Experiments with Feshbach molecules in a
Bose-Fermi mixture
byMichele Lynn Olsen
B.A. Physics, State University of New York at Geneseo,
2001

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulllmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2008

iii
Olsen, Michele Lynn (Ph.D., Physics)
Experiments with Feshbach molecules in a Bose-Fermi mixture
Thesis directed by Dr. Deborah S. Jin

In this thesis, I present studies of the creation of ultracold heteronuclear
molecules from a dilute gas mixture of 87 Rb and 40 K. Using a Fano-Feshbachresonance, atom pairs are associated into molecules via time-dependent magnetic
elds. Slow magnetic-eld sweeps are used to make large samples of highly vibrationally excited KRb molecules, while rapid magnetic-eld pulses are used tocreate quantum superpositions of atom and molecule states. Rabi- and Ramseytype experiments demonstrate coherent oscillations between atoms and molecules.
This new type of quantum superposition involves particles of dierent statistics(i.e. bosons and fermions) and demonstrates atom-molecule coherence without a
Bose-Einstein condensate (BEC). These atom-molecule oscillations may provide a
unique way to probe the many-body behavior of a strongly interacting Bose-Fermi
mixture.

iv

Acknowledgements

The work presented in this thesis could not have happened without thecontributions of many dierent people. First, I have to thank my advisor, Debbie,
for giving me the opportunity to work on this project and for all her help over
the years. In addition to experimental and theoretical guidance, Debbie worked
side-by-side with me in the lab for a while as a post post-doc, and even rewrote
our experiment control software. More importantly, though, her uncanny ability
to reduce a problem to its key elements has taught me how to attack almost
any problem. I can only hope that some of her determination, enthusiasm, andphysical intuition has rubbed o. I would also like to thank her for a bit of personalunderstanding. She was very supportive when I hit my breaking point with theexperiment after months and months of frustration and personal struggles andmade every eort to help.
As part of the ultracold atom group, I have also benetted from two additional advisors, Eric Cornell and Carl Wieman. They were never at a loss forsuggestions on how to improve a particular experiment or how to x a laser or
how best to present data in a poster or talk. Their years of experience with ultracold experiments have built a seemingly limitless pool of knowledge and intuition.
They are among the few people you actually take seriously when they suggest
taking a hammer or wrench to your vacuum system (okay, the ion pump)I haveto admit, that was kind of fun. While I never particularly relished giving talks at

v
the group seminars, I am thankful for the emphasis Carl, Eric, and Debbie placedon giving good talks and clearly presenting scientic data.
During my time in the lab, I've been fortunate to have worked with somewonderful people. Jon Goldwin and Shin Inouye were the grad student and postdoc on the experiment when I joined the lab. Jon and Shin were patient with all
my questions and taught me everything from how to align a laser beam to how
to design experiments. I learned from them how to run a lab (and sometimes
how not to). Shortly after Jon and Shin moved on to bigger and better things,
Tyler Cumby joined our lab as a grad student. Though Ty and I haven't always
seen eye to eye, his quiet dedication has been instrumental in the results of thisthesis. John Perreault joined us as a post-doc two years ago. John came on theexperiment just when I needed him most. He was a breath of fresh air at a time
when the experiment was stuck in the muck. His new perspective helped us get
the experiment back on its feet and moving forward. John's experience with interferometry proved indispensable in our most recent work, but more importantly,
his easy-going nature and optimism made working in the lab fun again. A newgrad student, Ruth Shewmon, joined the lab only a few months ago. Already, I
can see that she will be a valuable addition, and I hope that I have left her withan experiment that will be productive for years to come.
There are several other people who worked on the experiment with me anddeserve special thanks. Brett De Paola was a Visiting Fellow who worked in ourlab during my rst year at JILA. He helped me build and temperature stabilize
my rst laser (which is still in operation today). Jian-ying Lang, a grad student
who worked with me for a while, helped me redesign, rebuild, and reoptimize theapparatus. I've never worked with anyone with as strong a work ethic as Jianying. Jesper Bertelsen spent a few months in the lab as an exchange student andhelped with some of our early eorts at molecule creation. I'd also like to thankvi
Cindy Regal, who took time away from her own experiment to help me build
my new vacuum system. Thanks also go to two undergrads who worked with us,
Bonna Newman and Margot Phelps.
I would also like to thank a few additional members of the Tri-group whohave made contributions large and small. Paul Haljan taught me about imaging
and camera controls and rst introduced me to the worlds of Frame Transfer andKinetics Mode. Scott Papp could always be counted on for advice (sometimes
unsolicited) on how to improve various aspects of the experiment. Josh Zirbel
and Kang-Kuen Ni oered new expertise on Rb and K and a bit of friendly
competition in making molecules. Countless other grad students and post-docs
have impacted the workings of the experiment and made JILA a fun place to workthrough parties and lunches and ski trips and chatting in labs and hallways andsemi-annual grudge matches with the Ye group.
One of the greatest aspects of JILA is its sta.

Without the membersof the electronic shop, machine shop, scientic reports oce, supply oce andadministration, none of the experiments we do would be possible. In particular,
I'd like to thank Tom Foote, Hans Green, and Todd Asnicar of the machine shop.
Tom machined the forms for our IP trap, and Hans and Todd built me not one,but two collection cells, complete with getters and ampoules, and the second one
in a turn-around time of a week or two! In the electronics shop, I'd like to thankJames Fung-A-Fat and Terry Brown for always answering my questions about the
many servos we use, and Carl Sauer for painstakingly mode-matching microwave
coils for me. I'd also like to thank Julie Phillips in the SRO for help and support
with our most recent paper.
Throughout the years, there have been a handful of people that I could
always count on for a laugh, a place to hang out, or a shoulder to cry on. Ian
Coddington, Rachel Doriese, Chris Jones, Etienne Gagnon, Odele Hofmann Brian

vii
Laughman, Nathan Miller (now Tomlin), Kerry Tomlin, Nick Wagner, Mike and
Eva Ziemkiewicz all deserve special thanks. Chris, Etienne, Brian, Nathan, Nick,and I all started out in the physics program together. We bonded over interminable
homework sets of classical and quantum mechanics in the tower and have been
friends ever since. From camping and ski trips and bull-riding birthday parties
and volleyball to countless pint nights, trivia nights, game nights, girl's nights andlet's-go-grab-dinner nights, they've all made my grad school years unforgettable.
Special thanks go to Etienne, my roommate of 4 years, for putting up with mymesses and being someone to come home to, and Chris, for being my condante
over TV and cocktails (a few too many, maybe?), and Kerry, what can I say?
she's my best friend. And, of course, I have to thank Nick, my eternal partnerin adventure. I've never known Nick to turn down a day of skiing or hiking, andin our quasi-annual vacations, we've visited 15 national parks and monuments, 3countries, one Indian reservation and who knows how many miles of highway. I
cannot imagine my life without any of these fantastic friends.
Last, but denitely not least, I'd like to thank my family, Sheila, Alan, andMelissa, for all their love and support through the years.

viii

Contents

Chapter
1 Introduction

1

1.1

Fun things to do with Bose-Fermi mixtures . . . . . . . . . . . . .

2

1.2

The state of aairs . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Atom-molecule coherence . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.1

Quantum superpositions . . . . . . . . . . . . . . . . . . .

7

1.3.2

Atom-molecule superpositions in a Bose-Fermi mixture . .

8Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4

2 Experimental improvements

11

2.1

Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

New vacuum system. . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2.1

Large, controllable K MOTs . . . . . . . . . . . . . . . . .

17

2.2.2

Model of a two-species MOT . . . . . . . . . . . . . . . . .

19

2.3

Keeping all those atoms. . . . . . . . . . . . . . . . . . . . . . .

26

2.3.1

Fiber coupling the MOT light . . . . . . . . . . . . . . . .

26

2.3.2

Spin state purication . . . . . . . . . . . . . . . . . . . .

30

2.3.3

Quadrupole trap to IP trap transfer . . . . . . . . . . . . .

34

2.4

Too much of a good thing? . . . . . . . . . . . . . . . . . . . . . .

35

2.5

Magnetic-eld curvature compensation . . . . . . . . . . . . . . .

41

ix
2.6

Fast magnetic-eld pulse circuit and eddy current correction . . .

42

2.7

High-eld absorption imaging . . . . . . . . . . . . . . . . . . . .

48

2.8

Isolating the track controller that spews noise all over the lab

53

. .

3 Creating molecules with a Fano-Feshbach resonance

58

3.1

Fano-Feshbach resonances . . . . . . . . . . . . . . . . . . . . . .

58

3.2

Locating the resonances with atom loss . . . . . . . . . . . . . . .

60

3.3

KRb Feshbach molecules . . . . . . . . . . . . . . . . . . . . . . .

63

3.3.1

Magnetic-eld sweeps . . . . . . . . . . . . . . . . . . . . .

65

3.3.2

Density dependence of molecule creation sweep rates

. . .

69

3.3.3

Why we no longer use a 1075 nm broadband laser . . . . .

70

3.4

Molecule imaging . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

3.5

Locating the resonance with molecule sweeps . . . . . . . . . . . .

74

4 Semiclassical Monte Carlo simulations of molecule conversion

76

4.1

Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . .

77

4.2

Two-component systems . . . . . . . . . . . . . . . . . . . . . . .

78

4.3

40

K-87 Rb Bose-Fermi mixture in particular . . . . . . . . . . . . .

80

4.3.1

Number ratio . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.3.2

Gravitational sag . . . . . . . . . . . . . . . . . . . . . . .

89

4.3.3

Molecule conversion in expansion . . . . . . . . . . . . . .

92

5 Coherent atom-molecule oscillations

96

5.1

Rabi and Ramsey oscillations in the atom + photon system

. . .

96

5.2

Criteria for observing oscillations . . . . . . . . . . . . . . . . . .

102

5.3

Ramsey oscillations . . . . . . . . . . . . . . . . . . . . . . . . . .

104

5.3.1

Empirical optimization . . . . . . . . . . . . . . . . . . . .

105

5.3.2

Frequency dependence on B . . . . . . . . . . . . . . . . .

108

x

5.4

5.5

5.3.3

Amplitude dependence on density . . . . . . . . . . . . . .

110

5.3.4

Ramsey oscillations in expansion. . . . . . . . . . . . . .

112

Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

5.4.1

Ramsey oscillations without Rabi oscillations? . . . . . . .

114

5.4.2

Rabi oscillations in expansion . . . . . . . . . . . . . . . .

118

5.4.3

Amplitude dependence on B . . . . . . . . . . . . . . . . .

121

5.4.4

Frequency dependence on B . . . . . . . . . . . . . . . . .

121

Oscillation decoherence/dephasing . . . . . . . . . . . . . . . . . .

123

5.5.1

Non-technical mechanisms . . . . . . . . . . . . . . . . . .

123

5.5.2

Technical mechanisms

131. . . . . . . . . . . . . . . . . . . .

6 As yet unsubstantiated Monte Carlo predictions

134

6.1

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

6.2

Dependence of coherence times on temperature

136

6.3

Dependence of coherence times on quantum degeneracy. . . . . . . . . .
. . . . .

138

7 Conclusions and Outlook

142

Bibliography

144

Appendix
A Fitting dephased oscillations

154

B Nearest-neighbor distance and density-weighted density for Gaussian clouds
with dierent sizes, numbers, and aspect ratios.

161

xi

List of Figures

Figure1.1

Bosons and fermions at zero temperature.

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

2

2.1

CAD drawing of the new vacuum system. . . . . . . . . . . . . . .

15

2.2

CAD drawing of the new vacuum system (top view).

. . . . . . .

16

2.3

Failed MOT cell with Rb ampoules. . . . . . . . . . . . . . . . . .

18

2.4

Expected pressure dependence of single-species MOT numbers. . .

22

2.5

Expected dependence of the Rb MOT number on the partial pressures of Rb and K. . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

23

Expected dependence of the K MOT number on the partial pressures of Rb and K. . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.7

Measured MOT numbers when running only the K getter.

25

2.8

Hyperne structure of the relevant ground and excited states of 40 K

. . . .

and 87 Rb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Fiber coupling the MOT light. . . . . . . . . . . . . . . . . . . . .

28

2.10 Eect of optical molasses on the temperature of the Rb MOT. . .

31

2.11 Rb spin states loaded into the quadrupole trap. . . . . . . . . . .

33

2.9

2.12 Schematic of the cloud shapes in the quadrupole and Ioe-Pritchard
traps.

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

35

2.13 Temperature dependence of the fraction of atoms transferred to the
IP trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

xii
2.14 Quadrupole to IP trap transfer. . . . . . . . . . . . . . . . . . . .

37

2.15 Evaporation in the magnetic trap.

39. . . . . . . . . . . . . . . . .

2.16 False-color absorption images of a degenerate Fermi gas and a BoseEinstein condensate.

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

40

2.17 Magnetic-eld curvature cancellation. . . . . . . . . . . . . . . . .

43

2.18 Measured axial magnetic-eld inhomogeneity.

. . . . . . . . . . .

44

2.19 Circuit used to generate fast magnetic-eld ramps. . . . . . . . . .

45

2.20 Magnetic-eld ramp determined from the current through the auxiliary coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.21 Decaying magnetic eld after a magnetic-eld ramp. . . . . . . . .

47

2.22 Schematic of the eddy current model. . . . . . . . . . . . . . . . .

48

2.23 Magnetic-eld ramps that compensate for eddy elds. . . . . . . .

49

2.24 Magnetic-eld ramps with eddy current compensation. . . . . . .

49

2.25 Decaying magnetic eld after a fast magnetic-eld ramp. . . . . .

50

2.26 Magnetic-eld ramps that compensate for eddy elds. . . . . . . .

51

2.27 Measured magnetic-eld deviation after a fast ramp.

. . . . . . .

51

2.28 Laser frequencies for trapping, pumping, probing Rb and K. . . .

54

2.29 Schematic of the high-eld imaging conguration. . . . . . . . . .

57

2.30 Noise broadcast by the Soloist cart controller. . . . . . . . . . . .

57

3.1

Schematic of a Fano-Feshbach resonance. . . . . . . . . . . . . . .

59

3.2

Magnetic-eld dependence of the scattering length a at a FanoFeshbach resonance. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

61

Magnetic-eld dependent atom loss at Fano-Feshbach resonances
between 40 K atoms in the |9/2, −9/2i state and 87 Rb atoms in the

3.4|1, 1i state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Molecule association with magnetic-eld sweeps. . . . . . . . . . .

66

xiii
3.5

Molecule creation via magnetic-eld sweeps through a Fano-Feshbachresonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

68

Measured density dependence of adiabatic magnetic-eld sweep
speeds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.7

KRb molecular potentials. . . . . . . . . . . . . . . . . . . . . . .

72

3.8

Determination of the Fano-Feshbach resonance position via molecule
dissociation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

Temperature dependence of the molecule conversion in single-species
experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

75

79

Temperature dependence of the molecule conversion in a bosonicmixture of 85 Rb and 87 Rb. . . . . . . . . . . . . . . . . . . . . . .

81

4.3

A simplied energy level diagram for rf association. . . . . . . . .

82

4.4

Temperature dependence of the molecule conversion in a BoseFermi mixture of 87 Rb and 40 K. . . . . . . . . . . . . . . . . . . .

84

4.5

Eect of number ratio on molecule conversion for xed TC . . . . .

86

4.6

Eect of number ratio on molecule conversion for xed TF . . . . .

87

4.7

Adiabatic molecule conversion with an excess of fermions. . . . . .

90

4.8

Eect of gravitational sag on molecule conversion. . . . . . . . . .

91

4.9

Molecule conversion in expansion for two identical Bose gases. . .

94

4.10 Molecule conversion in expansion for a mixture of Rb and K. . . .

94

5.1

Energy level diagram for an atom in a radiation eld. . . . . . . .

98

5.2

Eigenstates of the dressed atom. . . . . . . . . . . . . . . . . . . .

98

5.3

Ramsey's method of separated oscillatory elds and analogous systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

5.4

Empirical optimization of Ramsey oscillations. . . . . . . . . . . .

106

5.5

Ramsey-type atom-molecule oscillations. . . . . . . . . . . . . . .

109

xiv
5.6

Magnetic-eld dependence of the measured oscillation frequency forthe molecule number in Ramsey-type double-pulse experiments. .

5.7

Density dependence of the amplitude of the molecule oscillationsin Ramsey-type experiments. . . . . . . . . . . . . . . . . . . . . .

5.8

113

Empirical optimization of pulse time for a cloud after 1 ms of expansion from the trap. . . . . . . . . . . . . . . . . . . . . . . . .

5.9

111

115

Time dependence of the molecule number in a single-pulse experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

5.10 Rabi-type atom-molecule oscillations. . . . . . . . . . . . . . . . .

119

5.11 Rabi-type atom-molecule oscillations with varying density clouds.

120

5.12 Magnetic-eld dependence of the peak-to-peak amplitude and frequency of the single-pulse oscillations. . . . . . . . . . . . . . . . .

122

5.13 Schematic of the energy eigenvalues of a trapped atom pair in thevicinity of a Fano-Feshbach resonance.

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

124

5.14 Illustration of dephasing. . . . . . . . . . . . . . . . . . . . . . . .

126

5.15 Calculated oscillations for typical in-trap conditions with coherence
times limited by the energy distribution of the pairs.

. . . . . . .

128

5.16 Calculated oscillations for typical expansion conditions with coherence times limited by the energy distribution of the pairs. . . . . .

129

5.17 Atom motion in the trap and in expansion. . . . . . . . . . . . . .

131

5.18 Magnetic-eld dependence of the coherence times of Ramsey oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1

Expected temperature dependence of the coherence time of atommolecule oscillations. . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

133

137

Expected dependence of the atom-molecule oscillation coherencetime on quantum degeneracy.

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

140

xv
6.3

Expected dependence of the nearest-neighbor atom-molecule oscillation coherence time on quantum degeneracy. . . . . . . . . . . .

141

A.1 Frequency distribution due to an inhomogeneous magnetic eld. .

156

A.2 Calculated distribution of relative kinetic energies, E , of nearestneighbor pairs in a 200 nK Boltzmann gas. . . . . . . . . . . . . .

158

A.3 Calculated atom-molecule oscillation for a 200 nK gas with a Boltzmann energy distribution at B = 546.135G.

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

158

A.4 Calculated distribution of the relative kinetic energies E of atom
pairs associated into molecules in the Monte Carlo simulation. . .

159

A.5 Calculated atom-molecule oscillation for pairs with the distribution
shown in Fig. A.4.

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

159

B.1 Illustration of the calculation of the typical nearest-neighbor distance for a mixture with dierent numbers of atoms.

. . . . . . .

162

B.2 Distribution of nearest-neighbor distances dnn for a simulated Gaussian cloud. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

B.3 Density dependence of the calculated nearest-neighbor distance. .

165

Chapter 1
Introduction

All of the light and matter in the universe can be divided into two categories:
bosons and fermions. They are distinguished from one another by their quantum
mechanical spins. Bosons have integer values of spin, while fermions have halfinteger values of spin. Examples of bosons in nature are photons and phonons,
while electrons, protons, and neutrons are fermions. While the distinction between
these two categories is not necessarily obvious in our daily lives, whether particles
have whole- or half-integer spin determines their quantum mechanical behavior.
At suciently low temperatures, a group of identical bosons will form a BoseEinstein condensate (BEC), in which all of the particles occupy the ground state.
This macroscopic occupation of a single quantum state is prohibited in groups ofidentical fermions. Instead, each particle must occupy its own state, leading to aFermi sea with singly lled energy states up to the Fermi level. This leads toa vast dierence in energy between groups of fermions and bosons at T = 0 (seeFig. 1.1).
In this thesis, I present experiments with an ultracold mixture of bosons(87 Rb) and fermions (40 K) at an interspecies Fano-Feshbach resonance. I will emphasize experiments that utilized the resonance to create heteronuclear molecules
and ultimately a coherent superposition of atoms and molecules. At the time
this work began, experiments with ultracold atomic gases at Fano-Feshbach reso-

bosons2

fermionsEF

Figure 1.1: Bosons and fermions at zero temperature. Identical bosons form a
BEC in which all of the particles occupy the ground state. Fermions form a Fermisea, in which one fermion occupies each energy state up to the Fermi energy, EF .

nances abounded and were making great headway into understanding the natureof interactions in BECs in bosonic samples [1, 2, 3, 4, 5] and exploring pairingin the BCS-BEC crossover in fermionic systems [6, 7]. Many of these singlespecies experiments involved the association of molecules with a Fano-Feshbachresonance [8]. There were, however, relatively few experiments exploring interactions between atoms with dierent nuclei at ultracold temperatures [9, 10, 11],and none had yet created heteronuclear molecules via Feshbach association. These
heteronuclear experiments allow for the study of interactions between atoms withdiering quantum statistics, that is, bosons and fermions and, therefore, open up
a host of phenomena to study.

1.1

Fun things to do with Bose-Fermi mixtures
The Bose-Fermi mixture is more than just the sum of its parts. Alone, ourspin-polarized Fermi gas is non-interacting at ultralow temperatures, while thebosons are weakly repulsive (aRb = 98.98 a0 ) [12, 13], which leads to a stable
BEC in suciently cold clouds. When mixed together, the interplay between the

3inter- and intraspecies interactions can lead to a wide variety of phenomena. If
the interspecies interaction is strong and attractive, this attraction can overcome
the boson-boson repulsion and cause the system to collapse [14, 9, 15] as in thebosenova observed in condensates with attractive interactions [16, 17]. A strong,
repulsive interspecies interaction will cause phase-separation [18, 19] in which thetwo species are forced to occupy physically separate regions of the trap. In between
these two extremes, however, there are predicted to be some subtle and some exoticeects of the interspecies interaction.
The eective interactions between the bosons can be altered by the presence of fermions. Repulsive Bose-Fermi interactions are expected to enhance analready repulsive boson-boson interaction, resulting in a lower critical temperature for Bose-Einstein condensation, while attractive interactions should reducethe boson-boson repulsion and shift the transition to higher temperatures [20].
In highly elongated traps, changing the eective boson-boson interactions byabruptly changing interspecies interactions is expected to create two-component
solitons in degenerate mixtures [21, 22, 23]. Just as the presence of fermions
can alter the boson-boson interactions, the presence of the bosons can aect thefermion-fermion interactions. With strong enough interspecies interactions (ofeither sign), phonons in a BEC can induce an eective fermion-fermion attraction, which may lead to Cooper pairing in a Fermi gas that would otherwise be
non-interacting [24, 25] or enhance the attraction of a two-component Fermi gas,
resulting in an increase in the critical temperature for pairing [26, 27].
Clearly, trapped Bose-Fermi mixtures are rich physical systems to study,
but the addition of a periodic trapping potential, such as an optical lattice opens
up a host of additional avenues of study. The system of a Bose-Fermi mixture on alattice is predicted to exhibit a multitude of novel phases including superuid andinsulating phases in addition to density waves and Fermi liquids [28] and in low-4dimensionality, a supersolid phase [29, 30]. Boson-mediated Cooper pairing is also
expected to occur in the lattice, oering a controllable system directly analogous tosuperconductivity in solid-state systems, which may shed additional light on high-TC superconductors [31, 32]. The addition of disorder to a lattice is expected tolead to Anderson-like localization of the bosons [33] and quantum spin glasses [34].
Bose-Fermi mixtures in optical lattices oer a playground for experimentalists thathas only begun to be explored with observations of localization of bosons due tosmall impurities of fermions [35] and shifts of the Mott Insulator transition due
to Bose-Fermi interactions [36].
The nal exciting avenue for Bose-Fermi mixtures I will mention is the subject of this thesisthe study of heteronuclear molecules. We have used an atomic
Bose-Fermi mixture to create highly vibrationally excited molecules and study
quantum superpositions of atoms and molecules [37]. These superpositions mayoer a probe of the many-body behavior of a strongly interacting Bose-Fermi mixture. Additionally, heteronuclear molecules can have a signicant electric dipole
moment when in the ro-vibrational ground state. Therefore, over the past several
years, there has been much interest in transferring Feshbach molecules to their
ground state to create a dense, ultracold sample of polar molecules. Such a sample was recently created by Ni et al. [38], which may be used as qubits in novel
quantum computation schemes [39], as probes of quantum phase transitions [40],or in quantum simulations of condensed matter systems [41].

1.2

The state of aairs
In the summer of 2004, we had recently discovered four Fano-Feshbach res-onances between 87 Rb and 40 K [11]. We had a knob with which to tune theinterspecies interactions and were poised to begin experiments investigating someof the phenomena discussed in 1.1 with one major technical stumbling block: low

5potassium number. From day to day, we could not guarantee that we would haveany potassium at the end of our evaporation, and even our best conditions left us
with only 30,000 potassium atoms. This was, of course, problematic for an experiment studying interactions between species. This issue was exacerbated by thefact that we had few diagnostics with which to optimize the potassium number.
We had uorescence imaging of the atoms in the MOT and absorption imaging in
the magnetic trap. The latter requires optically dense samples, which were only
achievable in the Fermi gas near the end of evaporation. Thus, if the potassium
number dropped too low or the temperature became too high, the atom gas would
become too dilute and its optical depth would simply be below the noise oor ofour detection during the entirety of the magnetic-trap evaporation. There are
numerous steps between collection of the atoms in the MOT and evaporation to
microKelvin temperatures, some of which could be optimized with uorescencein the MOT chamber, but some steps required absorption imaging. Without asignal to optimize, we were (in)eectively shooting arrows in the dark.
The rst order of business, then, was to rebuild the experiment with an eye
to capturing and keeping more potassium atoms without allowing the rubidium
number to suer. Much of my thesis work has gone to building a new vacuum
system and improving the many experimental steps required to create an ultracold(or even quantum degenerate) mixture of 87 Rb and 40 K. The result is a machinethat over the past year has reliably generated nearly quantum degenerate mixtures with more than half a million potassium atoms and comparable numbers of
rubidium atoms and allowing for the rst studies of coherent oscillations between
atoms and molecules in a Bose-Fermi mixture. My hope is that the machine
will continue to operate stably for years to come, enabling the study of some ofthe other phenomena mentioned in 1.1 and possibly some yet-to-be-theorized
applications of Bose-Fermi mixtures.

