Few-body collisions in a quantum gas mixture of 40 K and87

Rb atoms
byRuth Shewmon Bloom
B.S., Massachusetts Institute of Technology, 2008

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2014This thesis entitled:
Few-body collisions in a quantum gas mixture of 40 K and 87 Rb atoms
written by Ruth Shewmon Bloom
has been approved for the Department of Physics

Deborah Jin

Eric Cornell

Date

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

iii
Bloom, Ruth Shewmon (Ph.D., Physics)
Few-body collisions in a quantum gas mixture of 40 K and 87 Rb atoms
Thesis directed by Deborah Jin

This thesis describes experiments on few-body interactions in a mixture of ultracold bosonic
87 Rb and fermionic 40 K atoms. Ultracold atoms are celebrated as a platform to explore fundamental

quantum physics because their internal states, external potentials, and interactions with each othercan be controlled by straightforward electrical and optical tools. In the case of Bose-Fermi mixtures,control comes at the cost of stability when strong interactions give rise to inelastic collisions that
destroy the atomic samples. I present detailed measurements of the cross sections for these inelastic
processes and discuss how they fit into an emerging pattern of resonances that is leading to bettermodels of few-atom interactions. These phenomena have their roots in universal Efimov physics,
which is a paradigm of quantum three-body interactions that was originally inspired by exotic
nuclear states. I also discuss several methods to control the formation of KRb Feshbach molecules
within the finite lifetime of the interacting K and Rb atomic mixture.

Dedication

To Paul Shewmon, who inspires me to be a better scientist.

Acknowledgements

This thesis only lists one author on the title page, but it really includes the work of many
people. I’ve had the pleasure of working closely with some fantastic teammates in the lab while
we ran the experiments that are presented here. John Perreault was our post-doc when I firstarrived at JILA. His deep knowledge of optics and atomic physics was an invaluable resource. Ty
Cumby, who was the senior graduate student during most of my PhD, had a remarkable patienceand attention to detail. He taught me how to run quantum gas experiments, and we had a lotof fun working together. We were later joined by Ming-Guang Hu, whose powerful enthusiasm isdriving our group forward to a new generation of experiments. I’m sure the lab will accomplish
great things under his leadership.
Of course, none of this work would have been possible without our fearless leader, Debbie
Jin. She is equally at home in the lab’s tangled jungle of wires or in contemplating deep questions
on the quantum nature of matter. It’s amazing how much time Debbie makes for her students,
and I really appreciated the insightful advice she has given on every aspect of my work – fromdiagnosing technical problems to designing experiments, editing the written results, presenting
them at conferences, and more. She has a way of making complex problems seem simple. I
couldn’t have asked for a better research adviser.
Our lab is part of a larger group that is jointly run by Debbie Jin and Eric Cornell. Eric isa great leader and role model, and his love of physics is contagious. It’s also been wonderful tohave so many people in the group to bounce idea off of and borrow equipment from, and I’m gladthat I had the chance to learn about the inner workings of so many great experiments at our big

vimeetings.
JILA is a fun place to work because the community is so inter-connected. A huge number ofpeople from JILA contributed to the work in this thesis, more than I can list here, so I’d like to takea moment to thank everybody. In particular, Tracy Keep in the instrument shop did a masterful jobof building the magnetic coils that are central to our experiment. Jose D’Incao patiently answeredso many of my questions about scattering theory, for which I am also grateful. The JILA electronics
shop, especially Terry Brown and Carl Sauer, designed some incredible circuits for our lab.
Finally, I want to thank my family. This PhD has been hard work, but your support makes
every burden lighter. I love you.

Contents

Chapter
1 Introduction1

1.1

Bosons and Fermions near absolute zero . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Universality and ultra cold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Universal few-body interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

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

52 Magnetically tuned interactions

8

2.1

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

2.2

The interspecies 40 K-87 Rb Fano-Feshbach resonance . . . . . . . . . . . . . . . . . . 11

2.3

Inelastic loss: the cost of tunable interactions . . . . . . . . . . . . . . . . . . . . . . 13

3 An improved electromagnet system for trapping and manipulating atoms

8

17

3.1

The magnetic coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2

The high-current control system

3.3

Power dissipation and water cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Procedure: from room temperature to ultra cold

34

4.1

40 K and 87 Rb atoms

4.2

Initial cooling stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3

The optically trapped mixture

4.4

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

viii
5 Efficient magnetoassociation of KRb Feshbach molecules

48

5.1

How magnetoassociation works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2Counting the molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3

Dependence on the magnetic-field sweep rate . . . . . . . . . . . . . . . . . . . . . . 55

5.4

Saturated molecule number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 More KRb molecule results

65

6.1

Molecules formed by three-body recombination . . . . . . . . . . . . . . . . . . . . . 65

6.2

Radiofrequency photoassociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.3

Excess molecule kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7 The Efimov Effect

80

7.1

Echoes of nuclear physics in quantum gases . . . . . . . . . . . . . . . . . . . . . . . 80

7.2

Inelastic collisions as a signature of Efimov states . . . . . . . . . . . . . . . . . . . . 84

7.3

How predictable are three-atom interactions? . . . . . . . . . . . . . . . . . . . . . . 89

8 K-Rb-Rb collisions

95

8.1

Atom-molecule loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.2

Three-body recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.3

Comparison to other experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

9 Concluding remarks

110

9.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

9.2

Looking forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Bibliography

113

ix

TablesTable
2.1

Table of scaling laws for inelastic scattering in 40 K+87 Rb. . . . . . . . . . . . . . . . 15

3.1

Fields from the new magnetic coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

8.1

Universality and three-body parameters in K–Rb mixtures . . . . . . . . . . . . . . . 106

Figures

Figure1.1

Feshbach molecule size compared to influenza virus . . . . . . . . . . . . . . . . . . .

4

1.2

Scattering processes studied in this thesis . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1Interactions between pairs of ultracold neutral atoms. . . . . . . . . . . . . . . . . .

92.2

An interspecies 40 K-87 Rb Fano-Feshbach resonance . . . . . . . . . . . . . . . . . . . 12

2.3

Weakly bound states of a 40 K-87 Rb mixture . . . . . . . . . . . . . . . . . . . . . . . 14

3.1

The magnetic trap and bias coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2

Measured magnetic-field geometries for the IP trap . . . . . . . . . . . . . . . . . . . 21

3.3

Bias magnetic-field calibration from RF spectroscopy . . . . . . . . . . . . . . . . . . 23

3.4

An electrical schematic for the new magnet system . . . . . . . . . . . . . . . . . . . 25

3.5

Circuit diagram for the current servos . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6

Connections between servo, IGBT, and Hall effect current sensor . . . . . . . . . . . 27

3.7

Water cooling for the new magnet system . . . . . . . . . . . . . . . . . . . . . . . . 29

3.8

Heating and power dissipation in the magnetic coils . . . . . . . . . . . . . . . . . . 32

4.1

The 40 K and 87 Rb D2 lines with hyperfine structure. . . . . . . . . . . . . . . . . . . 36

4.2

Schematics of the resonant 40 K and 87 Rb laser systems. . . . . . . . . . . . . . . . . 38

4.3

Schematic of the vacuum system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4

Schematic of the optical trap and magnetic bias coils. . . . . . . . . . . . . . . . . . 42

4.5

Dual-resonant microwave antenna for 87 Rb. . . . . . . . . . . . . . . . . . . . . . . 45

xi
4.6

Low-magnification imaging system and example absorption image . . . . . . . . . . . 47

5.1

Magnetoassociation of Feshbach molecules . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2

Feshbach molecule association as a function of magnetic sweep rate . . . . . . . . . . 51

5.3

Timing sequences for molecule imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4

Optical trap release timing for measuring molecules . . . . . . . . . . . . . . . . . . . 54

5.5

Feshbach molecule association in the fast-sweep regime . . . . . . . . . . . . . . . . . 57

5.6

Bosons vs. fermions in an optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.7

Molecule conversion efficiency for displaced clouds . . . . . . . . . . . . . . . . . . . 61

5.8

Saturated Feshbach molecule association . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1

Timing diagram for measurements of Feshbach molecules formed by three-body recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2

Population of Feshbach molecules versus time near the Feshbach resonance . . . . . 68

6.3

Timing diagram for RF association of KRb Feshbach molecules . . . . . . . . . . . . 72

6.4

Radiofrequency photoassociation of KRb Feshbach molecules . . . . . . . . . . . . . 73

6.5

Feshbach molecule expansion from an optical trap . . . . . . . . . . . . . . . . . . . 75

6.6

Feshbach molecule kinetic energy vs T /TF . . . . . . . . . . . . . . . . . . . . . . . . 77

6.7

Expansion of molecules with and without Rb atoms . . . . . . . . . . . . . . . . . . 79

7.1

Energy spectrum for three atoms with resonant interactions . . . . . . . . . . . . . . 81

7.2

Links between three rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.3

Efimov scaling factor in different systems . . . . . . . . . . . . . . . . . . . . . . . . 85

7.4

Efimov theory lines for α for 40 K–87 Rb–87 Rb . . . . . . . . . . . . . . . . . . . . . . 87

7.5

Efimov theory lines for β for 40 K87 Rb + 87 Rb . . . . . . . . . . . . . . . . . . . . . . 90

7.6

Measured three-body parameters in homonuclear atom gases . . . . . . . . . . . . . 93

8.1

Timing diagram and raw data for an atom-molecule loss rate measurement . . . . . 97

8.2

An Efimov resonance in Rb-KRb collisions . . . . . . . . . . . . . . . . . . . . . . . . 99

xii
8.3

Timing diagram and raw data for a three-body recombination rate measurement . . 101

8.4

Measured three-body recombination rate coefficient vs magnetic field . . . . . . . . . 104

8.5

Measured three-body recombination rate coefficient vs scattering length . . . . . . . 105

8.6

Measured three-body parameters in heteronuclear mixtures . . . . . . . . . . . . . . 108

Chapter 1

Introduction

A prism disperses white light into a rainbow, but the light emitted by atoms splits into discretelines of color. These spectra inspired Niels Bohr to deduce the quantized structure of the atom [1].
One hundred years later, atoms are still helping us discover the quantum world. Techniques likelaser cooling and forced evaporation can bring a gas of atoms to ultra cold temperatures nearabsolute zero [2, 3]. Here, atomic wave functions spread out to overlap each other and quantum
mechanics plays out on a macroscopic scale. This thesis is about measurements of the collisions
between a few atoms in such an ultra cold gas.

1.1

Bosons and Fermions near absolute zero
Most of the particles and excitations in the known universe are categorized as either Bosons

or Fermions. Fermions have half-integer spins and the Pauli exclusion principle forbids them fromoccupying the same quantum state. Protons, neutrons, and electrons are all Fermions and Pauli
exclusion dictates the way these particles arrange themselves to create atoms and materials. On
the other hand, Bosons have integer spins and multiple Bosons can occupy the same quantum state.
The photon is a familiar example of a bosonic particle, and 4 He is a composite Boson famous for
its superfluidity at low temperatures.
Bose and Fermi gases have distinctly different behaviors in the limit of zero temperature.
The relatively friendly Bosons pile together into the quantum state with lowest energy, forminga Bose-Einstein Condensate (BEC) [4]. Atomic BECs have shown interesting quantum properties

2
like matter wave interference [5] and superfluidity [6]. Identical Fermions spread across the lowest available energy levels with one particle per quantum state, forming a Degenerate Fermi Gas(DFG) [7]. The DFG is essential to the behavior of matter all around us – electrons in a metal
form a DFG, as do neutron stars.
Mixtures of Bosons and Fermions, which are the subject of this work, can add up to more than
the sum of their parts. The superfluid transition temperature in (bosonic) liquid 4 He is raised by
the presence of (fermonic) 3 He [8]. Conversely, bosonic lattice excitations mediate Cooper pairingof electrons in solids, which leads to superconductivity [9]. As for cold atoms, Bose gases are often
used to sympathetically cool Fermi gases to lower temperatures. At these temperatures, stronglyinteracting ultra cold Bose-Fermi mixtures are expected to host quantum phase transitions andnovel pairing phenomena [10].

1.2

Universality and ultra cold atoms
Gases of ultracold atoms have promising applications in communications, information pro-cessing, and sensors, but I would argue that their most fascinating application is in the emulationof other quantum systems. The collective dynamics of strongly interacting quantum particles is a
frontier for modern science that could unlock the secrets of superconductors, dense nuclear matter,astrophysics, and more. Unfortunately, the behaviors of quantum systems are notoriously difficult to model1 . Feynman famously proposed that we circumvent this problem by using one wellcontrolled quantum system to simulate another quantum system’s behavior [11].
A remarkable level of control has already been demonstrated in ultra cold atom experiments,
thanks to a growing set of experimental tools [12]. For example, a magnetic Fano-Feshbach resonance was used to control interactions between fermionic atoms to realize Cooper pairing and the
BCS-BEC crossover transition in a Fermi gas [13]. The optical lattice, formed by an interference
To simply write down the full quantum state of N interacting particles requires 2N numbers. The cost ofcomputing quickly gets out of control as particle number increases. If every one of the protons in the known universe
were used as memory in a giant classical computer, then it could only store the complete description of a few hundred
quantum particles.
1

3
pattern in light, made it possible to explore the quantum phase transition from a Mott insulator
to a superfluid state in a Bose gas [14]. High-resolution optics were later used to manipulate anddetect these states one atom at a time [15]. In other experiments, inventive configurations of opticaland magnetic fields are being used to try to realize topologically non-trivial states of matter [16].
These tools are put to good use in ultra cold atoms because of a property called universality,
where very different quantum systems often exhibit very similar behavior. In spite of the length andenergy scales that appear to separate them, an ultra cold atomic Fermi gas near a Fano-Feshbachresonance has much in common with a solid-state superconductor. Dilute gases of ultra cold atoms
can serve as a test bed for quantum models and concepts that apply to other universal systems
that are less easy to access experimentally [17].

1.3

Universal few-body interactions
The work in this thesis focuses on universal few-body interactions between ultra cold atoms.

The most straightforward system in this category consists of two atoms with contact interactions.
A Fano-Feshbach resonance occurs at specific magnetic fields where the energy of these two atoms
is the same as a molecular bound state that has a different magnetic moment [18]. The scattering
length, which parameterizes two-body interactions, diverges near Fano-Feshbach resonances. Experimenters can dial in magnetic field values near the resonance to achieve repulsive or attractive
interactions that vary by orders of magnitude. This magnetic tuning means that a single ultra cold
atom experiment has the flexibility to operate over a huge parameter range rather than relying on
scattering properties that were fine-tuned by Nature (which must be done in nuclear physics, forexample).
For each Fano-Feshbach resonance there is an associated bound state called a Feshbach
molecule, whose binding energy is also controlled by the magnetic field [19]. These molecules
are an important first step in the formation process of ultra cold dipolar molecules, which promise
to dramatically extend the range of quantum models that can be simulated in ultra cold atomic
gases [20]. Feshbach molecules also have interesting properties in their own right, for example their

4

100 nmRb

Figure 1.1: Feshbach molecules are quantum-mechanical bound states that can attain gigantic
spatial extent. The influenza virus (left) contains hundreds of millions of atoms yet it has acomparable size to some of the diatomic 40 K87 Rb Feshbach molecules that are studied in this thesis(right). The virus cartoon is reproduced from www.cdc.gov.

huge size as illustrated in Fig. 1.1.
After two-body interactions, the logical next step is to consider three-body interactions. The
quantum three-body problem has a universal solution in the limit of strong interactions, which iscalled the Efimov effect [21]. This effect consists of an infinite series of three-body bound statescalled Efimov trimers. Amazingly, the trimers follow a discrete scale invariance such that for eachbound state there exists one larger by a scaling factor that depends on the relative masses of theconstituent particles. Much like Feshbach molecules, their spatial extent can be gigantic. Although
Efimov’s theory was intended for the internal structure of exotic nuclei, the first clear observationof these effects occurred in the collisions between ultra cold Cesium atoms [22]. Efimov states incold atoms occur near Fano-Feshbach resonances, where interactions are strong [23]. They give riseto inelastic three-body collisions that eject atoms from the ultra cold gas. Measurements of theseatom loss rates have revealed an intricate series of resonances in several different atomic species,
confirming key predictions of Efimov’s work and related theories [24].
These phenomena are fascinating, but they also have an important practical side-effect: atom
loss rates limit the available time to perform other quantum physics experiments. A quantitative
understanding of few-body collisions is therefore essential to the design of experiments that rely on

5strong interactions in ultra cold atomic gases. For this reason, it was exciting to discover that thelocations of Efimov loss resonances in different single-species quantum gases followed a simple andcompletely unexpected pattern [25, 26, 27]. This was an opportunity to refine and simplify existingmodels of few-body interactions, and a promise that measurements in one atomic species could be
used to predict very specific details about the behavior of another. It remains to be seen how far
this success can be extended to systems containing multiple species of atoms or systems with strong
background interactions that effectively weaken the influence of Fano-Feshbach resonances.

1.4

Overview
This thesis explores few-body physics in a Bose-Fermi mixture of 87 Rb and 40 K atoms withresonant interspecies interactions. To that end, Fig. 1.2 outlines the scattering processes that aremeasured. The rates of these few-body scattering processes can reveal a lot about the way atoms
see each other. Whereas related phenomena are well-studied in single-species quantum gases, thesame level of detail has not yet been realized in quantum gas mixtures like 87 Rb–40 K.
The extra complexity of atomic mixture experiments has so far made them less popular thantheir single-species counterparts. In the study of few-body collisions, however, the Bose-Fermi
mixture offers some advantages: the fermionic two-body Feshbach molecule has a longer lifetime
than similar molecules in Bose gases; the two types of atoms can also be measured independently togain more information about who participates in collisions; and Pauli exclusion prevents collisions
between identical 40 K atoms so that there are fewer types of collisions to analyze. More broadly,
the mass-imbalanced Bose-Fermi mixture provides an opportunity to test the generality of theories
that were honed on experiments with only one type of atom.
The remainder of the work is organized as follows.
Chapter 2 is about tuning the interspecies scattering properties with DC magnetic fields. This
includes a brief overview of the physics of Fano-Feshbach resonances and the inelastic loss processes
that accompany them. Chapter 3 introduces a high-current electromagnet system that is used tocreate these magnetic fields. Chapter 4 goes on to describe how samples of 40 K and 87 Rb atoms are6a) Feshbach molecule association:
Rb
Rb

K
K

b) three-body recombination:
Rb
Rb

Rb

K

Rb

K

c) Rb + molecule vibrational relaxation:

Rb
Rb

Rb

Rb

K

Rb

K

K

d) K + molecule vibrational relaxation:

Rb
K
K

K

Figure 1.2: Scattering processes studied in this thesis. Energy for Feshbach molecule association(a) is provided by time-dependent magnetic fields or RF photons. In either case, the association
process is coherent and the resulting ultra cold Feshbach molecules remain trapped. In contrast,
three-body recombination (b) is an inelastic collision that forms a molecule and releases an amountof kinetic energy equal to its binding energy. This energy is usually sufficient to eject the resulting
molecule and atom from the trap. Feshbach molecules exist in a highly excited vibrational state
and they can relax to a more deeply bound state by colliding with 87 Rb (c) or 40 K (d). This releases
enough kinetic energy to eject atom and molecule from the trap. The rates of processes (b) and(c) are modified by the Efimov effect due to the existence of 87 Rb–87 Rb–40 K Efimov trimer states.

7
prepared for experiments near absolute zero temperature, and how thermodynamic quantities likenumber and temperature of the atoms are extracted from photographs using standard absorption
imaging techniques.
The next two Chapters describe the formation of ultracold fermionic 40 K87 Rb Feshbach
molecules in the atomic gas (see Fig. 1.2a). Chapter 5 shows our systematic exploration of therate and ultimate efficiency of molecule creation by magnetic-field sweeps near a Fano-Feshbachresonance. Chapter 6 shows alternative methods that were employed to make Feshbach molecules.
These measurements combine to give a recipe for efficient molecule production, however they also
reveal a many-body suppression of pairing and an unexpected source of heating that prevent theformation of a degenerate Fermi gas of molecules.
Chapter 7 reviews the three-body Efimov effect and the previous measurements from ultra
cold atom experiments around the world. Chapter 8 focuses on our own measurements of inelastic
collision rates in the atomic 40 K and 87 Rb mixture, as depicted in Fig. 1.2 (b-d). These results
pinpoint the Efimov state spectrum in the 87 Rb–40 K mixture for the first time, and the location
appears to deviate from the pattern established by measurements in single-species atomic gases.
Chapter 9 reviews the major results and concludes with a discussion of future experiments.

Chapter 2

Magnetically tuned interactions

A typical ultracold atomic gas is a million times less dense than air, and yet this ethereal substance is routinely made into a strongly interacting quantum system for physics experiments. Suchcontrol is made possible by magnetic Fano-Feshbach resonances, which can be used to dynamicallytune the interaction strength between atoms by orders of magnitude. This phenomenon occurs
when a weakly bound molecular state’s energy is degenerate with two free atoms, and is reviewed
here in Section 2.1. Section 2.2 introduces the 40 K-87 Rb interspecies Fano-Feshbach resonancethat is central to the work in this thesis. Finally, Section 2.3 describes an important side-effectto tunable interactions near Fano-Feshbach resonances. Specifically, as pairwise interactions become stronger, few-body correlations are also enhanced and the rates of atom loss due to inelastic
collisions accelerate dramatically. Inelastic collisions pose a challenge to experiments that seek tocreate stable, strongly interacting quantum gases. However, they also create an opportunity to
study universal few-body interactions, which is an interesting and surprisingly complex problem inquantum physics.

