Ultracold 23Na6Li Molecules in the Triplet Ground
State
by

Timur Michael Rvachov
B.A.Sc., University of Toronto (2010)
Submitted to the Department of Physics
in partial fulllment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2018
c Massachusetts Institute of Technology 2018. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
February 9, 2018
Certied by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scott A. Hughes
Professor of Physics
Interim Associate Department Head

Ultracold 23Na6Li Molecules in the Triplet Ground Stateby
Timur Michael Rvachov
Submitted to the Department of Physics
on February 9, 2018, in partial fulllment of therequirements for the degree ofDoctor of Philosophy

Abstract
This thesis describes experiments in spectroscopy and formation of triplet 23 Na6 Li
molecules from an initial mixture of ultracold 23 Na and 6 Li. The production of quantum degenerate molecules with long-range dipolar interactions is a long-standing goalin low temperature physics. NaLi is a fermionic molecule with an electric dipole moment of 0.2 Debye and a magnetic dipole moment of 2 µB in its triplet ro-vibrational
ground state. The formation of an ultracold molecule with both electric and magnetic dipole moments allows for novel opportunities in control of ultracold molecular
reactions and studies of quantum many-body systems with dipolar interactions.
This experimental work consists of two parts. The rst is a thorough spectroscopic
study of the excited and ground triplet potentials of NaLi using one- and two-photon
photoassociation spectroscopy. We present the spectroscopic positions and strengthsof transitions to nearly all vibrational states in the excited c3 Σ+ and ground a3 Σ+
potentials of NaLi. This is the rst spectroscopic observation of triplet potentials in
NaLi and the rst demonstration of photoassociation in the Na-Li system.
The second part utilizes our spectroscopic results to coherently form an ultracoldgas of NaLi molecules. Starting with an ultracold Na-Li mixture, we use magnetoassociation to form weakly bound Feshbach molecules. The Feshbach molecules arethen transfered to the ro-vibrational triplet ground state using a two-photon stimulated Raman adiabatic passage (STIRAP) technique, forming 3 × 104 molecules at
a density of 5 × 1010 cm−3 and temperature of 3 µK. The molecules are long-lived
with a measured lifetime of 5 seconds, which highlights their fermionic nature and low
universal inelastic loss rate. The utility of the molecule's magnetic moment is demonstrated by performing electron spin resonance spectroscopy to measure the hypernestructure of the molecule.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

2

To my family

Acknowledgments
It is impossible to adequately express in writing my gratitude to those who havesupported me during my PhD. My MIT experience has at times pushed me to myintellectual, physical, and emotional limits, which I would not have been able toovercome alone. Looking back, I can remember moments of joy and moments of pain,
all of them encased in an enormous wreath of personal growth.
Prof. Wolfgang Ketterle, thank you for your mentorship. I came to MIT wholly
unaware of what it takes to be an experimental physicist, and I leave with an understanding and deep-rooted respect for scientic progress. I thank you not only foryour lessons in physics, but also for your great patience that has allowed me to learn
lessons that can not be taught with words, but only with experience. With you, I
have learned how to face failure, fall down, get back up, and to repeat this process
as many times as necessary. Your ability to simply connect ideas as a physicist andyour stalwart personality have set an example I will continue to follow.
Prof. Martin Zwierlein, thank you for dealing with my playful nature in a positive
manner and always encouraging me to grab physics by the horns. Discussions withyou have always left me inspired.
I would like to thank the BEC3 team, both past and present. You have beenwith me nearly every working day, and each of you has personally shaped my growth
as a scientist. Caleb Christensen, Ye-Ryoung Lee, Jae-Hoon Choi (Go Michigan!),Myoung-Sun Heo and Tout Wangyour welcoming guidance during my early years
in graduate school put me in a position to succeed. Dylan Cotta, thank you for giving
me, Tout, and Colin Kennedy a warm welcome to Los Angeles during my rst trip to
DAMOP. Derek Kita, Chenchen Luo, Yijun Jiang and Li Jingthank you for youragency in repairing the ever-evolving machine and plugging the holes whenever there
were leaks, both physically and metaphorically. I want to particularly thank Pascal
Notz for his friendship during a dicult year in BEC3 when the oven chamber hada meltdown. It was an experimentally frustrating period lled with setbacks andrepairs, and I do not know how we would have managed without your tireless work4

ethic and optimism.
Of the more recent team members, I would like to thank Hyungmok Son, Sepehr
Ebadi, Juliana Park, Ariel Sommer, and Alan Jamison. This thesis would be twochapters shorter were it not for your hard work and support. Sepehr, thank you not
only for building lasers, but for coming to our group with an alacrity and energy thatmade working with you an absolute pleasure. Hyungmok, countless times I have reliedon your companionship as we ran the experiment into the early morning. Many nights
would have ended short of success without your persistence. Alan, you've been on
the journey from the rst signal of photoassociation to the last click of the Submit
button. You've ignited a ame of progress in BEC3 which has lit the way for our
success. Julie, thank you for asking me questions, because as tired as I looked all thetime I actually really enjoyed the conversation. And Ariel, thank for coming back tomake sure I was okay. I am grateful to all of BEC3 for your friendship and support.
I would like to thank the Ketterle and Zwierlein groups, the entire 2nd oor ofbuilding 26 and the greater MIT-Harvard CUA community. You have provided awarm environment for scientic and personal growth. Aviv Keshet, thank you forwelcoming me to the Cicero club, an inuence that cannot be understated given myimmediate future in the tech industry. Will Lunden and Li Du, thank you for joining
the Cicero club. Tarik Yefsah and Julian Struck, thank you for keeping the curtain
border more loving and less aggressive than it could have been. Peter Park andSebastian Will, thank you for letting us copy your experiment. Colin Kennedy, Cody
Burton, Woo Chang Chung and Wenlan Chen, thank you for letting us steal near-IR
optics and a 1319 nm laser. Junru Li, Boris Shteynas, Furkan Top, and Chris Lee,
thank you for letting us steal everything else. Boris, somehow you and I would always
nd each other in the hallway at 3am; I'm not sure why I'm thankful for this, but I
am. My Canadian brothers, Alexei Bylinskii, Dorian Ganglo and Boris Braverman,
thank you for helping me navigate US tax regulations (oh and Boris thanks for the
op-amps too). I would like to thank Lee Liu, Alex Lukin, Prof. John Doyle and Prof.
Mikhail Lukin for organizing and participating in various kinds of chaos during the
CUA retreats.
5I want to thank my classmates, with whom I started the ultracold atoms journey,
Ivana Dimitrova, Jesse Amato-Grill, and Lawrence Cheuk. From studying for thequalifying exams, late nights in the hallway, to lab-skipping ski trips and more, wehave compassionately shared struggles and successes since day one.
Though at times there was heavy lifting to do in the lab, my physical limit was
truly pushed while playing with the MIT Men's Club Hockey Team. I thank all of
the team members past and present for an unforgettable experience both as athletes
and as friends. The coachesDave Hunter, Brad Holschuh, Tom Hopkins, and Nick
Sislerthanks for sticking with me while I stared at my feet and ate it at the redline. MIT Hockey has had such a positive impact on me both on and o the ice that
it is dicult for me to imagine who I would be without it.
I have been fortunate to have made great new friends at MIT, and to be supported
by many old ones. I would like to thank Andy and Dave Pritchard not only for thesailing in Rhode Island, but even more so for your caring and guidance. Your concise
and honest advice was like a lighthouse in the darkness. Maddy Aby, thank you
for the adventures in the snow and the sea. Jannik Beusse, thank you for at times
literally carrying me. Phil Russ, Sean Yamana, and Mladen Ciric: we have known
one another since highschool, but no matter how long we are apart I nd we still
understand each other just the same.
I would like to especially thank my two best friends, Niklas Jepsen and Nikolay
Perunov. Russian Nickfrom Tang Hall to Baystate Road to NYCthrough thegood times and the bad your honesty and goodwill were always appreciated. I wantto thank the entire Jepsen family: Niki, Maui, Karin, and Søren. You have welcomed
me into your family with warmth and love. Thank you for the adventures and theuplifting pull so many times when I needed it most.
This thesis is dedicated to my family: my two brothers, Alesha and Marat, andmy parents, Lila and Misha. I was born in Vinnitsa, Ukraine, and I nd myself now
graduating with a PhD from MIT. You have worked and sacriced to give me every
opportunity to succeed. Thank you for your love and support.

6

Contents
1 Introduction

14

1.1

Interacting atoms and molecules at absolute zero. . . . . . . . . . .

16

1.2

Production of cold molecules . . . . . . . . . . . . . . . . . . . . . . .

20

1.3

NaLi and the bi-alkali landscape . . . . . . . . . . . . . . . . . . . . .

24

1.4

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

29

2 Experimental apparatus
2.1

2.2

2.3

2.4

2.5

32

Ultracold Na, Li mixture experiment overview . . . . . . . . . . . . .

33

2.1.1

Cold atom production. . . . . . . . . . . . . . . . . . . . . .

33

2.1.2

Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.1.3

Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Oven and gate valve replacement . . . . . . . . . . . . . . . . . . . .

38

2.2.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.2.2

Improved oven and cleaning procedure . . . . . . . . . . . . .

40

Magnetic eld stabilization . . . . . . . . . . . . . . . . . . . . . . . .

43

2.3.1

Background: do we need active stabilization?

. . . . . . . . .

43

2.3.2

Stabilization circuit . . . . . . . . . . . . . . . . . . . . . . . .

45

2.3.3

Feshbach molecule sweep coil

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

49

Laser systems for NaLi molecules . . . . . . . . . . . . . . . . . . . .

50

2.4.1

Spectroscopy lasers . . . . . . . . . . . . . . . . . . . . . . . .

50

2.4.2

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

52

Future improvements . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

7

3 Photoassociation spectroscopy

58

3.1

Experimental approach . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.2

Finding a rst signal . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.3

Spectroscopic line positions. . . . . . . . . . . . . . . . . . . . . . .

66v ∗ = 11 ne structure . . . . . . . . . . . . . . . . . . . . . . .

70Line strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.3.1
3.4

4 Two-photon spectroscopy

77

4.1

Experimental approach . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.2

Choosing a pathway to the ground state . . . . . . . . . . . . . . . .

82

4.3

Dark-resonance spectra: line positions. . . . . . . . . . . . . . . . .

84

Rotational splittings . . . . . . . . . . . . . . . . . . . . . . .

87Line strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

4.4.1

Autler-Townes spectra . . . . . . . . . . . . . . . . . . . . . .

89

4.4.2

Dark-resonance lineshapes . . . . . . . . . . . . . . . . . . . .

91

Alternative paths to the triplet ground state . . . . . . . . . . . . . .

94

4.5.1

Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.5.2

Free-to-bound STIRAP . . . . . . . . . . . . . . . . . . . . . .

96

4.3.1
4.4

4.5

5 Formation of triplet ground state molecules

100

5.1

Experimental approach . . . . . . . . . . . . . . . . . . . . . . . . . .

101

5.2

Feshbach molecules revisited . . . . . . . . . . . . . . . . . . . . . . .

103

5.3

Molecular spectroscopy at high magnetic elds . . . . . . . . . . . . .

107

5.4

STIRAP transfer to the ro-vibrational ground state . . . . . . . . . .

112

5.5

Studies of ultracold NaLi triplet ground state molecules . . . . . . . .

118

5.5.1

Lifetime and trapping. . . . . . . . . . . . . . . . . . . . . .

118

5.5.2

Estimating the loss coecient . . . . . . . . . . . . . . . . . .

122

5.5.3

Electron spin resonance spectroscopy . . . . . . . . . . . . . .

127

6 Summary and outlook
6.1

131

Collisional studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

131

6.2

Simplifying experiments. . . . . . . . . . . . . . . . . . . . . . . . .

133

6.3

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

A Magnetic eld stability circuit calibration

137

B Dispersion coecients for Na, Li, and the NaLi molecule142

C Triplet ab initio potentials

143

9List of Figures
1-1

NaLi molecular potential overview . . . . . . . . . . . . . . . . . . . .

1-2

Dipole moments and universal loss rates for triplet heteronuclear bi-

22

alkali dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2-1

Ultracold atoms machine diagram . . . . . . . . . . . . . . . . . . . .

34

2-2

Hyperne states of Na and Li in a magnetic eld

. . . . . . . . . . .

35

2-3

Histogram of most common times to take experimental data . . . . .

39

2-4

Improved oven design . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2-5

Cleaning the oven skimmer . . . . . . . . . . . . . . . . . . . . . . . .

42

2-6

Feshbach coil current stabilization circuit . . . . . . . . . . . . . . . .

46

2-7

Feshbach eld noise measurement at 750 G . . . . . . . . . . . . . . .

48

2-8

Diagram of the STIRAP laser setup . . . . . . . . . . . . . . . . . . .

54

2-9

Beatnote linewidth of STIRAP lasers . . . . . . . . . . . . . . . . . .

56

3-1

Photoassociation to NaLi triplet molecular potentials . . . . . . . . .

62

3-2

First observation of photoassociation . . . . . . . . . . . . . . . . . .

64

3-3

Leroy-Bernstein expansion predictions. . . . . . . . . . . . . . . . .

65

3-4

A sample PA spectrum of the v ∗ = 11 excited vibrational state . . . .

67

3-5

PA line structure in the c3 Σ+ potential . . . . . . . . . . . . . . . . .

69

3-6

Polarization dependent PA of the v ∗ = 11 state . . . . . . . . . . . . .

70

3-7

Comparison of electronic structures in NaLi and O2 . . . . . . . . . .

72

3-8

v ∗ = 11 PA line strength . . . . . . . . . . . . . . . . . . . . . . . . .

74

3-9

PA line strengths for all observed vibrational states . . . . . . . . . .

75

10

4-1

Molecular potential energy level diagram for two-photon spectroscopy

79

4-2

Simulated dark-resonance and Autler-Townes spectra . . . . . . . . .

81

4-3

Franck-Condon factors for optimal STIRAP transfer . . . . . . . . . .

83

4-4

Sample dark-resonance spectra . . . . . . . . . . . . . . . . . . . . . .

84

4-5

Comparison of a3 Σ+ vibrational binding energies to theoretical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-6

86

Dark-resonance spectrum of rotationally excited NaLi ground state
molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

4-7

Autler-Townes spectrum of the ro-vibrational ground state . . . . . .

89

4-8

Bound-to-bound line strengths between vibrational states in a3 Σ+ andc3 Σ+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

4-9

Free-to-bound electromagnetically induced transparency

98

5-1

Molecular potentials and transitions for formation of NaLi triplet ground. . . . . . .

state molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

5-2

Optical dipole trap geometry at high magnetic elds . . . . . . . . . .

102

5-3

Feshbach molecule formation and lifetime . . . . . . . . . . . . . . . .

104

5-4

Centrifugal distortion of the Feshbach molecule wavefunction . . . . .

106

5-5

Zeeman splitting of the c3 Σ+ , v ∗ = 11 excited molecular state . . . . .

108

5-6

Transition diagram between the Feshbach state (a3 Σ+ , vg = 10) andintermediate excited state (c3 Σ+ , vg = 11) at high magnetic elds. . .

109

5-7

Upleg transition spectroscopy at 745 G . . . . . . . . . . . . . . . . .

110

5-8

Two-photon spectroscopy using Feshbach molecules . . . . . . . . . .

111

5-9

STIRAP from Feshbach molecules to the ro-vibrational ground state .

114

5-10 STIRAP eciency as a function of pulse duration . . . . . . . . . . .

115

5-11 Imaging molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

5-12 Initial measurements of the NaLi ground state liftime . . . . . . . . .

1195-13 NaLi triplet ground state molecule formation sequence with thoroughfree atom clean-out . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

120

5-14 Franck-Condon factors for optical excitations to the b3 Π potential of
the NaLi molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

5-15 Long lifetime in a pure sample of isolated NaLi triplet ground state
molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

5-16 Parametric modulation of the 1319 nm ODT to determine molecular
trap frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

5-17 Time-of-ight distributions of dissociated molecules . . . . . . . . . .

124

5-18 Loss coecients in NaLi collisions . . . . . . . . . . . . . . . . . . . .

127

5-19 Electron spin resonance spectroscopy of the NaLi molecules . . . . . .

129

5-20 Hyperne structure and transitions in the NaLi triplet ground state
molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

A-1 Modulation bandwidth of series IGBT vs. parallel transistor . . . . .

138

A-2 Magnetic eld stabilization test using injected noise . . . . . . . . . .

139

A-3 Hall probe noise performance. . . . . . . . . . . . . . . . . . . . . .

140

A-4 Fluxgate magnetometer noise performance . . . . . . . . . . . . . . .

141

12

List of Tables
1.1

Comparison of ultracold atomic and condensed matter systems . . . .

17

1.2

Experimental eorts to produce dipolar bi-alkali molecules . . . . . .

27

2.1

Lasers used for manipulation of NaLi molecules . . . . . . . . . . . .

51

4.1

NaLi a3 Σ+ vibrational state binding energies . . . . . . . . . . . . . .

85

4.2

Comparison of Feshbach resonances to PA . . . . . . . . . . . . . . .

96

B.1 Properties of the NaLi singlet and triplet ro-vibrational ground states

142

C.1 Table of ab initio potential values for the lowest three triplet potentials
of NaLi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

143

Chapter 1
Introduction
The eld of low temperature physics is lled with striking discoveries and unexpected
applications. In 1877, physicists Louis Paul Cailletet of France and Raoul Pictet of
Switzerland independently produced oxygen in its liquid state [1]. At the time, oxygen was among the permanent gases 1 , a group of substances which were believed tonever exist in a liquid or solid state. Cailletet's experiment used a highly pressurizedchamber of gaseous oxygen that was allowed to rapidly expand to a large volume.
This resulted a momentary temperature drop to approximately -200 ◦ C (70 K), andthe observation of a visible, liquid mist of oxygen2 . The following decades produceda urry of scientic accomplishments, including the discovery of the noble gases, andpunctuated by the liquefaction of Helium by the Dutch physicist Heike Kamerlingh
Onnes in 1908. Onnes' experiments produced liquid Helium at record low temperatures of 1.5 K, and marked the beginning of Helium as an industry standard ofcryogenic cooling. Only three years later, Onnes would use Helium as a coolant tolower the temperature of Mercury metal below 4.2 K, at which point its electricalresistance was measured to rapidly drop to zero. This was the rst observation ofsuperconductivity (or, the superuidity of the conducting electrons), which would become one of the focal points in modern physics, with mysteries extending to present
1 Including hydrogen, carbon monoxide, methane, and nitric oxide. The noble gases were not yet
discovered.

2 Pictet's experiment used an entirely dierent method of producing liquid oxygen at high (300atm) pressures [1].

14

day research [2]. Ironically, Onnes' prior experiments already produced the liquid
Helium itself in a superuid state, though this would go largely unnoticed for two
more decades [3].
As time would tell, these discoveries of low temperature physics had enormous
impact in both science and engineering. For example, liquid oxygen (or LOX as
it's known in the aerospace industry) is a common oxidizer in space-going rockets.
Superconductivity is employed in almost every magnetic resonance imaging (MRI)machine, where Niobium-Titanium3 coils are used to produce large magnetic elds of∼ 3 Tesla. In the scientic community, the aforementioned discoveries at the turn ofthe 20th century sparked an experimental race to absolute zero. Could zero kelvinbe reached? What new material properties would one nd at absolute zero? In 1995,
a team of scientists from the University of Colorado and MIT achieved temperatures below 1 µK in dilute vapors of alkali atoms [4, 5], eectively reaching absolute
zero4 . These atoms exhibited superuidity, similar to the low temperature eects
in the electrons of Mercury and liquid Helium but with signicantly smaller energy
scales. These atomic systems birthed the new eld of ultracold physics and quantum
simulation, which will be described in more detail shortly.
The formation of ultracold alkali gases at absolute zero was a momentous scienticachievement, but it was not without its limitations. The techniques used to produce
these gaseslaser cooling and evaporation (described in Chapter 2)were very material specic. It was possible to create millions of alkali atoms at nanokelvin temperatures and densities of ∼1013 atoms/cm3 , but not much else (compare these numberswith a day-to-day example: a milliliter of water, which contains 1022 molecules at adensity 1021 molecules/cm3 ). Cooling anything else to these temperatures was a signicant challenge, if not impossible. For example, the oxygen molecule (O2 ), which
played such a crucial role in the eld of low temperature physics, currently cannot becooled to the nanokelvin regime due to the limitations of laser cooling. Nonetheless,
3 Niobium-Titanium is a commonly used superconductor in industrial applications which is superconducting below 10 K.

4 While technically the temperature T = 1 µK is still above zero, it was below any other relevanttemperature scales in the system, thus the atoms behaved as if T = 0.

15the success of ultracold atoms experiments has intrigued scientists to continue thepursuit of the ultracold temperatures with other (non-alkali) atoms and molecules.
The work of this thesis describes the formation of ultracold NaLi molecules. In
a broad sense, it is a small step in expanding the types of materials which can bestudied in the ultracold regime. It is a direct continuation in the science, art, andtradition of low temperature physics. While I will go on to describe the specic,short term motivations for such a study, they should all be taken with a grain of salt.
In 1877 liquid oxygen was discovered, and in 1969 used as fuel to reach the moon.
Onnes' discovery of superconductivity in 1911 was used last year to obtain an MRI
image of my knee to diagnose an athletic injury. The long term impact of scienceis impossible to predict. The formation of ultracold NaLi molecules is unlikely toleave any such profound consequences, but it is nonetheless a step forward in low
temperature physics. Will the relatively young eld of ultracold atoms and molecules
grow to produce life changing events? To me, it already has.

1.1 Interacting atoms and molecules at absolute zero
The seminal works of 1995 demonstrated the cooling of gases of 87 Rb (University ofColorado [4]) and 23 Na (MIT [5]) in an isolated ultrahigh vacuum (UHV) to a regime
known as quantum degeneracy. Quantum degeneracy occurs when the thermal de
Broglie wavelength of each atom, λdB , begins to exceed the interparticle spacing,n−1/3 , where n is the gas number density5 that is, the atoms begin to overlap. 87 Rb
and 23 Na are both bosonic6 , and at quantum degeneracy undergo a phase transition toa Bose-Einstein condensate (BEC). The interactions between the alkali atoms resultin the BEC itself to be a superuid, the same eect observed in liquid Helium andthe superconductivity of electrons in a metal. Ostensibly, these systems are wildlydierentone a dilute vapor of atoms in a UHV chamber, the other electrons ina chunk of metal in a Helium refrigeratorbut they have in common that their
5 Equivalently, quantum degeneracy is reached when the phase space density is approximately
3

unity, nλdB ≈ 1.

6 Bosonic species have integer total spin, while fermionic species have half-integer total spin.

16

constituent particles follow the laws of quantum mechanics, and thus it was natural
to use ultracold systems to simulate the physics of electrons in a solids.
Quantum simulation with ultracold systems is a rapidly growing eld dedicatedto exploring problems in interacting, many-body quantum systems that are too computationally complex to simulate on a computer [6]. Table 1.1 shows a comparison ofenergy scales and techniques to describe the correspondence between a dilute ultracold
gas and a chunk of metal.

Particles of interest
Number density
Environment
Fermi temperature (TF )Interaction energy
Simplest probing techniques

Ultracold dilute gases
Atoms or molecules1012 cm−3 (∗)
Magnetic or optical trap (∗)
Optical lattices (∗)
2 µK (∗)
100 nK (∗)
Direct spatial
imaging

Electrons in a metal
Electrons
1022 cm−3
Crystal lattice of nucleii

5 × 104 K

10 K
Transport (resistance)measurements

Table 1.1: Comparison of ultracold atomic and condensed matter systems.
All numbers are order-of-magnitude approximations, and vary between dierent
atoms/materials. The (∗) label indicates which parameters are experimentally simple
to vary, demonstrating the high degree of control possible in ultracold systems whichmakes them suitable as quantum simulators.
An example of a notable success in quantum simulation is the BEC-BCS crossover[7]: In the years following BEC of bosons, fermionic alkali atoms such as 40 K [8] and6

Li [9] were cooled to quantum degeneracy. In the presence of interactions, these

fermionic species showed exactly the same superuidity mechanism as in low temperature superconductors, known as Bardeen-Cooper-Schrieer (BCS) pairing. A key
component of this work was the ability to change the strength of atom-atom interactions using a magnetically tunable Feshbach resonance [10]. Such tunability is notpossible in condensed matter systems, and this has allowed ultracold fermionic atoms
to go beyond simply simulating known eects in metals, as demonstrated in experiments measuring the superuid transition in strongly interacting (unitary) Fermi
gases [11]. The ability for ultracold gases to simulate new and unexplored physics
is particularly important in development of new theories. A modern open problem
17

is copper-oxide ceramics which exhibit superconductivity at relatively high temperatures of ∼ 100 K. Such materials are greatly intriguing towards engineering a roomtemperature superconductor, yet a microscopic theory of their superconductivity is
still a mystery [12]. One hopes that an ultracold simulation of a novel material could
help in nding the answer.
Using the BEC-BCS crossover as an example, it is clear that interparticle interactions are a crucial ingredient in determining the kinds of physics one can study inan ultracold system. Neutral ultracold atoms experience a van der Waals interaction,
given by the potential VvdW = − C6 /r 6 where C6 is known as a dispersion coecient,and r is the separation between two atoms. This is a short-range interaction, whichmeans the 1/r 6 potential rapidly drops to zero at large internuclear separations suchthat the interaction is well approximated by a delta function7 VvdW ∝ δ(r). On theother end of the spectrum are long-range interactions, such as the Coulomb interaction between two ions8 , VCoulomb = C1 /r . The Coulomb interaction is of obvious
importance in modern trapped cold ion experiments [13], however for the purposes ofmany quantum simulations the Coulomb interaction is too strong: C1 is is so largethat the Coulomb repulsion localizes the atoms to form a classical Wigner crystal
without any wavefunction overlap between neighboring atoms, negating any quantum eects. Thus there is an interest in creating a tunable, long-range interaction
which could be used to observe novel phenomena in quantum many-body physics.
A candidate for a tunable strength long range interaction is that of two permanent
dipoles, which can be either electric or magnetic:

Vd-d =

A ~ ~
~
~
d
·
d
−
3(
d
·
r̂)(
d
·
r̂)
2
1
2
1
4πr3

(1.1)

where r, r̂ is the interparticle separation and unit vector, d~ are the dipole momentsof the two particles, and the coecient A = −1
0 , µ0 for electric and magnetic dipole
7 The aforementioned experiments on the BEC-BCS crossover using tunable Feshbach resonanceschanged the interaction strength by changing the prefactor to this delta function, which is a usefulfeature to simplify theoretical calculations. For details, see [10].

8 This is an uncommon notation for brevity; here C = kq q
1

and q are the two interacting charges.

18

1 2 where k is the Coulomb constantinteractions, respectively. An quantum degenerate gas with such interactions could
be used to study dipolar eects in ultracold systems [14], with eventual larger goalsof identifying new mechanisms for superuid pairing applicable to solids [15, 16].
In the world of ultracold science, recent experiments to realize the dipole-dipole
interactions have taken two approaches: (i) schemes in atomic systems, and (ii) the
production of ultracold dipolar molecules, which is a primary motivation for the workof this thesis. With atoms, signicant progress with dipolar interactions has been
made with ultracold samples of highly magnetic atomic species, such as Chromium(d = 6 µB [17]), Erbium (d = 7 µB [18]), and Dysprosium (d = 10 µB [19]). Recent workwith magnetic atoms has shown the emergence of a magnetic ferro-uid caused by theunderlying dipolar interactions [20]. Alongside using magnetic interactions in atoms,
there are proposals to produce electric dipole interactions by dressing the atoms witha Rydberg state [21]. Rydberg states9 have strong electric dipole moments, but theirshort lifetime results in a trade-o between dipolar interaction strength and loss, andthe use of Rydberg dressing in many-body physics has thus far been limited [22].
The work of this thesis is motivated by the alternative approach to producing an
ultracold dipolar gas: the use of molecules with a permanent electric dipole moment.
The simplest case is a heteronuclear dimer, where the diering electron anities of
the two atoms skew the electron cloud distribution in the molecule, thus producing
a dipole. In atomic units, dipole moments are measured relative to an electron: theelectric dipole moment is measured units of ea0 where e is the electronic charge and a0
is the Bohr radius, while magnetic dipole moments are measured in Bohr magnetonsµB , the magnetic moment of an electron. In this thesis we will often use the Debye tomeasure electric dipole moments, which is dened as 1 D ≡ 10−18 statC·cm ≈ 0.39 ea0 .
The formation of electrically dipolar molecules is largely motivated by the strong
nature of electric dipole interactions: If we take the ratio of eq. 1.1 with two electric
(el)

(mag)

dipoles of ea0 to that of two magnetic dipoles of µB , we nd Vd-d /Vd-d= 4/α2where α is the ne structure constant. This ratio is approximately 105 , indicating
9 A Rydberg state is a highly excited electronic state of an atom, typically to states with principal
2

quantum number n > 30. The dipole moment of such a state scales as d ∝ n .

19the inherent advantage of using electric dipoles to produce strong interactions. Thisadvantage is apparent in specic examples of ultracold systems: the interaction energy
between two of the most magnetic atoms, Dy, with 10 µB magnetic dipole moments, isequivalent to two molecules with 0.1 D electric dipole moments. The largest electric
dipole moments of heteronuclear molecules are typically ∼ 4 D [23], and since thedipolar interaction energy scales as d2 (see eq. 1.1) this results in signicantly stronger
dipolar eects in molecules than possible with magnetic atoms.
Another advantage of ultracold molecules is that they can also be used to study
an entirely dierent kind of quantum many-body problem: Chemical reactions. Here
we momentarily abandon the idea of using long range elastic dipole-dipole collisions
as a kernel for observing novel quantum phases and focus on short range chemical
reactions. Accurately calculating the dynamics of just two colliding dimers10 is a
non-trivial problem in physical chemistry [24]. The high state purity and control inultracold molecular systems makes them a perfect test-bed for studies of quantumcontrolled chemical processes [25].
Having showered praise over the prospects of ultracold dipolar molecules, we now
turn to their primary disadvantage: producing an ultracold, stable gas of dipolar
molecules is dicult. The remainder of this chapter will overview the challenges andsolutions in producing cold molecules, and the entirety of this thesis is dedicated tosolving the problem of producing ultracold NaLi molecules.

1.2 Production of cold molecules
Even the simplest of moleculesdimers made of two atomsare notoriously dicultto cool to quantum degeneracy. The diculty stems from the two extra degrees of
freedom associated with a bound two-body system: vibration and rotation. Thusfar, laser cooling has been the most eective way to cool atoms from a hot vapor(∼ 600 K) to the sub-mK regime. Laser cooling relies on the atoms having a closed

10 Two colliding dimers = A collision of four nucleii and a dozen or so electrons, depending on thespecies.

20

set of electronic transitions, i.e. transitions which decay to a small set of states which
can be re-excited by a only a few dierent laser frequencies (a cycling transition is the
special case where the only possible excitation decay is to the initial state, requiring
only one laser). Alkali atoms are the ideal case with only one laser color necessary
for laser cooling11 . In contrast, the plethora of vibrational states in a molecule makeit nearly impossible to nd a closed set of transitions12 . This is illustrated in Fig. 1-1,
the molecular potential energy curves and states of the NaLi dimer. Loosely speaking,the left half of this diagram depicts the numerous molecular states, and the right half
free-atom states, of which there are only several. With these considerations in mind,
the eld of cooling molecules has split into two predominant approaches: (i) direct
cooling, which focuses on methods for directly manipulating hot molecules to reachcolder temperatures and (ii) ultracold association, which uses the robust techniques
for creating degenerate atomic gases in two-species experiments, and then coherently
fusing the two dierent atoms into molecules after they are already near absolute
zero.
Currently, the most successful approach to direct cooling of molecules is using
pre-cooled molecules from a buer gas source, followed by laser cooling and trappingin a magneto-optical trap (MOT) [27]. Buer gas cooling uses cryogenic Helium toproduce a source of molecules at ∼ 1 K temperatures [28]. The subsequent laser cooling requires substantial modications from the techniques used in alkali atoms, andhere I highlight some key dierences: The choice of molecular species is constrained
to those with vibrational state branching ratios that form a near-closed transition
amenable to laser cooling (currently in use are CaF [26, 29], SrF [30, 27] and YO[31]). The rotational state branching can be closed completely by using a rotating(N = 1) ground state with laser excitation to a non-rotating (N 0 = 0) excited state,
however this results in the existence of Zeeman dark states transparent to the laser
cooling light. Pumping into these Zeeman dark states can be eliminated by rapidly
11 Hyperne eects actually result in two laser frequencies, but they are typically only ∼ 1 GHzapart which is technically easy to implement.

