Tunable dipolar interactions and collisional shielding in aquantum gas of polar moleculesbyKyle Matsuda
A.B., Harvard University, 2016

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
2022

Committee Members:
Jun Ye, Chair
Ana Maria Rey
John L. Bohn
Eric A. Cornell
J. Mathias Weberii
Matsuda, Kyle (Ph.D., Physics)
Tunable dipolar interactions and collisional shielding in a quantum gas of polar molecules
Thesis directed by Prof. Jun Ye

Ultracold polar molecules open new directions in quantum science owing to their complex
internal structure and long-range dipolar interactions. Creation of a quantum-degenerate moleculargas, described in the first part of this thesis, is an important step toward unlocking the rich potentialof molecular systems. However, simultaneously controlling both elastic and inelastic molecularinteractions has remained an outstanding experimental challenge. Strong dipolar interactions havegenerally led to even stronger molecular losses, preventing the creation of deeply-degenerate andstrongly dipolar molecular gases.
In this thesis, we demonstrate several methods for tuning molecular interactions with external
electric fields. First, by tightly confining the molecules into a two-dimensional (2D) geometry andapplying a strong electric field, we induced repulsive dipolar interactions that suppressed molecularlosses while enhancing the elastic collision rate, allowing for direct evaporation of the molecules
to below the Fermi temperature in 2D. Second, at particular electric field strengths, we observed
resonant collisional shielding of rotationally excited molecules. At these resonances, the molecularloss rate could be tuned over nearly three orders of magnitude, and could be suppressed by up toa factor of 10. Third, we leveraged our highly controllable electric field to address and select single
2D layers of molecules, which allowed the observation of chemical reactions induced by interlayer
dipolar interactions. Finally, we studied the intralayer interactions between molecules placed ina coherent superposition of rotational states, a first step toward engineering the collective spin
dynamics in this system with long-range interactions.

Dedication

To Rich and Leslie, George and Irene, and Fujio and Amy.

Acknowledgements

Jun, thank you for everything — your support and guidance over the years has meant so muchto me. While you have shared an immense amount of physics insight and engineering knowledge,
more important than any technical lesson was seeing your infectious enthusiasm and optimism forscience. I hope to emulate your passion and fearlessness in solving problems wherever I go.
Thank you to my wonderful colleagues on the KRb experiment. Jake Covey built the newexperiment and got us all up to speed on running the machine. For the first several years, the KRbteam was postdocs Luigi De Marco and Giacomo Valtolina along with grad students Will Tobias
and me, and later joined by postdoc Jun-ru Li. We had a lot of fun working together in those firstfew years, even throughout all of the challenges we faced in getting the new machine working. It’s
hard to express here just how much it meant to work with this team, so I’ll keep it brief. Luigi,
Giacomo, and Jun-ru, you were such great mentors and I feel extremely lucky to have learned fromyou. All of you took me under your wing in different ways, and I am so grateful. Will and I started
on KRb at the same time and spent six years working together. Will, I learned a ton from working
with you, both from your electronics and laser knowledge as well as your careful and methodical
approach to thinking through scientific problems. We went through a lot of ups and downs together
on the experiment, but we made it! The project is in great hands with the new team of Cal Miller,
Annie Carroll, and Jake Higgins, and I’m excited to see what they will go on to do next.
Thank you also to the other members of the Ye group, for your scientific help and for yourfriendship, especially Will M., Ross, Lingfeng, Lindsay, Aki, Sara, Ed, Toby, Alex, Colin, Kyungtae,
John, Dhruv, Tanya, Kia, Will C., Lee, Qizhong, Chuankun, Jutta, Yewei, Justin, Kameron, Dave,v
Hao, Roman, Xin, and Vandy. Special thanks to Lee Liu, who was kind enough to read an earlyversion of Chapter 2 of this thesis and provide helpful suggestions.
We had the opportunity to discuss and collaborate with some of JILA’s amazing theorists,
including Ana Maria Rey, Thomas Bilitewski, Peiru He, John Bohn, Reuben Wang, and JoseD’Incao. It was a privilege to collaborate with external theory groups as well — Goulven Quéménerand Lucas Lassablière from Université Paris-Saclay, and Peter Zoller and Andreas Kruckenhauser
from Innsbruck. Thank you also to Eric Cornell and his group for many helpful discussions while
we were getting the machine up and running. JILA also has a fantastic support structure of bothtechnical and administrative staff — thank you to Terry Brown, Carl Sauer, Felix Vietmeyer, James
Fung-A-Fat, Hans Green, Kim Hagen, James Ulrich, J.R. Raith, Corey Keesling, Brian Lynch, Amy
Allison, Agnieszka Lynch, Krista Beck, Beth Kroger, and so many others for your help.
Thank you to John Doyle at Harvard for introducing me to experimental AMO physics as anundergrad. Before I joined John’s lab, I wasn’t sure about majoring in physics, let alone going tograd school. I had a wonderful experience in John’s lab learning from grad students Ivan Kozyryev
and Louis Baum. They were great teachers and so generous with letting me get involved in theexperiment. John’s incredible passion for cold molecule research also inspired me to seek out
molecule experiments for my graduate research.
Outside of physics, thank you to the members of the wonderful jazz scene at CU and in
Boulder and Denver with whom I had the pleasure of playing music over the past several years.
I feel lucky to have found such a welcoming musical community here. Thank you to the friends
who helped make my time in Colorado fun (in addition to those mentioned above), especially AJ,
Cecilia, Claudia, Isabel, Max, Kat, Clare, Pete, Scott, Adam, Franz, Gary, BK, and Jonathan.
Thank you to my parents Rich and Leslie and brother Dan for your unconditional love andsupport. I am fortunate to have a big, crazy, and wonderful extended family — thank you to my
grandparents, aunties and uncles, and cousins who have been there throughout everything. Most
importantly, thank you to my partner Angela. This thesis wouldn’t have been possible without
your endless patience, encouragement, and support.

Contents

Chapter
1 Introduction

1

1.1

Making ultracold molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Ultracold molecular losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Our work: Controlling interactions with electric fields . . . . . . . . . . . . . . . . .

62 Ultracold polar molecules and dipolar interactions

9

2.1

Energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2

Molecular Hamiltonian in the ground electronic and vibrational state . . . . . . . . . 10

2.3

Dipole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1

Calculating the matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2

Induced dipole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.3

Transition dipole moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.4

AC Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4

Dipolar interactions – Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5

Dipolar interactions – In the bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5.1

General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.2

Reactive losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.3

Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5.4

Resonant shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vii
2.6

2.7

Dipolar interactions – In the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.1

Two-particle calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6.2

Calculating the couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6.3

Beyond the p = 0 term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Apparatus and experimental sequence

43

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2

Optical trap setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3

Electric field control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4

3.5

3.3.1

Back-of-the-envelope requirements . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.2

Computing the field distribution . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.3

Useful electrode linear combinations . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.4

Minor imperfections of the electrode geometry . . . . . . . . . . . . . . . . . 58

3.3.5

Controlling the electrode voltages . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.6

FPGA control of AD5791 DACs . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.7

Cabling and connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.8High voltage rack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Radio-frequency fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4.1

Amplitude-shaped pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.2

Limitations of the rods: polarization purity and AC field homogeneity . . . . 69

CCD calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Degenerate Fermi gas of molecules in 3D

75

4.1

Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2

Properties of the molecular Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.1

TOF thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2.2

Reduced loss rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

viii
4.2.3
4.3

4.4

Density fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Feshbach ramp thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.1

Collisional damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3.2

Dependence on the Feshbach ramp rate . . . . . . . . . . . . . . . . . . . . . 92

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Dipolar evaporation of molecules in 2D
5.1

94

Producing a 2D sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.1

Large spacing lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1.2

Layer counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.1.3

Loading procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2

Loss suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3

Elastic dipolar collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4

Direct evaporation of molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5

5.4.1

Combined optical + electric potential . . . . . . . . . . . . . . . . . . . . . . 111

5.4.2

Evaporation trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4.3

2D Fermi gas of molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6 Resonant shielding of molecules in 2D and 3D
6.1

120

Physical mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1.1

Two-channel picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.2

Shielding in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3

Probing the collision anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.4

6.3.1

Contribution of mL states vs θ . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.3.2

Shielding in mL = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Shielding in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4.1

Elastic collisions at the shielding field . . . . . . . . . . . . . . . . . . . . . . 135

ix
6.4.2
6.5

Direct evaporation of molecules in 3D . . . . . . . . . . . . . . . . . . . . . . 138

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7 Single-layer control of molecules in a 1D optical lattice

141

7.1

Addressing a single layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.2

Magic angle electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.3

Interlayer dipolar spin-exchange interactions . . . . . . . . . . . . . . . . . . . . . . . 147

7.4

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8 Dipolar density shifts in 2D
8.1

152

Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8.1.1

More discussion of χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8.1.2

Validity of the spin model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.2

Dynamical decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.3

Density shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.4

Tuning the interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.5

Dynamically reversing the evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.6

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9 Conclusion and Outlook

168

Bibliography

171

Appendix
A KRb energy levels in external fields

189

A.1 Molecular Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
A.2 AC Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
A.3 Comparison to experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

x

TablesTable
2.1

Coupling strengths for equation (2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

Some handy dipole matrix elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1

Polarizabilities for K, Rb, KRb*, and KRb at 1064 nm . . . . . . . . . . . . . . . . . 48

4.1

Atom and molecule conditions for Figure 4.2 . . . . . . . . . . . . . . . . . . . . . . 78

Figures

Figure2.1

Cartoon of molecular energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Energies of N = 0 and 1 states as a function of E, including hyperfine structure . . . 13

2.3

Energy and induced dipole moment as a function of the applied electric field

2.4

Composition of field-dressed eigenstates as a function of E . . . . . . . . . . . . . . . 19

2.5

Transition dipole moments vs E

2.6

Effect of hyperfine structure on transition dipole moments . . . . . . . . . . . . . . . 21

2.7

Cartoon of differential light shifts in a non-magic trap . . . . . . . . . . . . . . . . . 22

2.8

Limiting cases of α(θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.9

p-wave barrier and dipolar anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 29

. . . . 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.10 Adiabatic energy curves vs d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.11 Adiabatic energy curves from quasi-2D to 2D . . . . . . . . . . . . . . . . . . . . . . 32
2.12 aD and ED vs d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.13 Energies of rotational state pairs vs  . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.14 Ramsey fringes for two interacting molecules . . . . . . . . . . . . . . . . . . . . . . 40
2.15 Electric field dependence of J⊥ and Jz . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1

Schematic of the JILA KRb machine . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2

Atomic level diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3

Relevant atomic hyperfine states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

xii
3.4

Optical trap geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5

Plug, ODT2, LSL, and imaging optics layout . . . . . . . . . . . . . . . . . . . . . . 49

3.6

ODT1 and side imaging optics layout . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.7

Geometry of the electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.8Some useful electrode linear combinations . . . . . . . . . . . . . . . . . . . . . . . . 57

3.9

Simplified block diagram of the electric field control . . . . . . . . . . . . . . . . . . 59

3.10 Corona sphere for HV connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.11 Transfer function of the HV cabling . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.12 Heating in the optical lattice from HV amp vibrations . . . . . . . . . . . . . . . . . 63
3.13 High voltage rack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.14 Block diagram of the K/KRb RF setup . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.15 Example Ramsey fringe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.16 AC rod configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.17 Probing N = 0 ↔ 1 while dressing 1 ↔ 2 . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.18 Calibration of the amplitude shaping setup . . . . . . . . . . . . . . . . . . . . . . . 70
3.19 Comparison of lineshapes using square and Blackman pulses . . . . . . . . . . . . . . 70
3.20 Inhomogeneous AC electric field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.21 Calibration of Andor iXon Ultra 888 CCD . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1

Cartoon of molecule production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2

Molecule production from a degenerate Bose-Fermi mixture . . . . . . . . . . . . . . 78

4.3

TOF thermometry of the molecular DFG . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4

Molecular loss measurements at T = 65 and 300 nK . . . . . . . . . . . . . . . . . . 83

4.5

Reduction of the chemical reaction rate at low T /TF . . . . . . . . . . . . . . . . . . . 84

4.6

Comparison of T /TF from fluctuations and TOF expansion . . . . . . . . . . . . . . 87

4.7

STIRAP-induced slosh of KRb* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.8

Collisional damping from K-KRb* elastic collisions . . . . . . . . . . . . . . . . . . . 91

xiii
4.9

Summary of K-KRb* elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.10 KRb T /TF versus Feshbach ramp rate . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1

Cartoon of intra- vs inter-band collisions . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2

Optical trap geometry for 2D work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3

Band mapping K and KRb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4

LSL optics setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.5

Parametric heating in the LSL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.6

Rb density distribution in the ODT vs LSL . . . . . . . . . . . . . . . . . . . . . . . 102

5.7

Rb layer distribution from matter-wave imaging . . . . . . . . . . . . . . . . . . . . . 104

5.8

Matter-wave magnification to align the LSL . . . . . . . . . . . . . . . . . . . . . . . 107

5.9

Dipolar suppression of loss in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.10 Crossover from 3D to quasi-2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.11 Elastic dipolar collisions in quasi-2D . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.12 Combined optical and electric potential . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.13 Evaporation sequence and trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.14 2D Fermi gas of molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.15 Expansion energy of the 2D molecular Fermi gas . . . . . . . . . . . . . . . . . . . . 117
6.1

Vshield (Rb ) as a function of |E| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.2

Two-channel picture of the shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3

Angular dependence of the shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.4

Large variation of β near the shielding resonance . . . . . . . . . . . . . . . . . . . . 128

6.5

Measurement of the shielding resonances . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.6

Collisions in quasi-2D with tilted E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.7

Collisions of |0, 0i in tilted E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.8

Summary of 2D shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.9

Resonant shielding in 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

xiv
6.10 Thermalization measurements at |Es |. . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.11 Evaporation of molecules in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.1

Layer-resolved spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.2

Spectroscopy on many layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.3

Rotational coherence of |0i and |1i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.4

Cartoon of dipolar exchange induced losses . . . . . . . . . . . . . . . . . . . . . . . 147

7.5

Losses induced by interlayer dipolar exchange . . . . . . . . . . . . . . . . . . . . . . 149

7.6

Dependence of the interlayer dipolar exchange on ∆ . . . . . . . . . . . . . . . . . . 150

8.1

Cartoon of the spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2

Dependence of χ on |E| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.3

Contrast decay with and without dynamical decoupling . . . . . . . . . . . . . . . . 160

8.4

Dynamical decoupling sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.5

Measurement of the dipolar density shift . . . . . . . . . . . . . . . . . . . . . . . . . 162

8.6

Tuning χ with |E| and α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.7Internal state dependence of χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.8

Reversing the spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

A.1 Microwave spectrum at |E| = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.2 Microwave spectrum at |E| = 1 kV/cm . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Chapter 1

Introduction

Controlling interactions is a central theme in quantum science. In quantum computation,
controlled interactions between individual quantum bits (or “qubits”) generate entanglement, thekey resource needed to achieve a quantum advantage over classical computers [1, 2, 3]. In quantum
sensing and metrology, interactions can be harnessed to create highly entangled states such as
squeezed states, which can have tremendously increased sensitivity over classical states [4, 5, 6].
And in quantum simulation, interacting systems can be used to create synthetic materials in thelab that are more easily tuned and probed than their real life counterparts, including novel statesof matter that have never been observed in the solid state [7, 8, 9].
Modern atomic physics has demonstrated the profound insights that can be gained by controlling the interactions between atoms, nature’s building blocks, to the highest degree. Quantumdegenerate gases, where the average spacing between particles is on the order of their de Broglie
wavelength, are the experimental platform for many of these explorations. In these gases, normally subtle effects like quantum statistics play a leading role — ultracold bosons condense intoa Bose-Einstein condensate (BEC) [10], where all of the atoms in the gas act in unison, while
ultracold fermions form a degenerate Fermi gas (DFG) [11], mimicking the behavior of electrons
in metals. The exquisite control afforded by ultracold gases has enabled beautiful experiments, including the direct observation of a quantum phase transition between a superfluid and an insulatingstate [12, 13]; the engineering of a smooth crossover in a fermionic gas between a Bardeen-CooperSchrieffer (BCS) superfluid, where the atoms are paired in momentum space as in conventional

2superconductors, and a BEC of loosely-bound Feshbach molecules, where the atoms are paired inreal space [14, 15, 16, 17]; and the creation of incredibly precise atomic clocks [18] that can sense
the gravitational redshift over a 1 mm scale [19].
These amazing advances depended on the ability to tame and control atomic interactions.
Even creating a quantum-degenerate gas, often the starting point for current experiments, requires favorable atomic properties to prevent undesired losses and to maintain thermal equilibrium
throughout the cooling process [20]. Fortunately, the dominant inelastic processes in alkali atoms
could be brought under control, which opened a path to the creation of BEC in 1995 [21]. In addition, the elastic collisions of the atoms were found to be highly tunable via Feshbach resonances[22], which allowed the interactions to be changed from attractive to repulsive and from zero to
extremely strong, giving a powerful method of controlling the many-body physics of the gas [7].
The scientific richness of ultracold atomic systems inspires the following question: What if
we consider more complicated quantum objects?
Molecules, with their chemical diversity and increased complexity over atoms, have the potential to open an even wider scientific frontier. Even simple diatomic molecules possess complicatedinternal structure and strong electric dipolar interactions that are desirable for applications including precision measurement, quantum metrology, and quantum simulation [23, 24, 25, 26, 27].
Cold (. 1 K) and ultracold molecules (. 10 µK) have already been used in the lab to shed lighton questions from molecular scales (chemistry) all the way up to cosmic ones (particle physics).
The production of cold molecules, which have only a few internal states populated, has enabledcontrolled chemistry [28, 29] and the study of fundamental quantum processes like scattering resonances [30, 31]. In the ultracold regime, where the relative motion of two colliding molecules isquantized, effects like quantum statistics can control the chemical reaction rate in the gas [32].
Starting from a well-defined initial quantum state, one can perform state-to-state chemistry andobserve how the product quantum state depends on that of the reactants [33, 34, 35]. At theother end of the spectrum, cold molecules have been used to probe the violation of fundamental
symmetries — in this case, the product of charge and parity reversal (CP) — by setting limits on

3
the electric dipole moment of the electron [36, 37]. These tabletop measurements are a probe of
CP-violating physics at energy scales comparable to those probed by the LHC, and are enabled, inpart, by using specific quantum states of carefully chosen molecular species [38, 39].
If we can cool molecules deep into the quantum-degenerate regime, where experiments withatoms have uncovered such a wealth of quantum many-body phenomena, we should be in for a treat.
In contrast to the contact interactions of atoms, polar molecules exhibit long-range and anisotropic
dipolar interactions [24, 26]. Fascinating new phases of matter have been predicted to emerge asa result, including spin liquids [40], Wigner crystals [41], p-wave superfluidity and unconventional
pairing [42, 43], and more. When confined to individual optical lattice sites, molecules can be used
to simulate spin Hamiltonians with highly tunable couplings [44, 45, 46]. With molecules in thequantum regime, it is possible to envision engineering tunable molecular quantum matter that is
tailored for certain precision measurements, in the spirit of recent work on optical clocks [47, 48].
As one example, dipolar interactions of molecules in 2D are predicted to generate spin-squeezed
states that could be used for precision metrology [49].
Realizing the full promise of ultracold molecules requires gaining full control of molecularinteractions. Nature has given us a good challenge here: taming molecules is a much trickier
task than taming atoms. The very complexity that makes molecules fascinating to study also
makes them difficult to control, even for the simple diatomic molecules discussed in this thesis. In
particular, controlling the interactions of molecules has proven difficult, preventing the creation ofquantum-degenerate gases until recently [50, 51, 52, 53, 54]. In the next sections, we discuss how
ultracold molecules are produced and the challenges in controlling their interactions.

1.1

Making ultracold molecules
Nearly every ultracold atom experiment starts with a laser cooling stage that brings atoms

from room temperature (or higher) down to typically several hundreds of microkelvin. Unfortunately, laser cooling is impractical for a generic molecule. Laser cooling relies on repeated cycles of
optical excitation and spontaneous emission to exert a force. Any population that decays into an-

4other internal state, outside of the main cooling transition, must be repumped using an additionallaser wavelength. For commonly used atoms, a single repump laser is enough to close the cycle andscatter hundreds of thousands of photons. Molecules have so many more internal states than atoms
that the number of repump lasers needed can quickly become impractical. In recent years, there
has been rapid progress in identifying molecules with convenient structure for laser cooling. Thesemolecules have an excited electronic potential with a very similar shape to the ground electronic
potential, which means that the chance of changing vibrational states during a spontaneous emission event is low (in molecule jargon, “the Franck-Condon factors are highly diagonal”), drasticallyreducing the number of repump lasers needed [55]. By a clever choice of the rotational states used
for cooling, leakage of molecules into higher rotational states can also be prevented [56]. With theseadvances, magneto-optical traps (MOTs) of diatomic molecules such as SrF [57], CaF [58, 59], andYO [60] have been produced, and even more complex polyatomic molecules are starting to be laser
cooled [61, 62, 63]. Despite these exciting advances, samples produced by direct laser cooling haveso far been limited to phase space densities (PSDs) several orders of magnitude away from quantum
degeneracy [64].
Before the advent of direct laser cooling of molecules, an alternative approach was pioneered
at JILA: simultaneously cool two different species of atoms down to near quantum degeneracy, andthen coherently associate pairs of atoms into molecules. This cooling method has the advantagethat molecules inherit the ultralow temperature of the parent atoms, and thus are created at a
PSD approaching quantum degeneracy. The association process typically consists of two steps:(1) the magnetic field is swept through a Feshbach resonance, adiabatically converting pairs ofcolliding atoms into loosely bound molecules [22], and (2) lasers are used to coherently convert the
Feshbach molecules into rovibrational ground state molecules via a process called stimulated Raman
adiabatic transfer (STIRAP) [65]. This approach led to the successful formation of a quantum gasof 40 K87 Rb molecules in 2008, at a temperature just above quantum degeneracy [66]. Since then,many other molecule species have been produced near quantum degeneracy, such as RbCs [67, 68],

5
NaK [69, 70, 71], NaRb [72], NaLi [73], and NaCs [74, 75].1

1.2

Ultracold molecular losses
With the creation of near-quantum-degenerate KRb came the surprising observation of ul-

tracold chemical reactions, even at ultralow temperatures. The KRb + KRb system is chemicallyreactive, with KRb + KRb → K2 + Rb2 exothermic by about 14 K [32], and these bimolecularchemical reactions were found to be the dominant loss process of the gas. With the molecules
colliding in a single partial wave, this system was an exciting platform for studying fundamental
connections between basic quantum mechanics and chemistry. In particular, the quantum statisticsof the particles was extremely influential on the loss rate — KRb molecules in the same internal
state, being identical fermions, must collide with odd motional angular momentum to have an overall antisymmetric wavefunction. In the ultracold regime, they collide in the lowest p-wave (L = 1)
channel and must tunnel through the centrifugal barrier to get close enough to undergo a chemical
reaction. However, if the molecules are put into a statistical mixture of internal states, then they
can undergo s-wave collisions, which have no barrier. Thus, by controlling a single nuclear spinstate, the chemical reaction rate could be changed by two orders of magnitude [32].
Those experiments demonstrated how long-range potential barriers, despite being much lower
than typical chemical energy scales, play a key role in ultracold chemical reactions. Later studies
showed that dipolar interactions could also be used to change the height of the centrifugal barrier,hence changing the loss rate [79]. Owing to the anisotropic nature of the dipolar interaction,
attractive head-to-tail collisions dominated the behavior and led to a sharply increasing loss withincreasing interaction strength. In conjunction with these experiments, theorists developed models
to explain the observed loss rates. They found that the loss of KRb was “universal”, in the sensethat the strong short-range reactive loss simplified understanding of the collisions, making the
behavior depend only on the long-range potentials [80, 81, 82, 83].
1

A drawback of this method is that it has so far been limited to molecules consisting of two alkali atoms(“bialkalis”), though alkali + alkaline-earth combinations are under investigation [76, 77, 78].

6
While the reactive loss provided an interesting and fruitful research direction, it also represented a roadblock for studying quantum many-body physics with the molecules. One approach tosuppressing the loss was to load the molecules into a deep three-dimensional (3D) optical lattice[84]. Since each lattice site had at most one molecule due to fermionic statistics, the chemical
reactions were shut off, enabling the observation of many-body spin dynamics due to resonant
dipolar spin exchange between molecules [46, 85]. Molecules loaded into optical tweezers [86, 74]
are similarly protected from reactive losses.
However, to create the most general and versatile molecular system, one would like theability to control the losses even in a bulk system. Many proposals require a deeply-degenerate gasof molecules, where the thermal energy is low enough for the interactions to be dominant. Ideally,
one could use evaporative cooling [20], one of the workhorse techniques of atomic physics, to further
cool the gas into deep degeneracy. However, successful evaporation depends on a favorable ratio ofelastic (“good”) to lossy (“bad”) collisions, which was hindered by the strong reactive losses.
This motivated other research groups to choose molecular species such as RbCs, NaK, andNaRb, in which this chemical pathway is endothermic [87]. In principle, these molecules should
be non-reactive, but in practice they have mostly exhibited (near-)universal loss [67, 69, 88, 89].
Investigation of the microscopic mechanism for the loss is still ongoing. Some experiments see
evidence for losses of collisional complexes [90, 34] due to the optical trapping light [91, 92, 93],
while others see no effect [94, 95]. Despite these challenges, there has been exciting progresson control of short-range atom-molecule collisions, with the recent observation of atom-molecule
Feshbach resonances by two groups [96, 97, 98].

1.3

Our work: Controlling interactions with electric fields
In 2018, we were able to create the first quantum-degenerate gas of polar molecules by startingfrom a deeply-degenerate atomic mixture (Chapter 4). Similar results were also achieved by the
Munich group in NaK in 2021 [53]. However, in both cases the molecules were produced at zero
electric field and hence had zero dipolar interaction, and the dominant interaction in the gas was7still short-range losses. Subsequently, we turned our focus to the long-range dipolar interactions asa means of controlling the molecular collisions. By engineering long-range barriers to prevent themolecules from reaching short range, one can suppress the molecular losses while also turning onelastic dipolar collisions.
We were able to demonstrate two complementary methods for controlling the molecular collisions. First, by tightly confining the molecules along the direction of the applied electric field, we
realized a quasi-two-dimensional (quasi-2D) system where the attractive head-to-tail dipolar collisions were suppressed [99, 100, 101, 102]. In 2D, the repulsive side-to-side collisions stabilized andthermalized the gas, allowing for efficient evaporative cooling of the molecules to below the Fermi
temperature (Chapter 5). Second, we observed resonant electric field shielding of the molecules[103, 104]. By tuning the strength of the applied electric field, dipole-coupled molecular states
could be made resonant, leading to a suppression of the loss in both 2D and 3D (Chapter 6).2
Both of these methods rely heavily on a large and tunable external electric field to polarize themolecules and control their dipolar interactions. The in-vacuum electrodes in our second-generation
JILA KRb machine [106], combined with precision analog control electronics [107], allow us toapply stable electric fields of up to 15 kV/cm on the molecules. Additionally, we can apply flexible
electric field gradients to control the potential energy landscape experienced by the molecules, andmanipulate the direction of the molecular dipoles by tilting the electric field vector. These newtechnical capabilities (Chapter 3) were an essential ingredient for the experiments described in thisthesis.
Since the dipolar interaction is long ranged, spatial control over the molecular distribution
also provides a means of controlling interactions in the gas. Leveraging our precise electric field
control, we were able to spectroscopically select individual 2D layers of molecules using an electricfield gradient. This layer-by-layer control of the molecular internal state allowed us to observe andcontrol dipolar spin exchange interactions between separated layers (Chapter 7). Most recently,
2

For another complementary approach, see the microwave shielding work by the Harvard [105] and Munich [54]groups, which also leverages the long-range dipolar interactions between molecules.

8
we have been spectroscopically studying the effects of dipolar interactions in the gas using Ramsey
spectroscopy, where the interactions manifest at lowest order as a density-dependent frequency shiftof the molecular transition (Chapter 8).

Chapter 2

Ultracold polar molecules and dipolar interactions

Here, we give a short primer on ultracold polar molecules and their dipolar interactions,focusing on the aspects that are most relevant for the experiments in this thesis. We will primarily
describe the behavior of 1 Σ molecules like KRb, though some of the general features may be
applicable to molecules with more complicated internal structure. First, we give a quick overview
of the molecular structure focusing on the most relevant states for our purposes, the rotational
states of the ground electronic and vibrational manifold. Next, we show how to calculate theinduced and transition dipole moments for the rotational states, and discuss their behavior as afunction of the external electric field. We also discuss how to control the AC polarizability of themolecules, an important ingredient for achieving long coherence times in an optical trap. Finally,
we conclude by describing some features of the dipolar interaction between molecules.

2.1

Energy scales
A major motivation for studying polar molecules is their rich internal structure [23, 27].

Compared to atoms, molecules have additional internal degrees of freedom, namely vibrations androtations, that can be harnessed for quantum applications.1 The experiments presented in thisthesis are primarily concerned with the molecular rotations, which are long-lived and interact via
long-range and anisotropic dipolar interactions.
1

An illustrative example is the 3 ∆1 state used for the electron electric dipole moment searches [36, 37]. In thoseexperiments, a careful choice of molecule and of its internal state provides numerous benefits: enormous (∼ GV/cm)
internal electric fields that can be controlled by small (∼ V/cm) fields in the lab, very low magnetic field sensitivity,and hyperfine states that act as an internal co-magnetometer [38, 39].

10

Figure 2.1: Cartoon of molecular energy scales.

To discuss the internal states of the molecule, it is helpful to invoke the Born-Oppenheimer
approximation, which says that we can treat the electronic and nuclear degrees of freedom separately[108]. The approximation is based on the observation that electronic energy scales are typically far
beyond vibrational or rotational energies, so the nuclear motion may be treated as adiabatic withrespect to the electronic states. While the BO approximation can break down in some interestingcases, particularly where two electronic potentials intersect, it is appropriate for the states of KRb
that we will work with in this thesis. Within the BO approximation, we can cleanly separate theelectronic, vibrational, and rotational wavefunctions: the electronic potentials (∼ 100s of THz)
support a number of bound states, which are the vibrational states (∼ 10s of THz); the vibrational
wavefunctions determine the average separation of the nuclei, which determines the moment ofinertia of the molecule and sets the rotational energy scale (∼ 1s of GHz). This hierarchy ofenergies is depicted schematically in Figure 2.1.

2.2

Molecular Hamiltonian in the ground electronic and vibrational state
The experiments in this thesis are performed in the ground X 1 Σ+ (v = 0) electronic andvibrational state of KRb. In this relatively simple state, with no electronic spin or orbital angularmomentum, and no vibrational quanta, the molecule can be imagined as two atoms connected bya rigid stick (the bond). Neglecting hyperfine states, the only degree of freedom left is the rotation

11of the molecule, described quantum mechanically by the rotational angular momentum N and its
projection mN onto the ẑ axis. The energies are given by the rigid rotor Hamiltonian,
Hrot = Bv N2 = Bv N (N + 1),(2.1)

where Bv = ~2 /2I is the rotational constant, and I is the moment of inertia of the molecule in the
ground vibrational state.
Unfortunately, the picture given by equation (2.1) is a bit too simple. Generally, the atoms
also have nuclear spin, which can complicate the spectrum due to couplings with the rotational
states. In our case, 40 K has IK = 4 and 87 Rb has IRb = 3/2, which give a total of 9 × 4 = 36
hyperfine states. We typically use the uncoupled basis |N, mN , mK , mRb i to describe the system,with mK and mRb the nuclear spin projections for K and Rb, respectively.
Taking into account the rotational and hyperfine structure and external magnetic and electricfields B and E, one obtains the following Hamiltonian [109],
H = Hrot + HS + Hhf + HZ

(2.2)

Hrot = Bv N2(2.3)

HS = −d · E

(2.4)

Hhf =

X

Vi · Qi +i=K,Rb

HZ = −gr µN N · B −X

ci N · Ii + c3 IK · T · IRb + c4 IK · IRb(2.5)

gi µN Ii · B(1 − σi ).

(2.6)

i=K,RbX
i=K,Rb

The couplings are given in Table 2.1, and full matrix elements are given in Appendix A. For
simplicity, we will assume throughout this chapter that B = Bẑ and E = Eẑ.
The hyperfine Hamiltonian consists of several terms. The first term describes the interactionof the electrons with the electric quadrupole moment Qi of the nucleus, which depends on the
nuclear spin Ii . The VK · QK term couples ∆N ≤ 2, |∆mN | ≤ 2, and ∆mN = −∆mK (andsimilarly for VRb · QRb ), although in practice states with different N are too far in energy tohave an appreciable mixing by the interaction. The second term in the Hamiltonian describes theParameter
Bv(eqQ)K(eqQ)Rb
cK
cRb
c3
c4
gr
gK
gRb
σK
σRb

Value
1.1139514(5) GHz
0.452(9) MHz−1.308(9) MHz−24.1 Hz420.1 Hz−48.2 Hz−2030.4 Hz
0.0140−0.3241.834
1321 × 10−63469 × 10−6

12

Table 2.1: Coupling strengths for equation (2.2). Values in the first section are experimental
values from Ref. [110]. Values in the second section are theoretical values from Ref. [109].

interaction between the rotation of the molecule and the individual nuclear spins, and the final two
terms describe the tensor and scalar spin-spin interactions. The Zeeman Hamiltonian contains theinteraction of the rotation and screened nuclear spins with B.
The hyperfine effects become very important at low electric field, leading to a maze of avoided
crossings for the N > 0 states as the field strength E is increased. Figure 2.2 shows the energiesof the N = 0 and 1 states in the low E region. While the N = 0 states (left panel) do not show
any avoided crossings, the N = 1 states (right panel) show a number of avoided crossings due tothe mixing from the electric quadrupole terms. Some of the avoided crossings are fairly narrowand difficult to ramp through adiabatically, so for some of the work in this thesis (particularly
Chapter 6) it was helpful to produce the molecules at higher E above the crossings. At low E, thestrong mixing from Hhf can also cause changes in properties like the transition dipole moments anddifferential polarizabilities between states.
Thankfully, we often do not need to consider the full hyperfine structure of the molecule. At
large E, the energies are dominated by HS and the nuclear spins are essentially decoupled from therotations, so the hyperfine structure is largely irrelevant when working within a single (mK , mRb )manifold. For most of the work described in this thesis, we will work in this regime. In the next

13
2230

0

Energy (MHz)

Energy (MHz)

2229

-3

2228

2227

-6

2226
0

200

400
E (V/cm)

600

0

200

400
E (V/cm)

600

Figure 2.2: Energies of N = 0 (left) and N = 1 (right) states as a function of E, including
hyperfine structure. A magnetic field of B = 545.9 G is also present. Both fields point along the ẑ
direction.

section, we focus on the rotational states and their dipolar characteristics, ignoring the hyperfine
structure unless otherwise noted.

2.3

Dipole moment
Polar molecules provide a very convenient experimental “handle”: their electric dipole mo-ment d. Classically, d describes how the molecule responds to E, according to U = −d · E.
Physically, d comes from the polarization of the electron distribution, which points along the internuclear axis in the frame of the molecule. At E = 0, the dipole moment averages to zero in thelab frame, but as we turn on E, the molecules start to align along the field, leading to an induced
dipole moment in the lab frame.
To see this quantum mechanically, we need to study the behavior of
H = Hrot + HS = Bv N2 − dz E,

(2.7)

14
where E = Eẑ and dz is the dipole operator along ẑ. At E = 0, H is rotationally symmetricand hence the eigenstates |N, mN i have zero induced dipole moment, hN, mN |dz |N, mN i = 0. By
turning on E, we mix adjacent rotational levels N and N ±1, leading to new field-dressed eigenstates|Ñ , mN i that have a non-zero induced dipole moment, hÑ , mN |dz |Ñ , mN i =6 0. We define |Ñ , mN i
as the field-dressed eigenstate that adiabatically connects to the bare rotational state |N, mN i as
E → 0. Note that mN is still a good quantum number when E k ẑ.
Since the neighboring rotational states are mixed by E, there must be a dipolar coupling
between the bare states, hN + 1, mN |dz |N, mN i 6= 0, which we call the transition dipole moment
between the two states. In addition to coupling to a static electric field, the transition dipole
moment allows us to drive Rabi oscillations between the states using near-resonant AC (microwave)
electric fields. A resonant AC field E(t) = ẑ Eeiωt causes strong mixing of the two states even forsmall field amplitude E, since the energy gap between the states (on the order of Bv ) is bridged by
the photon energy ~ω. In contrast, mixing the states with a static field requires that the couplinghN + 1, mN |dz |N, mN i × E is large enough to overcome the energy difference between the states.
As we later discuss, the transition dipole moment also leads to dipolar spin-exchange interactions,whereby a molecule in |N, mN i and another in |N 0 , m0N i may exchange rotational states.
2.3.1

Calculating the matrix elements

Here we give a short derivation of the matrix elements of the dipole operator, following the
explanation given in section 8.4.2 of Brown and Carrington and using the angular momentum
machinery from section 5.5 of that text [108]. The physical picture is that the molecule always has
a dipole moment that points along the internuclear axis in the molecule (body-fixed) frame, witha strength given by the permanent dipole moment dperm . We just need a way to rotate into theframe of the molecule.
How we transform a physical quantity between the lab and molecule frame depends on its
associated angular momentum (the rank k of its spherical tensor representation). More formally,
we need the Wigner rotation matrix, which tells us how states of different angular momentum

15
(k)

transform under rotations. Let Dpq (ω)∗ be the rank k rotation matrix, with ω representing theEuler angles for rotating from the molecule frame to the lab frame. With these definitions, theindex p is associated with the lab frame while the index q is associated with the molecule frame.
(k)

To rotate a rank k tensor Tq (A) from the molecule frame to the lab frame, we contract it with(k)

Dpq (ω)∗ [108],
Tp(k) (A) =

X

(k)
Dpq
(ω)∗ Tq(k) (A).