6

1.3

Atom-molecule coherence
Upon achieving the initial goal of increasing the number of atoms at theend of the experiment, we set out to study the interspecies interactions in our
Bose-Fermi mixture. Our rst step was to make KRb molecules from the ultracold
mixture. Due to their internal atomic structure, the interactions between 87 Rb and40

K atoms are resonantly enhanced at particular values of an externally appliedmagnetic eld, known as Fano-Feshbach resonances [11, 42, 19]. In the vicinity ofthese resonances, the sign and strength of the interactions between two colliding
atoms can be experimentally controlled by tuning the strength of an external
magnetic eld. Colliding atoms can be made to attract or repel one another oreven bind together into molecules. Through precise control of an external magnetic
eld, we are able to transfer a mixture of ultracold atoms into weakly-boundmolecules. These so-called Feshbach molecules are highly vibrationally excitedand have sizes that are comparable to the interparticle spacing in the atomicgas. Unlike the bosonic molecules formed in single species experiments (boson +boson = boson and fermion + fermion = boson), our molecules are fermions since
they are formed from the union of one bosonic and one fermionic atom. Because
they are made up of two dierent atomic species, these molecules are expected
to have an appreciable electric dipole moment when in their vibrational groundstate. Recently, much work has been focused on transferring Feshbach molecules
into the ro-vibrational ground state to generate an ultracold, dense sample ofdipolar molecules [38, 43]. In our experiment we instead chose to study the nearthreshold Feshbach molecules where we could exploit the similarities between themolecules and unpaired atoms to create coherent superpositions between atoms
and molecules [44, 45, 37].

7

1.3.1

Quantum superpositions

In the mid-20th century, molecular beam experiments by Rabi and Ramsey
demonstrated some of the earliest examples of quantum control [46, 47, 48, 49].
They showed that an oscillating magnetic eld can resonantly couple two atomic
states with dierent magnetic quantum numbers. Transitions between the twostates can be induced by appropriately tuning the frequency, amplitude, and duration of the oscillating eld. Additionally, applying a judiciously chosen oscillating
magnetic eld can result in a coherent superposition of the two quantum states,
where each particle simultaneously occupies both states. This method of couplingtwo quantum states with oscillating elds was originally developed for the measurement of nuclear magnetic moments but has found a multitude of applications
including neutron beam experiments [50], Josephson junctions in superconducting
devices [51], and atomic clocks [52], to name a few. In fact, the coupling of two
internal states of an atom with radio frequency (rf) or microwave photons is atool routinely used in atomic physics experiments.
In 2002, Donley et al. demonstrated the rst quantum superposition between atoms and molecules [44]. In this experiment, a BEC of 85 Rb atoms wascoherently coupled to diatomic molecules, creating pairs of atoms that were both
two separated atoms and molecules at the same time. The result was oscillationsin the atom population at a frequency that corresponded to the binding energyof the molecules. Instead of coupling the atom and molecule states with an oscillating magnetic eld, the superposition was created by pulsing the magnetic
eld to a value near a Fano-Feshbach resonance. More recently, Syassen et al.
used a Fano-Feshbach resonance to create a quantum superposition of atoms andmolecules from pairs of bosonic atoms prepared in the ground state of a tightly
conning optical lattice potential [45]. Long-lived Rabi oscillations between the8atoms and molecules were observed. In more direct analogy with the experimentsof Rabi and Ramsey, atoms and molecules have also been coherently coupled via
two-color photoassociation in BECs [53, 54]. Each of these experiments beganwith identical bosons occupying a single quantum state. This circumstance leads
to the question: Can an atom-molecule superposition be demonstrated with anincoherent ensemble of atoms such as non-condensed bosonic atoms, fermionic
atoms [55, 56, 57], or a mixture of dierent atomic species [58, 59]?

1.3.2

Atom-molecule superpositions in a Bose-Fermi mixture

In the case of a Bose-Fermi mixture, the participants of an atom-moleculesuperposition represent three dierent populations with dierent energy distributions and quantum statistics: bosonic atoms, fermionic atoms, and fermionicmolecules. Even at zero temperature the atom pairs would not occupy a single
energy state due to the energy distribution of the fermionic atoms (see Fig. 1.1).
The same is true of the molecules. This distribution of energies leads to a distribution of oscillation frequencies among the atom pairs in the cloud. Despite this,we have observed collective oscillations between atoms and molecules in our BoseFermi mixture, as will be discussed in detail in Chapter 5. These experiments
represent the rst observations of coherent superpositions of atoms and fermionicmolecules.

1.4

Overview of this thesis
This thesis details the steps that were necessary to create and understandFeshbach molecules and superpositions of atoms and molecules in a Bose-Fermi
mixture of 87 Rb and 40 K. While the basic techniques for trapping and cooling thismixture can be found in the thesis of my predecessor, Jon Goldwin [60], much of
the apparatus has been upgraded and optimized. In Chapter 2, I will highlight the9main changes that enabled signicant improvement in the eciency of creating
our nearly quantum degenerate mixture. This chapter is intended for the benetof future students and post-docs working on this apparatus or someone building
a similar machine.
In Chapter 3, I will discuss the Fano-Feshbach resonance, the critical toolthat has enabled all of the experiments detailed in this thesis. Fano-Feshbachresonances provide the knob with which we can control the interactions between
colliding atoms and allow us to coax unpaired atoms into forming molecules.
They also provide the coupling between atoms and molecules that allowed forthe creation of atom-molecule superpositions. In this chapter, I will introduce a
simple model of Fano-Feshbach resonances and how they can be used to create
molecules in general and discuss some details specic to our system.
In Chapter 4, I will outline a semiclassical Monte Carlo calculation for the
association of molecules using a Fano-Feshbach resonance. This model is an extension of that used to successfully predict molecule conversion fractions in bothbosonic and fermionic single-species experiments [61, 62] and bosonic heteronuclear experiments [63]. I will show that the model can also successfully accountfor the conversion fraction in Bose-Fermi mixtures [64] and discuss some of theparameters for optimizing the molecule fraction.
The main experimental results of this thesis are presented in Chapter 5,
where I will discuss the observation of coherent atom-molecule oscillations in both
Rabi- and Ramsey-type experiments.

I will highlight the dierences between

oscillations in our system and those observed in bosonic systems and introduce
potential applications of these and similar superpositions.
Chapter 6 details some predictions about the coherence times and envelopesof the atom-molecule oscillations that have yet to be tested experimentally. These
predictions arise from using the Monte Carlo calculation to predict not simply

10
the number of molecules, but also the relative kinetic energy between the atoms
that form the molecules. The results show some counter-intuitive dependences ontemperature and degeneracy.
Finally, in Chapter 7, I will conclude with a summary of the work presented.
I will also provide some thoughts on the goals for the experiment for the near future
and possible long-term directions to follow.

Chapter 2
Experimental improvements

The apparatus used for the experiments discussed in this thesis has been
described in detail in Jon Goldwin's thesis [60]. Since 2005, we have made many
improvements to the system that have allowed for ecient production of ultracold
and degenerate mixtures of 40 K and 87 Rb. In this chapter, I will not describe everychange that was made to the apparatus. Instead, I will begin with an overview
of the experimental steps that take a room-temperature gas of free atoms toa (nearly) degenerate Bose-Fermi mixture with tunable interactions in under aminute and a half. I will then discuss in detail the changes to the experiment thatbrought about the most signicant improvements and a few additional capabilities
that have enabled the studies of Fano-Feshbach resonances and atom-molecule
oscillations discussed in Chapters 3 and 5.

2.1

Experimental sequence
We begin our experiments by simultaneously collecting Rb and K atoms

from a room-temperature vapor via a two-species magneto-optical trap (MOT).
By overlapping the beams required for trapping Rb atoms with those for K atoms,
we are able to conne the two species in roughly the same position in space. We
typically trap 4 × 108 atoms in the Rb MOT and 4 × 107 atoms in the K MOT.
Each species is then prepared in a single spin state (|f = 2, mf = 2i for Rb, and12|9/2, −9/2i for K) and transferred into a quadrupole magnetic trap as discussed
in detail in 2.3.2 below. Here, f is the total atomic spin, and mf is the projectionof the spin along the magnetic eld. The vapor pressure in the MOT cell limits
the lifetime of a trapped atom to a few seconds. To achieve longer lifetimes, wetransfer the atoms to another cell, termed the science cell, which is connected to
the MOT cell via a long (60 cm), narrow (1 cm diameter) tube. This tube allows
for dierential pumping between the two cells, and the lifetime of the trapped
mixture can be several hundred seconds in the science cell. To transfer the atoms
to the science cell, we physically move the coils that generate the quadrupole trap
(a.k.a. the cart coils), which are mounted on a linear translation stage.
Once in the science cell, the atoms are transferred to a Ioe-Pritchard (IP)type magnetic trap as detailed in 2.3.3. Here, the mixture is cooled via sympathetic cooling. That is, the Rb gas is cooled by forced rf evaporation, and the Kgas is cooled by thermalizing collisions with the Rb atoms. Typically, we stop theevaporation when we reach a temperature of 2µK, leaving us with 1 × 106 atoms
in each species.
We use Fano-Feshbach resonances between K and Rb atoms in their lowest
energy states to control the interspecies interactions (see Chapter 3). In thesespin states, the atoms are not magnetically trappable. Therefore, we transfer the
mixture to a far-o-resonance optical dipole trap (FORT) generated by a single
focused laser beam with a 1/e2 radius 18 µm and a wavelength of 1090 nm. The
atoms are then transferred to the |9/2, −9/2i atomic Zeeman state of K and the|1, 1i state of Rb via rf and microwave adiabatic rapid passage, respectively.
To achieve a quantum degenerate mixture (or nearly so), the mixture mustbe further cooled. We simultaneously evaporatively cool the two species by decreasing the power in the optical-trap beam. This lowers the trap depth andcauses high energy atoms in both species to be ejected from the trap. At the13
end of the evaporation, the trap is adiabatically recompressed. At the end of thissequence, we can have a nearly pure BEC with 65,000 atoms with a degenerate
Fermi gas of 180,000 atoms at 0.2 TF , where TF is the Fermi temperature of theK gas. However, for most of the experiments presented in this thesis, we stop theevaporation before the onset of Bose condensation to avoid the associated highdensities.
During the optical trap evaporation, a nearly uniform magnetic eld is applied and increased to a eld near one of the Fano-Feshbach resonances (usually
542 G). The interspecies interactions can then be manipulated by varying themagnetic eld around the resonance. The nal properties of the atoms (number,
temperature, spin state, etc.) at the end of this sequence are then measured withspecies-selective absorption imaging, which is discussed in 2.7.

2.2

New vacuum system
Prior to 2005, our experiments suered from having too few K atoms at theend of the experimental sequence. From day to day, we could not guarantee that
we would be left with any K atoms at the end of the IP trap evaporation, andour best conditions left us with only 30,000 K atoms loaded into the optical trap.
This problem of low atom numbers at the end of the experiment was compoundedby the fact that we had few diagnostics with which to optimize the K number.
We were able to detect the K atoms in the MOT with uorescence imaging,but once the atoms are transferred into the science cell, we rely on absorption
imaging to detect them. This requires optically dense ensembles of atoms, andif the K number was too low or the temperature too high, the atoms would beindistinguishable from background noise in the images.
A seemingly simple solution to these problems was to increase the numberof K atoms in the MOT. However, we observed little to no improvement in our14starting conditions when we tried to increase the output of our K dispensers. These
dispensers, which we refer to by the unfortunate misnomer getters, are made inhouse at JILA and are based on the design of the alkali metal dispensers fromSAES Getters, Inc. [65]. The K getters contain commercially available enriched
KCl (5% 40 K, which is large compared with the 0.012% natural abundance of the40

K isotope) and calcium, a reducing agent. When current is passed through thegetter, ohmic heating enables a chemical reaction that releases the K atoms fromthe alkali salt. Controlling the current, therefore, controls the pressure in the
MOT cell, and, consequently, the number of atoms trapped by the MOT.
In the summer of 2005, we decided that we had to replace the K getters,
which, of course, meant breaking vacuum. In addition to improving the K atom
number and its controllability, this aorded us the opportunity to make a few
improvements to the vacuum system as well. We added a gate valve in middle ofthe transfer tube. This enables us to break vacuum on either the MOT-cell sideor the science-cell side while leaving the other under vacuum (a capability thatwas immediately useful!). We also swapped all the bolts along the transfer tube
for higher quality 316 stainless steel bolts. The previous vacuum system had been
fastened together with 304 stainless steel bolts, which are nominally nonmagnetic
but were generating magnetic elds of up to 10 G. These were likely the causeof a magnetic speed bump over which the cart had to pass slowly to prevent
severe atom loss [60]. Also, the titanium sublimation pump was moved closer tothe science cell to enable more rapid pumping. Schematic drawings of the vacuum
system in its current incarnation are shown in Figs. 2.1 and 2.2.
Initially, we were concerned that the Rb getters in the old system had contaminated the K getters. A new MOT cell was made with a K getter arm and two
Rb ampoules instead of Rb getters. To achieve the ultralow pressures required forour experiments, we baked the entire chamber at temperatures between 200 and15

Figure 2.1: CAD drawing of the new vacuum system.

Ti sub pumpcellsciencegate valve

track

40 l/s

20 l/scart

MOT cellion pumps"getters"

16

Figure 2.2: CAD drawing of the new vacuum system (top view).

17350◦ C for 9 days. All the metal parts were wrapped with tape heaters, and anoven made from re bricks was built around each glass cell to minimize thermal
gradients. When we cooled down the system and opened the oven surrounding the
MOT cell, we found that the Rb had reacted with the borosilicate glass, turning
both ampoules red and causing one to rupture as shown in Fig. 2.3. During thebake, the oven temperature near the ampoules was 275◦ C. This corresponds toa Rb vapor pressure of 0.7 Torr [66], which is more than 3 orders of magnitude
too small to break the glass. Presumably, the chemical reaction that turned theampoules red also stressed the glass.
We promptly closed the new gate valve, which sealed o the science cell, andreplaced the MOT cell with another, this time reverting to the original design oftwo getter arms and no ampoules. The MOT side of the vacuum chamber wasthen baked at 250◦ C for 7 days. Upon cooling the system and nding no cracks
in the glass cells, we began to test our new system.

2.2.1

Large, controllable K MOTs

Since the main goal of the new vacuum system was to achieve higher numbersof atoms in the K MOT, we chose to rst operate without running the Rb getters.
This way, we could ensure that the K getters worked properly and that we could
control the number of atoms in the MOT without having to consider the eectsof another species. By controlling the current through the K getter, we were able
to achieve K MOTs with 1.5 × 108 atoms with exponential ll times of 3.5 s. Thisis an order of magnitude improvement over the 107 -atom K MOTs we had worked
with in the past. Satised that our new K getters worked and that we could trap
large numbers of K atoms, we were then ready to introduce Rb into the system.

18

(a)

(b)

Figure 2.3: Failed MOT cell with Rb ampoules. A photograph of the MOT cellwith a K getter arm and two Rb ampoules taken after baking out the vacuum
system is shown in (a). The cell was baked at 275◦ C. (b) shows a zoomed-in viewof the ampoules. The Rb has reacted with the glass, causing it to turn red. Prior
to the bake, the glass was clear, and a puddle of silver Rb was visible. During thebake, one ampoule ruptured, and some Rb escaped and reacted with the oxygenin the air (white blob on upper ampoule).

19

2.2.2

Model of a two-species MOT

In addition to trapping large numbers of atoms in each species, we would
like to independently control the population in each MOT. To do so, we must
understand the behavior of the K MOT in the presence of Rb and vice versa.
Here, I present a simple model of the two-species MOT to illustrate how thenumber of atoms in each species depends on the total pressure in the MOT cell.
The model considers only elastic collisions between atoms, and, therefore, ignores
the light-assisted inelastic collisions that have been observed to reduce the numberof atoms in the K MOT [67].
Let us rst consider each MOT individually. The number of atoms N in asingle-species MOT can be described by [68]dNi (t) = L − ΓNi (t),
dt

(2.1)

where L is the loading rate, Γ is the loss rate of atoms from the MOT, and thesubscript i ={Rb, K} represents the species of atoms in the MOT. The loading
rate is proportional to the pressure of trappable atoms Pi in the cell

L = αi Pi .

(2.2)

Here, αi is a proportionality constant that depends on the geometry, laser intensities, and detunings of the MOT and the species being trapped. Atom can be lostfrom the MOT when an untrapped atom collides with a trapped atom. Therefore,
the atom loss rate depends on the total pressure [69]Γ = βi · (Pi + Pbg ) ,(2.3)

where Pbg is the pressure of background atoms, i.e. all species that are not being
trapped. βi is a proportionality constant, and we have assumed that an atom isjust as likely to be lost if it collides with another atom of its own species as with20
any other species. In equilibrium, the number of atoms in a MOT is

Ni =

L
αi Pi
αi
1
=
=
.
Γ
βi · (Pi + Pbg )βi 1 + Pbg

(2.4)

Pi

Now, let us consider the case of ideal getters in a perfect vacuum for Rb
and K, in turn. We will assume that an ideal, but unenriched Rb getter emits
3 85 Rb atoms for every 87 Rb atom, and an ideal, 5% enriched K getter emits 19
39

K (or 41 K) atoms for every 40 K atom. For the Rb MOT, Pbg = 3PRb , and for

the K MOT, Pbg = 19PK . Here, PRb and PK are the pressures of 87 Rb and 40 K,
respectively. The equilibrium number of Rb atoms in a MOT when this ideal
getter is red into a perfect vacuum is

NRb =

1 αRb
1
αRb
=
.
3PRb
βRb 1 + P
4 βRb

(2.5)

Rb

Likewise, ring an ideal K getter into a perfect vacuum results in

NK =

αK
1
1 αK
=
.
19PK
βK 1 + P
20 βK

(2.6)

K

For this case, in which the background pressure is proportional to the trappable
atom pressure, we nd the somewhat surprising result that the equilibrium atom
number in the MOT is independent of pressure. To compare the number of atoms
in the two MOTs, we make some assumptions about the proportionality constants

αi and βi . First, we assume the probability that a collision with a backgroundatom ejects a trapped atom from the MOT is the same for the two species, i.e.

βRb = βK = β . Next, we will assume that the loading of the Rb MOT is 3 times
more ecient than the K MOT 1 , so αRb = 3αK = 3α. The number in each MOT,
then, is

NRb =

3α
1 α
and NK =
.
4β
20 β

(2.7)

1 This assumption is based on the empirical observation that K MOTs trap fewer atoms than
Rb MOTs with similar intensities, detunings, and geometries.

21
In this idealized case, we nd the total number of Rb atoms is fteen times larger
than the number of K atoms, due to the relative fraction of trappable atoms.
Now, let us consider a more realistic case: ideal getters red into an imperfect vacuum with an initial pressure Pvac . For now, we will still consider eachspecies individually. We nd the equilibrium number of atoms in each MOT is

NRb =

3α
PRb
α
3
=
β PRb + 3PRb + Pvac
β 4 + PPvac

(2.8)

α
PK
α
1
=
.
β PK + 19PK + Pvac
β 20 + PPvac

(2.9)

Rb

NK =

K

The number of atoms in each MOT multiplied by β/α is plotted as a function of
the trappable atom pressure in Fig. 2.4. Not surprisingly, we see that the numberin the K MOT is lower than in the Rb MOT. We also see that the number in both
species reach 90% of the maximum number at the same total pressure. Due tothe output of the getters, the partial pressure at this saturation point is 5 times
lower for K than Rb.
Now, let us consider the two-species MOT, in which the number in eachspecies depends on the outputs of both getters. Now, PRb and PK are partialpressures. The background pressure for the 87 Rb MOT is made up of everythingthat is not 87 Rb, Pbg = P85Rb + P40K + P39K + Pvac = 3PRb + 20PK + Pvac , andthe background pressure for the 40 K MOT is Pbg = P85Rb + P87Rb + P39K + Pvac =4PRb + 19PK + Pvac . Now, the number in each species is
PRb
1
3α
3α
=
20PK +Pvac
β PRb + 3PRb + 20PK + Pvac
β 4+ P(2.10)

α
PK
α
1
=
=
,
4PRb +Pvac
β PK + 19PK + 4PRb + Pvac
β 20 +
P(2.11)

NRb =

Rb

NK

K

as shown in Figs. 2.5 and 2.6. Running either of the getters will increase the number of atoms in the corresponding MOT but will decrease the number of atoms in
the other MOT. This is because running the Rb getter increases the background

22

(a)

(b)

Figure 2.4: Pressure dependence of single-species MOT numbers. The numberof 87 Rb atoms in the MOT multiplied by β/α as a function of the 87 Rb partialpressure PRb scaled by the initial pressure Pvac is shown in (a). A similar plot for
a 40 K MOT is shown in (b). The dashed lines represent 90% of the maximum
number of atoms in the MOTs, which dier by a factor of 15 between the two
species. Both species reach 90% of the maximum when the total pressure is 10
times the initial pressure, but the partial pressure of trappable atoms diers dueto the output of the ideal getters as described in the text.

pressure for the K MOT and vice versa. Even in this idealized illustration, controlling the number of atoms in each MOT requires striking a balance between
the partial pressures of Rb and K.
In the experiment, this situation is further complicated. Josh Zirbel andcoworkers demonstrated that the K getters have a small (200 ppm) Rb contamination [70]. This has a negligible eect on daily operation, but we have foundthat over time, both getters become contaminated, presumably due to atom adsorption. Running either getter releases both Rb and K. Figure 2.7 shows thenumber of atoms in each species as a function of pressure when running a 5-A
current through only the K getter. The number of K atoms follows the expected
dependence, but we would expect a decrease in the number of Rb atoms as in the
top plot in Fig. 2.5. Instead, both increase, indicating that both K and Rb are
being released. This may indicate that the current running through the K getter

23

0.7
0.6

N

Rb

0.5
0.4
0.3
0.2
0.1

0.2

0.4

0.6

0.8

N

1.0

Rb

0.7

3.0

0.6

2.5

0.5

Rb

2.0

P

1.5

1.0

0.1

0.2

0.3

0.4

0.1

0.2

0.4

P

0.6

0.8

1.0

0

K

N

Rb

0.5

0.3

0.2

0.5

0.6

0.4

Figure 2.5: Dependence of the Rb MOT number on the partial pressures of Rb
and K. The contour plot shows the number of atoms in the Rb MOT NRb scaledby β/α calculated with Eq. 2.10 as a function of the partial pressures of 87 Rb and
40
K, PRb and PK in units of Pvac . The left-hand plot shows NRb as a function of
the Rb pressure for xed K pressures, corresponding to the vertical lines on thecontour plot. The top plot shows NRb as a function of the K pressure for xedRb pressures, corresponding to the horizontal lines on the contour plot.

24

0.05

0.04

N

0.03

K
0.02

0.01

0.2

0.4

0.6

0.8

N

1.0

K

0.05

3.0

2.5

0.04

2.0

Rb

0.03

P

1.5

0.02
1.0

0.01

0.01

0.02

0.03

0.2

0.4

P

0.6

0.8

1.0

0

K

N

K

0.04

0.05

0.5

Figure 2.6: Dependence of the K MOT number on the partial pressures of Rb
and K. The contour plot shows the number of atoms in the K MOT NK scaled byβ/α calculated with Eq. 2.11 as a function of the partial pressures of 87 Rb and
40
K, PRb and PK , in units of Pvac . The left-hand plot shows NK as a function of
the Rb pressure for xed K pressures, corresponding to the vertical lines on thecontour plot. The top plot shows NK as a function of the K pressure for xed Rb
pressures, corresponding to the horizontal lines on the contour plot.

25

(a)

G

(b)

G

Figure 2.7: Measured MOT numbers running only the K getter. The numbers of
atoms in the Rb (a) and K (b) MOT are plotted as a function of the exponential
ll rate Γ, which is proportional to the total pressure. The pressure is varied byrunning a current of 5 A through the K getter for varying durations. Contrary
to the expectation of the model presented in the text, the number in both MOTs
increase with increasing pressure, indicating that both species are released by
the K getter. The lines are the result of a simultaneous t to the data to N =Γ−ΓNmax Γ bg , which is equivalent to Eq. 2.4 with Nmax = α/β and Γbg = βPbg .
The t yields a background ll time of 1/Γbg = (9 ± 2) s and a maximum numberof Nmax = (5.2 ± 0.3) × 108 atoms in the Rb MOT and (5.9 ± 1.1) × 107 atoms in
the K MOT.

does not heat the getters enough to induce the chemical reactions that should
preferentially release K, but instead only heats the assembly enough to release
adsorbed atoms. Higher currents (> 6.2 A) are necessary to release enough K tovisibly coat the glass getter arm.
In the experiment, we typically operate with 4 × 108 atoms in the Rb MOT
and 4 × 107 atoms in the K MOT with exponential ll times of 1/Γ = 2.5 s. We
achieve these conditions by pulsing on the Rb getter at 4.5 A for 90 s each morning.
To maintain a steady pressure throughout the day, we run 3.75 A through the Kgetter with the Rb getter o. The K getter is turned o at night to extend its life.