12 There are also many possible rotational states, however a scheme of forming a cycling transition

for rotational states has been developed; see for example [26].

21

Li 22P + Na 32S

20
3 +

c Σ

400

v*=11
15

300

10

200

5

100

a 3Σ +

Energy (THz)

Energy (103 cm-1)

b 3Π

vg=0
0

0

Li 22S + Na 32S
Ng
3

− 100

−5

2

1
vg, Ng=0

X1 Σ +

− 10

4

6

8

EN=BvN(N+1)

10

12

14

− 200

16

18

20

Internuclear distance (a0)Figure 1-1: NaLi molecular potential overview. The ground (a3 Σ+ ) and rst excited(c3 Σ+ , b3 Π) triplet potentials as well as the ground singlet potential (X 1 Σ+ ) are drawn(the excited singlet potentials have been omitted for clarity). To give a sense ofscale, the vibrational states of the singlet potential are drawn (blue) with a typicalspacing of ∼ 5 THz. The inset shows further rotational state splittings, with energy
level spacing of EN = Bv N (N + 1) where N is the rotational quantum number
and the rotational constant Bv ≈ 10 GHz in NaLi. The lowest vibrational state of
the triplet potential (a3 Σ+ , vg = 0) and an excited vibrational state (c3 Σ+ , v ∗ = 11)
wavefunctions are drawn. A notable feature of the ground potentials is that the triplet
potential is much shallower than the singlet. While this diagram is specic to NaLi,
the molecular potentials are qualitatively similar for all other heteronuclear bi-alkali
combinations.
switching the MOT magnetic eld direction at radio-frequency (RF) speeds, requiring
the added complexity of high-speed in-vacuum MOT coils. Recent results with direct
cooling have produced molecular MOTs with 105 molecules at densities of 106 cm−3and temperatures of 300 µK [26]. These molecule numbers are a factor of 104 smaller
than possible in an atomic MOT, but they are competitive with cold molecule pro22

duction rates from ultracold association experiments. Aside from laser cooling, there
are alternative direct cooling methods such as Stark and Zeeman decelerators, which
use electrically (magnetically) dipolar molecules and rapidly switching electric (magnetic) elds to slow a hot beam of molecules [32]. Recent results with these techniques
have gone beyond beam-line experiments, demonstrating magnetic trapping and evenobservation of evaporative cooling in a sample of dipolar OH molecules [33].
Although direct cooling has made signicant advances in recent years, the mosteective method to produce near-quantum degenerate gases of molecules has been
ultracold association. This method was rst demonstrated in 2008 by two groups
with Cs2 [34] and KRb [35], and has since seen ubiquitous use with other species ofultracold atoms. This method begins with an ultracold mixture of two species whichis associated to a weakly-bound molecule by using a Feshbach resonance [10]. Thishighly vibrationally excited molecule is then transfered to its ro-vibrational groundstate through a two-photon process known as stimulated Raman adiabatic passage(STIRAP). The entire procedure is state selective and reversible, and does not signicantly heat the molecules. The University of Colorado group working with thefermionic KRb molecule has progressed the eld by leaps and bounds, with demonstration of molecular formation [35], chemical reactions [36, 37], and dipolar interactions
in a lattice [38]. Their experiments have produced ∼ 104 molecules at temperaturesof ∼ 300 nK. Towards forming a quantum degenerate gas of molecules, these numbers

correspond to phase space densities of nλ3dB ≈ 0.25 (T /TF = 2, where TF is the Fermi
temperature), a record in the eld of ultracold dipolar molecules.
With all the progress in molecule production using ultracold association, there is
still much room for improvement. A primary issue is that the fraction of atoms which
are converted into molecules is low, typically limited by Feshbach association eciency
to < 10 %. This poor formation eciency results in a loss of phase space density,
requiring further cooling of the molecules themselves. A problem arises because themolecules typically do not have favorable collisional properties (i.e. fast inelastic loss),thus making it impossible to use evaporation as is commonly done in atomic systems
at ultracold temperatures. Similar collisional issues could exist with the molecules
23employed in direct cooling (CaF, OH, etc.).

Many groups around the world are

circumventing this problem by improving molecule formation eciency, and ensuringthat the molecules never collide with each other. In ultracold experiments, the use ofan optical lattice to isolate pairs of atoms can signicantly improve Feshbach molecule
formation eciency by eliminating atom-molecule collisions [39]. Another recent idea
is to use optical microtraps to catch individual molecules [40], which can be applied
to both molecules directly cooled in a MOT or those associated from ultracold atoms.
There is certainly no shortage of ideas in the eld of ultracold molecule formation,and both direct cooling and ultracold association methods are likely to produce many
new results in the near future.

1.3 NaLi and the bi-alkali landscape
We now turn our attention to the world of ultracold association with heteronuclear
bi-alkali species. This thesis describes the formation and initial studies with thetriplet ground state of the 23 Na6 Li molecule. As we'll see, NaLi is a peculiar moleculecompared to other bi-alkali species and is particularly well suited for studies of thetriplet state. It should be noted that there are eorts to form heteronuclear ultracold
molecules from non-alkali species, such as alakline-earth atoms [41]. While thesemixtures have interesting properties, they currently have a roadblock to molecule
formation due to the absense of workable Feshbach resonances in alkaline-earth species[42]. Formation of molecules with mixtures involving the highly magnetic species (Cr,Er, Dy) is an intriguing possibility, but is thus far largely unexplored.
There are a number of parameters to consider when discussing heteronuclear bialkali molecules:• Quantum statistics (fermionic or bosonic): This is determined by the total
spin (nuclear and electronic) of the molecule; integer spin makes a boson, half
integer a fermion. This is particularly important when considering ultracold
collisions, since fermions in the same quantum state can only collide via pwave collisions which are frozen out at ultracold temperatures [43]. Thus when24considering inelastic loss due to molecule-molecule collisions at ultracold temperatures, typically fermionic molecules have much longer lifetimes than theirbosonic counterparts.

• Electronic spin state (magnetic moment): The two valence electrons on each alkali atom can be paired in either a triplet (aligned) or singlet (anti-aligned) spinconguration. The spin conguration determines the symmetry of the valence
two-electron wavefunction, and thus greatly aects the interatomic potential
between the two atoms. The triplet potential is always much shallower thanthe deeply bound singlet potential, as shown in Fig. 1-1. This dierence between
singlet and triplet potentials is common to all bi-alkali pairs. Furthermore, singlet molecules have anti-aligned spin, thus having no magnetic moment (withthe exception of the nuclear mangetic moment, which is a factor of 103 weakerand negligible for most experimental applications). In the triplet state the valence electrons are aligned, and thus triplet molecules have a magnetic momentof 2 µB .

• Collisional stability: The ratio of elastic to inelastic two-body collisions inmolecules is an important property for sample lifetime and prospects of evaporative cooling. One inelastic loss pathway is reactivity, where a species AB(A, B refer to alkali elements) may be energetically allowed to undergo the exchange reaction, AB + AB → A2 + B2 . Other inelastic pathways may exist if themolecule is not prepared in its ground state: For example, collisions of triplet
ground state molecules could result in inelastic loss from decay to states in themuch deeper singlet manifold (see Fig. 1-1). Thus in principle, singlet groundstate, non-reactive molecules are the most desirable candidates for long sample
lifetimes. However, as we will discuss shortly, recent experimental evidence has
not corroborated this hypothesis.

• Permanent electric dipole moment strength: The electric dipole moment is the
key property of a molecule which we would like to use for quantum simulation, thus molecules with a larger dipole moment are preferred. Homonuclear
25

molecules have zero permanent dipole moment due to their symmetry. Heteronuclear bi-alkali molecules have a wide range of dipole moment strengths,with typically larger values for pairs of atoms with a large mass imbalance. Singlet states also typically have a larger dipole moment, as they are more deeply
bound (i.e. shorter equilibrium distance, which can be seen in Fig. 1-1) which
results in a more polarized electron cloud due to the dierences in short-range
forces in the two atomic nucleii. For example, LiCs is one of the most dipolarbi-alkalis with a dipole moment of 5.5 D in the singlet state, and 0.5 D in the
triplet state [44, 23]. Note that the term permanent dipole moment is a slightmisnomerthese molecules must be polarized with an external electric eld toobserve the long range 1/r 3 dipole-dipole eets of eq. 1.1.
The rst formation of an ultracold dipolar molecule was fermionic 40 K87 Rb preparedin its singlet ro-vibrational ground state13 , a reactive molecule with a dipole momentof 0.5 D [35]. The lifetime of a pure molecular sample is often compared to theuniversal loss rate [43], which is a theoretical upper bound to the inelastic collision
rate. The universal model simplies inelastic collisions by assuming unit probabilityof loss at short-range, thus its only parameters are the strength of van der Waals (C6 )attraction and the molecular mass (described in Sec. 5.5.2). In the case of KRb, theloss rate was consistent with the universal rate, indicating that two-body reactions
played a signicant role in limiting the molecule lifetime [36].
Since the seminal work of KRb, other groups have formed singlet ro-vibrational
ground state molecules of bosonic RbCs [45, 46], NaRb [47] and fermionic NaK [48](Table. 1.2).

The motivation for using these species of molecules was their large

electric dipole moments and non-reactivity, which would in theory make for easyto-use molecules in quantum simulation experiments.

However, measurements ofthe molecule lifetime in these species showed unexpected results. The lifetimes of thebosonic species (RbCs and NaRb) were measured to be surprisingly short (∼ 100 ms),
considering the molecules' chemical stability [45, 47]. The NaK molecule exhibited
13 Their initial experiment also demonstrated forming the molecule in the triplet ro-vibrational
ground state, however the triplet molecule was not used for further studies to to its weaker dipole
moment of only 0.05 D.

26

Reactive
Non-reactive

SingletKRb (fermion, 0.5 D, 2008 [35])Triplet
NaLi (fermion, 0.2 D, this thesis)

RbCs (boson, 1.2 D, 2014 [45, 46])
NaK (fermion, 2.7 D, 2015 [48])
NaRb (boson, 3.2 D, 2016 [47])Table 1.2: Experimental eorts to produce dipolar bi-alkali molecules. We classify
each molecule by its spin state, reactivity, and give the permanent electric dipole andyear of rst formation.
signicantly longer lifetimes (∼ 1 s) due to its fermionic character, however the measured lifetime was still shorter than expected for unknown reasons [48].
These results indicate the presence of a novel inelastic loss process in molecule
collisions. The prevailing theory is that of sticky collisions [49]: Pairs of colliding
molecules form collision complexes, i.e. they orbit one another for a prolonged duration (e.g. ∼ 100 ms complex lifetime in RbCs), which then increases the likelihood
for a third molecule to join the collision complex and result in three-body loss. The
lifetime of a collision complex is proportional to the density of states in the system,
thus molecules with higher mass would suer more loss from sticky collisions. This
theory is thus far qualitatively consistent with recent results (especially in the heavier molecules such as RbCs), but conclusive evidence for complex formation is stilllacking.
With these recent advancements and roadblocks in the eld of ultracold molecules,
we now turn our attention to our molecule of interest: fermionic 23 Na6 Li. Both 23 Na
and 6 Li are workhorse species in the eld of ultracold experiments, seeing prolic use
since the rst BEC and degenerate Fermi gas experiments. Would their dimer makefor an interesting molecule? In the absolute (singlet) ground state, the NaLi molecule
is reactive with a predicted electric dipole moment of 0.5 D [23]. However, due to
its light mass, fermionic nature, and small van der Waals interactions, the universal
reaction rate is expected to be the smallest out of all the bi-alkali pairs (Fig. 1-2), withestimated lifetimes of > 1 s at densities of 1012 cm−3 [43]. Such a favorable universal
lifetime indicates that NaLi molecules will be experimentally usable in any internal
state (since the short-range loss mechanism is irrelevant), including the triplet ro27

vibrational ground-state. In the triplet ground state, NaLi has a predicted electric
dipole moment of 0.2 D [44], along with the magnetic dipole moment of 2 µB fromthe aligned electron spins. Fig. 1-2 shows a comparison of the universal loss rates
and dipole moments for all bi-alkalis in their triplet ground state, showing NaLi asa reasonably good candidate. While past experiments have focused on the singlet
ground state due to its stability (a hypothesis which is now under question due tothe sticky collisions), we chose to focus on the triplet ground state of NaLi, which
has an inherently slow inelastic loss rate.
0.5

LiCs

Dipole moment (Debye)

0.4

LiRb
LiK

0.3

0.2

LiNa
KCs

0.1

NaK

NaRb

KRb

RbCs

0.0

10-13

NaCs

10-12
10-11
10-10
K, Universal loss coefficient at 200 nK (cm3/s)

Figure 1-2: Dipole moments and universal loss rates for triplet heteronuclear bi-alkali
dimers. The dipole moments for triplet ro-vibrational ground state molecules wereobtained from [44], and the universal loss rates were calculated using the methods
described in Sec. 5.5.2 with C6 coecients from [50]. Blue circles indicate fermionicspecies which must collide in the p-wave channel, and red squares indicate bosonicwhich can undergo s-wave collisions and thus have signicantly higher universal loss
rates.
Triplet NaLi has both an electric and mangetic dipole moment. While the electric
dipole moment of triplet bi-alkalis is smaller than that of singlets, the extra magnetic
dipole moment opens up a variety of new experimental possibilities. In quantum
28

simulation, certain lattice models require a non-zero spin which can interact withthe electic dipole to simulate a variety of many-body Hamiltonians [51]. In quantum
chemistry, the non-zero magnetic moment can be used to tune collisional resonances
in molecules, analogous to Feshbach resonances observed in atomic systems [52]. Furthermore, NaLi is the lightest bi-alkali, resulting in a number of interesting opportunities. First, the light mass results in the lowest inelastic universal loss rate and also thelowest rate of sticky collisions among the bi-alkalis [49]. Collisional experiments with
NaLi could elucidate the open problems of inelastic collisions in ultracold molecules.
Second, the NaLi molecule has weak spin-orbit coupling14 , thus NaLi in the triplet
ground state should have weak coupling to the singlet manifold. The suppressionof inelastic scattering to singlet states may allow for sympathetic evaporative cooling of the NaLi molecule using collisions with Na atoms entirely within the triplet
manifold: while the exchange reaction 2NaLi → Na2 + Li2 is possible, the reaction
NaLi + Na → Na2 + Li is theoretically forbidden so long as the bi-alkali molecules

involved remain in their triplet states15 [53]. Sympathetic cooling of molecules withatoms has not yet been demonstrated, and would be a signicant step forward toreaching quantum degeneracy with dipolar molecules.

1.4 Thesis outline
The preceding discussion illustrates that triplet NaLi is a new experimental systemwith numerous prospects, which will hopefully push the eld of ultracold science,chemistry, and condensed matter physics for years to come. This thesis describes theexperiments in understanding the NaLi molecular structure in its triplet state, and theuse of this newfound understanding to form isolated, ultracold NaLi molecules. The
method of ground state molecule formation was to use association around a Feshbach
14 The molecule spin-orbit coupling is expected to be weak because the small spin-orbit couplingin the constituent Na, Li atoms.

15 Note that trimer formation, i.e.

NaLi + Na → Na2 Li, is energetically possible, however thisshould not happen in a two-body collision because energy conservation with only one body in thereaction products is not possible.

Thus such processes could only occur in three-body collisions,which are much less frequent.

29

resonance followed by STIRAP [34, 35]. This work stands on top of prior research on
the NaLi molecule in our group, particularly the formation of NaLi Feshbach molecules
described in [54] and the thesis of Tout Wang [55]. Thus, the focus of this work is
the optical molecular transitions necessary for the STIRAP, coherently transferring
the molecules to the ro-vibrational ground state, and nally initial studies using thetriplet ground state NaLi molecule.
Chapter 2 is an overview of the ultracold atoms experiment, with a focus on recent
modications for the purpose of formation and manipulation of molecules. Chapter 3
describes our rst experiments with optical transitions in the NaLi molecule, where
photoassociation (PA) was employed to perform spectroscopy on the excited triplet
states of NaLi. Chapter 4 builds upon the PA experiment to perform two-photon
spectroscopy of the triplet ground state potential and spectroscopically identify atwo-photon transition to the ro-vibrational triplet ground state. Chapter 5 puts allof these pieces together: We revisit our prior Feshbach molecule formation experiments, and implement the coherent STIRAP pulse sequence to transfer NaLi to its
triplet ground state. Chapter 5 also describes initial experiments with the triplet
NaLi molecule, including the measurement of its lifetime and manipulation of its
magnetic dipole moment. Chapter 6 is a summary of the work, describing both shortand long term prospects in the NaLi system.
Throughout this thesis we will heavily use molecular physics terminology. A common resource for a physics-perspective on molecules is the text of Herzberg [56]. A
concise introduction is also summarized in the thesis of Tout Wang [55] and I havealso found fruitful discussions on molecular sub-structure in the book of Brown andCarrington [57]. The notation in this thesis is as follows: v denotes a molecule's
vibrational quantum number (with vg referring to those in a ground electronic state,
and v ∗ an excited electronic state), and similarly N denotes a molecule's rotational
angular momentum quantum number. In most cases, the NaLi molecule we will consider are Σ states in Hund's case (b) (i.e., the angular momentum projection onto theinternuclear axis is zero, Λ = 0). In this case the total electronic spin (S ), the molecular rotation (N ), and the total angular momentum (J = N + S ) are good quantum
30

numbers.
We will often reference ab initio molecular potentials: These are a set of numerical potentials provided to us by the group of Olivier Dulieu, which are explicitly
tabulated in Appendix C. There are numerous comparisons of experimental data (especially Franck-Condon factors) to the results from these ab initio potentials; thisrefers to numerical integration of the two-body (Na, Li) Schrödinger equation withthese potentials to nd bound states16 . Upon completion of the NaLi molecular spectroscopy (Chapters 3 and 4), we were able to calculate improved potential curves
which are given [58] and [59]. The primary improvement of these potentials is molecular binding energies at the < 1 % level, thus comparisons of Franck-Condon factors
to either the improved or ab initio potentials yields nearly identical results.
The work of this thesis has resulted in three publications: the spectroscopy of
triplet NaLi molecular states [58, 59] and the formation of ultracold NaLi triplet
ground state molecules [60].

16 E.g. the potentials, state positions, and wavefunctions in Fig. 1-1 were drawn using the Numerov-

Cooley algorithm with ab initio data.

31

Chapter 2
Experimental apparatus
The experimental apparatus, colloquially known as BEC3, was built in 2001 to
produce 23 Na Bose-Einstein condensates and later upgraded in 2007 to co-produce 6 Li
degenerate Fermi gases. On occasion the machine is called The Science Chamber,which refers to the two-chamber design of BEC3, utilizing a main chamber for
ultracold atom production, and a second science chamber to perform experimentson the atoms (Fig. 2-1). The science chamber is a separate vacuum system isolated
from the main chamber by a gate valve, and provides much higher optical accessdue to the absence of coils and optics necessary for cold atom production in themain chamber. Such a design was necessary in experiments necessitating frequent
breaking of vacuum and was used in the pioneering work on atom-chip technology[61]. However, for all BEC3 experiments in the last decade (including this thesis),the main chamber alone has been sucient.
The primary purpose of this chapter is to give an overview of signicant upgrades to the apparatus which were necessary for the spectroscopy and productionof ultracold molecules, intended for an audience familiar with the techniques of ultracold atoms. Those looking for a primer on cold atom production are referred to
the Varenna notes [62, 63]. Details of the BEC3 design (technical drawings, coil geometries) are given in the thesis of Ananth Chikkatur [64]. The upgrade to produce
ultracold Li is described the thesis of Tom Pasquini [65]. Aside from the upgrades
discussed below, an overview of the current machine is well described by the thesis
32

of Gyu-Boong Jo [66]. While the Science Chamber has sat idle for a decade, future
work with NaLi molecules will likely require its use; these prospects, along with otherpotential upgrades are described at the end of this chapter.

2.1 Ultracold Na, Li mixture experiment overview
2.1.1 Cold atom production
BEC3 is an MIT standard ultracold atoms machine, outlined in Fig. 2-1. A thoroughdescription of the MIT-style ultracold atoms machines can be found in [67], andhere we discuss key points of our NaLi machine and introduce typical experimental
parameters and terminology. We use a dual species Na/Li oven to produce a hot
atomic beam (T ≈ 450 C), which is subsequently slowed to ∼ 100 uK using a spinip Zeeman slower. The Na is cooled and conned in a dark-SPOT magneto-opticaltrap (MOT) [68], which gives ∼ 1010 Na atoms at a temperature of ∼ 500 µK. The
Li uses a conventional MOT (with each of the 6 MOT arms containing both MOT
laser cooling and repumping laser frequencies) to produce 107 atoms at a similar
temperature to the Na. The MOTs are simultaneously loaded for ∼ 4 s, after which
the atoms are optically pumped into the stretched states |F = 2, mF = 2i for Na and|F = 3/2, mF = 3/2i for Li, where F, mF is the total angular momentum quantum
number and projection onto the magnetic eld axis, respectively. These states arecaught in a Ioe-Pritchard magnetic trap and the sample is cleaned to produce
a collisionally stable mixture: any other magnetically trapped states of Na in theupper hyperne manifold (ie. |F = 2, mF = 1, 0i) are transfered with an RF sweep
to the untrapped state |F = 1, mF = 1i, and expelled from the trap (note that the
population in lower hyperne trapped state |F = 1, mF = −1i should be very small
from the pumping process). We do not perform a similar cleaning procedure for Li,
since it constitutes a much smaller fraction of the initial magnetically trapped mixtureand does not signicantly contribute to collisional losses.
The magnetically trapped mixture is cooled for 30 seconds using forced RF evapo-

33

Dual species oven

Magnetic trap geometry

Na cup Li cup

an

em

Ze
r
we
slo

Compensation
coil

Bottom imaging
beam

Side
imaging beam

Imaging
viewport
Ioffe-Pritchard
coil package

Transport
viewportzyScience chamber

Main chamber

Slower beam
viewportx

Bottom imaging beam

Figure 2-1: A not-to-scale schematic of the experiment geometry. A dual species ovensources a hot Na and Li beam, which is decelerated by a Zeeman slower and trapped in
the main chamber. Ioe-Pritchard coil packages provide magnetic elds for the MOT,
magnetic trapping, and Feshbach eld (∼ 800 G) stages of the experiment. Shimcoils wrapped around the chamber are used for minor eld adjustments (< 10 G).
The atoms are commonly imaged from the top (with the imaging beam entering fromthe bottom of the chamber), or from the side (with the imaging beam entering fromthe opposite side). The science chamber was not in use, except occasionally for abeampath to make a crossed optical dipole trap.
ration of Na on the |F = 2, mF = 2i ↔ |F = 1, mF = 1i transition at ∼ 1.8 GHz. The
Li is not evaporated, but rather sympathetically cooled by the Na. At the end of thisprocess, the machine produces a typical mixture of 5 × 106 atoms in each species, at atemperature of T = 2 µK, with a total cycle time of 40 seconds. The magnetic trap iscigar shaped, with radial trapping frequency ωr,Na = 290 Hz, axial trapping frequency

ωr,Na = 13 Hz for Na, and corresponding Li trapping frequencies scaled by the massp
mNa /mLi ≈ 2. The nal peak densities were typically nNa ≈ 1 × 1013 cm−3
ratio

and nLi ≈ 5 × 1012 cm−3 . The species number balance is easily modied by varying
34

the duration of MOT loading (e.g. 2 s of Na MOT pre-load, followed by 3 s of dual
species MOT loading). A good benchmark for machine performance is pure Na BEC
number without the presence of any Li, for which 1 × 107 is typical.
The magnetic trap described above is the starting point for the photoassociation(PA) and two-photon spectroscopy experiments described in Chapter 3 and 4. For
the production of ground state molecules (Chapter 5), we required the use of a NaLi
Feshbach resonance in the Na |F = 1, mF = 1i and Li |F = 1/2, mF = 1/2i hypernestates that are not magnetically trappable (often named the Na |1i and Li |1i states
at high magnetic elds, as depicted in Fig. 2-2). Thus, an optical dipole trap (ODT)
23 Na

F=2
ing

l
Coo

1000

F = 3/2

0mS = -1/2

-2000

-3000

Mo

l. Fo

0

o
Co

0

mS = -1/2

-200

|4

F =1

400

800

rma
tion

mS = +1/2

ling

200

mS = +1/2

-1000

|6
|5
|4

400

Energy (MHz)

Energy (MHz)

2000

6Li

|8
|7
|6
|5

F = 1/2

|3
|2
|1

|3

Mo

-400

l. F
o

rm
atio

n

1200

0

Magnetic Field (G)

50

100

150

200

250

|2
|1

300

Magnetic Field (G)Figure 2-2: Hyperne states of Na and Li in a magnetic eld. The Na (left) andLi (right) hyperne states split into two manifolds according to their electron spinorientation relative to the external magnetic eld. For ultracold atoms production, aNa |8i and Li |6i mixture is used (thick red lines). For experiments involving molecule
formation, RF pulses are used to transfer the atoms to the low eld seeking Na |1i
and Li |1i states (blue lines).
was necessary, which allowed for the use of the magnetic eld for Feshbach molecule
formation, rather than trapping. To load into the ODT the RF evaporation was
cut short leaving the sample at T ≈ 10 µK, and the ultracold mixture was then adiabatically transfered into a 5 W, 1064 nm ODT with a beam waist of 30 µm, andtrap depth of 200 µK (while this depth may seem excessive given the sample tem35

perature, in practice the high power made ODT alignment much easier). During the
ODT transfer the gradient coils (responsible for radial magnetic connement) were
ramped o, leaving the ODT to provide tight radial connement. The axial trapping
force from the ODT Rayleigh range is weak (7 Hz for Na at full ODT power), andthe majority of axial trapping was still from magnetic connement, produced by thecurvature and anti-bias coils which were kept on. Both species were then transferedto their lowest Zeeman sublevels (ie. the |1i states) with simultaneous RF LandauZener sweeps, and immediately following this the axial magnetic eld was ipped tocreate an axially conning potential for the high-eld seeking states (see Sec. 2.3.5 of[63]). The magnetic eld was then ramped a few Gauss above the target Feshbachresonance at 745 G, where the sample is further evaporatively cooled by lowering
the ODT power to 0.3 W in 2.5 s. After this procedure, typical atom numbers were1 × 106 each of Na and Li at temperatures of 2 µK. The corresponding peak densities

were nNa = 4 × 1012 cm−3 and nLi = 3 × 1012 cm−3 .

2.1.2 Imaging
The ultracold sample was imaged in absorption from two possible directions: (i) Bottom imaging, with the imaging beam entering from below the chamber and producing
a top-view of the cloud, showing its elongated aspect ratio, and (ii) Side imaging,with the imaging beam entrance along the long-axis of the cloud, hence producing
symmetric-looking images with high optical density. We typically used a magnication of 2 for both congurations. Both Na and Li were imaged simultaneously on thesame camera, which allowed for quick successive image acquisition with a 5−10 msdelay (depending on the image resolution and camera readout speed). The bottom
imaging conguration looked down the direction of gravity and allows for time-ofight (TOF) times of up to 170 ms, and was the preferred conguration for both
atom number counting and temperature measurements. Side imaging was primarily
used for beam alignment (ODT, STIRAP beams), however it was also exceedingly
useful in observing weak molecular signals (see Chapter 5), owing to the high opticaldensities when imaging along the long axis of the trap.
36

2.1.3 Stability
An important aspect for all ultracold atoms machines is stability. This was especially
important for our experiments in photoassociation spectroscopy (Chapter 3 and 4),
which required repetitive data collection for nearly one year. It was very dicult toperform PA spectroscopy on any day during which the machine produced unstable
atom numbers. The majority of stability issues were caused by laser cooling problems,and here I will outline some very simple modications which greatly reduced atom
number uctuations and drift:• Fiberized injection lock seeding. The Li laser cooling system used two injection
locks to boost the laser power, which were seeded with light from an external
cavity diode laser (ECDL) locked to an atomic reference. Rather than sending
a free-space beam from the ECDL to seed the injection lock diodes, we used
an optical ber for each injection locked diode, hence modularizing the setup.
This made a drastic dierence in injection lock stability, especially because
tuning the ECDL grating would change its beam pointing direction. By using
an intermediate ber before the injection lock, all pointing uctuations resultin intensity uctuations in the seed light to which the injection locks are muchless sensitive.

• Simple laser table enclosures. We replaced the old curtain-style laser table enclosures with a rigid housing, consisting of sliding plexiglass panels. Just having
this simple, passive box around the laser table greatly reduced the amount of
day-to-day drift in optics alignment. The long-term goal is to actively temperature stabilize this enclosure, but this has not yet been implemented.

• The Li MOT uorescence trigger. The Li MOT uorescence was monitored
on a photodiode during MOT loading, and the experiment sequence would
continue only when a threshold level of uorescence was achieved. This trigger
would compensate for poor performance when the Li laser cooling system was
unstable, allowing us to focus on ultracold atoms physics rather than mundane
laser issues.
37• Counting total atom number. In experiments with only Na or Li, shot-to-shot
number uctuations of 10 % were typical. In experiments with a Na and Li mixture, 30 % number uctuations in both species were typical. These uctuations
were anti-correlated due to the changes in heat load for the Na as a sympathetic
coolant of Li during evaporation. When a higher than normal amount of Li wasloaded into the magnetic trap, this resulted in more Na being evaporated as it
tried to cool the Li (and vise versa for a lower than usual Li load). To combat this issue, in experiments where an eect in both Na and Li was expected(e.g. photoassociation loss, Feshbach resonances), a much cleaner signal could
be seen by monitoring the total number of both species1 .
The improved experimental stability with these adjustments was crucial in certain
data-intensive experiments, such as photoassociation spectroscopy. Fig. 2-3 shows ahistogram of data collection times over a 9 month period in BEC3, in comparison toa similar plot from the thesis of Dan Stamper-Kurn [69]. The larger volume of data
in BEC3 is perhaps a testament to the stability improvements to BEC machines over
the years (though, it is amusing that shape and mean of the histogram is the same).

2.2 Oven and gate valve replacement
2.2.1 BackgroundOur method of ultracold atoms production used forced evaporation of Na as a sympathetic coolant for Li. This required the number of atoms in the Na MOT to bea factor of ∼ 103 higher than those of Li, and correspondingly the oven ux and usage rate of Na was signicantly higher than Li. The oven was initially loaded with25 g of each Na and Li, with normal operating cup temperatures of 360 C for Li and
300 C for Na. Under these conditions the Na cup was emptied and replaced annually(known as an oven change), while the Li was rarely ever depleted (at least it wasnever an issue during my PhD). Where did all of the sodium go? The beam ux
1 Of course, when conrming an observation that required the presence of both species, we would
double-check with each species individually.

