneous density. While
the total width including the beyond mean-field correction does appear to agree betterwith the data, the uncertainty of our data is not adequate to differentiate between the
different theoretical predictions shown.
At the time of this writing our understanding of the data and theory presented
here is still incomplete. Please see our forthcoming paper for the final details andconclusions of these measurements.

7.9

Systematic effects on the Bragg line shift

The data in Section 7.8 appears to demonstrate the onset of beyond mean-field
effects to the condensate excitation spectrum. The observed Bragg line shift is significantly smaller than Bogoliubov theory predicts. A less interesting alternative explanation of the data is that the density was not as large as expected, and therefore a smaller
mean-field shift accounts for our observations. We performed a number of experiments
designed to test the condensate density in the strongly interacting regime. As discussed
below, we investigated the rate of inelastic losses and condensate heating. We also used
Bragg spectroscopy to monitor the effects of condensate heating. Finally, we studied
the temperature dependence of our Bragg resonance line shift.
7.9.1

Measurements of BEC fraction from absorption images

We have experimentally characterized 85 Rb condensate melting and atom number
loss due to inelastic collisions during Bragg spectroscopy at an average density of 8.5 ×
1013 cm−3 . The same experimental sequence described in Section 7.8 is used to compress
the condensate density. We then increase the scattering length to some value between 50
a0 and 900 a0 in 0.12 ms. Instead of applying a Bragg pulse, we release the atoms fromthe optical trap to let the gas expand, and then take an absorption image. We fit theimage to a two-component gaussian to extract the number of atoms in the condensate
and the non-condensed gas. The results of this experiment are shown in Figure 7.19.
Prior to any compression of the condensate density, the fraction of the total gas in thecondensate was greater than 85%. We verified that the condensate fraction does notchange during the density compression sequence.
The absorption images clearly show that a bimodal distribution of condensate
and thermal gas always exists, even with a = 900 a0 . This means that a large fractionof the remaining atoms are a part of the condensate following the hold in the stronglyinteracting regime. However, the reduction in BEC fraction at large a is disturbing.
A loss of particle number in the condensate during Bragg spectroscopy would leadto a lower density, which might explain our observations of a smaller line shift. The
measured atom number loss shown here is included in the theoretical predictions that

1.0

4

0.8

BEC fraction

5

4

NBEC (x10 )CHAPTER 7. BRAGG SPECTROSCOPY

3
2
1
0
0

200 400 600 800 1000 1200

a (a0)

182

initial BEC
fraction

0.6
0.4
0.2
0.0
0

200 400 600 800 1000 1200

a (a0)

Figure 7.19: (Left) Measured atom number and (Right) measured condensate fraction
after the condensate density compression and scattering length ramp for the Bragg
pulse as a function of a. The solid line is the predicted atom number loss based on
the measured three-body recombination rate of 85 Rb in Ref. [6]. The measured atom
number loss is used to estimate the density during the Bragg pulse for the experiments
shown in Fig. 7.18.

CHAPTER 7. BRAGG SPECTROSCOPY

183

are compared to our data is Fig. 7.18. The amount of atom loss that occurs during
the Bragg pulse is roughly consistent with previous observations [6], but the loss is
not large enough the explain the discrepancy between the Bogoliubov theory and ourobservations.
We discovered another small factor the limits the density of the condensate during
the compression sequence. The detailed shape of the magnetic-field ramps is importantin achieving the largest compression. There are small discrepancies between the desired
field ramps and what actually happens due to the finite bandwidth of the bias coil servo
electronics. A small scattering-length dependent correction to the density of less than
9% has been applied.
7.9.2

