Quantum Controlled Collisions and Magnetic
Trapping of Ultracold NaLi MoleculesbyJuliana Park
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2023
© Massachusetts Institute of Technology 2023. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
January 13, 2023

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lindley Winslow
Associate Department Head of Physics

2

Quantum Controlled Collisions and Magnetic Trapping of
Ultracold NaLi MoleculesbyJuliana Park
Submitted to the Department of Physics
on January 13, 2023, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Na6 Li is a fermionic molecule that has weak singlet-triplet mixing, making it asuitable system to study the triplet rovibrational ground state (𝑎3 Σ+ , 𝑣 = 0, 𝑁 =0). It is notable for its both non-zero electric (0.175 Debye) and magnetic (2𝜇𝐵 )dipole moments and small two-body scattering rate, as predicted by the universal
model for cold collision. Additionally, 23 Na6 Li is the lightest bi-alkali molecule, andthe theoretical simulation of collisions is relatively feasible compared to other heavy
molecules which makes it a promising benchmark system for theoretical quantum
scattering calculations.
This thesis describes three experiments and a numerical/theoretical work on 23 Na6 Li
molecules in the triplet ground state. The first two experiments are on molecular Feshbach resonances: in spin-polarized 23 Na6 Li+23 Na collisions and in 23 Na6 Li+23 Na6 Li
collisions. The first experiment focuses on the spectroscopic study of Feshbach resonances in two possible spin-polarized 23 Na6 Li+23 Na collisions from near 0 to 1400
Gauss. This allows learning about the molecular interaction potential surface andintermediate collision complexes, benchmarking theory, and controlling reactive collisions. The second experiment is a report of an unpredicted 𝑝-wave Feshbach resonancein 23 Na6 Li+23 Na6 Li collisions and interpretations. The resonance occurs for moleculesin the lower stretched hyperfine state near an open-channel degeneracy. The collision
loss rate is enhanced by more than two orders of magnitude from the 𝑝-wave universal
value at the background to near the 2D unitarity limit.
In addition to the search for magnetically tunable resonances, 23 Na6 Li moleculesin the triplet potential are suitable for magnetic trapping. The third experimentdescribes building an improved experimental setup that allows magnetic trapping of
23
Na6 Li molecules and studying various collisions in the magnetic trap by quantum
state control of molecules and atoms. The molecular density is a factor of 105 higher
than that reported for magnetically trapped ultracold molecules, and the temperatureis ≈ 1𝜇𝐾. This condition enables observation of both atom-molecule and moleculemolecule collisions in the ultracold regime and sympathetic cooling of 23 Na6 Li byevaporative cooling of 23 Na in the magnetic trap.
23

3

Lastly, this thesis presents the numerical and theoretical approach to finding awindow for an all-optical creation of molecules using Raman transitions from 23 Na
and 6 Li atoms to 23 Na6 Li molecules. All-optical creation of molecules in which themagnetic association step near a Feshbach resonance is eliminated is expected tobroaden the horizon of ultracold molecules to a larger pool and to eliminate the needfor high magnetic fields.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

4

Acknowledgments
The time I spent as a Ph.D. candidate at MIT was full of tension, wonder, andexcitement. I learned how much I can hate or love being an experimental physicist
while trying to be a good one. I realized that the excitement of being a scientist that
I had thought I lost long ago was not gone after all.
I am extremely grateful to Wolfgang Ketterle for offering me the opportunity toexperience the whole world of AMO physics. Being his student and working next door
to his first BEC machine has been a great honor. He was patient when I was being
too careful or slow and guided me to become a good researcher. I enjoyed listening
to his new ideas and often admired his insights and intuitions.
Only in my third year, I started to understand that some experiments are notdesigned for one or more ultimate scientific goals. The BEC3 machine, the NaLi
experiment, is one of them. With the given toolbox, one rather finds what to do
next. I had difficulty finding what to explore with NaLi molecules and my thesis
topic. In my fourth year, when I came and talked to Wolfgang about this concern,
he was not worried because he believed that new things and ideas will come as wego forward. Back then, I felt a bit of relief from his words, but I was also not fully
convinced, and talked about it to Alan, a former research scientist in Ketterle’s group.
In the end, he was right that we will find exciting new things to do. What I achieved
during my Ph.D. was possible thanks to his belief in BEC3.
When I first joined the BEC3 team, we were a team of five. Ariel Sommer, who wasa postdoctoral researcher, taught me how to code with Python, and Timur Rvachov,who was a senior graduate student, often handed me a random broken electronic box
to fix, which trained me to deal with everyday problems in the lab. Although theyleft about a year after I joined, I learned a lot from them. Alan had been like myco-supervisor. He gave me advice when I needed it when he was at MIT and also after
he left for the University of Waterloo as a professor. Hyungmok Son, who is a former
graduate student, spent the most time in the lab with me, keeping the 20-year-old
lab from falling apart. My time here would not have been the same without them.
5

As the rest of the team members left, in the last couple of years of my graduate
studies, I had to mostly work alone in the lab. I am grateful to have had the opportunity to connect and collaborate with Tijs Karman and Timur Tscherbul during
this period, who allowed me to dive into the field of quantum chemistry and mademe feel that I was not alone. Although Yukun Lu left for the BEC2 lab and Dypole
lab, he was very helpful at certain moments when I was completely stuck in the lab.
Li, Yiqi, Enrique, Airlia, Michael, and Cedric have been great friends. Every time I
had to vent due to the amount of stress I had in the lab, I went to their lab or officejust to relax and calm myself.
I would like to express my deepest appreciation to my relatives in America. My
grandparents Sunsoon Kang and Sunhee Kang, aunt Miran, Clara, and Jean, anduncle Thomas, Woosung, and Jaedeok have been like my parents in America. Thanks
to them and my cousins Andrew, Calvin, Daniel, Hannah, and Crispin, it did not feellike I was studying abroad but rather felt like I finally came back to my hometown.
Last but not least, I could not have undertaken this journey without the support of
my husband Wonyl Choi, and my parents in South Korea, Choonkil Park and Miyoung
Kang. Their belief in me has allowed me to face and overcome my challenges, andpursue my goals.

6

Contents

1 Introduction

17

1.1

Formation and properties of NaLi . . . . . . . . . . . . . . . . . . . .

18

1.2

Cold collisions in molecular systems . . . . . . . . . . . . . . . . . . .

22

1.2.1

Universal loss model for reactive collisions . . . . . . . . . . .

23

1.2.2

Fabry-Perot interferometer model for collision resonances . . .

24

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

1.3

2 Experimental apparatus
2.1

2.2

2.3

31

Instrumentation for control . . . . . . . . . . . . . . . . . . . . . . . .

31

2.1.1

Digital control . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

2.1.2

Analog control . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Coils and magnetic field . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.2.1

Curvature cancellation . . . . . . . . . . . . . . . . . . . . . .

36

Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.3.1

STIRAP lasers . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.3.2

Imaging & cleaning . . . . . . . . . . . . . . . . . . . . . . . .

40

2.3.3

Optical trapping . . . . . . . . . . . . . . . . . . . . . . . . .

41

3 Spectrum of Feshbach resonances in NaLi+Na collision

43

3.1

Experimental protocol and results . . . . . . . . . . . . . . . . . . . .

45

3.2

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.3

Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

7

4 Degeneracy-induced magnetic resonance in NaLi+NaLi collision

59

Measurement & Experimental Result . . . . . . . . . . . . . . . . . .

62

4.1.1

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

62

4.1.2

Model for molecular decay & density calibration . . . . . . . .

63

4.1.3

Density and temperature dependence on molecular decay rates

67

4.1.4

Loss rate constant. . . . . . . . . . . . . . . . . . . . . . . .

69

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.2.1

Hyperfine structure of NaLi . . . . . . . . . . . . . . . . . . .

72

4.2.2

Three-state model of 𝑝-wave resonant scattering . . . . . . . .

76

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.3.1

Collision complex lifetime . . . . . . . . . . . . . . . . . . . .

83

4.3.2

Estimation of coupling strength 𝛾1. . . . . . . . . . . . . . .

86

4.3.3

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4.1

4.2

4.3

5 Magnetic trapping and cooling of molecules

89

5.1

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

89

5.2

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

91

5.2.1

Decay models . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5.2.2

Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5.3.1

𝑝-wave collisions between molecules . . . . . . . . . . . . . . .

97

5.3.2

𝑠-wave collisions between molecules . . . . . . . . . . . . . . .

98

5.3.3

Collisions between NaLi and Na . . . . . . . . . . . . . . . . .

99

5.3.4

Sympathetic cooling of NaLi with Na . . . . . . . . . . . . . .

99

5.3.5

Inelastic collision and thermalization rates . . . . . . . . . . . 101

5.3

5.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Free to bound Raman transfer
6.1

105

Double Raman transfer via Feshbach molecule state . . . . . . . . . . 107
6.1.1

Coupling strengths of Raman beams . . . . . . . . . . . . . . 110

6.1.2

Li 2S to 2P transition strengths . . . . . . . . . . . . . . . . . 110
86.1.3

Molecular and atomic losses . . . . . . . . . . . . . . . . . . . 116

6.1.4

Effects of atomic densities and collisional loss . . . . . . . . . 120

6.2

Direct transfer to the ro-vibrational ground state . . . . . . . . . . . 125

6.3

Free to intermediate vibrational state 𝑣 . . . . . . . . . . . . . . . . . 131

6.4

Comparison of three all-optical methods . . . . . . . . . . . . . . . . 132
6.4.1

AC stark shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7 Summary and Outlook

143

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.2

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.1

Mysteries of non-universal reactive collisions . . . . . . . . . . 145

7.2.2

Towards a deeply degenerate gas of NaLi . . . . . . . . . . . . 146

7.2.3

Creation of stable qubits with rotational states . . . . . . . . . 147

A149
A.1 Circuit designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

B

157
B.1 Coupled-channels calculations . . . . . . . . . . . . . . . . . . . . . . 157

C165
C.1 Quasi-2D threshold law for 𝑝-wave collision . . . . . . . . . . . . . . . 165
C.2 Long-range coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
C.2.1 Coupling matrix elements between degenerate open channels|𝑎𝑎⟩ and |𝑎𝑏1 ⟩ due to rotationally excited states . . . . . . . . 168
C.3 Harmonic oscillator model . . . . . . . . . . . . . . . . . . . . . . . . 173
C.4 Three-state 𝑇 -matrix model of 𝑝-wave resonant scattering near degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
C.4.1 Derivation of 𝐹𝛼 (𝑘) matrix elements. . . . . . . . . . . . . . 177

C.5 Analysis on the two-body inelastic rate expression . . . . . . . . . . . 180

9

10

List of Figures
1-1 Photo-induced collisional loss. . . . . . . . . . . . . . . . . . . . . . .

25

1-2 Fabry-Perot interferometer model for reactive collisions. . . . . . . . .

27

2-1 Schematic of old digital control instrumentation circuit for single channel. 33
2-2 Schematic of newly designed digital control instrumentation circuit for
single channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2-3 Circuit diagram of curvature field cancellation . . . . . . . . . . . . .

38

2-4 STIRAP lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3-1 Rovibrational ground state energy level diagram of NaLi in the triplet
potential (𝑎3 Σ+ ) at zero bias field. . . . . . . . . . . . . . . . . . . . .

46

3-2 Collisional loss spectrum of NaLi molecules with Na atoms as a functionof magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3-3 Feshbach resonance in NaLi + Na upper stretched state near 884 G. .

50

3-4 Cumulative probability of resonance spacing from Na+NaLi collisions53

3-5 Distribution of scaled reduced widths for scattering resonances . . . .

54

4-1 Resonant molecular loss as a function of magnetic field and Moleculardecay curves both at 334.92 G resonance and away from the resonance. 61
4-2 Mean density estimation in the 3D to quasi-2D crossover regime . . .

65

4-3 Density correction factor . . . . . . . . . . . . . . . . . . . . . . . . .

66

4-4 Density dependency of molecular decay rate at 334.92 G . . . . . . .

67

4-5 Threshold laws of molecule-molecule collisions . . . . . . . . . . . . .

68

4-6 Enhanced two-body loss rate coefficient of molecules. . . . . . . . . .

71

11

4-7 Long-range coupling . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4-8 Hyperfine structure of NaLi(𝑎Σ+ ) in an external magnetic field. . . .

74

4-9 Hyperfine structure of NaLi(𝑎Σ+ ) near 334.92 G. . . . . . . . . . . .

75

4-10 Schematic of the resonance model with two open channels and a p-wave
bound state trapped behind a centrifugal barrier. . . . . . . . . . . .

78

4-11 Transmission probability of a cavity as a function of mirror couplingstrengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4-12 Schematic of the resonance model with more than one orthogonal collisional path.

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

81

4-13 Intermolecular potential of the NaLi (𝑎3 Σ+ ) molecules. . . . . . . . .

85

4-14 The coupling between the incident open channel |1⟩ with the p-wave
bound state 𝛾1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5-1 Experimental sequence for magentic trapping of molecules . . . . . .

92

5-2 NaLi energy level diagram and 𝑝-wave collisions . . . . . . . . . . . .

93

5-3 𝑠-wave collisions between molecules . . . . . . . . . . . . . . . . . . .

98

5-4 Collisional cooling of NaLi in a magnetic trap . . . . . . . . . . . . . 100
6-1 Ti:Sapphire laser setup for Raman transfer from free atoms to the
Feshbach state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6-2 Four photon transfer of sodium and lithium atoms to the rovibrational
ground state of NaLi molecules in the 𝑎3 Σ * + potential via Feshbach
molecule state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6-3 Franck-Condon factors (FCF) calculated from ab initio potentials. . . 111
6-4 Atomic and molecular photon scattering losses . . . . . . . . . . . . . 114
6-5 Results of a simplified loss model for free-to bound Raman transition.

115

6-6 Numerical calculation of free-to bound Raman transition to the Feshbach state via 𝑣 * = 49 with atomic photon scattering. . . . . . . . . . 118
6-7 Numerical calculation of free-to bound Raman transition to the Feshbach state via 𝑣 * = 49 with atomic photon scattering. . . . . . . . . . 119
12

6-8 Molecular photon scattering losses by the upleg beam driving transition
to nearby excited states in the 𝑐3 Σ+ potential from the Feshbach state. 121
6-9 Raman transfer with atomic and molecular decay as a function of Δ𝑀
via different 𝑣 * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6-10 Illustration of a Raman transition from free atoms to the rovibrational
ground state of NaLi 𝑎3 Σ+ . . . . . . . . . . . . . . . . . . . . . . . . 126
6-11 Maximum transfer to the ground state of NaLi 𝑎3 Σ+ potential as afunction of intermediate excited vibrational state 𝑣 * .

. . . . . . . . . 128

6-12 Franck-Condon factors between vibrational states in 𝑎3 Σ+ and 𝑐3 Σ+potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6-13 Maximum transfer to an intermediate vibration state 𝑣 in NaLi 𝑎3 Σ+
potential as a function of intermediate excited vibrational state 𝑣 * . . . 130
6-14 Maximum transfer to a vibration state 𝑣 in NaLi 𝑎3 Σ+ potential as afunction of atomic density. . . . . . . . . . . . . . . . . . . . . . . . . 133
6-15 Illustration of a lambda type two-photon transition with multiple excited states near an intermediate excited state. . . . . . . . . . . . . . 136
6-16 Free-to-bound transfer efficiencies and two-photon detuning from both𝑐3 Σ+ and 𝑏3 Π potentials. . . . . . . . . . . . . . . . . . . . . . . . . . 142
A-1 Schematic of a PCB design for digital control instrumentation. . . . . 152
A-2 Board of a PCB design for digital control instrumentation. . . . . . . 154
A-3 Schematic of PCB design for analog signal buffer. . . . . . . . . . . . 155
A-4 Board of PCB design for analog signal buffer. . . . . . . . . . . . . . 156
A-5 Circuit diagram of Li MOT fluorescence trigger and 60Hz trigger. . . 156
B-1 Loss rates as a function of B-field and scaling of the spin-independent
three-body interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 160
B-2 Coupled-channel calculations with representative magnetic field scans

161

B-3 Coupled-channel calculations with representative magnetic field scans

162

C-1 Radial dependence of NaLi-NaLi interactions. . . . . . . . . . . . . . 170
13

C-2 Matrix elements of NaLi-NaLi interactions . . . . . . . . . . . . . . . 171
C-3 Schematic of the resonance model with two open channels and a p-wave
bound state trapped behind a centrifugal barrier . . . . . . . . . . . . 178
C-4 Coupling matrix elements between the open and closed channels . . . 179

14

List of Tables
1.1

Heteronuclear bialkali molecules . . . . . . . . . . . . . . . . . . . . .

21

2.1

Magnetic field coils and power supply units. . . . . . . . . . . . . . .

36

3.1

The Feshbach resonance position, width and peak loss rate constant .

49

6.1

Rotational constants of excited vibrational states (𝑐3 Σ+ , 𝑣 * ) . . . . . 122

6.2

Limiting factors of free-to-bound Raman transition. . . . . . . . . . . 134

6.3

AC Stark shifts and laser requirements. . . . . . . . . . . . . . . . . . 139

6.4

AC Stark shifts and laser requirements including NaLi 13 Π potential.

A.1 Parts of a PCB design for digital control instrumentation.

15

141

. . . . . . 150

16

Chapter 1
Introduction
The field of ultracold polar molecules is driven by the success of ultracold atoms which
provided many exciting and diverse applications such as atomic clocks, precision testsof fundamental physics, quantum simulations of many-body quantum phenomena,and new quantum information architectures. The underlying foundation is the adventof optical and magnetic trapping of atoms, various cooling methods including laser
cooling and evaporative cooling, which have been the most effective ways to cool
atoms to the submilli-Kevin and nano-Kelvin regimes, respectively, techniques tocontrol the quantum states both at the internal and external level, and to manipulateinteractions between atoms.
Extending these types of control to ultracold polar molecules offers exciting new
directions of research. Molecules have a richer energy-level structure that includes
rotation, vibration energy levels, and a relatively strong anisotropic interaction. In
addition, they can have a long-range electric dipole-dipole interaction when they arepolarized. Due to their additional properties, they can offer opportunities to study
quantum state controlled chemistry [1, 2], quantum simulation [3–5], and quantum
information processing [6–9] in a different regime.
However, even the simplest molecules in the quantum degenerate regime are extremely difficult to prepare because of the extra degrees of freedom. Due to therotation and vibration energy levels, a closed set of electronic transitions normallydoes not exist, and it prevents us from performing laser cooling of molecules. Col17

lective efforts have been made to obtain molecules in a single quantum state andunderstand the interactions among them. There are two main streams toward ultracold molecules, which laid the foundation for achieving atom-like control of ultracold
molecules: (i) direct cooling of molecules that are chemically prepared and (ii) ultracold assembly of precooled atoms. Cooling of chemically synthesized molecules
became possible using direct laser cooling. This method is restricted to only a handful of molecules that form a near-closed transition, that is, CaF, SrF, and YO [10–12].
The ultracold association method was first demonstrated in 2008 by two groups with
Cs2 and KRb and has since been widely used to create other diatomic molecules aswell [13–19].

1.1

Formation and properties of NaLi

The focus of this thesis is on the 23 Na6 Li molecules in their triplet ground state thatare formed by the association of precooled Na and Li atoms. Although direct coolinghas made significant advances in recent years [20–22], the highest density that canbe reached is more than a few orders of magnitude lower than that of the ultracold
association and the typical temperature that is achieved is tens of 𝜇K (never reachedthe nano-Kelvin regime). So far, ultracold association has been the most effective
method for producing near-quantum degenerate gases of molecules [23]. Furthermore,
it is natural to produce bialkali molecules using the ultracold association method,as the experimental apparatus, named BEC3, was designed for the production ofultracold Na and Li atom mixtures1 .
One of the most successful recipes for achieving ground-state molecules by ultracold assembly has two combined steps. The first step is the magnetic associationof cold atoms to weakly bound molecules near a Feshbach resonance. The second
step is the efficient formation of ground-state ultracold molecules via the stimulated
Raman adiabatic passage (STIRAP). Although this recipe allowed the production of
BEC3 was built in 2001 to produce 23 Na Bose-Einstein condensates and later upgraded in 2007
to coproduce 6 Li degenerate Fermi gases.
118

cold molecules as low as tens of nano-Kelvin, it also has some drawbacks. It requires
a Feshbach resonance to be at an experimentally accessible bias field and to be strong
compared to any decoherence rate for an efficient magnetic association. The Na +Li mixture has very narrow (∼mG) resonances only at high fields (>700 G), andusing these resonances for molecule formation is technically challenging. Formationof weakly bound NaLi Feshbach molecules was first reported in 2012 [24] by the same
BEC3 lab. Even after the success of the experiment, achieving the ground state of
NaLi was also demanding because there was no prior spectroscopic data available for
STIRAP. A detailed spectroscopic study of the excited and ground triplet potentials
of NaLi was carried out using one- and two-photon photoassociation spectroscopy,
providing the spectroscopic positions and strengths of the transitions to most of thevibrational states in the excited 𝑐3 Σ+ and ground 𝑎3 Σ+ potentials of NaLi [25, 26],
and the first formation of an ultracold gas of NaLi molecules in the triplet groundstate was achieved using the spectroscopic results in 2017. Initially, 3 × 104 molecules
were formed at a density of 5 × 1010 cm−3 and a temperature of 3 𝜇K [13]. This was
further improved, which is described in this thesis and in the thesis of Hyungmok
Son [27].
Several parameters are important when discussing heteronuclear bialkali molecules.
The two valence electrons on each alkali atom can pair in either a triplet or a singlet
spin configuration. The triplet potential is much shallower than the deeply boundsinglet potential, and thus energetically can be unstable. However, NaLi is uniquecompared to other heteronuclear bialkaline molecules and is particularly suitable forstudying the triplet state. NaLi in the triplet ground state has weak coupling to thesinglet manifold due to the weak spin-orbit coupling and thus is meta-stable. Indeed,
they were measured to have a long lifetime (∼ 4 s) at a density of 5 × 1010 cm−3 anda temperature of 3 𝜇K. Singlet molecules have anti-aligned electron spins and thus
have a very small magnetic moment from the nuclear spin, while triplet molecules
have aligned electron spins and thus have a magnetic moment of 2 𝜇𝐵 . The non-zeromagnetic moment enables magnetic trapping of molecules and can be used to tune
collisional resonances in molecules.
19

Furthermore, although it has the smallest permanent electric dipole moment
among all singlet ground state molecules, it has the fourth largest permanent electric
dipole moment among triplet ground state molecules, as listed in Table 1.1. NaLi in
the triplet ground state has both magnetic and electric dipole moments. These dipole
moments provide ∼ 1/𝑟3 character between molecules 𝑖 and 𝑗 as
𝑉𝑖𝑗𝑑𝑑 =

1 𝜇⃗𝑖 · 𝜇⃗ 𝑗 − 3(⃗𝜇𝑖 · ⃗𝑒𝑖𝑗 )(⃗𝜇𝑖 · ⃗𝑒𝑖𝑗 )
3
2𝜋𝜖0
𝑟𝑖𝑗

(1.1)

where 𝜇𝑖 is the dipole moment and ⃗𝑟𝑖 is the position vector of the molecule 𝑖. Here,⃗𝑟𝑖𝑗 = |⃗𝑟𝑖 − ⃗𝑟𝑗 |, and ⃗𝑒𝑖𝑗 is the unit vector in the direction of ⃗𝑟𝑖𝑗 . Note that the electric
dipole interaction is typically much stronger than the magnetic dipole interaction.
For example, for the electric dipole of value 𝑑 ≈ 1𝐷 has a four orders of magnitude stronger interaction compared to the magnetic dipole of value 1𝜇𝐵 . Tunablelong-range interactions between spins through the electric dipole-dipole interaction
in ultra-cold polar molecules can offer new possibilities for quantum simulation [3].
NaLi is fermionic since the total spin 𝐹 (nuclear and electronic) of the molecule ishalf-integer. The states converge to 8 hyperfine thresholds in the zero-field limit due⃗ + 𝐼⃗Na + 𝐼⃗Li |, whereto the conservation of the total angular momentum 𝐹 = |𝐹⃗ | = |𝑆
⃗ = 1 is the total electron spin of NaLi (𝑎3 Σ+ ) and 𝐼Na = 3/2 and 𝐼Li = 1 are
𝑆 = |𝑆|the nuclear spins of Na and Li. With the method explained above, NaLi molecules
are initially formed in the |𝐹 = 7/2, 𝑚𝐹 = 7/2⟩ hyperfine state, which is a stretchedstate in which all nuclear and electron spins are aligned to the direction of the bias
field. Fermionic molecules including NaLi can only collide via 𝑝-wave collisions thatare suppressed by the centrifugal barrier and therefore have much longer lifetimes
than their bosonic counterparts if they exist.
Chemical reactivity is another property that needs to be considered when it comesto molecular collisions. Any collision that transfers internal energy into relative kinetic
energy is likely to eject both collision partners from the trap. If both species are in
their absolute ground state (singlet), inelastic collisions are impossible, but there
remains the possibility of reactive collisions through atom exchange 𝐴𝐵 + 𝐴𝐵 →20

Molecule

PDM (Debye)

PDM* (Debye)

𝐶6 (a.u.)

LiNa

0.57

0.175

3709

LiK

3.58

0.312

411682

LiRb

4.18

0.359

884704

LiCs

5.59

0.462

30409406

NaK

2.78

0.0269516606NaRb

3.31

0.05941507089

NaCs

4.69

0.09146946696KRb

0.62

0.0540

17720

KCs

1.84

0.101

469120RbCs

1.25

0.0344

180982

Table 1.1: Permanent dipole moment of molecules in the ground state of the 𝑋 1 Σpotential (PDM) [28], permanent dipole moment of molecules in the ground stateof the 𝑎3 Σ potential (PDM* ) [29], and long-range dispersion coefficients in atomic
units [30]. permanent dipole moment

𝐴2 + 𝐵2 2 . For alkali-metal dimers 𝐴𝐵(𝑎3 Σ+ ) in the rovibrational ground state of thetriplet potential, the following reaction processes may lead to collisional losses:
2𝐴𝐵(𝑎3 Σ+ ) → 𝐴2 (𝑎3 Σ+ ) + 𝐵2 (𝑎3 Σ+ )(1.2)

2𝐴𝐵(𝑎3 Σ+ ) → 𝐴2 𝐵 + 𝐵(1.3)

2𝐴𝐵(𝑎3 Σ+ ) → 𝐴2 (𝑋 1 Σ+ ) + 𝐵2 (𝑇 )(1.4)

2𝐴𝐵(𝑎3 Σ+ ) → 𝐴𝐵(𝑋 1 Σ+ ) + 𝐴𝐵(𝑇 )(1.5)

where 𝑇 is 𝑋 1 Σ+ or 𝑎3 Σ+ . Collisions of triplet ground state molecules could result ininelastic loss from decay to states in the much deeper singlet manifold as well mediatedby nonadiabatic spin-dependent couplings. Thus in principle, singlet ground state,
non-reactive molecules are the most desirable candidates for long sample lifetimes.
Collisions between two NaLi molecules in the triplet ground state are highly re2

For singlet alkali-metal dimers in levels near the potential minimum, trimer formation reactions
2𝐴𝐵 → 𝐴2 𝐵 + 𝐵 or 𝐴 + 𝐴𝐵2 are always energetically forbidden [31].

21active, although spin-dependent couplings are weak and reactions involving transitions to lower-spin states (Eq. 1.4 and Eq. 1.5) are suppressed. The reaction processes within the quintet potential (Eq.1.2 and Eq.1.3) are exothermic [29] for NaLi.
However, the collisions between NaLi and Na are peculiar, unlike collisions between
two NaLi. Collisions of 23 Na6 Li(a3 Σ+ ) with Na are generally chemically reactive(23 Na6 Li(a3 Σ+ ) + Na → Na2 (X1 Σ+g ) + Li + heat). However, the dissociation energies for 23 Na6 Li and 23 Na2 in the lowest triplet state 𝑎3 Σ+ are 208.0826(3) cm−1 [25]and 163.7(12) cm−1 [32], respectively. Therefore, the chemical reaction in the fully
spin-polarized atom–molecule system is strongly suppressed due to the conservationof the total spin [29, 33], and stable long-lived states can be supported. Sympathetic
cooling of NaLi molecules with Na atoms has been demonstrated in a 1-dimensional
1,596-nm optical lattice [34], which is a significant step forward to reaching quantum
degeneracy with molecules.

1.2

Cold collisions in molecular systems

At low temperatures of micro- and nanokelvin, the collision occurs at low partial
waves, and collision physics can be reduced to a few well-defined parameters. For
many systems, these parameters are the 𝑠-wave scattering length and the two-bodyloss coefficients. However, because of the strong anisotropic interactions at shortrange and multiple decay channels that are possible in collisions involving molecules,
it is much more difficult to describe them. In relatively simple systems, such aspairs of alkali-metal atoms or light atom-molecule systems, it is possible to solve the
Schrödinger equation using coupled-channel methods. However, for heavier and more
complex systems, the number of channels needed for convergence is too large, andcoupled channel approaches become prohibitively costly. In this regime, considerable
success has been achieved with effective single-channel methods that take into accountshort-range inelastic/reactive loss with a single parameter [35, 36].
22

1.2.1

Universal loss model for reactive collisions

In particular, a single channel model by Zbigniew Idziaszek and Paul S. Julienne [35]
introduces a complex potential to represent the long-range potential (van der Waals
and centrifugal potential at 𝑟 ≳ 𝑅0 ) and short-range loss dynamics (at 𝑟 < 𝑅0 ):2

𝑙(𝑙+1)𝑈𝑙 (𝑟) = − 𝐶𝑟66 + ℏ 2𝜇𝑟− 𝑖 𝛾(𝑟)and an energy-dependent complex scattering length22

˜ 𝑅0 denotes the range of short-range forces 𝛾(𝑟) simulates all short-range𝑎˜=𝛼
˜ − 𝑖𝛽.
coupling at 𝑟 < 𝑅0 to exoergic non-threshold exit channels that result in loss from theentrance channel. The van der Waals (vdW) potential is characterized by the length𝑎¯ = 4𝜋𝑅6 /Γ( 14 )2 where 𝑅6 = 12 (2𝜇𝐶6 /ℏ2 )1/4 . Long-range dispersion coefficients 𝐶6of bialkali molecules are given in Table 1.1. The model uses the analytic frameworkof generalized multichannel quantum defect theory (MQDT) and the two quantum
defect parameters: phase parameter 𝑠 = 𝑎/¯𝑎 and loss parameter 𝑦 where 𝑎 is thescattering length. These parameters represent the effects of complex short-range
(︁ )︁
∫︀ 𝑟
∫︀
1
𝑖 𝑟 𝑘(𝑥)𝑑𝑥√
dynamics on the wave function Ψ(𝑟) ∼ √ 1 𝑒−𝑖 𝑘(𝑥)𝑑𝑥 − 1−𝑦
𝑒
. The
1+𝑦
𝑘(𝑟)

𝑘(𝑟)

first term represents the flux of incident particles, whereas the second term gives
the flux reflected from the short-range potential. The quantum defect parameter0 ≤ 𝑦 ≤ 1 is related to the probability of irreversible loss of incoming scattering
flux from the entrance channel due to dynamics at short-range. For 𝑦 = 1 there is aunit probability loss, which corresponds to a special “universal” case and for 𝑦 = 0
there is no loss (incident and reflected fluxes are equal). For arbitrary 𝑦 the complex
scattering length for 𝑠-wave and 𝑝-wave collisions are given by:
1 + (1 − 𝑠)2
𝑖 + 𝑦(1 − 𝑠)𝑦 + 𝑖(𝑠 − 1)𝑎˜𝑙=1 (𝑘) = −2¯𝑎1 (𝑘¯𝑎) 2
𝑦𝑠 + 𝑖(𝑠 − 2)𝑎˜𝑙=0 (𝑘) = 𝑎 + 𝑎
¯𝑦

(1.6)
(1.7)

when 𝑘 |˜
𝑎| ≪ 1 and 𝑘𝑅6 ≪ 1. In the case of 𝑦 = 1 the elastic and inelastic orreactive collision rates depend only on the quantum scattering by the long-range
potential at distances 𝑅 ≳ 𝑎¯. Any incoming scattering flux that penetrates inside𝑎¯ experiences no reflection back into the entrance channel as it is lost to inelas23

tic/reactive channels. Elastic rate constant 𝐾 𝑒𝑙 = 2𝑔 ℎ𝑘|˜𝑎(𝑘)|2 𝑓 (𝑘) and inelastic𝜇˜(𝑘) where 𝑘 2 = 2𝜇(𝐸 − 𝐸0 )/ℏ2 with 𝐸 denoting
rate constant 𝐾 𝑙𝑠 = 2𝑔 𝜇ℎ 𝛽(𝑘)𝑓the total energy, 𝜇 the reduced mass, and 𝐸0 the threshold energy. The function
[︁
]︁−1
2
2
˜
𝑓 (𝑘) = 1 + 𝑘 |˜𝑎(𝑘)| + 2𝑘 𝛽(𝑘)≈ 1 is the saturation factor which deviates from
1 when 𝑘 |˜
𝑎| ≪ 1 is not satisfied. Then universal rate constants (𝑦 → 1) for elastic
and inelastic collisions are given as:
ℎ 2
𝑒𝑙
𝐾𝑙=0
= 4𝑔 𝑘¯
𝑎
𝜇
ℎ 2
𝑙𝑠
𝐾𝑙=1
= 4𝜎𝑔 𝑘¯
𝑎 (𝑘¯
𝑎)4
𝜇 1

ℎ
𝑙𝑠
𝐾𝑙=0
= 2𝑔 𝑎
¯
𝜇
ℎ
𝑙𝑠
𝐾𝑙=1
= 2𝜎𝑔 𝑎¯1 (𝑘¯
𝑎) 2
𝜇

(1.8)
(1.9)

Here, 𝑎¯1 = 𝑎¯Γ( 41 )6 /(144𝜋 2 Γ( 43 )2 ) ≈ 1.064¯
𝑎. The factor 𝑔 = 1 except that 𝑔 = 2 whenboth particles are identical species in identical internal states, and the factor 𝜎 = 3 for𝑙 = 1 for molecules in the universal limit. This result implies that reactive collisions
undergo universal loss, which is greater for larger 𝐶6 and therefore NaLi is long-lived
due to the small vdW coefficient despite its reactivity.
Ironically, most reactive collisions and even many non-reactive collisions involving
one or more dimers that have been studied are known to go under fast collision loss andhave loss rates saturated or close to their universal value [41–49]. There are proposed
explanations for the fast loss of non-reactive molecules including formation of a longlived collision complex during collision which is referred to as “sticky collision" [37,38]
and photoassociation of intermediate complexes by the optical trap [39,40] as depicted
in Fig.1-1. As new ideas emerge, collisional studies directed to intermediate complexes
became an active research. The lifetime of the intermediate complex in RbCs + RbCs
collision was estimated by shining pulsed optical excitation light [50] and long-lived
intermediate complex was observed in KRb molecular systems [40, 51].

1.2.2

Fabry-Perot interferometer model for collision resonances

The previously introduced single-channel model that uses the analytic framework of
MQDT may not be so intuitive, although it has simple results. Therefore, we now
24photo-excitation“stiky”collsion

short
range

close
range

long
range

Figure 1-1: Photo-induced collisional loss. The collisions between two molecules(green and red sphere) occur in a potential which is the attractive vdW potential−𝐶6 /𝑅6 at long-range 𝑅>𝑅long (𝑅 is the inter-particle distance) and a strongly repulsive potential at short-range 𝑅<𝑅short . Most experiments on collisions involving
one or more dimers show fast collision loss that are saturated or close to their universal value and it is proposed that the fast loss of non-reactive molecules involve longlived intermediate complexes so called “sticky collision" [37,38]. These complexes arephoto-excited by optical trap, and near universal loss can be observed [39, 40].

25

focus on the simple picture of an optical Fabry-Perot interferometer with two mirrors
developed in Ref. [33]. This picture is analogous to cold collisions that are explainedby two reflectors, where the two reflections are from quantum reflection by the longrange vdW potential and reflection at short-range by the repulsive core potential as
in Fig.1-1. The flux transmitted by the inner mirror M2 in Fig.1-2 represents inelastic
loss where mirror M1 represents long-range reflection (quantum reflection or reflection
by the centrifugal barrier) and mirror M2 represents reflection at short-range. For an
incoming flux 𝐼, the total transmission 𝑇 through both reflectors is given by the flux
transmitted in the absence of the second mirror (𝐼 · |𝑡1 |2 ) times the interference term(𝐶):(︂
𝑇 = (𝐼 · |𝑡1 | )2

1 − |𝑟2 |2|1 − 𝑟2 𝑟1 𝑒−𝑖𝜑 |2)︂≡ (𝐼 · |𝑡1 |2 )𝐶(1.10)

Here, 𝑟𝑖 and 𝑡𝑖 are the amplitude reflection and transmission coefficients for the mirror
Mi and 𝜑 is the round-trip phase. The first term represents the universal loss since
it assumes no reflection by the inner mirror (𝑟2 = 0). The inner reflection is directly
related to the quantum defect parameter 𝑦 introduced in the single channel model as:√
𝑟2 = (1 − 𝑦)/(1 + 𝑦), 𝑡2 = 2 𝑦/(1 + 𝑦), which is 1 for complete transmission and 0
for 100 % reflection. Long-range reflection approaches unity at very low energies and𝑟1 ∼ 1. In this limit, 𝐶(𝑦, 𝜑) = 2𝑦/(1 − cos 𝜑 + 𝑦 2 (1 + cos 𝜑)) and at 𝜑 = 0 contructive
interference leads to an enhancement of 𝐶 = 1/𝑦 and at 𝜑 = 𝜋 destructive interference
leads to a minimum transmission with 𝐶 = 𝑦.
This Fabry-Perot interferometer model can explain collisional resonances as well.
When a new bound state is added to the interparticle potential and the close-range
potential is modified, the Fabry-Perot is tuned over one full spectral range. This takes
the scattering length 𝑎 over its full range between ±∞. The Fabry-Perot phase 𝜑 isdirectly related to the quantum defect parameter 𝑠 by: cos 𝜑 = 1−2/(1+(1−𝑠)2 ) andtherefore we obtain 𝐶(𝑦, 𝑠) = 𝑦(1 + (1 − 𝑠)2 )/(1 + 𝑦 2 (1 − 𝑠)2 ) which exactly reproduces
the results from Eq.1.6. The single channel model explained earlier can be extended
for a Feshbach resonance, which occurs when the incoming scattering channel becomes
resonant to a bound state by an external magnetic field. The Feshbach resonance at
26

(a)

Μ2

Μ1

loss

(b)

incoming
flux

outgoing
flux
short-range

long-range

Μ2

Μ1

loss

incoming
flux

outgoing
flux
long-range

short-range

Figure 1-2: Fabry-Perot interferometer model for reactive collisions. The
scattering dynamics can be fully described by quantum reflection off the vdW potential at 𝑅∼𝑅long and reflection and transmission at 𝑅∼𝑅short . Transmission into
short-range implies inelastic/reactive loss [35,52–54]. This situation is fully analogous
to an optical Fabry-Perot interferometer with two partially reflective mirrors (M1 and
M2 ). Reactive loss is proportional to the flux transmitted through both mirrors. Depending on constructive and destructive interference between multiple reflections, theloss can be highly enhanced (on resonance, Fig. (a)) or suppressed (Fig. (b)) than
the transmission through mirror M1 only, which represents the universal loss.