38

Number of Shots

2500

BEC3, 9 months (2016)
BEC1, 1 year (1998)

2000
1500
1000
500
0

8am 10am12pm 2pm 4pm 6pm 8pm 10pm12am 2am 4am 6am

Figure 2-3: Histogram of most common times to take experimental data. In blue is ahistogram of experiment cycles from BEC3 during the period of Aug 22 2016 - May
15 2017, during which we worked on Feshbach molecule formation, STIRAP to thetriplet ground state, and studies of the triplet ground state molecules. In orange issimilar data from 1999 in BEC1.
to the main chamber is at most 1012 atoms/second; even if the oven was on all year
atoms × 3 × 107 seconds
that would account for 1012 second
year





= 1 mg of Na deposited into themain chamber (and even this is a huge overestimate, since the oven shutter is mostly
closed). The answer (as anyone who has looked inside a Na oven viewport wouldknow), is that only a small solid angle of the beam makes it down the slower tube,
and the rest is deposited onto the walls of the oven chamber itself.
During a routine oven change in late 2013, this became a problem. While baking
the oven after a successful Na oven change, all of the sodium that was collected on thewalls liqueed, and began to ow and migrate to unwanted areas of the oven. Thissodium clogged the oven shutter mechanism, destroyed the ion gauge, and cracked
a vacuum viewport (which miraculously still held vacuum). Furthermore, the gate
valve separating the oven from the Zeeman slower was already broken, hence there wasalways an ominous worry that a vacuum failure in the oven would propagate to themain chamber. For these reasons we decided to replace the oven entirely. During thistime we were lucky to have with us a highly talented Master's student, Pascal Notz,
39and the diagnosis of the oven issue, new oven design, replacement procedure, andperformance testing are all very well documented in his thesis [70]. In the following, I
will only summarize key modications to the oven design, comment on its performanceto date, and discuss possible future improvements.

2.2.2 Improved oven and cleaning procedure
A schematic of the improved oven is shown in Fig. 2-4. The goal of the new design
was for an oven with a similar form factor to the previous design (so it would approximately t in the same location), but with signicantly less risk of similar melt-down
failure and lower possibility of vacuum issues propagating to the main chamber. The
old faulty gate valve was replaced with two all metal gate valves. It has become an
empirical fact in both the Ketterle and Zwierlein groups that gate valves seem to suffer damage from the alkali atomic beams, although the exact mechanism is not clear.
The current solution is redundancy, preferably using gate valves from two dierent
manufacturers.
The prior oven shutter was based on a rotary bearing feed-through design whichbecame clogged with solid sodium. This was replaced with a much simpler shutter
consisting of a at sheet of copper attached to a exible vacuum bellows, which moves
in and out of the atomic beam-path (for details, see Sec 4.6 of [70]). The bellows based
design has no moving parts (aside from the exible bellows itself), and should function
even if encrusted by sodium.
The last signicant modication was to extend the skimmer such that its end
hanged over top of an empty vacuum cup, in an attempt to provide a simple, minimally invasive way to clean the oven and slow down the build up of stray sodium.
The skimmer is the rst object the atomic beam encounters out of the nozzle, andtends to accumulate large globules of sodium. In the old oven, the skimmer needed
to be removed and manually scraped to remove the sodiuman unpleasant processthat usually resulted in damage to the skimmer. The new oven uses a cleaning process which does not require manually scraping vacuum components. The skimmer isheated externally above the melting point of sodium, such that the liqueed globules
40Oven Shutter
Bellows Flange

Figure 2-4: Not-to-scale schematic of the new oven. Two gate valves are used toseparate the slower from the oven, a pneumatic gate valve (MDC GV-1500V-P-11)
and a manual hand-operated valve (VAT 48132-CE01). The oven shutter is transverseto the skimmer and is not shown. Inset: A close up of the skimmer and new alkali
collection mechanism. The red arrow depicts the intended path of liquid Na ow
during oven cleaning. Initial attempts to clean the oven using this method resulted
in Na spilling onto the bottom window. The blue line depicts where a glass shield
can be placed to prevent Na from splashing onto the window. Figure adapted from[70].
of sodium ow down o of the skimmer and into the alkali collection cup, which caneasily be replaced or recycled back into the oven (Fig. 2-4, inset).
The new oven was successfully installed during the middle of 2014, and has been
41

Figure 2-5: Photos of the oven through the bottom viewport. a) Normal oven operation: The atomic beam uorescence is visible as it passes through the skimmer.
Above the skimmer the beam is very bright. Below the skimmer the oven shutter isvisible, crossing the beampath. b) Na accumulation on the bottom oven window during cleaning. First attempts at melting Na o of the skimmer (as shown in Fig. 2-4)
resulted in Na spilling onto the window. A stream of solidied Na is visible in the
background. c) A second attempt at melting Na o of the skimmer with a shortenedskimmer. A stream of solid Na is visible. This Na would melt, ow, and re-solidify toform a waterfall. In the foreground (out of focus, bottom left) a spot of Na is visibleon the window. The shortened skimmer resulted in most of the Na accumulating in
the alkali collection cup, but some still managed to splash onto the window.
functioning as expected. During the installation procedure, we also found the slower
window (on the main chamber) to have been coated with sodium such that lighttransmission through the viewport was only 25 %, and it was replaced as well2 . Since
installation of the improved oven, we have attempted to employ the new skimmer
cleaning method, but with only partial success. While slowly heating the skimmer,
the sodium globules liqueed and slid down the skimmer, but the sodium stream
overshot the collection cup, and landed on the bottom viewport! (Fig. 2-5 b) Having
little intuitive understanding of the viscous, adhesive, and cohesive properties of liquidsodium, we designed the skimmer to be too long. We immediately pulled out theskimmer, cut it short, and tried the process again (Fig. 2-5 c). The second attempt
worked signicantly better; the majority of the sodium now fell from the skimmer
and onto the bottom of the vacuum ange, which upon further heating caused thesodium to ow into the collection cup. Nonetheless, a small amount of sodium still
2 The slower window needs to be headed to ∼ 200 C to prevent coating with sodium from theatomic beam, and we suspect the window heating band failed.

42managed to hop over the collection cup and reach the viewport again. We did notobserve how exactly this happened. The movement of sodium was erratic, sometimes
slowly moving when only partially liqueed (forming solid-trails such as Fig. 2-5 b),but owing very rapidly when a globule was fully liqueed. The ow events were so
rapid that we only caught a few glimpses of them by eye, and I suggest the use of arecording video camera in future such experiments.
Despite a small splash of sodium left on the bottom viewport, the procedure
worked and the end result was a clean skimmer. A simple solution to the windowsplash problem is to insert a long glass slide into the alkali collection cup to act as atransparent shield (a clear backboard, to borrow basketball terminology), as shownin the inset of Fig. 2-4. Along with this modication, the new oven is a simpler andcleaner source of hot sodium and lithium compared to past designs, and I hope it will
facilitate experiments in BEC3 for many years to come.

2.3 Magnetic eld stabilization
2.3.1 Background: do we need active stabilization?
The formation of NaLi Feshbach molecules requires active stabilization of magnetic
elds up to ∼ 800 G, with eld noise less than 1 part in 105 . Our prior work on NaLi
Feshbach molecules only succeeded in formation of molecules using the 745.4 G resonance that has a width of 10 mG [71, 72, 54]. This is the narrowest resonance aroundwhich Feshbach molecules have been formed to date, and the method of magnetic eldstabilization used in these experiments is described in Caleb Christensen's thesis [73].
Unfortunately, since Caleb's departure from the group, the specic knowledge andphysical implementation of his magnetic eld stabilization scheme was lost. Thus,
our initial attempts to re-form Feshbach molecules as a stepping stone towards thetriplet ground state did not use any active magnetic eld stabilization. Our intuition
that this would work was as follows:• The large inductance of the Feshbach (anti-bias) coil should lter out noise
43

uctuations from the driving power supply. The Feshbach coil has an inductanceof 250 µH and resistance of 50 mΩ, making a low pass lter for the current witha 30 Hz corner frequency.

• On an oscilloscope trace, we could see that most of the uctuations come from
60 Hz wall-plug noise, which was eliminated simply by synchronizing the experiment to the 60 Hz wall supply phase (i.e. prior to any magnetic eld sensitive step of the experimental sequence, the computer would wait until a peakof the 60 Hz oscillation before proceeding).

• A large amount of uctuations comes from the noisy analog setpoint which issent to the controlling power supply, often in the case of 60 Hz noise due toground loops. This was eliminated using a programmable digital setpoint froma device physically close to the high-current power supply.

• Any outside magnetic eld uctuations (e.g. magnetic eld noise at the atoms
from a nearby ion-gauge, or other sources external to the Feshbach circuit)would have to add in quadrature with the large 750 G bias eld, and be suppressed unless co-axial to the large eld.
With these simple modications, we measured that our eld noise was ∼ 10 mG RMS
without any active stabilization. This should have been just low enough to makeFeshbach molecules around the 10 mG wide resonance, but all of our attempts failed.
Given that we knew our eld stability was questionable, we decided to implement anactive stabilization circuit, which is described in the next section. But even with theactive eld stabilization, we could not make molecules. We later discovered that our
Feshbach molecule issues were unrelated to the magnetic eld, but rather an issue withthe free-atom clean-out light pulse (see Sec. 5.2). It turns out that the aforementioned
simple magnetic eld stability guidelines were sucient to form Feshbach molecules,
without any signicant performance decline in comparison to active stabilization.
Nonetheless, here I will present the details of our active magnetic eld stabilization
scheme. It is a signicant improvement over prior stabilization schemes, both in the44noise performance and in the methodology used to measure and characterize noise.
With this scheme we reach 1 mG RMS noise in the 102 −104 Hz band, at bias elds
up to 800 G.

2.3.2 Stabilization circuit
The circuit diagram of the Feshbach coil is shown in Fig. 2-6. We used a 20V, 500A
commercial power supply (TKD-Lambda, ESS 30-500-2-D-0806), for which we measured the RMS current uctuations to be ∼ 0.1 %3 . When operating at 200 A (roughly
corresponding to 700 G), this results in 200 mA of RMS noise current that needs to be
suppressed. Prior schemes used a Hall probe to measure the current through the coils,
and PID feedback to a high-power insulated gate bipolar transistor (IGBT) in series
with the coil to actively suppress noise uctuations. While this scheme has worked
in the past [65, 73], it has a number of draw backs: (i) One must be careful to always
ensure a low voltage drop across the IGBT to keep power dissipation low and prevent
the IGBT from melting (e.g. at 200 A, a 1 V drop results in 200 W dissipation). (ii)
The IGBT is a nonlinear device designed for switching rather than analog performance, and thus requires frequent PID adjustment if used at dierent bias points.
(iii) The IGBT is a large high-power device with a very large gate capacitance4 , andit has much lower modulation bandwidth than smaller transistors (see Appendix A
for comparison). An alternative approach, inspired by the methods of laser frequency
stabilization, is to split the feedback path into two: Low speed, high dynamic range
and high speed, low dynamic range. In the context of magnetic eld stabilization, weused the power supply voltage control as the slow feedback path (full dynamic range,
100 Hz bandwidth), and a low power transistor in parallel with the coil as the fast
feedback path (0.2 A dynamic range, 30 kHz bandwidth) as in Fig. 2-6 a. The use ofa tweeter transistor in parallel to bleed o current from the coil path is an elegant
way to reduce the problem of high-current stabilization into the regime of small signal
3 This measurement was without any of the simple modications listed in the previous section. It
is the worst-case noise to expect from the power supply.

4 This is fundamentally because the conductors in high power devices need to have large surface

area to accommodate a large current, and capacitance scales with area.

45

On/Off TTL

a)

IGBT
15A fuse

INA11420V
500A

NTE2541

Fast PID

Feshbach(anti-bias)
coils
L=250μH
R=50mΩ

LM6171 R1=0.1Ω

R2=1Ω
Ultrastab current
sensors

Slow PID

Feedback

Monitor

RSupply=1mΩb)Constant Voltage
SupplyTweeter
Transistor

RCoil=50mΩ
IT

ICoil

Figure 2-6: Feshbach coil current stabilization circuit. a) The full circuit diagram.
The current through the Feshbach coils was measured through a uxgate current
sensor (LEM IT-700S). This signal was split into two feedback paths. The slow
feedback was directly to the power supply voltage control using a commercial PID
controller (SRS SIM960). The fast feedback path was AC coupled and used a low
noise pre-amplier (SRS SR560) for feedback gain. This signal was passed into an
instrumentation amplier (INA114) for isolation, and an op-amp (LM6171) to drive
the tweeter transistor (NTE2541, which is a Darlington pair and requires little drive
current, however the gate op-amp was necessary when testing other transistors). R1
was used as a current-sense resistor for testing. R2, the 15A fuse, and a metal-oxide
varistor (MOV) were used in order to prevent the tweeter transistor from burning during voltage surges when switching current through the coils. The IGBT (Mitsubishi
CM1000HA-24H) was used only to switch the total current. A separate current sensor
was used to measure the current through the Feshbach coils. b) A simplied diagramof the Feshbach stabilization circuit. This simple model can be used to understand themodulation limitations of the tweeter transistor. It is missing the coil inductance andother reactive impedances from the power supply. Modeling the reactive componentsof the circuit was done empirically with a network analyzer.

46

electronics. A practical benet from this scheme is during its construction: burninga low power transistor costs only a few dollars, compared to hundreds of dollars forhigh-power elements. There is one caveat to this, which is that the large power supply
driving the coil is fundamentally a constant voltage supply5 and inhibits the abilityof the tweeter transistor to modulate the coil current. Consider Fig. 2-6 b in the caseof RSupply = 0; in this case no amount of modulation current IT would modify the coilcurrent. Using a simple application of Thevenin's laws we nd that the coil current
is modulated by the transistor by δICoil ≈ (RSupply /RCoil )δIT , which in our case is 2 %of δIT . Unfortunately, this means that in order to modulate away 0.2 A of current
noise, we need a 10 A transistor, requiring the use of slightly higher power electronics(the NTE2541 transistor we used is rated to 25 A, 120 W). The Feshbach coil current
was measured using a uxgate magnetometer (LEM IT-700s), which measures current from the magnetic eld around the current carrying cable and does not directly
connect to the coil circuit. We have found the uxgate magnetometers to have far
superior noise characteristics than Hall probes (see Appendix A). The current sensor was split into two feedback paths, connecting to commercial ampliers and PID
controllers (Fig. 2-6 a). In order to prevent accidentally injecting any noise into thecircuit (via e.g. ground loops), a separate current sensor was used to monitor the
feedback performance and the results are shown in Fig. 2-7.
The active stabilization using the tweeter transistor reduces the eld noise by∼ 10 dB up to 10 kHz, above which the performance becomes slightly worse dueto the expected servo bumps. The majority of the noise is at 60 Hz and higher
harmonics, which can be removed by syncing the experiment to a 60 Hz line trigger,
as previously mentioned. Thus, while running the atomic experiment we expect oureld stability to be below 1 mG RMS in the 102 −104 Hz band. Without stabilization
the RMS magnetic eld noise is 10 mG, which is consistent with our observation that
active stabilization with the tweeter transistor was not necessary to make Feshbach
5 Whether the supply is constant voltage or constant current depends on the frequency bandconsidered.

For example, at high enough frequencies the cable inductance would make a current

source (i.e.

RSupply = jωL becomes large). However, for the relatively low frequencies consideredhere the supply is well modeled as an ideal voltage source with a purely resistive output impedance.

47

Noise amplitude (dB/Hz1/2)
RMS Field Noise (mG)

-100
-120
-140
-160

Const. V ModePID On

10 1

10 0

101

10 2

10 3

10 4

10 5

10 6

f (Hz)

Figure 2-7: Feshbach eld noise spectrum measurement at 750 G. a) The noise spectrum of the current through the Feshbach coils. Black lines mark 60 Hz and higher
harmonics. The active eld stabilization reduces noise up to 5 kHz, however servobumping is observed at higher frequencies. This spectrum was measured by amplifying the uxgate current measurement using a commercial pre-amplier (SRS SR560),
and taking the Fourier transform of the oscilloscope trace. The noise amplitude iscalculated
as 20 log (Inoise /IDC ) where Inoise is the noise spectral density in units of√
A/ Hz and IDC is the DC current, approximately 230 A to produce a 750 G bias
eld. b) The RMS magnetic eld noise at 750 G (IDC = 230 A), calculated by integrating the noise spectrum from 1 Hz up to f . There is a signicant noise contribution
from the 60 Hz peak, and another signicant bump at 16 kHz which is the switchingfrequency of the uxgate current sensors.
molecules (note that we still required the slow feedback to the coil power supply).
While this feedback scheme was more than sucient for our purpose of Feshbach
molecule formation, there is still room for improvement. The fast feedback bandwidth is not limited by transistor speed (which is in excess of 1 MHz), but rather
LC resonances formed by the large Feshbach coil inductance and stray capacitance inthe high current power supplies (see Appendix A). Another issue is the linearization48

of the tweeter transistor, which would allow for better PID performance when thefull dynamic range of the transistor is necessary. In principle, this is easily achievedby placing the transistor (NTE2541) in the feedback path of the driving operational
amplier (LM6171)6 , but in practice this requires careful placement of shunt capacitors to ensure the linearized circuit doesn't oscillate. We gave only modest eort insolving these issues, and ultimately they were unimportant to the physics of ultracold
molecules.

2.3.3 Feshbach molecule sweep coil
The eld stabilization circuit previously described does well in order to make a noisefree DC bias eld near the Feshbach resonance of 745 G, however it was not designed
for fast eld sweeps that are necessary to make Feshbach molecules. In the past, theLandau-Zener magnetic eld sweeps for Feshbach molecule formation were performed
with one of the external compensation7 coils coaxial with the Feshbach coils. The
compensation coil was driven by a commercial low current supply of ∼ 10 A (producing ∼ 10 G) with an o-the-shelf noise stability of 0.01 %. During development andtesting of the high-current Feshbach coil, we noticed that modulating the current in
the compensation coil would inductively inject the modulation into the high-current
Feshbach circuit. The compensation coil is in Helmholtz conguration, co-axial toand surrounding the Feshbach coil, thus it makes sense that there is a large crossinductance between the two coils. While this eect found use in injecting noise into
the Feshbach circuit to test the active stabilization feedback, it limited the speed atwhich we could sweep the magnetic eld using the compensation coil. Intuitively, theinductance of the Feshbach coil tries to keep the eld constant, and changes in the
total eld ux through the coil are counter-acted by an induced current.
For these reasons, we decided to make a single small coil with a very small areathat would be placed as close as possible to the atoms to do very ne eld sweeps,
which we called the Mok coil (named after Hyungmok Son, the graduate student
6 See for example The Art of Electronics by Horowitz and Hill, p. 257 example CC.
7 Sometimes called shim coils.

49who wound it). The small area ensured a small cross inductance to the Feshbach coil,such that we couldn't nd any measurable current cross-modulation between the twocoils. The Mok coil was 1 inch in diameter with 7 turns of 32 American wire gauge(AWG) copper wire, inserted into one of the bucket windows as close to the atoms as
possible. This coil produced 35 mG/A at the atoms, allowing for 0.5 G eld sweeps
with modest power current supplies. In all Feshbach formation experiments, the large
stable DC bias eld was produced by the Feshbach coils, and the molecule formation
sweeps were performed by sweeping the Mok coil.

2.4 Laser systems for NaLi molecules
Unlike alkali-atoms, molecules have a broad range of ro-vibrational energy levels withexcitation spectra spanning hundreds of nanometers. We required new laser systems
for spectroscopy, formation, and optical trapping of the NaLi molecules. Table 2.1
overviews the laser systems used for the manipulation of NaLi molecules.
In the following sections I discuss the laser systems used for PA spectroscopy
and coherent transfer to the NaLi triplet ground state. For optical trapping of NaLi
molecules, we required the use of longer wavelength lasers than our standard 1064 nm
ODT to observe long lifetimes, and this is presented in detail in Chapter 5. The
1319 nm laser is a typical, out-of-the-box system without any signicant modications. In most cases, the lasers in Table 2.1 were introduced to the atomic/molecular
cloud along the long axis of the trap (see Fig. 2-1, inset).

2.4.1 Spectroscopy lasers
Our initial exploration of excited molecular states in NaLi was using PA spectroscopy,
which was performed using an external cavity diode laser (ECDL) at 671 nm. Using
an ECDL for spectroscopy was dicult: The mode-hop free tuning range is at best

3 GHz with a maximum tuning range of ∼ 5 nm. Searching for a single PA transition
over hundreds of GHz required frequent adjustments of the ECDL, and searching forother PA transitions required occasionally switching laser diodes to access dierent
50

Table 2.1: Lasers used for manipulation of NaLi molecules
Laser Name

Wavelength

Power

Purpose670 TiSaph
(M2 SolTiS)
Standard TiSaph
(M2 SolsTiS)
Downleg ECDL
(Homebuilt)
Mephisto670-725 nm

1.5 W

725-900 nm

2W

819 nm

20 mW

PA spectroscopy, high vibrational states of c3 Σ+
PA spectroscopy,
STIRAP upleg
STIRAP downleg

1319 nm

1W

ODT for molecules

wavelengths. Another issue was the spontaneous emission pedestal8 , which would
resonantly excite the free Li atoms and result in heating of the ultracold sample.
This pedestal was ltered out by passing the ECDL output 3 times through a heated
Li vapor cell. With this method, we found the rst 18 PA lines (v ∗ = 51−34, spanning
the range 671-694 nm) using an ECDL (see Chapter 3.).
The remaining PA lines were found using two Ti:Sapphire lasers from M-Squared
Lasers: one spanning the 725-900 nm range typical of the Ti:Sapphire gain medium,and a newer model which could reach wavelenths down to 671 nm (which in principle
could be used to laser cool Li). The Ti:Sapphire lasers are easily tunable, higher
power devices which allowed for a much faster pace of performing PA spectroscopy.
The dierence was staggering: It took 4 months to nd 18 PA lines by swapping
diodes in and out of an ECDL, and 10 days to nd the next 29 lines using theTi:Sapphire lasers9 . The PA lines initially observed using the ECDL were re-scanned
using the Ti:Sapphire lasers. ECDLs typically have a broad spectral pedestal due
to amplied spontaneous emission (ASE) which is not spectrally ltered by their low
nesse cavity. The Ti:Sapphire laser (with its much higher nesse ring cavity) doesn't
have this pedestal issue, and allowed for acquisition of much cleaner PA data.
8 An ECDL provides light with a ∼ 100 kHz linewidth, but in addition emits weak pedestal lightspanning a few nanometers which corresponds to the broad amplied spontaneous emission proleof the diode.

9 The deepest PA lines, v = 0, 1, 2 were outside the range of our Ti:Sapphire laser, and we are

grateful to the Fermi1 group of Prof. Zwierlein for letting us borrow their Ti:Sapphire laser which
could tune down to 1000 nm.

51

In all spectroscopic studies the wavelength was recorded using a HighFinesse WS-7
wavemeter. The wavemeter is nominally accurate to 60 MHz, however our experiencehas found it to be reliable down to 20 MHz, which is the spectroscopic uncertainty
quoted in our measurements. This was determined by testing the wavemeter using abroad range of wavelengths from locked lasers to atomic references from Li (671 nm),Na (589 nm), and Rb (780 nm). The wavemeter would drift ≤ 30 MHz daily, andthis was corrected for by daily calibration with Li laser cooling light.
The Ti:Sapphire lasers were tunable from a computer interface, and using theapplication programming interface (API) provided by the wavemeter and Ti:Sapphirelasers we implemented a soft-lock to digitally lock the laser to a desired wavemetervalue. We estimate the soft-lock provided a relative stability of 1 MHz, and greatly
increased the quality of PA data (especially PA line strength data, where the laser
needed to be constantly on PA resonance; see Chapter 3). We could also implement
this softlock on an ECDL using an Arduino microcontroller to convert digital feedback
commands to an analog signal for the ECDL piezo control.

2.4.2 STIRAP lasers
The formation of ro-vibrational ground state molecules from a highly excited Feshbachstate used the technique of two-photon stimulated Raman adiabatic passage (STIRAP) [74, 75] . This technique uses two laser pulses designed to keep the molecules
in a coherent dark statea state without any probability of electronic excitation andthus no loss from spontaneous emission. The physics of STIRAP is discussed in detailin Chapter 5, and here we only concern ourselves with the technical requirements ofthe two lasers.
For the molecular potentials of NaLi, we decided to use an 833 nm (upleg) and
819 nm (downleg) laser frequencies to couple the NaLi Feshbach and ground states
via the v ∗ = 10 excited vibrational state in the c3 Σ+ potential. The frequencies werechosen for a very strong downleg Rabi coupling with a compromised upleg couplingstrength (see Chapter 5). The weak upleg transition strength was compensated forby using the high power Ti:Saphhire laser tuned to 833 nm, while the downleg only52

required modest powers for which we used a home-built ECDL at 819 nm, following
the robust design of the Steck group at the University of Oregon [76].
The STIRAP transfer eciency depends on the strengths of the two laser couplings, and their relative phase stability. A simple estimate of the laser phase constraint is as follows: The typical Rabi frequencies achievable in a bound-to-boundmolecular transition are Ωmax /2π = 1 MHz, and since STIRAP is an adiabatic process
it requires pulse durations longer than Rabi frequency, for example τpulse  10 Ω−1max →τpulse ∼ 2 µs (the factor of 10 is somewhat arbitrary here, however it turns out to be areasonable value; see eq. 13 of [75]). Ideally there would be no phase uctuations overthe duration of the pulse, hence we require the relative linewidth of the two lasers to
be narrower than the Fourier limited bandwidth of the pulse, i.e. δωlasers < 100 kHz.
A more careful calculation requires the specic details of the two laser pulse shapes
along with a well dened noise model, which can be found in [77]. While the simple calculation presented here provides some intuition for the technical constraints
in the STIRAP procedure, our target for relative laser stability was signicantly lessthan this 100 kHz estimate for two reasons: (i) We did not know a priori what Rabi
frequencies are experimentally achievable in the NaLi system, and even considered
the possibility of a free-to-bound upleg transition for which we expected orders ofmagnitude weaker upleg Rabi frequencies (see Chapter 4), and (ii) A survey of prior
STIRAP experiments with ultracold atoms typically used pulse durations of < 1 ms
with lasers locked to 1 kHz. Erring on the side of caution, our goal was a similar
relative linewidth for the two STIRAP lasers.
There are two common methods to make a narrow lock for two lasers with a large
frequency dierence (in this case, the frequency dierence was 6 THz): phase lockingdierent teeth of a frequency comb, and frequency locking to a narrow cavity. While
the seminal work with KRb was performed using the former technique [78], we decided
to use a narrow cavity lock because of the simplicity of using a passive optical element
and prior experience with this technique in the NaK experiment in the Zwierlein group[79]. The two lasers (at 819 nm and 833 nm) are simultaneously passed through avery stable optical cavity, and locked to dierent interference maxima (see Fig. 2-8).
53

We used a commercial optical cavity made of ultra low expansion (ULE) glass witha nesse of 3 × 104 (linewidth of 15 kHz), and free spectral range of 1.5 GHz. The
ULE cavity was expected to drift in absolute frequency stability by ∼ 1 MHz/year,
however in approximately 6 months of use for ground state molecule formation norecalibration was necessary10 . This cavity allowed for locking lasers over a rangeof 810-880 nm to relative linewidths of < 1 kHz (depending on the locking signalto-noise). Note that strictly speaking this is not a phase lock of the lasers, but anarrow frequency lock. This means that if the laser frequencies are compared forsome duration of time, their phase variation will be small (because the lasers have avery narrow frequency), however there isn't a deterministic relationship between theinitial phases of the two lasers at the beginning of the time interval. For STIRAP,
only the phase-noise characteristics are important, and the relationship between theinitial phase of the two lasers plays no role in STIRAP eciency (see Sec. E3 of [75]).
Thus, a narrow frequency lock with the ULE cavity was sucient.
Similar for Ti:Sapphire

ECDL Feedback Scheme

Ti:Sapphire Feedback Scheme
4

Diode

ECDL

Lag-Lead &
JFET

HV
Amp

Fast
PI2D

Slow
I

amp

D2-125Laser Servo

ULE Cavity

Fiber EOMPZTDichroic
1.5 GHzFG
80 MHzFG

Combiner

Fast Piezo

Lag

Amp
Spli�er
Phase adj.

Phase Detector

Photodiode

Ti:Sapphire
Slow PZT

Coupler-10 dBSpectrum
Analyzer

OutputAmp
LB-1005
Servo

PI

Ti:S
Error IN

Figure 2-8: Diagram of the STIRAP laser setup. The ECDL and Ti:Sapphire were
locked to a ULE cavity. The lasers could be tuned relative to the cavity up to
1.5 GHz using a ber EOM to oset the lock frequency. The ber EOM also generated
sidebands to generate a Pound-Drever-Hall lock signal. Figure reproduced from [80].
The technical details and test measurements of the STIRAP lasers can be foundin the thesis of Sepehr Ebadi [80], a talented and tenacious undergraduate student
10 Our experiment is sensitive to the two-photon detuning rather than absolute frequency stability.
Considering the wavelength 833 nm → 360 THz, and the locked frequency dierence of 6 THz, we
expect the two-photon detuning to drift by 6 THz/360 THz × 1 MHz/year = 16 kHz/year.

54

we were privileged to work with for a year. Here, I only summarize key points. For
both lasers, we used high frequency ber electro-optic modulators (EOM) fed with
two frequencies to provide a tunable oset from the cavity (up to 1.5 GHz), and thePound-Drever-Hall (PDH) sidebands (∼ 30 MHz) for generating a lock signal. The
free-running unlocked linewidth is 50 kHz for the upleg Ti:Sapphire, and 200 kHz forthe downleg ECDL [76]. We needed to narrow both lasers to below 1 kHz linewidth. A
rule of thumb for laser narrowing is that the feedback bandwidth must be larger thanthe laser linewidth (this is true for typical laser noise models [81]). Thus the ECDL
required a much faster feedback path than the Ti:Sapphire laser. For the ECDL
feedback we used a 10 MHz Vescent Photonics D2-125 controller with a phase compensation circuit to provide fast current modulation via a transistor parallel with thelaser diode. The Ti:Sapphire feedback was simpler, using a 10 MHz Newport LB1005
controller11 to modulate the piezo-electric elements built into the Ti:Sapphire laser.
Note that in this case the modulation bandwidth is limited by the piezo actuation
speed, typically < 100 kHz. Although the ECDL was technically more challenging tolock than the Ti:Sapphire laser, we found its locked state to be much more robust
against noise. For example, moderate level noise (e.g. talking near the laser) would
routinely unlock the Ti:Sapphire, but had no aect on the ECDL. This was improvedby wrapping all Ti:Sapphire electrical and water cooling connections with sorbothane
at locations where they came in contact with the optical breadboard. A further minor improvement was achieved by enclosing the entire laser with sound-proof foam.
These modications improved the Ti:Sapphire lock stability (only loud noises suchas hard clapping would unlock the laser). Despite these improvements, the ECDL
lock was still more robust, typically insensitive to acoustic noise and unlocking only
if something was accidentally dropped onto the laser table.
The relative laser linewidth of the ECDL and Ti:Sapphire laser was estimated bymeasuring the electrical noise spectrum of the lock signal (for details, see [80]), andthe beatnote between the two lasers. By locking the two lasers to adjacent teeth of the
11 While the nominal bandwidth of the Newport controller is the same as the Vescent Photonics
product, we found the Vescent Photonics controller to have better performance.

55

Figure 2-9: Relative linewidth measurement of the ECDL and Ti:Sapphire STIRAP
lasers. The two lasers were locked to adjacent cavity mode separated by 1.5 GHz,and by overlapping the lasers on a fast photodiode we measured the corresponding
beatnote. This measurement shows a 100 Hz full-width half maximum linewidth overa 2 s integration time.
cavity, the 1.5 GHz beatnote was monitored on a fast photodiode (see Fig. 2-9). From
the frequency jitter of the beatnote, we observed a relative laser linewidth of 100 Hz.
There are two reasons why this measurement should only be regarded as a lower boundon the relative linewidth. First, the beatnote measurement was performed with thelasers diering in frequency by only 1.5 GHz, while the locking frequency for use in
the STIRAP experiment is 6 THz, thus if the linewidth is limited by any accordiontype vibrations of the cavity, then this eect is amplied by a factor of ∼ 1000 when
the lasers are locked to their working frequencies. Second, passing the locked laser
light through an optical ber can lead to ∼ kHz scale broadening from the mechanicalvibrations in the ber [82]. In our experiment both lasers are passed through the same
single-mode polarization maintaining optical ber, but at orthogonal polarizations to
produce opposite circular polarizations for STIRAP (see Chapter 5). Since the twopolarization maintaining axes of the ber have indices of refraction typically dieringby < 1 %, we assume the residual broadening from the ber is on the order of 10 Hz.
When locking the two lasers at their working wavelengths (819 nm and 833 nm),
we could no longer perform an optical measurement of the linewidth, but rather we56were able to estimate the absolute laser linewidths to be 400 Hz from measurementsof the electrical jitter of the lock signal [80]. Thus, from a combination of opticaland electric-based linewidth measurements, we quote the relative linewidth of our
STIRAP lasers to be < 1 kHz.