Bragg spectra after ramping to large a

We have used Bragg spectroscopy to directly probe the condensate momentum
distribution after the scattering length is set for Bragg spectroscopy. The density ofa 85 Rb condensate is compressed using the sequence of scattering length ramps described in Section 7.8. Following the density compression, the scattering length isquickly ramped to either 600 a0 or 900 a0 , under these conditions the condensate isstrongly interacting. For each of these scattering lengths we acquired Bragg spectra as
a function of time after the scattering length ramp is complete. The condensate density
immediately begins to drop once the scattering length ramp is initiated and the Bragg
spectra reflect the lower density with a line shift close to zero. In particular, we areinterested in the width of the Bragg resonance since this quantity should be sensitive
to any unexpected energy generated in the strongly interacting regime. We will refer tothis unexpected energy as heating but we note that there is likely not enough time forthe gas to rethermalize.
Figure 7.20 shows the measured Bragg linewidth as a function of time immediately
following the scattering length ramp. In each plot of the figure there are two theoreticalpredictions for the Bragg width based on Eqn. 7.16. The blue curves represent thepredicted Bragg width in the absence of any heating, and the red curves represent thepredicted width in addition to some heat. We varied the amount of heat so that themodel fit the data. For the 600 a0 (900 a0 ) data 6 nK (20 nK) of heat was required tomake our model fit well. At the shortest times the data is somewhat more consistent
with the zero-heating model, while at longer times the Bragg width clearly indicatesthat some energy has been generated in the gas. Since we lack a detailed understandingof the Bragg lineshape in the strongly interacting regime it is difficult to conclusively
assign a cause to the extra width observed in the Bragg lineshapes.
In a related experiment we attempted to characterize the extra energy observedin the Bragg spectra. The density of the condensate was increased using our standard
sequence and the scattering length was ramped to 900 a0 in 0.12 ms. The scattering
length was held at this value for 0.1 ms before it was lowered to 150 a0 in 0.1 ms. Bragg
spectroscopy was performed at this smaller scattering length to test for heating effects
during the 0.1 ms hold at 900 a. In particular we are interested in the Bragg width
which was measured to be 1.65 kHz. If no heating occurs during the hold at 900 a0 ,then the width should be approximately 1.1 kHz.

CHAPTER 7. BRAGG SPECTROSCOPY

184

6

6

900 a0

4
20 nK
3
2
1
0
0.0

0.5

1.0

4
3

1

2.0

0
0.0

Bragg pulse time (ms)

6 nK

2

0 nK

1.5

600 a05width (kHz)width (kHz)

5

0 nK
0.5

1.0

1.5

2.0

Bragg pulse time (ms)Figure 7.20: Measured Bragg linewidth as a function of time after the scattering length
was increased to 900 a0 (left) and 600 a0 (right). The blue curves are a prediction basedon our model for the Bragg width, and the red curves add some condensate heating to
the prediction.

CHAPTER 7. BRAGG SPECTROSCOPY
7.9.3

185

The temperature dependence of the Bragg line shift

We studied the temperature dependence of the Bragg line shift in a regime ofdensity and scattering length in which beyond mean-field effects are important. At finite
temperature a condensate has two components, the gas of atoms not in the condensed
state which are characterized by a temperature, and the gas of atoms which all occupy
the lowest quantum state of the trapping potential. Both of these components are
sampled by Bragg spectroscopy. Therefore our measured Bragg spectra are actually a
combination of the two components. Bragg scattering from the thermal gas experiences
no mean-field shift and the Bragg resonance is centered at the free particle energy of 85 Rb, i.e. the line shift is zero. Obviously there is a line shift for the condensate component.
If the fraction of atoms in the condensate is small, the thermal gas response to the Bragg
light should systematically shift the Bragg resonance towards the free-particle energy.
So far in this chapter, we have made predictions for the Bragg lineshape based on a
zero temperature BEC. In this section we estimate the systematic shift of the Bragg
resonance caused by the thermal gas.
To model the effects of finite temperature on our measured Bragg spectra, we
simulated the Bragg lineshape of the condensate and the thermal gas with gaussians ofdifferent center and width.11 The width of the thermal gas gaussian is related to thetemperature of the gas,12 while the width of the condensate gaussian is estimated fromour experimental data. The thermal gas gaussian is centered at the free-particle energyof 85 Rb. We combine the two gaussians and fit the result to extract the combined center
position. If the condensate fraction is very small, the combined center should be close
to the free-particle energy. For large condensate fraction the opposite limit applies.
Figure 7.21 shows the measured Bragg resonance center position as a functionof condensate fraction.13 Here the condensate fraction is measured after Bragg spectroscopy so that the value quoted is a lower limit. The data was acquired with a densityof approximately 1014 cm−3 and at a scattering length of 500 a0 . The curve in the figureis the result of our of a fit to the data based on the simulation described above. The
reasonable agreement between our data and the model is encouraging. It indicates thatthe size of the systematic correction to our measured Bragg line shift is small comparedto the correction of the Bogoliubov excitation spectrum due to beyond mean-field effects. In particular, we find that the line shift is typically changed by less than 1 kHz as
the condensate fraction increases from 0.5 to 1. In Figure 7.18 the data points shownas small filled circles indicate an estimate of the zero temperature Bragg line shift.
11