27

(︂𝐵=𝐵res acts as a lossless phase-shifter for the Fabry-Perot phase 𝑠(𝐵) = 𝑞 1 −Δ)︂where, 𝑞 characterizes the background scattering phase far away from the𝐵 − 𝐵resFeshbach resonance, and Δ is the width of the resonance.
The universal limit from the single-channel model assumes no short-range reflection and is given by all the flux that has not been reflected at long-range. The loss
rates can be modified only if the background loss rate is below the universal limit(𝑦 < 1) and the backreflected flux at short-range destructively or constructively interferes with quantum reflection via Feshbach resonances. Spin-polaried NaLi+Na withlow reactivity has the background loss rate more than an order of magnitude lower
than the universal limit. Therefore, it can be a possible system for the realizationof quantum chemistry controlled by an external field. In principle, due to its chemical reactivity, NaLi+NaLi is not a good candidate to observe magnetically tunablechemical reactions using Feshbach resonances. However, as we shall discuss shortly,
the experimental evidence has not corroborated this hypothesis.

1.3

Thesis outline

The preceding discussion illustrates that the ground state NaLi in the triplet potential
is an experimental system that has pushed the field of ultracold physics and quantum
chemistry. Since the first signal of the triplet ground-state NaLi molecules [13, 25,26] achieved on the first year of my Ph.D studies, I was involved in building thecontrol circuits, STIRAP laser system, and imaging laser system at a high magnetic
field for multiple hyperfine states of Na and Li atoms, which was necessary for two
important experiments: collisional cooling of NaLi with Na in an optical lattice [34],and control of reactive collisions by quantum interference [33]. These two experiments
are elaborated in the thesis of Hyungmok son [27].
This thesis focuses on three experiments that I led in the last two years of myPh.D studies after the two experiments mentioned above, and numerical/theoreticalwork on a new approach of creating ground-state NaLi using a free-to-bound Raman
28

transition. Chapter 2 provides an overview of the experimental apparatus, focusing on
recent modifications for the purpose of the formation and manipulation of molecules.
Chapters 3, 4 and 5 discuss the three experiments: the spectrum of Na + NaLi
Feshbach resonances, a Feshbach resonance in NaLi + NaLi collisions, and magnetic
trapping and cooling of NaLi. These experiments use magnetic quantum controlof NaLi in the triplet potential to understand molecular collisions and reactions at
ultracold temperatures. Chapter 6 discusses two-photon Raman transfer from Na
and Li atoms to NaLi molecules. Chapter 7 is a summary of the work with future
prospects.

29

30

Chapter 2
Experimental apparatus
The experimental apparatus or the BEC3 machine was built in 2001 and was initiallydesigned to produce 23 Na Bose-Einstein condensates. In 2007, the 6 Li atomic source
was added to the apparatus and modified to study 6 Li degenerate Fermi gases. After
upgrading, long efforts were made to create 23 Na6 Li molecules from ultracold atoms.
The further development of the apparatus that was necessary for molecular spectroscopy [25, 26] and the first production of 23 Na6 Li in the triplet ground state [13]is given in the thesis of Timur Rvachov. This chapter focuses on significant improvements to the apparatus after the first signal of the ground state molecules, necessary
for studying molecular collisions using magnetic quantum control.

2.1

Instrumentation for control

The control of the experiment is accomplished through the integration of commercial
and home-built apparatus. We use Cicero word generator for our experimental control
software, which can send messages to devices through RS-232, GPIB connections,and National Instruments (NI) digital and analog output cards. The purpose of thissection is to describe the instrumentation developed to generate reliable TTL signals
from the NI digital output and a circuit to buffer NI analog signals.
31

2.1.1

Digital control

To switch devices on and off during the experimental sequence, we use National
Instruments (NI) PXI-6533 digital output cards, in which output signals are sent
to home-built instrumentation to generate TTL compatible digital control signals.
This TTL compatible signal generating device is used to constitute two parts: "anopto-isolator box" and "a switch board". The opto-isolator electronic box isolates theground of the National Instrumentation cards from the rest of the system using a logic
gate optocoupler (FOD0720) and converts NI output low voltages (< 0.4V) or highvoltages (> 2.4V) to 0V or 5V via a digital buffer. After this stage, the converted
signals are sent to the "switch board" circuitry, which allows for either computeror manual control of the digital output voltage. It has two toggle switches and two
LEDs per channel. The first switch allows us to switch between computer and manualcontrol mode, and the second switch, which is enabled only when the first switch is
set to manual control mode, allows us to toggle between high and low output voltages.
Two LEDs indicate computer or manual mode and high or low output voltage. This
digital control instrumentation had a few problems, and redesigning was necessary for
future molecule experiments. First, all the wires were hand-soldered and very old andwould often fall apart. Second, it was not designed to isolate between different output
channels, which caused cross-talk among them. Lastly, the LEDs were powered byan output signal itself, and the brightness of the LED significantly depends on theamount of current drawn by the device being controlled. In many cases, LEDs on asingle "switch board" that contains eight output channels simultaneously light on andoff during experimental runs as a result. Figure 2-1 is a simplified schematic of theold circuit design, and figure 2-2 is a simplified schematic of the new circuit design fora single digital channel. Parts inside the green dashed box in figure 2-1 were includedin the old "switch boards" and the rest were in a separate "opto-isolator box". In the
new design, eight duplicates of every part in figure 2-2 were built in a single "switch
board". This was only possible by making the circuit compact by employing printed
circuit boards (PCBs). All output signals share the same ground which corresponds
32

Signal from NI Card
VDD1

VCC1(+5V)

Optocoupler

VDD2

VI

NC

*

VO

GND1

GND2

VCC2(+5V)

DPDT

VCC

OE
A

Y

GND
Buffer IC
Signal Out

SPDT

Figure 2-1: Schematic of old digital control instrumentation circuit for single
channel. This is a simplified schematic for single control output. Only the parts
inside the green dashed box were included in the old "switch boards" and the rest
were in a separate "opto-isolator box".
to ’GND2’ in figure 2-1 for the old design, while each output channel has individual
floated ground by having a separate DC to DC converter for the new design. (SeeAppendix A.1 for the full eight-channel PCB schematic and board design.)

2.1.2

Analog control

Analog control voltages (ranging from 0V to 10V) are used to vary laser frequencies
and intensities and magnetic fields in an experimental sequence. These voltages are
generated by PXI-6733 analog output cards and a newly developed low-noise systemusing high-precision DACs and floating power supplies by Will Lunden [55]. We use
the low-noise system to vary the intensity of our 1D optical lattice and the magnetic
bias field for Feshbach resonances. For the rest of the analog control, we use thevoltage signals generated by the analog output cards that are buffered by home-built
33

Signal from NI CardVI

VCC(+5V)

VO

DC

DC

VO

0V

VI

VO

DC
GND

L7805

GND1

GND

VCC(+5V)

VI

VI

DC

L7805

VDD1

Optocoupler

VDD2

VI

NC

*

VO

GND1

GND2

VO

GND2
0V

DPDT
Signal Out

VCC

OE
A

Y

GND
Buffer IC

SPDT

Figure 2-2: Schematic of newly designed digital control instrumentation
circuit for single channel. This is a simplified schematic for single control output.
In this new design eight duplicates of all the parts shown are built in a single "switch
board".

34

buffering circuits. For this, we use mainly two types of instrumentation amplifier(INA): either INA117 or INA111. INA117 has a large common-mode input rangeof ±200V and a protected input voltage of ±500V which is necessary to control the
output current of high-power power supply units. However, INA117 has a relatively√
large output voltage noise of, for example, 550 nV/ Hz at 10 kHz. On the other hand,
INA1111 has a small common-mode rejection voltage of ±10 V but low output voltage√
noise, for example, 10 nV/ Hz at 10 kHz which is even lower than the low-noise
system developed using high-precision DACs and floating power supplies. A circuit
diagram and the printed board layout for buffering analog signals using INA111 areprovided in Chapter A.1.

2.2

Coils and magnetic field

The ability to control the magnetic field in the main vacuum chamber where mostof the science happens is essential for cooling and trapping atoms and molecules.
For example, to prepare an ultracold Na/Li mixture in the lowest Zeeman energy
states (in the |𝐹, 𝑚𝐹 ⟩ basis, where 𝐹 is the total angular momentum and 𝑚𝐹 is the
projection of 𝐹 on the magnetic quantization axis, these states correspond to |1, 1⟩
for Na, and |1/2, 1/2⟩ for Li) for the magnetic association of NaLi Feshbach molecules
1, we produce an ultracold mixture of 23 Na and 6 Li in their upper-stretched hyperfine

states (|2, 2⟩ for Na and |3/2, 3/2⟩ for Li) using evaporative and sympathetic coolingin an Ioffe–Pritchard magnetic trap [57] at a bias field of ∼ 2 G. The mixture islater transferred to the lowest Zeeman states using a Landau–Zener (LZ) magnetic
field sweep in an optical trap (see Chapter 2.3.3 for details). The magnetic trap
requires two sets of four cloverleaf coils that create radial gradient fields in the 𝑥
and 𝑧 directions, and curvature coils that produce a parabolic field curvature in the𝑦 direction. Since the curvature coils also produce a substantial bias field along
Ultracold mixture of 23 Na and 6 Li in their lowest Zeeman states (|1, 1⟩ for Na and |1/2, 1/2⟩ forLi) is required for the formation of NaLi Feshbach molecules. Our previous work on NaLi Feshbach
molecules uses the 745.4 G 𝑑-wave Feshbach resonance between Na and Li in the lowest Zeeman
energy states, which has a width of 10 mG [56].
135Power supply unit
CurvatureBiasy-compy-trap-bias
spin-flip
Feshbach
MOK

Connected coils
curvature & antibias
curvaturecompensationcompensationcompensation
antibias
MOK coil

Bias field direction
(+)
(+)
(+)
(-)
(-)
(-)
(+)

Table 2.1: Magnetic field coils and power supply units.
𝑦, a single power supply unit("Curvature") is connected to the curvature coils andthe antibias coils, which can produce a relatively homogeneous bias field in the −𝑦direction (see Table 2.1 for details).
In addition, to study magnetically tunable scattering resonances, one should also
be able to stabilize magnetic fields at high fields. For example, the formation of NaLi
Feshbach molecules requires active stabilization of magnetic fields up to ∼ 800 G,
with field noise less than 1 part in 105 and the experiments described in chapters 3
and 4 require active stabilization of magnetic fields up to ∼ 1500 G. For producing
a high magnetic bias field, we use antibias coils powered by the "Feshbach" power
supply unit.

2.2.1

Curvature cancellation

Although antibias coils are meant to produce a homogeneous bias field, they can create
significant curvature fields (forming a saddle point) when the magnetic bias field bythese coils is high. Unfortunately, there is a mismatch between the position of thesaddle point of the field by antibias coils and the center of the Ioffe-Pritchard trap,
where the atoms are trapped for evaporative and sympathetic cooling. Therefore,
at a high bias field, atoms and molecules can feel a large gradient field if they areplaced at the center of the Ioffe-Pritchard trap. The displacement is estimated to
be about 2.5 mm in the 𝑥 direction and a few hundred 𝜇m in the 𝑦 direction. Thisestimation was done by monitoring the acceleration of atoms near the magnetic trap
center which is driven by the curvature fields. To prevent atoms and molecules from36

feeling a large gradient field, after evaporative and sympathetic cooling of the atoms
in the magnetic trap, we had been applying an additional gradient field in the 𝑥
direction using a compensation coil (x-compensation) that shifts the magnetic trap
center to near the saddle point. This leads to heating and poor transfer efficiencyof the atoms from the shifted magnetic trap to an optical trap2 . One approach tosolving this problem is to cancel the curvature created by the antibias coils. When themagnetic curvature is canceled out, atoms or molecules will not see a huge magnetic
field gradient, even though they are not positioned at the saddle point.
The goal is to have the curvature created by the current through the antibias coils
canceled out by that of the current through the curvature coils as shown in figure 2-3.
The radius of antibias coils is larger than the spacing between sets of coils on the left
and right sides. However, the spacing is larger than the radius for the curvature coils.
Therefore, the signs of the curvature created by these two sets of coils are opposite
to each other when the currents flow in the same direction. For this purpose, anadditional power supply was connected anti-parallel to the "Bias" power supply unit.
A rough condition to cancel out the curvature was found by maintaining the bias
field near 745 G and monitoring Na atoms in the state |2, 2⟩ (low-field seeker) and Li
atoms in the state |1/2, 1/2⟩ (high-field seeker) simultaneously. When the 210 amps
pass through the antibias coils and the 22 amps pass through the curvature coils, themagnetic bias field is near 745 G while the curvature field in the 𝑦 direction (axial
direction) is close to zero. However, this condition did not allow us to eliminate
inhomogeneity in the radial direction (𝑥𝑧 plane) because the saddle point created by
the curvature coils is also displaced by a few 𝜇m in the radial direction from thatcreated by the antibias coils.

2

In principle, this should not affect the transfer efficiency. However, since the displacement
is significant, the shifted magnetic trap is no longer an approximate harmonic trap but can havemultiple local minima.

37

Curvature
cancellation PS

Solid state relay

Curvature coils

175V

Anti-bias coils

TTL

UltrastabFeedback

TTL
IGBT

IGBT

Feshbach PS

Bias PS
~4
TTL

IGBT

18V
Curvature PS

Figure 2-3: Circuit diagram of curvature field cancellation.

38

2.3

Laser systems

2.3.1

STIRAP lasers

The formation of NaLi molecules in the triplet ro-vibrational ground state (𝑣 =0, 𝑁 = 0) from the Feshbach state (𝑣 = 10, 𝑁 = 2, 𝑚𝑁 = −2) is carried out using thetechnique called the stimulated Raman adiabatic passage (STIRAP). This techniqueuses two laser pulses designed to keep the molecules in a coherent dark state that isdecoupled from the intermediate excited state. For the molecular potentials of NaLi,
we decided to use 833 nm (upleg) and 819 nm (downleg) laser frequencies to couple
the NaLi Feshbach and ground states via the 𝑣 * = 11 excited vibrational state in
the 𝑐3 Σ+ potential, which are chosen to have strong downleg Rabi coupling with acompromised upleg coupling strength. Figure 2-4 is an energy level diagram showing
the STIRAP transitions. The relatively strong downleg transition requires only a few
100 𝜇W power and we use a home-built 818 nm external cavity diode laser (ECDL)3
while the weak upleg transition requires higher power and we use an injection locksetup with a high power laser diode4 seeded by a home-built 833 nm ECDL.
The two STIRAP lasers are locked to a commercial optical cavity made of ultralow expansion (ULE) glass with a linewidth of 15 kHz. This cavity allowed for locking
lasers over a range of 810-880 nm to relative linewidths of < 1 kHz. However, weobserve that STIRAP transfer from the Feshbach state to the ground state efficiency
drops significantly for STIRAP pulses longer than 50 𝜇s. This is probably due tothe upleg laser, which has much higher power than the downleg laser, acting on thedownleg path (transition between 𝑣 = 0, 𝑁 = 0, 𝑚𝑁 = 0 and 𝑣 * = 11, 𝑁 = 1, 𝑚𝑁 =−1). The strong upleg laser can induce AC Stark shift of the ground state on theorder of 10 kHz, leading to large two-photon detuning5 .
3

We use the ECDL design developed by the Steck group at the University of Oregon for the
STIRAP ECDLs
4
We use a laser diode mount and a 650 mW 830 nm single mode laser diode from Thorlabs(model: LD830-SE650) for the injection lock.
5
Problem of two-photon detuning due to AC stark shift becomes much more serious for free-tobound STIRAP described in Chapter 6.

39

Figure 2-4: STIRAP lasers.

2.3.2

Imaging & cleaning

To reliably detect molecules and atoms, we need resonant laser beams for atoms at
high bias field ( 745 G). Initially, imaging systems for Na in state |𝐹 = 1, 𝑚𝐹 = 1⟩
(upper-stretched state |𝐹 = 2, 𝑚𝐹 = 2⟩) and Li in state |𝐹 = 1/2, 𝑚𝐹 = 1/2⟩ (upperstretched state |𝐹 = 3/2, 𝑚𝐹 = 3/2⟩) were built by past members of the BEC3 teamto image atoms near 745 G (at low magnetic field near 0 G). In order to study
collisions between atoms and molecules, imaging/cleaning laser systems for Na instate |𝐹 = 2, 𝑚𝐹 = 1⟩ and |𝐹 = 2, 𝑚𝐹 = 2⟩ at high field6 were implemented later by
setting up a double-pass acousto optical modulator (AOM) for each from the Na state|𝐹 = 1, 𝑚𝐹 = 1⟩ high field imaging beam. In addition, an imaging beam for Na in
the lower-stretched state |𝐹 = 2, 𝑚𝐹 = −2⟩ was implemented to study collisions in
the lower-stretched NaLi + Na mixture in the quartet potential. After the magneto
optical trap (MOT) stage, the Na slower beam is frequency shifted by a −70 MHz
6

Although we colloquially call 745 G as "the high field", it is not truly the high-field regime for
Na in the ground state. 𝑚𝐽 is not a good quantum number, and the state |𝐹 = 1, 𝑚𝐹 = 1⟩ does
not have a well-defined quantum number 𝑚𝐽 = −1/2 but also has a small fraction of 𝑚𝐽 = 1/2
component. Therefore, the Na 𝐷2 transition from the ground state |𝐹 = 1, 𝑚𝐹 = 1⟩ to the excitedstate |𝐹 ′ = 0, 𝑚𝐹 ′ = 0⟩ using 𝜎− polarized light is not a closed cycling transition, but the excitedstate can decay to state |𝐹 = 2, 𝑚𝐹 = 1⟩ as well.

40single-pass AOM and sent to the imaging path for state |𝐹 = 2, 𝑚𝐹 = −2⟩ imaging
at high field.

2.3.3

Optical trapping

The Na/Li mixture is transferred from a magnetic trap to the combination of two
coaxial optical traps: a 1,064 nm optical dipole trap (ODT) (30 𝜇m waist, 10 W
power) and a 1,596 nm 1D optical lattice (50 𝜇m waist, 2 W power). After 0.4 s offorced evaporation in the 1,064 nm optical dipole trap, the trap is switched off andthe science is carried out in the 1,596 nm 1D lattice. The use of 1,596 nm light for thetrap improves the lifetime of the NaLi molecules by suppressing spontaneous photon
scattering compared to the more common 1064 nm trap. Later, instead of using a
combination of the two coaxial optical traps, the mixture is transferred to a 1,550 nm
1D optical lattice (30 𝜇m, 10 W power) and forced evaporated in the lattice. The
1D lattice provides strong axial confinement to overcome the anti-trapping producedby magnetic curvature even without shifting the magnetic trap or canceling out thecurvature created by antibias coils7 .

7We realized that we do not need to shift the position of the magnetic trap center with a compensation coil in order to well-confine the atoms and molecules only when we performed the third
experiment described in this thesis i.e., magnetic trapping and cooling of NaLi molecules.

41

42

Chapter 3
Spectrum of Feshbach resonances in
NaLi+Na collision
Collisional resonances that are electromagnetically tunable have become an established tool for modifying interactions between ultracold atoms and have been the
key for many applications, from magnetic association of loosely bound molecules toquantum simulations [58, 59]. For ultracold molecular systems, tunable collisional
resonances are a powerful measure for understanding interaction potentials, probe
collision complexes, and control chemical reactions [60].
In the case of cold collisions between alkali-metal atoms, the number of resonant
states remains typically small, and the resonances are usually tractable. However,in the case of cold collisions involving molecules, due to strong and anisotropic interactions, ro-vibrational excited states can also contribute to resonant states, andtherefore resonances themselves may not be well separated and are difficult to identify. Due to the large density of states of molecular systems [37, 39] it has been achallenge to perform rigorous scattering calculations and different methods enabling
efficient scattering calculations have become an active field [61, 62]. Despite these
efforts, due to the extreme sensitivity of the low-temperature observables to detailsof potential energy surfaces, [63, 64] there has been very little success in predicting
Feshbach resonances in molecular collisions.
Rather than performing exact quantum calculations, using simple statistical short43

range models while treating the physics of long-range scattering within multichannel
quantum defect theory is one feasible alternative approach that has been pursued forthe description of molecular resonances [37–39,65]. However, the validity of statisticalshort-range models is controversial and, therefore, detailed experimental and further
theoretical studies are needed.
Here we use a combined experimental and theoretical quantum chemistry approach, which, for the first time, sheds some light onto the nature of deeply boundcomplexes and the bottom-line mechanism of atom-molecule Feshbach resonances.
There are two approaches to understanding and characterizing resonances in collisions involving ultracold molecules. Resonances supported by long-range states of
the collision complex may be trackable, since the quantum numbers of the separated
atoms and molecules are approximately preserved. For example, resonant states incollisions involving Feshbach molecules [66–69] and in collisions between ground state
NaK and K [70, 71] were successfully analyzed in this way. In contrast, resonances
supported by collision complexes in which different quantum numbers are mixed due
to a strong anisotropic electronic interaction, as in the case for NaLi + Na [33], itis more difficult to understand the nature of the bound complexes. In these terms,
patterns of resonances can provide information on the interaction potential and resonant states. In either case involving tightly bound molecules, the underlying couplingmechanism has not yet been characterized.
We report the map of Feshbach resonances in collisions between triplet ro-vibrational
ground-state NaLi and Na, both prepared in the maximally spin-stretched state. NaLi
+ Na in the quartet potential is the lightest bialkali+atom system that has been realized and is chemically stable. Collisions of 23 Na6 Li(a3 Σ+ ) with Na are generally
chemically reactive (23 Na6 Li(a3 Σ+ ) + Na → Na2 (X1 Σ+g ) + Li + heat). However, thechemical reaction in the fully spin-polarized atom–molecule system is strongly suppressed due to the conservation of the total spin [29, 33], and stable long-lived states
can be supported. The system is suitable for modeling molecular scattering resonances because of the relatively small density of states and number of electrons, andtherefore more accurate quantum calculations are feasible compared to other heavier
44molecular systems.

3.1

Experimental protocol and results

Our experiments use triplet ground-state 23 Na6 Li (𝑎3 Σ+ ) molecule and Na atom mixtures in the spin-polarized quartet potential, which are chemically stable. The Hamiltonian that includes hyperfine coupling and Zeeman energy for triplet ground-state
23

Na6 Li is given as
𝐻 = 𝐻HF + 𝐻Zeeman⃗ · 𝐼⃗Na + 𝑎2 𝑆
⃗ · 𝐼⃗Li + 𝜇𝐵 (𝑔𝑠 𝑆⃗ + 𝑔𝐼 𝐼⃗Na + 𝑔𝐼 𝐼⃗Li ) · 𝐵⃗= 𝑎1 𝑆
Na
Li
ℏ

(3.1)

where 𝑎1 = 433.2(1) MHz and 𝑎2 = 74.6(1) MHz [13]. There are 36 states in the
ground rotational state manifold (𝑁 = 0). The states converge to 8 hyperfine thresholds in the zero-field limit due to the conservation of the total angular momentum⃗ + 𝐼⃗Na + 𝐼⃗Li |, where 𝑆 = |𝑆|⃗ = 1 is the total electron spin of NaLi
𝐹 = |𝐹⃗ | = |𝑆(𝑎3 Σ+ ) and 𝐼Na = 3/2 and 𝐼Li = 1 are the nuclear spins of Na and Li. The hyperfine
splitting due to the Na nucleus is significantly larger than that due to the Li nucleus,⃗ + 𝐼⃗Na | = 1/2, 3/2, and 5/2 is an approximately good quantum number thatso 𝐹1 = |𝑆characterizes the largest-scale hyperfine splittings in the zero-field limit. The energy
level diagram of NaLi in the triplet ground state (𝑎3 Σ+ , 𝑣 = 0, 𝑁 = 0) at zero bias
field is shown in Fig. 3-1.
There are a total of two possible states in the spin-polarized quartet potential, the“upper stretched state" where all nuclear and electron spins are aligned to the bias
magnetic field direction (|𝐹, 𝑀𝐹 ⟩NaLi + |𝐹, 𝑀𝐹 ⟩Na = |7/2, 7/2⟩NaLi + |2, 2⟩Na ) and the“lower stretched state", where all nuclear and electron spins are anti-aligned to thefield direction (|7/2, −7/2⟩NaLi + |2, −2⟩NaLi ). Here, 𝐹 is the total spin (electron andnuclear) and 𝑀𝐹 is the 𝐵- field projection of 𝐹 . The molecular and atom mixture in
the “upper stretched hyperfine state" with typical numbers of ∼3×104 and ∼3×105
respectively is produced in a 1, 596-nm one-dimensional optical lattice following themethod described in ref. [33, 72]. The mixture of “lower stretched state" is produced45

F = 7/2
F1 = 5/2

7/2

mF = 7/2

100 MHzF = 5/2

5/2

mF = 5/2

F = 3/2

3/2

mF = 3/2

1132 MHzF1 = 3/2

F = 5/2
F = 3/2
F = 1/2

5/2

mF = 5/2
mF = 3/2
mF = 1/2

1/2

mF = 1/2

1/2

3/2

561 MHzF = 1/2
F1 = 1/2

F = 3/2

mF = 3/2

3/2

Figure 3-1: Rovibrational ground state energy level diagram of NaLi in
the triplet potential (𝑎3 Σ+ ) at zero bias field. Quantum number 𝐹 = |𝐹⃗ | =⃗ + 𝐼⃗Na + 𝐼⃗Li | is the total angular momentum, where 𝑆 = |𝑆|⃗ = 1 is the total|𝑆electron spin of NaLi and 𝐼Na = 3/2 and 𝐼Li = 1 are the nuclear spins of Na and⃗ + 𝐼⃗Na | = 1/2, 3/2, and 5/2 is an approximately
Li. In the zero field limit, 𝐹1 = |𝑆
good quantum number that characterizes the largest-scale hyperfine splittings as thehyperfine splitting due to the Na nucleus is significantly larger than that due to the Li
nucleus. Molecules are formed in the low field seeking spin-polarized hyperfine state(|𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩) and is indicated in blue.

46

by coherent transfer from the sample of “upper stretched state" using a magnetic fieldsweep in the presence of radio frequency waves at a low bias field [73]. For this process,the bias field is dropped from 745 G where “upper stretched state" is prepared to a
low field near 8 G in 15 ms. After the state preparation, the bias field is ramped toa target value in 15 ms. We determined the collisional lifetimes of the atom-molecule
mixtures by holding the sample for a variable time at the target magnetic field.

The loss of NaLi molecules with Na atoms is measured as a function of the bias
field for both spin-polarized states. The number of molecules left after the bias field is
swept down by 12.6 G in each target field for 200 ms and normalized by that without
the field sweep is shown in Fig. 3-2 from near zero to 1420 G in steps of 8.76 G.
A fine scan that is at least a factor of 10 finer is performed in the fields where a
noticeable loss is found from the coarse scan in Fig. 3-2. We identified 8 resonancesin the upper stretched state and 17 resonances in the lower stretched state indicatedwith red vertical lines in Fig. 3-2.
The approximate spectral information of each resonance is estimated by fitting
the observed loss features from a finer search to a simple Lorentzian function witha slope (the slope is to consider the overall change in the background loss and theeffect of nearby other resonances). An example of a resonant loss feature near 884
G from a finer scan and a Lorentzian fit is shown in Fig. 3-3(a). The result of the
fit is summarized in Table 3.1. The resonances are labeled with numbers in orderof increasing resonance position in Gauss for two different initial collision channels,
respectively. For three resonances 4,5 and 6 in the upper stretched state, the loss
rates are mapped as a function of the bias field for a more accurate estimation of theresonance position and width, as shown in Fig. 3-3(b). For these three resonances,the center and width obtained by a Lorentzian fit to the loss rates instead of the lossfeature are summarized in Table 3.1. We confirmed that the approximate widths fromobserved loss features can be broader due to saturation compared to the more accurate
widths from the loss rate measurements with resonance 5 in the upper stretched state47Normalized NaLi number

1.0
0.8
0.6
0.4
0.2
0.00
1.0

Normalized NaLi number

(a)

200

400

600

800

1000

1200

1400

200

400

600

800

1000

1200

1400

(b)

0.8
0.6
0.4
0.2
0.00

Bias field (G)Figure 3-2: Collisional loss spectrum of NaLi molecules with Na atoms asa function of magnetic field a. in upper stretched hyperfine states andb. in lower stretched hyperfine states. Y-axis is the number of NaLi left after
sweeping down the bias field by 13 G at each field for 200 ms, normalized by the
number of NaLi without the field sweep. Each data point represents three to six
measurements with and without the sweep respectively. The number of Na atoms
in each pancake for each measurement was around 115, and the number of NaLi
molecules without the field sweep was about 35. We observed total of 25 resonances:
8 in the upper spin-polarized mixture and 17 in the lower spin-polarized mixture. Red
vertical lines indicate the position of Feshbach resonances. The errorbar represents
one standard deviation of the mean. Grey lines are interpolated data with a piecewise
cubic polynomial to guide the eye.

48

Collision channel : |7/2, 7/2⟩NaLi + |2, 2⟩Na
𝐵0 (G)Δ𝐵 (G)𝛽 (cm3 s−1 )1203.7(2)5.3(7)

1.5(6) × 10−112

237.9(1)6.3(4)

7.1(1) × 10−113372.0(7)20(3)

2.3(5) × 10−114657.8(3)5.2(8)

1.0(1) × 10−105884.3(3)10(2)6(1) × 10−11

6

978.2(2)4.9(3)

7.4(5) × 10−1071029.7(3)10(2)

1.5(3) × 10−1181135.2(8)15(3)

3.0(5) × 10−11

Collision channel : |7/2, −7/2⟩NaLi + |2, −2⟩Na
𝐵0 (G)Δ𝐵 (G)𝛽 (cm3 s−1 )182.5(1)

1.3(2)

1.0(3) × 10−102145(1)

18(4)5.3(9) × 10−113309(1)

13(4)

1.9(4) × 10−114361.82(5)1.0(2)

4.5(8) × 10−115449.8(2)3.1(5)

6(2) × 10−116478(3)

35(13)

3.4(6) × 10−117

561.8(2)2.4(6)

8(2) × 10−118590.7(2)9(1)8(2) × 10−119642.3(5)3(2)

2.9(4) × 10−11

10661.5(2)3.9(9)

2.4(5) × 10−11

11722.5(1)2.7(3)

8(4) × 10−1112860.8(2)2.1(6)

2.4(6) × 10−1013963.3(2)6(1)2(1) × 10−10141030.1(2)3.3(8)

9(3) × 10−11151083.3(3)4(1)5(1) × 10−11

161176.3(3)3.2(9)

8(2) × 10−11171269.2(1)2.6(3)

3.3(5) × 10−11

Table 3.1: The Feshbach resonance position, width and peak loss rate constant. Resonance positions and widths are obtained by Lorentzian fit to loss features,
except for resonances 4 to 6 for the incoming collision channel |7/2, 7/2⟩NaLi + |2, 2⟩Na .
The center and width are obtained by Lorentzian fit to the field-dependent loss rates
instead for these three resonances.

49

0.8
0.6
0.4
0.2
0.0
820

Loss rate (1/s)

101

(a)
840

Molecule number

Normalized NaLi number

1.0

860

880

900

920

880

900

920

30
20
10
0
0.00

0.05

0.10

0.15

Time (sec)

100

(b)
820

840

860

Bias field (G)Figure 3-3: Feshbach resonance in NaLi + Na upper stretched state near
884 G. a. NaLi number as a function of bias field where the NaLi + Na mixture areheld for 100 ms normalized to the number without the hold. b. Loss rate of NaLi
molecules with about 100 Na atoms in each pancake as a function of bias field. The
dashed blue lines are the fits to a simple Lorentzian function with a slope. The inset
is the molecule number per pancake as a function of hold time near the center of theresonance (∼ 884.3 G).

50

near 884 G. The width obtained from a Lorentzian fit to the loss feature of resonance
5 is 17(1) G, which is broader (about 50 %) compared to that from the loss ratemeasurement, which is 10(2) G.
The decay curves are taken as a function of the hold time near the center ofloss features as in the inset of Fig. 3-3(b) for the peak loss rate constants 𝛽 andare summarized in the last column of Table 3.1. Sodium densities are calibrated
by comparing the measured decay rates with the decay rate of the mixture in a
non-stretched spin state which is predicted to occur with the 𝑠-wave universal rate.
This method was validated in ref. [33] by confirming that the measured loss rate in|7/2, 7/2⟩NaLi +|1, −1⟩Na collision is, within the uncertainty, the 𝑠-wave universal rate,
which is well known for our system [74].
The upper and lower stretched Na + NaLi loss rate constants away from resonances at low field near 0.5 G are 7.5(2.2) × 10−12 cm3 s−1 and 6.7(2.0) × 10−12 cm3 s−1 ,respectively, and at high field near 540 G are 4.5(1.4) × 10−12 cm3 s−1 and 3.5(1.0) ×
10−12 cm3 s−1 , respectively. These background loss rates are more than an order ofmagnitude smaller than the universal loss rate constant for Na + NaLi 𝑠-wave collisions, which is 1.7 × 10−10 cm3 s−1 . The loss rate constants of the two spin-polarized
mixtures match both at low-field and high-field within the uncertainty. For bothstates, the background loss rates are lower at the higher magnetic field.

3.2

Analysis

One would expect the upper stretched state to have more possible decay channels
due to its higher energy. However, we have observed similar rates of backgroundloss and Feshbach enhanced losses for both states, and even more resonances for the
lower stretched state. This implies that Zeeman energies do not play a major roleand that dipolar relaxation is not a dominant decay mechanism for the collisional
complex. Instead, the decay is probably dominated by short-range chemical reactions
or inelastic loss, which are expected to be similar for both spin-stretched initial states.
In the range up to 1500 G, we observed 8 resonances in the upper stretched state51

and 17 in the lower stretched state. This observation provides the density of stableatom-molecule bound states in our system. The linewidths of the resonances range
from about 1 G to about 30 G and the spacing between resonances is on the order of
100 G. If the linewidths are interpreted as due to the finite lifetime of the resonant
state, this implies atom-molecule complexes with lifetimes in the range of 10 to 350
ns. One possible interpretation of the spacing between resonances is the rotational
structure of the intermediate complex. The spacing between resonances of the orderof 100 G corresponds to single spin-flip energy differences of 280 MHz and a rotational
constant with a moment of inertia for a complex of size around 30 𝑎0 . Note that therotationally excited states of the NaLi triplet molecules 𝑁 = 1 are at 9.25 GHz and𝑁 = 2 are at 27.75 GHz [13] corresponding to a double spinflip energy at 1650 G and5000 G, respectively.
We quantify the resonance statistics of 17 resonances from Na+NaLi collisions
in the lower stretched state provided in Table 3.1 using the Brody parameter 𝜂,which is a standard measure of chaos. For non-chaotic systems, in which resonances
have no correlations, the distribution of nearest-neighbor spacings is given by the
Poisson distribution, 𝑃P (𝑠) = 𝑒−𝑠 . On the other hand, for chaotic systems, which
emerge when the mean spacing between bound states is comparable to the couplingstrength, the repulsion between energy levels occurs. In this regime, the distributionof nearest-neighbor spacings is given by a Wigner-Dyson distribution, 𝑃WD (𝑠) =𝜋 2
𝜋𝑠𝑒− 4 𝑠 [75, 76]. The two distributions are smoothly interpolated through the Brody
2
𝐴

𝜂+1

parameter 𝜂 as 𝑃B (𝑠) = 𝐴𝑠𝜂 𝑒− 𝜂+1 ·𝑠 , known as the Brody distribution, where 𝐴 =(︁ )︁𝜂+1
𝜂+2(𝜂 + 1) · Γ 𝜂+1[77, 78]. Here, 𝜂 = 1 and 𝜂 = 0 lead to the Wigner-Dyson andPoisson distribution, respectively. The cumulative probability function of the Brody
distribution is given as
𝐹𝐵 (𝑠) = (𝜂 + 1) · [1 − 𝑒−𝛼𝑠

𝜂+1

].

(3.2)

Fig. 3-4 shows the cumulative probability of resonance spacing of the 17 resonancesfrom Na+NaLi collisions in the lower stretched state. The best fit of the data to Eq.
3.2 gives 𝜂 = 1.1(1), which shows the statistical signature of quantum chaos.
52

Cumulative propability

1.0

best fit
=1
=0

0.5

0.0

0

50

100

Spacing (G)

150

200

Figure 3-4: Cumulative probability of resonance spacing from Na+NaLi collisions in
the lower stretched state. The black line is the best fit to the cumulative probability
function of the Brody distribution given as Eq. 3.2 and the blue (red) dashed line isthe cumulative distribution with 𝜂 = 1 (𝜂 = 0).

53Porter-Thomas

12

Count

10
8
6
4
2
00

1

2

3

4

Scaled reduced width5

Figure 3-5: Distribution of scaled reduced widths for scattering resonances in thelower stretched state Na+NaLi mixture which are listed in Table 3.1. The PoterThomas distribution of scaled reduced widths is in black line.

However, statistical conclusions from only 17 resonances are speculative. On thebasis of our quantum-scattering calculations (see Chapter B.1), the Wigner-Dyson
statistics are not expected because each resonance can be assigned by 𝑀𝑆 , 𝐽, 𝑀𝐽
quantum numbers. (Resonances with different values for these quantum numbers do
not see one another.) In addition, it is difficult to conclude that the broad trend of the
distribution of resonance widths follows the Porter-Thomas distribution of resonancewidths, 𝑃PT (¯𝛾 ) = 𝛾¯ −1/2 𝑒−¯𝛾 /2 , which is also a statistical character of quantum chaos.
Here, 𝛾¯ is the normalized velocity width divided by the mean value of all normalized
widths [79]. Fig. 3-5 shows the histogram of the scaled reduced resonance width
distribution and the Porter-Thomas distribution. Due to the small statistical sample,
it is difficult to compare the broad trend of the distribution with the Porter-Thomas
distribution.
Here, we have focused on analyzing the spectrum of Feshbach resonances solely
based on the experimental observations. However, there are some open questions that54

are not answered with immediate interpretations, and some of the interpretations
may need further study for justification. First, the mechanism of coupling between
the initial scattering channel and the loss channels or resonant closed channels that
leads to back ground loss or resonances is not explained. Hypothetically, spin-flip
transitions can be induced by intramolecular spin-spin, spin-rotation, atom-moleculemagnetic dipole-dipole interactions, and others. What are the dominant interactions
that are responsible for the resonances and the background loss, and the detailed
mechanism, is still an open question. Second, there is insufficient information to
explain the qualitative difference between the upper stretched states and the lower
stretched states. One may expect the observed difference in the number of resonances
between two spin-stretched initial states to result from the difference in the density ofstates because resonant coupling can probe the chemical structure of the complex, andthe number of resonances is closely related to the density of states. However, only after
further investigation can we make a decisive statement on the qualitative difference
between the two states. Lastly, the nature of the collisional complex is inconclusive.
The validity of the simple interpretation of the spacing between resonances as therotational energy splitting of the complex, which leads to a complex size of around
30 𝑎0 is debateable. In addition, the analysis does not provide any other informationon the nature of the complex. For example, one may ask what the good quantum
numbers are of these collisional complexes. To unravel these open questions, we use
a combined experimental and theoretical quantum chemistry approach. The detailsof quantum calculation are presented in Chapter B.1.