2.5 Future improvements
I have discussed some of the key improvements and added subsystems necessary forthe experiments we have performed on NaLi molecule formation. Future experimentswith molecules in BEC3 are likely to become more complex, requiring reliable formation of ultracold molecules everyday. With this in mind, I believe the most benecial
future upgrades to BEC3 should focus on stability. Removing injection locks fromthe laser cooling lasers, using ber (rather than free-space optics) to deliver the MOT
beams, and improving the Na laser cooling seed-laser mode quality are just some ofmany small upgrades that would improve stability. BEC3 is nearly a two-decade oldmachine, however I am very optimistic that with continual upgrades it will continue
to deliver world-class performance.
Aside from stability, a large upgrade to BEC3 will be to once again utilize thescience chamber. The spectroscopy and production of ultracold NaLi did not require
the use of in-vacuum optomechanical devices, and the optical access was (barely)sucient. However, future experiments will likely necessitate transport of the NaLi
mixture into the science chamber. Formation and manipulation of molecules within
an optical lattice will require more optical access (even if the MOT optics are placedon ipper mirrors as in newer experiments such as BEC5, the bulky Ioe-Pritchard
coil packages are obstructive). Experiments to polarize the dipole moment of themolecules will require directional microwave horns, or even in-vacuum DC electriceld plates. It is my hope that after the initial hurdle of dual species transport tothe science chamber, the improved optical access will pay dividends when setting up
new, more complex experiments.

57

Chapter 3
Photoassociation spectroscopy
The use of STIRAP for coherent transfer to the NaLi triplet ground state requires
knowledge of both excited and ground state molecular potentials. The NaLi Feshbach molecules are in the vg = 11, Ng = 2 state of the ground triplet (a3 Σ+ ) potential[72, 54], thus in order to STIRAP to the lowest state of this potential we needed asuitable excited molecular potential of triplet character. Prior to this work, there were
no observations of any triplet excited states of NaLi. Past NaLi excited state spectroscopy experiments used hot (T ∼ 400 C) Na and Li mixtures [83, 84, 85], which contained free Na, Li atoms, and a thermal ro-vibrational distribution of NaLi molecules(although some experiments such as [84] used supersonic expansion to cool a moleculebeam down to 2 K, such a technique does not eectively cool the ro-vibrational degrees of freedom). In bi-alkali molecules the ground singlet potential is much deeper
than the triplet, hence the initial thermal distribution was entirely of singlet character, and nding transitions to excited triplet states was forbidden by selection rules.
This is particularly challenging in NaLi, which has a large vibrational energy spacing
and small spin-orbit coupling, thus singlet-triplet intercombination lines from accidental state crossings are rare. Recent experiments to nd intercombination lines to
NaLi triplet potentials were unsuccessful [85].
The limitation of a singlet character initial state for spectroscopy is readily overcome in an ultracold atoms experiment. The ultracold atoms machine gives hypernestate control over the atoms' collisional state, and in the NaLi system a triplet initial58

state can be directly accessed by forming Feshbach molecules, which are of triplet
character. Thus from the outset we had two options for excited state spectroscopy:(i) start from Feshbach molecules and look for bound-to-bound molecular transitionsor (ii) use photoassociation (PA) from an initial triplet collisional state. The useof Feshbach molecules has the advantage that bound-to-bound transition strengths
are a factor of ∼ 109 stronger than PA, giving a much stronger signal. Spectroscopyusing an initial Feshbach molecule state was done in many prior bi-alkali experimentswith success [35, 86, 87]. However, in the case of NaLi our prior measurement of the
Feshbach molecule lifetime was 1 ms, which is prohibitively short for spectroscopy,
and so we chose to use PA. PA spectroscopy has been previously used in a number ofultracold mixed species experiments [88, 89, 86, 90, 91], and is particularly useful inexperiments involving alkaline-earth species where the prospect of Feshbach molecule
formation is dicult [92, 42, 88]. As will be discussed in Chapter 5, when we experimentally revisited NaLi Feshbach molecule formation we found a much improved
lifetime of 200 ms, and in retrospect using the Feshbach molecules for spectroscopy
may have been a better option. The upshot of our misunderstanding of the Feshbach
molecule lifetime is that our PA measurements were the rst observation of PA in the
NaLi system.
In performing PA spectroscopy, we were given point-wise ab initio potential data
by Olivier Dulieu, to whom we are grateful (see Appendix C). From these ab initio
potentials we could calculate that v ∗ = 11, 12 of the c3 Σ+ potential would have thelargest Franck-Condon factors to the triplet ground state, and nding these states wasof particular interest. After our study of PA spectroscopy, we were able to calculate
improved potential ts which can be found in [58].

3.1 Experimental approach
Photoassociation (PA) is the formation of a molecule from initially unbound free
atoms using a photonic excitation (for a full review, see [93]). If a pair of colliding
Na and Li atoms is illuminated with light resonant to an excited molecular state,

59the pair can absorb a photon and form an electronically excited NaLi molecule. This
excited molecule will then rapidly decay to lower lying molecular states (Fig. 3-1).
The ultracold atoms experiment images only the free Na and Li atoms, hence a PA
event will appear as simultaneous loss of an Na and Li atom. For a bulk Na and Li
mixture, this process can be described by the rate equation:ṅNa = ṅLi = −K̄(v ∗ )InNa nLi(3.1)

where nNa , nLi are the free atom densities and I is the PA laser intensity. K̄(v ∗ ) is
the intensity normalized PA rate coecient, which depends on the PA laser detuning,
weakly depends on temperature (∝ T −1/2 ), and specic properties of the excited statev ∗ (see Sec. 3.4).

The form of eq. 3.1 gives us a rough guide for how to perform a PA spectroscopy
experiment: We want to squeeze the atoms to the highest densities possible, illuminate them with as high-intensity PA light as possible, and look for simultaneous Na
and Li loss while changing the PA laser frequency. Furthermore, we are interested
in nding PA to triplet excited molecular states, thus the initial collisional stateshould have Stot = 1. This condition was easily fullled since evaporatively coolingin the magnetic trap leaves the atoms in the stretched states, Na |F = 2, mF = 2i
and Li |F = 3/2, mF = 3/2i. The two-body collisional state has the quantum numbers |J = 1, mJ = 1i where J = N + S and N = 0 since the atoms undergo only s-wave
collisions at ultracold temperatures.
The magnetic trap frequencies were ωr,Na = 2π × 325 Hz, ωz,Na = 2π × 25 Hz forNa.

The atoms were cooled at a temperature of T = 3.7 µK which corresponds

to T /Tc = 2.7 and T /TF = 1. Forming an Na BEC was purposely avoided since itwould collapse the Na cloud to a small volume with poor spatial overlap with the
Li atoms. These trap parameters corresponded to densities of nNa = 9 × 1012 cm−3and nLi = 3 × 1012 cm−3 . It was possible to achieve higher densities by running morecurrent to the gradient coils to make a tighter magnetic trap, however three-bodyloss quickly became a limiting factor. The magnetic trap produced a small bias eld,60

Bz = 1.2 G.
The atoms were illuminated with the PA laser propagating along the long axis of
the trap, such that the beam could be focused to a small spot and produce the largest
PA intensities. Typical PA beam spot sizes had a 250 µm waist, which was purposely
larger than the typical cloud diameter of 50 µm, allowing for easier alignment1 . The
PA light was linearly polarized, which maximized the chances of observing PA byallowing for σ + and σ − selection rule transitions. The PA laser power was initiallyonly 6 mW when we used an ECDL laser, but later we were able to use a Ti:Sapphirelaser to send up to 600 mW to the atoms, allowing for much faster PA spectroscopy(see Sec. 2.4.1). When searching for a vibrational PA line, we would use the maximum
laser power (600 mW) and sweep the laser at ∼ 1 GHz over an exposure time of
3 seconds. Once a PA feature was found, ner scans were performed at much lower
laser intensities to avoid line broadening and possible Stark shifts. After PA exposure,the surviving free Na and Li atoms would be counted via absorption imaging.
It is worth comparing our method of PA spectroscopy using an ultracold mixture
to another technique accessible with our experiment: PA directly from a dual species
MOT. In such an experiment the two MOTs are illuminated with PA light and trap
loss is measured from the MOT uorescence2 [96, 97, 98, 99, 42]. We successfully
implemented such a scheme to briey study Li2 PA, and found a clear MOT-loss
signal using a lock-in amplier with ∼ 1 s integration times. The primary advantage
of such a technique is experimental simplicity and improved duty cycle, potentiallyoering a factor of 30 in search speed compared to the ∼ 30 s cycle time of ultracold
atoms production. But there are a number of disadvantages which led us to use
the ultracold mixture. First, the highest achievable MOT densities are ∼ 1010 cm−3 ,which would result in a factor of 100 lower PA rate than in an ultracold mixture.
On top of this, theoretical calculations predicted very low NaLi PA couplings due to
1 When rst looking for a PA signal, we would occasionally worry that the PA beam was too small
and misaligned with the atoms from chromatic aberrations in the imaging path which was used foralignment. Hence, we made the PA spot size slightly larger than necessary.

2 There are variations of this experiment which ionize the photoassociation products for detection(e.g. [95]), however this would need a signicant modication of our current setup which does nothave an in-vacuum microchannel-channel plate detector.

61

600

e 3Σ +
2 2S + 3 2P

d 3Π

500

2 2P + 3 2S

400

v*=51

Energy (THz)v*=11Rapid decay300

c 3Σ +
200

PA laser
b 3Π

100
5

T=4uK Scattering Wavefunction

0

2 2S + 3 2S

a 3Σ +

−5
5

10

15

20

25

30

35

40

45

Internuclear distance (a 0)

Figure 3-1: Photoassociation to NaLi triplet molecular potentials. The ground a3 Σ+and rst excited triplet potentials of NaLi are drawn. All triplet molecular potentials
connecting to an excited free Na atom are repulsive, and contain no molecular states.
In the ground potential, the free Na and Li atoms collide with an ultracold scattering
wavefunction. A photoassociation laser can excite the two atoms into an excitedmolecular state. The v ∗ = 51 state in the c3 Σ+ potential was the rst state we observedwith PA spectroscopy. The v ∗ =11 state in the c3 Σ+ potential is the intermediate state
which was used for STIRAP. After PA excitation, the excited molecular state quickly
decays to other bound states or high kinetic energy scattering states, resulting in trap
loss [94].
poor free-to-bound Franck-Condon overlap arising from the small C6 coecients in
Na and Li [100]. With these considerations, we expected the high-density ultracold
mixture to give a higher probability of nding a PA line. Another issue with using a
MOT is the background homonuclear PA lines. Heteronuclear MOT PA spectroscopy
in LiYb suered from observing much stronger Li2 PA near-dissociation, which had
to be carefully subtracted to see the heteronuclear signal [42]. In our ultracold spinpolarized Na and Li mixture, homonuclear Li2 PA is suppressed due to the fermionicnature of Li which prevents s-wave collisions at low temperatures. While Na2 PA is

62

still possible, it would be rare since the NaLi molecular potentials we are interested
in do not connect to the excited 32 P Na asymptote.

3.2 Finding a rst signal
In searching for NaLi photoassociation, we chose to use look for excited molecularstates near the Li D line asymptote. Using the excited Na asymptote was never an option, as all connecting triplet potentials were repulsive and without any bound states(Fig. 3-1). Searching for near-dissociation states gave us a smaller search window,
since the vibrational level spacing near-dissociation is only ∼ 100 GHz, as opposed to∼ 5 THz for deeply bound excited states. Furthermore, near-dissociation states typically have the strongest PA transitions because of their large size: The Franck-Condon
factor is dominated by the contribution at the turning points of the bound molecular
wavefunction (called the Condon points), and states with larger turning points would
better overlap with the large, slowly varying ultracold scattering wavefunction. The
disadvantage to searching for near-dissociation PA lines was o-resonant Rayleighscattering of the PA light to the Li atoms, which heats the ultracold sample andlimits the maximum PA exposure time.
With these ideas in mind, we began the search for a PA signal 40 GHz below theLi D1 line, where we could achieve 1 second exposure times using the full power ofan ECDL (∼ 6 mW). The raw data for the rst PA signal we observed is shown in

Fig. 3-2, which we later determined to be v ∗ = 51 of c3 Σ+ , shown in Fig. 3-1. The
clear signature of PA was the simultaneous loss of both Na and Li species under theillumination of light, which was double checked by observing no eect using an initialsample of Na or Li only. The eect of Rayleigh scattering is clearly visible at highPA frequencies, which resulted in Li number loss and partial Na loss due to collisions
between the species. A ner scan of this PA feature reveals line splitting and nersub-structure; these eects are discussed in the following section and here we only
discuss the vibrational structure.
After observing this single PA feature, we continued downward in PA frequency

63

1.5

Na atom number

Na

1.0
0.5
0.0

Li atom number

Li

1.0
0.5

Total atom number

0.0

Total

2

1

0

446.72446.73446.74446.75446.76446.77446.78PA frequency (THz)

Figure 3-2: First signal of photoassociation. The arrows point to a loss in both Na
and Li number, indicative of PA. This state would later be identied as v ∗ = 51 in thec3 Σ+ potential. The signal is visible in both Na and Li, however the total atom givesa much cleaner signal. The loss of Li due to Rayleigh scattering is clearly visible for
PA laser frequencies closer to the Li D1 line. The atom numbers are normalized totheir peak values.
looking for deeper states. Our prediction of where to search for the next state wasdetermined using the Leroy-Bernstein near-dissociation expansion, which provides asimple model for highly-excited vibrational state spacing based on the WKB approximation [101]:"
Eb = − A

∗
(vD
− v∗)

#3

1/6

C6

r
2π Γ(7/6), A ≡ 2h̄µ Γ(2/3)(3.2)

where Eb is the vibrational state binding energy, µ is the reduced mass, C6 is the van∗der Waals coecient, and vD
is a free tting parameter which can be interpreted as

the fractional vibrational quantum number at dissociation. After the observation of∗
one PA line, we used a theoretical value for C6 [102, 103] to nd the parameter vD
andobtain a prediction for the position of the following state. After nding two lines, we64

would then t their positions to eq. 3.2 to get a better prediction for the subsequentline, and thus bootstrap the PA spectroscopy.

Note that the C6 values for all

molecular potentials dissociating to the Li 22 P asymptote are the same [102, 103],
and from the Leroy-Bernstein analysis alone it was not yet possible to tell which
excited potential we were probing (either the b3 Π or c3 Σ+ potential). This would bedetermined later, when a large collection of spectroscopic lines could be compared to

ab initio calculations3 . The ab initio potentials are notoriously inaccurate for neardissociation states, and we relied primarily on the Leroy-Bernstein analysis to make
spectroscopic predictions for highly excited vibrational states.
Fig. 3-3 shows the result of tting eq. 3.2 to the binding energies of highly vi3 It is possible to determine the excited molecular potential from selection rule dierences between

Σ and Π states, or perhaps by a careful examination of the PA ne-structure. At the time, we weremore interested in nding deeper triplet PA lines for use in STIRAP.

b)
54.5
54.0

v*D

0

− 10

53.5
53.0
35

40

45

50

− 20

C6 ( 103 au)

Vibrational state binding energy (THz)

a)

− 30

− 40

30

35

40

45

50

30

v*=48

20
10

35

40

45

50

Lower fitting bound (v*)

v*

Figure 3-3: Leroy-Bernstein expansion predictions. a) Experimental PA line positions are shown with blue circles. A Leroy-Bernstein t (eq. 3.2) of the 4 highest
vibrationally excited PA lines (red outlines) is shown as a gray line. The resulting
Leroy-Bernstein t (grey line) shows worse deviation from measured values at deeper∗
vibrational states. b) Fitting parameter values (vD, C6 ) for dierent depths of tted∗
data (from v = 51 down to a lower tting bound). The ts become unphysical in∗
the gray area where vD> 54, since that would imply the existence of another boundstate. Based on the Leroy radius (see text), the most physically meaningful ts were
for the data points 48 ≤ v ∗ ≤ 51.
65

brationally excited states, which was used to predict the positions of deeper boundlevels. The tting parameters become less meaningful when tting a deeper rangeof bound states (Fig. 3-3 b). The range of validity for a Leroy-Bernstein analysis is
2 1/2
2 1/2
given by molecular states larger than the Leroy radius, RLR = 2 hrNa
i + hrLii,where r refers to the atomic radii of the Na and Li atoms [104]. In NaLi we estimate RLR ≈ 25a0 based on hydrogen wavefunctions with a simple quantum defectmodication. By calculating outer-turning points from ab initio potentials we ndthat only the states v ∗ ≥ 48 are larger than RLR , and using a Leroy-Bernstein t onthese states gives C6 = 16880 au, in agreement with theoretical predictions [102, 103].
Near-dissociation ts of deeper bound states are possible with higher order corrections to the Leroy-Bernstein formula as discussed in [105]. In our case, it was dicultto extract meaningful physical parameters with higher order corrections since the C6
contribution alone predicts deeper bound states than we observed (see Fig. 3-3 a),
and additional potential terms such as −C8 /r 8 that further deepen the potential do
not improve the t quality. Instead we simply appended eq. 3.2 with a Taylor series,∗∗−v ∗ )4 +. . ., which provided a useful tting function in search−v ∗ )3 +a4 (vD
Eb = a3 (vDof bound states but without physically meaningful parameters. As we searched towards deeper vibrational states (v ∗ < 30) we began using the ab initio potentials as
guidance rather than a near dissociation approach4 .

3.3 Spectroscopic line positions
As the spectroscopic search proceeded to deeper bound states, near-dissociation expansions became poor predictors of PA line position, and we relied on the ab initio
potentials. At this point it became very clear we were observing PA to the c3 Σ+potential5 , and we could denitively assign vibrational quantum numbers to the ob4 Finding PA line positions using ab initio potentials was done by either scaling the potentialfunctions or spline-tting the error between already found PA lines and ab initio predictions. The

exact method didn't matter, since after we found the rst ∼ 15 lines any smooth t to the data
produced a reasonable prediction.

5 Of course, we compared the near-dissociation state spectra to ab initio potentials very early inthe search process as well, and had hints that we were probing the66

3Σ potential.

served spectra. The ab initio potentials were accurate to 2 % for deeply bound states(v < 49), which corresponds to a ∼ 600 GHz prediction uncertainty. Such an uncertainty was still much too large to nd PA lines quickly (the typical single-shot sweep
width was 1 GHz exposed over 3 s for broad searches with the Ti:Sapphire laser), andnarrower search windows of 30 GHz were obtained by tting a smooth spline to theerror in the ab initio predictions.
Using this method, we were able to nd 50 out of 54 vibrational states in thec3 Σ+ potential. Their vibrational quantum number assignment was determined bycomparison to the ab initio predictions. Their rotational quantum number is N ∗ = 1,
which is determined by the selection rule ∆N = 1 for Σ ↔ Σ transitions. Fig. 3-4

shows a sample spectrum from the v ∗ = 11 PA line. The spectra were t to LorentzianAtom Number

1.0

J=1

0.5

J=0

J=2
0

-300

-200

-100

0

100

200

300

PA Frequency (MHz)Figure 3-4: A sample PA spectrum of the v ∗ = 11 excited vibrational state. Threenely split lines were observed, which were later determined to be dierent angularmomentum (J ) states of the excited molecule. The loss spectra was t to Lorentzianlineshapes.
lineshapes and the centers of the ts were used to determine the spectroscopic line
position. The linewidth of the PA resonances were determined by the Lorentzian
width for spectra taken with low PA loss, such that broadening from the nonlinear
density dependence of the PA strength (i.e.

the n2 dependence in eq. 3.1) could

be neglected. This yielded an average PA linewidth of ΓPA = 9(3) MHz for all PA
lines6 . These ts, along with their resonant PA frequencies, are shown in Fig. 3-5.
6 The PA linewidth could vary by factors of ∼ 2 for dierent excited vibrational states [106],67

All spectra displayed ne-structure on the order of 500 MHz, and the center of thevibrational structure was determined somewhat arbitrarily by visual inspection. The
absolute laser frequency measurement was accurate to ±10 MHz (see Chapter 2). The
vibrational state binding energies were determined by subtracting the laser frequency
from 446.789528 THz, which is the excitation frequency to the Li D1 line that connects
to the c3 Σ+ potential [107]. We observed no AC Stark shifts when taking PA spectrawith dierent laser intensities. From the peak tting uncertainty we estimate an
upper bound for the AC Stark shift for all lines to be < 0.1 MHz/W · cm−2 .

Certain vibrational states were not found during PA spectroscopy. v ∗ = 52, 53

were too close to the Li D line, resulting in rapid one-body heating of the sample
from the PA laser7 . v ∗ = 12 is missing because the necessary PA frequency coincides
with a water vapor absorption line, preventing the Ti:Sapphire laser from lasing.

v ∗ = 11, 12 are two of the states with strong coupling to the triplet ground state, but
having also found v ∗ = 11 we did not put in the eort to remove the humidity fromthe Ti:Sapphire cavity (with either vacuum or N2 gas) to search for v ∗ = 12. Lastly,v ∗ = 28 was not found after searching only a modest 30 GHz range around its expected
position. A possible explanation is the weak strength of this PA line (discussed in
the next section). Another possible explanation is a perturbative shift of the line due
to coupling to a dierent nearby molecular state8 .
Based on our observed PA spectra, we performed a potential t of the data which
improved upon the ab initio potentials. The results and methodology of the potential
tting can be found in [58].
however we did not have sucient data at low PA loss to measure the linewidth for each vibrational
state.

7 This in principle can be remedied by increasing the density of the atoms. The PA rate scales

linearly with density, thus at high enough densities the resonant PA rate could be faster than oresonant Rayleigh scattering.

8 The likelihood of this could be estimated from the ab initio potentials by looking for any othermolecular states that may be coincidentally nearby v

68

∗

= 28.

v

Center (THz)

53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0

Not foundNot found446.745507
446.666470
446.531930
446.329506
446.048142
445.678325
445.212270
444.642947
443.965166
443.174343
442.267134
441.240062
440.091042
438.818723
437.421235
435.898401
434.250830
432.476554
430.578101
428.555535
426.411331
424.145084
421.758990
Not found416.634052
413.897877
411.048349
408.087388
405.015460
401.834767
398.546558
395.152132
391.652683
388.049361
384.343117
380.533970
376.627938
372.618441
368.510306
Not found359.997987
355.595434
351.095576
346.498950
341.805843
337.016514
332.131063
327.149863
322.072749
316.899819
311.631057
306.266375

-400

-200

0

200

400

PA frequency (MHz)Figure 3-5: Spectroscopic frequencies and substructure for all observed states in thec3 Σ+ potential. On the left are the resonant PA laser frequencies, which correspond tozero detuning in the structure diagram. The structure diagram shows the tted datato all observed vibrational states. Since the observed PA linewidth can be broadened
by density depletion, the structure diagram shows the results with linewidths set
to ΓPA = 9 MHz, the average measured value for all vibrational states. The substructure splitting becomes larger for lower v ∗ , which is consistent with ne structureinteractions which become stronger for shorter internuclear separations.

69

3.3.1 v∗ = 11 ne structure
The v ∗ = 11 state was expected to have the highest Franck-Condon overlap with the
NaLi triplet ground state, and for this reason it would serve as the intermediate
state for STIRAP (details in Chapter 4). Thus, we needed an understanding of theobserved substructure in this state (Fig. 3-4). Experimentally, we could gure out
the quantum numbers by performing PA with dierent polarizations. Starting with a|J = 1, mJ = 1i, Ng = 0 collisional state, the excited state structure could be uniquelydetermined by PA with dierent polarizations as shown in Fig. 3-6. Using this method,

J=0

170 MHz

J=2
178 MHz
J=1

σ+

π
J=1

-1

0
mJ

1

Figure 3-6: Polarization dependent PA of the v ∗ = 11 state. The J ordering of theexcited state was determined by applying dierent polarizations of PA light. The
J = 2 state could be uniquely determined, as it is the only one which responds to σ +polarization. The J = 0 state could then be determined as it is the only one which
would not respond to π light. Note that in order to produce π polarization, the PA
beam was introduced to the atoms through a dierent beam path (the same beam
path as for the bottom imaging).
we found the excited state had the peculiar energy ordering of J = 1, 2, 0. This ispeculiar compared to the level splitting in alkali atoms, and cannot be explainedby a simple S · N type coupling. Searching the literature revealed that a similar
splitting was rst observed in O2 [108, 109], for which the triplet state is the absolute
ground state9 . This structure is explained mostly by a dipole-dipole (magnetic dipole)
9 Triplet absolute ground states are not often found in nature (NaLi is certainly not one of them),
and O2 is a rare example. It turns out this plays a strong role in certain chemical reactions which70interaction between the valence electrons, along with a smaller contribution from spinrotation coupling. The Hamiltonian for these interactions is given by




~
~
~
~
~ ·N
~
Hexc = α 3(S1 · r̂)(S2 · r̂) − S1 · S2 + β S(3.3)~1,2 are the individual electron spins, S~ =S~1 + S~2 , r̂ is the unit vector between
where Sthe two electrons, and α, β are coupling constants. Note that we have not explicitly
written out the r −3 dependence of the magnetic spin interaction, and the spatial average of this interaction term is contained in the α constant. There are two unknowns(α, β ): we observe two splittings, and solving this model yields α = 182.1(1) MHz,and β = 16.2(1) MHz, where the uncertainty is purely from measurement error.
The v ∗ = 11 NaLi ne structure is similar to O2 , however it has the opposite orderfor the J states. This dierence can be intuitively understood based on the natureof dipole-dipole interactions and the dierent types of bonding in NaLi compared toO2 . The dipole-dipole interaction is attractive in the head-to-tail conguration, andrepulsive in the side-by-side conguration. A triplet state has both electrons pointing
in the same direction, and the position of each electron is determined by the bondgeometry which is a σ bond in NaLi and a π bond in O2 . This is illustrated in Fig. 3-7,
showing that the same spin conguration can have dierent dipole-dipole interactionenergies in the NaLi and O2 molecule.
Finally, it is worth commenting on the structure of other vibrational states of
the c3 Σ+ potential and the possibility hyperne eects. It likely that vibrational
states close v ∗ = 11 have a similar J = 1, 2, 0 ne structure (Fig. 3-5), but we havenot carefully taken spectra with dierent PA polarizations for these lines and cannot
make a denitive identication of quantum numbers. Some lines (such as v ∗ = 1)
show similar three pronged structure along with ner doubling. This doubling islikely from hyperne structure, which we have thus far not considered. Note that wehave slight evidence that a similar doubling occurs in the v ∗ = 11 line, having observeda slight doublet structure in the J = 2 loss feature when we were probing PA withalso follow spin selection rules [110].

71

In both cases:
J=N+S1+S2=0

N=1

N=1
S1

Na
S1

Li

O

S2

O

S2

π-bondσ-bondFigure 3-7: Comparison of electronic structures in NaLi and O2 . Here, we graphicallyshow the electron spin and rotational angular momentum orientations which result in
the J = 0 state of both triplet NaLi and triplet O2 . The angular-momentum vectors
are drawn for intuition purposes onlythey are not well dened and J is the only goodquantum number. In NaLi, the σ bond results in repulsive dipole-dipole interactions
between the electrons, which explains why the J = 0 state has the highest energy(see Fig. 3-6). In O2 , the π bond results in attractive head-to-tail interactions, which
lowers the energy of the J = 0 state. Hence, the ordering of the J states in triplet O2
is opposite to that of triplet NaLi.

π polarization (but we did not take further measurements). Such small splittings
are consistent with hyperne eects. As a point of reference consider the atomic Li
excited state hyperne structure, which is only ∼ 5 MHz. We expect similarly smallvalues in the NaLi molecule. One may ask the question: how can we proceed withstate-selective STIRAP without a clear determination of the hyperne structure? The
details of the states involved in STIRAP will be discussed in Chapter 5, but the shortanswer is by choosing the correct polarization of light, we will rely on stretched-statetransitions which do not aect the nuclear spins. For example, the initial colliding
Na and Li atoms have aligned nuclear spins, and if we perform PA with σ + polarized
light, we excite the molecule to a |J = 2, mJ = 2i state and expect the nuclear spinstate to be the same.

72

3.4 Line strengths
The PA strength was measured for each found vibrational state in the c3 Σ+ potential.
This was done by measuring the atom loss under illumination with resonant PA light
as a function of PA pulse area. The total atom number density can be solved fromeq. 3.1,
1−R
1 + ReαIt
1+R
1 − ReαIt1=, if n ≡ nNa (0) = nLi (0),1 + n(0)K̄It

n(t)
=
n(0)



(3.4)

where It is the PA pulse area, R = nNa /nLi (0) is the density ratio, α ≡ nNa (0)(1−R)K̄ ,and in the second equality we show that the solution simplies greatly in the case ofequal Na and Li densities10 . nNa , nLi refer to average densities, which can be obtained
from the atom number and an eective volume Ve = NNa NLi /RnLi nNa dV , where

NNa , NLi are the Na and Li total atom numbers, respectively11 . This loss model canbe linearized if we assume the PA loss is small,n(t)2nNa nLi
≈1−
K̄It,
n(0)
nNa + nLi t=0(3.5)αIt  1.

Using these loss models, the PA line strength was measured by tuning the PA laser
on resonance and exposing the atoms for a variable pulse area. Fig. 3-8 shows such ameasurement for the central feature in the v ∗ = 11 line. The PA loss was t to eq. 3.4
and eq. 3.5 (for the region where n(t)/n(0) > 0.7), to extract the intensity normalized
loss coecient, K̄(v ∗ ).

The pulse area was varied by keeping the exposure time

constant and ramping up the PA intensity. This procedure is slightly counterintuitive,
since Fig. 3-8 looks like loss as a function of time, but it is not. The reason for
10 Thisresultisobtainedusing

L'Hôpital's

rule

on

the

indeterminate

term,

1−R
lim
. Equivalently, one can easily solve eq. 3.1 with nNa (0) = nLi (0).
R→1 1 − R exp (nNa (0)(1 − R)K̄It)

11 This denition accounts for dierences in density proles of the two species to give an eective

overlap volume. In this experiment, both species were not quantum-degenerate and the gravitationalsag was negligible. Thus to a good approximation nNa (~r) ∝ nLi (~r), and Ve ≈ R
refers to the number and density of either species.

73

N2
where N, nn2 dV

Total atom number (arb.)

1.2
1.0
0.8
0.6
0.4
0.2
0.0

0

50

100

150

200

250

300

PA pulse area (W/cm2 · s)Figure 3-8: v ∗ = 11 PA line strength. The total atom number loss as a function ofpulse area was measured with the PA laser on resonance to the v ∗ = 11 excited state.
The data is t to a full two-body model (eq. 3.4, red dashed line) and a linear model(eq. 3.5, green dashed line).
varying the PA intensity rather than exposure time was to sample PA loss at low
laser intensities and avoid PA saturation, which can suppress the measured loss rate[111].
Such loss measurements were taken for each observed vibrational state for thecentral feature in its substructure (i.e. the zero-detuning line as dened in Fig. 35), and the results are shown in Fig. 3-9. K(v ∗ ) were measured from the average ofthe full loss curve (eq. 3.4) and the linear model (eq. 3.5) ts. The uncertainty in
the K(v ∗ ) measurements has several contributions: (i) the variance between the twotting methods, (ii) the inhomogeneous density variation throughout the trap, and(iii) a time-dependent density drop for long (> 3 s) hold times, which was attributed
to technical heating of the sample (we suspect this was due to 60 Hz noise on the
power supplies which near-resonantly coupled to the axial trap frequencies, ωz,Na =2π ×25 Hz and ωz,Li = 2π ×50 Hz). The density increase from optical dipole trappingfrom the PA laser was considered in the analysis, but found to be negligible. In thelimit of weak PA, the PA Rabi frequency ΩPA can be obtained from the loss rate,nK̄I = Ω2PA /ΓPA where ΓPA = 2π × 9 MHz is the measured excited state linewidth.
74

10− 210− 3

10− 10

10− 4

10− 1110− 5

10− 1210− 6

10− 1310− 7

10− 1410− 8

10− 1510− 9

10− 1610− 17Ω2PA /I (MHz2/(W · cm− 2 ))K̄ (cm3 s− 1 /(W · cm− 2 ))10− 9

10− 10
0

10

20

30

40

50

Vibrational number v*
Figure 3-9: PA line strengths for all observed vibrational states. The density normalized PA loss coecients were measured for each observed vibrational line. Solid blue
points indicate measured PA strengths, while hollow red circles show the expected
strength of PA lines we were not able to nd. The data is t to Franck-Condon
factors calculated from ab initio potentials. The free-to-bound Rabi frequency, ΩPA ,was calculated from the PA loss coecients assuming an excited state linewidth of
9 MHz and density of n = 5 × 1012 cm−3 .
Fig. 3-9 shows the expected free-to-bound Rabi couplings for densities of n = 5 ×1012 cm−3 for both species.