In our model we assume that the response of the thermal gas to the Bragg scattering pulse is thesame as that of the condensate. This seems to be a reasonable assumption since the Bragg width of
the thermal gas is smaller than the free particle energy. If the Bragg width of the thermal gas were onorder or larger than the free-particle energy we would expect a significant reduction in the thermal gas
response.
p
12
The Bragg width of a thermal gas is 2kπ kB T /m.
13
We produce the largest values of condensate fraction in our experiments by stopping the opticaltrap evaporation very close to the bottom of the trap. In addition we typically hold the optical trap
depth at a very low value for a few hundred ms following evaporation to let thermal atoms fall out of
the trap.

CHAPTER 7. BRAGG SPECTROSCOPY

186

5
14

-3

line shift (kHz)n = 10 cm , a = 500 a0

4
3
2
1
0
0.00

0.25

0.50

0.75

1.00

BEC fraction (N0/Ntot)
Figure 7.21: Measured Bragg line shift as a function of the condensate fraction. We thefraction is zero the temperature of the gas is slightly below the transition temperature.
Our data indicate a significant increase in the Bragg lineshift as the condensate fraction
is increased. The curve represents the prediction of our model where the lineshift at
100% condensate fraction is 18.3 kHz.

CHAPTER 7. BRAGG SPECTROSCOPY

7.10

187

Bragg spectra at lower density

We have also acquired Bragg spectra over an even wider range of scattering length,
but for these measurements the density of the condensate was lower. A 85 Rb BEC was
created with an static average density of 2.0×1013 cm−13 and a scattering length 144 a0 .
To perform Bragg spectroscopy the scattering length was quickly increased in 0.12 ms
to a value in the range 144 a0 to more than 3000 a0 . The condensate density begins todrop as soon as the scattering length is changed. Therefore the Bragg duration was set
to be 0.25 ms. The average density during the Bragg pulse is calculated using the GPequation model. The character of the excitations ranges from free-particle-like at smalla (k ξ = 12 at 100 a0 ) to intermediate between phonon-like and free-particle-like at largea (k ξ = 2.5 at 2500 a0 ). Figure 7.22 shows the measured Bragg line shift and width as afunction of scattering length. Also in the figure we show a set of theoretical predictions
for the Bragg line shift and the width. The same theory that has been described above
is used, but with this data we can compare over a larger range of scattering length.
There are three potential problems with this data related to entering the stronglyinteracting regime. At a scattering length of 2000 a0 or greater we observe atom number
loss during Bragg spectroscopy. A possible source of this loss may be a projection ofcorrelated atoms pairs into molecules near the peak of the Feshbach resonance. Also
beyond 2000 a0 the two-body molecule binding energy associated with the Feshbachresonance is less than 10 kHz. With this small binding energy, the scattering length
ramp may not be slow enough compared to the timescale of many-body physics inthe condensate. Finally, our measurements of the Bragg resonance width exceed thepredicted value based on the known density and scattering length.