3.3

Interpretation

In this section, we will summarize the key points understood by the combined experimental and theoretical quantum chemistry approach rather than going throughthe details of the quantum calculations. Even state-of-the-art quantum chemistrycalculations cannot predict the position of resonances because of the uncertainty inthe interaction potentials. However, they can provide a deep understanding of the55

relevant states and their couplings.
Background loss. The inelastic losses observed in Na + NaLi collisions are due
to transitions to lower lying states — these are the singlet states of Na2 or NaLi,
and in the case of the upper stretched state, also lower Zeeman states. Similar
losses observed for the upper and lower stretched states imply that Zeeman relaxation
is not dominant. The loss is therefore due to transitions from the quartet to the
doublet potential which occur at close range. If the input state already has a doublet
component (e.g. by using Na in a different hyperfine state), the loss rate is much
faster since the reaction can occur directly in the doublet potential and does not needa spinflip process, which is weak.
Feshbach resonance. Feshbach resonances increase the wavefunction at close range
by coupling the open channel to closed channels. Our simulations show that the closed
channels are 3-body complexes in the quartet state with rotational excitation, eitherof NaLi (quantum number 𝑁 ), or rotation of Na around NaLi (quantum number 𝐿).
Doublet states have only a minor role. These complexes have a typical size of 20 − 40
𝑎0 . The input channel has 𝑁 = 0, 𝐿 = 0. The input state and the complex state
involve different 𝑀𝑆 , which is necessary for a magnetic Feshbach resonance. Moleculareigenstates in different Zeeman levels can be coupled by a strong anisotropic electronicinteraction since the different molecular Zeeman states have different rotational state
decompositions due to mainly spin-spin coupling and also spin-rotation coupling. As
a result, the complexes have mechanical rotation 𝐽 = 2. Note that spin-spin coupling
depends on the orientation of the molecular axis and, therefore, provides tensorial
coupling between the spin and the mechanical angular momentum. The enhanced
loss near Feshbach resonances is either due to enhanced loss via the open channel orby a short lifetime of the admixed closed channel. At least for the strong resonances,
it is the former.
Number of resonances. Our simulations can explain why the number of observed
resonances is smaller for the upper stretched state. Simulations with decay to lower
Zeeman states removed show a similar number of resonances. This implies that thedifference in the number of resonances is not due to a different density of states56

but is due to the rapid decay of some of the intermediate complexes which broaden
the resonances and makes them unobservable. Those complexes decay to lower-lying
Zeeman states via spin-spin coupling.

57

58

Chapter 4
Degeneracy-induced magnetic
resonance in NaLi+NaLi collision
One of the powerful methods used in cold atoms is Feshbach resonances, which canhappen when a molecular bound state in the closed channel approaches the thresholdof the scattering state in the open channel. They act as fingerprints of molecularinteraction potentials because of their high sensitivity to the shape of the molecularpotential energy surface and shed light on expanding our understanding of collision
dynamics.
However, there have been only a few discoveries of magnetically tunable Feshbach resonances in a molecular system. Exploration of Efimov physics was carriedout by observation of Feshbach resonances in collisions of loosely bound Cs2 andNaRb Feshbach molecules [66–69,80]. There are two atom-molecule systems in which
magenetically tuneable Feshbach resonances are observed. Scattering resonances in
K+NaK collision identified an intermediate complex as a loosely bound state between the molecule and the atom [71] and in Na + NaLi collision probed three-bodytightly bound states [33]. Feshbach resonances in collisions between two ground-state
molecules have not yet been identified before.
Here we report the observation of a pronounced, isolated Feshbach resonance incollisions between fermionic NaLi molecules in their ro-vibronic ground state. The
magnetically tunable resonance is extremely narrow (∼ 25 mG) and enhances the59loss rate by more than two orders of magnitude, providing strong evidence for astable and long-lived collision complex. The existence of long-lived complexes ina molecular system of high reactivity, such as NaLi, is unexpected and has strong
implications for controlling ultracold chemistry via scattering resonances. The longlived state revealed by our experiments is coherently excited, whereas so far, all otherobservations of collisional complexes are compatible with an incoherent population
described by rate equations.

The observed resonance is special in two regards: In simple models, resonantly
enhanced losses are only possible if the background loss rate is much smaller than
the so-called universal limit. A loss rate near the universal limit implies almost
complete inelastic loss at short-range and should suppress any long-lived resonant
state. However, we observe loss rates close to the universal limit outside the narrow
resonance. Second, the NaLi + NaLi Feshbach resonance is observed at a finite
magnetic field where two open channels become degenerate, and it is possibly a new
type of Feshbach resonance with a mechanism different from the Feshbach resonances
observed in ultracold atomic systems. This mechanism is unique to molecule-molecule
collisions, as the required single-molecule degeneracies cannot be realized in atomic
systems at practicable field strengths. On the contrary, two-molecule degeneracies
used to engineer shielding interactions in ultracold KRb + KRb and CaF + CaF
collisions, are commonly found in molecular collisions. Our results suggest that the
new type of degeneracy-induced magnetic resonance could be ubiquitous in ultracold
molecular physics, offering an exquisite new mechanism for tuning intermolecularinteractions with external electromagnetic fields. We present simple models which
explain the observed behavior.
60

(a)NaLi number(a.u.)

1.0

0.6

334.8

(b)

NaLi Number

NaLi number
per pancake

30

334.9

20

335.0

Bias field (G)334.92 G
334.82 G
745 G

30
20
10
0 0

1

2

Hold time (s)

10

0.00

0.02

Hold time (s)0.04

Figure 4-1: a. Resonant molecular loss as a function of magnetic field. The
remaining molecule number after holding for 30ms at a target field is plotted near334.9 G. Molecule numbers are normalized by the molecule number without the 30
ms hold. Each data point represents three to six measurements with and without 30
ms hold respectively, and the error bars are 1 standard deviation. The black dashedline is a Gaussian fit. The blue (green) vertical line indicates the field where themolecular decay curve in blue (green) is obtained in Fig. 4-1(b). b. Moleculardecay curves at 334.92 G resonance and away from the resonance. The
main plot shows the molecular decay curve near the center of the resonance within 50
ms. The upper right subplot shows decay curves away from the resonance at 334.82
G in green squares and at 745 G in red triangles and also near the resonance at 334.92
G with blue circles from 0 to 2 s. Dashed lines are fits to a simple model for two-bodyloss using mean square regression.

61

4.1

Measurement & Experimental Result

4.1.1

Experimental sequence

We prepare ground-state 23 Na6 Li (𝑎3 Σ+ ) molecules in their lower stretched hyperfine
state, where all nuclear and electron spins are anti-aligned to the bias field direction.
For the state preparation, we first produce triplet ground state molecules in the upper
stretched hyperfine state, where all spins are aligned along the bias field, via magnetic
association followed by stimulated Raman adiabatic passage (STIRAP) near 745 G
in a 1 dimensional (1D) 1596 nm optical lattice [13, 33, 72]. The bias field is dropped
from 745 G to a low field near 8G in 15ms for coherent microwave state transferfrom the upper stretched state (|𝐹, 𝑚𝐹 ⟩ = |7/2, 7/2⟩) to the lower stretched state(|𝐹, 𝑚𝐹 ⟩ = |7/2, −7/2⟩). After the state preparation, field is ramped to a target
value in 15ms. The search for scattering resonances is done for the bias field range of
40.5 G < 𝐵 < 1401.6 G. After waiting for a certain time for the molecules to collide
at a target field, we return the field to ∼ 8 G for reverse state transfer (|7/2, −7/2⟩ →|7/2, 7/2⟩) for detection. We used absorption imaging of lithium atoms dissociatedfrom molecules to measure the number and temperature of molecules, and thus thefield was brought to 745 G. An extra time of 15 ms < t < 30 ms after each magnetic
field ramp was sufficient for the bias field to settle within the range of the magnetic
field inhomogeneity across the molecular sample.
Within more than 1000 G range in bias field, only one Feshbach resonance isobserved as shown in Figure 4-1(a). While the number of molecules decreases by
about 40% in 600 ms at most bias field values, more than half of the molecules
are lost after 30 ms hold near 334.914(10) G (Fig.4-1(b)). In order to verify thatthe enhanced loss is due to a resonance in collisions between two fermionic NaLi
molecules in the same hyperfine state, we investigate three aspects: the effect of
impurities, the density dependence and temperature dependence on molecular decay
rates. We confirm that the loss enhancement is not due to impurities by examining
the decay curves at the resonance. About 30 molecules per lattice site decreases down
to about 7 molecules in 40 ms as shown in Figure 4-1(b) and are depleted to the level62

that is barely detectable in less than a hundred milliseconds. A little fraction ofcontamination of the molecular sample that may exist can only cause depletion of thesample up to the contamination level mainly through s-wave collisions. This confirmsthat the loss enhancement comes from two identical fermionic molecules and not fromother unwanted states.

4.1.2

Model for molecular decay & density calibration

Exploration of the other two aspects: density dependence and temperature dependence on molecular decay rate requires careful modeling of molecular decay and density calibration. We model the two-body loss with a differential equation that takes
into account the time dependence of temperature: [23]3𝑇˙ (𝑡)𝑛(𝑡)˙= −𝛽(𝑇 (𝑡))𝑛2 (𝑡) − 𝑛(𝑡)
,
2
𝑇 (𝑡)

(4.1)

where 𝛽 is the two-body loss rate constant, 𝑛 is the mean density, and 𝑇 is thetemperature of the molecules. Anti-evaporation causes an increase in temperatureof up to 50% within a molecular decay time near 334.9 G. We fit the measured
temperatures to a linear function of time, 𝑇 (𝑡) = 𝐻𝑡 + 𝑇𝑜 , where 𝐻 is the heating
rate and 𝑇𝑜 is the initial temperature. Both away from and near the resonance, theloss rate coefficient has a temperature dependence that can be expressed as 𝛽 =𝛽0 (𝑇 (𝑡)/𝑇0 ), where 𝛽0 is the initial loss rate coefficient when the temperature is 𝑇0 .
To determine the rate coefficient from equation Eq. (4.1) requires accurate knowledgeof the molecular density. The mean molecular density can be expressed with theeffective number of particles, 𝑁 eff and the mean volume, 𝑉eff , of molecules for a single
pancake, as 𝑛 = 𝑁 eff /𝑉eff .
We obtained the effective particle numbers for a single pancake from the measured
number of molecules, 𝑁 tot , and the distribution of the numbers over the pancakes.
The observed axial profile of NaLi follows a Gaussian form with width 𝜎=450(60)𝜇m,
so we assume a Gaussian distribution of the number of particles per pancake. As
the average weighted over a Gaussian, the effective particle number per pancake is63√𝑁 eff =𝑁 tot · 𝑎/(2 𝜋 · 𝜎), where the lattice constant is 𝑎=𝜆/2 and 𝜆=1596 nm.
The trap volume of each pancake, 𝑉eff , is determined from the measured molecule(0)

temperature and trap frequencies. For a purely harmonic trap, one obtains 𝑉eff =𝜔¯ −3 (4𝜋𝑘𝐵 𝑇 /𝑚)3/2 where the geometric mean of the NaLi trap frequencies is 𝜔
¯ =(𝜔𝑥 𝜔𝑦 𝜔𝑧 )1/3 .
However, there are two corrections which we determine separately: (1) the confinement in each pancake is moderately anharmonic, and (2) the system is in the crossover regime between quasi-2D and 3D, 𝑘𝐵 𝑇 ∼ ℏ𝜔𝑧 .
First, the anharmonicity of the trapping potential leads to a modified mean volume
(1)

𝑉eff

∫︀
(1)
1/𝑉eff =𝑑𝑉 e−2𝛽𝑈 (𝑟,𝑧)
(4.2)

∫︀
[ 𝑑𝑉 e−𝛽𝑈 (𝑟,𝑧) ]2

where 𝑈 (𝑟, 𝑧) is the potential of a single lattice site. We use the same trap model
validated in ref. [33] to determine 𝑈 (𝑟, 𝑧) and integration limits. With typical con(1)(0)

ditions for molecular loss measurements, the mean volume 𝑉eff is larger than 𝑉eff

by less than 25%. However, some measurements at lower density required weakeroptical traps for which the anharmonicity correction is larger and had to be taken
into account for proper density calibration.
Second, the tight confinement in the lattice direction makes the classical thermal
distribution for harmonic trapping in the pancakes invalid for low molecular temperatures. We estimate the corrected volume as∑︀
2 𝛽(−𝑖− 12 )ℏ𝜔𝑧 2𝑑𝑉 ( ∞)
𝑖=0 𝜌(𝑟)|𝜑𝑖 (𝑧)| e∫︀∑︀∞1[ 𝑑𝑉 𝑖=0 𝜌(𝑟)|𝜑𝑖 (𝑧)|2 e𝛽(−𝑖− 2 )ℏ𝜔𝑧 ]2∫︀(2)
1/𝑉eff =(4.3)

where 𝑟 is the radial coordinate, 𝑧 is the axial coordinate along the beam direction,𝛽 = (𝑘𝐵 𝑇 )−1 , 𝜑𝑖 (𝑧) is the 𝑖th eigenstate of the axial harmonic oscillator and 𝜌(𝑟) is
the classical thermal distribution in the radial direction, which is a Gaussian function.
The black line in Figure 4-2 is the result of Eq. 4.3 using the typically radial trap
frequency (𝑓𝑥 , 𝑓𝑦 ) = (450, 640) Hz as a function of the ratio of thermal energy to(0)axial zero-point energy. The red dashed line is 1/𝑉eff and the blue dashed line is the(2𝐷)

inverse of the quasi-2D volume 1/𝑉eff, assuming that all atoms in the 0𝑡ℎ eigenstate
64Mean Density (N eff × cm 3)

1011

(2)
1/Veff(2D)
1/Veff(0)
1/Veff

1010

9
1010
1

kBT/

z

100

Figure 4-2: Mean density estimation in the 3D to quasi-2D crossover regime
as a function of the ratio of the thermal energy to axial zero-point energy.
(0)Black line is the result of Eq. 4.3, red dashed line is the estimated 1/𝑉eff and blue
dashed line is the inverse of the quasi-2D volume where it is assumed that all atoms
are in the 0𝑡ℎ eigenstate of the axial harmonic oscillator, 𝜑0 . Calculation is with thetypically radial trap frequency (𝑓𝑥 , 𝑓𝑦 ) = (450, 640) Hz.

of the axial harmonic oscillator, 𝜑0 :∫︀

(2𝐷)
1/𝑉eff =𝑑𝑉 [𝜌(𝑟)|𝜑0 (𝑧)|2 ]2∫︀.
[ 𝑑𝑉 𝜌(𝑟)|𝜑𝑖 (𝑧)|2 ]2(4.4)

For all the experimental conditions which is indicated with gray shaded area in Fig.
(0)

(2)

4-3, 𝑉eff differs from the more accurate 𝑉eff by less than 20% 1 . Since it is unclear(1)

how this correction might interact with the larger correction from 𝑉eff , we include(2)

𝑉eff only as an enlargement in the uncertainty.

1

(2𝐷)

(2)

(0)

(2)

Difference between 𝑉eff and 𝑉eff is much larger than that of 𝑉eff and 𝑉eff . This implies thatthe system is closer to 3D than quasi-2D in terms of molecular density

65Mean density ratio

2.2

Density Correction Factor

2.0
1.8
1.6
1.4
1.2
1.0
10 1

(2)/V (0)
Veff
eff
(2)
(2D)
Veff /Veff

kBT/

z

100

Figure 4-3: Density correction factor. Gray shaded area indicates the conditions(2)(0)
in which the experiments were carried out. Ratio of 𝑉eff to 𝑉eff is in red and ratio(2)(2𝐷)of 𝑉eff to 𝑉eff is in blue.

66

=3

NaLi initial decay rate (s 1)

102

=2
101
=1

0
1010
10

1011

NaLi initial density (cm 3)

1012

Figure 4-4: Density dependency of molecular decay rate at 334.92 G. The
initial decay rates are plotted as a function of initial molecule mean density. The(𝛾−1)green, blue and red dashed lines are the fit results for 𝛾 = 1, 2 and 3 in 𝑅𝑜 = 𝐴𝑛𝑜with 𝜒2𝑟𝑒𝑑 = 13.50, 1.43 and 20.42 respectively.

4.1.3

Density and temperature dependence on molecular decay rates

To understand the loss mechanism, the molecular decay rate is measured as a function(𝛾−1)

of initial density and modeled with an equation 𝑅𝑜 = 𝐴𝑛𝑜

as shown in Figure

4-4. 𝑅𝑜 is an initial decay rate, 𝑛𝑜 is an initial molecular density, 𝛾 is a parameter
determined by the number of particles involved in collision loss and 𝐴 = 𝛽𝑁𝑜 where 𝑁𝑜
is an initial number of molecules. In general, the loss rate constant 𝛽 is temperaturedependent, and to treat 𝐴 as a constant we controlled the initial temperatures of themolecules to be the same within 15 %. 𝛾 = 1.82(9) gives the best fit, confirming thatthe resonant loss is due to two-body collisions. Molecule densities are estimated fromthe lattice trap frequencies and molecular temperature, as explained in Chapter 4.1.2.
We map the temperature dependence of the molecular decay rate constant and67Rate Coefficient (cm3s 1)

10 9

334.92G
745G

10 10
10 11
10 12
1.0

2.0

3.0

Initial Temperature (uK)

4.0 5.0

Figure 4-5: Threshold laws of molecule-molecule collisions. Initial rate coefficients are plotted as a function of initial temperature of NaLi molecules. Bluedata points are measurements near the center of the resonance and red data points
are measurements away from resonance near 745 G. Dotted lines are the fits to thepolynomial function 𝛽 = 𝐶𝑇 𝑙 .

68compare it with the Wigner threshold law [81]. We generate molecular gases atdifferent temperatures by varying the initial temperature of the Na and Li atomic
mixture. The initial temperature of the molecule ranging from 0.74(8) uK to 3.88(36)uK and from 1.33(7)uK to 4.40(25) uK is achieved away (745G) and at resonance(334.92 G), respectively. We model the temperature scaling with the polynomial
function 𝛽 = 𝐶𝑇 𝑙 and obtain 𝐶 = 1.8(7) × 10−10 𝑐𝑚3 /(𝑠𝐾) and 𝑙 = 1.5(4) near the
resonance in blue dotted line and 𝐶 = 1.7(5) × 10−10 𝑐𝑚3 /(𝑠𝐾) and 𝑙 = 0.98(27) asshown in Fig. 4-5. This p-wave Wigner threshold law [81] like behavior implies thatthe loss is caused by collisions between two identical fermions2 .

4.1.4

Loss rate constant

We map out the measured loss rates as a function of the magnetic bias field. Naturalcomparisons for the observed decay rates are the unitary limit and the universal loss
rate. As the 1D optical lattice creates a trapping potential tightly confined in theaxial direction, our experiments are done in the crossover between 2D and 3D physics.
𝑢𝑛𝑖𝑡.
The p-wave unitarity limit in a three dimensional trap, 𝛽3𝐷=6 𝜇ℏ 𝜆𝑑𝐵 , where 𝜆𝑑𝐵 is√︀the de Broglie wavelength given as 𝜆𝑑𝐵 = 2𝜋ℏ2 /𝑘𝐵 𝜇𝑇 and 𝜇 is the reduced mass

which is half of the NaLi molecule mass, 𝜇 = 𝑚NaLi /2 3 . The unitarity limit in a
√
𝑢𝑛𝑖𝑡.
𝑢𝑛𝑖𝑡.
quasi-2D trap, 𝛽2𝐷=4 𝜇ℏ ( 𝜋𝑙0 ). 𝛽2𝐷
scales linearly to the oscillator length in thetightly confined direction, 𝑙0 = (ℏ/𝑚NaLi 𝜔𝑧 )1/2 , instead of the de Broglie wavelength,
𝜆𝑑𝐵 (see Chapter C.1 for details). Here, 𝜔𝑧 is the trap angular frequency in the tightly
confined direction. We use an approximate value of NaLi-NaLi long-range dispersion
coefficient (𝐶6 = 5879 a.u) obtained by summing all 𝐶6 coefficients between two
constituent atoms [83]. The universal loss rate constant for 𝑝-wave (𝑠-wave) collisions
2

The system is in the quasi-2D to 3D crossover regime, and therefore, the Wigner threshold law
in both quasi-2D and 3D regime had to be derived for proper comparison with the experimental
result. We confirm in C.1 that 𝛽 ∝ 𝑇 in both regimes. ⟨
⟩
⟨
⟩
3

𝜋ℏ
𝑢𝑛𝑖𝑡.
Inelastic loss rate constant is given by 𝛽3𝐷= 𝑔𝜎 𝑚𝑢
1

1
𝑘𝑟𝑒𝑙

in ref. [82] where1
𝑘𝑟𝑒𝑙

is thermally1averaged 𝑘𝑟𝑒𝑙 = 𝜇ℏ 𝑣𝑟𝑒𝑙
, 𝑔 = 2 for indistinguishable particles, and the degeneracy factor 𝜎 = 3 for
√︁
⟨
⟩
2𝜇
1
𝑝-wave collision. The thermal averaging of the inverse of relative velocity 𝑣𝑟𝑒𝑙=
𝜋𝑘𝐵 𝑇 , and√︀
𝑢𝑛𝑖𝑡.
therefore, 𝛽3𝐷=6 𝜇ℏ 𝜆𝑑𝐵 , where 𝜆𝑑𝐵 is the de Broglie wavelength given as 𝜆𝑑𝐵 = 2𝜋ℏ2 /𝑘𝐵 𝜇𝑇 .

69

𝑢𝑛𝑖𝑣
𝑢𝑛𝑖𝑣
is 𝛽𝑙=1/𝑇 = 1.2 × 10−12 𝑐𝑚3 /𝑠 · 𝜇K (𝛽𝑙=0= 1.85 × 10−10 𝑐𝑚3 /𝑠).

The rate constant, 𝛽, increases by more than two orders of magnitude as thebias field approaches 334.918(5) G from near 𝑝-wave universal value to above 𝑠-wave
universal value. Loss rate coefficients are plotted as a function of magnetic field andfitted to a Lorentzian function for two temperatures, 1.8 𝜇K and 4.2 𝜇K, in Fig.4-6.
The loss rate constant contrast is ∼ 155 for 1.8 𝜇K and ∼ 230 for 4.2 𝜇K. The rate
constants at the peak are far below the 3D unitarity limits but approach near the2D unitary limits. The width of the resonance is as narrow as or narrower than theinhomogeneity of the magnetic field across a molecular sample, ∼ 25 mG, given thefact that the Lorentian widths from the fits for both 1.8 𝜇K and 4.2 𝜇K are also ∼ 25
mG. The result of 4.2 𝜇K shows an overall higher loss rate constant compared tothat of 1.8 𝜇K as expected from the p-wave threshold law at the resonance, 𝛽 ∝ 𝑇 ,mentioned earlier.

4.2

Analysis

We will now develop a model that addresses our major experimental findings. The
universal limit assumes that the loss rate is given by all the flux which has not been
quantum reflected, i.e. it has tunneled through the centrifugal barrier located at(︀ 6 )︀1/4∼ 2.2𝑅vdw with the van der Waals (vdW) length 𝑅vdw = 21 2𝜇𝐶≈ 66𝑎0 . Ratesℏ2higher than the universal limit observed here are only possible (1) if losses occur atlong-range (outside the 𝑝 wave barrier) as depicted in Figure 4-7, or if substantial backreflection from short-range (behind the centrifugal barrier) destructively interfereswith quantum reflection.
A single narrow resonance between two NaLi molecules in their lower stretchedhyperfine state is located at a magnetic field where there are multiple other hyperfine
states that are energetically close by. We first rule out the possibility (1) by showingthat these states are not directly coupled strongly to the initial scattering state at
long-range. We then focus on the second possibility (2). Nearly degenerate states
may have indirect coupling mediated by a bound state of an intermediate collision70

Two-body loss rate constant (cm3s 1)

10 8

10 9

unit.
3D (4.2 K)unit.
3D (1.8 K)unit.
2D (4.2 K)unit.
2D (1.8 K)
univ.
l = 1 (4.2 K)
univ.
l = 1 (1.8 K)
univ.
l=0

10 10

10 11

10 12
334.8

334.9

Bias field (Gauss)

335.0

Figure 4-6: Enhanced two-body loss rate coefficient of molecules. Initial twobody loss rate coefficients which are indicated with blue squares and green circles,
are plotted as a function of the bias field. Data points are obtained using a simple
two-body loss model for initial temperatures ∼ 4.2 uK (blue squares) and ∼ 1.8 uK(green circles). Solid lines are Lorentzian fits to the data points. The dotted redline is the 𝑠-wave universal value and dash-dot lines are p-wave universal values for
𝑇 = 4.2 uK (blue) and 𝑇 = 1.8 uK (green). Dashed horizontal lines are the 3D
unitarity limits and solid horizontal lines are the 2D unitarity limits.

71

Potential Energy (uK)

40

400

l(l + 1)/R 2

200
0

20

0

p waves wave50 100

150

a+b
0

0

Rb

C6/R 64080

R (nm)

a+a
120

160

Figure 4-7: Long-range coupling. a+a is the initial collision channel where a is thelower stretched state. a+b is another collision channel coupled to a+a at long-range
𝑅𝑏 = 100 nm where b is a near-degenerate hyperfine state.
complex. Our analysis shows that such long-range coupling is too weak to play arole in the observed molecular resonance (see chapter 4.2.1 for details). This suggeststhat the intermediate bound state does not suffer from short/close range loss and theresonance arises in an open channel.

4.2.1

Hyperfine structure of NaLi

There are a total of 9 hyperfine states crossing the lower stretched state at bias fields
between 320 G and 340 G. There are three states that cross at fields within a few
hundred mG of the resonance. Due to the uncertainty of the hyperfine constants,
the uncertainty of their energies relative to the stretched state is ±400 kHz, andtherefore each of them is a possible candidate to cross the stretched state at or near
the observed Feshbach resonance at 334.92 G.
The state indicated with an red "a" in Figure 4-9 is the lower stretched state72

|𝐹 = 7/2, 𝑚𝐹 = −7/2⟩. The three states close in energy near the resonance are indicated as 𝑏1 , 𝑏2 and 𝑏3 in Figure 4-9. The most likely candidate to couple to thestretched state is the state 𝑏1 = |𝐹 = 5/2, 𝑚𝐹 = −3/2⟩ since its nuclear spin character
overlaps with the stretched state by 50%, whereas for the other two states, non-zero
nuclear spin overlap less than 0.1% may arise from small intra-molecular spin-spin(−551 MHz) and spin-rotation (14.5 MHz) couplings. More explicitly, The Zeeman
states can be expanded in the fully uncoupled basis set|𝑁, 𝑚𝑁 , 𝑆, 𝑚𝑆 , 𝐼𝑁 𝑎 , 𝑚𝐼𝑁 𝑎 , 𝐼𝐿𝑖 , 𝑚𝐼𝐿𝑖 ⟩(4.5)⃗ and of its
where |𝑁, 𝑚𝑁 ⟩ are the eigenstates of the rotational angular momentum 𝑁
projection on the 𝐵-field axis, |𝑆, 𝑚𝑆 ⟩ are those of the electron spin angular momentum, and |𝐼, 𝑚𝐼𝑖 ⟩ are the eigenstates of the nuclear spin angular momenta of Na andLi (𝑖 = 𝑁 𝑎, 𝐿𝑖). The leading terms in the expansion of the Zeeman states involved in
the avoided crossings near 𝐵 = 334.92 G are, neglecting small (≲ 1%) contributions
from the 𝑁 = 2 rotationally excited states,73|𝑎⟩ = |𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = − ⟩ = |0, 0, −1, −1, − ⟩ ,223|𝑏1 ⟩ = |𝐹1 = 3/2, 𝐹 = 5/2, 𝑚𝐹 = − ⟩23
1
1≈ 0.71 |0, 0, 1, −1, − ⟩ − 0.57 |0, 0, 0, −1, − ⟩ − 0.40 |0, 0, −1, −1, ⟩ ,22
2
1|𝑏2 ⟩ = |𝐹1 = 3/2, 𝐹 = 5/2, 𝑚𝐹 = − ⟩21
13≈ 0.76 |0, 0, 0, −1, ⟩ − 0.55 |0, 0, 1, −1, − ⟩ + 0.31 |0, 0, −1, −1, ⟩ ,22
21|𝑏3 ⟩ = |𝐹1 = 3/2, 𝐹 = 5/2, 𝑚𝐹 = − ⟩2
3
1≈ 0.91 |0, 0, 0, −1, ⟩ − 0.38 |0, 0, 1, −1, ⟩ .
2
2

(4.6)

Additionally, the difference in 𝑚𝐹 to the lower stretched state, is the smallest forstate 𝑏1 which is 2 4 .
Spin flip processes can, in principle, happen at long-range outside the 𝑝-wave
4

State 𝑏1 lies ∼ 100 kHz above the lower stretched state at 334.92 G in figure 4-9. Given theuncertainty of the hyperfine constants, this value can be as high as 480 kHz or as low as −300 kHz

73

Zeeman shift (MHz)

100

150

200

3000
2000
1000
0

250

1000
2000
3000

300300

320

340

0

200

360

B field (G)

400

600

380

800

1000

400

Figure 4-8: Hyperfine structure of NaLi(𝑎Σ+ ) in an external magnetic field.
The blue dashed vertical line indicates the position of the Feshbach resonance (∼334.92 G). The subplot show the Zeeman shifts from 0 to 1000 G whereas the main
plot is zoomed into where there are 9 near degenerate states.

74213.0

Zeeman shift (MHz)

a
213.4

b1
b2

213.8

b3
214.2

334.4

334.6

334.8

B field (G)

335.0

335.2

Figure 4-9: Hyperfine structure of NaLi(𝑎Σ+ ) in an external magnetic field.
𝑎 in red is the lower stretched hyperfine state of NaLi molecules. 𝑏1 , 𝑏2 and 𝑏3 in blue
are other hyperfine states that are energetically close to state 𝑎 near 334.92 G.

75

barrier due to the dipole-dipole interaction. However, due to the selection rules of themagnetic dipolar interaction (|Δ𝑚𝑆 | = 1), a single spin flip cannot provide coupling
between the near-degenerate hyperfine states of interest |𝑎⟩ and |𝑏1 ⟩, which correspond
to 𝑚𝑆 = 1 and 𝑚𝑆 = −1, respectively. Therefore, only weak higher order spinflips5 or
spin-rotation couplings exist. A loss rate constant higher than 10−10 cm3 s−1 requires
a coupling strength on the order of 100 kHz at long-range (∼ 100 nm)6 . However, thestrongest higher-order process has a coupling strength around 0.05 kHz. We therefore
assume that we have a 𝑝-wave resonance enabled by high reflectivity at close range.

4.2.2

Three-state model of 𝑝-wave resonant scattering

To explain the observed Feshbach resonance, it is necessary to assume the presenceof a short-range loss mechanism. A minimal model for such a mechanism involves
a bound state (state 3) coupled to the open channels. The simplest model for the
Feshbach resonance near state degeneracies that involves enhanced loss contains two
open channels |1⟩ and |2⟩ coupled to a stable bound state, channel |3⟩, where |1⟩ is
the incident scattering channel and |2⟩ is the loss channel where particles are in adifferent internal state from |1⟩ as depicted in Figure 4-10(a) [84]. This is analogous
to an incident light coupled in at one side and coupled out at the other side of a single
mode cavity. The transmission through a cavity as illustrated in Figure 4-10(b) isexpressed in a simple form with the mirror coupling strengths 𝛾¯𝑖 as:
𝑇 =𝐼𝛾¯1 𝛾¯2.
2(𝜔𝑜 − 𝜔) + [(¯𝛾1 + 𝛾¯2 )/2]2(4.7)

Here, 𝛾¯𝑖 = − 2 ln 𝑟𝑖 /𝜏𝑅𝑇 where 𝑟𝑖 is the reflection coefficients of two mirrors, M1 andM2 and 𝜏𝑅𝑇 is the round-trip time of the flux travelling in the cavity 7 . 𝜔0 is the5
At the position of the 𝑝-wave barrier (𝑅𝑏 = 100 nm), the interaction between two spins withmagnetic moments 2𝜇0 , where 𝜇0 is the Bohr magneton, is 𝑉 𝑚𝐷𝐷 = 0.052 kHz. Higher order spinflip
must involve an intermediate state |𝑘⟩, which is off-resonant by its Zeeman energy Δ𝑎𝑘 ≃ 1𝜇0 ×300 G≃ 400 MHz. This further reduces the magnetic dipolar coupling between the open channels by thefactor (𝑉 𝑚𝐷𝐷 /Δ𝑎𝑘 )−1 = 7.7 × 106 to much less than 1 mHz.
6
The peak of the barrier is at 241/4 · 𝑅vdw ≈ 7.8 nm where the van der Waals length 𝑅vdw =(︁)︁1/4
1 2𝜇𝐶6, and the outer turning point 𝑅𝑏 is at 100 nm at 3.35 𝜇K temperature.
22ℏ7Transmission of flux through a Fabry–Pérot cavity with two mirrors, M1 and M2 , normally has the Airy distribution expressed in terms of the incoming light flux, 𝐼, mirror reflec-

76angular frequency of the cavity mode, 𝜔 is the angular frequency of the light, and 𝐼
is the incoming flux8 .
Assuming that the input mode is either (nearly) resonant (𝜔 − 𝜔0 ≪ 𝛾¯1 ) or far-off
resonant (𝜔 − 𝜔0 ≫ 𝛾¯1 ) with the Fabry-Perot cavity, tuning the coupling term 𝛾¯2 now
leads to a pronounced resonance-type feature in the Fabry-Perot transmission. For
small 𝛾¯2 , transmission is proportional to 𝛾¯2 . However, for large 𝛾¯2 , the quality factor of
the resonance is reduced, there is less and less built up of light inside the resonator, andthe transmission decreases with 𝛾¯2 . For resonant coupling, the maximum transmission
is at 100% (𝑇 = 𝐼) when 𝛾¯2 = 𝛾¯1 while for off-resonant coupling, the maximum
transmission is at the reduced value by a factor of 𝛾¯1 /(𝜔 − 𝜔0 ) when 𝛾¯2 = 2(𝜔 − 𝜔0 )as shown in Figure 4-11(a). For on (or off) resonant coupling, the full width at half
maximum (FWHM) of the transmission spectrum as a function of 𝛾¯2 is expressed as√√
FWHM = 4 2¯
𝛾1 (of 2 𝜔 − 𝜔0 ), and the transmission line is narrower or sharper forsmaller 𝛾¯1 (or 𝜔 − 𝜔0 ) as shown in Figure 4-11(a) and (b).
We suggest that this mechanism is responsible for the observed resonance. For
this, we assume the existence of a coupling between the incoming channel and a longlived collision complex with energies that do not tune relative to one another withthe magnetic field. This may seem to call for a remarkable coincidence, but moleculemolecule collisions are predicted to have a high density of resonances. The decay
strengths of the channel |3⟩ to the open channels 𝛾𝑖 , where 𝑖 = 1, 2, are represented
by the mirror coupling strengths 𝛾¯𝑖 . The incoming scattering flux and collision loss ofmolecules are represented by the incoming flux of light and the transmission throughthe second mirror in the optical filter picture, respectively.
The transmission of a Fabry–Pérot exhibits the analytical result of the inelastic
loss rate extended from the p-wave interaction model in Ref. [86] by Prof. Timurtion and transmission coefficients, 𝑟𝑖 and 𝑡𝑖 , and single-pass phase shift that light exhibits, 𝜑 as
𝑡21 𝑡22
𝑇 = 𝐼 (1−𝑟1 𝑟2 )2 +4𝑟. This Airy distribution is well approximated to a Lorentzian spectral line2
1 𝑟2 sin 𝜑shape near a resonance (𝜔 ∼ 𝜔𝑜 ) for highly reflective cavities (𝑟12 𝑟22 < 0.1) as Eq.4.7 [85].
8
The process is fully analogous to an harmonic oscillator with resonance 𝜔0 and damping rate𝛾¯2 driven at frequency 𝜔 via a frictional input coupling 𝛾¯1 . The ratio of the power dissipated via 𝛾2
to quarter of the nominal drive power (drive power at zero amplitude of the harmonic oscillator) is
exactly given by Eq. 4.7. (see appendix)

77

Figure 4-10: a. Schematic of the resonance model with two open channels
and a p-wave bound state trapped behind a centrifugal barrier. Channel |1⟩
and |2⟩ are the open channels that are coupled to a closed channel |3⟩, Channel |1⟩
corresponds to the initial scattering channel where two NaLi molecules are in the lower
stretched hyperfine state. Channel |2⟩ corresponds to another open channel wheremolecules are in a different hyperfine state and energetically close to the incident
scattering state. b. Schematic of a Fabry–Pérot cavity. Two mirrors M1 andM2 couple light into and out of the Fabry–Pérot resonator created by these mirrors
with coupling strengths 𝛾¯𝑖 . When the spacing between the mirrors is tuned to form acavity mode that is resonant to the incoming light, transmission loss is enhanced byconstructive interference of the light amplitude inside the cavity.

78

Figure 4-11: a. Transmission probability of a cavity for different detunings,
𝜔 − 𝜔0 . For given coupling strengths of the first mirror, 𝛾¯1 = 1 (in arbitrary units),
transmission probability reaches unity for resonant coupling (blue) while maximizes
to 𝛾1 /(𝜔 − 𝜔0 ) for far off-resonant coupling (black). b. Transmission probabilityof a resonant cavity for different first mirror coupling, 𝛾¯1 . For given couplingstrengths of the first mirror, 𝛾¯1 =1, 3 or 9, transmission probability reaches unity when𝛾¯2 = 𝛾¯1 at a resonance. The line width of the transmission probability as a functionof 𝛾¯2 is narrower for smaller 𝛾¯1 .

79Tscherbul, to whom we are grateful. The details of the derivation of the inelastic loss
rate are included in Chap. C.4. The decay strengths from channel |3⟩ to |1⟩ (|2⟩)
are proportional to 𝛾1 ∝ 𝜆1 𝑘 3 (𝛾2 ∝ 𝜆2 𝑘 ′3 ) where 𝜆1 (𝜆2 ) is the open-closed channel
coupling and 𝑘 (𝑘 ′ ) is the open channel wavevector. 𝑘 3 scaling of the decay strength
comes from the threshold behavior of 𝑝-wave inelastic collision rate (∝ 𝑘 2 ) and thedensity of states of an open channel (∝ 𝑘). Using this scaling, we obtain the inelastic
loss rate:𝑔2 ∝𝑘 −1 𝛾1 𝛾2(𝐸 − 𝐸0 )2 + [(𝛾1 + 𝛾2 )/2]2(4.8)

where 𝐸 is the incident energy and 𝐸0 is the resonant energy. The numerator scaleswith 𝑘 2 instead of 𝑘 3 , which is different in structure from equation (4.7)9 . For particle
collisions, the initial channel is in a single wavevector, so we see threshold 𝑝-wave
scaling (𝑘 2 ) for the input coupling. Inelastic decay from the closed channel proceeds
into a continuum of states, which adds the third power of 𝑘 for the density of states inthat continuum. The experimentally observed 𝑘 2 dependence of the loss rate constant
implies that the 𝛾1 never dominates the other terms in the denominator of 𝑔2 .
The model presented so far can explain a resonant enhancement of the loss as afunction of the magnetic field. However, unlike in the case of spin-protected NaLi+Na[33] or non-reactive NaK+K Feshbach resonances [71] where the background loss
rates were observed to be far below the universal values, NaLi+NaLi collision isreactive and the loss rate is saturated to near 𝑝-wave universal value (𝑦 ∼ 1) at thebackground. Despite the fact, the loss rate is enhanced from the background by morethan two orders of magnitude at the resonance. This peculiar enhanced loss feature
was unexpected based on existing simple models for reactive collisions in molecularsystem. Collision of ultracold molecules is often described with the quantum-defect√√model used in Ref. [35] where quantum-defect parameter 𝑦 = (1 − 𝑅)/(1 + 𝑅)
is directly related to the probability of incoming scattering flux being reflected backat short range 𝑅. This single-channel model was extended to describe the Feshbachresonance in collision between the NaLi molecule and Na atom and it was shown
9

In the appendix, Eq. 4.8 is derived using a T matrix formalism.