(2.8)

q
(k)

Now we can see how to calculate the dipole matrix elements: we use Dpq (ω)∗ to rotate into themolecular frame, where the dipole operator is just a number dperm . In other words, the dipole
matrix elements are just those of the rotation matrix times dperm .
(k)

The matrix elements of Dpq (ω)∗ are given on page 167 of Brown and Carrington [108],p(k)hJ, mJ , Ω|Dpq(ω)∗ |J 0 , m0J , Ω0 i = (−1)mJ −Ω (2J + 1)(2J 0 + 1)



0
0
k J 
 J k J  J×,−Ω q Ω0−mJ p m0J

(2.9)

where J is the total angular momentum, mJ is its projection onto ẑ in the lab frame, and Ω is the
projection of J onto the internuclear axis. The terms in parentheses are Wigner 3j symbols.
We restrict our attention to the rotational states in the ground electronic state of KRb, so
J = N and Ω = Ω0 = 0, which gives2p(k)hN, mN |Dpq
(ω)∗ |N 0 , m0N i = (−1)mN (2N + 1)(2N 0 + 1)



0
0
k N 
N k N   N
×

.
0
0 q 0
−mN p mN

(2.10)

Recall that the bottom row of a 3j symbol must sum to zero. This means that for the states we(k)care about, the only nonzero matrix element of Dpq (ω)∗ is the one with q = 0.
2
A brief explanation of why J = N and Ω = Ω0 = 0: The ground state of KRb is 1 Σ+ . The 1 refers to theelectronic spin multiplicity, 2S +1, indicating that S = 0, and Σ refers to the projection Λ of electronic orbital angularmomentum L onto the internuclear axis. Analogous to how we write S, P , D, ... for atomic orbitals, spectroscopists
write Σ, Π, ∆, ... for Λ = 0, 1, 2, .... Since S = 0 and Λ = 0, this implies that Ω = 0. Finally, J = N because S and
L are zero.

16
Using this, we can calculate the dipole matrix elements in the lab frame,
hN, mN |Tp(1) (d)|N 0 , m0N i =

1
X

(1)
hN, mN |Dpq
(ω)∗ Tq(1) (d)|N 0 , m0N i

q=−1
(1)

(1)

= hN, mN |Dp0 (ω)∗ Tq=0 (d)|N 0 , m0N i(1)= dperm hN, mN |Dp0 (ω)∗ |N 0 , m0N ip= dperm (−1)mN (2N + 1)(2N 0 + 1)



0
0
1 N 
N 1 N   N
×

.
0
0 0 0
−mN p mN

(2.11)In the second line, we used the fact that only the q = 0 term contributes for the rotational states in(1)

the same electronic state. Finally, in the third line, we used Tq=0 (d) = dperm (the dipole moment
points along the internuclear axis in the frame of the molecule). Table 2.2 lists the dipole matrix
elements between a few of the lowest rotational levels.
From equation (2.11), we can directly read off the dipole selection rules. The top rows of the
3j symbols must satisfy the triangle inequality, so |N − N 0 | ≤ 1. In addition, the first 3j symbol iszero if N = N 0 , so we see that the dipolar coupling changes N by ±1. The bottom rows must sum
to zero, so the second 3j symbol implies mN − m0N = p. Since p ranges from −1 to 1, the dipolar
coupling changes mN by 0 or ±1.
We will often use a slightly more ergonomic notation for the spherical components of the(1)

dipole operator, dp = Tp (d). In terms of the Cartesian components, these are

2.3.2

d0 = dz ,(2.12)

1d±1 = ∓ √ (dx ± idy ).
2(2.13)

Induced dipole moment

With the dipole matrix elements in hand, we can analyze the behavior of equation (2.7)
as a function of E. We again assume that E = Eẑ, so only the p = 0 term of equation (2.11)
is relevant and mN is preserved. It is also convenient here to introduce the dimensionless fieldMatrix element
h1, 0|d0 |0, 0i
h1, ±1|d±1 |0, 0i
h2, 0|d0 |1, 0i
h2, ±1|d±1 |1, 0i

17Value in units√ of dperm
1/√3
1/√ 3
2/ √15
1/ 5

Table 2.2: Some handy dipole matrix elements.

strength  = E/Ec , where Ec = Bv /dperm is the characteristic field strength for the molecule(Ec = 3.9 kV/cm for KRb).
At  = 0, the eigenstates are just the bare rotational states |N, mN i, with energies EN =Bv N (N + 1). For small , we can calculate the Stark shift in second-order perturbation theory,
giving the following simple expression for the energy shifts of the lowest states |0, 0i and |1, 0i,
2Bv
6(2.14)

2
Bv ,10(2.15)(2)∆E0,0 = −(2)

∆E1,0 =and induced dipole moments(2)∂∆E0,0(2)= dperm
d0,0 () = −(2)

d1,0 () = −∂E(2)∂∆E1,0∂E3= − dperm .
5(2.16)(2.17)

At higher , these simple expressions break down as E causes significant mixing of the rotational states. Figure 2.3 shows the energies and induced dipoles (d = hÑ , mN |d0 |Ñ , mN i) of theseveral lowest rotational levels as a function of  (solid lines), obtained by diagonalizing equation(2.7) using states up to Nmax = 8. The dotted lines are the results from second-order perturbation theory. Note that the stretched states with |mN | = Ñ all have induced dipoles that increase
monotonically with , since they are the lowest-energy states in each mN manifold. All of the otherstates have an initially negative induced dipole, before eventually turning positive. This means,for example, that the direct dipolar interaction strength (∝ d2 ) between |0̃, 0i molecules increases
monotonically with E, while the interaction between |1̃, 0i molecules reaches a local maximum near
 = 1.75 (roughly E = 7 kV/cm for KRb).

18
0.8

8

Induced dipole / dperm

0.6Energy / Bv

4

0

-4

0.4

|0̃,0⟩
̃
|1,0⟩

0.2

|1̃, ± 1⟩|2̃,0⟩|2̃, ± 1⟩|2̃, ± 2⟩

0.0

-0.2
0

2

4

6

8

10

ϵ

0

2

4

6

8

10

ϵ

Figure 2.3: Energy and induced dipole moment of the N ≤ 2 states as a function of the applied
electric field. The dotted lines show the second-order perturbation theory results, valid for small .

Finally, we can plot the state composition of the field-dressed eigenstates |0̃, 0i (solid) and|1̃, 0i (dashed) in terms of the bare rotational states as a function of  (Figure 2.4A). The state
mixing can also be visualized by plotting the wavefunctions of the states at large  as shown in (B).
From the mean z-position of the wavefunctions, we can directly see the sign and magnitude of theinduced dipole. Notably, at  = 2, the dipole moment of |1̃, 0i is negative and more probability isconcentrated at z < 0, while at  = 10, the situation is reversed.

2.3.3

Transition dipole moments

In the previous section, we looked at the behavior of the induced dipole moments, which are
the diagonal matrix elements of dp . Next, we consider the behavior of the off-diagonal elements(transition dipole moments) hÑ 0 , m0N |dp |Ñ , mN i. The transition dipole moments allow us to resonantly couple different rotational states with AC electric fields and lead to dipolar spin-exchangeinteractions of the molecules (see section 2.6).
We note here that the distinction between the induced and transition dipoles is somewhat
artificial — after all, they are both matrix elements of the same operator dp . When applying a static

19

Figure 2.4: (A) The composition of |0̃, 0i and |1̃, 0i in terms of the bare rotational states as afunction of . (B) Plots of the wavefunctions of |0̃, 0i and |1̃, 0i at several values of . The shaded
area represents the z = 0 plane.

field E, all neighboring states |Ñ , mN i and |Ñ + 1, mN i are mixed according to the competition
between the Stark term and the rotational energy, leading to an induced dipole moment of eachstate. In contrast, by applying an AC electric field E(t) = Ee−iωt , we can resonantly bridge theenergy gap between a particular pair of states |Ñ , mN i and |Ñ + 1, mN + pi to create an isolated
two-level system. Since there is a transition dipole moment between the states, their populations
undergo Rabi oscillations at an angular frequency Ω = hÑ + 1, mN + p|dp |Ñ , mN iEp /~. Here, Ep=0
(Ep=±1 ) is the amplitude of π (σ ± ) polarization in E.
Having just found the field-dressed eigenstates in the previous section, we can evaluate thetransition dipole moments hÑ 0 , m0N |dp |Ñ , mN i as a function of . Owing to the state mixing shownin Figure 2.4, we would expect that the transition dipole moments for Ñ → Ñ ± 1 become weakeras the electric field is increased. On the other hand, the initially forbidden couplings Ñ → Ñ ± 2(and higher) become weakly allowed as the states become very mixed. This is illustrated by Figure
2.5, which shows the behavior of transition dipole moments from |0̃, 0i as a function of .

Transition dipole / permanent dipole

20
0.6

π transitions
0|⟨1̃,0|d |0̃,0⟩|0|⟨2̃,0|d |0̃,0⟩|0.4

0|⟨3̃,0|d |0̃,0⟩|

σ

0.2

0.0
0

5
ϵ

±

transitions|⟨1̃, ± 1|d±1|⟨2̃, ± 1|d±1|0̃,0⟩||⟨3̃, ± 1|d±1|0̃,0⟩||0̃,0⟩|

10

Figure 2.5: Transition dipole moment between |0̃, 0i and |Ñ , 0i or |Ñ , ±1i as a function of .

Since the transition dipole moments tend to be strongest near E = 0, they can be significantly
affected by the nuclear electric quadrupole couplings, which are most important at low E. Figure 2.6
shows a calculation for the Ñ = 1 states of KRb at E ≤ 1000 V/cm. The states are colored based√on their transition dipole moment with |gi = |0̃, 0, −4, 1/2i, normalized by dperm / 3. Without
hyperfine structure, we would expect to only see one allowed transition for each of p = 0, ±1.
However, as shown in the plot, mixing due to the nuclear electric quadrupole coupling causes nonnegligible coupling to a number of other states, especially in the region below 500 V/cm where
there are a number of avoided crossings (see also Figure 2.2). Though these mixings complicate
the spectrum, they also enable control of the hyperfine degrees of freedom through their coupling
to the rotational states [111].

2.3.4

AC Polarizability

To conclude our discussion of the response of the molecules to external fields, we consider
the effect of far off-resonant optical fields. Far red-detuned optical beams exert a dipole force
on the molecules due to coupling with electronically excited states, giving an AC Stark shift
U (r) = −αI(r). Dipole traps formed by red-detuned laser beams are a standard technique for

21

π

σ⁺

0

σ⁻

Energy (MHz)

-1
2230
-2
2226
-3

2222

log10(transition dipole)

2234

-4

0

500
E (V/cm)

0500
E (V/cm)

0500
E (V/cm)

Figure 2.6: Effect of hyperfine structure on transition dipole moments. The plots show the energies
of states in Ñ = 1 for E ≤ 1000 V/cm and B = 545.9 G, with both fields along ẑ. The states arecolored by their transition dipole moment with |0̃, 0, −4, 1/2i. Transitions with relative strength≤ 10−4 are all colored the same.

ultracold atoms [112], and our optical traps for molecules are based on the same principles. However, molecules have a more complicated light shift due to their anisotropic response to the electricfield, which causes a polarization-dependent AC Stark shift [110, 113].
Understanding the AC Stark shift is key for creating long-lived coherent superpositions of
the molecular rotations. If the ground and excited states have different AC polarizabilities, theyexperience different trapping potentials and the resonant transition frequency varies with the position in the trap, shown schematically in Figure 2.7. This inhomogeneous broadening leads todecoherence across the molecular sample. Beyond single particle effects, differential light shifts canalso suppress interactions, for example, by detuning the dipolar spin-exchange interactions between
molecules in a 3D optical lattice [46].
A similar problem also arises in atomic systems, most notably in optical lattice clocks where
differential light shifts are an important source of systematic uncertainty [18]. The solution forclocks is to find a “magic wavelength” where the polarizabilities of the ground and excited states
happen to match [114]. In molecules, it is generally harder to find a magic wavelength where22

U10(x)

U00(x)

ν0

x

nmol

Figure 2.7: Cartoon of differential light shifts in a non-magic trap. The trapping potentials forthe two states, U00 (x) and U10 (x), are different, leading to a variation in the resonant transition
frequency as a function of the position in the trap.

losses from off-resonant excitation are also low, due to the dense spectrum of higher excited states[115, 116]. Instead, we can exploit the anisotropic part of the polarizability to find a “magicpolarization angle” where the ground and excited states are matched [110, 113].
Let us be more concrete and examine the AC Stark Hamiltonian from Ref. [117],√
1
6
Hac = − (αk + 2α⊥ )Ilaser −(αk − α⊥ )T (2) (,  ) · C2 (ω)Ilaser .
3
3(2.18)

Here, Ilaser is the light intensity and  its polarization vector. We assume that the light is linearlypolarized, as is the case in our experiments; otherwise, there would also be a vector AC Stark term[113]. The molecular parameters αk and α⊥ characterize the frequency-dependent response parallel(2)

and perpendicular to the internuclear axis, respectively, and C2p (ω) = Dp0 (ω)∗ is the rotation
matrix into the molecular frame.3 Note that the only off-diagonal matrix elements come fromC2 (ω), which couples states of different N and mN , weighted by the components of T (2) (,  ). For
typical experimental values of Ilaser , the AC Stark shifts (on the order of the trap depth, typicallyseveral µK) are much smaller than the energy splittings between states of different Ñ (order of
Bv ∼ 100 mK), and we can safely neglect the N -changing part. The explicit form of the matrix
elements in terms of 3j symbols can be found in Appendix A.
To gain some intuition, let us focus on the states we use most often, N = 0 and 1, at E = 0.
Following Ref. [110], we will assume that we have solved for the molecular eigenstates at Ilaser = 0
3Here, ω represents the Euler angles for rotation to the molecular frame, not to be confused with the angularfrequency of the optical field (the dependence of which is assumed to be included in αk and α⊥ ). Ref. [117] uses
the notation C2 (α, β), where α and β represent the Euler angles, but we use C2 (ω) in this section to match with the(2)earlier definition of Dpq (ω)∗ and avoid symbol clash with the polarizability α.

23before adding Hac as a perturbation. The ground state is isotropic, so its polarizability α00 is
independent of the polarization,1
α00 = (αk + 2α⊥ ).
3(2.19)The excited state polarizabilities are determined by the following matrix elements,21
h1, 0|α|1, 0i = (αk + 2α⊥ ) − (αk − α⊥ )(1 − 3 cos2 θ)
3
15
1
1
h1, ±1|α|1, ±1i = (αk + α⊥ ) + (αk − α⊥ )(1 − 3 cos2 θ)3√
15
2
h1, 0|α|1, ±1i = ±(αk − α⊥ ) sin θ cos θ51
h1, ±1|α|1, ∓1i = − (αk − α⊥ ) sin2 θ,
5(2.20)(2.21)(2.22)(2.23)

where θ is the angle between  and ẑ.
The importance of the off-diagonal matrix elements depends on the energy splittings between
the states. In the simplest case, shown schematically in Figure 2.8A, the three N = 1 states are
split enough in energy that only the diagonal elements matter. These can be made equal to α00 at
the “magic angle” where cos2 θ = 1/3 (θ ≈ 54.7 deg). From equation (2.18), it turns out that thisis true of every diagonal matrix element, not just for N = 1, since they all involve only the p = 0(2)

component Tp=0 (,  ) ∝ (1 − 3 cos2 θ).
This scenario tends not to be very realistic since, aside from hyperfine structure, the |1̃, ±1i
states are degenerate. Consequently, their mixing by Hac is usually important. Figure 2.8B shows
the limiting case where |1̃, ±1i are exactly degenerate but still separated in energy from |1̃, 0i. In thebasis of |1̃, −1i and |1̃, 1i, Hac is well-approximated by a simple two-level system with eigenstates
given by symmetric and antisymmetric superpositions of |1̃, ±1i. As the energy splitting between|1̃, −1i and |1̃, 1i is gradually increased from zero, the green and pink curves in (B) approach
the green curve in (A). In the intermediate regime, higher-order terms in Ilaser (sometimes called“hyperpolarizability”) also contribute, since the light shifts the energies of the states and hence
also changes the degree of mixing.
Having discussed some limiting cases, we now mention two regimes of particular relevance

24

A
|0,0⟩
|1,0⟩
|1,±1⟩

Polarizability

α₀₀

B
|0,0⟩
|1,0⟩
|1,−1⟩ − |1,+1⟩|1,−1⟩ + |1,+1⟩

α₀₀
0

45
90 135θ (degrees)

180

Figure 2.8: Limiting cases of α(θ). (A) No mixing: the energy splittings of the N = 1 states arelarger than any AC Stark shifts. The observed polarizabilities are just the diagonal terms of Hac .
(B) Maximum mixing of |1, ±1i: |1, ±1i are degenerate and split in energy from |1, 0i. The two|mN | = 1 states are strongly mixed by the optical field, resulting in symmetric and antisymmetriceigenstates. Both (A) and (B) are computed assuming that the eigenstates at Ilaser = 0 are purerotational states |N, mN i, and that αk > α⊥ > 0.

for our experiment. At low E, the N = 1 states have small splittings of ∼ 100 kHz due to the
nuclear electric quadrupole interaction. Since the optical trap depth can be on the order of 100
kHz, AC Stark shifts can mix all three states. The quadrupole interaction also causes some mixing
with states of different hyperfine character. The behavior at E = 0 was studied experimentallyin KRb in Ref. [110], and subsequently in RbCs [118], NaK [119] and NaRb [120], as well as
in 2 Σ CaF molecules [121]. At slightly larger E, the Stark shifts suppress the nuclear quadrupole
couplings, and the hyperfine structure decouples from the rotational states. In NaK, this decouplingat moderate E (300 V/cm) was used to suppress the hyperpolarizability term for |1̃, 0i, leading toan enhancement of the coherence time between |0̃, 0i and |1̃, 0i [119].
At high E, |1̃, 0i is much higher in energy than |1̃, ±1i due to the different DC Stark shifts ofthe states, so the off-diagonal terms are not important for |1̃, 0i. The most notable feature is thatthe state mixing from E can change the strength of the anisotropic term. As shown theoretically in
Ref. [113], this leads to an electric field strength (“magic field”) where the polarizabilities of |0̃, 0i

25
and |1̃, 0i are matched for all θ. For KRb, the magic field is predicted to occur at E ≈ 10 kV/cm[113].
Finally, we should mention that magic wavelengths have recently been found for NaK [122] byutilizing a narrow electronic transition [123]. Further theoretical work suggests that this technique
can also be extended to simultaneously achieve magic conditions for several rotational states [124].
This concludes our summary of the dipolar characteristics of rigid-rotor molecules in an
external electric field. Next, we consider the dipolar interaction between two molecules.
Before moving on, a brief word on notation. So far, it has been helpful to distinguish between
the bare and field-dressed states to discuss wavefunction overlaps, matrix elements, etc. However,in experiments, we do not have access to the bare rotational states at any nonzero electric field.
For describing the dipolar phenomena that we observe at E > 0, we will often only care about the
behavior of the field-dressed eigenstates. Because of this, from now on we will drop the tilde on
the field-dressed eigenstates, writing them as |N, mN i when the meaning is clear from context.

2.4

Dipolar interactions – Introduction
Perhaps the most intriguing aspect of ultracold molecules is their dipolar interactions. The

long-range and anisotropic nature of these interactions makes polar molecules an exciting setting
for studying many body physics [24, 26, 27]. Classically, two dipoles interact according toVdd =d1 · d2 − 3(d1 · r̂)(d2 · r̂),
4π0 r3(2.24)

where r is the vector separating the dipoles.
In this thesis, we study a range of dipolar phenomena, including dipolar collisions of molecules
in a bulk (2D) geometry, resonant shielding arising from dipolar coupling between pairs of internal
states, as well as dipolar spin-exchange interactions between separated ensembles of molecules. Allof these can be be more or less understood at an intuitive level from a single equation: Vdd expressedin terms of spherical tensor operators [45],√26 X
Vdd (r, θ, φ) = −(−1)p C2,−p (θ, φ)Tp(2) (d1 , d2 ),4π0 r3p=−2(2.25)

26
where θ and φ are the polar and azimuthal angles of r, respectively, and we have assumed that E
points along ẑ. As a reminder, the spherical tensor components are [108]

1
(2)
−1
−1 +1
T0 (d1 , d2 ) = √ 2d01 d02 + d+1
1 d2 + d1 d2
6

1
(2)
0
0 ±1
T±1 (d1 , d2 ) = √ d±1
1 d2 + d1 d2
2
(2)

±1
T±2 (d1 , d2 ) = d±1
1 d2 .

(2.26)(2.27)(2.28)

Here, dpi is just the same dipole operator we studied in section 2.3, acting on the i-th molecule.
The angular part is C2,p (θ, φ) = (4π/5)1/2 Y2,p (θ, φ), where Yl,m (θ, φ) is a spherical harmonic [45].
The interaction Vdd depends not only on the internal rotational states of the interacting
molecules (through d1 and d2 ), but also on their relative motion (through r, θ, and φ). As a result,the qualitative behavior of Vdd can be strongly influenced by the external trapping geometry, which
determines the accessible motional states. In the next sections, we will consider the behavior of
equation (2.25) in two opposite limits where the molecules are either free to move in 3D (“bulk”)
or pinned to sites in an optical lattice. We will focus primarily on the dipolar interactions between
ultracold molecules — other dipolar systems, such as magnetic [125, 126] and Rydberg [127, 128]
atoms are exciting areas of active research, but are not discussed in detail here.

2.5

Dipolar interactions – In the bulk
In a bulk 3D geometry, two colliding molecules can be characterized by their collision wavevec-

tor k and relative angular momentum state |L, mL i. The collision energy is determined by thetemperature T , which sets the average relative velocity of the collision partners. The angular momentum L contributes a centrifugal barrier in the collisions, and there is also an isotropic van der
Waals interaction with coefficient C6 , giving a total interaction of [81]V (r, θ, φ) =~2 L(L + 1) C6− 6 + Vdd (r, θ, φ),2µr2rwhere µ is the reduced mass of two colliding molecules.

(2.29)

27
We need to calculate the matrix elements of Vdd to understand equation (2.29) quantitatively.
To simplify things, we’ll make two assumptions: the molecules are initially in the same field-dressedrotational state |N, mN i, and their internal states are not changed by the interaction. The first
assumption is generally true in our experiments since we prepare all of the molecules in a single
internal state; otherwise, the molecules are distinguishable and are quickly lost due to s-wave
collisions (which have no centrifugal barrier) [32]. The second assumption is generally true since
the excited rotational states are typically not energetically accessible — the rotational splitting
Bv ∼ 50 mK is many orders of magnitude higher than our collision energy (set by T ∼ 300 nK).4
Since mN 1 and mN 2 cannot change, only the p = 0 term of equation (2.25) is relevant, andwe getp
1
mL
hL, mL |Vdd |L0 , m0L i = −(−1)(2L + 1)(2L0 + 1)
4π0 r3



0
0
2 L 
L 2 L   L2× 2(dN,mN ) ,
0 0 0−mL 0 m0L

(2.30)

where dN,mN = hN, mN |d0 |N, mN i is the induced dipole. The interaction no longer depends on
the internal states of the molecule, except through dN,mN , so we can drop the subscripts andparameterize the interactions with the single parameter d (the induced dipole moment). Also, notethat the second 3j symbol enforces mL = m0L .
If the molecules are prepared in the same internal state, then we also have to account for theirquantum statistics. For identical fermions (bosons), only odd (even) L are allowed. The centrifugal
term in equation (2.29) suppresses the interactions for higher L, so we will primarily be interested
in the behavior of the lowest channel (L = 1 for fermionic 40 K87 Rb).

2.5.1

General considerations

Let us start by considering a few limits. At E = 0, the induced dipole d is zero and thus Vdd =0. Here, the system is spherically symmetric and both the total angular momentum N1 + N2 + L
4As we will discuss later, the resonant shielding effect occurs when we can engineer the breakdown of this second
assumption.

28
and its projection are conserved. As E is turned on, only cylindrical symmetry about ẑ remains,which leaves the projection mN 1 + mN 2 + mL the only good quantum number — in other words,
Vdd mixes partial waves. Since C2 (θ, φ) couples states with L0 = L ± 2, we end up with two sets ofcoupled partial waves consisting of the even and odd values of L.
We can also use a fixed value of |d| > 0 and consider the limits with r. It is helpful to startby extracting some characteristic length scales from V (r). This is done by setting each term equal
to the kinetic energy at some separation r = aD,vdW = 1/k, giving the dipolar length [129]
aD =

µ d2
~2 4π0(2.31)and the van der Waals length avdW = (2µC6 /~2 )1/4 , with k the collision wavevector. At large
separations r  aD , avdW , the interaction V (r) is zero and L and mL are good quantum numbers.
On the other hand, at r ∼ aD , the states are significantly mixed by Vdd , causing changes in both
the reactive losses and elastic scattering of the molecules.

2.5.2

Reactive losses

Two KRb molecules can be lost in a collision by undergoing an exothermic chemical reaction
2KRb → K2 + Rb2 , which releases about 15 K of energy that propels the products from the trap
[32, 79, 34]. The reaction rate can be elegantly described by a quantum threshold model [81], which
relates the reaction rate to the probability of tunneling through a potential barrier to reach shortrange (r  avdW ) where a reaction can occur. Such a barrier is naturally present for fermionic
KRb due to the centrifugal p-wave (L = 1) barrier. Turning on Vdd alters the height of the barrier
and hence changes the probability of tunneling to short range.
Let us first consider the situation with no dipolar interactions. The behavior of V (r) for
d = 0 and L = 1 is shown schematically in Figure 2.9A. There is a p-wave barrier formed by thecompetition between the centrifugal and van der Waals terms, whose height Vb is much higher
than T (Vb = 24 µK for KRb). This barrier prevents the molecules from reaching short range(gray region in Figure 2.9A), where the behavior depends sensitively on the details of the unknown

29

Figure 2.9: (A) Schematic diagram of the p-wave barrier in the absence of dipolar interactions.
(B) Relative orientation of the molecules for L = 1 collisions with different values of mL .

short-range potential. For KRb, the experimentally observed loss behavior [32, 79] is well-describedby using a perfectly absorbing short range potential (the “black hole model”), which is attributed
to the exothermic chemical reactions [82, 83, 81]. As a result, the molecular loss rate is “universal”(determined only by the long range behavior) [82] and set by the rate of tunneling through thep-wave barrier, which occurs with probability (Ecol /Vb )L+1/2 (Ecol is the collision energy) [81].
Turning on Vdd changes the height of Vb and alters the loss rate. To compute the change
in Vb , we can make an adiabatic approximation by diagonalizing V (r) at many values of r andstitching together the results [81]. This produces adiabatic energy curves as a function of r, shownin Figure 2.10. For mL = 0 (A), the height of the barrier decreases monotonically with increasingd. In contrast, for mL = ±1 (B), Vb is enhanced for moderate d > 0, but lowered for larger d.
The qualitative trends can be understood from the form of Vdd . At small d, the mixing
between different L is not too strong, and we only need to consider the diagonal matrix elements.
Here, the main feature can be directly seen from equation (2.30): owing to the overall (−1)mL +1
factor, the interaction is attractive for mL = 0 but repulsive for mL = ±1. This makes senseclassically because an mL = 0 collision corresponds to a head-to-tail approach of the molecules,
while an mL = ±1 collision is a side-to-side approach (shown schematically in Figure 2.9B). At
larger d, where the off-diagonal elements become important, the barriers of all three L = 1 states
decrease owing to mixing with L = 3 (dashed curves in Figure 2.10). This can be understood from

30

A

B
40

40

0D
0.05 D
0.15 D
0.25 D
0.35 D

Energy (μK)

30

20

10
0

0D
0.05 D
0.15 D
0.25 D
0.35 D

30

20

10

L=1
L=3

0

-10

-10

200

600

1000

1400

r (a0)

200

600

1000

1400

r (a0)

Figure 2.10: Adiabatic energy curves vs d, calculated for KRb. (A) mL = 0. (B) mL = ±1. These
use C6 = 16130 a.u. [130]. To avoid cluttering the graph, the L = 3 curves (dashed lines) are only
shown for d = 0, 0.15, and 0.35 D.

perturbation theory: L = 1, being lower in energy than L = 3, is repelled downward in energy by
the coupling, hence reducing the barrier.
In 3D, the reduced barrier for mL = 0 leads to a sharply increasing loss rate when d isincreased above a critical value. (For KRb, this occurs around d = 0.1 D.) Above this corner, theloss rate increases rapidly as d6 [79, 81], presenting a challenge for studying the dipolar interactionsof molecules in 3D. However, the anisotropy of the dipolar interaction means that we can control
the loss behavior by controlling the external confinement. In particular, tight confinement in theẑ-direction effectively removes the mL = 0 collisions. Since Vb is initially enhanced for mL = ±1,
this leads to a suppression of the loss rate in 2D with increasing d, until the mixing with L = 3
becomes significant [131, 99, 100, 101, 102, 52]. Our experiments along this direction are describedin Chapter 5.

2.5.2.1

Competing length scales in 2D

Here, we make a brief digression to discuss the 2D loss suppression in more detail. We assume
molecules are strongly confined into 2D by a harmonic potential along ẑ. Further, we assume that31
the molecules occupy only the lowest oscillator state in the z-direction, which also implies thatkB T  ~ω, where ω is the trap frequency along ẑ.
The discussion in the previous section provides a simple pertubative explanation of the dipolar
loss suppression in 2D. There, we used the 3D form of the interaction from equation (2.30), andassumed that the 2D confinement just controls which incoming mL states are allowed. Since thecollisions still look essentially 3D, we refer to them as being “quasi-2D.” However, the long-range
nature of the dipolar interaction leads to the breakdown of the quasi-2D picture when d becomesvery large — specifically, when the dipolar length aD becomes larger than the harmonic lengthpaho = ~/µω in the tightly-confined direction. To distinguish from quasi-2d, the aD  aho limit issometimes referred to as “true 2D.” The quasi-2D picture suggests that once d becomes sufficiently
large, the loss rate in 2D will only increase with larger d owing to coupling with L = 3. On thecontrary, full theoretical treatments predict that the loss rate eventually becomes highly suppressed
when aD  aho [100, 101].
To gain more intuition, we calculate the adiabatic energy curves for fermionic KRb in Figure
2.11, following Ref. [100]. Compared to the 3D calculation from Figure 2.10, we add an additional
harmonic confinement (1/2)µω 2 z 2 to equation (2.29). First, consider the top panel of Figure
2.11, which corresponds to a moderate dipolar strength in the quasi-2D regime (aD = 400a0 and
aho = 2160a0 ). Here, from our previous argument, we expect to see suppression of loss in themL = ±1 collisions. Only the states satisfying fermionic symmetry have been plotted [100]. The
blue solid curves correspond to mL = ±1 collisions, while the red dashed curves correspond to
mL = 0 collisions. At long distance, the curves connect to harmonic oscillator states of the relative
motion, with relative oscillator quanta n and energy hν(n + 1/2) (where ν = ω/2π) [100].
We will focus on the behavior of the n = 0 state, which corresponds to collisions between two
molecules in the ground harmonic oscillator state.5 The n = 0 state connects to L = 1, mL = ±1
for r < aho . Thus, as seen in the plot, the dipolar interaction leads to an enhanced barrier for
two molecules in the ground state of the 2D potential. On the other hand, n = 1, corresponding
to an antisymmetric superposition of molecules in the ground and first harmonic oscillator states,

32

Figure 2.11: Adiabatic energy curves from quasi-2D to 2D, calculated for KRb in |0, 0i. The
vertical trapping frequency is ν = ω/2π. Blue solid curves are mL = ±1; red dashed curves are
mL = 0. The lowest 8 curves satisfying fermionic symmetry are shown. Top panel: d = 0.15 D,
ν = 20 kHz, corresponding to aD = 400a0 and aho = 2160a0 . Middle panel: d = 0.5 D, ν = 20
kHz, corresponding to aD = 4500a0 and aho = 2160a0 . Bottom panel: d = 0.5 D, ν = 200 kHz,
corresponding to aD = 4500a0 and aho = 680a0 .

33connects to L = 1, mL = 0. As shown theoretically in Ref. [100] and experimentally in Ref. [102],
this leads to a enhanced loss rate between molecules in different harmonic oscillator states. For thework in Chapter 5, it was thus crucial to load the molecules into the lowest band of the 1D lattice
to prevent these lossy mL = 0 collisions.
In the middle panel, d has been increased to 0.5 D, while keeping ν = 20 kHz (aD = 4500a0
and aho = 2160a0 ). We can see that the barrier for n = 0 has decreased owing to coupling withn = 2 (which connects to L = 3, mL = ±1). Thus the quasi-2D picture still holds in this regime,
leading to a decrease in the barrier for the lowest mL = ±1 channel at large d. This is the same
qualitative behavior as seen in 3D (Figure 2.10).
Finally, in the bottom panel, d has been kept at 0.5 D but ν is increased to 200 kHz. Here, the“true 2D” regime is starting to be reached, and the barrier in n = 0 has become clearly larger thanin the middle panel. From the standpoint of perturbation theory, this makes sense: the coupling
between n = 0 and 2, which causes the suppression of the barrier for n = 0, is weakened as theenergy spacing (∝ hν) is increased. Theoretical calculations show that the n = 0 barrier becomes
further enhanced as ν is increased [131, 100]. For rigorous calculations of the loss rates in thequasi-2D to 2D crossover, see Refs. [99, 100, 101, 131, 132].
For the experiments in this thesis, we remained in the quasi-2D regime with a maximum
dipole of d = 0.3 D (aD = 1600a0 ) and ν = 17 kHz (aho = 1830a0 ). We could possibly start to
explore the crossover by using a much deeper optical lattice confinement, which might allow us toreach a higher elastic to inelastic collision ratio [131]. Experiments on other molecules with larger
dipole moments may also start to probe the true 2D regime. For bosonic molecules, it is necessary
to reach the true 2D regime to observe the 2D dipolar loss suppression since there is initially nobarrier in the collisions [100, 101].
5

Technically, n labels the relative oscillator state, so n = 0 only ensures that the molecules are in the same
single-particle oscillator state along z. However, for concreteness, we will stick to the most experimentally relevantcase of both molecules in the ground oscillator state.

34
2.5.3

Elastic scattering

In addition to undergoing reactive losses, the molecules can also scatter elastically, with rate
Γ = hnσel vi. Here, n is the molecular density, σel is the elastic cross section, and v is the relative
velocity between collision partners. The brackets h·i denote the thermal average over the sample.
In atomic physics, we typically think of interactions between ultracold spin-polarized fermionsas being very weak. For identical fermions colliding via contact interactions, the collision partners
must tunnel through the p-wave barrier to scatter, leading to a cross section that becomes verysmall at ultralow temperatures (σel ∝ T 2 ) [133]. The strategy used to create the first atomic Fermi
gas, and still the most commonly used today, is to use a spin mixture of two different hyperfine
states, thus enabling s-wave collisions to rethermalize the gas [134].
However, identical dipolar fermions can have strong, energy-independent elastic collisions
at ultralow temperatures. This has enabled the evaporative cooling of spin-polarized gases ofstrongly-magnetic fermionic atoms [135], and of KRb [52] and NaK [54] molecules. Within the
Born approximation, valid at low collision energies, σel is energy-independent and depends only
on aD for identical dipolar fermions, yielding σel = (32π/15)a2D [129, 136]. This cross section isuniversal in that it depends only on the single parameter aD . Note that in 3D, the reactive losses
scale as d6 , while the elastic rate scales as a2D ∝ d4 , meaning that the loss dominates in 3D ingeneral. On the other hand, in 2D the loss is initially suppressed while σel is increasing. This leads
to a favorable ratio γ = σel /σreactive of around 200 for KRb at d = 0.2 D, enough to experimentallyobserve evaporative cooling [52].
Finally, we should explain what is meant by the condition that the collision energy is “low”.
As originally explained in Ref. [129], the dipolar length also defines a characteristic energy scale
ED = ~2 /µa2D , which sets the highest collision energy Ecol where σel is energy-independent. Physically, this can be interpreted as the requirement that the scattering is in the long-wavelength
regime, 1/k  aD , which implies Ecol  ED . For Ecol  ED , σel is no longer energy-independentand falls off as 1/k [129].

35
6000
10⁻⁴
10⁻⁵

2000

0
0.0

→

←
0.1

0.2

0.3

0.4

ED (K)aD (a0)

10⁻³
4000

10⁻⁶

10⁻⁷

0.5

d (D)

Figure 2.12:
KRb.

aD (blue, left axis) and ED (orange, right axis) vs d, calculated with the mass of

The trends of aD and ED as a function of d are shown in Figure 2.12. In our experiments
on KRb, we have stayed in (or near the edge of) the universal regime, with ED = 570 nK at ourhighest dipole moment of d = 0.35 D (achieved in 2D [52]) compared to hEcol i/kB = T ≈ 250 nK.
Experiments at higher electric field or in other molecular species with larger dperm will start toprobe the edge of the universal regime.

2.5.4

Resonant shielding

So far, we have only seen the effect of the induced dipole moments. We made this restriction
somewhat artificially by assuming that the molecule internal states do not change during a collision,
though this condition tends to naturally be satisfied since the energy splittings between different
rotational states are usually far larger than the other relevant energy scales (collision energy anddipolar interaction energy).
Relaxing this constraint, we can expand Vdd in the basis |N1 , mN 1 ; N2 , mN 2 ; L, mL i to get
hN1 , mN 1 ; N2 , mN 2 ; L, mL |Vdd |N10 , m0N 1 ; N20 , m0N 2 ; L0 , m0L i




√
p=2
0
0
L 2 L X
L
2
L 
p
6
p
mL
0 + 1) 
=−
(−1)
(−1)
(2L
+
1)(2L




4π0 r30 0 0 p=−2−mL −p m0L
× hN1 , mN 1 ; N2 , mN 2 |Tp(2) (d1 , d2 )|N10 , m0N 1 ; N20 , m0N 2 i.

(2.32)

We can see that Vdd couples different internal states |N1 , mN 1 ; N2 , mN 2 i via the transition

36

Figure 2.13: Energies of rotational state pairs |N1 , mN 1 ; N2 , mN 2 i as a function of  = E/Ec .
|1, 0; 1, 0i has crossings with |0, 0; 2, 0i and |0, 0; 2 ± 1i near  = 3.2. While |0, 0; 2 ± 2i (gray) also
crosses |1, 0; 1, 0i near  = 2.8, the states do not have a strong dipolar coupling since |1, 0i ↔ |2, ±2i
is not dipole-allowed.

dipole moment, although the coupling must overcome the energy cost of changing the internal
state. If two pairs of internal states |N1 , mN 1 ; N2 , mN 2 i and |N10 , m0N 1 ; N20 , m0N 2 i can be tuned
into resonance using E, we might create a situation where the transition dipole moment plays
a dominant role in the collisions. This is indeed possible if we initially put the molecules in arotationally excited state, |1, 0; 1, 0i. As shown in Figure 2.13, the energy of |1, 0; 1, 0i crosses thatof |0, 0; 2, 0i and |0, 0; 2 ± 1i near  ≈ 3.2. These states are strongly coupled through their transition
dipole moments, since2(2)h0, 0; 2, 0|T0 (d1 , d2 )|1, 0; 1, 0i = √ h0, 0|d0 |1, 0ih2, 0|d0 |1, 0i
6
1(2)
h0, 0; 2, ±1|T±1 (d1 , d2 )|1, 0; 1, 0i = √ h0, 0|d0 |1, 0ih2, ±1|d±1 |1, 0i.
2(2.33)(2.34)In the vicinity of the crossings, these couplings are near-resonant and the potential experienced by
two colliding |1, 0i molecules is strongly modified. In particular, the loss rate can be suppressed byan order of magnitude in KRb (“resonant shielding”) [137, 138]. This effect is the subject of theexperiments in Chapter 6.