26

2.3

Keeping all those atoms
Having achieved our goal of increased K atoms in the MOT, we wanted toensure that the improved initial conditions would translate to larger numbers at
the end of the experiment. This meant minimizing losses in the experiment, withparticular emphasis on the transfer of the atoms between the various traps. In
this section, I describe three important improvements that allowed us to maintainthe larger K numbers throughout the experiment.

2.3.1

Fiber coupling the MOT light

To trap atoms in the MOTs, we use two frequencies of laser light for eachspecies. The trap light is red detuned of the f = 9/2 → f 0 = 11/2 transitionin 40 K and the f = 2 → f 0 = 3 transition in 87 Rb, while the repump light is
red detuned of the f = 7/2 → f 0 = 9/2 and f = 1 → f 0 = 2 transitions (seeFig. 2.8). To improve the quality of our MOT beams, we have ber coupled the
K and Rb trapping beams as well as the Rb repumping beam as shown in Fig.
2.9. Previously, we had used a single tapered amplier to amplify all of the beams
and found that the K and Rb light emerged with dierent divergences, making
it impossible to eciently ber couple both colors. The amplier for the K lightis now unnecessary, because enough power for trapping the K atoms is generated
by a 120-mW injection-locked laser [70]. The Rb and K beams are coupled into asingle ber, and we are able to achieve 70% and 50% eciency for the K and Rb
light, respectively. This scheme delivers 140 mW of Rb light ( 5% repump), 45
mW of K trap light, and 17 mW of K repump light to the 3-beam retro-reected
MOT.
Because the ber-coupled MOT beams have Gaussian proles, ease of alignment and its reproducibility have improved.

More importantly, though, the27102 MHzf = 5/2 (54.5)f = 7/2 (30.6)f = 9/2 (-2.3)
f = 11/2 (-45.7)

P

4

3/2

f = 3 (194)

P

5

3/2

496 MHzf = 2 (-73)
f = 1 (-230)f = 0 (-302)

780 nm

767 nmf = 2 (2563)

1286 MHzf = 7/2 (714.3)
f = 9/2 (-571.4)

S

4

1/2

S

5

1/2

6835 MHzf = 1 (-4272)
40

K

I = 4

87

Rb

I = 3/2

Figure 2.8: Hyperne structure of the relevant ground and excited states of 40 K
and 87 Rb with the splittings calculated from Ref. [71].

28
to MOT

from K
IjLL

fromtaperedfrom K

amplifier

ECDL

Figure 2.9: Fiber coupling the MOT light. The red lines represent the Rb trap andrepump beams. The Rb beams emerge from a tapered amplier and are combinedwith the K trap beam (solid blue line), which is generated by an injection-locked
laser (IjLL). All three beams are coupled into the same ber, and the output isdirected toward the MOT after combining with the K repump beam (dashed blueline) which is generated by an external cavity diode laser (ECDL).

structure-free beams allow for lower temperatures and purer spin compositionsin the quadrupole trap. Ideally, we would like to load cold, spatially small, andspin-polarized clouds into the quadrupole trap. The Rb MOT, however, is large
in spatial extent due to strong radiation trapping [72]. A spatially large cloud will
gain excess potential energy when transferred to the quadrupole trap, which will
result in a higher temperature gas in the trap. To reduce the size of the Rb cloud,
we apply a compressed MOT stage, or CMOT, in which we reduce the intensityof the repumping beam for 50 ms [73, 60]. Before the MOT beams were bercoupled, we found that the positions of the MOT, CMOT, and quadrupole trap
diered, presumably due to structure in the MOT beams. In transferring fromone stage to the next, the atoms acquired additional kinetic energy due to theshifting trap centers, negating some of the benets of the CMOT. Now, we ndthe traps are well overlapped, eliminating heating due to excess motion.
In addition to an improved CMOT stage, the Gaussian MOT beams haveenabled the incorporation of a polarization gradient cooling stage, or optical molasses, for the Rb atoms [74]. For the molasses, the MOT magnetic-eld gradient29
is abruptly switched o, the Rb trap light frequency is shifted 40 MHz red of thecycling transition, and the MOT shim elds are changed. The shim elds are
tuned by watching the expansion of the Rb MOT in real time using a small CCD
camera hooked up to a TV monitor. The elds are set so the MOT expands slowly
and uniformly over several seconds. In the experiment, however, the duration of
the molasses stage is only 2 ms. Figure 2.10 shows a measurement of the relative
temperature of the Rb atoms with and without the molasses stage. The atoms
are loaded into a variable strength quadrupole trap, held in the trap for 400 ms,
and recaptured into the MOT. The fraction recaptured is determined by comparing the measured MOT uorescence before and immediately after the hold in thequadrupole trap. An atom with zero velocity will be trapped when the trapping
force of the quadrupole eld overcomes the force of gravity. Atoms with larger
velocities (higher temperatures) will require greater trapping forces.
We can compare the fraction of atoms held by the quadrupole trap as afunction of the trap strength to a simple model that calculates the fraction of
atoms with energies less than the trap depth for a cloud with a given temperature.
We assume the trap depth U for a given atom is given by

U = µβW
W− mg ,
22(2.12)

where µ is the magnetic moment of the atom, m is the mass of the atom, β is themagnetic-eld gradient, g is the acceleration due to gravity, and W is the widthof the cross-sectional width of the MOT cell. We also assume that the momentum
distribution of atoms in the MOT is given byn(p) =

N2

3/2(2πmkb T )e−p /(2mkb T ) ,(2.13)

where N is the number of atoms in the MOT, T is the temperature, and kb is
the Boltzmann constant. For an atom to be trapped by the quadrupole trap, we

30
require

p2
U
≤ ,
2m
2

(2.14)

which arises from the Virial Theorem for a linear trap (potential energy = 2 ×
kinetic energy). We can compare the measured MOT recapture fraction to thecalculated fraction of atoms that satisfy Eq. 2.14 to estimate the temperatureand spin purity (see 2.3.2) of the cloud loaded into the quadrupole trap. Using
the model, we estimate a temperature of 300 µK without the molasses and atemperature of 200 µK after the molasses stage. Applying the model to data taken
before the MOT beams were ber coupled indicates temperatures as high as 600µK. Therefore, by ber coupling our MOT light, we were able to make a factor of
3 improvement in the temperature of the atoms loaded into the quadrupole trap.
After the CMOT and molasses stages, the atoms are cold (≤ 200 µK) but ina somewhat random distribution of spin states. Before loading the atoms into thequadrupole trap, we add a stage of optical pumping to spin-polarize each species
as discussed below. Note that an optical pumping pulse has been applied the data
shown in Fig. 2.10.

2.3.2

Spin state purication

Ideally, we would like to load the quadrupole trap with a pure sample of Rb
atoms in the |2, 2i state and K atoms in the |9/2, 9/2i state. Any other mixture ofspin states will be able to undergo spin-ip transitions that can heat the mixture
and expel atoms from the trap. Because we load 10 times more Rb than K atoms
into the quadrupole trap, we have strict requirements on the spin purity of the Rb
atoms. A 10% spin impurity in Rb could obliterate the entire K population. To
prepare as pure a sample as possible, after the molasses stage, optical pumping
light is applied to both species for <1 ms. A combination of repumping light and31

Figure 2.10: Eect of optical molasses on the temperature of the Rb MOT. Plotted is the fraction of Rb recaptured into the MOT after a 400 ms hold in thequadrupole trap as a function of the trap gradient. The atom clouds represented
by the red points are loaded into the quadrupole trap after the CMOT stage. The
blue points show atom clouds that have been further cooled by a 2 ms opticalmolasses, resulting in a steeper onset of trapping. The dashed lines are the resultof a simple model for the fraction of atoms that can be trapped in the quadrupole
trap. The blue line is the expected fraction for a 200 µK cloud with 80% of theatoms in the |2, 2i state, while the red line is for a 300 µK cloud with 50% of theatoms in the |2, 2i state. The solid lines are the dashed lines shifted by 2.5 G/cm
to match the data. This shift may be due to an additional eld, such as a MOT
shim that is kept on during the transfer. All the curves have been scaled by 75%
to account for the lifetime in the quadrupole trap.

32
Zeeman pumping light (σ + polarized light on the f = 2 → f 0 = 2 Rb transition
and f = 9/2 → f 0 = 9/2 K transition) transfers the atoms into the stretchedstates.

Since the dierent spin states have dierent magnetic moments, theyrequire dierent quadrupole trap strengths to be trapped. Therefore, the methodof measuring the fraction of atoms recaptured from varying strength traps can beused to measure the eciency of the optical pumping in addition to monitoring
the temperature of the cloud. Figure 2.11 compares the fraction of Rb atoms
recaptured into the MOT with and without optical pumping as a function of trap
strength. The Rb |2, 2i atoms have a magnetic moment of µ = µB and can betrapped at 16 G/cm, while Rb |2, 1iand |1, −1i atoms have a magnetic momentµ = 21 µB and require gradients greater than 32 G/cm. Here, µB is the Bohrmagneton. Without optical pumping, all the Rb spin states are populated, andwe see two clear steps as atoms with the dierent magnetic moments becometrapped. We estimate that after optical pumping, more than 90% of the captured
Rb atoms are in the |2, 2i state. The eciency of the optical pumping is likely
limited by the optical depth of the gas. Note that the data in Fig. 2.10 indicatethat the optical pumping eciency increases after the molasses stage, which islikely due to the decrease in optical depth of the expanded gas.
The Rb recapture data shown in Fig. 2.11 suggest an additional mechanism
for further purifying the spin mixture, which was rst implemented by Scott Papp[75]. By loading the atoms into a quadrupole trap with a gradient < 32 G/cm,
only atoms in the |2, 2i state can be trapped. In the experiment, we load theatoms into a trap with a gradient of 29 G/cm in the vertical direction, wait 400
ms to ensure that unwanted spin states fall out of the trap, then increase the trap
strength to 98 G/cm for the transport along the vacuum system. While the initialtrap lters out all but one spin state of Rb, 7 spin states of K can be trapped.
Thus, we must rely solely on optical pumping to purify the K atoms. However,33

Figure 2.11: Rb spin states loaded into the quadrupole trap. The fraction of Rb
atoms recaptured into the MOT after a 400-ms hold in the quadrupole trap as
a function of the trap gradient is plotted for atoms clouds with (red points) andwithout (blue points) optical pumping. The curves are the results of a model forthe atom fraction loaded into the quadrupole trap for a temperature of 100 µK
and scaled by 77% to account for the lifetime in the quadrupole trap. Without
optical pumping, the majority of the atoms are not in the desired |2, 2i state.
With optical pumping, 90% of the atoms are in the |2, 2i state. Therefore, there
is only a small increase in the recaptured fraction above 32 G/cm, the gradientat which |2, 1i and |1, −1i atoms begin to be trapped.

34
a small impurity in the K spin mixture will result in correspondingly small atom
losses.
Because it is sensitive to temperature, spin state, and number, this techniqueof measuring the fraction of atoms recaptured from the quadrupole trap has proved
to be a crucial tool for optimizing the initial stages of the experiment. For example,
we are able set the duration of the optical pumping pulse so that each species iswell polarized but the temperature is not increased. We are also able to ensurethat the molasses reduces the temperature of the Rb gas without sacricing atoms
in either species. As a result, we are able to load into the quadrupole trap a spinpolarized gas comprised of > 70% of the atoms in the Rb MOT and > 60% of theatoms in the K MOT at a temperature of 100 µK.

2.3.3

Quadrupole trap to IP trap transfer

One point in the experiment where we are particularly susceptible to atom
loss is in the transfer from the quadrupole trap to the IP trap. As shown in Fig.
2.12, the shapes of equilibrium clouds in the two traps are severely mismatched.
In the quadrupole trap, the cloud is shaped like an m&m, while in the IP trap,
it is cigar-shaped. Because the cloud is initially too large in the radial direction
and too small in the axial direction, if we abruptly turn o the quadrupole trap
and turn on the IP trap, breathe modes are induced along three dimensions.
While we cannot tighten the IP trap along the axial direction, we can weaken theconnement in the radial direction to make a round(ish) trap. By increasing thenal trap strength of the quadrupole trap, we are able to match the radial cloudsizes in the traps, eliminating radial breathing. However, we are still left with asizable mismatch along the axial dimension of the traps. When a cloud is loaded
into the IP trap, the axial size begins to breathe about the equilibrium size in
the IP trap (axial rms size, σz = 1.3 mm). If the atoms are too energetic, they

35
quadrupole traps /s = 0.5zr

IP traps /s = 5

y

z

r

z

x

Figure 2.12: Schematic of the cloud shapes in the quadrupole and Ioe-Pritchard
traps. The traps are both cylindrically symmetric about the same axis, but dier
by a factor of 10 in aspect ratio (σz /σr ).

will splatter onto the science cell walls, which are only 5 mm away from the trap
center (< 4σz ). Figure 2.13 shows the measured fraction of Rb atoms transferred
into the IP trap as a function of temperature. To calculate the expected fraction
transferred to the IP trap, we use a model similar to that described in 2.3.1. We
calculate the fraction of atoms whose energy is less than the trap depth of the IP
load trap set by the science cell walls. The data show reasonable agreement withthis simple model of the transfer and demonstrate nearly 100% transfer with ourcoldest clouds. Once the atoms are transferred, we increase the radial connement
to the full trap strength with a 5:1 aspect ratio. Figure 2.14 shows the current
through the quadrupole and IP traps for the optimized transfer.

2.4

Too much of a good thing?
All our eorts to improve the K number in the early stages of the experiment

led to a signicant increase in the number of K atoms loaded into the IP trap.
Prior to the new vacuum system, we could not measure the number of atoms in36

Quadrupole trapmK)

temperature (Figure 2.13: Temperature dependence of the fraction of atoms transferred to the
IP trap. The measured fraction of Rb atoms transferred from the quadrupole trap
to the compressed IP trap is plotted as a function of the temperature in the 400 A
quadrupole trap (Q-trap). The solid line is the result of a model of the transfer.

37

0.5

y (cm)y (cm)

0.5

-0.5

-0.5
-0.5

z (cm)

0.5-0.5

z (cm)

0.5

Figure 2.14: Quadrupole to IP trap transfer. Plotted are the currents for the
optimized transfer from the quadrupole trap to the IP trap. The quadrupole trap
current I is increased to decrease the radial size of the cloud before the transfer.
The gradient of the quadrupole trap in the axial direction is equal to I ×0.49G/(cmA). The IP trap is rounded by bleeding current from the IP bias coils (see Fig.
2.17). The red curve shows the current passing through the bias coils, while theblue curve shows the current passing through the IP radial and pinch coils. The
lower plots show calculated magnetic-eld contours of the initial IP trap (left) andthe trap after compression (right) for x = 0.

38
the K cloud with absorption imaging until it was cooled below 10 µK. Now, we
are able to image the K cloud immediately after loading into the IP trap and at
any time during the subsequent evaporation. We estimate that this corresponds
to at least a factor of 5 increase in the optical depth of the K cloud loaded into
the trap, and if we assume the K cloud is in thermal equilibrium with Rb, thattranslates to at least a factor of 5 more K atoms. Beyond the improvement in thenumber, an important consequence is that we are now able to probe both species
at every stage of the experiment.
With such a plethora of K to play with, it seemed the world was our oyster.
Soon, however, we encountered a new problem that had previously been inconceivable: too much K. Sympathetic cooling of the K atoms relies on the heat capacityof the Rb gas being greater than that of the K gas [76, 77]. Away from quantum
degeneracy in both species, the heat capacity is given by Ci = kB Ni , where kB is
the Boltzmann constant and Ni is the number of atoms in species i = {K, Rb}.
This implies that the number of Rb atoms must be greater than the number of K
atoms for ecient cooling. Figure 2.15 shows the number of atoms in each species
as a function of the temperature before and after the experimental improvements
detailed above. Here, we see clearly the increase in the K number. At 1 µK, theK number is more than 10 times higher in the After plot than in the Before.
Before the improvements, the low K number allowed the Rb to reach quantum
degeneracy before the evaporation became inecient. Now, the Rb evaporationeciency begins to sharply decrease around 3 µK, where the number of atoms
in the two species become equal. At 3 µK, both clouds are well above quantum
degeneracy with T /TF = 2 and T /TC = 6. Here, TF is the Fermi temperature forthe K atoms and TC is the critical temperature for Bose-Einstein condensation of
the Rb atoms.
Most of our experiments require the atom gases to be quantum degenerateBeforeAfter

39

Figure 2.15: Evaporation in the magnetic trap. The two plots show the numberof atoms N as a function of temperature T before and after the experimental
improvements detailed in 2.2 and 2.3. The open red circles in the After plotshow the number of Rb atoms when no K atoms are loaded into the trap (Rb
only), while the lled red circles represent the Rb number in the presence of K
(lled blue triangles). The lled circles with rings in the Before plot indicate that
the Rb gas is a BEC. The dashed line is a t of the Rb only data in the Afterplot to N = A



T
µK

b

and is plotted in the Before plot for ease of comparison.

The t yields A = (2.1 ± 0.3) × 106 and b = 0.86(2).

or nearly so. To achieve a more degenerate mixture, after evaporating to a few
microKelvin in the magnetic trap, we transfer the atoms to a far-o-resonanceoptical dipole trap. This trap is generated by a single focused laser beam with a

1/e2 radius of 18 µm and a wavelength of 1090 nm. By decreasing the power of the
optical-trap beam, we simultaneously evaporate both species. This eliminates theheat capacity limitations imposed by sympathetic cooling, and we only require
thermalizing interspecies collisions to keep the K cloud in thermal equilibrium.
Thus, in the optical trap we are able to achieve quantum degenerate mixtures.
Figure 2.16 shows time-of-ight absorption images of a nearly pure Rb BEC with

6.5 × 104 atoms and a K cloud of 1.8 × 105 atoms at T /TF = 0.2 released fromthe same trap. Here, we determine T /TF from a surface t to a Thomas-Fermi
distribution [78].

40

K

Rb

Figure 2.16: False-color absorption images of a degenerate Fermi gas and a BoseEinstein condensate. The left image shows a cloud of 1.8 × 105 K atoms at 0.2
TF after 10 ms of expansion from the optical trap. The right image shows a Rb
BEC with 6.5 × 104 atoms after 20 ms of expansion. The calculated radial trap
frequencies are 290 and 180 Hz for K and Rb, respectively, and the calculated
axial frequencies are 3.1 and 1.6 Hz.

41
All of the experimental improvements detailed above have allowed us toachieve quantum-degenerate Bose-Fermi mixtures in the optical trap. However,for the experiments discussed in the rest of this thesis, we evaporate to temperatures slightly above TC to avoid the high Rb atom densities that accompany the
BEC transition.

2.5

Magnetic-eld curvature compensation
In this and the following sections, I will discuss several new capabilities thatwere added to the experiment for the purpose of studying atoms and molecules
near KRb Fano-Feshbach resonances. These include improved control over the
homogeneity of the magnetic eld, the ability to rapidly varying the strength ofthe eld, and the ability to detect the atoms and molecules near the resonances.
Accessing the KRb Fano-Feshbach resonances requires large magnetic elds(> 450 G). We generate these elds with the cart coils with the direction of thecurrent through one of the coils reversed from the quadrupole trap conguration.
Because the coils are not in a proper Helmholtz conguration (where the radius of
the coils is equal to their separation), the eld is not uniform, but has a minimumin the axial direction. This is problematic for two reasons: (1) the Fano-Feshbachresonances we use are between high-eld-seeking atomic states of Rb and K, and if
the anti-trapping force of the magnetic eld overwhelms the trapping force of the
optical trap, which can occur during evaporation, the atoms are ejected from thetrap. (2) Observing atom-molecule oscillations with long coherence times requires
the magnetic eld to be uniform across the atom/molecule cloud (see Chapter 5).
To cancel the curvature of this bias eld, we apply an additional eld generated by the axial coils of the Ioe-Pritchard trap. The IP coils generate a eld in
opposition to that from the cart coils. Since the curvature depends on the current
through the coils, an appropriately chosen current will cancel the magnetic-eld

42
curvature, as shown in Fig. 2.17. Figure 2.18 shows the measured magnetic eldalong the axial direction of the cloud for 376 A through the cart coils and 4.76
A through the IP pinch and bias coils. The current through the IP coils was
empirically optimized to minimize the magnetic-eld variation across the atom
cloud. The gradient implies that the IP and cart coils' elds are not centered on
one another in one or more directions. Perfect cancellation requires the elds fromthe two sets of coils to be perfectly centered on one another in three dimensions,and we have found that the centers are oset by 1.1 mm in the vertical direction.
Eorts are currently under way to correct this and achieve more uniform magnetic
elds.

2.6

Fast magnetic-eld pulse circuit and eddy current correction
Many of our experiments, particularly those discussed in Chapter 5, require

rapid variations in the magnetic eld near the Fano-Feshbach resonances. The
cart coils that primarily generate the magnetic eld have large inductances andare controlled by a servo with a bandwidth of 500 Hz. To achieve fast magnetic
eld ramps, we have installed a low inductance auxiliary coil pair controlled bya high-bandwidth servo. Two 1-inch diameter, single-loop coils separated by 32
mm are mounted concentrically with the IP axial coils. These auxiliary coils areconnected to a capacitor charged to high voltage by a 160-V power supply as
shown in Fig. 2.19 [3]. The auxiliary coils can generate a eld of up to 50 G for
5 ms, and the bandwidth of the servo allows for magnetic-eld ramp durations as
short as 5 µs. When added to the eld generated by the cart coils, the auxiliary
coils can generate rapid magnetic-eld sweeps in the vicinity of the Fano-Feshbachresonances.2
2 Note that the auxiliary coils also add inhomogeneity to the total magnetic eld which must
be compensated for with the IP axial coils as above, but this correction was neglected in the
past.

43

13 mm

(a)

19 mm

37 mm

8.4 mm
5.2 mm

2 mm
21.5 mmz

2 mm
6.5 mm12.8 mm31.2 mm

76 mm

(d)
(b)

(c)

Figure 2.17: Magnetic-eld curvature cancellation. A schematic of the coils used
to generate a nearly uniform magnetic eld is shown in (a). The cart coils, shownin orange, generate a large eld with a small curvature. The axial coils of theIoe-Pritchard (IP) trap, shown in blue, generate a eld with the opposite curvature. The magnetic eld along the axis of the coils for the cart coils (calculated
for a current of 388 A) and the IP axial coils (4.56 A) are shown in (b) and (c),respectively. The total eld (assuming that the eld centers are perfectly overlapped) is shown in (d). Notice the dierent scale. The current through the IP
coils was chosen to minimize the magnetic-eld variation over ±1 mm. With thiscancellation method, the magnetic-eld variation can in principle be reduced to
0.2 mG. A typical rms cloud size along z at the end of evaporation in the opticaltrap is 200 µm.

44Figure 2.18: Measured axial magnetic-eld inhomogeneity. The magnetic eldalong the axial direction of the cloud is measured using rf on the Rb |1, 1i → |1, 0i
Zeeman transition for currents of 376 A through the cart coils and 4.76 A throughthe IP pinch and bias coils. The line is a linear t to the data, yielding a gradientof 0.17 ± 0.7 G/cm.

45
Current sensor

50 k

W

160V

6.8 mF

auxiliary

+

coils

LED

G

E

C

IGBT

Figure 2.19: Circuit used to generate fast magnetic-eld ramps. A Hall eectcurrent sensor (Danfysik Ultrastab 866) monitors the current, and an insulated
gate bipolar transistor, or IGBT, (Powerex CM200HA-24H) controls the current
through the auxiliary coils.