2.1

Magnetic Fano-Feshbach resonances
The interaction strength between atoms varies by orders of magnitude as a function of mag-netic field near a Fano-Feshbach resonance. Leveraging this effect to control two-body interactions,ultra cold atom experiments can access a huge variety of few- and many-body phenomena. Magnetic Fano-Feshbach resonances are an important tool for atomic physics. As such, a number ofEnergy

Energy9

Separation

Magnetic field

Figure 2.1: Left: Schematic interaction potentials between pairs of ultracold atoms. Right: Thebound state and pair of free atoms have different magnetic moments. When their energies cross, a
Fano-Feshbach resonance occurs (red arrow).

great review articles already cover the subject in detail [12, 18, 19]. This section highlights a few
properties that are relevant to the observations discussed in the rest of the thesis.
A pair of ultra cold atoms can interact through a potential like the one shown in Figure
2.1 (left). For small internuclear separation, there is a hard core repulsion. At large separation,interactions are dominated by a 1/r6 tail characterized by the van der Waals length, rvdW . Twopotentials are drawn in the figure, one in which we consider a pair of free atoms (the ‘entrance
channel’) and another with a bound state having similar energy (the ‘closed channel’). Consider
the free atoms, which are in a state with nonzero magnetic moment (right, Figure 2.1). As themagnetic field is varied, their energy changes too. The bound state, formed by atoms in a different
Zeeman sub level, has a different magnetic moment. Coupling between the entrance and closed
channels mixes the two states as their energies approach each other. A magnetic Fano-Feshbachresonance occurs when the energies are degenerate, and the strength of atom-atom interactions
diverges.
And what does this atom-atom interaction look like? In the limit of zero temperature,kinetic energy vanishes. As a consequence, atoms cannot climb centrifugal barriers and collisions
with nonzero angular momentum are suppressed. Spherically symmetric s-wave collisions dominate.

10
The interatomic scattering potential can then be renormalized as a hard sphere with radius a, suchthat the elastic collision cross section is σel = 4πa2 . The interspecies scattering length a near aninterspecies Fano-Feshbach resonance follows
a(B) = abg 1 −∆B − B0(2.1)

where B is the magnetic field, abg is the background scattering length, ∆ is the resonance width,and B0 is the location of the Fano-Feshbach resonance [28]. Larger |a| means stronger interaction,a < 0 is attractive, and a > 0 is repulsive. Although the work in this thesis is focused on FanoFeshbach resonances that enhance s-wave interactions, many resonances also exist for higher angularmomenta.
A two-body bound state called a Feshbach molecule is also associated with each resonance[29, 19]. The Feshbach molecule’s binding energy, E, is determined by the scattering length, a,near a resonance.
E=~22µ(a − ā)2

(2.2)

where ~ is Planck’s constant divided by 2π, µ is the reduced mass of the atoms forming the molecule,and ā = 0.96 rvdW is the average scattering length of the system. This bound state exists whena > 0, with the binding energy approaching zero as a approaches infinity. Feshbach molecules
belong to a universal class of ‘halo states’, which are named for their enormous size that can exceed
the scale of inter-particle (or inter-atomic) forces [29]. Feshbach molecules play a key role in the
BCS-BEC crossover in gases of ultra cold Fermions [30], and also serve as a starting point for thecreation of ultra cold polar molecules [20].
Fano-Feshbach resonances are classified according to their strength [18]. For strong ‘entrance
channel dominated’ resonances, the universal formulae 2.1 and 2.2 dominate the scattering properties over a large range of a. Near weaker ‘closed-channel dominated’ resonances, a competes
with other interaction length scales like R∗ = ~2 /(2µabg δµ ∆). Here, δµ is the differential magnetic
moment between the entrance and closed channels. These ideas are summarized by the parametersres = ā/R∗ , where sres  1 belongs to an entrance channel dominated resonance and sres  1

11belongs to a closed-channel dominated resonance. The former type usually has ∆ > 1 G, leadingthem to be called ‘broad’ Fano-Feshbach resonances. These are the most useful resonances froman experimental perspective, because the scattering length is less sensitive to magnetic-field noise.
Entrance channel dominated Fano-Feshbach resonances also provide the best access to universal
scattering phenomena.
A huge diversity of Fano-Feshbach resonances exists across the Alkali atoms, which impacts
the efficiency of evaporative cooling as well as the types of experiments available to each species [18].
For example, the Fermion 6 Li has the unusual configuration of three strong overlapping FanoFeshbach resonances between the three lowest energy spin states, meaning that all possible pairwiseinteractions are simultaneously enhanced. Although 40 K’s strongest Fano-Feshbach resonance ismuch weaker, this Fermion has better options for detection and spectroscopy thanks to the existenceof a nearby spin state that is unaffected by any resonance. The Boson 133 Cs has a strong resonancenear zero magnetic field, meaning that the gas must be cooled at a higher field to prevent thedestructive inelastic collisions that accompany Fano-Feshbach resonances (see Section 2.3). This
inconvenience is more than offset by the existence of several extremely strong resonances at higher
magnetic fields [31]. In contrast, 87 Rb is a particularly easy Boson to cool but it possesses no strong
Fano-Feshbach resonances at all.

2.2

The interspecies 40 K-87 Rb Fano-Feshbach resonanceFortunately, there are several interspecies Fano-Feshbach resonances between the 40 K atoms

and 87 Rb atoms. The strongest of these resonances has sres = 1.9, and is located at B0 = 546.62
G with ∆ = 3.04 G and abg = −187 a0 as measured by Ref. [32], where a0 is the Bohr radius.
Figure 2.2 plots the scattering length as a function of magnetic field. This resonance is between the|f, mf i = |1, 1i and |9/2, −9/2i states of 87 Rb and 40 K, where f is the total angular momentum
and mf its projection on the magnetic-field axis. These are the lowest energy states in a magnetic
field, and so the mixture is protected against inelastic spin-changing two-body collisions.
These resonance parameters set requirements for the electromagnet systems and their cali-

12

4 0

K - 8 7 R b s c a tte r in g le n g th ( a 0 )2 0 0 01 0 0 00

-1 0 0 0-2 0 0 0

5 4 2

5 4 4

5 4 6

B (G )

5 4 8

5 5 0

5 5 2

clear Feshbachof 40 K and 87 R
point for futur
polar molecule
more deeply b
microwave fieldbasic propertie
typical size andmagnetic-field
timate proper
binding energy
VI.

PR

The propert
size, are deter
Figure 2.2: Top: The 40 K-87 Rb interspecies scattering length a as a function of magnetic field
closFIG. 5:Fano-Feshbach
Energy of theheteronuclear
molecules,
relaB near a magnetic
resonance,
calculatedFeshbachusing parameters
measuredby Ref. dominant[32].
with these cha
Bottom: The
KRbmolecule−E,
as a function
B, adapted from[33]. The solid
tive
to Feshbach
the Rb |1,
1⟩ + energy,
K |9/2,−9/2⟩
atomicofthreshold,plottedline is a coupled-channel
calculationofthemoleculebindingenergythat
includesnon-universal
con- this is, of
as a function of magnetic field. The data (solid circles) agree
andtributions fromclosed-channel
wave function.
line indicates the universal
well thewith
our calculation
based Theon adashedfull coupled-channel
the- formula
is magnetic-fie
for molecule binding energy (Equation 2.2), which is accurate at high scattering lengths.
ory (solid line). Near the Feshbach resonance one expects acontributions
universal relationship between the s-wave scattering length
then be determ
and the binding energy. The black, dashed line shows thement relative
predicted binding energy using the universal prediction [31].
numerical deriv
Inset: Same as the main plot but looking at the region close
shown in Fig.
to the Feshbach resonance. We fit the inset data to obtain a

13bration. To access the large field B0 (within the spatial constraints of our experiment) requires
hundreds of Amps of current through coils of water-cooled hollow-core copper tubing. Chapter 3
describes the implementation of a new magnet system for this purpose. The resonance width ∆is also relatively small, which makes the scattering length quite sensitive to magnetic-field fluctuations. For example, formula 2.1 predicts a scattering length a = 2420 a0 when the magnets areturned on to a field of 546.4 G near the Fano-Feshbach resonance. The lab’s ambient magnetic fieldchanges by about 5 mG when a nearby freight elevator changes floors; this increases the scattering
length to 2480 a0 , which is a noticeable shift. To avoid additional noise, the electromagnet current
control is constructed to achieve better than 10 ppm fractional instability.
For our purposes, the most useful range near the Fano-Feshbach resonance is where theinterspecies scattering length aKRb dominates over all other length scales and interactions areuniversal. This is bounded from below by the interspecies van der Waals length rvdW = 72 a0 .
Figure 2.2 shows how the KRb molecule binding energy deviates from Eqn. 2.2 at low scattering
lengths. For three-body 87 Rb–87 Rb–40 K interactions, the nearly magnetic-field independent 87 Rb–
87 Rb scattering length a
RbRb = 100 a0 also plays a role. The useful range for aKRb is bounded from√above by either the thermal deBroglie wavelength λdB = h/ 3µkB T or the mean inter-particle
spacing n1/3 , whichever is smaller. Here, h is Planck’s constant, µ = 27.4 amu is the reduced massof 40 K and 87 Rb, kB is the Boltzmann constant, and T is the temperature. For a typical gas mixture
at T = 500 nK and mean density n = 5 × 1012 cm−3 , we have λdB = 13000 a0 and n1/3 = 11000 a0 .

2.3

Inelastic loss: the cost of tunable interactions
A gas of ultra cold neutral atoms is really in a metastable state, beneath which lies a zooof two-body bound states (Fig. 2.3). Inelastic scattering into one of these states converts binding
energy to kinetic energy, which can propel atoms out of the trap entirely. At least three atoms
are required for a collision to form a bound state while conserving energy and momentum. The
odds of such a few-body encounter are predicted to increase dramatically with the magnitude of
the interspecies scattering length, a (Table 2.1). Although gases with only one or two types of

14

CoG

-1
-2

A(-1)

A(-1)
D(-3)B(-2)
A(-2)
C(-3)

bb(-1)
aa(-1)

1)
cc(-

)

(-2

qc

-3

)

-3

)

rf(

-2
rb(

E/h (GHz)

-1
B(-2)
-2

)
(-3
qe
-3)
pd(

D(-3)
cc(-

2)
a

-10
A(-3)
D(-4)
C(-4)

-20

bb(-2)-3
A(-2) aa(-2) g(-3)
C(-3)
-4
0

aa(-3)S-4T

rb(-4)

B(-4)
A(-4)

bf(-3)
ce(-3)
dd(-3)

40

rb(-3)B(-3)

E/h (GHz)0

B (mT)

80

120

-30
0

aa(-4)

400
B (mT)

800

40 K40 87
Figure 11: Boundenergyoffor boundthe Mtot = states−7/2 s-wave
of 87
theRb
K
Rb 13: Boundwith
Figurestate energy
for the M
states (left)of the 40 K87 Rb
Figure2.3: state
A zoo
inboundthestates
mixturebindingenergies
0 tobound3 GHz
tot = −7/2 s-wave
molecule down to 4 GHz binding energy. The zero of energy is the energy Eα of theaa down to 30 GHz binding energy. The zero of energy is the energy Eα of the aa
molecule
andGHz
Inelastic
into
these
bound
statesleads
toin Figure
atom11.loss,since
a lineschannel0at toeach30B field.
The (right).
level are labeledaccording toscattering
the channel index
ofchannel
their
at eachB field. Labels
are thesame asThe shorthorizontal
dominant spin component and by the vibrational quantum number n countingnext
downto the B = 0 axis labeled by vibrational quantum number n show the energies oftypicaloptical
trap
for
our
experiments
can
only
hold
atoms
with
a
kinetic
energyofh×20
kHzfrom the dissociation limit of the channel. The Roman letters indicate the groupinthe levelsof the X1 Σ+ and a3 Σ+ adiabatic Born-Oppenheimer potentials relative to
Figure
6 withThe
which magnetic
the B = 0 channels
are associated.
energygravity (CoG) of the isseparated
multiplet at
E/h =of4.843439
orless.
Feshbachresonance that wetheuse
incenter
ourof experiments
at a atommagnetic
field
GHz. The X1 Σ+ and a3 Σ+ potentials support N = 100 and 32 vibrational levels
54.6 mT. Adapted from [34].
respectively. The normal vibrational quantum number v counting up from v = 0 as
the lowest level is v = N + n.

35

37

15
Table 2.1: Three-body inelastic scattering processes in the 40 K-87 Rb mixture near an interspecies
Fano-Feshbach resonance and their universal scaling with interspecies scattering length [35]. In
these reactions, 40 K87 Rb denotes the Feshbach molecule state and (40 K87 Rb)’ denotes a moredeeply bound state like the ones in Figure 2.3. The processes are categorized by quantum statistics
where ‘B’ are identical 87 Rb atoms (Bosons), ‘F’ are identical 40 K atoms (Fermions), and ‘X’ aredistinguishable atoms in a third state. In practice, the role of ‘X’ can be played by a 40 K atom ina different spin state.

Process
87 Rb+40 K+40 K
40 K+87 Rb+87 Rb
40 K87 Rb+40 K
40 K87 Rb+87 Rb
40 K87 Rb+X

→
→
→
→
→

40 K87 Rb+40 K
40 K87 Rb+87 Rb(40 K87 Rb)’+40 K(40 K87 Rb)’+87 Rb(40 K87 Rb)’+X

Type
B+F+F Three-body recombination
B+B+F Three-body recombination
F+BF Atom-molecule relaxation
B+BF Atom-molecule relaxation
X+BF Atom-molecule relaxation

Predicted Scaling
a6
a4
a−3.12a1
a−1

Fermions are protected by the Pauli Exclusion principle, inelastic collisions and atom loss canrapidly deplete strongly interacting Bose gases, Bose-Fermi mixtures, or Fermi mixtures with morethan two distinguishable states. The 40 K-87 Rb mixture’s most prevalent inelastic loss mechanisms
are three-body recombination and atom-molecule vibrational relaxation.
In three-body recombination, three atoms collide to produce a two-body bound state, witha kinetic energy equal to the binding energy distributed between the molecule and third atom.
This process can form a deeply bound molecule at negative scattering lengths and either a deeply
bound molecule or a Feshbach molecule at positive scattering lengths. In either case, all threeatoms are typically propelled out of the sample. A Fano-Feshbach resonance amplifies threebody recombination for any triplet of atoms where at least two pairwise interactions are strong.
In fact, the resonant loss of atoms as a function of magnetic field is one of the most common
signals used to search for Fano-Feshbach resonances in quantum gases [18]. In the case of the
40 K-87 Rb mixture, B+F+F recombination
Rb
K
K
Rb K
K
is suppressed by the


Rb K
Rb
Pauli exclusion principle and B+B+F recombination Rb Rb K
dominates
the inelastic loss.

Atom-molecule vibrational relaxation occurs in quantum gas mixtures of atoms and Fesh-16bach molecules. When an atom collides with a molecule, the molecule can relax to a more deeply
bound state like the ones in Fig. 2.3. This process releases the difference in the molecule’s binding energy as kinetic energy, invariably propelling atom and molecule out of the trap. As is thecase with three-body recombination, quantum statistics play a role in determining which molecule
relaxation collisions are enhanced near an interspecies Fano-Feshbach resonance. F+BF relax

K
Rb K
Rb K
K
ation
is suppressed by the Pauli exclusion principle and B+BF


Rb K
Rb
relaxation Rb Rb K
dominates near the Fano-Feshbach resonance.
Three-body recombination and atom-molecule relaxation rates constrain the possible exper-

iments that can be performed at strong interactions. This is particularly true for three-bodyrecombination, which scales as a4 and quickly overwhelms thermalizing elastic collisions that only
scale as a2 . This means that 40 K-87 Rb gases at high scattering lengths can be depleted by inelastic
loss before even reaching thermal equilibrium. Chapters 5 and 6 of this thesis will describe theoptimization of Feshbach molecule formation, which is another dynamical process that competes
against these inelastic loss timescales. A quantitative understanding of the loss rates is useful tothese experiments. On a more positive note, the rates of three-body recombination and atommolecule relaxation are sensitive probes of three-body correlations. Chapters 7 and 8 will explore
how the presence of three-body bound states modulates the scaling laws in Table 2.1, giving riseto sequences of resonances that occur near most Fano-Feshbach resonances. Before I describe anyof these results, however, the next two Chapters will summarize how we prepare our quantum gas
mixtures for experiments.

Chapter 3

An improved electromagnet system for trapping and manipulating atoms

Electromagnets are essential to the work in this thesis. Large, homogeneous, and stable
magnetic fields enable tunable interactions near the interspecies 40 K-87 Rb Fano-Feshbach resonanceat 547 G. Magnetic traps are the workhorses of the cooling sequence, which collects atoms from a
hot vapor and creates a gas at less than one millionth of a degree above absolute zero temperature.
Reliable atom state preparation, spectroscopy, and imaging also depend on stable magnetic fields.
This Chapter introduces an electromagnet system that improves the magnetic trapping and control
capabilities of our apparatus. Optical access is also enhanced, which paves the way for new methods
to image and manipulate the atoms in future experiments. Section 3.1 describes the electromagnetic
coils and the fields they individually produce. Section 3.2 is about the high-current control system.
In Section 3.3, water cooling and thermal dissipation are discussed.

3.1

The magnetic coils
The idea to build a new magnetic trap was sparked when a worn piece of electrical insulation

suddenly caused bursts of electrical current to arc across an air gap into the grounded surface of theold trap’s supporting structure. The failure occurred deep inside of the trap assembly, which had
to be painstakingly deconstructed and surgically repaired. Although the old trap was temporarily
revived, a safer and more robust mechanical structure was clearly needed. We took advantage ofthe redesign to improve other aspects of the trap as well. The old copper support structure hada large electrical inductance that limited the speed of the magnet system. We replaced it with a18
rigid Garolite G-10 phenolic support that has negligible electrical conductivity. The old system
only ran cooling water through the support structure. Now, the chilled water runs through thehollow cores of the copper wires themselves, which provides more uniform thermal dissipation witha smaller spatial footprint. To improve experiments near Fano-Feshbach resonances, the new design
also features a pair of Helmholtz coils that we can turn on independently of the other trap coils
to provide a homogeneous field while the atoms are confined in an optical trap. The design ofthese new coils was detailed in Ty Cumby’s thesis [36]. Tracy Keep from the JILA instrument
shop precisely wound the coils and mounted them to their rigid frame. Here, I will describe their
testing and implementation. This section describes characterization of the fields produced by thecoils when they are conducting a few hundred Amps of electrical current, which is their normal
mode of operation during an experiment.
The new magnet system is arranged in a Ioffe-Pritchard configuration [37] as shown in Fig. 3.1.
It consists of three pairs of coils, and it is designed to trap the atoms at a magnetic-field minimum.
The tight trapping radial directions are labeled x and y, and the weak axial trapping direction
along the axis of the Pinch and Bias coils is labeled z. Near the center, the Pinch coil pair creates22

a magnetic field whose z-component follows Bz = BP0 + 21 γ(z 2 − x +y2 ). The field offset BP0 andcurvature γ are both proportional to the electrical current in the Pinch coils. A pair of Bias coils
in Helmholtz configuration contributes a homogenous Bz = −BB0 in the z direction, the strengthof which is likewise proportional to the current through the Bias coils. The dominant contribution
from the Ioffe coils is a linear quadrupole field in the radial directions, Bx = βx and By = −βy.
These fields combine to form a harmonic trap with potential U = 12 mωx2 x2 + 12 mωy2 y 2 + 12 mωz2 z 2in the region near the center. Following Pritchard [37], the trap frequencies are approximately
s 

µ β2
γ
−
(3.1)
ωx = ωy =
m B0
2
r
µ
ωz =
γ
(3.2)
m
for a magnetic moment µ, mass m, and magnetic-field minimum B0 = BP0 −BB0 . In an experiment,
the 87 Rb and 40 K atoms are both trapped in states that have µ = µB , where µB is the Bohr

0

1 0

2 0

-4

-4

-2

-2

y [m m ]0z [m m ]

0

2

2

4

4

x

y

z

Figure 3.1: Left: Calculated magnitude of the trap magnetic fields for x = y = 0 (top) and for x = z = 0 (bottom). The atoms are trappedat the field minimum at the center. Right: Arrangement of coils in the Ioffe-Pritchard magnetic trap. The Pinch coils create an axial fieldwith a curvature and the opposing Bias coils subtract a constant field to bring the value at the center of the trap down to 4 Gauss. The
Ioffe coils create a radial quadrupole field to confine the atoms in the radial directions. The Bias coils can also be turned on alone, which
generates a large homogeneous magnetic field.

0

2 0

4 0

6 0

8 0

1 0 0

B [G ]B [G ]

3 0

19

20
magneton. For typical magnetic trap currents, the trap frequencies are ωx = ωy = 125 Hz andωz = 18 Hz.
To better understand the magnet system, we constructed a test setup to measure β, γ, BP0 ,and BB0 . One coil pair at a time was connected to a high-current power supply and a watercooling line while an F.W. Bell Gauss/Tesla meter on a three-axis translation stage was used tomap out the field between the coils. Table 3.1 lists the results and Figure 3.2 shows some of themeasurements. We measured the Ioffe coil pair’s magnetic-field gradient to be 89.7(2) G/cm in theradial y direction near the trap center, at a current of 90.0 A. The Pinch field was mapped out
in the x-z plane for near the center of the trap at a current of 100 A. A two-dimensional surface
fit to the equation Bz = BP0 + 21 γ(z 2 − x2 /2) gave γ = 106(2) G/cm2 and BP0 = 334(1) G. At acurrent of 160 A, the Bias coils made a field of 524(1) G. All of the measured values were close to
the predictions based on models of the coils [36].
The scattering properties of ultracold 40 K and 87 Rb atoms are particularly sensitive to magnetic fields during phases of the experiment where only the Bias coils are turned on. A goodcalibration of this field is especially important. Figure 3.3 shows a map of the Bias field for
an electrical current of 163 A, near the interspecies Fano-Feshbach resonance, as measured byspatially resolved RF spectroscopy on an elongated 40 K cloud in an optical trap. We used the|f, mf i = |9/2, −9/2i → |9/2, −7/2i transition resonance to identify the magnetic-field values. The
dashed line indicates a 0.079(3) G/cm gradient along the axial z direction. The solid line indicatesa curvature of 3.2(6) × 10−7 G/cm2 , which corresponds to a confining trapping frequency of 0.01 Hz
for the 40 K atoms in the |9/2, −9/2i state that we use for our experiments. Although the Bias coils
and leads were carefully designed [36] to generate no magnetic-field gradient, this value is just overhalf of the stray gradient that existed with the previous Bias system. It is possible that the residual
gradient is caused by a magnetized object near the atoms. This could be tested by measuring thegradient at a different Bias field to see if it is proportional to the Bias current. In any case, thegradient is acceptably small for the applications in this thesis. External gradient-compensatingcoils may be required in future work.