Comparison of the PA strength measurements with theoretical values can be done
to various degrees of rst-principles rigor. A full treatment of PA theory can befound in [111, 112], with insights closer to experiments in [113, 114]. Here we outline
a simple approach, which allows us to use Franck-Condon factors from the ab initio
potentials to t the strength data12 . We begin with the PA rate coecient, which
12 This allows us to ignore the normalization and thermal averaging of the scattering wavefunctionin the free-to-bound transition, as well as the transition dipole matrix element of the molecule which
is not trivial to calculate. As will be described, these parameters are all lumped into a single tting
constant.

75

must be proportional to the square of the coupling matrix element,K̄(v ∗ ) ∝ |hψmol (v ∗ )|D(R)|ψsc (E)i|2∗

2∝ |hψmol (v )|ψsc (E)i|(3.6)

if D(R) = D0

where |ψmol (v ∗ )i is the excited molecular wavefunction, |ψsc (E)i is the ultracold twobody scattering wavefunction, and D(R) is the coupling dipole moment, which varies
as a function of internuclear distance in the molecule, R. In the second line of eq. 3.6
we've made the assumption that the dipole moment is a constant D0 , which allows
us to model K(v ∗ ) purely by the Franck-Condon factors, |hψmol (v ∗ )|ψsc (E)i|2 . Using
the ab initio potentials, we numerically calculated the free-to-bound Franck-Condon

factors and t the strength data with the proportionality factor in eq.3.6 as the only
free parameter (Fig. 3-9). This simple model produces an excellent t to the data,
being remarkably accurate over 4 orders of magnitude in the PA line strengths. Thereis a simple interpretation of the PA strengths: The Franck-Condon factors are large forlarge excited molecules near-dissociation which have good overlap to the even larger
ground state scattering wavefunction, and the overlap only decreases as the excitedstate molecules become smaller and more deeply bound. The sharp dips in the PA
strengths are from nodes in the scattering wavefunction that overlap with the outerturning point of the excited state molecule, producing a very weak Franck-Condon
factor. The Franck-Condon t deviates slightly at low vibrational quantum number(v < 6), which may be explained by our assumption of a constant dipole moment:
While the dipole moment D(R) does not vary strongly with internuclear separation(typically by a factor of ∼ 2 over all sizes of bi-alkali molecules, R = 5 − 50 a0 ), thevariation is strongest at short R, which exactly corresponds to deeply bound states[106, 115].

76

Chapter 4
Two-photon spectroscopy
Following the spectroscopy of excited triplet NaLi potentials discussed in the previous
chapter, the natural next step was to perform two-photon spectroscopy of the grounda3 Σ+ potential. The goal was to spectroscopically nd the triplet ro-vibrational
ground state (vg = 0, Ng = 0 of the a3 Σ+ potential). As with the excited triplet NaLi
potentials, the ground potential was never previously observed in conventional photon
uorescence or absorption spectroscopy. Of all the bi-alkali species, NaLi was the only
mixture without uorescence spectra of the triplet ground states [116, 117, 118, 119,
120, 121, 122, 123, 124]. The reason for non-observation of the triplet ground statesis once again the inability to prepare a triplet-character initial state in experiments
involving hot gases, and the small mixing between triplet and singlet states in NaLi[85] (see Chapter 3). Observations of the triplet potential have been made possible
with ultracold atoms experiments. The highest vibrational state of the a3 Σ+ potential
have been observed through Feshbach resonances to the vg = 10 vibrational state [71,54, 72]. To reach the next vibrational state with Feshbach spectroscopy (vg = 9) would
require elds in excess of 1 Tesla, making Feshbach spectroscopy of deeper boundstates experimentally infeasible. Our focus was two-photon spectroscopy, which was
the obvious approach given that the end goal was to perform STIRAP.
Although the lack of spectroscopic data was a challenge, spectroscopy and STIRAP to the triplet ground states of NaLi has a number of favorable features. The
triplet ground potential is very shallow (∼ 6 THz deep) and NaLi is a light molecule77with a low density of states, resulting in only 11 bound vibrational states in the triplet
ground potential. Knowing the position of the least bound state from Feshbach spectroscopy, we could use a boot-strapped approach and start near-dissociation as wehave done with the PA spectroscopy1 . Another advantage is that we can stay entirely
within the triplet manifold, allowing for a simpler choice of intermediate excited state
for the two photon spectroscopy (Fig. 4-1 a). Prior STIRAP experiments focusing on
the singlet ground state were restricted to excited molecular states with singlet-triplet
mixing (detailed discussions in [86, 125, 87]). In triplet NaLi it is much simplerwechoose an excited state which has the strongest Franck-Condon for STIRAP, i.e. withstrong overlap to the ro-vibrational triplet ground state. In fact, nding a suitableintermediate state for a potential transfer to the singlet ground state of NaLi could
be dicult. NaLi vibrational level spacings are large and the spin-orbit coupling is
the weakest among the bi-alkalis, making accidental degeneracies between a singlet
and triplet molecular states rare2 .
With our interest focused only on the triplet ro-vibrational ground state, we used

ab initio calculations to determine that the v ∗ = 11 state of c3 Σ+ was the ideal intermediate state for STIRAP. This choice of intermediate state will be discussed in
more detail in Sec. 4.2, after a brief overview of the experiment setup.

4.1 Experimental approach
Experimentally, two-photon spectroscopy was a simple modication of the PA experiment described in Chapter 3. All of the experimental parameters (temperature,density, trap geometry, etc.) are the same, except we introduce another laser beam,
known as the downleg laser, to search for bound-to-bound molecular transitions(Fig. 4-1). The entrance collisional state, an excited molecular state |v ∗ i, and the aground vibrational state |vg i make a three-level system. There are two kinds of mea-

1 Note that the ground molecular states are so small in size that Leroy-Bernstein ts are nottechnically valid. This will be discussed further in Sec. 4.3

2 Rare, but not forbidden. It is possible to use the improved potentials obtained in this thesis tolook for singlet-triplet mixed excited states, though I have not made such an eort.

78a) 16b)Energy ( × 103 cm −1)

14

3 +

c Σ

2 2P + 3 2S

v*=48, N*=1

∆ AT

v*=11, N*=1

12
10

PA

Ω downΩ downLi+Na, T=4μK

0-0.1

c 3Σ +v*=11, N=1*Ω PA

2 2S + 3 2S
a 3Σ +-0.2
vg=0, N g =0

T=3µK

-0.3
5

10

15

20

25

30

Internuclear distance (a0)a 3Σ + vg=0, N=0

Figure 4-1: Molecular potential energy level diagram for two-photon spectroscopy.
a) The relevant molecular potentials for spectroscopy of the triplet a3 Σ+ potential.
Drawn are the Na and Li ultracold collisional wavefunction, the excited intermediate wavefunctions, and the ro-vibrational triplet ground state wavefunction. b) An
example three level system formed by the collisional state, the v ∗ = 11 intermediate
state, and the ro-vibrational ground state. The downleg coupling dresses the intermediate state into two levels, resulting in an Autler-Townes splitting. This three-levelsystem is qualitatively the same for other combinations of intermediate and groundvibrational states.
surements we performed on this system: (i) dark-resonance spectroscopy, which wasmost useful in spectroscopically nding the ground vibrational levels, and (ii) AutlerTownes spectroscopy, which was most useful in characterizing the bound-to-boundtransition strengths.
Dark-resonance spectroscopy consists of keeping the PA laser on resonance witha known free-to-bound PA transition, and sweeping the downleg laser in search for a
two-photon resonance. When the downleg laser is not resonant, we observe atom loss
from the resonant PA laser coupling to |v ∗ i. However, when the downleg laser comesclose to a bound-to-bound resonance, it causes an AC Stark shift to the excited state|v ∗ i, pushing it away from PA resonance (Fig. 4-1 b). This causes a suppression of PA
which is observed as a rise in atom number at a particular downleg laser frequency,
and in turn signals that we have found a bound-to-bound transition. This is known79

as dark-resonance because the collisional state becomes dark to the PA light3 . The
binding energy of the ground molecular state is the dierence between the PA anddownleg laser frequencies on resonance. The width of the dark-resonance spectrum
is proportional to the downleg laser intensity (Width ∝ I/Γ), which is discussed in
more detail in Sec. 4.4.
An Autler-Townes spectrum is the opposite conguration, where the downleg laser
is kept on resonance and the PA laser frequency is swept. In this case the eect ofthe downleg laser is no longer a perturbative AC Stark shift, but rather a dressingof the excited state into two levels, |vg i ± |v ∗ i, which are split by the Rabi frequencyΩdown (Fig. 4-1). If Ωdown  Γ, the PA laser frequency can be swept to probe thesedressed states and nd two distinct loss features. The spacing between the two lossfeatures is exactly Ωdown (if the downleg is exactly on resonance), making for a clear
measurement of the downleg Rabi frequency.
Dark-resonance and Autler-Townes spectra are two spectroscopic methods toprobe the same physics of a three-level system. This connection is shown in Fig. 4-2,
where the steady-state optical Bloch equations for a three-level system were numerically solved for various PA and downleg coupling strengths and detunings4 [126]. The
excited molecular state undergoes rapid decay to unobserved states, thus the numerical solution for the excited state population also determines the loss lineshapes. Boththe dark-resonance and Autler-Townes spectra can be understood by taking appropriate cuts through Fig. 4-2 a. The electromagnetically induced transparency (EIT)feature of Fig. 4-2 b will be experimentally discussed in Sec. 4.5.2 and in Chapter 5with an initial state of Feshbach molecules. Note that these steady-state solutions
used a slightly dierent excited state decay model than the one in our system: Fora non-zero steady state solution to exist, the optical Bloch equations of [126] use aclosed system which has an excited state that decays back to the two ground states
3 This is the opposite spectroscopic signal compared to PA experiments, where a loss in atoms
indicated resonance.

4 The master equation in [126] has the form ρ̇ = 1 [H, ρ] − D where ρ is the density matrix,ih̄
H is the Hamiltonian, and D is the Lindblad form disspiation term.
correspond to solving for ρ when ρ̇ = 0.

80

The steady-state solutions

a)

b)
10

Ωup = 0.1ΓΩdown =5 Γ

5Downleg detuning (Γ)

Downleg detuning (Γ)

10

0

-5

-10
-10

-5

0
5
PA detuning (Γ)Ωup = 0.1ΓΩdown =1 Γ

5

0

-5

-10
-10

10

-5

5
0
PA detuning (Γ)

10Loss (arb.)
0

0.5

1

Figure 4-2: Simulated dark-resonance and Autler-Townes spectra. The excited (intermediate) state population is plotted as a function of PA and downleg detunings.
Since the molecular excited state rapidly decays to unobservable states, the excitedstate population is proportional to loss. All detunings and Rabi frequencies are inunits if Γ. a) The loss spectra for large downleg couplings. The dark-resonance feature is visible when varying the downleg detuning while the PA is on resonance. The
Autler-Townes doublet is visible when varying the PA detuning while keeping thedownleg on resonance. b) The loss spectra for small downleg couplings, showing anarrow suppression of loss on two-photon resonance corresponding to the formation ofa coherent dark state between the collisional state and ground molecular state. Thisnarrow suppression of loss is also known as electromagnetically induced transparency(EIT).
of the three level system. In our two-photon molecule spectroscopy experiment thisis not the case, since the excited molecular state typically decays to other unobserved
molecular states. This results in slight deviations between the simulation and ourobservations; for example, Fig. 4-2 b shows a slight decrease in loss for resonant PA atlarge downleg detunings, which corresponds to optical pumping into |vg i in a closed
system. This does not happen in the experiments, as resonant PA would result incomplete loss of the atomic sample at steady-state.
The majority of two-photon spectra were taken in the dark-resonance congura81

tion, in order to nd the 11 bound vibrational states. It is also possible to determine
line strengths from the dark-resonance spectra, and this is discussed in more detailin Sec. 4.4. The downleg laser was an additional Ti:Sapphire (either at ∼ 820 nm or∼ 670 nm depending on which intermediate state was used) which shared the same
optical ber and beampath as the PA laser, hence it had the same spot size (250 µm
waist). The PA laser was the Ti:Sapphire laser (used in Chapter 3) when using anintermediate state near v ∗ = 11 (833 nm), and an ECDL (Toptica DL100) when using an intermediate state near v ∗ = 48 (670 nm). The simultaneous exposure to PA
and downleg light had similar durations as in the PA spectroscopy, typically a few
hundred milliseconds, until 50 % resonant PA loss was observed. The downleg laser
was linearly polarized to allow for both σ + and σ − selection rule transitions. The
rotational selection rules for Σ ↔ Σ potential transitions are ∆N = 1, hence from a

N ∗ = 1 rotational state it is possible to access both Ng =0, 2 ground states. (As areminder, the PA light only couples to N ∗ = 1 states from an initial N = 0 s-wave
scattering wavefunction.)

4.2 Choosing a pathway to the ground state
Prior to performing two-photon spectroscopy, we used ab initio potentials to identify
potential STIRAP pathways to the triplet ground state. We required an intermediate
excited state with strong coupling to both the Feshbach molecules and to the triplet
ground states. Fig. 4-3 shows the Franck-Condon factors calculated from ab initio potentials for dierent excited states of the c3 Σ+ potential coupled to the ro-vibrational
ground state and the Feshbach state. Since the ro-vibrational ground state wavefunction is a small Gaussian-like wavepacket (see Fig. 4-1), its Franck-Condon factors to
excited states exhibit a very sharp peak. The Feshbach Franck-Condon factors showless drastic variations due to the large size of a Feshbach molecule, and look qualitatively the same as the previously studied PA strengths5 (Fig. 3-9). In a two-photon
5 This similarity makes intuitive sense, as both the Feshbach state and a collisional state wavefunction look very similar at short range.

82

Feshbach state(vg=10)Franck-Condon Factor

vg=0

Product

Excited state vibrational level, v*
Figure 4-3: Franck-Condon factors for optimal STIRAP transfer. Ab initio potentials were used to calculate bound-to-bound Franck-Condon factors from the Feshbach
molecule (orange) and the ro-vibrational triplet ground state (blue) to excited vibrational states of the c3 Σ+ potential. The two-photon Rabi coupling is proportional tothe product of these two Franck-Condon curves, and is shown in green.
process involving the Feshbach molecules, an intermediate excited state, and the rovibrational ground state, the two-photon coupling strength would scale as the product of the individual Franck-Condon couplings, Ω2two-ph ∝ |hψmol (vg = 0)|ψmol (v ∗ )i|2|hψmol (v ∗ )|ψmol (vg = 10)i|2 , which is also shown in Fig. 4-3. The two-photon couplingis dominated by the sharpness of the vg = 0 ↔ v ∗ transition strength. By inspectionof Fig. 4-3, we chose the v ∗ = 11 as the most likely candidate for STIRAP6 , and forthis reason the v ∗ = 11 state is extensively used throughout this chapter.
6 v ∗ = 12 was also a good choice, however we never found this state during PA. See Fig. 3-5.

83

4.3 Dark-resonance spectra: line positions
We found all 11 vibrational states of the a3 Σ+ potential using using dark-resonancespectroscopy, all with molecular rotational quantum number Ng = 0. We used a similar search method as with PA spectroscopy, rst looking for weakly bound states andthen moving down the potential. The v ∗ = 48 intermediate state was used to take
spectra of states vg = 9, 10, and the v ∗ = 11 state was used for spectra of vg = 0 − 9.
Fig. 4-4 shows sample dark-resonance spectra for the least bound state vg = 10, Ng = 0
and the ro-vibrational ground state vg = 0, Ng = 0. In all dark-resonance spectra we

Total atom number (arb.)a)b)

vg=10 via v*=48

vg=0 via v*=11

1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5
-0.5

0

0.5

-5

Downleg detuning (GHz)05Downleg detuning (GHz)

Figure 4-4: Sample dark-resonance spectra. The total atom number was monitored
during resonant PA excitation, and a suppression of PA is observed when a downleg
vibrational transition is found. a) A sample dark resonance spectrum for the leastbound vibrational state. b) A sample dark-resonance spectrum for the ro-vibrational
ground state. Both plots are t to the lineshape model of eq. 4.4. The center of the tdetermined the vibrational binding energy, and the width can be used to determine
bound-to-bound transition strength (Sec. 4.4).
observed only one feature without any substructure. For spectra using the lower intermediate state, v ∗ = 11, we knew the angular momentum state of the PA transition
to be J = 2, mJ = 2 (see Sec. 3.3.1), thus it made sense that only the corresponding
stretched ground state Ng = 0, J = 1, mJ = 1 is coupled by the downleg laser, resulting84

in only one dark-resonance feature7 . For measurements using the higher intermediate
state v ∗ = 48, we did not make a careful determination of the PA substructure andcannot conclusively determine why only one dark-resonance feature was observed in
the vg = 9, 10 spectra. This did not concern us, since our primary goal was to uncover
the vibrational states of the a3 Σ+ potentials and understand the vg = 0 state.
Table 4.1 gives the laser frequencies on two-photon resonance for each observed
vibrational state, as well as its binding energy. The vibrational quantum numbers vg

vg
10
9
8
7
6
5
4
3
2
1
0

PA Frequency (THz)
446.32951
446.32951
359.99798
359.99798
359.99798
359.99798
359.99798
359.99798
359.99798
359.99798
359.99798

Downleg Frequency(THz)
446.33886
446.40187
360.23086
360.51511
360.93362
361.48861
362.17836
363.00098
363.95290
365.03180
366.23614

Binding Energy (GHz)
9.35(1)72.36(1)232.88(4)517.13(3)935.64(3)1490.63(3)2180.38(6)3003.00(4)3954.92(6)5033.82(3)6238.16(1)Table 4.1: NaLi a3 Σ+ vibrational state binding energies. The uncertainty stems from
tting dark-resonance features. In the case of vg = 10, 9, 0, Autler-Townes spectra(see text) were used in binding energy determination. Autler-Townes spectra produce
much narrower features, thus the uncertainty is limited by the wavemeter to 10 MHz.
were assigned based on comparisons with the ab initio potential, and all of our binding energy observations were within 3 % of the theoretical predictions [85]. We also
searched below the ro-vibrational ground state without nding any dark-resonance,conrming that we observed the vibrational ground state8 , vg = 0. The only priormeasurement of the a3 Σ+ potential was with Feshbach spectroscopy, which observed
the least bound state at a binding energy of 9.35335(50) GHz, in agreement with
our observation [72]. Selection rules on the downleg transitions allowed for angular
7 This is only approximately correct. I mentioned earlier that both PA and downleg lasers have+ σ − ) polarizations, hence the PA should also couple to the non-stretched J = 2, mJ = 0−−
state with σpolarization. However, in our measurements we found the σtransition to be muchlinear (σ+weaker than the stretched state transition (which is consistent with Clebsch-Gordon coecients),
thus I ignore this eect here.

8 This search was 200 GHz wide, centered around the extrapolated line position based on a85

Prediction Error, Emeas-Ecalc (GHz)10
Ng=0 position if observation is Ng=20

Observed position

Ng=0

− 10
− 20
− 30
− 40
− 50
− 60
− 70

Ng=0
0

2

4

6

Vibrational number vg

8

10Figure 4-5: Comparison of a3 Σ+ vibrational binding energies to theoretical calculations from the ab initio potential. Plotted is the dierence between measured binding
energy and theoretical binding energy, Emeas − Ecalc , showing that the theoretical predictions generally predicted deeper bound states compared to the observations. Solidblack circles show the observed line positions. The rotational quantum number Ng = 0
is certain for vg = 0, 10. Hollow red circles show the hypothetical position of the Ng = 0
rotational state if one of the observations was a Ng = 2 state (the crossed out data
points for vg = 0, 10 indicate they cannot be true). Based on the smoothness of theobserved data (solid black), we conclude that all of our observations are Ng = 0 states.
(Consider the contrary by swapping a hypothetical and observed for a given vg thiswould result in a signicant kink in the resulting black curve.) The rotational splittings here are calculated from the ab initio potential by adding an Ng = 2 centrifugal
barrier to the potential.
momentum states of Ng = 0, 2. In NaLi, the ∆Ng = 2 rotational splitting is ∼ 0.4 % of

the binding energy, which is much too small to directly rely on ab initio potentials forquantum number assignment. Instead, Ng was determined as follows: for the leastbound state vg = 10, we knew Ng = 0 by comparison with Feshbach resonance spectra[72]. For the vibrational ground state, vg = 0, we explicitly found the Ng = 2 stateusing dark-resonance spectra above the Ng = 0 state (discussed in the following section). For all other vibrational states in-between (vg = 1 − 9), we inferred that Ng = 0

by comparing their binding energy to the ab initio prediction, and not observing any
3+harmonic-oscillator t of the bottom of the a Σpotential, i.e. at ∼ 7.4 THz binding energy.

86discontinuous jumps in the prediction error (Fig. 4-5).
Using the measured binding energies of these vibrational states, we constructed
an improved potential curve using a full tting procedure, described in [59]. Unlike
the potential tting of the excited c3 Σ+ potential in Chapter 3, the Leroy-Bernsteinnear-dissociation ts are not a valid model for the states in the a3 Σ+ potential because
all of the vibrational states are too tightly bound. The Leroy radius gives a shortrange length scale beyond which near-dissociation expansions become invalid (seeChapter 3). For NaLi, the Leroy radius is 25a0 , and the only state larger than thisin the a3 Σ+ potential is the least-bound state with an outer turning point of 27a0 .
Even though there aren't enough states for a proper near-dissociation t, the LeroyBernstein formula (eq. 3.2) was nonetheless useful as a tting function during initialstages of two-photon spectroscopy, however the resulting values of vD and C6 were
unphysical9 .

4.3.1 Rotational splittings
We performed dark-resonance spectroscopy on the lowest vibrational state in thea3 Σ+ potential to explicitly observe the rotational ground state (Ng = 0) and the rotationally excited Ng = 2 state. Fig. 4-6 shows the dark-resonance spectrum of therotating molecule. As opposed to the rotational ground state dark-resonance spectrum, multiple features spanning 200 MHz were observed. This is to be expectedfrom the extra multiplicity in an Ng = 2 rotational state, which is likely complicatedby S · N and S1 · S2 coupling in a triplet molecule as observed in the excited states(Sec. 3.3.1). We attempted to identify the quantum numbers in the ground state byusing dierent rotational states of the v ∗ = 11 intermediate state, and observing theresulting observed ground states in accordance with selection rules. The results arethus far inconclusive10 . The data is presented as-is for the benet of future workwith NaLi molecules, which may require a thorough understanding of the rotational
9 It's likely we could have used any tting function with reasonable success, but it was nice to use

a function with some physical reasoning behind it. For example, in Fig. 4-5 the ab initio prediction
error was on occasion t to a power series to predict positions of neighboring vibrational levels.

10 There is no fundamental reason for thissimply further work is required.

87

3
Atom number (arb)

J=0

PA: σDownleg: σ+

2

1

0
− 100

0

100

J=1

PA: σDownleg: σ+

J=2

PA: linear
Downleg: linear

J=2

PA: σ+
Downleg: σ-

200

300

Binding Energy Detuning (MHz)Figure 4-6: Dark-resonance spectrum of rotationally excited NaLi, vg = 0, Ng = 2.
Dierent angular momentum (J ) states of the v ∗ = 11 intermediate state (see Fig. 36) were used in attempting to determine the quantum numbers of the Ng = 2 groundstate. Dierent laser polarizations were used; linear refers to σ + + σ − polarization. The spectra show only the small step-size ne scans, and in the bottom three
spectra no structure was observed where there are no datapoints (i.e. at detunings> 100 MHz). Zero detuning corresponds to 27.7 GHz above the Ng = 0 state, andhigher detuning means lower molecule binding energy.
splittings to polarize the molecule dipole moment using a resonant microwave eld
[127, 128, 129].
Even without a clear understanding of the ne structure in the vg = 0, Ng = 2
state, we can estimate the rotational constant of the molecule. The splitting between
the Ng = 0 and Ng =2 state is measured to be 27.7(1) GHz, in agreement with the

ab initio prediction of 27.74 GHz.

The error in our measurement stems from the

spread of the ne structure in the Ng = 2 spectra and the inability to properly denea center-of-mass position without knowing the quantum numbers involved.

This

splitting can be used to determine the rotational constant in a simple rigid rotor
model, Erot = BN (N + 1), from which we obtain B = 4.63(2) GHz for this vg = 0
state.
88

4.4 Line strengths
4.4.1 Autler-Townes spectra
We have so far discussed the spectroscopic positions of the two-photon dark-resonancespectra. Fits of the dark-resonance lineshapes can also be used to measure the boundto-bound downleg transition strengths. Before moving forward with careful analysisof the dark-resonance lineshapes, it is worth considering the other spectroscopic conguration possible in a three-level system: observation of the Autler-Townes splitting by holding the downleg laser on resonance and varying the PA laser frequency(Fig. 4-1 b). When the downleg laser is resonant, the Autler-Townes splitting is exactly the downleg Rabi frequency. This makes for a very straight-forward methodof measuring bound-to-bound coupling strengths. Furthermore, the limiting case ofAutler-Townes spectroscopy with a weak downleg laser can result in EIT, which wewill discuss in the next section.
Fig. 4-7 shows a sample Autler-Townes spectrum of the ro-vibrational ground statefor various powers of the downleg laser. The downleg coupling splits the original PAb)3Ω down (×102 MHz)a)
Ntotal (arb.)

32 W/cm2

2

6 W/cm2

1

No downleg

− 100

0

2

1

0

100

0

100

200

Downleg intensity (W/cm2)

PA detuning (MHz)Figure 4-7: Autler-Townes spectrum of the ro-vibrational ground state. a) Shown
are PA spectra of the central feature in the v ∗ = 11 vibrational state, using dierent
intensities of downleg light resonant to the vg = 0 ↔ v ∗ = 11 transition. The presenceof downleg coupling splits the PA feature into two loss features separated by the
Rabi coupling Ωdown . b) The Autler-Townes splitting as a function of downleg laser√
intensity, tted to Ωdown ∝ Idown , where Idown is the intensity of the downleg laser.

89

feature into two, corresponding to the two diagonalized states of the bound-to-boundsystem |v ∗ i ± |vg i. The states are split by the downleg Rabi frequency, which is proportional to the square root of the downleg laser intensity, Ωdown ∝√Idown as shownin Fig. 4-7 b. The Rabi splitting achieved in the vg = 0 ↔ v ∗ = 11 transition are on
the scale of hundreds of MHz with only a few mW of laser power. This large splitting
is to be expected, as we've specically chosen the v ∗ = 11 intermediate state for its
high Franck-Condon overlap to the ro-vibrational ground state. Such Autler-Townes
spectra were taken only for the vg = 0, 9, 10 states. Aside from measuring the downleg

Rabi frequency, they also allowed for a more precise determination of the vibrational
state binding energies. Any downleg detuning would show up in the Autler-Townes
splitting as a generalized Rabi frequency, Ωgen =pΩ2down + ∆2down and also cause aslight asymmetry in the Autler-Townes splitting relative to the original single PA feature (Fig. 4-7 a shows the resonant symmetric splitting, but o-resonance the AutlerTownes peaks simply move left or right relative to the original PA feature, as canbe seen in the simulated spectra of Fig. 4-2). By changing the downleg frequency
until this asymmetry disappeared, we could more precisely determine the vibrationalbinding energies compared to the broad dark-resonance spectra. This precision isreected in the uncertainties in Table 4.1 for vg = 0, 9, 10, which is limited only by the
wavemeter performance11 to ± 10 MHz. The reason Autler-Townes spectra were more
precise than dark-resonance spectra in determining the molecular binding energies ispurely technical. The dark-resonance spectra were taken with the highest possible
downleg intensity (large Ωdown ) to maximize the probability of nding a ground vibrational state, hence the dark-resonance lineshapes were very broad (∼ Ω2down /Γ) incomparison to the two Autler-Townes features with widths of Γ. In principle the samelevel of precision could be achieved with dark-resonance spectra if a smaller downleg
coupling was used (as in the narrow feature of Fig. 4-2 b).
11 As will be discussed in the next section, an Autler-Townes spectrum with very small downleg

coupling can yield kHz level resolution of the ground state, however this is only relative precision,
and we are still reliant on the wavemeter when reporting absolute values.

904.4.2 Dark-resonance lineshapes
While Autler-Townes spectra provide an elegant perspective into the physics of threelevel systems, such spectra can only be studied when all of the resonant transition
frequencies are known. Since our primary work was to discover these unknown transition frequencies, most of the collected data was in the dark-resonance conguration12 .
We now turn our attention to determining Ωdown from a dark-resonance spectrum.
In this case, the PA laser provides a loss mechanism which is suppressed by the AC
Stark shift in the vg ↔ v ∗ transition. The loss induced by PA has been discussed inSec. 3.4 ; with balanced Na and Li densities and weak PA loss, we use the linearized
PA loss model, eq. 3.5, to get:n(t)≈ 1 − n(0)Kt,
n(0)

(4.1)

where K ≡ K̄I is the density normalized loss coecient, and n(t) = nNa = nLi is theatom number density. The PA loss coecient is given by [111]

K∝

ΓΩPA
1
≈
2
∆2PA + [(ΩPA + Γ) /2]∆2PA + (Γ/2)2(4.2)

where ΩPA , ∆PA are the PA Rabi frequency and detuning, respectively, and Γ is theexcited state linewidth (measured in Chapter 3 to be Γ = 2π×9 MHz). We have madethe approximation that the PA Rabi frequency is well below saturation (ΩPA  Γ),which is justied by the PA line strength results of Sec. 3.4, where typically ΩPA ≈ 2π×

1 kHz13 . The PA is tuned on-resonance making ∆PA zero. However, we now consider
a downleg coupling which causes an AC Stark shift to the PA transition,

1
∆PA =
2

q

2

2



∆down + Ωdown − ∆down≈Ω2down4∆down(4.3)

12 Taking Autler-Townes spectra for all vibrational lines was a time-consuming possibility, but our
focus was on studying the ground state and we chose to move on.

13 One doesn't need the exact results of Sec. 3.4 to see this.

predicts a loss rate of

2π × 1 kHz.