Figure 2.20 shows a magnetic-eld ramp generated by the auxiliary coils
determined from the measured current through the coils. The calculated ramp
rate is 5.5 G/µs. However, the actual magnetic-eld ramp rate is much slower
due to induced eddy currents. Figure 2.21 shows the total magnetic eld after adownward magnetic-eld ramp spanning 7.4 G with a modest ramp speed of 7.6
G/ms. At the end of the ramp, the expected change in magnetic eld is undershot
by nearly 20%, and the eld takes more than 5 ms to decay to the correct value.
We believe this deviation is due to eddy currents induced in the copper forms
that house the IP coils. Such large eddy elds were unacceptable for nearly all of
the experiments discussed in this thesis, particularly the atom-molecule oscillation
experiments discussed in Chapter 5, which require precisely controlled elds with
ramp speeds nearly 20 times faster. While replacing the IP trap forms with others
made of non-conducting material may be an option for the future, we chose toactively compensate for the eddy currents with the current through the auxiliary

46
coils.
We model the eddy currents with a RL circuit that couples to the auxiliary
coil circuit through a mutual inductance as shown schematically in Fig. 2.22. The
magnetic ux through the eddy current circuit (right-hand circuit in Fig. 2.22) isgiven by

Φ2 = I1 M21 − I2 L2 ,(2.15)

where M21 is the mutual inductance, I1 is the current through the auxiliary coils,
and I2 is the current through the eddy circuit. By applying Faraday's Law andKirchho's Law to the eddy current circuit, we nd

I2 R2 + E2 = I2 R2 −

dI2
dI1
dΦ2
= I2 R2 + L2
− M21
= 0.
dt
dt
dt

(2.16)The magnetic elds produced by each circuit are proportional to their respective
currents (B1 = C1 I1 and B2 = C2 I2 ). Thus, we can recast Eq. 2.16 in terms ofthe magnetic elds as

dB2
R2
dB1
B2 +
=η
,
L2
dt
dt

(2.17)

where η = CC2 ML12 . We can determine η and L2 /R2 from measurements similar to12

that shown in Fig. 2.21.
For a linear ramp of the auxiliary magnetic eld, the solution to Eq. 2.17
for the induced eld is given by

B2 (t) = (B2 (0) − ζ) e−t/τ + ζ,

(2.18)

1
where the exponential decay time is τ = L2 /R2 and ζ = −τ η dB
. If B2 (0) is zero,dt

which can be ensured in the experiment, then ζ is the amplitude of the undershootof the eld. We nd that η is 0.295 ± 0.011.
By solving Eq. 2.17 over an entire ramp sequence, we can engineer thecurrent in the auxiliary coils such that the auxiliary eld and the eddy eld sum
to the desired ramp as shown in Fig. 2.23. The measured magnetic eld and47

Figure 2.20: Magnetic-eld ramp determined from the current through the auxiliary coils. The current implies a magnetic-eld ramp rate of 5.5 G/µs.

546.9
552

546.6

549

B (G)

546.3

546
546.0

-2

0

2

4

545.7
545.4
0

1

2

3

4

5

time (ms)Figure 2.21: Decaying magnetic eld after a magnetic-eld ramp. The total magnetic eld at the atoms, measured with rf π -pulses on the Rb |1, 1i → |1, 0i atomic
Zeeman transition, is plotted as a function of time after the end of a 7.6 G/ms
ramp with the auxiliary coils. The red line is a t to a decaying exponential
B = A · e−t/τ + Bf , yielding an undershoot of A = (1.35 ± 0.03) G and a decay time of τ = (0.92 ± 0.03) ms. The inset compares the t to the expectedmagnetic-eld ramp, shown in blue.

R2

R1

L1

M21

48

L2

I1

I2

Figure 2.22: Schematic of the eddy current model. The left-hand circuit represents
the auxiliary coil circuit shown in detail in Fig. 2.19, which is coupled to the eddy
current circuit (right-hand circuit) by the mutual inductance M21 . Shown is the1
direction of the induced current I2 for decreasing auxiliary-coil current I1 ( dI
< 0).
dt

deviation from the desired 7.6 G/ms ramp with the eddy current compensation
are shown in Fig. 2.24. During the entirety of the ramp, the magnetic eld iswithin 80 mG of the expected eld, and after the ramp ends, the eld settles towithin 20 mG of the nal value in only 50 µs. When we perform ramps with speedsof 140 G/ms, we measure additional eddy elds that decay with a timescale of 90µs as shown in Fig. 2.25. We model and correct for these smaller and faster eddy
elds in the same manner and nd the coupling parameter η to be 0.090 ± 0.005.
Figure 2.26 shows the calculated magnetic eld for these fast ramps. With bothlevels of eddy current correction in place we are able to engineer ramps that span
2.4 G with speeds of 140 G/ms to within 35 mG of the desired elds as shown in
Fig. 2.27.

2.7

High-eld absorption imaging
While the uorescence from atoms in the MOT is a useful diagnostic for cal-culating the number and temperature of the atoms in the MOT cell as discussed
above (see 2.3.2), once the atoms leave the MOT cell, we rely on absorption
imaging to extract information about the clouds.

Detailed discussions of ab-

49
15

B (G)

10

5

0

-5
-15

-10

-5

0

5

10

time (ms)Figure 2.23: Magnetic-eld ramps that compensate for eddy elds. The black
curve shows the calculated auxiliary magnetic eld that cancels the induced eddy
eld (blue line) to generate a trapezoidal ramp of the total eld (red line).

(a)

(b)

Figure 2.24: Magnetic-eld ramps with eddy current compensation. The measured
total magnetic eld (•) and the calculated magnetic-eld ramp (red line) are shown
as a function of time in (a). The dierence between the measured and calculated
eld is shown in (b). The dashed vertical lines represent the hinge points of the
ramp.

50

Figure 2.25: Decaying magnetic eld after a fast magnetic-eld ramp. The totalmagnetic eld at the atoms, measured with rf π -pulses on the Rb |1, 1i → |1, 0i
atomic Zeeman transition, is plotted as a function of time after the end of a 140
G/ms ramp with the auxiliary coils. The red line is a t to a decaying exponential
B = A·(1−e−t/τ )+Bf , yielding an undershoot of A = (310±20) mG and a decay
time of τ = (90±10) µs. The inset compares the t to the expected magnetic-eld
ramp, shown in blue.

51
8

6

B (G)

4

2

0

-2

-500

0

500

1000

time (

1500

2000

ms)

Figure 2.26: Magnetic-eld ramps that compensate for eddy elds. The black
curve shows the calculated auxiliary magnetic eld that cancels the sum of theinduced eddy elds (blue and green lines) to generate a trapezoidal ramp of thetotal eld (red line).

Figure 2.27: Measured magnetic-eld deviation after a fast ramp. Plotted is thedierence between the measured magnetic eld and the total calculated eld (redline in Fig. 2.26) after the 140 G/ms upward ramp.

52
sorption imaging can be found, for example, in Refs. [79] and [80]. In brief, acollimated beam of resonant laser light is directed toward the atom cloud, and its
shadow is imaged onto a Roper Scientic back-illuminated CCD camera (MicroMax 1024B). The light integrates through one dimension of the cloud, rendering a
two-dimensional image from which we can extract the atom number, temperature,and density.
For much of the experimental sequence, we are able to interrogate the atoms
with low-eld imaging. In this technique, all trapping elds are turned o, andthe atoms are allowed to ballistically expand for 5 to 20 ms, depending on their
number and temperature. A circularly polarized laser beam is directed toward
the atoms along the axis of the vacuum system (in through the MOT cell, andout through the science cellsee Fig. 2.2). For imaging Rb, the light is tuned to
the f = 2 → f 0 = 3 atomic cycling transition, and for K, the light is tuned to thef = 9/2 → f 0 = 11/2 transition. A small (3 G) magnetic eld is directed along
the axis of the probe beams to maintain quantization. By switching the directionof this eld, we are able to image either the negative or the positive mf atomic
Zeeman states. Using the camera's kinetics mode, we are able to take images ofboth the K cloud and the Rb cloud separated by only a few milliseconds, allowing
us to probe both species in a single experimental run.
Many of our experiments involve manipulating the magnetic eld in thevicinity of interspecies Fano-Feshbach resonances at elds above 500 G. An accurate measure of the number and temperature of the atoms and any molecules
formed at the resonance requires the ability to image the clouds at these large
magnetic elds. Due to the Zeeman eect, the K |9/2, −9/2i → |11/2, −11/2i
cycling transition at these elds is shifted from the zero-eld transition by morethan 600 MHz, and the Rb |2, 2i → |3, 3i transition is shifted by more than 700
MHz. Figure 2.28 shows how the light for this high-eld imaging is obtained from

53
the Rb and K lasers. At low elds, the atomic Zeeman states are nearly degenerate, and are not distinguished by the probe light. However, at these large elds,
the transitions for imaging the two lowest-energy Zeeman states in K, |9/2, −9/2i
and |9/2, −7/2i are separated by > 65 MHz, and the transitions for imaging the|2, 2i and |2, 1i states of Rb are separated by nearly 200 MHz. Thus, imaging athigh elds enables spin-selective imaging of both species. For experiments with
the Fano-Feshbach resonances, the Rb atoms are not in the f = 2 state, but inthe |1, 1i state. To access the f = 2 → f 0 = 3 cycling transition, we transfer the
Rb atoms to the |2, 2i state via microwave adiabatic rapid passage. For this, wecommissioned a coil from the JILA shop that is impedance-matched at 8.0 GHz[63]. The coil is also impedance-matched at 6.8 GHz, the frequency needed toprepare the Rb atoms in the |1, 1i state at low elds.
While we are able image to both species at high elds, our current imaging
conguration is not optimal. As shown in Fig. 2.29, the probe beams are perpendicular to the large bias magnetic eld. The beams, therefore, cannot be made tobe σ + or σ − polarized with respect to the eld. The best option for this geometry
is to linearly polarize the beams. This results in a factor of two decrease in theobserved optical depth, i.e. a factor of two decrease in our signal. The optimal
conguration would be a circularly polarized imaging beam along the magnetic
eld, resulting in an automatic factor of two increase in signal from the polarization and an additional increase in the optical depth due to integrating throughthe long direction of the atom cloud. Eorts are currently under way to install asecond imaging axis along the magnetic eld (dashed lines in Fig. 2.29).

2.8

Isolating the track controller that spews noise all over the lab
One problem that plagued our new and improved experimental setup fromtime to time was rf noise that was picked up by the IP trap. When measuring the

54
118 MHz55-75 MHz56 MHzRb trap
ECDL

-1

+2

+2

to Rb

lock toRb repump

Rb IjLL

f = 2 to f'=2,3

ECDL

optical260 MHzcrossoverlock toRb

f = 1 to f'=1,2
crossover+1

Amplifier

78 MHzpumping

to Rb
low B

+1

imaging

250 MHz+2

to Rb MOT
to Rb highto Rb

B imaging

optical pumping

211 MHz

to K MOT,K repump

opticalECDL

pumping+2

230-250 MHz+2
to beat, Dn = 610 MHz

lock to39

K f=2
to K high B
imaging
to K low B

193 MHz

100-130 MHz

imaging

K trapECDL+2+1

K IjLL

to K
MOT

to K
opticalpumping

Figure 2.28: Laser frequencies for trapping, pumping, probing Rb and K. The
K repump and Rb external cavity diode lasers (ECDLs) are each referenced toatomic transitions using saturated absorption spectroscopy [81], while the K trap
ECDL is referenced to the K repump laser via an oset lock [75]. The green boxes
are acousto-optic modulators (AOMs) with the numbers 1 and 2 representing asingle- or double-passed AOM and the + and - indicating whether the light isshifted blue or red. The loops represent optical bers.

55
pick-up with a spectrum analyzer, we observed noise peaks around 2 and 5 MHz.
These frequencies are particularly detrimental to atoms in the magnetic trap asthey are able to drive Zeeman transitions to untrapped states in low-energy atoms.
For comparison, the trap bottom corresponds to 2.1 MHz for Rb and 0.9 MHz for
K. The result of this rf noise was a severe reduction in the trap lifetime from 200 s
to 40 s, rendering the magnetic-trap evaporation, which takes 60-70 s, hopelesslyinecient.
This rf noise would disappear occasionally, often coinciding with changes
to the IP trap current servos or with changes to the IP trap circuit itself.3
Inevitably, though, the noise would rear its ugly head after a few hours, days, orweeks. Since we believed we were inuencing the noise with changes to the IP
trap circuitry, we thought the rf must have been originating in the power supply,
the servos, or a ground loop in the system. After many months of this mysterious
rf coming and going, we were able to determine the source: the Aerotech Soloist,
the controller for the track for the quadrupole coils.
Shortly after the overhaul of the vacuum system, due to a malfunction in its
communications with the experiment control computer, we replaced the BA Intellidrive that had previously controlled the track with Aerotech's latest controller,
the Soloist. This new controller, while having the advantage of more user-friendly
controls, has the disadvantage that it broadcasts harmonics of 80 kHz up to 25
MHz as shown in Fig. 2.30. The noise was transmitted via the various inputs
and outputs of the Soloist to the control computer, the TTL controls for all theequipment in the lab, the power lines, and, of course, the IP trap. To isolate thecontroller, we built a mu-metal enclosure around the Soloist. However, the transmission lines to and from the Soloist also required isolation. We improved theshielding of the TTL inputs, and the coaxial cable that carries the trigger signal
3 Note that the data in the 2.4 were taken during one such happy time.

56
for the cart has been moved from the cable tray above the optics table, where thenoise is easily transmitted to nearby cables, to below the table, where no othercables pass. Additionally, we now use a ber-optic USB cable to transmit signals
to and from the control computer, and the AC power passes through a noise suppressor to prevent the Soloist from contaminating the AC for other equipment in
the lab. These steps to isolate the track controller have eliminated the rf pick-up
at the IP trap and the associated lifetime problems, allowing for the experiments
discussed in the rest of this thesis.

57

Bprobe lightcameraFigure 2.29: Schematic of the high-eld imaging conguration (top view). Currently, atom clouds (red oval) are imaged with probe beams perpendicular to themagnetic eld. The dashed lines represent a proposed new imaging axis along themagnetic eld.

80 kHzCENTER 18 MHz

SPAN 36 MHz

CENTER 2 MHz

SPAN 3 MHz

CENTER 1.12 MHz

SPAN 500 kHz

Figure 2.30: Noise broadcast by the Soloist cart controller. Shown are spectrum
analyzer traces of rf picked up by a 1-in diameter coil placed in close proximity to
the Soloist. Harmonics of 80 kHz are observed up to 25 MHz.

Chapter 3
Creating molecules with a Fano-Feshbach resonanceThe critical tool that has enabled all of the experiments detailed in thisthesis is the interspecies Fano-Feshbach resonance. Magnetic-eld tunable FanoFeshbach resonances provide the knob with which we can control the interactions between colliding atoms and allow us to coax unpaired atoms into forming
molecules. I will begin this chapter with a brief introduction to Fano-Feshbachresonances and how they aect atomic scattering properties. I will then describe
how we are able to exploit the resonances to create and probe KRb molecules.

3.1

Fano-Feshbach resonances
A full calculation of atom interactions at a Fano-Feshbach resonance re-quires detailed knowledge of interatomic potentials and the coupling between
many atomic energy states [82, 83, 84, 85]. For example, a Fano-Feshbach resonance involving 40 K atoms in the |9/2, −9/2i state and 87 Rb atoms in the |1, 1i
state can couple the atoms to eleven other pair states. However, the atomic interactions near resonance are typically dominated by the coupling between only
two states and can be described by a two-level Hamiltonian [86, 87]. Here, I will
use this simplied two-level picture to illustrate the features of Fano-Feshbachresonances.
The potential energy of two colliding S -state alkali-metal atoms can be59E

closed channel

DE

E

HF

E

res

open channel

E

res

(b)

(a)

E

open channel

R

B

Figure 3.1: Schematic of a Fano-Feshbach resonance. (a) A Fano-Feshbach resonance results from coupling between the threshold of an open channel of theinteratomic potential and a bound state in a closed channel. At zero magnetic
eld, the dissociation thresholds are split by the hyperne energy ∆EHF . (b) An
external magnetic eld causes the energy of each state to vary according to the
Zeeman eect, which can lead to level crossings.

generically described by a potential that is repulsive at small internuclear distances, R, and has a van der Waals attraction that goes as 1/R6 at large R, as
shown schematically in Fig. 3.1a. Hyperne interactions cause the energy levels of
the possible collision channels (labeled by the quantum numbers f and mf of thepair) to split. An atom pair entering an elastic collision in a lower channel withkinetic energy less than the hyperne splitting ∆EHF cannot exit the collisionin the upper channel due to energy conservation. Thus, a channel is said to beclosed if the energy of the colliding atoms is less than that channel's dissociation threshold and open otherwise. In ultracold atom gas experiments, collisionenergies are . 1 µK, while hyperne energy splittings can be 10s to 100s of mK.
Therefore, the dierent channels are well-resolved by the colliding atoms.
It is possible for a bound state in a closed channel to be nearly degenerate with the dissociation threshold of an open channel, as drawn in Fig. 3.1a.
As illustrated in Fig. 3.1b, due to the Zeeman eect the energy dierence between channels can be magnetic-eld dependent. A Fano-Feshbach arises when

60
the dissociation threshold of an open channel, which is coupled to a closed channel, becomes nearly degenerate with a bound state in that closed channel. Theinterchannel coupling is provided by the Coulomb, or exchange, interaction, which
couples together dierent hyperne states at short internuclear distances [85, 88].
When two atoms collide, the strong electrostatic interactions can overcome thehyperne interaction, allowing spin ips to occur. This interchannel coupling creates a dressed molecule state, which is a linear superposition of the open andclosed channel states. The magnetic-eld at which this dressed molecular stategoes through threshold is known as the Fano-Feshbach resonance position B0 andis in general shifted from the eld at which the bare states cross [86].
As we know from the theory of low energy scattering (see for example, Ref.
[89] and [90]), when a bound state is introduced into a scattering potential, thes-wave scattering length diverges. The scattering length near a Fano-Feshbachresonance is given by [88, 91]a(B) = abg 1 −∆B − B0,(3.1)

where abg is the background scattering length for atoms in the open channel, andthe magnetic-eld width of the resonance ∆ is dened as the dierence between

B0 and the eld at which a = 0. Figure 3.2 shows the magnetic-eld dependenceof the calculated interspecies scattering length for the Fano-Feshbach resonanceused in the majority of the experiments in this thesis.

3.2

Locating the resonances with atom loss
It has been shown both theoretically [93, 94, 95, 96, 97] and experimentally[91, 98, 99, 100, 101] that the increased scattering lengths for elastic collisions at
Fano-Feshbach resonances are accompanied by increased inelastic collision rates.
The predominant inelastic collisions in ultracold atom gas experiments are spin

61

B0

abg

D

B
Figure 3.2: Magnetic-eld dependence of the scattering length at a Fano-Feshbachresonance. The interspecies scattering length is calculated for the Fano-Feshbachresonance between 40 K atoms in the |9/2, −9/2i state and 87 Rb atoms in the |1, 1i
state at B0 = 546.76 G [11, 92, 19, 64, 37]. The calculation uses ∆= -3.6 G [64]and abg = −185 a0 [92], where a0 is the Bohr radius.

62
relaxation, in which one or both atoms spin ip into a lower energy state, andthree-body recombination, in which three atoms collide, allowing two to form amolecule with the third carrying away the binding energy. Both of these mechanisms can result in heating and loss of atoms from the trap. Thus, one signature ofa Fano-Feshbach resonance is magnetic-eld dependent atom loss. We originallyidentied four Fano-Feshbach resonances between 40 K atoms in the |9/2, −9/2istate and 87 Rb atoms in the |1, 1i state by measuring the number of atoms in
the trap after a long (1 s) hold at a variable magnetic eld [11]. These initial
experiments were performed with a relatively high temperature (14 µK) and anumber ratio of NRb /NK = 10.
Upon achieving the improved conditions discussed in 2.4, we set about
remeasuring the positions of the Fano-Feshbach resonances by observing losses.
We prepared the mixture in an optical trap with radial and axial frequencies of
340 and 4.4 Hz for Rb and 495 and 6.3 Hz for K. The number of atoms in eachspecies varied from day to day from 4 − 7 × 104 K atoms and 6 − 9 × 104 Rb atoms
at a temperature of 320 nK, corresponding to 0.8 − 0.9 TF and 2.0 − 2.3 TC . Here,TF is the Fermi temperature for the K gas, and TC is the critical temperature for
Bose-Einstein condensation for Rb. Figure 3.3 shows the locations of three FanoFeshbach resonances between 40 K and 87 Rb in their lowest energy states. For eachresonance, the number of atoms remaining in each species after a variable holdtime at Bhold was measured. The hold time was chosen such that for all values of

Bhold no more than 80% of the initial number in each species was lost from thetrap. Additionally, it was conrmed for each resonance that loss was observed inboth species and that no losses were observed if only one species was present. In
Figs. 3.3b and c, the atom loss feature at the Fano-Feshbach resonance appears
asymmetric. We attribute this asymmetry to losses incurred when the magnetic
eld is ramped across the resonances. A schematic of the time-dependent magnetic

63
eld is shown in Fig. 3.3a.
Figures 3.3d and 3.3e show number loss and corresponding heating at the
Fano-Feshbach resonance at 515 G. Both show a double-peaked structure, indicating that this resonance is a p-wave resonance [102].1 The existence of a p -wave
resonance in this mixture had been predicted in Ref. [102], but which of the four
originally observed Fano-Feshbach resonances was the p -wave resonance became
the subject of some confusion. Due to the high temperature of the initial observations [11], the doublet structure could not be resolved. Our calculations suggested
that the 495 G resonance was a p -wave resonance [11], while a subsequent calculation from the LENS group constrained by the observation of many additional
resonances in several spin states ascribed p -wave character to the 515 G resonance[92]. This later identication was conrmed by S. Ospelkaus et al., with the observation of the doublet structure of the 515 G resonance through a measurement
similar to that shown in Fig. 3.3d, but in ner detail [19].

3.3

KRb Feshbach molecules
If experiments can be performed on timescales short compared with thosefor the atom losses and heating at a Fano-Feshbach resonance, the divergenceof the s-wave scattering length (Fig. 3.2) and the introduction of a bound state(Fig. 3.1) can be exploited. Some of the initial experimental work with FanoFeshbach resonances employed magnetic-eld control of atomic interactions toimprove evaporation eciency in 85 Rb [99] and 133 Cs [103] and to control thesize of and induce collapse in a BEC [104, 16]. In recent years, however, much
experimental work has been directed toward the production of large samples ofultracold molecules via magnetic-eld sweeps across Fano-Feshbach resonances
1 For the data in Fig. 3.3, we were limited to steps of 40 mG by the resolution of the DAC
that controlled the magnetic eld. This limitation was later eliminated by the introduction ofthe Fast B circuit discussed in 2.6.

64

(a)
magnetic field

thold
Bhold

Bstart

time

(b)
(d)
Bhold (G)(e)(c)
Bhold (G)

Bhold (G)Figure 3.3: Magnetic-eld dependent atom loss at Fano-Feshbach resonances between 40 K atoms in the |9/2, −9/2i state and 87 Rb atoms in the |1, 1i state. The
resonances are approached from below as shown in (a). For each resonance, Bstartis at least 5 G below the peak of the loss. Plotted in (b)-(d) is the measured atom
number in one species after a variable hold time at a magnetic eld Bhold in thevicinity of three Fano-Feshbach resonances between 40 K and 87 Rb atoms in theirlowest energy states. The hold times are 100 ms in (b), 40 ms in (c), and 400
ms in (d). Shown in (e) is the increase in temperature associated with the atom
loss at resonance. The lines are guides to the eye. In all the plots, the error bars
represent the statistical error of repeated data points.

65[105, 8]. This magneto-association technique has been shown to convert as muchas 90% of a sample of ultracold atoms into highly vibrationally excited Feshbach
molecules [61].

3.3.1

Magnetic-eld sweeps

Since Fano-Feshbach resonances arise from the coupling of an atom pairstate to a bound state, it is possible to associate free atoms into molecules with
time-dependent magnetic elds. A variety of methods have been used to create
Feshbach molecules. These include applying rapid magnetic-eld pulses [44] andoscillating magnetic elds [62] near a resonance. The simplest technique for associating atoms into Feshbach molecules is to sweep the magnetic eld throughthe resonance [105]. As shown schematically in Fig. 3.4, atoms initially preparedon the high-eld side of the resonance can be converted to molecules by sweeping
the magnetic eld downward in time. Molecules can be created if the ramp speed
is slow compared to the two-body physics of the Fano-Feshbach resonance. If theramp is too fast, however, the atoms will remain in the atom pair state after the
ramp. It is this method of using adiabatic magnetic-eld sweeps that we employed
to begin creating KRb Feshbach molecules.
There are several Fano-Feshbach resonances between K |9/2, −9/2i atoms
and Rb |1, 1i atoms (see Fig. 3.3, Ref. [92]). We choose to create KRb molecules
using the resonance near 547 G because it is the widest accessible resonance between these two states. To make molecules, we prepare a mixture of 8 × 104 Rb
atoms and 2.5 × 105 K atoms in an optical trap with radial trapping frequencies
of 560 Hz for Rb and 730 Hz for K and axial frequencies of 7.2 Hz for Rb and10.4 Hz for K. The temperature of the mixture is 300 nK, which corresponds to
1.2 TC and 0.3 TF . For the 547 G resonance, the molecule state exists only on thelow-eld side of the resonance. A magnetic-eld sweep to create molecules must,

66E

2-atom state

0

B

0

molecule state

B
Figure 3.4: Molecule association with magnetic-eld sweeps. At the FanoFeshbach resonance, the energy of the 2-atom state and the molecule state aredegenerate. By preparing the atoms on the high-eld side of the resonance, thensweeping the eld downward through resonance, atoms can be converted into
Feshbach molecules.

