A Quantum Gas of Polar MoleculesbyKang-Kuen Ni
B.S. Creative Studies, University of California, Santa
Barbara, 2003

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2009

iii
Ni, Kang-Kuen (Ph.D., Physics)
A Quantum Gas of Polar Molecules
Thesis directed by Professor Dr. Deborah S. Jin

Ultracold polar molecular gases promise new directions and exciting applications in precision measurements, ultracold chemistry, electric-field controlledcollisions, dipolar quantum gases, and quantum information sciences. This thesis presents experimental realization of a near quantum degenerate gas of polar
molecules, where the phase-space density of the gas achieved is more than 10 orders of magnitude higher than previous results. The near quantum degenerate
gas of polar molecules is created using two coherent steps. First, atoms in an ultracold gas mixture are converted into extremely weakly bound molecules near a
Fano-Feshbach resonance. Second, the weakly bound molecules are transferred to
the ro-vibronic ground state using a coherent two-photon Raman technique. The
fact that these ground-state molecules are polar is confirmed with a spectroscopic
measurement of the permanent electric dipole moment. Finally, manipulation of
the molecular hyperfine state is demonstrated; this allows molecules to be populated in a single quantum state, in particular, the lowest energy state. Withan ultracold gas of molecules, full control of molecular internal state, and electricfield as a new handle, ultracold molecular collisions, including ultracold chemical
reactions and dipolar collisions, are studied.

Dedication

For Ruey-Shiang, Wei-Tou, Kangyu, and Tillv

Acknowledgements

First, I would like to thank my advisors, Debbie Jin, Jun Ye, and Carl
Wieman. Carl convinced me to come to JILA and started up the KRb project withanother graduate student, Josh Zirbel. Although he was only around during myfirst three years of graduate school when we were still building the main apparatus,
I have learned tremendous amount of technical skills as well as scientific ways ofthinking from him. Here, I would like to share a private email from Carl on
his thought as what one should learn in graduate school and hopefully it also
summed up the excellent education I got here in JILA. “The thing that I believe
is absolutely essential is that the students leave graduate school able to function
as independent scientists. This means understanding and evaluating what topics
in their field are worthwhile (or not), and being able to design and carry out good
research projects. For my students, that includes getting a lot of expertise in avariety of experimental technologies and techniques, being able to design and build
complex apparatus and make it work, being able to carry out and analyze goodexperiments, be able to present their work well both in speaking and writing,and being able to work eﬀectively with collaborators. And of course there islearning to work hard in a focused eﬃcient manner. These are all the aspects ofwhat it takes to be a good scientist, and I believe that if a student masters thisworking on something like a cold atom experiment with me, they should be able
to move successfully into many other areas of research or teaching later on as theychoose...”vi
I am privileged to continue and finish my thesis with Debbie. She became
my advisor during the period that I have learned and grown the most scientificallyin grad school. Debbie has an amazing intuition in physics. She never let all thetechnical and even conceptual details get in the way of understanding the simplest
and the big picture ideas while still being extremely rigorous and careful. I thankher for being patient and critical of my writings, my presentations, and my being
as a student and a researcher. I am alway amazed by how much she is available tous. There were countless times when we would chat from experimental problems
to things in daily life for hours and hours.
I am also very fortunate to work and to learn from Jun. His optimistic andenthusiastic spirits have encouraged me numerous times when I was feeling pessimistic or frustrated. In addition, he has an infinite bank of technical knowledge
and physics intuitions that always help to solve our problems. He is one of the
few advisors who can still turn knobs on an optical table and make things better!
I have benefited as a physicist and a person from spending long hours a day
for many years with Josh Zirbel and Silke Ospelkaus. Josh has infinite patientteaching me how to align optics, build circuits, prepare vacuum, and diagnose
problems. We have spent many years together in the dark basement constructing our apparatus. Although we built the apparatus together, most of the credit
should go to him for leading the eﬀort and setting the high standards for making
things work well in the lab. The robust KRb system is the key to our success thatboth Josh and I as well as other KRb members have benefited greatly from. I
am very excited that both of us have survived grad school after years of discouragements and frustrations from all the technical and the scientific challenges andnow be ready for anything in the future.
This thesis would not have been accomplished in the “timely” fashion without the collaboration with Silke. Silke joined Josh and I at the time when wevii
were working toward our first result. She immediate taught us how to work eﬃciently toward fruitful results. Over the next two and a half years, I have learned
tremendous amount from her and enjoy working with her greatly. Together with
Silke, there is nothing we cannot accomplish!
I would also like to thank other members on the project, Avi Pe’er, Marcio Miranda, Brian Neyenhuis, and Dajun Wang, who contributed a great deal
in making this thesis happen. Avi had been thinking about making moleculeseven before joining our collaborated project and immediately elevated my understanding of working toward making deeply bound molecules. Marcio joined us
with his expertise of working with a frequency comb. He had always helped us
advancing toward our next goal. Brian had joined us a few years back and helpedwith numerous projects allowed us to be always progressing. Dajun joined witha great amount of knowledge of KRb spectroscopy and helped us make our way
through our own vast amount of spectroscopy data. Silke and I have now left thiswonderful experiment in the hands of Marcio, Brian, and Dajun, who I am sure
will soon bring more exciting news from the project.
Our project would not have been successful without collaborations withmany theorists. Svetlana Kotochigova and Paul Julienne have helped us find away to make absolute ground-state molecules. They have been extremely niceand generous with their time to be always answering our questions with pages of
emails promptly. Goulven Quéméner and John Bohn are in-house theorists who
happen to be the world experts on polar molecules and ultracold collisions. I
thank them for their theoretical insights and for never saying “no” to any thing
we asked them to calculate. I enjoy discussing and talking to them with all the
exciting happenings with KRb even outside the dedicated three continuous hours
every week. I also like to thank Dominic Meiser who is an expert on Mathematica
for always helping me with anything Mathematica.

viii
JILA has been a wonderful place to work. With the collaborative atmosphere in JILA, we have numerous weekly meetings. One of them is the “trigroup” meeting. I thank Eric Cornell and all the members of the tri-group, who
are the most supportive fellow group members, for always sharing their expertise, equipments, and most importantly lunch discussions. I thank the Ye groupmembers, especially Brian Sawyer and Ben Stuhl who work on Stark decelerating
molecules, for sharing expertise and helping me simulate electric fields that are
relevant for our experiments. I also thank friends in graduate school, in particular Stephanie Meyer, David Hume, Adam Pearlstein, Brian Seaman, Rajiv Bhat,
Meret Kramer, Murray Holland, Ying-ju Wang, and Wengang Zhang.
Members of JILA electronic and machine shops and IT department, in particular, Terry Brown, Carl Sauer, Mike Whitmore, James Fung-A-Fat, Blaine
Horner, Hans Greene, Tracy Keep, Todd Asnicar, Tom Foote, Ariel Paul, Kim
Hagen, and J. R. Raith, are the professional help who have contributed to ourwork. I thank all the JILA staﬀ who had made it easy with all the purchasing,traveling, computer related issues, etc. In particular, I thank Krista Beck for her
assistance over the years while being an assistant to Carl and Debbie.
My family have provided me with constant support even when they are far
away. My father (Dr. Ni #1 in my family) has always been enthusiastic about
discussing physics through long-distance phone calls and helped me stay motivatedthroughout the years. My mother has provided infinite support and always has
great confidence in me. My sister, who has became Dr. Ni #7 in my family lastyear was in grad school with me throughout the same time. Going through grad
school together encouraged me during many periods of low time.
Finally, I would like to thank Till Rosenband for his Mathematica and ANSYS wizardry. I thank him for his unconditional support and his company fornumerous adventures together.

Contents

Chapter
1 Introduction

1

1.1

Why Ultracold Polar Molecules? . . . . . . . . . . . . . . . . . . .

1

1.2

Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Dipole-Dipole Interaction . . . . . . . . . . . . . . . . . . . . . . .

8

1.3.1

Ultracold Collisions of Dipoles . . . . . . . . . . . . . . . .

10

1.3.2

Dipolar Quantum Gas . . . . . . . . . . . . . . . . . . . .

15

Contents of this Thesis . . . . . . . . . . . . . . . . . . . . . . . .

16

1.4

2 Challenges for Making Ultracold Polar Molecules

17

2.1

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

17

2.2

Direct Cooling of Molecules . . . . . . . . . . . . . . . . . . . . .

18

2.3

Photo-association . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.4

Fully Coherent Scheme . . . . . . . . . . . . . . . . . . . . . . . .

23

3 Apparatus

26

3.1

K-Rb Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.2

Electric-field Plates . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.3

Molecular Spectroscopy Lasers . . . . . . . . . . . . . . . . . . . .

32

3.3.1

Widely Tunable Ti:sapphire Laser . . . . . . . . . . . . . .

32

3.3.2

Diode Lasers . . . . . . . . . . . . . . . . . . . . . . . . .

33

x
3.3.3

Ti:sapphire Comb . . . . . . . . . . . . . . . . . . . . . . .

34

3.3.4

Phase Locking . . . . . . . . . . . . . . . . . . . . . . . . .

35

4 Feshbach Molecules
4.1

40

Heteronuclear Feshbach Molecules . . . . . . . . . . . . . . . . . .

41

4.1.1

Conversion Eﬃciency . . . . . . . . . . . . . . . . . . . . .

43

4.1.2

Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.1.3

Electric Dipole Moment . . . . . . . . . . . . . . . . . . .

46

4.2

Enhancement Factor . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.3

Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

5 KRb* Potentials
5.1

54

Hund’s Coupling Cases . . . . . . . . . . . . . . . . . . . . . . . .

54

5.1.1

Hund’s Case (a) . . . . . . . . . . . . . . . . . . . . . . . .

56

5.1.2

Hund’s Case (b) . . . . . . . . . . . . . . . . . . . . . . . .

56

5.1.3

Hund’s Case (c) . . . . . . . . . . . . . . . . . . . . . . . .

57

Near-Threshold States . . . . . . . . . . . . . . . . . . . . . . . .

57v � = −14 of 2(0− ) . . . . . . . . . . . . . . . . . . . . . . .

62

5.3

Bridging to Deeply Bound Molecules . . . . . . . . . . . . . . . .

62

5.4

23 Σ+ and the Ω = 0− and Ω = 1 Components . . . . . . . . . . .

63

5.4.1

v � = 10 of 23 Σ+ (Ω = 0− ) . . . . . . . . . . . . . . . . . . .

65

5.4.2

v � = 23 of 23 Σ+ (Ω = 1) . . . . . . . . . . . . . . . . . . .

70

5.2

5.2.1

6 Coherent Two-Photon Raman Transfer

72

6.1

Three-level System . . . . . . . . . . . . . . . . . . . . . . . . . .

73

6.2

Dark Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

6.3

Detuned STIRAP, v = −3 . . . . . . . . . . . . . . . . . . . . . .

79

6.4

Extending STIRAP to Deeply Bound States . . . . . . . . . . . .

81

xi
7 Triplet Rovibrational Ground-State Molecules(N = 0, v = 0 of a3 Σ+ )

87

7.1

Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

7.2

Hyperfine-rich Ground-state Structure . . . . . . . . . . . . . . .

89

7.3

Stark Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . .

93

7.4

Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

8 Absolute Ro-Vibronic Ground-State Polar Molecules(N = 0, v = 0 of X 1 Σ+ )

98

8.1

Transfer Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.2

Two-photon Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 100

8.3

Stark Spectroscopy Analysis . . . . . . . . . . . . . . . . . . . . . 102

8.4

Population Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.5

Trapped Molecular Temperature and Density . . . . . . . . . . . . 109

9 Identifying and Manipulating Molecular Hyperfine States

114

9.1

Identify Hyperfine States . . . . . . . . . . . . . . . . . . . . . . . 116

9.2

Manipulate Molecular Hyperfine States . . . . . . . . . . . . . . . 117
9.2.1

State Transfer to the Lowest Hyperfine State . . . . . . . . 117

9.2.2

More Hyperfine Spectroscopy . . . . . . . . . . . . . . . . 120

10 Ultracold Chemistry and Inelastic Collisions

123

10.1 Atom-Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . 124
10.2 Universal Loss Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10.3 Molecule-Molecule Collisions . . . . . . . . . . . . . . . . . . . . . 129
10.3.1 p-wave Collisions . . . . . . . . . . . . . . . . . . . . . . . 131
10.3.2 s-wave Collisions . . . . . . . . . . . . . . . . . . . . . . . 135

xii
11 Electric Dipole-Dipole Collisions

137

11.1 Working with Electric Dipoles . . . . . . . . . . . . . . . . . . . . 137
11.1.1 E-field Gradients . . . . . . . . . . . . . . . . . . . . . . . 138
11.1.2 Fringing Fields and a Non-symmetric Ground Plane . . . . 140
11.2 Inelastic Dipolar Collisions . . . . . . . . . . . . . . . . . . . . . . 143
11.2.1 Anisotropy of Losses . . . . . . . . . . . . . . . . . . . . . 145
11.2.2 Anisotropic Heating. . . . . . . . . . . . . . . . . . . . . 147

11.3 Elastic Dipolar Collisions . . . . . . . . . . . . . . . . . . . . . . . 150
11.4 Cross-dimensional Rethermalization . . . . . . . . . . . . . . . . . 151
12 Conclusions and Future Directions

155

12.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Bibliography

158

Appendix
A Sample Mathematica Code168

B Tapered Amplifier Design

172

Chapter 1
Introduction

1.1

Why Ultracold Polar Molecules?
The brand new field of ultracold polar molecules is motivated by the successof ultracold atoms. Ultracold atomic gases have provided many exciting and diverse applications including atomic clocks, precision tests of fundamental physics,
sensors of tiny forces, and new architectures for quantum information. Thesesystems are miniature laboratories for studying ultracold collisions, resonances,and few-body physics. They also provide model systems where one can explore
many-body quantum phenomena such as Bose-Einstein condensation, superfluidity, and Fermi superfluidity. The underlying basis for these many uses is thatultracold atomic gases can be trapped, interrogated for long times, and precisely
controlled at the quantum level, in both the atoms’ internal (i.e. electronic, fine,and hyperfine) and external degrees of freedom (i.e. motional state in the laboratory frame). Extending this type of control to an ultracold gas of polar molecules
opens exciting new research directions because of the richer energy level structureof molecules, the possibility for reactive collisions, and the relatively strong andlong-range electric dipole-dipole interaction between polar molecules.
The rich energy level structure of molecules, which includes rotational andvibrational levels, opens new possibilities for precision measurements [1, 2, 3].
One example is the search for a permanent electric dipole moment of the electron,

2
where polar molecules can provide much larger internal electric fields than could
be directly applied to electrons in the lab or inside an atom. The more complex
internal states of polar molecules will also enable the development of new tools forcontrolling and manipulating ultracold gases. In the arena of ultracold collisions,polar molecule gases oﬀer electric-field tuning of interactions [4], access to new
types of resonances [5], and ultracold chemistry [6]. For quantum information
applications, the relatively strong dipole-dipole interaction, which can be switchedon and oﬀ as desired with an applied external electric field, oﬀers a means forcontrolled creation of entangled states [7, 8, 9]. The dipole-dipole interaction,
which is long-range as opposed to the contact interaction of ultracold atoms, could
also be exploited in quantum gases of polar molecules to realize novel phases ofmatter [10, 11, 12].
1.2

Overview of the Thesis
This thesis tells the story of our quest to make and study a quantum gas ofground-state polar molecules. This has been a long-standing goal in the ultracold
gas community since the creation of quantum degenerate atomic gases more thana decade ago [13, 14]. The work described in this thesis demonstrated creation of
an ultracold gas of ground-state polar molecules [15] that is more than 10 ordersof magnitude higher in phase-space density than previously achieved and is only afactor of 15 in phase-space density away from quantum degeneracy. Phase-spacedensity is defined as nΛ3 , where n is the peak density of the gas and Λ is thethermal de Broglie wavelength.
In 2002, Carl Wieman, Debbie Jin, and Jun Ye at JILA began planning acollaborative ultracold molecule experiment that would use their respective expertise. This was during the exciting time when the Wieman group was using
a Fano-Feshbach resonance to form Rb2 Feshbach molecules in a Bose-Einstein3
condensate and seeing coherent atom-molecule oscillations [16], and the Jin group
had just observed the first evidence of K2 Feshbach molecules in a quantum degenerate Fermi gas [17]. It was thought that as a first step toward ultracold polar
molecules, we should develop the technology of making and understanding heteronuclear Feshbach molecules. Then, we would team up with the Ye group to finda way of transferring Feshbach molecules to the absolute ground state. Shortly
after formulating the idea, Josh Zirbel and I began constructing our apparatus
in the summer of 2003. A detailed description of the apparatus is in J. Zirbel’s
thesis [18].
Making a quantum gas of polar molecules is challenging in part due tothe same complex internal structure that makes molecules an interesting andricher system compared to an atomic system. There are, in general, two coolingapproaches. One is to directly cool molecules to low translational temperatures.
However, all the cooling methods demonstrated so far can only reach down to themK regime with a trapped gas density of 108 cm−3 or below. This corresponds
to a phase-space density of the order 10−13 [19, 20, 21]. The other approach is
to associate pairs of ultracold atoms into tightly bound molecules. The maindiﬃculty with this approach is to find a way to eﬃciently convert atoms that areusually very far apart to tightly bound molecules without allowing the released
binding energy to heat up the gas. In 2005, Sage et al. [22] demonstrated a scheme
where they could make about 10 RbCs molecules in their absolute rovibrational
ground-state from 108 laser-cooled atoms in a single transfer cycle, with a gas
phase-space density achieved on the order 10−14 . Their result highlighted a generalscheme as well as the challenges to convert ultracold atoms into tightly boundmolecules.
To make a quantum gas of polar molecules in their absolute ground state(lowest vibrational, rotational, and electronic ground state), we perform a two-4
step coherent transfer process. In the first step, we make weakly bound Feshbach
molecules from a quantum degenerate mixture of K and Rb atoms. In the second
step, we shrink the size of the molecules to reach the absolute ground state using asingle-step of two-photon Raman transfer. The first step is described in J. Zirbel’s
thesis [18]. Although I will cover the whole story in this thesis, I will focus more on
the second step, which is surprisingly eﬃcient. This “surprising” result was built
on a step-by-step learning process that started after making and understanding
KRb Feshbach molecules in 2006 (in Hamburg) [23] and 2007 (in our group) [24].
The work presented in this thesis was done in collaboration with many people, inparticular with Silke Ospelkaus.
Once weakly bound Feshbach molecules are available, it is necessary to use
a fully coherent transfer technique to manipulate their internal state to reach theabsolute ro-vibronic ground state. Coherent transfer ensures that the molecules all
populate a single internal quantum state and that the ultracold gas is not heatedin the process. In 2007, Winkler et al. [25] demonstrated robust and highly
eﬃcient state transfer of homonuclear Rb2 Feshbach molecules to their second
least bound state (vibrational level v = −2 and binding energy Eb = h · 637
MHz) using a coherent two-photon Raman transfer technique. We implemented
the same technique in our heteronuclear system (which would allow us eventually
to make polar molecules), and demonstrated transfer of KRb Feshbach molecules
to another near-threshold vibrational state, v = −3 (h · 10 GHz) [26]. The eﬃcienttransfer relies on (1) identifying a suitable electronically excited intermediate statethat has large Franck-Condon factors (FCFs) to both the initial and the target
states and (2) maintaining phase coherence of the two laser beams that drive thetransfer process. For transfer from Feshbach molecules to another near-threshold(weakly bound) molecular state, a suitable intermediate state is relatively easy
to find, although one must consider possible constraints from selection rules. It