37

2.6

Dipolar interactions – In the lattice
We next consider the behavior with the molecules pinned in sites of a deep 3D optical lattice.

Here the motion is completely frozen out, so the angular part C2 (θ, φ) just contributes a geometricfactor, and the dynamics occur in the internal states of the molecules. Without the relative motion,
two molecules in the same single-particle eigenstate |N, mN i do not show interesting dynamics, andso we need to consider two-particle states with different internal states |N1 , mN 1 ; N2 , mN 2 i. Thissystem can be mapped onto a quantum magnetism model (the “XXZ” model) with long-range andanisotropic interactions that are tunable by experimental control parameters such as external DC
or AC microwave fields [45, 24], and can exhibit a wide range of many-body phenomena including
dipolar spin liquids [40], spin-orbit coupling [139], and unconventional superfluidity [42].
We first consider two molecules localized in space and separated by a distance r, initiallyprepared in some superposition of the single-molecule states |0i ≡ |N = 0, mN = 0i and |1i ≡|N = 1, mN = 0i. Assuming that there are no other states nearby in energy, the evolution under
Vdd remains in the subspace of two-particle states spanned by {|00i, |01i, |10i, |11i}, and only thep = 0 term of Vdd is relevant. With these constraints, we can simplify equation (2.25) to obtainVdd =1 − 3 cos2 θ 2
d0 |00ih00| + d21 |11ih11| + d0 d1 |10ih10| + d0 d1 |01ih01|
4π0 r3
+d201 |01ih10| + d201 |10ih01| ,(2.35)(2.36)

where d0 = h0|d0 |0i and d1 = h1|d0 |1i are the induced dipole moments, and d01 = h1|d0 |0i = h0|d0 |1i
is the transition dipole moment between |0i and |1i.
We often talk about this system using the language of magnetism by forming a spin-1/2
system | ↓i ≡ |0i and | ↑i ≡ |1i. By rewriting equation (2.36) in terms of spin-1/2 operators S1and S2 , we obtain1 − 3 cos2 θVdd =4π0 r3J⊥ + −Jz S1z S2z + V + W (S1z + S2z ) +(S1 S2 + c.c.)2,(2.37)

where Jz = (d0 − d1 )2 , V = (d0 + d1 )2 /4, W = (d20 − d21 )/2, and J⊥ = 2d201 [45]. For many particles

38
confined in an optical lattice, the above equation generalizes to [45]1XJ⊥  + −
z z
z
z
Vdd =
V (ri − rj ) Jz Si Sj + V ni nj + W (ni Sj + Si nj ) +Si Sj + c.c.
,22(2.38)

i6=jwhere V (r) = (1 − 3(r̂ · ẑ)2 )/4π0 r3 and ni is the number of molecules on site i.
We typically work in a deep lattice where the tunneling between sites is negligible. In thiscase, the V (W ) term turns into a static site-to-site energy shift (effective magnetic field). If, forsimplicity, we further assume that the lattice has unit filling, then the V and W terms are uniform
over the whole lattice, andJ⊥  + −1Xz z
V (ri − rj ) Jz Si Sj +Si Sj + c.c.
.
Vdd =
2
2

(2.39)

i6=j

The Jz term gives a long-range and anisotropic Ising interaction, while the J⊥ term gives
the dipolar spin-exchange (“flip-flop”) interactions, where a molecule in | ↑i and one in | ↓i tradeinternal states. The dynamics of equation (2.39) were studied experimentally in Refs. [46, 140]
using Ramsey spectroscopy at E = 0, where Jz = 0 but J⊥ is strong. The main observation wasa modulation of the Ramsey contrast as a function of time, with multiple frequency components
set by the dipolar energy shifts J⊥ × V (r) between different pairs of lattice sites. The behavior of
the Ramsey contrast was also controlled by the choice of internal state, which affects the strengthof the interactions, and the molecule density [140]. In the experiments of Chapter 7, we also studied the spin-exchange interactions between separated ensembles of molecules prepared in different
rotational states.

2.6.1

Two-particle calculation

The many-body dynamics of equation (2.39) is of course very complicated, but we can gain
some intuition by doing a “toy” two-particle calculation. For two particles, we can write equation(2.39) in the following form,
Vdd /~ = −Jz
J⊥(|10ih10| + |01ih01|) +(|01ih10| + |10ih01|) ,2
2(2.40)

39
where we have absorbed some factors into Jz and J⊥ for simplicity, and shifted the zero point of theenergy by −Jz /4. We can diagonalize Vdd by first noting that we have two eigenstates |00i and |11i
with zero energy. The remaining bare states |01i and |10i form a two-level system with the sameenergy offset −Jz /2 and coupling J⊥ /2, so they form symmetric and antisymmetric eigenstates |±iwith energies E± = (1/2)(−Jz ± J⊥ ).
Having calculated the eigenstates of Vdd , we can now write down what happens in a Ramsey
sequence. Let us assume that the molecules are initialized in |00i. Since |00i is an eigenstate of
Vdd , nothing happens until the first pulse of the Ramsey sequence, with area θ, puts them in an
initial superposition


 

θ
θ
θ
θ
cos |0i + sin |1i ⊗ cos |0i + sin |1i
2
2
2
2
θ
θ
1
= cos2 |00i + sin2 |11i + √ sin θ|+i.
2
22|ψ(t = 0)i =(2.41)(2.42)The molecules then evolve for some time T under Vdd ,ei(Jz −J⊥ )T /2θθ√sin θ|+i.
|ψ(t = T )i = cos2 |00i + sin2 |11i +222(2.43)At the two-particle level, the dynamics depends only on the difference χ ≡ J⊥ − Jz .
Finally, a pulse of area (π −θ) and variable phase φ closes the interferometer, and we measure
the populations in |0i and |1i. After some algebra, these are


1
χT
χT
2
P0 (T, φ) = sin θ 1 − cos θ cos φ sin+ sin φ cos
2
2
2(2.44)

P1 (T, φ) = 1 − P0 (T, φ).
From equation (2.44), we can see a few interesting limiting cases. First, consider θ = π/2
where the populations are initially balanced. Here, P0 (T, φ) = (1/2)(sin φ cos χT /2). In the noninteracting case, P0 (T, φ) = (1/2) sin φ, so we see that the interactions lead to a modulation of
the contrast C(T ) ∝ cos χT /2. If instead θ 6= π/2, then the fringes also shift with a frequency∝ (χ/2) cos θ. This frequency shift is linear in the population imbalance cos θ ∝ N0 − N1 and is thetwo-particle analogue of the mean-field shift studied in Chapter 8. Figure 2.14 shows the calculatedRamsey fringes P0 (T, φ) for two different Bloch sphere tipping angles (A) θ = π/2 and (B) π/4.

40Figure 2.14: Ramsey fringes for two molecules interacting via Vdd . (A) θ = π/2. There is nofrequency shift but the contrast oscillates at angular frequency χ/2. (B) θ = π/4. There is afrequency shift of (χ/2) cos θ.

For two particles, the dynamics can be perfectly cancelled using a dynamical decoupling
sequence called the WAHUHA sequence. Ref. [141] gives a detailed explanation of the two-particle
case. In an optical lattice with many particles, the WAHUHA sequence removes the dominant
oscillation in the contrast, which comes from the nearest-neighbor coupling; however, the wide
range of coupling strengths coming from further interaction partners results in an eventual decayof the contrast [46].

2.6.2

Calculating the couplings

The values of the couplings J⊥ and Jz depend on the states that make up our spin-1/2 system| ↓i, | ↑i. They can also be tuned over a wide range by changing the external electric field, whichchanges the induced and transition dipole moments [45, 140]. Figure 2.15 shows the trends of J⊥
and Jz as a function of  for the spin systems formed by {|0, 0i, |1, 0i} (solid lines), {|0, 0i, |1, ±1i}
(dashed), and {|0, 0i, |2, 0i} (dotted). Note that {|0, 0i, |1, 0i} can be tuned to points where eitherof J⊥ or Jz is dominant. On the other hand, J⊥ is dominant over the entire plotted range for{|0, 0i, |1, ±1i}; conversely, Jz is dominant for {|0, 0i, |2, 0i}. Finally, note that J⊥ < 0 for the{|0, 0i, |1, ±1i} system, so switching from there to the {|0, 0i, |1, 0i} system in the middle of a

41

Figure 2.15: Electric field dependence of J⊥ and Jz .

measurement can reverse the evolution of the many-body state [49].
More generally, any externally controlled mixing of the states can be used to control theinteraction strengths — for example, by applying a microwave drive to the molecules, we can makeone or both of | ↓i and | ↑i a coherent superposition of bare states. To calculate the interaction
parameters, we just need to repeat the analysis of equation (2.36), using our choice of | ↑i and| ↓i in place of |0i and |1i. Several interesting cases are worked out fully in Ref. [45]. A similareffect also occurs naturally due to the avoided crossings at low electric field (Figure 2.2) that mix
the N = 1 states, with each avoided crossing producing a sharp dip in J⊥ . For the density shiftmeasurements in Chapter 8, it was important to take these mixings into account when calculating
the expected size of the interaction shift.

2.6.3

Beyond the p = 0 term

So far, we have focused solely on the p = 0 term of equation (2.25), since the p 6= 0 terms
are off-resonant and hence highly suppressed. However, in some cases they can be brought toresonance. For example, if the |1, ±1i states are made degenerate, then the pair states |0, 0; 1, 1i(2)

and |1, −1; 0, 0i are also degenerate and interact resonantly via T±2 (d1 , d2 ). This is the idea behindthe spin-orbit coupling proposal of Ref. [139]. The interaction gives a spin-orbit coupling because
the total angular momentum projection mN 1 + mN 2 + mL is conserved by Vdd , so if mN 1 + mN 2

42changes by p, then mL necessarily changes by −p. A similar effect can also occur in the resonant
shielding when coupling |1, 0; 1, 0i and |0, 0; 2, ±1i, which interact via the p = ±1 terms.

2.7

Summary
We started this chapter by discussing the relevant energy scales of our diatomic molecules,

eventually focusing on the rotational levels of the ground electronic and vibrational state. We
showed the full Hamiltonian describing the rotational states in external magnetic and electric fields,
including hyperfine couplings, and the complicated energy spectrum that can arise, particularly at
low electric fields. Next, we described the induced and transition dipole moments of the molecules,
and their changes under an external electric field, as well as the effects of the optical trapping light.
Finally, we described aspects of the dipolar interactions between molecules, highlighting a few casesthat are especially relevant to the work in this thesis.

Chapter 3

Apparatus and experimental sequence

The experiments in this thesis were conducted on the second-generation JILA KRb machine.
The main feature of the new apparatus is the in-vacuum electrode assembly, which allows us toapply large and controllable electric fields to the molecules. The design and initial testing of theapparatus is described in detail in Jacob Covey’s thesis [106]. This chapter aims to give an updated
overview with a focus on several of the important features that were used to conduct the experimentsin this thesis. A complementary discussion, with emphasis on other aspects of the experiment, canbe found in William Tobias’s thesis [107].

3.1

Overview
Figure 3.1 shows a drawing of the machine, with several regions highlighted for reference inthe following discussion. The experiment consists of the following stages:• Laser cooling: K and Rb atoms are collected in a vapor cell magneto-optical trap (MOT,
blue region in Figure 3.1). The relevant atomic transitions are shown in Figure 3.2. Afterseveral laser cooling stages, including Λ-enhanced gray molasses cooling for K [142, 143]
and Rb [144], atoms are optically pumped to the low-field-seeking stretched states (|F =9/2, mF = 9/2i for K and |2, 2i for Rb) and loaded into a magnetic quadrupole trap. At
this point, we typically have around 3 × 107 K and 6 × 108 Rb at 100 µK.
• Magnetic transfer and plugged quadrupole evaporation: The magnetic quadrupole coils aremounted on a moveable stage (“Cart” in Figure 3.1), which translates down a differential

44

Figure 3.1: Schematic of the JILA KRb machine as viewed from above. SC: Science cell. The ion
pumps (toward the top of the image) are not shown. The magnetic transfer cart is shown in the
middle of its motion.

45
pumping tube (orange region) to the science cell (green region), where all subsequent stagesof the experiment are performed. After transport, a blue-detuned plug beam is turned
on to prevent Majorana spin-flip losses [145]. Radio-frequency (RF) evaporation of Rb
is performed on the |2, 2i → |1, 1i transition, and K is sympathetically cooled throughcollisions with Rb. At this point, we typically have 4 × 106 K and 6 × 106 Rb at 4 µK.
• Optical evaporation: K and Rb are loaded into a crossed optical dipole trap and evaporated
by lowering the trap depth. The favorable background K-Rb scattering length (−185a0 ,where a0 is the Bohr radius) allows for efficient sympathetic cooling of K through collisions
with Rb. Midway through the evaporation, the magnetic field is ramped to 30 G and theatoms are transferred to the Feshbach states (|9/2, −9/2i for K and |1, 1i for Rb) using
RF adiabatic rapid passage (ARP). Figure 3.3 depicts the relevant atomic hyperfine states.
After evaporation, the magnetic field is ramped to 555 G, above the broad interspecies
Feshbach resonance at 546.6 G. The atom number and temperature depends on the final
evaporation cut, but typical conditions are 5 × 105 K and 1.5 × 105 Rb at around 150 nK.
• Molecule formation and detection: Loosely-bound Feshbach molecules are produced byramping the magnetic field through the interspecies Feshbach resonance. The molecules arecoherently transferred to their ground rovibrational state using stimulated Raman adiabatic
passage (STIRAP). Remaining unpaired atoms are removed from the trap using resonant
light. For detection, the molecules are mapped back into atoms by reversing the coherent
association process, at which point they are detected using standard absorption imaging
techniques.
The atomic mixture preparation stages have been described in many group theses [146, 147,
141, 106]. Most of the laser cooling stages and laser setups are still similar to the first-generation
apparatus [141], except for the new Λ-enhanced gray molasses and tunable STIRAP setups, which
are described in detail in Will Tobias’s thesis [107]. The plugged quadrupole setup is fairly similar
to the Mainz K-Rb mixture experiment, and the theses of Thorsten Best [148] and Sebastian Will

46

Figure 3.2: Atomic level diagrams for 87 Rb (left) and 40 K (right). GMC: gray molasses cooling.
EOM: electro-optic modulator. The beams labeled “GMC (EOM)” are generated from sidebandson the “GMC (main)” beams using high frequency EOMs at 6.835 and 1.286 GHz.

Figure 3.3: Relevant hyperfine states of 87 Rb (left) and 40 K (right). The red arrows indicate thetransitions used for state preparation. For Rb, |2, 2i → |1, 1i is used for the RF evaporation in
the plugged quadrupole trap and for state preparation at 30 G. For K, transfer from |9/2, 9/2i to|9/2, −9/2i is performed with 9 consecutive ARPs at 30 G.

47[149] were very helpful in optimizing our procedure.
The early stages of the experimental sequence consist mostly of standard techniques fromthe ultracold atom community. We briefly describe the optical trap setup around the science cell,and then focus on some of the subsystems most relevant for the molecule experiments: the electricfield control, RF setup, and CCD setup.

3.2

Optical trap setup
Optical dipole traps (ODTs) and lattices are ubiquitous in the AMO community; see Refs.

[112, 150] for a detailed introduction. We use traps generated by a high power fiber laser (Nufern
NuAmp 50W) at λ = 1064 nm, which is far red-detuned from any atomic or molecular electronictransitions in our system. This results in a conservative trapping potential,
U (r) = −αI(r),(3.1)

where α is the (species- and wavelength-dependent) polarizability and I(r) is the position-dependent
intensity of the trapping beam. Table 3.1 gives the values of α for K and Rb, as well as some of
the KRb states used in this work. Note that the polarizability of excited rotational states of KRb
also depends on the polarization angle of the trapping light (see section 2.3.4 for an introduction).
Figure 3.4A shows a schematic of the ODTs and imaging paths in the science cell. The plug
beam (760 nm, 20 µm waist) is focused on the center of the quadrupole trap to prevent spin-flip
losses. After plugged quadrupole evaporation, we load the atoms into a crossed ODT formed by two
Gaussian beams at 1064 nm (labeled ODT1 and ODT2). The setup has changed several times over
the years, and the following discussion uses the beam parameters used in the lab as of this writing.
ODT1 propagates along −x̂ with waists w1,y = 35 and w1,z = 130 µm. ODT2 propagates along −ẑ
with waists w2,y = 35 and w2,x = 175 µm. Both beams are elliptical with the smaller waist against
gravity (−ŷ), and together they generate the trapping potential shown in Figure 3.4B. Note thatthe experimental coordinate system uses ŷ as the vertical direction.
To create a degenerate Bose-Fermi mixture, the atoms are evaporated in the ODT by reducing

48

SpeciesK
Rb
KRb* (Feshbach)KRb (|0, 0i at |E| = 0)α1064 /h (Hz/(W/cm2 ))28
32
αK + αRb [151]
55 [110]

ω/ωK
1
0.72
0.83
0.79

ER /kB at λ = 1064 nm (nK)
211
97
67
67

Table 3.1: Polarizabilities for K, Rb, KRb* (Feshbach molecules), and KRb at 1064 nm.

Figure 3.4: Optical trap geometry. (A) Schematic of the plug, optical traps, and imaging pathsin the science cell. LSL: large spacing lattice. (B) Crossed ODT potential formed by ODT1 andODT2, calculated for K in the y = 0 plane. The ODT powers for this calculation are the initialvalues used for loading from the quadrupole trap. Gravity is not included in this calculation.

49

Figure 3.5: Plug, ODT2, LSL, and imaging optics layout. For more details on the LSL opticalsetup, see section 5.1.1.

Figure 3.6: ODT1 and side imaging optics layout.

50

Figure 3.7: Geometry of the electrodes. (A) The science cell with in-vacuum electrode assembly
visible inside. (B) Dimensions of the electrode assembly in mm, viewed axially (along ẑ). Adaptedfrom Ref. [106].

the optical power over several seconds in two exponential ramps. The large spacing lattice (LSL)and vertical lattice (VL, not shown) traps are used for the 2D experiments and are described in
more detail in Chapter 5. The optical layouts for generating the beams in Figure 3.4 are shown in
Figures 3.5 and 3.6.

3.3

Electric field control
As introduced in Chapter 2, a key technique for manipulating the molecules is applying an

external electric field E. In this section, we will discuss a few high-level considerations, followed bysome technical details of the setup. See Will Tobias’s thesis [107] for a complementary discussion.
In particular, Will’s thesis has an in-depth analysis of the electric field stabilization, which isimportant for understanding the limits to the molecule coherence from E.
The main feature of the second-generation JILA KRb apparatus is the in-vacuum electrodeassembly shown in Figure 3.7. The assembly consists of two fused silica plates coated with atransparent conductor (indium tin oxide (ITO)) and four tungsten rods. The transparent platespreserve optical access from the top and bottom, and help to create a homogeneous field. In themost basic field configuration, the plates generate a parallel-plate capacitor field along ŷ, while
the rods compensate for field gradients arising from edge effects. By independently controlling thesix electrodes, we can generate a flexible field configuration with control over the field strength

51|E|, its gradients in the x̂ and ŷ directions, and the angle θ in the x-y plane. In-vacuum bias
tees on the rods allow for simultaneous application of DC and microwave voltages to the rods. In
particular, this allows us to drive rotational transitions of the molecules using the near-field electricfield pattern generated by the rods. Jacob Covey’s thesis [106] has more details on the design andinstallation of the electrodes.

3.3.1

Back-of-the-envelope requirements

Let us start by discussing the requirements on the electric fields. We will frame this discussionin terms of two types of experiments that are explored in this thesis:• Collisions of the molecules at a fixed E (introduced in section 2.5). We should be able
to change the field strength |E|, tilt the field angle θ, and controllably apply or shim out
gradients. For the evaporative cooling experiments of Chapter 5, we also needed a tunablecurvature of |E| to controllably lower the trap depth.
• Ramsey spectroscopy to probe dipolar many-body physics in the gas, for example to study
the XXZ model [45, 140] (introduced in section 2.6). The interactions can be controlledwith the same field parameters (|E| and field angle). The molecular coherence will belimited by inhomogeneous broadening (e.g., field gradients) and time-varying field noise.
Of course, both of these benefit from a very stable and tunable E, though the precise requirements
are different.

3.3.1.1

Field strength and angle

Let us first consider what electric field strengths we need. For collisional experiments, wewould like to be able to access a wide range of induced dipole moments to tune interactions in
the gas. Owing to the large characteristic field of KRb (Ec = 3.9 kV/cm), we need to turn on alarge field of ∼ 12 kV/cm to get a sizable dipole moment of even 60% of dperm in the |0, 0i state(Figure 2.3). The resonant shielding explored in Chapter 6 requires a field of ∼ 12.5 kV/cm. A

52
similar field magnitude is needed to access the regime of the XXZ model where Jz is dominant
over J⊥ (Figure 2.15). The first-generation KRb machine had electrodes mounted outside of theglass cell, and experiments at large |E| were limited by charging of the glass cell, which resulted
in a non-repeatable field strength [141, 106]. Our in-vacuum electrodes allowed us to apply large
electric fields (up to about 15 kV/cm used in this thesis) without any major problems.
The capability to rotate the angle θ of E also allows one to manipulate the molecular interactions, especially when combined with confinement into reduced dimensions. For example, byconfining the molecules in a 2D plane and changing θ, the dipolar interactions can be changed fromall repulsive (θ = 0) to a combination of attractive and repulsive (θ = π/2). We used this capability
to control the collisions (Chapter 6) as well as the mean-field interaction (Chapter 8).

3.3.1.2

Gradients

Homogeneity of the electric field is also important to consider. At the most basic level, straygradients can cause molecules to spill from the trap. The force from an electric field gradientalong x is Fx = d(∂|E|/∂x), where d is the induced dipole moment. Putting this in experimentally
relevant units, we get∂|E|Fx (kB × nK/µm) ≈ 2.4 × [d (D)] ×
∂x



V
cm2


.

(3.2)

We can very roughly estimate that the gradients will start to become important when the force
times the extent of the cloud is comparable to the temperature. For a typical cloud size of ∼ 20 µm
and T = 200 nK, and d = 0.2 D, this gives ∼ 20 V/cm2 as the typical scale of gradient that willcause problems. These numbers are representative for KRb in |0, 0i at |E| = 4.5 kV/cm. Gradients
become more and more challenging to deal with as |E| increases, since both d and the electrodevoltages are increasing (so the fractional sensitivity is higher).
In Ramsey experiments, gradients across the cloud also contribute an inhomogeneous broadening that lead to decay of the contrast. We can rewrite our expression for Fx in more convenient53
units for the spectroscopy,∂|E|
Fx (h × Hz/µm) ≈ 50 × [ddiff (D)] ×
∂x



V
cm2


,

(3.3)

where ddiff is the differential dipole moment between the two states being probed. The coherencetime is roughly the energy scale of the inhomogeneity over the cloud. Considering ddiff = 0.2 D,cloud size of 20 µm, a gradient of ∼ 10 V/cm2 already gives an inhomogeneity of ∼ 2 kHz andlimits the coherence to ∼ 1/(2π × 2 kHz) ≈ 0.1 ms. Flipping this around, the sensitivity of theRamsey contrast also makes it the best signal to optimize when trying to shim out gradients of E.
For simplicity, we have only considered linear gradients here, although field curvatures will cause
similar issues.
Electric field gradients can also detune dipolar spin-exchange interactions [46, 152]. In KRb,
these interactions have an energy scale of ∼ 100 Hz over 532 nm (a typical lattice spacing) for the{|0, 0i, |1, 0i} system. A field gradient causes a spatially-varying energy shift between |0, 0i and|1, 0i that suppresses the interaction. From equation (3.3), a gradient of just a few V/cm2 cansignificantly detune the spin-exchange over several lattice sites.

3.3.1.3

Noise

Finally, we must consider noise on the electric field. We will make the distinction between
two different regimes of noise:• Low-frequency (“DC”) noise that occurs on timescales longer than our typical moleculeinterrogation time τ (τ ∼ 10 ms for a Ramsey experiment). This manifests as a nonrepeatable |E| from shot to shot.
• “AC” noise near the Rabi frequency Ω of our pulses or, if using dynamical decoupling, thedelay time between pulses.
Let us first consider the simpler case of DC noise. DC fluctuations of |E| are largely irrelevant
for collision experiments, since the induced dipole moment does not change rapidly with |E|. One

54
important exception is the shielding resonances studied in Chapter 6, which occur near 12.5 kV/cm
and are on the order of 20 V/cm wide. This gives a requirement on the fractional stability of ∼ 1600
parts per million (ppm).
Spectroscopy experiments place a much more stringent requirement on the shot-to-shot stability of |E|. Consider a Ramsey experiment probing two states with differential dipole moment
ddiff for dark time τ . Then the shot-to-shot variation in the phase is δφ = ddiff × δ|E| × τ /~, where
δ|E| is the shot-to-shot variation of |E|. Writing this in convenient experimental units, we getδφ (deg) ≈ 0.18 × |E|



V
cm






δ|E|
×
(ppm) × [ddiff (D)] × [τ (ms)].
|E|

(3.4)

For concreteness, suppose |E| = 1 kV/cm, where we have a relatively small differential dipole
ddiff = 0.08 D for |0, 0i ↔ |1, 0i for KRb. Plugging in numbers, we see that for τ = 1 ms and afractional stability of δ|E|/|E| = 1 ppm, the shot-to-shot phase deviations are already quite largeat δφ ≈ 14 deg. At higher |E|, ddiff can be significantly larger, making the Ramsey phase even moresensitive to δ|E|. Our electric field stabilization is designed to limit δ|E| to the 10 ppm level (seesection 3.3.5), and so we typically see that the phase of the Ramsey oscillation becomes scrambled
after a few hundred µs.1
If the fluctuations of |E| are slow compared to τ , then we can cancel them completely witha single spin echo pulse. However, if the fluctuations occur on a timescale comparable to τ , thenwe will need to add additional spin echo pulses to form a dynamical decoupling sequence [154]. Insuch a sequence, a train of equally-spaced echo pulses forms a bandpass filter for the noise centeredat 1/2τp , where τp is the pulse spacing [155, 156]. For the measurements in Chapter 8, where |E|
was as large as 9 kV/cm, it was essential to use dynamical decoupling to maintain phase coherenceof the molecules with the microwaves. Using the methods of Ref. [155], we could also measure
the power spectrum of |E(t)| through the noise in the molecule spectroscopy, and saw that the1

Note that shot-to-shot fluctuations of |E| cause the relative phase of the molecules and the microwaves to vary
shot-to-shot, but they do not affect the coherence between molecules in the sample. By randomizing the phase of theRamsey pulses and measuring the variance of the populations in the two states, we can still extract the moleculemolecule coherence without requiring a perfectly stable |E| (e.g., in section 7.2). This is somewhat analogous to asituation that can occur in optical lattice clocks where the atom-atom coherence exceeds the atom-light coherence
due to laser phase noise [153].

55
measured noise was in reasonable agreement with the expected field fluctuations calculated fromdirect measurements of the electrode voltages with an FFT machine (see Figure 3.11 of Ref. [107])
From this discussion, we can see that the high sensitivity of the molecules to E places stringent
requirements on the electric field stabilization. One approach to easing these requirements is to
reduce the bandwidth of the system, for example by adding low pass filters before the electrodes. A
downside is that very rapid changes of E are no longer possible. This is not a huge loss — although
rapid changes of |E| could be an interesting way to quench or modulate the dipolar interactions,microwave pulses can also be used to similar effect with less experimental effort. As a result, it often
makes sense to limit the bandwidth of E as much as possible. At very low bandwidths, timescales
associated with the molecular motion and collisions (at or below the trap frequency) can become
relevant; for maximum flexibility, the bandwidth of E should still remain far above these frequency
scales.2 Experimentally, we found that a bandwidth of a few kHz was a good compromise forstrongly suppressing AC noise while still being much larger than the typical transverse trapping
frequencies of ∼ 40 Hz.

3.3.2

Computing the field distribution

The discussion in the previous section underscores the level of flexibility and control needed for
E. A convenient model for the field distribution makes it easy to optimize the electrode voltages forany particular experiment. Will Tobias did COMSOL simulations to numerically find the potential
Vi φ(i) (x, y) from applying voltage Vi on the i-th electrode with all others grounded.3 By summingthese, we obtain the total electric potential for any combination of electrode voltages. To makethis even easier to work with, we fit the discretized electric potentials to a polynomial in x and y(7th order happened to work well) to obtain a closed-form approximation to the electric potential,
V (x, y) =

6
X
i=1

Vi φ(i) (x, y) ≈

6
X
i=1

Vi

7
X

(i)

αjk xj y k

(3.5)

j,k=0

2
For example, many of our calibrations involve kicking the molecules in the trap using an electric field gradient,which is not possible if the change in E is slower than the trap frequency. Methods like Stern-Gerlach imaging ofmolecules [107] are also difficult to implement if E is too slow.

56This made it very fast and easy to take derivatives to obtain the electric field and its gradients. Notethat we assume the potential is homogeneous along the ẑ-direction (the long axis of the electrodes).
Experimentally, we found good agreement of the COMSOL model with measurements of the actual|E(x, y)| obtained by rotational spectroscopy of the molecules at several positions in the sciencecell (see Figures 3.13 and 3.14 of Ref. [107]).
We use equation (3.5) to optimize the electrode values, with the goal of approximating thedesired E(x, y) based on several target parameters:• the field strength |E|,
• gradients ∂|E|/∂x and ∂|E|/∂y,• curvatures ∂ 2 |E|/∂x2 and ∂ 2 |E|/∂y 2 ,• and angle θ = arctan(Ex /Ey ) from the ŷ axis.
All target parameters are calculated at the geometrical center of the electrodes, which is where
the molecules are located (to the best of our knowledge). The optimization is currently very basic,
consisting of a least squares “fit” of the calculated field parameters to the target parameters byvarying the electrode voltages.
Note that we have six target parameters, but only five degrees of freedom (since changing allelectrodes by an offset voltage does not change E). This means that, in general, our optimization
procedure cannot perfectly match the desired E parameters except for in configurations of highsymmetry (such as a homogeneous field oriented along ŷ). To get around this, our optimization
routine allows us to ignore individual parameters in the fits, which we often need to do when e.g.,tilting the field angle θ. For the 1D lattice experiments (along ŷ), we often neglect the curvature∂ 2 |E|/∂y 2 , since the force from the small residual field curvature is negligible compared to the tightlattice confinement along ŷ.

57

Figure 3.8: Some useful electrode linear combinations.

3.3.3

Useful electrode linear combinations

The optimization routine makes it relatively easy to calculate the required electrode voltages
for a desired configuration of E. However, it is not very helpful for developing an intuition forwhat the electrodes are doing. To get a better physical picture, we often decompose the electrodevoltages in terms of certain linear combinations. These are illustrated in Figure 3.8. The electrodes
are labeled by their spatial arrangement: LP/UP stands for “lower/upper plate”, LE for “lower
east rod”, UW for “upper west rod”, etc.
The first combination, labeled “offset”, should not affect E. The remaining combinations
make the most sense in the context of perturbations from a homogeneous field along ŷ. To create abias field along +ŷ, we start by applying voltages ±V to the LP and UP, respectively (“bias” in the
figure). If the rods were somehow removed, the electric field strength would be roughly that of aparallel plate capacitor, |E| ≈ 2V /l, where l = 6 mm is the separation of the plates. The lower andupper rods are set to ±γV , where γ is a scaling factor (“rod scale”) that controls the curvatures∂ 2 |E|/∂x2 and ∂ 2 |E|/∂y 2 . For our geometry, we find that γ = 0.4225 cancels both curvatures,and we refer to this as the “flat field” configuration.4 Starting from the flat field, perturbing withthe “rod offset” (“h grad”) combination adds a linear y- (x-) gradient. Adding some “east-west”
component starts to tilt the field direction away from ŷ and toward x̂.
34φ(i) (x, y) is basically the Green’s function for our electrode geometry.
Intuitively, this choice of γ makes the rod voltages approximately equal to what the potential would be at their

58
3.3.4

Minor imperfections of the electrode geometry

Small imperfections in the electrode geometry can cause gradients of |E| in the nominally flat
field configuration. When we first started using the electric field, we observed a small horizontal
gradient that led to loss of the molecules. The gradient scaled linearly with the field strength,with a magnitude of ∼ 10 V/cm2 at 1 kV/cm bias, which would be consistent with a geometricimperfection of the electrodes, and was easily shimmed experimentally by adjusting ∂|E|/∂x to
maximize the molecule number after a hold time. For the experiments in Chapters 7 and 8, weshimmed the horizontal and vertical gradients more precisely by maximizing the Ramsey coherenceof the |0, 0i ↔ |1, 0i transition.
A more difficult gradient to fix would be a nonzero ∂|E|/∂z, since we do not have any way
to shim along that direction (the long axis of the electrodes). Such a gradient could occur if, forexample, the ITO plates were not parallel. We have observed what seem like small z-gradients on
the order of several V/cm2 at a bias of |E| = 4.5 kV/cm. A z-gradient of this magnitude is smallenough that it is difficult to measure reliably, given that we also have no way to controllably change
its magnitude. More systematic studies will be needed to gain a clearer picture of the limitations
posed by z-gradients in our system.

3.3.5

Controlling the electrode voltages

As mentioned in section 3.3.1, we require roughly 10 ppm stability of the field up to bandwidths of several kHz. This is challenging considering that the electrode voltages can be up to
several kV. A block diagram of the high voltage (HV) stabilization system is shown in Figure
3.9. The desired voltage waveforms Vi (t) are calculated using the optimization routine described
previously, and sent via USB from our computer control software to an FPGA evaluation board
from Opal Kelly (XEM6001). The FPGA programs six 20-bit precision digital-to-analog converters(DACs, Analog Devices AD5791) to output scaled-down waveforms −Vi (t)/2000, which provide they-positions due the plates at ±V , if the rods were removed. This also means that the parallel plate approximation
for |E| at the center of the electrodes is fairly good (at the ∼ 1% level) in the flat field configuration.

59

Figure 3.9: Simplified block diagram of the electric field control. For a detailed schematic of theservo, see Refs. [157, 107].

control voltages for the servos. The servos were designed by Terry Brown in the JILA electronics
shop. Each servo consists of an error amplifier, which compares the HV after a precision voltage
divider with the DAC signal, and a loop filter with three proportional-integral (PI) stages and adifferentiator (D) stage. The servo output is sent to an amplifier (Trek 10/10B-HS or Matsusada
AMT-5B20-LC), whose output goes through a 1 MΩ current-limiting resistor and is sent to theelectrode. The Trek (Matsusada) amplifiers have a maximum output of 10 (5) kV, which gives amaximum field of |E| = 33 kV/cm in the flat field configuration.
Achieving good stability requires a great deal of care with the analog electronics. Will Tobias
took the lead on most of the analog aspects, in particular the ±5V reference and DAC output
filtering, and his thesis contains a detailed discussion [107]. See also Jamie Shaw’s undergraduate
honors thesis for some initial testing of the high voltage servos [157], and Appendix A of Will’s
thesis for a detailed noise analysis of the full system [107]. I worked primarily on the FPGA design
and the high voltage cabling and connections, which are described in the following sections.

3.3.6

FPGA control of AD5791 DACs

The computer control system interfaces with the AD5791 DACs via an Opal Kelly XEM6001
evaluation board, which contains a USB microcontroller and Xilinx Spartan 6 FPGA. The XEM6001
was chosen due to familiarity, as it is used extensively in the rest of our control system. For

60
simplicity, each of the 6 DACs is controlled with an independent process in the FPGA. At the startof each experimental run, the desired sequence of (time, voltage) pairs for each DAC channel issent from the control system using Opal Kelly’s Python SDK and loaded into a RAM block on the
FPGA. Each pair is interpreted by the FPGA firmware as the endpoint of a voltage ramp, and theintermediate ramp points are computed as the sequence runs in order to save space in the FPGARAM. After data transfer, the FPGA waits until it receives a trigger from the main TTL board,at which point it starts to execute the sequence. The AD5791 DACs expose a serial peripheral
interface (SPI) bus that is written to by the FPGA to control the DAC output voltage.
Unfortunately, we found that distortions from the ribbon cabling between the FPGA and the
DACs limited the fastest reliable SPI clock speed to 4 MHz, significantly lower than the specified
maximum clock rate of 25 MHz for the AD5791. This results in a minimum update time of about
8 µs, which is still much faster than the HV response (discussed in the next section). The limited
RAM on the FPGA limits the number of sequence points to 210 = 1024 on each channel. To avoid
wasting RAM, the computer control code concatenates adjacent sequence blocks together wherethe voltage is constant before loading the sequence onto the FPGA.