80Figure 4-12: a. Schematic of the resonance model with more than one
orthogonal collisional path. The incoming flux is split into two or more (orthogonal) components, one part (10%) has the resonant behavior, whereas the other part(90%) has non-resonant universal loss. b. Schematic of a Fabry–Pérot cavity
with a polarizing beam splitter (PBS). Two mirrors form a resonator where
p-polarization component of light amplitude is enhanced near a resonance. There are
two paths for light to travel that are decoupled by a PBS. s-polarization component
is reflected by the PBS and rejected.

81

that the scattering loss rate that is suppressed below the universal value by a factorof 𝑦 can be enhanced by a Feshbach resonance at most above the universal value
by 1/𝑦 [33]. With the extended single channel model, this large enhancement which
requires 𝑦 ≪ 1 cannot be explained for NaLi+NaLi collisions which are reactive(2NaLi→Na2 +Li2 +heat) and have 𝑦 ∼ 1.
One possible explanation is that the incoming flux is split into two or more (orthogonal) components, one part has the resonant behavior described above, whereas
the other part has non-resonant universal loss (i.e. full transmission at short range)
as depicted in Fig.4-12(a). For example, if one tenth of the scattering flux is coupled to the resonance, but its loss is enhanced by a factor of 1000, the total loss canbe 100 times the universal limit. Outside the resonance, the majority of the flux
(here assumed to be 90 %) provides the background loss near the universal rate. At
resonance, the enhancement of the collision loss rate is inversely proportional to 𝛾2 ,which can be easily obtained from the Fabry–Pérot picture. With sufficiently weak
open-closed channel couplings, large enhancement in collision loss is allowed despite
close to unity loss of scattering flux that reaches short/close range.
This model can be pictured with a polarizing beam splitter(PBS) placed insidea Fabry–Pérot cavity as illustrated in Figure 4-12(b). As an extension of a simple
cavity picture described earlier, the mirror M1 represents quantum reflection by thelong-range vdW potential, and M2 represents the reflection of a stable bound state
by coupling to another near degenerate hyperfine state. There are two optical paths
which are decoupled at the PBS by the light polarization. Light that enters throughthe outer mirror M1 has fraction, 𝜅𝑝 , of p-polarization which represents the longlived bound state and passes through the PBS traveling toward mirror M2 . The
other fraction, 𝜅𝑠 , corresponding to s-polarization represents short-lived states andis reflected at the PBS resulting in complete rejection. This perfect removal of theflux represents short/close range loss with unity probability (𝑦 = 1). The total lossof incoming flux is composed of the transmission of p-polarization through M2 andthe rejection of the s-polarization. Transmission through M2 , 𝑇𝑝 is given as equation(4.7) with an extra factor 𝜅𝑝 and loss or s-polarization is given as 𝑇𝑠 =𝐼𝜅𝑠 𝛾¯1 𝜏𝑅𝑇 /2.
82

When 𝜅𝑠 ≫ 𝜅𝑝 off-resonant transmission is dominated by the complete rejection of
the s-polarization component of light, which is analogous to the molecular collision
loss at the background being close to the universal value.

4.3

Discussion

The simplest model for an atom-like p-wave Feshbach resonance in the NaLi+NaLi
collision assumes two resonances at a certain magnetic field. It is unusual to have threestates that are near degenerate by coincidence, and one may suggest two possibilities:
the existence of many stable bound states or a resonant bound state having the samemagnetic moment as the initial state. The presence of dense stable bound states that
may couple to the incident channel allows having Feshbach resonances all throughwhile scanning the magnetic field. Similarly, if the magnetic moment and the energyof a stable bound state match that of the initial state, the Feshbach resonance isinsensitive to magnetic field change, and the system will always be on resonance. For
these cases, the energy difference between two hyperfine states is tuned by changing
the magnetic field, while always being at a Feshbach resonance. The coupling of abound state to a degenerate loss channel is tuned by an external magnetic field andallows the Feshbach resonance to be visible.

4.3.1

Collision complex lifetime

One may estimate the collision complex lifetime to be ∼ 28.6 𝜇s from the width of
the scattering loss feature assuming that magnetic field scan sweeps through a 𝑝wave Feshbach resonance like previously observed Feshbach resonances 10 . However,according to our model, the observed width may not reflect the width of the longlived state but would depend instead on the ratio of coupling strengths for the two
open channels. A more reliable estimate uses the classical round-trip time for azero-energy particle in the combined centrifugal and van der Waals (vdW) potential,
10

Difference in magnetic moment between the initial state and the bound state is assumed to be
1 Bohr magneton for this estimate.

83

2

ℏ
1
is determined by thewhich is equal to ℏ/𝐸vdW . The vdW energy, 𝐸vdW = 2𝜇
2
𝑅vdw
(︀
)︀
1/4
6
vdW length 𝑅vdw = 12 2𝜇𝐶≈ 66𝑎0 , where 𝜇 is the reduced mass of two NaLi
ℏ2molecules and 𝐶6 is the NaLi-NaLi long-range dispersion coefficient.
We consider the threshold limit with the molecule-molecule potential in Eq. 4.9
and a hard wall at an intermolecular distance 𝑅 = 0.
𝑉 (𝑅) = −𝐶6 𝑙(𝑙 + 1)ℏ2
+
𝑅6
2𝜇𝑅2

(4.9)

Classically, the round-trip time of a particle in this potential between two turning
points, the inner turning point 𝑅 = 𝑅𝑖 = 0 and the outer turning point 𝑅 = 𝑅𝑜 =(𝐶6 𝜇/ℏ2 )1/4 , is given as:∫︁ 𝑅¯
𝑡𝑅𝑇 = 201𝑑𝑅 = 2
𝑣

∫︁ 𝑅¯
0

1
√︁ √︁
2
𝜇

𝐶6
ℏ2
− 𝜇𝑅
2
𝑅6

𝑑𝑅

√︂
∫︁ 1
¯4
𝑅
𝜇
√︁
=2
𝑑𝑋
2𝐶6 0
1
− 1
𝑋6

𝑋2

√ 𝜇3/2 𝐶61/2 ∫︁ 1
1
√︁
𝑑𝑋
= 2
2
ℏ
1
0
− 1
𝑋6

𝑋2

(4.10)√ 𝜇3/2 𝐶61/2 ∫︁ 1
𝑋3
√
= 2
𝑑𝑋
ℏ2
1 − 𝑋40√ 𝜇3/2 𝐶61/2 1 ∫︁ 1 1
√ 𝑑𝑌
= 2
ℏ2
4 0
𝑌
1/2
𝜇3/2 𝐶ℏ= 1/2 62 =
.
2 ℏ
𝐸vdw
¯ and,
Here, for the integrand on the first line of Eq. 4.10, substitute 𝑋 = 𝑅/𝑅
for the integrand on the fourth line of the equation, substitute 𝑌 = 1 − 𝑋 4 . For
collisions between two identical NaLi molecules, this classical round trip time 𝑡𝑅𝑇 isapproximately 5.67 ns.
The lifetime of the collision complex is finite due to losses resulting from coupling
to other channels (nonzero mirror transmissions in the cavity model). The decay of aradiation mode of a cavity can be expressed as 𝐼(𝑡) = 𝐼0 𝑒−𝑡/𝑡𝐶 . For reflectivity 𝑅, the
84

1000

Potential Energy (uK)5000l(l + 1)/R 2
Ri

Ro
10

500

4.2K
0

1000

89.3

136.4

100

150

10

1500
2000 0.0

1.8K

C6/R 6
2.5

5.0

20 0

7.5

10.0

R (nm)

50

12.5

15.0

17.5

Figure 4-13: Intermolecular potential of the NaLi (𝑎3 Σ+ ) molecules. The
𝑝-wave centrifugal barrier is also shown. The upper and lower bounds on the experimental collision energies (4.2 𝜇K and 1.8 𝜇K) are marked by green horizontal lines.
The outer turning points for these collision energies are 𝑅𝑏 = 89.3 and 136.4 nm,
respectively.

85

time to make 𝑁 reflections is 𝑡 = 𝑁 𝑡𝑅𝑇 where 𝑡𝑅𝑇 is the round trip time. After 𝑁
reflections, the radiation intensity is 𝐼0 𝑒−𝑁 𝑡𝑅𝑇 /𝑡𝐶 = 𝐼0 𝑅𝑁 which gives for the lifetime𝑡𝑅𝑇
𝑡𝑐 = 𝑙𝑛(1/𝑅)≈ 𝑡𝑅𝑇 /(1−𝑅). Therefore, the photon lifetime in a cavity is the round-trip

time times the resonant enhancement divided by 4 11 . Using this analogy and theobserved loss enhancement factor of 230, we obtain an estimate for the lifetime of thecomplex of 320 ns.
We consider this to be a low estimate. Even if the 𝑝-wave resonance is at resonance,
due to thermal broadening, the maximum loss enhancement is reduced by a factorℏ𝛾1 /𝑘𝐵 𝑇 ≈ 0.1 which implies a 10-fold longer lifetime (see chapter 4.3.2 for details).
In addition, if we assume that the loss enhancement is possibly not 230, but 2300 for
10% of the incident flux (model with polarization beam splitter), then the lifetime
would be another factor of ten higher and could be tens of microseconds.

4.3.2

Estimation of coupling strength 𝛾1

The coupling between channel 1 and channel 3, 𝛾1 , can be estimated from the reflection coefficient for quantum reflection and the round trip time behind the 𝑝-wave
barrier. For this, we use the optical cavity analogy, where the first mirror couplingis 𝛾¯1 = −4 ln 𝑟1 /𝜏𝑅𝑇 . With the classical round trip time 𝑡𝑅𝑇 ∼ 5.67 ns and 𝑝-wave
quantum transmission 𝑇𝑝 12 , one can estimate 𝛾1 ≈ − ln (1 − 𝑇𝑝 )/𝑡𝑅𝑇 ≈ 2𝑇𝑝 /𝑡𝑅𝑇 ≈2¯
𝑎1 𝑎¯2 𝑘 3 /𝜏𝑅𝑇 , where 𝑎¯ = 4𝜋𝑅vdW /Γ( 41 )2 , 𝑎¯1 = 𝑎¯Γ( 41 )6 and 𝑘 is the collision wavevector. 𝛾1 is calculated as a function of the collision energy, as shown in Fig.4-14(a). We
can confirm that 𝛾1 is much smaller than the thermal energy for our experimental
conditions (𝑇 ≃ 4.2 𝜇K) as shown in Fig.4-14(b).
11Quantum-defect parameter 0 ≤ 𝑦 ≤ 1 is related to the probability of irreversible loss of incoming
scattering flux from the entrance channel due to chemical reaction or inelastic loss at short range.
√
Based on the single-channel Fabry-Perot model introduced in chapter 1.2.2, 𝑡2 = 2 𝑦/(1 + 𝑦) where𝑡2 represents the short-range reflectivity of the scattering flux and the factor 1/𝑦 is the maximum
enhancement possible above the universal limit with the tuneable round trip phase of the cavity.
This relation leads to 𝑡22 = 1 − 𝑅 = 4𝑦/(1 + 𝑦)2 ≈ 4𝑦 for 𝑦 ≪ 1.
12
The short-range transition matrix 𝑇 reduces to the quantum transmission coefficient 𝑡1 atthreshold [87] and the universal loss rate coefficient is directly related to the transition matrix
2
as 𝐾 = 2𝑔 𝜋ℏ
𝜇𝑘 |𝑇 | . By comparing this relation with the known universal loss rate coefficient [88], we
obtain the probability of quantum transmission for 𝑝 wave collision, 𝑇𝑝 ≈ 2¯
𝑎1 𝑎
¯2 𝑘 3 .

86

(b)

30

10 1
20

1/kT

1=-2ln(r1)/tRT (2× kHz)

(a)

10 2

10

00

2

4

6

8

Collision energy (uk)

3
10 10
1

10

102

103

104

Collision energy (nK)

Figure 4-14: The coupling between the incident open channel |1⟩ with thep-wave bound state 𝛾1 as a function of collision energy. a. 𝛾1 estimatedby taking the classical round trip frequency 1/𝜏𝑅𝑇 times minus the log of quantum
reflection probability |𝑟1 |2 is plotted as a function of collision energy. b. The ratio
between the estimated ℏ𝛾1 and the thermal energy is plotted as a function of thecollision energy.

874.3.3Conclusion

The most surprising aspect of our results is the existence of a long-lived collisioncomplex in NaLi+NaLi collisions, which allows barrier-free chemical reactions in all
channels [29]. A long-lived complex is necessary for the existence of a high-Q resonance, such as we observed, independently of any detailed model. Magneticallytuneable resonances between two ground state molecules have yet to be discoveredin other systems, although it has been uncovered that long-lived four-body collisioncomplexes can form in other molecular systems including Rb2 Cs2 [50] and K2 Rb2 [40].
These other observations are most likely related to “sticky” collisions [37–39, 89] connected with a high density of states. These complexes are formed incoherently andlead to a loss rate at the universal rate. In contrast, an enhancement above the universal rate (as observed here) is only possible when the flux reflected at short range
destructively interferes with the quantum reflected flux and, therefore, requires fullphase coherence.
There exists a challenge of answering the question why Feshbach resonances are
unseen in other systems. Feshbach resonances may be too dense, or intermediate
collision complexes may undergo a fast loss from higher-body collision or photoinduced processes in other systems. In these cases, Feshbach resonances are almost
impossible to detect, as they are unresolved or broadened, although they exist. So
far, the Feshbach resonance in NaLi+NaLi collisions is unique. It is created by along-lived collision complex which does not go through fast collision loss or photoinduced loss and it is detected by the coupling of the complex to a nearly degenerate
hyperfine state.

88Chapter 5
Magnetic trapping and cooling ofmolecules
Trapping ultracold molecules in conservative traps is essential for applications—suchas quantum state-controlled chemistry, quantum simulations, and quantum information processing. These applications require high densities or phase-space densities.
We report magnetic trapping of NaLi molecules in the triplet ground state at highdensity (≈ 1011 cm−3 ) and ultralow temperature (≈ 1 𝜇K). Magnetic trapping atthese densities allows studies on both atom-molecule and molecule-molecule collisions
in the ultracold regime in the absence of trapping light, which has often lead to undesired photo-chemistry. We measure the inelastic loss rates in a single spin sample
and spin-mixtures of fermionic NaLi as well as spin-stretched NaLi+Na mixtures. We
demonstrate sympathetic cooling of NaLi molecules in the magnetic trap by radio frequency evaporation of co-trapped Na atoms and observe an increase in the molecules’
phase-space density by a factor of ≈ 16.

5.1

Introduction

Ultracold molecules offer new opportunities for quantum state controlled chemistry[1, 2], for quantum simulations [3–5], and for quantum information processing [6–9].
For more than two decades, various methods were developed with the goal to trap89

molecular samples at densities high enough to study molecular collisions. Using buffer
gas cooling [90–93] or Stark or magnetic deceleration [94–96], molecules were trappedin magnetic traps in the temperature range of tens or hundreds of millikelvin. In
only one case (O2 ) were densities high enough (estimated at ≈ 1010 cm−3 ) to observe
molecular collisions [97]. Dipolar molecules can be trapped with electric forces. Collisions between CH3 F molecules were observed in an electrostatic trap at densitiesof 107 cm−3 [98], and elastic collisions of OH were observed in an electromagnetic
trap at temperatures around 60 mK [99], though none of these studies reached theultracold temperature regime. Ultracold laser-cooled molecules (in the 50 - 100 𝜇K
range) were transferred into magnetic traps at densities ≲ 106 cm−3 [20,100], too low
for the observation of intermolecular collisions.
Optical traps have the advantages of trapping non-magnetic molecules and providing tight confinement, which has allowed the study of collisions involving molecules
created by ultracold assembly or direct laser cooling. For example, collisional resonances have been observed in atom-molecule mixtures [33,70,71] and between molecules[73] in optical traps. The disadvantage of optical traps is the small trapping volume
and the presence of intense laser light, which can induce photochemistry. This has
become a major concern recently after many experiments have found fast collisional
losses even for non-reactive molecules, possibly due to “sticky collisions” connectedwith long-lived complexes [37–39,89,101]. Recent experiments to test these proposals
suggest that optical traps can cause short lifetimes of molecules and are not truly
conservative [40, 50, 102–104] emphasizing the need for “laser-free” trapping.
Here, we report magnetic trapping of triplet NaLi molecules in the rovibrational
ground state with high density (≈ 1011 cm−3 ) and ultracold temperature (≈ 1𝜇K).
The typical density is a factor of 105 higher compared to previous experiments withmagnetically trapped ultracold molecular gases. Inelastic losses are detected in singlecomponent and in spin-mixtures of a fermionic NaLi molecular gas.
Another major long-standing goal has been magnetic trapping of molecules together with atoms for sympathetic cooling of molecules to achieve higher moleculardensities or phase-space densities [105,106]. Magnetic co-trapping of NH and N [107],90

and more recently of CaF and Rb [108] and O2 with Li atoms [109] has been achieved.
However, so far only inelastic collisions were observed [107,108], with atomic densities
far too low for sympathetic cooling.
Here we demonstrate sympathetic cooling of molecules in a magnetic trap. We use
a spin-stretched NaLi+Na mixture, which had been studied in ref. [72] and observe
an increase in the phase-space density (PSD) of the molecular gas by more than anorder of magnitude after radio frequency (RF) evaporation of Na atoms.

5.2

Experimental methods

The experiments are carried out with NaLi molecules in the triplet ground state(𝑎3 Σ+ , 𝑣 = 0, 𝑁 = 0) created by means of ultracold assembly of Na and Li atoms in
the lowest-energy Zeeman states following the procedure described in ref. [33, 72, 73].
The procedure has been improved so that the Na/Li mixture is transferred directlyfrom the initial magnetic trap into a 1550 nm 1D optical lattice without the use of
extra optical traps (previous configurations required a 1064 nm optical dipole trap
and a 1596 nm 1D optical lattice). After 0.4 seconds of forced evaporation of the atom
mixture, 105 molecules at 1.8 𝜇K temperature are formed in the maximally stretchedlow-field seeking hyperfine state (|𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩) using Feshbach association and STIRAP transfer to the triplet ground state. Here, 𝐹 is the total
angular momentum including electron and nuclear spins, 𝑚𝐹 is the 𝐹 projection
to the quantization axis, and 𝐹1 is a good quantum number in the zero-field limit
combining the electron spin and the Na nuclear spin [13].
After ramping down the bias field from ∼ 745 G, where the association of molecules
occurs, to a low magnetic bias field in 20 ms, molecules are transferred from the opticallattice to an Ioffe-Pritchard magnetic trap with a bias field of 0.56 G in 100 ms. The
molecules are trapped for various hold times and transferred back to the optical latticein 100 ms for detection at high field (∼ 745 G). The number of molecules is counted byabsorption imaging of the Li atoms from the recaptured and dissociated molecules.
The experimental sequence is illustrated in Figure 5-1. The transfer efficiency of
91

I

II

Bias field (G)STIRAP Clean

Lattice
Intensity

IVIII

100 ms
Lattice/magnetic
trap ramp

Magnetic
trapping

100 ms
Lattice/magnetic
trap ramp

20 µs 500 µs
Feshbachassociation
1 ms

I

III

IV

Imaging
ReverseSTIRAP

RF/Microwave

II

20 µs

III

II

Feshbachdissociation

I

Time

Figure 5-1: Top figures are the illustration of particles confined in different traps. I.
Na/Li mixture is confined in a 1D lattice potential (indicated with orange dashedline). II. NaLi molecules are trapped in the same 1D lattice. III. NaLi molecules
are trapped in a hybrid trap created by a 1D lattice and a magnetic trap (indicatedwith black line). IV. Molecules are purely confined by a magnetic trap. The middle
plot is the experimental sequence to produce and isolate NaLi molecules (time axis
is not to scale). Molecules are formed via Feshbach formation and STIRAP in a
1D lattice, and the free atoms are removed using resonant light pulses at high field.
After the magnetic bias field is dropped to low field, the lattice is ramped down andthe magnetic trap is turned on in 100 ms. In the magnetic trap RF or microwave isapplied to the molecules for thermometry or preparation of molecular spin mixtures.
For detection, molecules are transferred back to the 1D lattice, the molecule formation
process is reversed, and the dissociated free atoms are imaged. The bottom row is
the lattice intensity as a function of time along with particle and trap type indicatedusing index I-IV.

92

(a)

(b)

(c)

Figure 5-2: (a) Rovibrational ground state energy level diagram of NaLi molecules inthe triplet potential (𝑎3 Σ+ ) at zero bias field. Molecules are initially formed in the low
field seeking spin-polarized hyperfine state (|𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩) indicatedin blue. States in red and green are the other hyperfine states used for creating
molecular spin-mixtures. (b) Number decay of low field seeking spin-polarized NaLi
molecules (|𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩) in a magnetic trap. (c) The number ofmolecules left in the magnetic trap as a function of the RF-knife frequency. The
dashed line is a fitting function for a temperature of 1.06 𝜇K. The inset is the 𝑚𝐹
energy levels of NaLi in |𝐹1 = 5/2, 𝐹 = 7/2⟩ near the center of a magnetic trap (inblack lines). Exemplary density profile of molecules in the top hyperfine state is inblue dashed line.

93molecules from the optical trap to the magnetic trap is close to 100 % whereas therecapture efficiency back to the optical trap is only about 50 % due to a smaller
optical trap volume.
We investigate mainly three types of inelastic collisions: 𝑝-wave collisions in single spin component molecular gas, 𝑠-wave collisions in molecular spin-mixutures, andlow-reactivity atom-molecule collisions i.e. NaLi+Na in the low field seeking spinstretched states. For this, molecules in the hyperfine state |𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩
are held by the magnetic trap in the single spin state, or together with molecules in another hyperfine state or Na atoms in the upper spin-stretched state (|𝐹 = 2, 𝑚𝐹 = 2⟩).
The energy level diagram of NaLi molecules in the triplet ground state is shown in Fig.
5-2(a). We focus on the number decay of NaLi in state |𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩
in the magnetic trap.

5.2.1

Decay models

The differential equations that describe the decay of molecules in the upper stretchedstate in the presence of another type of particle 𝑖 are given as:
2
𝑁˙ NaLi = −(𝐾𝑖 /𝑉ov )𝑁𝑖 𝑁NaLi − (𝛽0 /𝑉eff,NaLi )𝑁NaLi(5.1)

𝑁˙ 𝑖 = −(𝐾𝑖 /𝑉ov )𝑁NaLi 𝑁𝑖 − (𝛽𝑖 /𝑉eff,𝑖 )𝑁𝑖2 ,(5.2)

where 𝑁NaLi represents the number of NaLi in the upper stretched state and 𝑁𝑖
represents the number of particles of type 𝑖 that are magnetically trapped with themolecules. Here, 𝑉ov is the volume of the regime where the 𝑖-type particles overlap
with the upper stretched molecules, 𝑉eff,𝑖 is the mean volume filled with 𝑖 particles,
𝛽0 (𝛽𝑖 ) is the two-body molecular (particle 𝑖) loss rate coefficient, and 𝐾𝑖 is the loss
rate coefficient for the collisions of 𝑖 +NaLi pairs.
We solve the given differential equations for three different conditions: 𝑁𝑖 = 0,
𝑁𝑖 ≈ 𝑁NaLi , and 𝑁𝑖 ≫ 𝑁NaLi . The two-body loss in the single spin component(|𝐹1 = 25 , 𝐹 = 27 , 𝑚𝐹 = 72 ⟩) molecular gas, i.e. 𝑁𝑖 = 0, is described by the second term94

of Eq. 5.1 only. The analytical solution is given as 𝑁NaLi (𝑡) = 𝑁0 1+𝛽0 𝑁10 𝑡/𝑉eff 1 , where
𝑁0 is the initial NaLi number.
For collisions in two-component mixtures that we study experimentally, the decayof the NaLi number is well described by Eq. (5.1) and (5.2) with the second terms
approximated to zero since 𝛽0 , 𝛽𝑖 ≪ 𝐾𝑖 . With this approximation, the analytic
solution becomes2 :𝑁NaLi (𝑡) =𝐷

𝐶𝑒𝐷Γ𝑡 − 1𝐷
𝑁𝑖 (𝑡) = − 1 −𝐷Γ𝑡
𝑒
−1
𝐶

(5.3)
(5.4)

where Γ = 𝐾𝑖 𝑁𝑖 (0)/𝑉ov is the loss rate, 𝐶 = 𝑁𝑖 (0)/𝑁NaLi (0), and 𝐷 = 𝑁𝑖 (0)−𝑁NaLi (0).
In a regime where 𝑁𝑖 (0) ≫ 𝑁NaLi (0) in a two-component mixture, 𝑁𝑖 (𝑡) can be approximated to 𝑁𝑖 (0) throughout and Eq. 5.3 is reduced to a simple exponential decay,
𝑁NaLi (𝑡) = 𝑁0 𝑒−𝐾𝑖 𝑡/𝑉ov . In the experiment, the Na atom number was more than a factor of 10 larger than the NaLi molecule number, so the decay of NaLi is well describedby the exponential function.
Now we discuss the volumes 𝑉eff and 𝑉ov in Eqs. (5.1) and (5.2). Assuming aharmonic trap, one obtains:𝑉eff,𝑖 = 𝜔¯ 𝑖−3 (4𝜋𝑘𝐵 𝑇𝑖 /𝑚𝑖 )3/2
𝑁𝑖 𝑁NaLi
𝑑𝑉 𝑛𝑖 𝑛NaLi
[︂(︂
)︂ (︂
)︂]︂ 32𝜋𝑘𝐵 𝑇𝑖𝜇Na 𝜇Na 𝑇NaLi 2
−3
=𝜔
¯ Na
+
,
𝑚𝑖
𝜇𝑖
𝜇2 𝑇𝑖

𝑉ov = ∫︀

where the geometric mean of the sodium trap angular frequencies, 𝜔¯ Na = (𝜔𝑥 𝜔𝑦 𝜔𝑧 )1/3
1

In order to account for the additional loss while and after the transfer back to an optical trap,
the 𝛼𝜏 term should be added to 𝛽0 𝑁0 𝑡. 𝛼 is the ratio of decay rate in the optical trap to that in themagnetic trap and 𝜏 is the time molecules spend in the optical trap. Although the optical trap istighter than the magnetic trap, since only 50% of the molecules are transferred back to the opticaltrap, 𝛼 is typically not much greater than 1. In addition, 𝜏 is short (effectively ≈ 50 ms undermaximum trap depth). Therefore, this additional loss term can be neglected under the experimental
conditions.
2The 𝛼𝜏 term should be added on top of 𝐷Γ𝑡 in Eq. 5.1 in order to account for the additionalloss while and after the transfer back to an optical trap. Similarly, this additional loss term can beneglected under the experimental conditions.

95

= 2𝜋×(282·282·14.8)1/3 Hz ≈ 2𝜋 × 106 Hz. Here, 𝑛𝑖 is the density, 𝑇𝑖 is the temperature, 𝑚𝑖 is the mass and 𝜇𝑖 is the magnetic moment of an 𝑖 particle. The geometric mean of the trap angular frequencies for NaLi in the upper stretched state,𝜔¯ NaLi =¯𝜔Na · (𝑚Na /𝑚NaLi ) · (𝜇NaLi /𝜇Na ), where the mass ratio, 𝑚NaLi /𝑚Na ≈ 29/23 andthe magnetic moment ratio, 𝜇NaLi /𝜇Na ≈ 2.

5.2.2

Thermometry

We determine the temperature of a NaLi gas in the magnetic trap from its density
profile. To find this, we let NaLi molecules in the |𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩ hyperfine state undergo radio frequency (RF) induced spin-flips in a magnetic trap thathas a trap bottom of 0.56 G. RF is swept from some high value (> 600 kHz) to a lower
value 𝑓 in 600 ms 3 . This RF can induce 7 simultaneous transitions, by one quanta of𝑚𝐹 each, from state |𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩ to |𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = −7/2⟩,
allowing the molecules to escape from the trap. The number of molecules left in themagnetic trap is determined by the final RF value (RF-knife frequency). The maximum possible energy of the molecules is 𝐴 = 72 ℎ(𝑓 − 𝑓0 ) where 𝑓 is the RF-knife
frequnecy and 𝑓0 is the transition frequency from |𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩ to|𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 5/2⟩ at the bottom of the magnetic trap.
Near the center, the magnetic field of the Ioffe-Pritchard magnetic trap varies
quadratically with the distance from the origin. The states of the particles are enumerated by a set of quantum numbers [𝑛𝑥 , 𝑛𝑦 , 𝑛𝑧 ] in a general three-dimensional
harmonic trap potential 𝑉 (𝑥, 𝑦, 𝑧) = 21 𝑚(𝜔𝑥2 𝑥2 + 𝜔𝑦2 𝑦 2 + 𝜔𝑧2 𝑧 2 ) and the energy of a particular state is given by 𝜖 = ℏ(𝑛𝑥 𝜔𝑥 + 𝑛𝑦 𝜔𝑦 + 𝑛𝑧 𝜔𝑧 ) + 𝜖0 , where 𝜖0 = 21 ℏ(𝜔𝑥 + 𝜔𝑦 + 𝜔𝑧 )is the zero-point energy in this harmonic trap. For 𝜖 ≫ 𝜖0 , the number of states
2with energy between 𝜖 and 𝜖 + 𝑑𝜖 is estimated as 𝑔(𝜖)𝑑𝜖, with 𝑔(𝜖) = 2ℏ𝜖3 𝜔¯ 3 where𝜔¯ = (𝜔𝑥 · 𝜔𝑦 · 𝜔𝑧 )1/3 is the geometric mean of the trap angular frequencies. Therefore,
3

The thermometry is carried out when there are only NaLi molecules in the upper-stretched state,and 𝑠-wave collisions are impossible. Therefore, 600 ms is far too short for molecules to thermalize
during the RF sweep.

96the particle number with energy between 0 and 𝐴 is∫︀ 𝐴𝑔(𝜖)𝑒−𝛽𝜖 𝑑𝜖
𝑁 (𝐴) =𝑁tot ∫︀ 0∞𝑔(𝜖)𝑒−𝛽𝜖 𝑑𝜖0]︀𝑁tot [︀ −𝛽𝐴=𝑒{−𝛽𝐴 · (𝛽𝐴 + 2) − 2} + 2
2

(5.5)
(5.6)

where 𝛽 = (𝑘𝐵 𝑇 )−1 is the Boltzmann factor. Fitting the data in Fig. 5-2(c), which is
the NaLi number as a function of the RF-knife frequency 𝑓 , to Eq. 5.6 provides theestimate for the molecule temperature of 1.06 ± 0.18 𝜇K.
For the temperature estimation of NaLi in the other hyperfine state either state|𝐹1 = 5/2, 𝐹 = 5/2, 𝑚𝐹 = 5/2⟩ or state |𝐹1 = 3/2, 𝐹 = 5/2, 𝑚𝐹 = 5/2⟩ of a molecular spin mixture, we assume that the average total kinetic energy and the average
total potential energy are equal in the harmonic trap by the virial theorem. After molecules are transferred from the upper-stretched state to the other hyperfine
state, the average potential energy of NaLi is reduced by the magnetic moment ratio
𝜇NaLi* /𝜇NaLi momentarily, while the kinetic energy remains the same. Here, NaLi
represents the upper stretched state and NaLi* indicates the other hyperfine state.
The average potential and kinetic energy redistribute to be equal. With this simple
model, the temperature is estimated as 𝑇NaLi* ≈ 𝑇NaLi (𝜇NaLi* /𝜇NaLi + 1)/2. The Na
temperature is measured directly from the time-of-flight (TOF) absorption imaging
out of the magnetic trap at low field.

5.3

Results

5.3.1

𝑝-wave collisions between molecules

We first observe 𝑝-wave inelastic collisions of NaLi in the upper spin-stretched state.
The number of molecules in the magnetic trap decays by more than 50% starting from6 × 104 in a few seconds, as shown in Fig. 5-2(b), while the typical vacuum lifetime
is greater than 20 seconds. The molecular temperature is 1.06 ± 0.18 𝜇K, which ismeasured by applying an RF-knife of various frequencies as shown in Fig. 5-2(c).
97Molecule number (×103)

40|5/2, 7/2, 7/2 + |5/2, 5/2, 5/2|5/2, 7/2, 7/2 + |3/2, 5/2, 5/2

30
20
10
00.0

0.2

0.4

0.6

0.8

1.0

Hold time (sec)

1.2

1.4

Figure 5-3: Number decay of low field seeking spin-polarized NaLi molecules(|𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩) when magnetically trapped together with moleculesin state |𝐹1 = 5/2, 𝐹 = 5/2, 𝑚𝐹 = 5/2⟩ (in red circles) and with molecules in state|𝐹1 = 3/2, 𝐹 = 5/2, 𝑚𝐹 = 5/2⟩ (in green squares).
The initial density is 1.7 × 1011 cm−3 —a factor of 105 greater than the experiments
carried out in ref. [20, 100]. The loss rate coefficient is calculated as (3.6 ± 1.4) ×
10−12 (cm3 /s)(TNaLi /𝜇K) from the best fit to a two-body loss model described above.
Within the uncertainty, this value is consistent with the value reported in ref. [33],which was measured near 980 G.

5.3.2

𝑠-wave collisions between molecules

Next, we observe 𝑠-wave inelastic collisions by creating a spin-mixture of moleculesin two different hyperfine states. A transition from |𝐹1 = 5/2, 𝐹 = 7/2, 𝑚𝐹 = 7/2⟩ to|𝐹1 = 5/2, 𝐹 = 5/2, 𝑚𝐹 = 5/2⟩ is driven by 100 MHz RF until the molecules form anear 50/50 spin mixture in the magnetic trap. Similarly, the mixture of |5/2, 7/2, 7/2⟩
and |3/2, 5/2, 5/2⟩ is prepared with a 1133 MHz microwave drive. Due to magnetic field inhomogeneity, the superposition of the two spin states decoheres withinℎ≲ 500𝜇s, where Δ𝜇 is the magnetic moment difference between the𝑇2 ≈ Δ𝜇·𝛿𝐵

two hyperfine states, and 𝛿𝐵 ≈ 10 mG is the magnetic field inhomogeneity, during
98and after an approximate 𝜋/2 pulse. The loss rate coefficients are calculated to be(3.6 ± 1.5) × 10−11 cm3 /s and (1.1 ± 0.4) × 10−10 cm3 /s respectively by fitting thedata in Fig. 5-3 to a loss model described in the Methods section. These values are
significantly lower than the 𝑠-wave universal value estimated using the approximate
𝑢𝑛𝑖𝑣
NaLi-NaLi long-range dispersion coefficient (𝐾𝑙=0= 1.85 × 10−10 cm3 /s).

5.3.3

Collisions between NaLi and Na

For atom-molecule collisions, we load together about 4 × 105 Na atoms and about
4 × 104 NaLi in the upper stretched hyperfine states into the magnetic trap. The
temperature of the atoms is ≈ 2𝜇K from the TOF imaging. Since the Na number
is much greater than the NaLi number we fit the molecule decay in Fig. 5-4(a) to asimple exponential decay curve. The measured loss rate constant 𝛽 = (6.3 ± 1.4) ×
10−12 cm3 /s is about a factor of 30 lower than the universal value (1.72 × 10−10 cm3 /s)
which is well known for this system [74].

5.3.4

Sympathetic cooling of NaLi with Na

With the low-reactivity atom-molecule mixture, we demonstrate sympathetic coolingof molecules in the magnetic trap. Here, via RF controlled evaporation, we have
independent control over the molecule and atom trap depths. It is possible to coolspin-polarized NaLi using collisions with Na, also spin-polarized in the same directionas NaLi, because NaLi has a favorable ratio of elastic to inelastic collisions 𝛾 with Na.
At high field, it was measured to be 𝛾 ≈ 300 [72]. After loading the atom-molecule
mixture into the magnetic trap, we slowly evaporate Na atoms out of the trap with amicrowave sweep. We perform a microwave sweep to remove all Na from the magnetic
trap in 1 second, which is chosen to be similar to the lifetime of NaLi with Na andlonger than the thermalization time among Na (∼ 670 ms) and between Na and NaLi(∼ 90 ms) (see Chap.5.3.5 for details).
We compare the temperature of NaLi after the Na evaporation to that of NaLi froman identical mixture loaded to the magnetic trap but with sudden removal of Na using
99

Molecule number (×103)(a) 40
30
20
10
00.0

(b)
Molecule number (×103)

NaLi+Na mixture

0.2

0.4

0.6

0.8

Hold time (sec)

1.0

1.2

20

10

Na suddenly removed
Na slowly evaporated

0 460 480 500 520 540 560 580 600

RF knife frequency (kHz)

Figure 5-4: (a) Molecule number as a function of hold time in a magnetic trap whentrapped together with Na. The atom-molecule mixture is in the low-field seeking
spin-stretched state. Na number is about 4 × 105 and the temperature is ≈ 2𝜇K.
(b) Molecule number as a function of RF-knife frequency. Data in red circles is withslow evaporation of Na and data in black squares is with Na suddenly removed after
loading into a magnetic trap. The density profile obtained from the red (black) data
implies a temperature of 0.8(1) 𝜇K ( 2.3(3) 𝜇K). The fits in dashed lines use a fit
function described in Chap.5.2.2.

100

a resonant light pulse after the loading. The two temperatures are estimated by the
RF-knife frequency scan described earlier (Fig.5-4(b)). Evaporation of Na leads to atemperature of NaLi molecules of 0.8(1) 𝜇K, substantially lower than the temperature
without the evaporation (2.3(3) 𝜇K), while the molecule number is decreased by 30%.
This corresponds to an increase in PSD of the molecular gas by a factor of ≈ 16.