Fermi's golden rule applied to PAΩ2PA /Γ, and typical experiments show loss in ∼ 100 ms, yielding ΩPA =91

where Ωdown , ∆down are the downleg Rabi frequency and detuning, respectively. In thesecond step we've assumed Ωdown  ∆down ; even though our downleg Rabi frequencies
are quite large (typically 100 MHz), the range of the dark-resonance spectra is evenlarger (a few GHz, as in Fig. 4-4), which makes this assumption valid14 . Putting together eq. 4.3, eq. 4.2 and eq. 4.1, we get the dark-resonance lineshape tting function

N (t)n(0)Kt
=1−
2
2
N (0)
1 + (Ωdown /(2∆down Γ))
A
= 2
2 +B
∆down + (Ω2down /(2Γ))(4.4)

where N (t) is the remaining total atom number and A, B are tting parameters15 .
Thus, the dark-resonance lineshape is a Lorentzian with width Ω2down /Γ. The darkresonance spectra for each vibrational state was t to eq. 4.4 to determine Ωdown ,as shown for vg = 0, 10 in Fig. 4-4. The bound-to-bound transition strengths for all
vibrational states obtained using this dark-resonance tting procedure as well as thesimpler Autler-Townes spectra are shown in Fig. 4-8. The dark-resonance spectra
shows good agreement with the Autler-Townes results, verifying the validity of ourtting model.
The strength data is compared to theoretical Franck-Condon ts (the same model
as eq. 3.6 in Sec. 3.4) showing good agreement. As expected, using the v ∗ = 11 intermediate state yields the strongest coupling to the ro-vibrational ground state, since
the outer turning point of v ∗ = 11 is at the same internuclear separation as the vg = 0
ground state. Interestingly, the coupling strength to the next state, vg = 1, is 100 times
weaker. This eect has a very satisfying explanation: the vg = 1 state wavefunction isanti-symmetric with a single-node, therefore producing a very small overlap integral
with the outer-turning point of the excited v ∗ = 11 state. Another interesting featurein the strength data is the sharp drop in Franck-Condon factor for the vg = 8 state
14 The purpose of this assumption is to have a simpler expression in eq. 4.4. Numerically, the exact
form of eq. 4.3 doesn't signicantly change the tting results.

15 More precisely in eq. 4.4, A = n(0)KtΩ2down2Γ2eters oat.

92

, and B = 1 − n(0)Kt, but we let these param-Ω2down /I (MHz2/(W/cm2))

10 3

10 2

10 1

10 0
0

2

4

6

8

10

3 +

a Σ vibrational state, vg
Figure 4-8: Bound-to-bound line strengths between vibrational states in a3 Σ+ andc3 Σ+ . Hollow circles (triangles) show the coupling strengths measured from darkresonance spectra with the v ∗ = 11 (v ∗ = 48) intermediate state by tting with eq. 4.4.
Solid circles (triangles) show line strengths measured by Autler-Townes spectroscopy.
The dashed lines are ts from ab initio calculations of Franck-Condon factors. Notethat the ab initio calculations are drawn as continuous lines for visual clarity, andthey do not have any meaning for non-integer vg .
using the v ∗ = 48 intermediate state, which explains why our attempts to observe thevg = 8 state using the ∼ 670 nm transition failed, necessitating the use of the deeper
bound v ∗ = 11 intermediate state.

The line strength data from both dark-resonance and Autler-Townes spectra conrmed our hypothesis that the v ∗ = 11 state of the c3 Σ+ potential was a good candidate for STIRAP. Moving forward, we knew that using a weak (mW) downleg laser
and a strong upleg16 laser (300 mW) we could produce Rabi frequencies on the orderof 10 MHz, which is more than sucient for STIRAP transfer within a few dozen µs(see Sec. 2.4.2 for STIRAP pulse time considerations).
16 No longer called the PA laser, since the Feshbach-to-(v ∗ = 11) transition is bound-to-bound.

934.5 Alternative paths to the triplet ground stateWithin this chapter we have charted the course to the NaLi triplet ground state. In
the next chapter, we apply this knowledge to coherently transfer Feshbach molecules
into the ro-vibrational ground state, a technique which has become a sort of industry
standard in ultracold molecules [35, 34]. But before moving forward, it is worth
considering other methods of reaching the triplet ground state using our new foundknowledge of NaLi triplet potentials and transition strengths.

4.5.1 Pumping
One possible method of forming triplet ground state NaLi is by optical pumping, i.e.
using a photoassociation excitation followed by uncontrolled spontaneous decay, withthe hope that some fraction decays into the ro-vibrational ground state. Such schemes
have been successfully used in dual-species MOTs to form singlet ground state RbCs
[124, 130], LiCs [131], NaCs [132], KRb [133] molecules and triplet RbCs [134] withformation rates of ∼ 104 molecules · s−1 at temperatures of ∼ 100 µK. A unique variantof these experiments was the use of PA-pumping to access a highly excited vibrational
state followed by STIRAP to the ro-vibrational ground state [135], thus avoiding theuse of a Feshbach resonance. In all cases the ground state molecules were detectedusing resonantly enhanced two-photon ionization (REMPI) with a microchannel plate
detector. These experiments oer the simplicity of working directly out of a MOT(no evaporative cooling, no Feshbach molecules) and the use of only one molecule
formation laser (the PA beam). Their disadvantage is cleanliness: all ground state
molecules are produced in an untrapped state surrounded by a cloud of atoms in
the MOT. This makes it dicult to isolate a pure sample of single-state molecules,
and hence the experiments typically report steady-state formation rates rather thantrapped molecule number or lifetimes (since both of those quantities are very small).
In any case, it is intriguing to estimate the performance of triplet NaLi in such an
experiment. Due to the light mass of NaLi and the shallow nature of the triplet a3 Σ+
potential, there are only 11 triplet vibrational states. Thus a spontaneous decay from94

a randomly chosen excited molecular state has a ∼ 10 % chance of falling into thevg = 0 state we're interested in. Of course, we can be more precise given the content
of this chapter: careful inspection of Fig. 4-8 shows a vibrational branching ratio ofb = 50 % from the v ∗ = 11 excited molecular state to vg = 0. The total ground state
molecule formation rate using PA excitation can be estimated fromR(vg = 0) = bn2 IV K̄(v ∗ = 11),(4.5)

where R is the molecule formation rate, b = 0.5 is the downleg branching ratio, n
is the MOT density17 , I is the laser intensity, V is the volume over which the laser
illuminates the MOT, and K̄(v ∗ = 11) is the PA rate coecient which is given in Fig. 39. One must remember the temperature dependence of PA, K̄ ∝ T −1/2 , and that thedata in Fig. 3-9 was taken at T = 4 µK. Assuming a MOT density of n = 1010 cm−3 ,T = 100 µK, a 1 W PA laser with a 0.5 cm diameter MOT18 , we nd the ground state
production rate to be R(vg = 0) = 5×104 s−1 , on par with previous experiments usinga MOT.
We can also estimate the formation rate if we employed the same pumping scheme
but with an initial near-quantum degenerate Na and Li mixture used throughout thisthesis. In this case the total atom number is N = 107 , the density is n = 5×1012 cm−3 ,T = 4 µK, and we suppose a 1 W PA beam illuminating a cigar shaped cloud witha 30 µm waist and 0.5 mm length. This yields a formation rate of R(vg = 0) = 2 ×

108 s−1 , which would deplete the atomic sample in 20 ms to form 5 × 106 molecules.
While this seems promising, it does not solve the problem of creating an impure
molecular sample. Indeed, if we consider the typical universal loss rates (see Chapter 5

for details) for collisions of NaLi molecules with Na or Li to be Kloss ≈ 5×10−10 cm3 /s,
the molecule lifetime would be ∼ 3 ms, which poses an obvious problem for such aformation scheme. This illustrates the importance of having a clean molecular sample,
and highlights why coherent, state-selective techniques such as Feshbach formation
17 There are two MOTs involved, hence an overlap density would be required for a more exact
calculation.

18 The laser beam size drops out of the equation, since IV = P L where P is the laser power and

L is the length of the MOT.

95

followed by STIRAP are desired. Nonetheless, the discussion presented here serves asa starting point in considering new molecule formation schemes in which the problem
of forming a clean molecular sample needs to be addressed.

4.5.2 Free-to-bound STIRAP
Chapter 5 will discuss in detail the use of Feshbach molecule formation and STIRAP
to reach the NaLi triplet ro-vibrational ground state. Here, we briey take the time toquestion one step of this process: is Feshbach molecule formation necessary? Feshbachformation provides a crucial coupling between free atoms and a molecular boundstate, but why not simply use photoassociation? The short answer is that Feshbach
molecule couplings and lifetimes are much more favorable compared to PA (Table 4.2).
Comparing the free-to-bound coupling strength to the molecular state linewidth Ω/γ

Coupling Strength (Ω)

Feshbach formation∼ 1 MHz typical (e.g. [136])
10 kHz in NaLi [54]Photoassociation≤ 100 kHz(See Fig. 3-9)

Molecular State Lifetime (γ −1 )

1 ms

10 ns

Ω
γ

103 (10 in NaLi)

0.01

Table 4.2: Comparison of Feshbach resonances to PA. The Feshbach state lifetime
is taken to be a typical value in a molecular sample containing other atoms, whichis the kind of environment in which free-to-bound formation would be performed.
Isolated Feshbach molecules have signicantly longer lifetimes in accordance with theuniversal loss rate [43].
makes it clear that coherent formation is typically easier using a Feshbach resonance(here, Ω is the free-to-bound coupling matrix element and γ is a limiting decoherence
linewidth). However, the technique of STIRAP transfers population between groundstates of a three level system while avoiding the short-lived excited state. Thus therelevant decoherence timescale is no longer limited by the short spontaneous decayof an electronically excited molecular state, but is likely much longer, γ −1  10 ns.

96Determining the decoherence limit γ is not an obvious procedure19 , but it makes
plausible the prospect of coherent control (Ω/γ > 1) using PA in a STIRAP scheme.
Eliminating the use of Feshbach resonances using such a PA-STIRAP scheme
would greatly simplify many experiments. Finding strong Feshbach resonances formolecular formation among dierent atomic species is a bit of a lottery: homonuclear
6

Li exhibits a 300 G wide resonance around which 100 % molecule formation is pos-sible [137].

23

Na40 K has a 30 G resonance with 15 % molecule formation eciency[136]. NaLi had the unfortunate luck-of-the-draw to have 2 mG wide resonances, withformation eciencies of only 5 % (the narrowest resonances used to make molecules todate [54]). Alkaline-earth species have no Feshbach resonances in their ground state20 ,and Feshbach resonances using a meta-stable atomic state exhibit loss too rapid forany practical use [92]. A PA-STIRAP scheme gives freedom to choose an excitedintermediate state, which could allow PA-STIRAP to be more widely applicable toother atomic species compared to Feshbach association.
This experimental simplicity of the PA-STIRAP scheme was appealing, despite theweak coupling strengths of PA. Even if poor molecule formation eciencies of 0.1 %were achievable, this would correspond to the formation of 104 molecules, on par with
Feshbach experiments. Recently such a PA-STIRAP scheme was demonstrated in a
BEC of Sr, where the high densities in a BEC allowed for signicantly improved Rabi
frequencies (ΩPA = 2π × 3 MHz) and a Sr2 molecule formation eciency of 9 % [138].
Another method to improve strength of the PA coupling is to produce very tightconnement for two atoms using an optical lattice, forcing the free atoms to within asmall volume and increasing the free-to-bound Franck-Condon factor [139].
With all these considerations in mind, we decided to attempt a PA-STIRAP transfer to vg = 0 using the v ∗ = 11 intermediate state. By this time we had already constructed the ULE cavity setup that locked the PA and downleg lasers to < 1 kHz
19 Identifying the source of decoherence in a one-photon PA process is simple, since spontaneous
decay is orders of magnitude faster than any other process. In a two-photon STIRAP process things
become more complicated, as ground state molecular lifetimes and other decoherence processes suchas magnetic-eld jitter, sample temperature, laser linewidth, etc. all have similar timescales on theorder of kHz.

20 This is not from any kind of lottery: Their ground state is a singlet 1 S

no magnetic tunability.

97

0 state, which exhibitsrelative linewidth (see Sec. 2.4.1). With this high degree of coherence between the
two lasers, the Autler-Townes spectra (Fig. 4-7) could taken with very low downleg couplings, Ωdown < Γ (here, Γ is the excited state linewidth, 9 MHz as measured
with PA spectroscopy), resulting in electromagnetically induced transparency (EIT)[140]. Fig. 4-9 shows one such EIT spectrum using PA. The narrow feature indicatesthe formation of a coherent superposition of the triplet ro-vibrational ground state
and the collisional scattering wavefunction, cos θ|ψsc i − sin θ|ψmol i where the mixing
angle is given by tan θ = ΩPA /Ωdown (see Sec. III D of [140]). Assuming innitely

Total atom number (arb.)

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3

−15

−10

−5

0

5

10

15

PA detuning (MHz)Figure 4-9: Free-to-bound electromagnetically induced transparency. A PA spectrumof v ∗ = 11 using a PA power of 300 mW (500 W/cm2 ) and downleg power of 5 µW
(0.01 W/cm2 ). A sharp EIT feature with 200 kHz FWHM is observed. Note thefeature is o-center from the PA loss minimum simply because the downleg laser was
not exactly on resonance in this particular dataset. In this way we could determine
relative detunings in our system down to the kHz level.
narrow lasers and a long ground state molecule lifetime, the width of the EIT feature is given by (Ω2down + Ω2PA )/4Γ [141]. In Fig. 4-9, the couplings are estimated tobe Ωdown = 2π × 2 MHz (from measurements in Fig. 4-8), and ΩPA = 10 kHz (frommeasurements in Fig. 3-9), resulting in an expected EIT width of 100 kHz, roughly
consistent with the measured width of 200 kHz in Fig. 4-9. This implies the forma98

tion of a coherent state, |ψsc i − 0.005|ψmol i, which corresponds to a population ofNmol = Natoms · (0.005)2 = 250 ground state molecules.

Eorts to increase the molecular fraction were not successful. 300 mW was the
maximum PA power available which limited the PA Rabi frequency to 10 kHz, andfurther reducing the downleg coupling only resulted in the disappearance of the EIT
feature. Attempts at forward and reverse STIRAP pulses (discussed in more detailin Sec. 5.4) using a wide range of pulse durations and powers did not show any evidence of molecule formation. With a maximum PA coupling of 10 kHz, we expected
to use STIRAP pulses longer than 1 ms for adiabatic transfer, although we also attempted shorter pulses where non-adiabatic ineciencies in the STIRAP process were
expected. It is possible that an unidentied source of decoherence limited the STIRAP pulse length; as will be discussed in Chapter 5, the STIRAP pulse lengths using
Feshbach molecules were limited to at most 200 µs for reasons we have not fully understood. Another important consideration in PA-STIRAP is that the reverse STIRAP
transfer from a molecular state to free atoms follows a Wigner threshold law from thecontinuous density of collisional states, and thus the reverse STIRAP pulse necessary
for molecule detection is not 100 % ecient [138]. This is signicantly dierent inmolecules formed using a Feshbach resonance, which can be dissociated to free atoms
with perfect eciency because the dissociation process is a non-coherent, sponta-

neous process. Thus it is possible that 1 % molecule formation with PA-STIRAP isundetectable due to a compounded 1 % (or worse) reverse-transfer eciency.
With all of these open questions regarding PA-STIRAP, we chose to move forwardwith Feshbach formation followed by STIRAP. In this way, we hoped to learn how to
perform STIRAP in a bound-to-bound system, and perhaps in the future come backto the less intuitive case of free-to-bound STIRAP.

99

Chapter 5
Formation of triplet ground state
molecules
In the last two chapters, we have discussed experiments to spectroscopically identify
a two-photon pathway from Feshbach molecules to the ro-vibrational triplet groundstate of NaLi. The Feshbach molecules themselves were an ongoing project when I
joined the Ketterle group in 2010, with successful formation in 2012 [54]. In thischapter, all of these pieces are put together to make a pure sample of ultracold triplet
ground state NaLi molecules. Fig. 5-1 shows the relevant molecular potentials andstates used in coherently forming the molecules. Sec. 5.2 describes the formationof weakly bound Feshbach molecules, where we discovered a longer-than-expectedFeshbach molecule lifetime. These molecules were then coupled to the ro-vibrational
ground state with two-photons: an upleg laser (replacing the PA transition of thelast two chapters) up to the v ∗ = 11 intermediate state, and a downleg laser down
to the ro-vibrational ground state.

The upleg and downleg laser frequencies are

discussed in Sec. 5.3, where the spectroscopic results of Chapter 2 and 3 are applied
to the specic case of Feshbach molecules at high magnetic elds. In Sec. 5.4 we
describe the transfer and detection of molecules in the triplet ro-vibrational groundstate.
Sec. 5.5 will delve into studying properties of the new molecular system we've
created. Triplet ground state NaLi molecules are fermionic with a non-zero magnetic
100

16
14

2 2P + 3 2S

c 3Σ +

12

−1

)

10

3
Energy ( × 10cm

8

b3 Π

6

Ω up
833 nmΩ down
819 nm0-0.1

2 2S + 3 2S

a 3Σ +-0.2-0.3

X 1Σ +

-0.4
5

10

15

20

25

Internuclear distance (a0)Figure 5-1: Molecular potentials and transitions for formation of NaLi triplet groundstate molecules. The a3 Σ+ ground potential and c3 Σ+ excited potential are shown inred. The wavefunctions of Feshbach molecules, the electronically excited intermediate,
and the triplet ground state are drawn, along with the laser transitions employed for
STIRAP.
moment of 2 µB and an electric dipole moment of 0.2 D [23, 44]. Since the triplet
NaLi is both reactive with itself and not in the absolute ground state, the lifetime ofa molecular sample will be of particular interest: Sec. 5.5.1 and 5.5.2 will describe thedetails of trapping the molecules on the timescale of several seconds, and compare ourresults to a universal collision model. Lastly, Sec. 5.5.3 will demonstrate the novel
feature of a magnetic dipole moment in an ultracold molecule by performing electron
spin resonance spectroscopy (ESR).

5.1 Experimental approach
In contrast to the PA and two-photon spectroscopy, the experiments presented here
were performed in a single beam optical dipole trap (ODT). This is necessary because
101Feshbach resonances exist only between non-magnetically trappable states of atomic
Na and Li, necessitating the use of optical connement. The starting point of all
experiments performed in this chapter are an ultracold Na and Li mixture in theirlowest Zeeman sub-levels1 , conned in an ODT at a bias eld of Bz = 745.55 G (seeFig. 2-2).

The cloud has a cigar-shaped geometry similar to that of experiments

performed in the magnetic trap (Fig. 5-2). The radial connement is produced byHigh field resonant
clean / Imaging beamσ- 819 nm

yxz

1064nm / 1319nm
ODT

Bz

σ+ 833 nm

Figure 5-2: Optical dipole trap geometry at high magnetic elds. The dual species
Na and Li mixture was held in a single beam ODT formed by a far o-resonant laser
at 1064 nm or 1319 nm. The radial connement was produced by the ODT, and axial
connement was produced by both the ODT and the magnetic eld curvature of the
Feshbach coils (producing a eld maximum). The STIRAP beams were co-axial to
the ODT beam. A transverse two-color beam resonant with free Na and Li atoms wasused for both imaging the atoms and removing them from the trap when isolation ofmolecules was desired.
the ODT, with a trapping frequency of ωr, Na = 500 Hz for Na. The axial connement
is primarily from the Feshbach coils which produce a eld maximum to conne thehigh-eld seeking Na and Li |1i states, with a trapping frequency of ωz, Na = 13 Hz.

Note that all of the molecules we discuss in this chapter are low eld seeking, and areaxially anti-trapped by the magnetic eld. This will be particularly important when−1attempting to measure molecular lifetimes beyond timescales of ωz,
Na (Sec. 5.5.1).

Typical atom numbers were 1 × 106 of each Na and Li at temperatures of 2 µK, withdensities of ∼ 3 × 1012 cm−3 for both species. The corresponding peak densities werenNa = 4 × 1012 cm−3 and nLi = 3 × 1012 cm−3 .

Another important distinction in this experiment compared to those discussed
1 I.e. the |1i states.

102

in prior chapters is how molecule formation is detected. The PA experiments in
Chapter 3 showed molecule formation by observing atomic loss originating from theuncontrolled spontaneous emission following PA to an excited molecular state. Conversely, all molecule formation experiments in this chapter were performed with stateselective and reversible methods. In an ultracold mixture of atoms and molecules, themolecules were isolated by illuminating the sample with resonant light to clean away
free atoms (while leaving the molecules unaected2 ), and then counted by reversing
the molecule formation procedure and imaging the dissociated atoms (see Fig. 5-3 a).
Thus all measurements of molecule number in this chapter are from measurements of
their constituent free atoms. Since the molecules of interest here are dimers, a sample of dissociated molecules should yield equal amounts of Na and Li. While we didobserve this to be true, in many measurements the dissociated Li number alone wasused to count molecule number. This was because the free-atom imaging was done
at high magnetic elds, in which Li exhibits a much higher-quality cycling imaging
transition compared to Na3 , which results in better signal-to-noise in images of Li.

5.2 Feshbach molecules revisited
Feshbach resonances provide a means of coherently forming weakly bound moleculesfrom free atoms [142]. Formation of NaLi Feshbach molecules was rst achieved in ourgroup in 2012 [54, 55]. In comparison to other bi-alkali species, NaLi has very narrow(∼ mG) resonances at high elds (> 700 G), and using these resonances for molecule
formation is experimentally dicult. The narrow widths of these Feshbach resonances
are due to a weak spin-orbit coupling between the scattering and molecular states,
which ips the electronic spins on the colliding Na and Li pair (∆S = 2) while forming
a rotating (Ng = 2) molecule [72]. For a number of years this coupling mechanism was
2 As will be discussed in the next section, the Feshbach molecule state in particular was notentirely transparent to the atom clean-out light, and it was important not to over-illuminate thesample with clean-out light to avoid adverse eects.

3 Imaging the |1i state of either Na or Li at low elds is never a cycling transition due to groundstate hyperne coupling.

High magnetic elds reduce this eect, however since Li has a much

lower hyperne constant than Na, it exhibits a cycling transition even at ∼ 700 G, while Na needs

signicantly higher elds.

103

poorly understood [143, 72], and consequently the formation of NaLi molecules wasa dicult, multi-year eort [73]. In the following, I describe our brief re-visit ofexperiments in formation of Feshbach molecules as a stepping stone towards reaching
the triplet ground state.
Our approach was to replicate the experiment of our prior report [54]. The experimental sequence is shown in Fig. 5-3 a. The Feshbach coils were used to produce a large
bias eld of 745.55 G, just above the Feshbach resonance. A small shim coil was used
to lower the bias eld and perform a Landau-Zener sweep to form Feshbach molecules.
We found optimal Feshbach molecule formation eciency with a 300 mG/ms sweep
for 1 ms, which resulted in formation of 3 × 104 Feshbach molecules, corresponding

Bias field (G)

a)

b)

747

Feshbachformation

746

Clean

745

Dissociate,image

1ms 100µs Variable holdTime

Feshbach mol. num. ( 103)

to a formation eciency of 3 %4 .

20

τ=20 ms
10

0

0

20

40

60

Hold time (ms)80

Figure 5-3: Feshbach molecule formation and lifetime. a) A schematic of the Feshbach
molecule formation and detection sequence. A Landau-Zener formation sweep was
followed by resonant clean-out of the free atoms to isolate the Feshbach molecules.
Raising the eld back above the Feshbach resonance results in dissociation of themolecules to free atoms, which could then be imaged. b) The Feshbach molecule
lifetime measured with such a sequence, showing a 20 ms lifetime.
Our prior work with Feshbach molecules reported very fast loss timescales (< 1 ms)
if the remaining free-atoms were not quickly removed from the sample. We used a
300 µs resonant clean-out light pulse to remove the |1i state Na and Li atoms,

with intensities of INa = 80 µW/cm2 , ILi = 15 µW/cm2 . The clean-out pulse adversely
4 Our prior work [54] found a slightly higher formation eciency of 5 %.

It is likely we could

optimize the Feshbach molecule formation eciency by varying the density, however our primary
focus was to move on with STIRAP.

104

aects the Feshbach molecules, and it is crucial that the clean-out beam is perpendicular to the long axis of the cigar-shaped cloud (Fig. 5-2). In this geometry the beam
illuminates a low optical density of atoms, allowing for the use of low laser beam
intensities. Our initial attempts to form Feshbach molecules used an axial clean-out
beam with considerably higher intensities (∼ 1 W/cm2 ) that were necessary to punchthrough the high axial optical density within a few hundred µs, but this beam also
destroyed the Feshbach molecules themselves. The clean-out pulse is not perfect, andleaves approximately 5 % of the atoms in the trap in a Zeeman dark statethe eectsof this will be further described in Sec. 5.5.1 during the measurement of the groundstate molecule lifetime.
Feshbach molecule lifetimes were measured by holding the molecules for a variable
duration before dissociating and imaging. If the molecules were kept in the presenceof either species (i.e. the clean-out pulse was applied at the end of the variable holdin Fig. 5-3 a), a short < 0.5 ms lifetime was observed in agreement with our priorreport [54]. However, we found the isolated Feshbach molecule lifetime to be 20 ms(Fig. 5-3 b), a factor of 20 longer than we had previously observed5 . We suspect thereason for this discrepancy was the 1064 nm Nd:YAG laser used in the 2012 experiment to produce the ODT was unknowingly producing multi-mode, broad (∼ 2 nm)
spectrum light which resonantly excited a bound-to-bound transition in the Feshbach
molecules6 . This laser has since been repaired and its single-mode spectrum conrmed
using the 60 MHz resolution wavemeter. Without optical excitation of the Feshbach
molecules from the ODT, they now exhibited a lifetime that was limited by collisions
with the remaining free-atoms left from the imperfect clean-out pulse. Observationof a 20 ms Feshbach molecule lifetime was a fortuitous surprise. Although we wereprepared to work with 1 ms lifetimes7 , the extended lifetime provided some breathing
room in case the experiments did not go according to plan.
Having discussed the formation of Feshbach molecules, we now briey discuss
5 Of course, lifetime measurements depend on the number density in the sample, but in this casewe were careful to replicate the same density as in our prior experiment.

6 This was a dicult problem to diagnose, since the multi-mode laser worked very well for trapping

the atomic species and only disturbed molecular states.

7 STIRAP is a relatively fast procedure, requiring only a few hundred µs.

105their properties pertaining to two-photon STIRAP. As previously mentioned, NaLi
Feshbach molecules are unique among the bi-alkalis in that they are rotating withquantum numbers vg = 10, Ng = 2, mN = − 2 [72]. This means the polarizations of
the two photons used in STIRAP must be chosen to remove the two units of angularmomentum from the molecule to place it in the rotational ground state (discussed
in the following section). Another relevant consideration is the eect of molecularrotation on transition strengths. The two-photon spectroscopy of Chapter 4 studied
only the vg = 10, Ng = 0 state which has a binding energy of only 9.35 GHz. The
Feshbach state Ng = 2 has a binding energy of 5.85 GHz, which is more than a mere
perturbative shift of the Ng = 0 state, prompting the question of how might centrifugal
distortion change the Franck-Condon factors. Fig. 5-4 shows the result of numericallysolving for the least bound (vg = 10) wavefunction using ab initio potentials withand without an Ng = 2 centrifugal barrier. The results show negligible dierences
between the Feshbach wavefunction and the Ng = 0 state, thus we did not expect any
signicant changes to the transition strengths measured with two-photon spectroscopy
in Chapter 4.

Wavefunction amplitude (a0-1/2)

0.3
0.2
0.1
0
-0.1

0

20

40
60
Internuclear distance (a0)80Figure 5-4: Centrifugal distortion of the Feshbach molecule wavefunction. The Feshbach molecule wavefunction as calculated from ab initio potentials for Ng = 0, 2 rotational states, showing only a minor eect from centrifugal distortion.

1065.3 Molecular spectroscopy at high magnetic elds
The spectroscopy of Chapters 3 and 4 were performed with an initial collisional stateof Na |8i and Li |6i atoms at a bias eld of Bz = 1.2 G. In Sec. 4.2 we concluded thatthe v ∗ = 11 state of the c3 Σ+ potential was the ideal intermediate state for STIRAP.

In this section, we applied these ndings to perform spectroscopy from an initial
Feshbach molecule state at a bias eld of Bz ≈ 745 G. Note that we will ignore theeect of nuclear spins (hyperne coupling), until a brief discussion at the end of thissection.
First, we consider the eect of the magnetic eld on the intermediate state v ∗ = 11
which was discussed in detail in Sec. 3.3.1. The Hamiltonian for the excited state
structure is given by eq. 3.3 with an additional electronic spin Zeeman term, which
we re-write here for clarity

Hexc = Hd-d + Hspin-rot + HZeeman


(5.1)
~1 · r̂)(S~2 · r̂) − S~1 · S~2 + (16.2 MHz)S~ ·N~ − µB Sz Bz= (182.1 MHz) 3(Swhere we've inserted the measured coupling strengths (see Sec. 3.3.1), and have ig-~ ·B~ as this coupling is
nored any rotational magnetic coupling terms of the form N
typically weak. Fig. 5-5 shows the results of numerically diagonalizing eq. 5.1: At
magnetic elds above 200 G, the excited state structure splits into three diverging
manifolds, corresponding to the three projections of the triplet electronic molecular states onto the magnetic eld axis. This separation of energy scales between theincreasing electronic Zeeman interaction and constant terms Hd-d and Hspin-rot is analogous to the Breit-Rabi formula for hyperne eects in alkali atoms (see for example
Fig. 2-2). The high-eld rotational quantum numbers are deduced by conservation ofangular momentum along the magnetic eld axis (mtotal = mS + mN is conserved).
Unlike the Breit-Rabi solution, the high-eld ordering of the mN states in Fig. 5-5 ismuch less intuitive due to the electrons' dipole-dipole coupling, Hd-d .
Having understood the Zeeman splittings in the excited molecular state, we now
turn our attention to laser spectroscopy of the Feshbach molecules. Fig. 5-6 shows the
107

800

mS=1

mN
1
-1
0

mS=0

mN
0

600
400

Energy (MHz)

J=0
200

J=2

0
− 200

-1
1
J=1

− 400
− 600

mS=-1

− 800
0

50

100

150

mN
-1
1
0

200

Magnetic field (G)Figure 5-5: Zeeman splitting of the c3 Σ+ , v ∗ = 11 excited molecular state. The excitedstates split into three manifolds due to the total electronic spin alignment relative tothe external magnetic eld, and are further split by the weaker dipole-dipole andspin-rotation coupling in the molecule. The angular momentum projection quantum
numbers are deduced from angular momentum conservation.
relevant states in a transition from the Feshbach molecules to the excited v ∗ = 11 state
at various magnetic elds. The Feshbach molecule splittings were carefully studied by
the Tiemann group [72], and here we use their results without explicitly solving theircoupling Hamiltonian. Molecular electric dipole transition selection rules between Σstates in Hund's case (b) require the molecular rotation to change by ∆N = ± 1,

∆mN = ± 1, 0. Thus the initial Feshbach state at 745 G with quantum numbers

vg = 10, mS = 1, Ng = 2, mN = − 2 can only couple to the excited v ∗ = 11, mS = 1,

N ∗ = 1, mN = − 1 state with σ + polarized light. The laser frequency of this transition

can approximately deduced from the PA transition to the v ∗ = 11, J = 2 low-eld
excited state at 359.99798 MHz measured in Chapter 3 and adding the additional
Feshbach molecule binding energy of 5.85 GHz, yielding an expected upleg transition

108E (GHz)
4
v*=11
0
N*=1
-4

PA
0

|6 +|8

Uplegfrom FB mol.
1000

2000

0
Feshbachresonance|1 +|1
-4
Ng=2
vg=10
Ng=0

-8
0

1000
Magnetic field (G)2000

Figure 5-6: Transition diagram between the Feshbach state (a3 Σ+ , vg = 10) and intermediate excited state (c3 Σ+ , vg = 11) at high magnetic elds. (Top) The excitedstate Zeeman splitting of Fig. 5-5 is shown at high magnetic elds. (Bottom) The free
atom Zeeman shifts (dashed lines) and the a3 Σ+ , vg = 10 molecular Zeeman shifts(red lines for hyperne states of Ng = 2, blue lines for hyperne states Ng = 0) are
shown (adapted from Fig. 3 of [72]). The free-to-bound PA transitions studied in
Chapter 3 and the bound-to-bound transition starting with Feshbach molecules areshown with red arrows. The large number of lines in the bottom diagram are fromhyperne states which were considered in the analysis of [72]. Such hyperne eects
in the excited state (top diagram) would be too small to see on the scale of thesegures.
frequency of 360.0038 THz8 . Since both the Feshbach and excited state molecules are
8 In this simple explanation I've ignored the ∼ 200 MHz splitting between the v ∗ = 11, J = 0, 2

states in Fig. 5-5, however this is inconsequential since the high eld states mS = 1, mN = ± 1 end

up nearly degenerate. In any case, a careful examination of Fig. 5-5 and Fig. 5-6 will give the expected109Feshbach molecule number ( 103)40

σ- pol.