21

F ie ld in y d ir e c tio n ( G )

2 0

1 0

0
x p o s itio n :1 9 .8 m m
1 7 .3 m m
1 4 .8 m m
1 2 .2 m m
9 .7 m m

-1 0

-2 0
4

5

6

7

8

9

y m ic r o m e te r p o s itio n ( m m )x p o s itio n :7 .1 m
1 2 .2 m
1 7 .3 m
2 2 .4 m
2 7 .5 m

F ie ld in z d ir e c tio n ( G )

3 6 0

3 4 0

m
m
m
m
m

3 2 0

3 0 0
0

2

4

6

8

1 0

z m ic r o m e te r p o s itio n ( m m )Figure 3.2: Top: Measurements of the gradient from the Ioffe coils at 90 Amps (points) with linear
fits to extract the field gradient (lines). Bottom: Measured curvature from the pinch coils at 100
Amps. The lines show the result of a surface fit to the data. For both plots, the positions include
arbitrary offsets due to the initial positions of the micrometers.

22
Table 3.1: Measured parameters of the fields from the new magnetic coil system. I denotes thecurrent through the relevant coil. Error bars are statistical.
Parameter
Ioffe radial gradientPinch axial curvature
Pinch axial field
Bias axial field
Bias axial gradient
Bias axial curvature

Variableβ/Iγ/I
BP0 /I
BB0 /I
βB0 /I
γB0 /I

Model [36]
1.00
1.06
3.48
3.32

Measured value
1.00(2)
1.06(2)
3.34(1)3.306(1)4.8(2)×10−4
2.0(4)×10−9

Units
G/cm/AG/cm2 /A
G/A
G/A
G/cm/AG/cm2 /A

RF spectroscopy is also the tool of choice for calibrating the field’s absolute value. A combination of measurements on the 87 Rb |2, 2i → |1, 1i microwave transition and the 40 K |9/2, −9/2i →|9/2, −7/2i transition over a range of magnetic-field values gives the calibration, BB0 = 0.36(5)G +I × 3.306(3)G/A where BB0 is the Bias coil field in units of G, I is the current in units of A, andthe error bars are statistical. The offset is presumably caused by the Earth’s magnetic field, which
is oriented along the axis of the Bias coils.

3.2

The high-current control system
The electrical currents are another key part of the magnet system. At various stages of theexperiment, the magnetic fields from different coils need to be ramped on and off in repeatable sequences. We accomplish this by placing an Insulated Gate Bipolar Transistor (IGBT) in series witheach current path as shown in Figure 3.4 and controlling the gate voltages with servo electronics.
The priorities for the new control electronics are low noise, low sensitivity to RF interference, andreliability. This section describes the transistors, sensors, cabling, and servos that make it all work.
The magnet coils are wired in two parallel paths with the Bias coils in one path and the
Ioffe and Pinch coils in the other (Fig. 3.4). The entire high-current system is powered by one
Agilent 6690A power supply, which can provide up to 440 A at 15 V. The power supply provides
a constant voltage and the transistors control the currents through each path. There are two mainmodes of operation. First, to form a magnetic trap, both current paths are turned on to about

23

K d e n s ity

M a g n e tic fie ld ( G )5 3 8 .8 2 5

5 3 8 .8 2 0

5 3 8 .8 1 5

0

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 01 2 0 0

1 4 0 0A x ia l p o s itio n ( u m )Figure 3.3: Using RF spectroscopy on a gas of 40 K atoms, we measure the Bias magnetic field as afunction of axial position. A fit to the central linear region gives a field gradient of 0.079(3) G/cm
for an average magnetic field close to the interspecies 40 K-87 Rb Fano-Feshbach resonance (black
dashed line). A fit to the entire data set gives a field curvature of 3.2(6) × 10−7 G/cm2 . Zeroposition on the x axis represents the edge of the imaging camera.

24
200 A. Second, most experiments are eventually performed in an optical trap with a homogenous
magnetic field close to the interspecies Fano-Feshbach resonance. This requires 160 A of current
through the Bias coils while the Ioffe and Pinch coils are turned completely off.
Previous high-current electromagnets in our group had pushed power transistors to theirlimits such that a failure occurred every few months, one of which even started a small fire. To
handle high current loads it was also necessary to connect several transistors in parallel. Progressin the power electronics industry has made these problems obsolete. There is now a variety ofhigh-power transistors to choose from that can individually handle the maximum output of ourpower supply. For the same price point and physical size, Metal Oxide Semiconductor Field EffectTransistors (MOSFETs) are typically faster and can handle the highest currents, but they are also
more easily damaged by transient voltage swings. Insulated Gate Bipolar Transistors (IGBTs) are
popular in arc welding applications because they can typically survive several thousand volts acrossthe conduction path. Since the inductive backlash of quickly turning off our electromagnets pushes
hundreds of Volts onto the transistors, and because the sizable inductance and capacitance of themagnet coils prevents us from taking advantage of a MOSFET’s fast response, we use IGBTs.
Specifically, the SKM400GA-12V IGBT model by Semikron is used for IGBTs 1 and 2 inFigure 3.4. They each have three terminals: Gate (G), Emitter (E), and Collector (C). A positive
G-E voltage turns the device on, allowing current to flow from C to E. A 30 Ohm resistor is attachedto each gate to smooth out the turn-on behavior of the circuit. The transistor is rated for VCE
up to 1.2 kV, so no additional protection circuitry is needed for the high-current path. Like a
MOSFET, however, the IGBT is easily damaged by transient gate voltages so a 20 V back-to-back
TVS diode is placed between terminals G and E. These IGBTs enter a linear mode of operationthat is well-controlled by a servo when VCE > 2 V, and so we choose the output voltage of thehigh-current supply such that VCE is at least 3 V on each IGBT. Since the magnetic trap operates
with 200 A of current through each current path, each transistor dissipates 600 W of power. To
protect the IGBTs from thermal overload, we mount them to a water-cooled plate as discussed in
Section 3.3.

25

Bias coil A

Pinch coil A

Bias coil B

Pinch coil B

Servo Hall
probe 2Servo 2+Isolation
amp-

IGBT 2C

Ioffe coil A

Ioffe coil B

GE

Servo Hall
probe 1Servo 1
MonitorHall probe 2

NI 6733-PXI DACIGBT 1
MonitorHall probe 1C

GE

Figure 3.4: Electrical schematic for the new magnet system. Solid lines indicate the high-current
path. The current through each path is controlled by a SKM400GA-12V transistor (IGBTs 1 and 2).
A feedback loop consisting of a Hall effect current sensor (Danfysik IT-600s) and a servo controls
the Gate-Emitter voltage of each IGBT to achieve the desired current. A LabVIEW program
controls the NI PXI-6733 DAC, which in turn outputs analog voltages to control the power supply
voltage and the current through each path. We use out-of-loop ‘monitor’ hall probes, which are
identical to the in-loop ‘servo’ hall probes, to measure the current without making any connection
to the ground of the control system.

26
VctrlVctrl+

+

x1

x1
+

VsenseVsense+

+

PID

VG

+

VE

x10
Vmonitor

x1

QQ116A3

Figure 3.5: Diagram of the feedback electronics (‘Servo 1’ and ‘Servo 2’ in Figure 3.4) used to control
the magnetic trap currents. The circuit was designed by Terry Brown and James Thompson, anda complete schematic can be found under the reference number ‘QQ116A3’ in the JILA electronics
shop.

The Gate-Emitter voltage of each IGBT is controlled by a QQ116A3 coil driver servo fromthe JILA electronics shop, which we configure for unipolar operation. Figure 3.5 shows a simplified
schematic. There are two differential voltage inputs to the servo: a sense voltage, Vsense , that is
proportional to the measured electrical current; and a control voltage, Vctrl , which is generated by
a National Instruments PXI-6733 DAC installed in our main experiment control computer. Eachof these voltages goes through a buffer as soon as it enters the servo. The sense buffer has gain of
10 and the control buffer has gain of 1. The buffered signals are added to each other to generate
an error signal, which is fed into a Proportional-Integral-Differential (PID) filter. The PID output
is then fed to an output buffer stage. This two-terminal output connects to the IGBT’s Gate andEmitter with a twisted pair of wires. Because the servo’s Emitter connection is isolated from theservo ground, the high-power and low-noise paths are also isolated from each other. A secondary
output, Vmonitor , samples and buffers the sensor input so that we can monitor the current.
We measure the current in each path using a Hall effect current sensor1 . A precision 2.5
Ohm resistor measures the sense current, and the resulting sense voltage is connected by a twisted
1

Each current sensor is a IT 600-s ULTRASTAB probe, which produces a sense current equal to 1/1500 of themeasured current. Although this probe is rated to a much higher current than we require, it was chosen for its
superior resistance to thermal drift.

9 <

+15 V

5 <

-15 V

4 <

Ground

1 <
6 <

iS

From servo’s
power supply

QQ116A3

HiroseRM12BRD-6S
Connector

2.50 Ω

LEM Ultrastab
IT600s

27

15 Ω

5
6

1
2

Vsense+Vsense-Current

VG
VE

Figure 3.6: Connections between a servo, IGBT, and Hall effect current sensor. Green dashed lines
show grounded shielding around each cable.

shielded cable to the servo’s sense input. Since the servo’s sense buffer amplifies the signal by afactor of ten, the control voltage into the servo commands a current of 30 A/V. Hall effect current
sensors are useful for measuring high currents because they can be electrically isolated from thecircuit being measured. One downside is that they can consume significant power when sensing
larger currents. The original design placed the sense resistor to measure the Hall current sensor
output inside of the servo, but we found that the heat dissipated in the resistor caused drifts in the
servo electronics. Instead, the sense resistor should ideally be placed on a heatsink near the probe.
Similarly, the power supply for the Hall probe, which had previously been connected through theservos, was rerouted through a separate box to avoid thermal drifts.
Nearly all of our experiments use pulses of radio and microwave radiation to manipulate or
detect the atoms’ states. These signals can interfere with sensitive laboratory equipment, whichpick up the radiation and rectify it into DC drifts. RF rectification is a particularly tricky problem
for magnetic-field servos, where a small DC drift in Vsense or Vctrl changes the magnetic field at theatoms. This can noticeably alter the atoms’ interactions with each other or entirely disrupt theradiation’s interaction with the atoms by shifting the Zeeman resonance. The new control system
attacks RF rectification in four ways. First, each servo is constructed in its own well-shielded

28
aluminum NIM module. We sandblast the anodization on the backside of the front panel to ensure
a good electrical connection to the rest of the module as well as to the grounded rack. The NIM
rack itself is mounted away from RF amplifiers and synthesizers. Second, we use a dedicated power
supply for the servos and Hall probes to minimize the paths by which RF can enter the system.
Third, signal paths outside of the servo are transmitted by twisted pairs of wires with groundedshielding (Figure 3.6). The one exception is the control voltage signal, which is transmitted over a
coaxial BNC cable. All cables are kept as short as possible. Fourth, RF choke filters are installed
wherever any signal or power supply connects to the servo. The filters, which are similar to Π
filters, are screwed directly into the grounded case of the servo (complete diagrams are included in
the servo’s schematic in the JILA electronics shop, under reference QQ116A3). These strategies
appear to be successful. We measure RF rectification from a relatively high power square pulse on
the 40 K |9/2, −9/2i → |9/2, −7/2i transition near 80 MHz (Rabi frequency Ω = 2π × 15 kHz), andfind that the radiation shifts the measured magnetic field by less than 10ppm as measured by theatoms.

3.3

Power dissipation and water cooling
Under magnetic trap operating conditions, the high-current power supply outputs 400 A at

8V. With this much power dissipated in the magnetic trap system and servo electronics, heating is a serious concern. Runaway heating can occur because copper’s resistance increases withtemperature. The new electromagnet system is designed for ‘walk-away safe’ steady-state power
dissipation: under normal conditions the system can be turned on for an arbitrary amount of time,and if an anomalous temperature or cooling water flow rate is detected then the system turns itself
off. This gives experiments maximum flexibility in terms of duty cycle, plus the magnet system
is protected from programming errors, cooling failures, or other accidents. A combination of coil
design, water cooling, low-resistance cabling, choice of control electronics, and safety interlocks areused to accomplish this.
Power dissipation in the magnetic coils themselves is particularly important because they

29

Bias coil A(16 turns)

Bias coil B(16 turns)

Building-widechilled water
systemPinch coil A(16 turns)

Ioffe coil A(2 turns)

Flow ratemonitor

Cablingheatsink

Pinch coil B(16 turns)

Ioffe coil B(2 turns)

IGBT cooling
plate

SafetyinterlockFigure 3.7: Schematic of the electromagnet water-cooling circuit. Water is split into four parallelpaths to improve the efficiency of cooling. The heated water from the coils is recycled through awater cooling plate and used to cool the IGBTs. A Proteus 800-series flow rate monitor is connected
to a safety interlock so that the high-current power supply is turned off in the event of a watercooling failure. All water connections through non-conducting paths are made with flexible poly-flo
tubing and push-to-connect plumbing fittings.

30
are the most time-consuming parts to replace if something goes wrong. Each coil is wound fromcopper tubing with a square cross section that is 1/8 inch per side [36]. Water flows through thecenter of the tubing in a round channel to carry away heat. Previous designs of similar magnet
coils at JILA involved 3/16 inch tubing, however the smaller 1/8 inch cross section was chosen to
make the coils as small and close to the atoms as possible. This choice has paid off in terms fieldstrength for a given operating current. For example, to reach the Fano-Feshbach resonance with
the new Bias coils requires less than half of the current that was needed with the previous larger‘Helmholtz’ coils. To simultaneously connect electricity and water, leads at the end of each coil aresoldered to 1/4” round copper tubing, which passes through a copper block to which high-current
wires are bolted. This configuration has the nice side effect of water-cooling the ends of the highcurrent cabling as well.
Process cooling water flows through the coils in four parallel paths as shown in Figure 3.7.
This scheme ensures that cold water is delivered to each coil. At room temperature each Bias coilhas a measured electrical resistance of 9.3 mΩ, whereas each Pinch coil has 6.1 mΩ and each Ioffe
coil only has 1.5 mΩ. We measure these resistances between points near the junctions between
square and round copper tubing at the ends of the coils. Since each coil is wound from the sametubing, and each water cooling path has a similar electrical resistance and electrical current, theparallel water paths also have similar steady-state power dissipation and resistance to water flow.
However, the Bias coils have the highest resistance and operating temperature, so we use them tocharacterize the water cooling system.
Figure 3.8 shows the steady-state operating temperatures of the Bias coils for the rangeof electrical currents that can be produced by the Agilent 6690A power supply. We take thesemeasurements under typical laboratory conditions, where the process cooling water is suppliedwith 65 psi of pressure and a temperature of 13 C. We measure the temperatures at the hottest
points on the coils, which are at the ends of the copper tubing where water flows out of each coil.
The electromagnet system has two modes of operation. In trap mode, it draws a total of 400 A,
where 200 A flows through each of the parallel paths shown in Figure 3.4. When the Bias coils

31
are turned on alone to manipulate optically trapped atoms near the interspecies Fano-Feshbachresonance, they draw 160 A. Under both conditions, we can see that the coils’ temperatures heatto near room temperature.
In addition to the coils, about 2 kW of power is also dissipated in the servo IGBTs andhigh-current wiring in the steady-state magnetic trap. We mount the control IGBTs together ona single water-cooling plate. We were unable to destroy any transistors in testing steady-state
dissipation of up to 1 kW per (water-cooled) transistor, and neither of the IGBTs has failed after
a year of regular use in the experiment. At this point, it seems safe to conclude that the poweris adequately dissipated at the IGBTs. Somewhat unsurprisingly, the hottest parts of the highcurrent systems are those without any water cooling, namely the leads. Current is delivered fromthe power supply to the coils by flexible braided copper 4/0 welding cables and copper bus bars.
The wires themselves have a resistance of 49 µΩ/ft at room temperature. As measured on our
installed cabling at room temperature, a crimp joint that attaches a copper lug to the end of a
4/0 welding cable has an average resistance of 50 µΩ. The Y-connections where the power supply
cables are split into the two current paths (see Figure 3.4) contain three such crimps in close
proximity, plus additional resistance where the lugs are bolted together. This concentration ofhigher resistances leads to localized heating which, in turn, increases the resistance and therefore
leads to more heating. Although we did not observe any effect on the stability of the current, thesteady-state temperatures of these joints make them painful to the touch. Therefore, for safety,
we heat sunk the junctions to small cooling blocks connected to the exhaust of the process coolingwater (Figure 3.7).
Due to the high currents flowing through the system, a water cooling failure can quickly
damage the coils and IGBTs or even start a fire. We implemented a safety interlock to turn the
power supply off in the event of such a problem. The most common cooling failures are buildingwide pressure drops in the chilled water supply. A Proteus 800 series flow meter trips the interlock
when the water flow rate falls below a safe value. Other types of failures, like clogs in certain tubes,
programming errors, or loosened connections, don’t effect the overall flow rate but have caused

32

T e m p e ra tu re (C )3 5

2 5 0 A3 0
2 5

2 0 0 A

2 0

1 5 0 A

1 5
1 0 A

1 0
0

1 0 0

1 0 0 A
5 0 A
2 0 A

2 0 0

3 0 0

4 0 0

S te a d y s ta te T in c r e a s e ( C )H o ld T im e ( s )8 0

P r e d ic tio n

7 0
6 0
 

5 0

= 0 .0 0 0 3 6 (1 )C /A 2 x I2

4 0
3 0
2 0
1 0

 

0
0

5 0

1 0 0

1 5 0

2 0 0

= 0 .0 0 0 3 3 (3 ) C /A 2 x I2

2 5 0

3 0 0

3 5 0

4 0 0

4 5 0

5 0 0

I (A )

Figure 3.8: Top: Temperatures of the Bias coils vs. time for a variety of currents flowing throughthe Bias coil pair. A steady-state temperature is achieved after about 45 seconds, which is similar
to duration of the magnetic trap stage of a typical experiment. Open symbols show one coil’s
temperature, closed symbols show the other. The temperature difference between the two coils
could be caused by a small kink or burr that impedes the flow of cooling water, or it could simply
be a calibration difference between the two sensors. Bottom: Steady-state Bias coil temperaturevs. current follows the quadratic dependence predicted in Ty Cumby’s thesis [36].

33
catastrophic overheating in previous systems within our group. A series of Sensata/Airpax 67L050
airfoil temperature sensors are attached to each coil and the high-current leads near each transistor.
If any measured temperature rises above 50 C, then the safety interlock is triggered. We chose thiscutoff because it is significantly higher than the normal operating temperatures – it is not clearthat any damage would actually occur to the system at 50 degrees.

Chapter 4

Procedure: from room temperature to ultra cold

Our quantum gas machine has been in operation for more than a decade since it was originallyconstructed by Jon Goldwin [38]. The design incorporates popular techniques from experimentswith Bose-Einstein condensates, including a vapor-cell Magneto-Optical Trap [39, 3, 40], sympathetic cooling in a magnetic trap [41, 42], and a far-detuned optical trap [43] that sets the stage
for our measurements. The apparatus is constantly being updated, with changes catalogued in
graduate student theses [44, 36]. For my part, updates to the apparatus include the magnet systemfrom the previous Chapter as well as improved laser systems, microwave delivery, optical trapping,and imaging. For completeness and to define experimental parameters relevant to our data, I will
describe these updates while reviewing our typical sequence for preparing and measuring a quantumgas.
To that end, Section 4.1 introduces the main characters: the Boson 87 Rb and the Fermion 40 K.
These atoms are first collected from a dilute room-temperature vapor inside of a vacuum chamber
and then cooled to a temperature of 1 microKelvin by the four-stage process that is described in
Section 4.2. This ultracold gas is subsequently transferred into the focus of an intense infraredlaser beam in Section 4.3. Within this optical trap, we further cool the atoms to a few hundred
nanoKelvin. Experiments are performed on the atoms at a magnetic field near the interspecies
Fano-Feshbach resonance at 547 G. Finally, the optical trap is snapped off and the atoms drift
outwards into the vacuum. Resonant light illuminates the gas, which casts a shadow onto a CCD
camera, which transfers the data to a computer for analysis (Section 4.4). Because the measurement

35
process scatters atoms to the outer reaches of the vacuum chamber, we create a new sample forevery data point.

4.1

40

K and 87 Rb atoms

40 K and 87 Rb are an all-star team in the world of quantum gases. 87 Rb atoms formed the first

Bose-Einstein condensate [4] and 40 K was the first atomic degenerate Fermi gas [7]. In the ultra cold
regime, the interactions between different 40 K atoms or between 40 K and 87 Rb atoms can be tuned
by magnetic Fano-Feshbach resonances, which yields a rich phase diagram that includes interactiondriven BEC collapse and phase separation [45, 46]. More recently, 40 K-87 Rb interactions in opticallattices were used to shift the superfluid to Mott insulator quantum phase transition temperaturein 87 Rb atoms [47]. 40 K-87 Rb Feshbach molecules were the also first ultra cold Fermionic molecules
to be created in the lab [48]. This development led to the association of ultra cold ground-statepolar 40 K-87 Rb molecules [20], which promise to bring controlled quantum chemistry and stablelong-ranged interactions to the world of quantum gases.
Both 40 K and 87 Rb are Alkali metals, which means that they each have a single unpaired
valence electron with an energy level structure similar to an oversized Hydrogen atom. Figure 4.1
outlines the states relevant to our work. The D2 lines for these atoms are located at 767 nm
and 780 nm respectively. We use laser light that is resonant with these transitions to manipulate
and measure each species. Figure 4.2 shows schematics of these laser systems for 40 K and 87 Rb.
Beam shaping optics, wave plates, polarizers, optical isolators, shutters, and mirrors are omitted
for clarity. The 40 K lasers are locked to the F = 2 → F 0 line in the 39 K saturated absorption
spectrum, and the 87 Rb lasers are locked to the F = 1 to F 0 = 1, 2 crossover peak in the 87 Rb
saturated absorption spectrum. Although the laser systems are drawn separately for each species,
we combine the laser beams into a smaller number of paths before they hit the atoms: the K ‘MOT’
beam and Rb ‘MOT trap’ beam are coupled into the same optical fiber before they are used to trap
and cool the atoms; the ‘OP Zeeman’ and ‘OP Repump’ beams for both species are combined intoa single path used for optical pumping after the MOT stage; and the ‘Low B’ and ‘High B’ imaging

36

f’=3 (194 MHz)f’=5/2 (+55.2 MHz)f’=7/2 (+31.0MHz)f’=9/2 (-2.3 MHz)f’=11/2 (-46.4 MHz)

4P
3/2

5P
3/2

f’=2 (-73 MHz)

RepumpTrap, probe, ZP

780nm

RepumpTrap, probe, ZP

767nm

f’=1 (-230 MHz)f’=0 (-302 MHz)

f=2 (+2563 MHz)4S

f=7/2 (+714.3 MHz)

1/2

5S

1/2

f=9/2 (-571.5 MHz)

f=1 (-4272 MHz)

40K

87Rb

Figure 4.1: The D2 lines for 40 K (left) and 87 Rb (right). Thick vertical red arrows indicate theprimary transitions that we use for trap, cool, optically pump, and image the atoms. The thin
vertical red arrows show the repump transitions, and dashed lines show spontaneous decay paths
to the other groundstate. Hyperfine splittings for both species are indicated at zero external
magnetic field and are not to scale [49, 50]. 40 K has nuclear spin I = 4 and 87 Rb has I = 3/2.