3.3.7

Cabling and connections

The HV cabling and connections are loosely modeled off the techniques used on the Stark
decelerator experiments in JILA. Thanks to Dave Reens of Jun’s group and Yomay Shyur of Heather
Lewandowski’s group for their help. Our experimental voltage and current requirements are far
lower than those of Stark decelerators, so some of this is likely overkill.
The output of each HV amplifier is an Amphenol Alden A- (Trek amps) or B- (Matsusadaamps) series connector. We use Amphenol Alden A- and B-series cables (> 15 kVDC rated) forthe connections within the HV rack from the amplifier output to the servo sense input. The HV
feedthroughs on the vacuum chamber use PowerBoot cables from MDC Vacuum Products (20 kV
rated). Each cable is fed through a 1/2” or 3/4” diameter Tygon tube surrounded by a grounding
braid. The Tygon tubing helps to maintain a greater distance between the cable core and the61

Figure 3.10: Corona sphere for HV connection.

grounding braid to further increase the dielectric breakdown of the cable. The grounding braid isconnected to the either the optical table or the HV rack ground to provide a low impedance pathin case of a discharge.
The high voltage cabling requires several homemade tees and adapters between the two cabletypes. Sharp corners or points on any of the conductors should be avoided, as these can cause
regions of high electric field that lead to corona discharge [158]. Soldering HV components can thus
be quite cumbersome as large solder balls are needed to reduce sharp edges. For greater flexibility
in the connections, we use corona spheres to make these connections (Figure 3.10).5 The corona
sphere maintains a smooth outer surface to suppress corona discharge. The cables are inserted
through holes drilled in the sphere and clamped down by a set screw to make a good electricalconnection. At our highest HV operation of about 5 kV (so far), we have not observed any signsof corona discharge.
The cabling from the high voltage amplifiers to the vacuum feedthroughs has an important
impact on the eventual electric field bandwidth. For safety (of both grad students and the apparatus), we have 1 MΩ resistors on the HV output to limit the current to ≤ 10 mA. Combinedwith the capacitance of the cabling, this creates an RC filter that limits the switching time of5

Instructions courtesy of Dave Reens: https://www.instructables.com/High-Voltage-Corona-Nuts/

62

A

B
10
LP voltage (V)Amplitude (dB)

0

-10

-20

-30

5

0

10¹

10²
10³
Frequency (Hz)

10⁴

-0.5

0.0

0.5
1.0
Time (ms)

1.5

2.0

Figure 3.11: Transfer function of the HV cabling. The 1 MΩ resistors on the HV output combinewith the capacitance of the cabling to form an RC filter. (A) Measurement of the transfer function of
the 1 MΩ resistor + cable. The measured corner frequency is 1.3 kHz, from which we extract a cablecapacitance of C = 120(20) pF. (B) Rise time of the LP voltage as measured on an oscilloscope.
The fitted exponential rise time is 140 µs, in good agreement with the expected 1/RC = 120 µs
expected from (A). In (B), the cable is connected to the electrode, which is also expected to add asmall capacitance, though we estimate it is ∼ 1 pF.

the electrodes. As discussed in section 3.3.1.3, this is actually beneficial since it reduces the servo
bandwidth required for E. Figure 3.11A shows a measurement of the transfer function of one of the
HV cables. We extract a cable capacitance of C = 120(20) pF, which is in reasonable agreementwith the estimated capacitance from the radii of the cable and the shielding braid [107]. Figure
3.11B shows the rise time of the LP electrode voltage, which has an exponential rise time consistent
with the corner frequency measured in (A). We calibrated all 6 electrode setups with this method
and found similar results.

3.3.8

High voltage rack

The HV amplifiers were initially set up on the floor of the lab under the main optical table ofthe experiment to minimize the lengths of cables needed. Unfortunately, this led to excess heatingin our 1D optical lattice. The HV amplifiers have large fans that produce vibrations, air currents,and heat. This unwanted environmental noise showed up as heating of the atoms and molecules.

63

A

B
0.5

off: 20(4) nK/s
Temperature (μK)

Temperature (μK)

0.5

on: 40(2) nK/s
0.4

0.3

0.2

off: 18(2) nK/s
on: 17(5) nK/s

0.4

0.3

0.2
0

2

4
Time (s)

6

8

0

5
Time (s)10

Figure 3.12: Heating rate of a K Fermi gas in the optical lattice from HV amp vibrations. (A)Before moving the amps to the rack. The heating rate with the HV amps off (blue) is much lower
than with them on (orange). (B) After moving the amps to the rack. The heating rate does notshow any effect from the HV amps.

To characterize this, we produced a spin-polarized Fermi gas of K, loaded it into the 1D lattice, andmeasured the temperature as a function of hold time (Figure 3.12A). With the HV amps poweredoff (blue points), we measured a heating rate of 20(4) nK/s, slightly higher than the ∼ 10 nK/s
expected from off-resonant scattering. With the HV amps on (orange points), the heating rate
went up to 40(2) nK/s. This heating was a major barrier to successfully evaporating the molecules.
Our solution was to move the HV electronics into a reconfigurable server rack next to theexperiment (Figure 3.13). We mounted the rack directly into the floor using concrete anchors andalso placed a thick layer of Viton rubber between the rack and floor to damp the vibrations of the
supplies. The fans are mounted on the back of the supplies, so we put panels on the back and sidesof the rack to block the air currents from interfering with other laser setups in the lab. We also
attached acoustic damping material to the insides of the panels to further suppress vibrations. We
left the top and front of the rack open, which keeps the inside of the rack from becoming very hot
but also lets some noise and air currents escape. These changes were apparently good enough to
fix the vibration problem, since we measured no effect of the HV amps on the heating rate after

64

Figure 3.13: High voltage rack holding 6 HV amps, servos, and stable DACs. This picture wastaken just after installation of the rack, so many BNC and HV connections are missing.

65

Figure 3.14: Block diagram of the K/KRb RF setup.

moving the supplies (Figure 3.12B). Having all the HV electronics in the same rack also made iteasy for us to try several different grounding schemes to improve the noise performance (see Ref.
[107] for details). In the future, we should add an air-water heat exchanger inside of the rack, which
would allow us to fully enclose it to further reduce the temperature and air current fluctuations.

3.4

Radio-frequency fields
Radio-frequency (RF) fields are an important tool for manipulating ultracold atoms andmolecules. We use them for state preparation of the atoms using adiabatic rapid passage (ARPs),
as well as for spectroscopy of the molecules. A block diagram of the RF setup for K and KRb
is shown in Figure 3.14. The Rb RF system is fairly simple and is not shown in the diagram.
The main RF sources are built from a direct digital synthesis (DDS) chip from Analog Devices(AD9910 evaluation board). The DDS evaluation board provides a cheap and flexible synthesizer
with frequency (0 to 400 MHz), amplitude, and phase control, which is programmed over serial
peripheral interface (SPI) from an Arduino. The Arduino is also connected to a computer overUSB, which allows integration with our computer control system.
Our system consists of two Arduino/AD9910 setups:(1) “K/KRb DDS”: used for K state preparation at 10 and 80 MHz, and for driving N = 0 ↔ 1

66

Figure 3.15: Example Ramsey fringe. (A) Schematic of the Ramsey measurement. Two pulsesof area π/2 are spaced by a time T . The first pulse is performed at zero phase offset, which is
programmed in (for example) DDS profile 1. The second pulse is performed at a variable phaseoffset ∆φ, which is programmed in profile 3. The amplitude is controlled with a switch (ZASWA-250DR) to produce square pulses. Amplitude-shaped pulses can also be produced; see section 3.4.1.
(B) Experimentally measured Ramsey fringe on the |0, 0i ↔ |1, 0i transition of the molecules. Thisdata was taken at a short time (T = 10 µs) before there is any measurable decoherence of the fringe.

transitions of the molecule;(2) “N = 2 DDS”: used for driving N = 1 ↔ 2 transitions of the molecule.
The three different frequencies are represented by the colored lines in Figure 3.14. Both DDSs havea switch after the output to turn the RF on and off. The K/KRb DDS additionally has a switch
to toggle between the low frequency (K) and high frequency (KRb) paths. The K transitions arewithin the frequency range of the DDS output, which is sent directly to a 20 W amp (Mini Circuits
LZY-22). After this amp, there is a high power switch (Mini Circuits MSP2TA-18) that toggles
between a non-resonant coil for the ARPs at 30 G and a resonant 80 MHz coil for hiding unpaired
K near the Feshbach resonance. The 80 MHz antenna has three loops and is impedance matchedusing a variable capacitor.
Both of the N = 0 ↔ 1 and 1 ↔ 2 frequencies are higher than the DDS can reach (> 2.2 and> 4.4 GHz respectively), so each DDS is mixed with a synthesizer to achieve the desired frequency.
The advantage of this setup over simply using the synth output directly is that the DDS’s output
frequency, amplitude, and phase can be changed much faster than the synth’s. A key feature of the

67

Figure 3.16: AC rod configurations. For simplicity, the in-vacuum bias tees are omitted for thehorizontal configuration.

AD9910 is the ability to set preprogrammed “profiles”, each of which specifies a frequency, output
power, and phase offset. Switching between profiles is fast (∼ µs) and phase coherent, which allows
us to do Ramsey spectroscopy and more complicated dynamical decoupling sequences such as XY8[154] (Chapter 8). As an example, Figure 3.15 shows how a simple Ramsey sequence is implemented
with the AD9910 profiles. The profiles are switched with a TTL signal (not shown) in the middleof the sequence to set the phase offset ∆φ for the second pulse. The precise timing of the pulses iscontrolled by the switch directly after the output of the DDS (Mini Circuits ZASWA-2-50DR, 5 ns
typical rise/fall time).
The N = 0 ↔ 1 and 1 ↔ 2 frequencies are combined with a power splitter/combiner and sent
to a broadband amp (CTT AGM/060-4343, 43 dBm output). The output of the amp is connectedto the AC port of the in-vacuum bias tee on the rod electrodes. We can manually switch between
the two configurations shown in Figure 3.16, which produce either mostly ŷ (vertical) or mostly x̂
(horizontal) polarization. At our maximum power, we observe π-pulse times of 0.5 to 10 µs for the|0, 0i ↔ |1, 0i transition with the rods connected in the vertical configuration. The variation in the
Rabi frequency is due to the frequency-dependent couplings to the rods from the bias tees [106].
Some limitations with this setup are described in section 3.4.2.
We had initially used a switch to combine the N = 0 ↔ 1 and 1 ↔ 2 frequencies, which gives

68

Figure 3.17: Probing N = 0 ↔ 1 while dressing 1 ↔ 2. (A) Diagram of the involved rotational
levels. The molecules are initially prepared in |0, 0i. (B) Lineshapes for |0, 0i ↔ |1, 0i without(blue) and with (orange) dressing on |1, 0i ↔ |2, 0i at E = 4.56 kV/cm.

a higher power at the molecules than the power splitter. However, the splitter has the advantagethat multiple tones can be applied at the same time. Figure 3.17 shows how we can probe on the|0, 0i ↔ |1, 0i transition while dressing with |1, 0i ↔ |2, 0i. The molecules are initially preparedin |0, 0i and are excited to the |1, 0i state using a microwave pulse. Scanning the frequency ν ofthis pulse without the dressing produces a single peak in the transferred fraction (blue points in
Figure 3.17). Turning on an additional dressing field resonant with the |1, 0i ↔ |2, 0i transition
splits the peak into two peaks separated by 2Ω12 (orange points), where Ω12 is the Rabi frequencyof the dressing field. By detuning both drives from one-photon resonance, we can also drive Rabi
oscillations directly between |0, 0i and |2, 0i without populating |1, 0i. Extending this capability to
be able to apply many tones simultaneously may be helpful in the future, for example to control
the dipolar interaction using microwave dressing [45] or to implement synthetic dimensions using
the rotational states [159].

3.4.1

Amplitude-shaped pulses

The envelope of the RF pulse can add undesired frequency components to the output. For
most of the work in this thesis, the pulse envelope was not important and we used square pulses
for simplicity. However, for the layer selection (Chapter 7), we used amplitude-shaped pulses toavoid frequency sidebands that could potentially drive molecules in adjacent layers. To implement

69
this, we added another mixer (Mini Circuits ZFM-2, not shown in Figure 3.14) just before the
ZX05-83LH-S+ mixer on the N = 0 ↔ 1 path. A function generator programmed with the desiredenvelope was connected to the IF port of the new mixer and externally triggered to fire the pulse.
To avoid distortion of the pulse, it was important to calibrate the LO and IF powers to remain in
the linear regime of the mixer (Figure 3.18A).
For the layer selection, we used Blackman-shaped pulses,
A(t) = 0.42 − 0.5 cos

2πt
τ




+ 0.08 cos

4πt
τ


,

(3.6)

where τ is the total time of the pulse. Note that A(t) reaches a maximum of 1 at t = τ /2. The pulseRτ
area is 0 dt A(t) = 0.42τ , so compared to a square pulse of the same amplitude, the Blackman
pulse needs to be longer by a factor of 1/0.42 = 2.38 to obtain the same pulse area. An example
RF pulse with a Blackman envelope is shown in Figure 3.18B. Figure 3.19 shows a comparison of
RF lineshapes on the molecules using square (blue) and Blackman (orange) pulses. The prominent
sidebands from square pulses are removed by using Blackman pulses.

3.4.2

Limitations of the rods: polarization purity and AC field homogeneity

In principle, driving the microwave fields directly on the rods produces the near-field E(t)and gives a clean microwave polarization. However, we find in practice that the homemade bias
tees (see Ref. [106]) do not provide an equal coupling onto all of the rods, and we can drive all three∆mN = 0, ±1 transitions with similar Rabi frequencies in the nominally ŷ-polarized configuration(Figure 3.16). So far, we have not had a problem addressing individual transitions by resolving
them in frequency rather than with polarization. However, well-controlled polarization would be
required for some microwave dressing schemes [160] or microwave shielding [161, 162, 105, 54].
We have also observed that the Rabi oscillations of the molecules decay over ∼ 25 cycles(Figure 3.20A), which we attribute to an inhomogeneous Rabi frequency Ω caused by the unequal
coupling onto the rods. This explanation is consistent with the observation from Figure 3.20B thatthe oscillation decays slower with decreasing Ω, up until the point where we start to become limited

70
0

A

B
0.25
Output (V)RF (dBm)

-10

-20

-30

measured
0.00

calculated
-0.25

-40
0.01

0.10
IF (V)

1.00

-25

0

25

50 75 100 125
t (μs)

Figure 3.18: Calibration of the amplitude shaping setup. (A) Calibration of the mixer. The
part (ZX05-10L) calibrated in this plot was slightly different from what we eventually used (ZFM2+) in the layer selection measurement. The input at the LO port was 3 dBm at 75 MHz. (B)Blackman-shaped pulse output from the mixer, as measured on an oscilloscope (blue). The input
at the LO port was 3 dBm at 21 MHz and the max IF voltage was 0.6 V. The Blackman envelope
from equation (3.6) with τ = 100 µs is shown for comparison (orange). The envelope is scaled tomatch the data.

Transferred fraction

0.8

squareblackman
0.6

0.4

0.2

0.0
790

800
810
820
830
Frequency - 2248 MHz (kHz)840

Figure 3.19: Comparison of lineshapes using square (blue) and Blackman (orange) pulses. Thelineshapes were obtained by driving |0, 0i → |1, 0i at |E| = 1 kV/cm. The maximum pulse amplitudes were set equal and the duration of the Blackman pulse was chosen to make the pulse areas
equal. The square pulse data is fit to a Rabi lineshape (blue line), and the Blackman pulse data is
fit to a Gaussian lineshape (orange line).

71

B

1.0

Fraction |1,0⟩

Ω/2π = 45(1) kHz

0.5

1/e decay = 250(70) μs
0.0

Rabi contrast at 700 μs

A

1.0

0.5

0.0

0

100

200
300
Time (μs)

400

0

25
50
Ω/2π (kHz)

Figure 3.20: Inhomogeneous AC electric field. (A) Rabi oscillations at |E| = 4.5 kV/cm. The
oscillations show decay after ∼ 25 cycles, indicating an inhomogeneous Rabi frequency across thecloud. (B) Rabi oscillation contrast at the end of a 700 µs pulse as a function of the average Rabi
frequency Ω. The contrast is improved for lower Ω, until inhomogeneities of E start to dominate
at the lowest Ω.

by inhomogeneous broadening of the lineshape due to E-field gradients.6 In simulations, we foundthat the measured decay is consistent with a ∼ 2% variation of Ω over the 2σ width of the cloud.
This data was taken in the vertical configuration of Figure 3.16, where we do expect to have anonzero gradient of Eac owing to the asymmetry between the voltages of the upper and lower rods.
However, we calculate that the expected vertical gradient of Eac is about a factor of 5 too small toaccount for the observed inhomogeneity of Ω. A factor of 2 difference in coupling efficiency between
the two upper rods, or a fairly large phase shift between them, would explain the observed decayof the Rabi oscillations by creating a horizontal gradient of Eac .
For this reason, it was important to use the XY8 sequence [154] for the experiments in Chapter
8, which cancels errors in the pulse area to lowest order. The limitations from the unequal coupling
could probably be further mitigated by individually phase-shifting and attenuating the signals foreach rod to shim the RF field. According to Jun, the JILA eEDM experiment has done extensive
work on shimming the RF fields on the eight electrodes that generate their rotating electric field,
6

The data was taken before carefully shimming this particular field configuration.

72
so they will be a good resource if we decide to pursue this further.

3.5

CCD calibration
The atoms and molecules are imaged onto a CCD camera using standard absorption imaging(see Refs. [163, 164] for a detailed introduction). We use two Andor iXon Ultra 888 electronmultiplying (EM)CCD cameras; the axial (ẑ) and side (−x̂) imaging paths are combined on thefirst camera (see Figure 3.5), and the top (−ŷ) imaging goes to the second camera. An absorption
measurement consists of three images: (1) probe light with atoms (“shadow” frame Sshadow ); (2)probe light without atoms (“light” frame Slight ); (3) no probe light without atoms (“dark” frame
Sdark ). The optical density (OD) frame is obtained by
OD = − ln

Sshadow − Sdark.
Slight − Sdark(3.7)

For low probe light intensity, the OD on a pixel of area A is converted to the number of atoms N
on the pixel using N = (A/σ) × OD, where σ = 3λ2 /2π is the resonant absorption cross section at
wavelength λ. For higher light intensity, one must account for saturation of the imaging transition[165, 166].
An important noise source in absorption imaging is the photon shot noise of the probe beam.
For the density fluctuation measurement in section 4.2.3, it was important to remove the photon
shot noise contribution to accurately determine the atom and molecule number fluctuations. A
detailed description of the image analysis procedure is given in Refs. [51, 107]. Here, we will
focus on one important step, the calibration of the CCD gain. To understand why the CCD gain
is important, consider what happens during an exposure. Photons incident on the CCD arrayare converted to photoelectrons with some quantum efficiency q. If the number of photons is
Poisson-distributed with mean Nphoton , then the number of electrons is also Poisson-distributedwith mean Ne = qNphoton [163]. The photoelectrons are then converted to CCD counts according
to a (deterministic) gain factor g that depends on the camera model and settings. This results ina mean number of counts Nc = gNe , but a variance ∆Nc2 = g 2 ∆Ne2 = g 2 Ne = gNc . Thus, Nc is

73

Figure 3.21: Calibration of Andor iXon Ultra 888 CCD. (A) Conventional mode (no EM gain).
The linear fit gives Var(Sdiff ) = 0.589(2)S + 7.77(9). (B) EM mode, with a nominal EM gain of 50.
The linear fit gives Var(Sdiff ) = 12.184(25)S − 77(14).

not Poisson-distributed, but instead is “stretched out” by a factor of g. Measuring the slope of∆Nc2 /Nc gives us g and allows us to determine the photon shot noise.
We decided to check the value of g against the specified value from the data sheet following
the procedure in Ref. [163]. To do this, we took 60–100 absorption images without atoms present
and varied the probe light intensity between a few different values. From the shadow and lightframes of each image, we formed the difference image Sdiff = Sshadow − Slight . We chose a small
analysis region near the center of the probe where the intensity of the beam is fairly uniform,and calculated the variance Var(Sdiff ) over the pixels in this region. If S is the average numberof counts per pixel in the analysis region (after subtracting the background Sdark ), then Var(Sdiff )should satisfy
Var(Sdiff ) = 2gS + 2(rg)2 ,(3.8)

where g is the camera gain and r is the readout noise in electrons. The factor of two on bothterms comes from the subtraction of Sshadow and Slight . The second term is missing an additional
factor of 2 compared to Ref. [163] because Sdiff was formed from (Sshadow − Slight ) instead of[(Sshadow − Sdark ) − (Slight − Sdark )]; the latter contains an extra readout noise contribution fromthe dark frames.7
7

It looks like the readout noise from Sdark should cancel out, since it is perfectly correlated between (Sshadow −74
According to equation (3.8), we can extract g and r from a linear fit of Var(Sdiff ) versus
S. The results are shown in Figure 3.21. For conventional CCD mode (no EM gain) in (A),we extracted a gain of g = 0.295(1) counts/electron and readout noise of r = 6.69(4) electrons,
reasonably close to the specified values of g = 1/3.5 = 0.286 counts/electron and r = 6.86 electrons.
In units of CCD counts, the readout noise is rg = 1.97(1).
We also calibrated the camera with a nominal EM gain of 50 in (B). The EM amplifier has
its own shot noise which doubles the first term in equation (3.8) [167]. The readout noise comes
after the EM amplifier and thus is unaffected. This leads to a modified equation
Var(Sdiff ) = 4AgS + 2(rg)2 ,(3.9)

where A is the EM gain. Accounting for this extra noise, we obtained Ag = 3.046(6) counts/electron
for the data in Figure 3.21B. Using the specified value of g = 1/16.4 counts/electron, we extracted
A = 49.9(1), which agrees with the nominal EM gain setting.
The fitted offset in Figure 3.21B is slightly negative, which we attribute to the readout noise
being too small to measure. This is understandable since the purpose of the EM amplification is tomake the readout noise negligible. Plugging in the specified values of g = 1/16.4 counts/electron
and r = 25.1 electrons, we estimate a readout noise contribution of 2(rg)2 = 4.5, which is muchless than the smallest measured Var(Sdiff ) of ∼ 500 in Figure 3.21B.

Sdark ) and (Slight − Sdark ). However, this actually depends on whether fast kinetics is being used. If Sshadow and Slightare obtained from different regions of the CCD, then readout noise on Sdark in those two regions is uncorrelated,giving the additional factor of 2 in the variance.

Chapter 4

Degenerate Fermi gas of molecules in 3D

Creation of a quantum-degenerate molecular gas has been a long-standing goal for the ultracold molecule community. Unfortunately, the complexity of ultracold molecules made this challenging. The highest PSD molecular gases have been formed by associating the molecules fromnear-degenerate atomic gases using Feshbach resonances. However, low conversion efficiency ofatoms into molecules and high losses on the Feshbach resonance prevented attempts to push thistechnique further to create degenerate molecular gases. Without methods to control the inelastic
losses of the molecules while maintaining a high elastic collision rate, direct evaporative cooling of
the molecules to higher PSD was also unfeasible.
In our first experiments in the new JILA KRb machine, we started by trying to optimize the
Feshbach association stage, encouraged by the much improved atom conditions compared to theprevious apparatus. By starting from a deeply-degenerate Bose-Fermi mixture of 40 K and 87 Rb,
with many more fermions than bosons (NK /NRb ≈ 8), we were able to mitigate the losses on the
Feshbach resonance and produce a gas with N = 30 × 103 molecules at T = 50 nK and T /TF = 0.3[50], where TF is the Fermi temperature of the molecular gas. We also observed an intriguing
suppression of the reactive loss rate as the molecular gas became degenerate, which is still not
fully understood [168, 169]. Finally, we measured the density fluctuations in the gas, which aresuppressed by fermionic statistics as the gas becomes degenerate [51]. The density fluctuations
gave an alternative measurement of T /TF that agreed with our measurements of the momentum
distribution.

76
Though our observables were consistent with the molecular gas being initially created inthermal equilibrium, the dynamics while crossing the Feshbach resonance are a complicated competition between inelastic three-body losses and thermalizing elastic collisions. We also studied
the elastic collisions between atomic K and the Feshbach molecules (KRb*) and estimated thatsix elastic collisions occurred per molecule during the Feshbach ramp, facilitating thermalization ofKRb* [51].
This chapter is based on Refs. [50, 51]. Since the writing of Ref. [50], there has been agreat deal of exciting progress on degenerate molecular gases. Our group demonstrated direct
evaporation of molecules in 2D to T /TF < 1 [52] (Chapter 5). In 2021, the Munich group createda DFG of NaK molecules in the Feshbach association stage [53], and subsequently through direct
evaporation using microwave shielding [54]. A BEC of homonuclear Cs2 Feshbach molecules in 2D
was also demonstrated by the Chin group in Chicago [170].

4.1

Experimental procedure
The starting point of the experiments is a Bose-Fermi mixture in an optical dipole trap

(ODT) formed by two crossed beams at λ = 1064 nm. In Ref. [50], the beams crossed at 45◦ andthe trap frequencies were (ωx , ωy , ωz ) = 2π × (45, 250, 80) Hz for K. In Ref. [51], the beams crossed
at 90◦ , and the trap frequencies were (ωx , ωy , ωz ) = 2π × (60, 240, 60) Hz for K. Trap frequencies
for Rb, KRb*, and KRb are obtained by scaling the above trap frequencies by 0.72, 0.83, and 0.79,
respectively. The experiment coordinate system is defined with ŷ pointing counter to gravity. The
magnetic field B is oriented along ŷ.
In addition to the ODT, we also used a shallow (30ER for KRb) lattice along ŷ duringmolecule production. Before adding the lattice, we noticed a significant slosh of the molecules inŷ after production, which caused the cloud to heat up. We initially attributed this to differential
gravitational sag, which would cause a separation between the Rb and KRb equilibrium positionsin the ODT. However, we later realized that the slosh was caused by the momentum kick from the
1
STIRAP transfer, which imparts a momentum of kSTIRAP = 2π λ1up − λdown, where λup = 970

77

Figure 4.1: Cartoon of molecule production. Starting from an overlapping DFG of K (bluespheres) and BEC of Rb (red spheres) in the ODT, Feshbach molecules are produced by sweeping
the magnetic field B across the interspecies Feshbach resonance. Feshbach molecules are thentransferred to the ground rovibrational state using STIRAP. Reproduced from Ref. [50].

nm and λdown = 690 nm are the wavelengths of the two STIRAP beams (see section 4.3.1 for moredetails). The shallow lattice completely damped the oscillations, and the lattice was ramped off
immediately after molecule production.
The molecule production process is illustrated schematically in Figure 4.1. We started withthe atomic mixture at B = 555 G, above the broad interspecies K–Rb Feshbach resonance at
B0 = 546.62 G (width of 3.04 G) [171]. KRb* was created by adiabatically sweeping B across the
Feshbach resonance from 555 G to 546.1 G. The Feshbach sweep was typically performed in 5 ms(see also section 4.3.2). After the Feshbach sweep, molecules were transferred to the ground stateusing STIRAP [66] with a typical efficiency of 85%. All of the experiments in the chapter wereconducted at zero electric field (|E| = 0). Remaining unpaired atoms were removed immediately
after STIRAP using resonant light, except for in the K–KRb* collision measurements (section
4.3.1).
By changing the final evaporation cut in the ODT before molecule production, we varied theinitial atom numbers NK and NRb and temperature T , as well as the resulting molecular condition.
The results are shown in Figure 4.2. The three columns of Figure 4.2 correspond to the threetemperature conditions listed in Table 4.1. At the highest atom number and temperature, Rb was
uncondensed (left panel of Figure 4.2). Only about 15% of Rb was converted into molecules, atypical Feshbach conversion efficiency for non-degenerate atomic clouds [172, 151], but owing to
the high total atom number we created up to N = 105 molecules at T /TF = 1. At the coldest

78

Figure 4.2: Molecule production from a degenerate Bose-Fermi mixture. The plots show verticalslices through the measured atomic (top row) and corresponding molecular (bottom row) distributions in TOF. Insets show the 2D absorption images corresponding to the plotted slices. At
the highest atomic evaporation cut (left column), we produce as many as N = 105 molecules at
T /TF = 1.0(1). At the lowest evaporation cut (right column), we produce around N = 3 × 104
molecules at T /TF = 0.33(3). The atom and molecule numbers and temperatures for these measurements are reported in Table 4.1. Reproduced from Ref. [50].

T (nK)
230
110
50

NRb
6 × 105
2 × 105
7 × 104

T /Tc
>1
0.9
0.5

NK
1.2 × 106
1 × 106
5 × 105

T /TF,K
0.3
0.2
0.1

NKRb
1.0(1) × 105
5.0(5) × 104
3.0(5) × 104

T /TF,KRb
1.0(1)
0.54(3)0.33(3)

Table 4.1: Atom and molecule conditions for Figure 4.2. Reproduced from Ref. [50].

79
cloud, Rb was about 85% condensed and we converted up to 50% of the Rb, producing a molecular
gas at T /TF = 0.3 (right panel of Figure 4.2). The quoted T /TF is obtained from fitting the cloud
to a Fermi-Dirac distribution, described in more detail in the next section. At each evaporation
cut, the trap was recompressed to the trap frequencies reported above before producing molecules.
This trap depth was found experimentally to improve the density overlap of K and Rb, without
increasing the density so much that three-body losses were excessive.

4.2

Properties of the molecular Fermi gas
In this section, we describe our measurements of the molecular Fermi gas. We first measured

the molecular distribution in time-of-flight (TOF) expansion to determine the temperature of themolecules. For a degenerate gas, the shape of the distribution is determined by quantum statistics,
allowing us to extract the T /TF of the molecular cloud. We also measured the rate of chemical
reactions in the gas [32], and observed a reduction in the chemical reaction rate as the gas became
degenerate. Finally, we measured the suppression of density fluctuations in the gas owing to Fermi
statistics.

4.2.1

TOF thermometry

Let us start by writing down some of the basic equations for harmonically-trapped ideal
Fermi gases [173, 174]. A classical gas (Boltzmann statistics) has a Gaussian density distribution,


N
x2
y2
z2
ncl (x, y, z) =exp − 2 − 2 − 2 ,2σx 2σy2σz(2π)3/2 σx σy σz

(4.1)

q
where σi = kB T /mωi2 is the thermal size of the cloud along i ∈ {x, y, z}. If the particles obey
Fermi statistics, they instead follow a Fermi-Dirac distribution,nFD (x, y, z) =




N
x2
y2
z2
Li
−ζ
exp
−
−
−
/Li3 (−ζ),
3/2
2σx2 2σy2 2σz2(2π)3/2 σx σy σz

(4.2)

80
where ζ = eβµ is the fugacity, with β = 1/kB T the inverse temperature and µ the chemical potential.
The function Lin (−z) is a polylogarithm. The fugacity is related to the degeneracy of the gas byLi3 (−ζ) = −1.
6(T /TF )3(4.3)

In a 3D harmonic trap, TF can also be calculated fromTF =~ω(6N )1/3 ,
kB

(4.4)

where ω = (ωx ωy ωz )1/3 is the geometric mean trapping frequency.
To measure the temperature of the cloud, we quickly switch off the ODT to let the gas
expand in TOF. After a time t of expansion from a harmonic trap, the density distribution has theqsame shape but with the sizes scaled according to σi → σi 1 + ωi2 t2 [175]. Absorption imaging
integrates the signal along the probe axis (here assumed to be ẑ), so we fit to the column-integrated
distributions


x2
y2
ncl,int = A exp − 2 − 2
2σx 2σy



y2
x2
nFD,int = A × Li2 −ζ exp − 2 − 2/Li2 (−ζ),
2σx 2σy

(4.5)
(4.6)

where A is the peak density after time-of-flight. For both the classical and Fermi gas, we extract
T from the fitted sizes σi according toT =mωi2 σi2.
kB (1 + ωi2 t2 )(4.7)

Note that at long times t  1/ωi , the ωi dependence drops out and T = mσi2 /kB t2 — in otherpwords, σi expands linearly with velocity kB T /m. For the Fermi gas, we can also independently
extract T /TF from the fitted fugacity ζ according to equation (4.3).
Figure 4.3A shows a density profile of the molecular DFG after TOF expansion, obtainedby an azimuthal average of the full 2D column-integrated absorption image. As shown by the fit
residuals (bottom panel), the Fermi-Dirac fit (blue) captures the shape of the cloud better thanthe classical Gaussian fit (red). From the fitted fugacity, we extract T /TF = 0.31(2) for this cloud.

81

Figure 4.3: TOF thermometry of the molecular DFG. (A) Fit of the molecular density after 10 ms
of TOF. The data points (black) are obtained by azimuthally averaging the cloud. The Fermi-Diracfit (blue) captures the shape of the cloud better than the Gaussian fit (red), as illustrated by the
fit residuals (lower panel). A Gaussian fit to the outer wings of the cloud (green) overestimates
the density at the center of the cloud. (B) The deviation δU/Ucl of the internal energy of the gas
from the classical prediction as a function of T /TF . The trends of KRb (purple circles) and K (bluesquares) both agree well with the calculation for an ideal Fermi gas (black line). Reproduced from
Ref. [50].

82
A Gaussian fit to the outer wings of the cloud (green) captures the temperature of the cloud but
drastically overestimates the density at the center of the distribution.
As the gas becomes deeply degenerate, the average energy per particle U saturates at(3/4)kB TF instead of following the classical equipartition value of Ucl = 3kB T . Therefore, thedeviation δU = U − Ucl from the classical prediction gives a measure of the degeneracy of the gas.
To estimate U , we use U = 3kB Tgauss , where Tgauss is the temperature obtained from equation(4.7) by fitting the entire cloud to a Gaussian distribution. In Figure 4.3B, we plot the measured
δU/Ucl = 1 − T /Tgauss as a function of T /TF . The internal energy for KRb (purple circles) shows asignificant deviation below T /TF ≈ 0.6, in agreement with the calculated result for an ideal Fermi
gas (black line). For comparison, we also plot the same quantity measured on our 40 K atomic Fermi
gas (blue squares), which shows the same trend.

4.2.2

Reduced loss rate

As introduced in section 2.5.2, and experimentally studied in Refs. [32, 79, 34], KRb molecules
experience losses at ultralow temperatures owing to chemical reactions. Since the molecules are
identical fermions, they collide predominantly in p-wave collisions (L = 1) at ultracold temperatures. The chemical reaction rate is limited by tunneling through the p-wave barrier to reach
short range, which for non-degenerate gases results in the two-body loss rate coefficient β scaling
linearly with T . In the absence of heating, the loss is well-modeled by a simple differential equation,
dn/dt = −βn2 , which has solutionn(t) =n0,
1 + βn0 t(4.8)

where n is the average molecular density,
1
n=
N

Z

dx dy dz ncl (x, y, z)2 =N
8π 3/2 σx σy σz

,

(4.9)

n0 is the initial density, and β is the two-body loss rate coefficient that describes the reactive
losses. According to equation (4.8), one can estimate β by finding the time τ = 1/βn0 where

83

Figure 4.4: Molecular loss measurements at T = 65 nK (blue squares) and T = 300 nK (redcircles). Solid lines are fits to the solution of equation (4.10). Reproduced from Ref. [50].

the density has dropped by half. One can also estimate β by the initial slope of the loss, using
dn/dt ≈ −(βn0 )n, which gives a rate Γ = 1/τ = βn0 .
For a gas in 3D, anti-evaporative heating leads to a more complicated density decay thandescribed by equation (4.8). Owing to the density dependence of the loss, chemical reactions mainly
occur in the densest, coldest part of the gas. As a result, the losses cause an effective heating of thegas, referred to as “anti-evaporation” [79], which further lowers the density n. We approximated thetemperature evolution by a linear heating rate h, so T (t) = T0 + ht. Experimentally, we measured
heating rates of h = 10 – 30 nK/s, which are somewhat larger than the value h = βn0 T0 /12
predicted by a simple anti-evaporation model [79].
By measuring the average molecular density n after a variable hold time, we obtain loss
measurements like the two shown in Figure 4.4. To fit the decay in the presence of heating, we
model the loss of the molecules with the following differential equation,dn
3n dT= −βn2 −
,
dt
2T dt

(4.10)

Since β ∝ T , we set β = bT in the differential equation to account for changes in the loss rate due
to heating. The coefficient b can be calculated using multichannel quantum defect theory (MQDT),
which gives b = 0.8(1) × 10−5 cm3 s−1 K−1 for KRb [32].

84

Figure 4.5: Reduction of the chemical reaction rate at low T /TF . Reproduced from Ref. [50].

The preceding discussion is geared toward thermal gases, but we might expect the behavior
to change in quantum degeneracy. At low T /TF , the collision energy is no longer proportional toT , but rather is determined by the Fermi temperature TF . Therefore, as T → 0, we might expect
the loss rate to saturate at β ∼ b × (Eavg /kB ), where Eavg /kB = 3TF /4 is the average energy in
the Fermi gas at T = 0 [168].
Experimentally, we saw the opposite trend from what is suggested by this simple physicalargument. As shown in Figure 4.5, the fitted loss rate β/T decreased as the initial T /TF of thecloud was lowered. At T /TF > 0.6 (red points), we measured a constant β/T as a function of theinitial T /TF , in agreement with the classical prediction. The average of those points (black line)
gave β/T = 0.84(6) × 10−5 cm3 s−1 K−1 , in good agreement with the MQDT prediction (red line).
As the gas became degenerate, at T /TF < 0.6 (blue points), we observed a drop in the fitted β/Tby nearly a factor of 4. Based on our physical intuition, we thought that this might be due tothe reduction in density fluctuations as the gas becomes degenerate. The blue line in Figure 4.5
shows the density fluctuations hδn2 (r)i/hn(r)i scaled by the MQDT value for b. However, later
theoretical work found that the density fluctuations play a negligible role in the parameter regimeof our experiment [168].
The data points in Figure 4.5 were obtained by fitting loss curves like the ones in Figure 4.4

85
to the solution of equation (4.10)1 ,
3/2

n(t) =

n0 hT0.
√√√3/2(T0 + ht)(2n0 T02 ( T0 − T0 + ht)b + h( T0 + ht + 2bn0 tT0 ))(4.11)

The average density n was approximated by fitting the entire cloud in TOF to a Gaussian andcalculating n according to equation (4.9) using the expansion temperature Tgauss determined fromthe fit. For our experimental range of T /TF , we checked numerically that this procedure reproduces
n within a few percent of the exact value obtained by integrating the FD distribution (see also Ref.
[176]). The temperature T0 was set to the measured initial temperature of the gas using the FD
fit. Note that with this definition, the drop in β/T is even more unexpected, since if β were todepend only on TF and not T , as in the simple intuitive picture discussed previously, then β/Tshould diverge as T → 0. We also tried setting T0 to the expansion energy Tgauss of the cloud as aproxy for the energy density of the gas, but saw a similar trend (see Figure 3 of Ref. [168]).
The observation of suppressed losses in the degenerate regime has so far stimulated two
related theory works [168, 169]. Ref. [168] found that interactions between molecules and collisional
complexes could cause a similar trend to what we observed, although the required molecule–complex
cross section is extremely large (even exceeding the s-wave unitary limit). Ref. [169] found new
universal relations for gases with two-body losses by computing the contacts of the system, andfound a deviation from β ∝ T behavior at low temperature. More investigation is needed todetermine if this effect can explain our measurement. With the creation of a DFG of NaK molecules[53, 54], it will be interesting to see if a similar loss suppression at low T /TF is observed in a different
molecular system.

4.2.3

Density fluctuations

We also measured the density fluctuations of the molecular DFG in Ref. [51]. Owing toFermi statistics, the density fluctuations become suppressed as the gas becomes degenerate. This
1

Note that there is a small typo in the Supplementary Material of Ref. [50]. The last term in the denominatorof the equation corresponding to (4.11) was missing a factor of t. The equation was correct in the code that we used
to analyze the loss curves of the molecules in Ref. [50].