5.3.5

Inelastic collision and thermalization rates

To investigate the limiting factor for sympathetic cooling, we compare the three
relevant time scales in a Na+NaLi mixture: the inelastic collision and thermalization
rates between Na and NaLi, and the thermalization rate among Na atoms. The
experiment was done with about 33 × 103 NaLi and 420 × 103 Na at a temperature of≈ 2𝜇K in a magnetic trap. The initial densities of Na and NaLi are 1.25 × 1011 cm−3
and 3.3 × 1010 cm−3 , respectively. The initial inelastic collision rate is experimentallymeasured to be Γinel ≈ 1.2 s−1 as shown in Fig. 5-4(a).
Next, we estimate the thermalization rate between Na and NaLi. In a massimbalanced system, the factor 3/𝜉 quantifies the approximate average number ofcollisions per particle required for thermalization, where 𝜉 = 4𝑚Na 𝑚NaLi /(𝑚Na +𝑚NaLi )2 ≈ 0.987 [72,110]. Thus, the relation between the thermalization rate and theelastic scattering rate is given by Γth ≈ Γel /(3/𝜉). In our system, where the particle
number is largely imbalanced, we can write the thermalization rate asΓth ≈(𝑁Na /𝑉ov )𝜎el 𝑣rel
,
3/𝜉

(5.7)

where 𝑁Na /𝑉ov is the average density of Na atoms seen by NaLi√︂molecules, 𝜎el is the
(︁
)︁
𝑇NaLi
𝑇Na
elastic scattering cross-section and the relative velocity 𝑣rel = 8𝑘𝜋𝐵 𝑚
+
.
𝑚NaLi
Na
The 𝑠-wave elastic scattering cross-section between Na and NaLi is given by 𝜎el ≈4𝜋𝑎2 , where 𝑎 is the scattering length. Using an approximate value for the scattering
length 𝑎 = 263(66)𝑎0 , where 𝑎0 is the Bohr radius [72], the thermalization rate isestimated to be ≈ 11 s−1 .
Similarly, the thermalization rate among Na is given as Γ̃th =101(𝑁Na /𝑉eff,Na )˜𝜎el 𝑣˜rel
3≈

1.5 s−1 , where the 𝑠-wave elastic scattering cross-section between Na atoms 𝜎
˜el ≈
8𝜋˜
𝑎2 , where the scattering length between Na atoms 𝑎
˜ = 85(3)𝑎0 [111]. The relative√︁𝑇Na
velocity 𝑣˜rel = 16𝑘𝜋 𝐵 𝑚. We see that Γinel ∼ Γ̃th ≪ Γth , and the sympathetic coolingNaof NaLi is limited by the slow Na thermalization rate.

5.4

Summary

In summary, we have shown magnetic trapping of molecules with a factor of 105 higher
density compared to ultracold molecules previously studied in a magnetic trap. We
have measured the inelastic collision rates for two selected molecular spin mixtures
and a spin-stretched Na+NaLi mixture that serve as prototypes for future studies onstate-dependent molecule-molecule and atom-molecule collisions in the magnetic trap.
Quantitative analysis of molecular collisions in the magnetic trap is much simpler thanin optical traps, because the magnetic trap is well-approximated by a harmonic potential whose trap frequencies are determined by molecule magnetic moments, which
are typically well known. In contrast, optical traps can be highly anharmonic near
the top of the trap, and ratios of trap frequencies (unless directly measured) can bedifficult to determine because of the unknown molecular ac polarizability.
Our collisional studies show that NaLi molecules with various collision partners
have loss rates far below the universal limit. Although all collision systems are reactive, the absorption probability at close range is much smaller than one. This is wellunderstood for the collisions in the spin-stretched Na+NaLi mixture [29,33] where thequartet potential in the input channel is non-reactive, and inelastic collisions are only
possible via spiflips. However, this explanation does not apply to the s-wave moleculemolecule collisions studied here. Also, a molecule-molecule Feshbach resonance has
been observed for a strongly reactive input potential [73]. These observations suggestthat collisional resonances and collisional complexes should occur more generally, andmotivate more systematic studies of collisions involving NaLi and collision partners
in various hyperfine states, and also for other molecules [112]. We also demonstrated
sympathetic cooling of NaLi by RF evaporation of Na atoms, increasing the PSD of102

the molecules by a factor of ≈ 16. This can be further optimized, but is eventuallylimited by the slow Na thermalization rate, which is significantly slower than the rateof elastic collisions between Na and NaLi. Using a second trap for Na that enhances
the thermalization rate should allow cooling into the quantum-degenerate regime.

103

104

Chapter 6
Free to bound Raman transfer
Progress toward ultracold molecular system has been made with various species byeither synthesis of preliminarily cooled atoms or direct cooling of molecules chemically
produced. One of the most successful recipes for achieving ground state molecules(without direct laser cooling of molecules) has two combined steps. The first stepis the magnetic association of cold atoms into weakly bound molecules near a Feshbach resonance. The second step is the efficient formation of ground-state ultracold
molecules via the stimulated Raman adiabatic passage (STIRAP). Although thisrecipe allowed the production of cold molecules as low as tens of nano Kelvin, it also
has some drawbacks. It requires a Feshbach resonance to be at an experimentally accessible bias field and to be strong compared to any decoherence rate for an efficientmagnetic association. The Na + Li mixture has very narrow (∼mG) resonances onlyat high fields (>700 G), compared to other diatomic alkali metal mixtures, and using
these resonances for molecule formation is technically challenging. However, we haverelied on one of these resonances at 745G to generate ground-state molecules. Even
with a low magnetic association efficiency of about 5%, having a large initial atomic
mixture number allowed a competitive number of ground-state molecules, typically
5 × 104 .
All-optical creation of molecules in which the magnetic association step near a
Feshbach resonance is eliminated can broaden the horizon of ultracold molecules toa larger pool and simplify some of the experimental complication. There have been105

efforts toward all-optical creation of molecules. However, there are only a couple ofdifferent molecules that have been successfully formed in their ground state. Recent
experiments on Sr2 molecule formation have shown the all-optical creation of an
ultracold gas of alkaline-earth metal dimers in their absolute ground state [113, 114].
In addition, all optical creation of RbCs molecules using photoassociation (PA) was
demonstrated using spontaneous emission [115–119].
In the following subsections, three different approaches to the all-optical formationof NaLi molecules in the ground state of the 𝑎3 Σ+ potential are discussed in a chainof thought that gradually develops on top of a simplified model. Therefore, the full
picture is revealed only at the end of the chapter. The work described in this chapter
heavily utilizes ab initio calculations of the NaLi triplet potentials (𝑎3 Σ+ , 𝑐3 Σ+ , and𝑏3 Π) provided to us by Prof. Olivier Dulieu, to whom we are grateful. The table of
ab initio potential values for the lowest three triplet potentials of NaLi is provided inAppendix C of ref. [120].
Here, we model the NaLi free-to-bound Raman transitions as an effective threelevel system. The most complete knowledge of the system is contained in the statevector |𝜓⟩ = 𝑐𝑖 |𝑖⟩ + 𝑐𝑒 |𝑒⟩ + 𝑐𝑓 |𝑓 ⟩, which is a superposition of three orthogonal statevectors: the initial state |𝑖⟩, the intermediate excited state |𝑒⟩ and the final state |𝑓 ⟩.
Here, we do simple modeling and numerical calculation of the Schrodinger equationthat describes our system (effective three-level system with additional loss terms andenergy shifts) given by:
⎡ ⎤
⎤⎡ ⎤
⎡
Ω𝑑𝑜𝑤𝑛 (𝑡)
0
𝑐𝑖
0
𝑐
2
⎥ ⎢ 𝑖⎥
⎥
⎢
2
2
𝑑 ⎢
⎢ ⎥
⎥
⎢
⎢ Ω𝑑𝑜𝑤𝑛 (𝑡)⎥Ω+Ω𝑢𝑝Ω𝑢𝑝 (𝑡)⎢ 𝑐𝑒 ⎥ = 𝑖 ⎢ 2⎥ ⎢ 𝑐𝑒 ⎥Δ− 𝑑𝑜𝑤𝑛
4Δ
2
𝑑𝑡 ⎣ ⎦⎣⎦⎣ ⎦Ω𝑢𝑝 (𝑡)
𝑐𝑓
0
𝛿
𝑐𝑓
2
⎡
⎤⎡ ⎤
𝑅𝐴
0
0
𝑐
⎥ ⎢ 𝑖⎥1⎢⎢⎥⎢ ⎥− ⎢ 0 Γ𝑀 + 𝑅𝑀 0 ⎥ ⎢ 𝑐𝑒 ⎥
2⎣
⎦⎣ ⎦
0
0
𝑅𝐶
𝑐𝑓

(6.1)

where the two Rabi frequencies are Ω𝑢𝑝 and Ω𝑑𝑜𝑤𝑛 and Δ and 𝛿 are single- and two106

photon detunings set by the laser frequencies, respectively. We can define the effective
detunings Δ̄ ≡ Δ + Δ′ and 𝛿¯ ≡ 𝛿 + 𝛿 ′ where Δ′ and 𝛿 ′ are detunings caused by the
Raman beams acting on other nearby atomic or molecular transitions, and creating
light shifts. The term in blue compensates the AC Stark shifts caused by two Rabi
couplings in an isolated three-level system. Γ𝑀 is the linewidth of the intermediate
excited state. The terms in red and orange are additional decay terms that arise
specifically in our system. 𝑅𝐴 and 𝑅𝑀 are decay rates of the initial atomic state
and the final molecular state by photon scattering. 𝑅𝐶 is the atom-molecule collision
loss rate. We first focus on optimizing the Raman transition with only atomic andmolecular photon scattering (𝑅𝐴 , 𝑅𝑀 ) setting Δ′ and 𝛿¯ to zero and with complete
Stark shift compensation (with term in blue). We then further explore the effects ofcollisional loss and light shifts.

6.1

Double Raman transfer via Feshbach moleculestate

The most staightforward approach to forming a ground state molecule is to replace
the well-established method of forming Feshbach molecules, magnetic association neara Feshbach resonance, with a two-photon Raman transfer from the ultracold atomic
mixture to molecules in the Feshbach state (𝑣 = 10, 𝑁 = 2). Stable ground-state
molecules are eventually created from optically created Feshbach molecules by thestandard STIRAP sequence. The Feshbach state 𝑣 = 10, 𝑁 = 2 has a binding energyof 5.85 GHz, which can be bridged with an electro-optic modulator (EOM) from theenergy of free atoms. This allows using a single laser source with an EOM driven
with 5.86 GHz radiofrequency (RF) to achieve a two-photon 𝜋 pulse. There are a
few advantages to this method. First of all, by having an EOM in addition to asingle laser source, it is easier to achieve a much longer coherence time between two
Raman beams compared to having two independent lasers locked to a stable reference
cavity. Second, only one additional laser is required to create Feshbach molecules, and107

Figure 6-1: Ti:Sapphire laser setup for Raman transfer from free atoms to the Feshbach state.

the existing STIRAP lasers can be used to form the ground-state molecules from the
Feshbach molecule. Lastly, it can be checked whether the first step (optical formationof Feshbach molecules) was successful by imaging atoms that are dissociated from a
Feshbach molecule near the ∼ 745 G Feshbach resonance.
However, this approach has limitations. The Frank-Condon factors (FCF) of twopaths: a. FCF between free atomic states and the intermediate excited molecularstate (upleg path), and b. FCF between the Feshbach state and the intermediate
excited molecular state (downleg path) differ by more than two orders of magnitude,
and the binding energy of the Feshbach state is only 5.85 GHz; the Raman beamthat bridges the atomic state to one of the excited molecular bound states (upleg)
also acts strongly on the other path (downleg path), causing significant off-resonant
Rayleigh scattering of Feshbach molecules. Additionally, this also causes large AC
stark shifts of the molecular bound states leading to fast decoherence during theRaman transition. We will ignore the effect of AC stark shifts for now, since inprinciple, it can be compensated with additional laser beams. However, it turns out
to be one of the most critical aspects in evaluating the approach.
108

Figure 6-2: Four photon transfer of sodium and lithium atoms to the rovibrational ground state of NaLi molecules in the 𝑎3 Σ+ potential via Feshbach
molecule state. Free sodium and lithium atoms are transferred to the Feshbach state(𝑣 = 10, 𝑁 = 2) in the 𝑎3 Σ+ potential by two photon Raman transfer mediated by
an excited molecular state (𝑣 * = 49) in the 𝑐3 Σ+ potential. Feshbach molecules are
eventually transfered to the rovibrational ground state of NaLi molecules in the 𝑎3 Σ+
potential either using two photon Raman pulse or STIRAP mediated by 𝑣 * = 11.
The two red arrows indicate the first two photon couplings and the two black arrows
indicate the second two photon couplings.

1096.1.1

Coupling strengths of Raman beams

A newer model of Ti:Sapphire laser from M-Squared Laser spanning 670-725 nm
range1 can be used for Raman transfer of free atoms to the Feshbach state. With thecurrent continuum-to-bound Raman experimental setup using the Ti:Sapphire laser
and a fiber EOM from EOSpace (PM-0S5-10-PFU-PFU-670), we are able to get up
to 300 mW for Upleg and 100 uW for Downleg, respectively. Assuming a beam waistof 50 𝜇 m, which is comparable to the size of 1596 nm lattice and a 1064 nm dipole
trap, the intensities of the two Raman beams are given as:
300mW= 3.82 × 106 mW · cm−2 = 3.82 × 103 W · cm−2𝜋(50 × 10−4 cm)2
100uW
𝐼𝑑𝑜𝑤𝑛 =
= 1.27 × 103 mW · cm−2 = 1.27 W · cm−2𝜋(50 × 10−3 cm)2
𝐼𝑢𝑝 =

(6.2)

The Rabi coupling strength between the free atomic state and 𝑣 * = 49 by the upleg
beam of intensity 𝐼𝑢𝑝 is estimated to be Ω𝑢𝑝 ≈ 2𝜋 × 31.1kHz for the atomic densityof 𝑛 = 5 × 1012 cm−3 for both sodium and lithium atomic clouds. Given the FranckCondon factor2 difference between the upleg and the downleg as shown in Fig.6-3, theRabi frequency of the downleg (Ω𝑑𝑜𝑤𝑛 ) is roughly 1000 times higher than the Rabi
frequency of the upleg for individual excited vibrational states when 𝑣 * is smaller
than 50.

6.1.2

Li 2S to 2P transition strengths

States in the 𝑐3 Σ+ potential have electronic states dissociating into Na(3𝑠)+Li(2𝑝),and therefore, free-to-bound upleg and downleg pulses can cause near-resonant lightscattering by Li 𝐷1 and 𝐷2 transitions leading to atomic loss 3 . This effect can belarge for highly excited vibrational states as an intermediate excited state, and it isdominated by the upleg pulse, since its intensity is normally a few orders of magnitude
1

A newer model of Ti:Sapphire laser from M-Squared Laser spanning 670-725 nm range wasoriginally used for finding photoassociation (PA) lines of high vibrational states of 𝑐3 Σ+ NaLi.
2
Fanck-Condon factors are calculated from ab initio potentials.
3
Na 𝐷1 and 𝐷2 transitions are far off-resonant to upleg and downleg optical frequencies (tens of
THz or more), and the caused loss is negligible compared to the loss caused by Li transitions.

110Figure 6-3: Franck-Condon factors (FCF) calculated from ab initio potentials. Free-to-bound (𝑣 * ) FCFs are indicated in red, Feshbach state (𝑣 = 10, 𝑁 = 2)
to excited state FCFs are indicated in blue and the ground state to excited state FCFs
are indicated in green. The red dashed circles are identifying the two FCFs relevantto the first two photon transfer: free-to-bound (atoms to 𝑣 * = 49) FCF and 𝑣 * = 49
to 𝑣 = 10 FCF. The black dashed circles are identifying the two FCFs relevant to thesecond two photon transfer: FCF between 𝑣 = 10 and 𝑣 * = 11, and FCF between𝑣 * = 11 and 𝑣 = 0.

111

larger than that of the downleg pulse as estimated in Eq. 6.2.
Here, we first quantify the strengths of these atomic transitions driven by theupleg pulse. In a two-level system with a coherent drive by near-resonant light, theoptical Rabi frequency is given by Ω = ⟨𝑏| 𝜇^ · E |𝑎⟩ /ℏ = 𝜇𝑏𝑎 𝐸0 /ℏ where 𝜇𝑏𝑎 is theelectric-dipole transition matrix element for states 𝑎 and 𝑏, and 𝐸0 is the electric field√strength of the incident optical field. We can write this as Ω = 4.37𝜇𝑏𝑎 𝐼 with Ωin unit of 2𝜋×MHz, 𝜇𝑏𝑎 in Debye, and 𝐼, the light intensity, in mW/mm2 . However,for a full description, we need to consider all the hyperfine states. The interaction
between the internal states of the atom and an external, near-resonant optical field
is quantified through the electric dipole transition matrix elements. These matrix
elements describe how the internal states of the atom couple to each other via an⃗ interaction with the near-resonant field. If we start in a hyperfine
electric dipole (⃗𝜇 · 𝐸)
state represented by |(𝐽𝐼)𝐹 𝑚𝐹 ⟩ and couple to a state |(𝐽 ′ 𝐼 ′ )𝐹 ′ 𝑚𝐹 ′ ⟩, then the matrix
element for this transition is given by ⟨(𝐽 ′ 𝐼 ′ )𝐹 ′ 𝑚𝐹 ′ | 𝜇^ |(𝐽𝐼)𝐹 𝑚𝐹 ⟩, where 𝜇^ is theelectric dipole operator. We can use the Wigner-Echart Theorem to represent thismatrix element in terms of a 𝑚-independent reduced matrix element ⟨𝐽 ′ | |^𝜇(1)| |𝐽⟩,
Wigner 3-J and 6-J symbols as:√︀′′⟨(𝐽 ′ 𝐼 ′ )𝐹 ′ 𝑚𝐹 ′ | 𝜇^ |(𝐽𝐼)𝐹 𝑚𝐹 ⟩ = 𝛿𝐼 ′ 𝐼 (−1)𝐹 +𝐹 +𝐽 +𝐼−𝑚𝐹 +1 (2𝐹 ′ + 1)(2𝐹 + 1)
⎫
⎛
⎞⎧
′
′
′⎬
⎨
(6.3)
𝐽
1 𝐽
𝐽 𝐼 𝐹
⎠
×⎝
⟨𝐽 ′ | |^𝜇(1)| |𝐽⟩⎩𝐹 1 𝐽 ⎭−𝑚 ′ 𝑞 𝑚𝐹𝐹

where 𝜇^(1, 𝑞) is an irreducible spherical tensor operator for the electric dipole operator with 𝑞 = −1, 0, 1 for electric dipole radiation with polarization 𝜎 − , 𝜋 and 𝜎 + ,respectively.
For example, let us consider a free-to-bound Raman transition from Na and Li
atoms with their electron spins aligned to the bias field direction in the groundlevels, |𝑆Na = 1/2, 𝑚𝑆Na = 1/2, 𝑆Li = 1/2, 𝑚𝑆Li = 1/2⟩, to the NaLi Feshbach state(|𝑣 = 10, 𝑁 = 2, 𝑚𝑁 = −2, 𝑆 = 1, 𝑚𝑆 = 1⟩) in the 𝑎3 Σ+ potential via an intermediate excited state (|𝑣 * = 49, 𝑁 = 1, 𝑚𝑁 = −1, 𝑆 = 1, 𝑚𝑠 = 1⟩) in the NaLi 𝑐3 Σ+ po112

tential. This process requires that the upleg pulse be polarized 𝜎− and the downleg pulse be polarized 𝜎+, as illustrated in Fig. 6-2. With the Upleg intensity
mentioned above which is 𝐼𝑢𝑝 = 3.82 × 104 mW · mm−2 , optical Rabi frequency,
for example, of Li 𝐷1 transition from the ground upper stretched hyperfine state|𝐽 = 1/2, 𝐹 = 3/2, 𝑚𝐹 = 3/2⟩ to the excited state |𝐽 ′ = 1/2, 𝐹 ′ = 3/2, 𝑚𝐹 ′ = 1/2⟩ is√√Ω = 4.37 × (− 32 ) × (−8.433 Debye) × I ≈ 3.4 GHz 4 .

The photon scattering rate of a two-level atomic system 𝑅𝑠𝑐𝑎𝑡𝑡 = Γ𝜌𝑏𝑏 where 𝜌𝑏𝑏 isthe fraction of the population in the excited state 𝑏 and Γ is the linewidth of state 𝑏.
Using the solution of optical Bloch equations that describes the excitation of a two2/2
level atom by radiation, the scattering rate is expressed as 𝑅𝑠𝑐𝑎𝑡𝑡 = Γ2 Δ2 +ΩΩ2 /2+Γ
2 /2
2

Ω
which is approximated to 4Δ
2 Γ when detuning Δ is much greater than the Rabi

frequency Ω and the linewidth Γ. Therefore, the total atomic scattering rate 𝑅𝐴 ,considering all near-resonant hyperfine states, is perturbatively given by:⃒2⃒∑︁ ⃒ ⟨(𝐽 ′ 𝐼 ′ )𝐹 ′ 𝑚𝐹 ′ | 𝜇^ · E |(𝐽𝐼)𝐹 𝑚𝐹 ⟩ ⃒⃒⃒𝑅𝐴 =⃒ Γ𝐹 ′ ,𝑚𝐹 ′⃒2Δ𝐹 ′ ,𝑚

(6.4)

𝐹′

𝐹 ′ ,𝑚𝐹 ′where Γ𝐹 ′ ,𝑚𝐹 ′ is the natural linewidth Δ𝐹 ′ ,𝑚𝐹 ′ is the photon detuning of an excitedstate |𝐹 ′ , 𝑚𝐹 ′ ⟩. From Eq.6.3 and Eq.6.4, the photon scattering rate by Li 𝐷1 line isabout 381 Hz and by 𝐷2 line is about 177 Hz leading to total scattering rate of about
558 Hz 5 .
113

Figure 6-4: Atomic and molecular photon scattering losses. High intensity
upleg pulse with 𝜎− polarization in green can drive Li 𝐷1 and 𝐷2 transitions leading
to photon scattering. Strong downleg Rabi coupling Ω𝑑𝑜𝑤𝑛 causes photon scattering
through the molecular transition (𝑣 = 10 → 𝑣 * = 49).

114

M/ down(a) 10

4

103
102
101

(b) 105t (ms)

103
101
10 1

Loss|tot(c) 102
101
100
0

10

20

v*

30

40

50

Figure 6-5: Results of a simplified loss model for free-to bound Raman
transition. (a) Optimum
ratio of the single photon detuning to the downleg Rabi√︁𝑀
𝐴frequency ( ΩΔ𝑑𝑜𝑤𝑛= ΓΓ𝑀𝐴 Δ) as a function of excited vibrational state 𝑣 * picked as
Ω𝐴an intermediate state. (b) Raman 𝜋 pulse duration in ms obtained with the optimum
ratio in (a). (c) Minimum total photon scattering loss given by Eq. 6.6 as a functionof excited vibrational state 𝑣 * .

115

6.1.3

Molecular and atomic losses

Near-full-contrast Raman transitions require a single Rabi cycle time to be shorter
than the laser coherence time (∼ 1 ms). For this, it is necessary to have Ω𝑑𝑜𝑤𝑛
much greater than Ω𝑢𝑝 in order to provide a sufficient two-photon rabi frequency
Ω𝑒𝑓 𝑓 = Ω𝑢𝑝 Ω𝑑𝑜𝑤𝑛 /Δ with weak upleg. However, in this case ,Ω𝑑𝑜𝑤𝑛 ≫ Ω𝑢𝑝 , photon
scattering loss of the molecule population by the downleg pulse can be significant aswell in addition to the atomic losses by the Li 𝐷1 and 𝐷2 transitions as mentioned
in the previous section. The goal of this section is to find the condition of minimum
photon scattering loss created by these two effects illustrated in Fig. 6-4.
Molecular and atomic photon scatting losses in half the Rabi cycle 𝑡 = 𝑡𝜋 =4𝜋Δ𝑀
6are given by:Ω𝑢𝑝 Ω𝑑𝑜𝑤𝑛

Ω2𝑑𝑜𝑤𝑛𝑡=𝑡𝑅 𝜋 · Γ𝑀 · 𝑥√· Γ𝑀 · 𝑡 −−−→
2
4Δ𝑀
𝐹𝐶
2
Ω
𝜋 · Γ𝐴
𝑡=𝑡𝑅√
Loss|A = 𝐴2 · Γ𝐴 · 𝑡 −−−→4Δ𝐴
𝐹 𝐶 · Δ2 · 𝑥

Loss|M =(6.5)

𝐴

where the relative FCF, 𝐹 𝐶 is defined by Ω2𝑢𝑝 = 𝐹 𝐶 · Ω2𝐴 , Γ𝑀 is the linewidth of themolecular excited state (∼ 9 MHz) and 𝑥 = ΩΩ𝐴𝑑𝑜𝑤𝑛. Here, Ω𝐴 is the effective atomic·Δ𝑀
Rabi frequency7 and Δ𝑀 is the single photon detuning of the two-photon Raman
transition. We regard scattering of a single photon as a loss, since we want to stay
below the recoil temperature. To minimize the total loss Loss|tot = Loss|M + Loss|A ,𝐷1 line reduced matrix element ⟨𝐽 ′ = 1/2| |^𝜇(1)| |𝐽 = 1/2⟩ = −8.433 Debye and 𝐷2 line reduced
𝜇(1)| |𝐽 = 1/2⟩ = 11.925 Debye [121].
matrix element ⟨𝐽 ′ = 𝑒/2| |^5
The estimate assumed the upleg frequency is resonant to the free to 𝑣 * = 49 transition. (Δ𝐷1 ≈258 GHz and Δ𝐷2 ≈ 268 GHz) Since the excited-state hyperfine splittings are much smaller thanthe photon detunings in this case, the effective Rabi frequency of 𝐷1 transitions Ω𝐷1 and that of𝐷2 transitions Ω𝐷2 can be approximated to the [︁⃒sum of⃒ all relevant]︁ ⃒ individual Rabi frequencies.
⃒
⃒ ⃒
⃒
√ ⃒⃒
⃒
⃒ √ ⃒ ⃒ ⃒Therefore, Ω𝐷1 ≈ ⃒Ω1/2,3/2,1/2 ⃒ + ⃒Ω1/2,1/2,1/2 ⃒ = ⃒− 32 ⃒ + ⃒ 13 ⃒ × ⃒4.37 × (−8.433 Debye) × I⃒ ≈
⃒
⃒ ⃒
⃒ ⃒
⃒ [︁⃒⃒ 1 ⃒⃒ ⃒⃒ 1 √︁ 2 ⃒⃒ ⃒⃒ 1 ⃒⃒]︁
4.158 GHz and Ω𝐷2 ≈ ⃒Ω3/2,5/2,1/2 ⃒+⃒Ω3/2,3/2,1/2 ⃒+⃒Ω3/2,1/2,1/2 ⃒ = ⃒− 6√⃒ + ⃒− 3 5 ⃒ + ⃒ 2√10 ⃒ ×
2
⃒
⃒
√
⃒
⃒
⃒4.37 × (11.925 Debye) × I⃒ ≈ 2.940 GHz . The three numbers as subscripts of Ω s indicate the 𝐽 ′ ,4

𝐹 ′ and 𝑚𝐹 ′ quantum numbers in order6
The duration of a 𝜋 pulse 𝑡𝜋 is defined by 𝑡𝜋 · Ω𝑒𝑓 𝑓 = 𝜋 where two photon Rabi frequencyΩ ΩΩ𝑒𝑓 𝑓 = 𝑢𝑝4Δ𝑑𝑜𝑤𝑛 .
7Since Δ𝐷1 ≈ Δ𝐷2 = Δ𝐴 (Δ𝐷1 , Δ𝐷2 ≫ |Δ𝐷1 −Δ𝐷2 |) and Γ𝐷1 ≈ Γ𝐷2 , we can define the effective∑︀2
atomic Rabi frequency Ω𝐴 by Ω2𝐴 = 𝐹 ′ ,𝑚𝐹 ′ |⟨(𝐽 ′ 𝐼 ′ )𝐹 ′ 𝑚𝐹 ′ | 𝜇^ |(𝐽𝐼)𝐹 𝑚𝐹 ⟩| .

116

we take the derivative of Loss|tot with respect to 𝑥 to find the optimal value of 𝑥. We
come to the conclusion that in this model, the optimal value 𝑥 does not depend on the
√︁
Γ𝐴 1
𝐷
FCF if the pulse duration is fixed to half the Rabi cycle and 𝑥 = Ω𝐴Ω·Δ
=
Γ𝑀 Δ𝐴
𝑀
gives the minimum total loss. From this relation, we can find the optimal ratio
between the molecular single-photon detuning Δ𝑀 and the downleg Rabi frequency√︁Δ𝑀
𝐴Ω𝑑𝑜𝑤𝑛 as Ω𝑑𝑜𝑤𝑛 = ΓΓ𝑀𝐴 Δ. The optimal ratio is plotted as a function of 𝑣 * in Fig.
Ω𝐴
6-5(a). For a given excited molecular state, the minimum scattering loss achieved by
the optimum value 𝑥 is given as:
2𝜋
Loss|tot =
ΔA

√︂

ΓA ΓM
.
FC

(6.6)

Therefore, using a lower lying exit state that has a larger Δ𝐴 than that of 𝑣 * = 49 fora Raman transfer has a smaller scattering loss, as plotted in Fig. 6-5(c). However, themaximum upleg and downleg Rabi frequencies that we can achieve become smallerwith low 𝑣 * molecular states, and 𝑡𝑝𝑖 starts to become much larger than 1 ms (Fig.
6-5(b)). If we do not have a limited laser coherence time, using, for example, 𝑣 * = 44
as an intermediate excited state is a better choice. Figures 6-6 and Figure 6-7 arenumerical calculations of the Schrödinder equation describing the three-level systemwith an extra decay term that accounts for the atomic photon scattering using anintermediate state 𝑣 * = 49 and 𝑣 * = 44 respectively. With 𝑣 * = 49, the maximum
population transfer efficiency to the Feshbach state is ∼ 1.2 % at around 0.25 ms,
while with 𝑣 * = 44, the maximum population transfer efficiency to the Feshbach stateis ∼ 11.5 % at around 15 ms, which is much longer than the laser coherence time (∼1 ms). In case the Raman pulse duration is limited to 1 ms, the transfer efficiency
reaches only ∼ 0.3 %, which is worse than what we can achieve with 𝑣 * = 49.
So far, we have been optimizing a rather simplified process: Raman transition in athree-level system with an additional atomic loss term. However, unlike alkali atoms,
molecules have many vibrational and rotational states, which may interfere with thesimplified process. For example, there are more than 100 rovibrational states within
the 𝑣 * = 49 branch. Despite the upleg free-to-bound Rabi frequency being only about
117

Figure 6-6: Numerical calculation of free-to bound Raman transition to the
Feshbach state via 𝑣 * = 49 with atomic photon scattering. (a) Time evolutionof the populations in the final Feshbach state (in blue), intermediate excited state (inred), initial free atomic state (in dashed green) and the total population (in black)are plotted. The total population decreases over time because of atomic loss termfrom Li 𝐷1 and 𝐷2 transitions is included in the calculation. Optimum ratio of thesingle photon detuning to the downleg Rabi frequency is chosen in the numericalcalculation. (b) Transferred population to the Feshbach state is plotted as a functionof pulse duration. The transfer is maximized to ∼ 1.2 % at around 0.25 ms.

118

Figure 6-7: Numerical calculation of free-to bound Raman transition to the
Feshbach state via 𝑣 * = 44 with atomic photon scattering. (a) Time evolutionof the populations in the final Feshbach state (in blue), intermediate excited state (inred), initial free atomic state (in dashed green) and the total population (in black)are plotted. The total population decreases over time because of atomic loss termfrom Li 𝐷1 and 𝐷2 transitions is included in the calculation. Optimum ratio of thesingle photon detuning to the downleg Rabi frequency is chosen in the numericalcalculation. (b) Transferred population to the Feshbach state iS plotted as a functionof pulse duration up to 40 ms. The transfer is maximized to ∼ 11.5 % at around15 ms. (c) The transfer to the Feshbach state only reaches to ∼ 0.3 % if the pulse
duration is limited to 1 ms.

119

30 kHz at most, the upleg can act strongly on the downleg path driving transitions
between the Feshbach state to some of these other rovibronic states (e.g. 𝑣 * = 49,
𝑁 = 3, 𝑚𝑁 = −3)8 . Additionally, even small fraction of upleg polarization impurity(𝜎+ component) can cause fast molecular scattering through the Feshbach state to𝑁 = 3, 𝑚𝑁 = −1 states in the 𝑐3 Σ+ potential. Figure 6-8 is an illustration of theupleg beam acting on the downleg path through nearby excited rovibronic states in
the 𝑐3 Σ+ potential. For example, the upleg beam of 3.82 × 104 mW/mm2 intensity
leads to a molecular scattering rate of more than 20 kHz with Δ𝑀 ≈ 1.45 GHz 9 .
The photon scattering rate of about 22 kHz is caused by the molecular transition
between the Feshbach state and 𝑣 * = 49, 𝑁 = 3, 𝑚𝑁 = −3 and scattering rate ofmore than 1 kHz can be caused by only 1% impurity of the upleg beam polarization(𝜎+) through the Feshbach state to the 𝑣 * = 49, 𝑁 = 1, 𝑚𝑁 = −1 transition. Usinga highly excited vibrational state 𝑣 * as an intermediate state tends to have a higher
additional molecular scattering loss by the upleg acting on the downleg path, since
the higher 𝑣 * has a smaller rotational constant 𝐵 (see Table 6.1) and, therefore, theoptical frequency of the upleg is closer to these molecular transition frequencies. Even
without considering molecular photon scattering losses by the upleg beam acting on
the downleg path, population transfer efficiency was less than 2 % (see Figs. 6-6 and
6-7). The inclusion of additional molecular decay in the numerical calculation further
reduces the transfer efficiency.

6.1.4

Effects of atomic densities and collisional loss

One way of reducing the loss by atomic and molecular light scattering to obtain a
non-negligible population transfer from the free atomic state to the Feshbach stateis to shorten the Rabi cycle time or in other words, increase the two-photon Rabi
𝑑𝑜𝑤𝑛frequency (Ω𝑒𝑓 𝑓 = Ω𝑢𝑝 × Ω4Δ). We saw previously that simply increasing Ω𝑑𝑜𝑤𝑛 /𝑀rather increases molecular loss by strong Ω𝑑𝑜𝑤𝑛 and there is an optimum value for8

The rotational selection rules for Σ ↔ Σ potential transitions are Δ𝑁 = 1, hence from a 𝑁 = 2
Feshbach state it is possible to access 𝑁 = 1, 3 excited states.
9*√︁ With downleg intensity given in Eq. 6.2 and 𝑣 = 49, optimum single photon detuning Δ𝑀 =
Γ𝑀 Δ𝐴Γ𝐴 Ω𝐴 × Ω𝑑𝑜𝑤𝑛 ≈ 1.45 GHz.

120

Figure 6-8: Molecular photon scattering losses by the upleg beam driving
transitions to nearby excited states in the 𝑐3 Σ+ potential from the Feshbachstate. High intensity upleg pulse with 𝜎− polarization indicated with a dashed green
arrow can drive transitions from the Feshbach state (𝑣 = 10, 𝑁 = 2, 𝑚𝑁 = −2) tovibrational states with 𝑁 = 3, 𝑚𝑁 = −3 in the 𝑐3 Σ+ potential (green dashed circles).
Small fraction of 𝜎− polarization component due to polarization impurity of the upleg
can drive transitions from the Feshbach state (𝑣 = 10, 𝑁 = 2, 𝑚𝑁 = −2) to excitedstates in the 𝑐3 Σ+ potential with 𝑁 = 1, 𝑚𝑁 = −1 or 𝑁 = 3, 𝑚𝑁 = −1 (green dotted
circles). These transitions lead to molecular photon scattering losses indicated with
red dashed wiggles in addition to those caused by the strong downleg Rabi frequency.

121

𝑣 * 𝐵 constant (GHz)
0
8.83
8.71
1
2
8.59
8.48
3
4
8.36
8.24
5
8.12
6
7
8.00
7.88
8
9
7.76
10
7.64
11
7.52
7.40
12
7.27
13
14
7.15
7.02
15
16
6.89
6.76
17
6.63
18
19
6.50
20
6.38
6.24
21
22
6.08
5.93
23
24
5.79
25
5.66

𝑣 * 𝐵 constant (GHz)
26
5.51
27
5.36
28
5.20
29
5.02
30
4.86
31
4.67
32
4.51
33
4.32
34
4.14
35
3.96
36
3.77
37
3.57
38
3.38
39
3.16
40
2.96
41
2.75
42
2.53
43
2.31
44
2.10
45
1.88
46
1.65
47
1.42
48
1.21
49
0.994
50
0.905

Table 6.1: Rotational constants of excited vibrational states (𝑐3 Σ+ , 𝑣 * ). The
rotational constants are calculated from the 𝑐3 Σ+ ab initio potential.