20

0
40

30

σ+ pol.
20
0
− 100

0

100

Upleg detuning (MHz)0

360.0025

360.0030

360.0035

360.0040

360.0045

Upleg laser frequency (THz)

Figure 5-7: Upleg transition spectroscopy at 745 G. (Top) Feshbach molecule spectrum after a 50 mW, 200 µs exposure to the upleg laser with σ − polarization, showing no signicant loss features. (Bottom) Feshbach molecule spectrum after a 2 mW,
200 µs exposure to the upleg laser with σ + polarization, showing a clear loss feature.
The inset shows a ner scan (4 µW, 300 µs exposure), revealing only a single loss feature. Note that the σ − spectrum shows a slight resonant dip, however this is simply
due to polarization imperfections in the excitation beam and the use of higher power
upleg laser light.
in the mS = 1 electronic state, the large Zeeman shifts are common mode.
Fig. 5-7 shows the results of upleg spectroscopy performed with Feshbach molecules
at 745 G. As with PA spectroscopy, loss of Feshbach molecules due to uncontrolled
spontaneous decay from the excited state signals the nding of a bound-to-bound resonance. As expected, we found only one spectroscopic line in the upleg spectroscopy
at 360.00381(1) THz, which responds only to σ + polarization. In contrast to the
PA spectroscopy, the upleg transition strength was signicantly stronger due to thebound-to-bound transition. Typical upleg laser powers were µW with exposure timesof only a few hundred µs, which are a factor of 109 smaller pulse areas than used in
the PA spectroscopy.
upleg at a frequency of 360.0038 THz.

110

a)

b)

Upleg σ
vg=0
Ng=0

1

-1

0

0
mN

1

580 mW/cm2

4- 819 nmvg=10
Ng=2 -20

Downleg σ+ 833 nm-1

2

Feshbach molecule number (arb.)

v*=11
N*=1

3

150 mW/cm2

2

24 mW/cm2

1.00

1
0.25

7 mW/cm2
0

− 20

− 0.5
0.0
0.5
Upleg Frequency (MHz)

0

20

40

Upleg Frequency (MHz)Figure 5-8: Two-photon spectroscopy using Feshbach molecules. a) An energy level
diagram showing the molecular rotation angular momentum states. b) An AutlerTownes spectrum of the ro-vibrational ground state, taken at various downleg powers
with a 100 µs exposure time and 17 mW/cm2 upleg power. At low downleg intensities
a narrow EIT feature is observed (red data points) with a width of 75kHz.
Using this intermediate transition we performed two-photon spectroscopy to ndthe downleg transition frequency to the ro-vibrational triplet ground state, vg = 0,

Ng = 0. The laser frequency of the downleg transition at high magnetic elds was
expected to be very similar to the result of low-eld two-photon spectroscopy (Chapter 4), once again due to the common-mode Zeeman shifts in the mS = 1 electronic
state. We found the downleg transition to the ro-vibrational ground state at a laser
frequency of 366.23611(1) THz 9 . Fig. 5-8 shows the Autler-Townes spectrum of thisthree-level system and the transition to a sharp EIT feature. This two-photon spectroscopy required σ − polarized downleg light, which is consistent with a mN = −1 → 0

9 This is 30 MHz lower than the measured frequency in Sec. 4.3, which is explained by the slightne-structure dierence between the excited mS = 1,mN = −1 state and the mS = 1,mN = 1 state at
745 G (as shown in Fig. 5-5, although the x-axis does not extend to 745 G). Recall that the mS = 1,

mN = 1 is exactly the stretched J = 2, mJ = 2 state used for the two-photon spectroscopy at loweld in Chapter 4.

111

transition from the excited state down to the ro-vibrational ground state (Fig. 5-8 a).
Lastly, we mention the eect of hyperne coupling. In all of the aforementioned
molecular transitions (including magneto-association around the Feshbach resonance),
the nuclear spin states of the Na and Li nucleii are mI,Na = 3/2, mI,Na = 1, respectively. As described in the previous section, NaLi Feshbach molecule formation uses
a magnetic coupling between the electron spins and the orbital angular momentum
which does not aect the nuclear spins. In the case of the one- and two-photon
molecular spectroscopy discussed here, small excited state hyperne splittings on theorder of ∼ MHz surely exist10 . While these splittings are small, they would be easily
resolvable at low magnetic elds since an EIT spectrum provides < 100 kHz resolution (Fig. 5-8 b). However, the use of a high magnetic eld environment decouples
the hyperne interaction, and thus the nuclear spins are unaected during photonic
transitions. This is a subtle point, as the non-observation of hyperne splittings in the
two-photon spectroscopy experiments in Chapter 4 was obviously not due to highmagnetic elds, but rather the use of stretched angular momentum states. Thus, for
one reason or another, we have been able to conveniently ignore nuclear spin states andhyperne couplings. As we move the discussion forward towards the triplet groundstate NaLi molecule, it should be noted that hyperne eects in electronic groundstates are signicantly larger than in electronic excited states, and these eects will
become the focal point of Sec. 5.5.3.

5.4 STIRAP transfer to the ro-vibrational groundstate
The procedure of stimulated Raman adiabatic passage (STIRAP) was pioneered forsystems of ultracold molecules in KRb [35] and Cs2 [34], and has since been applied
to a number of heteronuclear bi-alkali combinations [45, 46, 48, 47]. STIRAP itself
is a widely applicable technique of coherently transferring between quantum states
10 This energy scale is estimated from the hyperne splittings of free an excited free atomic Li,
3which is the dissociation limit of excited NaLi c Σ+112potential.

while avoiding spontaneous emission [75]. One may ask why a two-photon transfer
is even necessarywhy not simply couple the Feshbach and ground states directlywith a single photon? There are two issues with this approach. The primary limitation is that despite the non-zero electric dipole moment of the NaLi molecule, thedipole transition is small due to the poor wavefunction overlap between the Feshbach and ground states, though in this case the Franck-Condon-type matrix element~ gnd i, where thecontains a dipole moment operator, i.e. hψFesh |d~nuc |ψgnd i ∝ hψFesh |R|ψnuclear dipole moment operator, d~nuc , is proportional to the internuclear separation,~ (see eq. 2.12 of [55]). The second limitation is technological: strong coherent raR
diation sources at the 6 THz resonant transition frequency do not exist11 . Therefore
to transfer molecules between dierent vibrational states in the same electronic potential we must employ an intermediate excited state and the frequency dierence of
two infrared lasers in a two-photon transition.
In our case, STIRAP allows for coherent transfer of Feshbach molecules to thetriplet ground state via the v ∗ = 11 excited state, which has a short spontaneous
decay lifetime of ∼ 30 ns. The excited state is never populated and the system stays
in the dark eigenstate (assuming zero two-photon detuning)|ψdark i = cos θ|ψFesh i + sin θ|ψgnd i(5.2)

where the mixing angle is given by the ratio of upleg and downleg couplings, tan θ =Ωup /Ωdown (for a derivation, see Sec. III D of [140]). By changing the relative upleg and downleg couplings, we can move the population from purely in the Feshbach state (Ωup = 0) to purely in the ro-vibrational ground state (Ωdown = 0). By
considering these limiting cases we nd that STIRAP requires a somewhat peculiar
pulse sequence, where transfer to the ground state is achieved by rst turning on
the downleg followed by increasing the upleg intensity as in Fig. 5-9 b (this is the
opposite pulse order as one would intuitively expect when sequentially transferring|ψFesh i → |ψexc i → |ψgnd i, hence it's been colloquially called the counter-intuitive

11 THz sources are an active eld of research. For recent developments in the 1-6 THz band, see[144].

113

N ( ×10 3 )

30
20
10
0

Na
Li
0

20

30

40

20

30

40

Upleg
Downleg40Ω(MHz)

10

20

0

0

10

STIRAP time (µs)

Figure 5-9: STIRAP from Feshbach molecules to the ro-vibrational ground state.
The Feshbach molecule number is shown at various points during a forward and reverse STIRAP pulse sequence. The disappearance and re-appearance of the Feshbach
molecules indicates a reversible transfer to the ro-vibrational ground state.
pulse sequence). Changes to the upleg and downleg coupling must be done slowly−1(τpulse  Ω−1up , Ωdown ) to keep the transfer adiabatic and not excite any population in

the short-lived excited state.
We applied the counter-intuitive (STIRAP) pulse sequence to our sample of Feshbach molecules with the upleg and downleg lasers on two-photon resonance to within
10 kHz (as determined by the EIT spectrum). Since the STIRAP process is coherent
and reversible, a successful STIRAP transfer to the ground state was conrmed by thedisappearance of Feshbach molecules and a later re-appearance if the pulse sequence
was reversed (we call this forward and reverse STIRAP transfer). This eect isshown in Fig. 5-9.
We found optimal transfer using a 40 µs round-trip forward and reverse STIRAP
pulse for which 74 % of the initial Feshbach molecules were recovered, corresponding to
an 86 % one-way transfer eciency. The pulse sequence was only crudely optimized byvarying pulse durations and initial laser intensities. The pulse-shapes were intended
to be linear ramps in laser intensity, however technical control limitations of laser

114

intensity on short (µs) timescales resulted in the pulses of Fig. 5-9.
The upleg and downleg Rabi frequencies were calibrated by upleg loss measurements and Autler-Townes spectra, respectively (as in Chapters 3 and 4). The entiretyof the STIRAP transfer happened in a ∼ 5 µs time window at a Rabi frequency cross-ing Ωdown ≈ Ωup ≈ 5 MHz, which satised the adiabatic criteria with 5 µs  Ω−1 ≈0.03 µs. Increasing the STIRAP pulse duration only worsens the transfer eciency(Fig. 5-10), further indicating that the transfer eciency is not limited by adiabaticity.

Round trip efficiency (%)

80

60

40

20

0

100

200

300

400

Forward and reverse STIRAP pulse time (μs)

Figure 5-10: STIRAP eciency as a function of pulse duration. The STIRAP eciency only decreased with longer pulse durations, possibly revealing a timescale foruncontrolled decoherence.
The cause of decreased STIRAP eciency for longer pulses is likely some form of
uncontrolled decoherence. The relevant time-scale for this decoherence is the transfer
duration (i.e. when Ωup ≈ Ωdown ), which we can see from Fig. 5-9 is approximately
1/8th of the total forward-reverse pulse time. Thus it seems there are issues maintaining coherence at times in excess of 50 µs corresponding to a 3 kHz broadening.
We have not made a careful study of this decoherence, and here I present a brief
discussion of possible eects that may limit the transfer duration:• The laser linewidths are an obvious suspect, and using a number of dierent estimation methods (see Chapter 2, or [80]) we expect the relative laser linewidth115

to be < 1 kHz. This timescale limit is only a factor of 3 away from the observed
decoherence, thus the laser linewidths are the primary suspect in limiting the
STIRAP coherence time.

• A slim possibility is magnetic eld noise: the triplet state transitions considered here have mostly common mode Zeeman shifts, with the exception of a

∼ 10 kHz/G transition shift from ne structure eects. But even a conservative
estimate of eld stability of 30 mG RMS (see Chapter 2) would only result in
300 Hz of broadening.

• The lifetimes of either the Feshbach state or ro-vibrational ground state couldin principle impose a transfer time limit. However, as measured in Sec. 5.2 theFeshbach molecule lifetime was in excess of 20 ms, and as will be discussed in
the following section the ro-vibrational ground state lifetime was similarly longin the 1064 nm ODT.

• The ultracold sample temperature, T ≈ 2 µK, corresponds to a 40 kHz Zeeman
broadening and a 25 kHz Doppler broadening for NaLi molecules. However,both of these eects should be common mode: As in the case of magnetic eldnoise, all Zeeman broadening is common to the molecular states with an mS = 1
projection of the triplet state. The Doppler shifts are common to both uplegand downleg laser beams (since they are co-propagating), thus they do notsignicantly aect the two-photon detuning during STIRAP and should also be
negligible.
This decoherence eect notwithstanding, the 86 % eciency obtained with a 20 µs
pulse was easily sucient to study the ground state molecules with reasonable signalto-noise (SNR). Fig. 5-11 shows a typical image of NaLi ground state molecules,
which were detected by imaging free atoms after Feshbach association, forward andreverse STIRAP, and Feshbach dissociation. Two possible imaging geometries wereused: bottom and side imaging. The bottom images showed faint traces of atoms,
however numerical integration of the image along the axial trap direction revealed

116

Feshbach fielda)Side imaging beamσ+yz

Bottom imaging beam
linear polarization = σ + + σ -

x

g

b)
z

Na

x

Li
1mm
Optical density
0

c)

0.5

1.0

Na

1.5

Li

y

0.1mm

x

Figure 5-11: Imaging molecules after free atom dissociation. a) Two possible imaging
geometries were used; bottom and side imaging. b) Bottom images, showing a faintsignal in both Na and Li atom number, which can be numerically integrated andtted to Gaussian proles to accurately count molecule number. c) Side images,
which clearly show the molecular signal in both Na and Li. All images were taken at
745 G. Note that these particular images are after a round-trip STIRAP to the triplet
ground state, however images of Feshbach molecules (without ground state transfer)
were of similar appearance.
clear Gaussian peaks. Side imaging allowed the imaging light to naturally observe ahigher optical density and produce clearer images. Although both Li and Na images
contained the same atom number, Li images typically produced a clearer signal due
117to its better quality cycling transition at 745 G. Single-axis integrated bottom images
and side images both yielded similar SNR, typically ∼ 20.

5.5 Studies of ultracold NaLi triplet ground state
molecules
5.5.1 Lifetime and trapping
Measurement of the lifetime of triplet ground state NaLi molecules was the rstdenitive test of whether these molecules would be of any practical use. Long lifetimes
make for an experimentally usable system and ultimately determine what kind ofphysics can be studied with the NaLi molecule.
NaLi is a fermionic molecule with exothermic two-body reactions, thus from theoutset we expected the lifetime to be limited by the universal inelastic collision rate[43], which is approximately12 Kuniv ≈ 10−11 cm3 /s. With an initial atomic densityof 1012 cm−3 and a ∼ 1 % ground state molecule formation eciency, we expected
molecule densities of nmol ≈ 1010 cm−3 and lifetimes of (nmol Kuniv )−1 ≈ 10 s. The restof this section will describe the experimental adventure to verify this simple calculation.
The molecular lifetime measurement was performed by simply adding a variable
hold time between the forward and reverse STIRAP pules. This experimental sequence and corresponding lifetime measurement are shown in Fig. 5-12. The measured lifetime, 28 ms, was considerably shorter than the expected 10 s, but there
were a number of issues with the experimental sequence. First, the clean-out pulse
was imperfect, and would optically pump the Na and Li to other hyperne Zeeman
states. This optical pumping eect was particularly bad for Na since at 745 G its
imaging/cleaning transition is not cycling. This left approximately 5 × 104 Na atoms
in the upper F = 2 Zeeman manifold (see Fig. 2-2), which were discovered by direct
12 A detailed discussion of universal collisions will be given towards the end of this section. For
now, we follow only rough calculations to give insight into the experimental procedure.

118

b)

747

745

Dissociate,image

Feshbachformation
STIRAP Clean

746
1ms

Molecule Number (103 )Bias field (G)

a)

20µs

300µs

ReverseSTIRAP

20µs

Variable hold

Time

20

τ=28 ms

10

0

0

20

40

60

Hold time (ms)Figure 5-12: Initial measurements of the NaLi ground state lifetime. a) The experimental sequence to form triplet ground state molecules and measure their lifetime.
Note that we waited until forming ground state molecules before removing free atoms
with clean-out light. The ground state molecules were not at all aected by clean-out
light, thus allowing for a longer duration clean-out pulse. b) The measured triplet
ground state lifetime with such an experimental procedure, showing an unusuallyshort lifetime of 28 ms.
absorption imaging at low magnetic elds after a clean-out pulse. These leftover Na
atoms would remain in the optical trap and result in bad collisions with the NaLi
molecules.
The clean-out issue was xed by suddenly turning o the magnetic eld and adding
a secondary 15 ms clean-out pulse using the Li and Na laser cooling (MOT) light.
The advantage of this approach was that it was thorough; we could be condent thatno free atoms would be able to hide in dark states from the high-intensity MOT light
at low magnetic elds. The disadvantage was that the sudden eld drop of ∼ 800 G
gave a kick to the magnetically sensitive molecules, and furthermore the Feshbach eldwould have to be later turned back on to perform reverse STIRAP for detection13 .
Despite this disadvantage, lowering the eld would have been necessary to keep the
NaLi molecules trapped, as they were in low eld seeking states and the Feshbach bias
13 Another technical disadvantage to switching an inductive coil with 200 A of current o and on
rapidly was that this caused some pretty serious voltage spikes which seemed to aect a lot of otherelectronics in the lab due to grounding issues. The computer monitors would slightly icker duringthe eld shuto.

These spikes also thoroughly tested the high-voltage capability of the magnetic

eld stabilization circuit described in Chapter 2.

119

Bias field (G)

747
746

ReverseSTIRAPSTIRAP Clean

Dissociate,
image

745
20µs 300µs
20100

1 ms Feshbachformation

MOT
Clean

20µs

15 ms Variable holdTime

Figure 5-13: NaLi triplet ground state molecule formation sequence with thoroughfree atom clean-out. After STIRAP to the ground state and a clean-out pulse at
745 G, the magnetic eld was immediately dropped to 13 G and the sample wasfurther cleaned of free atoms with MOT light. The lifetime of the molecules wasmeasured at low eld using a variable hold time. Detection of molecules required thereversal of STIRAP and Feshbach dissociation.
eld produces an axial eld maximum14 . This full clean-out and eld-drop sequenceis shown in Fig. 5-13.
Even with this improved lifetime measurement sequence, the NaLi molecule lifetime was extended to only 100 ms, which brings us to the second lifetime limitation: the 1064 nm ODT laser. Although 1064 nm is not enough excitation energy toreach the c3 Σ+ potential studied in Chapter 3, ab initio calculations indicate that the1064 nm ODT could drive excitations to the signicantly deeper b3 Π potential. Fig. 514 shows the calculated Franck-Condon factors for transitions to the b3 Π potential forboth the NaLi ro-vibrational ground state and Feshbach state. Ab initio calculations
reveal that 1064 nm is 1 THz detuned from driving the NaLi triplet ground state(vg = 0) to the v ∗ = 16 state of the b3 Π potential. A rough calculation using the ODTparameters of 0.3 W and a 30 µm waist gives a scattering rate of 5 Hz15 , consistent
with the measured lifetime. Fig. 5-14 also shows that the Franck-Condon factor dropso rapidly towards longer wavelength, and with this in mind we decided to switch
14 An alternative would be to use a crossed-optical dipole trap for axial connement, an idea wetoyed with but ultimately abandoned because of lack of ODT laser power.

15 In the limit of large detuning, the scattering rate is R = (FCF) Γ I/Isat

2 4(∆/Γ)22, where Γ = 2π×6 MHz,2

the detuning is ∆ = 1 THz, I = 0.3 W/(30 µm) , Isat = 6 mW/cm , and the Franck-Condon Factor
near the 1064 nm transition can be read o of Fig. 5-14 to be (FCF) = 5 × 10

120

−3

.

10 − 1

Feshbach Transition FC Factors
Gnd Transition FC Factors

Franck-Condon factor

10 − 310 − 510 − 710 − 910 − 1110 − 131064 nm

800

1000

1319 nm

1200

1400

1600

1800

Wavelength (nm)Figure 5-14: Franck-Condon factors for optical excitations to the b3 Π potential of the
NaLi molecule. Ab initio potentials were used to calculate Franck-Condon factors
as a function of laser excitation frequency for an initial NaLi molecule in the triplet
ground state (green circles) and the Feshbach molecular state (red crosses). The plot
indicates that both ground state molecule and Feshbach molecules are susceptible toheating from a 1064 nm ODT. For the ground state molecules, a 103 reduction in
Franck-Condon factor is possible by using a 1319 nm ODT.
the ODT to a 1319 nm laser16 which would reduce the bound-to-bound transition
rate by a factor of 1000. The experimental sequence still used the 1064 nm ODT for
ultracold atomic mixture preparation, but the mixture was transfered to a 250 mW,
32 µm waist, 1319 nm ODT before the molecule formation sequence of Fig. 5-13. The
resulting NaLi triplet state lifetime in the 1319 nm ODT is plotted in Fig. 5-15. The
molecule number exhibited a sharp loss in the rst 300 ms due to magnetic kicks
from the sudden eld shut-o and turn-on in the experimental sequence. These kicks
excited a center-of-mass axial sloshing of the molecules in the ODT resulting in aperiodic loss pattern with a frequency of 7 Hz, consistent with the axial trapping
frequency of the 1319 nm ODT. After this initial loss, the molecule number stabilized and exhibited a long lifetime on the order of seconds. We assumed a two-body
16 There is nothing particularly special about the wavelength 1319 nm.

We are lucky to workwithin the Center of Ultracold Atoms, where an unused 1319 nm laser can be found lying around.

121

Molecule number (× 103 )

25

20

20

10

15

0
0.0

0.2

10

0.4

t (s)

5
0

0

1

2

3

4

5

Hold time (s)Figure 5-15: Long lifetime in a pure sample of isolated NaLi triplet ground state
molecules. After an initial loss of molecules due to the sudden magnetic eld drop,
the NaLi molecules exhibit a slow decay with a tted two-body loss timescale of
4.6 s. Inset: At short times (< 0.3 s), the molecule number drops while showing a
7 Hz oscillation, consistent with axial sloshing of the molecules in the single beam
ODT.
loss mechanism and used the same tting function as in PA experiments, eq. 3.4, to
extract a loss timescale17 of τ = 4.6 s.
This long timescale was an exciting result, because it immediately allowed us toprobe the molecules for long durations to understand their properties. Not having
a short lifetime constraint makes these experiments easy. In the following sections,
we will describe the rst experiments we performed to discover properties of our new
triplet ground state NaLi molecules.

5.5.2 Estimating the loss coecient
In order to compare our lifetime measurement to theoretical predictions, we mustnormalize the result by the molecule density. In order to nd the molecule density,
we must determine the trapping parameters and temperature of the sample. The
trap frequency was determined by parametrically modulating the 1319 nm ODT and
17 Use eq. 3.4, with n−1

Na = nLi ≡ nmol and τ ≡ (nmol K̄I)

122.

Molecule number ( × 103)

10
8
6
4
2
0

0

1

2

3

4

5

Modulation frequency (kHz)
Figure 5-16: Parametric modulation of the 1319 nm ODT to determine molecular
trap frequencies. A loss in the molecule number is observed at 1.54 kHz, indicatinga radial trapping frequency of 770 Hz for the NaLi molecules.
looking for a loss resonance at twice the radial trapping frequency, 2fr,mol . Fig. 5-16
shows the results of modulating the trap for 1 s at 10 % of the trap depth, revealing aradial trap frequency of fr,mol = 770 Hz. This value is consistent with a simple modelof NaLi that assumes the molecule to have a single transition with unity oscillator
strength at 830 nm and a linewidth 6 MHz, which gives a radial trap frequency of

fr,mol = 800 Hz, axial trap frequency of fz,mol = 7 Hz, and trap depth of Umol = 21 µK(here we have used the well known equations for a Gaussian beam ODT; see eq. 10
and Sec. IV.A of [145]). Such a model is valid since the dominant Franck-Condon
factors for the triplet ground state NaLi molecule are at approximately 830 nm transitions (819 nm for c3 Σ+ and 840 nm for b3 Π potentials, see Fig. 4-3 and Fig. 5-14,
respectively), and the Franck-Condon factors obey a sum rule which adds to unity[146]. Since the c3 Σ+ and b3 Π potentials dissociate to to the excited 2 P1/2 and 2 P3/2
transitions in Li, respectively, this simplied molecular transition model assumes thesame transition strengths as the combined D1 and D2 transitions in a free Li atom.
The 7 Hz axial trapping frequency obtained from this analysis is consistent with theoscillations observed in the lifetime measurement (Fig. 5-15, inset).
The temperature of the molecules was measured by imaging time-of-ight (TOF)123

Integrated signal (arb.)

a)

10

Na
TOF
0 ms

0

Li
TOF

0 ms

0

1 ms
− 10

0.5 ms
− 10

2 ms
3 ms
200

400

1.5 ms

− 20

− 20
0

1 ms

600

0

200

Distance (μm)

mω2 σ 2 / kB (µK)

b)

300

400

600

Distance (μm)

mω2 σ 2 / kB = T ( 1 + ω2 t 2 )T = 3 µK

200

100

Na
Li

0
0

20

40

60

80

ω2 t 2

Figure 5-17: Time-of-ight distributions of dissociated molecules. After a 1 s holdtime at low eld, the molecules were dissociated and allowed to expand in time-ofight. a) The integrated density distributions of side images of Na and Li at various
TOF times. The distributions were t to Gaussian proles, as expected for a thermalgas. Note the Li expands faster than Na due to its light mass. b) The Gaussian
widths as a function of TOF time used to extract the sample temperature. ω is theradial trapping frequency for the relevant species (ωNa = 0.5 kHz, ωNa = 1 kHz), σ isthe measured Gaussian prole radius, and m is the species mass.
distributions of the dissociated atoms to obtain Tmol = 3 µK (Fig. 5-17). This measurement is consistent with the calculated ODT trap depth of 21 µK18 , however
one must be careful to consider how the process of molecule dissociation can affect the sample temperature. The STIRAP process is a coherent, single-body eectwhich only changes the internal (vibrational) degree of freedom, and does not aect
molecular motion19 . However, Feshbach dissociation can non-trivially aect the tem18 A typical rule-of-thumb is that the sample temperature will be 10 % of the trap depth.
19 The STIRAP beams actually do give the molecules a small momentum kick from absorbing an124perature. Consider an NaLi molecule moving with kinetic energy and temperature(mNa + mLi )v 2 = kB Tmol , which upon perfect dissociation will leave two free atoms
with energies mLi v 2 < mNa v 2 < kB Tmol . Thus the TOF measurement of free atoms
may be underestimating the molecular sample temperature20 . A competing eect isthat a diabatic magnetic eld jump for Feshbach dissociation can result in coupling
the molecules to higher energy scattering states which heats the sample [147, 148].
The molecules may overshoot the resonance and add relative kinetic energy between
the Na and Li atoms, which in the center-of-mass frame results in more heating of thelighter Li atoms. Our observations show the same temperature for both dissociatedNa and Li (Fig. 5-17 b), which could mean the two aforementioned eects cancel one
another. We did not pursue a careful study of these eects any further, but rather
placed conservative error bounds on the measured temperature Tmol = 3+3−1.5 µK.
Assuming the NaLi molecules obey a classical thermal in-trap distribution, we canuse the measured temperature and trap frequencies to calculate the peak molecule
density to be nmol = 5 (4) × 1010 cm−3 , from which we can determine the degeneracy
parameter T /TF ≈ 100. With these measurements, we nd the 4.6 s molecule lifetime

corresponds to a density normalized loss coecient of Kmeas = 1.6 ± 1.2 ×10−11 cm3 /s.
We have also performed measurements of the atom-molecule lifetimes (by omitting
the resonant clean-out pulse for one of the species21 ), resulting in < 1 ms lifetimes
corresponding to much faster loss coecients of Kmeas,atom-mol ≈ 3 × 10−9 cm3 /s. The
density normalized loss rates can be compared to theoretical predictions, specicallythe universal model for ultracold collisions [43].

The universal model assumes aworst-case scenario where each two-body collision has 100 % probability of losing
both collision partners at short-range, resulting in a loss rate governed only by theupleg photon and emitting a downleg photon. However, in our case the beams are co-propagating,and this momentum dierence corresponds to negligible added energy of 86 pK.

20 If the atomic sample is allowed to thermalize then both species will have the temperature 1 T ,2 molfrom an application of the equipartition theorem on two degrees of freedom. But in our experiment
we held the sample for 10 ms after dissociation which is not enough time for a mixture of Na and10

Li at densities of n ∼ 10−3cm

to equilibrate.

21 Since the atom-molecule lifetime were so short, we did not need to perform the double clean-out

pulse and eld drop as we did to measure the long lifetimes.
Fig. 5-12 a.

125

Thus the sequence was similar tostrength of the long range (van der Waals) attraction. The universal model is thequantum analogue of the classical Langevin collisional model (commonly used for ion
collisions [149]). The density normalized loss coecients for s- and p-wave collisional
channels are given by

KL=0 = g

4πh̄
ā
µ

KL=1 (T ) = 1513 ā3 kB T /h

(5.3)

where g = 2(1) for (non-)identical particles, µ is the reduced mass of the collisionpartners, and ā = (2π/Γ(1/4)2 ) (2µC6 /h̄2 )1/4 is the van der Waals length. The C6
values for molecule-molecule and molecule-atom collisions are well approximated bysumming over the known C6 values for the atom-pairs which make up the collisionpartners (e.g. C6NaLi+Li = C6Na+Li + C6Li+Li ) as has been previously done in [150, 151]
(see Appendix B for specic values of coecients in the NaLi molecule). There is alsoa contribution to the C6 from the dipole-dipole interactions of NaLi, however this is
only a < 1 % eect [43]. At the ultracold temperatures considered here, we expect

KL=1  KL=0 , since p-wave collisional partners must tunnel through the centrifugal
barrier before suering the eects of any short-range loss. This is the primary reason
we expected NaLi to have a long lifetime regardless of any possible loss mechanisms:
the NaLi molecule is fermionic with the smallest C6 coecient of the bi-alkalis, thus
exhibiting very slow inelastic collision rates given by KL=1 .
Fig. 5-18 compares the measured loss coecients to predictions based on the universal model. The atom-molecule loss coecients are in close agreement with theuniversal predictions for s-wave collisions, implying that atom-molecule collisions are
maximally inelastic. This is expected, since the atoms and molecules are in opposite
spin states (mS = −1/2 and mS = 1, respectively), which immediately allows for spinchanging collisions. The molecule-molecule loss rate is signicantly lower than thes-wave universal rate due to their fermionic nature and Pauli blocking of s-wave collisions. The low molecule-molecule loss rate is a conrmation that we have prepareda single-state sample of triplet ground state NaLi molecules. However, the rate is not
as low as we would expect, still being a factor of ∼ 3 faster than the p-wave universal
126

Kloss (cm3 / s)10− 9

10− 10

10− 1110− 12

MeasuredUniv. KL=1
Univ. KL=0

NaLi Only

NaLi+Na

NaLi+Li

Figure 5-18: Loss coecients in NaLi collisions. The measured density normalized losscoecients for molecule-molecule and molecule-atom collisions are shown with black
circles. Hollow red circles are the expected universal loss rates for s-wave collisions,
KL=0 . The green triangle is the p-wave universal rate prediction, KL=1 . The atommolecule collisions show a loss rate consistent with the s-wave universal prediction.
The molecule-molecule collisions exhibit a rate much lower than the s-wave universal
prediction, indicating Pauli blocking of s-wave collisions. The error bars stem fromthe temperature uncertainty and density averaging in the harmonic trap.
rate. We suspect the higher loss rate is still due to single-body scattering of the1319 nm ODT. Although the 1319 nm ODT allowed for a factor of 30 improvement
in lifetime over the 1064 nm ODT, it may still drive bound-to-bound transitions tothe b3 Π potential that extends down to 1800 nm transition wavelengths (Fig. 5-14).

5.5.3 Electron spin resonance spectroscopy
The uniqueness of triplet NaLi molecules is their long lifetime while having bothelectric and magnetic dipole moments. In prior ultracold bi-alkali formation experiments, it was typical to measure the electric dipole moment via Stark spectroscopy[35, 45, 46, 48, 47], which showed good agreement with theoretical predictions. In
contrast to these works, here we focus on the aspect of NaLi molecules that has previously not been studied in the world of ultracold bi-alkali molecules: the magnetic
dipole moment.
127

We studied the magnetic dipole transitions of the NaLi molecule by electron
spin resonance (ESR) spectroscopy.