7.11

Future work

A variety of future experiments might be possible with the our system. It might
be interesting to study the effects of the molecular state associated with the Feshbachresonance on the Bragg spectrum. This could be accomplished by driving two-photon
Bragg transitions between a pair of colliding condensate atoms and a bound Feshbach
molecule. Since Bragg transitions conserve both energy and momentum, the resonancecondition for a Bragg transition is modified by the molecule binding energy to be E(p) =h̄2+Vint where M is the reduced mass, p is the momentum of the molecule,p2 /2 M − 2 M
a2and Vint represents the effect of atom-molecule interactions. These atom-molecule Bragg
spectra would represent a different way to create Feshbach molecules and it might allow
the interactions between atoms and Feshbach molecules to be studied.
It might also be interesting to study the excitation spectrum of a condensate forsmaller momentum excitations than we have done so far. One complication of studying
these low momentum excitations is that the size of our signal for Bragg spectroscopy is
reduced. Since the total momentum imparted to the condensate is smaller, the amplitude of condensate COM oscillations in the trap is reduced. The signal-to-noise ratioof the measurements presented in this chapter may be large enough to perform initial
measurements of phonon-like excitations. It is also possible to detect the momentum imparted to the condensate by directly counting the the number of photons scattered fromthe Bragg laser beams [175]. By dramatically lowering the intensity of one Bragg beam,
and increasing the other by a corresponding amount, the change in photon number that

CHAPTER 7. BRAGG SPECTROSCOPY

188

2line shift (kHz)

(a)
1

0

-1
0

1000

2000

3000

a (a0)
6(b)width (kHz)5
4
combined width

32collisional width
pulse width

1
inhomogeneous width

0
0

1000

2000

3000

a (a0)

Figure 7.22: (a) Bragg line shift as a function of scattering length with an averagedensity of approximately 2.0 × 1013 cm−3 . The theory lines, in order of smallest tolargest shift, correspond to the following theories calculated for our trapped gas using
a local density approximation: Hartree-Fock-Bogoliubov with n a3 and k a finite but
k ξ À 1 (black solid line), Beliaev first approximation with k a finite, n a3 = 0 and k ξany value (red dashed line), Bogoliubov with n a3 = 0 and k a = 0, but k ξ any value(blue dotted line), and Beliaev second approximation with n a3 finite, k a = 0, andk ξ any value (magenta dash-dotted line). (b) Bragg width as a function of scattering
length at a density of 2.0 × 1013 cm−3 . The lines show the contributions of the pulse
duration (green dashed line), the inhomogeneous density of the condensate (red dashedline), and the width due to collisions during Bragg spectroscopy (blue dshed line). The
black solid line is a combination of three contributions following Eqn 7.16.

CHAPTER 7. BRAGG SPECTROSCOPY

189

corresponds to the number of atoms that make a Bragg transitions can be detected. We
expect that the signal-to-noise ratio of this method could be approximately 20.

Chapter 8
Conclusion

This thesis has presented my work with an ultracold gas of 85 Rb and 87 Rb atoms.
The second and third chapters of this thesis are dedicated to describing the new apparatus that was constructed. In designing and assembling the system, I tried to makethe apparatus as robust and versatile as possible. I hope this will enable a wide rangeof new experiments using quantum-degenerate Bose gases with tunable interactions tobe performed. Where relevant I have tried to write this thesis in such a way that piecesof the apparatus can be reproduced in other quantum gas experiments at JILA.
The fourth chapter contains a recipe for robust creation of 85 Rb condensates withup to 80,000 atoms, a significant improvement over the previous record. We are further
able to create either a singles species 87 Rb condensate or a dual BEC of the two species.
This has required a thorough study of sympathetic and simultaneous cooling of 85 Rb
and 87 Rb. Our versatile system has opened the door to a variety of experiments utilizing
widely tunable interactions.
I have had the chance to perform a set of experiments using the tunability ofinterparticle interactions via magnetic-field Feshbach resonances. These include studiesof the first heteronuclear Feshbach molecules [155], of tunable phase separation in a
two-species BEC [176], and of the excitation spectrum of a strongly interacting 85 Rb
condensate using Bragg spectroscopy [177]. The Bragg spectroscopy work has been themost exciting physics to work on; we have seen initial evidence of long-sought beyond
mean-field corrections to the excitation energy of a superfluid. And I hope the work
described in this thesis represents only the start of a series of experiments that will
elucidate strongly correlated Bose gases.

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