5turns out that the most important thing is to find an isolated line that makes thesystem an “ideal” three-level system.
For transfer to near-threshold molecular states, maintaining phase coherence of the two Raman beams, which have a beat frequency of tens of GHz or
below is straightforward using good electronics and commercial photodiodes andsynthesizers. However, it is a challenge to extend the technique for our next stepof reaching more deeply bound states. To allow transfer to any arbitrary deeper
bound molecular state and even to the absolute ground state, we implemented astabilized optical frequency comb that spans the wavelength range from 532 nm
to 1064 nm. The comb laser provides a stable reference that allows us to maintainphase coherence of any two or more lasers in this large wavelength range.
With a Raman laser system capable of reaching deeply bound molecularstates, the next issue is finding a suitable intermediate state that allows us tobridge the enormous diﬀerence between the wave functions of Feshbach moleculesand deeply bound molecules. Although our ultracold gas apparatus typicallyyields spectroscopy data with extremely high sensitivity and accuracy, it has arelatively low duty cycle compared to conventional spectroscopy experiments.
Therefore, we needed to choose a scheme carefully and used all available datafrom conventional spectroscopy to narrow down our search. Given our previous
experience in searching for intermediate states near threshold, it appeared to bea daunting task to follow the vibrational series (for both excited state and groundstate) from threshold to the deeply bound regime. For example, the ground state
potentials alone support 132 vibrational levels.
It was after discussions with our theoretical collaborators, Svetlana Kotochigova and Paul Julienne, that we understood both the advantages and thechallenges for instead directly searching for a low vibrational (low-v) ground state.
The advantage is that conventional hot molecule spectroscopy has precise infor-

6
mation specifically for the low-v states [27, 28]. Their data allows an accurate
construction of the molecule potential in the region where it is, to a good approximation, harmonic, which yields a relatively small uncertainty of the absolute
ground-state location. Therefore, if we could identify a suitable intermediate
state with good FCFs to both the initial Feshbach molecules and our desired
low-v state, the search for the target state would have a finite range to cover.
Identifying such an intermediate state in the “forest” of excited-state potentials
also required significant theoretical calculations and a full knowledge of what experimental spectroscopy was already available.
Working in the low-v regime, we first demonstrated transfer to the triplet
rovibrational ground state. Encouraged by these results, we then pursued transfer
to the absolute ro-vibronic ground state. We chose a scheme that used an intermediate state that was similar to the photoassociation work by Sage et al. [22].
This choice was partially based on the demonstration by Sage et al. for RbCs
and partially based on the fact that related spectroscopy data was available [27].
Although the predicted transition strengths were relatively small, we were hopeful that we could achieve reasonable coupling strengths because we were working
with a near quantum degenerate gas where the cloud size is very small and thecoupling Raman lasers can be focused to a very tight spot for a high intensity.
In 2008, to the surprise of the community and to our own delight, we wereable to perform a single-step of STImulate Raman Adiabatic Passage (STIRAP)
to transfer 90% of Feshbach molecules to the absolute ground state. At nearly
the same time, two other groups reported using a similar technique to transfer
homonuclear Cs2 and Rb2 Feshbach molecules into their respective deeply boundstates [29, 30]. Since then, we have identified and manipulated the hyperfine stateof the ground-state molecules. In particular, we can create molecules that not only
populate a single hyperfine state, but we also have the ability to put them in any

7desired hyperfine state. The robust production of absolute ground-state molecules
in any desired hyperfine state aided in our first exploration of the basic collisional
properties of ground-state KRb molecules. We investigated inelastic/reactive forboth atom-molecule and molecule-molecule collisions. For atom-molecule collisions, we could prepared all species, K, Rb, and KRb, in the lowest energy states.
We observed KRb loss due to ultracold chemical reactions with K. When either Rb
or KRb were in an excited hyperfine state, we observed hyperfine-state-changing
collisions. We then removed all the atoms and began to explore molecule-molecule
collisions. The KRb molecules we create are indistinguishable fermions preparedat a temperature of a few hundreds of nano-Kelvin, which is below the heightof the p-wave barrier of kb ·24 µK [31]. Molecules collisions via s-wave are notallowed by the quantum statistics and collisions via p-wave are suppressed by thebarrier. We observed molecule collisions via tunneling through the p-wave barrier
followed by a near unit-probability of chemical reactions at short-range. For distinguishable fermions (molecules prepared in two diﬀerent spin states) and s-wave
collisions, the collision rates are enhanced by a factor of 10-100.
Understanding basic collisional properties of molecules allowed us begin to
explore one unique feature of polar molecules – their large and tunable electric
dipole moment. In the presence of an external electric field, the molecules begin
to polarize along the field direction and the dipole-dipole interactions between
molecules begin to play an important role in collisions. In addition, depending on
the orientation of dipoles, “head-to-tail” collisions between dipoles are attractiveand “side-by-side” collisions are repulsive. The attractive (or the repulsive) interaction eﬀectively lowers (or raises) the p-wave collisional barrier and subsequently
enhances (or reduces) the chemical reaction rate for KRb-KRb. We explored thisanisotropic nature of dipolar collisions by looking at both the inelastic loss andthe corresponding temperature evolution of the molecular gas.

8
1.3

Dipole-Dipole Interaction
Going back to the motivations for ultracold polar molecules, in many appli-

cations, the specific chemical nature of molecules is not critical, and the essential
ingredient is having a reasonable large permanent electric dipole moment. The
−
→−
→
−
→
−
→
→
−
d1 ·b
r )(d2 ·br)dipole-dipole interaction, in SI units, has the form d1 ·d2 −3(, where d1 and
34π�o r→−−d2 are the two dipole moments, �o is the permittivity of vacuum, and →r is theseparation between the two dipoles. The dipole-dipole interaction has two features that are very diﬀerent from typical interactions between ultracold atoms.
First, the dipole-dipole interaction falls oﬀ as 1/r3 and therefore has a long-range
character. In contrast, the interparticle potentials between atoms and/or nonpolar molecules can be adequately described as a “contact interaction,” which
can to an excellent approximation be simply treated as a delta-function. Second,the dipole-dipole interaction is spatially anisotropic, and, as can be seen from theexpression above, the interaction can be either attractive or repulsive depending−on the orientation of the two dipoles with respect to the intermolecular vector, →r.
In contrast, the interaction between atoms at ultracold temperatures is typicallylimited by finite angular momentum barriers to spatially isotropic s-wave interactions, i.e. where the relative orbital angular momentum is zero. This special
property of polar molecules permits the experimentalists to control the relative
spatial orientation, and thereby tune the intermolecular interaction with external
electric fields.
Samples of ultracold gases with dipole-dipole interactions have been realizedusing chromium atoms in a Bose-Einstein condensate [32]. Chromium has an exceptionally large magnetic dipole moment of 6 µb , where µb is the Bohr magneton.
In these pioneering experiments, the anisotropic character of the dipole-dipole interaction has been beautifully revealed [33, 34]. One of the key advantages of9polar molecules is that the electric dipolar interaction can be much stronger. For
example, the interaction between two polar molecules with a typical electric dipole
moment of order of a Debye, where 1 D = 3.34 · 10−30 C·m, is approximately equal
to the interaction between magnetic dipoles with a dipole moment of 100 µb . Thesemuch stronger interactions will in particular allow the long-range character (1/r3 )of the dipolar interaction to be manifested in experiments. This long-range nature of the dipole-dipole interaction is exploited in proposals aimed at realizing
new types of many-body Hamiltonians and novel phases of matter with ultracold
polar molecules. A review of theoretical work on this subject can be found in Ref.
[10, 35].
To get a sense of the range of dipolar interactions, it is convenient to givea characteristic length scale, called the dipole length lD = M d2 /(4π�0 �2 ) [36],where M is the reduced mass, d is the dipole moment of the molecules, and �
is Planck’s constant divided by 2π. The dipole length is essentially a measure of
the intermolecular separation at which the dipole-dipole interaction energy equals
the kinetic energy that corresponds to a de Broglie wavelength at that separation.
To get appreciable long-range interactions in a gas of polar molecules, we wantlD to be comparable or greater than the average interparticle spacing. For atoms
interacting via magnetic dipoles, this dipole length is typically on the order ofa few tens of Bohr radii, a0 , or less. For example, Rb with a magnetic dipole
moment of 1 µb has a dipole length of lD = 1 a0 = 5 · 10−11 m, while Cr witha magnetic dipole moment of 6 µb has lD = 23 a0 . This is more than 100-foldsmaller than the mean interparticle distance even in ultracold atomic gas samples
at the highest achievable densities (n) obtained thus far, for which n−1/3 ≈ 5000
to 20000 a0 . For polar molecules interacting via electric dipoles, however, thislength scale can be as large as 104 to 106 a0 , and therefore could be comparable
to or even much larger than the interparticle spacing in an ultracold gas. For

10example, KRb with a dipole moment of 0.566 D has a dipole length lD of 6000
a0 , while LiCs with a dipole moment of 5.5 D has lD of 6 · 105 a0 . In addition,
the complex internal structure of molecules oﬀers unique possibilities for shielding
and controlling molecular interactions by means of AC and DC electric fields [37].
Dipole-dipole interactions in an ultracold gas of polar molecules can include
both two-body and many-body eﬀects. Two-body eﬀects are due to processes
where one needs only consider two molecules interacting with each other in isolation. These include, for example, ultracold collisions and ultracold chemistry,where the fact that the gas is actually made up of many molecules simply enhances the observed total rate of these processes. In contrast, many-body eﬀects
are those that are not simply the sum of two-body processes. One example is themean-field energy of a dipolar Bose-Einstein condensate [38]. In general, observing many-body eﬀects will likely require a quantum degenerate sample of polar
molecules, such as a Bose-Einstein condensate or a quantum degenerate Fermi
gas.
1.3.1

Ultracold Collisions of Dipoles

Collisions in the ultracold regime have a number of novel features. With
their low energies and long interaction times, ultracold collisions are sensitive
probes of the interaction potential and can often be controlled with the application of modest external fields. In addition, the quantum statistics of the colliding
particles, for example that of identical fermions or identical bosons, can play animportant role. Molecular gases could introduce a new type of ultracold collision,
namely chemically reactive collisions, and polar molecules oﬀer the unique possibility for electric-field control of collisions. Understanding these two-body processes
will be important for exploring many-body phenomena in ultracold gases of polar
molecules.

11
For a given collision energy, interparticle collisions classically occur over arange of impact parameters with corresponding angular momentum barriers. For
atoms, the ultracold temperature regime is often characterized by the requirementthat collisions occur in only one angular momentum partial wave, namely the swave, the only one for which no centrifugal barrier exists. Similarly, for molecules
at no external electric field when their dipoles have not yet been activated, theultracold regime can be defined by requiring that only s-wave collisions occur.
(Of course for indistinguishable fermions, s-wave collisions are not allowed.) Thisrequirement means that collision energies in the gas should be much less than theheight of the p-wave centrifugal barrier, which is, for example, 24µK [31] for KRb
molecules. In the presence of an external electric field, however, the molecules willtend to polarize and the dipole-dipole interaction becomes important. This has
the eﬀect of mixing diﬀerent angular momentum partial waves, and even thoughultracold collisions may occur in a single scattering channel, that channel can nolonger be simply described by a single partial wave.
The long-range nature of the interaction between polar molecules gives rise
to a very diﬀerent and fascinating temperature dependence of ultracold collisions for polar molecules compared to neutral atoms. The following discussion
is summarized from the theoretical insights and results from Bohn et al. [36].
For species with a short-range potential (cf., delta function approximation for
ultracold atom-atom potentials), the elastic collision cross section becomes a constant, independent of temperature in the ultracold regime. This threshold regimeof temperature-independent elastic scattering cross section is a consequence when
the range of the interaction is much smaller than the inverse scattering wavevector,
1/k. Although this makes the theoretical description of the collisions really simple, the elastic cross section is also capped to a certain value. As discussed above,for dipole-dipole scattering, the range of the interaction can be characterized by12
a dipole length, lD . The threshold regime for scattering then occurs only whenlD < 1/k, which means that the relative kinetic energy of two colliding molecules isless than ED = �6 (4π�o )2 /(M 3 d4 ). Take KRb as an example (shown in Fig. 1.1),
when fully polarized, this threshold regime corresponds to a temperature of order
80 nK. As the induced dipole moment of KRb decreases, hence the decrease of lD ,the temperature-independent elastic cross section also decreases. The long-range
dipolar elastic collisional cross section in the threshold T -independent regime is
proportional to the forth power of the dipole moment (∝ d4 ). One consequence isthat because dipole length is large, the elastic cross section is huge. For fermionicKRb molecules, where dipolar collisions mixes odd partial waves, the cross sectiond2
is 2 · 16π· lD= 6.7 · 10−8 cm2 · ( Debye)4 . Another important consequence is that15

for temperatures higher than ED /kB (kB is the Boltzmann constant), the elastic
scattering cross section will be σ = 8πlD /(3k) and therefore increasing with decreasing temperature as T −1/2 . This corresponds to a temperature independent
rate constant for elastic collisions of kelastic = σv = 8πlD �/(3M ), which for KRb
with a dipole moment of 0.566 D would yield kelastic ≈ 2.5 · 10−9 cm3 /s independent of the temperature down to 80 nK. This means that KRb polar molecules at
80 nK are predicted to have elastic cross sections on the order of σ ≈ 5 · 107 Å2 ,a scale much bigger than a typical size of a molecule!

13

Quasi-Universal Dipolar Scattering

17

Figure 1.1: Universal
dipolarcollision
cross section
v.s.of collisional
Figure elastic
5. Elastic
cross section
for scatteringof pairs
fermionic 40energy
K87 Rb molecules(for spin-polarized
fermionic
KRb
polarThis
molecules)
with
an induced
dipole mo-calculation
in identicalinternalstates.
calculation
is basedon the “universal”
thatincludesonly dipole-dipole
In eachthe radius
is rescaled soment of 0.566 D,
0.283
D, 0.142
D, and 0.07interactions.
D [36]. The
crosscasesectionshowsr0 twothatthes-wavescattering
lengthvanishes.
The
top
curve
is
the
cross
sectiondiﬀerent regimes as a function of collisional energy (temperature). The cross sec- for fullypolarized
molecules
with dipole moment!µ" = a0.566
Debye. regime
The dipole
is halved fortion increases withdecreasing
temperatureand reaches
thresholdwhen
eachsuccessively
lower
curve.
the de Broglie wavelength is comparable or larger than the dipolar interaction
length scale. In addition, the dipolar elastic cross section is orders of magnitudelargerthan a typical cross section for s-wave scattering of ultracold atoms.
[12] J. L. Bohn, Phys. Rev. A 62, 032701 (2000).
[13] B. C. Sawyer, B. L. Lev, E. R. Hudson, B. K. Stuhl, M. Lara, J. L. Bohn, and J. Ye, Phys. Rev.
Lett. 98, 253002 (2007).
[14] B. R. Johnson, J. Comput. Phys. 13, 445 (1973).
[15] M.S. Child, Molecular Collision Theory (Dover, New York, 1974).
[16] B. Deb and L. You, Phys. Rev. A 64, 022717 (2001).
[17] A. Derevianko, Phys. Rev. A 67, 033607 (2003); erratum Phys. Rev. A 72, 039901 (2005).
[18] D.-W. Wang, New. J. Phys. 10, 053005 (2008).
[19] S. Yi and L. You, Phys. Rev. A 63, 053607 (2001).
[20] A. V. Avdeenkov and J. L. Bohn, Phys. Rev. A 71, 022706 (2005).
[21] B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, (Addison-Wesley, 1983).
[22] R. J. Glauber, Lectures in Theoretical Physics, ed. W. E. Brittin et al. (Interscience Publishers,Inc., New York), Vol. 1 (1959).
[23] Thomas F. Gallagher, Rydberg Atoms, (Cambridge University Press, New York, 1994), see page

14
Now, let us consider a question relevant to the experiments, namely what isthe density required to observe elastic dipole-dipole collisions. To be experimentally sensitive to collisional eﬀects, such as loss of trapped molecules for inelastic
collisions or rethermalization of the gas for elastic collisions, requires that the
typical time between collisions, given by one over the collision rate, is less than
the interrogation time. Therefore, for a rough estimate, if we have a time of 1s to observe a trapped sample of polar molecules with a dipole moment of 1 D,
kelastic ≈ 8 ·10−9 cm3 /s and this implies a minimum number density of about n ≈1.3·108 cm−3 is required to observe elastic dipole-dipole collisions. This is a generalprediction for elastic collisions of polar molecules. For any specific experiment,
other important considerations will include the eﬀects of quantum statistics, contributions from the short-range part of the scattering potential, and the possibilityof scattering resonances [5, 4], which can futher increase these collision rates byorders of magnitude. Additionally, although the above discussion mainly focusedon elastic processes, investigating inelastic collisions will be the crucial first step
because it will determine the lifetime of trapped samples of polar molecules. In
particular, chemically reactive collisions between polar molecules (or between polar molecules and atoms) represent an intriguing new frontier of encounters to be
explored in the ultralow temperature regime. By detecting inelastic, or reactive,
scattering through trap loss measurements, one can explore how these processes
depend on temperature, electric field, quantum statistics, the internal state of thetrapped molecules, and the geometry, or dimensionality, of the trapping potential.
In summary, long-range interactions at this ultracold regime can determine therates for short-range reactions.

15
1.3.2

Dipolar Quantum Gas

Ultracold gases can be used to realize interesting many-body quantum systems where the interactions between particles, along with particles’ quantum
statistics, govern the macroscopic behavior of the system. In an ultracold gasof polar molecules, the interparticle interactions are relatively strong, spatiallyanisotropic, and long-range. Proposals taking advantage of these interactions,
here I listed only a limited number of examples, include studies of Bose-Einsteincondensates or Fermi gases with dipolar interactions [10, 35], experimental realization of exotic condensed matter Hamiltonians with polar molecules confined in an
optical lattice [12], architectures for manipulating quantum information [7, 8, 9],and creation of self-assembled dipolar crystals [39].
These proposals typically require molecules with large dipole moments ina gas with relatively high density (small inter-particle separation) and ultralow
temperature. The highest densities and lowest temperatures are achieved for gasesthat are quantum degenerate. In particular, in a Bose-Einstein condensate of polar
molecules, even a relatively small dipole moment is suﬃcient to realize a situation
where the dipole-dipole interaction strength far exceeds the kinetic energy. Moregenerally, the quantum degeneracy of a gas can be characterized by its phasespace density, P SD = nΛ3 where n is the peak density and Λ is the thermalde Broglie wavelength. For a quantum degenerate gas, P SD ≥ 1. For a given
gas temperature and density, the size of the dipole moment sets the interaction
strength and in general, one wants a larger dipole moment to more easily observe
interaction eﬀects. As an extreme example in the strongly interacting limit, itis predicted that a gas of polar molecules confined in a two-dimension pancake
geometry can form a dipolar crystalline phase [39, 40]. For a gas density of 1012
cm−3 and molecular mass of about 100 amu (atomic mass unit), this crystalline

16
phase requires that the dipole moment be greater than 3 Debye [39].
1.4Contents of this Thesis
Chapter 2 will present previous results from other groups and the challenges

in trying to produce an ultracold gas of polar molecules. Chapter 3 will introduce some key aspects of the apparatus – the K-Rb machine and the Raman
laser system. Chapter 4 covers the basics of the heteronuclear KRb Feshbach
molecules that serve as the starting point for the state transfer that eventually
makes polar molecules. Chapter 5 presents our understanding and mapping of
the KRb electronic excited states that we use to transfer Feshbach molecules todeeply bound states. Chapter 6 discusses our demonstration of near-thresholdstate transfer using a two-photon Raman technique. We extend the technique forstate transfer to the triplet rovibrational ground state in Chapter 7 and reach theabsolute ro-vibronic ground state in Chapter 8. In Chapter 9, a general schemeof molecular hyperfine state manipulation is presented. With full control of theinternal state quantum numbers of polar molecules, we study their collisions withatoms, indistinguishable molecules, and distinguishable molecules in Chapter 10.
In Chapter 11, we study molecular dipolar collisions by tuning of an external
electric field. Finally, Chapter 12 summarizes the thesis and gives an outlook for
future directions.