37
beams for both species are also combined into a single imaging beam path. Before the beams are
combined, we use shutters and Acousto-Optic (AO) modulators to independently switch different
beams on and off.
Recent years have seen a few improvements to our lasers. The saturated absorption locks forboth atoms’ repump lasers now generate an error signal by modulating the laser phase at 10 MHz
with electro-optic modulators, which provides more bandwidth and stability than the previous
method of modulating the laser current at 300 kHz. We also replaced the 87 Rb repump laser,which was previously a commercial ECDL, with a DBR from Photodigm. The DBR puts out
enough power that the Rb repump light can now be sent directly to the atoms in the MOT, whichbypasses the tapered amplifier and shortens the optical path. New beam-shaping and polarization
optics have also been employed to improve the imaging beams, which are described in more detailin Section 4.4.
As for the atoms themselves, the entire preparation and measurement of the quantum gas
occurs in an ultra-high vacuum environment. The vacuum system contains two glass chambers, the(relatively) high-pressure collection cell and the lower-pressure science cell (see Figure 4.3). These
two cells are separated by a long tube to maintain differential pressure between the two chambers.
Each experiment begins with a dilute vapor of K and Rb atoms in the collection cell. Rb atoms
are dispensed by a commercial getter source from SAES. K atoms come from a similar home-built
dispenser [51]. Since the natural abundance of 40 K is only 0.012 percent, our K source uses enrichedmaterial that contains 5 percent 40 K [44]. Whereas the atom dispensers were originally pulsed every
morning to fill the collection cell with alkali atoms, we now run them at a lower constant current
to get more consistent day-to-day atom numbers.

4.2

Initial cooling stages
To transform the warm vapor into an ultra cold quantum gas, we subject the 40 K and 87 Rb

to multiple stages of cooling. Although the apparatus can produce a quantum-degenerate mixtureof a Bose-Einstein Condensate and a Degenerate Fermi Gas, the experiments in this thesis cut the

38
AO

AO

K trapECDL
BeatnotelockPD

K vapor cell

EO

PD

FP cavity

PD

TA
OP Zeeman

10 MHzLow B imaging

AO
Saturatedabsorption
lock

MOT

AO

K repump
ECDL

High B imaging

OP Repump

AO

Saturatedabsorption
lock

PD

AO

Rb vapor cell

AO
Rb trap
ECDL
FP cavity

PD

AO
Rb vapor cell

EO

PD

TA
10 MHzSaturatedabsorption
lock

Rb trapslave

Low B imaging

High B imaging
OP Zeeman

MOT Trap
MOT Repump

Rb repump
DBR
AO

OP Repump

Figure 4.2: Top: Our 40 K laser system. Laser light is generated by a pair of Extended Cavity
Diode Lasers (ECDLs) and amplified by a Tapered Amplifier (TA), the spectral output of which
is monitored on a scanning Fabry-Perot (FP) cavity. To repump laser phase is dithered at 10
MHz by an Electro-Optic (EO) modulator before the beam passes through a K vapor cell. The
transmitted light is measured on a photodiode (PD) and then demodulated to generate an error
signal. Bottom: Our 87 Rb laser system is similar to the 40 K system except that the 87 Rb repump
light is generated by a Distributed Bragg Reflector (DBR) laser diode, and then it is transmitted
directly to the atoms without being amplified by a TA. Another difference is that the 87 Rb trap
laser is locked to the Rb vapor cell by modulating the laser current directly at 300 kHz rather thanusing an EO to modulate the laser phase.

39

Ion
pump
20 L/sIon
Pump
40 L/sCollection cell

Science cell
Magnetic Transport

Figure 4.3: A cartoon of the vacuum system. A dilute vapor of K and Rb is dispensed into thecollection cell (left) where the atomic sample is formed at the beginning of an experiment. The
sample is then transferred through a narrow differential pumping tube into the science cell whereexperiments take place (right).

40
evaporative cooling short just before the onset of quantum degeneracy. This maximizes the spatial
and momentum overlap between the 40 K and 87 Rb clouds and simplifies our analysis of data on theinteractions between the two species. A huge variety of experiments are possible once the quantumgas mixture is prepared. The preparation and cooling of the gas is a repeatable sequence that ismore or less the same for every data point.
The 40 K and 87 Rb atoms are initially captured by a three dimensional dual-species vaporcell MOT in the vacuum chamber’s collection cell. Next, we spatially compress the 87 Rb MOT
by reducing the repump power and detuning the trap light to reduce the photon scattering rate.
At the same time, we jump the 40 K trap light towards resonance for improved Doppler cooling.
Next, we zero the magnetic field using several shim coils and the 87 Rb atoms are further cooled bypolarization gradient cooling. A quantization field is then applied along with a pulse of circularly
polarized optical pumping light (the ‘OP Zeeman’ and ‘OP Repump’ beams in Figure 4.2) to bring
40 K and 87 Rb into the low-field-seeking |f, m i = |9/2, 9/2i and |2, 2i states respectively.
f

At thispoint, we load the atoms into a quadrupole magnetic trap (Qtrap), which is generated by a pair ofhigh-current electromagnet coils. The Qtrap coils are mechanically translated 81 cm to the otherend of the vacuum chamber to deliver the atoms to the science cell. At this point, we typically havea few million 40 K atoms and a few times 108 87 Rb atoms at a temperature near 400 microKelvin.
Once the atoms are in the science cell, we briefly reduce the quadrupole field so that any
residual 87 Rb atoms in the |2, 1i state fall out of the trap. Next, we load the purified gas into the
Ioffe-Pritchard (IP) magnetic trap that was described in Chapter 3, and the Qtrap coils returnto their home at the collection cell on the other end of the chamber. This trap is approximately
harmonic in all three directions, with trap frequencies ωx = ωy = 125 Hz and ωz = 18 Hz. The
atoms are trapped at the minimum of the magnetic field. In the IP trap, we cool the gas byforced evaporation on the 87 Rb atoms, which sympathetically cools the 40 K atoms through elastic
collisions between the two atom species [38, 42]. We apply an RF field, which is resonant with|f, mf i = |2, 2i → |2, 1i and |2, 1i → |2, 0i transitions at the edge of the atom cloud, to remove the“hot” 87 Rb atoms. The |2, 0i atoms have no magnetic moment and so they fall out of the magnetic

41
trap. The atom cloud shrinks as it cools and we sweep the RF frequency downwards to remove
atoms closer and closer to the center of the magnetic potential. After evaporation in the magnetic
trap we typically have about one million 40 K atoms and two million 87 Rb atoms at a temperaturenear 1 microKelvin.

4.3

The optically trapped mixture
After an ultra cold mixture is produced in the magnetic IP trap, we transfer the atoms

into a far-detuned Optical Trap (OT) formed by light from a 10 W multimode fiber laser with awavelength of 1090 nm. This wavelength is red-detuned from the strong D1 and D2 lines, so thatthe 40 K and 87 Rb atoms both feel an attractive potential that is proportional to the intensity of thelight. The OT potential is independent of hyperfine or Zeeman states, which opens the possibility ofexperiments with different spin mixtures. This trap also enables access to tunable interactions near
Fano-Feshbach resonances at high magnetic fields, which are the crucial to the work in Chapters 5,
6, 7, and 8.
Following Grimm et al. [43], the potential U for OT light with intensity I is given by
3πc2 Γ
U =−
2ω03



1
1
+
ω0 − ω ω0 + ω



I

(4.1)

where Γ is the transition line width, c is the speed of light, ω0 is the resonant frequency, and
2πcω = 1090
nm is the frequency of the light. In the limit of large detuning, the D1 and D2 lines

in alkali atoms can be combined into a single effective ‘D’ line whose resonant frequency is theaverage of the frequencies for the D1 and D2 lines. Using parameters from [49], the 87 Rb D
2πc
2πc
40 K transition has ω =transition has ω0 = 787.6
0
nm and Γ = 2π × 5.9 MHz. Similarly, the768.4 nmand Γ = 2π × 6.0 MHz [50]. Plugging these numbers into Eq. 4.1, we can see that the ratio of the
optical potentials is UK /URb = 0.88. In practice, gravity pulls the atoms down away from the OT
center. The heavier 87 Rb atoms sag further and so they sit in a slightly different part of the beam
intensity profile.
Figure 4.4 shows the geometry for an experiment in the optical trap. The horizontal OTFastB coil

FastB coilg

Qtrap / transport coil

IP Bias coil

Horizontal OT

Science cell

IP Bias coil

Qtrap / transport coilTransfer tube tocollection cellb.

Vertical OTgFigure 4.4: The arrangement of the optical traps, glass science cell, and magnetic bias coils used for an experiment at high magnetic fields(not to scale). The arrows on magnetic field coils show the direction of electric current, and the arrows on laser beams show the directionof propagation. Subfigure (a) shows the top view and subfigure (b) shows a side view. The vertical OT beam shown in (b) is only turned
on for some experiments. The Qtrap magnetic transfer coils (dashed lines) are only present during the transfer into the IP trap. Duringexperiments at high fields, the transfer coils rest near the collection cell at the other end of the vacuum chamber.

a.

42

43
beam has a 1/e2 radius of 19 µm. Alone, it forms an elongated trap with an aspect ratio of 100
that is used in the majority of experiments. The vertical OT beam has a 1/e2 radius of 200 µm.
The trap formed by the combination of horizontal and vertical beams has an aspect ratio of about
20, which is useful if a spatially compressed cloud is desired. The Bias magnetic field is oriented
along the axis of the horizontal optical trap beam. This field is actually the sum of two Helmholtz
pairs: the strong Bias coils (which are also part of the IP trap) and the lower-inductance FastB
coils.
The 40 K and 87 Rb atoms are initially loaded into the optical trap in the low-field-seeking|9/2, 9/2i and |2, 2i states. To prepare the atoms for experiments near the interspecies FanoFeshbach resonance, we transfer the to the high-field-seeking |9/2, −9/2i and |1, 1i states. During
these transfers, the Bias coils provide a quantization axis with a 28 G magnetic field. For the
40 K atoms, an RF pulse with a linear chirp from 9.7 MHz to 8.0 MHz drives an Adiabatic Rapid

Passage (ARP) across the mf = 9/2 → 7/2 → 5/2 → 3/2 → 1/2 → −1/2 → −3/2 → −5/2 →−7/2 → −9/2 states [52]. The 40 K atoms are transferred into the |9/2, −9/2i state with about 90
percent efficiency. For the 87 Rb atoms, a microwave pulse with a linear chirp between 6.893 GHz
and 6.896 GHz drives an ARP from |2, 2i to |1, 1i with about 98 percent efficiency.
With the installation of the new electromagnet system (Chapter 3), the radio and microwave
antennae which were attached to the old magnet system needed to be replaced. Of these, the 87 Rb
microwave antenna was the most challenging since it needed to have dual resonant frequencies at
6.9 GHz and 8.0 GHz. These frequencies of interest are resonant with the |2, 2i → |1, 1i transition
at magnetic fields of 28 G and 545 G respectively. Figure 4.5 shows the antenna at an early stage of
construction and its final measured reflection coefficient. The antenna, which was designed by Carl
Sauer in the electronics shop, consists of copper traces on both sides of a thin phenolic substrate. It
is mounted between the magnetic bias coils near the science cell, about 6 mm above the position of
the atoms. The emitted microwave radiation is centered on the circular aperture, which is aligned
above the atoms. Two small pieces of copper tape (not shown) are soldered to the long straightsection of the lead on the top side. We adjusted their positions and orientations to stub tune the44resonant frequencies.
Once the atoms are prepared in their high-field seeking states in the optical trap, the Bias
coils ramp the field up to 550 G. The power in the horizontal OT beam is gradually lowered sothat the most energetic atoms fall out and the sample is evaporatively cooled. Next, we ramp the
OT power back up by a factor of two or so to ensure that the atoms are confined in the harmonic
region at the center of the trap. At this point, the quantum gas is fully prepared for the sortsof experiments that are described in the next four Chapters. Most of these take the form of DC
magnetic field sequences to manipulate atomic interactions, radio / microwave pulses to control
and measure the atoms’ states, or deformations of the optical potential, which help us to learn
about the scattering properties of the atoms under different conditions.

4.4

Imaging
All of the data in this thesis comes from pictures of quantum gas mixtures. From these images

we extract the number (N ), temperature (T ), and density (n) of each atom species selectively. We
track the evolution of these quantities to extract the scattering rates in Chapter 8. In the next
Chapter, we apply these techniques to image 40 K87 Rb Feshbach molecules. They are also used todiagnose and optimize the preparation or manipulation of the atoms. For any of these applications,
we initiate the imaging sequence by quickly snapping the optical trap off. After the clouds have
expanded into the chamber for a few milliseconds, we perform destructive absorption imaging withpulses of resonant light [53]. This method measures the optical depth of a shadow cast by theatoms, which is imaged using the optics depicted in Figure 4.6.
The resulting pictures reveal the 2D column density profile n(x, y) in the strong radial ydirection and weak axial x direction of the optical trap. We fit these to a Gaussian surface:22− x2− y2Nn(x, y) =e 2σx 2σy .
2πσx σy(4.2)

Based on the cylindrical symmetry of the trap, we assume that the shape of the density distribution
along the imaging axis is the same as it is in the measured radial y direction. Formula 4.2 is valid

45

Top / signal

In situ

Bottom / ground

1 cm

Freespace

Figure 4.5: A dual-resonant antenna for 87 Rb. Left: the magnitude of the power reflection coefficient S11 as measured by a network analyzer. A lower reflection coefficient presumably implies
more efficient transmission of radiation to the atoms, since there are no dissipative elements inthe system. Right: Photographs of the top and bottom sides of antennae before stub tuning. An
edge-mounted SMA jack is soldered to the lower end of the antenna (as oriented in the picture).
The center SMA pin is connected to the copper trace on the top side and the ground pins are
soldered to the ground plane on the bottom side.

46
for a harmonically confined thermal gas in the Maxwell-Boltzmann limit. For an ideal gas in
kB T
kB T
2
2
this limit, the in-trap RMS cloud sizes are given by σx,0= mω2 and σy,0 = mω 2 where kB isxy

the Boltzmann constant and the trap frequencies are defined such that the trap potential has theform U = 12 mωx2 x2 + 12 mωy2 y 2 + 12 mωz2 z 2 . When the trap is switched off at time t = 0, a thermal
noninteracting gas will undergo ballistic expansion such that
σ(t) =r

σ02 +kB T 2
t .
m

(4.3)

in each direction. We use this formula to extract the temperature from the measured sizes of the
expanded clouds. In some of our data the 40 K gas is quantum degenerate, with T /TF as low as
0.2. For this degeneracy, a Gaussian fit to the density profile can measure the number of atoms
but not its temperature [54]. Since the Fermi gas is in thermal equilibrium with a 87 Rb Bose gasthat still follows a Maxwell-Boltzmann distribution, we use the 87 Rb cloud size as a measurementof temperature for both atom species in these situations. For T /TF > 0.6, we find that thetemperatures measured from the expanded 87 Rb and 40 K clouds agree with each other.
For all data points, we image the 87 Rb atoms with light tuned to the cycling f = 2 → f 0 = 3
transition, and we image the 40 K with light tuned to the cycling f = 9/2 → f 0 = 11/2 transition.
However, there are a few differences in the imaging procedure depending on the specific experiment.
For diagnostic images of early experiment stages, the magnetic-field quantization axis is provided
by an auxiliary coil that points along the imaging axis. In this case, the probe light is circularly
polarized. For imaging the atoms at high magnetic fields, the Bias field sets the quantization axis,
and is perpendicular to the imaging axis. The probe light in this case is linearly polarized and theatoms’ measured OD is consequently reduced by a factor of two. Furthermore, experiments at low
magnetic fields are insensitive to the Zeeman state whereas high field imaging selectively images
40 K atoms in the |9/2, −9/2i state and 87 Rb atoms in the |2, 2i state.

Finally, since high-field

experiments are performed on 87 Rb atoms in the |1, 1i state, a microwave ARP transfers 87 Rb
atoms into the |2, 2i state immediately before imaging.

47

f = 300 mm

1024-BR
CCD Camera

f = 100 mm

Laser
illumination

40K, false color

Figure 4.6: A cartoon of the low-magnification imaging system. Resonant laser light propagatesalong the magnetic transport axis of the vacuum system. The atoms’ shadow is imaged on a CCD
camera with a magnification of three. A sample OD image of 40 K atoms is shown with false color
indicating the optical depth.

Chapter 5

Efficient magnetoassociation of KRb Feshbach molecules

Magnetic Fano-Feshbach resonances can be used to convert an ultra cold gas of atoms into
an ultra cold gas of molecules. These weakly bound Feshbach molecules act as a starting point
for experiments from superfluidity to quantum chemistry, and so the physics of how these pairs
form has been an active topic of research [19, 18]. Previous work has mostly focused on bosonicFeshbach molecules formed out of two fermionic or two bosonic atoms, whereas less is known
about the association of fermionic Feshbach molecules from one Boson and one Fermion. This
Chapter presents our detailed measurements of fermionic 40 K87 Rb in the atomic gas mixture of
40 K and 87 Rb.

Section 5.1 describes how Feshbach molecules are created when a magnetic fieldsweeps through the Fano-Feshbach resonance. Section 5.2 reviews our tools for detecting andimaging the molecules. In Section 5.3, a quantitative model based on the Landau-Zener paradigm
accurately predicts the rates of molecule formation in the limit of small molecule numbers. Whenthe atomic sample is substantially depleted, molecule production saturates. Section 5.4 shows how
the maximum number of 40 K87 Rb molecules falls short of a theoretical prediction for experimentsthat begin with a degenerate Fermi gas of 40 K atoms.
The bulk of the material in this Chapter is published in Reference [55]. In particular, Sections
5.3 and 5.4 are published nearly verbatim in that work. Sections 5.1 and 5.2 also contain material
from the paper, but they are expanded to include more detail and background information.

49

5.1

How magnetoassociation works
One way to make Feshbach molecules in an atomic gas is to sweep the magnetic field acrossa Fano-Feshbach resonance. This process, called magnetoassociation, has proven to be one of themost robust methods of Feshbach molecule formation and is effective for both wide and narrow
Fano-Feshbach resonances as well as for gases of fermionic atoms, bosonic atoms, or mixtures of two
atomic species [19, 18]. Figure 5.1 illustrates the two-body energy spectrum for Feshbach molecules
in a gas of 40 K and 87 Rb atoms. In the limit where the atoms’ kinetic energy is much greaterthan the harmonic oscillator level spacing in the trap, there is a continuum of free-particle states
on both sides of the Fano-Feshbach resonance. The two-body Feshbach molecule state has zerobinding energy at the resonance, and increasing binding energy as the field is lowered. For theseexperiments, we use the broadest 40 K–87 Rb Fano-Feshbach resonance at B0 = 546.618 G.
To transfer free atoms into the Feshbach molecule state, we ramp the magnetic field downwards from 550 G through the resonance to 545.7 G, which is from right to left in Figure 5.1. The
ramp rate Ḃ is key to the magnetoassociation process as shown in Figure 5.2. Let f be defined as
the number of molecules divided by the lesser of the initial numbers of atoms of 40 K and 87 Rb, N< .
If Ḃ is fast compared to the timescales of two-body adiabaticity, then f is small. In this fast sweep
limit, molecule association is essentially the independent sum of many two-body pairing processes
between nearby atoms. Section 5.3 characterizes this regime. As the field sweeps get slower andmore adiabatic, f increases and saturates. Notice that f saturates well below unity, which is explored in Section 5.4. In order to extract information about the fast-sweep and saturated regimes
independently, we fit the molecule conversion efficiency as a function of the magnetic-sweep rate|Ḃ| to the following formula:f = f0 1 − e−Γ/(f0 |Ḃ|) .

(5.1)

Here, f0 is the saturated molecule conversion fraction and f = Γ/|Ḃ| in the fast-sweep limit. Before
moving on to the measurements of these quantities, I will first describe how we detect and image
the Feshbach molecules.

50Free atoms

𝐵
Feshbachmolecule

FeshbachresonanceE

Figure 5.1: Schematic two-body energy diagram for 40 K and 87 Rb as a function of magnetic fieldnear an interspecies Fano-Feshbach resonance. Zero on the vertical axis represents the thresholdfor a continuum of free two-body states. The Feshbach molecule state (dashed line) exists at fields
below the Fano-Feshbach resonance. In magnetoassociation, we sweep the field from right to left
on these axes.

51

0 .1 5f

0 .1 00 .0 50 .0 0

0 .0

0 .1

.

0 .2

1 /|B | (m s /G )Figure 5.2: The fraction of the minority atom species converted to molecules, f , vs the inverse
magnetic-sweep rate, 1/|Ḃ|, across the Fano-Feshbach resonance. The dashed line shows a fit to
Eq. (5.1), which gives Γ = 6.5(5) G/ms and f0 = 14(2)%, or 3.8(5) × 104 molecules, for this data.
The initial gas consisted of 2.6 × 105 40 K atoms and 3.4 × 105 87 Rb atoms at T = 490 nK. This data
was published in Reference [55].

52
𝐵 − 𝐵0𝐵 − 𝐵0𝐵

𝐵t

PrepareAssociate
Clean
Dissociate

Image

t

PrepareAssociate

Clean

Image

Figure 5.3: Schematic timing sequences for creating and imaging molecules. The colored bars along
the bottom label the various stages of the sequence. Left: Imaging dissociated molecules. Right:
Direct molecule imaging.