122

Figure 6-9: Raman transfer with atomic and molecular decay as a functionof Δ𝑀 via different 𝑣 * . The plots are the numerical calculation of free to Feshbachstate Raman transfer with additional atomic and molecular decay driven by the upleg
pulse. The left (right) plot is the result without (with) a molecule-atom collisionalloss term. The calculation is done assuming upleg beam polarization purity of 99%
and atomic densities of 5 × 1014 cm−3 . The upleg beam power is set to 300 mW andthe downleg beam power is set to 100 𝜇W with beam waists of 50 𝜇m.
Ω𝑑𝑜𝑤𝑛 /Δ𝑀 for minimum total loss. Therefore, it is necessary to increase Ω𝑢𝑝 to
actually benefit from larger Ω𝑒𝑓 𝑓 . Since FCF is proportional to the larger density
among the two atomic clouds (Na and Li) 10 , this can be achieved by preparing
atomic clouds with higher densities.
However, Feshbach molecules are chemically reactive with Na atoms or Li atoms,
and it is natural to assume that the molecule+atom collisional loss rates are close¯ 𝑁 𝑎 𝑛𝐿𝑖 where 𝐼 is the
Photoassociation (PA) rate equation is given by 𝑛˙ 𝑁 𝑎 = 𝑛˙ 𝐿𝑖 = −𝐾𝐼𝑛¯
PA laser intensity and 𝐾 is the intensity normalized PA rate coefficient. By defining 𝐷 ≡¯ 𝐿𝑖 (𝑛𝐿𝑖 + 𝐷).
𝑛𝐿𝑖 (0) − 𝑛𝑁 𝑎 (0) = 𝑛𝐿𝑖 (𝑡) − 𝑛𝑁 𝑎 (𝑡), the rate equation is expressed as 𝑛˙ 𝐿𝑖 = −𝐾𝐼𝑛
𝐷
The solution of the differential equation is given by 𝑛𝑁 𝑎 (𝑡) = [𝑛 (0)/𝑛 (0)]𝑒𝐷𝐾𝐼𝑡
and 𝑛𝐿𝑖 (𝑡) =
¯
−1
𝐿𝑖
𝑁𝑎
𝑛(𝑡)
−𝐷
1
¯ and there. 𝑖) If 𝑛𝑁 𝑎 (0) = 𝑛𝐿𝑖 = 𝑛(0),=≈ 1 − 𝑛(0)𝐾𝐼𝑡¯¯−𝐷 𝐾𝐼𝑡10[𝑛𝑁 𝑎 (0)/𝑛𝐿𝑖 (0)]𝑒

𝑛(0)

−1

1+𝑛(0)𝐾𝐼𝑡¯ = Ω2 /Γ𝑃 𝐴 . 𝑖𝑖) If 𝑛𝑁 𝑎 (0) ≫ 𝑛𝐿𝑖 ,fore FCF is proportional to 𝑛(0) of the relation 𝑛(0)𝐾𝐼𝑃𝐴¯𝑛𝐿𝑖 (𝑡)−𝑛𝑁 𝑎 (0)𝐾𝐼𝑡¯ and therefore FCF is proportional to 𝑛𝑁 𝑎 (0).
≈ 1 − 𝑛𝑁 𝑎 (0)𝐾𝐼𝑡
𝑛𝐿𝑖 (0) ≈ 𝑒

123

to the universal values 11 . Universal loss rate constants are 𝐾NaLi−Na ≈ 1.53 ×
10−10 s−1 cm3 with Na atoms and 𝐾NaLi−Li ≈ 3.425 × 10−10 s−1 cm3 with Li atoms.
Figure 6-9 is the result of numerical calculations of the Raman transition with atomic
and molecular decay as a function of Δ𝑀 through different excited states 𝑣 * withstronger Ω𝑢𝑝 achieved by denser atomic clouds (5 × 1014 cm−3 ). The left graph isthe result without including the molecular loss from molecule + atom collisions andthe right graph includes the collisional losses of molecules by Na and Li atoms. The
calculation is done assuming a polarization purity of the upleg beam of 99% andatomic densities of 5 × 1014 cm−3 . The upleg beam power is set to 300 mW andthe downleg beam power is set to 100 𝜇W with beam waists of 50 𝜇m. With denser
atomic clouds, much higher population transfer can be achieved especially with low
𝑣 * (for example, more than 30% via 𝑣 * = 11) if collisional losses do not exist, as you
can see in the left plot. Once, collisional losses are taken into account, as transfer
with low lying 𝑣 * tends to have a longer 𝑡𝜋 , experiences a longer time for collisionalloss as well, and therefore, stronger suppression of population transfer, as shown in
the right graph. Therefore, higher densities may not be always helpful because of theuniversal rate limited collisional loss of NaLi with surrounded atoms 12 , and only afew percent transfer is achieved with 𝑛𝑁 𝑎 = 𝑛𝐿𝑖 = 5 × 1014 at most with 𝑣 * = 48. It
is important to optimize both the atomic densities and 𝑣 * to maximize the numberof Feshbach molecules.
In summarity, atomic scattering is suppressed by increasing the detuning from theatomic resonance, e.g. going to smaller 𝑣 * . However, this increases molecular scattering and gives an optimum detuning. Density can speed up the molecule formation,
but then runs into the limit of collisional loss. For the optimum choice, all three loss
11

It is experimentally confirmed by us that NaLi Feshbach molecule+atom (with Na state|𝐹 = 2, 𝑚𝐹 = 2⟩ and |𝐹 = 1, 𝑚𝐹 = 1⟩, and Li state |𝐹 = 1/2, 𝑚𝐹 = 1/2⟩) collision rates are 𝑠-wave
universal rate limited. However, ground-state molecules in their stretched state live long with Na
atoms also in their stretched state (when they are aligned) unlike Feshbach molecules. (The collisional loss rate constant is about 2 orders of magnitude smaller than the universal value.)12
We will see later that, indeed, 𝑛 = 5 × 1014 gives more transfer than 𝑛 = 5 × 1015 . The loss
with Li atoms is about twice larger than with Na atoms, and preparing the Li density 10 times
lower while keeping the Na density high may lead to a higher transfer efficiency. However, this will
require loading 10 times less Li atoms, and the transferred molecule number will be less since a 10
time lower Li number does not increase the transfer efficiency by more than a factor of 10.

124

processes contribute about equally.

6.2

Direct transfer to the ro-vibrational ground state

The second approach is to perform a single Raman transfer from an ultracold gas of
atoms directly to the molecular ro-vibrational ground state. This approach allows fora single Raman transfer to create ground-state molecules, and hence, the experimental
complexity can be reduced. The FCF of the ground state 𝑣 = 0 to an excited state 𝑣 *transition becomes comparable to or even smaller than that of the free to 𝑣 * transition
for a high 𝑣 * as in Fig. 6-3, and it is natural to choose a rather lower lying 𝑣 * asan intermediate excited state for the Raman transition. As an example, Figure 6-10
illustrates a free-to-ground direct Raman transition via an intermediate excited state𝑣 * = 26, 𝑁 = 1, 𝑚𝑁 = −1 using 𝜎− polarization for both upleg and downleg beams.
As in the Raman transition from free to the Feshbach state, high intensity upleg canact on the downleg path causing light scattering. The binding energy of the groundstate (6.2381 THz) is large compared to that of the Feshbach state (5.85 GHz), andtherefore, the rate of this molecular light scattering can be tuned to much smaller
value than we saw in the free-to-Feshbach state Raman transfer. Upleg frequency
can be tuned to have relatively large detuning from the ground state to excitedmolecular state transition lines. However, achieving a sufficiently high two-photon
Rabi frequency can be extremely challenging because of even smaller FCFs.
Direct Raman transfer from free to the 𝑣 = 0 ground-state molecules can bedivided into mainly two different situations: Either i) Ω𝑑𝑜𝑤𝑛 ≫ Ω𝑢𝑝 or ii) Ω𝑑𝑜𝑤𝑛 ≲ Ω𝑢𝑝
unlike the free-to-Feshbach state transfer where Ω𝑑𝑜𝑤𝑛 was always greater than Ω𝑢𝑝 .
For case i) Ω𝑑𝑜𝑤𝑛 ≫ Ω𝑢𝑝 , which is achieved with an intermediate state 0 ≤ 𝑣 * ≤ 31
when the maximum upleg power is set to 300 mW and the downleg power to 10 mW,
the molecular photon scattering by the strong downleg cannot be neglected, while theatomic photon scattering by the upleg is relatively small, since the upleg frequency
is far detuned from the Li 𝐷1 and 𝐷2 lines. Close-to-perfect Raman transfer to theground state could be possible using one of the excited states deep in the excited 𝑐3 Σ+125Figure 6-10: Illustration of a Raman transition from free atoms to the rovibrational ground state of NaLi 𝑎3 Σ+ . 𝑣 * = 26, 𝑁 = 1, 𝑚𝑁 = −1 in the 𝑐3 Σ+potential is chosen as an intermediate excited state as an example. 𝜎− polarization
for both upleg (green arrow) and downleg (blue arrow) beams are used. High intensity upleg pulse with 𝜎− polarization indicated with a dashed green arrow can drive
transitions from the ground state (𝑣 = 0, 𝑁 = 0, 𝑚𝑁 = 0) to vibrational states with
𝑁 = 1, 𝑚𝑁 = −1 in the 𝑐3 Σ+ potential. Small fraction of 𝜎+ due to polarization
impurity of the upleg can drive transitions from the ground state to excited states in
the 𝑐3 Σ+ potential with 𝑁 = 1, 𝑚𝑁 = 1.

126

potential if laser coherence time was not limited and collisional losses were negligible.
Eliminating collisional loss could possibly be achieved by having only one Na and
Li atom per each 3D optical lattice site or optical tweezer. However, as the opticalfrequency difference between the upleg and downleg is several THz, one cannot use asingle laser source for both pulses, and the coherence time of these beams are expected
to be not much smaller than 1 ms. Here, we will assume that the laser coherence timeis only 50 𝜇s 13 . Under these conditions, the maximum transfer occurs when Loss|tot ≈
Ω2

4𝜋Δ𝑀
down
Loss|M ≈ 4Δ
· ΓM · t𝜋 is minimized. If we set the 𝜋 pulse duration 𝑡𝜋 = Ω𝑢𝑝
to
2
Ω𝑑𝑜𝑤𝑛
M

be at most the coherence time 𝑡𝐶 , we find that Loss|tot ≈ Ω24𝜋·tC · ΓM , and therefore,up

less loss for larger upleg FCF. On the other hand, for case ii) Ω𝑑𝑜𝑤𝑛 ≲ Ω𝑢𝑝 , which isachieved with an intermediate state 32 ≤ 𝑣 * ≤ 52, the molecular loss by the downleg
pulse becomes significantly smaller than the atomic loss if far detuned (|Δ𝑀 | > Γ𝑀 ).
Ω2

Similarly, the total loss is expressed as Loss|tot ≈ Loss|A = 4ΔA2 · ΓA · tC , and thereforeA*the loss is smaller for lower 𝑣 . However, we are more limited by small two-photon
Rabi coupling strength, and unable to be in the far detuned limit. Therefore, the total(︁)︁Ω2up +Ωdown
1
1
2
loss Loss|tot ≈ Loss|M =· ΓM · tC ≈ 4𝜋 Ω2 + Ω2
· ΓM , and less loss
2
up
down
(︁ 4ΔM)︁
through 𝑣 * with smaller Ω12 + Ω21. A strong correlation between the maximum
𝑢𝑝

𝑑𝑜𝑤𝑛

transferred population to the ground state and the FCFs can be seen in Fig. 6-11.
Assuming 30 𝜇m beam waists for upleg and downleg beams instead of 50 𝜇m allows
at least a factor of few higher transfer efficiency for transitions with relatively small
upleg FCF as it allows further transfer within 50 𝜇s. However, as will be discussed
later in the chapter, this will create larger inhomogeneity, causing larger two-photon
detuning during the Raman transition.

127

Figure 6-11: Maximum transfer to the ground state of NaLi 𝑎3 Σ+ potential as a function of intermediate excited vibrational state 𝑣 * . Transfer
efficiency to the ground state was numerically calculated for atomic densities of𝑛𝑁 𝑎 = 𝑛𝐿𝑖 = 5 × 1014 cm−3 with the upleg power of 300 mW and downleg powerof 10 mW. Molecule+atom collisional loss is ignored only in the blue plot. Blue andred plots are the results with the upleg and downleg beam waists of 50 𝜇m and thegreen plot is the result with 30 𝜇m. Polarization purity of the two Raman beams are
assumed to be 95 % and the pulse duration is limited to 50 𝜇s.

128

Figure 6-12: Franck-Condon factors between vibrational states in 𝑎3 Σ+ and𝑐3 Σ+ potentials. They are calculated from ab initio potentials. Free-to-bound (𝑣 * )FCFs of 𝑛 = 5 × 1014 cm−1 are indicated in black.

129

Figure 6-13: Maximum transfer to the ground state of NaLi 𝑎3 Σ+ potential
as a function of intermediate excited vibrational state 𝑣 * . Transfer efficiency
to the ground state was numerically calculated for atomic densities of 𝑛𝑁 𝑎 = 𝑛𝐿𝑖 =5 × 1014 cm−3 with the upleg power of 300 mw and downleg power of 10 mW except
for 𝑣 = 10. The blue and purple data are with the downleg power of 100 𝜇W. Beam
waist, polarization purity and pulse duration of the two Raman beams are assumed
to be 30 𝜇m, 95 % and at most 50 𝜇s respectively except for 𝑣 = 10. Pulse duration
to 𝑣 = 10 is limited to 1 ms.

130

6.3

Free to intermediate vibrational state 𝑣

The goal of this section is to optimize the free-to-bound transition by exploring Raman transitions from free to an intermediately bound vibrational state in the 𝑎3 Σ+
potential. As mentioned in the previous sections, for Raman transition from free toFeshbach state (a high-lying vibrational state in 𝑎3 Σ+ potential), high atomic andmolecular photon scattering by the upleg acting on the downleg path is the main
limiting factor while for transfer to the ground state 𝑣 = 0, 𝑁 = 0 (a low-lying vibrational state), molecular photon scattering by weak upleg Rabi frequency is the main
limiting factor. Collisional loss is a common factor in both cases.
Choosing a vibrational state 𝑣 with a large binding energy can decouple the downleg and upleg paths by creating a large frequency difference of the two Raman beams.
However, low-lying vibrational states 𝑣 tend to have a small FCF with a highly excited
vibrational state 𝑣 * in the 𝑐3 Σ+ potential and therefore require the use of low-lying 𝑣 *that has a small Frank-Condon factor (FCF) with the atomic scattering wavefunction.
The calculation of FCF between different 𝑣 and 𝑣 * is plotted in Fig.6-12.
Transfer efficiency to a bound state 𝑣 from free atoms including atomic and molecular photon scattering loss and collsional loss is numerically calculated and plottedin Fig.6-13. The duration of the Raman pulse is restricted to at most 50 𝜇s for𝑣 < 10 and 1 ms for the final state 𝑣 = 10. Atomic densities are assumed to be𝑛𝑁 𝑎 = 𝑛𝐿𝑖 = 5 × 1014 cm−3 , the upleg and downleg power are set to 300 mW and 10
mW except for 𝑣 = 10. The blue and purple data for 𝑣 = 10 in Fig.6-13 are with adownleg power of 100 𝜇W. The beam waist, polarization purity and pulse duration of
the two Raman beams are assumed to be 30 𝜇m, 95 % and at most 50 𝜇s, respectively,
except for 𝑣 = 10. The transfer efficiency to the 𝑣 = 8 state reaches more than 50% through 𝑣 * = 48, which is greater than what can be achieved by transfer to the13

In principle, one can achieve longer coherence time by using cavity lock to a stabilized cavity.
However, we initially thought our STIRAP lasers for coherent transfer of NaLi in the Feshbach state
to the ground state is in the order of 50 𝜇s from the fact that the transfer efficiency drops for longer
STIRAP pulses. However, we later noticed that this is possibly due to AC Stark shifts in the orderof 10 kHz causing large two-photo detunings. Although we assumed that the coherence time of
two Raman beams for free-to-bound transfer is only 50𝜇s, this is probably one of the worse we canachieve131

Feshbach state, which is ∼ 25% or to the ground state, which is ∼ 6%.

6.4

Comparison of three all-optical methods

We have explored three approaches: four-photon Raman transfer via Feshbach state,
direct two-photon transfer to the rovibrational ground state, and four-photon transfer
via intermediate vibrational state. The first approach is the most obvious approach,and therefore the first approach we considered, which is simply replacing a Feshbachresonance with a Raman transition. This approach allows using a single laser sourcefor both Raman beams, and having long coherence time (∼ 1 ms or more). However,the large photon scattering rate prevents one from taking advantage of this longcoherence time. The second approach, direct transfer to the ground state, is the most
simple among the three. Photon scattering is relatively small, due to the large binding
energy of the ground state. However, the transfer efficiency is mainly limited by weak
Rabi coupling strength or short two-photon coherence time. The analysis of thelast approach is to find the middle ground of the first two approaches. The transfer
to state 𝑣 = 8, 𝑁 = 0 gave the best performance, and here we only focus on thisparticular state among different paths of the third approach. Figure 6-14 is the resultof the three approaches for different atomic densities. Transfer efficiency increases forhigher densities until collisional loss becomes significant. For example, for transfer to𝑣 = 0, 𝑁 = 0, the transfer efficiency is rather lower when 𝑛 = 5 × 1015 cm−3 thanwhen 𝑛 = 5 × 1014 cm−3 . Table 6.2 shows the list of limiting factors for efficient
Raman transfer, comparing the three approaches for different atomic densities.

132

Transfer efficiency

100

10 1

10 2

3
10 10
12

1013

1014

v = 0, N = 0 (v * = 26)v = 8, N = 0 (v * = 47)v = 10, N = 2 (v * = 48)
1015
1016

Atomic density (cm 3)

Figure 6-14: Maximum transfer to a vibration state 𝑣 in NaLi 𝑎3 Σ+ potential
as a function of atomic density. Transfer efficiency to three different bound states
are numerically calculated for different atomic densities with the upleg power of 300
mW and downleg power of 10 mW except for 𝑣 = 10. 𝑣 = 10, 𝑁 = 2 is with thedownleg power of 100 𝜇W. Beam waist, polarization purity and pulse duration of the
two Raman beams are assumed to be 30 𝜇m, 95 % and at most 50 𝜇s respectively
except for 𝑣 = 10. Pulse duration to 𝑣 = 10 is limited to 1 ms.

133

134

𝑣 = 0, 𝑁 = 0(via 𝑣 * = 26)
5.613 %, 16.844us
Weak Rabi Coupling
Collisional Loss
≫ Atomic Scattering,
Molecular Rayleigh Scattering
5.081%, 50us
Weak Rabi Coupling> Collisional Loss
≫ Atomic Scattering,
Molecular Rayleigh Scattering
3.609 %, 50 us
Weak Rabi Coupling≫ Collisional Loss
≫ Atomic Scattering
Molecular Rayleigh Scattering
0.504%, 50us
Weak Rabi Coupling≫ Collisional Loss
Atomic Scattering,
Molecular Rayleigh Scattering𝑣 = 8, 𝑁 = 0(via 𝑣 * = 48)
49.991 %, 2.510 us
Weak Rabi Coupling
Collisional Loss
> Atomic Scattering,
Molecular Rayleigh Scattering
46.688 %, 11.001 us
Weak Rabi Coupling
Collisional Loss
> Atomic Scattering,
Molecular Rayleigh Scattering
43.122 %, 19.000 us
Weak Rabi Coupling> Collisional Loss
> Atomic Scattering> Molecular Rayleigh Scattering
19.180%, 50us
Weak Rabi Coupling> Atomic Scattering≫ Collisional loss,
Molecular Rayleigh Scattering𝑣 = 10, 𝑁 = 2(via 𝑣 * = 47)
24.686%, 2.712us
Molecular Rayleigh Scattering
Weak Rabi Coupling
Collisional Loss
≫ Atomic Scattering
12.414 %, 6.484us
Molecular Rayleigh Scattering
Weak Rabi Coupling> Collisional Loss
≫ Atomic Scattering
7.943 %, 8.101 us
Molecular Rayleigh Scattering
Weak Rabi Coupling≫ Collisional loss
≫ Atomic Scattering
1.146%, 14.341us
Weak Rabi Coupling
Molecular Rayleigh Scattering≫ Atomic Scattering
Collisional Loss

Table 6.2: Limiting factors of free-to-bound Raman transition. Three different cases, transfer to the ground state,intermediate vibrational state 𝑣 = 8 and Feshbach state, are analyzed for four different atomic densities. The analysis isperformed with a Raman beam waist of 30 um and a upleg power of 300 mW. The downleg power (maximum pulse duration) is
10 mW (50 us) for free to 𝑣 = 0 and 𝑣 = 8 transitions and 100 𝜇W (1 ms) for free to 𝑣 = 10 transitions. The maximum transfer
efficiencies and the time it takes to reach that value are colored blue. The strongest contributions on limiting the transfer are
in bold text.

5 × 1012

5 × 1013

1 × 1014

5 × 1014

Density(cm−3 )

6.4.1AC stark shifts

So far, all calculations have been done with perfect compensation of AC Stark shifts
driven by the two Raman beams. Although there are ways to experimentally compensate for these light shifts [113], it can be challenging to completely cancel themout, and single-photon and two-photon detunings may vary during a Raman transfer.
Therefore, it is important to analyze the size of light shifts caused by the Raman
beams and the effects on free-to-bound transfer efficiencies. AC stark shifts can be
broken down mainly into two parts: AC Stark shifts on the downleg path and on
the upleg path. Here, we define the AC Stark shift on the downleg (upleg) path asa change in the energy difference between the intermediate excited state, 𝑒, and thefinal state (initial state), 𝑓 (𝑖), as illustrated in Figure 6-15. There are three leading
terms of the AC Stark shift on the downleg (up) path: The light shift of state 𝑒 and𝑓 (𝑖) driven by the downleg (upleg) coupling between these two states, the light shiftof state 𝑒 by the upleg (downleg) coupling between state 𝑒 and 𝑖 (𝑓 ), and the lightshift of state 𝑓 (𝑖) by the upleg (downleg) coupling between state 𝑓 (𝑖) and nearby
excited states, 𝑒′𝑗 s. However, the light shift of state 𝑖 by the downleg is relatively
small among these terms, and the two AC Stark shifts are approximated as:
AC Stark shift on the downleg path[︃]︃′𝑓,𝑒′𝑗 2𝑖,𝑒 22∑︁ (Ω𝑓,𝑒𝑗 )2
(Ω
)
)
(Ω𝑓,𝑒
(Ω
)
𝑢𝑝
𝑢𝑝
𝑑𝑜𝑤𝑛
≈ 𝑑𝑜𝑤𝑛 +
+
+
2Δ𝑀
4Δ𝑀
4Δ
4(𝐸𝐵 + Δ𝑀 )
𝑀
𝑗

(6.7)

AC Stark shift on the upleg path
2
2
(Ω𝑖,𝑒
(Ω𝑓,𝑒
𝑢𝑝 )
𝑑𝑜𝑤𝑛 )
≈
+
4Δ𝑀
2Δ𝑀

(6.8)

where, Ω𝑦,𝑧
𝑥 is the coupling strength between state 𝑦 and 𝑧 by a Raman beam 𝑥, Δ𝑀
is the single-photon detuning, and 𝐸𝐵 is the binding energy of the final state 𝑓 . The
common factors in Eq. 6.7 and 6.8 serve as changes in single-photon detuning during
135

Figure 6-15: Illustration of a lambda type two-photon transition with multiple excited states near an intermediate excited state. Initial state 𝑖 is coupled
to an excited state 𝑒 by a upleg beam with coupling strength of Ω𝑖,𝑒
𝑢𝑝 . Final state 𝑓
is coupled to an excited state 𝑒 by a downleg beam with coupling strength of Ω𝑓,𝑒𝑑𝑜𝑤𝑛 .
′
There may be non-zero coupling between state 𝑖 or 𝑓 and other excited states 𝑒𝑗 s with𝑦,𝑒′

coupling strength of Ω𝑥 𝑗 where 𝑥 = 𝑢𝑝, 𝑑𝑜𝑤𝑛 and 𝑦 = 𝑖, 𝑓 . Δ𝑀 is the single-photon
detuning, Δ′𝑗 is the upleg (downleg) detuning from transition 𝑖 (𝑓 ) to 𝑒′𝑗 , and 𝐸𝐵 is
the binding energy of the final state, 𝑓 .

136

a Raman transition, and the effective single-photon detuning is given by:Δ̄𝑀 ≈ Δ𝑀 +

2
2
(Ω𝑖,𝑒
(Ω𝑓,𝑒
𝑢𝑝 )
𝑑𝑜𝑤𝑛 )
+
.
4Δ𝑀
2Δ𝑀

(6.9)

On the other hand, the difference between Eq. 6.7 and 6.8 serve as the change in
two-photon detuning, and if the system is on two-photon resonance in the limit of
weak light coupling, the effective two-photon detuning is given by:[︃]︃
′𝑓,𝑒′𝑗 2𝑓,𝑒𝑖,𝑒 22∑︁ (Ω𝑓,𝑒𝑗 )2
)
(Ω
(Ω
)
)
(Ω
𝑢𝑝
𝑢𝑝
𝑑𝑜𝑤𝑛
𝛿¯𝑀 ≈ 𝛿𝑀 + 𝑑𝑜𝑤𝑛 −
+
+
4Δ𝑀
2Δ𝑀
4Δ
4(𝐸𝐵 + Δ𝑀 )𝑀
𝑗

(6.10)= 𝛿𝑀 + 𝛿¯𝑑𝑜𝑤𝑛 + 𝛿¯𝑢𝑝
where 𝛿¯𝑑𝑜𝑤𝑛 and 𝛿¯𝑢𝑝 are defined as:𝑓,𝑒′2∑︁ (Ω 𝑗 )2
(Ω𝑓,𝑒
𝑑𝑜𝑤𝑛
𝑑𝑜𝑤𝑛 )¯+
𝛿𝑑𝑜𝑤𝑛 ≡
4Δ𝑀
4Δ
𝑀
𝑗
𝑓,𝑒′

2
∑︁ (Ω𝑢𝑝 𝑗 )2
(Ω𝑖,𝑒
𝑢𝑝 )
.
𝛿¯𝑢𝑝 ≡ −
+
2Δ𝑀
4(𝐸
+
Δ
)
𝐵
𝑀
𝑗

(6.11)The effective single-photon and two-photon detunings are calculated for three different
free-to-bound transitions: transfer to 𝑣 = 0, 𝑁 = 0 via 𝑣 * = 26, transfer to 𝑣 =8, 𝑁 = 0 via 𝑣 * = 47 and transfer to 𝑣 = 10, 𝑁 = 2 via 𝑣 * = 48. The results
are in Table 6.3. AC Stark shift by the upleg acting on the downleg path plays amajor role in creating two-photon detuning. To what extent AC stark shifts should
be compensated in order to maintain the transfer efficiencies obtained from complete
compensations can be estimated by comparing the calculated shifts with the Raman
pulse duration. In other words, effective two-photon detuning must be much less than
the inverse of the Raman pulse duration to prevent a significant decrease in transfer.
For this, both Raman beams and additional laser beams that may be used for lightshift compensation must be intensity, polarization, and frequency stabilized to meetthis condition. Laser intensity fluctuations and light shift compensation fluctuation
are required to be much less than 2.5 %, 0.9 % and 0.15 % for transfer to state 𝑣 = 0,137

𝑁 = 0, state 𝑣 = 8, 𝑁 = 0, and 𝑣 = 10, 𝑁 = 2, respectively.
Meeting the estimated laser requirements can be already demanding even though
we only considered nearby excited molecular states in NaLi 𝑐3 Σ potential. Including
also the states in the NaLi 13 Π potential is important for an accurate analysis. As the
AC Stark shift by the upleg acting on the downleg path takes a dominant part of thetwo-photon detuning, we now focus only on 𝛿¯𝑢𝑝 . The calculations of 𝛿¯𝑢𝑝 by 𝑐3 Σ+ and𝑏3 Π+ are listed in Table 6.4. 𝛿¯𝑢𝑝 by 𝑐3 Σ+ and 𝑏3 Π+ are comparable in size and add up
to create even larger two-photon detuning. Therefore, laser intensity fluctuations andlight shift compensation fluctuation are rather required to be much less than 1.5 %,0.39 % and 0.08 % for transfer to state 𝑣 = 0, 𝑁 = 0, state 𝑣 = 8, 𝑁 = 0, and 𝑣 = 10,
𝑁 = 2 respectively. For example, if the two Raman beams have a Gaussian profile
with a beam waist of 30 𝜇 m and the atomic clouds are in an isotropic harmonic
trap leading to density of 1014 cm−3 and 4 𝜇m cloud size, without any light shiftcompensation, the free-to-bound transfer efficiencies become 0.1 %, 0.4 %, and 0.05
% instead of 5.1 %, 46.7 %, and 12.4 % as listed in Table 6.3 and 6.4.
An intermediate excited state, which allows the upleg wavelength to be at a ‘magic
wavelength’ (causes a significantly smaller AC Stark shift), is worth exploring at thispoint. The calculations of 𝛿¯ taking into account the potentials 𝑐3 Σ+ and 𝑏3 Π are given
in Fig. 6-16. For example, when transferring free atoms to the 𝑣 = 0, 𝑁 = 0 state,choosing an excited intermediate state 𝑣 * = 3 or 𝑣 * = 12, which requires a upleg
wavelength of ∼ 930.822 nm, gives a smaller AC Stark shift compared to choosing𝑣 * = 26. 𝛿¯ is approximately 0.25 MHz and 0.6 MHz respectively, and without any
light shift compensation, under the same condition explained above, the free-to-boundtransfer efficiency is 1.8 % and 0.7 % instead of 5%, and 3.2% respectively. These arehigher than 0.1 %, which we can achieve with 𝑣 * = 26.

138

139

∼ 100 kHz(∼ 10 kHz)∼ 300 kHz(∼ 30 kHz)∼ 1.5 MHz(∼ 1 MHz)

𝑣=0
10 mW
𝑁 =0
(100 𝜇W)(𝑣 * =26)

𝑣=8
10 mW
𝑁 =0
(100 𝜇W)(𝑣 * =48)𝑣 = 10100 𝜇W𝑁 =2(10 𝜇W)(𝑣 * =47)∼ 1.5 MHz(∼ 1 MHz)∼ 500 kHz(∼ 50 kHz)∼ 100 kHz(∼ 10 kHz)𝛿¯𝑑𝑜𝑤𝑛∼ 100 MHz∼ -12 MHz∼ -800 kHz𝛿¯𝑢𝑝

12.4 %(6.5 𝜇s)46.7 %(11 𝜇s)5.1 %(50 𝜇s)Transfer
Efficiency(Time 𝑡𝑅 )𝛿𝐼 ≪ 0.15% (5.85 GHz)
672.108 nm, 672.099 nm
a.We have all the lasers that we need fortwo step Raman transfer𝛿𝐼 ≪ 0.9% (232.9 GHz)
671.684 nm, 671.334 nm
a.Can use current STIRAP lasers for thesecond Raman transfer
b.Need an extra laser for the Downleg
and a way to lock it𝛿𝐼 ≪ 2.5% (6.238 THz)
724.315 nm, 713.561 nm
a.Do not need second Raman transition
b. Need an extra laser for the Downleg
and a way to lock it

Laser Requirements

Table 6.3: AC Stark shifts and laser requirements. Calculation is done with atomic densities of 1 × 1014 cm−3 , upleg powerof 300 mW, Raman beam waists of 30 𝜇m, and a maximum pulse duration of 50 𝜇s. The laser intensity fluctuation 𝛿𝐼 should
be much less than the ratio of 1/𝑡𝑅 to 𝛿¯ to maintain the transfer efficiencies calculated with perfect light shift compensation.

Δ̄𝑀 − Δ𝑀

Downleg
Power

Final
State

In conclusion, all-optical transfer from free atoms to the NaLi rovibrational groundstate of the 𝑎3 Σ+ potential is not straightforward. A simple replacement of thewell-established method of creating loosely bound Feshbach molecules with a twophoton Raman transition will lead to a transfer efficiency of less than 0.1%. However,careful selection of a Raman transition path (use of ‘magic wavelength’) or using
additional lasers for light shift compensation may achieve a few percent or higher
transfer efficiency to the ground state.

140

Final
State

Upleg
Power
(Downleg
Power)

𝑣=0
300 mW𝑁 =0(10 mW)(𝑣 * =26)𝑣=8300 mW𝑁 =0(10 mW)(𝑣 * =48)𝑣 = 10300 mW𝑁 =2(100 𝜇W)(𝑣 * =47)

𝛿¯𝑢𝑝 (𝑐3 Σ+ )∼ -800 kHz∼ -12 MHz∼ 100 MHz𝛿¯𝑢𝑝 (𝑏3 Π)Transfer
Laser
Efficiency
Requirements(Time)∼ -500 kHz

5.1 %(50 𝜇s)𝛿𝐼 ≪ 1.5%
724.315 nm,
713.561 nm(6.238 THz)

∼ -11 MHz46.7 %(11 𝜇s)𝛿𝐼 ≪ 0.39%
671.684 nm,
671.334 nm(232.9 GHz)∼ 85 MHz12.4 %(6.5 𝜇s)𝛿𝐼 ≪ 0.08%
672.108 nm,
672.099 nm(5.85 GHz)

Table 6.4: AC Stark shifts and laser requirements including NaLi 𝑏3 Π potential. Calculation is with atomic densities of 1 × 1014 cm−3 , Raman beam waistsof 30 𝜇m, and maximum pulse duration of 50 𝜇s.

141

Figure 6-16: Free-to-bound transfer efficiencies and two-photon detuningfrom both 𝑐3 Σ+ and 𝑏3 Π potentials. (a) Numerically calculated transfer efficiencies from free atoms to a bound state via different intermediate excited state 𝑣 * areplotted for three different final bound states. (b),(c) and (d) are total effective twophoton detuning 𝛿¯ including all the states in 𝑐3 Σ+ and 𝑏3 Π potentials. Data in square,
circle, and triangle correspond to transfer to state 𝑣 = 0, 𝑁 = 0, state 𝑣 = 8, 𝑁 = 0,
and state 𝑣 = 10, 𝑁 = 2 respectively.

142

Chapter 7
Summary and Outlook
7.1

SummaryThis thesis described collision studies on 23 Na6 Li molecules in the triplet ground state.
It explained a spectroscopic investigation of Feshbach resonances in two possible spinpolarized 23 Na6 Li+23 Na collisions from near 0 to 1400 G magnetic bias field using
combined experimental and theoretical quantum chemistry approach. By focusing on
analysis of the Feshbach spectrum solely on the basis of the experimental observations,
we could rule out some of the possible coupling mechanism for the resonances, andestimate the number, lifetime, and size of stable NaLi-Na complexes. Together withcoupled-channel calculations, although they cannot predict the position of resonances,
for the first time, questions on the dominant interactions that are responsible for the
resonances and the background loss, the qualitative difference between the two spinstretched states, and the nature of the complex could be answered.
It also described the first observation of a Feshbach resonance between rovibrational ground-state molecules at ultracold temperatures. This discovery was madeby careful search through a magnetic bias-field range of 1500 G with triplet NaLi
molecules in the lower-stretched hyperfine state. The resonance rises from a background loss at the universal limit, which is not compatible with the most commonly
studied models of ultracold collisions. The more than hundred-fold enhancement ofloss observed implies the existence of a remarkably long-lived collision complex in a143system with barrier-free reactions in all channels [29] and requires full phase coherence for the flux reflected at short range destructively interferes with the quantum
reflected flux. This result is surprising and raises many questions for future work.
In addition to the exploration of magnetically tuneable scattering resonances,
23

Na6 Li molecules in the triplet potential are suitable for magnetic trapping. Withan improved experimental setup that allows magnetic trapping of 23 Na6 Li molecules,
quantitative studies on various collisions in the magnetic trap was demonstrated byquantum state control of molecules and atoms. The molecular density is a factorof 105 higher than that of other magnetically trapped ultracold molecules [20, 100],and the temperature is ≈ 1𝜇𝐾. This enabled observation of both atom-moleculeand molecule-molecule collisions in the ultracold regime and sympathetic cooling of
23

Na6 Li by evaporative cooling of 23 Na in the magnetic trap.
This new method allows studies of molecular collisions in a photon-free environ-ment. Experiments probing photo-induced loss for ultracold molecular systems in
optical traps have been reported using chopped optical dipole traps or repulsive box
potentials made with blue-detuned light. However, even for the longest dark times
and the lowest intensities, photo-induces loss could not be completely suppressed, andloss rates consistent with universal loss were observed [50, 103, 104]. For magneticallytrapped molecules, photo-induced losses can be studied with arbitrarily small lightintensities.
Lastly, this thesis described numerical and theoretical work on finding a condition
for all-optical creation of molecules using Raman transitions from 23 Na and 6 Li atoms
to 23 Na6 Li molecules. The well-established method for creating ultracold bialkali
molecules is by ultracold assembly which requires a Feshbach resonance to be at anexperimentally accessible bias field and magnetic field stabilization within the widthof the Feshbach resonance for an efficient magnetic association. All-optical creationof molecules is expected to broaden the choice of ultracold molecules to a larger pooland eliminate going to high magnetic fields. However, we discovered that all-opticaltransfer from free atoms to the NaLi rovibrational ground state of the 𝑎3 Σ+ potential
is not straightforward. Simple replacement of the well-established method of creating144

loosely bound Feshbach molecules with a two-photon Raman transition will lead toa transfer efficiency of less than 0.1%. With careful selection of a Raman transition
path (use of ‘magic wavelength’) or using additional lasers for light shift compensation,
we expect to achieve transfer efficiency comparable to that of the scheme with the
Feshbach resonance.

7.2

Outlook

7.2.1

Mysteries of non-universal reactive collisions

For chemically stable collisions, one would expect loss rates below the universal limit
and for highly reactive collisions, loss rates at the universal limit. Many experiments
found loss rate constants at or close to the universal value predicted for reactive
collisions even for non-reactive molecules. It has been proposed that this is possibly
due to “sticky collisions” connected with long-lived complexes [37–39, 89, 101] andintense trap light, which can induce photo-chemistry. Recent experiments [40,50,102]
showed that optical traps can cause short lifetimes of molecules and are not truly
conservative, but others [103,104] showed inconclusive results, and the reason for fastloss in non-reactive systems still remain as an open question.
In addition, we have now observed several types of highly reactive collisions whereloss rates do not match the universal limit. Our collision studies in a magnetic trap
show that NaLi molecules with various collision partners have loss rates that deviate
from the universal limit although they are all reactive. First, 𝑝-wave loss rate between
NaLi molecules in the upper-stretched state was measured to be a factor of 2 higher
than the universal limit in a 1,596 nm 1D lattice [33] although molecules in thelower-stretched state decay at the universal limit [73] as expected. This was initiallythought to be originating from an error in the trap volume estimation of the 1D
optical lattice or additional loss due to the trapping light or magnetic force at highfield. However, we observed the same factor of 2 in a light-free magnetic trap with low
bias field near the trap bottom. Second, 𝑠-wave collisions in NaLi spin mixtures and145

NaLi+Na mixture have loss rates much lower than the universal limit. Although allcollision systems are reactive, the absorption probability at close range is much lower
than one. This is well understood for the collisions in the spin-stretched Na+NaLi
mixture [29, 33] where the quartet potential in the input channel is non-reactive,
and inelastic collisions are only possible via spiflips. However, this explanation does
not apply to the 𝑠-wave molecule-molecule collisions. This is a new open questionthat is not explained with existing models. In addition, a Feshbach resonance has
been observed between molecules in a strongly reactive input potential [73]. Theseobservations suggest that collisional resonances and collisional complexes should occur
more generally and motivate more systematic studies of collisions involving NaLi andcollision partners in various hyperfine states, and also for other molecules [112].

7.2.2

Towards a deeply degenerate gas of NaLi

Sympathetic cooling of NaLi by evaporation of Na atoms has been demonstrated
in this work and also in [34]. In particular, this thesis described the microwave
evaporation of Na in a magnetic trap to cool NaLi molecules. This increased thephase space density of the molecules by a factor of ≈ 16. This can be further improved
but is eventually limited by the slow Na thermalization rate, which is significantly
slower than the rate of elastic collisions between Na and NaLi. Using a second trap
for Na that enhances the thermalization rate should allow cooling into the quantumdegenerate regime.
Another approach toward a deeply degenerate gas of NaLi is to have somewhat
lossy evaporation, but compensate for this by increasing the numbers of trapped atoms
and molecules. In Chapter 6, we explored the possibility of all-optical formation of
NaLi in the triplet ground state. Although this method cannot increase the atomto-molecule conversion efficiency but can only do a similar job, it allows complete
elimination of optical trapping. Combining the magnetic trapping that we saw in
Chapter 5 and all-optical formation of molecules can be a new direction to achievea higher number of molecules, since multiple transfer steps from one trap to anotherthat cause significant loss of atoms or molecules can also be eliminated.
146

7.2.3Creation of stable qubits with rotational states

The work of this thesis mainly focused on molecular collisions and quantum chemistry. However, ultracold molecules are also recognized as a new platform for quantum
computing and quantum information processing. Qubit coherence times are criticalto the performance of any robust quantum computing platform. With state-of-the-art
technology, the most stable rotational qubits that have been reported have a coherence time of 100 ms [122]. Even with careful tuning of the angle between the opticaltrap polarization and the applied magnetic field to a ’magic’ angle that greatly reduces the differential polarizability, the dominant decoherence mechanism was still
the differential polarizability between two qubit states. With minor technical upgrades to our apparatus, stable microwave-induced dipoles or rotational state qubits
can be achieved. In the light-free magnetic trap, we expect to produce dipoles witha coherence time that is even longer than the record long coherence time.
For example, rotational qubits √12 (|𝑁 = 0, 𝑚𝑁 = 0, 𝑚𝑆 = 1, 𝑚𝐼Na = 3/2, 𝑚𝐼Li = 1⟩
+ |𝑁 = 1, 𝑚𝑁 = 1, 𝑚𝑆 = 1, 𝑚𝐼Na = 3/2, 𝑚𝐼Li = 1⟩) with collision energy of 1 𝜇K are
estimated to have a coherence time of ∼ 1.2 second in a magnetic trap 1 . However, thedominant decoherence mechanism of rotational qubits for NaLi in the triplet state is
the mixing between different electron spin states due to the intramolecular spin-spinℎcoupling2 . This leads to a coherence time of ≈ 200 ms (𝑡𝑐 ≈ Δ𝜇×Δ𝐵×10−4 ≈ 200 ms)

which is, without careful adjustment for suppressed differential polarizability, still
longer than the longest previously observed coherence time.