Chapter 2
Challenges for Making Ultracold Polar Molecules

2.1

Introduction
Molecules are intriguing systems where their internal energy levels, which

include hyperfine, rotational, vibrational, and electronic levels, span a large spectrum ranging from kHz to hundreds of THz. However, because of this complex
level structure, cooling molecules, both in their translational motion and theirinternal states, is diﬃcult. Experimental eﬀorts aimed at producing cold or ultracold polar molecules generally follow one of two diﬀerent approaches: one isto directly cool molecules to an ultralow temperature; the other is to start fromultracold atoms, and then associate them into tightly bound molecules. Taking
examples from diﬀerent cooling methods, I have summarized previous progress
and the result from this thesis (the last row) in Table 1.
MethodDipole (D)buﬀer gas coolingstark deceleration
photo-association
coherent transfer

NH (1.39)
OH (1.67)
RbCs (1.3)
KRb (0.574)

T
550 mK
50 mK100 µK
160 nK

Trapped Density
108 cm−3
106 -107 cm−3
104 cm−3
1012 cm−3

PSD
10−14
10−13
10−14
0.06

Ref
[19]
[20, 21][22]
this thesis

Table 2.1: Summary of experimental results for production of cold and ultracold
polar molecules. Selected examples of the molecule species with its electric dipole
moment (calculated or measured) and the achieved gas temperature, trapped
number density, and phase-space density are given. (The photo-association result
was for an untrapped sample of molecules.)

18
2.2

Direct Cooling of Molecules
The most straightforward approach for producing ultracold polar molecules

would simply be to directly cool them to ultralow temperatures, just as is done
for ultracold atomic gases. However, the powerful technique of laser cooling[41, 42, 43], which launched experimental investigation of ultracold atomic gases,
is much more challenging for molecules due to complex internal state structures.
The basic issue is that achieving low temperature necessitates momentum recoil
accumulated from thousands to millions of excitation and spontaneously emitted
photons. This in turn requires eﬃcient cycling in a “closed” 2 or 3 level system,
which is generally not feasible in electronic excitation of molecules. Recently, laser
cooling of molecules via coherent scattering inside high-finesse optical cavities has
been proposed [44, 8, 45, 46], but a necessary ingredient for successful implementation is an initial sample of cold molecules with a suﬃciently high density toenable collective eﬀects [46]. Free space laser cooling has also been proposed [47],including the possibility of constructing a general purpose magneto-optic trap fora certain class of polar molecules [48].
Many general molecule cooling techniques developed over the last 10 years
have proved to be very successful in producing a diverse set of cold polar molecules,
including but not limited to CaH, CaF, OH, NH3 , ND3 , H2 CO, NH, PbO, andYbF. These so-called “direct cooling” techniques include imbedding and coolingmolecules in a cryogenically cooled helium buﬀer gas [49, 19, 50], slowing moleculesfrom a supersonic jet through Stark deceleration [51, 21], magnetic [52, 53] oroptical deceleration [54], velocity filtering of a molecular beam [55], production ofcold molecules via kinematic collisions [56], or backward motion of a molecularbeam source nozzle [57]. Direct cooling of polar molecule gases provides access totemperatures in the range of 10’s to 100’s of mK, which is suﬃciently cold to allow

19
molecules to be confined in electro-static [20], AC electric field [58], and magnetic
traps [59, 19]. These cold species can be used for precision measurements [3],collisional studies [60, 21, 61], and cold chemistry [6, 62]. To further reduce themolecular temperature, methods such as the proposed laser cooling approaches
discussed above, cooling via evaporation, or sympathetic cooling with ultracold
atoms might prove to be viable. Two preconditions for their success will be i) a
favorable ratio of elastic and inelastic collisions and ii) a suﬃciently dense sample
of molecules accumulated inside a trap. With further improvements that permit
lower temperatures and higher trapped gas densities, it looks promising for dipoledependent cold collision dynamics in directly cooled polar molecule gases to beaccessible for study in the near future.
2.3

Photo-association
An alternative to directly cooling polar molecules is to start from ultracold

atoms and convert them pairwise into deeply bound molecules. This takes advantage of the fact that atoms can be laser cooled to temperatures of 100 µK or
below. However, the challenge here is to eﬃciently bring the atoms together toform tightly bound molecules while avoiding any heating of the gas due to therelease of the binding energy, which usually corresponds to a temperature of a few
thousand Kelvins (Fig. 2.1). Heating in general means that many energy states
are populated, including both external and internal states.
One way to accomplish this is photo-association [63], where one uses lightto drive a transition between a scattering state of two atoms and an electronically
excited molecular state. The electronically excited molecules then quickly decay
by spontaneous emission to electronic ground-state molecules. The advantage ofthis incoherent optical process, as opposed to collisional formation of molecules, isthat the binding energy is removed in the energy diﬀerence between the absorbed

20

6000K
Figure 2.1: Forming ultracold molecules from ultracold atoms is challenging because a large binding energy is released when two atoms form a tightly boundmolecule. For example, the binding energy of the ro-vibronic ground-state KRb
molecules equals to a temperature of 6000K (4000 cm−1 ), which is 1010 times
larger than a typical temperature of an ultracold atom gas.

and emitted photons rather than being released as kinetic energy that heats theultracold gas. In photo-association, ideally only a small amount of heating ofmolecular external motion remains due to the recoil of the molecule as it emitsa photon in its spontaneous decay; this heating typically corresponds to a few
hundreds of nano-Kelvin. However, typically many internal states are populated
at the end of a photo-association process.
Use of photo-association to make ultracold polar molecules in their rotational, vibrational, and electronic ground state was first demonstrated by Sage
et al. [22]. In their experiment, optical fields were applied to transfer Rb andCs atoms in a continuum of scattering states to the absolute ground state of the
RbCs molecule using the two steps illustrated in Fig. 2.2. A two-step process wasnecessary because of the extremely small wavefunction overlap between two free
atoms and a tightly bound molecule. For example, even at a density of 1012 cm−3 ,21which is a couple orders of magnitude higher than that of laser-cooled atom gases,
the typical distance between two atoms is 20,000 a0 . In contrast, a tightly bounddiatomic molecule has an internuclear separation of 10 a0 or less.
The first step used by Sage et al. was photo-association followed by spontaneous decay to produce large, weakly bound electronic ground-state moleculesin high vibrational (high-v) states. In the spontaneous decay, many high-v levelsof the triplet ground state were populated, with the biggest branching ratio being
7% into v = 37 [22]. The second step was then to apply two laser fields for a
pump-dump transfer process using absorption and stimulated emission. This steptransferred molecules in the v = 37 of the triplet ground-state to v = 0 of thesinglet ground-state through an intermediate electronic excited state.
This photo-association technique produced absolute ground-state moleculesthat were at the ultracold temperature of 100 µK in their translational motion.
However, the creation process was ineﬃcient. In this proof-of-principle experiment, each laser pulse sequence only produced 10 absolute ground-state moleculesfrom the initial 108 atoms. In addition, molecules in many other higher energy
vibrational and rotational levels were produced at the same time. Nevertheless,this experiment highlights the general scheme as well as the challenges in converting ultracold atoms into tightly bound molecules. Other experiments haveused photo-association of laser-cooled atoms to create ultracold polar moleculesin bialkali systems such as KRb [64, 65], NaCs [66], and LiCs [67].
To make this incoherent process more eﬃcient in the production of deeply
bound ground-state molecules, one obviously desires to have an enhanced couplingstrength from the initial scattering atom state to a suitable excited molecular state
as well as an increased Franck-Condon factor from that excited molecular state
to a deeply bound vibrational level in the electronic ground potential. An enhanced phase-space density of the initial atomic gas can dramatically enhance

22

PA

3!+

X1!+
v=0
Figure 2.2: Schematic of a two-step optical transfer process described in Sage etal.[22] that produced RbCs in the absolute ground state from laser-cooled Rb andCs atoms. The first step is to create weakly bound molecules (in high vibrational
levels) by photo-associating (PA) atoms into excited-state molecules that thendecay back to the ground electronic state by spontaneous emission (wiggly line).
These weakly bound molecules provide a better wavefunction overlap for the second transfer to the absolute ground state. The second step is to use a stimulated
emission pump-dump process (dashed arrows), to enhance optical transfer fromweakly bound molecules to the absolute ground state. Typically this process created 10 molecules in the absolute ground state when starting from 108 atoms ina single two-step transfer.

23
the transition rate for the first step. But optimization of the overall eﬃciency islimited by the molecular structure. Furthermore, this incoherent process will not
be able to produce molecules populating only a single target state. Techniques
and proposals have emerged to enhanced the transition strengths using accidental
resonances between molecular states, either in the excited electronic potentials[68, 69] or in the ground potential [70]. A recent proposal suggests that simply
performing photo-association of atoms near a Fano-Feshbach resonance is suﬃcient to drive them into a deeply bound molecular state either through one-photon
or two-photon transitions [71, 72]. Although resonant-assisted photo-association
enhances the transition by orders of magnitude, the overall transition rate is stillrather weak. Using either one-photon or two-photon processes to eﬃciently drive
atoms to a deeply bound state still appears challenging. An additional issue withthis approach is that the initial state is a scattering state (unless the initial atomic
gas is quantum degenerate), and the time scale for two atoms to come close enoughto be photo-associated, which can be of order a few ms, can eﬀectively limit thetransition Rabi frequency.
2.4

Fully Coherent Scheme
One extremely powerful solution to all the challenges outlined above is touse a transfer process that is fully coherent and therefore even reversible. In
particular, we can replace each incoherent step of the photo-association methodby a coherent transfer step. The advantages of a fully coherent method are i)
it is adiabatic, and therefore results in no heating and ii) it is single targetstate selective, therefore the process can be much more eﬃcient. However, toavoid strong transfer field to induce non-linear eﬀect, one or two coherent steps
may not be suﬃcient to pair atoms together and transfer them to the absoluterovibrational molecular ground state. In that case, a general solution is to use a24
carefully designed train of coherent pulses that can achieve a complete populationtransfer to a single target state via coherent accumulation of small population foreach pulse [73]. But it turns out that for KRb, and this should also be true forother heteronuclear bi-alkali molecules in general, one can find a single excitedelectronic state that has relatively strong transition strengths and makes a singleor two-step transfer possible.
In the following chapters, I will describe in detail the complete scheme of
the fully coherent method that we use to produce a near quantum degenerate gasof ultracold polar molecules in their ro-vibronic ground state [15]. Here I provide
a quick summary. As opposed to a typical photo-association experiment discussed
in the previous section, we start with an ultracold atomic gas at a high phasespace density. This high phase-space density is achieved using the now standard
techniques of laser cooling followed by forced evaporation [74, 75]. At the end of
the evaporation, we have a quantum degenerate gas of K and Rb atoms, each ina single hyperfine state, at a temperature of several hundred nK. The atoms are
confined in a far-detuned optical dipole trap, which can also confine the resulting
KRb molecules.
We then use two coherent steps for production of ultracold polar molecules.
First, we use a magnetic-field tunable Fano-Feshbach resonance, to convert atoms
pairwise into molecules. (Strictly speaking, this step is not fully coherent because
the initial state is a thermal ensemble and there is loss during the conversion.)
Secondly, these extremely weakly bound molecules are then optically transferred
into the ro-vibronic ground state using a two-photon coherent process. The basic
approach of making molecules near a Fano-Feshbach resonance and then using
coherent optical manipulation of the molecules’ internal state has also been used
to produce ultracold gases of homonuclear (i.e., non-polar) molecules [30, 29]. In
the next chapters, I will discuss in more detail the two steps in this transfer that25
makes it possible to map the high phase-space density of an ultracold atomic gas
into tightly bound polar molecules in the ultracold regime.

Chapter 3
Apparatus

The main workhorse of the experiments presented in this thesis has twoparts, the atoms/molecules machine and a spectroscopy/Raman laser system.
3.1

K-Rb Machine
The detail of the hardware and the software features of the K-Rb machine

can be found in J. Zirbel’s thesis [18]. Here I will briefly discuss the methods that
we use to prepare ultracold gas of 40 K and 87 Rb atoms, and describe a few specificadditions to the apparatus since Zirbel’s thesis.
The first step of our experiment is to prepare an ultracold mixture of K and
Rb atoms. To do this, we start by trapping and cooling K and Rb simultaneously
in a dual-species magneto-optical trap (MOT). We typically have K atom number
NK = 1 · 107 (w/o the presence of the Rb MOT) and Rb atom number NRb =2 - 4·109 in the MOT. The MOT loading time is on the order of 5 s. We thenphysically move atoms through a 0.7-m transfer region to a diﬀerentially pumped
higher vacuum region of the chamber. The transfer eﬃciency is 10-20%. The
atoms are then loading into a standard Ioﬀe-Pritchard (IP) magnetic trap. At
this stage, we perform forced evaporative cooling on Rb by driving hot Rb atoms
from the |F = 2, mF = 2� state to the untrapped |1, 1� state. At the same time,
the K atoms gas is sympathetically cooled by thermal contact with the directly

27
cooled Rb gas. The K atoms are in the |9/2, 9/2� state. Here, F denotes the total
atomic spin and mF is the spin projection along the magnetic-field direction. We
also remove any Rb atoms in the |2, 1� state to prevent trap loss due to collisions
with K atoms in the |9/2, 9/2� state. At the end of the IP trap evaporation, we
typically have 2 - 3·106 Rb atoms and 6 - 7·105 K atoms at 1µK. This gas mixture
is then transferred into an optical dipole trap (OT). (The system is capable ofhaving up to 5·106 Rb atoms in the presence of the same number of K atoms
simply by increasing Rb MOT laser power (and hence the Rb MOT number), but
we found that in the later stages, higher Rb number limits the K number.)
For our experiments aimed at making ro-vibronic ground-state molecules,
we used a single-beam optical dipole trap with a waist of 40 µm (Fig. 3.1(a)). The
trap laser beam was derived from either a 1064 nm single frequency fiber laser
or a 1090 nm multi-mode fiber laser. After loading atoms in the optical trap, wetransfer the K and Rb atoms into their respective lowest hyperfine states, K in
|9/2, −9/2� and Rb in |1, 1�, at a magnetic field of 31.29 G. The transfer uses
adiabatic rapid passage by a frequency swept rf and microwave fields. We thenincrease the magnetic field to 532.30 G for further evaporation before ramping thefield close to 546 G for Feshbach molecule creation. Evaporation in the opticaltrap is accomplished by lowering the laser power to cool the gas to a few hundredsof nanoKelvins. In the later experiments (Chapter 10 and 11), we changed our
dipole trap to a crossed beam geometry with two intersecting elliptical-shaped
beams (Fig. 3.1(b)). Each beam has a waist of 40µm×200µm. (When using
the crossed beam optical dipole trap, we also shape our Raman laser beam to be
elliptical. The 690 nm Raman laser is focused to a waist of 38.5µm×205µm, andthe 970 nm Raman laser is focused to 43.5µm×205µm waist.) A summary of ouroperating condition is given in Table. 3.1.
Finally, I wish to mention that we were able to shorten the experimental

Rb atoms
K atoms
Feshbach MoleculesMOT
2 - 4·109
1 ·107 (w/o Rb)

IP # and Temp.
2 - 3·106 at 1 µK
6 - 7·105 at 1 µK

28
OT # and Temp
3·105 at 300 nK
3·105 at 300 nK
3 - 5·104 (Ch. 4)

Table 3.1: A benchmark of experimental operating condition.

cycle time by a factor of 2 since Zirbel’s thesis: the evaporation time in the IP
trap was shorten from 60 s to 35 s and the evaporation time in the OT went from40 s to 3-4 s. The significant reduction of OT evaporation time was accomplishedsimply by aligning the trap so that the beam had a smaller tilt away from being perpendicular to the direction of gravity. This also improved our atom andFeshbach molecule numbers in the OT by a factor of 2 or 3.

29

optical dipole trap

probe

(a)

Raman lasers
from two fibers

crossed dipole trap

probe

(b)

Raman lasers
from two fibers

homogeneous B field
uniform E field parallel to gravity
Top View (gravity direction into the page)

Figure 3.1: Top view of a simplified layout of our experiment (a) using a single
beam optical dipole trap and (b) using a crossed beam dipole trap. In bothconfigurations, a uniform magnetic field is provided by a pair of Helmholtz coils.
The Raman laser beams for molecular internal state manipulation co-propagate
along the magnetic-field direction. The imaging beam or “probe” is perpendicular
to the magnetic-field direction. Gravity in this picture points into the page.

30
3.2

Electric-field Plates
One new addition to the apparatus was a pair of electric-field plates. Be-

cause these were incorporated after the main apparatus was built, the plates wereconstrained to be located outside of our glass-cell (“science cell”) vacuum chamber(Fig. 3.2).

grounded microwave coil

~ 0.25 cm

indium tin oxide coating
1.35 cmscience cell (glass)wall thickness: 0.125 cm

KRb

1.0 cm
~ 2.3 cmtransparent E-field plate(4 cm x 2.37 cm x 0.09 cm)
nearby grounded coils

gravity

Figure 3.2: Schematic cross-section view of the apparatus near the “science cell”(not drawn to scale). One anti-gravity coil and an rf coil are left out from thedrawing.

The E-field plates are cut from #CH-50IN-S209 transparent plates withindium tin oxide coating purchased from Delta Technologies. The plates transmit
80-90% of light in the wavelength range of 600 nm to 1100 nm. (Plates with
5-10% better transmission properties are also available from the same companynow.) The plates are 4.0 cm long, 2.37 cm wide, and 0.09 cm thick and the innerwalls are separated by 1.35±0.02 cm with a phenolic holder. The coating surfaces
face each other. The pair of plates fits around our glass cell, which comes from
Starna Cells and is made out of Borofloat (a type of pyrex glass with dielectric31
constant ∼ 4.8) with a single wall thickness of 0.125 cm. The E-field at the positionof “our molecules” is calculated from the geometry and the applied voltages on theplates in addition to considering the glass walls as dielectric material between theplates. For an applied voltage diﬀerence of 1V across the plates, the corresponding
uniform E-field between plates pointing along the gravity direction is
1V
1.35 cmV·= 0.868 .
1.35 cm 1.35 cm − 2 · 0.125cm + 2 · 0.125cm/4.8
cm

(3.1)

This calculated field has an uncertainty from the measured separation between
plates (1.5%), the dielectric constant of the glass material (1%), the thickness ofthe glass wall (unknown), and the fringe field that came from the finite size ofthe plates (0.7%). Together, we estimated the total uncertainty of the calculated
field to be 3-5%. This does not include possible free floating electrons or patch
charges. So far, we do not have an independent way of calibrating the electricfield.
Obviously, the biggest concern with placing the plates outside the vacuum
chamber is that dielectric breakdown can occur at a relatively low applied voltage.
(The air breakdown field is 30 kV/cm.) Therefore, we tried to insulate surrounding
metal by covering them with many layers of Kapton tape. One of the closest metal
planes is a microwave coil that is laid out on a copper circuit board and is only
0.16 cm from the E-field plate surface and 0.25 cm from the E-field coating surface
where the voltage is applied (Fig. 3.2). With 3 - 5 layers of Kapton tapes, weonly saw immediate discharge between the microwave coil and the E-field plate
above ∼ 6 kV applied voltage diﬀerence. We also tested the breakdown voltage
by putting a grounded wire wrapped in 2 layers of Kapton tape directly on thecoating surface, which is essentially a test of the breakdown of the Kapton tape.
For this test, we saw discharge happened at ∼ 4 kV. (This wire is the same as what
we used for the magnetic-field coils that surround the E-field plates.) In addition,

32
we have also tested the plates with a mocked up glass cell in between. For oursystem, we are confident that we can apply up to 10 kV (+5 kV on one plate and-5kV on the other) across the field plates before any breakdown happens. This
corresponds to an electric field of ∼ 9 kV/cm.
However, we run into another limit before the dielectric breakdown occurs.
Namely, when the electric field is turned up too high, we saw our glass cell (perhaps) being polarized and does not relax after the field is oﬀ. This usually happensat an applied voltage of 6.0 kV (5.2 kV/cm) and was seen using sensitive Stark
spectroscopy on KRb (Chapter 8.3).
3.3

Molecular Spectroscopy Lasers
To manipulate molecular internal ro-vibronic states, we used many diﬀerent

tunable lasers to map out relevant electronic excited states and ground states of40

K87 Rb. These lasers include a home-built tunable Ti:sapphire (Ti-Sa) laser, an

ELS VersaDisk tunable around 1030 nm, and home-built diode lasers in either
Littman or Littrow configurations at various wavelengths.
3.3.1

Widely Tunable Ti:sapphire Laser

A widely tunable cw Ti-Sa laser is ideal for performing molecular spectroscopy in the wavelength range that is relevant for finding suitable intermediate
states to bridge the large wavefunction mismatch between Feshbach molecules
and any desired deeply bound state. The laser we use was built by Lisheng Chen
before/during the same time when we were building the K-Rb machine. An excellent discussion of the layout of our Ti-Sa laser and the elements for large andstable frequency tuning can be found in Chapter 2.2 of Chen’s thesis [76]. Since
Lisheng had already graduated by the time we started to work with the Ti-Sa, it
was postdoc Avi Pe’er who revived the Ti-Sa laser and helped to incorporate it33
into our experiment.
The wavelength tuning range of the Ti-Sa is mostly set by the cavity mirrors
and the normal lasing wavelength of a Ti-Sa crystal. We first used the Ti-Sa laser
to scan for KRb* states near the Rb D1 line (∼ 795 nm). For this purpose, weused a set of mirrors with coating that allows a tuning range from 730 nm to 850
nm. The wavelength is tuned using optical elements in the cavity, including athick etalon, a thin etalon, and a birefringent filter (see Chen’s thesis [76]). In
this wavelength range, the Ti-Sa laser is oﬀset-locked to a temperature stabilized
Fabry-Perot cavity that has a mode spacing of 250 MHz. The Ti-Sa linewidth is< 20 kHz and the absolute long-term frequency stability is better than 2 MHz.
We later changed the mirrors inside the cavity such that the Ti-Sa was lasingnear 1 µm with a tuning range of ± 50 nm. This change is ideal for the searchof one-photon transitions from Feshbach molecules to KRb* states in the excitedpotential 23 Σ+ . For this wavelength range, we directly lock the Ti-Sa laser to the
Ti-Sa frequency comb (see discussion below).
3.3.2

Diode Lasers

We use numerous home-built external cavity diode lasers in the Littrow
configuration for our K-Rb machine; these use non-AR-coated diodes from Hitachi
and Sharp. But for molecular spectroscopy, we use AR-coated (coated either
in JILA or commercially from Eagleyard) diode lasers mostly in the Littman
configuration. This configuration, although it usually gives less available laser
power, has the advantage that the output does not steer within a large frequency
tuning range. For our current setup, we manipulate the internal states of KRb
using a pair of diode lasers in the Littman configuration as our “Raman lasers.”