5.2

Counting the molecules
Much like atoms, 40 K87 Rb Feshbach molecules are probed by resonant absorption imaging

after a brief expansion from the optical dipole trap. Figure 5.3 describes two different methods tomake the molecules visible on the atomic transitions. In both cases, we prepare the atoms at afield of 549.9 G, which is above the 546.62 G interspecies Fano-Feshbach resonance. We sweep themagnetic field down through the resonance to 545.7 G to form molecules. A typical speed for thisadiabatic magnetoassociation is 3 G/ms. To clean out the unpaired 40 K atoms, we ramp the fieldin 50 µs down to 544.7 G and then transfer unpaired 40 K atoms from the |9/2, −9/2i state to the|9/2, −7/2i state using RF Adiabatic Rapid Passage (ARP). The binding energy of the molecules
here is h×3 MHz [33] and the ARP for atoms does not affect the molecules. The atoms become
invisible to probe light once they are in the |9/2, −7/2i state.
At this point, we have two options. The most straightforward of these is to ramp the field backthrough the Fano-Feshbach resonance to dissociate the molecules. Since all unpaired 40 K atoms
were rendered invisible by the ARP, the usual imaging of 40 K atoms only counts the atoms that werepreviously part of molecules. Unfortunately, images of dissociated molecules are potentially prone

53
to systematic errors from heating as the field is ramped back through the strong interactions at theresonance. An alternative strategy, which was employed for the data discussed in the remainder ofthis Chapter, is to image the molecules directly. By ramping the magnetic field back up to 546.0 G,
we reduce the binding energy of the molecules so that they become resonant with the 40 K imaging
transition. The molecule number measured with this technique is 85±5% of the number measured
after dissociating the molecules and imaging the resulting atoms. Therefore, when imaging themolecules below the Fano-Feshbach resonance, a multiplicative factor is applied to correct the
number for this measurement inefficiency.
To minimize the loss of molecules due to inelastic collisions, we can release the gas fromthe optical trap immediately after the molecules are formed. The sudden turn-off of the opticaltrap initiates an expansion of the gas that rapidly lowers the densities and switches off inelastic
collisions. Figure 5.4 shows the measured molecule number and kinetic energy as a function of
the time that the optical trap is turned off, trelease . Here, trelease = 0 is defined as the time whenthe magnetic field crosses the Fano-Feshbach resonance. For trelease > 0 we observe a heating anddecrease in the molecule number with increasing trelease . We attribute this to inelastic collisions
as the molecules spend more time in a relatively high density atom-molecule gas mixture in the
trap. For trelease < 0, the magnetic field crosses the Fano-Feshbach resonance after the gas starts
expanding. For short expansion times, where −0.3 < trelease < 0, the measured number and energyof the molecules is constant within the measurement uncertainty. For longer expansion times,
corresponding to trelease < −0.3, the rapidly dropping atom gas density reduces the efficiency ofmagnetoassociation according to the model in Section 5.3. Thus, we find that trelease = 0 gives themost accurate measurement of the number and temperature of molecules that are created. While
this Chapter focuses on the number of molecules created, the kinetic energy of the molecules willbe the topic of Section 6.3 in the next Chapter.
Another issue with quantifying the molecule formation process is that inelastic collisions cancause loss and heating of the atoms even before molecules are formed. In particular, three-bodyrecombination is an inelastic collision process for three atoms that produces a molecule plus an atom,

54

t (m s )4
3

B - B 0 (G )

2
1
0
-1
-2

E K R b / k B ( µK )

0 .8
0 .6
0 .4
0 .2
0 .0

(1 0 4)

6

K R b

4

N

2
0
-1 .0

-0 .5

0 .0

0 .5

1 .0

t r e le a s e ( m s )Figure 5.4: Top: A typical magnetic-field sweep sequence for molecule association. Here, t = 0
is defined as the time when B crosses the Fano-Feshbach resonance. Middle: measured molecule
kinetic energy, from the RMS cloud size after time-of-flight expansion between 2.5 ms and 3.5 ms,
vs the time, trelease , at which the optical trap is suddenly turned off to release the gas. The red
dotted line shows the initial atom temperature, T = 380 µK. Bottom: Measured molecule number
vs trelease . For comparison, the dashed line shows an exponential decay with a time constant of 2.7
ms. This data was published in Reference [55].

55both with excess kinetic energy. The rate for three-body recombination increases dramatically near
the Fano-Feshbach resonance, which I will explore in much more detail in Chapters 7 and 8. Fora Bose–Fermi mixture, the dominant three-body recombination process removes two bosons andone fermion from the trap. Characterizing the initial atom conditions by the peak 87 Rb density
squared times the peak 40 K density, we estimate the fractional atom loss to be below 10 percent
as long as this density product remains lower than 2×(1013 cm−3 )3 . To isolate our measurementsfrom inelastic loss, the initial atom conditions for data in Sections 5.3 and 5.4 are chosen to satisfy
this condition. However, it is worth noting that our largest molecule clouds are produced fromhigher atom densities even though inelastic loss complicates the formation process. For example,
7 × 104 40 K87 Rb molecules were measured in a mixture whose density product n2Rb nK was initially
3 × (1013 cm−3 )3 . The atom gas in this case was prepared at 560 nK, with 6.5 × 105 40 K atoms at
T /TF = 0.51 and 2.1 × 105 87 Rb atoms at T /Tc = 2.0.

5.3

Dependence on the magnetic-field sweep rate
This section focuses on the fast-sweep regime of molecule creation, where a simple extension ofa two-body picture predicts that the molecule number increases linearly with the sweep duration [56,19]. Here, the molecule number follows a Landau–Zener-like behavior [57, 19] with the transition
probability into the molecular state given by P = 1 − e−2πδLZ ≈ 2πδLZ . For two particles in abox with volume V, the Landau–Zener parameter δLZ depends on the Fano-Feshbach resonanceparameters and is given by [56]δLZ =1 2π~ abg ∆
V µ
Ḃ

(5.2)

where ~ = h/2π and µ is the two-body reduced mass. Interestingly, this result applies for thermal
atoms (fermions or bosons) as well as for atoms in the motional ground-state, such as atoms in a
Bose–Einstein condensate or optical lattice [56]. Using classical probability theory, Eq. (5.2) canbe generalized to multiple particles simply by multiplying P by the total number of available atom
pairs to get the number of molecules [56]. Using a local density approximation, the molecule density

56
at position r is then given by
nmol (r) = n1 (r)n2 (r)2π

2π~ abg ∆
.
µ
Ḃ

(5.3)

Integrating Eq. (5.3) over the trapped atom gas distribution, and dividing by N< , givesf ≈ hn> i2π
where hn> i = N1<R

2π~ abg ∆
Γ
=
µ
|Ḃ|
Ḃ

(5.4)

n1 (r)n2 (r)d3 r describes the density overlap of the two atom clouds.

Plugging the relevant Fano-Feshbach resonance parameters into Eq. (5.3) yields
Γ/hn> i = (2.8 ± 0.1) × 10−12 cm3 G/ms.

(5.5)

This prediction for Γ, which is shown as a solid line in Fig. 5.5, agrees well with our measurements ofΓ as a function of the density overlap of the initial atom gas, hn> i. From this it seems reasonable toconclude that magnetoassociation in the perturbative regime of fast magnetic-field sweeps behaves
as expected in our Bose–Fermi mixture. Systematic errors in the measured atom number or molecule
number would also change the measured value of Γ. The agreement between data and modelin Figure 5.5 suggests that our imaging process accurately counts the number of molecules. To
accomplish this agreement, it is crucial for the model to include a maximum molecule fraction
f0 which is below unity. In the next section we turn our attention to this saturation, which isimportant for experiments where one would like to create larger numbers of fermionic Feshbach
molecules.

5.4

Saturated molecule number
Feshbach molecule conversion efficiencies as high as 90 percent have been observed in homonu-

clear Fermi gases [58]. In these experiments, it was found that the adiabatic molecule fraction f0
is determined by the phase-space density in the parent gas of atoms [58]. Monte-Carlo simulations
that select atom pairs within a phase-space distance criterion γ accurately predict f0 over a broad
range of initial gas conditions. The phenomenological criterion γ was determined by a fit to the

57

Γ(G

/ m s )

1 0

5

0

0

1

2

3
〈n > 〉 ( 1 0 1 2 c m

4
-3

5

)

Figure 5.5: The initial molecule conversion efficiency divided by the magnetic-field sweep rate, Γ, vsthe density overlap of the two atom clouds.The prediction for Γ using the 40 K–87 Rb Fano-Feshbachresonance parameters (Eq. (5.5)) is shown as a solid line. Here, the atomic densities are varied
by associating molecules at different times during expansion from the optical trap (up to 1.2 ms),
as well as by adjusting the evaporation trajectory and optical trap frequencies. This data waspublished in Reference [55].

58
data. The same pairing criterion also accurately predicted the saturated molecule number in thermal and degenerate Bose gases [58] and for heteronuclear molecules in a Bose-Bose mixture of 87 Rb
and 85 Rb [59] as well as a Fermi-Fermi mixture of 6 Li and 40 K [60]. Intuitively, this Stochastic
Phase Space Sampling (SPSS) calculation predicts a sharp increase in the molecule fraction as
the atoms are brought into the quantum degenerate regime, approaching nearly unit conversion
efficiency at zero temperature.
In contrast, the efficiency of molecule creation has been quite low in Bose–Fermi mixtures like
40 K–87 Rb. The problem of pairing in a strongly interacting Bose–Fermi mixture has been a topic

of theoretical interest [10, 61, 62, 63]. On a more practical note, fermionic Feshbach molecules arealso an essential step in creating a gas of ultracold polar 40 K87 Rb molecules. With the long-range
anisotropic interactions of polar molecules, physicists could access novel quantum phases of matterand pave the way for many new experiments. However, the inefficiency of Feshbach molecule
production limits the initial phase space density of the polar molecular gas. As a consequence,no experiment has yet managed to create a quantum-degenerate Fermi gas of polar molecules inthe six years since Feshbach molecules were first adiabatically transferred into the rovibrational
ground state [20]. In addition to 40 K–87 Rb, fermionic Feshbach molecules have also been createdin 23 Na–40 K and 6 Li–23 Na mixtures [64, 65], and these molecules could conceivably be transferred
into their ground state as well. In all of these experiments, the fraction of minority atoms converted
to molecules was less than 30 percent.
With a phase-space density criterion for pairing, it is not hard to see why a Bose–Fermi
mixture has fewer eligible atom pairs for creating molecules. At zero temperature all of the bosons
condense into the ground state of the trap whereas the (spin-polarized) fermions must each occupy
different trap states as shown in Figure 5.6. At high temperatures both species are spread out
across a broad distribution of momenta and positions. In either limit, most of the fermions andbosons are too far apart in phase space to pair. It is the regime between these limits, where theFermi gas is quasi-degenerate but the Bose gas has a temperature above the BEC transition, thatthe highest molecule conversion efficiencies are predicted by the phase-space density pairing (SPSS)

59

BosonsFermions

Figure 5.6: Schematic distribution of atoms in an optical trap at low temperatures. The fundamental challenge of pairing bosons with fermions lies in their low-temperature behavior. The
bosons condense into the trap’s ground state whereas the fermions must each occupy different trap
states. Fermi-Fermi and Bose-Bose pairs, on the other hand, see improved phase-space overlap at
low temperatures and so more pairs are formed.

60
model. Although previous measurements of molecule creation in the 40 K–87 Rb mixture followed
the predictions of the SPSS calculation, these were limited to a regime of relatively high T /TF andlow number of the fermionic atoms. The measurements in this section pursue the initial conditions
for which the greater conversion efficiency was predicted.
Another challenge for heteronuclear Feshbach molecules is that differences in the masses,
polarizabilities, and/or magnetic moments of the initial atoms can cause the two atom clouds tohave different equilibrium positions in the trap. To measure the significance of this effect on themolecule conversion efficiency, we intentionally separated the 40 K and 87 Rb clouds along the weakaxial direction of the optical trap by applying a magnetic-field gradient before molecule formation.
The results of this measurement are shown in Fig. 5.7, where we plot the saturated molecule
conversion efficiency against the spatial offset of the two atom clouds. The conversion efficiency
drops when the spatial separation is significant compared to the root-mean-squared (RMS) size ofthe 87 Rb gas in the trap, σRb . For the data in this Chapter, the spatial offset of the atom clouds
in the axial direction is less than 1.2 σRb and the calculated offset in the vertical direction due todifferential gravitational sag is less than 0.3 σRb . These relative displacements of the atom clouds
were accounted for in all modeling of the pairing process.
The top panel of Figure 5.8 shows our measurements of the molecule creation efficiency f0
as a function of T /TF for the initial 40 K atom gas. Here, we varied the magnetic-field sweep rate,using the findings of Sec. 5.3 to ensure that the sweeps were sufficiently slow that the molecule
number was in the saturated regime. For this data, the number of 40 K atoms is larger than thenumber of 87 Rb atoms by a factor between 1.6 and 7. Rb radial (axial) trap frequencies range from360 to 550 Hz (4 to 6 Hz), and initial atom temperatures are between 250 nK and 770 nK. This
broad range of initial conditions is used to provide a stronger test for the SPSS calculation and also
to make sure that the 87 Rb gas is above the BEC transition temperature for all measurements evenwhen T /TF is low. The absence of a condensate improves phase-space overlap between the two
species and minimizes the inelastic loss processes that increase with 87 Rb density. As was observedfor homonuclear molecules, the conversion efficiency increases for higher initial atomic phase-space

61

0 .2 0f

0 .1 50 .1 00 .0 50 .0 00

1

2

3

C l o u d s e p a r a t i o n ( σR b )Figure 5.7: Molecule conversion efficiency, f , as a function of the separation between the 40 K and
87 Rb clouds in units of the width, σ , of the 87 Rb cloud, which is the smaller of the two atom
Rbclouds. The solid curve shows an empirical fit to a Gaussian with an RMS width of 1.8(1) σRb .
For this data, molecules are created from 2.8 × 105 87 Rb atoms and 2.1 × 105 40 K atoms at T =490 nK and T /Tc =1.6. This data was published in Reference [55].

62densities.
In the bottom panel of Figure 5.8, we compare these results against the Stochastic PhaseSpace Sampling (SPSS) calculation developed by Hodby et al. [58]. This semi-classical model isbased on the idea that two atoms are able to pair if they lie within a volume of relative phase-space
set by an adjustable parameter γ:µ |vrel | |rrel | <

h
γ
2

(5.6)

where vrel is the relative speed of the two atoms and rrel is their separation. Another property of
the calculation is that atoms are only allowed to pair once (i.e., paired atoms are removed fromfurther consideration). To determine the parameter γ in our system, we fit the molecule creation
efficiency data from Fig. 5.8 to the SPSS calculation. First, a Monte-Carlo calculation generates
atom distributions matching the initial conditions for each measurement. These atoms are thenrandomly paired according to the criterion in Eq. (5.6) to determine the molecule creation efficiency,
as described in [36]. A fit to the data with initial T /TF > 0.55 returns a value of γBF = 0.38(3).
The error bar is dominated by a 10 percent systematic uncertainty in the measured conversion
fraction. This measured γBF for 40 K–87 Rb is consistent with the values γF = 0.38(4) found in 40 K
and γB = 0.44(3) measured in 85 Rb [58]. Thus, Feshbach molecule formation appears to operate inmuch the same way for bosons, fermions, and Bose–Fermi mixtures as long as the parent atoms’
distributions are well overlapped in the case of the two-species mixture.
As T /TF is lowered, the bosonic 87 Rb gas continues to shrink while Pauli pressure forces the
40 K atoms into a relatively broad distribution of momenta and positions.

The SPSS simulation

fully takes into account the differences between Bose and Fermi-Dirac distributions in this regime.
Even so, the bottom panel of Fig. 5.8 shows that the SPSS calculation with γBF = 0.38 significantly
overestimates the conversion efficiency for atom clouds with low T /TF . No single value of the pairing
criterion γ fits the entire temperature range of our data. In contrast, previous measurements forhomonuclear Feshbach molecules in 40 K agreed with SPSS predictions for T /TF as low as 0.05 [58].
The failure of the SPSS model to describe our measurements in the quantum-degenerate regime

63

1 .0
0 .8

f

0 .6
0 .4
0 .2
0 .0
0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

0 .8

1 .0

In itia l T /T Ff - fS P S S

0 .0

-0 .2

-0 .4
0 .0

0 .2

0 .4

0 .6
In itia l T /T F

Figure 5.8: Top: Measured conversion efficiency, f , vs the initial T /TF of the 40 K atoms. The
atom number ratio ranges from 1.6 ≤ NK /NRb ≤ 2 (squares), 2 ≤ NK /NRb ≤ 4 (circles), or
4 ≤ NK /NRb ≤ 7 (triangles). Molecule conversion efficiency increases for higher atomic phasespace density (lower T /TF ). Bottom: The difference between the SPSS calculation and the data,f − fSPSS , vs the initial T /TF of the 40 K atoms. This data was published in Reference [55].

64suggests that this model is inadequate for describing the production of fermionic molecules fromquantum-degenerate atom gas mixtures.
The phenomenological SPSS model is popular because of its intuitive simplicity and (untilnow) successful track record. Nevertheless, the pairing criterion γ is not derived but instead is
extracted from experiments. This has motivated the development of other, possibly more rigorous, theoretical models. One theory, based on the assumption that chemical equilibrium is maintained through the entire magnetic field ramp, has successfully duplicated the SPSS results with nofree parameters for Bose-Bose and Fermi-Fermi pairing in both thermal and quantum-degenerategases [66]. This model was later extended to include heteronuclear molecules with all combinations of quantum statistics [67]. A more recent theoretical paper compared various incarnationsof the SPSS model against this chemical equilibrium theory for all possible combinations of quantum statistics, revealing that the models differ for fermionic molecules in a way that depends on
the ratio of the number of atoms from each species [68]. Since the chemical equilibrium models
automatically include the quantum statistics of the molecules, it would be interesting to compare
these theories against the data of Figure 5.8. Another possibility is that heteronuclear molecule
formation involves higher partial waves due to the mismatch in the trap energy levels between the
two atom species [69]. A different theoretical approach includes few-body interactions and theirinfluence on the Bose–Fermi pairing process [70], predicting a suppression of pairing that could be
related to our observations. Chapters 7 and 8 of this thesis will explore few-body interactions in
the 40 K–87 Rb mixture specifically.

Chapter 6More KRb molecule results

This Chapter will dive deeper into the world of fermionic Feshbach molecules. Magnetoassociation, which was studied in the previous Chapter, is only one of several ways to generate molecules.
Section 6.1 highlights the competing timescales of inelastic collisions that can both produce anddestroy Feshbach molecules in a gas of atoms. Just by holding at a magnetic field below theFano-Feshbach resonance, in some cases we can create as many molecules in the gas as we do bymagnetoassociation. In Section 6.2, we use radio frequency photons to directly transfer the atoms
into the Feshbach molecule state. Finally, Section 6.3 reports measurements of the temperatureof the molecules using time-of-flight expansion after the optical trap is switched off. These experiments reveal a surprising excess of molecular kinetic energy, which is independent of the methodby which the molecules are produced.

6.1

Molecules formed by three-body recombination
Inelastic collisions are notorious enemies of ultra cold atom experiments: they release heatand eject atoms from traps, which diminishes the effectiveness of evaporative cooling and can makeit essentially impossible for many strongly interacting gases to be studied in equilibrium. Thissection casts inelastic collisions in a slightly more positive light: at large interspecies scattering
lengths, inelastic collisions in the 40 K–87 Rb mixture actually produce trappable 40 K87 Rb Feshbach
molecules. Recall that three-body recombination results in a diatomic molecule, with kinetic energy
equal to the binding energy shared between the molecule and a third atom. When this binding

66
energy is sufficiently small, as is the case with Feshbach molecules at a high scattering length, thereaction products remain trapped and molecules accumulate in the gas.
Figure 6.1 shows a magnetic-field sequence designed to explore these dynamics by imaging
the molecules themselves. As with the data in the previous Chapter, we initially prepare a gas of
40 K and 87 Rb atoms in the |f, m i = |9/2, −9/2i and |1, 1i states.
f

The molecules are dissociatedand imaged above the Fano-Feshbach resonance, similar to the timing in the left part of Figure5.3. A small imaging background, typically about 104 atoms, is left over by an imperfect transfer
of unpaired K atoms into the invisible |9/2, −7/2i state during the ‘clean’ stage of the imaging
sequence. We measure the size of this effect by preparing a spin-polarized sample of 40 K atoms
only with no 87 Rb present. Since no Feshbach molecules can be made in this mixture, all 40 K
atoms that can be imaged after the cleaning sequence are attributed to the imaging background.
We subtract this background1 from our measurements of molecule number.
Figure 6.2 shows the molecule number vs thold measured at a few different scattering lengths
near the Fano-Feshbach resonance. For all of these scattering lengths, the Feshbach moleculebinding energy is less than 1.2 microKelvin. At short hold times near the resonance, a fast rate ofthree-body recombination leads to an accumulation of molecules. At longer times, the molecules
decay by vibrational relaxation. The single-beam optical trap for these experiments has a periodof 160 ms in the weak axial direction and 1.7 ms in the tight radial direction for the 87 Rb atoms.
There are initially 5 × 105 40 K atoms with an average density of 8 × 1012 /cm3 and 6 × 105 87 Rb
atoms with an average density of 1.4 × 1013 /cm3 . The two species are prepared at a temperatureof 700 nanoKelvin, which is 1.5 times the BEC transition temperature of the 87 Rb atoms and 0.6
times the Fermi temperature of the 40 K atoms.
Two types of three-body recombination events occur during these experiments. The first
type creates a molecule with large binding energy, and it therefore imparts a large kinetic energy
to the resulting molecule and atom. Two 87 Rb atoms and one 40 K atom are lost from the trap for
1

Since our method of preparing a 40 K gas without 87 Rb atoms results in a gas with as many as 50% more 40 K
atoms, we could be overestimating the imaging background.

67

5 5 1

M a g n e tic fie ld ( G )

5 5 0
5 4 9
5 4 8
5 4 7
5 4 6

t h o ld

5 4 5

0

1

2

3

t (m s )Figure 6.1: Timing sequence for measurements of Feshbach molecules formed by three-body recombination. The 40 K and 87 Rb atoms are prepared at a weakly interacting scattering length far
below the interspecies magnetic Fano-Feshbach resonance. We quickly ramp the field to a stronglyinteracting value where three-body recombination produces trapped Feshbach molecules. After avariable hold time, thold , we switch the optical trap off. At this point we also ramp the magnetic
field away from the Fano-Feshbach resonance so that a resonant RF pulse can drive the unpaired
40 K atoms into the invisible |9/2, −7/2i state. Finally, we ramp the magnetic field across the FanoFeshbach resonance, which dissociates the Feshbach molecules. The 40 K atoms that were previously
part of Feshbach molecules are imaged on the cycling |9/2, −9/2i → |11/2, −11/2i transition.