1

NaLi has rotational 𝑔-factor 𝑔𝑟 ≈ 0.06 based on simple analysis provided in [123] which divides
the nuclear and electron contributions to 𝑔𝑟 and provides how they scale with mass. Using thisestimate value for 𝑔𝑟 , coherence time of a superposition state ℎ/(𝑔𝑟 × 𝜇𝑁 × Δ𝐵) ≈ 1.2 s (where themagnetic field inhomogeneity, Δ𝐵, is approximately 20 mG in this case.)2
spin-rotation coupling contribution is relatively small.
Approximatetely, the 𝑁 = 0
upper-stretched state,|𝑁 = 0, 𝑚𝑁 = 0, 𝑚𝑆 = 1, 𝑚𝐼Na = 3/2, 𝑚𝐼Li = 1⟩,
is mixed withthe state |𝑁 = 2, 𝑚𝑁 = 2, 𝑚𝑆 = −1, 𝑚𝐼Na = 3/2, 𝑚𝐼Li = 1⟩ by
0.02%.
𝑁 = 1 upperstretched state,|𝑁 = 1, 𝑚𝑁 = 1, 𝑚𝑆 = 1, 𝑚𝐼Na = 3/2, 𝑚𝐼Li = 1⟩ is mixed with state|𝑁 = 3, 𝑚𝑁 = 3, 𝑚𝑆 = −1, 𝑚𝐼Na = 3/2, 𝑚𝐼Li = 1⟩ by 0.01%.

147

148

Appendix A

A.1

Circuit designs
Part

Value

MFG

C(IN)

22 × 10−6 F

FK11X7R1C226M

C(IN_OC)

10−7 F

SR205E104MAR

C(OUT_OC)

10−7 F

SR205E104MAR

CH(IN_L)

CHOKE

PE-1812ACC101STS

CH(IN_R)

CHOKE

PE-1812ACC101STS

CH(OUT_L)

CHOKE

PE-1812ACC101STS

CH(OUT_R)

CHOKE

PE-1812ACC101STS

DC-DC(IN)

5V - 9V Converter

PDS1-S5-S9-S

DC-DC(OUT)

5V - 9V Converter

PDS1-S5-S9-S

F(SIG)

2K

FMP100JR-52-2K2

L(IN)

3.3 × 10−7 F

AR205F334K4R

L(OUT)

3.3 × 10−7 F

AR205F334K4R

LPF

15 × 10−9 F

RCER72A153K0DBH03A

L_IN

680 × 10−6 H

DC630R-684K

OUT

SMA Connector

5-1814832-1

R(BI_B)

1 kΩ

FMP100JR-52-1K

R(BI_F)

1 kΩ

FMP100JR-52-1K

R(IN)10−7 F

SR205E104MAR
149

Table A.1 continued from previous page
Part

Value

MFG

R(OUT)

10−7 F

SR205E104MAR

R(S)

2 kΩ

FMP100JR-52-2K2

R(TRAN)

2 kΩ

FMP100JR-52-2K2

R0

470 Ω

FMP100JR-52-470R

T1

2N3904

2N3904BU

U$1

OPTOCOUPLER

FOD0720

U$2

ICBUFFER

SN74AHCT1G125DCKR

U$5

SIGNAL_JUMPER

1729089VCC

SMA Connector

5-1814832-1

VREG(IN)

5V VOLTAGEREG

L7805CV

VREG(OUT)

5V VOLTAGEREG

L7805CV

Table A.1: Parts of a PCB design for digital control
instrumentation.

150

IN
P$2

P$3

P$1

P$4

P$3

GND

GND

0V

V_IN

+V

0V

P$2

+V

P$1

P$2

P$3

P$1

P$4

P$3

L( OUT)
P$4

VIN

CMC

DC-DC Converter (5V->9V)

P$4

3.3*10-7F

CMC

IN

R( OUT)
10-7F

F(SIG)
2K

VDD1

10-7F

OUT

OUT

Voltage regulator

*

GDN1

R( I N)
10-7F

ICBUFFER
P$1
P$4VCCOE
P$4P$1

A P$2

P$2

GNDP$3 P$5P$3 P$5Y

R(S)
2K
P$1

P$1
P$2
P$3

P$3

DPDT

P$2

2K

1K
R(BI_B)

P$6

1K BI1
LED3

2N3904
T3

R(TRAN)

R(BI_F)

P$4

P$4
P$5
P$6

P$5

470
P$1

R0

LED4
MO1

SPDT

P$2

P$3

P$3

GND

GND*4

LPF

P$2

15*10-9F

P$1

OUT

OUT

0V

P$2

+V

P$1

P$2

P$3

P$1

P$4

P$3

P$4

3.3* 10-7F

IN

OUT

OUT

R( OUT) 1

Voltage regulator

CMC

10-7F

VDD1

VI
C(IN_OC)1
VI

*
*

2K

VDD2VDD1

+V

VDD2

0V

V_IN

NC

GND

C(OUT_OC)1

VIN

NC

GND

DC-DC Converter (5V->9V)

F(SIG)1

10-7F

P$3

P$4

VO

P$4

CMC

VO

P$3

P$1

GND2GND1

P$2

L( OUT) 1
P$1

GND2

IN
P$2

Optocoupler

GDN1

10-7F

SMA Connector

ICBUFFER
P$1
P$4
OE
VCC
P$4P$1

A P$2

P$2

GNDP$3 P$5P$3 P$5Y

R(S)1
2K

DPDT

P$2

P$1

P$1
P$2
P$3

P$3

P$4
P$5
P$6

P$4

R(BI_F)1P$6

1K
R(BI_B)1
1K BI1
LED1

2N3904
T1

2K

R(TRAN)1

P$5

470
P$1

SPDT

P$2

R1

LED2
MO1

P$3

P$3

GND

GND*4

LPF1

P$2

15*10-9F

P$1

OUT

OUT

0V

P$2

+V

P$1

P$2

P$3

P$1

P$4

P$3

P$4

3.3* 10-7F

IN

OUT

OUT

R( OUT) 2

Voltage regulator

CMC

10-7F

VDD1

VI
C(IN_OC)2
VI

*
*

2K

VDD2VDD1

+V

VDD2

0V

V_IN

NC

GND

C(OUT_OC)2

VIN

NC

GND

DC-DC Converter (5V->9V)

F(SIG)2

10-7F

P$3

P$4

VO

P$4

CMC

VO

P$3

P$1

GND2GND1

P$2

L( OUT) 2
P$1

GND2

IN
P$2

Optocoupler

GDN1

10-7F

SMA Connector

ICBUFFER
P$1
P$4
OE
VCC
P$4P$1

A P$2

P$2

GNDP$3 P$5P$3 P$5Y

R(S)2
2K

DPDT

P$2

P$3

P$4

R(BI_F)2

P$6

1K
R(BI_B)2
1K BI1
LED5

2N3904
T2

2K

R(TRAN)2

P$1

P$1
P$2
P$3

P$4
P$5
P$6

P$5

470
P$1

SPDT

P$2

R2

LED6
MO1

P$3

P$3

GND

GND*4

LPF2

P$2

15*10-9F

P$1

OUT

OUT

P$2

P$3

P$1

P$4

P$3

3.3* 10-7F

IN

OUT

OUT

Voltage regulator

R( OUT) 3
10-7F

VDD1

VI
C(IN_OC)3VI**

ICBUFFER
P$4
P$1
OE
VCC
P$4P$1

A P$2

P$2

GNDP$3 P$5P$3 P$5Y

R(S)3
2K
P$2

P$1

P$1
P$2
P$3

P$3

P$4
P$5
P$6

P$5

P$4

R(BI_F)3

P$6

1K
R(BI_B)3
1K BI1
LED7

470
P$1

P$1
P$2

SPDT
P$3

P$3
GND*4

P$2

GND

P$4

DPDT

CMC

VDD2VDD1

P$1

VDD2

P$2

+V

C(OUT_OC)3

0V

NC

+V

NC

0V

V_IN

F(SIG)3

10-7F

GND

LPF3

VIN

VO

GND

DC-DC Converter (5V->9V)

VO

P$3

P$4

GND2GND1

P$4

CMC

15*10-9F

P$3

P$1

2N3904T4

P$2

L( OUT) 3
P$1

GND2

IN
P$2

Optocoupler

GDN1

10-7F

SMA Connector

2K

P$1

OUT

OUT

Voltage regulator

R(TRAN)3

P$2

IN

L( I N)
3.3* 10-7F

P$4

*

P$3

VI

P$4

CMC

VO

P$3

P$1

VDD2VDD1

P$2

NC

P$1

GND2GND1

P$2

+V

VDD2

0V

C(OUT_OC)+V

Optocoupler

0V

V_IN

NC

GND

VO

VIN

DC-DC Converter (5V->9V)

GND2

GND

P$4

GND

P$3

GND

P$4

CMC

GND

-

P$3

P$1

GND

+

P$2

GND

P$1
-

GND

GND*4

+

GND

Inductor

C(IN)

GND

22*10-6F

GND

GND

OUT

10-7F

IN
P$2

OUT

GND

SMA Connector

VIC(IN_OC)

151

LED8
MO1

R3

2K

P$7

P$8

P$9

P$7

P$8

P$9

P$4

P$5

P$6

P$4

P$5

P$6

P$1

P$2

P$3

P$1

P$2

P$3

Signal input (Jumper)

GND

OUT

OUT

P$4

P$3

P$4

3.3* 10-7F

IN

OUT

OUT

R( OUT) 4

Voltage regulator

CMC

10-7F

*
*

VDD1

10-7F
P$3

P$1

VI

GDN1
P$2

NC

P$1

VO

P$2

+V

GND2GND10V

2K

VDD2VDD1+V

Optocoupler

0V

V_IN

NC

GND

GND

VIN

VO

GND

DC-DC Converter (5V->9V)

VDD2

P$3

P$4

GND2

P$4

CMC

10-7F

P$3

P$1

F(SIG)4

C(OUT_OC)4

P$2

L( OUT) 4
P$1

GND

IN
P$2

VI
C(IN_OC)4

SMA Connector

ICBUFFER
P$1
P$4
OE
VCC
P$4P$1

A P$2

P$2

GNDP$3 P$5P$3 P$5Y

R(S)4
2K

DPDT

P$2

2K

P$3

P$4

P$4
P$5
P$6

P$6

R(BI_F)4
1K
R(BI_B)4
1K BI1LED9

2N3904
T5

R(TRAN)4

P$5

P$1

P$1
P$2
P$3

470
P$1

SPDT

P$2

R4

LED10
MO1

P$3

P$3

GND

GND*4

LPF4

15*10-9F

P$1
P$2

OUT

OUT

P$4

P$3

P$4

3.3* 10-7F

IN

OUT

OUT

R( OUT) 5

Voltage regulator

CMC

10-7F

*
*

VDD1

10-7F
P$3

P$1

VI

GDN1
P$2

VO

P$1

GND2GND1

P$2

+V

2K

VDD2VDD1

0V

NC

0V

+V

Optocoupler

GND

V_IN

NC

VIN

VO

GND

DC-DC Converter (5V->9V)

GND

P$4

VDD2

P$3

GND2

P$4

CMC

10-7F

P$3

P$1

F(SIG)5

C(OUT_OC)5

P$2

L( OUT) 5
P$1

GND

IN
P$2

VI
C(IN_OC)5SMA Connector

ICBUFFER
P$4
P$1
OE
VCC
P$4P$1

A P$2

P$2

GNDP$3 P$5P$3 P$5Y

R(S)5
2K

DPDT

P$2

2K

P$3

P$4

P$4
P$5
P$6

P$6

R(BI_F)5
1K
R(BI_B)5
1K BI1
LED11

2N3904
T6

R(TRAN)5

P$5

P$1

P$1
P$2
P$3

470
P$1

SPDT

P$2

R5

LED12
MO1

P$3

P$3

GND

GND*4

LPF5

P$2

15*10-9F

P$1

OUT

OUT

P$3

P$4

3.3* 10-7F

IN

OUT

OUT

R( OUT) 6

Voltage regulator

10-7F

*
*

VDD1

10-7F
P$4

CMC

VI

GDN1
P$3

P$1

2K

VDD2VDD1

P$2

NC

P$1

VO

P$2

+V

GND2GND1

0V

Optocoupler

0V

+V

NC

GND

V_IN

GND

VIN

VO

GND

DC-DC Converter (5V->9V)

VDD2

P$3

P$4

GND2

P$4

CMC

10-7F

P$3

P$1

F(SIG)6

C(OUT_OC)6

P$2

L( OUT) 6
P$1

GND

IN
P$2

VI
C(IN_OC)6SMA Connector

ICBUFFER
P$4
P$1
OE
VCC
P$4P$1

A P$2

P$2

GNDP$3 P$5P$3 P$5Y

R(S)6
2K

DPDT

P$2

P$4

R(BI_F)6P$6

1K
R(BI_B)6
1K BI1
LED13

2N3904
T7

2K

P$3

P$4
P$5
P$6

P$5

R(TRAN)6

P$1

P$1
P$2
P$3

470
P$1

SPDT

P$2

R6

LED14
MO1

P$3

P$3

GND

GND*4

LPF6

15*10-9F

P$1
P$2

OUT

OUT

3.3* 10-7F

IN

OUT

OUT

Voltage regulator

R( OUT) 7
10-7F

*
*

VDD1

10-7F
P$3

P$4

VI

GDN1
P$4

CMC

2K

VDD2VDD1

P$3

P$1

NC

P$2

P$1

VO

P$2

+V

GND2GND1

0V

Optocoupler

+V

NC

0V

V_IN

VO

GND

VIN

DC-DC Converter (5V->9V)

VDD2

GND

L( OUT) 7
P$4

GND2

P$3

10-7F

P$4

CMC

F(SIG)7

C(OUT_OC)7

P$3

P$1

GND

P$2

P$1

GND

IN
P$2

VI
C(IN_OC)7SMA Connector

ICBUFFER
P$1
P$4
OE
VCC
P$4P$1

A P$2

P$2

GNDP$3 P$5P$3 P$5Y

R(S)7
2K

DPDT

P$2

P$4

R(BI_F)7

P$6

1K
R(BI_B)7
1K BI1
LED15

2N3904
T8

2K

P$3

P$4
P$5
P$6

P$5

R(TRAN)7

P$1

P$1
P$2
P$3

470
P$1

P$2

SPDT

LED16
MO1

R7

P$3

P$3

OUT

GND

GND*4

LPF7

P$2

15*10-9F

P$1

OUT

SMA Connector

Figure A-1: Schematic of a PCB design for digital control instrumentation.

152

153
LED8
MO1

LED7
BI1

L(OUT)3
10-7F

470
R3

R(BI_B)31K

2K
F(SIG)4
R(BI_F)3
1K

C(OUT_OC)3

R(TRAN)42K

R(OUT)5

10-7F

R(OUT)4

10-7F

LPF4

10-7F

3 2 1 T5
15*10-9F
2N3904C(IN_OC)4
2K
R(S)4

LED10
MO1
470
R4

L(OUT)4

2K
F(SIG)5
R(BI_F)4
1K
LED9
BI1
R(BI_B)41K

3.3*10-7F
10-7F
C(OUT_OC)4

R(TRAN)5
2K

LPF5

10-7F

2K
R(S)5

15*10-9F
C(IN_OC)5
3 2 1 T6
2N3904

LED12
MO1

2K
F(SIG)6
R(BI_F)5
LED11 1K
BI1
R(BI_B)51K

3.3*10-7F
L(OUT)5
470
R5

10-7F
C(OUT_OC)5

R(TRAN)62K
10-7F

R(OUT)6LPF615*10-9F
10-7F
3 2 1 T7
2N3904C(IN_OC)6
2K
R(S)6

LED14
MO1

LED13
BI1
R(BI_B)61K

3.3*10-7F

470
R6

2K
F(SIG)7
R(BI_F)6
1K

L(OUT)6

10-7F
R(TRAN)72K
C(OUT_OC)6
10-7F

R(OUT)7

2K
R(S)7

LPF7
10-7F
15*10-9F
C(IN_OC)7
3 2 1 T8
2N39043.3*10-7F
LED16
MO1

LED15
BI1

L(OUT)7470
R7

R(BI_F)7
1K
R(BI_B)71K

10-7F

C(OUT_OC)7

154
Figure A-2: Board of a PCB design for digital control instrumentation.
3.3*10-7F

L(IN)

10-7F

R(IN)

F(SIG)
2K

C(IN)

10-7F

R(OUT)

2K
R(S)

22*10-6F

10-7F

R(TRAN)
15*10-9F C(IN_OC)
2K

3 2 1 T3
2N3904LPF

3.3*10-7F
LED4
MO1

2K
F(SIG)1
R(BI_F)
LED3
1K
BI1
R(BI_B)
1K

10-7F

L(OUT)470
R0

C(OUT_OC)

R(TRAN)1
2K

10-7F

10-7F

R(OUT)1

C(IN_OC)1

15*10-9F
2K
R(S)1

LPF1
3 2 1 T1
2N3904
10-7F

3.3*10-7F

LED2
MO1

LED1
BI1

C(OUT_OC)1

L(OUT)1470
R1

R(BI_B)11K

2K
F(SIG)2
R(BI_F)1
1K

R(TRAN)2
2K
10-7F
10-7F

R(OUT)2

C(IN_OC)2
15*10-9F
2K
R(S)2

3 2 1 T2 LPF2
2N3904

LED6
MO1

2K
F(SIG)3
R(BI_F)2
LED5 1K
BI1
R(BI_B)21K

3.3*10-7F
10-7F

L(OUT)2

C(OUT_OC)2

470
R2

10-7F

R(OUT)3

C(IN_OC)3
R(TRAN)3
15*10-9F
2K

10-7F
2K
R(S)3

LPF3
3 2 1 T4
2N39043.3*10-7F
LED8
MO1

LED7
BI1

L(OUT)3

Figure A-3: Schematic of PCB design for analog signal buffer.

155

a fk =QNSNO_ ~ì Ñçêã =`gO

Figure A-4: Board of PCB design for analog signal buffer.

Figure A-5: Circuit diagram of Li MOT fluorescence trigger and 60Hz trigger.

156

Appendix B

B.1

Coupled-channels calculations

This section describes quantum mechanical coupled-channel calculations for Na-NaLi
scattering which were carried out by Tijs Karman. The calculations include theelectronic interactions, the Zeeman interaction with the magnetic field, the magnetic dipole-dipole interaction, and the spin-rotation and spin-spin couplings. The
electronic interaction is partitioned into two-body interactions that are taken fromexperiment [25,124,125], and a non-additive three-body interaction that is calculatedwith state-of-the-art coupled-cluster calculations that are described in a forthcoming
paper [126]. We assume that the molecular bond length is fixed at the equilibrium
position of the triplet potential, which naively cannot explain the chemical reactions
that form singlet NaLi or Na2 molecules at the low-spin potential. In our coupledchannel calculations, we model these by imposing an absorbing boundary condition at
𝑅 = 4.5 𝑎0 , which can be reached on the low-spin potential, but not on the high-spin
potential, which is highly repulsive at these short distances.
For simplicity we start by ignoring hyperfine and vibrational degrees of freedom.
The scattering wavefunction is expanded in the basis of fully coupled channel functionsof the form|(𝑁 𝐿)𝐽(𝑠mol , 𝑠atom )𝑆; 𝒥 ℳ⟩ =∑︁

⟨𝐽𝑀𝐽 𝑆𝑀𝑆 |𝒥 ℳ⟩

𝑀𝐽 ,𝑀𝑆× |(𝑁 𝐿)𝐽𝑀𝐽 ⟩ |(𝑠mol 𝑠atom )𝑆𝑀𝑆 ⟩ (B.1)
157

where ⟨𝐽𝑀𝐽 𝑆𝑀𝑆 |𝒥 ℳ⟩ is a Clebsh-Gordan coefficient. The quantum number 𝑁
represents the rotational angular momentum of the NaLi molecule, and 𝐿 the angularmomentum associated with the end-over-end rotation, which are Clebsch-Gordan
coupled to a total mechanical angular momentum 𝐽 with the 𝐵-field project 𝑀𝐽 .
Similarly, 𝑠mol = 0 or 1 denotes the NaLi molecular electronic spin and 𝑠atom = 1 /2
is the atomic electronic spin, and 𝑆 the total electronic spin with 𝐵-field projection
𝑀𝑆 . In the coupled basis, 𝐽 and 𝑆 are subsequently coupled to a total angularmomentum 𝒥 and a magnetic field projection ℳ = 𝑀𝐽 +𝑀𝑆 . ℳ is strictly conserved,
whereas, for a large enough magnetic field, 𝑀𝑆 becomes a good quantum number
and therefore 𝑀𝐽 = ℳ − 𝑀𝑆 is also a good quantum number. Due to the large
singlet-triplet splitting in the NaLi molecule, 𝑠mol = 0 or 1 is also a good quantum
number. For a separated atom and molecule, 𝑚𝑠mol and 𝑚𝑠atom would separately
become good quantum numbers, but at chemically relevant distances the exchange
splitting between the doublet and quartet interaction potentials is dominant, so that𝑆 = 1/2 and 3/2 are good quantum numbers. Hence, we can effectively consider each
|𝑆 𝑀𝑆 ⟩ state separately, with only perturbatively weak couplings between them. For
each of these spin channels, there are strong and anisotropic interactions that couple
different 𝑁 and 𝐿 channels but conserve 𝐽 and 𝑀𝐽 . The initial channel corresponds
to 𝑠-wave collisions in the spin-stretched ground state, |(00)00⟩ |3/2 3/2⟩.
First, we consider the sensitivity of the scattering rates to the interaction potential
shown in Figure B-1(a). Here, we scale by a factor 1 + 𝜆 the non-additive three-bodypart of the interaction potential, that is, the part that is computed ab initio and isuncertain up to an estimated 10 %. By modifying the potential by only 0.1 % we findthat the resonances start to shift so that realistically their positions are completely
undetermined, and when the scaling reaches several percent we tune across magnetic
field-independent resonances, indicating that the background scattering length is undetermined. Next, shown in Figure B-1(b), we again scale the three-body interaction
but now only for the low-spin doublet potential, leaving the high-spin quartet potential unchanged. In this case, we find that several of the resonances are now completely
independent of the scaling of the low-spin potential up to 𝜆 = 0.1, indicating that are158these resonances are supported by the high-spin potential, i.e., the resonance states
have well defined 𝑆 = 3 /2 .
This can be understood as no stable resonance states are supported by the chemically reactive low-spin potential.
The analysis above indicates that the ab initio prediction of resonance positions isbeyond the capability of state-of-the-art calculations. Although the coupled-channels
calculations cannot predict the positions of the resonances, we can still use thesecalculations as a “numerical experiment” to investigate the nature of the resonancestates, the coupling mechanisms, and the observed differences between the two spinstretched states.
We perform coupled-channels scattering calculations with the interactions scaledby 1 + 𝜆 and analyze the typical behavior observed for different 𝜆 between −0.1 and+0.1. Representative magnetic field scans can be seen in Fig. B-2. We observe approximately 10 resonances for the high-field seeking bottom spin-stretched state andonly around 5 resonances in the low-field seeking top spin-stretched state, respectively. This is in qualitative agreement with the experiment which observes 17 and 8
resonances, respectively. In the companion paper we show that based on the densityof states we would expect to see approximately 10 resonances for either spin-stretchedstate [126]. We furthermore show that including molecular vibrations would increase
the density of state by approximately 50 %, which can partially explain the lower
number of observed resonances compared to experiment. We show that including hyperfine interactions increases the density of resonances somewhat. With these effects
in mind, one could almost claim quantitative agreement with experiment regarding
the density of resonances, although again it is not possible to ab initio predict theirpositions.
One may expect that the high-field seeking state supports more magnetically tunable resonances because the other Zeeman states correspond to closed channels – andhence can support Feshbach resonances – even in the rotational ground state, whereas
closed channels for the low-field seeking state occur only for excited rotational states.
We investigate this in our coupled-channels calculations by artificially excluding the159

Figure B-1: Loss rates as a function of B-field and scaling of the spin-independent
three-body interaction by a factor 1+𝜆 for 𝑁max = 20. This shows scalings within theuncertainty of at least several percents, the background loss and resonance positions
are undetermined. Several B-independent resonances are observed, where the 𝜆scaling tunes the initial spin-stretched potential such that it supports a resonance,i.e., a bound state near zero energy. Hence, prediction of resonance positions requires
knowledge of the interaction potentials to an accuracy that cannot realistically beachieved by ab initio calculations.

160

10-9

High-field seeker

Low-field seeker-10k (cm3/s)

10

10-11
10-12
-13

10

Full calculation
Perturbative spin couplings
Perturbative anisotropy
Anisotropy off

-14

10

10-15

00

14

00

12

00

10

0

80

0

60

0

40

0

20

0

B (G)

Figure B-2: Representative magnetic field scans. Results for the low and high-fieldseeking states in orange and blue solid lines. The high-field seeking state typicallyshows around 10 resonances below 1500 G, whereas the low-field seeker shows aroundhalf as many resonances. This qualitative difference between the two states is also
observed experimentally. We investigate whether the anisotropy of the electronicinteraction and the spin-spin and spin-rotation coupling act perturbatively, by scaling
down these couplings by a factor of two and scaling the resulting cross section up by
a factor four. Agreement with the full calculation indicates the spin-spin and spinrotation coupling act perturbatively, whereas the interaction anisotropy does not.
The coupling mechanism however does involve the anisotropy as turning this off
entirely produces a much smaller cross section dominated by the magnetic dipoledipole interaction (dash-dotted line). When both anisotropy and dipole-dipole areturned off, the calculated cross section is zero.
non-initial Zeeman states in the rotational ground state, see Fig. B-3. Somewhat
surprisingly we find that excluding channels from the calculation does not reduce
the number of resonances for the high-field seeking state, but rather increases theobserved number of resonances for the low-field seeker, where the excluded channels
correspond to asymptotically open channels. In the presence of these open channels
some of the resonances decay rapidly such that they are not resolved, leading to alower number of observable resonances, explaining the observed qualitative difference
between the low and high-field seeking states.
Finally, we investigate numerically the coupling mechanism that gives rise to theobserved resonances. As argued above, each resonance state can essentially be assigned a molecular and a total electron spin quantum number 𝑠mol = 1 and 𝑆 = 3 /2 ,
161

10-9

k (cm3/s)

10-10
-11

10

10-12
10-13
10-14

00

14

00

12

00

10

0

80

0

60

0

40

0

20

0

B (G)

Figure B-3: Representative magnetic field scans. Qualitative differences in the
typical number of resonances for the low and high-field seeker disappear when we
exclude from the calculation channels corresponding to non-initial Zeeman states in
the rotational ground state. Excluding these channels does not reduce the numberof resonances for the high-field seeker, but rather increases the typical number ofresonances for the low-field seeker. In this case, asymptotically open channels are
removed from the calculation, which otherwise would lead to fast decay that renders
some of the resonances unresolved.

as only the non-reactive quartet spin state supports stable resonance states. The
resonance is magnetically tunable only if the Zeeman state 𝑀𝑆 is changes. To couple states with Δ𝑀𝑆 ̸= 0 a spin-dependent coupling must be involved, such as the
spin-rotation and spin-spin coupling. From the tensor rank of these couplings we candetermine they couple states with the selection rules, 𝐽 = 0 → 1 and |Δ𝑀𝑆 | ≤ 1, and𝐽 = 0 → 2 and |Δ𝑀𝑆 | ≤ 2, respectively. Since 𝐽 is approximately a good quantum
number, both mechanisms give rise to distinct and independent resonances. Since
the density of states of the collision complex scales with 2𝐽 + 1 and the differential magnetic moment is higher for larger |Δ𝑀𝑆 | transitions, we conclude that most– aproximately 75 % – of the resonances are due to spin-spin coupling. Spin-spin
coupling does not occur for 2 Σ molecules, and hence is a somewhat unique couplingmechanism for NaLi(3 Σ). To confirm that the role of the spin-dependent interactionsin the coupling mechanism we have performed coupled-channels calculations where
we reduced these couplings by a factor two. The resulting cross section multiplied162

by four is shown as the crosses in Fig. B-2. The agreement with the full calculation indicates scaling with the square of the coupling strength that is expected for
perturbatively weak spin-dependent couplings. This confirms we can fully assign theresonances approximately good quantum numbers 𝑠mol = 1, 𝑆 = 3 /2 , each resonancealso has definite 𝑀𝑆 constrained by Δ𝑀𝑆 ≤ 1 and ≤ 2 selection rules, and 𝐽 = 1 or 2,
for spin-rotation and spin-spin coupling, respectively, whereas the 𝑁 and 𝐿 quantum
numbers are strongly mixed due to the anisotropic interaction at short range.
The coupling mechanism involving the spin-rotation and spin-spin coupling also requires an anisotropic interaction potential. The physical picture is that the anisotropicinteraction with the atom can reorient the molecule, and because the spin is coupled
to the molecular axis by spin-rotation and spin-spin coupling, this can lead to Zeeman transitions. To confirm this picture we have performed calculations that exclude
interaction anisotropy, shown as the dash-dotted line in Fig. B-2. The resulting crosssection is smaller and results from long-range Zeeman relaxation by the magnetic
dipole-dipole coupling. If both the magnetic dipole-dipole interaction and the interaction anisotropy are switched off, the cross section in our model vanishes. The
role of the interaction anisotropy is non-perturbative, however, as can be seen fromcomparison between the solid and dotted line in Fig. B-2, which compares the crosssection from the full calculation to four times the cross section obtained with the interaction anisotropy halved. The crucial and non-perturbative role of the anisotropic
electronic interaction implies that the spectrum of resonances cannot be described bya simplified model that accounts only for the isotropic long-range 𝑅−6 interaction,
contrary to previous observations of ultracold atom-molecule resonances [71].
We make a direct comparison to the scattering calculations with the density ofstates computed quantum mechanically using the same channel basis as used in the
scattering calculations in a forthcoming paper [126]. The total number of states fromthe quasiclassical estimate is close to the number of resonances from the scattering
calculations, but not in perfect agreement with it, because of the light masses andrelatively weak interactions in the spin-stretched state. We see most of the 3-bodystates are between 20 and 40 𝑎0 which is in agreement with the complex size estimated163

from the simple interpretation of the spacing between resonances as rotational energy
splitting of the collision complex.

164

Appendix C

C.1

Quasi-2D threshold law for 𝑝-wave collision

Two particles confined axially in an external harmonic trap can be described by thestationary Schrödinger equation [127]:]︂[︂ℏ+ 𝑈 (𝑟) + 𝑉𝑡𝑟 (𝑟) Ψ(𝑟) = 𝐸Ψ(𝑟).
−
2𝜇

(C.1)

Here, 𝑉𝑡𝑟 (𝑟) = 12 𝜇𝜔 2 𝑧 2 , 𝑈 (𝑟) is the interparticle interation and 𝜇 is the reduced
mass. The energy in quasi-2D consists of harmonic oscillator energy and the 2D
2 2

2 2

kinetic energy given by ℏ2𝜇𝑘 = 𝐸 = 21 ℏ𝜔 + ℏ2𝜇𝑞 where 𝑞 is the wavenumber in the 2D
plane. Provided that the van der Waals (vdW) length scale is much smaller than the√︀(︀ 6 )︀1/4
oscillator length 𝑎¯ ≪ 𝑙𝑜 , where 𝑎¯ = Γ(2𝜋1 )2 2𝜇𝐶
and 𝑙𝑜 = ℏ/(𝜇𝜔), so that theℏ24trapping potential can be regarded as constant in the interaction range, the scattering
process can be described by the 𝑠 or 𝑝-wave pseudopotential using the 3D scattering
length or the scattering volume as:𝜕
2𝜋ℏ2 𝑎3𝐷𝑠𝛿(𝑟) 𝑟𝜇𝜕𝑟
2 3𝐷𝜋ℏ 𝑉𝑝 (𝑘)
𝜕3 3 ⃗⃗𝑈𝑝 (𝑟) =
∇𝛿(𝑟)
𝑟 ∇
𝜇
𝜕𝑟3
𝑈𝑠 (𝑟) =(C.2)

Here, 𝜇 is the reduced mass, 𝐶6 is the vdW long-range dispersion coefficient, andthe scattering length and volume are defined with the 3D phase shift 𝜂𝑙 by 𝑎3𝐷
𝑙 (𝑘) =
− tan 𝜂𝑙 /𝑘 and 𝑉𝑙3𝐷 (𝑘) − tan 𝜂𝑙 /𝑘 3 respectively. The relation between the 2D and 3D
scattering length for 𝑝-wave collision is derived to be:165

(︂ √ )︂2 (︂ 2 )︂2
3 𝜋
𝑞2|𝑙𝑝 |2 |𝑎3𝐷
𝑝 |
2
2𝑙𝑜
𝑘
(︂ √ )︂ (︂ 2 )︂3 𝜋𝑞
𝛽𝑝2𝐷 =|𝑙𝑝 |2 𝛽𝑝3𝐷
2𝑙𝑜
𝑘2

2
|𝑎2𝐷
𝑝 | =

(C.3)

√ 3 2 −1and 𝑊 (0) ≈ 0.328 1 from the solution ofwhere 𝑙𝑝 = (1 − 𝑎3𝐷
𝑝 × 𝑊 (0)/( 𝜋𝑙𝑜 𝑘 ))the Schrödinger equation C.1 to connect the 2D and 3D scattering lengths given inref. [127].
𝑙𝑠
Reactive scattering rate constant in quasi-2D is given as 𝐾2𝐷= 𝑔 4ℏ
𝛽 2𝐷 𝑓2𝐷 where𝜇𝑓2𝐷 = (1 + |𝑎2𝐷 |2 + 2𝛽 2𝐷 )−1 [127]. From the relation between the loss rate constantand the scattering lengths given by Eq.C.3, the quasi-2D loss rate constant for 𝑝-wave
collision can be expressed with 3D scattering parameters as:4ℏ 2𝐷· 𝛽𝑝 · 𝑓2𝐷𝜇[︂(︂ √ )︂ (︂ 2 )︂
]︂
3 𝜋
4ℏ
𝑞
2 3𝐷
=𝑔 ·
|𝑙𝑝 | 𝛽𝑝 · 𝑓2𝐷
𝜇2𝑙𝑜𝑘2(︂ √ )︂ (︂ 2 )︂ [︂
]︂
3 𝜋
𝑞
4ℏ
2 (𝑠 − 1)𝑦𝑠 − (𝑠 − 2)𝑦2¯
𝑎1 (𝑘¯𝑎)· |𝑙𝑝 |2 𝑓2𝐷
=𝑔 ·
𝜇
2𝑙𝑜
𝑘2
(𝑦𝑠)2 + (𝑠 − 2)2(︂ √ )︂ [︂
]︂
4ℏ
3 𝜋
2
2 (𝑠 − 1)𝑦𝑠 − (𝑠 − 2)𝑦
=𝑔 ·
𝑞 (2¯
𝑎1 𝑎
¯ )
· |𝑙𝑝 |2 𝑓2𝐷
𝜇2𝑙𝑜(𝑦𝑠)2 + (𝑠 − 2)2

𝑙𝑠
𝐾2𝐷
=𝑔

(C.4)

supported by 3D complex scattering length of 𝑝-wave collision expressed with two
dimensionless quantum defect parameters 𝑠 and 𝑦 [88] as 𝑎3𝐷= 𝛼𝑝3𝐷 − 𝑖𝛽𝑝3𝐷 =𝑝𝑦+𝑖(𝑠−1), and thus 𝛽𝑝3𝐷 = 2¯
𝑎1 (𝑘¯𝑎)2 (𝑠−1)𝑦𝑠−(𝑠−2)𝑦. Eq. C.4 implies that the−2¯
𝑎1 (𝑘¯𝑎)2 𝑦𝑠+𝑖(𝑠−2)(𝑦𝑠)2 +(𝑠−2)23D 𝑝-wave scattering rate is modified by the factor |𝑙𝑝 |2 and 𝑞 2 /𝑘 2 with a wavenumber√𝑘¯𝑝 = 32𝑙𝑜𝜋 2 . As 3D loss rate constant is proportional to temperature or 𝑘 2 , this 𝑘 2 factor cancels out with 1/𝑘 2 of 𝑞 2 /𝑘 2 in the second line of Eq. C.4 leading to the relation𝑙𝑠
𝐾2𝐷∝ 𝑞 2 · |𝑙𝑝 |2 𝑓2𝐷 , and recovering the linearly dependent loss rate on temperature.

(︁ √ )︁2𝜋
2
2
2𝐷
Scattering length for 𝑠-wave collision is given by |𝑎2𝐷=
|𝑙𝑠 |2 |𝑎3𝐷=𝑠 |𝑠 | and 𝛽𝑠𝑙𝑜(︁ √ )︁
√
𝜋
2 3𝐷
3𝐷
−1
|𝑙𝑠 | 𝛽𝑠 where, 𝑙𝑠 = (1 + 𝑎𝑠 /( 𝜋𝑙𝑜 ) × 𝑙𝑛(𝐵ℏ𝜔/𝜋𝜖)) and 𝐵 ≈ 0.9049 [27].
𝑙𝑜√2𝑠-wave scattering rate is modified by the factor |𝑙𝑠 |2 with a wavenumber 𝑘¯𝑠 = 𝜋 [27].
1

𝑙𝑜

166

C.2

Long-range coupling

Here we show that the long-range coupling between the two hyperfine states |𝑎⟩ and|𝑏1 ⟩ involved in the crossing shown in 4-9 is too weak to explain the observed loss rates.
To explain the observed Feshbach resonance, it is, therefore, necessary to assume thepresence of a short-range loss mechanism. A minimal model for such a mechanism
involves a bound state (channel |3⟩) coupled to the open channels (|𝑎𝑎⟩ and |𝑎𝑏1 ⟩).
Long-range coupling would occur outside the centrifugal 𝑝-wave barrier of the
input channel. The peak of the barrier is at 241/4 · 𝑅vdw ≈ 7.8 nm where the van(︀ 6 )︀1/4
der Waals length 𝑅vdw = 21 2𝜇𝐶, and the inner turning point is at 100 nm atℏ23.35 𝜇K temperature. In the Born approximation, for a potential of average strength
𝑉0 in an effective volume 4𝜋𝑅03 /3, the total low energy elastic scattering cross section
𝜎 = 4𝜋(2𝜇𝑉0 𝑅03 2) (identical to the solution for a spherical square well potential with3ℏ2

radius 𝑅0 ). Applying this relation to the observed nearly unitarity limited crosssection of 2.95 × 10−11 cm2 corresponding to the loss rate constant of 10−10 cm3 s−1
at 1 𝜇K and using the position of the inner turning point to estimate 𝑅0 = 100 nm,
provides a coupling matrix element 𝑉0 of 16 kHz. This is the required value forcoupling outside the 𝑝-wave barrier to be compatible with the observed loss rates.
For inelastic collisions with final wavevector 𝑘 ′ , the rate has an additional factor 𝑘 ′ /𝑘due to the density of states, but for large 𝑘 ′ , the matrix element will decrease with𝑘 ′ , so our rough estimate for the required spatial coupling matrix element should also
apply to inelastic collisions.
First, we show that magnetic dipolar interactions [128] which lead to spin exchange
and dipolar relaxation and often limit the lifetime of magnetically trapped atoms, arevery weak outside the barrier. At the position of the 𝑝-wave barrier (𝑅𝑏 = 100 nm),the interaction between two spins with magnetic moments 2𝜇0 , where 𝜇0 is the Bohrmagneton, is 𝑉 𝑚𝐷𝐷 = 0.052 kHz, which is already small. However, due to theselection rules of the magnetic dipolar interaction (|Δ𝑀𝑆 | = 1), a single spin flip
cannot provide coupling between the near-degenerate hyperfine states of interest |𝑎⟩and |𝑏1 ⟩, which correspond to 𝑀𝑆 = 1 and 𝑀𝑆 = −1, respectively (see above).
167

Therefore, the coupling must involve an intermediate state |𝑘⟩, which is off-resonantby its Zeeman energy Δ𝑎𝑘 ≃ 1𝜇0 × 300 G ≃ 400 MHz. This further reduces themagnetic dipolar coupling between the open channels by the factor (𝑉 𝑚𝐷𝐷 /Δ𝑎𝑘 )−1 =7.7 × 106 to much less than 1 mHz. We can thus rule out the magnetic dipolarinteraction as a source of the observed loss.
Spin flips in collisions of 3 Σ molecules can also occur via coupling of the incident
channels |𝑎⟩ and |𝑏1 ⟩ to excited rotational states [129,130]. This mechanism is similar
to that of magnetic dipolar relaxation discussed above, with the excited rotational
states (𝑁 ≥ 1) playing the role of the intermediate Zeeman states. A distinctive
feature of this mechanism is that it is mediated by the anisotropy of the electrostaticinteraction between 3 Σ molecules (which couples the 𝑁 = 0 incident states to 𝑁 ≥ 1closed-channel states of the same 𝑀𝑆 ) and the spin-rotation and spin-spin interactionsin the 𝑁 ≥ 1 manifolds, which couple states of different 𝑀𝑆 .
Below, we quantify this molecular spin relaxation mechanism by estimating themagnitude of the anisotropic coupling between the degenerate open channels |𝑎⟩ and|𝑏1 ⟩ due to the excited rotational states at 𝑅 = 100 nm. We find that the strongest
coupling due to the electric dipole-dipole interaction is only 0.05 kHz, and is therefore
too small to explain the observed loss rate.