67
therefore, begin above the resonance. We prepare the atoms on the high-eld sideof the resonance by rst slowly increasing the magnetic eld to 542.1 G during theoptical trap evaporation, then rapidly sweeping the magnetic eld up through theresonance to 549.9 G at a speed of 140 G/ms. To create molecules, we then sweep
the magnetic eld down through the resonance to 543.6 G as shown in Fig. 3.5a.
To see the eect of the magnetic-eld sweep, we measure the number of remaining
Rb atoms using resonant absorption imaging. The gas is released from the trap
and imaged after 2 ms of expansion. The Rb probe light is not sensitive to themolecules. Therefore, molecule creation results in a perceived loss of atoms. The
measured number of Rb atoms remaining after the sweep through resonance isshown as a function of the ramp speed in Fig. 3.5b. For these data, the 1/e ramp
speed for molecule creation is Ḃ1/e = (26 ± 7) G/ms.
The observed atom loss from magnetic-eld sweeps through resonance isconsistent with molecule creation, but as we have seen in Fig. 3.3, Fano-Feshbachresonances can also cause real atom losses. To verify that we are adiabaticallycreating molecules, we can check that the apparent atom loss is reversible. We
sweep the magnetic eld through resonance at a speed of 3.3 G/ms, then reverse
the sequence as shown in Fig. 3.5b. The sweep back up through resonance dissociates the molecules, causing them to reappear as unbound atoms that we arethen able to image. Figure 3.5d compares the measured number of Rb atoms
after such a double-sweep experiment with the number left after a single sweep.
The number of atoms remaining after two sweeps is larger than after only one
sweep, indicating that some of the atoms were not lost but transferred to themolecule state after the rst sweep. The number of molecules recovered by the
upward sweep equals the dierence between the two curves in Fig. 3.5d. For the
shortest hold time, we recover 3 × 104 molecules. By varying the time between
the two sweeps, we can measure the lifetime of the molecules. At the hold eld

68

B

B

(a)

(b)

B = 3.3 G/ms

549.9 G

B0

B0543.6 G

t
(c)

t
(d)

inverse ramp speed

Figure 3.5: Molecule creation via magnetic-eld sweeps through a Fano-Feshbachresonance. A schematic of the time dependent magnetic eld for creating
molecules is shown in (a). The measured number of Rb atoms NRb remaining after the molecule creation ramp as a function of the inverse ramp speed(1/Ḃ) is shown in (c). Before the resonance is crossed, the measured number of
Rb atoms is NRb = 8 × 104 . The line is a t to NRb = A e−β/Ḃ + y0 , yielding
A = (4.0 ± 0.6) × 104 and β = (26 ± 7) G/ms. Note that the number of Rb
atoms remaining after the fastest sweeps indicates an initial loss of 2 × 104 Rb
atoms, which may be due to the rst resonance crossing. (b) shows a schematic of
the magnetic-eld sweeps used to measure the molecule lifetime in (d). (d) The
blue points show the measured number of Rb atoms after a slow sweep back up
through resonance as a function of the hold time at B = 546.27 G. To accountfor any atom loss at the hold eld, this number is compared with the number ofatoms remaining after the molecule creation sweep and hold (red points). Subtracting the two curves, which are exponential ts to the data, yields the numberof molecules recovered by the second sweep Nmol = (3.1 ± 0.2) × 104 and the 1/elifetime τ = (2.0 ± 0.4) ms.

69
of B = 546.27 G, we observe a decay in the number of atoms (red points in Fig.
3.5d), and to accurately measure the molecule lifetime, we must subtract thisatom loss. For the molecules in Fig. 3.5d, the 1/e lifetime is (2.0 ± 0.4) ms.
Comparing the initial number of Rb atoms, 8 × 104 , with the number measured after one and two sweeps, we observe losses that are not recovered by thesecond sweep. We believe this is due to the enhanced 3-body and atom-molecule
collision rates associated with the resonance [106, 107].

3.3.2

Density dependence of molecule creation sweep rates

Theoretical calculations for the adiabatic association of molecules with magneticeld sweeps in single-species experiments have shown that the 1/e ramp speed
depends linearly on the atom gas density [86, 108]. Measurements of the adiabatic ramp speed for varying densities in 85 Rb experiments are consistent with alinear dependence [61]. We have measured the 1/e ramp speeds for a variety of
Rb and K densities as shown in Fig. 3.6. We vary the density by changing both
the relative number in the two species and the strength of the optical trap. We

pR
hnK i hnRb i, where hni i = N1i n2i (r)d3 r is thedene the two-species density as 2

average number density of species i. This generalization from the single-species
case arises from the fact that the number of possible pairs in a single species
is N (N − 1)/2 ≈ N 2 /2, while the number of possible pairs in the mixture is

NK NRb [109]. The measured 1/e ramp speeds are inconsistent with the expected
linear dependence on the two-species density, but instead show a quadratic dependence on the density. This suggests that our generalization to two species isincorrect. However, these measurements were performed observing only atom loss
(not directly imaging moleculessee 3.4 below). Our measurements of molecule
number are always smaller than the observed atom loss, indicating atom and/or
molecule loss during the sweep. Therefore, the measured sweep speeds may be

70
a convolution of molecule creation and loss rates. Future eorts will be focusedon understanding (and minimizing) loss mechanisms, and studying the density
dependence of molecule creation rates.

3.3.3

Why we no longer use a 1075 nm broadband laser

Prior to the results shown in Fig.

3.5, our initial experiments creating

molecules were performed in an optical trap formed from a multi-longitudinalmode 1075 nm ber laser. We were able to observe loss with a single downward
sweep through the Feshbach resonance, while no loss was observed with a single
upward sweep.

This was suggestive of molecule formation, but double sweep

experiments similar to that shown in Figs. 3.5c and d yielded small numbersof atoms returning with lifetimes much shorter than 1 ms. Zirbel and coworkers
demonstrated that the short molecule lifetime was due to the optical trap light [70].
We believe the trap light excites the Feshbach molecules to the lowest vibrational
states of the 3(1) excited state potential, as shown in Fig. 3.7. We now use a
multi-longitudinal mode ber laser at a wavelength of 1090 nm for the optical
trap. The energy of a photon from this laser is too small to excite bound-boundtransitions to the 3(1) state. While the 2(1) states are accessible, these are notexpected to be strong transitions [110]. With this laser forming the optical trap,
we have observed molecule lifetimes as long as 10 ms, which are likely limited bycollisions with unbound atoms [106].

3.4

Molecule imaging
The fact that we are able to recover missing atoms with additional magnetic-

eld sweeps indicates that we are, in fact, creating molecules with the initial
downward sweep through the resonance. This method of comparing the numberof missing atoms after a molecule association sweep with the number recovered

71

Figure 3.6: Measured density dependence of adiabatic magnetic-eld sweep
speeds. The number of atoms remaining after magnetic-eld sweeps through the
Fano-Feshbach resonance were measured as a function of the sweep speed as inFig. 3.5c, for mixtures with varying densities. The 1/e sweep speed Ḃ1/e is plottedp

as a function of the two-species density nKRb = 2 hnK i hnRb i. The line is a t ofthe data to Ḃ1/e = A · nbKRb , yielding A = (3.7 ± 0.3) × 10−27 cm6 and b = 2.1 ± 0.2.

72

Figure 3.7: KRb molecular potentials. Laser light with a wavelength of 1075 nm
can excite transitions from the Feshbach molecule state to vibrational states inthe 3(1) or 2(1) excited molecular potentials. (This gure was produced using thepotential energies of the KRb molecule from Ref. [111].)

73
with a dissociation sweep will underestimate the molecule number if there are any
additional atom or molecule losses as the Fano-Feshbach resonance is traversed
again. Additionally, if the fraction of atoms converted to molecules is small, an
accurate measurement of the molecule number is dicult due to typical shot-toshot atom number variations of 7%. To avoid these limitations, we image themolecules directly.
We detect the molecules with spin-selective absorption imaging using lighttuned to the K |9/2, −9/2i → |11/2, −11/2i cycling transition [112, 64]. This lightis sensitive to molecules and unbound K atoms. To avoid imaging the unboundatoms, we use a 30-µs pulse of rf tuned to the K atomic Zeeman transition totransfer the atoms to the previously unoccupied |9/2, −7/2i Zeeman state with
99% eciency. We typically image the molecules at B = 546.04 G. Because theirbinding energy of h×140 kHz is large compared to the spectral width of the rf
pulse, the molecules are unaected by the pulse. A 50-µs pulse of probe light isthen applied after 0 to 5 ms of expansion from the trap. Presumably, the rstphoton incident upon a molecule causes it to dissociate, while subsequent photons
scatter o the newly unbound K atom.
We are able to image the molecules at magnetic elds within 1 G of the
Fano-Feshbach resonance.

This range is large compared to that expected forthe case of homonuclear molecules at a similar resonance. This is because theenergy dierence between the molecular ground and excited electronic potentials
varies more slowly with internuclear separation R for the case of heteronuclear
molecules than for the homonuclear case. For homonuclear molecules, the excitedelectronic potential varies as 1/R3 , while the ground potential varies as 1/R6 . For
heteronuclear molecules, both potentials vary as 1/R6 , allowing for a wider rangeof binding energies over which the molecules can absorb imaging light tuned tothe atomic resonance.

74

3.5

Locating the resonance with molecule sweeps
As we have seen in 3.2, we can locate a Fano-Feshbach resonance by observ-

ing atom losses, but a more precise determination of its position can be obtainedby observing the Feshbach molecules. To measure the position of the resonance,
we observe the dissociation of molecules in a low density mixture [6] with themagnetic-eld sequence shown in Fig. 3.8a. Molecules are created with a slow
magnetic-eld sweep through the resonance. The optical trap is then abruptly
turned o to allow the atom/molecule mixture to ballistically expand and thedensity to drop. The magnetic eld is then increased to a value Bhold near the
Fano-Feshbach resonance, then decreased to B = 546.04 G to image any remaining
molecules. If Bhold is above the resonance, the molecules dissociate. Performing
this measurement on the expanded gas ensures that the atom density is suciently
low that no molecules are formed with the nal ramp through resonance and thatmany-body interactions are negligible. The observed number of molecules decreases sharply at B0 = (546.66 ± 0.06) G as shown in Fig. 3.8b, and in good
agreement with Refs. [64] and [37]. This method oers a much more precise measurement of the Fano-Feshbach resonance than the 3 G wide loss feature observedin Fig. 3.3.

75

B

(a)

Bhold

B0
t

0

(b)

Bhold (G)Figure 3.8: Determination of the Fano-Feshbach resonance position via molecule
dissociation. A schematic of the magnetic-eld sequence for creation and dissociation of molecules is shown in (a). The measured number of molecules Nmol as afunction of Bhold is shown in (b). A t of the data to an error function yields theFano-Feshbach resonance position B0 = (546.66 ± 0.06) G, where the uncertainty
is given by the 10%-90% width.

Chapter 4
Semiclassical Monte Carlo simulations of molecule conversion

In the previous chapter, I showed that we can associate ultracold atoms
into molecules with adiabatic magnetic-eld sweeps through a Fano-Feshbach resonance. This technique had become widely used in single-species experiments
by 2003, but how the nal molecule fraction depends on experimental conditions
was not studied in detail experimentally until 2005. Hodby et al. proposed thatmolecule conversion via an adiabatic sweep should depend on the phase space density of the atomic gas [61]. An adiabatic sweep smoothly alters the atom pairs'
wavefunctions, but does not change the number of radial nodes in the two-atom
wavefunction. Thus, Hodby et al. argued, for a pair of atoms to be converted intoa molecule, they must have originally been suciently close in phase space forthe pair wavefunction to be converted into that of a bound molecule. They developed a semiclassical Monte Carlo simulation to test this hypothesis and foundthat the experimentally measured molecule conversion eciency in the cases ofidentical bosons and a two-component Fermi gas are described equally well by
this model. The simulation was later extended and shown to successfully predictmolecule conversion in a two-species Bose gas [63]. In this chapter, I will discuss
the extension of the simulation to include Bose-Fermi mixtures for the purpose ofdesigning experiments that maximize the fraction of molecules created.

77

4.1

Monte Carlo simulation
The Monte Carlo simulation was developed to describe the fraction of atoms

converted to molecules in the limit of slow magnetic-eld sweeps. It does notaddress the dynamics of molecule formation, nor does it describe the timescalesof the adiabatic sweeps. The simulation operates on three simple assumptions:(1) Any atom can form a molecule with any other atom in the single-species case,or any non-identical atom in the case of two-component samples, if the two aresuciently close in phase space, (2) Once a molecule is formed, the two atoms
remain paired, and (3) The system is staticthere are no losses, no collisions,and no motion over the timescale of the conversion. This implies that threebody recombination is so slow that it can be ignored, and the only mechanism
for molecule formation is magnetoassociation. The simulation rst generates anensemble of particles whose randomly assigned positions and momenta follow thecorrect distributions for trapped bosons and/or fermions. The simulation program
searches for a partner for each atom in turn, forming a molecule if the pair of atoms
satises the phase space criterion|∆rrel ∆vrel µ| <

γ
h,
2

(4.1)

where ∆rrel is the spatial separation of the two atoms, ∆vrel is their relative
velocity, µ is reduced mass, h is Planck's constant, and γ is an experimentally
determined constant.1
The conversion fraction predicted by the simulation was compared to experiments with an uncondensed Bose gas of 85 Rb atoms and a two-component Fermi
gas of 40 K atoms taken over a wide range of phase space densities of the gas [61].
Figure 4.1, reproduced from Ref. [61], shows the measured molecule fraction and1 Equation 4.1 diers from Eq. 3 in Ref. [61] by a factor of 2 because I have made thesubstitution m → 2µ.

78
prediction from the Monte Carlo simulation. The molecule conversion eciency
increases with increasing phase space density (decreasing T /TF in Fig. 4.1 b) as
expected, and the experimental data agree very well with the simulations. The
results are best t values of γ = 0.44 ± 0.03 for the boson data and γ = 0.38 ± 0.04
for the fermion data, indicating that molecule association is governed by the same
physics in both cases.

4.2

Two-component systems
The 85 Rb and 40 K experiments showed that molecule formation via adia-

batic sweeps depends on the phase space density of the atom gas, and in bothcases, the conversion fraction approaches 100% as the quantum degeneracy of theatom gas increases. There are, however, factors in addition to the phase spacedensity that can aect the conversion eciency in two-component systems. The
40

K experiments were performed with a nearly 50/50 mixture of two spin statesthat experienced the same trapping potential, leading to identical spatial andmomentum distributions for the two spin components. If the two components
have dierent spatial proles, quantum degeneracies, or numbers of particles, theoverall eciency of molecule conversion will change. In the 40 K experiment, it
was observed that by changing the ratio of the two spin states, the fractional
conversion of the spin state having fewer atoms could be increased [113].
The Monte Carlo simulation has been extended to include heteronuclear
mixtures of bosons and compared to experiments in which Feshbach molecules
were associated from an ultracold mixture of 85 Rb and 87 Rb [63]. Using the valueof γ determined from the single-species boson experiment [61], the simulation
correctly predicted the molecule conversion eciency. While previous experiments
had shown monotonic increases in molecule conversion with phase space density(see Fig. 4.1), in this experiment, the measured molecule fraction follows a peaked

79

(a)

(b)

85

Rb

40

K

Figure 4.1: Temperature dependence of the molecule conversion in single-species
experiments. The measured fraction of 85 Rb atoms converted into molecules as afunction of phase space density for an uncondensed sample is shown in (a), andthe fraction of 40 K atoms converted into molecules as a function of the degeneracy of the two-component Fermi gas is shown in (b). The solid circles (•) areexperimental data, while the lines are the best ts to Eq. 4.1 for the Monte Carlosimulation. (Figure reproduced from Ref. [61].)

80
curve as function of quantum degeneracy as shown in Fig. 4.2. The temperature of87

Rb is varied across the BEC transition, but the 85 Rb gas remains uncondensed

due to a smaller number of atoms. The fraction of 85 Rb atoms converted intomolecules increases as the 87 Rb gas cools, approaching TC , the critical temperaturefor Bose-Einstein condensation, but then decreases as the 87 Rb condensate fraction
increases. Above TC , the spatial proles and momentum distributions of the two
clouds are similar, and as in the single-species case, higher phase space density
results in a larger conversion fraction. Below TC , a signicant fraction of the87

Rb atoms are in the condensate. The uncondensed 85 Rb cloud has a spatial

extent and momentum distribution that are much larger than the BEC's. This
reduced overlap causes reduced molecule conversion despite the increased quantum
degeneracy.

4.3

40

K-87 Rb Bose-Fermi mixture in particular

In a Bose-Fermi mixture, we expect the molecule conversion to behave similarly to the 85 Rb-87 Rb mixture experiment discussed above. If the temperatureof the mixture is brought below TC for the bosons, there will be a reduction in
the spatial overlap of the boson and fermion clouds, causing a reduction in themolecule conversion eciency. We have adapted the Monte Carlo simulation toaccommodate a mixture of bosons and fermions and have compared the predicted
molecule fraction to data from another 40 K-87 Rb experiment at JILA [64].
In this experiment, the molecules are not created by magnetic-eld sweeps,
but by rf association [112].

For this process, the atoms do not begin in the

Fano-Feshbach resonant state.

87

Rb atoms in the |1, 1i state and 40 K atoms

in the |9/2, −7/2i states are prepared on the low-eld side (molecule side) of
the Fano-Feshbach resonance at B0 = 546.76 G; this resonance aects collisions
between Rb |1, 1i atoms and K |9/2, −9/2i atoms Rf photons tuned to the K

81

Figure 4.2: Temperature dependence of the molecule conversion in a bosonicmixture of 85 Rb and 87 Rb. The measured fraction of 85 Rb atoms converted tomolecules as a function of the 87 Rb degeneracy is shown. Throughout the experiment, the 85 Rb gas remains uncondensed. The solid circles (•) are experimental
data, and the lines represent the the molecule fraction predicted by the Monte
Carlo simulation using γ = 0.44 ± 0.03. (Figure reproduced from Ref. [63].)

82
|1,1

+

|9/2, -7/2

|1,1

+

|9/2, -9/2

rf

E

b

Fano-Feshbach molecule

Figure 4.3: A simplied energy level diagram for rf association. Atoms start in
the |1, 1i + |9/2, −7/2i state and are transferred into the molecular state via an rf
photon detuned from the K |9/2, −7/2i → |9/2, −9/2i Zeeman transition by anamount equal to the molecule binding energy Eb .

|9/2, −7/2i → |9/2, −9/2i atomic Zeeman transition can create atom pairs whose
interactions are controlled by the nearby resonance. An rf photon blue detuned
from the |9/2, −7/2i → |9/2, −9/2i transition by the molecule binding energy
can create a Feshbach molecule. A schematic of the energy levels involved in theassociation is shown in Fig. 4.3.
To create molecules, a mixture of 3 × 105 Rb atoms and 1 × 105 K atoms
are assembled in a trap with radial trapping frequencies of 211 and 136 Hz andaxial trapping frequencies of 4 and 2 Hz, for K and Rb, respectively, at a temperature of T /TC = 0.8 and a magnetic eld of B = 546.17 G. A transverse rf
eld is applied with a frequency that is swept from 80.132 to 80.142 MHz in 2 ms.
These frequencies correspond to the energy splitting between the |9/2, −7/2i and|9/2, −9/2i plus 90 to 100 kHz, spanning the 95 kHz molecule binding energy. The
sweep creates Feshbach molecules via rf adiabatic rapid passage, and as discussedin 3.4, the number of molecules created can be measured with direct imaging.
The fraction of K atoms converted to molecules as a function of temperature
is shown in Fig. 4.4. The temperature is varied by holding the mixture in the trap

83
for a variable time after the evaporation ends. Instabilities in the optical trap
strength and position cause heating, resulting in an increase in temperature while
the number of atoms remains constant. As expected, above TC , the molecule
fraction increases as the temperature decreases due to the increased phase spacedensity, but decreases below TC as the spatial overlap decreases. The solid line in
Fig. 4.4 shows the molecule fraction predicted by the Monte Carlo simulation using

γ = 0.44, and the dashed lines represent the uncertainty in the conversion due touncertainties in the measured numbers, temperature, and trapping frequency. The
predictions from the model are scaled by 0.70 to account for observed molecule
losses due to collisions with unbound atoms on the timescale of the rf sweep.
While the Monte Carlo simulation was originally developed to describe
molecule association by adiabatic magnetic-eld sweeps through resonance, we
expect that it will also describe Feshbach molecules created via rf association. The
model was shown to correctly predict the fraction of atoms converted to molecules
through a similar technique in 85 Rb by Thompson et al. [62]. In that experiment,
atoms were prepared in the Fano-Feshbach resonant state, and molecules wereassociated by modulating the magnetic eld at a frequency equal to the moleculebinding energy. Here, we see that the simulation does correctly predict the fractionof atoms converted to molecules via rf association and for the case of a Bose-Fermi
mixture.
Since we are able to accurately predict the molecule conversion eciency
with the Monte Carlo simulation, we can use it to design experiments to optimize
the molecule fraction. The data in Fig. 4.4 were taken with a 3:1 Rb to K number
ratio in a weak optical trap with signicant gravitational sag between the two
atomic gases. Now, we ask: how do these and other experimental parameters
impact the molecule fraction?

84

Figure 4.4: Temperature dependence of the molecule conversion in a Bose-Fermi
mixture of 87 Rb and 40 K. The fraction of K atoms converted to molecules is plotted
as a function of T /TC for the Rb atoms. The solid circles (•) are experimental
data, while the lines are the result of the Monte Carlo simulation for NRb /NK = 3,
taking into account the eects of gravitational sag and uncertainties in the trap
frequencies and atom numbers. This gure diers from Fig. 3 in Ref. [64] in thatthe data above TC are shifted to higher temperature due to correcting an error in
the calculated trapping frequencies. Also, the simulated conversion fraction has
been scaled to correct for the atom-number dependence in the simulation, which
results in lower conversion fraction below TC and larger conversion fraction above
TC for larger atom number. The simulation was originally run for 16 000 atoms,
while the experiment was performed with 400 000 atoms.

85

4.3.1

Number ratio

Let us consider the eect of the number ratio on molecule conversion. For the
purpose of elucidation, we consider a mixture of equal numbers of two Boltzmanngases (atoms that are suciently hot that we need not consider whether theparticles are bosons or fermions), which we label A and B . Let us assume thatmolecule conversion is less than one hundred percent. Now, if we double thenumber of particles in B while keeping the temperature xed, we have doubled
the number of possible partners for a given atom in species A. This is likely
to increase the number of pairs that satisfy the phase space criterion Eq. 4.1,
resulting in a larger fraction of species A converted into molecules.
As we have seen above, to associate a signicant fraction of atoms intomolecules, the atom gas must be nearly quantum degenerate. Therefore, we musttake into account whether the atoms are bosons or fermions, and whether one
species Bose condenses. Figures 4.5 and4.6 show the results of the Monte Carlosimulation for a trapped mixture of fermionic 40 K and bosonic 87 Rb. As expected,
the fraction of atoms of one species converted to molecules depends on the numberof atoms in the other species. For all cases, the molecule fraction is maximized
when T = TC as we have seen in Figs. 4.2 and 4.4. With an excess of fermions at

TC for the bosons (see Figs. 4.5 a and b), the simulation predicts 80% of the Rb
population to be converted into molecules with a number ratio of NK /NRb = 3,
while a number ratio of NK /NRb = 4 yields 90% conversion eciency. For an
excess of bosons, however, the molecule conversion fraction for the K atoms does
not exceed 70% even with number ratios as high as NRb /NK = 25.
To understand this dierence we must consider the phase space density ofeach species individually. If the number of bosons at TC is xed and the numberof fermions increases, the number of possible partners for the bosons increases,

86

Below T

Above T

C

(b)

(c)

(d)

N

K

fixed

N

Rb

fixed

(a)

C

Figure 4.5: Eect of number ratio on molecule conversion for xed TC . Plotted
are the results of the Monte Carlo simulation for molecule conversion for various
temperatures and number ratios of K and Rb conned in an isotropic trap. (a)and (b) show the fraction of Rb atoms converted into molecules as a function of
the ratio of K to Rb atoms with the Rb number xed. Here, TC is constant, but
T /TF decreases with increasing NK . For clarity, (a) shows the conversion fraction
for T ≤ TC , while T ≥ TC is shown in (b). (c) and (d) show the fractional
conversion of K atoms into molecules while varying the Rb number. Here, TC and
T /TF increase with increasing NRb . Each closed circle (•) represents a single runof the Monte Carlo simulation, and the lines, which are double-exponential ts,are intended only as a guide to the eye.

87

(b)

(c)

(d)

NK fixed

NRb fixed(a)Figure 4.6: Eect of number ratio on molecule conversion for xed TF . Plotted
are the results of the Monte Carlo simulation for molecule conversion for various
temperatures and number ratios of K and Rb conned in an isotropic trap. The
fraction of Rb atoms converted into molecules as a function of the ratio of K to
Rb atoms with the Rb number xed is shown in (a) and (b). Here, T is constant,and T /TC increases with increasing NK . The fraction of K atoms converted intomolecules as a function of the ratio of Rb to K atoms with the K number xedis shown in (c) and (d). Here, T is constant, and T /TC decreases with increasing
NRb . The lled symbols represent points where T /TC > 1, and the open symbols
represent points where T /TC < 1. The lines are guides to the eye.