34
3.3.3

Ti:sapphire Comb

A stable optical frequency comb is crucial in maintaining a well-definedphase relation between our Raman lasers, which are very diﬀerent in frequency.
The basics of state-of-the-art frequency combs can be found in many JILA theses,
e.g. [77]. The purpose of this section is only to describe how we incorporate thecomb into our system and perform some basic diagnostics and characterizations.
More detail will be available in the next thesis from our project by Marcio Miranda, who has been the main person responsible for operating and maintaining
the comb.
In our experiment, we use a home-built Ti-Sa frequency comb. When the
Ti-Sa is mode-locking, the laser output from the cavity is a train of pulses that are
equally spaced in time. The operating condition for the laser is measured to have
650 - 750 mW time-averaged laser power and a spectrum centered around 800 nm
with a full width half max (FWHM) of ∼ 30 nm (Fig. 3.3). Directly out of thelaser cavity, we then broaden the spectrum (shown in Fig. 3.4) using a non-linear
supercontinuum fiber (Crystal Fibre FemtoWHITE 800) with a coupling eﬃciencyof ∼ 30%. This broadened spectrum conveniently covers the wavelengths we areinterested in with a certain spectral holes that can be avoided with compromises
among all the wavelengths of interest. This is accomplished by fine adjustmentsof the alignment and the polarization into the non-linear fiber.
The comb serves as a frequency ruler in the optical domain. Our comb isprecisely stabilized in the following way: The repetition rate, frep ≈ 756 MHz,
determined by the comb cavity length is maintained by referencing one comb
tooth to a stable 1064 nm Nd:YAG. The Nd:YAG (linewidth < 1 kHz) is locked
to a stable cavity in Jan Hall’s lab. The error signal from frep is fedback to the
PZT of a cavity mirror to actively adjust the comb cavity length. The comb oﬀset

35

-20
FWHM=30 nm @ 800nm

Power(dB)(dB)
Relative
Power

-30
-40
-50
-60
-70
-80
-90

resolution bandwidth=10nm-100
500 600 700 800 900 1000 1100 1200

Wavelength (nm)Figure 3.3: Ti-Sa comb spectrum, measured directly out of the comb cavity. The
time-averaged power of the laser is ∼700 mW. The laser spectrum centered at 800
nm with a full width half max of 30 nm.

frequency fco , which can be tuned by the pump laser power into the cavity, isusually around 50-300 MHz. The fco beat is derived from beating the 532 nm
part of the comb spectrum and the doubling of the 1064 nm part of the spectrum,in a so-called self-referencing configuration. The fco is then locked to a stable rf
source that is in turn referenced to a 10 MHz commercial quartz clock (Wenzel
Associates). The error signal of fco is fedback to an acoustic-optical modulator toactively adjust the pump power into the cavity.
3.3.4

Phase Locking

As I have mentioned earlier, maintaining the phase coherence of the Raman
lasers is crucial for coherent transfer molecules between internal states. Therefore,
we have to insure the two Raman lasers to not only maintaining their frequencies,

36

Power
(dB)(dB)
Relative
Power

-30
-35

970nm,-40 dBm

532nm,-41 dBm-40
1064nm,-40 dBm-45-50

690nm,-46 dBm-55-60

resolution bandwidth=10nm

500 600 700 800 900 1000 1100 1200

Wavelength (nm)Figure 3.4: Full spectrum of the frequency comb in our system after broadened
through a non-linear supercontinuum fiber. Operating with this spectrum, we
typically get Raman laser (690 nm and 970 nm) beats, YAG (1064 nm) beat,and fco (532 nm and doubling of 1064 nm) on the order of 30 - 35 dB (100 kHz
bandwidth) from the noise floor.

but also have a definite and predictable phase evolution throughout the duration of
the transfer. Our coherent transfer manipulation usually takes a few µs, therefore
the relative laser linewidth has to be on the order of 10 kHz for the transfer (thatuses a few MHz Rabi frequency coupling) to be eﬃcient. The phase coherence of
the Raman lasers are maintained through phase-locking (with current feedbackof Raman laser diodes) to our stable frequency comb. In our setup, most of thelaser power from the Raman laser system is directly sent to our vacuum chamber
for molecular internal state manipulation. We do not send the comb light to themolecules (Fig. 3.5). The complete electronics of the locking configuration isshown in Fig. 3.6.
From an operational point of view, our locking procedure is the following. To

37lock a Raman laser at a desired frequency, we first check the laser frequency on awavemeter (Bristol Instrument 621 series). The wavemeter is internally frequency
referenced to a HeNe laser. We found that the long-term drift of our wavemeteris < 30 MHz and the short-term stability is ∼ ± 5 MHz. The absolute frequencyof our wavemeter reading as compared to the frequency referenced to a wellcharacterized comb is usually about 30-50 MHz too high. Since our comb has arepetition frequency that is relatively high at ∼ 756 MHz, our 10 MHz wavemeterstability is more than suﬃcient for us to determine the closest comb tooth that ourlaser beats with. Knowing the closest comb tooth’s absolute frequency (from frep
and fco ) allows us to lock our laser to a desired frequency by setting the specificcomb-to-diode beat using a synthesizer (Fig. 3.6). The diode laser frequency iscoarsely tuned by a piezoelectric transducer controlling the grating laser feedback
and is finely tuned by the diode current feedback.

PLL

PLL

Frequency Comb

970 nmRaman Laser
To KRbmolecules

690 nmRaman Laser

Figure 3.5: Our Raman lasers, which we use to manipulate molecular internal
states, are phase-locked to a stable frequency comb. PLL stands for phase-lock
loop.

38
comb
infrared filter
Raman
laser via 20-m
PM fiberselective wavelength(10 nm bandwidth)

ZFM-2
L

filter rejecting frep

R

EOT photodioderf signal sends from the comb to theET-2030A/ET-3000A Raman table via 20-m BNC cable

synthesizer(control beat freq)

I

150 MHzband pass filter

feedback

10 MHzreferenced

amplifier, ZKL-2
inmonitoring spectrum analyzer

CPL

coupler, ZFDC-10-5
outL
150 MHz, synthesizer

ZFM-2
R
I

Loop filter (fast servo box)

Figure 3.6: Layout of our Raman laser phase lock electronics. Rf components arelabelled by MiniCircuits part numbers.

39
Our coherent transfer manipulation of molecular internal states requires
time-varying couplings between internal states. These are accomplished by intensity ramp of the Raman lasers. To control the laser intensity (shown in 3.7),
we send the laser through an acousto-optic modulator (AOM). We use the -1
diﬀracted order of the beam and control the rf power that goes into the AOM in
the following way: We first directly program our desired intensity ramps (usually
linear ramps) into an arbitrary waveform generator. The generator gives a dc
voltage signal and is then mixed with an rf frequency source of 98.5 MHz before
the mixed rf signal is amplified and sent to drive the AOM. This varying rf power
directly control the intensity of the Raman laser that is seen by our molecules.
10 MHzreferenced

ZAD-2HL

Agilent 33220A arbitrary
waveform generator
set for 50! load

100 !

I
need towork at DC

98.5 MHzsynthesizerR

RF switch
ZASWA-2-50DR+
Amp-1th - experiment
diode/Ti-Sa
Raman laserAOM

0th - Bristol wavemeter andthe comb for phase locking

Figure 3.7: Layout of electronics for our Raman laser intensity ramp. Rf components are labelled by MiniCircuits part numbers.

Chapter 4
Feshbach Molecules

As was first introduced in Chapter 2.4, pairing atoms to form Feshbach
molecules was the crucial first step to making a quantum gas of polar molecules.
Feshbach molecules are exotic, extremely loosely bound molecules that only exist near a Fano-Feshbach resonance. These are scattering resonances that allow
experimenters to control interactions in a quantum gas of atoms [78, 79, 80, 81].
This powerful tool has played a key role in a number of exciting developments inultracold atom gases, such as realization of the BCS-BEC crossover in Fermi gases[82], the controlled collapse of a BEC [83], and the observation of bright matterwave solitons [84]. Here I will discuss one particular application, namely creationof Feshbach molecules, which is relevant to our experiment. As illustrated in Fig.
4.1(a), these resonances occur when the energy of a pair of atoms in one hyperfine
scattering (open) channel is the same as the energy of a molecular bound state ina diﬀerent hyperfine (closed) channel. Because the molecule and the pair of free
atoms can have diﬀerent magnetic moments, atom scattering can be tuned in andout of resonance using precisely controlled magnetic fields. This allows atom pairs
to be converted into weakly bound molecules through an adiabatic magnetic-fieldsweep across a resonance (Fig. 4.1(b))[85]. This technique of magneto-association
has been used for creation of homonuclear Feshbach molecules in studies of atomic
Bose-Einstein condensates and quantum degenerate Fermi gases [16, 86]. Creation

41of these homonuclear Feshbach molecules can be very eﬃcient and has been shownto depend on only a single parameter, the phase-space density of the initial gas[87].

bound state
in “closed” channel
EnergyEnergy“closed”

R“open”
B field ramp

colliding atoms
in “open” channel

B
(a)

(b)

Figure 4.1: (a) A Fano-Feshbach resonance occurs when the energy of a pair ofatoms in one hyperfine (“open”) channel is the same as a molecular bound state
in a diﬀerent hyperfine (“closed”) channel. The system can be tuned into andout of resonance by changing the strength of an applied external magnetic field.
(b) An appropriate magnetic-field ramp across a Fano-Feshbach resonance canadiabatically convert pairs of atoms into weakly bound Feshbach molecules.

4.1

Heteronuclear Feshbach Molecules
To make polar molecules in the end, the starting Feshbach molecules haveto be heteronuclear. Heteronuclear 40 K87 Rb Feshbach molecules were first createdby rf-association in an optical lattice [23]. Subsequently, we characterized theirproperties [88] and made collisional measurements [24] using the same rf association technique but in a single-beam optical dipole trap. The details of thesemeasurements can be found in J. Zirbel’s thesis [18]. One of the diﬃculties withrf association in our system is that the strong rf field can couple into our electronics. In particular, we find that the rf aﬀects our magnetic field servos and cause

42
the magnetic field to change slightly. This makes the number of created Feshbach molecules irreproducible. Therefore, for all the experiments discussed in thisthesis, we instead used a time-varying magnetic-field sweep across resonance toadiabatically convert pairs of atoms to Feshbach molecules. This method produces
a few 104 molecules similar to the rf association technique, but is more robust day
to day.
The Fano-Feshbach resonance we use for molecule creation has for its open
channel the K|F = 9/2, mF = −9/2�+Rb|1, 1� state (labeled “aa” using P. Julienne’s convention [89]) while the closed channel is dominated by the K|7/2, −7/2�+Rb|1, 0� state (labeled “rb”) at a resonant field of 546.7 G [90, 91, 92]. P. Julienne’s labeling convention is simply to label hyperfine states at a high magnetic
field according to their energies. The lowest energy state is labelled “a” and thesecond lowest energy state is labelled “b”, etc. Since we work with molecules
composed of K and Rb, the first label refers to the hyperfine state of K and thesecond label refers to Rb. The Feshbach molecule has a total angular momentum
projection along the magnetic-field axis mF = −7/2.
Before molecule creation, we first prepare an ultracold mixture of Rb and K
atoms, with each species having on the order of a few 105 atoms, at a temperaturearound 200 nK in a single-beam optical dipole trap. A description of preparing
the ultracold mixture can be found in Chapter 3. A few 104 Feshbach molecules
are then created using a single linear field ramp from 8 G above the resonance toa field around 545.90 G in 4 ms. The field is provided by a set of Helmholtz coils(∼ 9 cm in diameter). Because of imperfections in the current servo, the magnetic
field immediately after the ramp overshoots 100 - 200 mG too low before the field
settles to the final value (stable to 20 mG) within 2 ms. The molecule binding
energy at 545.90 G is h·240 kHz. The molecular size is about 300 a0 , and the closed
channel fraction is ∼ 35% [88]. The mean distance between atoms in an initial gas

43density of 1013 cm−3 is a factor of 3000 bigger than the molecule size. The gas ofFeshbach molecules typically has an expansion energy corresponding to 350 nK,
while the initial atom gas temperature is ∼ 200 nK. We have never investigatedthe heating systematically. However, it could be due to an “anti-evaporation”
process from strong three-body loss near the Fano-Feshbach resonance. As wevaried the initial atomic gas conditions and changed optical dipole trap waist
many times over the years in which this thesis work was conducted, the Feshbach
molecule expansion energy varied in the range between 150 nK to 700 nK.
4.1.1

Conversion Eﬃciency

Making a large number of Feshbach molecules is a prerequisite for our subsequent experiments, and here I share some experience of optimizing heteronuclear
Feshbach molecule creation. As has been mentioned, in the case of homonuclear
Feshbach molecule creation, the phase-space density of the initial atomic gas isa single parameter that determines the saturated conversion eﬃciency for a slow
ramp[87]. For a degenerate Fermi gas, up to 90% conversion eﬃciency from atoms
to molecules has been demonstrated [87]. For heteronuclear 85 Rb87 Rb molecules
where the constituent atoms are both bosons [93], it has been shown that only thephase-space density of one gas determines the eﬃciency provided that the cloudspatial overlap is good.
For 40 K87 Rb Feshbach molecules, the best eﬃciency demonstrated so far ina single-beam optical trap is 25% using rf association [88]. According to a phenomenological model that fits our data well, the relatively low eﬃciency comesfrom two competing factors, namely, the phase-space density and the spatial overlap of the 40 K and 87 Rb clouds. There is a mismatch in the cloud sizes for thefermionic K gas and bosonic Rb gas when the gases are well in the quantum degeneracy regime due to Pauli pressure for the fermions. In addition, the equilibrium

44

Figure 4.2: An image of an expanded cloud of Feshbach molecules that shows a
low density region in the center of the cloud. Sine the trap frequency is low in thehorizontal direction, we probe the in-trap position of the cloud in the horizontal
direction after a short time-of-flight expansion. The hole in the density profile isdue to large inelastic loss in the region of high Rb density.

positions of the two clouds in a weak optical trap are displaced in the verticaldirection due to the balance of the optical trapping force and the gravitational
force, which is larger for the heavier Rb atoms. The best molecular conversion
eﬃciency was found close to T /Tc = 1 experimentally with our parameters, where
Tc is the temperature of the onset of a Bose-Einstein condensation [88].
In addition to phase-space density and spatial overlap of the clouds, KRb
Feshbach molecules suﬀer strong inelastic collisions, especially with Rb atoms [24].
This becomes an important issue when using a long duration adiabatic field ramp
to make molecules. In Fig. 4.2, I show an example of Feshbach molecules witha large loss at the center due to a high Rb density gas. To avoid strong loss due
to Rb, we choose to operate at higher gas temperature where the Rb density islower. In the end, we typically work with a conversion eﬃciency of 10-20%.
4.1.2

Lifetime

Feshbach molecules are in a high vibrationally excited (high-v) state andcan be vibrationally quenched in ms time scales from collisions with residual Rb
and K atoms in our trap. Each inelastic collision results in a large energy releaseand subsequently, the molecule is lost from our trap. To prolong the lifetime

45
of Feshbach molecules, we remove residual atoms using a combination of rf or
microwave fields to transfer these atoms to a state with (quasi-)cycling transition
followed by resonant light pulses to heat the atoms out of the optical trap [18].
The Rb(|1, 1�) removal procedure is the following. We first spin flip Rbfrom |1, 1� to |2, 2� using a pulse of microwave near 8 GHz and then apply a shortlight pulse that drives the |2, 2� to |F � = 3, m�F = 3� cycling transition. Because|1, 1� to |2, 2� has a large magnetic field sensitivity (2.3 MHz/G at 546 G) and
our field ramp usually takes 2 ms to settle after reaching the final field, we use an
Adiabatic Rapid Passage(ARP) of duration 200 µs where we sweep the microwave
frequency to perform the spin flip. Each ARP has an eﬃciency higher than 95%.
We perform the procedure (ARP + blast light pulse) three times in a 1 ms time4

Molecule Number (10 )span starting at 1 ms after the field ramp.

6
5
4
3
2

!=7.6(3)ms

1
0

10

20

Hold Time (ms)30

Figure 4.3: A typical KRb Feshbach molecule lifetime. Looking at the KRb
lifetime v.s. field behavior (not shown here) as compared to [24], we believe thelifetime is limited by residual undetected K atoms. However, this is suﬃciently
long to perform state transfer to more deeply bound molecular state.