86effect was previously studied experimentally in atomic Fermi gases [177, 167]. At high temperature
2 /N = 1. Here, σ 2 and(T /TF > 1), the density fluctuations have a Poissonian distribution, with σNN

N are the variance and mean, respectively, of the molecule number within a certain probe volume.
However, for a deeply-degenerate Fermi gas (T /TF  1), the density fluctuations are suppressed
2 /N = (3/2)(T /T ).
by the Pauli exclusion principle and σNF
2 /N is given by [167]More quantitatively, the ratio σN2
Li1/2 (−ζe−V (x,y,z)/kB T )σN,=
Li3/2 (−ζe−V (x,y,z)/kB T )N(4.12)

2 /N is a local quantity that
where V (x, y, z) is the 3D harmonic trapping potential. Note that σN

depends on the position in the gas. Near the center of the gas, the local fugacity ζe−V (x,y,z)/kB T
is maximized, and the fluctuations are maximally suppressed. Near the edge of the cloud, the2 /N = 1. After integrating through the imaging axis (in thisfluctuations approach shot noise, σN

case ŷ), we obtain the column-integrated expression
2
σN
Li1 (−ζe−V (x,z)/kB T )=.
Li2 (−ζe−V (x,z)/kB T )N(4.13)

Since the trapping potential is harmonic, expansion for time t in TOF simply rescales the coordiq
nates by xi → xi / 1 + ωi2 t2 .
We measured density fluctuations in K, KRb*, and KRb gases. Measuring the fluctuationsof non-degenerate and degenerate atomic K gases allowed us to benchmark our experimental procedure. Comparing degenerate KRb* and KRb gases revealed the effect of STIRAP, which producesa small number of holes in the KRb distribution owing to its non-unity efficiency.
The procedure for the fluctuation measurement on the molecules was very similar to thatused in Ref. [177] on a 6 Li DFG. The measurement consisted of taking 50–60 absorption imagesof atomic or molecular clouds in TOF, prepared with the same sequence. Before imaging, themolecules were dissociated into K atoms by adiabatically ramping across the Feshbach resonancein reverse, in order to accurately determine the molecule number. Each image was subdivided into
bins that formed detection sub-volumes. For a given bin i, the j-th image gave a measurement

87

Figure 4.6: Comparison of T /TF from fluctuations and TOF expansion. Measurements were
performed on DFGs of K (blue square), KRb* (orange circle), and KRb (red triangle). The dashedline indicates equal T /TF from the two methods. Reproduced from Ref. [51].

of the local particle number Nij in that bin on the j-th realization of the gas. By looking at the
2 and N for bin i. Fitting equation (4.13) over
distribution of Nij over images j, we could obtain σN

the bins gave a measurement of ζ, which we converted to T /TF with equation (4.3).
2 /N is ideally unity for a non-degenerate gas, but in practice, it depends on a
The ratio σN

number of technical factors. The absolute atom or molecule number calibration from absorption
2 /N , so calibrating the effective absorption cross section
imaging directly affects the measured σNand saturation intensity are crucial [165, 166]. The finite resolution of the imaging system can cause
smearing of the signal over multiple bins, which reduces the observed fluctuations [177, 167]. Photon
shot noise also adds additional fluctuations to the raw images that must be removed to determine
the density fluctuations. These experimental details are discussed in detail in Will Tobias’s thesis[107] (also see Christian Sanner’s thesis [163]). See section 3.5 for details on the CCD calibration,
which was needed to determine the photon shot noise contribution.
By measuring the density fluctuations for degenerate gases of K, KRb*, and KRb, we could
compare the T /TF measured from fluctuations and the value measured from FD fits to the cloudin TOF expansion (Figure 4.6). Overall, we obtained good agreement for the two methods for all
species. We did observe a higher level of fluctuations for KRb than KRb*, consistent with increased
number fluctuations from a non-unity STIRAP efficiency. At T /TF = 0.4, the number of thermal

88holes in an ideal Fermi gas is still significant, and we calculate that the lowest-energy state in
the trap has an occupation fraction of 0.77. Therefore, fluctuations caused by non-unity STIRAP
transfer (85% typical efficiency) do not greatly exceed the thermal fluctuations, and the STIRAP
transfer brings the cloud slightly out of equilibrium. However, for lower T /TF , thermal fluctuations
are further suppressed and the relative contribution of STIRAP-induced holes could become verysignificant. See Ref. [107] or the Supplementary Material of Ref. [51] for more details.

4.3

Feshbach ramp thermalization
In the previous sections, we described our measurements on the KRb DFG. The measure-ments of the TOF distribution and fluctuations were consistent with a gas in thermal equilibrium.
However, the dynamics of crossing the Feshbach resonance can be quite complicated, owing to thecompetition of strong elastic collisions and rapid three-body losses near resonance.
The K-Rb-KRb* mixture has been previously studied experimentally [178, 179]. Near the
Feshbach resonance, the constituents of the loosely-bound Feshbach molecules are indistinguishable
from the free atoms, leading to a fermionic suppression of inelastic collisions of KRb* with K anda bosonic enhancement of those with Rb [178]. To minimize inelastic Rb-KRb* losses, we used an
initial Bose-Fermi mixture with NK  NRb . The highest molecule number was produced when
the remaining unpaired Rb was no longer condensed after molecule production. After forming
molecules, the peak density of K was approximately 10 times larger than Rb, so thermalization ofKRb* is expected to occur predominantly through collisions with K.
In this section, we describe measurements of the K-KRb* elastic collision cross section as afunction of B near the Feshbach resonance. We also measured the dependence of the molecular
T /TF on the Feshbach ramp rate. We found that the molecular degeneracy saturated as a functionof the Feshbach ramp rate, indicating that elastic collisions predominate over inelastic ones.

89
4.3.1

Collisional damping

To measure the elastic cross section, we studied the damping of KRb* center-of-mass oscillations in the ODT due to collisions with K. The damping rate is proportional to the elastic collisionR
rate Γ = nσvrel , where n = (1/NK + 1/NKRb* ) nK nKRb* d3 x is the overlap density of the twopspecies, σ is the K–KRb* elastic collision cross section, and vrel = (8kB /π)(TK /mK + TKRb* /mKRb* )is the thermally-averaged relative velocity of the two species [180, 181, 182, 183].
We prepared a mixture of K and KRb* in the ODT with the following sequence. First, wecreated KRb in the ODT with the Feshbach ramp and STIRAP. The weak lattice is not used forthis measurement. Next, pulses of resonant light were used to remove all of the Rb and most of
the K. A second STIRAP pulse was used to transfer KRb back to KRb*, producing a mixture of
NKRb* = 2 × 104 at TKRb* = 300 nK and NK = 1.5 × 105 at TK = 600 nK.
An initial center-of-mass oscillation in ŷ could be selectively excited on KRb* by changing
the timing between the STIRAP pulses. As mentioned earlier, each STIRAP sequence imparts amomentum of kSTIRAP = 2π (1/λup − 1/λdown ), where λup = 970 nm and λdown = 690 nm are thewavelengths of the two STIRAP beams. The STIRAP beams propagate along ŷ. By changing
the timing of the two STIRAP pulses relative to the KRb vertical trap period 2π/ωy = 5.2 ms,
the momentum recoils from the pulses could be made to add up or cancel, giving control over theamplitude of the KRb* oscillation (Figure 4.7). For the oscillation damping measurement, we chosea delay between the STIRAP pulses such that the oscillation amplitude of KRb* was maximized.
After producing the K-KRb* mixture, we ramped B to a target value and measured theKRb* oscillations in ŷ. RF spectroscopy on K was used to calibrate B. No oscillation of K was
initially excited by the STIRAP pulses, and we did not observe any induced oscillation of K fromcollisions with KRb*. Example oscillations at B = 546.1 G are shown in Figure 4.8. Startingwith no K (top panel, n = 0), the oscillations become increasingly damped with increasing overlap
density n.
From the exponential damping time τ , we could extract the elastic collision cross section σ.

K R b * o s c illa tio n a m p litu d e ( a r b .)

90
3

2

1

0
6

8
1 0
1 2
T im e b e tw e e n S T IR A P p u ls e s ( m s )Figure 4.7: STIRAP-induced slosh of KRb*. The solid line is a fit to a sinusoid with the frequency
fixed at 190 Hz (the measured ωy /2π for KRb). Reproduced from Ref. [51].

The damping of the oscillations is described by the following coupled differential equations [183],
4 mKRb NKRb∗
Γ(ẏK − ẏKRb∗ )3 M
N
4 mK NK2ÿKRb∗ = −ωKRbΓ(ẏK − ẏKRb∗ ),∗ yKRb∗ +3 M N
2ÿK = −ωKyK −(4.14)(4.15)

with M = mK +mKRb and N = NK +NKRb∗ . For the experimental number ratio (NK /NKRb* ≈ 7.5)
and collision rate, and for initially stationary K, the motion of the two species is nearly uncoupled.
Therefore, we approximated the dynamics by2ÿKRb∗ = −ωKRb∗ yKRb∗ −4 mK NKΓẏKRb∗ ,3 M N(4.16)

which gives a damping time τ of1
2 mK NK
=
Γ.
τ
3 M N

(4.17)

We corrected the experimentally measured damping time τexp to remove the small backgrounddamping τ0 observed at n = 0, according to 1/τ = 1/τexp + 1/τ0 . From τ , we could extract σ andp2 a2 ), where k =the s-wave K–KRb* scattering length aad , using σ = 4πa2ad /(1 + kth
2µkB T /~2thadis the thermal collision wavevector and µ is the K–KRb* reduced mass [22]. A universal prediction
gives aad = 1.09a for the mass ratio mK /mRb = 0.46 of our system, where a is the K–Rb scattering
length [184].

91

Figure 4.8: Collisional damping from K-KRb* elastic collisions at B = 546.1 G, varying theoverlap density n. The K-KRb* scattering lengths |aad | = 684(110)a0 and 648(88)a0 extracted
from the middle and bottom panels, respectively, are in good agreement. Reproduced from Ref.
[51].

Figure 4.9: Summary of K-KRb* elastic collisions extracted from collisional damping measurements. Reproduced from Ref. [51].

92
0.6

T/TF

0.5

0.4

0.3

0.1

1.0

10.0

Feshbach ramp rate (G/ms)

Figure 4.10: KRb T /TF versus Feshbach ramp rate. Reproduced from Ref. [51].

By measuring the damping rate as a function of B, we obtained a measurement of |aad | vs
B (Figure 4.9). The dashed line is a fit |aad | = ca to a single parameter c, which describes a
scaling between aad and the K–Rb scattering length a. The best fit gave c = 0.74(5). We do notnecessarily expect that the universal prediction aad = 1.09a holds for the regime of our experiment,
owing to the high collision energy and relatively low scattering lengths studied here. Nonetheless,we measured a large K-KRb* elastic cross section near resonance. From the measured K-KRb*elastic cross section, we estimated that about 6 elastic collisions occur during the Feshbach ramp,
facilitating thermalization. More details on this calculation can be found in the Supplementary
Material of Ref. [51].

4.3.2

Dependence on the Feshbach ramp rate

Varying the Feshbach ramp rate provides an additional method for studying the competition
between elastic and inelastic processes [14]. For a very low ramp rate, the inelastic collisions near
the Feshbach resonance will result in significant molecule loss; conversely, for a very high ramp rate,
the gas may not have enough time to thermalize. We measured this experimentally by monitoring
the T /TF of the KRb gas a function of the Feshbach ramp rate (Figure 4.10). For intermediate
ramp rates of 0.5–3 G/ms, the T /TF reaches a minimum at 0.3. For very slow ramp rates (< 0.593G/ms), inelastic losses near the Feshbach resonance reduce the molecule number resulting in a
sharp increase in T /TF . We also observe a gradual rise in the T /TF as the ramp time becomesshorter than the trap oscillation period, while the molecule number remains constant, suggestingthat thermalization is hindered for fast ramp speeds.

4.4

Conclusion
In this chapter, we presented experiments on a DFG of molecules. We characterized thedegeneracy of the gas both through the momentum distribution in TOF expansion and by measuring
the suppression of density fluctuations owing to Fermi statistics. The chemical reaction rate in thegas also became suppressed for T /TF < 0.6, a phenomenon that is still not fully understood. Finally,
we discussed measurements of the K-KRb* elastic collision cross section and the dependence of the
KRb T /TF on the Feshbach ramp rate, which allowed us to characterize the thermalization processes
during molecule production.
The capability to make a molecular DFG directly at the molecule association stage opens newpaths in molecular quantum science. However, an important ingredient that was still missing at
the time of Refs. [50, 51] was the ability to turn on elastic dipolar collisions between ground-state
KRb while suppressing reactive losses. In the next chapters, we describe two general techniques
for doing so.

Chapter 5

Dipolar evaporation of molecules in 2D

In the previous chapter, we produced a degenerate Fermi gas of molecules from two deeply
degenerate atomic clouds. However, these initial experiments were ultimately limited by our inability to observe elastic interactions between ground state molecules in the 3D gas. In the absenceof an external electric field E, the primary interactions between spin-polarized KRb are p-wave
collisions, which are expected to be weak [82]. The rapid dipolar loss in 3D [79] prevented us from
ramping up a strong electric field and studying elastic dipolar collisions in the Fermi gas.
As a result, we decided to go to quasi-2D, where previous KRb experiments had demonstrated
a reduction of loss at moderate |E| [102]. As discussed in section 2.5.2, reactive losses are suppressedin 2D thanks to the enhanced barrier from dipolar repulsion in the mL = ±1 collisions [99, 100, 101].
We thought we could go beyond Ref. [102] in several areas: we could access higher |E| to make amore complete map of the loss as a function of d, and with more control over spurious field gradients;
we could try to measure the elastic dipolar collisions directly, which had not been previously
observed in ultracold molecules; and we could improve the initial PSD of the molecular gas andachieve a high enough density to see evaporative cooling.
More generally, dipolar interactions lead to intriguing phenomena in 2D. The interplay of the
1/r3 power law of the interaction with the dimensionality leads to the possibility of true long rangeorder in 2D, in contrast to the usual BKT transition for short-range interactions [185]. In addition,
the combination of strong external confinement and the dipolar anisotropy enables control of thesystem via the orientation of the dipoles (set by E). For example, if E is oriented perpendicular

95
to the 2D plane, then all pairs of molecules have the same sign of interaction. But if E is rotatedinto the 2D plane, then molecules experience both attractive and repulsive interactions. This canhave a strong effect on the collisions of the molecules (Chapter 6), as well as on the collective spin
dynamics when the molecules are driven by a microwave field (Chapter 8).
A high PSD in 2D is necessary for many of these explorations, so trying to demonstrateefficient evaporation of molecules in 2D is a natural first step. Luckily, 2D is not only favorablefrom a collisional perspective, but also from the basic kinematics of the gas. As described in
section 4.2.2, p-wave losses in 3D lead to anti-evaporation, where the coldest and densest portionof the cloud experiences the fastest loss [79]. This effect arises from the competition of the pwave threshold law (β ∝ collision energy) with the 3D density distribution in a harmonic trap andresults in detrimental heating of the gas. In 2D, the density of states is lower and as a result, theory
predicts that there should be no anti-evaporation in 2D for p-wave interactions [186].1
This chapter is based on the work in Ref. [52]. First, we describe some technical aspects of
preparing a 2D sample of molecules. We next present measurements of the suppressed loss rateand enhanced elastic collision rate that arises from the repulsive dipolar collisions in 2D. The long
lifetime of the molecules and strong elastic collisions facilitate direct evaporation of the molecules
to as low as T /TF = 0.6, realizing a long-lived and strongly dipolar 2D molecular quantum gas.

5.1

Producing a 2D sample

The experiments in this chapter are performed in the quasi-2D regime. A quasi-2D Fermi√
gas satisfies EF , kB T  ~ωy , where ωy is the vertical trap frequency and EF = ~ωr 2N , withωr the geometric mean trap frequency in the transverse direction. This condition means that the
motional energy scales are lower than the harmonic oscillator spacing in ŷ, so the particles populate
only the lowest oscillator level in the ŷ direction while having a thermal distribution in the radial
direction. If the confining potential is a deep optical lattice, then this is equivalent to requiring
1

This also implies that in a 1D system, or a homogeneous (box) trap, losses can reduce the average kinetic energyof the gas.

96

Figure 5.1: Cartoon of intra- (A) vs inter-band (B) collisions. (A) Two molecules in the lowest
band collide with mL = ±1 due to the strong vertical confinement. (B) A molecule in the lowest
band and another in the first excited band can collide with mL = 0. The mL = 0 collisions have a
reduced barrier and thus dominate the loss between molecules in different bands.

that the particles occupy only the lowest band.2
The 2D dipolar loss suppression relies directly on the quasi-2D nature of the gas. Twomolecules in the ground band collide with mL = ±1, and enjoy an enhanced barrier to reacting
at short range owing to the dipolar repulsion. On the other hand, collisions between molecules in
the ground band and first excited band can have mL = 0, and thus have an increased loss rate
due to dipolar attraction. This can be seen intuitively by considering the wavefunctions along they-direction in the harmonic oscillator approximation, as shown in Figure 5.1. More rigorously, thiscomes from the symmetry requirement that the total wavefunction of a colliding pair of identicalfermions must be anti-symmetrized. For collisions with mL = 0, the relative wavefunction issymmetric in the in-plane coordinates, so the molecules must be antisymmetric in their oscillator
states in ŷ, which is not possible if the molecules are both in the ground band [100, 102]. Thiscan also be seen from the adiabatic energy curves in 2D (see Figure 2.11 or the aforementioned
references).
In our experiment, the strong confinement into 2D is created by a 1D optical lattice (Figure5.2) formed by two λ = 1064 nm beams (green beams in Figure 5.2) propagating with an angleof 11◦ from the ±ŷ axis. At the crossing point, this creates a lattice with spacing alatt = 540 nm
2
In the context of dipolar interactions, quasi-2D can have another meaning: aD < aho , so the collisions “look 3D”(see section 2.5.2.1). In this chapter,we achieved a maximum dipole of d = 0.3 D, corresponding to aD = 1610a0 .
p
The harmonic length was aho = ~/µωy = 1830a0 , so our experiments were also quasi-2D with respect to the dipolar
collisions.

97

Figure 5.2: Optical trap geometry for 2D work. The atoms are initially trapped in the ODT(orange), then loaded into a single layer of the LSL (red) to squeeze the distribution in the ydirection, and finally loaded into the science lattice (green), which confines the atom into 2D
planes with a lattice spacing of 540 nm. Adapted from Ref. [52].

98

Figure 5.3: Band mapping K and KRb. (A) Image of a K DFG after band mapping from the VL.
The width v0 t of the Brillouin zone after time of flight t is shown for scale, where v0 = ~(π/alatt )/mis the velocity at the edge of the Brillouin zone. (B) Band-mapped image of a K DFG after
parametric heating. Population is clearly visible in the 2nd excited band. The vertical scales in(A) and (B) are the same. (C) A 2D gas of KRb after band mapping from the VL. The image isthe average of 9 measurements. Note that the vertical scale of (C) is not the same as (A) and (B).

that we call the vertical lattice (VL). The beams are angled from the vertical to avoid spurious
superlattices formed by reflections off the ITO plates, and are round with waists of 125 µm. We
typically operate around ωy /2π = 17 kHz, so ~ωy /kB = 800 nK, which is larger than the typicaltemperature of T = 250 nK. The lattice frequency ωy is directly measured using parametric heatingof the molecules [187]. To characterize the band populations in the VL, we use the standard
technique of band mapping [188, 187]. This is done by ramping down the VL power over ∼ 500 µs,
which shuts off the lattice adiabatically with respect to the band spacing. As a result, the band
populations of the atoms or molecules are directly visible after time-of-flight, as shown in Figure
5.3.
To produce molecules in 2D, we first load the atomic mixture from the ODT (orange beams
in Figure 5.2) into the large spacing lattice (LSL, red beams). The LSL beams form a lattice alongŷ with 8 µm spacing, which compresses the cloud to improve mode-matching with the VL. Finally,
the atoms are loaded into the VL, and molecules are produced with the Feshbach field sweep and
STIRAP. In principle, we could have produced molecules in 3D before loading into the lattice, but
the molecular losses and lack of thermalizing collisions at |E| = 0 make the timescales challenging.

99

Figure 5.4: LSL optics setup. All of the lenses are achromatic doublets, except the 30 cm cylindricallens. This diagram is simplified to show the most important optics; the full beam path is shown in
Figure 3.5.

Additionally, the initial opening of the band gap when loading the lattice is less favorable for themolecules than the atoms. The initial size of the band gap is given by the photon recoil energy
K
2 /2m, where k
ER = ~2 klattlatt = 2π/λ. At λ = 1064 nm, the recoil energies are ER = 210 nK
KRb = 67 nK for K and KRb, respectively. Only particles with energy less than E can be
and ERR

loaded into the ground band, which imposes a stringent requirement that T, TF < ER /kB . Since
KRb < E K , this makes it significantly harder to load the molecules directly into the VL [187].
ER
R

In the following sections, we describe the effect of the LSL on the VL loading, the measurementof the VL layer populations, and the detailed 2D loading procedure.

5.1.1

Large spacing lattice

The LSL is formed by two beams that cross at a shallow half-angle of α = 4◦ , which produces
a lattice with λ/(2 sin α) = 8 µm spacing. The optical setup is shown in Figure 5.4 and is basedon the accordion lattice design used in Ref. [189]. The two LSL beams are split from a single
input beam using a pair of polarizing beam splitter (PBS) cubes.3 After the cubes, all optics arecommon between the two beams. Next, the beams go through a 1-to-1 telescope. The second lensof the telescope is mounted on a micrometer stage, which we adjust to set the crossing point of
the beams on the atoms in the z-direction. To avoid compressing the cloud unnecessarily along thehorizontal direction, the beams are elliptical at the position of the atoms, with a 50 µm waist in
the vertical (y-direction) and 330 µm waist in the horizontal (x-direction). The beam shaping is

100

Figure 5.5: Parametric heating in the LSL. The vertical size after TOF of a single LSL layer of
Rb is plotted as a function of the modulation frequency of the LSL. Inset: a single layer of KRb in
the LSL, imaged in situ.

done with the 30 cm cylindrical lens in Figure 5.4, which is oriented horizontally and positioned
about 45 cm away from the final focusing 15 cm (spherical) lens. As a result, the beam is roughly
collimated in the horizontal direction after the final lens, but focusing in the vertical direction.
We found that the passive phase stability of the system was good enough that we only needed
to optimize the LSL loading one or twice a day. At first, we used a thin piece of glass on one of thetwo beams before the final lens to adjust the relative phase of the beams (and hence the positionof the fringes on the atoms). We eventually found that the step size on our Newport picomotor
mirrors was fine enough to optimize the loading using the overall pointing of the two beams, andwe removed the glass plate.
We characterized the LSL trap frequency using parametric heating. Figure 5.5 shows thevertical size σy of Rb in TOF as a function of the intensity modulation frequency. Rb was initiallyloaded into a single LSL layer for this measurement. The fitted peak is 2.4 kHz, slightly lower thanthe calculated value of 2.8 kHz based on the measured beam sizes and powers. For calibration
purposes, this measurement was done at a higher LSL trap depth than that used for the molecule
experiments. The inset shows a single layer of molecules in the LSL, imaged in situ in the LSL +3

We originally tried to use a Wollaston polarizer to create the two LSL beams, following the scheme of Ref. [190].
However, we found that getting the beams to the desired aspect ratio was more challenging with the Wollaston setup
than the PBS setup.

101
VL. To ensure that molecules were produced in a single LSL layer, we also used expansion fromthe LSL in TOF with matter-wave imaging (see section 5.1.2.1 and Figure 5.8).
Loading the VL directly from the ODT subdivides the cloud into many layers, drasticallyreducing the PSD in each layer. The LSL spacing is large enough that cloud in the ODT can beloaded into a single LSL layer, which allows us to compress the cloud along ŷ before loading into the
VL. Since Rb cloud is much smaller than the K DFG, we can estimate the extent of the moleculesby calculating the size of the Rb cloud before loading the lattice. The density distribution of athermal gas in a harmonic trap is


N
y2
z2
x2
nthermal (x, y, z) =exp − 2 − 2 − 2 ,2σx 2σy2σz(2π)3/2 σx σy σzwhere σi =

(5.1)

q
kB T /mωi2 . The density distribution for a BEC isµ
nBEC (x, y, z) =g

where Ri =



x2
y2
z2
1− 2 − 2 − 2 ,
Rx Ry
Rz

(5.2)

q
2µ/mωi2 is the Thomas-Fermi radius, µ = (~ωho /2)(15N a/aho )2/5 is the chemical

potential, and g = 4π~2 a/m is the interaction coupling constant. Here a ≈ 100a0 is the Rb s-wave
scattering length, ωho = (ωx ωy ωz )1/3 is the geometric mean trap frequency, and aho is the oscillator
length associated with ωho [10].
Our typical starting conditions in the ODT were around 1.2 × 105 Rb at T = 90 nK with
30% BEC fraction. The ODT trap frequencies are (ωx , ωy , ωz ) = 2π × (40, 180, 40) Hz. Figure 5.6
shows the calculated density distribution along ŷ after integrating in the x- and z-directions (blueline), which can be compared to alatt (green line). This shows that Rb is spread into many layers
when loading directly from the ODT. In contrast, using the LSL with a weaker ODT creates a
combined trap with (ωx , ωy , ωz ) = 2π × (25, 600, 25) Hz. Typical conditions in the LSL were 9 × 104
Rb at T = 110 nK with 30% BEC fraction loaded into a single LSL layer. The distribution in the
LSL (orange line) is much narrower than in the ODT, allowing us to squeeze the atoms into fewer
layers.

Lattice potential (arb.)

Integrated density (10⁴/μm)

102
ODT

4

LSL

3

2

1

0
1.0
0.5

0.0
-6

-4

-2

0
y (μm)

2

4

6

Figure 5.6: Rb density distribution in the ODT (blue) and LSL (orange), compared with the
VL spacing (green). Loading the lattice from the LSL subdivides the cloud into fewer lattice sites
than loading from the ODT. The plotted density profiles (along ŷ) are obtained by integrating the
calculated 3D density in the two transverse directions (x̂ and ẑ). The Rb number, temperature,and trap frequencies for the calculation are given in the text.

5.1.1.1

Limits of 2D in the LSL

We briefly considered trying to make a 2D sample of molecules in the LSL without using
the science lattice. This would be technically challenging, since it would require us to increase the
LSL power significantly if we wanted to reach the same ωy as the VL, though it would allow us tostart with all of the molecules contained in a single site. However, we quickly realized that there
is a more fundamental problem with this approach. Namely, the quasi-2D requirement imposes a
maximum particle number for the Fermi gas, p
1 ωy 2
EF = ~ωr 2Nmax < ~ωy =⇒ Nmax <
,
2 ωr

(5.3)

where ωr is the transverse trapping frequency. This imposes a limitation on the maximum particle
number based solely on the geometry of the lattice beams. Plugging in the numbers for our LSL,

103
we find that this would limit us to Nmax ∼ 104 particles per layer. In our current scheme of loading
the atoms into 2D first, then making molecules, this would severely limit the initial K number.
Considering the low conversion efficiency of K to KRb, this is an especially harsh limitation. Other
2D Fermi gas experiments tend to use a smaller spacing lattice to increase ωy /ωr and Nmax [191],
but we are already using close to our maximum numerical aperture (NA) on the LSL path. As aresult, using the LSL as the only source of 2D confinement was not a viable strategy.

5.1.2

Layer counting

The LSL helps to compress the atoms into only a few layers of the VL, at the cost of making
it harder to measure the population distribution in the VL layers. At alatt = 540 nm, the VL layers
are too closely spaced to optically resolve. Even the vertical width of the cloud in the LSL + VL
is below our imaging resolution of ∼ 2 µm, which means we cannot infer the VL layer populations
from the density profile of the cloud in situ. As a result, imaging in situ does not provide muchquantitative information, and we instead measure the molecules in TOF after band mapping toaccurately determine the total molecule number N and temperature T .
The molecule signal in TOF is a sum over several unequally populated layers, and we needto account for this to quantitatively understand our measured loss and thermalization rates, aswell as the fitted T /TF of the cloud. To do so, we introduce an effective number of layers τ ,which we divide the total number N by to estimate the average density in each layer. The precise
definition of τ depends on the physical quantity we are measuring, and its value is calculated fromour measurement of the Rb layer distribution in the VL (described in the next section).

5.1.2.1

Matter-wave focusing

For the work in this chapter, we were not yet able to spectroscopically address individual
layers of the science lattice (as in Chapter 7). To measure the layer populations, we instead useda matter-wave focusing technique to magnify the in situ density distribution in TOF [192, 193].
Specifically, we shut off the science lattice diabatically but leave the ODT on for an additional

104

Figure 5.7: Rb layer distribution from matter-wave imaging. (A) Averaged image of 20 measurements of the Rb layer distribution in the VL using matter-wave magnification. (B) Extracted layer
distribution for the data in (A). Adapted from Ref. [52].

time Tho /4, where Tho is the trapping period of the ODT in the y-direction. As the atoms evolvein the harmonic potential of the ODT, their distribution in phase space rotates. At Tho /4, theinitial position distribution along ŷ is mapped onto the momentum distribution. At this point, weturn off the ODT and let the atoms expand in time-of-flight, which magnifies the in-situ atomic
distribution.
Our measurement of the Rb layer distribution is shown in Figure 5.7A. Since the molecules
are created within the confines of the Rb distribution, we estimate the final KRb distribution
from that of Rb (though the precise distribution of molecules may be slightly different owing to thecomplicated interplay of loss and molecule formation when crossing the Feshbach resonance). These
data were taken with the same number and temperature of Rb as in the molecule experiments.
The layer measurement on Rb involved a few technical challenges. We found that the K–Rb
interactions during the phase space rotation caused a significant reduction in the layer contrast, so
we removed K for the measurement in Figure 5.7. We checked this explanation by repeating themeasurement with the K-Rb mixture at the zero-crossing of the Feshbach resonance, and foundthat the layer contrast was restored. In addition, the center position of the fringes was not stable
shot-to-shot. Since the VL is formed by two separate beams, its phase can drift if the relative phaseof the two beams is not actively stabilized (see Ref. [107]). The VL phase was not stabilized forthese measurements, and along with small pointing fluctuations of the ODTs, this made the center

105of the layers move from shot to shot. As a result, we had to align the images manually by shifting
the fringes into alignment. Figure 5.7B shows the extracted layer distribution from the data in (A),obtained by integrating the image in the transverse direction and fitting the resulting 1D profile
with a sum of Gaussians with centers displaced by the layer spacing.

5.1.2.2

Determining τ

We would like to estimate several physical quantities, such as the Fermi temperature TF or
two-body loss rate coefficient β, averaged over the layer distribution in Figure 5.7. In TOF, wePmeasure the total molecule number N =i Ni , where Ni is the number in the ith layer, andtemperature T . We therefore need to find an effective number of layers τ such that substituting
N/τ in our formulas gives the layer-averaged quantity of interest. This explanation is still somewhat
nebulous but will hopefully be made clearer by the examples below.
For a fixed molecule distribution, the definition of τ depends on the physical quantity being
measured. This is because some of these quantities depend non-linearly on the number in each layer.
A simple (linear) quantity of interest is the average number of molecules per layer hNi i, where thebrackets h·i denotes the average with respect to the layer distribution. Using τ = N/hNi i, we
obtain τ = 4.6(2) for the data in the Figure 5.7B.
We also want to measure the layer-averaged Fermi temperature, which is nonlinear in N .
Consider the 2D Fermi temperature in the ith layer,
TF,i =
where ω =

~ω p
2Ni ,
kB

(5.4)

√

ωx ωz is the geometric mean trapping frequency. The layer-averaged quantity is√
hTF,i i = (~ω/kB )h 2Ni i. By equating hTF,i i with the expression for TF in a single layer but with ap√reduced number N/τ , TF = (~ω/kB ) 2N/τ , we obtain τ = N/h Ni i2 . Thus, by using a different

definition of τ , we can absorb the nonlinear dependence on Ni into the effective layer number τ .
√Extracting τ = N/h Ni i2 from the data in Figure 5.7B, we find τ = 4.9(2). Note that the√two estimates N/hNi i and N/h Ni i2 give similar values since the layer distribution in the LSL

106+ VL is fairly narrow. We used τ = 5(1) to calculate the molecule TF in Ref. [52], where theadded uncertainty accounts for systematic errors from the non-uniform conversion of Rb to KRb
and differences in the evaporation efficiency between the layers.
We also measured the loss rate coefficient β to observe the 2D dipolar loss suppression. In
the layered system, the two-body loss is described by the following differential equation,dni= −βn2i ,
dt

(5.5)

with the solutionni (t) =ni (0),
1 + βni (0)t(5.6)

where ni (t) is the average density in the ith layer. We do not have direct access to the individualP
layer densities ni (t), but we can extract n(t) = i ni (t) by measuring the total number N and T .
We would like to find an effective number of layers τ such that we can fit n(t)/τ to a layer-averaged
two-body loss equation,d(n/τ )≈ −β(n/τ )2 ,
dt

(5.7)

to accurately extract β.
To determine τ for the loss measurement, we numerically simulated the decay over time of
the non-uniform molecule distribution from Figure 5.7B, and compared it to the decay of a gas
with a uniform layer distribution and the same N and T . We defined τ as the number of layersof the uniform distribution for which the loss rate matched that of the non-uniform distribution.
Owing to the nonlinear nature of the decay in time, this definition of τ also depends on the total
measurement time. We modeled the system using the theoretically expected β over our typicalmeasurement time of several seconds, and found τ = 8(1) for the layer distribution in Figure 5.7B.

5.1.3

Loading procedure

We started with the atomic mixture in the ODT at a magnetic field of 555 G, with trap
frequencies of (ωx , ωy , ωz ) = 2π × (40, 180, 40) Hz for Rb. The atoms were next loaded into the107

Figure 5.8: Matter-wave magnification of (A) Rb, (B) K, and (C) KRb clouds released from theLSL. All of the Rb is loaded into a single layer, and K is loaded primarily into the central layer.
The KRb image in (C) was taken with the LSL intentionally misaligned to show two clear layersof molecules. Note that the region size and color scale are different for all three images.

LSL, with Rb occupying only a single LSL layer, and the ODT was decompressed to reach trap
frequencies of (ωx , ωy , ωz ) = 2π × (25, 600, 25) for Rb in the combined trap. Finally, the atoms
were loaded into the VL and the ODTs were adjusted to reach transverse trap frequencies of(ωx , ωz ) = 2π × 40 Hz for KRb. Molecules were then created with the Feshbach field ramp and
STIRAP. To ensure that the molecules were created in a single LSL layer, we used the same matterwave focusing technique discussed in the previous section, but with the LSL playing the role of the
VL. This generates images similar to those in Figure 5.8.
In optimizing the loading sequence, we learned that it was important to keep the LSL on
after loading the VL to prevent the atoms from tunneling. If the atoms are allowed to tunnel, thenumber of layers of the molecules will increase. The tunneling element J is approximately given by√

J/ER = 1.39666s1.051 e−2.12104 s ,(5.8)

where s is the lattice depth in units of ER [194]. At our typical VL depth (ωy = 2π × 17 kHz
for KRb), we calculate (JK , JRb , JKRb ) = h × (220, 13, 0.2) Hz. Owing to its small mass, K isparticularly likely to tunnel. The site-to-site energy shift in the VL from gravity (h × 530 Hz forK) was not enough to fully shut off the tunneling. Keeping the LSL on provided an additionalsite-to-site shift that completely suppressed the tunneling of the atoms. The LSL was then ramped
off after making molecules.

108

Figure 5.9: Dipolar suppression of loss in 2D. (A) Loss measurement of molecules in quasi-2D at
d = 0.2 D. The data is fit to equation (5.7) to extract the two-body loss rate coefficient β. (B)Summary of β vs d. Reproduced from Ref. [52].

5.2

Loss suppression
Having produced a 2D molecular gas, we next studied the dipolar loss suppression by mea-suring the molecular loss rate as a function of |E|. The typical starting conditions for these measurements were N = 20 × 103 at T = 250 nK, with τ = 5 and T /TF = 1.5. The trap frequencies
were (ωx , ωy , ωz ) = 2π × (40, 17 × 103 , 40) Hz for KRb.
Before measuring the loss rates, we needed to cancel uncompensated electric field gradients,
particularly in the loosely trapped x-direction. Strong gradients cause additional one-body loss
that could cause us to overestimate the loss rate coefficient β. To fix the gradients, we started bycentering the molecules geometrically in the electrodes (discussed in detail in Will Tobias’s thesis[107]). Next, we shimmed the horizontal gradient ∂|E|/∂x at each field configuration to maximize
the molecule number left after a set hold time (typically 100 ms).
After doing this, we were able to observe the 2D dipolar suppression. An example loss
measurement at d = 0.2 D is shown in Figure 5.9A. Stabilized by repulsive dipolar collisions, themolecules have a very long lifetime and are still detectable after 10 s. The data was fit to equation(5.7) to determine the two-body loss rate coefficient β (solid line). The dependence of β on d is

109

Figure 5.10: Crossover from 3D to quasi-2D. The measured β (gray circles) and heating rate(orange squares) as a function of the vertical trap frequency ωy . Reproduced from Ref. [52].

shown in part (B). Strikingly, at d = 0.2 D, β was suppressed by nearly a factor of 5 below thezero field value. The increase of the loss rate for d > 0.25 is consistent with a quasi-2D picture of
the dipolar collisions, in which the mixing with L = 3 reduces the barrier at large d [81, 100] (seealso section 2.5.2).
We also explored the crossover from 3D to quasi-2D at d = 0.2 D by varying ωy /2π from 200
Hz to 17 kHz (Figure 5.10). This was done by lowering the VL power, while adjusting the ODTpower to maintain (ωx , ωy ) = 2π × (40, 40). The initial gas temperature T = 250 nK was roughly
constant as a function of ωy , and the gas was in 3D for small ωy . In this regime, the loss rate β(grey) was strongly enhanced by attractive dipolar collisions, similar to previous 3D measurements[79]. As ωy was increased, the measured β quickly dropped, reaching a plateau for ωy > 2π × 7
kHz, corresponding to the quasi-2D regime where ~ωy > kB T . In addition, we observed that theheating rate (orange) in 2D was zero, in agreement with theoretical predictions [186], and in stark
contrast to the enormous anti-evaporative heating (∼ µK/s) observed in 3D [79].