88
increasing the likelihood of making molecules (see Figs. 4.5a and b). Since thetemperature is xed and the number of fermions increases, the phase space densityof the fermions increases. This increased phase space density results in an increasein the molecule fraction as in the simple example above. If, instead, we have a xednumber of fermions at TC with varying boson number (see Figs. 4.5c and d), thesituation becomes slightly more complicated. As the number of bosons increases,
the number of possible partners for the fermions increases, but TC increases aswell. Since the number of fermions is xed, the phase space density of the fermionsdecreases due to the increased temperature. It is this competition between theincreased number of possible partners and the decrease in the phase space densityof the fermions that leads to lower conversion fraction for an excess of bosons
than an excess of fermions. Alternatively, we could x the number of fermionsand the temperature and vary the number of bosons as in Figs. 4.6c and d. In thiscase, the phase space density of the fermions remains constant. As the number ofbosons increases, the number of possible pairs increases as does the phase spacedensity of the bosons, leading to an increase in molecule conversion. If we began
at TC as in the previous example, any increase in number will cause the bosons
to condense leading to a reduction in spatial overlap. Here, it is the competition
between the increased number of partners and the reduced spatial overlap that
leads to lower conversion fraction for an excess of bosons.
From the simulations, we nd that by operating near TC for the bosons witha modest excess of K atoms, we can expect the majority of the Rb atoms to beconverted into molecules. We have compared this prediction with the number ofatoms that disappear from detection in adiabatic sweep experiments. The mixture
is prepared in an optical trap with radial trapping frequencies of 560 Hz for Rb
and 730 Hz for K and axial frequencies of 7.2 Hz for Rb and 10.4 Hz for K. With

NRb = 7 × 104 and a number ratio of NK /NRb = 3.6 ± 0.5, the temperature of the

89
mixture, 350 nK, corresponds to 1.4 TC for the bosons and 0.4 TF for the fermions.
Figure 4.7 shows experimental data for the number of Rb atoms remaining aftera magnetic-eld sweep through resonance as a function of the inverse ramp speed.
If we assume all the missing atoms have been converted to molecules, the observed
molecule fraction for adiabatic sweeps agrees well with the simulation. Here, wend that (66 ± 6)% of the Rb atoms are converted to molecules. Note, however,that fewer than 20% of the K atoms are converted to molecules. From this point
forward, I will use the term molecule fraction to mean the fraction of the lowernumber species that is converted to molecules since the maximum possible numberof molecules is set by the lower-number species.

4.3.2

Gravitational sag

The calculations in 4.3.1 assumed that the two species are spatially centeredon the same point. In the experiment, the Rb and K atoms experience a dierent
gravitational sag, causing the cloud centers to be separated in the vertical direction
by an amount ∆y = g



1
2
ωRb


− ω12 . Here, g is the acceleration due to gravity andKωK and ωRb are the angular trapping frequencies in the vertical direction for K and
Rb, respectively. Figure 4.8 shows the predicted molecule fraction as a functionof the separation of the cloud centers. The molecule conversion begins to drop
steeply when the cloud separation becomes comparable to the rms widths of theclouds. For the data in Fig. 4.4, the gravitational sag was nearly twice the rms
widths of the clouds, resulting in a 30% reduction in the peak molecule conversion
eciency when compared with the case with no gravitational sag. However, theexperiments in Fig. 4.7 and all of the experiments discussed in Chapters 3 and5 were performed with clouds whose dierential gravitational sag was less thanhalf the rms cloud size, where the eects of sag on the molecule conversion can besafely neglected.

NRb

90

Figure 4.7: Adiabatic molecule conversion with an excess of fermions. Initially, a
mixture is prepared with NK /NRb = 3.6 ± 0.5 at T /TC = 1.4 (green points). The
red points show the number of Rb atoms remaining after a magnetic-eld sweep
through resonance. The black solid line shows the Monte Carlo prediction for thenumber of atoms remaining after adiabatic molecule conversion, while the dashedlines represent the uncertainty in the molecule number due to the uncertainty inthe initial atom numbers. The observed loss corresponds to (66 ± 6)% of the Rb
atoms.

91

Figure 4.8: Eect of gravitational sag on molecule conversion. The upper plotshows the simulated molecule fraction as a function of the cloud separation for
NK /NRb = 1 (red circles) and NK /NRb = 3 (black circles) at T = TC for atoms
conned in an isotropic trap. The lower plots show the simulated vertical prolesof the integrated number density for Rb (red circles) and K (blue triangles) withcloud separations of 3 and 7 µm. The rms sizes of the clouds are 3 and 4 µm for
the Rb and K clouds, respectively.

92

4.3.3

Molecule conversion in expansion

Some of the experiments that will be discussed in Chapter 5 required atomic
gases with low densities. While there are many ways to reduce the density of asample, one method we employed was to release the atoms from the trap, allowing
the clouds to ballistically expand before creating molecules. When a cloud isreleased from the trap, the atoms y apart with the momentum they had in thetrap, rapidly increasing the spacing between particles (and, thus, reducing thedensity). This also has the eect of locally cooling the cloud. That is, as thecloud expands, it separates itself into velocity classesthe atoms that remain
near the center of the cloud have small velocities, while the atoms that are on theoutskirts of the cloud have the largest velocities. If we were to divide the cloud
into shells, each shell would contain a smaller distribution of velocities than theinitial cloud, eectively cooling the sample.
What impact does expansion have on the molecule fraction? The criterion
for molecule creation requires ∆rrel ∆vrel be small. The distribution of ∆rrel ofall possible pairs increases as the cloud expands, while the distribution of ∆vrel
remains the same. We might naïvely expect that the molecule fraction should
simply be inversely proportional to the mean interparticle spacing. However, itis the product ∆rrel ∆vrel that determines if a molecule is made. In expansion, an
atom's position and momentum are correlated, so it becomes increasingly likely for
two neighboring atoms (atoms with small ∆rrel ) to have similar velocities (small∆vrel ). For the cases we have investigated with the Monte Carlo simulation, themolecule fraction decreases as the cloud expands, but the local cooling in the gas
causes it to decrease more slowly when compared with a simple dilation of thecloud. Figure 4.9 shows the expected molecule fraction as a function of expansion
time for two Bose gases with equal masses and identical position and momentum

93
distributions. The solid circles show the molecule fraction for a cloud of atoms
whose positions follow the formxi (t) = xi0 +

pi0
· t,
m

(4.2)

where xi0 and pi0 are the atom's position and momentum in the trap, with thesubscript i = x, y, z representing the cardinal directions. m is the mass, and t is
the expansion time. To identify the eect of local cooling in expansion, the Monte
Carlo simulation was also performed for a simple dilation of the cloud (open circles
in Fig. 4.9)

xi (t) = xi0q
1 + ωi2 t2 ,(4.3)

where ωi are the angular trapping frequencies. The density of the atom cloud
retains the proper dependence on expansion, but the momentum remains uncorrelated with position. Clearly, local cooling in expansion maintains a larger fraction of pairs with small relative velocity despite the increase in the interparticle
spacing.
If, instead of identical atoms, we have two components with dierent masses
and magnetic moments, we must additionally take into account the reduction in
spatial overlap from the gases expanding at dierent rates. This is the case in
our mixture of K and Rb. For expansion times long compared to the trappingK
period, the ratio of the sizes will become σσK ≈ σσ0K ω, where σ0K,Rb represents
ωRb

0Rb Rbthe in-trap rms size of the atom cloud. The K trapping frequencies in the opticaltrap are typically 40 − 50% greater than the Rb trapping frequencies due mainly
to the smaller mass but also due to additional trapping forces provided by theinhomogeneous magnetic eld. For a mixture with similar sizes in the trap, theK cloud will rapidly become larger than the Rb cloud as they expand. Figure

94

sr =10 s0r

Figure 4.9: Molecule conversion in expansion for two identical Bose gases at T =TC . The solid circles (•) show the fraction of molecules converted in the Monte
Carlo simulation as a function of expansion time from a trap with a radial trapping
frequency of 690 Hz and an axial trapping frequency of 9.0 Hz. The atom clouds
expand according to the atoms' momenta in the trap, causing the position andmomentum of each atom to be correlated. For comparison, the open circles (◦)
show the molecule fraction for a dilation of the atom cloud, where the positionof each atom is scaled by a constant factor, irrespective of its momentum. The
density for a given expansion time is the same as that of the solid circles, but the
momentum and position of a given particle are uncorrelated. At the dashed line,
the radial size has increased tenfold.

srK =10 s0rK
srRb =10 s0rRb

Figure 4.10: Simulation of molecule conversion in expansion for equal numbers of
Rb and K at T = TC . The K atoms experience trapping frequencies that are 1.4
times greater than the Rb trapping frequencies. The two atom clouds, whose sizes
are initially similar, expand at dierent rates, causing a reduction in the spatial
overlap at longer expansion times.

95
4.10 shows the expected molecule fraction for a gas with equal numbers of 40 K
and 87 Rb at T = TC . In comparison with Fig. 4.9, the expected molecule fraction
decreases more rapidly with expansion time due to the mismatch of the spatial
extent of the two species. After 2 ms of expansion from the trap, the density of
the clouds has decreased by two orders of magnitude, while the molecule fraction
has only decreased by a factor of 2.3. Therefore, we can access a wide range ofdensities with comparatively small eects on the molecule conversion by releasing
the atoms from the trap before making molecules.
From the Monte Carlo simulations, we now have guidelines for maximizing
the molecule fraction: (a) the temperature of the mixture must be near to TC ,(b) there should be more fermions than bosons, (c) the trap should be suciently
tight in the vertical direction such that the dierential gravitation sag is smaller
than the rms size of the cloud, and (d) if we are willing to accept a hit in molecule
conversion, we can allow the mixture to expand from the trap to access lower
densities.

Chapter 5
Coherent atom-molecule oscillations

Armed with the knowledge of how to make large samples of molecules, wewere then ready to look for coherent atom-molecule oscillations. After all, more
molecules should translate to larger amplitude oscillations. In this chapter, I
present our studies of atom-molecule coherence. Starting with an ultracold mixture of 87 Rb and 40 K atoms, we create superpositions of atom pairs and molecules.
The superpositions are generated by pulsing an applied magnetic eld near aninterspecies Fano-Feshbach resonance to coherently couple atom and moleculestates. Rabi- and Ramsey-type experiments show oscillations in the molecule
population that persist as long as 150 µs and have up to 50% contrast. The frequencies of these oscillations are magnetic-eld dependent and are consistent withthe predicted molecule binding energy. This new type of quantum superposition
involves particles of dierent statistics (i.e. bosons and fermions). In the same
spirit of experiments that showed that the coherence properties of a BEC are notnecessary to observe spatial matter-wave interference [114], we show that coherent
atom-molecule superpositions can be created without a BEC.

5.1

Rabi and Ramsey oscillations in the atom + photon system
The pioneering experiments of Rabi and Ramsey demonstrated some of theearliest examples of quantum control [47, 48, 49]. These experiments showed that97
by resonantly coupling two quantum states with an external eld, a superposition
can be created and coherently manipulated. Probably the clearest example thistype of quantum coherence is the system of a two-level atom in a near-resonantradiation eld. This system is treated rigorously in many textbooks (see, forexample, Ref. [89] and [115]), so I will only highlight some of the key points here.
One particularly intuitive description of the atom + photon system is the socalled dressed atom picture in which the eigenstates of the atom are modied,
or dressed, by a quantized radiation eld. We consider the Hamiltonian

H = Ha + Hrad + Hint ,(5.1)

where Ha is the bare atom Hamiltonian, Hrad is the radiation energy, and Hint
is the atom-eld interaction that couples the atomic levels. A schematic of theenergy level diagram of the the rst two terms is shown in Fig. 5.1. The energy
levels of the bare atom are shifted by n × ~ω , where n is the number of photons
in the radiation eld. If the photon energy ~ω is close to the energy dierence
between the ground and rst excited state ~ω0 , the interaction matrix Hint mixes
the two states, creating the eigenstates |1i and |2i, which are hybrids of the barestates |e, ni and |g, n + 1i as shown in Fig. 5.2.
The interaction matrix can be written in the {|g, n + 1i, |e, ni} basis as




 δ
Hint = ~ 

Ω
2

Ω
2

0


,

(5.2)

where the on-resonant Rabi frequency Ω is a measure of the coupling between
the two states, which depends on the intensity of the eld, and δ = ω − ω0 is
the detuning of the radiation frequency from the transition frequency.1

By

diagonalizing Hint , we nd the dressed states
1 Here, time dependence has been eliminated by invoking the rotating wave approximation.

98

E
4
3
2

e

1

g

4
3

e

2

Ñd

Ñw0
g

Ñw

1

n+1

n

Radiation Field Quantum Number

Figure 5.1: Energy level diagram for an atom in a radiation eld. The energy
levels of the bare atom are shifted by the energy in the radiation eld, n × ~ω .
When the photon energy approaches the level splitting ~ω0 , the states |e, ni and|g, n + 1i become nearly degenerate. (Figure adapted from Ref. [116].)

E|g, 1 ñ|1ñ|e, 0 ñ

ÑW
|2ñ

0

d

Figure 5.2: Eigenstates of the dressed atom. Plotted are the energies of the
dressed states |1i and |2i as a function of the detuning of the radiation eld fromthe atomic resonance (solid lines). The dashed lines represent the energies ofthe bare states |e, 0i and |g, 1i. On resonance, the energy dierence between the
coupled states is ~Ω.

99|1i = sin θ|g, n + 1i + cos θ|e, ni(5.3)|2i = cos θ|g, n + 1i − sin θ|e, ni,(5.4)

where tan 2θ = − Ωδ , and the eigenenergies are E = ~2 δ ±√Ω2 + δ 2 . As shownin Fig. 5.2, the dressed states |1i and |2i form the branches of a hyperbola whose
asymptotes are the bare states |e, ni and |g, n + 1i, and their energy dierenceis ∆E = ~Ωeff , where Ωeff =√Ω2 + δ 2 . The dressed atom picture results in an

avoided crossing. If an atom begins in the ground state in a radiation eld thatis suciently detuned from resonance, it can be transferred to the excited state
by adiabatically sweeping the radiation frequency through resonance. Precisely
on resonance, the atom is in an equal superposition of ground and excited states.
The probability of a transition from the ground state after a time t with the eldat a xed detuning isΩsin2
P (t) =Ωeff

Ωeff t
2


.

(5.5)

The probability of nding the atom in the excited state oscillates in time, and thefrequency corresponds to the energy dierence between the dressed states. Withan ensemble of atoms, the population in the excited state is N × P (t), where N is
the number of atoms. On resonance, the population oscillates with 100% contrast.
Away from resonance, the amplitude of these Rabi oscillations decreases, and thefrequency increases.
The observation of Rabi oscillations can be a very precise method for measuring the energy dierence between coupled energy states.

However, precise

frequency measurements require long interrogation times. For experiments, thismeans that the intensity and frequency of the radiation as well as any externalelds must be constant in time. Ramsey's method of separated oscillatory elds

100
reduces these stability requirements by reducing the time the driving eld interacts
with the atoms.
Ramsey's method uses two short interaction pulses separated in time (orspace) by a time of free evolution that can be long compared to the interaction
time as shown schematically in Fig. 5.3a.

If the time t1 is chosen such thatthe atoms experience a π/2 pulse (that is, they undergo one quarter of a Rabi
oscillation), the atoms will be in an equal superposition of ground and excitedstates after the pulse. During the time t2 , the states evolve freely, acquiring a
relative phase due to the energy dierence between the two states. The second

π/2 pulse interferometrically reads out the phase, with the nal population ineach state depending sinusoidally on the phase dierence acquired during t2 . Thissequence is analogous to a Mach-Zehnder interferometer (see Fig. 5.3c). The rstpulse acts as a beamsplitter, and the relative phase acquired during t2 is analogous
to the path length dierence of the two arms of the interferometer. The second
pulse can be thought of as the second beamsplitter, which recombines the twopaths, with the intensity at the detector depending on the interference of the two
beams. If the pulses in a Ramsey experiment are not π/2 pulses, the resulting
oscillations will have less than 100% contrast, as in the case of an interferometerwith beamsplitters that are not 50/50.
As we have seen above and in Chapter 3, the systems of an atom in aradiation eld and two atoms colliding near a Fano-Feshbach resonance can bothbe described by an avoided crossing between two coupled energy states. In theformer system, two internal states of an atom are coupled by a near-resonant
oscillating eld.

In the latter system, the state of two free atoms is coupled

to that of a bound molecule by a magnetic eld.

As in the atom + photon

system, atoms near a Fano-Feshbach resonance can undergo Rabi and Ramsey
oscillations. The population can oscillate between unbound atom pairs and bound(b)Magnetic Field(a)Magnetic Field

101

t1

tevolve

ttop

t1
ttop

B

0

Btop

Bevolvetevolve(c)

Figure 5.3: Ramsey's method of separated oscillatory elds and analogous systems. (a) Ramsey's method. Two oscillatory elds are applied for a short time
t1 , separated by a free evolution time tevolve . (b) Schematic of the magnetic-eldpulse sequence to observe Ramsey-like oscillations between atoms and molecules
near a Fano-Feshbach resonance (B0 ). (c) A Mach-Zehnder optical interferometer.

102
molecules. Figure 5.3b shows a schematic of a magnetic-eld sequence used in our
experiment to induce Ramsey oscillations between atoms and molecules. Here, thetwo oscillating elds are replaced by two pulses toward resonance. As discussed
in Refs. [117] and [8], the rst pulse creates a coherent superposition of atoms
and molecules by providing overlap between the interparticle spacing (which istypically 10 000 a0 in experiments) and the mean dimer sizeahri = ,
2

(5.6)

where a is the s-wave scattering length, which diverges at the resonance. The
superposition evolves during the time tevolve according to the energy dierence
between the atom and molecule states at Bevolve . The second pulse overlaps thestates again, such that the nal population in each state depends on the relative
phase between the atom and molecule states at the end of tevolve .

5.2

Criteria for observing oscillations
I have argued above that the system of two atoms near a Fano-Feshbach res-onance is analogous to that of a two-level atom in a radiation eld. Theoretically,
Rabi- and Ramsey-type atom-molecule oscillations can be observed, but what are
the criteria for experimental observation? Previous experiments that observed
atom-molecule oscillations began with identical bosons prepared in a single energy state in either a BEC or a tightly conning optical lattice [44, 45, 54]. We
pose the questions: Can an atom-molecule superposition be demonstrated withan incoherent ensemble of atoms, such as non-condensed bosonic atoms, fermionic
atoms [55, 56, 57], or a mixture of dierent atomic species [58, 59]? As demonstrated below, the answer to this question is yes, but there are several criteriathat must be met to observe atom-molecule oscillations.
For any Rabi or Ramsey experiment, we require a detection technique that

103
distinguishes between the two coupled states. As discussed in Chapter 3, we canselectively image either unbound atoms or molecules. We also need the coherencetime of the oscillation to be long compared with its period. If the coherence timeis too short (or the period too long), the oscillations will be overdamped, and no
oscillations will be observed. It is this criterion that can make a BEC or opticallattice advantageous, though not necessary. A pair of atoms will oscillate with afrequency dened by the energy dierence between the atom pair and moleculestates. If every atom pair has the same energy, and all the molecules also sharea single energy state, then all of the pairs will oscillate with the same frequency.
Then the coherence time is likely to be limited by collisions, molecule loss, or
spatial inhomogeneities in the magnetic eld. In our Bose-Fermi mixture, theatom pairs are described by a distribution of energies. This leads to a distributionof oscillating frequencies, which will cause the macroscopic population oscillation
to dephase in time. Narrower energy distributions should lead to longer coherence
times, implying that mixtures with low absolute temperatures will have the longest
coherence times.
Additionally, we require magnetic-eld ramps that are suciently fast to
induce oscillations. If magnetic-eld ramps such as those drawn in Fig. 5.3b
are too slow, an atom pair state will adiabatically follow an eigenstate toward
and away from resonance, and no oscillations will be observed. Diabatic ramps
are necessary to project the atom pair state into multiple states for the nalpopulation to oscillate. The magnetic-eld ramp rate required for diabaticity
depends on the density of the sample [86, 118, 61]. As discussed in 2.6, ourmagnetic-eld ramp durations cannot be shorter than 5 µs, and a typical ramp
to induce atom-molecule oscillations spans 0.5 G, resulting in a typical maximum
ramp speed of 100 G/ms. With these ramp speeds, we have found that we only
observe oscillations in mixtures with peak densities . 3 × 1013 cm−3 (see 5.3.3

104
and 5.4.2).

5.3

Ramsey oscillations
In this section, I discuss the experimental observation of Ramsey-type os-

cillations in our Bose-Fermi mixture. We began with a Ramsey experiment (asopposed to the conceptually simpler Rabi experiment) because in the 85 Rb experiment that rst demonstrated atom-molecule oscillations, Rabi-like oscillations
were only observed over a very limited range of magnetic elds [44].
The details of the experimental apparatus are described in Chapter 2. Here,I describe the experimental conditions for our observations of Ramsey oscillationsin the trapped mixture, which henceforth will be referred to as the typical intrap conditions. We prepare an ultracold Bose-Fermi mixture of 87 Rb atoms in
the |1, 1i state and 40 K atoms in the |9/2, −9/2i state. The atoms are conned
in a far-o-resonance optical dipole trap formed by a single laser beam with awavelength of 1090 nm focused to a 1/e2 radius of 18 µm. The gas mixture isevaporatively cooled by decreasing the power of the optical-trap beam. Following
the evaporation, the optical trap power is adiabatically increased so that the nalmeasured trap frequencies are 350 Hz for Rb and 490 Hz for K in the radial
direction and 5.2 Hz for Rb and 8.1 Hz for K in the axial direction. In this trap,
we have 8 × 104 Rb atoms and 2 × 105 K atoms at a temperature of 200 nK,
which corresponds to 1.2 TC and 0.3 TF , where TC is the critical temperature for
Bose-Einstein condensation of the Rb gas and TF is the Fermi temperature of
the K gas. These conditions correspond to the criteria for large molecule fraction
discussed in the previous chapter. The peak densities of the Rb and K clouds
are n0Rb = 1.1 × 1013 cm−3 and n0K = 1.3 × 1013 cm−3 , respectively. During theevaporation, we turn on an applied magnetic eld of 542 G, preparing the mixture
near to the interspecies Fano-Feshbach resonance at B0 = 546.76G.

105

5.3.1

Empirical optimization

We began looking for oscillations with a magnetic-eld sequence similar tothat used in the 85 Rb double-pulse experiment [44] with the parameters scaled
to account for the dierent density, background scattering length, and resonancewidth in our mixture. The sequence is shown schematically in Fig. 5.3b. At theend of the sequence, we image the molecule cloud with the technique discussed in3.4. After signicantly reducing the density of atoms in the trap (for a discussionsee 5.3.3 below), we were able to observe oscillations in the molecule number
with amplitudes of nearly 5 000 molecules. To maximize the molecule oscillation
amplitude, we then set about empirically optimizing each of the magnetic-eldramp parameters.
Figure 5.4 shows some of the optimizations of the magnetic-eld sequence.
In these experiments, the amplitude of the oscillation in the molecule population
was measured with Bevolve = 546.1 G, and each parameter (Bstart , Btop , ttop , andall the ramp speeds) was varied in turn. Note that a multidimensional optimization in parameter space was not performed.2

Although, in retrospect, such asearch, particularly between ttop and Btop , would have ensured a global maximum
in the oscillation amplitude. From the optimizations, we can make several observations. The dependence on Btop (Fig. 5.4b) indicates that for a hold time of

15 µs, the largest amplitude oscillation is not at the Feshbach resonance, but at
546.6 G. If we assume that the maximum occurs when the pulse corresponds to aπ/2 pulse, this would imply a Rabi frequency of 2π×17 kHz at 546.6 G, but, as weshall see below, we were unable to observe Rabi opping under these conditions.
The decay of the amplitude as the pulse approaches the resonance may, instead,
2 By this, I mean xing the value of one parameter, say t

top , and varying the value of another,say Btop , then choosing another ttop and varying Btop , and so on, until an optimum is found in
the 2D space.

(a)Magnetic Field

106

ttop

ttop
B0

Btop

Bevolve= 546.1 G

Btop

Bstart(c)(b)

ttop = 15

ms

Btop = 546.58 G

(d)

(e)

Figure 5.4: Empirical optimization of Ramsey oscillations. A schematic of themagnetic-eld pulse sequence is shown in (a) with the optimization parameterslabeled. In (b)-(d), the oscillation amplitude was measured over two to four
periods of the oscillation at Bevolve . Plotted in (b) is the amplitude as a functionof the pulse height with ttop = 15 µs. (c) shows the dependence on ttop , thehold time at Btop = 546.58 G. The amplitude as a function of the starting eldfor the rst pulse is shown in (d). As a representative plot for the ramp speed
optimization, the amplitude as a function of the inverse ramp speed for the third
ramp is shown in (e). The vertical lines in (b) and (c) indicate the position of the
Fano-Feshbach resonance and its uncertainty.

107
be due to increased collisions causing decoherence on the timescale of the pulse.
The interspecies collision rate is given byΓKRb = hnKRb iσhvi.
Here, hnKRb i= N1<

R

(5.7)

nK (r)nRb (r)d3 r, where nK and nRb are the number densityof K and Rb, respectively, and N< is the number of atoms in the species withfewer atoms. hvi =p
8kb T /(πµ) is the mean relative velocity between K and

Rb atoms at a temperature T , where µ is the K and Rb reduced mass and kb is
the Boltzmann constant, and σ = 4πa2 is the interspecies collision cross section
with the scattering length a given by Eq. 3.1. Above 546.6 G, ΓKRb exceeds 30
kHz, and we expect that any particle colliding with an atom pair would destroy
the atom-molecule superposition. Similarly, the time dependence in Fig. 5.4c may
indicate a Rabi frequency or increased decoherence due to spending too much timenear the resonance.
In Fig. 5.4d, the amplitude of the oscillation is shown as a function of themagnetic eld at which the rst pulse starts. We nd that the pulse must start
below 546 G for maximum amplitude. This corresponds to a detuning of 0.8 G,
which is smaller than the peak-to-zero width of the Fano-Feshbach resonance,∆ = 3.6 G [64]. Though, the decrease in amplitude near the resonance is most
likely due to slower ramp speeds. Recall from 2.6 that we are limited to ramp
speeds of Ḃmax =Btop −Bstart. Below 546 G, the ramp speeds are 140 G/ms (7µs/G),
5 µs

but at 546.6 G, the maximum ramp speed is 30 G/ms (30 µs/G). In Fig. 5.4e,
we see that slower ramps (larger inverse ramp speeds) induce smaller oscillations.
For the 0.7 G ramp in Fig. 5.4e, the maximum ramp speed is 140 G/ms, which is

5× faster than the 1/e ramp speed required for adiabatic molecule conversion atthese densities (see Fig. 4.7). When these ramp speeds become comparable, theoscillation amplitude drops to about half the maximum value.