46
The K(|9/2, −9/2�) removal involves two spin flips and an optical transition.
The two spin flips drive |9/2, −9/2� first to |9/2, −7/2� and then to |9/2, −5/2�both using rf-π-pulses with durations on the order of 10 µs. Finally, a light pulse of
50 µs resonant with the |9/2, −5/2� to |11/2, −5/2� transition is applied to remove
K from the trap. This is a 96%-closed cycling transition. The light is 140 MHz
red-detuned from the Feshbach molecule-free atom (K |F � = 11/2, m�F = −11/2�)transition.
An alternative of K removal in the later experiments, we “hide” molecules
in a deeply bound vibrational state and then turn on the 100%-closed cycling
light on the |9/2, −9/2� to |11/2, −11/2� transition to directly remove residual K
atoms.
After the K and Rb removal procedures, the number of residual atoms is lessthan the minimum number we can detect with our absorption imaging (< 3000in the single-beam optical trap and < 1000 in the crossed-beam optical trap). A
typical lifetime for trapped Feshbach molecules is then 7 ms at 545.88 G as shownin Fig. 4.3 (using the first K removal procedure), which is more than suﬃcientfor further internal state manipulation of the molecules.
4.1.3

Electric Dipole Moment

We can create a high phase-space-density gas of heteronuclear Feshbach
molecules. However, because the average separation of the two nuclei in eachmolecules is relatively large, on the order of 300 a0 , the two valence electrons
are only extremely weakly shared by the nuclei; therefore, the molecular electric
dipole moment is negligible. Ab initio calculations of the permanent electric dipole
moment (Fig. 4.4) as a function of the internuclear separation from Kotochigova
et al. [94] give a dipole moment of 5 · 10−11 D for KRb Feshbach molecules. Only
when the internuclear separation is smaller than 10 a0 , does the dipole momentDipole Moment (Debye)

47
0

1
10

X1! v=0
-4

10

-8

10

Feshbach
Molecules

-12

10

10

100

Internuclear Seperation (a 0)

Figure 4.4: Calculated KRb permanent electric dipole moment as a function ofinternuclear separation [94]. Feshbach molecules have a size of the order ∼ 300 a0
and a negligible dipole moment of ∼ 10−11 D. To make molecules polar, we needto shrink the the internuclear separation, e.g. absolute ground-state molecules
have a dipole moment of ∼ 1 D.

approach its maximum value of nearly 1 D. Therefore, to make polar moleculesin the next step we want to manipulate the molecules’ internal state to reacha deeply bound level where the molecule is small and has a significant electric
dipole moment. In thinking about this next step, it should also be noted that inspite of the fact that Feshbach molecules are unusually large dimers, the magnetoassociation step nevertheless reduces the internuclear separation from a few tensof thousand a0 in the initial atom gas to a few hundred a0 in the molecules.
This facilitates further molecule state manipulation by providing a much larger
wavefunction overlap to deeply bound states.

48
4.2

Enhancement Factor
The importance of this first step of magneto-association can be seen in ameasured enhancement of the optical excitation rate to a KRb* (KRb electronically excited) state. Here, I present measurements of this enhancement not only
for a near-threshold KRb* state, but also for a deeply bound KRb* state. Thisis relevant to the use of optical transitions to bring Feshbach molecules either toa lower vibrational state near the atomic threshold in one case (see Chapter 6)
or finally to the lowest ro-vibronic ground state (Chapter 8). The comparison, as
illustrated in Figure 4.5, is to apply a light pulse to drive a bound-bound transition from Feshbach molecules or a free-bound (photo-association) transition fromatoms to an electronically excited state to make KRb* molecules. The excitedstate molecules then subsequently decay, with the rate Γ, and are lost from thecorresponding Feshbach molecule or atom signal. This KRb* molecule creation
rate can be characterized by an exponential time scale τ relevant to the transition Rabi frequency Ω as τ = Γ/Ω2 . To back out the transition-dipole-included
Franck-Condon factor (FCF) of the bound-bound or free-bound transition, we
normalize the coupling Rabi frequency by the laser intensity, Ω2 /I.
In Figure 4.6, we compare the FCF enhancement of bound-bound to freebound transitions for a near-threshold excited state (v � = −9 of the 2(1) potential,
Eb = h · 43 GHz). In Figure 4.7, the FCF enhancement comparison is done for thesame deeply bound excited state that we use for the coherent optical transfer to theabsolute ground state, as described in Chapter 8. In Figure 4.6, the enhancement2mfactor is ΩΩm2 /I≈ 500; while in the case of a deeply bound excited state (Figure/Iaa4.7), the enhancement factor is even higher, ≈ 5000! This large enhancement
factor reveals several orders of magnitude advantage when initiating the final
population transfer step from a weakly bound vs free atom precursor state.

49bound-boundΓfree-bound

Ωm

Γ

Ωa

10,000 a0

300 a0Figure 4.5: Schematic of the comparison of optical excitation rates for boundbound and for free-bound transitions. In both cases, the target bound state is thesame. The transition strengths are characterized by a Rabi frequency Ω. Once
molecules are in KRb excited electronic state, they decay spontaneously with acommon rate Γ. The depletion of molecules/atoms can be characterized by an
exponential time scale, e−t/τ , where τ = Γ/Ω2 . By measuring the time scale, we
can compare Ω for the two cases.

4.3

Detection
Finally, before discussing the next step in creating absolute ground-state

molecules (or any deeply bound ground-state molecules), we need to consider
molecule detection. To detect molecules, we use a method that is very standard
in an ultracold atom experiment but may not be the most intuitive method forpeople who work with “chemist’s molecules”.
To probe atoms, we use time-of-flight (TOF) absorption imaging. Here, thegas is suddenly released from the trap and then imaged after an expansion time(TOF) that is typically a few to a few tens of ms. For the imaging, a pulse ofresonant laser light illuminates the expanded gas and a CCD camera records the
shadow image of the cloud. The total number of atoms and their momentum
distribution can be obtained from these images.

50

3

(a)

(b)

16
Natom/104

τ = (3.9 ± 0.7) · 10−3 ms

40

12Nmol/104mol
2 Ṅt=0= (640 ± 120) · 107 /s

1

0.02

t/ms

0.04

0

0.06

Rb
K

8
τ = (1.9 ± 0.7) ms

4

0
0

87

atom
Ṅt=0
= (3.6 ± 1.2) · 107 /s

0

2

6
t/ms

4

8

10

Figure 4.6: Comparison of the optical excitation rates for (a) Feshbach moleculesand (b) atoms to an excited state near the atomic threshold, v � = −9 of 2(1)(the notation that labels the molecular state will be introduced in Chapter 5).
This comparison was taken using the same laser power in both cases. From thetime scale, we see the enhancement factor for starting with Feshbach molecules as
2compared to starting with atoms is ∝ ΩΩm2 = ττma ≈ 500.

2.0
2

5

Im=1.54 W/cm

Rb/K Number (10 )4

Molecule Number (10 )a1.5!m=0.093(6) ms

1.0
0.5
(a)

0.0

0.00

0.05

0.10

0.15

0.20

Pulse Length (ms)

0.25

1.5

Ia=3.9 W/cm2

! a=190(30) ms

1.0

K
0.5

Rb

(b)

0.0

0

200

400

600

800

Pulse Length (ms)1000

Figure 4.7: Comparison of the optical excitation rates (a) from Feshbach molecules
to a deeply bound excited state (v � = 23 of the excited 23 Σ+ potential) and(b) from atoms to the same excited state. The measured depletion of themolecule/atom population is due to excited KRb* formation followed by spontaneous decay and loss from the system. The KRb* formation rates were characterized by the exponential time scales, τm and τa , respectively. The enhancement factor for starting from Feshbach molecules instead of atoms near the Fano-Feshbacharesonance is ττma /I≈ 5000.
/Im

51
For molecules, however, imaging, or even just detecting molecules, with lightabsorption is much more challenging by the complicated internal state structure
and lack of a clean two-level cycling transition. An extremely useful feature ofheteronuclear Feshbach molecules, which have a tiny binding energy at magnetic
fields near resonance, is that they can be imaged using light resonant with an
atomic cycling transition [23]. At an applied magnetic field of 546 G, which isless than 1 G detuned from a K-Rb resonance, KRb Feshbach molecules canbe imaged using light resonant with the K atom cycling transition. When amolecule absorbs a photon, it dissociates, after which subsequent photons canscatter oﬀ the resulting free K atom. This gives a strong absorption signal, butdoes not distinguish between Feshbach molecules and any leftover unbound K
atoms. To image only the Feshbach molecules, we remove selectively unbound K
atoms (|9/2, −9/2�) using rf transition as described in section 4.1.2. In our system,
we typically image Feshbach molecules with an eﬃciency of 85% as compared to
K atoms at the same magnetic field. In Figure 4.8, we show a series of TOF
absorption images of the Feshbach molecule gas. For each absorption image ofthe gas, we fit the optical depth (OD) of the gas as two gaussians in the horizontal
and the vertical directions,
2

2

2

2

OD(x, z) = ODpeak · e−x /2σx · e−z /2σz .

(4.1)

The cloud sizes, σx and σz , give information of the expansion energy of the gas:
σx2 =

kb T
kb T
[1 + (ωx t)2 ] and σz2 =[1 + (ωz t)2 ],
2
mωx
mωz2

(4.2)

where ωx and ωz are the trap frequencies in the x- and the z-directions. In thecase when the expansion time is long compared to the trap period, the expansion
energy can be directly extracted from kb T = (m σ 2 /t2 ). The cloud size and its

52

TOF = 1 ms

TOF = 3 ms

TOF = 6 ms

TOF = 9 ms

Figure 4.8: A series of time-of-flight images for KRb Feshbach molecules createdwithin 1G of the resonance field. From these images, we can extract the total number of molecules (5.3 · 104 ) and the expansion energy (kb ·330 nK) of the molecular
gas.

53
peak optical depth give information of the total molecule number,
N = σx σz ODpeak (1 + 4(

∆ 2) )/σ0 /(eﬀ),
Γ

(4.3)

where ∆ is the laser detuning, Γ is the natural linewidth of the transition, σ0 is thescattering cross section, and eﬀ is the detection eﬃciency. From these images (Fig.
4.8), we determine that we create 5.3 · 104 Feshbach molecules, at an expansion
energy corresponding to a temperature of 330 nK.
This detection technique cannot be directly applied to deeply bound molecules.
However, by combining detection of Feshbach molecules with eﬃcient manipulation of the molecules’ internal states, we can bring ground-state molecules back to
the Feshbach state and then image. This give us information of molecule number
and their momentum distribution. This technique is very powerful for studying
the quantum behavior of a gas of polar molecules.

Chapter 5
KRb* Potentials

Producing a large number of ultracold heteronuclear KRb Feshbach moleculeswith T /TF as low as 1.4 is only the first step toward the creation of ultracold polar
molecules. This is because the internuclear separation of Feshbach molecules isstill too large to have any significant electric dipole moment (Fig. 4.4). To makeultracold polar molecules, we need to shrink the size of the Feshbach molecules
significantly. We achieved this by transferring KRb molecules into a tightly boundinternal state using a two-photon Raman technique (discussed in the next chapter). The success of such a technique relies on the identification of one or moresuitable electronically excited intermediate states that bridge the very diﬀerent
size wavefunctions for the initial and final molecular states. To look for a suitableintermediate state, we started by understanding and spectroscopically probing the
KRb excited electronic potentials (KRb*). This chapter contains relevant information concerning particular excited states and is not a comprehensive study of
KRb*.
5.1

Hund’s Coupling Cases
To study KRb excited potentials, we first need to be acquainted with somemolecular notation. Because molecules have rich internal structure, molecularstates are often described by a large set of quantum numbers. Figuring out a set of

55

15

12.8

(a)

4S+5P3/2

1 1"

10Energy (103 cm-1)

(b)

12.6

23!+
21!+

5

1 3"

4S+5P1/2

12.4

15

2(1)
3(1)
4(1)

10

0

-

2(0 )
+
+

3(0 )

12.2

a3!+
X1!+

X1!+

10

3(0 )
2(0 )

5

-5

-

4S+5S

a3!+

0

20

30

20

25

30

-5
Internuclear
0Separation
10
20 (a
300)
Figure 5.1: KRb electronic potentials (from ab initio calculations in [94] ). (a) The
potentials are shown for electronic ground-state potentials that asymptotically go
to the 4S+5S atomic threshold and electronic excited potentials that connect tothe 4S+5P threshold. Molecular potentials in the short range are labelled bymolecular spectroscopic notations described in the section of Hund’s case a. (b)A zoomed diagram of the excited potentials at large internuclear separations. Dueto strong spin-orbit coupling in this regime, the potentials are labelled by their
total angular momentum projection onto the internuclear axis, Ω, Hund’s case c.

56
the “good” quantum numbers is essential for understanding molecular potentials
and their spectra. A very useful and general guideline is the Hund’s cases basedon couplings between electronic and rotational motions [95]. Here, I will only
mention the relevant Hund’s cases for this work, namely case (a), (b), and (c).
5.1.1

Hund’s Case (a)The most commonly encountered Hund’s case is Hund’s case (a). In thiscase, molecular electronic orbital angular momentum L is strongly coupled to theinternuclear molecular axis and the electronic spin S in turn is strongly coupled
to L through spin-orbit coupling. Projections of L and S onto the molecular axis
are Λ and Σ. Λ + Σ = Ω is the total angular momentum projection onto themolecular axis. Heteronuclear molecular potentials at a small internuclear separation are usually Hund’s case (a) (Fig. 5.1(a)) and can be labelled using molecular+/−

spectroscopic notation as 2Σ+1 ΛΩ , where +/− refers to reflection symmetry of
the spatial electronic wave function through the plane containing the internuclear axis. For Λ = 0, 1, 2, etc, the corresponding spectroscopic notations are
Σ, Π, ∆, etc. The spectroscopic notations are sometimes preceded with a letteror numbered-label such as X, a, b, 1, 2, etc. The potential labelled by X denotes
the absolute lowest electronic potential of the molecule. Any higher lying electronic potentials from X are labelled sequentially by a, b, c , etc for the electronic
ground potentials (or 1, 2, 3, etc for the electronic excited potentials) that haveotherwise the same spectroscopic notation.
5.1.2

Hund’s Case (b)Hund’s coupling case (b) describes molecules with Λ = 0, e.g. X 1 Σ+ anda3 Σ+ of KRb. X 1 Σ+ described the lowest (X) electronic state of KRb that has
S = 1, Λ = 0. In this case, rotations of molecular nuclei, N , are decoupled from

57
S and the nuclear spin (I).
5.1.3

Hund’s Case (c)Hund’s case (c) usually describes molecules with a large internuclear separation, where strong spin-orbit coupling between L and S dominates over theircoupling to the molecular axis individually. In this case, Ω is a good quantum
number but Λ and Σ are not. Therefore, potentials near the dissociation limit(Fig. 5.1(b)) are labelled by Ω. This notation can also be used to label potentials
at short range (Fig. 5.1(a)) when Ω is emphasized (Ω is always a good quantum
number for Hund’s case (a) molecules), however, additional quantum numbers
will be required.
5.2

Near-Threshold States
Molecular spectra, although complex, usually have some regularities in thespacing between spectra lines. The largest periodic spacing between lines of theexcited potentials near the atomic threshold 4S+5P corresponding to the spacingof vibrational states. At this large internuclear distance, vibrational series follow
potentials that are labelled in the Hund’s case (c). Theses potentials, 2(1), 3(1),4(1), 2(0)+ , 2(0)− , etc, shown in Fig. 5.1(b) are characterized by their total
angular momentum projection, Ω. The first number 2, 3, 4 labels the progressionof the potentials according to their energy for the same Ω in the parentheses and+/- labels the reflection symmetry of the total electronic wave function, including
spin, through a plane containing the internuclear axis. Excited molecular statesnear threshold are expected to have a decent Franck-Condon overlap (apart fromconsidering wavefunction nodal patterns) with Feshbach molecule wavefunctions
even when their size is about 10 times smaller than the Feshbach molecules.
To look for these vibrational series, we scanned a tunable Ti-Sa laser (Chap-58
ter 3.3.1) to drive bound-bound transitions from Feshbach molecules. After transitions to KRb*, molecules spontaneously decay and are lost from our absorption
signal. Using loss spectra, we located excited vibrational series. All the frequencies of the levels obtained are referenced to the transition from the 4S+5S atomic
threshold with K in |9/2, −9/2� and Rb in |1, 1� to 4S+5P1/2 threshold with K
in |9/2, −9/2� and Rb in |2, x� at 546 G. Empirically, this “reference” transition
was observed at 377111.684(4) GHz which was measured using a wavemeter. We
performed a scan from ∼ 30 GHz below the 4S+5P1/2 atomic threshold to ∼ 800
GHz (∼ 25 cm−1 ) below.
To identify which of the observed vibrational levels are associated with theΩ = 1 or the Ω = 0 potentials, we looked at the hyperfine structure of the levels.
Because we conducted our search at a high magnetic field, for each Ω = 1 level,
we expected to see rich hyperfine structure that has spacings comparable to thatof rotational structure; while for each Ω = 0 level, we expected a much simpler
spectrum. A typical structure-rich spectrum of a Ω = 1 vibrational level (v � = −9,
where the minus sign refers to counting the vibrational quantum number fromthe atomic threshold) is shown in Fig 5.2. Each line dip corresponds to loss of
Feshbach molecules due to the laser driving a bound-bound transition. A typical
spectrum of Ω = 0 (v � = −14) is shown in Fig. 5.3 where both 0+ and 0− areseen. The structure associated with Ω = 0+ /0− are rotational lines.
To identify the particular potential, e.g. 2(1) but not 3(1), we compared
our observed vibrational spectra to S. Kotochigova’s prediction based on her potentials in [94]. Although the exact location of the calculated series may not be
accurate, the spacings between vibrational levels are usually good. To confirm ourassignment, we analyzed the vibrational spacings further. LeRoy and Bernstein[96] and Stwalley [97] had independently shown that vibrational spacings near
the dissociation limit (i.e. near the atomic threshold) are determined only by the59

4

Number (10 )

1.5
1.0
0.5
0.0
-46

-45

-44
-43
-42
Detuning From Threshold (GHz)-41

search for isolated -200GHz below Rb 5P1/2 threshold4

molecule number (10 )Figure 5.2: A typical hyperfine-rich spectrum of a vibrational level of the 2(1)potential near the atomic threshold. The spectrum is shown for the v � = −9 level,
where the minus sign refers to counting the vibrational quantum number from theatomic threshold, using π-polarization laser light.

1.8

1.2

1.5
1.0

0.6

0.5
0.0

0.0

0+
-216.0-215.50--215.0-214.5-214.0-213.5-213.0

Energy below threshold (GHz)-212.5

Freq (MHz)
-5

0

5

-212.0Figure 5.3: A Typical spectrum a 2(0) vibrational level (v � = −14). Here weobserve both the 0+ and the 0− components. Since their structure is similar, we
identified them through their diﬀerence in vibrational spacing with adjacent levels(Fig. 5.4(b)). (Inset) Detailed scan of the 0− component that reveals a doublet
structure.

long-range potential. Therefore, we can relate the vibrational quantum numberv � (counting from the threshold) to their binding energy, Ev� , by a semiclassicalequation [96, 97, 98]√
2 π Γ(1/2 + 1/n) �(n−2)/2nv =−2µ Cn1/n Ev�+ v0�
h(n − 2) Γ(1 + 1/n)
�

(5.1)

60

Vibrational Quantum Number

-5
2(1)

-10
-15
-20
(a)

-25

2(1), C 6=0.81(2) x 10

5

-5

+

2(0)

-10

-

2(0)

-15
-20
-25

+

5

-

5

2(0) , C 6=1.03(5) x 10(b)22(0) , C 6=0.85(2) x 10

3

4

5

1/3

6

7

1/3

8

9

10

E v (GHz )Figure 5.4: Vibrational series of (a) 2(1), (b) 2(0+ ), and 2(0− ). By fitting withequation 5.1, we can extract the C6 coeﬃcient that describes the potentials in thenear-dissociation regime. In addition, for (a), we distinguished two levels (in red)that have Ω = 1 structure but belong to a diﬀerent potential, 3(1).