68

6 x 1 0 4

(a )

N

K R b

4 x 1 0 4
2 x 1 0 4
0
6 x 1 0 4

(b )

N

K R b

4 x 1 0 4
2 x 1 0 4
0
6 x 1 0 4

(c )

N

K R b

4 x 1 0 4
2 x 1 0 4
0
6 x 1 0 4

(d )
4

N

K R b

4 x 1 02 x 1 0 4
0

0

1

2

3

4

5

6

7

8

H o ld tim e ( m s )Figure 6.2: The dynamic population of fermionic Feshbach molecules in 40 K–87 Rb mixtures. Theinterspecies scattering length is 5100 a0 for (a), 3400 a0 for (b), 2500 a0 for (c), and 1700 a0 for
subplot (d). The lines show fits to Equation 6.4, where the only free parameter is the fraction of
three-body recombination events that produce Feshbach molecules, F .

69each event. The second type creates a Feshbach molecule, which is trapped for the combinationsof optical trap strength and magnetic field that we study here. After a recombination event,one 87 Rb atom and one 40 K atom are converted into one 40 K87 Rb Feshbach molecule, but all
participants in the reaction remain trapped. The total rate of three-body recombination events ischaracterized by the rate coefficient α, which is defined by Equation 7.1 and measured in Figure8.5. At a given magnetic field, a fraction F of these recombination events make Feshbach molecules
and the remaining fraction (1 − F ) produce deeply bound molecules. In addition to three-bodyrecombination, the molecule population is also affected by inelastic collisions with 87 Rb atoms (ourmeasurements in Figure 8.2 show that the rates of inelastic molecule collisions with 40 K atoms aremuch smaller than they are with 87 Rb atoms, so we will leave 40 K collisions out of this discussion
for simplicity). Each of these collisions removes a 40 K87 Rb molecule and a 87 Rb atom from thetrap. They are characterized by the loss rate coefficient β, which is defined by Equation 8.1 andmeasured in Figure 8.2.
Putting all of these pieces together, we get a set of coupled differential equations to model
the population of atoms and molecules in the trap as a function of the hold time, t at a magnetic
field near the Fano-Feshbach resonance:ZṄKRb (t) = d3 r F α n2Rb (r, t) nK (r, t) − β nKRb (r, t) nRb (r, t)ZṄRb (t) = d3 r (F − 2) α n2Rb (r, t) nK (r, t) − β nKRb (r, t) nRb (r, t)Z
ṄK (t) = d3 r α n2Rb (r, t) nK (r, t)

(6.1)
(6.2)
(6.3)

Here, nRb (r, t) and nK (r, t) are the atom number densities and nKRb (r, t) is the number density ofthe molecules as a function of position and hold time.
To gain intuition about the evolution of the molecule population, let’s simplify these equations
with a few approximations. First, we approximate the atom clouds as sharing the same gaussian
cloud sizes σx , σy , and σz . The equilibrium ratio of 40 K cloud size to 87 Rb cloud size is 1.07 in
our optical trap, which means that the error from this approximation should be small. Second, we
neglect the depletion of the majority population of 87 Rb atoms. With these approximations, plus

70
an initial condition with NK0 40 K atoms and zero 40 K87 Rb molecules, the approximate solutions
to Equations 6.1 - 6.3 are,NKRb (t) =

A0
F NK0
A0 − B 0

NRb (t) = NRb0
NK (t) = NK0 e−A0 t

e−B0 t − e−A0 t



(6.4)
(6.5)
(6.6)

where the constant A0 is equal to α hn2Rb in and the constant B0 is equal to β hnRb in , and thequantities in angle brackets are density-weighted averages. Equation 6.4 highlights the competition
between two timescales: one is set by the rate of loss due to atom-molecule collisions (B0 ) and theother by the rate of molecule production due to three-body recombination (A0 ). The quantity F NK0
is the maximum possible number of molecules that can be created by three-body recombination
when 40 K is the minority species.
We fit the data in Figure 6.2 the Equations 6.4–6.6, where F is the only free parameter. The
results are F = 0.12(2), F = 0.19(1), F = 0.31(2), and F = 0.38(4) for subfigures (a), (b), (c),and (d) respectively. To calculate the rate constants A0 and B0 , we use our own measurements
of atom and molecule loss rates from Chapter 8. Unfortunately, it is clear from the disagreement
between the fits and the data in Figure 6.2 that the approximations behind Equations 6.4 were notjustified for higher scattering lengths. The depletion and heating of the 87 Rb atom population by
three-body recombination appears to lengthen the lifetime of the molecules. For a more accurate
model, we could solve Equations 6.1–6.3 numerically. In any case, the peak molecule numbers in
Figure 6.2 are similar to those from magnetoassociation (see Chapter 5). This is possible because
the timescales for three-body recombination are so much faster than those for atom-molecule loss inthe 40 K–87 Rb mixture at high scattering lengths. However, since this method of molecule formation
adds kinetic energy to the molecules and is particularly sensitive to fluctuations in the magnetic
field or atom density, we do not routinely use it as a starting point for other experiments withmolecules.

71

6.2

Radiofrequency photoassociation
Photoassociation with Radio Frequency (RF) photons is another option for creating Feshbach

molecules in a gas of atoms. RF association has been well studied in the 40 K–87 Rb system, withearly measurements of 40 K87 Rb molecule creation efficiency [33] and lifetime [71] as well as acharacterization of the interspecies scattering length via the molecules’ binding energy [32]. In thissection, we revisit RF photoassociation because it avoids complications associated with crossing themagnetic Fano-Feshbach resonance. In Section 6.3, we investigate the temperature of molecules
formed by magnetoassociation and by RF association, and find larger than expected expansion
energies in both cases. In this section, we describe our procedure for RF association.
First, we prepare a quantum gas mixture of 40 K atoms in the |9/2, −7/2i state and 87 Rb atoms
in the |1, 1i state at a magnetic field of 540 G. Since the |9/2, −7/2i state does not participate in
the Fano-Feshbach resonance at B0 = 546.618 G, the 40 K and 87 Rb atoms only interact weakly,
with the background interspecies scattering length of -187 a0 . To associate molecules, we follow
the timing diagram in Figure 6.3. We sweep the magnetic field to 546.1 G in 0.3 ms. We use a
Toptica VFG-150 synthesizer to generate a pulse of RF with frequency ν. The RF power has aGaussian envelope with a 1/e full width of 400 microseconds. This pulse is subsequently amplified
by a Minicircuits ZHL-5W-1 amplifier and broadcast to the atoms using a resonant antenna.
Figure 6.4 shows the resulting RF spectrum. The leftmost peak is due to the bare |9/2, −7/2i →|9/2, −9/2i transition at ν0 = 80.0392(4) MHz for the 40 K atoms. The resonant frequency ν0 corresponds to a magnetic field of 546.13 G, where the interspecies scattering length is 980 a0 . The
shorter asymmetric peak on the right side is from the photoassociation of Feshbach molecules, witha peak value of NKRb ≈ 30, 000 molecules. The distance between these peaks corresponds to thebinding energy of the KRb Feshbach molecule state. Equation 2.2 predicts a molecule binding
energy of 80 kHz at this magnetic field, which would correspond to a resonant frequency of 80.12
MHz on the x axis of Figure 6.4.

72

𝑅𝐹 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒t
𝐵 − 𝐵0tPrepareAssociate

Image

Figure 6.3: Timing diagram for RF photoassociation of fermionic Feshbach molecules. The 40 K
and 87 Rb atoms are prepared in the |9/2, −7/2i and |1, 1i states. We ramp the magnetic field toa value within 1 Gauss of the Fano-Feshbach resonance and hold for a few milliseconds to allow
small magnetic-field transients to damp out. A gaussian RF pulse, blue-detuned from the 40 K
|9/2, −7/2i → |9/2, −9/2i transition, then drives 40 K atoms into the 40 K87 Rb Feshbach moleculestate. Next, the optical trap is switched off and the gas of atoms and molecules expands. Finally,
the molecules are imaged directly using the |9/2, −9/2i → |11/2, −11/2i optical transition.

73

N u m b e r in |9 /2 ,- 9 /2 >4 x 1 0 5
K

3 x 1 0 5

R b

ν0

2 x 1 0 5

1 x 1 0 5

0
0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

1 4 0

1 6 0

ν- 8 0 (M H z )Figure 6.4: RF spectrum for 40 K atoms prepared in the |9/2, −7/2i state, in a mixture with 87 Rb
atoms in the |1, 1i state. The bare 40 K |9/2, −7/2i → |9/2, −9/2i transition is measured with a 40
µs Gaussian RF pulse (left peak, hollow triangles) and the 40 K87 Rb photoassociation transition ismeasured with a 400 µs Gaussian RF pulse (right peak, filled diamonds). The inset shows schematicenergy levels for the experiment.

74

6.3

Excess molecule kinetic energy
Up to this point, we have focused on the number of fermionic pairs in the 40 K–87 Rb mixture.

Since many experiments seek to create degenerate gases of Feshbach molecules, the temperatureof these pairs is also an essential quantity. One previous experiment created fermionic 40 K87 Rb
molecules and measured their temperature to be 300 nK, whereas the temperature of the atomgas was initially 150 nK [33]. Later, a different group created fermionic 23 Na40 K molecules with atemperature of 500 nK, even though the temperature of the atoms was initially only 220 nK [64].
In contrast, an experiment with bosonic 6 Li40 K molecules found that the molecules had the same
temperature as the gas of atoms that they were created from [60].
Since conservation of momentum should cause a Feshbach molecule to inherit the centerof-mass momentum of its parent atoms, it is surprising that the fermionic molecules do not also
appear to inherit the temperature of the atom gas. In this Section, our measurements show thatthe temperature of the 40 K87 Rb molecules is consistently higher than that of the atoms, over abroad range of initial densities and temperatures, and that this behavior is consistent regardlessof whether molecules are formed by magnetoassociation or by RF photoassociation. We measure
the temperature of the molecules from their kinetic energy during time-of-flight expansion in one
radial trap direction, which is shown in Figure 6.5. Using the 87 Rb atoms as a thermometer, wepredict the expansion velocities of the 40 K atoms as well as the 40 K87 Rb molecules. Whereas the
40 K atoms match the expected velocity quite well, the molecules in this measurement expand fasterthan our prediction by a factor of 1.2.
2
Figure 6.6 summarizes the molecule expansion energy along one direction, E = 12 mhvKRbi,from measurements similar to Figure 6.5 over a broad range of initial atom conditions. At T /TF <0.5, where the 40 K atoms have extra average kinetic energy compared to the Maxwell-Boltzmann
distribution, we modify our calculation of the expected molecule expansion energy (grey solid line).
First, we calculate the kinetic energy of the 40 K atoms numerically from the thermal distribution of
an ideal non-interacting degenerate Fermi gas following [54] (blue dashed line), at the temperature

75

R b a to m s
K a to m s
K R b m o le c u le s

R a d ia l c lo u d s iz e ( u m )

5 0
4 0
3 0
2 0
1 0
0
0

1

2

3

4

5

T im e o f flig h t ( m s )Figure 6.5: Measured RMS cloud widths σ for 87 Rb atoms (red circles), 40 K atoms (blue triangles),and 40 K87 Rb molecules (black diamonds). The 87 Rb expansion data fit to a line through the origin
with slope vRb = 8.71(7)µm/ms (red dash-dotted line), which corresponds to T = 800 nK. The
40
blue dashedp line shows the predicted expansion velocity for K atoms at the same temperature,vK = vRb mRb /mK , which fits the data quite well. In contrast,
the measured expansion velocitypfor the molecules is higher than the predicted vKRb = vRb mRb /mKRb (grey solid line). For thisexperiment, the molecules are created following the timing diagram in figure 5.4 with trelease = 0.
The initial average in-trap densities are 5 × 1012 / cm3 for the 40 K atoms and 7 × 1012 / cm3 forthe 87 Rb atoms. The optical trap has a weak axial trapping frequency f = 7 Hz and two tightradial trapping frequencies of 600 Hz for the 87 Rb atoms. For these conditions, the 40 K atoms areat T /TF = 0.7.

76
measured from the expansion energy of the 87 Rb gas. Next, we use this calculation to predict the
distribution of 40 K87 Rb molecule energies. Center-of-mass momentum is conserved by the molecule
formation process, so that mKRb~vKRb = mK ~vK + mRb~vRb where the mx refers to mass and ~vx refers
to velocity for each species x. Solving for the root-mean-squared molecule velocity of the molecules,
we get2hvKRbi=

2 i + m2 hv 2 i
m2K hvK
Rb Rb
m2KRb

(6.7)

where the average of the dot product between the 87 Rb and 40 K velocities is presumed to be
2 i with the Maxwell-Bolzmann
zero. Equation 6.7 combines the numerically calculated value for hvK
2 i = 3k T /m , to predict the average kinetic energy of the molecules, which is plotted
formula, hvRb
BRb

as a thick grey line in Figure 6.6.
On average, the measured energy for the molecules in Figure 6.6 is 1.6(2) times the simple
prediction in Formula 6.7. Figure 6.6 clearly shows that the extra 40 K87 Rb molecule energy in
our measurements is present regardless of the initial degeneracy of the 40 K gas, or the densityof 87 Rb atoms that can undergo collisions with the molecules. Furthermore, the fact that the
RF photoassociated and magnetoassociated 40 K87 Rb molecules have the same measured expansion
energy rules out any heating that might have occurred as the magnetic field crossed the FanoFeshbach resonance. Another difference between these two methods is that the unpaired 40 K atoms
remain in the resonantly interacting |9/2, −9/2i state after magnetoassociation whereas they arein the non-resonant |9/2, −7/2i state after RF photoassociation. Therefore we can also rule out
interactions with unpaired resonant 40 K atoms as the cause of the mysterious extra energy.
To further exclude 87 Rb atoms as culprits, we measured the expansion of 40 K87 Rb moleculesfrom the trap after the unpaired 87 Rb atoms were removed from the trap. Figure 6.7 shows the expansion of molecule clouds with and without the 87 Rb removal sequence, which combines Adiabatic
Rapid Passage to the |2, 2i state with resonant light on the cycling |2, 2i → |3, 3i transition. Since
the experiment is performed in a crossed dipole trap, we monitor the expansion in the tight radial
direction as well as the weak axial direction. In both directions, there is a negligible difference in

77

2 .5

E / (1 .5 k B T )

2 .0
1 .5
1 .0
0 .5

K R b F e s h b a c h m o le c u le s :M a g n e to a s s o c ia te d
R F p h o ta s s o c ia te d

0 .0
0 .2

0 .4

0 .6

0 .8

1 .0

T 0/T F
2 .5

E / (1 .5 k B T )

2 .0
1 .5
1 .0
0 .5

K R b F e s h b a c h m o le c u le s :M a g n e to a s s o c ia te d
R F p h o ta s s o c ia te d0 .0
02 x 1 0 12

4 x 1 0 12

6 x 1 0 12

I n i t i a l <n R b > ( c m

8 x 1 0 12
-3

1 x 1 0 13)

Figure 6.6: The measured kinetic energies for molecules formed by magnetoassociation (black
diamonds) and RF photoassociation (purple squares). Top: For different intial T /TF of the 40 K
atoms. The expansion energy of the 87 Rb atoms is used to measure the temperature (red dashdotted line), from which we predict the expansion energy of the 40 K atoms (blue dashed line) and40 K87 Rb molecules (grey solid line). Bottom: The same data are plotted against the initial averagedensity of 87 Rb atoms. Because the temperature appears to be independent of the density of 87 Rb
atoms, we conclude that inelastic collisions with these atoms cannot account for all of the increased
temperature of the molecules.

78
the expansion rates for molecules with and without the 87 Rb atoms. This result implies that theextra expansion energy is not being caused by interactions between the unpaired atoms and themolecules during expansion. In addition, the measured expansion velocities are 6.2(3) µm/ms in
the axial direction and 6.6(1) µm/ms in the radial direction, which are consistent with each other.
This rules out the possibility that the fast radial expansion in Figure 6.6 could be a signature ofhydrodynamic expansion in the strongly interacting gas, where one would have slower expansion
in the axial direction.
At the time of this writing, the temperature of the KRb molecules remains difficult to explain.
In the remaining Chapters of this thesis, I will present measurements of three-body effects thatindicate a strong interaction between KRb molecules and Rb atoms. It is possible that theseinteractions are affecting the physics of molecule formation in the 40 K–87 Rb gas mixture. Anotherpossibility is that the Fermi statistics of the molecules somehow plays a role.

79

E x p a n d e d c lo u d s iz e ( u m )

6 0

4 0

2 0

0
0

1

2

3

4

5

6

7

T im e o f flig h t ( m s )Figure 6.7: Expansion of 40 K87 Rb molecules from a crossed dipole trap. Squares (larger cloudsizes) indicate the measured axial cloud sizes and diamonds (smaller cloud sizes) indicate the measured radial cloud sizes. The solid black points show the expansion of molecules in the presences of
unpaired 87 Rb atoms whereas the pink points show the expansion of the molecules after unpaired
87 Rb atoms have been removed from the trap. The grey bands show the expected molecule expansion velocities based on Equation 6.7 including the distribution of the quantum-degenerate 40 K
atoms with initial T /TF = 0.3. The range of the bands reflects the difference between the 87 Rb
temperature measured before and after molecule formation.

Chapter 7

The Efimov EffectThis Chapter lays the ground work for our measurements of three-body collisions, which arepresented in Chapter 8. First, Section 7.1 reviews the Efimov effect, which is becoming a standard
framework for understanding few-body physics in quantum systems. The Efimov effect is based ona fascinating solution to the problem of three strongly interacting quantum particles: an infinite
series of three-body bound states that obey a discrete scale invariance. Although the Efimov effectwas originally proposed in the context of nuclear theory, gases of ultra cold atoms are a nearly ideal
platform to explore this effect because the interactions can be controlled by magnetic fields near
Fano-Feshbach resonances. The Efimov states in these gases mediate resonant inelastic collisions,which is a major source of data on few-body interactions as described in Section 7.2. Finally,
Section 7.3 discusses emerging patterns in the measured spectra of Efimov states across different
atomic species as well as predictions of how the 40 K–87 Rb mixture might fit in.

7.1

Echoes of nuclear physics in quantum gases
The three-body problem is generically difficult; even the classical orbits of three planetswith gravitational interaction need to be solved numerically, and, in many cases, the system ischaotic [72, 73]. In 1970, Vitaly Efimov insightfully observed that the problem of three resonantlyinteracting quantum particles can be reduced to an effective 1/R2 potential, and derived analytic
solutions for the three-body bound states [74]. Figure 7.1 sketches the energies of Efimov’s states.
This solution has become an important paradigm in the study of few-body interactions (review

81

1

𝑎−

1

Free atoms
𝜆𝑎−

…

…

1

𝜆𝑎∗

1

𝑎∗

1

𝑎

Dimers +Atoms
Trimers

E

Figure 7.1: A cartoon of the energy spectrum for three atoms with resonant interactions. The twobody scattering length a diverges at the Fano-Feshbach resonance at the origin. E = 0 represents
the threshold for a continuum of free three-atom states. A two-body Feshbach molecule state exists
at positive scattering lengths (black dashed line). An infinite series of universal three-body Efimov
states (red lines) is present when the interaction strength diverges. The Efimov states become
degenerate with the threshold of free atom energies at negative scattering lengths λn a− wheren =∈ {0, 1, 2, ...}. At positive scattering lengths λn a∗ , the Efimov states are degenerate with thethreshold of energies for a Feshbach molecule plus a free atom.

82
articles include References [23, 75, 24, 76]). More broadly, the Efimov states set a length scale
when others diverge at the Fano-Feshbach resonance. In addition, the Efimov states potentially
impact many-body physics in unitary quantum gases [77, 78].
The Efimov three-body bound states have intriguing properties. Where the scattering length
a diverges, they obey a discrete scale invariance: for each state there exists one even larger bya factor of λ [74]. They can even exist when the two-body potential doesn’t support a boundstate, which has drawn comparisons between Efimov states and Borromean rings (Figure 7.2). Like
Feshbach molecules, the Efimov states reach a spatial extent that spans far beyond the scale ofinter-particle forces [29]. One recent paper observed evidence of a three-atom Efimov state on theorder of 1 µm in size [79]. The states are also universal, in that their properties are determined solely
by the two-body scattering lengths and a three-body parameter κ∗ that determines the locationof the first Efimov state. This independence from microscopic detail means that Efimov’s solution
can be applied to both nuclear structure and interactions between ultra cold atoms in spite of thevast difference between their energy scales [76].
The scaling factor λ = eπ/s0 is determined by a transcendental equation that only depends on
the number of resonant pairwise interactions between the constituent particles and the ratio between
the masses of these particles [80]. When only one of three pairwise interactions is resonant, there
is no Efimov effect all. For three equal-mass particles where all three pairwise interactions areresonant, we have λ = 22.7. This is comparable to the tuning range over which a is the dominant
length scale in most quantum gas experiments near Fano-Feshbach resonances. When only two
pairwise interactions are resonant in three-body systems of equal-mass particles, the density of
Efimov states drops to λ = 1986 [81]. The mass ratio can have a huge impact on λ. For heavylight-light combinations with resonant interspecies interactions the value of λ increases sharply,
making the observation of an Efimov state extremely unlikely. On the other hand, heavy-heavylight combinations see an enhanced density of Efimov states. Our 40 K–87 Rb mixture has λ = 123.
The recently achieved 133 Cs-133 Cs-6 Li quantum gas mixture has λ = 5.5, making it possible to see
the Efimov state sequence repeat itself three consecutive times [82, 83]. Figure 7.3 illustrates the

83

Figure 7.2: A topological analogy to the Efimov bound states. The triplet on the left is in a
‘Borromean’ configuration such that no single pair of rings is linked. Similarly, Efimov statesremain bound at negative scattering lengths where there is no two-body Feshbach bound state.
The triplet on the right is in a ‘tango’ configuration, where each pair of rings is linked. This
matches the situation at positive scattering lengths where both two-body Feshbach and three-body
Efimov bound states exist.