108
The optimized pulse sequence begins after a slow (3 G/ms) ramp to 545.80
G and consists of two trapezoidal pulses with 15 µs holds at Btop = 546.58 G,
separated by a variable hold time tevolve at Bevolve . The outer ramps have speeds of
140 G/ms, while the inner ramps have speeds of Ḃmax . This optimized sequence
results in oscillations with peak-to-peak amplitudes of 104 molecules, which is
10% of NRb . At Btop , the calculated molecule size is 1 800 a0 ; this is comparable to the typical distance between nearest-neighbor Rb and K atoms, which isapproximately hnKRb i−1/3 = 10 500 a0 .
Figure 5.5 shows the measured molecule population Nmol after the optimized
double pulse for two dierent values of Bevolve . The right-hand axes show the number of molecules normalized by the number of molecules created by an adiabatic
sweep through resonance Nmax . The number of molecules oscillates as a functionof tevolve , as expected for an atom-molecule superposition. Note, however, thatthe peak-to-peak amplitudes of the oscillations are only 30% of Nmax .

5.3.2

Frequency dependence on B

The data in Fig. 5.5a and 5.5b show markedly dierent oscillation frequencies. We have measured the oscillation frequency for various values of Bevolveand nd that the frequency corresponds to the predicted binding energy of themolecules. Figure 5.6 shows the measured oscillation frequency νosc as a function of magnetic eld. The solid curve is a t to the universal prediction for themolecule binding energy near the resonance [119],

Eb =

~2
.
2µKRb (a − ā)2

(5.8)

Here, µKRb is the 40 K and 87 Rb reduced mass, ā = 68.8 a0 is the van der Waals
length of the long range potential, and a = abg [1 − ∆/(B − B0 )] is the s-wave
scattering length with abg = −185 a0 [92]. From the t, we extract the resonance

109

(a)

Bevolve = 546.08 G(b)Bevolve = 546.33 G

Figure 5.5: Ramsey-type atom-molecule oscillations. The molecule number Nmolis shown as a function of the hold time at Bevolve = 546.08 G in (a) and 546.33
G in (b), which correspond to detunings from the Fano-Feshbach resonance of∆B = 0.68 G and 0.43 G, respectively. The right-hand axes show the measured
molecule number normalized by the maximum number of molecules associatedby an adiabatic sweep through resonance, Nmax . The lines are ts to Nmol =22
Ae−t /(2σ ) sin(2πνosc t−φ)+y0 , yielding an oscillation frequency νosc = (136.5±1.3)
kHz and rms damping time σ = (32 ± 5) µs for the data in (a) and νosc = (36 ± 2)
kHz and σ = (22 ± 4) µs for the data in (b).

110
position B0 = (546.76 ± 0.06) G and width ∆ = (−3.6 ± 0.1) G, in agreementwith Ref. [64]. The width is larger than the theoretical width quoted in Ref. [92].

5.3.3

Amplitude dependence on density

Prior to the observations of the atom-molecule oscillations discussed above,
we attempted to induce Ramsey oscillations in mixtures with peak densities of

5 × 1013 cm−3 , resulting in no observable oscillations. In the 85 Rb experiment, it
was observed that when the density of the BEC was increased by an order ofmagnitude, the fractional amplitude of the oscillations signicantly decreased [3].
We reasoned that we might be operating with prohibitively large densities, so
we reduced the density of our trapped mixture to the conditions described above
and were immediately rewarded with oscillations. Figure 5.7 shows the observed
oscillation amplitude as a function of the peak K density. For these data, themixture is initially prepared in the typical in-trap conditions. To vary the density,
the power in the optical trap is then increased, resulting in the larger trapping
frequencies. The number in each species and ratio of Rb and K densities remain
relatively constant while the density varies. Note that the temperature of the gases
increases with increasing density, but the measured T /TC and T /TF of the sample
remain constant. Since molecule conversion via magnetic-eld sweeps depends
on the phase space density of the sample and not on the absolute temperature[61], we might expect that the amplitude of oscillations induced by magneticeld pulses should also be independent of temperature. Therefore, we ascribe thedecrease in oscillation amplitude to the increase in density.3

There are a numberof possible explanations for the amplitude of the oscillations to decrease as thedensity increases. The criterion for adiabaticity depends on density [61, 86], so
3 We did not measure N

max as a function of density, but as in the single-species case [61], we

expect Nmax to be density-independent.

111

osc

(kHz)

300

200

100

0
545.8

546.0

546.2

546.4

546.6

B

evolveFigure 5.6: Magnetic-eld dependence of the measured oscillation frequency forthe molecule number in Ramsey-type double-pulse experiments. The circles (•)
correspond to double-pulse experiments performed with the atoms conned in the
optical trap, while the triangles (N) represent experiments performed after 1 msof expansion from the trap. The solid line is a t to the universal prediction
for the molecule binding energy given by Eq. 5.8 yielding the resonance position
B0 = (546.76 ± 0.06) G and width ∆ = (−3.6 ± 0.1) G.

112
as the density increases, the fast ramp speeds become less and less diabaticand are less able to create the atom-molecule superposition necessary to observe
oscillations. For these data, the magnetic-eld ramp speeds are 100 G/ms, whichis 3.8 times faster than the 1/e ramp speed (β = (26 ± 7) G/ms) for adiabatic
molecule creation at a K density of 1 × 1013 cm−3 (see Fig. 3.5c). At a density of

3 × 1013 cm−3 , we expect the adiabatic ramp speed to be 3 × 26 G/ms = 78 G/ms,
which is comparable to the ramp speed of the pulses. As we have seen in Fig.
5.4e, when these speeds become comparable the oscillation amplitude drops by afactor of two. Also, increased density causes an increase in the collision rate, which
could increase the rate of decoherence of the superposition. Another possible cause
for the reduced amplitude with increased density is an increase in the spread ofrelative velocities of the atom pairs due to the increased temperature (see 5.5.1).
If this were a signicant eect, we would expect the decreased amplitude to beaccompanied by shorter coherence times, but we observe similar coherence times.
Whatever the cause, it seemed clear that further lowering the atom densities was
likely to increase the oscillation amplitude.

5.3.4

Ramsey oscillations in expansion

In general, it can be dicult to signicantly alter the density of atoms in the
trap. The factor of 5 reduction in density we had already achieved was introducing
instabilities in the daily running conditions. To further reduce the density in thetrap seemed a tall order. One simple method of reducing the density of the atomic
gas is to allow the cloud to ballistically expand from the trap. With small changes
to the expansion time, the density can change by orders of magnitude.
For experiments performed in expansion, we were no longer concerned with
reducing the density in the trap and were able to load all of our atoms into theoptical trap. At the end of evaporation, we have 1 × 106 Rb atoms and 6 × 105 K

113

K peak density (10

13

-3

cm

)

Figure 5.7: Density dependence of the molecule oscillation amplitude in Ramseytype experiments. Plotted are the amplitudes of the molecule population oscillations for various densities achieved by varying the trapping frequency of the
optical trap.

114
atoms at 1200 nK in an optical trap with radial trapping frequencies of 690 and
970 Hz for Rb and K, respectively. The temperature corresponds to T /TC = 1.6
for Rb and T /TF = 0.7 for K. After 1 ms of expansion, the peak densities are

n0Rb = 3 × 1012 cm−3 and n0K = 7 × 1011 cm−3 . For these conditions, which will bereferred to as the typical expansion conditions, we nd Nmax is (9.2 ± 0.6) × 104 ,which is 15% of NK , and the 1/e ramp speed for adiabaticity is (9.1 ± 1.0) G/ms.
The double pulse sequence was again empirically optimized for these newconditions. Again we found that the optimum amplitude was achieved with ourfastest ramp speeds, but the optimum time spent near the resonance increased.
Figure 5.8 shows the oscillation amplitude as a function of ttop . At Btop = 546.58
G, we observe an optimum in the hold time at 50 µs. This implies a Rabi frequencyof 2π×5 kHz, in agreement with the expected frequency from the binding energy
curve in Fig. 5.6. The triangles in Fig. 5.6 correspond to experiments performed
after 1 ms of expansion using a hold time of 50 µs at the top of the optimized
pulses. The goal of performing experiments after allowing the gas to expand was
to increase the oscillation amplitude, but the observed amplitude did not increase.
This is likely due to a competition between larger oscillations due to lower densities
and smaller molecule fractions due to decreasing phase space density as the cloud
expands (see 4.3.3). As discussed below, the low densities achieved by releasing
the atoms from the trap did, however, enable the observation of Rabi oscillations.

5.4

Rabi oscillations

5.4.1

Ramsey oscillations without Rabi oscillations?

Since we can observe clear Ramsey oscillations, and Ramsey oscillations are
induced by two pulses that are ideally each one quarter of a Rabi oscillation,
shouldn't we be able to observe Rabi oscillations? Under this supposition, we115

10x103

amplitude

8x103
6x103
4x103
2x103
0

0

20

40
t

top

60

80

100

( s)

Figure 5.8: Empirical optimization of pulse time for a cloud after 1 ms of expansion
from the trap. The line is a parabolic t to identify the optimum ttop .

116began to look for Rabi oscillations with single pulses toward the Fano-Feshbachresonance. However, as shown in Fig. 5.9, single-pulse experiments under the same
conditions (typical in-trap conditions) as successful double-pulse experiments do
not yield oscillations in the molecule population. Instead, we observe a monotonic
increase in the molecule number that exponentially approaches a value less thanNmax . If we take this to be a critically damped oscillation, we can take thecalculated frequency as an upper limit on the oscillation frequency at Btop . For
the green data, whose Btop is similar to that used in the double pulse experiments,
we estimate a Rabi frequency of ≤ 2π×(5.1±0.7) kHz. This is in better agreementwith the 2π×3.5 kHz we would expect from the binding energy curve in Fig. 5.6
than with the 2π×17 kHz implied by the empirical pulse optimization (see Fig.
5.4). These data indicate that decoherence mechanisms on the timescale of 10'sof microseconds prevent the observation of Rabi oscillations. How, then are weable to observe Ramsey oscillations?
We can think of the 15-µs pulses in the Ramsey-type experiments as being
sub-π/2 pulses. Just as in the atom-photon system, if the pulse duration is shorter
than a π/2 pulse, the resulting superposition will not be a 50/50 mixture of two
states. Instead, the superposition will favor the initial state, which in our case is
the unbound atom pair state. When two of these sub-π/2 pulses are applied ina Ramsey-type experiment, the population in each state will still oscillate. The
two pulses will not be able to transfer the entire population out of the initialstate, though, resulting in oscillations with smaller amplitudes than in the ideal
case. This may explain why our Ramsey oscillations do not span the total possible
molecule number.

117

magnetic field(a)
B0 = 546.76 G
Btop
ttop

Bfast

Bend = 546.04 Gt(b)

Figure 5.9: Time dependence of the molecule number in a single-pulse experiment.
(a) A schematic of the magnetic pulse sequence. The ramp speed, Ḃfast is 140
G/ms. (b) The measured molecule number is shown as a function of the hold time
ttop for several dierent values of Btop . No oscillations in the molecule number are
visible. The lines are ts to Nmol = Ae−t/τ + y0 . The green data correspond to theheight of the two pulses in a typical double-pulse sequence in which Ramsey-type
oscillations are observed.

118

5.4.2

Rabi oscillations in expansion

While we are unable to observe Rabi oscillations in the trapped mixture, we
are able to observe oscillations in single-pulse experiments at the lower densities
enabled by expansion of the gas. A schematic of a single pulse is shown Fig. 5.9a.
After 1 ms of expansion from the trap, the magnetic eld is ramped from 545.80
G to Btop , held for a time ttop , and ramped back down, with both ramps having
speeds of 140 G/ms. The molecule population after a single pulse to Btop = 546.51
G as a function of the hold time ttop is shown in Fig. 5.10. Dening the contrast
as the ttop = 0 amplitude |A| divided by the nal level of the damped oscillation

y0 , we measure a contrast of 0.5 ± 0.2.
Since we can observe Rabi oscillations at some densities but not others, thevisibility of Rabi oscillations must depend sensitively on the density of the gas.
Figure 5.11 shows the molecule population as a function of hold time for gaseswith several dierent densities. The gases are prepared identically in the trap,
and the dierent densities are achieved by varying the expansion time from thetrap. If the density is too high, we recover the behavior of Fig. 5.9, but for the
lower densities, we observe oscillation contrasts comparable to that in Fig. 5.10.
Once the peak densities drop to ≤ 1012 cm−3 , we observe little change in theoscillation visibility when the density is further reduced. Presumably, at these
densities, the magnetic-eld ramps are suciently diabatic to have little impact
on the oscillation amplitude. We estimate that the magnetic-eld ramp speeds
are >100 times faster than the 1/e ramp speed for adiabaticity. Also, the collision
rates are becoming negligible (ΓKRb < 3kHz).

119

Figure 5.10: Rabi-type atom-molecule oscillations. The measured molecule number is shown as a function of the hold time ttop for a single pulse where Btop22
was 546.51 G. The line is a t to Nmol = Ae−t /(2σ ) cos(2πνosc t) + y0 , yielding afrequency νosc = (9.3 ± 0.5) kHz and rms damping time σ = (80 ± 20) µs.

120

Figure 5.11: Rabi-type atom-molecule oscillations with varying density clouds.
The measured molecule number is plotted as a function of the hold time at
Btop = 546.56G for single magnetic-eld pulses applied after varying expansion times from the trap, or times-of-ight (tof). The lines, which are ts to
Nmol = Ae−t/τ1 sin(2πνosc t − φ) + B · (1 − e−t/τ2 ) + y0 , are included only as a guideto the eye.

121

5.4.3

Amplitude dependence on B

The peak-to-peak amplitude of the oscillations in the molecule population
depends on the value of Btop as shown in Fig. 5.12a. As Btop is increased, we
observe fewer oscillations before they damp out, and for the data above Btop =546.6 G, only one period of the oscillation is observed. Therefore, we dene
the peak-to-peak amplitude as the dierence in molecule number between therst maximum and the subsequent minimum. We nd that the amplitude of theoscillations is peaked near the Fano-Feshbach resonance where we observe a peakto-peak amplitude that is 23% of Nmax . We observe Rabi-type oscillations forvalues of Btop as far as 290 mG from the resonance. At Btop = 546.47 G, theoscillation amplitude drops to 6% of Nmax . Here, the calculated molecule size is

1 100 a0 , which is 5% of the typical distance between nearest-neighbor K and Rb
atoms of 23 700 a0 .

5.4.4

Frequency dependence on B

The frequency of the Rabi-type oscillations also depends on Btop as shownin Fig. 5.12b. Below the resonance, the measured frequency agrees with theprediction for the molecule binding energy, while above the resonance, where we
only observe one period of the oscillation, the frequency saturates. This saturation
may be expected in a many-level system [120, 118]. Figure 5.13 shows a schematicof the energy levels of two atoms conned in a harmonic trap. Above resonance,
the molecule state adiabatically evolves into the ground state of the trap, andthe energy dierence between the trap states approaches a constant value. If, forexample, a diabatic magnetic-eld ramp to a value of Btop above the resonancecreates a superposition of pairs in the ground state and the rst excited state,
we would expect the observed oscillation frequency to approach the trap level

122

Figure 5.12: Magnetic-eld dependence of the (a) peak-to-peak amplitude and (b)frequency of the single-pulse oscillations. The solid line in (a) is a Gaussian tto the data, while the solid line in (b) is the molecule binding energy curve from
Fig. 5.6. The vertical lines represent the tted Fano-Feshbach resonance position
and uncertainty from Fig. 5.6.

123
splitting divided by h. This two-atom picture can be applied to a many-bodysystem through a rescaling of the two-atom oscillator frequency to match thedensity of the ensemble [120, 118]. The eective oscillator frequency is given by
[120]

ω0 =
where hni i = N1iR~
π (hnK ihnRb i)1/3 ,
2µKRb

(5.9)

n2i (r)d3 r is the average number density of species i.4 We notethat the observed saturation frequency of (1.7 ± 0.8) kHz agrees with the expected
many-body level spacing of 2ω 0 /~ = 1.2 kHz.
The results of the single-pulse experiments can be used to infer the best
pulses for double-pulse experiments. Maximum contrast should be achieved byusing pulses where the hold time at Btop gives one quarter cycle of the singlepulse oscillation, which is analogous to a π/2 pulse in Ramsey's experiments [49].
In fact, we nd that the empirically optimized pulse sequence for the expanded
clouds corresponds to this condition.

5.5

Oscillation decoherence/dephasing
For both the single-pulse and double-pulse experiments, we observe dampingof the atom-molecule oscillations within a few hundred microseconds (see Fig. 5.5,
5.10, and 5.11). There are many mechanisms for this loss of visibility of the fringes.
Some are intrinsic to the atomic gas, while others are technical. The coherence
times we observe appear to be limited by a combination of these mechanisms.

5.5.1

Non-technical mechanisms

Molecule lifetime
4 In Eq. 5.9, we have generalized to two species by replacing hni2 /2 with hn

K ihnRb i. This

substitution follows from the fact that the number of possible pairs in a single species is N (N −1)/2 ≈ N 2 /2, while the number of possible pairs in the mixture is NK NRb [109].

Energy

124

0

B

0

Magnetic Field

Figure 5.13: Schematic of the energy eigenvalues of a trapped atom pair in thevicinity of a Fano-Feshbach resonance. The solid lines represent the dressed trap
states, while the dashed line shows the bare ground state of the trap.

125
There are many reasons why an oscillation in an ensemble of atoms might
damp. Possibly the simplest cause of damping is loss of moleculesif the numberof molecules decays with time, the amplitude of the oscillation will decay as well.
The lifetime of Feshbach molecules depends on the density of the sample, theinternal states and quantum statistics of the constituent atoms, and the proximity
to the resonance [121, 122, 123, 124, 125, 126, 106]. For our mixture, we expect
molecule lifetimes longer than a millisecond over the range of densities and elds
probed [106] and have measured lifetimes as long as 10 ms. If the molecule lifetime
were the cause of the damping, as in the experiments of Syassen et al. [45], thetime-averaged number of molecules would decay. Instead, we observe damping inwhich the mean molecule number is constant. This type of damping is not due toa loss of molecules but is likely due to dephasing.

Energy distributions
Because we begin with an incoherent ensemble of atoms, the oscillations
are expected to damp due to the diering relative kinetic energies of the atom
pairs. Let us consider a simple illustration of this damping mechanism. Let us
assume we have an ensemble of 10 atoms, each of which can pair with one andonly one other atom, and each pair has a dierent relative kinetic energy. The
probability of nding a given atom pair in the molecule state Ppair after either
the single- or double-pulse sequence oscillates with a frequency determined by theenergy dierence between the atom and molecule states. A distribution of energies
implies a distribution of frequencies. If we were able to monitor the oscillation
frequency of each pair, we might observe oscillations such as those shown in Fig.
5.14. All the pairs begin oscillating in phase, but get out of sync, or dephase, due
to their diering frequencies. Now, let us, instead, measure the total populationin the molecule state as a function of time. The probability of nding the entire

126

(a)

(b)

Figure 5.14: Illustration of dephasing. (a) The probability of nding an atom
pair in the molecule state, Ppair , is shown as a function of time for ve ideal pairs
with 100% contrast and innite coherence times but dierent frequencies. (b) The
probability of nding all of the pairs in the molecule state, Pensemble , is plotted asa function of time.

population in the molecule state is given byPensemble =1
NpairsX

Ppair ,

(5.10)

where Npairs = 5 is the number of atom pairs. The total probability, Pensemble ,damps even if the coherence of the individual pairs persists, as shown in Fig.
5.14b.5
In the experiment, we measure the number of molecules as a function of timeand are only sensitive to the total number in the molecule state. In the absence ofother decoherence or dephasing mechanisms, the envelope of the oscillation in the
total population is determined by the distribution of frequencies of the oscillating
pairs. These oscillations in the molecule population may provide a probe of theinitial relative kinetic energy distributions of the atom pairs.
Using the Monte Carlo simulations discussed in Chapter 4, we can calculate
expected coherence times based on either of two assumptions about the distribu5 In the illustration, because there are only 5 atom pairs, the total probability would experience revivals of the oscillation. In the experiment, there is a near-continuum of relative kinetic
energies, so we expect no such revivals.

127tion of the relative kinetic energies of the oscillating pairs: (1) the distribution is
the same as that of the pairs converted to molecules via adiabatic sweeps or (2) the
distribution is the same as that of nearest-neighbor pairs. For trapped mixtures,the second assumption implies that the distribution reects the temperature of
the gas, and for expanded clouds, it reects the local temperature distribution.
Figure 5.15 shows expected oscillations for the typical in-trap conditions. Boththe estimate based on the energy of the molecules and that of nearest neighbors
predict coherence times longer than those observed experimentally (compare to
Fig. 5.5b), but the nearest-neighbor calculation predicts a coherence time only afactor of 2 longer. Figure 5.16 shows similar calculations for the typical expansion conditions. Once again, we nd that the coherence time estimated from theenergy distribution of the molecules is much longer than our observations, while
the nearest-neighbor energy distribution yields a result much closer to the experimental results (compare to Fig. 5.10). This may indicate that the requirements
for creating pairs in a superposition are dierent or more stringent than for associating molecules with an adiabatic magnetic-eld sweep. We might suspect thatfor the diabatic pulse to pair atoms, the interparticle separation must be similar
to that of the molecule. This would introduce a real-space criterion in addition to(or instead of) the phase-space criterion, Eq. 4.1. However, there are additional
decoherence mechanisms that are likely limiting the coherence times of both theRabi- and Ramsey-type oscillations we have observed in the experiment. Dephasing due to the relative kinetic energy of the pairs is independent of magnetic eld.
As we shall see below ( 5.5.2), the coherence times measured in the experiment
do depend on magnetic-eld, indicating that we are not yet probing the relative
kinetic energy distribution of the pairs.

Atom dynamics
Another factor that can aect the coherence of atom-molecule oscillations is

128

(a)

Rb

(b)

Figure 5.15: Calculated oscillations for typical in-trap conditions with coherence
times limited by the energy distribution of the pairs. The calculation in (a) uses
the relative kinetic energy distribution of molecules formed by adiabatic magneticeld sweeps, while the calculation in (b) uses that of nearest-neighbor atom pairs.

129

(a)

K

(b)

Figure 5.16: Calculated oscillations for typical expansion conditions with coherence times limited by the energy distribution of the pairs. The calculation in(a) uses the relative kinetic energy distribution of molecules formed by adiabatic
magnetic-eld sweeps, while the calculation in (b) uses that of nearest-neighboratom pairs.

130
the motion of the atoms in the trap. Collisions between molecules and unboundatoms or other molecules are likely to break up the molecules, which in general,
limit the molecule lifetime. However, the collision rate (Eq. 5.7) is negligible on
the timescale of the observed oscillations. For the in-trap Ramsey oscillations(Figs. 5.5 and 5.6), the calculated collision rate is < 5 kHz. There may, however,be more subtle interactions that cause decoherence.
We expect the likelihood of coupling between two atoms to depend on theirdistance from one another. Because the atoms are not stationary in the trap, but
are moving around, we might imagine partner-swapping events. Let us consider aK atom initially coupled to a Rb atom (Rb #1), with an uncoupled Rb atom (Rb#2) nearby. At some later time, it may be the case that Rb #2 is in closer proximity to the K atom than Rb #1, which may result in the K atom coupling insteadto Rb #1. The resulting superposition would likely have a dierent phase, contributing to the dephasing of the observed atom-molecule oscillations. For suchpartner swapping to aect on the coherence time of the observed oscillations,however, the timescale of the motion of the atoms must be short compared to theinterrogation time. As an estimate of this timescale, we use the Monte Carlo simulation discussed in Chapter 4 to calculate the time it takes for nearest-neighboratoms to separate by an additional distance d, which is the typical interparticle
spacing (see Appendix B for a discussion of the interparticle spacing). Fig. 5.17
shows the results of these calculations for both the trapped and expanded gas
experiments. For the typical in-trap conditions, the separation of nearly half ofthe pairs increases by d in only 20 µs, while for the typical expansion conditions,
it takes 40 µs. In both cases, the timescale for motion is comparable to the observed coherence times (see Figs. 5.5 and 5.10). The motion of the particles may
be contributing to the decoherence of our observed oscillations. This may also
explain why the contrast of the observed Ramsey oscillations is less than 1 both131
in trap and in expansion, where the pulses toward resonance are 15 µs and 50 µs
long, respectively.

(a)

(b)

Figure 5.17: Atom motion in the trap and in expansion. Plotted is the calculated
fraction of nearest neighbors that have separated by an additional distance equal
to the typical interparticle spacing for the typical in-trap conditions (a) and thetypical expansion conditions (b).