61where n = 6 for our van der Waals long-range potential, h is the Planck’s constant,Γ denotes the Gamma function, µ is the reduced mass, and v0� is a fitting oﬀset.
1/3

In Fig. 5.4, we plot the 2(1), 2(0)+ , and 2(0)− vibrational series v � vs Ev�

and see that our data rarely deviating from the linear fit. We also extracted C6from the fit using equation 5.1. This extracted C6 is about 10-20% lower fromthe theory calculations. A more accurate long-range potential is characterizedby C6 , C8 , and perhaps C10 . To get a better potential in the long-range, S.
Kotochigova adjusted her values for C6 and C8 (C10 was set to zero) by fitting
our data. Moreover, deviations from the linear relation in Fig. 5.4(a) allowed us
to distinguish vibrational levels (red points, 131.25 GHz- and 450 GHz-detuned)that have Ω = 1 structure but belong to the 3(1) potential, which asymptoticallygoes to the 4S+5P3/2 threshold. All observed levels and their assignments are
summarized in Table 5.1.
v�
-8
-9
-10
-11
-12
-13
-14
-15
-16
-17
-18
-19
-20
-21
-22

2(1) (GHz)
28.2
43.3
61.8
83.9
108.1
150.3
188
231
284
345.2
408.5
488.0
572.8

2(0)− (GHz)2(0)+ (GHz)

165
213.4
262.5
321.2
385
457.08
541.0
628.2
726.9
833.9

170
215.7
263.3
314.7
377.8
446.7
514.0
616.9
711.4
812.6

Table 5.1: Summary of vibrational levels observed near the atomic threshold of
4S+5P. The frequencies are given for the line center of power-broadened lineshapes
and were measured using a wavemeter.

62
5.2.1

v � = −14 of 2(0− )In the last section, we identified diﬀerent vibrational series in the excitedelectronic states near the dissociation limit. From these data, we picked v � = −14
of 2(0− ) as an intermediate state for two-photon transfer to near-threshold groundstates. In particular we chose a strong feature (indicated with an arrow) that is
isolated from other lines by ∼ 1 GHz. The isolation of the line is crucial forworking with a clean three-level system for an eﬃcient population transfer. A
detailed scan of the feature indicated by the arrow reveals a doublet substructure.
5.3

Bridging to Deeply Bound Molecules
Performing spectroscopy near the dissociation limit of the electronic excitedstates enabled us to choose a suitable intermediate state for near-threshold molecular state transfer (Chapter 6). However, these near-threshold excited states are
not suitable for bridging the large diﬀerence in the molecular wavefunction of Feshbach molecules and deeply bound states of the electronic ground potentials. In
Fig. 5.5, we plot the square of the total transition dipole moment (this includes
the electronic transition dipole moment square and the wavefunction overlap) for
transitions from vibrational levels of the 2(0)+ (Fig. 5.5a) and 3(1)(Fig. 5.5b)
potentials to both the near-threshold v = −1 and the deeply bound v = 0 of
a3 Σ+ , and the deeply bound v = 0 of X1 Σ+ . For the deeply bound levels, we nolonger number v from threshold, and v = 0 is in fact the lowest energy level ofthe potential. The transition dipole moments are calculated by S. Kotochigova
and are plotted as a function of their detuning energy from threshold. As we
can see, the near-threshold excited states are only good for near-threshold groundstates, e.g. the v = −1 state. Therefore, to find an appropriate molecular statethat can bridge v = −1 and v = 0 simultaneously, we have to choose a deeply

63bound excited state. (In fact, our initial molecular state will be weakly boundFeshbach molecules, which would have an even weaker transition dipole momentto the excited state as compare to the v = −1 case.) In the deeply bound regime(where the position of the inner turning point of v = −1 has good amplitude), we
found several excited-state potentials that could serve as the bridge. To pursue
molecule internal state transfer from Feshbach molecules to the triplet or the singlet rovibrational ground state, we chose to go through deeply bound vibrational
levels of the 23 Σ+ potential (also labeled as 3(1) and 3(0)− according to their Ωcomponents).
5.4

23 Σ+ and the Ω = 0− and Ω = 1 Components
Looking at the full set of electronic potential curves (Fig. 5.1), it is not

diﬃcult to see that there are certain “coincidences” where inner and outer turning
points of an excited state vibrational level could line up well to overlap both the
Feshbach molecule state (basically at Energy = 0) and the v = 0 state of either
a3 Σ+ or X1 Σ+ . Our choice was the vibrational levels associated with 23 Σ+ , whichhave relatively large calculated transition dipole moments to our desired molecular
ground states and have transition frequencies that are conveniently covered by ourwidely tunable Ti-Sa laser as well as commercial diode lasers. To experimentally
identify these states, since potentials in this region are usually constructed by
ab initio calculations, where there could be a large uncertainty in the predictedenergies of vibrational levels, we looked for available experimental spectroscopy
data as a guideline. The v � = 0 to v � = 13 of 23 Σ+ for 39 K85 Rb has been reportedby [27]. Our search was guided by Kotochigova’s ab initio calculation fitted toexperimental data from 39 K85 Rb [27], with the appropriate mass scalings.
The 23 Σ+ potential has two components in the short range that are distinguished by their quantum number Ω = 0− or 1, which in turn can be used

64
0.01

(a) 2(0)+v=-1 of a3Σ+|d|2 /(ea0)2

10�4
10�6
10�8
10�10
v=0 of a3Σ+

10�12

�7000

�6000

�5000

�4000

�3000

�2000

�1000

0

Excited-State Detuning Energy from Threshold (cm-1)|d|2 /(ea0)2

0.01(b) 3(1)v=-1 of a3Σ+v=0 of X1Σ+

10�4
10�6
10�8
10�10
10�12

v=0 of a3Σ+

�3500

�3000

�2500

�2000

�1500

�1000

�500

0

Excited-State Detuning Energy from Threshold (cm-1)

Figure 5.5: Predicted transition dipole moments from vibrational level in theelectronically excited potentials (a) 2(0)+ and (b) 3(1) to v = −1 and v = 0of the triplet ground-state potential a3 Σ+ and v = 0 of the singlet ground-state
potential X1 Σ+ (not shown for 2(0)+ ). The near-threshold excited states havegood transition dipoles only to the near-threshold electronic ground state, e.g.
v = −1, not v = 0. Therefore, to transfer from v = −1 to v = 0, we need tochoose an excited state in the deeply bound regime. For transfer to both the singlet
and the triplet rovibrational ground state, we chose two diﬀerent vibrational levels
that are associated with 3(1) (or associated with 3(0), which is the same potential
but characterized by Ω = 0). The larger transition dipole moment from 3(1)to v = 0 of X1 Σ+ seems too good to be true. It turned out to be a factor of 20
smaller (400 smaller in d2 ), but still had the same qualitative pattern as a functionof excited-state energy. However, the reduced magnitude was still suﬃcient toenable eﬃcient transfer (Chapter 8)! All transition dipole moments shown here
were calculated by S. Kotochigova.

65
to label the potentials as 3(0)− and 3(1) (Fig. 5.1). Since the two components
share the same potential, their vibrational level spacings and their wavefunction
overlap to the electronic ground states are similar. However, their electric dipole
transitions are not the same because of angular momentum selection rules. For
molecular state transfer to the triplet electronic ground state, both Ω = 0− andΩ = 1 will work. However, for transfer to the singlet electronic ground state,
only the Ω = 1 component will have singlet-triplet mixing due to perturbations
that come from the 11 Π (Ω = 1) potential [99]. Since 11 Π lies above 23 Σ+ , we
expect that the Ω = 1 component of 23 Σ+ to be pushed toward lower energy
relatively to the Ω = 0− component. The splitting between Ω = 0− and 1 is thesecond-order spin-orbit coupling that was calculated in [99]. Because only theΩ = 1 component will couple to the singlet electronic ground state, it is crucial tospectroscopically distinguish the two components. Again by using loss spectra of
Feshbach molecules after driving bound-bound transitions, we can identify theseseries. Table 5.2 summarizes all of our observed levels of 23 Σ+ .
5.4.1

v � = 10 of 23 Σ+ (Ω = 0− )During our first search for 23 Σ+ levels, we identified v � = 8 − 12 of 23 Σ+(Ω = 0− ). Because the Feshbach molecule wavefunction has lots of oscillationsin the small internuclear separation region, we observed strong variations in thetransition dipole moment as shown in Fig. 5.6. We chose the v � = 10 level as ourintermediate state for transfer to the triplet rovibrational ground state (Chapter
7). We have also scanned in more detail the v � = 10 level and observed several hyperfine lines that served as intermediate states to access diﬀerent hyperfine statesof the triplet rovibrational ground state (Fig. 5.7). For the triplet rovibrational
ground-state molecules, we measured their permanent electric dipole moment bymeasuring the level Stark shift when applying an electric field (Chapter 7.3). The

v�
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

297012Ω = 0− (GHz)290406.764
291758.052
293099.603
294431.403
295753.57
297066299604300890302158303425304685305949304756306033308352309601.4308438309707.6Ω = 1 (GHz)290359.5
291709.7
293049.5

66

Table 5.2: Summary of observed vibrational levels for both the Ω = 0− and 1
components of 23 Σ+ .

KRb* states are also Stark shifted with electric fields. In Fig. 5.8, the measured
Stark shifts of the line that has a transition frequency ∼ 293104.055 GHz fromFeshbach molecules are listed.
Finally, we calibrated the transition dipole moment from Feshbach molecules
to v � = 10 of the 23 Σ+ electronically excited potential by using both powerbroadened lineshapes and a measured one-photon excitation rate. We consider
our system as an open coupled two-level system shown in the cartoon drawingof Fig. 5.9(b). Feshbach molecules are driven to KRb* by a laser with Rabi
frequency Ω and KRb* then subsequently spontaneous decays with a rate γ. This
decay is assumed to populate levels that are outside of the two-levelsystem, hence
is lost. We describe our system mathematically by H = �2 

0Ω/2Ω/2 ∆ − iγ/2.

We first measured power-broadened lineshapes (Fig. 5.9(a)) by scanning

67

d /(ea0)

2

-5

10

-6

2

10

-7

10

8

9

10

v’

11

12

4

molecule number (10 )Figure 5.6: Nodal pattern of the transition from Feshbach molecules to diﬀerent
vibrational levels, v � , of the 23 Σ+ (Ω = 0) potential.

3

5

2

293104.055 GHz

4
3
2

1

1

293099.538 GHz

0

100

0

103.7 103.8 103.9 104.0 104.1 104.2 104.3 104.4 104.5

105

110

115

293xxx.x GHz
Figure 5.7: Detail of the structure of v � = 10 of 23 Σ+ (Ω = 0). The two lines weused to accessed diﬀerent hyperfine states of the triplet rovibrational ground states
are the ones with transition frequencies of 293099.538 GHz and 293104.055 GHz
from Feshbach molecules. These specific frequencies are derived from referencing
the Ti-Sa laser to a stable optical frequency comb. The coarse frequency scaleon the y-axis were reading from a wavemeter. (Inset) Detailed structure of thestrong line centered at 293104 GHz.

68

KRb* Stark Shift (MHz)

0
-2
-4
-6
-8
-10
-12
0.0

Field
(kV/cm)0.57

Stark Shift
(MHz)
-1.13

0.88

-2.50

1.06

-3.72

1.32

-6.03

1.76

-10.7

0.5

1.0

1.5

2.0

E-field (kV/cm)
Figure 5.8: Stark shift of the v � = 10 of 23 Σ+ (Ω = 0− ) state of KRb*. This linethat has a transition frequency ∼ 293104.055 GHz from Feshbach molecules atzero electric field.

the laser detuning ∆ from the resonance. For these measurements, we used three
diﬀerent laser intensities (corresponding to three diﬀerent Rabi frequencies Ω, with√I ∝ Ω) and a light pulse duration of 20 µs. The power-broadened lineshapes foran open system is diﬀerent from a closed system in that when Ω > γ, the lineshapeis not Lorentzian but rather has a flat bottom loss spectrum (see the red curvein Fig. 5.9(a)). This can be exactly modeled by solving the given Hamiltonian
for an open two-level system. The lineshapes give us information about ΩΓ . To
get Ω and Γ independently, we fixed ∆ = 0 and pulsed on the laser for various
times to watch the Feshbach molecule signal decay due to KRb* creation (Fig.
5.9(b)). The exponential time scale is ∝ ΩΓ2 . Together, we obtain Γ = 6.7 MHz(real frequency and not angular frequency) and Ω. Knowing the laser intensitythat corresponding to a specific Rabi frequency Ω and (dipole moment/(ea0 ))=√Ω/(109 )/(0.22068)/ I, where Ω is in angular frequency, and I is in W/cm2 , weobtained that the transition dipole moment to be 0.004(2) ea0 .

69

30 000

Molecule Number

25 000
20 000
15 000
10 000

(a)

5000
0
780

790

800

810

820

Frequency (MHz)

Molecule Number

30 00025 000∆

20 000Ωγ

15 000
10 000

(b)

5000
0

0

20

40

60

80

100

120

140

Pulse Duration (μs)
Figure 5.9: Calibration of transition dipole moment from (a) power-broadenedlineshapes and (b) one-photon molecular loss time scale.

70
5.4.2

v � = 23 of 23 Σ+ (Ω = 1)

For two-photon transfer from Feshbach molecules to the singlet rovibrational
ground state, we chose v � = 23 of 23 Σ+ (Ω = 1). Originally, we wanted to use ahigher level, v � ∼ 34, for the state transfer because the predicted transition dipole
moment to the singlet rovibrational ground-state was the largest. However, since
the vibrational levels become quite irregular in this regime due to perturbations
from three diﬀerent potentials, we decided to cease the search for higher v � andto try v � = 23. This worked!
Again this level has hyperfine structure. We have characterized three ofthe states and these are summarized in Table 5.3. Each line can only be driven
from Feshbach molecules using a specific circular polarization. We do not identify
or assign their hyperfine quantum number. In addition, we have calibrated thetransition dipole moment from Feshbach molecules using the method described in
the last section. This was performed using Raman lasers with 50/50 mixture of σ +and σ − light. We measured the transition dipole moment of the line corresponding
to transition frequency of 309602.851 GHz to be 0.005(2) ea0 . Finally, the Stark
shift of the same line is shown in Fig. 5.10.
Transition freq fromFeshbach Molecules

polarization

ground hyperfine-state coupling

309602.753 GHz
309601.737 GHz
309600.656 GHz

σ+
σ−
σ−

| − 4, 1/2�| − 4, 1/2�, | − 3, −1/2�, | − 4, −3/2�
| − 4, 1/2�, | − 3, −1/2�, | − 4, −3/2�

Table 5.3: Summary of three hyperfine lines of the v � = 23 level of the 23 Σ+(Ω = 1) potential. Each state can only be driven from Feshbach molecules using
a specific circular polarization light. In addition, each state can couple to specific hyperfine states (hyperfine state labeling notation is discussed in Chapter 9)of the ro-vibronic ground state through a two-photon transition from Feshbach
molecules.

KRb* Stark Shift (MHz)

71

120

Field
(kV/cm)Stark Shift
(MHz)

100

0.44
0.88
1.32
1.76
1.85

3.0
11.7
26.6
38.7
41.7

2.20

53.8

2.64

75.9

80
60
40
20
0

0

1

2

E-field (kV/cm)

3

Figure 5.10: Stark shift of the v � =23 of 23 Σ+ (Ω = 1). This specific line has atransition frequency ∼ 309602.753 GHz from Feshbach molecules at zero electricfield. The line is for guides of the eyes.

Chapter 6
Coherent Two-Photon Raman Transfer

Using coherent methods to transfer atoms to tightly bound molecules has
the advantage that the transfer is state-selective, eﬃcient, and can take away thebinding energy through the applied transfer fields (either magnetic fields, electricfields, or laser fields). In Chapter 4, I have already introduced the first coherent
transfer method, which creates Feshbach molecules from atoms through an adiabatic magnetic-field ramp across a Fano-Feshbach resonance. In this chapter, I
will introduce the second coherent method, which we use to shrink the moleculesize. The discussion follows work in [26, 100].
Making high vibrational level (high-v) molecules from a BEC of atoms using
a coherent Raman transition was first demonstrated with Rb2 by Wynar et al. in2000 [101]. Using a molecular signal that came from loss of atoms immediately
after the Raman pulse transfer, they were able to determine the binding energyof the molecules along with the condensate mean-field energy precisely. In 2007,
Winkler et al. [25] had extended the coherent transfer idea to a more robust
method, namely, STImulated Raman Adiabatic Passage (STIRAP) [102], andtransferred Rb2 Feshbach molecules to the v = −2 level (h·637 MHz bound) with
87% eﬃciency. Here, the minus sign on the vibrational level refers to counting thelevel from the atomic threshold, e.g. v = −1 would be the least bound level.
As STIRAP having been demonstrated for eﬃcient state transfer for homonu-

73
clear molecules to v = −2, we were certain that the same would work for our heteronuclear molecules, KRb. However, the final goal was to make polar molecules,
which means that the KRb molecules would have to be tightly bound. Furthermore, the v = 0 level (counting v here from the bottom of the potential) seems
special, since it is the lowest energy level and should be maximally stable. The
big challenge would then be applying the STIRAP technique to transfer molecules
between vibrational levels with very diﬀerent sizes.
Here, I will introduce the two-photon Raman transfer technique by first
discussing transfer from Feshbach molecules to a high-v level, in particular v = −3.
In subsequent chapters, I will describe how we extend the technique to reach theabsolute ground state.
6.1

Three-level System
Fully coherent manipulation is the key to eﬃcient transfer molecules between

diﬀerent vibrational levels. Our transfer technique uses three molecular states,labelled |i�, |e�, and |g�, that are coupled together by two laser fields, Ω1 and Ω2 ,as shown in Fig. 6.1. Since the excited state |e� is usually lossy and short-lived,
we introduce a decay rate γ for the state |e�. In general, states |i� (the Feshbach
molecule state) and |g� (the target vibrational state) can also decay, but this willbe ignored since the time scale is usually much longer than the transfer process
in consideration. The simplified Hamiltonian of the system, after making therotating-wave approximation, in the basis of |i�, |e�, and |g� isΩ1 (t)
0 
 0

�
H= 
Ω1 (t) 2∆ − iγ Ω2 (t) 

,
2

0
Ω2 (t)
2δ

(6.1)

74
where ∆ denotes the one-photon detuning and δ denotes the two-photon detuning
shown in Fig. 6.1. When the Raman resonance condition is met, namely δ = 0,
1 (t)one of the eigenstates of the system is cosθ |i� + sinθ |g� where θ = tan−1 ( Ω).
Ω2 (t)

This state is dark in that it does not couple to the lossy |e� level. This dark state isimportant because it allows, with an appropriate time dependence of the applied
coupling fields Ω1 (t) and Ω2 (t), the full population of |i� to be converted to |g�
without ever populating the lossy excited state |e�. This process is called STIRAP
and it provides fully coherent transfer between states |i� and |g� [102]. It is
important to note that we have assumed that there is a well-defined phase relation
between the two coupling fields. This assumption sets a challenging requirement
for lasers that couple states that are very diﬀerent in binding energy.

∆

γΩ

2

|e�

Ω1

δ

|g�

|i�

Figure 6.1: Open three-level system. The three states labelled by |i�, |e�, |g�, arecoupled by two laser fields, Ω1 and Ω2 . The two laser frequencies are detuned from|e� by ∆ (one-photon detuning). The frequency diﬀerence between two lasers is
detuned from the energy diﬀerence of |i� and |g� by δ (two-photon detuning). The
excited state |e� is short-lived with an decay rate γ that corresponds to a loss ratefrom the three-level system.