84
wide variety of λ values in three-body systems.
Another requirement for the Efimov effect to occur is that the resonant pairwise interactions
overwhelm other energy scales. Since the long-ranged Coulomb repulsion between charged particles
overwhelms the Efimov 1/R2 potential, nuclear Efimov states are most likely to exist in the socalled ‘halo nuclei’ where a neutral core is orbited by two loosely bound neutrons, such as 11 Li or
20 C [84, 29].

Though no definitive sign of a nuclear Efimov state has been observed at the timeof this writing, the prospect of observing them in halo nuclei continues to drive experimental andtheoretical research [85, 86]. The extreme mass ratio in these nuclei unfortunately leads to an
unfavorable density of Efimov states. The 11 Li (9 Li−n − n) nucleus has λ = 3 × 1018 , for example.
Whereas extraordinary fine tuning is required on Nature’s part to realize a nuclear Efimov states,
the scattering parameters in cold atoms can be easily tuned by physicists thanks to Fano-Feshbachresonances. Dozens of Efimov states have been catalogued from observations in ultracold atomic
gases since their initial observation in a gas of neutral Cesium [22]. For these experiments, Efimov
physics can be explored over scattering lengths that are between the finite range of the two-bodypotential (set by the van der Waals length for wide Fano-Feshbach resonances) and the smallest ofeither the thermal deBroglie wavelength or the inter-particle spacing. In practice this means thatthe atoms must be in the ultra cold regime to observe Efimov effects, and that it is possible toexplore a broader range of interaction strengths by reducing the temperature and density of theatomic gas [87, 79].

7.2

Inelastic collisions as a signature of Efimov states
All of the Efimov states to be identified so far in ultra cold atomic gases have been too short-

lived for direct observation. Instead, experimentalists typically rely on measurements of scattering
rates to point the way to the Efimov spectrum1 . When an Efimov state becomes degenerate withthe free atom threshold at negative scattering length a− , strong three-body correlations mediate aresonant increase in the rate of three-body recombination. Similarly, at positive scattering length a∗1

Another less common method is to photo associate the three-body states directly [88, 89, 90].

85
6Li-133Cs-133Cs

40K-87Rb-87Rb

λ = 4.9

λ = 123
133Cs-133Cs-133Cs

41K-41K-87Rb

λ = 22.7

λ = 3.5x105

9Li-n-nλ=3x1018

log10 𝜆
0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Figure 7.3: The length scaling factor λ = eπ/s0 for a few systems of interest. Heavy-Heavy-Light
mass systems have a high density of Efimov states, whereas Heavy-Light-Light mass systems haveextremely sparse Efimov states.

where the Efimov state’s energy intersects the threshold for an atom and a Feshbach molecule, there
is a resonant increase in the rate of inelastic atom-molecule collisions. These scattering lengths aremarked in Figure 7.1. There is also an interference minimum located at positive scattering lengtha+ , where the three-body recombination rate is reduced. These three Efimov features at a− , a∗ ,and a+ form a universal sequence that repeats itself every time the scattering length is changed bya factor of λ. The details of the universal functions for scattering rates near Efimov resonances arecovered in [23] and extended to heteronuclear mixtures in [81].
The first observation of an Efimov state was performed by cataloguing the three-body recombination rate in an ultra cold gas of identical 133 Cs atoms [22], a method that remains widely used
today. In three-body recombination, three atoms collide to form a diatomic molecule. An amountof kinetic energy equal to this binding energy is then shared between the resulting molecule anda third atom. This energy is usually sufficient to eject the atom and molecule from the trap. For
the 40 K–87 Rb system with resonant interspecies interactions, two channels exist for interspecies
three-body recombination; however, only 87 Rb+87 Rb+40 K supports Efimov resonances [23], while
87 Rb+40 K+40 K loss is suppressed by Fermi statistics. Therefore we are most interested in the for-mer process, which involves two bosons and one fermion. In order to compare measurements taken
under different experiment conditions, we define the density-independent three-body recombination

86
rate coefficient α byṄRb (t) = 2ṄK (t) = −2αZd3 r nK (r, t) n2Rb (r, t),(7.1)

where nRb (r, t) and nK (r, t) are the densities of 87 Rb and 40 K, respectively.
The recombination rate increases dramatically near Fano-Feshbach resonances, where α scales
as a4 [91]. The Efimov states modulate this overall dependence, leading to a universal pattern ofpeaks and valleys that has been observed in a variety of cold atom experiments [24]. For a < 0 in
the 40 K–87 Rb–87 Rb system [81], we haveα(a < 0) =Cαcoth(πs0 )sinh(2η∗ )~a4,
2 sin2 [s0 ln(a/a− )] + sinh2 (η∗ ) mK

(7.2)

whereas for positive a,α(a > 0) = Cαsin2 [s0 ln(a/a+ )] + sinh2 η∗
sinh2 (πs0 + η∗ ) + cos2 [s0 ln(a/a+ )] 4coth(πs0 )cosh(η∗ )sinh(η∗ )
~a
+
.
2
2
m
sinh (πs0 + η∗ ) + cos [s0 ln(a/a+ )]
K

(7.3)

Here, mK is the mass of 40 K (or, more generally, the lighter atom). The coefficients Cα and s0
are determined by the mass ratio and the number of resonant interactions. For 40 K–87 Rb–87 Rb,
Reference [81] calculates Cα = 354 and s0 = 0.6536. The decay factor η∗ accounts for the finite
lifetime of the Efimov states, and can be different for different atomic systems. The equation for
a > 0 contains two terms because there are two recombination channels: the first term is forrecombination that forms a Feshbach molecule state, and the second term is for recombination that
forms a deeply bound molecule.
The predicted loss resonances at at a− and a+ are quite dramatic, with a wide base and atypical height spanning orders of magnitude in α. Figure 7.4 plots Equations 7.2 and 7.3 for the
40 K–87 Rb–87 Rb system. Here, we choose the resonance locations based on previous measurementsin the 41 K–87 Rb–87 Rb system [92] as described in the next Section. The different colored solid lines
show a typical range of values for η∗ across cold atom experiments. As η∗ increases, the resonances
become broader. The overall value of α also increases with η∗ at negative scattering lengths, even87

1 0 -1 5
1 0 -1 7
1 0 -1 9
1 0 -2 3

α(c m

6

/s )

1 0 -2 1
1 0 -2 5
1 0 -2 7

η∗ =
η∗ =
η∗ =
η∗ =
η∗ =

1 0 -2 9
1 0 -3 1
1 0 -3 3 1-1 0-1 0 2-1 0 30 .2
0 .1
0 .0 50 .0 20 .0 1-1 0 4-1 0 5a (a 0)1 0 -1 5
1 0 -1 7
1 0 -1 9
1 0 -2 3

α(c m

6

/s )

1 0 -2 1
1 0 -2 5
1 0 -2 7

η∗ =
η∗ =
η∗ =
η∗ =
η∗ =

1 0 -2 9
1 0 -3 1
1 0 -3 3 1-1 0-1 0 2-1 0 3-1 0 40 .2
0 .1
0 .0 50 .0 20 .0 1-1 0 5a (a 0)Figure 7.4: Theoretical α vs. a for 40 K–87 Rb–87 Rb three-body recombination from Equations 7.2
and 7.3. Top: loss peaks at negative scattering lengths λn a− . Our measurements in Chapter
8 access the ranges within the dashed lines. Bottom: interference minima at positive scattering
lengths λn a+ .

88
in the regions between Efimov resonances. Notice the periodicity of the resonances, which repeat
each time the scattering lengths change by λ = 122.7.
Our experiments, described in Chapter 8, can measure the region inside of the dashed lines
in Figure 7.4, which spans about half of the period of the Efimov spectrum. Recall that the Efimov
model is valid when the scattering length overwhelms the other length scales of the system. Thisregion is bounded at low scattering lengths by the van der Waals length, rvdW = 72 a0 , and athigh scattering lengths by the thermal wave vector 1/k ≈ 3000 a0 . The trap lifetime for theatoms is about 45 seconds, which, for typical densities, makes it difficult to measure values of α
lower than 10−28 cm6 /s. The upper bound for α is set by the unitarity limit which is αsat =~192π 2 /(2µK-Rb-Rb k 4 ) ≈ 10−21 cm6 /s [87]. The maximum values for a and α both increase forcolder temperatures, however we are limited to temperatures above 200 nanoKelvin in order tomaintain a good overlap between the 40 K and 87 Rb clouds.
Another type of Efimov resonance occurs in the atom-molecule scattering rate when an Efimov
state becomes degenerate with the energy threshold for a free atom and a Feshbach molecule [93, 23].
These resonances are less well-understood than their three-body recombination counterparts, due
in part to the relatively high difficulty of preparing and measuring a gas of Feshbach molecules. In
our quantum gas mixture, the Efimov states have two 87 Rb atoms and one 40 K atom. Therefore
the Efimov resonance occurs in collisions between 87 Rb atoms and 40 K87 Rb Feshbach molecules.
This atom-molecule inelastic loss rate is characterized by a coefficient β, defined such that [81]ṄKRb (t) = ṄRb (t) = −βZd3 r nKRb (r, t) nRb (r, t),(7.4)

where nKRb (r, t) is the number density of the Feshbach molecules and nRb (r, t) is the number
density of the atoms.
For collisions between 87 Rb atoms and 40 K87 Rb Feshbach molecules, universal theory predicts
that β should increase with a modest scaling of a1 near the Fano-Feshbach resonance [35]. Aswith three-body recombination, this dependence is modulated by a universal sequence of Efimov

89
resonances [23]:β(a) = Cβsinh(2η∗ )~a.
2sin [s0 ln(a/a∗ )] + sinh (η∗ ) mK
2

(7.5)

In the 40 K–87 Rb mixture, Reference [81] calculates s0 =0.6536 and Cβ = 10.1.
Figure 7.5 plots Equation 7.5 for the same range of η∗ values that were used in Figure 7.4.
Once again, the Efimov peaks are both broad and tall, and become broader as the decay parameterη∗ increases. The dashed lines indicate the range accessible to our experiments. As with the threebody recombination measurements, the Efimov theory applies when 72 a0 < a < 3000 a0 . The lower
bound for β is set by the measured lifetime of molecules in a gas where all unpaired atoms havebeen removed. The unitarity limit sets the upper bound on β at βsat = ~π/(µRb-KRb k) ≈ 5 × 10−10
cm3 /s for typical temperatures of 300 nanoKelvin [94]. Observe that the Efimov resonances inthis case are likely to be saturated at their peaks, leading to a distorted shape. In Chapter 8, we
will account for this effect rather crudely by fitting a multiplicative factor to the amplitude of theuniversal curve.

7.3

How predictable are three-atom interactions?
Looking at the predicted loss rates in Figures 7.4 and 7.5, it is clear that the locationsof Efimov resonances can have a big impact on experiments that seek to leverage tunable interactions near Fano-Feshbach resonances. Fortunately, these locations turn out to be surprisingly
predictable. This Section summarizes recent measurements of the relationships between Efimov
resonance locations at different scattering lengths and even across different atomic species [95, 96,
97, 98, 25, 26, 99], which has driven the development of improved models for few-body interactions
between atoms [100, 27, 101, 102, 103]. Since most of the work to date has been performed withhomonuclear atomic gases, our measurements in Chapter 8 are motivated by the desire to extend
this understanding to mixed-species systems.
Universal theory based on the Efimov effect predicts relationships between the locations of theresonances at a− , a+ , and a∗ [23, 81]. In principle, the measurement of a single Efimov resonance90

1 0 -6

η∗ =
η∗ =
η∗ =
η∗ =
η∗ =

1 0 -7

β(c m

2

/s )

1 0 -80 .2
0 .1
0 .0 50 .0 20 .0 1

1 0 -9
1 0 -1 0
1 0 -1 1
1 0 -1 2 1
1 0

1 0 2

1 0 3

1 0 4

1 0 5

a (a 0)Figure 7.5: Theoretical β vs. a for 40 K87 Rb + 87 Rb molecule vibrational relaxation collisions from
Equation 7.5. Our measurements in Chapter 8 access the range within the dashed lines.

91
pins down the locations for the entire Efimov spectrum in a given system. For homonuclear systems,
the theoretical relationships between Efimov resonances associated with the same three-body boundstate are [23]:
|a− |
= 4.9
a+
a+
= 4.46a∗|a− |= 21.9a∗(7.6)

The first relation, between a− and a+ , has been confirmed by a number of experiments [22, 97, 104].
In contrast, the values for a∗ reported thus far [105, 106, 107] have not followed the predicted
locations relative to a+ and a− . This suggests that models of the atom-molecule loss resonances incold atoms may require additional input beyond the Efimov effect [103].
More generally, the Efimov relationships between resonance locations are a function of themass ratio and the number of resonant interactions. For our heteronuclear mixture of 40 K and 87 Rb
atoms, they are [81]:|a− |= 11.1a+a+= 21.7
a∗
122.7
|a− |
=
= 240.6
a∗
0.51

(7.7)

These relationships have been imposed between the resonances in Figures 7.4 and 7.5.
Whereas Efimov theory predicts a relationship between resonance locations, the so-called‘three-body parameter’ that sets the absolute resonance locations was initially understood to be a
non-universal quantity subject to each system’s unique interaction potential at short range [108,23, 109]. However, as measurements of Efimov resonances accumulated in single-species cold atom
experiments, an unexpected pattern emerged: Efimov loss resonances were consistently measuredata−,universal = −9.1 rvdW

(7.8)

92
for different nuclear spin states [25], different Fano-Feshbach resonances [26], and even different
atomic species [95, 96, 97, 98, 25, 26, 99].
Figure 7.6 collects worldwide measurements of Efimov resonance locations in single-species
cold atom experiments2 , where each point represents a measured Efimov resonance, following theformat of Reference [26]. Each vertical axis compares the measured Efimov resonance location to a‘universal value’ by combining Equation 7.8 with the universal relationships in Equations 7.6. For
each plot, the full span of the vertical axis represents a period of the Efimov spectrum, where avaries by λ = 22.7.
If the three-body parameter were determined by a detailed interplay between non-universal
short-ranged potentials, as was originally expected from the Efimov theory, then the points in Figure
7.6 would be scattered over the whole vertical range of the plots. Instead, the points in subfigures(a) and (b) tightly cluster around a simple prediction based on only the van der Waals length(Equation 7.8, dashed horizontal line), which implies that the three-body parameter is universal.
The points in subfigure (c), on the other hand, appear to depart from that pattern. The pink circle
points are measured by the types of atom-molecule loss resonances shown in Equation 7.5 andFigure 7.5. Since they are measured in the same systems where a− follows the universal three-bodyparameter, it seems likely that Equation 7.6 misses some important aspect of the way atoms andmolecules collide. The blue diamond points represent ‘avalanche peaks,’ which are small atom loss
resonances that appear near the universal a∗ location in several experiments [97, 98, 112]. The
mechanism by which atom loss occurs near a∗ in a gas of pure atoms remains poorly understoodat the time of this writing [113].
The unexpected pattern shown in Figure 7.6(a) stimulated theoretical research, which has
since shown that the van der Waals potentials of cold atoms screen out the details of the nonuniversal short-ranged potentials [114, 100, 27, 101, 102]. This leads to the universal three-bodyparameter in ultra cold atoms. Measurements in the three-body recombination rate for ultra cold2

One data point measuring a non-universal value of a− for 39 K in Ref. [97] is deliberately excluded because alater publication by the same group [110] claimed that the measurement was not repeatable.

93

L o g 2 2 .7 ( a - / a u n iv e r s a l)

0 .4

(a )
0 .2
0 .0
-0 .2
-0 .4
6

7

L i

3 9

L i

8 5

K

1 3 3

R b

C s

A to m ic s p e c ie s

L o g 2 2 .7 ( a + / a u n iv e r s a l)

0 .4

(b )
0 .2
0 .0
-0 .2
-0 .4
6

7

L i

3 9

L i

8 5

K

1 3 3

R b

C s

A to m ic s p e c ie s

L o g 2 2 .7 ( a * / a u n iv e r s a l)

0 .4

(c )
0 .2
0 .0
-0 .2
-0 .4
6

L i

7

L i

3 9

K

8 5

R b

1 3 3

C s

A to m ic s p e c ie s

Figure 7.6: Measurements of the three-body parameter in homonuclear systems by three-bodyrecombination (blue diamonds) and atom-molecule loss (pink circles). Each subplot corresponds toa different type of Efimov resonance: subplot (a) shows measurements of a− [111, 98, 104, 25, 97,
99, 22, 26, 79, 110]; subplot (b) shows measurements of a+ [104, 25, 97, 22]; and subplot (c) shows
measurements of a∗ [105, 97, 98, 106, 112]. The horizontal dashed lines show a prediction based ona three-body parameter that is universally determined by the van der Waals length.

94
4 He∗ have also been consistent with the same universal three-body parameter [115]. More recently,

a different universal model based on the van der Waals potential [103] was able to explain thelocations of both a− and a∗ for 133 Cs, including the deviation of their relationship from the simple
Efimov rules in Equation 7.6.
For homonuclear cold atom systems, a combination of universal models based on Efimov
physics and a large number of experimental measurements have driven the development of a detailed
quantitative understudying of three-body interactions. In contrast, only a handful of measurements
exist for mixed-species systems. The first was performed by the LENS group in a Bose–Bose mixtureof 41 K and 87 Rb atoms with resonant interspecies interactions [92]. They identified loss resonances
corresponding to a− = −246 a0 and a tentatively labelled avalanche peak at a∗ = 667 a0 in the 41 K–
87 Rb–87 Rb system.

One theory has predicted a universal three-body parameter for heteronuclear

mixtures, which would imply that LENS’s 41 K–87 Rb–87 Rb system and our 40 K–87 Rb–87 Rb systemhave nearly identical Efimov resonance locations [116]. The resonance locations in Figures 7.4 and
7.5 are based on LENS’s measurement of a− . These figures show that multiple Efimov features
would appear within our experimental window, if the two K–Rb mixtures had identical threebody parameters. Our measurements presented in Chapter 8 investigate this possibility. Afterour measurements were published, two independent groups measured Efimov loss resonances in
mixtures of 6 Li and 133 Cs atoms [82, 83]. They demonstrated a high density of Efimov statesconsistent with the predicted value of λ = 4.9, which made it possible to observe three consecutive
Efimov states for the first time in any experiment. This illustrates the rich behavior of few-bodyinteractions in quantum gas mixtures.

Chapter 8

K-Rb-Rb collisions

In this Chapter, I describe our measurements of inelastic collision rates as a function of theinterspecies scattering length, and then discuss our results in the context of the universal Efimov
effect. Similar measurements for atoms with the same mass have uncovered a wealth of insights into
few-body interactions. Extending this understanding to include mixtures of different atomic species
is important to build a more general picture. To that end, we search for evidence of 40 K–87 Rb–
87 Rb three-body states and compare our findings with theoretical predictions [116] as well as withpast measurements of the isotopically different 41 K–87 Rb–87 Rb mixture by the LENS group [92].
Section 8.1 presents our measurement of an Efimov resonance in the collisions between 40 K87 Rb
Feshbach molecules and 87 Rb atoms. In Section 8.2, we present measurements of the three-bodyrecombination rate coefficient α over the entire range of interspecies scattering lengths available
to our experiments, and find that the data exclude the presence of Efimov resonances in that
range. Finally, in Section 8.3 we analyze these findings in terms of a predicted universality of thethree-body parameter that sets the Efimov resonance locations.
The bulk of the material in this Chapter is published in Reference [117]. In particular,
Sections 8.2 and 8.1 are published nearly verbatim in that work. Section 8.3 also contains material
from the paper, but it is expanded to include comparisons against newer results from other groups.

96

8.1

Atom-molecule loss
I’ll begin by considering inelastic atom-molecule collisions, which eject the participating atomand Feshbach molecule from the trap. The inelastic collision rate is enhanced when a is tuned to avalue, denoted a∗ , where an Efimov trimer state is degenerate with the atom-molecule threshold [93,23]. Such Efimov resonances have been observed as broad loss features for 133 Cs [105] and 6 Li
dimers [106, 107] (see Figure 7.5 and the accompanying discussion in Section 7.2 for more on theseresonances). Part of our motivation to study atom-molecule collisions in the 40 K–87 Rb mixture camefrom previous measurements of the same mixture that were made by our group in 2008 [71]. They
found that collisions between 87 Rb atoms and 40 K87 Rb molecules intensified as they reduced thescattering length, which ran opposite to the expected trend [35]. Helfrich et al. later suggested thatthese experiments had encountered one side of an Efimov resonance [81]. Our new measurements,
which are the topic of this Section, extend these data to lower values of a where we do in fact observe
an atom-molecule resonance, which is the first one to be measured in a heteronuclear mixture.
We map out the molecule loss rates for three different collision partners: 87 Rb atoms in the|1, 1i state, 40 K atoms in the |9/2, −9/2i state, and 40 K atoms in the |9/2, −7/2i state. Efimov
resonances are only possible for the first case, and, in the last case, the collision partner does notparticipate in the Fano-Feshbach resonance. For these measurements, the atom gas mixture wastypically prepared at a = −300 a0 and T = 300 nK. We measured 87 Rb–40 K87 Rb collisions in the
single-beam optical dipole trap, whereas 40 K–40 K87 Rb collisions were measured with an additional
confining laser beam that increases the 40 K axial trap frequency. For data in this Chapter, the 40 K
atoms are at T /TF > 0.2 and the 87 Rb are at T /Tc > 1.1.
Figure 8.1 shows a timing diagram and an example of raw data from a 87 Rb–40 K87 Rb loss
measurement. First, the magnetic field is slowly swept across the Fano-Feshbach resonance at a rateof 3 G/ms to form 40 K87 Rb Feshbach molecules by magnetoassociation, following the procedures
discussed in Chapter 5. To prepare a pure sample of molecules and one particular atom species,
we then hold the magnetic field at 545 G while we use a combination of resonant light and RF

97

5 5 1

M a g n e tic fie ld ( G )

5 5 0
5 4 9
5 4 8
5 4 7
t h o ld

5 4 6
5 4 5

-2

-1

0

1

2

3

t (m s )1 2
1 0

N u m b e r / 1 0

4

8
6
4
2
0
0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

t h o ld ( m s )Figure 8.1: Top: Magnetic-field sequence for molecule production and loss measurements (green
solid line). The red dashed line marks the interspecies Fano-Feshbach resonance. Bottom: Loss ina gas of 87 Rb atoms and 40 K87 Rb molecules. Blue triangles show the number of 40 K atoms whichwere bound as 40 K87 Rb molecules during thold . Purple squares show 87 Rb atoms, including thosethat were free and those that were bound into 40 K87 Rb molecules.