110

Figure 5.11: Elastic dipolar collisions in quasi-2D. (A) Cross-dimensional thermalization measurements at d = 0.1 D (orange diamonds) and 0.21 D (gray squares). Error bars are 1 SE of 5
measurements. (B) Summary of measured thermalization rate Γth vs induced dipole moment d.
The filled point was measured at d = 0 D but artificially placed at 0.03 D to be representable on
the logarithmic scale. Reproduced from Ref. [52].

5.3

Elastic dipolar collisions
The long lifetime shown in Figure 5.9A gives ample time to measure the elastic collisions via across-dimensional thermalization measurement. The elastic collisions occur with rate Γel = nσel vth ,pwhere n is the average density in 2D, σel is the elastic cross section, and vth = πkB T /2µ is thethermally-averaged relative velocity of two collision partners, with µ the reduced mass of KRb.
The thermalization rate is given by Γth = Γel /α, where α is the number of collisions to thermalize.
Monte-Carlo simulations predict α = 8 in the temperature range of our measurements [186].
To measure Γth , we diabatically increased the power of one of the ODT beams to compress
the cloud along ẑ. This brought the cloud out of thermal equilibrium and created an imbalance inthe thermal energy on the two axes, Tz > Tx . Elastic collisions redistributed the excess energy inẑ to x̂, leading to an increase in Tx over time as the cloud thermalized. We fit the increase in Tx
to an exponential curve to extract the thermalization rate Γth .
Two representative thermalization measurements are shown in Figure 5.11A. Changing dfrom 0.1 D (orange) to 0.21 D (gray), we measured an increase of Γth by over a factor of 10. Figure
5.11B shows a summary of the measured Γth as a function of d, demonstrating that we can tune

111
the elastic collision rate over at least two orders of magnitude. We attribute the constant Γth below
d = 0.1 D to coupling between x̂ and ẑ from the trap anharmonicity, which limited the smallest
Γth we could measure.
By fitting the measured Γth for d ≥ 0.1 D (solid line), we extracted Γth ∝ d3.3±1.0 . The fitted
dependence is in good agreement with theory, which predicts a crossover between two regimes inthe scattering depending on kaD , where k is the collision wavevector and aD is the dipolar length[195]. For kaD  1 (low energy regime), Ref. [195] predicts σel ∝ ka2D ∝ kd4 . For kaD  1p√(semiclassical regime), the prediction is σel ∝ aD /k ∝ d/ k. Assuming a collision energy ofEcol /kB = 450 nK, we calculate that kaD ranges from 0.1 to 0.9 over the fitted range of d.
An important figure of merit for evaporative cooling is γ = σel /σinel , the ratio of good tobad collisions. To make a direct connection to our measurements, we can express γ equivalently as
γ = Γel /Γinel = αΓth /Γinel , where Γinel = n0 β is the initial loss rate and n0 is the initial average 2Ddensity. Using the theoretically predicted value of α = 8 [186], we extracted a maximum value ofγ = 200(60) at d = 0.2 D, indicating that efficient evaporation is possible in our system.

5.4

Direct evaporation of molecules
Thanks to the favorable collision ratio γ measured in the previous sections, we were able toperform efficient evaporation of the molecules to below the Fermi temperature in 2D. To perform
evaporation, we leveraged our electric field control to spill the hottest molecules out of the gas byapplying an anticurvature of |E| in the x-direction. The molecular T /TF after evaporation was
obtained from fitting the cloud in time-of-flight, and further supported by measuring the expansion
energy of the gas as a function of the final T /TF . The following subsections discuss the measurement
in more detail.

5.4.1

Combined optical + electric potential

Forced evaporative cooling is performed by lowering the trap depth to spill out the hottest
molecules. Typically, this is described in terms of the ratio η of the trap depth to the thermal energy

112
kB T of the gas (for non-degenerate gases). When holding the gas in the trap without evaporating,one typically needs to keep η ≈ 10 to avoid losing particles.
To evaporate, one lowers η to begin spilling hot particles. We could not lower the VL power
to lower η, as is typically done when evaporating in an optical trap, since we needed to maintaina high ωy for the 2D loss suppression to remain effective. Instead, we applied an anticurvatureof |E| along x̂ to controllably lower the trap depth of the combined optical and electric potential.
Calculated trapping potentials along the x-direction are shown in Figure 5.12A, showing how thetrap depth can be significantly lowered by the field curvature. Relative to lowering the VL power,
this approach also maintains higher transverse trapping frequencies in the center of the cloud. As
a result, the thermalization rate could be kept high while the trap depth was lowered.4
To quantitatively characterize the combined potential, we measured ωx in the combined
potential as a function of the applied field curvature at |E| = 5 kV/cm (Figure 5.12B). The
curvature was controlled by the ratio of r of the rod voltages to the plate voltages (see also section
3.3.3). From knowledge of the VL beam waists and the electrode geometry (section 3.3.2), we
calculated the expected behavior of ωx as a function of r (dashed line) and found good agreementwith our measurements (circles).

5.4.2

Evaporation trajectory

To evaporate the molecules, we slowly increased r to spill the hottest molecules from the trap
while letting the remaining molecules rethermalize. Figure 5.13A shows r as a function of time fora typical evaporation sequence, with the corresponding calculated trap depth plotted in (B). We
started with E in the “flat field” configuration (r = 0.4225), where the anti-trapping from E isnegligible. The sequence consisted of several ramps of r:(1) An initial fast ramp over 20 ms to start spilling the cloud. We optimized the endpoint of
4

In principle, using a linear gradient instead of a quadratic one could allow us to maintain an even higher elastic
collision rate. Ref. [196] used this scheme in a 3D trap to achieve very fast and efficient evaporation of an atomic Cs
gas. We never saw good results by using a gradient instead of anticurvature to evaporate, which we suspect is due tothe trap being more smoothly deformed when ramping up the curvature (as opposed to the gradient).

113

Figure 5.12: Combined optical and electric potential. (A) Combined potentials at the “flat field”
configuration (left) and at the end of evaporation (right). (B) Measurements of ωx in the combined
optical and electric potential as a function of the rod scale r. The experimental data (gray circles)
show good agreement with the calculated potential (dashed line). The solid gray line is a linear
fit to the data. The horizontal line shows the calculated value of ωx in the flat-field configuration(r = 0.4225). Adapted from Ref. [52].

114

Figure 5.13: Evaporation sequence and trajectory. (A) Value of rod scale r over the evaporation.
(B) Calculated trap depth of the combined optical and electric potential over the evaporation. (C)Evaporation trajectory at |E| = 6.5 kV/cm and d = 0.25. The fit (orange line) gives Sevap =1.06(15), significantly better than the threshold of Sevap = 2 (dashed line) needed to increase PSD.
(D) Calculated η for the evaporation curve shown in (C). (E) Summary of Sevap versus the induced
dipole moment d. The evaporation is most efficient near d = 0.2 to 0.25 D, where γ is the highest.
Adapted from Ref. [52].

115
this ramp to be the point where molecules just started to be lost.
(2) A two-stage evaporation ramp.
(3) A final 100 ms ramp to recompress the trap to its original depth. We compared the PSD
with and without evaporation at the same trap frequencies to remove the effects of adiabatic
cooling.
By interrupting the evaporation ramps midway through and recompressing to the flat field, we could
map out the evaporation trajectory. Figure 5.13C shows an evaporation trajectory at |E| = 6.5
kV/cm (d = 0.25 D), showing how the total molecule number N and fitted temperature T evolve
over the evaporation. The ratio of the measured molecule temperature T with the calculated
trap depth gives an estimate of η over the evaporation trajectory (Figure 5.13D). Our optimized
trajectory used η ≈ 4 for the initial stage of evaporation, which is in rough agreement with the
expectation from theoretical simulations [186]. This low value of η signals a fairly aggressive
evaporation curve, in comparison to typical values of η = 6–8 for alkali atoms in a harmonic trap[20]. It makes sense intuitively that we need to evaporate quickly, since evaporating too slowly will
eventually result in all molecules being lost by reactive collisions.
To characterize the evaporation efficiency, we calculated the evaporation slope Sevap , definedbySevap =∂ log N.
∂ log T

(5.9)

With this definition, a lower Sevap is a more efficient evaporation. Since PSD ∝ N T −2 for a
non-degenerate 2D gas, we must have Sevap < 2 to increase PSD in 2D. Fitting the evaporation
trajectory in Figure 5.13C, we obtain Sevap = 1.06(15) (solid line), significantly better than thethreshold of Sevap = 2 (dashed line). We optimized the evaporation trajectory at several values of|E| to measure Sevap as a function of d (Figure 5.13E). The best Sevap was achieved near d = 0.2–0.25
D where γ is maximized.

116

Figure 5.14: 2D Fermi gas of molecules. (A) Absorption images after band mapping from the
VL and 5.84 ms of TOF, for a gas at T /TF = 2.0(1) (top) and T /TF = 0.81(15) (bottom). (B)Profiles along x̂ after integrating out the y-direction. The profiles are well-fit by the Fermi-Dirac
distribution (equation (5.14), black lines). A Gaussian fit to the outer wings of the cloud (orange
lines) overestimates the density in the center of the colder cloud, signaling the onset of degeneracy.
Reproduced from Ref. [52].

5.4.3

2D Fermi gas of molecules

Since Sevap < 2, the evaporative cooling increases the PSD of the molecules and lowers T /TF .
Starting from an initial condition of T /TF = 1.5(1), we were able to cool the gas below T /TF = 1
and observe the onset of degeneracy in 2D. Our best evaporation gave N = 1.7(1) × 103 andT /TF = 0.6(2).
Figure 5.14A shows images of the molecular gas after band mapping and TOF for T /TF =2.0(1) (top) and T /TF = 0.81(15) (bottom). The colder cloud was produced by evaporation, andshows a clear decrease in the momentum spread along x̂. The momentum spread in the y-direction
is limited by the size of the Brillouin zone from the band mapping procedure, so we integrated out
the y-direction to obtain the 1D profiles in (B). The molecular T /TF is determined by fitting the
profiles in (B) to the Fermi-Dirac distribution (black lines). Fitting the outer wings of the cloud toa Gaussian (orange lines) produces a good fit for the hotter cloud, but significantly overestimates
the density in the center of the colder cloud. This is a hallmark of the onset of quantum degeneracy.
To characterize the change in the momentum distribution as the gas becomes degenerate, we
extracted the deviation δU = U − Ucl of the energy U of the fermionic gas from the classical energy

117

Figure 5.15: Expansion energy of the 2D molecular Fermi gas. The temperature T is determinedby the Fermi-Dirac fit to the entire cloud (gray circles) or a Gaussian fit to the outer wings (orange
squares). The gray solid and purple dashed curves show the calculated δU/U for the 2D and 3D
ideal Fermi gases, respectively. Reproduced from Ref. [52].

Ucl = 2kB T . We obtained U = 2kB Trel from the width of the momentum distribution by fitting
the whole cloud to a Gaussian profile and calculating the release temperature
Trel =

mωx2 σx2,
kB (1 + ωx2 t2 )(5.10)

where t is the TOF and σx is obtained from the Gaussian fit.
The results are shown in Figure 5.15. A large deviation of δU/U is observed below T /TF = 1,
in agreement with simulations (gray solid line). The temperature T , which determines the classicalenergy Ucl , was obtained directly from the Fermi-Dirac fit (gray circles) or by fitting a Gaussian
to the outer wings of the cloud (orange points). Both methods show good agreement with the
expected trend. Note that the effect is much stronger in 2D than in 3D (purple dashed line) owing
to the decreased density of states in 2D.

5.4.3.1

Fitting the 2D gas in time-of-flight

Here, we briefly discuss the relevant equations for fitting. The in-situ density distribution of
a 2D Fermi gas is




1
1
1
2 2
2 2
,
n(x, z) = − 2 Li1 − exp βth µ − mωx x − mωz z
22
λdb(5.11)

118
with λdb the thermal deBroghlie wavelength, βth = 1/kB T the inverse temperature, and µ the
chemical potential. In 2D, µ is defined by
N
=
τ



kB T
~ωr

2

Li2 (−eβth µ ).

The cloud is imaged along ẑ, so we see the column-integrated density
s




1
2π
1
2 2
.
nint (x) = − 2
Li3/2 − exp βth µ − mωx x
2
λdb βth mωz2

(5.12)(5.13)The imaged cloud after TOF also has a finite extent in the y-direction from the band mapping
procedure. We integrate out the y-direction, and fit the resulting 1D profile to22
nint (x) = A × Li3/2 −eq−x /2σx /Li3/2 (−eq ),(5.14)

where q = βth µ is the log fugacity [173]. From the fit, we can extract T /TF via


T
TF

2
=−

1
,
2Li2 (−eq )(5.15)and T fromT =mωx2 σx2,
kB (1 + ωx2 t2 )(5.16)

where t is the TOF.
We can also determine T by fitting the outer wings of the cloud. To do this, we first fit theentire cloud to extract its Gaussian width σ. Then we fit the cloud again, including only the region
beyond 1.5σ from the fitted center of the cloud, which we define as the outer wings of the cloud.
The second fit gives a new Gaussian width σout and corresponding temperature Tout via equation(5.16).

5.5

Conclusion
In this chapter, we utilized repulsive dipolar collisions in 2D to stabilize the gas against loss

while providing a large elastic scattering rate. The favorable ratio γ = 200(60) enabled direct
evaporation of the molecules to below TF in 2D. The techniques described here are general and119
predicted to work for the other fermionic bialkalis, as well as bosonic molecules at higher d and ωy[101].
The physics of dipolar gases in 2D is very rich and is ripe for exploration with a 2D Fermi gasof molecules. For example, the collective modes of the system can be modified as the elastic collision
rate is tuned by the dipole moment [197]. With the conditions achieved in this work, we should be
able to explore the crossover between the collisionless and hydrodynamic regimes. The work in thischapter focused on the direct dipolar interaction between the induced dipole moments, but dipolar
spin-exchange can also be brought into play by putting the molecules in a coherent superposition ofrotational states (Chapter 8). Owing to the interplay of interactions and dimensionality, interesting
phenomena such as spin-orbit coupling [139] or spin-squeezing [49] is predicted to occur. Direct
evaporation in 2D will be an important first stage to produce the low-entropy gases needed forthese explorations.

Chapter 6

Resonant shielding of molecules in 2D and 3D

In the previous chapter, we showed how the combination of dipolar interactions and tightconfinement into 2D could stabilize the molecular gas. This realized a long-lived gas of moleculeswith strong dipolar interactions, and enabled direct evaporation to below the Fermi temperature
in 2D. Those results utilized the induced dipole moment of the molecules to control the collision
properties.
In this chapter, we show how the transition dipole moment can enter into the collision process.
In particular, we observed resonant collisional shielding of molecules, which caused the loss ratein the |1, 0i state to be modulated by nearly three orders of magnitude near a field of |E| = 12.5
kV/cm. At some values of |E|, the loss rate could be suppressed by nearly a factor of 10, and themolecules were shielded from loss. Physically, the shielding arises from a near degeneracy of a pairof molecules in |1, 0i|1, 0i with another rotational pair |0, 0i|2, ±1i or |0, 0i|2, 0i, which occurs at a
certain value of |E|. The two kets denote the (symmetrized) rotational states of the two colliding
molecules. Since |1, 0i|1, 0i is coupled to both |0, 0i|2, ±1i and |0, 0i|2, 0i through the transition
dipole moment (section 2.5.4), the effective intermolecular potential is strongly modified when theserotational state pairs are near-degenerate.
An interesting feature of the shielding, compared to the 2D dipolar repulsion, is that it also
works in 3D. After our initial observation in 2D [137], we investigated the properties of the shieldingin 3D, including the elastic collisions at the shielding point, and used it to evaporatively cool themolecules in 3D [138].

121

6.1

Physical mechanism
The idea of shielding molecular collisions with a static E was introduced in pioneering theo-

retical work by Avdeenkov et al. in 2006 [103]. A simple intuitive picture comes from considering
the van der Waals interaction between two colliding molecules. In second order perturbation theory,
we obtain
Vvdw (r) =X |hY |Vdd (r)|Xi|2
Y

EX − EY.

(6.1)

Above, |Xi represents the incoming channel (internal states of the two molecules as well as partialwave), while Y runs over all other dipole-coupled collision channels. Since Vdd (r) ∝ 1/r3 , this gives
the usual Vvdw (r) ∝ 1/r6 scaling.
Equation (6.1) already hints at some of the main features of the shielding. Typically, theenergy denominators EX − EY are large1 , suppressing the dipolar coupling between |Xi and |Y i
except for at very close distances where Vdd becomes very strong. However, if we can engineer asituation where some specific state |Y 0 i becomes nearly degenerate with |Xi, then the coupling to|Y 0 i can become the dominant term in Vvdw at longer distances.
Let us define Vshield as the contribution to Vvdw from |Y 0 i,
Vshield =|hY 0 |Vdd (r)|Xi|2
,
∆E

(6.2)

with ∆E = EX − EY 0 . We will assume that ∆E is small, so the dipolar coupling is near resonant.
If ∆E < 0, then Vshield is an attractive potential. On the other hand, if ∆E > 0, then Vshield forms
a repulsive barrier. Since the reactive loss rate for KRb is directly linked to the rate of tunnelingthrough long-range potential barriers (see section 2.5.2 or Ref. [81]), this implies that the molecularloss can be controlled by ∆E. Also, note that the value of Vshield depends on the magnitude of thematrix element hY 0 |Vdd |Xi but not its sign. As long as ∆E > 0, this means that both side-to-sidecollisions (mL = ±1) and head-to-tail collisions (mL = 0) have a repulsive barrier. As a result, theshielding is expected to work in 3D, in contrast to the 2D dipolar loss suppression of Chapter 5.
1This is true for KRb and other Σ molecules, but molecules with Λ-doublets in the ground state (e.g., OH [198])
can have small energy denominators even at zero electric field.

122
In the absence of dipolar interactions, the p-wave barrier keeps the molecules from colliding
at short range. The peak of the p-wave (L = 1) barrier is located at a distance

Rb =

3
L(L + 1)

1/4


avdW =

3µC6
~2

1/4
,

(6.3)

where avdW is the van der Waals length, with C6 is the van der Waals coefficient and µ the reduced
mass. For KRb, Rb = 270a0 and the height of the barrier is Vb = 24 µK.
By calculating Vshield (Rb ) as a function of the energy denominator ∆E, we can estimate
the effect of the shielding. To make things more concrete, let us plug in |Xi = |1, 0i|1, 0i and|Y 0 i = |0, 0i|2, 0i, and ignore the motional angular momentum state for simplicity. The orderof magnitude of the coupling is hY 0 |Vdd (r)|Xi ∼ d21 d10 /4π0 r3 , where d21 = h2, 0|d0 |1, 0i andd1,0 = h1, 0|d0 |0, 0i are the transition dipole moments, and we get
1
Vshield (Rb ) =∆E

d21 d104π0 Rb3

2
.

(6.4)

Let us first calculate the effect at |E| = 0, where ∆E is large and Vshield should be very small.
Here, we have ∆E = 2Bv , where Bv is the rotational constant, and the dipole matrix elements are√√
d10 = dperm / 3 and d21 = 2dperm / 15 (see Table 2.2). Plugging in numbers for KRb, we obtain
Vshield = −550 nK at |E| = 0, which is small compared to Vb = 24 µK.
However, ∆E can be tuned by |E|. The dependence is shown in Figure 6.1A for KRb; notably,∆E = 0 at |Eres | = 12.505 kV/cm. The dipolar coupling strength d21 d10 also changes somewhat
with |E| due to rotational state mixing (Figure 6.1B). With these two quantities, we can calculate
how Vshield (Rb ) varies with |E| (Figure 6.1C). For ∆E > 0, Vshield (Rb ) is large and positive, forming
a repulsive shield for the molecules.
We stress here that the shielding resonances come from the energy degeneracy of the initial
scattering state and another free scattering state with different rotational angular momentum. This
is in contrast to Feshbach resonances, which arise from bound states. The shielding resonances are
essentially Förster resonances — for example, as studied in Rydberg atoms [199, 200, 201] — where
two dipole-coupled pairs of states are tuned to resonance with |E|.

123

Figure 6.1: Vshield (Rb ) as a function of |E|. (A) Energy difference ∆E = 2E|1,0i − E|2,0i − E|0,0i .
The dashed line is ∆E = 0. (B) The dipolar coupling strength d21 d10 . (C) The estimated barrier
Vshield (Rb ), zoomed into the region around |Eres |. Also shown is the height Vb = 24 µK of the p-wave
barrier (dashed line).

6.1.1

Two-channel picture

When ∆E becomes very small, the perturbative expression from equation (6.1) breaks down.
Full quantum scattering calculations are needed to accurately determine the behavior for any ∆E[104]. However, near ∆E = 0, the two coupled states |1, 0i|1, 0i and |0, 0i|2, 0i are most important,and one can approximate the system with a two-channel model to estimate the height of thebarrier [138, 202]. In this section, we give a brief description of the two-channel model, similar to
the explanation given in the supplement of Ref. [138]. For a more rigorous treatment, see the workby our theory collaborators in Ref. [202].
√
To simplify notation, let us define |1i ≡ |1, 0i|1, 0i and |2i ≡ (|0, 0i|2, 0i + |2, 0i|0, 0i)/ 2.
Note that both |1i and |2i are symmetrized in the molecular internal states. Then at each |E|, we
can write down a simplified Hamiltonian in the basis of |1i and |2i,

H12 (r) H11 (r)H(r) = ,
H21 (r) H22 (r) − ∆E

(6.5)

124
where
~2 L(L + 1) C6− 6 + h1|Vdd (r)|1i
2µr2
r2~ L(L + 1) C6
H22 (r) =− 6 + h2|Vdd (r)|2i2µr2r
H11 (r) =H12 (r) = h1|Vdd (r)|2i,

(6.6)
(6.7)
(6.8)

and ∆E = 2E|1,0i − E|0,0i − E|2,0i . The electronic van der Waals coefficient is used C6 = 12636
a.u. [203] to avoid double-counting the rotational contribution. To calculate hi|Vdd (r)|ji, we canuse equation (2.32),
hN1 , mN 1 ; N2 , mN 2 ; L, mL |Vdd |N10 , m0N 1 ; N20 , m0N 2 ; L0 , m0L i



√
0
0
L 2 L  L
2 L 
p
6
mL
0 + 1) 
=−
(−1)
(2L
+
1)(2L




4π0 r30 0 0−mL 0 m0L
× hN1 , mN 1 ; N2 , mN 2 |2d01 d02 |N10 , m0N 1 ; N20 , m0N 2 i.

(6.9)

Above, we have only kept the mN -preserving term since all the involved rotational states havemN = 0. For simplicity, we neglect the partial wave mixing and keep only terms with L = L0 = 1,
to obtain√
2 6
h1|Vdd (r)|1i =f (L, mL )d21
4π0 r3√
2 6
h2|Vdd (r)|2i =f (L, mL )(d0 d2 + d202 )4π0 r3√√
2 6
h1|Vdd (r)|2i =f (L, mL ) 2d21 d10 ,34π0 r(6.10)(6.11)(6.12)

where dN = hN, 0|d0 |N, 0i is the induced dipole moment of |N, mN = 0i and dN N 0 = hN, 0|d0 |N 0 , 0i
is the transition dipole moment between |N, 0i and |N 0 , 0i. The angular part is given by



2 L
L 2 L  L
f (L, mL ) = (−1)mL +1 (2L + 1) 

.
0 0 0
−mL 0 mL

(6.13)

For L = 1, we have f (1, ±1) = 1/5 and f (1, 0) = −2/5. Since the interaction preserves mL , we cansolve H(r) for each mL separately.

125

Figure 6.2: Two-channel picture of the shielding, showing adiabatic energy curves calculated fromequation (6.5). (A) |E| = 12.49 kV/cm (|E| < |Eres |). (B) |E| = 12.56 kV/cm (|E| > |Eres |). In
both plots, the solid (dashed) line represent |1i (|2i). The curves for mL = ±1 (blue) and mL = 0(red) show similar behavior.

Diagonalizing H(r) at each r yields the adiabatic energy curves shown in Figure 6.2. In
both plots, the solid lines represent |1i and the dashed lines represent |2i. For |E| < |Eres | (A), theeffective potential is strongly attractive and the p-wave barrier is completely wiped out. Conversely
for |E| > |Eres | (B), the potential is strongly repulsive and the molecules are shielded. Note thatthe qualitative behavior of mL = ±1 (blue) and mL = 0 (red) is the same, as we had argued fromthe form of Vvdw . We can also understand the behavior in terms of level repulsion between thestates — the upper state is pushed up in energy, thus experiencing a repulsive interaction, while theopposite is true for the lower state. The effect is stronger in mL = 0 since the dipolar interactionis twice as strong as in mL = ±1.
From equation (6.5), we can also estimate the width of the shielding features as a functionof |E|. The dipolar coupling becomes resonant when |H12 (r)| ≥ |∆E|. Since this criterion depends
on r, we use r = Rb as an estimate of the relevant length scale for a collision. Plugging in theparameters for KRb, we find that H12 (Rb ) ≈ h × 4.2 MHz at |Eres |. Given the differential Stark
shift of h × 215 kHz/(V/cm) between |1i and |2i at |Eres |, this implies a width of roughly 20 V/cm,
which is quite narrow compared to |Eres |.

126

Figure 6.3: Angular dependence of the shielding, demonstrating how the effective intermolecularpotential varies as a function of the relative orientation of the molecules at (A) |E| = 12.72 kV/cm(|E| > |Eres |) and (B) |E| = 12.50 kV/cm (|E| < |Eres |). Reproduced from Ref. [138].

As shown in Refs. [138, 202], one can also visualize the dependence of the effective potential
on the orientation of the molecules. This is done by leaving the angular part of Vdd as a functionof θ instead of expanding it in spherical harmonics as we did above. Here, θ is the polar angleof the vector r between the molecules. For the states considered here, we can do this by simply
replacing f (L, mL ) with (1 − 3 cos2 θ)/2 in the equations above. The resulting potential energy
maps are shown in Figure 6.3. From these plots, we can clearly see the gaps in the barrier at√
θ = arccos( 3) ≈ 54.7◦ where the dipolar coupling vanishes.
The shielding occurs when the molecules are prepared in the upper branch of the avoided
crossing (|E| > |Eres |). Consequently, inelastic collisions that result in decay of |1i to |2i areenergetically open and are enhanced very close to |Eres |. In our experiment, molecules that undergo
inelastic collisions are immediately lost due to reactive s-wave collisions. Thus, the optimal field forshielding is not simply the field with the largest barrier in the adiabatic curves, but rather where
the sum of the reactive and inelastic rates is minimized [104].
In the next sections, we will describe the experimental measurements of the shielding indetail. Before moving on, we have two more brief comments. First, while all of the discussion in
the section centered on the coupling between |1, 0i|1, 0i and |0, 0i|2, 0i, similar arguments hold for|0, 0i|2, ±1i, which is also coupled to |1, 0i|1, 0i via dipolar interactions. One interesting difference

127
is that the dipolar interaction also changes the motional state since the total angular momentum
projection must be conserved. Second, the resonant electric field shielding is very general, relying
only on the rigid rotor structure of the molecule. Theoretical calculations for the bosonic bialkali
molecules [204] and several 2 Σ molecules [205] all show a strong shielding. Hence, our experimental
results should be broadly applicable to other ultracold molecule species.

6.2

Shielding in 2D
Our initial study of the shielding was done in 2D [137]. The procedure for preparing a 2D

gas of molecules was similar to the previous 2D work. Typical starting conditions were N = 2 × 104
molecules in the |0, 0i state at T = 250 nK, corresponding to T /TF = 1.8. The molecules were inτ = 6 layers of the VL, with trap frequencies of (ωx , ωy , ωz ) = 2π × (34, 17.7 × 103 , 34) Hz. Since
kB T /~ωy ≈ 0.3, the gas was in the quasi-2D regime.
The molecules were produced at |ESTIRAP | = 4.5 kV/cm to sidestep the many avoided crossings of the |1, 0i state at low electric field (see Figure 2.2). STIRAP initially populates the |0, 0i
state, and we used a resonant microwave π-pulse at 2.6 GHz (Rabi frequency Ω = 2π × 200 kHz)to transfer the molecules to |1, 0i with a typical efficiency of 95%. Any molecules remaining in|0, 0i were quickly lost due to s-wave collisions with the majority |1, 0i state [32]. After the state
preparation, E was ramped to its target configuration in 60 ms. After a variable hold time t, E
was ramped back to ESTIRAP , and the molecules were transferred back to |0, 0i and dissociatedfor imaging. The average 2D molecular density n(t) was fit to the solution of the two-body loss
differential equation dn/dt = −βn2 to extract the loss rate coefficient β, same as in the 2D loss
measurements in the previous chapter. Since the shielding depends sensitively on |E|, we calibrated
the electric field strength to a few parts in 104 using microwave spectroscopy of the |0, 0i to |1, 0i
transition.
We first measured β for |1, 0i in the region around |Eres | = 12.51 kV/cm to look for the
anticipated sharp changes of β near the resonance with |0, 0i|2, 0i. The field was oriented alongŷ for these measurements, so the molecules were colliding with mL = ±1. We observed the loss

128

Figure 6.4: Large variation of β near the shielding resonance. (A) Loss measurements at |E| =12.50 kV/cm (green squares) and 12.67 kV/cm (orange circles). The field is oriented along ŷ. The
inset shows the 12.50 kV/cm data on an enlarged x-axis. (B) Calculated adiabatic energy curves
for mL = ±1 collisions of |1, 0i, corresponding to the values of |E| in (A). The black curve shows thep-wave + dipolar barrier at 12.67 kV/cm with the resonant dipolar coupling to |0, 0i|2, 0i removed.
Adapted from Ref. [137].

curves shown in Figure 6.4A, which show a two orders-of-magnitude change in β over a 1.4% change
in |E|. For |E| < |Eres | (green squares), the molecules were lost in tens of milliseconds, while for|E| > |Eres | (orange circles), we were able to detect molecules out to 20 seconds. Figure 6.4B shows
the adiabatic energy curves for mL = ±1 collisions of |1, 0i corresponding to the values of |E| usedin (A), calculated by our theory collaborator Goulven Quéméner. The black curve is the diabatic
contribution from the p-wave centrifugal barrier and the mL = ±1 dipolar repulsion (d = −0.08D), which ignores the effect of the resonant dipolar coupling. In the shielded case (orange line),
the molecules experience a 300 µK barrier in the effective potential, which strongly suppresses thetunneling to short range. Conversely, in the anti-shielded case (green), the molecules experience astrong attraction that completely overwhelms the diabatic barrier, leading to rapid loss.
From the physical picture described in the previous section, we would expect to see two
sharp features in β arising from the crossings of |0, 0i|2, ±1i and |0, 0i|2, 0i with |1, 0i|1, 0i (Figure
6.5A). These crossings occur at electric fields of |E1 | = 11.72 kV/cm and |E2 | = 12.51 kV/cm,
respectively. Measuring β as a function of |E| revealed the two sharp features shown in Figure

129

Figure 6.5: Measurement of the shielding resonances. (A) Energies of |1, 0i|1, 0i (gray), and|0, 0i|2, 0i and |0, 0i|2, ±1i (blue) as a function of |E|. The two state crossings at |E1 | and |E2 | areclearly visible. (B) Summary of β measurements near |E1 | and |E2 |. The data agree well with thetheoretical calculation (solid line) with no free parameters, using the experimentally measured T
and ωy . The field is oriented along ŷ for these measurements. Adapted from Ref. [137].

130
6.5B, showing the extreme tunability of the reactive loss by almost three orders of magnitude near
the crossings. In particular, we measured a factor of 8(3) suppression in β between the shielded(12.77 kV/cm) and background (10.13 kV/cm) loss rates, realizing a collisionally-shielded and longlived gas of molecules at a high electric field. The solid lines in (B) are theoretical calculations for
mL = ±1 collisions with no free parameters, using the experimental T and ωy . The calculations√
were performed in 3D and scaled to quasi-2D using β2D = β3D / πaho [102], where aho is the
harmonic oscillator length along ŷ.

6.3

Probing the collision anisotropy
The measurements described so far had E oriented along ŷ, so the molecules collided with

mL = ±1 as in the previous 2D work. Thanks to our highly controllable electric field (section 3.3),
we are also able to rotate the electric field in the x-y plane by an angle θ away from ŷ. Changing
θ, in conjunction with the 2D confinement, allows us to probe the anisotropy of the collisions. At
θ = 0◦ , E is perpendicular to the 2D pancakes (x-z plane) and the molecules only meet in sideto-side collisions (mL = ±1). However at θ = 90◦ , E lies within the pancakes and the molecules
have both head-to-tail (mL = 0) and side-to-side (mL = ±1) collisions, as shown schematically in
Figure 6.6A. Note that in this section, we define mL and mN as the projections of L and N onto
the quantization axis E, instead of onto the fixed axis ŷ. Since the dipoles are aligned along E, thismeans that the usual picture of a head-to-tail (side-to-side) approach corresponding to a mL = 0(±1) collision still holds as θ is varied.

6.3.1

Contribution of mL states vs θ

By rotating E, we can measure the loss rate β0 in the mL = 0 channel and compare it withthe loss rate β±1 in the mL = ±1 channels, which maps out the shielding as a function of therelative orientation of the molecules. A rigorous treatment of the collisions of tilted dipoles inquasi-2D can be found in Ref. [206]. In this work, we approximated the collisions by considering
only the lowest three angular momentum states mL = 0, ±1. Since the dipolar interactions are

131

Figure 6.6: Collisions in quasi-2D with tilted E. (A) Schematic diagram of the collisions. The
angular momentum states are defined with respect to E, which sets the quantization axis. At
θ = 0◦ , the molecules only experience mL = ±1 collisions. At θ = 90◦ , the molecules experience all
three of mL = 0, ±1. (B) Normalized contributions of the various mL states to the scattering as θ
is varied. The lines are the weighting factors from equations (6.17) and (6.18). (C) Measured loss
rate β of |1, 0i molecules at |E| = 7.09 kV/cm as a function of θ. Reproduced from Ref. [137].

relatively weak (except for in a narrow range of |E| near resonance2 ), the dipolar length aD is smallpcompared to the harmonic oscillator length aho = ~/µωy , and the collisions can be treated as
3D. For the largest induced dipole |d| ≈ 0.1 D studied here, we obtain aD = 180a0 compared toaho = 1800a0 . The dipolar interactions are weak enough that different L are not strongly mixed,
and hence we consider only the lowest manifold L = 1.
Rotating the field mixes the angular momentum states |L = 1, mL i together. The initiallyallowed states at θ = 0 are mL = ±1 owing to the 2D confinement. These states transform underθ 6= 0 according to the Wigner rotation matrices [108],
1 + cos θsin θ
1 − cos θ|1, 1i − √ |1, 0i +|1, −1i222
1 − cos θsin θ
1 + cos θ|L = 1, mL = −1i → |ψ−1 (θ)i =|1, 1i + √ |1, 0i +|1, −1i.
222|L = 1, mL = 1i → |ψ+1 (θ)i =(6.14)(6.15)

Roughly speaking, we want to figure out the probability PmL (θ) of scattering with mL as a function
2

We can estimate the effect of the dipolar coupling between the channels
directly on resonance by calculating√aD ∼ µd21 d10 /4π0 ~2 = 1040a0 , corresponding to a dipole moment d ∼ d21 d10 = 0.24 D.

132
of θ,
|h1, mL |ψ+1 (θ)i|2 |h1, mL |ψ−1 (θ)i|2+2
2
2
1 + cos θP−1 (θ) = P+1 (θ) =
4
2
sin θ
P0 (θ) =2

PmL (θ) =(6.16)(6.17)(6.18)The total loss rate β is the sum over the rates βmL in each mL , weighted by 2PmL (θ). The factorof 2 comes from the fact that the total loss rate is the sum over the individual loss rates in eachchannel, not the average. This yields the following expression for β,β(θ) = β0 sin2 θ + (β−1 + β+1 )1 + cos2 θ.
2(6.19)As a check, note that β(0◦ ) = β−1 + β+1 and β(90◦ ) = β0 + (1/2)(β−1 + β+1 ), which matches ourintuition from Figure 6.6A. The full dependence on θ is shown in Figure 6.6B, which plots P0 (θ)(red line) and P1 (θ) + P−1 (θ) (blue line). By combining measurements at θ = 0◦ and θ = 90◦ , we
can obtain β0 and β1 + β−1 separately.
Figure 6.6C shows the measured β for |1, 0i molecules at |E| = 7.09 kV/cm, as a functionof θ. At θ = 0◦ , β is minimized due to the repulsive mL = ±1 dipolar collisions (d = −0.12 at
this field). As θ is changed away from 0◦ , attractive mL = 0 collisions enter the scattering and β
increases. At θ = 90◦ , β is increased by an order of magnitude over θ = 0◦ .
Leaving the resonant shielding for a moment, let us briefly discuss the collisions of tilted
dipoles. The data in Figure 6.6C was taken at |E| = 7.09 kV/cm, where |d| is maximized for |1, 0i(in our range of accessible |E|), in order to maximize the change of β with θ. However, we should
see an even bigger effect of θ using the |0, 0i state, which has a much larger d. The combinationof a large d and tilted E is predicted to cause some interesting effects — for example, molecules
may be excited to higher bands in the lattice during a collision when their dipoles are tilted [206].
We have taken some preliminary data with |0, 0i that shows a rapid increase of the loss when the
dipoles are tilted by even a few degrees (Figure 6.7), though more systematic studies are needed to
say anything more.

133
10⁻⁵
N=0

β (cm²/s)

10⁻⁶0.05 D

0.14 D

10⁻⁷

0.19 D

0.28 D10⁻⁸045
θ (deg)90

Figure 6.7: Collisions of |0, 0i in tilted E. Points are experimental data, lines are fits to equation(6.19) with fitting parameters β0 and (β−1 + β+1 ). For larger d, a sharp rise in β is observed when
θ is changed away from 0◦ . The values of β at θ = 0◦ follow the trend in Figure 5.9B: as d isincreased, β is initially suppressed before increasing again at high d (owing to the quasi-2D natureof the system).

6.3.2

Shielding in mL = 0

As explained in the previous section we can extract β±1 ≡ β−1 + β+1 and β0 by combining
measurements at θ = 0◦ and 90◦ . We measured β±1 (A) and β0 (B) over a wide range of |E| tomap out the shielding resonances, shown in Figure 6.8. In the background region away from theresonances, we observed β0 > β±1 , consistent with the picture of dipolar attraction (mL = 0) or
repulsion (mL = ±1) from Ref. [52]. Near resonance, we observed sharp features at |E1 | and |E2 |in both angular momentum states. In particular, in the mL = 0 channel, we observed a variationof β0 by a factor of 1000(400) near the features, and a suppression of β0 by a factor of 23(10) at
the maximum shielding (11.84 kV/cm) compared to the background loss rate (at 11.32 kV/cm).
The experimental measurements agree well with theoretical calculations (solid lines) with no freeparameters.