75
For the first experiment, we demonstrated transfer of weakly bound Feshbach molecules with a size of the order 300 a0 (binding energy ∼ h · 240 kHz) toanother weakly bound state, v = −3, that has a size of 34 a0 and a binding energyof ∼ h · 10 GHz. To do this, we identified an intermediate state that bridges the
wavefunction diﬀerence. As we have already seen in Chapter 5.2, the vibrational
level structure for the KRb excited molecular potentials near the atomic 4S+5P
threshold is very complicated. In particular, the level that associated with Ω = 1,
e.g. 2(1), have rich hyperfine structure within a 3 GHz-span around each vibrational level at a magnetic field of 546 G. These nearby lines will make it diﬃcult tofind an isolated three-level system and allow oﬀ-resonance excitations. Therefore,
we pick an intermediate state that is associated with a potential that has Ω = 0,
which greatly simplifies the hyperfine structure. The particular intermediate state
we chose was v � = −14 of 2(0− ), which has a very simple structure (see Fig. 5.3)
with individual lines that are spaced by at least ∼ 1 GHz.
To transfer population from Feshbach molecules to a high-v level via the chosen excited intermediate state, some knowledge of selection rules is helpful. First,
our Feshbach molecules are 80% triplet (total electronic spin S = 1) in character[89], therefore, it is more favorable for transferring them into vibrational levels of
the triplet electronic ground potential, a3 Σ+ . In addition, molecules in a high-v
level are expected to have a hyperfine structure similar to the atomic hyperfine
structure at the same magnetic field (Fig. 6.2). Therefore, we label the high-v levels by the same atomic hyperfine quantum numbers. For example, KRb in v = −3
with K|9/2, −9/2�+Rb|1, 1� character is labelled aa(−3) (using the same notation
introduced in Chapter 4.1.1. or [89]). KRb in v = −4 with K|7/2, −7/2�+Rb|1, 0�
character is labelled rb(−4). Since our Feshbach molecules are ∼ 65% in the “aa”
open channel and ∼ 35% in the “rb(−2)” closed channel, the two-photon Raman
transition will mostly likely only couple strongly to high-v levels with “aa” or “rb”

76
character [88]. Therefore, we only looked for vibrational states associated withaa(v) and rb(v). In addition, the hyperfine quantum numbers of the intermediate
state can impose additional selection rules. Since the intermediate state hyperfine
quantum number is harder to assign, we test the intermediate states empirically.
40

87

K R b aa s-sym m etry bound state s
Feshbach molecules

0

A(-1)

E /h (G H z)

B(-2)
D (-3)
A(-2)
C (-3)-5)
qc( - 3rb ( - 3 )c c (- 3 )b b (- 3 )

B(-3)

-1 0

A(-3)

a a (-3 )

0400

B (G auss)

800

1200

Figure 6.2: KRb high-v molecular levels vs magnetic field. Molecular vibrational levels near the atomic threshold have a hyperfine structure derived
from atomic ones. Here all energy are referenced to the atomic threshold of
K|9/2, −9/2�+Rb|1, 1�. Each molecular level can be labelled by its hyperfine
quantum numbers and its vibrational quantum number. For example, KRb in
v = −3 with K|9/2, −9/2�+Rb|1, 1� character is labelled by aa(−3). Figure courtesy of P. Julienne (2007).

6.2

Dark Resonance
Before performing STIRAP to coherently transfer population from the Fes-hbach molecule state to a desired target vibrational level, we first need to precisely

77
determine the energy of the target vibrational state. We did this using two-photon
dark resonance spectroscopy. Recall that there exists a dark state, as discussed
1 (t)above, cosθ |i� + sinθ |g� where θ = tan−1 ( Ω), when the two-photon RamanΩ2 (t)

resonance condition is fulfilled. In the limit of Ω2 � Ω1 , the dark state becomes|i�, and therefore no loss is expected for |i� when δ = 0.
To search for a particular near-threshold vibrational level in the electronically ground potential, we fixed the frequency of the laser corresponding to Ω1
to resonantly drive Feshbach molecules to the electronically excited intermediate
state. Then we scanned the Ω2 laser frequency and monitored the populationof the Feshbach molecules, after pulsing on both lasers. For this case of nearthreshold transfer, the beat frequency of the two Raman lasers is measured usinga photodiode and then phase-locked to a stable microwave source. In the absenceof Ω2 , the Feshbach molecules disappear because they are resonantly excited bythe Ω1 laser field. These excited molecules will decay by spontaneous emission
into a large number of molecular and atomic states and are lost from our signal,
which comes exclusively from the weakly bound Feshbach molecules. However,when the second laser field, Ω2 , fulfills the Raman resonance condition, δ = 0,
the Feshbach molecules remain due to the formation of the dark state. Measuring
this dark resonance position then gives the energy diﬀerence between the initialstate |i� and the target state |g�. However, strictly speaking, the exact binding
energy is determined only when the Raman condition is fulfilled in the limit ofΩ1 � Ω2 . When Ω1 is not � Ω2 , the dark resonance position will be slightly
Stark-shifted by the corresponding to laser field Ω1 . Therefore, to precisely determine the binding energy, we performed dark spectroscopy with various Ω1 laser
powers and extrapolated the energy corresponded to zero Ω1 laser power. From
this measurement, the binding energy of aa(−3) was precisely determined to beh· 10.49238(15) GHz at 545.88(5) G [26]. We have also probed other states, such

78
as aa(−2) and rb(−3), and summarize their binding energies in Table 6.1.
v
aa(−2)rb(−3)aa(−3)rb(−4)

binding energy/h
3.1504(10) GHz
7.31452(15) GHz
10.49238(15) GHz
21.547(1) GHz

theoretical prediction/h
3.1510 GHz
7.3187 GHz
10.4976 GHz
21.5661 GHz

Table 6.1: Summary of the observed high-v levels and a comparison with thetheoretical predicted values from P. Julienne. (rb(−4) was measured using a
diﬀerent phase-locked Raman laser system involving referencing the two lasers toa frequency comb).

1.2

fraction

1
0.8
0.6
0.8

0.4

0.4

0.2
0
−15

−10

−5

0

0
−1

δ/MHz

5

10

0
15

1
20

δ/MHz

Figure 6.3: A typical dark resonance spectrum, shown for rb(−3). The asymmetryof the lineshape comes from the one-photon detuning not being exactly zero.
(Inset) A zoomed in of the dark resonance feature.

The two-photon dark resonance can also be used to characterize the transition coupling strength. Here, we take a scan of rb(−3) as an example. The darkresonance scan (Fig. 6.3) was achieved by scanning the Ω1 laser frequency while

79
fixing the laser frequency of Ω2 on resonance (∆ = 0). We observed a one-photon
loss feature with a Lorenzian width of ∼ 6 MHz and a narrow dark-resonancefeature by definition at δ = 0. The splitting of the two minima corresponds to
the Rabi frequency Ω2 .
6.3

Detuned STIRAP, v = −3
Once the precise location of aa(−3) was determined, we used STIRAP [102]to transfer KRb Feshbach molecules into aa(−3). As discussed above, when the
two-photon detuning δ = 0 is maintained, the population of |i� can be adiabatically transferred to |g� by an appropriate choice for the time dependence of thecoupling laser fields with corresponding Rabi frequencies Ω1 (t) and Ω2 (t). Figure
6.4(a) shows the counter-intuitive STIRAP pulse sequence used in the experiment.
First, we turned on Ω2 (corresponding laser intensity I2 ) to couple |e� and |g�.
While the intensity of I2 is ramped down from I2max to 0 within τp = 20 µs, theintensity of I1 (corresponding Rabi frequency Ω1 ) is ramped up from 0 to I1max ,thereby adiabatically transferring Feshbach molecules to the target state, without ever populating the lossy intermediate state |e�. At the end of the STIRAP
transfer, I1 can be turned oﬀ. Reversing the pulse sequence in time reverses thetransfer process from |g� to |i�.
Figure 6.4(b) shows STIRAP transfer of KRb from the weakly bound Feshbach molecule state to aa(−3) using the pulse sequence in Fig. 6.4(a). Since weonly detect population in the initial Feshbach molecule state, we also perform areverse STIRAP pulse sequence to bring the deeply bound molecules back to the
Feshbach molecule state for detection. For the data in Fig. 6.4(b), the round-triptransfer has an eﬃciency of 71%. Assuming equal eﬃciency each way, the one-way
STIRAP transfer eﬃciency is 84%.
So far, we have emphasized that eﬃcient STIRAP transfer requires main-80

I/I maxa) 1

I2 /I2max

0.8
0.6
0.4
0.2
0

I1 /I1max
τpb) 1
fraction

0.8
0.6
0.4
0.2
0

−30

−20

−10

0

10

tpulse/µs

2030Figure 6.4: Population transfer from Feshbach molecules to aa(−3) using a coherent two-photon STIRAP transfer technique. (a) Laser intensity pulse sequence
normalized to their individual intensity maximums. The counter-intuitive sequence is to turn I2 on first, ramp it down while ramping I1 up. The second
half of the sequence is for a reversed STIRAP process. (b) Population transfer
corresponding to the pulse sequence in a. The data is shown as the fraction of
population in the Feshbach molecule state. The dashed line is the corresponding
population in the target state, aa(−3). The lines are fits to the data using themodel based on equation 6.1.

81
taining δ = 0. But we have not yet discussed the one-photon detuning, ∆. Usinga transfer model based on equation 6.1, it is easy to show that any ∆ works provided that ∆ is not so large such that the laser coupling strengths between states
drop significantly for accessible laser powers. The most straightforward thing is toperform STIRAP at ∆ = 0. However, there are certain advantages of performing
STIRAP at ∆ �= 0. For example, since a STIRAP lineshape is in general muchnarrower than a dark resonance lineshape and we look for loss of initial-state population in our one-way STIRAP signal, performing detuned STIRAP (Fig. 6.6)
enables us to find the target state location more precisely. (In the case when
∆ = 0, one-way STIRAP lineshape will show loss which could be either real loss
or transfer to a “dark” state for the entire scanned range of δ.)
One interesting feature of detuned STIRAP is that the resonant position
can move depending on imbalanced Stark shifts from the coupling laser fields
and the sign of the detuning. These can all be understood in a model based on
the Hamiltonian given equation 6.1. Therefore, by performing detuned STIRAP
with both positive and negative ∆ and various intensity ratios of the coupling
lasers, we can determine the unperturbed δ = 0 location along with the laser
powers that give the coupling ratio Ω1 /Ω2 = 1 (Fig. 6.5). Using the model, we
extracted Ω2 /Ω1 = 1.7 ± 0.1 and ∆ = 50 ± 5 MHz for the lineshape shown in Fig.
6.6. Together with the previous measurement of Ω2 and I1max =3.7(1.5) W/cm2 ,I2max =1.0(0.4) W/cm2 , we obtained the eﬀective transition dipole moments of
d1 =0.050(15) ea0 and d2 =0.17(4) ea0 .
6.4

Extending STIRAP to Deeply Bound States
With coherent transfer from Feshbach molecules to a near-threshold vibra-

tional state demonstrated for a heteronuclear system, we moved on to finding
a way to reach a deeply bound state that has a large permanent electric dipole

82

Frequency (GHz)10.4932 at fixed !1
10.4928

red-detuned (83 MHz)

!1/!2=1
10.4919775GHz

10.4924
10.4920
10.4916
10.49120blue-detuned (70 MHz)

1

2

3

4

!2 (arb.)
Figure 6.5: AC stark shift measured for the detuned STIRAP resonance. The
resonant frequency diﬀerence of the two Raman lasers depends on the coupling
field strength Ω2 due to imbalanced laser Rabi frequency. By measuring the STIRAP resonance using blue-detuned and red-detuned STIRAP, we can determine
the exact energy diﬀerence of the initial and the target state energy (where thelines crosses). The crossing point happens when the two-photon Rabi frequency
ratio Ω1 /Ω2 = 1, which allows us to extract the transition dipole moment ratio
very accurately. To get the binding energy of aa(−3), we add the beat frequency
plotted here with the Feshbach molecule binding energy.

83

fraction in |i�

0.8

a)

double STIRAP
single STIRAP

0.6

0.6

b)

0.4

0.4
0.2

0.2
0
−400

−200

0

δ/kHz

200

400

−200

0

δ/kHz

200

0

Figure 6.6: Detuned-STIRAP lineshape (a) Fraction of total population in theinitial Feshbach molecule state v.s. two-photon detuning after both single STIRAP and round-trip STIRAP. (b) The corresponding population in the aa(−3)
state after a single STIRAP.

84
moment. There were three major challenges. The first was a technical challenge.
A “technical” limitation of our first demonstration experiment was that the two
Raman lasers were compared using a photodiode to record their beat, which wasthen phase-locked to a microwave source. This technique is only applicable overa limited frequency range. For example, the bandwidth of the photodiode waslimited to 20 GHz. To stabilize two lasers with a large frequency diﬀerence, evenone in the optical domain, one needs to have a stable reference that can bridge thelarge diﬀerence. Of course, we also need to have new lasers with the desired wavelengths. We used a home-built stable optical frequency comb for our reference.
The relevant comb parameters and its setup are described in Chapter 3.3.3.
Once the comb was implemented, we looked for the next vibrational level,rb(−4) ≈ h · 21GHz, and found it right away. However, using the same excitedintermediate state (v � = −14 of 2(0− )), the transition dipole moment to deeply
bound states decreases by orders of magnitude. The second challenge, as was
described in Chapter 5, was then to choose an intermediate state that can bridge
the large wavefunction mismatch between Feshbach molecules and a target low-v
state. The identification of a suitable intermediate state began with our theoretical collaborators, S. Kotochigova and P. Julienne, who calculated transition
dipoles for the vibrational levels of diﬀerent excited-state potentials to all the vibrational levels in the ground electronic state. We were especially interested in
the transition dipole moments from a single excited state to both v = −1 (counts
down from threshold) and v = 0 (counts up from absolute ground state). Basedon this, they made suggestions of a few possible “routes” to the low-v levels. We
chose a route that used a similar intermediate state as the one used in the photoassociation work by Sage et al. [22]. One main reason for this choice was there
were some spectroscopic data available in the relevant range for KRb of diﬀerent isotopes [27]. These relevant vibrational levels are the low-v levels associated85with the 23 Σ+ potential and are expected to have good wavefunction overlap to
the low-v levels of the electronic ground potentials. Our biggest concern was if
we could drive weakly bound Feshbach molecules directly to these low-v levelsof 23 Σ+ using our experimentally available laser power. Although being slightly
skeptical of the relatively large calculated dipole moments for these transitions,
we were optimistic because of an observation we made [18] when first trying tocreate Feshbach molecules. We had seen a “lucky” coincidence where our 1075 nm(linewidth of 1 nm) optical trap laser was causing loss of the Feshbach molecules.
This could be because the light was driving Feshbach molecules to perhaps v � = 0of the 23 Σ+ potential [18]. Now, we have experimentally identified the low-v seriesof 23 Σ+ as described in Chapter 5.
The final challenge was the potentially large uncertainty of the locationsof relevant low-v levels of the electronic ground potential. Our experiment, although it has a high sensitivity and an unprecedented resolution for molecular
spectroscopy, has a cycle time of a minute. Therefore, it is not suitable for a large
spectroscopy scan. For this reason, we have to understand the uncertainty of thepotential in diﬀerent regions of internuclear separations. (We also care about thesame thing for electronic excited-state potentials, which is discussed in Chapter5.) For the near-threshold region that was accessed in the last section, the uncertainty of the vibrational levels is 1%. If we scale this uncertainty directly tov = 0 of a3 Σ+ (h·7 THz) or v = 0 of X 1 Σ+ (h·125 THz), it would be 70 GHz or 1
THz (half of the vibrational spacing)! Luckily, Pashov et al. [28] constructed very
accurate KRb ground-state potentials for both the triplet and the singlet potentials using conventional spectroscopy of 39 K85 Rb low-v states and matching thiswith comprehensive Feshbach resonance data near the atomic threshold. Afterusing mass-scaling to calculate the vibrational series for 40 K87 Rb, the uncertainty
is expected to be ∼ 7 GHz for the triplet ground state and ∼ 10 GHz for the

86singlet ground state. One side note is that since not all the “intermediate”-v were
mapped out, the uncertainty in the intermediate region would be much higher.
After understanding the uncertainties of all v-states and solving the firsttwo challenges, we decided that instead of following a vibrational series from thenear-threshold region to the low-v region, we would directly jump to searchingfor a v = 0 state. In particular, since going to the singlet v = 0 level requires
additional singlet-triplet mixing of the excited electronic state, we went for v = 0of the a3 Σ+ potential first.

Chapter 7
Triplet Rovibrational Ground-State Molecules(N = 0, v = 0 of a3 Σ+ )After demonstrating molecular state transfer to near-threshold vibrational
levels, we decided to move quickly toward attempting to transfer Feshbach molecules
to a low-v state directly. In this chapter, I will show that by carefully choosing
an intermediate excited state, we can coherently transfer Feshbach molecules tothe rovibrational ground state of a3 Σ+ in a single-step of STIRAP. The discussion
here is based on our publication in [15].
7.1

SchemeOur goal here is to coherently transfer molecules from the initial Feshbach

molecule state, |i�, to the triplet rovibrational ground state, |g� (N = 0, v = 0 of
a3 Σ+ ) (Fig. 7.1). The strength of the transition dipole moment between statescomes from both the electronic transition dipole moment and the wave function
overlap, i.e. Franck-Condon Factor (FCF). Since both the initial and the final
states are predominately triplet in character, we chose a triplet intermediate excited state, |e� (v � = 10 of the electronically excited potential 23 Σ+ ), with a favorable wave function overlap. To choose such a state from a “forest” of excited-state
potentials is not an easy task. However, there are guidelines one can follow. Whendriving electronic transitions, it is typically a good approximation to assume that

88

12

4S+5P

10

23"+

|e� = (v � = 10)

3-1Energy (10 cm )86!
!22 (998 nm)
300284.474 GHz!
!11 (1023 nm)

0.4293104.055 GHz

0.2|i� = Feshbach Molecules

0.0

4S+5S

+

a 3"
-0.2
5

|g� = (v = 0)

10

X1"+
15

20

Internuclear Separation (a0)Figure 7.1: Diagram of the KRb electronic ground and excited molecular potentials and the vibrational levels involved in the two-photon coherent state transfer
to the triplet ground state. Here, the intermediate state |e� is the v � = 10 level ofthe electronically excited 23 Σ+ potential. The vertical arrows are placed at therespective Condon points of the up and down transitions. Note the diﬀerent energy scales for the excited and the ground potentials. The intermediate state hasfavorable transition dipole moments for both the up leg (|i� to |e�) and the downleg (|e� to |g�), where the initial state |i� is a weakly bound Feshbach moleculestate and the final state |g� is the rovibrational ground state (v = 0, N = 0) of thetriplet electronic ground potential, a3 Σ+ . This listed frequencies for the up anddown transitions are the most common transitions we used. (These frequencies
are the Raman laser frequencies to the molecules and are 98.5 MHz lower fromour laser locking point due to an AOM frequency shift, see Chapter 3.3.4. for an
operational note.)

89
the heavy nuclei do not move during the transition (Born-Oppenheimer approximation). Therefore, to estimate the strength of the transition dipole moment, welook at the wave function amplitude at the same internuclear separation. Nuclei
in a vibrationally excited state molecule spend most of their time at their innerand outer turning points, thus the wave function amplitude is the largest at these
positions. In contrast, nuclei in a vibrationally ground state molecule have thelargest amplitude at the center of the potential well. More rigorously, we findfavorable transitions using the classical Condon point argument. The Condon
point is the internuclear distance where the photon energy matches the diﬀerence
between the excited and ground-state potential energy curves. Following theseguidelines and calculations from our theoretical collaborators, S. Kotochigova and
P. Julienne, we began our experimental search and identified v � = 10 of 23 Σ+
as a suitable intermediate excited state. In our scheme (Fig. 7.1), the first laser
field, λ1 (1023 nm) connects our initial Feshbach molecule state, |i�, to the excitedintermediate state, |e�, and the second laser field, λ2 (998 nm), drives the downtransition to the ground vibrational level of the electronic ground a3 Σ+ potential(|g�).
7.2

Hyperfine-rich Ground-state Structure
To search for the triplet vibrational ground state (a3 Σ, v = 0), we performed

two-photon dark resonance spectroscopy as was introduced in Chapter 6.2 in thelimit of a strong Ω2 and a weak Ω1 . Based on the KRb potential published by
Pashov et al. [28], P. Julienne calculated the triplet v = 0 binding energy witha predicted uncertainty of 0.1% (7 GHz). Even this small uncertainty made theexperimental search challenging since our dark resonance features in the past
were at most tens of MHz wide. Nevertheless, we began our search by first fixing
the laser frequency of the λ1 to resonantly drive the transition from the initial

90

4
3

(A)
(a)

1

3
2

4

Molecule Number (10 )

2
1
0
-7185
4

-7180

-7175

Binding
Energy(GHz)
Energy/h
(GHz)

-7170

(B)

3
2
1

(b)

0
-30

-20

-10

0

10

20

30

! (MHz) +7177.688
GHzδ(MHz)

Figure 7.2: The v = 0 ground-state level of the triplet electronic ground potential,
a3 Σ+ . (a) Hyperfine and rotational states of the a3 Σ+ v = 0 ground-state molecule
at a magnetic field of 546.94 G, observed using two-photon spectroscopy andscanning the down leg frequency. The measured number of Feshbach molecules
is plotted as a function of the frequency diﬀerence of the two laser fields. We
show two sets of data, vertically oﬀset for clarity, obtained using two diﬀerent
intermediate states, which are ro-hyperfine states of the v � = 10 level of theelectronically excited 23 Σ+ potential. Peaks label 1 and 2 correspond to hyperfine
states in the rotational ground-state, while peak 3 corresponds to a rotationallyexcited state. (b) We precisely determine the energy and the transition dipole
moments for individual states using the two-photon spectroscopy where we scan
the up leg frequency. The measured number of Feshbach molecules is plotted asa function of the two-photon detuning. The dark resonance shown here is for the
triplet rovibrational ground state corresponding to peak 2 in (a).