98
pulses to selectively remove unpaired atoms from the trap. When measuring collisions with 40 K
atoms in the |9/2, −7/2i state, we use RF transfer the unpaired 40 K atoms into the |9/2, −7/2i
hyperfine state at this point. After preparing the atom-molecule sample, we ramp the field up to
the the value desired for the loss measurement, wait a variable hold time thold , turn off the opticaltrap, and then jump the magnetic field back across the Fano-Feshbach resonance to dissociate themolecules. We extract the number of molecules, the number of free atoms, and the in-trap cloudsizes from images of the resulting atom clouds after a few milliseconds of expansion from the trap.
In-trap radial sizes are calculated from the measured axial size and the trap aspect ratio.
From the Rb+KRb data, we extract the loss-rate coefficient β by fitting the measured lossof atoms and molecules to numerical solutions of Eqs. (8.1), which are reproduced here:
ṄKRb (t) = ṄA (t) = −βZd3 r nKRb (r, t) nA (r, t),(8.1)

where nKRb (r, t) is the number density of the Feshbach molecules and nA (r, t) is the number densityof the atoms. We account for heating by including the measured cloud sizes at each point in time.
The estimated systematic uncertainty in β, which is dominated by the uncertainty in the in-trap
densities, is ±20%. For our measurements of collisions between 40 K atoms and 40 K87 Rb molecules,
the depletion and heating of the much larger atom cloud are negligible during the loss measurementand Eq. (8.1) has the analytic solution NKRb (t) = NKRb (0)e−t/τ where 1/τ = hnK iβ, and hnK i isthe density-weighted atom density. The loss rate coefficient β for K–KRb collisions is extracted
by fitting the molecule number to an exponential decay as a function of time, with τ and theinitial molecule number as free parameters. This method give results that are consistent with moredetailed calculations like the ones we use for Rb+KRb collisions.
Figure 8.2 plots the measured β vs. a. The rate of molecule collisions with 40 K in either
hyperfine state decreases smoothly with increasing a, as was seen previously [71]. In contrast, forcollisions between 40 K87 Rb molecules and 87 Rb atoms, we observe a prominent resonance whose
location lies beyond the range of previous data [71] and is consistent with the suggestion of a∗ =200(50) a0 made by Ref. [81]. The solid line in Fig. 8.2 shows a fit to the universal formula for99

4 x 1 0 -1 0

β(c m

3

/s )

6 x 1 0 -1 0

2 x 1 0 -1 0

0
1 0

1 0 0

a (a 0)1 0 0 0Figure 8.2: Measured β vs. a for collisions of 40 K87 Rb molecules with 40 K atoms in the |9/2, −7/2istate (triangles), with 40 K atoms in the |9/2, −9/2i state (squares), and with 87 Rb atoms in the|1, 1i state (solid diamonds). Vertical error bars indicate one standard deviation of statistical error,
and horizontal error bars correspond to an upper bound of ±0.02 G variation in B across thecloud. A broad resonance is evident in the 87 Rb+40 K87 Rb loss and the solid curve shows a fit tothe universal T = 0 shape for an Efimov resonance Eq. (7.5). The horizontal dashed line indicatesthe estimated unitarity-limited β for the collisions between 87 Rb atoms and 40 K87 Rb molecules.
This data was published in Reference [117].

100
an Efimov resonance, Equation 7.5, which gives a resonance location a∗ = 230(10) a0 , a resonancewidth set by η∗ = 0.26(3), and a resonance amplitude Cβ = 3.2(2). Here, we exclude data at
a < 2rvdW = 144 a0 from the fit because the non-universal short ranged interaction potential
between the atoms has a strong influence in this range [18]. Whereas the values of a∗ and η∗ arefunctions of the particular interaction potentials of 40 K and 87 Rb atoms, the value of Cβ has been
predicted to equal 10.1 based on universal theory [81]. The discrepancy between theory and ourmeasured value of Cβ could be due to saturation at finite temperature: the horizontal dashed line
indicates the unitarity-limited value for β, which is βsat = ~π/(µRb-KRb k) ≈ 5 × 10−10 cm3 /s at
300 nanoKelvin [94]. This value is close to much of our data near the resonance center. It would
be interesting to calculate the saturation effect more accurately by averaging over the thermal
distribution of collision energies [87], or to repeat these measurements on a gas at an even lower
temperature. However, recent measurements of similar resonances in 133 Cs also find a non-universal
value for the amplitude Cβ , which suggests that a more detailed theoretical approach is generally
required for atom-molecule Efimov resonances [118].

8.2

Three-body recombination
Now we turn our attention to three-body recombination, which is widely used to reveal

Efimov physics in cold atoms. Three atoms can collide inelastically and form a diatomic molecule,which typically releases enough energy to eject all three atoms from the trap. This process isresonantly enhanced when an Efimov trimer state approaches zero binding energy at a negativea denoted a− . On the other side of the Fano-Feshbach resonance, interference effects lead tominima in the three-body rate coefficient α at a positive a denoted a+ . These features modulate an
overall a4 dependence of α that results in a large increase of atom loss rates near a Fano-Feshbachresonance [119, 23]. Figure 7.4 and the accompanying discussion in Section 7.2 give more details
on these resonances.
In the only previous observation of heteronuclear Efimov resonances, enhanced loss for 41 K–
87 Rb–87 Rb was observed at a = −246(14)a [92]. An additional narrow loss feature at 667(1) a
−
0
0

101

5 5 1

M a g n e tic fie ld ( G )

5 5 0
5 4 9
5 4 8
t h o ld
5 4 7
5 4 6

0

1

2

3

t (m s )Atom number

400 000
300 000
200 000
100 000
0

200

400

600

800

Hold time HmsL

1000

1200

1400

Figure 8.3: Top: An example timing diagram for a measurement of the three-body recombination
rate coefficient α at B = 548 G. Bottom: The measured number of Rb atoms (red points) andK atoms (blue points) vs thold . Solid lines show the result of the fit used to obtain α following
Reference [120].

102
indicated a possible atom-molecule resonance. We were interested in testing whether matching
features existed in our 40 K–87 Rb–87 Rb system. After the three-body parameter that sets Efimov
resonance locations was found to be universal in homonuclear systems, Wang et al. predicted asimilar pattern for heteronuclear systems [116]. Since the two K–Rb mixtures have the same interspecies van der Waals length and non-resonant Rb-Rb scattering length, as well as nearly identicalmass ratios, a universal three-body parameter would result in Efimov resonances at the same interspecies scattering lengths. Previous measurements of α for a 40 K–87 Rb mixture saw no features
beyond a4 scaling; however, the data were relatively sparse [71] and it had been hypothesized thatan Efimov resonance could have fit between the points [81]. This Section presents additional, moredetailed, measurements of α for 40 K–87 Rb that cover values of a where the 41 K–87 Rb features wereseen.
For these measurements, the atom gas mixture is prepared by evaporating at a field where|a| ≈ 300 a0 on the same side of the Fano-Feshbach resonance as the field for the loss measurement,
B. Figure 8.3 shows a timing diagram and an example of raw atom loss data. Once the gas isprepared, the field is then swept to B in a short time compared to the loss, and held for a variable
time thold . Finally, the optical trap is turned off and the field is swept back to the evaporation
value where the atoms are imaged. From our measured cloud sizes and atom numbers, we obtain
α which is defined byṄRb (t) = 2ṄK (t) = −2αZd3 r nK (r, t) n2Rb (r, t).

(8.2)

We employ an analysis similar to that described in Ref. [120], which uses the measured atom
densities at each time to account for heating of the gas. Whereas Ref. [120] used quantum degenerategases, we adapt the analysis to the Maxwell-Boltzmann density distributions that describe our atom
clouds. As a check of this analysis, we compare the results against a brute-force numerical solutionof the differential equations, which calculates heating rates from anti-evaporation of Rb atoms
rather than taking measured cloud sizes as a given input. We find the results to be consistent for
both methods, although the former method involves less computational overhead. Based on our

103ability to accurately extract the in-trap density of the atoms from expanded images, we estimate
the systematic uncertainty in α to be ±20%.
Figure 8.4 shows the measured α vs. B. We measure an increase of α by over four orders ofmagnitude near the s-wave resonance at B0 = 546.618 G. A second peak, on the order of 0.1 G wide,is evident near 547.4 G. As illustrated in the inset to Fig. 8.4, this feature is not consistent with the
expected shape of an Efimov resonance, and we identify it as a two-body d-wave Fano-Feshbachresonance [46, 121]. Because it is technically challenging to control a near a narrow resonance, we
exclude the data for B between 547.25 G and 547.50 G from further analysis and calculate a using
Equation 2.1 for the s-wave resonance alone. Namely, we set a = −187a0 (1+3.04G/(B−546.618G))
following Reference [122].
Figure 8.5 shows the measured α vs. a, using the same data presented in Figure 8.4. These
data are consistent with the more sparse measurements of Ref. [71]. For comparison, we also
plot the universal form for α as a function of a [81] with Efimov resonances corresponding to themeasurement of a− by the LENS group [92]. These functions are given in Equations 7.2 and 7.3,
and the location for the interference minimum at a+ is extrapolated from the value of a− using
Equation 7.7. We use η∗ = 0.02 for the plotted curves, which gives the best agreement to themeasured magnitude of α. The calculated shapes for Efimov features at this value of η∗ (orange
solid lines) are extremely large compared to any gaps or uncertainty in our data over the measured
range. We conclude that there are no Efimov features in α at negative a between -200 a0 and-3000 a0 or at positive a between 200 a0 and 3000 a0 .
We also used these data sets to measure the ratio of 87 Rb loss to 40 K loss. Within theuncertainties, this ratio does not show a dependence on a and the average value is 2.1(1), whichis consistent with 87 Rb+87 Rb+40 K three-body recombination being the dominant loss process. A
resonance feature with a different multistage process has also been proposed, in which three-bodyrecombination generates an energetic 40 K87 Rb Feshbach molecule that undergoes subsequent resonant elastic collisions with 87 Rb atoms, eventually ejecting multiple 87 Rb atoms from the trap. Thisso-called ‘avalanche mechanism’ loss has been hypothesized to exist near atom-molecule Efimov res-104

1 0 -2 1

1 0

1 0 -2 3

-2 4

α(c m

6

/s )

1 0 -2 5

1 0

5 4 7 .0-2 6

1 0 -2 8

5 4 5

5 5 0
B (G )

5 4 7 .5

5 4 8 .0

5 5 5

Figure 8.4: α vs B. There are two clear peaks near 546.6 G and 547.4 G: The first is the s-wave
Fano-Feshbach resonance that we use to control a, and the second, much narrower feature we
attribute to a d-wave interspecies Fano-Feshbach resonance. Vertical error bars show the statisticaluncertainty in α, and horizontal error bars show an upper bound of ±0.03 G for the variation in
B during the measurement. Inset: α vs. B near the feature at 547.4 G. The solid, dotted, anddashed lines show the universal shape of an Efimov resonance, Eqs. (7.2) and (7.3), for η∗ = 0.1,η∗ = 0.02, and η∗ = 0.004. This data was published in Reference [117].

105

1 0 -2 3

1 0 -2 5

α(c m

6

/s )

1 0 -2 4

1 0 -2 6
1 0 -2 7-1 0 2-1 0 3a (a 0)

1 0 2

1 0 3

1 0 -2 3

α(c m

6

/s )

1 0 -2 4
1 0 -2 5
1 0 -2 6
1 0 -2 7

a (a 0)Figure 8.5: α vs. a for a < 0 (upper panel) and a > 0 (lower panel). The dashed green lines show
an a4 dependence with no Efimov resonances. The solid orange lines show Equations 7.2 and 7.2 forthe three-body parameter that was measured in 41 K–87 Rb–87 Rb by the LENS group [92]. A narrow
two-body Fano-Feshbach resonance blocks experimental access to the range −1090 ≤ a/a0 ≤ −840,
which is indicated by the shaded region on the upper plot. This data was published in Reference[117].

106
Table 8.1: Comparison of K–Rb–Rb Efimov resonances to universal theory. The ‘Three-bodyparameter’ column lists measurements or predictions of Efimov resonance locations. The a− /a0 ,a+ /a0 , and a∗ /a0 columns use Equation 7.7 to calculate the universal locations of 40 K–87 Rb–87 Rb
Efimov resonances corresponding to the same Efimov bound state described in the first column.
In order to show potential resonances associated with other Efimov bound states, some rows are
scaled by factors of λ = eπ/s0 .
Three-body parametera∗ = 230 a0a∗ = 230 a0 /λ
a+ = 2800 a0
a+ = 2800 a0 /λ
a− = −246 a0
a− = −λ × 246a0a∗ = 667 a0a∗ = 667 a0 /λSource40 K+87 Rb Experiment [117]40 K+87 Rb Experiment [117]Universal 3BP Theory [116]Universal 3BP Theory [116]41 K+87 Rb Experiment [92]41 K+87 Rb Experiment [92]41 K+87 Rb Experiment [92]41 K+87 Rb Experiment [92]a− /a0-55000-450-30000-250-246-30000-160000-1300

a+ /a0
5100
41
2800
23
22
2700
14000
120

a∗ /a0
230
1.9
130
1.1
1.0
130
667
5.4

onances like ours in Figure 8.2, where elastic atom-molecule collisions are also enhanced [97]. Sucha peak was tentatively identified by the LENS data in the 41 K–87 Rb–87 Rb system [92]. However,our data show no sign of such a narrow loss resonance or of any increase in the ratio of lost 87 Rb
atoms to lost 40 K atoms near a∗ = 230 a0 . We have since made more detailed followup measurements to search for even narrower resonances, but we still find no evidence for avalanche collisions
in either the atom loss rate or the ratio of lost 87 Rb atoms to lost 40 K atoms [123].

8.3

Comparison to other experiments
Armed with the measurements in Figures 8.2 and 8.5, what can we now conclude? In thisSection we examine the data in the context of universal Efimov theory as well as predictions basedon the universality of the three-body parameter. Table 8.1 lists measurements and predictionsof K-Rb-Rb Efimov resonance locations. The relationships between different resonance locations
from universal Efimov theory, which are listed in Equation 7.7, are used to calculate corresponding
values of a− /a0 , a+ /a0 , and a∗ /a0 where resonances would hypothetically be observable in ourdata. Due to the large scaling factor, λ = 122.7, many of these predicted resonances are outside ofour measurement range and therefore not testable in our current experiments.

107First, we check the self-consistency of our own measurements. Given our observed resonanceat a∗ = 230(30) a0 (Figure 8.2), universal relations predict a+ = 5100 a0 and a− = −55, 000 a0
which are outside of our measurement range. The absence of an Efimov resonance in Figure 8.5
is therefore consistent with universal theory. However, if there were to exist a more deeply boundEfimov state, then scaling by λ = 122.7 would give a+ = 41 a0 and a− = −450 a0 . The latterfeature in α is clearly excluded by our measurements. This suggests that the observed resonanceat a∗ = 230 a0 belongs to the most deeply bound Efimov state in our system. However, it is worth
noting that the ratio of |a− | and a∗ measurements have not always agreed with universal Efimov
theory [105, 111, 106, 26].
Next, we compare our results for 40 K–87 Rb to the previous measurements for 41 K–87 Rb. The
two mixtures share the same isotope of Rb, they both have resonant interspecies interactions with
rvdW = 72 a0 [18, 124], and their mass ratios differ by only a few percent. If the three-bodyparameter is universally determined by the two-body potential, one would expect 41 K–87 Rb and
40 K–87 Rb mixtures to have the same Efimov resonance locations [116]. However, our measurements

exclude a− near -246 a0 , where the 41 K–87 Rb–87 Rb Efimov resonance was observed. The 41 K–87 Rb
work also identified a feature in α at 667(1) a0 as a possible atom-molecule resonance [92]. For the
40 K–87 Rb mixture, we find a∗ = 230(30) a0 with direct measurements of atom-molecule loss and

we do not observe a corresponding feature in α.
Wang et al. predict a universal three-body parameter for heteronuclear mixtures, and specifically predict the locations of K–Rb–Rb Efimov resonances to be a+ = 2800 a0 and a− < −30000 a0[116]. The corresponding atom-molecule loss feature would be at a∗ = 130 a0 . Our measurementsof α exclude a+ = 2800 a0 and we find a∗ = 230(30) a0 from atom-molecule loss; both results
suggest that the value of a+ is slightly higher than the prediction. Intriguingly, the LENS group’s
measurement of a− = −246 a0 in the 41 K–87 Rb mixture is consistent with the presence of a moredeeply bound Efimov state whose location is related to Wang et al.’s predictions by the universal
scaling factor λ. Furthermore, recent measurements of Efimov resonances in 6 Li-133 Cs mixtures by
two different groups have also been similar to the predictions by Wang et al. [82, 83]. In these mea-

108

0 .4

L o g λ( a - / a u n i v e r s a l )

0 .2

0 .0

-0 .2

-0 .4
6

L i- 1 3 3 C s - 1 3 3 C s

4 0

K -87R b -87R b

4 1

K -87R b -87R b

A to m ic m ix tu r e

Figure 8.6: Measured three-body parameters in heteronuclear mixtures (points) compared against atheoretical universal three-body parameter for heteronuclear systems (dashed line) [116]. Individual
measurements have been rescaled by multiples of λ = eπ/s0 to facilitate a comparison of multiple
Efimov states. Solid blue diamonds show measurements of a− via three-body recombination [92,82, 83], the solid pink circle shows a value of a− that would correspond to our measurement ofa∗ from Figure 8.2, and the empty blue diamond shows a tentative assignment of a− that would
correspond to an ‘avalanche’ atom loss feature [92].

109
surements, the Efimov series also included a state that was more deeply bound than the theoreticalprediction. Figure 8.6 summarizes these comparisons.
Overall, we do not find clear evidence for a universal three-body parameter. However, given
the large Efimov scaling factor (122.7), the observed differences are relatively small. In addition,
the clearest features seen for both K–Rb mixtures occur at scattering lengths that are only a few
times larger than rvdW , where non-universal finite-range effects can play a role [125, 126, 127]. Thisrole may be greater for 40 K–87 Rb, where the Fano-Feshbach resonance has an intermediate strengthparameterized by sres = 2.0, compared to the strong 41 K–87 Rb resonance with sres = 26 [18]. On
the other hand, a universal three-body parameter has been observed in homonuclear 7 Li atoms [25]at Fano-Feshbach resonances with sres = 0.9 and in 39 K atoms with sres as low as 0.1 [110]. Aninteresting question is whether the strength of the Fano-Feshbach resonance has more influence on
the locations of Efimov features in heteronuclear systems.

Chapter 9

Concluding remarks

9.1

Summary
In this Thesis, I explored few-body interactions in an ultra cold mixture of 87 Rb and 40 K

atoms. I began by demonstrating a magnetic trapping system for the atoms that can also provide astable, uniform magnetic field to improve our control over the atoms’ interactions near a magnetic
Fano-Feshbach resonance. Our experiments rely on this resonance to access universal phenomena
where the interspecies interactions are strong.
Chapter 5 studied the conversion of atoms to fermionic Feshbach molecules using magnetic
sweeps. By combining a degenerate Fermi gas of 40 K atoms with a thermal Bose gas of minority
87 Rb atoms, we are able to pair as much as 45 percent of the 87 Rb atoms into 40 K87 Rb molecules.

However, we still measure significantly fewer molecules than would be expected from a modelthat has successfully described bosonic molecule formation in Bose or Fermi gases. Based on thisdiscrepancy, it seems likely that the Fermi statistics of the 40 K–87 Rb molecules exerts a strong
influence on the pairing process. In Chapter 6 we found that the kinetic energy of these molecules
corresponds to a temperature that is about 1.5 times larger than that of the atoms. Together,
the surprisingly low number and high temperature of the molecules prevent the formation of adegenerate Fermi gas of Feshbach molecules in our experiments.
In Chapters 7 and 8, we measured the rates of three-body collisions in the 40 K–87 Rb gas
mixture as a function of the interspecies scattering length. These measurements reveal a resonancein the collisions between 87 Rb atoms and 40 K87 Rb molecules that is connected to the universal

111
Efimov series of three-body bound states. We also measure a notable absence of features in theatom loss rate, in defiance of some theoretical predictions. In combination with a growing numberof measurements in diverse heteronuclear systems, these results will hopefully lead to a more quantitative understanding of three-body collisions between cold atoms. In the meantime, three-bodyinteractions are now more thoroughly characterized in the popular 40 K–87 Rb mixture.

9.2

Looking forward
The range of useful scattering lengths for our experiments is presently limited by the density

and temperature of the atom samples. In part, the minimum temperature and density are limitedby the fact that our method of supporting the atoms against the pull of gravity also has the effectof compressing the gas. It would be possible to achieve lower atom temperatures and densities
on an apparatus in low-Earth orbit, which would open the door to quantum physics experiments
at extremely high scattering lengths. The Cold Atom Laboratory project at the Jet Propulsion
Laboratory is scheduled to launch an experimental apparatus that creates ultra cold samples of 40 K
and 87 Rb atoms on the International Space Station in 2016. This will be an exciting development
for our field.
For those of us who are trapped down here on Earth, there are still plenty of experiments
within reach of terrestrial 40 K–87 Rb experiments. In terms of few-body physics, there is another
Fano-Feshbach resonance that overlaps the one that we used for the experiments in this thesis, andcouples to the same K Zeeman state but to a different Rb Zeeman state [124]. A spin mixtureof 87 Rb atoms with a polarized gas of 40 K atoms could have two different resonantly enhanced
interspecies scattering lengths, which is a situation where interesting few-body phenomena havebeen predicted [109]. On the other hand, we could push our experiments to colder temperaturesand study many-body effects in fully quantum-degenerate gases. One particularly exciting new
development in our lab has been the implementation of a so-called ‘painted’ optical potential,
which rapidly modulates the position of an optical trap beam to create a time-averaged potential.
This can create a larger, more harmonic, potential than the optical traps that were used in this112
thesis. This flexible potential, whose shape can be changed dynamically during an experiment, sets
the stage for a variety of exciting measurements of many-body physics.

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