The ultracold molecular sample was held in

the 1319 nm ODT at low eld (∼ 10 G, during the variable hold in Fig. 5-13) and
exposed to radio-frequency (RF) pulses. We knew the initial molecular rotational
state to be Ng = 0, the spin state to be |S = 1, mS = 1i, and nuclear spin states|INa = 3/2, mI,Na = 3/2i and |ILi = 1, mI,Li = 1i. As in alkali atoms, we expected thesespins to be coupled via hyperne coupling of the form

~ · I~Li + aNa S~ · I~NaHHF = aLi S(5.4)

where aLi,Na are the hyperne coupling constants. This Hamiltonian is diagonalized by
the total angular momentum eigenstates F = S +INa +ILi . Thus our initial molecularstate after STIRAP was the stretched state |F = 7/2, mF = 7/2i, and the rst transition we probed with ESR spectroscopy was |F = 7/2, mF = 7/2i ↔ |F = 7/2, mF = 5/2i
with an expected linear Zeeman shift of ∼ 0.8 MHz/G. Fig. 5-19 shows a sample ESR
spectrum for this transition in the NaLi molecule. ESR spectra of the free atom
Na |F = 1, mF = 1i ↔ |F = 1, mF = 0i transition and the Li |F = 1/2, mF = 1/2i ↔

|F = 1/2, mF = − 1/2i transition were used to calibrate the magnetic eld22 . The

ESR spectra were taken using narrow Landau-Zener sweeps of the RF frequency totransfer molecules to the |F = 7/2, mF = 5/2i state which is not involved in the reverse
STIRAP transfer, thus the ESR resonance appears as molecule loss. A wide rangeof Rabi frequencies, 1-100 kHz, were employed over exposure times of 10-400 ms, depending on the desired spectrum resolution. Fig. 5-19 b shows the measured Zeeman
shift of the |F = 7/2, mF = 7/2i ↔ |F = 7/2, mF = 5/2i molecular transition, exhibiting a quadratic deviation from the low-eld linear Zeeman shift. By measuring thisdeviation, we could begin to calibrate the hyperne constants in numerical diagonalizations of eq. 5.4 and predict the positions of spin-ip transitions to other F
states. Fig. 5-20 shows the measured hyperne structure and spectra to all other
22 The free atom experiments were not done simultaneously with the molecular sample, i.e. they
were performed by repeating the full experimental sequence but by omitting the molecule formation
and resonant cleaning steps.

128

Number (arb.)a)Bz = 18.63 G

1.2
0.8
0.4

NaLi

Na

0.0
12.9

13.0

13.8

Li
13.9

17.0

17.1

b)

Resonance center (MHz)rf frequency (MHz)
15
10
5
0

0.0

5.0

10.0

15.0

20.0

Magnetic field (B)Figure 5-19: Electron spin resonance spectroscopy of the NaLi molecules. a) RF
spectra for the Na |F = 1, mF = 1i ↔ |F = 1, mF = 0i, NaLi |F = 7/2, mF = 7/2i ↔
|F = 7/2, mF = 5/2i, and Li |F = 1/2, mF = 1/2i ↔ |F = 1/2, mF = − 1/2i transitions
are shown, revealing a unique resonance which can identify the NaLi molecules. Suchspectra were taken at various mangetic elds, and the position of the NaLi resonancein shown in (b). The measured resonance positions (hollow circles) deviate from thelinear Zeeman shift (dashed blue lines). The black solid line is the result of numericallysolving eq. 5.4, showing excellent agreement with the data.
transitions which were accessible with magnetic dipole selection rules from our initialstate. These spectra are well t using the hyperne Hamiltonian eq. 5.4 with constants aLi = 74.61(3) MHz and aNa = 433.20(3) MHz. Interestingly, these values arealmost exactly half of the free atomic value for Na and Li. This is not unexpected,
as triplet NaLi is a very weakly bound molecule, thus each of the two electrons is localized to its respective atom and has the same atomic hyperne interactions. Since
we consider the total spin of the two electrons in our hyperne Hamiltonian, each of
the hyperne constants is half the free-atom value.
The use of RF to directly manipulate the NaLi molecule's magnetic moment is
a new technique in the eld of ultracold dipolar molecules. Fig. 5-19 a illustrates

129

F1 = 5/ 2

F
7/ 2
100 MHz
5/ 2
3/ 21132 MHz
5/ 2
3/ 2
1/ 2

F1 = 3/ 2

561 MHz
1/ 2
F1 = 1/ 2
3/ 2

b)Resonance center (MHz)

a)

110

F1=5/2
7,7
2 2

105

5,5
2 2

7,7
2 2

1160

100
95

F1=5/2

1180

3/2
5,5
2 2

1140

0

10
Bias field (G)

20

1120

0

10
Bias field (G)20

Figure 5-20: Hyperne structure and transitions in the NaLi triplet ground state
molecule. a) Energy level diagram of the NaLi hyperne states. The states accessible
from our initial state |F = 7/2, mF = 7/2i are marked in red. b) The Zeeman shifts
measured for the two accessible F changing transitions, t to a numerical diagonalization of the hyperne Hamiltonian, eq. 5.4.
that the ESR spectra of NaLi can be used to directly identify and probe the NaLi
molecule. For example, a careful measurement of the width of these spectra at low
Rabi frequencies could be used to directly measure the molecule temperature via
inhomogeneous Zeeman broadening in a magnetic trap, foregoing the complicationsof using dissociated atoms (as discussed in the previous section). This scheme isalso experimentally simple, allowing for the manipulation of the molecules' hypernestates without the use of an intermediate state in two-photon RF [129] or optical [48]transitions.

130

Chapter 6
Summary and outlook
The work of this thesis has described the experimental eorts in spectroscopy andpreparation of ultracold samples of NaLi molecules. Using PA spectroscopy, we havemapped out the vibrational states of the NaLi excited c3 Σ+ potential. This was therst measurement of the excited triplet states of the NaLi molecule, and the rstobservation of PA in a Na-Li mixture. By adding a second laser to this experiment,
we performed two-photon spectroscopy of the triplet ground (a3 Σ+ ) potential, andspectroscopically identied the triplet ro-vibrational ground state. Building on theprior work with Feshbach molecules in the Ketterle group [54], we formed Feshbach
molecules and transfered them to the ro-vibrational triplet ground state using a coherent two-photon STIRAP. Finally, we began to characterize some basic propertiesof the triplet ground state NaLi molecules, including their lifetime and the novel feature of a magnetic moment in a heteronuclear molecule [60]. We now take the time toreect on these results, and consider what place NaLi has in the future of ultracold
molecules experiments.

6.1 Collisional studies
The most notable property of the NaLi molecule that we have measured is its long
lifetime of 5 s at a density of 5 × 1010 cm−3 , which is expected to be even a factor of
10 longer if it were not limited by scattering from the trapping laser (as discussed in131Sec. 5.5.1). Such long lifetimes of the triplet NaLi molecule are almost contradictory
to its description: The triplet state is highly excited with many energetically allowed
singlet decay channels, with even more collisional loss channels available from thereactivity of NaLi molecule collisions. Of course, the long lifetime we have measured
does not imply the molecules are collisionally stable, but is rather a reection of theinherently low universal rates in the NaLi molecule, which fundamentally stems fromits small C6 coecient. Thus the NaLi molecule lifetime is impervious to possible new
two-body loss mechanisms, such as the sticky collisions which seem to be severely
limiting the lifetimes of singlet, non-reactive bi-alkali species [46, 48, 45] .
The observation of a long lifetime of the NaLi molecule approaching the universal
rate is an interesting result, however the ultimate goal is to observe elastic collisions with the NaLi molecule (i.e. a lifetime longer than the universal limit) which
could be used for evaporative cooling. Towards that end, the electric and magnetic
dipole moments available in the NaLi molecule can facilitate interesting collisional
studies. Consider for example a collisional study of a mixture of triplet NaLi withfree Na atoms. In Chapter 5, we have measured the lifetime of this mixture to bequite short (∼ 1 ms), however this was using Na atoms with an opposite spin ori-

entation1 to that of the molecule, thus spin-changing collisions were possible. If a
spin-polarized mixture of Na and NaLi was used the results could be quite dierent:
The reaction NaLi + Na → Na2 + Li is energetically forbidden so long as we consider only molecules in the triplet spin state [44]. While the use of spin-polarization
to avoid inelastic collisions to lower-lying spin states has worked well in atomic systems (consider for example, sympathetic cooling of 6 Li with 23 Na [9]), whether thiswill result in favorable molecule-atom collisions is still an open question. Collisional
spin-relaxation in 3 Σ molecules has been previously studied showing that moleculeswith a large rotational constant have lower inelastic spin-relaxation rates, a resultthat is favorable considering the light mass of NaLi molecules [152]. Furthermore, it
has been calculated that collisional spin-relaxation can be controlled in magnetically
1 The Na was in the |1i state, the lowest Zeeman hyperne state, and the molecules were in theirhighest (stretched) Zeeman state.

132

and electrically dipolar molecules by a careful application of external magnetic andelectric elds [153, 52]. With spin-relaxation (theoretically) under control, there is
still the question of sticky collisions: Inelastic rates from sticky collisions scale withmass [49], and since the collision NaLi + Na involves relatively light masses one would
not expect severe losses, however an experimental demonstration is still lacking. If
all of these eectsreactivity, spin-relaxation, and sticky collisionscan be brought
under control, then sympathetic cooling of triplet NaLi molecules using Na atoms
is a real possibility, which would be an enormous step forward to reaching quantum
degeneracy with dipolar molecules.
The preceding example is just one possibility of an evaporation scheme uniqueto the NaLi molecule. Another common proposal is to use a 2D trapping geometry
and a transverse electric eld to polarize the molecules such that the dipole-dipole
interactions are repulsive, which should stabilize against sticky collisions or othershort-range loss processes [154]. All such schemes nd themselves at the conuence ofmany dierent collisional eects which need to be carefully understood. The best-caseoutcome is a method of evaporative cooling of dipolar molecules, while the worst-caseis a series of collisional studies with unfavorable results. Nonetheless, understanding
these kinds of collisional properties is a key area of research at the forefront of what
every (bi-alkali, directly cooled, etc.) molecules experiment will need to understandin order to reach quantum degeneracy. As illustrated in the example of NaLi + Nacollisions, the triplet NaLi systemwith an electric and magnetic dipole momentiswell suited to study many dierent eects in cold collisions and quantum chemistrywhich are relevant to the entire eld of cold molecules.

6.2 Simplifying experiments
The diculty of dealing with molecular collisions is evident in the recent history ofthe eld: The rst formation of an ultracold dipolar molecule, KRb, was nearly 10
years ago, and a quantum degenerate sample of dipolar molecules has still not beenachieved. Collisional loss is certainly one problem, but another is the sheer complexity133

of these experiments. Dual-species quantum degenerate mixtures, Feshbach molecule
formation, narrow laser locks for STIRAPall of these technical burdens build ahigh entry barrier to dipolar molecule formation experiments2 . Thus, we turn thediscussion towards a dierent question: Can the NaLi system simplify experimentswith molecules?
One experimental advantage of NaLi is in the ultracold atom number prior tomolecule formation. The collisional properties of Na-Li mixtures are favorable, and
it is typical to form a quantum degenerate Bose-Fermi mixture of 5 million atoms inboth species, about a factor of 10 larger than other bi-alkali experiments [155, 136,
156, 157]. These initially large atomic samples allow a competitive number of groundstate molecules (typically 3 × 104 ), despite being bottlenecked by the poor Feshbach
molecule formation eciency of only 3 %. Indeed, the Feshbach resonances in NaLi arenarrow and the most dicult to work with among the bi-alkalis. However, as discussed
in Sec. 4.5.2, recent experiments in Sr2 molecule formation have demonstrated theuse of STIRAP from an initial collisional statei.e. without the use of Feshbach
molecules [138]. By increasing the upleg coupling using a highly-vibrationally excitedstate in the c3 Σ+ NaLi potential and squeezing the Na-Li mixture to high density,
it is possible that free-to-bound STIRAP could form weakly bound NaLi molecules
which could then be transfered using the typical pathway (Sec. 4.2) to the triplet
ground state3 . Even if the eciency of such a process is a few percent, the large
initial Na-Li atom number would allow for the formation of ∼ 104 molecules without
the technical complexity of using a Feshbach resonance. Furthermore, since triplet
NaLi molecules have a magnetic moment and thus magnetically trappable, such afree-to-bound STIRAP could be done entirely in a magnetic trap. The prospect ofstarting with an ultracold magnetically trapped Na-Li mixture and coherently forming
molecules with only a laser pulse would greatly simplify the experiment (compare to
2 We have not mentioned the diculties of direct cooling experiments, but they too have signicanttechnical overhead. For example, see [26].

3 Note that the rst (free-to-bound) STIRAP would be to a weakly bound vibrational state (witha binding energy on a few GHz), thus making it possible to use frequency shifting elements (such as
an electro-optic modulator) for light generation, rather than the burdensome cavity locks describedin Chapter 2.

134the current molecular production sequence, as was shown in Fig. 5-13).
Another even simpler method for producing triplet NaLi molecules is by opticalpumping (described in Sec. 4.5.1). Since the triplet ground potential is so shallowthat it only contains 11 vibrational states, PA followed by spontaneous decay has ahigh probability of decaying to the triplet ro-vibrational ground state. Thus it may
be possible to produce triplet ground state NaLi molecules simply by PA in a dualspecies MOT [158, 159]. Detection may be possible with direct imaging of moleculeson the strong downleg transition (a3 Σ+ , vg = 0 to c3 Σ+ , v ∗ = 11, see Sec. 4.2), a technique already demonstrated in KRb [160]. While such a scheme introduces issues
with incoherent production and a dirty sample, these problems could be circumvented using a combination of magnetic trapping and resonant clean-out of atoms
and unwanted molecular states (as we have done in Sec. 5.5.1).
Thus, we see that there are possibilities for a much simpler experimental realization of triplet NaLi molecules. These techniques are not only applicable in NaLi:
Although the technique of pumping is particularly well suited for obtaining groundstate triplet molecules, the free-to-bound STIRAP could be employed in any ultracold association experiment (especially those in which the Feshbach resonances are
impractical to use [92]). Nevertheless, the large atom numbers achievable in Na-Li
mixture experiments provide a unique advantage to any ultracold association scheme.

6.3 Concluding remarks
In the near future, it is clear that triplet NaLi molecules have a place in performinginteresting quantum chemistry studies and simplifying the experimental diculty ofmolecule production. What about the far future? When will a lattice of dipolar
molecules be available for quantum simulation? Here, the answer is much less clear.
Collisional issues have been the primary barrier towards forming high lling fractionsin optical lattice experiments [38], which highlights the importance of understanding
molecule collisions for all experiments going forward. An alternative style to producing a molecule-based quantum simulator is with individual assembly, where the recent

135advent of ultracold atom microscopy and high-resolution optical trapping could allow
for assembly of a many-body system molecule-by-molecule, thus avoiding collisions
altogether [161, 162, 40]. Still, microscopy experiments have their own set of technicalchallenges, such as how a dipolar molecule may behave at very short distances neara microscope objective.
Though there are clearly great challenges in the eld of cold molecules, there isalso no shortage of physics to be studied. The NaLi molecule in its triplet stateprovides a new system and a unique approach to pushing the eld forward. The workof this thesis has been only to form these molecules, placing them on the cusp offurther scientic discovery.

136

Appendix A
Magnetic eld stability circuit
calibration
Constructing the feedback circuit for creating magnetic elds with a relative stability of 10−6 required extensive testing of the feedback with dierent components(details in Chapter 2). Fig. A-1 shows the current modulation bandwidths with using
a series IGBT (Mitsubishi CM1000HA-24H) in comparison to a parallel transistor(NTE2541), used in a test circuit similar to Fig. 2-6 but without the inductive coil,instead with a non-inductive cable as the load. As expected, the parallel transistor isable to modulate current much faster than the IGBT, however when driving an inductive coil the modulation bandwidth of both devices was limited by LC resonances fromthe coil and some stray capacitance, limiting the modulation bandwidth to ∼ 20 kHz(this limitation is the cause of the servo-bumps in Fig. 2-7). Fig. A-2 shows the performance of the active noise cancellation circuit when tested with injected noise frominductive coupling to another coil. Fig. A-3 and Fig. A-4 compares the noise performance for current sensing using a Hall probe (FW Bell CLN-300 and LEM LF 310-S)
and uxgate magnetometer (LEM UltraStab IT 700-S). We found the uxgate magnetometer to have far superior noise performance to the Hall probes, and thus theuxgate sensor was chosen for the noise cancellation circuit. Note that Fig. A-3 shows
the noise performance of the CLN-300 Hall probe, but the performance of the LF 310S probe is similar (− 130 dB noise at 1 kHz) and a noise-spectrum can be found on137

its datasheet. In all gures, the decibel level is calculated as 20 log (Vnoise /VDC ) where√
Vnoise is the noise spectral density of the measurement device in units of V/ Hz and

VDC is a reference voltage which is given in the gure captions.

Phase (Degrees)

Amplitude (dB)

20
0
−20
−40
−60

0
−45
−90
−135
−180

Parallel BJT
Series IGBT

−225
101

102

103

104

105

106

107

Frequency (Hz)Figure A-1: Modulation bandwidth of series IGBT vs. parallel transistor. The amplitude and phase of the modulation produced by a series IGBT (red) and a paralleltransistor (blue) are shown when driving a non-inductive load (i.e., we replaced thecoil with a short cable.) The parallel transistor shows a much higher modulation
bandwidth up to 10 MHz. The notches in modulation at ∼ 50 kHz are of unknown
origin, however they were worsened when driving the (inductive) coil. Driving an
inductive coil introduced resonances at ∼ 20 kHz, severely limiting the modulation
performance of the parallel transistor, however we did not debug this issue further.

138

Amplitude (dB)
Suppression Factor

0
−10
−20
−30
−40
−50
−60
−70

Injected NoisePS Voltage Feedback
Transistor Feedback

PS Voltage Feedback
Transistor Feedback

102
101
100
10−1
101

102

103

104

105

Frequency (Hz)Figure A-2: Magnetic eld stabilization test using injected noise. Current modulation
was induced in the Feshbach coil using cross-inductive coupling from modulation in
the spin-ip bias coil to simulate noise (top, blue curve). The injected modulation wasthen suppressed using only the slow feedback to the Feshbach power supply voltage
control (red curves), and the full feedback utilizing both the slow path and the fasttransistor feedback (purple curves). The current was measured using an LEM UltraStab IT 700-S. The top diagram shows the measured relative modulation amplitude,
showing injected noise up to 10 kHz. In the top diagram, the reference level for the
decibel calculation is arbitrary (it was a reference signal from a network analyzer)
and is physically meaningless. The bottom diagram shows the noise suppression factor when using active feedback, calculated from the dierence in curves of the top
diagram. In the bottom diagram, a servo-bump is visible when using full transistor
feedback which worsens noise performance at ∼ 30 kHz.

139

− 90

Hall Probe 0A
Hall Probe 108A
Preamp Floor

Noise amplitude (dB/Hz1/2)

− 100
− 110
− 120
− 130
− 140
− 150
− 160
− 170
10 1

10 2

10 3

10 4

10 5

10 6

f (Hz)

Figure A-3: FW Bell CLN-300 Hall probe noise performance. The noise spectra of
the pre-amplier without a Hall probe connected (gray), with the Hall probe on but
measuring zero current (blue), and with the Hall probe measuring 108 A of current(red) are shown. All noise spectra are referenced to the Hall probe signal output
when measuring 108 A of current. We see that the noise in the current measurement
is dominated by the Hall probe, and any additional noise from the coil power supply
is only slightly visible in the Hall sensor.

140

− 90

Ultrastab 0A
Ultrastab 108A
Preamp Floor

Noise amplitude (dB/Hz1/2)

− 100
− 110
− 120
− 130
− 140
− 150
− 160
− 170
10 1

10 2

10 3

10 4

10 5

10 6

f (Hz)

Figure A-4: LEM UltraStab IT 700-S uxgate magnetometer noise performance.
The noise spectra of the pre-amplier without the uxgate magnetometer connected(gray), with the uxgate magnetometer on but measuring zero current (blue), andwith the uxgate magnetometer measuring 108 A of current (red) are shown. All noise
spectra are referenced to the uxgate magnetometer signal output when measuring
108 A of current. In this case the uxgate magnetometer shows impressive noise
performance, barely adding any noise to the system when measuring a perfect (zero)current. When the current is switched to 108 A, the uxgate magnetometer is able
to clearly measure the noise of the coil power supply.

141

Appendix B
Dispersion coecients for Na, Li, andthe NaLi molecule
The ground state C6 coecients from theoretical calculations for Na and Li are

C6,Li-Li = 1389 au, C6,Na-Na = 1556 au, C6,Na-Li = 1467 au [50]. The C6,mol for a molecule
has two contributions, from the electronic excitation (similar to atoms), and an additional term originating from rotational excitations of the permanent dipole mo(el)(rot)

ment, C6,mol + C6,mol .

The electronic contribution can be approximated by sum-ming over all pair-wise atomic dispersion coecients, i.e. C6,mol = C6,Li-Li + C6,Na-Na +2C6,Na-Li = 5879 au. From second order perturbation theory, the rotational contribu(rot)tion can be approximated as C6,mol = d4 /6B where d is the permanent dipole moment
and B is the rotational constant [43]. Table B.1 shows some relevant properties of
the NaLi molecule with data compiled from literature, ab initio potentials, and theaforementioned dispersion coecient approximations.
23

Na6 Li
Dipole moment (Debye)
Rotational Constant (vg = 0) (GHz)
Polarizability (au)Electronic C6 (au)
Dipolar (rotational) C6 (au)a3 Σ+

X 1 Σ+

0.18 [44]4.6
365 [163]5879
60.5 [23]11.4
237 [163]
5879
144

Table B.1: Properties of the NaLi singlet and triplet ro-vibrational ground states.

142

Appendix C
Triplet ab initio potentials
The work in this thesis heavily utilized theoretical calculations of the NaLi triplet
potentials. The ground (a3 Σ+ ) triplet potential is reported in the X -representation
analytic form in [85], and it is tabulated for convenience in Table C.1. This potential
is based on an ab initio calculation slightly corrected for the position of the leastbound vibrational (Feshbach) state. Prior to our experiments, the excited triplet
states of NaLi were unobserved and thus we relied purely on ab initio potentials,
also shown in Table C.1. This data was provided to us by Prof. Olivier Dulieu, towhom we are grateful. With the experimental results of Chapter 3 and Chapter 4, wewere able to improve upon these potentials, and the resulting potentials and tting
methodology is reported in [58] and [59].
Table C.1: Table of ab initio potential values for thelowest three triplet potentials of NaLi. All values are in
atomic units.
Internuclear Distance (a0 )
2.20
2.40
2.60
2.80
3.00
3.20
3.40
3.60

a3 Σ+

b3 Π

8.298786028 0.178444959
4.921037500 0.153316981
3.041868538 0.131343365
1.947723817 0.112170707
1.285362500 0.095472700
0.870649674 0.080999428
0.603234445 0.069509551
0.426266630 0.058019673
Continued on next page143

c3 Σ +
0.283498199
0.255513453
0.229375255
0.204847433
0.181869054
0.160572470
0.142238783
0.123905096

Table C.1  continued from previous page

Internuclear Distance (a0 )
3.80
4.00
4.20
4.40
4.60
4.80
5.00
5.20
5.40
5.60
5.80
6.00
6.20
6.40
6.60
6.80
7.00
7.20
7.40
7.60
7.80
8.00
8.20
8.40
8.60
8.80
9.00
9.20
9.40
9.60
9.80
10.00
10.20
10.40
10.60
10.80
11.00
11.20
11.40
11.60
11.80
12.00

a3 Σ+

b3 Π

0.306424210 0.049221901
0.223577634 0.042027238
0.165234976 0.036929826
0.123454347 0.031832413
0.093075166 0.028515972
0.070677026 0.026179881
0.053951581 0.024680354
0.041315011 0.023888606
0.031663974 0.023691731
0.024219000 0.023990920
0.018422235 0.024699563
0.013869604 0.025742048
0.010265125 0.027052795
0.007389717 0.028575214
0.005079631 0.030260658
0.003211354 0.032067342
0.001690912 0.033959384
0.000446231 0.035905955-0.000285653 0.037880504-0.000528270 0.039860147-0.000711634 0.041825100-0.000846497 0.043758260-0.000941536 0.045644834-0.001004219 0.047472096-0.001040898 0.049229153-0.001056870 0.050906843-0.001056517 0.052497640-0.001043440 0.053995617-0.001020597 0.055396427-0.000990410 0.056697295-0.000954856 0.057897006-0.000915548 0.058995868-0.000873788 0.059995637-0.000830625 0.060899380-0.000786897 0.061711330-0.000743263 0.062436660-0.000700236 0.063081271-0.000658210 0.063651528-0.000617479 0.064154033-0.000578255 0.064568070-0.000540685 0.064982107-0.000504865 0.065320280
Continued on next page144

c3 Σ +
0.108827179
0.095923772
0.086001547
0.076079322
0.068734094
0.062811473
0.058100788
0.054413934
0.051587457
0.049480342
0.047971395
0.046957101
0.046349562
0.046074445
0.046069069
0.046280700
0.046665093
0.047185228
0.047810200
0.048514260
0.049275983
0.050077559
0.050904201
0.051743649
0.052585767
0.053422203
0.054246118
0.055051955
0.055835250
0.056592483
0.057320934
0.058018586
0.058684025
0.059316362
0.059915160
0.060480370
0.061012283
0.061511471
0.061978744
0.062400238
0.062821732
0.063199892

Table C.1  continued from previous page

Internuclear Distance (a0 )
12.20
12.40
12.60
12.80
13.00
13.20
13.40
13.60
13.80
14.00
14.20
14.40
14.60
14.80
15.00
15.20
15.40
15.60
15.80
16.00
16.20
16.40
16.60
16.80
17.00
17.20
17.40
17.60
17.80
18.00
18.20
18.40
18.60
18.80
19.00
19.20
19.40
19.60
19.80
20.00
20.20
20.40

a3 Σ+

b3 Π

-0.000470846 0.065615658-0.000438646 0.065873512-0.000408258 0.066098602-0.000379652 0.066295181-0.000352785 0.066467020-0.000327601 0.066617416-0.000304038 0.066749252-0.000282025 0.066865038-0.000261491 0.066960919-0.000242359 0.067056799-0.000224556 0.067136240-0.000208007 0.067206636-0.000192639 0.067269171-0.000178379 0.067321875-0.000165159 0.067374578-0.000152913 0.067419077-0.000141577 0.067458999-0.000131089 0.067494905-0.000121394 0.067527273-0.000112435 0.067556518-0.000104161 0.067583001-0.000096524 0.067605942-0.000089477 0.067628883-0.000082979 0.067648792-0.000076988 0.067666963-0.000071466 0.067683578-0.000066379 0.067698797-0.000061693 0.067712757-0.000057378 0.067725583-0.000053404 0.067737384-0.000049746 0.067748256-0.000046377 0.067757904-0.000043275 0.067767551-0.000040418 0.067775801-0.000037803 0.067784051-0.000035330 0.067791405-0.000033046 0.067798227-0.000030934 0.067804563-0.000028979 0.067810455-0.000027168 0.067815937-0.000025489 0.067821045-0.000023931 0.067825807
Continued on next page145

c3 Σ +
0.063550963
0.063876380
0.064177605
0.064456114
0.064713369
0.064950804
0.065169815
0.065371740
0.065550563
0.065729387
0.065887457
0.066033133
0.066167414
0.066286411
0.066405408
0.066510761
0.066607995
0.066697789
0.066780740
0.066857416
0.066928322
0.066991488
0.067054655
0.067110904
0.067163030
0.067211358
0.067256192
0.067297803
0.067336440
0.067372339
0.067405707
0.067435661
0.067465615
0.067491576
0.067517537
0.067540869
0.067562623
0.067582241
0.067601859
0.067619544
0.067636069
0.067651513

Table C.1  continued from previous page

Internuclear Distance (a0 )
20.60
20.80
21.00
21.20
21.40
21.60
21.80
22.00
22.20
22.40
22.60
22.80
23.00
23.20
23.40
23.60
23.80
24.00
24.20
24.40
24.60
24.80
25.00
25.20
25.40
25.60
25.80
26.00
26.20
26.40
26.60
26.80
27.00
27.20
27.40
27.60
27.80
28.00
28.20
28.40
28.60
28.80

a3 Σ+

b3 Π

-0.000022484 0.067830250-0.000021138 0.067834399-0.000019887 0.067838278-0.000018722 0.067841907-0.000017636 0.067845304-0.000016624 0.067848486-0.000015680 0.067851470-0.000014797 0.067854268-0.000013973 0.067856896-0.000013202 0.067859152-0.000012481 0.067861407-0.000011806 0.067863663-0.000011173 0.067865918-0.000010580 0.067867850-0.000010023 0.067869670-0.000009501 0.067871384-0.000009010 0.067873001-0.000008548 0.067874525-0.000008115 0.067875964-0.000007707 0.067877323-0.000007322 0.067878605-0.000006960 0.067879817-0.000006619 0.067880962-0.000006298 0.067882045-0.000005994 0.067883043-0.000005708 0.067884040-0.000005437 0.067884934-0.000005182 0.067885828-0.000004940 0.067886652-0.000004712 0.067887434-0.000004495 0.067888105-0.000004291 0.067888775-0.000004097 0.067889446-0.000003913 0.067890116-0.000003739 0.067890787-0.000003574 0.067891361-0.000003418 0.067892045-0.000003269 0.067892563-0.000003128 0.067893055-0.000002994 0.067893524-0.000002867 0.067893970-0.000002746 0.067894395
Continued on next page146

c3 Σ +
0.067665959
0.067679474
0.067692128
0.067703980
0.067715087
0.067725500
0.067735270
0.067744438
0.067753048
0.067760440
0.067767831
0.067775222
0.067782614
0.067788945
0.067794906
0.067800521
0.067805815
0.067810806
0.067815515
0.067819960
0.067824156
0.067828120
0.067831865
0.067835407
0.067838667
0.067841926
0.067844847
0.067847768
0.067850459
0.067853011
0.067855196
0.067857381
0.067859565
0.067861750
0.067863935
0.067865711
0.067867486
0.067869261
0.067870685
0.067872108
0.067873531
0.067874954

Table C.1  continued from previous page

Internuclear Distance (a0 )
29.00
29.20
29.40
29.60
29.80
30.00
30.20
30.40
30.60
30.80
31.00
31.20
31.40
31.60
31.80
32.00
32.20
32.40
32.60
32.80
33.00
33.20
33.40
33.60
33.80
34.00
34.20
34.40
34.60
34.80
35.00
35.20
35.40
35.60
35.80
36.00

a3 Σ+

b3 Π

c3 Σ +-0.000002631-0.000002521-0.000002417-0.000002318-0.000002224-0.000002134-0.000002048-0.000001966-0.000001889-0.000001814-0.000001743-0.000001676-0.000001611-0.000001549-0.000001490-0.000001434-0.000001380-0.000001329-0.000001279-0.000001232-0.000001187-0.000001144-0.000001102-0.000001063-0.000001025-0.000000988-0.000000953-0.000000920-0.000000888-0.000000857-0.000000828-0.000000799-0.000000772-0.000000746-0.000000721-0.000000697

0.067894800
0.067895186
0.067895553
0.067895904
0.067896238
0.067896557
0.067896861
0.067897152
0.067897429
0.067897695
0.067897948
0.067898190
0.067898422
0.067898643
0.067898855
0.067899058
0.067899252
0.067899437
0.067899615
0.067899786
0.067899949
0.067900106
0.067900256
0.067900400
0.067900538
0.067900670
0.067900797
0.067900919
0.067901036
0.067901149
0.067901257
0.067901361
0.067901461
0.067901556
0.067901649
0.067901737

0.067876377
0.067877800
0.067879003
0.067880145
0.067881235
0.067882275
0.067883267
0.067884229
0.067885120
0.067884578
0.067885389
0.067886164
0.067886906
0.067887615
0.067888293
0.067888942
0.067889563
0.067890158
0.067890728
0.067891273
0.067891796
0.067892297
0.067892778
0.067893239
0.067893681
0.067894105
0.067894511
0.067894902
0.067895277
0.067895637
0.067895983
0.067896316
0.067896635
0.067896942
0.067897237
0.067897521

147

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