134

Figure 6.8: Summary of 2D shielding. (A) β±1 as a function of |E|. (B) β0 as a function of |E|.
The solid lines are theoretical calculations with no free parameters, using the experimental T andωy . The gray shaded region in (A) is the same as Figure 6.5B. Reproduced from Ref. [137].

6.4

Shielding in 3D
Figure 6.8 shows that the shielding is indeed effective in all three of mL = 0, ±1, as expectedfrom theoretical predictions [104] and the arguments in section 6.1. Next, we explored the shieldingin 3D and elastic collisions at the shielding field (|Es | = 12.72 kV/cm) [138]. For KRb, we observedthat the elastic collisions are nearly unaffected at |Es |, displaying the same anisotropy expected
for pure dipolar scattering [136, 207]. We were also able to perform efficient evaporative cooling at|Es | to increase the PSD of the molecules. These results demonstrate tunable dipolar interactions
in a long-lived 3D gas of molecules.
For the 3D loss measurements, we followed a similar procedure to the 2D shielding measurements, except the molecules were created in the ODT without the LSL or VL. Typical starting
conditions for the measurements were N = 15 × 103 molecules at T = 330 nK. The ODT trap
frequencies were (ωx , ωy , ωz ) = 2π × (45, 250, 45) Hz for |1, 0i at |ESTIRAP | = 4.5 kV/cm, with a135

Figure 6.9: Resonant shielding in 3D. (A) Measured 3D loss rate coefficient β as a function of|E|. The data agree well with a theoretical calculation (solid line) at T = 330 nK with no freeparameters. (B) Molecule loss measurement at |E| = 12.50 kV/cm. (C) Molecule loss measurement
at |Es | = 12.72 kV/cm. Reproduced from Ref. [138].

weak dependence on |E| [113]. We fit the average 3D molecular density n as a function of holdtime to a differential equation that incorporates two-body loss and heating [79, 50],dn
3n dT
= −βn2 −
,
dt
2T dt

(6.20)

where β is the two-body loss rate coefficient.
Figure 6.9 shows the measured β as a function of |E|. The shielding resonances at |E1 | and|E2 | are again strikingly sharp. At |Es | (orange point in (A), and inset (C)), the molecules had along lifetime ∼ 10 s, realizing a long-lived 3D gas at high electric field.

6.4.1

Elastic collisions at the shielding field

The long shielded lifetime of the molecules provided an opportunity to study the elastic collisions of KRb in 3D for the first time. One question we had was whether the shielding would

136
substantially modify the dipolar elastic collisions. For fermionic dipoles at low temperature, neglecting the resonant shielding, σel is given by the universal prediction σel = (32π/15)a2D , where aDis the dipolar length [129, 136]. The resonant shielding modifies the effective intermolecular potential and hence also affects the elastic collisions. Numerical calculations from Ref. [104] predict that
σel is sharply enhanced very close to the resonances, where the loss is also maximized, but is only
slightly modified from the universal value at |Es |. At |Es |, KRb in the |1, 0i state has d = −0.08
D, corresponding to a universal prediction of σel = 2.8 × 10−12 cm2 .
In addition to changing the total cross section σel , the shielding could also conceivably alter
the anisotropy of the collisions. We measured both σel and the collision anisotropy using crossdimensional thermalization measurements. The trap intensity was modulated at a frequency of 2ωy
to parametrically heat the molecules along the ŷ direction. We then monitored the temperatures Tx
and Ty along x̂ and ŷ as a function of the hold time. The molecules thermalized with a rate Γth =Γel /Ncoll , where Γel = nσel vth is the elastic collision rate and vth the thermally-averaged relative
velocity. The parameter Ncoll is the number of collisions required to thermalize, which characterizes
how efficiently the collisions redistribute energy between the axes. For contact interactions, Ncoll =2.7 for s-wave collisions and 4.1 for p-wave collisions. Since the dipolar interactions are anisotropic,
Ncoll depends on the relative angle θ between the excitation direction ŷ and the dipole orientation(parallel to E) [136]. Anisotropic rethermalization of dipoles was previously studied in fermionic[208] and bosonic [209] gases of strongly magnetic atoms.
We performed cross-dimensional thermalization measurements at |Es | and several values ofθ to measure the dependence of Ncoll on θ. Experimentally, θ was varied by rotating E in thex-y plane. The cloud was parametrically heated for 50–100 ms to achieve an initial temperature
imbalance of Ty ≈ 2.5Tx , then allowed to rethermalize during a variable hold time. Panels (A)and (B) of Figure 6.10 show thermalization measurements at θ = 45◦ and θ = 90◦ , respectively,
demonstrating the large variation in Γth that can be observed by rotating θ.
To analyze these data quantitatively, the measured n, Tx , and Ty were fit to the following set

137

Figure 6.10: Thermalization measurements at |Es |. (A) and (B) show cross-dimensional thermalization measurements at θ = 45◦ and 90◦ , respectively. (C) Summary of Ncoll vs θ. Reproduced
from Ref. [138].

of coupled differential equations,
n dTx
dn
n dTy
= −KL (Ty + 2Tx )n2 −−dt
2Ty dt2Tx dtdTyn2Γth= KL (−Ty + 2Tx )Ty −(Ty − Tx ) + cy
dt
4
3
n
dTx
Γth
= KL Ty Tx +(Ty − Tx ) + cx ,dt
43

(6.21)(6.22)(6.23)

which capture the complicated interplay of loss, anti-evaporation, and rethermalization in thesystem [79]. Above, cx and cy are background heating rates and KL represents the two-bodyploss per spatial direction. (In equilibrium, β = 3KL T .) In Γth , vth = 8kB (2Tx + Ty )/3πµ isdefined using the average temperature. The equations assume that the temperatures along the two
unmodulated directions, Tx and Tz , remain equal throughout the thermalization process.
By fitting the thermalization rates at each θ, we could extract Ncoll as a function of θ (Figure
6.10C). From the cross-dimensional thermalization measurement, we cannot independently determine σel and Ncoll , so we assumed the universal value for σel . We observed a strong dependence of+0.2
Ncoll on θ, a direct consequence of the anisotropy of the collisions. Near θ = 45◦ , Ncoll = 1.6−0.1
waslower than the value of 2.7 for s-wave collisions, signalling very efficient thermalization. The experimental data agrees well with theoretical calculations for purely dipolar scattering from Reuben
Wang and John Bohn (solid line) [207], demonstrating that the dipolar character of the elastic
collisions is preserved at |Es |.

138

Figure 6.11: Evaporation of molecules in 3D. The main panel shows the trajectory of N and
T during the evaporation. We extract a slope Sevap = 1.84(9) (solid line), indicating efficientevaporation. For comparison, a slope of Sevap = 3 (dashed line) does not increase PSD. The
inset shows the increase of PSD during the evaporation as the total molecule number is reduced.
Reproduced from Ref. [138].

6.4.2

Direct evaporation of molecules in 3D

From the measured loss rates, we extract a ratio of good to bad collisions γ = σel /σre = 12(1),where σre = β/vth is the cross section for reactive collisions, so elastic collisions dominate in thesystem. While this ratio is not as favorable as the γ = 200(60) that we found for |0, 0i molecules
in 2D, it was still high enough for us to see dipolar evaporation of the molecules in 3D for the firsttime. Figure 6.11 shows the evaporation trajectory, obtained by lowering the ODT power at |Es |and θ = 0◦ . The solid line shows the fitted Sevap = ∂ log N/∂ log T = 1.84(9), compared to theupper threshold of Sevap = 3 (dashed line) to increase PSD in 3D. The inset shows that the PSD
increased by roughly a factor of 4, from 0.014(1) to 0.06(2), corresponding to a reduction in T /TF
from 2.3(1) to 1.4(2).
The best T /TF reached by evaporation in 3D was still higher than the conditions of N =25 × 103 and T /TF = 0.3 in our previous 3D work [50]. We attribute most of this discrepancy to the139
additional technical challenges involved with preparing the molecules in |1, 0i at |Es |. In particular,
losses during the ramp from ESTIRAP to Es limit the initial molecule number for evaporation. Aftersome upgrades to our STIRAP lasers, we now have the ability to create molecules directly at Es(see Ref. [107]), which should allow us to improve the initial evaporation condition and reach alower T /TF .

6.5

Conclusion
The resonant shielding is a powerful technique for controlling molecular losses by leveraging

the transition dipole moments of the molecules. It is highly general in terms of molecular species[104, 205, 204] and geometry, working in 2D and 3D. For species with larger dipole moments thanKRb, the achievable γ is expected to be as high as 106 [204], which should facilitate highly efficientevaporation. Our measurements of Ncoll showed that the elastic collisions at |Es | are very similar
to those of pure dipoles, suggesting that the resonant shielding could be an excellent setting forstudying 3D dipolar phenomena such as Fermi surface deformation [210] and sound propagation[211, 212].
In the previous chapter, we discussed the 2D loss suppression from repulsive dipolar collisions,which realized a non-resonant collisional shielding. Together with the resonant shielding, thesetechniques form a powerful toolbox for controlling molecular losses while retaining strong elastic
collisions. In addition, the Harvard CaF [105] and Munich NaK [54] groups recently demonstrated
microwave shielding [161, 213], in which the N = 0 and N = 1 states are coupled using a bluedetuned microwave drive. The microwave photons supply the necessary energy to make the dipolarinteractions between these states near-resonant, similar to the role played by |E| in the resonant
shielding, resulting in shielding from loss. An advantage of microwave shielding, as opposed toresonant shielding, is that the additional degree of freedom (Rabi frequency and detuning, as
opposed to only the detuning for resonant shielding) allows one to also enhance the elastic collision
rate. Ref. [54] was able to leverage this to perform efficient evaporative cooling to a degenerate
Fermi gas in 3D. One advantage of the resonant shielding, however, is the ability to easily tune the140dipole orientation by tilting E. In contrast, for microwave shielding the quantization axis is set by
the microwave polarization, which may be more difficult to rotate.
There have also been exciting recent advances in controlling atom-molecule collisions. For
the NaLi + Na system, γ > 50 was achieved, enabling sympathetic cooling of NaLi by Na [214].
Additionally, atom-molecule Feshbach resonances have been observed in NaK + K [96] and NaLi
+ Na [98], even allowing the association of loosely-bound triatomic molecules [97]. For KRb, Ref.
[215] observed an anomalously long lifetime of KRb + Rb complexes, which may suggest that it ispossible to find Feshbach resonances in this system as well. However, the complexes were observed
to be destroyed by the 1064 nm trapping light, so such explorations might need to be done in a
box trap [94].

Chapter 7

Single-layer control of molecules in a 1D optical lattice

In the previous two chapters, we discussed two different methods for protecting the moleculesfrom unwanted reactive losses. With the losses under control, we renewed our focus on studying
2D dipolar physics. As previously mentioned, dipolar interactions in 2D provide a very interesting
setting for studying many-body physics. The interplay of the long-range interaction with reduced
dimensionality leads to new phenomena in 2D, such as condensation of rotational excitations [185,216, 217]. In addition, the combination of the dipolar anisotropy and strong confinement makes
the many-body behavior of the system highly tunable with the dipole orientation [45, 218]. Theseelements combine to lead to intriguing phenomena in 2D such as topological superfluidity [42, 43],
spin liquids [40], and spin-squeezing [49].
For many of these proposals, a single 2D layer of molecules is required. Since the dipolarinteractions are long ranged, molecules in adjacent 2D layers can interact, potentially competingwith the intralayer interactions. We can also flip this statement around — by controlling theinternal state of the molecules in a layer-resolved manner, we can gain control of the interlayerinteractions and thus enable a new level of control over the molecules.
In this chapter, we show how to prepare molecules in a single 2D layer by performing microwave spectroscopy in an electric field gradient. The gradient causes a layer-to-layer shift of the|0, 0i to |1, 0i transition, which allows us to individually address each layer. A similar technique isused in ultracold atom experiments to select 2D planes, for example in quantum gas microscopes[219, 220], but with a magnetic field rather than electric field. The capability of layer-resolved

142internal state control allows us to observe interlayer dipolar exchange interactions by putting adjacent layers in different (dipole-coupled) rotational states. The interlayer interactions manifest as
an enhancement of the reactive loss rate, allowing layer-to-layer control of the chemical reactions
mediated by the dipolar interactions.
This chapter is based on the work in Ref. [152]. We will focus mainly on the high-level results
here, but getting the layer selection to work well required solving a number of technical challenges
on the experiment. Will Tobias led much of these efforts and his thesis covers those aspects indetail [107].

7.1

Addressing a single layer
For the experiments in this chapter, we created molecules in 2D in the VL. The procedure wassimilar to Chapter 5, but without using the LSL. The LSL was removed to purposefully load manylayers in VL, in order to make the center of the layer distribution very homogeneous. This wasadvantageous for our studies of three isolated layers, where we wanted to make the molecule number
in each as similar as possible. Typical starting conditions for the experiments were N = 20 × 103
molecules at T = 350 nK in about 15–20 layers of the VL. Owing to the large number of populated
layers, T /TF ≈ 3.5 for these studies. Molecules were produced using STIRAP at |Ebias | = 1 kV/cm.
The combined trap frequencies of the ODT + VL were (ωx , ωy , ωz ) = 2π × (42, 17 × 103 , 48) Hz for|0, 0i molecules at |Ebias |.
The molecules were imaged in situ in the VL by dissociating the ground state moleculeswith STIRAP and imaging the Feshbach molecules. The VL layers are too closely spaced to be
optically resolvable, so our measurements gave the total number of molecules in each internal state,summed over the populated layers. Since STIRAP is state-selective, we could measure the numberof molecules in each rotational state |N i ≡ |N, mN = 0i, allowing us to normalize by the total
molecule number when desired. However, we found that the |0i imaging pulse caused a significant
loss of |1i molecules by excitation to a different intermediate state. Since we did not observe thesame loss in |2i, we solved this problem by shelving |1i in the |2i state during the |0i imaging pulse.

143

Figure 7.1: Layer-resolved spectroscopy. (A) Schematic of the molecules in several layers of the
VL. A strong vertical field gradient ∇|E| parallel to ŷ is used to create a site-to-site shift of the|0i ↔ |1i transition frequency. (B) Energy shifts of rotational states with |E| near |Ebias | = 1
kV/cm. (C) Spectroscopy of a trilayer of molecules. A gradient of ∂|E|/∂y = 6.4(2) kV/cm2 wasused, giving a layer-to-layer shift of ∆ = h × 14 kHz. Reproduced from Ref. [152].

The basic principle of the layer selection is shown in Figure 7.1A. The molecules are held in the
VL with a layer spacing of a = 540 nm along ŷ. We apply a bias electric field of strength |Ebias | andvariable tilt θ to cancel differential AC Stark shifts [113, 110]. In contrast to the previous chapter,the tilt angle θ is defined between E and the VL polarization (along x̂). By applying a gradient∇|E| = ŷ(∂|E|/∂y), we induce a site-to-site Stark shift of the rotational transition from |0i to |1i.
The DC Stark shifts of |0i, |1i, and |2i near 1 kV/cm are shown in (B), demonstrating the shiftof the |0i ↔ |1i transition with |E|. At |Ebias |, the differential dipole moment is (d0 − d1 )/h = 40
kHz/(V/cm).
The vertical gradient ∂|E|/∂y leads to a site-to-site transition shift∆=d1 − d0
h



∂|E|
a,
∂y

(7.1)

144

Fraction |1⟩

0.08

0.06
0.04
0.02

0.00
650

700

750
800
850
900
Frequency - 2248 MHz (kHz)

950

1000

Figure 7.2: Spectroscopy on many layers. A narrow microwave π-pulse is scanned on a sample with∼ 20 layers, at ∂|E|/∂y = 6.4 kV/cm2 . By fitting the spectrum to a sum of Gaussian lineshapeswith an overall Gaussian envelope, we extract the layer number τ = N/hNi i = 19(1). This data
was taken in slightly different conditions from Ref. [152] (which had τ ≈ 12).

where dN is the induced dipole moment of |N i at |Ebias | and (∂|E|/∂y)a is the change in the electricfield between two neighboring sites. We typically used a gradient of ∂|E|/∂y = 6.4 kV/cm2 , leading
to a site-to-site shift of ∆ = 14 kHz. Figure 7.1C shows a microwave spectrum of three isolated
layers (a “trilayer”) with a field gradient of ∂|E|/∂y = 6.4(2) kV/cm2 . Using a narrow microwave
pulse, we cleanly resolved the three layers spectroscopically, showing that we could individually
address and manipulate 2D layers.
This technique, combined with state-selective molecule removal, was used to initially prepare
the trilayer system probed in Figure 7.1 from the initial distribution with many populated layers.
Figure 7.2 shows a layer spectroscopy scan on an initial sample with 19 layers. The molecules were
initially in the |0i state and were transferred to |1i with a narrow microwave π-pulse. The quantity
on the y-axis is the number of |1i molecules divided by the total number of molecules. For Figure
7.2, we probed only every second occupied layer to reduce the data acquisition time. The data is
fit (solid line) to a sum of Gaussian lineshapes representing the signal from each VL layer, with an
overall Gaussian envelope to capture the layer distribution.
To prepare a trilayer as in Figure 7.1, we started with the molecules initialized in |0i and145
many layers populated (as in Figure 7.2). We transferred only the three central layers to |1i, andused resonant light from one of the STIRAP beams (690 nm) to blast away the untransferredmolecules in |0i. During the blast, the |1i layers were shelved in |2i to avoid excitation by the blast
light. By further combining layer selective pulses on |0i ↔ |1i, global π-pulses on |0i ↔ |1i and|1i ↔ |2i, and global optical blasts, we could prepare an arbitrary pattern of layers in |0i, |1i and|2i.
In this section, we glossed over many of the technical details required to make the layer
selection work. The phase of the VL must be well-stabilized to the electrode assembly to keep
the y-position (and thus transition frequency) of the layers stable from shot-to-shot. In addition,
the axis of the VL must be well-aligned to the direction of the electric field gradient. Owing to
the large radial extent of the cloud, we calculated that even small tilts of the VL of < 1◦ degree
would completely wash out the layer structure. And of course, |E| must be stable enough toallow the necessary spectroscopic resolution to see individual layers. These technical aspects of themeasurement are discussed in depth in Will Tobias’s thesis [107]. One other important technicaldetail, the use of Blackman shaped pulses to minimize RF sidebands on the layer selection pulses,
is discussed in section 3.4.1.

7.2

Magic angle electric field
An important prerequisite for the layer selection was reducing the inhomogeneous broadening

from the different lattice potentials felt by |0i and |1i. As discussed qualitatively in section 2.3.4,|0i and |1i experience a differential AC Stark shift from the trapping light that depends on theangle θ between the E and the light polarization. The differential shift leads to an inhomogeneous
broadening of the lineshape on each layer, making the layers more difficult to spectroscopicallyresolve. In our experiments, the VL polarization is fixed along x̂, but we can rotate E to change
θ. The polarizabilities of |0i and |1i are expected to be matched at the “magic angle” θm = 54.7◦[113, 110], corresponding to the condition cos2 θm = 1/3.
In this chapter, we used the experimentally determined value of θm = 57◦ , found by measuring

146

Figure 7.3: Rotational coherence of |0i and |1i. (A) Comparison of Ramsey fringes for θ = 90◦(top panel) and θ = θm (bottom panel). (B) Ramsey contrast decay at θ = θm . The decay is fit to22
a Gaussian envelope e−t /τ , which gives a coherence time of τ = 1450(80) µs. For comparison, thedashed line shows the fitted coherence decay for the non-magic (θ = 90◦ ) configuration from (A).
Reproduced from Ref. [152].

the shift of the |0i ↔ |1i transition as a function of the VL power at several values of θ. We also
rotated the polarizations of the ODT beams such that their polarizations formed an angle of θmwith E. This configuration was used in the layer selection measurements (Figures 7.1 and 7.2) tomaximize the layer contrast.
To characterize the differential AC Stark shift, we prepared a single layer of molecules andstudied the coherence with Ramsey spectroscopy. Ramsey fringes for short evolution times areshown in Figure 7.3A, showing a clear difference between θ = 90◦ (top panel) and θ = θm (bottom
22

panel). We fit both fringes to a sinusoid with a Gaussian envelope e−t /τ , where t is the evolution
time. At θ = 90◦ , the differential polarizability is large, leading to a short coherence time τ =310(30) µs. In contrast, τ is significantly increased at θm . Figure 7.3B shows the envelope of theRamsey contrast at θm as a function of evolution time t. For t > 600 µs, the Ramsey oscillation
phase became scrambled by slight shot-to-shot changes in |E| at the few 10s of ppm level. Thus,
to extract the coherence at long times, we randomized the phase of the second Ramsey pulse andused the variance of the measured |0i and |1i populations to extract the contrast. Fitting the

147

Figure 7.4: Cartoon of dipolar exchange induced losses. (A) Two adjacent layers are populatedwith molecules in the same state state |0i. Each layer experiences slow p-wave losses at rate βp .
(B) Two adjacent layers are populated with |1i and |0i, respectively. Interlayer dipolar exchangewith rate γ causes a small population admixture in each layer, which is quickly lost via s-wave
interactions at rate βs . The intralayer p-wave loss βp is not shown, but is also present in (B).

contrast decay, we extract τ = 1450(80) µs at θm , a factor of 5 improvement over θ = 90◦ (dashedline). After the experiments of Ref. [152], we recalibrated θm and found a slightly different value ofθm = 54◦ , closer to the theoretical value of 54.7◦ . We attribute the discrepancy to small backgroundgradients of E that were present when initially calibrating θm . With the updated value of θm , wewere able to achieve Ramsey coherence times of about 4 ms, described in the next chapter.

7.3

Interlayer dipolar spin-exchange interactions
As introduced in section 2.6, a molecule in |0i and another in |1i can undergo dipolar spin-

exchange and trade rotational angular momentum states [45, 46]. This interaction can be quite√
strong, since it arises from the transition dipole moment d01 between |0i and |1i (d01 = dperm / 3
at |E| = 0), and it is a resonant process since the pair states |0i|1i and |1i|0i have the same energy.
Our ability to prepare each 2D layer in a different rotational state allows us to explore and control
these interactions in a layer-resolved manner. For example, we can imagine preparing a systemwith several adjacent layers populated in |0i, except for the center layer which is initialized in|1i. Ignoring losses for the moment, we would expect that the initial rotational excitation on thecentral layer diffuses to the adjacent layers through dipolar exchange. Perhaps the rate of diffusion
could be changed with disorder (e.g., from an optical speckle pattern [221]) or a potential gradient[222, 223], allowing the study of localization in the system [224, 225].

148
In our bulk 2D layers, with no transverse lattices to suppress collisions, the most immediate
effect of the spin-exchange interactions is an enhanced loss rate. This is depicted schematicallyin Figure 7.4. Consider two adjacent layers in the state |0i, as in (A). Since all molecules are
identical, the collisions within the layer are p-wave and the losses are relatively slow with rate βp .
However, if an adjacent layer is populated with molecules in |1i, as in (B), then the layers arecoupled via dipolar exchange interactions. This leads to a small admixture of |1i impurities in the
predominantly |0i layer (and vice versa). Since impurities are distinguishable from the majority
population, they are quickly lost in s-wave collisions with rate βs . The spin-exchange rate γ isexpected to be much faster than βp but much slower than βs , so the overall loss rate is limited by
γ. This physical system was treated theoretically in Ref. [226] and in the Supplementary Materialof Ref. [152].
To probe this effect, we created a trilayer of molecules and studied the loss by measuring thenumber N of molecules remaining in the central layer after a variable hold time (Figure 7.5A), with∂|E|/∂y = 0. We fit the data to the solution of dN/dt = −βN 2 to extract the two-body loss rate
coefficient β. Note that β has different units here than in the rest of this thesis. For convenience, wewill refer to the trilayer configurations according to N1 N2 N3 , where Ni is the rotational quantum
number of the ith layer. For example, 101 refers to a central layer of |0i between two layers of |1i.
We started by measuring βp and βs as a baseline. To measure βp , we prepared 000 and 111
trilayers, and found βp = 2.99(17) × 10−3 s−1 for |0i and |1i (green squares). We found a slightly
different value βp = 1.78(24) × 10−3 s−1 for |2i, which we attributed to a slightly smaller van der
Waals coefficient for |2i, though more measurements are needed to understand this. To measure
βs , we prepared a 000 trilayer and then applied a π/2 pulse to create a 50/50 mixture of |0i and|1i. Fitting the decay of |0i, we measured βs = 2.0(3) × 10−1 s−1 (purple triangles in inset), about
two orders of magnitude higher than βp [32].
For a 101 trilayer (blue circles), we observed an order of magnitude faster loss than in 111(green squares). This demonstrates the strong enhancement of loss from the interlayer dipolarinteractions. Using our rotational state control, we could also probe configurations where spin-

149

Figure 7.5: Losses induced by interlayer dipolar exchange. (A) Comparison of losses with andwithout dipolar exchange. In a 000 (green) or 202 trilayer (orange), dipolar exchange is forbidden
and the molecule number decays slowly at βp . The loss rate for a 101 trilayer (blue) shows a marked
enhancement from interlayer dipolar interactions. Inset: loss measurement for a 50/50 mixture of|0i and |1i, used to obtain βs . (B) Density dependence of the dipolar exchange. The loss rate for alayer of |0i molecules shows a linear increase with the number of adjacent |1i layers. Reproduced
from Ref. [152].

exchange is forbidden due to selection rules. For example, neglecting the small state mixing at|Ebias |, the electric dipole transition between |0i and |2i is forbidden since N changes by two.
Accordingly, a 202 trilayer (orange diamonds) experiences no interlayer dipolar exchange, and thus
showed no enhancement of the loss rate above βp . By changing the layer configuration, we could also
vary the effective density of interacting molecules. Figure 7.5B shows that the exchange-induced
loss rate between |0i and |1i was approximately doubled when doubling the number of adjacentlayers, since the molecules in |0i have twice as many partners for exchange.
As mentioned before, dipolar spin-exchange is a resonant process. Thus, we were interested
to see whether it could be suppressed by reintroducing the site-to-site transition shift ∆ used forthe layer selection. Note that ∆ is precisely the detuning of the interlayer exchange from resonance.
To study this quantitatively, we fit the measured loss curves to a rate equation that models theeffects of βp , βs , and γ on the rotational populations in each layer, where γ is the average rate ofspin-exchange in the gas [152]. This allowed us to extract γ as a function of ∆.

150

Figure 7.6: Dependence of the interlayer dipolar exchange on the site-to-site shift ∆. (A) Spinexchange rate γ versus ∆ at θ = 90◦ and T = 334(30) nK. The 101 trilayer (blue circles) shows abroad feature in γ(∆), while the 202 trilayer (orange diamonds) are consistent with γ = 0 acrossthe range of measured ∆. (B) FWHM of γ(∆) for a 101 trilayer as T is varied, at θ = θm . The
insets show measurements of γ(∆) at T = 378(30) nK (circles, left inset) and 643(40) nK (squares,
right inset). Reproduced from Ref. [152].

An example measurement of γ(∆) is shown in Figure 7.6A. From a maximum value at ∆ = 0
of γ = 7.0(6) × 10−3 s−1 , γ was reduced with increasing |∆| for 101 (blue circles). Measurementsof the spin-exchange forbidden 202 system (orange diamonds) were consistent with γ = 0 overthe full range of measured ∆. Perhaps surprisingly, γ(∆) was quite broad for the 101 trilayer,with a full width at half maximum (FWHM) of 6.4(6) kHz, which is much larger than the ∼ 50
Hz interactions between neighboring layers [46]. The measured FWHM is similar to the moleculetemperature kB T /h ≈ 7 kHz, and follows an approximately linear trend with T (Figure 7.6B). Thissuggests that two molecules undergoing dipolar exchange can also change their motional states

151
in the harmonic trap, thereby changing their kinetic energy to compensate for ∆. Two-particle
calculations by Thomas Bilitewski and Ana Maria Rey support this physical picture [152]. Theinterlayer dipolar interactions in this system lead to a complicated interplay of dipolar exchange,
reactive losses, and external motion.

7.4

Conclusion
In this chapter, we showed how to prepare a single 2D layer of molecules from an initial dis-tribution of many layers. We utilized our electric field control to reduce inhomogeneous broadening
by minimizing the differential AC Stark shift with θ. The ability to control the rotational state of
the molecules with single layer resolution allowed us to study the interlayer dipolar interactions in
a controllable way. Changing the layer-to-layer detuning ∆ revealed a complex interplay of dipolarinteractions and harmonic mode-changing collisions.
Looking forward, the preparation of single 2D molecular systems opens a path to studying
a wide range of 2D dipolar physics, from collective oscillations [197] to spin squeezing [49]. WithE oriented perpendicular to the layer, all pairs of molecules in the system have the same sign ofdipolar interaction, which can lead to highly collective behavior. If the s-wave losses are suppressedby a shallow 3D optical lattice, the many-layer system could also be used to study transport andlocalization between planes [224]. Recently, theory [222] and experiments on trapped ions [223]have shown that a form of many-body localization can arise from a linear potential gradient (likeour layer selection gradient) in the absence of disorder, and perhaps in the future our system would
be well-suited for studying similar localization physics. Our theory collaborators Thomas Bilitewski
and Ana Maria Rey have also suggested that the layered system naturally realizes a multi-mode
squeezing Hamiltonian, similar to two-mode squeezing observed in spinor BECs [227, 228], which
could be an interesting future direction to explore.

Chapter 8

Dipolar density shifts in 2D

In the previous chapter, we used microwave spectroscopy to address single 2D layers in our
optical lattice. By putting neighboring layers in different rotational states, we were able to look at
the effect of interlayer dipolar interactions on the reactive loss.
In this chapter, we instead explore the intralayer interactions between molecules in a coherent
superposition of rotational states. By mapping rotational states | ↓i = |N = 0, mN = 0i and| ↑i = |1, 0i or | ↑0 i = |1, −1i onto a spin–1/2 degree of freedom, we realize a 2D quantum spin
system with long-range and anisotropic interactions. In contrast to earlier work where molecules
were pinned to sites of a 3D optical lattice [46], here the molecules are free to move in the 2D layers,
resulting in an itinerant system where coherent collective spin dynamics compete with collisions. At
lowest order, the dipolar interactions cause a mean-field shift of the transition frequency between| ↓i and | ↑i [229], which we detect using Ramsey spectroscopy. At lower T /TF , the dipolarinteractions in 2D are predicted to generate spin-squeezed states [49].
Long range, anisotropic, and highly tunable dipolar interactions make ultracold molecules
an attractive platform for explorations of quantum spin systems [24, 26, 27], which are central to
many questions at the intersection of condensed matter physics, materials science, and quantum
information [230, 231, 232]. Here, we leveraged our control of E to demonstrate aspects of thistunability. While Ref. [46] operated at E = 0, we studied the dynamics over a large range of |E|from 0 to 9 kV/cm, allowing us to vary the relative strength between the Ising and spin-exchangeinteractions and explore the full spin-1/2 Hamiltonian [140]. Additionally, we varied the angle α of153
E from ŷ to rotate the orientation of the dipoles in the 2D plane.1 This changed the interactions
from purely repulsive at α = 0◦ to (on average) attractive at α = 90◦ . By choosing | ↑i = |1, 0i
or | ↑0 i = |1, −1i, we could also change the sign and magnitude of the dipolar interactions. Taking
advantage of this fact, we dynamically reversed the sign of the couplings in the underlying spin
Hamiltonian by switching from | ↑0 i to | ↑i in the middle of a measurement, leading to a reversal of
the Ramsey phase accumulation. This is a key capability needed for performing a time-reversal or“untwisting” protocol on a spin squeezed state [49], which allows one to enjoy a metrological gainfrom spin squeezing without requiring detection at the single particle level [233].
This chapter is based on a manuscript which is still in preparation [234], and some of thiswork is still preliminary. I will mainly focus on giving a high-level summary of what we have seen
so far. The experiments were done in 2D using a similar procedure to Chapter 7. Typical starting
conditions were N = 20 × 103 at T = 400 nK in about 20 layers of the VL. The combined trap
frequencies in the ODT + VL were (ωx , ωy , ωz ) = 2π ×(45, 17×103 , 45) Hz for |0, 0i at |E| = 0. The
molecules were in the quasi-2D regime (kB T /~ωy = 0.5) and predominantly occupied the lowest
harmonic level in ŷ. Molecules were produced directly at the target E for each measurement using
our tunable STIRAP setup [107].

8.1

Spin Hamiltonian
Let us start by introducing the collective spin Hamiltonian used to analyze the system. Since

the molecules are free to move in the 2D planes, the single particle states are described by a spin parttimes a motional part. The motional eigenfunctions are given by the harmonic oscillator modes inthe trap. This is illustrated schematically in Figure 8.1. The interlayer interactions described in theprevious chapter are weak on the timescales used for these measurements, so we can approximate
neighboring 2D layers as being uncoupled.
In the limit of few elastic collisions, the motional states are effectively frozen and do not
1

Note that the field angle α is denoted θ in the previous chapters. We change notation here to avoid symbol clash
with the Bloch sphere polar angle θ.

154

Figure 8.1: Cartoon of the spin Hamiltonian. In a 2D layer, each single particle eigenstate can be
decomposed into the product of a spin-1/2 degree of freedom and a motional state in the harmonic
trap. The dynamics are controlled by |E| and the angle α between E and ŷ. Reproduced from Ref.
[234].

Figure 8.2: Dependence of χ on |E|. (A) Dipolar interactions between | ↓i and | ↑i. The moleculesinteract via their induced dipole moments (d↑ and d↓ ), as well as the transition dipole moment(d↓↑ = h↓ |d| ↑i = h↑ |d| ↓i). (B) Strengths of these dipolar interaction terms plotted as a functionof |E| for | ↓i = |0, 0i and | ↑i = |1, 0i of KRb. The dipolar coupling strength is given in units of
dp , the permanent dipole moment (0.574 D for KRb). The black line shows the dependence of χ,which controls the magnitude of the density shift, on |E|. Adapted from Ref. [234].

155
participate in the dynamics [49, 235, 236, 237]. In this “frozen mode” approximation, the dynamics
take place only in the spin degrees of freedom,
H=


i X
1 Xh z z z
Jij si sj + Jij⊥ sxi sxj + syi syj +szi hzi ,
2
ij

(8.1)

i

{x,y,z}where the indices i and j label the occupied harmonic oscillator modes in the trap. Above, si

isa spin-1/2 operator acting on a particle in harmonic mode i. This theory has successfully modeled
the spin dynamics of p-wave interacting fermions in optical lattice clocks [235, 236], and describeddynamical phase transitions in a 40 K Fermi gas [237]. For our system, the couplings Jijz , Jij⊥ , andhzi depend on the induced dipole moments d↓ and d↑ and the transition dipole moment d↓↑ (shownin Figure 8.2A), as well as the matrix element of the spatial part of the dipolar interaction withrespect to oscillator modes i and j [49].
Assuming that these matrix elements do not depend too strongly on i and j, the spin model
can be approximated by a collective spin Hamiltonian,
H = J⊥ S 2 + χSz2 + hz Sz ,where Sα =

(8.2)

α
i si is the collective spin operator obtained by summing the individual spin–1/2P

operators. The couplings J⊥ , χ = Jz − J⊥ , and hz are obtained by averaging the mode-dependent
couplings in equation (8.1) over the occupied harmonic modes. In contrast to the 3D latticesystem, the couplings Jijz and Jij⊥ are relatively homogeneous for the bulk system and thus thecollective approximation is expected to be valid [49]. The first term J⊥ S 2 protects the system fromdecoherence, since processes that would decohere a molecule necessarily change the value of S, andare thus suppressed by the many-body gap J⊥ [236]. The second term gives one-axis twisting withcoupling strength χ, which generates spin squeezing [4, 6, 238]. The third term gives an effective
magnetic field hz from the background field generated by the spins.
At the mean-field level, χSz2 ≈ 2χhSz iSz , which produces a spin-dependent external field thatshifts the transition frequency between the two spin states by an amount proportional to the spin

156
imbalance hSz i = (N↑ − N↓ )/2. For Ramsey spectroscopy, this manifests as a phase shift∆φ = N χT cos θ,(8.3)

where T is the Ramsey evolution time and cos θ = (N↑ − N↓ )/N = 2hSz i/N is the normalized
population imbalance (N = N↓ + N↑ is the total molecule number). Thus, χ is responsible for the
dipolar density shift in our system. Physically, we expect that ∆φ scales not only with the total
number N but with the average density n of molecules. This dependence is actually already present
in equation (8.3) since χ implicitly depends on the transverse trapping frequencies and temperature,
which determine the populated modes in equation (8.1). To make the density dependence more
explicit, we redefine χ such that∆φ= nχT cos θ,
2π

(8.4)

where now χ has units of Hz per 2D density.
The value of χ depends on the dipole moments according toχ ∝ −(d↓ − d↑ )2 + ηd2↓↑ ,(8.5)

and, as a result, χ is tunable by |E| (Figure 8.2B). The factor η depends on the rotational angularmomentum projections of | ↓i and | ↑i: η = 2 for the {|0, 0i, |1, 0i} system, while η = −1 for{|0, 0i, |1, ±1i}.2 At small |E|, the largest contribution to χ is from the transition dipole d↓↑ , whileat large |E|, the contribution of the induced dipoles d↓ and d↑ are stronger. Since χ depends on the
averaged relative orientation of interacting molecule pairs, it is also tunable by the angle α between
E and ŷ.
(2)2This can be seen from equation (2.25). The molecules are interacting through T0 (d1 , d2 ), which generates∓
spin-exchange between |0, 0i and |1, 0i (|0, 0i and |1, ±1i) through the 2d01 d02 (d±1 d2 ) term. This explains the factorof 2; the negative sign can also be obtained by calculating the matrix elements. Classically, two molecules interacting
in a superposition of |0, 0i and |1, 0i look like two dipoles oscillating along the quantization axis. Two molecules in asuperposition of |0, 0i and |1, ±1i look like dipoles rotating in the plane perpendicular to the quantization axis.
R 2π Owing
to the dipolar anisotropy, the time-averaged interaction in the second case picks up a factor of (1/2π) 0 dθ (1 −3 cos2 θ) = −1/2.

157
8.1.1

More discussion of χ

Before moving on to discuss the experimental measurements, we try to develop some intuition
for the form of χ. For concreteness, we assume | ↓i = |0, 0i and | ↑i = |1, 0i in this section. Let ↓and ↑ be the interaction energy per particle of | ↓i and | ↑i, respectively. First, consider a moleculein | ↓i. It feels an energy shift proportional to d2↓ N↓ (d↓ d↑ N↑ ) from interactions with | ↓i (| ↑i)through the induced dipole moment. It also feels an interaction from the transition dipole moment,d2↓↑ N↑ , giving a total interaction energy of↓ ∝ d2↓ N↓ + d↓ d↑ N↑ + d2↓↑ N↑ .