5.5.2

Technical mechanisms

Magnetic eld inhomogeneity
We believe the main technical decoherence mechanism in our experiment is
spatial inhomogeneity in the applied magnetic eld. A spatial variation in magnetic eld would cause the molecule binding energy to vary across the cloud. Given
the quadratic dependence of the molecule binding energy on magnetic eld nearresonance (Eq. 5.8), this results in more rapid dephasing for larger magnetic-elddetuning from the Fano-Feshbach resonance. Figure 5.18 shows the measured coherence time of Ramsey oscillations as a function of magnetic eld. The measured
coherence times of oscillations with Bevolve < 546.1 G are consistent with a 60 mGmagnetic-eld variation along the axial direction. If this magnetic-eld inhomogeneity were the only mechanism for decoherence we would expect the coherence

132time at Bevolve = 546.3 G to be 70 µs. Instead, we measure 18 µs, indicatingthat another decoherence mechanism dominates closer to the Fano-Feshbach resonance. The magnetic-eld dependence for Bevolve > 546.1 G suggests that thisadditional mechanism may be collisional, though the observed coherence times
are a factor of 14 shorter than the collision times 1/Γ we calculate with Eq. 5.7.
By improving the magnetic-eld curvature cancellation discussed in 2.5, weshould be able to improve the homogeneity of the magnetic eld. This will allow
for the investigation of the intrinsic dephasing due to the relative kinetic energiesof the atom pairs.

133µ dB a3(a - abg)2µ

1nKRb a2

Figure 5.18: Magnetic-eld dependence of the coherence times of Ramsey oscillations. The rms damping time is plotted as a function of Bevolve for Ramsey
oscillations induced in trapped clouds under typical in-trap conditions. The solidline shows the calculated damping time for a linear magnetic-eld gradient of 0.5
G/cm in the axial direction, which corresponds to a magnetic-eld variation of
δB =60 mG across the atom/molecule clouds. The dashed line is 1/ΓKRb calculated for hnKRb i = 5 × 1012 cm−3 and scaled by a factor of 14 to match thedata.

Chapter 6
As yet unsubstantiated Monte Carlo predictions

The Monte Carlo simulations discussed in Chapter 4 have proven to be a
powerful tool in predicting the number of molecules that can be created via adiabatic magnetic-eld sweeps through Fano-Feshbach resonances [61, 63], oscillating
magnetic elds near a resonance [62], and rf association [64]. However, our direct measurements of the number of molecules created by adiabatic magnetic-eld
sweeps in the Bose-Fermi mixture always yield conversion fractions that are lower
than we predict with the Monte Carlo simulations. We believe this to be due toenhanced losses incurred while crossing the resonance, and eorts are currently
underway to understand and minimize these losses. If we can eliminate such losses,
we may be able to use these simulations to predict more than just the number ofmolecules. Because the formation of molecules in the calculation depends on therelative momentum of the constituent atoms, we can also make predictions about
the distributions of the relative kinetic energies of the paired atoms. To date,
the ability of the Monte Carlo simulations to accurately predict the molecules'
momenta (and, consequently energy) has not been experimentally tested. The
atom-molecule oscillations we observe (see Chapter 5) may be able to probe theenergies of the pairs and oer a method to test this aspect of the simulations.
In the absence of technical or collisional decoherence mechanisms, the coherence time of atom-molecule oscillations should be determined by the relative

135
kinetic energy distribution of the pairs. By using the Monte Carlo simulation
to predict these energies, we can, therefore, make predictions about the coherence time and the shape of the observed oscillations. In this chapter, I discuss
how these quantities depend on the temperature and quantum degeneracy of theinitial atom gas.

6.1

Assumptions
In applying the results of the Monte Carlo simulations to atom-molecule

oscillations, we are making several assumptions. The rst assumption is that thefrequency at which an atom pair oscillates is determined by the energy dierence
between the atom pair and molecule state as νosc = (Eb + Erel )/h. Here, Eb isthe molecule binding energy and the relative kinetic energy is Erel = p2rel /(2µ),
where µ is the reduced mass and prel is the magnitude of the relative momentumof the atom pair. We are also assuming that the dephasing of the atom pairs
is the dominant mechanism for decoherence. That is, the oscillation in the total
population is given by1
Ntot (t) =
2

Z ∞


cos(2πνosc t) f (νosc ) dνosc + 1 ,

(6.1)

0

where f (νosc ) is the probability distribution of oscillation frequencies of the atom
pairs. Therefore, by calculating the probability distribution of relative kinetic
energies, we can determine the expected shape of the oscillation and vice versa.
We also make the assumption that the distribution of Erel for pairs coupledby diabatic magnetic-eld pulses is identical to either the distribution of Ereleither for pairs associated by adiabatic magnetic-eld sweeps (pairs satisfying thephase-space criterion, Eq. 4.1) or for nearest-neighbor pairs. Neither of these islikely to be strictly true in the experiment, but as we shall see below, we may
be able to use the temperature dependence of the oscillation coherence times to136
determine which, if either, of these distributions best reects the pairs coupled by
the magnetic-eld pulses.

6.2

Dependence of coherence times on temperature
In this section, we consider the eect of temperature on the coherence timesof the oscillations. As we have seen in Chapter 4, the number of molecules formed
via adiabatic magnetic-eld sweeps through a Fano-Feshbach resonance is expected to depend only on the quantum degeneracy of the mixture, and not on theabsolute temperature. However, for the same quantum degeneracy, a mixture at ahigher temperature will have a wider distribution of Erel . This could result in more
rapid dephasing of atom-molecule oscillations. Figure 6.1a shows calculated atommolecule oscillations for mixtures at a variety of temperatures. In each of thesesimulations, the number of atoms are NK = 15, 000 and NRb = 5, 000, and thetemperature T of the mixture is equal to TC and 0.27 TF . As we might expect,
the calculated coherence time, shown in Fig. 6.1b, is inversely proportional to
the temperature of the mixture. Therefore, to observe long-lived atom-molecule
oscillations in the experiment, we want to work with clouds with low absolutetemperatures.
The above calculations assumed the energy distribution of the oscillating
pairs to be the same as the energy distribution of molecules associated by adiabatic sweeps. If, instead, we assume that the oscillating pairs are described by theenergy distribution of nearest-neighbors, we nd a dierent shape for the oscillations and shorter coherence times. The 1/e coherence time for T = TC = 36nK is
220 µs for the nearest-neighbor calculation, in contrast to 3300 µs for the molecule
calculation (see Appendix A for a discussion). However, we still nd that for clouds
with the same quantum degeneracy, the expected coherence time is inversely proportional to the temperature.

137

(a)

(b)

Figure 6.1: Expected temperature dependence of the coherence time of atommolecule oscillations. (a) Calculated atom-molecule oscillations for atom clouds
with NK = 15, 000 and NRb = 5, 000 at a temperature T = TC . For each curve, we
run the Monte Carlo simulation for molecule creation to obtain the distributionof relative kinetic energies of the molecules, and the number of molecules as afunction of time is calculated with Eq. 6.1 with a molecule binding energy Eb =h×10kHz. While these oscillations do not t to exponentially decaying envelopes,
as an estimate of the coherence time for each, we calculate the time at which
the amplitude drops by 1/e. The calculated 1/e coherence times, τ , are plotted
as a function of temperature in (b). The line is a t to τ = A/T , yielding
A = (1.197 ± 0.008) ms·nK.

138

6.3

Dependence of coherence times on quantum degeneracy
In comparing the calculated energy distributions of nearest-neighbors andof molecules, we see that the phase-space criterion for molecule formation (Eq.
4.1) preferentially selects out pairs with low relative kinetic energy. This leads
to longer coherence times if the diabatically coupled pairs must also satisfy thiscriterion. Simply comparing the observed coherence times in the experiment (inthe absence of other decoherence mechanisms) to these predictions should give agood indication of whether either of these energy distributions is reected in theoscillating pairs. An additional method to determine if the diabatic magneticeld pulses preferentially select low-energy pairs is to study the dependence of thecoherence time of the quantum degeneracy of the mixture.
Figure 6.2a shows the calculated atom-molecule oscillations for clouds atvarying T /TC , with TC xed at 36 nK. Here, the envelope of the oscillation is determined by the distribution of energies of adiabatically-associated molecules. As
the temperature of the cloud drops below TC , we see that the oscillation envelopechanges, and the coherence time decreases with decreasing temperature (see Fig.
6.2b), and above TC , we observe little dependence on the temperature. This isin direct contradiction to the naive picture that a colder cloud ought to have anarrower distribution of energies. However, we can understand this by considering
how atoms are paired in the Monte Carlo simulation. To form a molecule, atoms
are required to have a small product of ∆rrel ∆prel . Above TC , the position andmomentum distributions of the Bose and Fermi clouds are similar to one another.
While the momentum distribution may be wide for both, it is possible to have
many pairs with a large center-of-mass momentum but small relative momentum.
In this case, the distribution of Erel will have a large contribution at low energies, which will result in long coherence times. In the case of a zero temperature

139
mixture, all the Rb atoms are in a BEC. The relative momentum between a Rb
atom and any K atom, then, is equal to the momentum of the K atom. Therefore,
the distribution of relative energies is simply the energy distribution of a T = 0
Fermi gas. Due to the Pauli exclusion principle, there can be no pile-up at low
energies. This wider distribution of relative energies results in shorter coherence
times, despite the lower temperature.
If, instead, we assume that the pairs coupled by the diabatic pulses havean energy distribution similar to that of nearest-neighbors, we nd that the coherence times of the atom-molecule oscillations follow the intuitive notion thatcolder clouds result in longer-lived oscillations. Figure 6.3 shows the results ofcalculations using the Monte Carlo simulation to identify nearest-neighbor pairs
and calculate their relative kinetic energies. Here, we have no preferential selection of low-energy pairs since the distribution of Erel of nearest neighbors is thesame as the distribution of Erel over the whole cloud. As the mixture gets colder,the distribution of Erel becomes narrower, and the coherence time increases.
Discerning the functional form of the envelope of an oscillation to determine the energy distribution of the oscillating pairs can be dicult in real experiments where shot-to-shot noise may limit the visibility of the fringes. However,by observing whether the coherence time increases or decreases as a function of
the quantum degeneracy of the mixture, we should be able to determine if theatom pairs coupled by the diabatic magnetic-eld pulses are subject to the same
phase-space criterion as molecules associated via magnetic-eld sweeps, oscillating
magnetic elds, and rf association.
By studying the coherence times of oscillations that are not limited by collisions or magnetic-eld inhomogeneities, we should be able to probe the distributions of the relative kinetic energies of the pairs coupled by the pulses and perhaps
gain insight into the coupling mechanism.

140

(a)

(b)

Figure 6.2: Expected dependence of the atom-molecule oscillation coherence timeon quantum degeneracy. (a) Calculated atom-molecule oscillations for atom clouds
with NK = 15, 000 and NRb = 5, 000 and TC = 36 nK. For each curve, we run theMonte Carlo simulation for molecule creation to obtain the distribution of relative
kinetic energies of the molecules, and the number of molecules as a function oftime is calculated with Eq. 6.1 with a molecule binding energy Eb = h × 10kHz.
The calculated 1/e coherence times are plotted as a function of temperature in
(b).

141

(a)

(b)

Figure 6.3: Expected dependence of the nearest-neighbor atom-molecule oscillation coherence time on quantum degeneracy. (a) Calculated atom-molecule oscillations for atom clouds with NK = 15, 000 and NRb = 5, 000 and TC = 36 nK. For
each curve, we run the Monte Carlo simulation to identify nearest neighbors andobtain their relative kinetic energies. The number of molecules as a function oftime is calculated with Eq. 6.1 with a molecule binding energy Eb = h × 10kHz.
The calculated 1/e coherence times are plotted as a function of temperature in(b).

Chapter 7
Conclusions and Outlook

In this thesis, I have described the renovation of our Bose-Fermi mixture
apparatus and the experiments creating ultracold fermionic molecules that followed. These experiments culminated in the creation of a quantum superpositionof atoms and molecules, which involves three chemically distinct species with different quantum statistics: bosonic Rb atoms, fermionic K atoms, and fermionicKRb molecules. In this superposition, two atoms are both a pair of free atoms

and a bound molecule at the same time! The superposition was not demonstrated
in two or only a few atoms but in tens of thousands of atoms simultaneously. Additionally, these atoms were not prepared in a single energy state, but occupied a
distribution of states. Despite this, coherent oscillations in the macroscopic atom
population have been observed. These oscillations may prove to be a direct probeof the temperature and quantum degeneracy of the molecules.
Currently, eorts are under way to eliminate magnetic-eld inhomogeneities,
which are likely the dominant mechanism for the dephasing of the atom-molecule
oscillations we have observed. In the absence of such technical causes for dephasing, the coherence times of the oscillations should be limited by the distributionof relative kinetic energies of the pairs participating in the superposition. This
will allow for the testing of my predictions for the oscillations' dependences ontemperature and degeneracy.

143
Similar oscillations have been predicted for quantum degenerate two-component
Fermi systems [55, 56, 57]. As in our experiments, the initial atomic gas would
not be prepared in a single energy state but would instead be described by a
Fermi-Dirac energy distribution. A particularly interesting application would beto study coherent oscillations between the superuid BCS state and a molecular
BEC in the BCS-BEC crossover [55, 56].
For the longer term, I expect this experiment to move away from molecules
and begin to study some of the more exotic phenomena predicted in Bose-Fermi
mixtures. With so many options, it's dicult to predict which path will be chosen,
but I think the observation of Boson-mediated Cooper pairing (with or without alattice) would be a great achievement.

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Appendix A
Fitting dephased oscillations

In Chapter 5, I posited that the damping of the observed atom-molecule
oscillations is likely caused by a distribution of oscillation frequencies of the atom
pairs. This distribution of frequencies may be due to the relative kinetic energies of
the atom pairs, but for much of the data, it is most likely due to the inhomogeneityof the magnetic eld.

We t the data to sines with Gaussian envelopes andextracted oscillation frequencies. In this appendix, I address the question of how
the measured frequency is inuenced by the shape of the distribution of oscillation
frequency.
We will begin with the case of an oscillation with a coherence time dened
purely by the inhomogeneity of the magnetic eld. For the purpose of illustration,
we will consider an atom cloud well above quantum degeneracy such that thepositions of the atoms are described by a Gaussian distribution. We also assume
that in the absence of the magnetic-eld inhomogeneity, each pair would oscillatewith a frequency ν0 , equal to the molecule binding energy divided by h (i.e. all
the atom pairs occupy a single energy state). In the experiment, we nd themagnetic eld varies linearly along the axial direction of the cloud with a gradientof 0.5G/cm.

As shown in Fig.

A.1a, this implies a Gaussian distribution ofmagnetic elds across the cloud. Figure A.1b shows the resulting distribution ofoscillation frequencies. Here, we have assumed ν0 =100 kHz corresponds to the155molecule binding energy at the center of the cloud. The distribution of frequencies
is also well-approximated by a Gaussian.1

The oscillation in the total populationis the sum of the oscillations of all the pairs



Z∞



Npairs 22
A sin(2πνt)e−(ν−ν0 ) /(2σ ) dν + 120

Npairs −t2 σ2 /2
Nmot (t) =esin(2πν0 t) + 1 .
2
Nmol (t) =(A.1)

In this case of a Gaussian frequency distribution around ν0 , the total population
oscillation will have a frequency of ν0 and be bounded by a Gaussian envelope.
Now, let us consider atom pairs in a uniform magnetic eld, but with a
distribution of relative kinetic energies. Once again, we assume a classical gas. If
the energy distribution of the pairs mimics that of nearest neighbors, the relative
kinetic energy E will follow the Boltzmann energy distribution (see Fig. A.2)√
P (E) = A Ee−E/(kb T ) ,(A.2)

where A is a normalization constant and kb is the Boltzmann constant. If weassume each atom pair will oscillate with a frequency ν = ν0 +νE , where νE = E/h,then the oscillation in the total population will be given by
Nmol (t) =N max  2√2
τ 3/2 πZ∞sin (2π(ν0 + νE )t)√νE e−νE /τ dνE + 10

Nmax
Nmol (t) =2!
sin 2πν0 t + 23 arctan(2πtτ )
+1 ,
3/4
1 + (2πtτ )2(A.3)

where τ = kb T /h. Figure A.3a shows the oscillation of a 200 nK cloud with

ν0 =100 kHz. In this case, the frequency of the oscillation in the total population
varies with time. We can rewrite the argument of the sine to make this explicit
3
3 arctan(2πtτ )
2πv0 t + arctan(2πtτ ) = 2πt v0 +,24πt
1 For larger variations in magnetic eld or very near the resonance, the quadratic dependenceof the binding energy on magnetic eld causes this distribution to become asymmetric.

156

(a)

(b)

Figure A.1: Frequency distribution due to an inhomogeneous magnetic eld. Using the Monte Carlo simulation discussed in chapter 4, we generate Gaussian
clouds with an rms size of 150 µm in the axial direction. We then calculate themagnetic eld and corresponding molecule binding energy (using Eq. 5.8) for eachpair's average position assuming a 0.5 G/cm gradient along the axial direction.
The calculated distribution of magnetic elds (a) and oscillation frequencies (b)are shown for a cloud is centered on B =546.135 G, where the molecule binding
energy is hν0 = h × 100 kHz. The red curves show Gaussian ts with rms widths
σB = 7.2 mG and σν = 2.9 kHz.

157

ν(t) = v0 +3 arctan(2πtτ )
.
4πt

(A.4)

Figure A.3b shows the time dependence of the frequency. Since all of the pairs
have frequencies ≥ ν0 , the initial frequency of the oscillation in the total population is also higher than ν0 . The pairs at the high energy tail of the Boltzmann
distribution (pairs with high oscillation frequencies) are the rst to dephase, andthe frequency decreases, approaching the zero-energy pair frequency, ν0 . If wewere to t this oscillation to a sine with a Gaussian envelope, we would underestimate the coherence time and measure a frequency blue-shifted from the moleculebinding energy by up to 23 τ = 23 kbhT .
Now, we consider a distribution of relative kinetic energies of the oscillating pairs that reects the energy distribution of molecules created via adiabatic
magnetic-eld sweeps. In this case, we must obtain the energy distribution from
the Monte Carlo simulation (see Chapter 4). Figure A.4 shows the distributionof relative kinetic energies of the atom pairs that become molecules for a mixture
with NK = 2.5NRb at 200 nK, which corresponds to 1.2 TC and 0.3 TF .
Figure A.5a shows the calculated oscillation with ν0 = 10 kHz. The distribution of relative kinetic energies leads to a variation of the oscillation frequency
in time. To extract the frequency as a function of time, we t the oscillation in
Fig. A.5a with a sliding window 100 µs wide. Figure A.5b shows the dierence
between the tted frequency and ν0 . As in the case of the Boltzmann energy
distribution above, the measured frequency approaches the zero-energy frequency
as the high-energy pairs dephase. The apparent dip around 700 µs is likely an
artifact of the tting. Initially, the frequency exceeds ν0 by 0.4 kHz, which is equal
to 0.1 kbhT . This is much smaller than the shift for the Boltzmann distribution, indicating that the adiabatic molecule formation process modeled with the Monte
Carlo simulation preferentially forms pairs with small relative kinetic energy.

158

Figure A.2: Calculated distribution of relative kinetic energies, E , of nearestneighbor pairs in a 200 nK Boltzmann gas.

(a)

(b)

Figure A.3: Calculated atom-molecule oscillation for a 200 nK gas with a Boltzmann energy distribution at B = 546.135G. The number of molecules as a functionof time, calculated with Eq. A.3 is shown in (a). The frequency ν , calculated with
Eq. A.4 is plotted in (b). The frequency of a pair with zero relative kinetic energy
is ν0 = 100 kHz.

159

Figure A.4: Calculated distribution of the relative kinetic energies E of atom
pairs associated into molecules in the Monte Carlo simulation. The points are
the results of a simulation of a mixture with NK = 2.5 NRb at 200 nK, whichcorresponds to 1.2 TC .

(a)

(b)

Figure A.5: Calculated atom-molecule oscillation for pairs with the distribution
shown in Fig. A.4. The calculated number of molecules as a function of time isshown in (a). Fits to the oscillation were performed, and the frequency νfit in a
sliding window 100 µs wide was extracted. The dierence between νfit and thezero energy oscillation frequency ν0 is shown in (b). Initially, the νfit exceeds ν0
by 0.4 kHz, which is 0.1kb T /h.

160
We know that in the experiment, we have magnetic-eld inhomogeneities

and a distribution of energies. If the magnetic eld limits the coherence time to,say, 30µs, then the measured frequency will be the zero-energy frequency (moleculebinding energy divided by h) shifted by the frequency of the energy-distributed
oscillation during the rst 30 µs, i.e. ∼ 32 kbhT and ∼ 0.1 kbhT for the above examples.
Therefore, the measured frequencies in Fig. 5.6 may be shifted from the moleculebinding energy by up to 32 kbhT = 6 kHz.

Appendix B
Nearest-neighbor distance and density-weighted density for Gaussian
clouds with dierent sizes, numbers, and aspect ratios.

Often working with a two-species experiment requires extending theory done
for single-species experiments. Sometimes this can be as simple as replacing an
atomic mass with twice the reduced mass, but at times it is necessary to know
the average density or nearest-neighbor distance. In this appendix, I address
the questions of how to dene these quantities in the two-species case. For this
discussion, both clouds are assumed to have Gaussian proles, but may containdierent numbers of atoms and have dierent rms widths and aspect ratios.
We begin with the question of how to dene the typical nearest-neighbordistance, which is relevant in comparison to the size of the molecules near a FanoFeshbach resonance (see chapter 5). At rst glance, this seems a simple task
generate Gaussian clouds, pick an atom, nd its nearest neighbor (of the opposite
species), calculate the separation between the two atoms, and repeat until all theatoms have been counted. In the case of two clouds with dierent numbers of
atoms, we encounter a hitch: on average, the distance to the nearest atom in the
opposite species depends on which species we are considering. This is illustrated
in Fig. B.1. Here, we have a larger number of red atoms than blue atoms. If wecalculate the average distance from a red atom to the nearest blue atom (option1), we will nd a much larger distance than if we calculate the average distance162

Figure B.1: Illustration of the calculation of the typical nearest-neighbor distancefor a mixture with dierent numbers of atoms. In Option 1, we calculate thedistance from a given red atom to the nearest blue atom. In Option 2, we calculate
the distance from a given blue atom to the nearest red atom.

from a blue atom to the nearest red atom (option 2). So, which do we choose?
In considering molecule creation, we might expect that an atom is likely to bindto its nearest neighbor. We would want to compare the distances between the
Rb and K atom of each nearest-neighbor pair to the interatomic distance in amolecule (Eq. 5.6). In option 1, a single blue atom can be the nearest neighbor to
many red atoms, but if it were to form a molecule, the blue atom would only pairwith one red atom. In option 2, the gas is separated into nearest-neighbor pairs
and lone red atoms, as would occur if every possible atom were to be associatedinto a molecule. In the following discussion, we will always calculate the nearestneighbor distance dnn from the cloud with fewer atoms, a.k.a. option 2.
If we properly dene the nearest-neighbor distance and the two-species density nKRb , we expect the relationship

dnn = C (nKRb )−1/3(B.1)

where C is a proportionality constant. Figure B.2 shows a typical distribution ofnearest-neighbor distances in a Gaussian cloud.
In a single-species experiment, there are typically two relevant measuresof density, the peak density n0 and the average, or density-weighted, density

163

Figure B.2: Calculated distribution of nearest-neighbor distances dnn for a simulated Gaussian cloud.

164

hni = N1Rn2 (r)d3 r. Here, n(r) is the number density of atoms. We generalize thisR
to the two-species case as hn12 i = N1< n1 (r)n2 (r)d3 r , where we have normalizedby N< , the number of atoms in the species with fewer atoms.
For a Gaussian cloud, the density is given by− 1 (r2 +λ2 z 2 )n(r) = n0 e 2σr2

(B.2)

,

where σr is the rms size in the radial direction, λ = σz /σr is the ratio of the axial
and radial sizes, and n0 = N λ/(2πσr2 )3/2 . For two Gaussian clouds with dierent
sizes, numbers, and aspect ratios (but the same center), the density-weighted
density ishn12 i =

1
2π

3/2


N2 λ1 λ212
σ1 + σ22



12 2
λ1 σ2 + λ22 σ12

1/2
,

(B.3)

where we have assumed that N2 ≥ N1 . Figure B.3 shows the most probablenearest-neighbor distance in simulated clouds as a function of the two-species
density hn12 i calculated with Eq. B.3. We nd the expected relation, Eq. B.1,
with the proportionality constant C ∼ 21 . Note that all of the simulations were
performed for the case where the cloud with fewer atoms is the spatially smaller
cloud and has a lower density (N1 ≤ N2 , σ1 ≤ σ2 in both the axial and radial
directions, and n1 ≤ n2 for all space).

165

Figure B.3: Density dependence of the calculated nearest-neighbor distance. The
most probable nearest-neighbor distance dnn is plotted as a function of the twospecies density-weighted density hn12 i (Eq. B.3) for simulated Gaussian clouds.
The black points indicate simulations where the number and sizes of the two clouds
are the same. In the simulations represented by the red points, the numbers, sizes,
and/or aspect ratios of the two clouds dier from one another. The line is a t to
dnn = C hn12 i−1/b with C = 0.51 ± 0.03 and b = 3.008 ± 0.019.