C.2.1

Coupling matrix elements between degenerate open channels |𝑎𝑎⟩ and |𝑎𝑏1 ⟩ due to rotationally excited states

Here, we estimate the matrix elements between the degenerate open channels |𝑎𝑎⟩ and|𝑎𝑏1 ⟩ due to long-range interactions between NaLi(𝑎3 Σ+ ) molecules. The interactions
are described by the multipole expansion [131]𝑉^ (R, r𝐴 , r𝐵 ) = (4𝜋)3/2∑︁^ 𝑟^𝐴 , 𝑟^𝐵 )𝑉𝜆𝐴 ,𝜆𝐵 ,𝜆 (𝑅, 𝑟𝐴 , 𝑟𝐵 )𝐴𝜆𝐴 ,𝜆𝐵 ,𝜆 (𝑅,(C.5)

𝜆𝐴 ,𝜆𝐵 ,𝜆^ 𝑟^𝐴 , 𝑟^𝐵 ) are the angular functions, 𝑉𝜆 ,𝜆 ,𝜆 (𝑅, 𝑟𝐴 , 𝑟𝐵 ) are the radial
where 𝐴𝜆𝐴 ,𝜆𝐵 ,𝜆 (𝑅,𝐴 𝐵^ = R/𝑅, and 𝑟^𝑖 = r𝑖 /𝑟𝑖 . To leading order, the exexpansion coefficients [131], 𝑅
pansion (C.5) contains the electric dipole-dipole, dipole-quadrupole, and quadrupole168

quadrupole interactions. We assume that the internuclear distances of NaLi molecules
are fixed at their equilibrium values. The rigid rotor approximation is expected to
be extremely accurate since the long-range NaLi-NaLi interactions at 𝑅 = 𝑅𝑏 (seebelow) are thousands of times smaller than the spacing between the ground and thefirst excited vibrational states of NaLi (ℏ𝜔10 = 40.2 cm−1 [132]). Since we are interested in long-range physics outside the 𝑝-wave barrier, we will also neglect the spin
dependence of the NaLi-NaLi interaction, which is significant only at very close range(𝑅 ≤ 10 𝑎0 ).
The radial expansion coefficients in Eq. (C.5) are expressed in terms of the multipole moments 𝑄𝜆𝑖 ,0 of the interacting molecules (𝑖 = 𝐴, 𝐵)
𝑄𝜆𝐴 0 𝑄𝜆𝐵 0
𝛿𝜆,𝜆𝐴 +𝜆𝐵
𝑅𝜆+1(−1)𝜆𝐴·[(2𝜆𝐴 + 1)(2𝜆𝐵 + 1)(2𝜆 + 1)]1/2[︂]︂1/2(2𝜆𝐴 + 2𝜆𝐵 + 1)!
·(2𝜆𝐴 )!(2𝜆𝐵 )!

𝑉𝜆𝐴 ,𝜆𝐵 ,𝜆 (𝑅, 𝑟𝐴 , 𝑟𝐵 ) =(C.6)

The leading terms for two interacting neutral polar molecules such as NaLi are√︂𝑑𝐴 𝑑𝐵 2
𝑉112 (𝑅) = − 3(dipole-dipole),
𝑅
3
𝑑𝐴 𝑄𝐵
20
(C.7)
(dipole-quadrupole),
𝑉123 (𝑅) = −4𝑅 √︂
𝑄𝐴 𝑄𝐵 14
𝑉224 (𝑅) = 20 5 20
(quadrupole-quadrupole).
𝑅5
where 𝑑𝑖 and 𝑄𝑖20 are the electric dipole and quadrupole moments of the 𝑖-th molecule.
Note that the long-range interaction (C.5) is spin-independent, and hence can only
couple the states of the same 𝑀𝑆 , 𝑀𝐼1 , and 𝑀𝐼2 . We use the accurate ab initio
value of the molecule-frame electric dipole moment 𝑑NaLi = 0.167 D [132], and an
approximate value of the electric quadrupole moment ΘNaLi = 10 a.u. based on thecalculated values for Na2 and Li2 from Ref. [133]. Our estimates are not sensitive tothe precise magnitude of ΘNaLi , since the dominant contribution at 𝑅 = 𝑅𝑏 is givenby the electric dipole-dipole interaction.
Figure C-1 shows the radial dependence of the long range interactions between
169

4× 10-4

p-wave barrier
DD
DQ
QQ
Potential Energy (µK)

Energy (K)3× 10-4

2× 10-4

1× 10-440
20~ l(l+1)/R²
4.2 µK0- C6/R⁶
0

40

89.3

80

R (nm)

136.4

120

0
20

40
80
60
Interparticle distance R (nm)100

Figure C-1: Radial dependence of the dipole-dipole, dipole-quadrupole, andquadrupole-quadrupole interactions of NaLi (𝑎3 Σ+ ) molecules. The 𝑝-wave centrifugal
barrier is also shown (dashed line). The upper and lower bounds on the experimental
collision energies (4.2 𝜇K and 1.8 𝜇K) are marked by green horizontal lines. The
turning points at the centrifugal barrier for these collision energies are 𝑅𝑏 = 89.3 and136.4 nm, respectively.

two NaLi molecules. While the dipole-dipole interaction dominates outside the 𝑝wave barrier (𝑅 ≥ 100 nm) both the dipole-quadrupole and quadrupole-quadrupole
interactions grow in magnitude as 𝑅 becomes shorter. At 𝑅 = 100 nm, the magnitudeof the electric dipole-dipole, dipole-quadrupole, quadrupole-quadrupole, and magnetic
dipole-dipole interaction terms in Eq. (C.7) are (in kHz): 𝑉𝐷𝐷 = −3.446, 𝑉𝐷𝑄 =−0.339, 𝑉𝑄𝑄 = 0.0273, and 𝑉𝑚𝐷𝐷 = 0.0519.
Equating the barrier energy plotted in Figure C-1 and the endpoints of the experimental range of collision energies (1.8−4.2 𝜇K), we obtain the corresponding
range of distances of closest approach of two NaLi molecules in the 𝑝-wave channel
as 𝑅𝑏 = 89.3−136.4 nm. For simplicity, we will use a value close to the middle ofthis interval, 𝑅𝑏 = 100 nm = 1890 𝑎0 , to estimate the magnitude of all long-range
coupling matrix elements.
Having parameterized the anisotropic long-range interaction between two NaLi
molecules (C.7), we now proceed to evaluate its matrix elements between the degenerate open channels |𝑎⟩ and |𝑏1 ⟩. The general matrix elements are given by:170

Matrix element (kHz)Matrix element (kHz)

2

0.04
0.02
0
-0.02
0

1

2
4
6
Open channel index

0

-1

0

100

200
300
Closed channel index

400

Figure C-2: Matrix elements of the NaLi-NaLi interaction at 𝑅 = 100 nm as afunction of the channel index labeling the basis states |𝛾𝐴 𝛾𝐵 𝑙𝑚𝑙 𝜂⟩. The initial channel
is |𝑎𝑎, 𝑙 = 1, 𝑚𝑙 = 0⟩ and the total angular momentum projection 𝑀tot = −7. The
channel index labels closed channels, in which one or both NaLi molecules are in
their 𝑁 ≥ 1 excited rotational states. Only the matrix elements with the absolutemagnitude exceeding 1 Hz are plotted. The magnetic field 𝐵 = 333 G is tuned near
the crossing between the |𝑎⟩ and |𝑏1 ⟩ hyperfine-Zeeman levels. Inset: Histogram of
direct coupling matrix elements between the incident channel and lower-lying open
channels, where both NaLi molecules are in the ground 𝑁 = 0 rotational states.

171⟨𝛾𝐴 𝛾𝐵 𝑙𝑚𝑙 𝜂| 𝑉^ (R, r𝐴 , r𝐵 ) |𝛾𝐴′ 𝛾𝐵′ 𝑙′ 𝑚′𝑙 𝜂 ′ ⟩(C.8)

where ⟨𝛾𝐴 𝛾𝐵 𝑙𝑚𝑙 𝜂| are the properly symmetrized basis states for two identical molecules(𝜂 = −1 for identical fermions), 𝛾𝐴 and 𝛾𝐵 refer to the internal hyperfine-Zeeman
states of the molecules, 𝑙 is the orbital angular momentum for the collision, and 𝑚𝑙 is
its projection on the space-fixed quantization axis defined by the external magnetic
field [130]. The initial scattering state of interest corresponds to 𝛾𝐴 = 𝛾𝐵 = 𝑎, 𝑙 = 1,
𝑚𝑙 = 0, and 𝜂 = −1.
The matrix elements are calculated by a straightforward extension of the proceduredescribed in Ref. [130] to include the hyperfine structure of both NaLi molecules (seethe Supplemental Material of Ref. [134] for more details about the basis functions).
Using a minimal basis including three lowest rotational states of each of the NaLi
molecules (𝑁 = 0−2) and two partial waves (𝑙 = 1, 3), leads to the total number ofcoupled channels 𝑁ch = 9908 for the total angular momentum projection 𝑀tot = −7.
We note that this basis set is expected to produce converged results at 𝑅 = 100 nm,
where the largest anisotropic long-range coupling |𝑉𝐷𝐷 | = 3.446 kHz is much smaller
than the spacing between the 𝑁 = 0 and 𝑁 = 1 rotational levels (2𝐵𝑒 = 8.4 GHz).
Figure C-2 shows the matrix elements between the incident channel |𝑎𝑎, 𝑙 = 1, 𝑚𝑙 = 0⟩
at 𝑅 = 100 nm and all final channels. The largest of these matrix elements 𝑉01 ≃1.5 kHz
is due to the long-range electric dipole-dipole coupling between the ground and excited(𝑁 = 1) rotational states. While these matrix elements do not directly couple the degenerate channels |𝑎⟩ and |𝑏1 ⟩, they do contribute to indirect second-order couplings
estimated below. There are also direct couplings between the incident channel andlower-lying relaxation channels (with single-molecule hyperfine-Zeeman state lower
in energy that the incident state |𝑎𝑎⟩), which occur between the 𝑙 = 1 and 𝑙 = 3
partial waves. These couplings are mediated by the intramolecular spin-spin interaction, which couples the different 𝑀𝑆 components of the 𝑁 = 0 and 𝑁 = 2 rotational
states [129]. As shown in the inset of Figure C-2, the largest of these direct couplings
is about 0.04 kHz, which is too small to be responsible for the observed loss.
It remains to consider the second-oder couplings between two degenerate channels
172

|𝑎⟩ and |𝑏1 ⟩ via rotationally excited states. These couplings are suppressed by thefactor (𝑉01 /Δ01 )−1 , where Δ01 = 2𝐵𝑒 is the energy of 𝑁 = 1 rotational states relative
to the ground rotational state. Using the values Δ01 = 8.4 GHz and 𝑉01 = 1.5 kHz,
we obtain (𝑉01 /Δ01 )−1 = 5.6 × 106 , and thus the second-order couplings are smaller
than 1 mHz, and can be neglected.

C.3

Harmonic oscillator model

Resonant dissipation energy in a driven harmonic oscillator via frictional input coupling with damping is fully analogous to the resonant transmission of flux througha Fabry-Perot cavity explained above. A harmonic oscillator with resonance 𝜔0 anddamping rate 𝛾2 driven at frequency 𝜔 via a frictional input coupling 𝛾1 is describedby the differential equation:𝑑2 𝑞(𝑡)
𝑑𝑞(𝑡)
𝑑
+ 𝛾2
+ 𝛾1 (𝑞(𝑡) − 𝑞𝑑 (𝑡)) + 𝜔02 𝑞(𝑡) = 0.
2
𝑑𝑡
𝑑𝑡
𝑑𝑡

(C.9)

Here, we assume the drive 𝑞𝑑 (𝑡) = 𝑞0 sin 𝜔𝑡. By rearranging the terms, we obtain the
more intuitive equation of driven harmonic oscillator with damping 𝛾 = 𝛾1 + 𝛾2𝑑2 𝑞(𝑡)𝑑𝑞(𝑡)𝑑+𝛾+ 𝜔02 𝑞(𝑡) = 𝛾1 𝑞𝑑 (𝑡).
2
𝑑𝑡
𝑑𝑡
𝑑𝑡

(C.10)

The complete solution of the Eq. C.10 is a superposition of the particular solution,
𝑞𝑝 (𝑡) and the homogeneous solution, 𝑞ℎ (𝑡). However, if there is any damping at all inthe system (that is, if 𝛾 > 0), then the homogeneous part of the solution goes to zerofor large t, and we are left with only the particular solution. Therefore, the steadystate solution is given as:
𝑞(𝑡) = 𝑞𝑝 (𝑡) =𝛾1 𝑞𝑑 𝜔cos(𝜔𝑡 − 𝜑)2(𝜔 − 𝜔02 )2 + (𝛾𝜔)2(C.11)

where 𝜑 is defined by tan 𝜑 = 𝛾𝜔/(𝜔 2 − 𝜔02 ). Rate of energy dissipated due to 𝛾2 ,𝑃𝑜𝑢𝑡 = 𝛾2 𝑞(𝑡)˙ 2 . Let us calculate the rate at which energy is being dissipated, which173

should be equal to the rate at which work is being done. Starting with Eq. C.9 and˙ + 𝛾2 𝑞˙2 = 𝛾1 (𝑞˙𝑑 − 𝑞)˙ 𝑞˙multiplying both sides by 𝑞,
˙ we obtain the equation [¨
𝑞 𝑞˙ + 𝜔02 𝑞 𝑞]that shows that [Rate of change in kinetic and potential energy] + [Rate at whichenergy is being dissipated by 𝛾2 ] = [Power transferred to the system (𝑃𝑖𝑛 )]3 .
By substituting for 𝑞(𝑡) and 𝑞𝑑 (𝑡) from any particular solution, we can prove thatthe average power transferred to the system ⟨𝑃𝑖𝑛 ⟩ and the rate of energy dissipationby 𝛾2 ⟨𝑃𝑜𝑢𝑡 ⟩ are⟨𝑃𝑖𝑛 ⟩ = ⟨𝛾1 (𝑞˙𝑑 − 𝑞)˙ 𝑞˙𝑑 ⟩⟨ (︃𝛾1 𝑞02 𝜔 3)︃⟩𝛾1 𝑞02 𝜔 2 cos2 𝜔𝑡 − √︀sin(𝜔𝑡 − 𝜑) cos 𝜔𝑡(𝜔 2 − 𝜔02 )2 + (𝛾𝜔)2
)︃
(︃
1 2 2
𝛾1 𝜔
sin 𝜑 .
= 𝛾1 𝑞0 𝜔 1 − √︀2
(𝜔 2 − 𝜔02 )2 + (𝛾𝜔)2=⟨︀⟩︀⟨𝑃𝑜𝑢𝑡 ⟩ = 𝛾2 𝑞˙2⟩⟨𝛾12 𝑞02 𝜔 4
2sin (𝜔𝑡 − 𝜑)= 𝛾2 2(𝜔 − 𝜔02 )2 + (𝛾𝜔)21
𝛾12 𝛾2 𝑞02 𝜔 2≈
8 (𝜔 − 𝜔0 )2 + (𝛾/2)2(C.12)(C.13)The last line of Eq. C.13 is obtained by taking the limit 𝜔 − 𝜔0 ≪ 𝛾. The ratio of
the power dissipated via 𝛾2 to a quarter of the nominal drive power (drive power atzero amplitude of the harmonic oscillator) has the exact form as Eq. 4.7.

C.4

Three-state 𝑇 -matrix model of 𝑝-wave resonant
scattering near degeneracies

This section provides a microscopic derivation for Eq. 6.3 which was developed byour theory collaborator Prof. Timur Tscherbul using an extended nonperturbative
𝑇 -matrix model of 𝑝-wave resonant scattering [86]. The model includes a single 𝑝3

The rate at which work is done by the drive can be decomposed into two parts: rate of loss as
friction in 𝛾1 and the power transferred to the system (𝑃𝑖𝑛 ). That is, 𝛾1 (𝑞˙𝑑 − 𝑞)˙ 𝑞˙𝑑 = 𝛾1 (𝑞˙ − 𝑞˙𝑑 )2 +𝛾1 (𝑞˙𝑑 − 𝑞)
˙ 𝑞˙

174

wave bound state (or closed channel) |3⟩ coupled to two open channels: the incident
channel |1⟩ and the outgoing inelastic channel |2⟩ with threshold energies 𝐸1 and 𝐸2(𝐸1 > 𝐸2 ). The open channels are separated by the energy gap Δ > 0 such that
𝐸1 = 𝐸2 + Δ. The total energy of the two-molecule system before the collision is
𝐸1 (𝑘) = 𝑘 2 /2𝜇 = 𝑘 2 /𝑚, 𝜇 = 𝑚/2 is the reduced mass of the two identical moleculesof mass 𝑚, 𝑘 is the wavevector in the incident open channel |1⟩, and we have set
𝐸1 = 0, so that 𝐸2 = −Δ. The incident scattering state in channel |1⟩ is a plane
wave |k⟩ with energy 𝐸1 (𝑘) = 𝑘 2 /𝑚 multiplied by the internal state vector of thecolliding molecules |𝛼⟩.

By summing the diagrammatic expansion for the 𝑇 -matrix, one obtains the following expression for the off-diagonal matrix elements between the open channels 𝛼
and 𝛼′ (𝛼, 𝛼′ = 1, 2) [86]𝑇𝛼,𝛼′ =𝐶 𝑘𝑘 ′ 𝐹𝛼 (𝑘)𝐹𝛼′ (𝑘 ′ ),𝐿3 𝐸 − 𝛿 − Σ1 − Σ2

(C.14)

where 𝐸 = 𝑘 2 /𝑚 is the collision energy in the incident channel |1⟩ and 𝛿 > 0 is
the energy of the bare p-wave bound state |3⟩, 𝐿3 is the quantization volume, and𝐶 = cos 𝜃, where 𝜃 is the angle between the incoming and outgoing wavevectors [86].
The functions 𝐹𝛼 (𝑘) quantify the coupling between the open and closed channels as a∫︀|𝐹𝛼 (𝑞)|2function of the wavevector 𝑘, and Σ𝛼 (𝐸)= 4𝜋(2𝜋)−3 𝑞 4 𝑑𝑞 𝐸−𝐸
are the open-closed
3𝛼 (𝑞)channel couplings in the energy space. These couplings are crucial as they determine
the resonance width. They can be evaluated by regularizing the diverging terms, andthen setting 𝐹 (𝑞) → 𝐹 (0), which results in the following expression
Σ𝛼 (𝐸) = 𝜆𝛼−𝑖𝑚(𝑚[𝐸 − 𝐸𝛼 (0)])3/2
12𝜋(C.15)(𝛼)− 𝛿0 − 𝜂𝛼 [𝐸 − 𝐸𝛼 (0)]where 𝜆𝛼 = |𝐹𝛼 (0)|2 are the 𝑞 → 0 limits of the open-closed channel couplings, andthe integrals(𝛼)𝛿0 = (6𝜋 2 )−1∫︁|𝐹𝛼 (𝑞)|2 𝑚𝑞 2 𝑑𝑞

175

(C.16)

2 −1∫︁

𝜂𝛼 = (6𝜋 )|𝐹𝛼 (𝑞)|2 𝑚𝑑𝑞(C.17)

depend on the exact form of the coupling matrix elements between the open andclosed channels. Note that (i) 𝐸1 (0) = 0 and 𝐸2 (0) = −Δ by definition, and (ii) the
first term on the right-hand side of Eq. (C.15) is purely imaginary (since we assume
Δ > 0) and thus gives rise to resonance width𝛾(𝐸, Δ)=(𝛾1 + 𝛾2 ) + 𝛾𝑑
= − 2Im(Σ1 + Σ2 ) + 𝛾𝑑

(C.18)

5/2=𝑚(𝜆1 𝐸 3/2 + 𝜆2 (Δ + 𝐸)3/2 ) + 𝛾𝑑6𝜋where we have introduced the intrinsic width 𝛾𝑑 of the 𝑝-wave bound state due to thecoupling to lower-lying inelastic channels other then |1⟩ and |2⟩ [see Figure C-3(a)].
While Ref. [86] assumes that Δ is much larger than all the other energy scales in the
problem, we do not make such an assumption here. Indeed, in our model, the new
Feshbach resonance occurs when Δ → 0.
Defining the resonance shift 𝛿0 = 𝛿01 + 𝛿02 and neglecting the dimensionless terms
𝜂𝛼 , which are expected to be very small compared to unity [86], we obtain from Eq.
(C.14)
𝑇𝛼,𝛼′ =

𝐶
𝑘𝑘 ′ 𝐹𝛼 (0)𝐹𝛼′ (0).
𝐿3 𝐸 − (𝛿 − 𝛿0 ) + 𝑖𝛾(𝐸, Δ)/2

(C.19)

Here, we have also assumed that the bound-continuum coupling matrix elements 𝐹 (𝑘)are well approximated by their zero-𝑘 values 𝐹𝛼 (0), which is a good approximation in
the limit 𝑘 → 0 (note, however, that this approximation starts to break down as 𝑘𝑅𝑏
approaches 1, where 𝑅𝑏 is the “size” of the 𝑝-wave bound state, as shown below). The√︀final wavevector in Eq. (C.19) 𝑘 ′ = 𝑚(Δ + 𝐸). We are interested in the two-bodyinelastic rate for the transition |1⟩ → |2⟩ at fixed collision energy

𝑔2 (𝐸, Δ) = |𝑇1,2 |2 𝜌2 (𝑘 ′ )≃1
𝑘 2 𝑚(Δ + 𝐸)3/2 𝜆1 𝜆2
𝐿6 [𝐸 − (𝛿 − 𝛿0 )]2 + [ 𝑚5/2 {𝜆1 𝐸 3/2 + 𝜆2 (Δ + 𝐸)3/2 } + 𝛾𝑑 ]2 /4
6𝜋

176(C.20)

where 𝜌2 (𝑘 ′ ) = (𝑚/2)3/2√︀2(Δ + 𝐸)/2𝜋 2 is the density of states in the final channel|2⟩ [135]. Eq. 6.3 is identical to Eq. (C.20) up to a constant overall scaling factor
and with the intrinsic decay width of state |3⟩ 𝛾𝑑 = 0. This provides a microscopic
justification for the Fabry-Pérot model. In particular, the Fabry-Pérot decay rates
5/2

5/2

may be expressed as 𝛾1 = 𝑚6𝜋 𝜆1 𝐸 3/2 and 𝛾2 = 𝑚6𝜋 𝜆2 (Δ + 𝐸)3/2 , providing insight
into their collision energy and Δ dependence.
We now discuss the main features of the expression for the two-body inelastic
rate (C.20). To this end, consider the expression for the resonance width 𝛾(𝐸, Δ)
in the denominator of Eq. (C.20) given by Eq.(C.18). In addition to the intrinsic
width 𝛾𝑑 , the width contains contribution from (i) the coupling between the incident
open channel |1⟩ with the 𝑝-wave bound state 𝛾1 ∝ 𝐸 3/2 ∝ 𝑘 3 , and (ii) the coupling
between open channels |1⟩ and |2⟩ through the bound state 𝛾2 ∝ (𝐸 + Δ)3/2 ∝ (𝑘 ′ )3 .
Equation (C.20) shows that the inelastic rate away from the resonance (when𝐸 − (𝛿 − 𝛿0 ) > 𝛾1 + 𝛾2 ) or for 𝛾𝑑 > 𝛾1 + 𝛾2 exhibits the standard 𝑝-wave scaling
𝑔2 ∝ 𝑘 2 (𝑘 ′ )3 [86]. This seems to apply to the experimental result, which does not see
a change of temperature scaling at the resonance. This is expected to be the common
situation, since Δ/𝐸 > 1 in the experimentally relevant regime. The normal 𝑝-wave
threshold scaling will be modified if the denominator of Eq. C.20 becomes energydependent, which requires the detuning 𝐸 − (𝛿 − 𝛿0 ) and 𝛾𝑑 to be small comparedto 𝛾1 + 𝛾2 . Under these (rather unlikely) conditions, the scaling changes to 𝑔2 (𝐸) ∝𝑘 2 (𝑘 ′ )3 /[𝜆1 𝑘 3 + 𝜆2 (𝑘 ′ )3 ].

C.4.1

Derivation of 𝐹𝛼 (𝑘) matrix elements

The p-wave bound state in channel |3⟩ is described by the wavefunction ⟨R|𝜓𝑏 ⟩ =𝜓𝑏 (R) of the intermolecular separation vector R. For a single p-wave bound state,^ where 𝑚u isthe radial and angular variables separate to give 𝜓𝑏 (R) = 𝑔1 𝑌1𝑚u (𝑅),the projection of the bound state’s angular momentum 𝑙 = 1 on a quantization axis
u. We assume that the bound state is coupled to both of the open channels |k, 𝛼⟩(𝛼 = 1, 2) via the coupling matrix elements ⟨k, 𝛼| 𝑉 |3𝑚𝑘 ⟩, where ⟨R|k𝛼⟩ = 𝑒𝑖𝑘𝑧 |𝛼⟩
are the scattering states in the open channels 𝛼. Expanding the incoming plane wave
177

3

Δ

γd

1
2

Figure C-3: a. Schematic of the resonance model with two open channels and a pwave bound state trapped behind a centrifugal barrier.
in spherical waves, 𝑒𝑖𝑘𝑧 =

𝑙
𝑙𝑖

∑︀

√︀

4𝜋(2𝑙 + 1)𝑗𝑙 (𝑘𝑅)𝑌𝑙0 (𝜃, 𝜑), we see that only the 𝑙 = 1

component couples to the p-wave bound state due to the orthogonality of the sphericalharmonics (and the assumption of isotropic bound-continuum coupling).
For practical purposes it is convenient to define the bound-continuum couplingsin the 𝑘-space [86]
∫︁
√
1 ∞ *
^
𝑔 (𝑅)𝑗1 (𝑘𝑅)𝑅2 𝑑𝑅
𝐹𝛼 (𝑘) = 𝑖 12 ⟨𝛼| 𝑉 |3⟩
𝑘 0 1(C.21)

where ⟨𝛼| 𝑉^ |3⟩ is a spin matrix element. The advantage og the 𝐹𝛼 (𝑘) matrix elements
is that they have a well-defined 𝑘 → 0 limit. Expanding 𝑗1 𝑘𝑅 ≃ 𝑘𝑅/3 + 𝑂((𝑘𝑅)5 ),we get√1
𝐹𝛼 (0) = 𝑖 12 ⟨𝛼| 𝑉^ |3⟩𝑘∫︁ ∞𝑔1* (𝑅)𝑅3 𝑑𝑅(C.22)

0

The approximation 𝐹 (𝑘) ≃ 𝐹 (0) is used in the previous section and in Ref. [86]to simplify the expression for the 𝑇 -matrix elements near threshold. To assess thequality of this approximation, we plot in Figure C-4 the coupling matrix element 𝐹 (𝑘)as a function of the dimensionless parameter 𝑞𝑅𝑏 , where 𝑅𝑏 gives the characteristic
size of the p-wave bound state [𝑔1 (𝑅) = 𝑅2 𝑒−𝑅/𝑅𝑏 ]. We observe that 𝐹 (𝑘) starts todeviate from unity at 𝑞𝑅𝑏 ≥ 0.01. Using 𝑅𝑏 = 100𝑎0 , we see that 𝐹 (𝑘) ≃ 𝐹 (0) is agood approximation for the incident collision channel (𝑘𝑅𝑏 ≃ 0.1 for 𝐸 = 10 kHz).
This is no longer the case when the open-closed splitting becomes large compared to178

Figure C-4: Coupling matrix elements between the open and closed channels plotted
as a function of the wavevector 𝑘 for two sample bound-state wavefunctions: 𝑔𝑙 (𝑅) =𝑅𝑙+1 𝑒−𝑅/𝑅𝑏 with 𝑙 = 1 (top curve) and 𝑙 = 2 (bottom curve). THe couplings arenormalized to unity at 𝑘 = 0.
the collision energy (Δ/𝐸 ≫ 1). We observe from Fig.C-4 that while for Δ/𝐸 = 30
(𝑘 ′ 𝑅𝑏 ≃ 0.1) the approximation is still good, it becomes too crude when the energy
release into the final channel exceeds the collision energy by a factor of ≥ 100. In
this situation (which include the one considered in Ref. [86]), the exact 𝑘-dependent
matrix element 𝐹 (𝑘) should be used.

179

C.5

Analysis on the two-body inelastic rate expression

In this section we focus on what we can learn about p-wave resonance from the twobody inelastic rate for the transition |1⟩ → |2⟩ at fixed collision energy which is
given as Eq. C.20 we simplify the form of the two-body inelastic rate as 𝑔2 (𝐸, Δ) =1 𝛾2
𝐺 Θ2 +(𝛾𝛾1 +𝛾
2 by redefining a few physical parameters with simplified notations
2 +𝛾𝑑 )

given as24𝜋 −1/2 −1/2
𝑚
𝐸
𝐿6
𝑚5/2
𝛾1 =
𝜆1 𝐸 3/2
6𝜋
𝑚5/2
𝛾2 =
𝜆1 (Δ + 𝐸)3/2
6𝜋
𝐺=(C.23)

Θ = 2[𝐸 − (𝛿 − 𝛿0 )].
We consider mainly the two cases: (i) sweeping through 𝛾2 or Δ to the optimum
point, and (ii) sweeping through Θ = 0 point for maximum inelastic loss. The inelastic
rate which varies as a function of 𝛾2 has it’s maximum value when 𝛾2 = 𝛾2𝑀 . This
optimum value 𝛾2𝑀 is found by taking the partial derivative of 𝑔2 with respect to 𝛾2
as
⃒
[︂
]︂
𝜕𝑔2 ⃒⃒
2𝛾2𝑀
1
= 𝐺𝛾1
−
𝜕𝛾2 ⃒𝛾 𝑀
Θ2 + (𝛾1 + 𝛾2𝑀 + 𝛾𝑑 )2 {Θ2 + (𝛾1 + 𝛾2𝑀 + 𝛾𝑑 )2 }2
2(C.24)=0→ 𝛾2𝑀 =

√︀(𝛾1 + 𝛾𝑑 )2 + Θ2

As 𝛾2 is tuned away from 𝛾2𝑀 the inelastic rate 𝑔2 decreases. The full width at half
maximum (FWHM) of the loss feature as a function of 𝛾2 is found by finding the twovalues of 𝛾2𝑚 in which 𝑔2 (𝛾2𝑚 ) = 21 𝑔2 (𝛾2𝑀 ). The two values are given as√︀
𝛾2𝑚 = 2 (𝛾1 + 𝛾𝑑 )2 + Θ2 + (𝛾1 + 𝛾𝑑 )
√︂[︁
]︁2
√︀
2
2
± 2 (𝛾1 + 𝛾𝑑 ) + Θ + (𝛾1 + 𝛾𝑑 ) − (Θ2 + (𝛾1 + 𝛾𝑑 )2 )180

(C.25)

The FWHM is the difference between the two points given in Eq.(C.25). In certain
limits, the FWHM can be simplified as
√︂[︁
]︁2
√︀
(𝐹 𝑊 𝐻𝑀 ) = 2 2 (𝛾1 + 𝛾𝑑 )2 + Θ2 + (𝛾1 + 𝛾𝑑 ) − (Θ2 + (𝛾1 + 𝛾𝑑 )2 )√√
Θ≫𝛾1 +𝛾𝑑−−−−−−→
2 3Θ = 4 3 [𝐸 − (𝛿 − 𝛿0 )]√Θ≪𝛾1 +𝛾𝑑−−−−−−→
4 2(𝛾1 + 𝛾𝑑 ).

(C.26)In the limit in which the intermediate bound state is far away from the incident
channel energetically compared to 𝛾1 and 𝛾𝑑 (Θ ≪ 𝛾1 + 𝛾𝑑 ), the maximum loss occurs
when 𝛾2 = Θ and the width indicates the detuning of the intermediate bound state.
In the limit in which the system is close to the Feshbach resonance Θ ∼ 0, themaximum loss occurs when 𝛾2 = 𝛾1 + 𝛾𝑑 and the width is proportional to 𝛾1 + 𝛾𝑑 .
From this analysis, it is difficult to know the absolute value of Θ or 𝛾1 + 𝛾𝑑 from theexperimentally obtained width as we only know the FWHM in the unit of Δ and notin the unit of 𝛾2 .

We now focus on what we can learn from the enhancement of the loss rate. We
assume the incident scattering flux is divided into two parts: fraction 𝜅𝑠ℎ𝑜𝑟𝑡 that iscoupled to states that undergo fast short-range loss and fraction 1 − 𝜅𝑠ℎ𝑜𝑟𝑡 = 𝜅𝑙𝑜𝑛𝑔
that is stable and long-lived. The inelastic loss rate at the background is close touniversal value as we also assume that 𝜅𝑠ℎ𝑜𝑟𝑡 ≫ 𝜅𝑙𝑜𝑛𝑔 . The loss rate at the background𝑔2 = 𝐺 · 𝜅𝑠ℎ𝑜𝑟𝑡 · 𝑡2 where 𝑡2 is the transmission probability of the incident flux at longrange. We define Γ such that 𝛾1 = 𝜅𝑙𝑜𝑛𝑔 Γ. Note that Γ is approximately proportional
to 𝑡2 in the limit of 𝑡 ∼ 1, 𝛾 ≈ 𝛼𝑡2 . The factor 𝛼 is unknown. The enhancement
factor in the case of optimizing 𝛾2 , is expressed as𝐶≡

𝑔2 (𝛾2𝑀 )𝑔 𝑏𝑎𝑐𝑘𝑔𝑟𝑜𝑢𝑛𝑑
[︂ 2
]︂
[︀
]︀
𝛾1 𝛾2𝑀
2 −1
= 𝐺 2
𝐺
·
𝜅
·
𝑡
𝑠ℎ𝑜𝑟𝑡
Θ + (𝛾1 + 𝛾2𝑀 + 𝛾𝑑 )2
𝜅𝑙𝑜𝑛𝑔
𝛾2
=
·𝛼· 2
𝜅𝑠ℎ𝑜𝑟𝑡Θ + (𝛾1 + 𝛾2𝑀 + 𝛾𝑑 )2181(C.27)

where 𝛾2𝑀 is the optimum value of 𝛾2 for maximum loss rate which was calculated in
Eq.(C.24). It is necessary to know what 𝛼 is in order to learn about 𝛾1 and 𝛾𝑑 fromthe experimentally observed enhancement factor, ∼ 100. We come back to the sameproblem that we encountered from the analysis of FWHM. 𝛼 is directly related to 𝜆1 ,and knowing the open-closed channel couplings 𝜆1 and 𝜆2 is necessary to estimate
the lifetime of the collision complex.
The enhancement of total loss in a Fabry–Pérot with a PBS for a given mirror
coupling strength 𝛾¯1 as illustrated in Figure 4-12(b) can be analyzes in a similarmanner. The loss has two components: transmission of p-polarized light through thesecond mirror, 𝑇𝑝 (𝜔0 , 𝛾¯2 ), which varies as a function of the cavity mode frequency
and mirror coupling strength 𝛾¯2 and reflection of s-polarized light at the PBS, 𝑇𝑠 . We
assume s-polarization component is much greater than p-polarization component of
the coupled incoming flux after the mirror M1 (𝜅𝑠 ≫ 𝜅𝑝 ) and consider two situations.
In the first situation, the spacing between two mirrors are fixed to create either aresonant or off resonant mode to the incoming light and 𝛾¯2 is scanned. In the second
situation both 𝛾¯1 and 𝛾¯2 are fixed but the Fabry–Pérot resonator mode is scanned
across the mode of the incoming light. The enhancement factors of the two settings,
𝐶1 and 𝐶2 , are given respectively as
𝑇𝑠 + 𝑇𝑝 (𝜔0 = 𝜔, 𝛾¯2 = 𝛾¯1 )𝜅𝑝2≈·
𝑇𝑠 + 𝑇𝑝 (𝜔0 = 𝜔, 𝛾¯2 ≫ 𝛾¯1 or 𝛾¯2 ≪ 𝛾¯1 )𝜅𝑠 𝛾¯2 𝜏𝑅𝑇
𝑇𝑠 + 𝑇𝑝 (𝜔0 = 𝜔)
𝜅𝑝
8¯
𝛾2
𝐶2 ≡
≈
·
𝑇𝑠 + 𝑇𝑝 (𝜔0 = 𝜔 + Δ𝜔)𝜅𝑠 (¯𝛾1 + 𝛾¯2 )2 𝜏𝑅𝑇
2
𝛾
¯1 =¯
𝛾2 𝜅𝑝
−−−→
·
.
𝜅𝑠 𝛾¯2 𝜏𝑅𝑇

𝐶1 ≡(C.28)

where |Δ𝜔| ≫ (¯𝛾1 + 𝛾¯2 )/2. The enhancement factors are inversely proportional tothe mirror coupling strength and even for small 𝜅𝑝 , large loss enhancement can beachieved with sufficiently weak mirror couplings.

182

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