(8.6)

These terms are depicted schematically in Figure 8.2A. Classically, the proportionality factor isgiven by the integral of the dipolar interaction V (r) times the molecular density distribution. A
similar argument for | ↑i gives↑ ∝ d2↑ N↑ + d↓ d↑ N↓ + d2↓↑ N↑ .

(8.7)

In Ramsey spectroscopy, we are only sensitive to the energy difference ↑ − ↓ ,↑ − ↓ ∝ d2↑ N↑ − d2↓ N↓ + d↓ d↑ + d2↓↑ (N↓ − N↑ ).

(8.8)

We make the substitutions
N↓ + N↑ N↓ − N↑
+
,
2
2


N↓ + N↑ N↓ − N↑
N↑ =
−
,
2
2
N↓ =

(8.9)
(8.10)

to obtainN↓ + N↑N↓ − N↑22− (d↓ + d↑ )+ d↓ d↑ + d2↓↑ (N↓ − N↑ )
2
2
 



N↓ − N↑ 
N↓ + N↑= (d2↑ − d2↓ )+−(d↓ − d↑ )2 + 2d2↓↑ .
22

↑ − ↓ ∝ (d2↑ − d2↓ )(8.11)(8.12)

The first term depends on the total number of molecules, so it is a constant shift that ends up being
canceled by our decoupling pulses. However, the second term is proportional to the imbalance hSz i158
and the dipolar dependence of χ from equation (8.5), directly giving us the mean field shift χhSz i.
This calculation can be generalized to the {|0, 0i, |1, ±1i} system by replacing the spin-exchange
terms d2↓↑ N↑ → (η/2)d2↓↑ N↑ in ↓ (and similarly for the corresponding term in ↑ ).
Some intuition can also be gained from basic two-particle Ramsey calculation in section 2.6.1.
Though that calculation assumed that the molecules were pinned in space, the form of χ was thesame as in equation (8.5). Figure 2.14B shows that there is a shift in the Ramsey fringes over time
for N↓ 6= N↑ , which is the two-particle analogue of the density shift studied in this chapter.

8.1.2

Validity of the spin model

Elastic collisions between molecules couple the motional states in the trap. Therefore, thereduction to the pure spin model in equation (8.1) depends crucially on the elastic collision rate
being low. However, molecules placed in a rotational superposition have strong dipolar interactions
that could potentially lead to a high elastic collision rate. Consider two molecules in a symmetricsuperposition,
|+i =

1
1
(| ↓i + | ↑i) ⊗ (| ↓i + | ↑i) = (| ↓↓i + | ↓↑i + | ↑↓i + | ↑↑i) ,2
2(8.13)

which is the state of a pair of molecules just after the initial Ramsey pulse (section 2.6.1). At|E| = 0, this gives a dipolar interaction strength ofh+|2d01 d02 |+i =1h↓↑ |d01 d02 | ↑↓i + h↑↓ |d01 d02 | ↓↑i = d2↓↑ .
2(8.14)

Comparing the value above to the same expression calculated for two induced dipoles d (which√
gives 2d2 ), the effective dipole moment for interactions of the equal superposition is d↓↑ / 2 [239].
√√
For |E| = 0, | ↓i = |0, 0i, | ↑i = |1, 0i, this gives a large dipole moment of d↓↑ / 2 = dperm / 6,
corresponding to a dipolar length of aD = 990a0 for KRb at |E| = 0. Of course, this analysis only
holds while the molecules are fully coherent; otherwise, the assumption that the molecules collide
in |+i is no longer valid.
Experimental measurements are ongoing in the lab to try to understand the role of elastic
dipolar collisions between molecules in the spin dynamics. Physically, we would expect that the spin

159
model is still valid at short times before the molecules have a chance to collide, even if it breaks
down at longer times. By scaling the elastic collision rate measured in Ref. [52] for the dipole√
moment (dperm / 6) and temperature used here (450 nK), we estimate an elastic collision rateΓ ∼ 100 s−1 at the highest molecular densities studied here. For the density shift measurements,
we use an interrogation time on the order of 1 ms, much shorter than the estimated collision time,
and thus we expect that the spin model is still a good description of the system.

8.2

Dynamical decoupling
To measure the density shift, we require sufficiently long coherence time to measure the phaseshift ∆φ. For our parameters, we estimate a frequency shift on the order of ∆ν = nχ ≈ 100 Hz.
Thus if we want to see a ∆φ of a few degrees, we require a coherence time on the order of about 1
ms. This is longer than the coherence time of a few hundred µs observed when the trap is non-magic[110, 152].
In the previous chapter, we showed that the coherence between | ↓i = |0, 0i and | ↑i = |1, 0i
could be increased to above 1 ms by bringing E to the experimentally-measured magic angleθm = 57◦ . After that work, we more carefully characterized the differential AC Stark shifts fromour lattice and found an updated value of θm = 54◦ , closer to the theoretically predicted 54.7◦ [113].
Figure 8.3 shows the decay of the Ramsey contrast as a function of time at |E| = 1 kV/cm and θm(gray circles), from which we extract a 1/e coherence time of τc = 4.0(4) ms. This measurement
was taken with no spin echo pulses, so τc could be limited by gradients of E or a small residual
differential AC polarizability.
Since the polarization of the VL is constrained to lie in the x-z plane, reaching magic trapping
conditions requires tilting E to θm . This poses a limitation for fully mapping out the behavior of χ.
For example, we would like to observe the dependence of χ on the field angle α, which necessarily
involves tilting the field away from the magic angle. We would also like to measure χ as a functionof |E|. At |E| = 0, the trap is non-magic since the quantization axis is set by the magnetic fieldalong +ŷ, and we observed that the Ramsey coherence was sharply reduced to 0.24(1) ms (black

160
1.0
1 kV/cm, magic, no DD

0 kV/cm, no DD

Contrast

0 kV/cm, XY8*3

0.5

0.0
0

10

20

30

Time (ms)Figure 8.3: Contrast decay between | ↓i and | ↑i with and without dynamical decoupling. At|E| = 1 kV/cm and θm (gray circles), the fitted 1/e coherence time with no spin echo pulses isτc = 4.0(4) ms. Moving the electric field to |E| = 0 (black triangles), away from the magic trapping
condition, decreases the coherence time to τc = 0.24(1) ms without decoupling pulses. At the
non-magic condition, using the XY8*3 DD sequence (green squares) improves the coherence time22
to τc = 17(1) ms for a low density of molecules. All fits are to e−t /τc . Adapted from Ref. [234].

Figure 8.4: Dynamical decoupling sequence. The standard Ramsey sequence (no DD) consists ofan initial pulse of area θ about ŷ (in spin space), followed by an evolution time T , and finally apulse of area π − θ about an axis n̂0 = x̂ cos φ + ŷ sin φ. To add DD, we insert one or more XY8sequences during T . One XY8 sequence (zoomed region) consists of 8 π-pulses about x̂ or ŷ spacedby time τ , for a total duration of 8τ . XY8*n refers to n XY8 sequences concatenated, for a totalinterrogation time of T = 8nτ . Reproduced from Ref. [234].

161
triangles in Figure 8.3).
To bypass this limitation, we implemented a dynamical decoupling (DD) pulse sequence [154].
DD sequences consist of many successive spin echo pulses, spaced by time τ . The pulse train forms
a bandpass filter that rejects noise outside of a narrow bandwidth around f0 = 1/2τ [155, 156]. DD
is a standard technique in fields such as NMR [240] and NV centers [241, 242] for increasing thecoherence time T2∗ limited by technical noise toward the intrinsic coherence limit T2 of the quantum
system. In our system, the DD sequence removes inhomogeneous broadening from differential AC
Stark shifts and gradients of E, as well as time-dependent noise on E.
Specifically, we implemented the XY8 pulse sequence [154, 243], depicted in Figure 8.4. The
XY8 sequence consists of an initial delay τ /2, followed by 8 π-pulses spaced by τ , and a final τ /2
delay at the end of the sequence. The total duration is 8τ . The Bloch sphere rotation axis of thepulses alternate between x̂ and ŷ to reduce sensitivity to errors in the pulse area. The sequence isalso time-symmetric for more effective noise suppression [244]. The XY8 sequence also serves as abuilding block for creating longer sequences (denoted XY8*n) by concatenation. Technical detailsof the RF system used to generate the pulses can be found in section 3.4.
Using the XY8*3 sequence (green squares in Figure 8.3) vastly improved the coherence timeat |E| = 0 by almost two orders of magnitude, from τc = 0.24(1) ms to 17(1) ms. The measurement
was taken at a low molecule density n = 0.14(2) × 107 cm−2 to avoid interaction effects. With acoherence time on the order of 10 ms, we could move on to systematically study the dipolar density
shift as a function of |E| and α.

8.3

Density shift
The measurement of the density shift proceeded in several stages. At a fixed density n, we

conducted Ramsey experiments between | ↓i and | ↑i using an XY8*3 DD sequence, with a typical
total evolution time of T = 1.2 ms. By scanning the phase φ of the final Ramsey pulse, we measured
Ramsey fringes like those shown in Figure 8.5A and determined the phase shift ∆φ for each fringe.
We took fringes at several different values of the initial pulse area θ to vary the strength of the162

Figure 8.5: Measurement of the dipolar density shift for the {| ↓i, | ↑i} spin manifold with a total
evolution time of T = 1.2 ms. (A) Example Ramsey fringes taken with an XY8*3 sequence at|E| = 0. The initial pulse areas are θ = π/4 (red), π/2 (black), and 3π/4 (blue). The phase shift∆φ is extracted for each fringe by a fit to a sinusoid. (B) Plots of ∆φ versus 2hSz i/N = cos θ fortwo different densities n = 1.4(1) × 107 cm−2 (black) and 0.65(7) × 107 cm−2 (gray). Solid lines
are linear fits. The slope of ∆φ/T versus 2hSz i/N gives the mean-field interaction shift ∆ν. (C)Mean-field interaction ∆ν versus average density n. For each n, a set of Ramsey fringes are taken
to obtain ∆ν. A linear fit to the data directly gives χ. Reproduced from Ref. [234].

mean-field shift, which is proportional to the population imbalance cos θ. The red, black, and blue
data in Figure 8.5A correspond to θ = π/4, π/2, and 3π/4, respectively. According to equation(8.4), the mean-field interaction ∆ν = (1/2π)∆φ/T is linear with 2hSz i/N = cos θ. Thus, fitting
the slope of ∆φ versus 2hSz i/N allows us to extract ∆ν. Figure 8.5B shows ∆φ versus 2hSz i/Nfor two different values of n. The lower density measurement (gray) shows a shallower slope, andthus lower ∆ν, than the higher density measurement (black).

163
In principle, we can extract χ from a measurement of ∆ν at a single density n by assumingthat equation (8.4) holds. Instead of making this assumption, we measured ∆ν at several valuesof n to experimentally determine the dependence on n (Figure 8.5C). We observed the expected
linear trend, and from a linear fit (solid line) we extract χ = 4.9(3) × 10−6 Hz/cm−2 .

8.4

Tuning the interactions
Using the procedure of the previous section, we measured χ as a function of E to demonstrate

the high degree of tunability of our molecular spin system. By varying |E|, with fixed α = 0◦ , wecould probe the effects of rotational state mixing on the mean-field interactions. As shown in
Figure 8.2B, dipolar spin-exchange is the dominant interaction at low |E|. As |E| increases, theincreasing strength of the direct (Ising) interactions from the induced dipoles competes with thespin-exchange to cause a reduction in χ, leading to χ = 0 near |E| ≈ 7 kV/cm. For even higher|E|, the direct interactions are dominant and χ is negative. We experimentally measured χ overa wide range of |E| to observe this trend (Figure 8.6A). In particular, at the highest field of 8.9
kV/cm, χ was negative, demonstrating the high degree of tunability with |E|. The solid line showsthe expected dependence of χ, given by A[−(d↓ − d↑ )2 + 2d2↓↑ ], which is fit to the experimental
data with the overall scaling factor A. The small dips in the fitted curve at low |E| are caused by
avoided crossings of | ↑i with other hyperfine states in N = 1, as calculated for KRb at B = 545.9
G and including the optical fields of the VL and ODT.
The combination of the reduced dimensions with the dipolar anisotropy can also be used tocontrol χ via the angle α between E and ŷ (the tightly confined direction). In Figure 8.6B, wemeasured χ as a function of α at fixed |E| = 1.02 kV/cm. At α = 0◦ , all pairs of molecules interact
repulsively, leading to a large and positive χ. In contrast, at α = 90◦ , the dipoles lie in the plane
and the molecules experience both attractive and repulsive interactions. Owing to the form of thedipolar anisotropy, the interactions are attractive on average, resulting in a negative χ. Assumingthat the trapping potential is symmetric in the x-z plane, the interaction strength has a simple
angular dependence A(3 cos2 α − 1) [234]. The solid line in Figure 8.6B is a fit to this functional

164
form.
The choice of internal states used to form the spin-1/2 system offers another knob for tuning
dipolar interactions. Figure 8.7 shows a measurement of the mean-field interaction between | ↓i and| ↑0 i = |1, −1i at |E| = 0 (green squares). The measurement of the {| ↓i, | ↑i} system from Figure
8.5C (black circles) is shown for comparison. Using | ↑0 i instead of | ↑i changes the magnitude ofχ and reverses its sign, and we measured χ0 = −2.3(8) × 10−6 Hz/cm−2 for the {| ↓i, | ↑0 i} system.
The experimental ratio of χ0 /χ is consistent with −1/2, the value expected from equation (8.5) at|E| = 0. Interaction control via the internal state allows us to rapidly change the coupling strengths
in the spin Hamiltonian, since the duration of an RF pulse is much shorter than the experimental
limit of several ms for smoothly changing |E| or α.

8.5

Dynamically reversing the evolution
The ability to switch the parameters of the spin Hamiltonian allows us to dynamically control

the evolution of the many-body state. In particular, by switching the excited state from | ↑0 i to
| ↑i in the middle of the Ramsey evolution, the spin Hamiltonian couplings are instantaneously
changed by a factor of −2 at |E| = 0. This reverses the many-body dynamics, a key ingredientfor studying quantities such as out-of-time-ordered correlators [245]. Such time-reversal schemes
are also useful for approaching the Heisenberg limit of measurement sensitivity without requiring
single particle detection [233].
Here, we demonstrate the reversal of the mean-field interaction shift, a first step towardrealizing such a many-body echo in our dipolar spin system. The measurement consisted of threestages (Figure 8.8A). The first and third stages consisted of evolution in a coherent superpositionof {| ↓i, | ↑0 i} or {| ↓i, | ↑i}, respectively, using XY8 to protect the spin coherence. The middle
stage consisted of a sequence of three pulses (denoted R in Figure 8.8A) that swapped the excitedstate population from | ↑0 i to | ↑i with a duration of 70 µs. We measured the phase difference ∆Φbetween the Ramsey fringes obtained for θ = π/4 and θ = 3π/4, in order to reject systematics suchas extra shifts caused by frequency switching of the microwave source.

165

Figure 8.6: Tuning χ with |E| and α for the {| ↓i, | ↑i} spin manifold. (A) Dependence on |E|.
Initially large and positive at low |E|, χ passes through zero near |E| = 7 kV/cm, and becomes
negative for higher |E|. The change in sign of χ is a direct manifestation of the competition
between the interactions arising from the transition and induced dipoles, see equation (8.5). The
solid line is the calculated χ(|E|) including hyperfine structure, scaled to the experimental data.
(B) Dependence on the field angle α. Owing to the dipolar anisotropy, χ depends strongly on theorientation of the electric field set by α. The solid line is a fit to A(3 cos2 α − 1). Adapted from
Ref. [234].

Figure 8.7: Internal state dependence of χ. The mean-field interaction is measured for {| ↓i, | ↑i}(black circles, same data as Figure 8.5C) as well as {| ↓i, | ↑0 i} (green squares) spin manifolds
at |E| = 0. The slope of the mean field interaction versus density gives χ for each system. The
measured for {| ↓i, | ↑i} is consistent with a factor of 2 larger |χ| than {| ↓i, | ↑0 i}, and opposite
interaction sign. Adapted from Ref. [234].

166

(a)
R:
①

"
!

t=0

R

XY8

χ2

②

#−!

XY8

%

χ1 = − 2χ2

ts

③

T

(b)
40

ΔΦ (Deg)

20

0

-20

-40
0.0

0.5

1.0

1.5

2.0

2.5

3.0

T (ms)
Figure 8.8: Reversing the spin Hamiltonian. (A) Experimental pulse sequence used to measure
the reversal. An initial pulse of area θ creates a superposition of | ↓i and | ↑0 i. After a hold time
ts , during which an XY8 sequence is used to suppress decoherence, the excited state population iscoherently transferred from | ↑0 i to | ↑i using a composite pulse R. The remaining T − ts hold timeis spent in a superposition of | ↓i and | ↑i, with another XY8 pulse sequence again suppressing
decoherence. A final pulse of area π − θ and variable phase φ is used to measure the populationsin | ↓i and | ↑i. (B) The differential phase shift ∆Φ measured between θ = π/4 and θ = 3π/4
versus the total time T . Green squares: phase accumulation in {| ↓i, | ↑0 i}. Black circles: phaseaccumulation in {| ↓i, | ↑i}. The black line is the piecewise linear fit described in the text. The
density is n = 1.1(1) × 107 cm−2 . Reproduced from Ref. [234].

167
The full sequence in (A) was used for each measurement of ∆Φ, though the lengths of thefirst stage (duration ts ) and third stage could be independently varied to map out the evolution of∆Φ before and after the reversal R as a function of the total time T (Figure 8.8B). For T ≤ 1.2
ms, we kept the third stage (in {| ↓i, | ↑i}) as short as possible and scanned the time ts of the firststage. This let ∆Φ accumulate from interactions in the {| ↓i, | ↑0 i} spin manifold (green squares).
For T > 1.2 ms, we kept the time of the first stage fixed at ts = 1.2 ms, and scanned the timeof the third stage. We observed a change in the slope of ∆Φ versus T from negative to positiveat T = 1.2 ms, indicating a reversal in the sign of the mean-field interaction. The data is well-fit
by a piecewise linear function where the slopes before and after the reversal have a ratio of −2, as
expected from Figure 8.7. These results demonstrate reversal of the spin Hamiltonian at |E| = 0,
paving the way for future studies of dynamic engineering of the many-body state.

8.6

Conclusion
In this chapter, we described initial explorations of the collective spin physics of dipolar

molecules in 2D. With the help of dynamical decoupling sequences to suppress single-particle decoherence, we observed the mean-field interaction shift from the dipolar interactions. By changing|E|, the field angle α, or the internal state, we could change the magnitude and sign of the density
shift. Finally, we showed that by changing the internal state we could reverse the sign of the couplings in the spin Hamiltonian, which we observed through a reversal of the mean-field shift. This
itinerant dipolar spin system opens many exciting future directions, which are discussed in moredetail in the following concluding remarks.

Chapter 9

Conclusion and Outlook

In this thesis, I described our work over the past six years on the JILA KRb experiment.
After making molecules in the new machine in early 2018, we were able to leverage the improved
atom conditions to create a DFG of molecules in the ODT at T /TF = 0.3 (Chapter 4). The Fermi
statistics were apparent from the distribution in TOF expansion and the density fluctuations of
the cloud. However, turning on a large electric field to polarize the molecules and induce strong
dipolar interactions led to rapid losses in 3D. We therefore explored two different methods forprotecting the molecules from loss while maintaining elastic collisions. Confining the molecules
into 2D, with the dipoles oriented perpendicular to the plane, allowed us to exploit the interaction
anisotropy to stabilize the gas while also turning on strong elastic dipolar collisions (Chapter 5).
This enabled direct evaporative cooling of the molecules through dipolar collisions to T /TF < 1 in
2D. Alternatively, by putting the molecules in an excited rotational state, we could observe resonant
collisional shielding of the chemical reactions near Förster resonances tuned by the external electricfield (Chapter 6). The resonant shielding was effective in both 2D and 3D, and enabled studies ofelastic dipolar collisions and direct evaporation in 3D. Both of these methods are highly generaland are expected to be applicable to other polar molecule species. Complementary work at Harvard[105] and Munich [54] also demonstrated suppressed loss rates and enhanced elastic collision rates
using microwave shielding.
With improved control over the collisions, we shifted our focus from two-body physics ofmolecular collisions toward the mean-field (and in the future, many-body) dynamics of the dipolar

169
gas. A single 2D layer is an ideal setting for studying the collective behavior of dipoles, so weleveraged our precise electric field control to spectroscopically address and select single 2D pancakes(Chapter 7). Our layer-resolved rotational state control led to the observation of a long-range
collision process where two molecules in adjacent layers could scatter by exchanging their rotational
state while also changing motional states in the trap, enabling control of the chemical reaction rate
through the internal state of molecules in a neighboring 2D layer. Finally, in our most recent work,
we have started to tune the collective spin dynamics of molecules prepared in a superposition ofrotational states (Chapter 8). Through the strength and orientation of the electric field or the
choice of rotational states, we showed that we can tune the magnitude and sign of mean-field
dipolar shifts in the system.
The 2D dipolar spin system promises to open many new scientific explorations. The meanfield shift studied in Chapter 8 is a first manifestation of the Sz2 term of the collective Hamiltonian,which also generates spin squeezing [49]. Thus, in the near future it will be exciting to start measuring the spin noise in the system. Perhaps one day, squeezing generated by dipolar interactions
could be used to enhance the sensitivity of next-generation precision measurements with molecularquantum gases [246]. One can also envision further engineering the Hamiltonian by using different
rotational states or adding microwave dressing to tailor the interactions [41, 45, 160]. Spin-orbit
coupling arising from dipolar interactions, either using the scheme of Ref. [139] or possibly by exploiting the shielding resonances, could also extend the already rich dynamics in 2D. Additionally,
the competition between the coherent spin dynamics and collisions in this itinerant spin systemprovides an interesting area for further exploration. At one extreme, the dipolar collisions can bestrong enough to enter the hydrodynamic regime in 2D [197]. At the other extreme, transverse
lattices can be added to protect the spin dynamics by fully suppressing reactive losses and elastic
collisions.
More generally, it will be exciting to look for new and creative ways to push the limits oncontrol of the molecules. Most of the experiments in this thesis relied on one main technical advance(in addition to our capability of preparing a quantum gas of molecules): the ability to produce a170large and highly controllable electric field for the molecules. It is fun to think about how ourelectric field control can be combined with other innovations from around the ultracold community
to further enhance our experimental capabilities. Single molecule control, either in tweezers [86, 74]
or in a quantum gas microscope [247], brings new opportunities. Our experiment was designed tobe compatible with high resolution imaging if we decide to explore this direction [106]. A magic
wavelength trap for the |0, 0i ↔ |1, 0i rotational transition, as demonstrated in NaK [122], could
also significantly simplify our experiments. For KRb, the corresponding transition is near 1020
nm [123]. Theory predicts that the magic condition could even be met for multiple rotational
states simultaneously [124], facilitating (for example) studies of synthetic dimensions encoded in
the ladder of rotational states [159]. Blue-detuned box traps [94] for KRb could facilitate more
collisional studies, for example in higher rotational levels or in the non-reactive Rb-KRb mixture[215], by removing the influence of the trapping light [91, 92]. Dipolar interaction strengths could
be boosted by sub-wavelength trapping [248] to facilitate pairing and superfluidity [249, 43], anddispersive imaging techniques could allow for sensitive and non-destructive detection of molecules[250, 251]. Adding periodic driving [252] could enable Floquet engineering of polar molecule systems
to realize novel states of matter [253]. Hybrid systems of polar molecules and Rydberg atoms also
promise new opportunities by combining the advantages of both subsystems [254, 255, 256]. The
future of molecular quantum science is bright!

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Appendix A

KRb energy levels in external fields

We will start by summarizing a few of the important equations in Chapter 2, following
Chapter 5 of Ref. [108].
(k)

Let Dpq (ω)∗ be the rank k rotation matrix, with ω the Euler angles for rotating from themolecule frame to the lab frame. This has matrix elements(k)hJ, mJ , Ω|Dpq(ω)∗ |J 0 , m0J , Ω0 i = (−1)mJ −Ω

J0



p
 J k
 J
(2J + 1)(2J 0 + 1) −Ω q Ω0
−mJ

k

J0

p m0J



.

(A.1)
The index p is associated with the lab frame while the index q is associated with the molecule frame.
This also gives a helpful mnemonic for remembering the matrix elements, since the molecule frame
index q appears alongside Ω and the lab frame index p appears alongside MJ in the 3j symbols.
(k)

To get a tensor Tq (A) from the molecule frame to the lab frame, we contract it with(k)

Dpq (ω)∗ ,Tp(k) (A) =

X

(k)
Dpq
(ω)∗ Tq(k) (A).

(A.2)

q
(1)

In terms of the Cartesian components of A, the rank 1 tensor Tp (A) is(1)

T0 (A) = Az ,

(A.3)

1
(1)
T±1 (A) = ∓ √ (Ax ± iAy ).
2

(A.4)

190
(2)

The rank 2 tensor Tp (u, v) is given by1(2)
T0 (u, v) = √ (2uz vz − ux vx − uy vy )61
(2)
T±1 (u, v) = ∓ (ux vz + uz vx ± i(uy vz + uz vy ))2
1(2)
T±2 (u, v) = (ux vx − uy vy ± i(ux vy + uy vx )) .
2
In terms of 3j symbols, the Wigner-Eckart theorem is


0
k j 
 j
(k)
0
hj, m|Tp(k) (A)|j 0 , m0 i = (−1)j−m  hj||T (A)||j i.
−m p m0

(A.5)
(A.6)
(A.7)

(A.8)

Here are a few useful reduced matrix elements:
hj||T (1) (J)||j 0 i = δjj 0p
j(j + 1)(2j + 1)−1j(2j − 1)  J 2 J √
hj||T (2) (J, J)||j 0 i = δjj 0
6−J 0 J−1eQ  I 2 I 
hI||T (2) (Q)||I 0 i = δII 0 .
2−I 0 I(A.9)(A.10)(A.11)

T (2) (Q) above is the nuclear quadrupole tensor, needed for calculating the nuclear electric quadupole
interaction.

A.1

Molecular Hamiltonian
Following Ref. [109], the molecular Hamiltonian in the presence of B and E is
H = Hrot + HS + Hhf + HZ

(A.12)

Hrot = Bv N2(A.13)

HS = −d · E(A.14)Hhf =X

Vi · Qi +i=K,Rb

HZ = −gr µN N · B −X

ci N · Ii + c3 IK · T · IRb + c4 IK · IRb(A.15)

gi µN Ii · B(1 − σi ).

(A.16)

i=K,RbX
i=K,Rb

191
Above, we have ignored the centrifugal distortion term Dv N2 N2 , which is negligible for the lowest
rotational states.
The physical meaning of individual terms of H and coupling constants for KRb are discussed
in section 2.2. Here, we will write down the relevant matrix elements. We work in the uncoupledbasis |N, mN , m1 , m2 i, where m1 (m2 ) is the K (Rb) nuclear angular momentum projection. The
rotation term is already diagonal in the uncoupled basis, and the Stark term is discussed in detailin section 2.3.1. The Zeeman term is already diagonal if B = Bz. However, it might be useful inthe future to vary the angle between B and E, so let us work out the general case for B. All of theterms of HZ have the same form (up to swapping N for Ii ), so we can just focus on the N · B term,
0hN, mN |N · B|N , m0N i =

1
X

(1)

(−1)p hN, mN |Tp(1) (N)T−p (B)|N 0 , m0N i

p=−1= (−1)N −mN



1
X

 N
(1)
(−1)p T−p (B) 
p=−1
−mN

= (−1)N −mN δN N 0p
N (N + 1)(2N + 1)

1

N0

p m0N
1
X



(1)
 hN ||Tp (N)||N i


 N
(1)
(−1)p T−p (B) 
p=−1
−mN

1

N0

p m0N



.

(A.17)The orientation of E can also be allowed to vary by generalizing the Stark term to
HS = −

X(1)(−1)p dp T−p (E).

(A.18)

p

The hyperfine Hamiltonian is the most complicated. The first term (nuclear electric quadrupole)192
is calculated explicitly in Brown and Carrington, pg. 476:
2
X

(2)

hN, mN , Ii , mi |(−1)p Tp(2) (Vi )T−p (Qi )|N 0 , m0N , Ii0 , m0i i

p=−2


p
qX
N
=
(−1)p (−1)−mN (2N + 1)(2N 0 + 1) 
2 p
0


2
Ii0 
 Ii
(2)
0
× (−1)Ii −mi  hIi ||T (Qi )||Ii i−mi −p m0i

2
0

N0



 N

0
−mN


p
(eqQi ) X
N
= δIi Ii0(−1)p (−1)−mN (2N + 1)(2N 0 + 1) 
2
p
0


−1
2
Ii   Ii 2 Ii  Ii
× (−1)Ii −mi  .
−mi −p m0i−Ii 0 Ii

2
0

(k)

N0

2

N0

p m0N







 N

0
−mN

Note that the N -dependent part is the matrix element of Dp,q=0 (ω)∗ =

2

N0

p m0N





p
4π/(2N + 1)Y2p (ω) =C2p (ω), which explains the correspondence between the form that is used in Ref. [109] versus Ref.
[117, 124]. For KRb, the c4 I1 · I2 term is the next largest, with matrix elements
hm1 , m2 |I1 · I2 |m01 , m02 i =

1
X

(1)

(−1)p hm1 , m2 |Tp(1) (I1 )T−p (I2 )|m01 , m02 i

p=−1



1
X



1 I1 
 I1(1)(−1)p (−1)I1 −m1  hI1 ||T (I1 )||I1 ip=−1−m1 p m01


1
I2 
 I2
(1)
× (−1)I2 −m2  hI2 ||T (I2 )||I2 i−m2 −p m02
=

=

1
X

(−1)p (−1)I1 +I2 −m1 −m2pp
I1 (I1 + 1)(2I1 + 1) I2 (I2 + 1)(2I2 + 1)

p=−1







1 I1   I2
1
I2 
 I1
×

.
0
0
−m1 p m1
−m2 −p m2

193
The nuclear spin-rotation term is
hN, mN , Ii , mi |N · Ii |N 0 , m0N , Ii0 , m0i i =X(1)(−1)p hN, mN |Tp(1) (N)|N 0 , m0N ihIi , mi |T−p (Ii )|Ii0 , m0i i

p



N0



1
 N

(1)
0
(−1)p (−1)N −mN  hN ||T (N)||N ip−mN p m0N


0
1
Ii 
 Ii
(1)
0
× (−1)Ii −mi  hIi ||T (Ii )||Ii i−mi −p m0i=

X= δN N 0 δIi Ii0

X(−1)p (−1)N +Ii −mN −mippp
× N (N + 1)(2N + 1) Ii (Ii + 1)(2Ii + 1)

1 N   Ii
1
Ii 
 N
×

.
0
0
−mN p mN−mi −p mi
The last term is the nuclear magnetic dipole-dipole tensor term. This term is negligibly small for
KRb [109], so we ignore it for simplicity.
The calculations in Chapter 2 that include hyperfine structure were performed with a solver,
BialkaliSpectrum.jl1 , written in Julia [257]. We have written a few automated tests for BialkaliSpectrum.jl, some of which are described below, though users should be aware that the code has
not yet been thoroughly tested and may contain bugs. The plots in Chapter 2 were created with a
Julia plotting library called Makie.jl [258].

A.2

AC Stark shift
Ref. [117] gives the following form for the AC Stark shift:√16
Hac = − (αk + 2α⊥ )Ilaser −(αk − α⊥ )T (2) (,  ) · C2 (α, β)Ilaser ,3
3

1(A.19)

https://github.com/kylematsuda/BialkaliSpectrum.jl. After writing BialkaliSpectrum.jl, we became aware of asimilar project Diatomic-py from the Cornish group [259].

194
(2)

where C2p (α, β) = Dp0 (ω)∗ is the rotation to the molecular frame, with α and β the Euler angles.
Expanding, we find matrix elements
hN, mN |T (2) (,  ) · C2 (α, β)|N 0 , m0N i = δN,N 0

X(−1)p Tp(2) (,  )hN, mN |C2,−p (α, β)|N, m0N ip= δN,N 0X(k)(−1)p Tp(2) (,  )hN, mN |D−p,q=0 (ω)∗ |N, m0N ip= δN,N 0 (−1)mN (2N + 1)

X(−1)p Tp(2) (,  )
p





N
×
0

2 N  N

0 0
−mN


2

N 
.
−p m0NAs a check, we will compare this formula against the explicit forms for N = 1 given in Ref.
[110]. From Ref. [110], the diagonal terms are
3αk + 2α⊥αk + 4α⊥sin2 θ +cos2 θ
55
2αk + 3α⊥αk + 4α⊥α|1, ±1i =h1, ±1|αsin2 θ +cos2 θ.
55
α|1, 0i =h1, 0|α(A.20)(A.21)

We will assume that  lies in the x-z plane, so  = (sin θ, 0, cos θ), and then the components of thetensor have simple expressions,1(2)
T0 (,  ) = √ (2 cos2 θ − sin2 θ)
6(2)

T±1 (,  ) = ∓ sin θ cos θ(2)

T±2 (,  ) =1
sin2 θ.
2(A.22)(A.23)(A.24)The diagonal components are2

1 2 1  1
2
2
h1, 0|T (2) (,  ) · C2 (α, β)|1, 0i = 3  √ (2 cos θ − sin θ)
6
0 0 0
√
6
=
(2 cos2 θ − sin2 θ)(A.25)15
1 2 1   1 2 1  1
2
2
h1, ±1|T (2) (,  ) · C2 (α, β)|1, ±1i = −3  √ (2 cos θ − sin θ)
6
0 0 0−1 0 11= − √ (2 cos2 θ − sin2 θ),
5 6(A.26)

195
so the full matrix elements are1
2
α|1, 0i = (αk + 2α⊥ ) + (αk − α⊥ )(2 cos2 θ − sin2 θ)
h1, 0|α315
3αk + 2α⊥αk + 4α⊥sin2 θ +cos2 θ
=551
1
α|1 ± 1i = (αk + 2α⊥ ) − (αk − α⊥ )(2 cos2 θ − sin2 θ)
h1, ±1|α315αk + 4α⊥2αk + 3α⊥sin2 θ +cos2 θ,=5
5which agree with the expressions from Ref. [110]! Now, let us check the off-diagonal terms,



1
2
1
1
2
1



h1, 0|T (2) (,  ) · C2 (α, β)|1, ±1i = ∓3  sin θ cos θ
0 0 0
0 ∓1 ±1√3=±
sin θ cos θ(A.27)
5



2
1 2
3 1 2 1  1
h1, ±1|T (2) (,  ) · C2 (α, β)|1, ∓1i = −  sin θ2
0 0 0
∓1 ±2 ∓1
r31
sin2 θ,(A.28)=−
25
so we get√2(αk − α⊥ ) sin θ cos θ51
α|1, ∓1i = − (αk − α⊥ ) sin2 θ.
h1, ±1|α5
α|1, ±1i = ±h1, 0|αThese also match the equations from Ref. [110].

A.3

Comparison to experiment
Here, we briefly describe a few of the checks we have done on the output from BialkaliSpec-

trum.jl. Basic tests based on symmetry are implemented, for example, requiring that the spectrum
is invariant under rotation of all fields (magnetic, electric, and optical) by the same angle.
Another automated test is to compare the calculated KRb spectrum at 545.9 G to the calculated values in Table II of Ref. [111]. Using the molecular parameters given in Ref. [111], we have196
good agreement (< 1 kHz, the precision given for the calculations in the paper) with all of the entries in the table, except for the last row, which we believe is a minor typo. (If we use |1, 0, −3, 1/2i
as the excited state instead of |1, 0, −3, 3/2i, we have agreement with the value reported in thetable. This would also make sense given the differential Zeeman shifts at 545.9 G.)
We also checked the calculated AC polarizability against the experimental values in Ref. [110].
We diagonalized the Hamiltonian from the main text of Ref. [110] and compared the polarizabilities
of N = 0 and 1 with the output of our calculation (with the correct hyperfine states). We get within
about 3.5% of the reported values over the full range of polarization angles compared to the of α.
The experimental values of α were fitted using a perturbative model: the hyperfine states were
only used to calculate the I = 0 energies of the states, but the rest of the model included only
the |N, mN i quantum numbers. This makes a small but noticeable difference as shown in the
Supplementary Material of Ref. [110], and likely explains why we cannot reproduce the AC Stark
shifts exactly with our calculation.
Finally, we also quickly checked the calculation by taking experimental data at |E| = 0(Figure A.1) and 1 kV/cm (Figure A.2). In both cases, we saw roughly the anticipated spectrum
from the calculation. We have also used the calculation to help find the |0, 0i → |1, 0i at a variety
of other fields, although at high fields the uncertainty of the experiment–calculation agreement isdominated by small uncertainties in the experimental calibration of |E|.

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Relative coupling

N |0,0⟩ (10⁴)

2.0
1.5
17 μs

1.0

40 μs

0.5
0.0

0.4
0.0
800
900
1000
1100
Frequency - 2227 MHz (kHz)

Figure A.1: Microwave spectrum at |E| = 0 and B = 545.9 G. The molecules are held in the
ODT + VL. Upper panel: Depletion from the ground state |0, 0, −4, 1/2i is measured as a functionof the microwave frequency, for two different pulse times 17 µs (blue) and 40 µs (orange). Lower
panel: Calculated transition frequencies and coupling strengths considering |E|, B, and the opticaltraps. The calculated coupling strength depends on the polarization of the microwaves, which is notknown (see section 3.4.2). We assumed that the microwave polarization was tilted 30 deg from thequantization axis for this calculation, which roughly reproduces the observed transition strengths.

2.0

N |0,0⟩ (10⁴)-30 dB, 280 μs

1.5-18 dB, 600 μs

1.0

0.5

0.0
850
900
Frequency - 2248 MHz (kHz)950

Figure A.2: Microwave spectrum at |E| = 1 kV/cm. Here, E was oriented with an angle of 36
deg from vertical. The molecules are held in the ODT + VL at B = 545.9 G. The blue data
points show microwave transfer from the initial state |0, 0i to |1, 0i of the same hyperfine character(mN is defined relative to E). The calculation predicted another hyperfine state of |1, 0i with a∼ 10× weaker coupling at about 120 kHz higher transition frequency. We were able to observe thisadditional state by increasing the power and duration of our pulse (orange points).