Feshbach molecule state to the v � = 10 intermediate state. The λ1 laser byitself causes complete loss of all the Feshbach molecules. We then varied thelaser frequency of the strong coupling laser (λ2 ), and monitored the initial state
population after pulsing on both laser fields simultaneously. When the Raman

91
condition is fulfilled, the initial state population reappears (Fig. 7.2a). To oursurprise, pretty much on our first shot, we saw a signal corresponding to the v = 0of the a3 Σ+ potential. It turns out that this is because the down transition dipole
moment is strong and there is rich hyperfine structure that broadens the v = 0line to be 30 to 50 GHz wide.
The measured binding energy of the triplet v = 0 molecules is h×7.18 THz(corresponding to 240 cm−1 ) at 545.94 G. We find that the v = 0 level has rich
hyperfine plus rotational structure at this magnetic field (see Fig. 7.2(a)). Because the accessible final states are influenced by selection rules of the two-photontransition and the quantum number of the intermediate state, we have performed
the two-photon spectroscopy using diﬀerent states of the v � = 10 intermediatelevel (more details on the structure of the intermediate state can be found in Fig.
5.7). In Fig. 7.2(a), the measured hyperfine spectrum of the v = 0 triplet groundstate is shown for two-photon spectroscopy going through two diﬀerent states ofthe v � = 10 electronically excited level. The feature labelled peak 1 is the lowest
energy state we observed in the v = 0 manifold, this state is about 15 GHz lessdeeply bound than the prediction using potentials from [28]. Interestingly, theconstruction of the potential [28] comes from mass scaling experimental data of39

K85 Rb starting only at v = 3 and higher. Therefore, it is conceivable that theextrapolation to v = 0 could introduce some error.
In addition to triplet v = 0 molecules, we have also observed similar groundstate hyperfine structure for the v = 1 and v = 2 levels of the a3 Σ state, with
v = 1 having a binding energy of h×6.675587(1) THz and v = 2 having a binding
energy of h×6.188766(1) THz (Here, I give binding energy corresponding to thelowest energy state that we observed.) Combining this information with [28], one
should be able to construct the most accurate KRb a3 Σ potential to date.
The quantum numbers (hyperfine and rotational) of the states of v = 0 could

v
0
1
2

binding energy
7.180420(1) THz
6.675587(1) THz
6.188766(1) THzv � = 10 to v transition dipole moment
0.004(2) ea0∼0.004 ea0∼0.004 ea0

92

Table 7.1: Summary of the lowest observed hyperfine state for v = 0, 1, 2.

in principle be identified by a careful comparison to theories, e.g. [99]. Here, I
identify the rotational quantum numbers of the three lowest energy triplet v = 0
states seen in the two-photon spectrum. The peaks labeled 1, 2, and 3 in Fig.
7.2(a) occur at a binding energy of h×7.180420(1) THz, h×7.177688(1) THz, andh×7.177263(1) THz, respectively. Peak 1 corresponds to the lowest hyperfine statein the rotational ground-state (N = 0), peak 2 is a diﬀerent hyperfine state with
N = 0, and peak 3 is the lowest energy hyperfine state with N = 2, where N is therotational quantum number. This identification is based on Hund’s coupling case(b), where spin and molecular rotation are essentially decoupled and the molecular
hyperfine structure can be understood from calculations using a separated atombasis with the rotational progression appearing as a constant shift to all hyperfine
levels. Because of parity selection rules for optical transitions, we observe only
states with even N . The calculated rotational constant is B = 0.5264 GHz, which
gives a predicted splitting between the N = 0 and N = 2 levels of 6B = 3.158
GHz, while the observed splitting of peak 1 and 3 is 3.155 GHz. This assignment
will be verified in the next section using Stark spectroscopy.
Finally, using a dark resonance measurement such as shown in Fig. 7.2(b),we have measured the strength of the |e� to |g� transition. Here, we fix the downleg (λ2 ) laser frequency and scan the up leg (λ1 ) laser frequency. From the widthof the dark resonance for the rovibrational triplet ground state (peak 2), we findthat we can drive the transition from v � = 10 to the triplet v = 0 state with a Rabi
frequency of 2π ·8 MHz. This measurement used an exceedingly low laser power of

93
60 µW focused to a beam waist of 55 µm. The transition dipole moment derived
from this measurement is 0.20(2) ea0 , which is only one order of magnitude weaker
than a typical atomic optical transition!
7.3

Stark Spectroscopy
A quantum gas made of heteronuclear molecules in a low-v state has many

exciting new features - one of them is that the particle possesses a significant
permanent electric dipole moment that oﬀers new possibilities for control of thequantum gas. KRb molecules in the triplet rovibrational ground state are predicted to have an electric dipole moment of 0.05(3) Debye (D) [94]. This is nineorders of magnitude larger than the calculated 5 · 10−11 D dipole moment of theinitial Feshbach molecules and only about one order of magnitude smaller than atypical polar molecule dipole moment of 1 Debye. To measure the dipole moment,
we performed DC Stark spectroscopy on the three lowest energy states observedin the two-photon spectrum (Fig. 7.2(a)). We applied a uniform DC electric fieldin the range from 0 to 2 kV/cm using a pair of transparent electric-field plates
that are separated by 1.35 cm outside the glass-cell based vacuum chamber. A
detailed description of the field plates is provided in Chapter 3.2. We measured
the Stark shift using the dark resonance spectroscopy discussed above. This twophoton spectroscopy measures the energy splitting between the initial and final
states, and because the initial state has a negligible dipole moment, any frequency
shift of the dark resonance can be attributed to the final-state Stark shift. For
these measurements we lowered the laser powers to give a dark resonance widthof 500 kHz. The measured Stark shifts vs electric field are shown in Fig. 7.3.
The eﬀect of a DC electric field is to couple states of opposite parity. For
the a3 Σ+ v = 0 molecules, the opposite parity states are even-N and odd-N
rotational states. In Fig. 7.3 we see that the two lowest energy states, which are

94
the rotational ground state N = 0, exhibit similar Stark shifts. From the measured
Stark shift, we fit the data by considering rotational state mixing up to N = 5(similar and more detailed analysis can be found in Chapter 8.3 for the singlet
vibrational ground state). This yields the molecules’ electric dipole moment tobe 0.052(2) D. The Stark shift of the third energy state, corresponding to peak3, is measured to be about 10 times smaller than that for the peak 1 and peak 2
states. This smaller Stark shift for the N = 2 state is consistent with an electricStark Shift (MHz)dipole moment of 0.052 D.

0.1
0.0
-0.1
0.0

N=2

-0.2

N=0

|mN|=0|mN|=1|mN|=2-0.4-0.6

a3! v=0

-0.8
-1.0

0.0

0.5

1.0

1.5

2.0

Electric Field (kV/cm)
Figure 7.3: Stark spectroscopy of triplet v = 0 molecules. Stark shifts of thelowest three states in the triplet v = 0 manifold in Fig. 7.2(a) are measured for
a DC electric field in the range from 0 to 2 kV/cm. The bottom panel shows the
Stark shifts of the two lowest energy states which are N = 0. A combined fit to theshifts of peak 1(solid circles) and peak 2 (open circles) with our main systematic
error from a 3% uncertainty of the electric field calculation give an electric dipole
moment of 0.052(2) D. The top panel shows the Stark shift of peak 3 (squares)
and the expected N = 2 curves calculated for an electric dipole moment of 0.052
D and diﬀerent |mN | projections.

95
7.4

Transfer
After our spectroscopy studies of the triplet rovibrational ground state, we

performed population transfer. We used peak 2 as the rovibrational groundstate target for our coherent state transfer, which is performed using the counterintuitive pulse sequence of STIRAP[102] that I have introduced in Chapter 6.
The STIRAP beams are co-propagating in order to minimize photon recoil. The
measured time evolution of the initial-state population during a double STIRAP
pulse sequence is shown in Fig. 7.4(a). The roundtrip transfer eﬃciency of 31%
implies a one-way transfer eﬃciency of 56%, which corresponds to 3·104 triplet v =0 N = 0 polar molecules at a peak density of 1012 cm−3 . This transfer techniqueallows us to reach a single quantum state without heating. The STIRAP eﬃciency
is limited by the final-state molecule lifetime, which we measured to be 170 µs asshown in Fig. 7.4(b). We believe the lifetime is most likely limited by collisions
with background atoms, which can induce spin flips and cause molecules to decay
into lower lying states in the singlet electronic ground potential. (Collisions andlifetimes of a simpler system, the absolute rovibrational ground-state molecules,
will be discussed in Chapter 10.) If so, the collisional decay rate could be reduced
either by perfecting the removal of the remaining atoms or by starting the molecule
production with atom pairs tightly confined in individual sites of an optical lattice.

96
1.0

(A)

Fraction

0.8
0.6
0.4
0.2
0.0

(a)

(A)

0.8

Fraction

-20

0

20

40

60

Time (µs)

80

100

120

4

1.0

0.6
0.4
0.2
0.0

2.0

1.0

0

20

40

60

Time (µs)

80

100

120

! = 170(30) µs

0.5

0

100

200

300

400

Hold Time (µs)

4

-20

(b)
(B)

1.5

0.0
-40

Molecule Number (10 )Molecule Number (10 )

-40

2.0

(B)

Figure 7.4: Time evolution of the initial state population during STIRAP statetransfer
and a measurement of the triplet rovibrational ground-state molecule
1.5
lifetime. (a) Here we monitor the Feshbach molecule population as we apply! = 170(30)µs
the 1.0
STIRAP pulse sequence.
Weaklybound Feshbach molecules are coherently
transferredintothetripletrovibrational
ground state after a 25 µs one-way STI0.5
RAP pulse sequence. The measured population completely disappears since thedeeply-boundmolecules are dark to the imaging light. After a 10 µs hold, we
0.0
0
100
200
300
400
then perform the Holdreversed
pulse sequence that coherently transfers theTime STIRAP(µs)
ground-state molecules back to Feshbach molecules. The molecule number after
the roundtrip STIRAP is 1.8 · 104 . Assuming equal transfer eﬃciency for the two
STIRAP sequences, we obtain one-way transfer eﬃciency of 56% and an absolute
number of triplet rovibrational ground-state polar molecules of 3.2 · 104 . (b) We
measure the triplet v = 0 lifetime by varying the hold time after one-way STIRAP
before transferring them back to Feshbach molecules for imaging. The lifetime ismeasured to be 170(30) µs.

97
The triplet work presented in this chapter provides not only a conceptual
but also a physical leap from where we started (see comparison Table 7.2). For along time, we doubted the feasibility of a single transfer step to eﬃciently bringnear-threshold molecules to a low-v state, such as the triplet rovibrational groundstate. However, looking back, although the experiment was very challenging, there
should have been no conceptual surprise of the result! Propelled by this success,we hoped that transferring molecules to the absolute rovibrational ground state
would be a rather straightforward extension. The key is again to identify a suitableintermediate state that not only bridges the larger wave function mismatch but
also allows the electronic spin state to change.

Binding Energy(GHz)
Size (a0 )Dipole Moment(Debye)Thesis Chapter

Feshbach moleculesa3 Σ(v = −3)a3 Σ(v = 0)

X 1 Σ(v = 0)0.00024

10.49238

7180

125313

∼ 300

34

11.2

7.7

5×10−11

1×10−40.052(2)0.57(2)4

6

7 (current)

8 (next)

Table 7.2: Summary of KRb ground-state vibrational level transfers we havedemonstrated. The work in this chapter showed a leap in the molecular binding
energy and the size (and the corresponding electric dipole moment) from theprevious demonstration of near-threshold transfer.

Chapter 8
Absolute Ro-Vibronic Ground-State Polar Molecules(N = 0, v = 0 of X 1 Σ+ )The success of using a single step of STIRAP to transfer a large fractionof molecules to the triplet rovibrational ground state motivated us to pursue ourlong-standing goal – making a high phase-space-density gas of absolute ro-vibronic
ground-state molecules (N = 0, v = 0 of X 1 Σ+ ). The additional challenges werei) to identify one or more intermediate states to bridge the even larger wavefunction mismatch between the Feshbach molecule state and the singlet rovibrational
ground state, ii) to have a non-trivial singlet-triplet mixing in the intermediate
excited state, iii) to build new lasers at the appropriate wavelengths, and iv) tophase-reference and stabilize the two continuous-wave coupling lasers, which havea much larger frequency diﬀerence.
Before our work, most people believed that multiple transfer steps would be
required to bridge the large wavefunction mismatch between Feshbach molecules
and the absolute ground state. However, as we demonstrated for KRb [15], thisturned out not to be necessary. For other bi-alkali heteronuclear molecules, we also
expect that a single excited electronic state can be identified that has relatively
strong transition strengths to both the initial Feshbach molecule state and theabsolute ground state [103]. This chapter follows discussions in [15, 104, 100].

99

12

1 1"

4S+5P

|e� = (v = 23)
�

10

23!+

!1 (970 nm)

3-1Energy (10 cm )8309602.752 GHz6!2 (690 nm)
434 922.454 GHz|i� = Feshbach Molecules

a3!+

4S+5S

0
-1

∆

-2
-3

γ Ω2

|e�

Ω1

δ

X1!+

-4

|g� = (v = 0)
5

10

|g�

|i�

15

20

Internuclear Separation (a0)Figure 8.1: Schematic of the transfer from KRb Feshbach molecules to the absoluterovibrational electronic ground state. The intermediate state was chosen to be thev � = 23 state of the nominally 23 Σ+ electronically excited potential for its good
wavefunction overlap to both the Feshbach molecules and the absolute groundstate molecules. The three levels are coupled by two lasers, λ1 (970 nm) andλ2 (690 nm), that are both referenced to a stable optical frequency comb. The
exact frequencies are given for our most commonly used pair of up and downtransitions. (These frequencies are the Raman laser frequencies to the molecules
and are 98.5 MHz lower from our laser locking frequency due to an AOM shift, see
Chapter 3.3.4.) (Inset) Schematic of a three-level system with relevant notations
introduced.

100
8.1

Transfer Scheme
An appropriate choice of the electronically excited molecule state allows apair of cw lasers to be used for a single-step STIRAP that directly accesses themolecule’s absolute ground state. Once the two lasers are properly stabilized, thelong lifetimes of the initial and final states ensure that the transfer process is very
eﬃcient and reaches a single rovibrational state (N = 0, v = 0 of X 1 Σ+ ). Furthermore, the fully coherent nature of the optical transitions, which eﬀectively drive
the weakly bound Feshbach molecules directly to the deepest bound electronic
ground state, avoids any heating due to random photon recoils.
For the creation of absolute ground-state KRb molecules, our scheme isshown in Fig. 8.1, where the wavelengths of the cw coupling lasers are 970 nm(λ1 ) and 690 nm (λ2 ). In our experiment, the two-photon beat can be maintained
to a few kHz linewidth by referencing each laser to a stable optical femtosecond
comb [105]. We used the vibrationally excited v � = 23 level of the nominally 23 Σ+
electronically excited molecular potential [15, 99]. This state has a small mixing
with a nearby 11 Π state, which make it possible to couple this excited state to
the 1 Σ absolute ground state. The upward transition dipole moment from the
Feshbach molecule state to this intermediate state was determined to be 0.005(2)ea0 (in Chapter 5.4.2).
8.2

Two-photon Spectroscopy
Before performing STIRAP to coherently transfer from the Feshbach moleculestate to the absolute ground state, we precisely determined the energy of the absolute ro-vibronic ground state. We did this using two-photon Raman spectroscopy
as was introduced in Chapter 6 and 7. To search for the absolute ground state,
we fixed the laser frequency λ1 to resonantly drive Feshbach molecules to the

101

4

Number (10 )4

Molecule Number (10 )432scanning λ2

1
0

0

100

200

300

scanning λ1
4
3
2
1
0
-20

! + (!
=125319702) (MHz)MHz!0-125319703-100! (MHz)

400

10

20

500

Figure 8.2: Finding the absolute ground-state level (N = 0, v = 0 of X 1 Σ+ )of KRb. The binding energy of the absolute ground state was identified using
two-photon dark resonance spectroscopy where the laser frequency λ1 was fixed
and the frequency λ2 was scanned. The absolute ground state was found to be ∼400 MHz less bound than the predicted location using potential in [28]. (Inset)
The coupling strength of the λ2 transition (corresponding Rabi frequency Ω2 ) was
determined by measuring the dark resonance with fixed λ2 laser frequency while
λ1 was scanned. The dark resonance fit that allowed us to extract Ω2 is shown as
the red line.

electronically excited intermediate state. Then we scanned the laser frequency
λ2 and monitored the population of the Feshbach molecules, |i�, after pulsing on
both lasers. When the second laser field fulfills the Raman resonance condition,
δ = 0, the Feshbach molecules remain due to the formation of a dark state. Fig.
8.2 shows the two-photon spectroscopy results. We found the ro-vibronic groundstate only ∼ 400 MHz less bound than the prediction based on the KRb potential from [28]! Our measurement precisely determined the binding energy of theabsolute ground state to be h·125.319702 THz (4180.22 cm−1 ) at 545.88G [15].
Knowing the energy of the absolute ground state, |g�, we can characterize
the coupling strength Ω2 (Rabi frequency) by fixing the laser frequency of λ2 on
the one-photon resonance (∆ = 0) and scanning the laser frequency λ1 . The inset
of Fig. 8.2 shows the measured population of the Feshbach molecule state, |i�,after the two-photon laser pulse. The frequency splitting between the two minimacorresponds to the Rabi frequency (Ω2 ) of the |g� to |e� transition. For this data,

102
the 7 MHz Rabi frequency was obtained using 6 mW of laser power focused to a
40 µm beam waist; this yields a transition dipole moment of 0.012(3) ea0 for theintermediate to the ground state transition.
8.3

Stark Spectroscopy Analysis
The same form of two-photon spectroscopy in the last section can also be

used to map out the energy splitting of states |i� and |g� as a function of an applied
electric field. In the case where the initial state |i� has a negligible dipole moment,such as a Feshbach molecule, the measured Stark shift comes solely from the final
state |g�. This allows us to measure one of the most exciting properties of thero-vibronic KRb molecules – their permanent electric dipole moment! It is worth
mentioning that many theoretical eﬀorts went into calculating the KRb permanentelectric dipole moment over the last decade. However, these calculations are
extremely diﬃcult and each theoretical group gives a diﬀerent result ranging from
0.5 Debye to 1.2 Debye. (A summary of theoretical results can be found in [106].)
To understand the eﬀect of an applied electric field, I wish to first point out
the diﬀerence between a magnetic dipole moment and an electric dipole moment.
A magnetic dipole moment is an intrinsic property of a spin. Even in the presenceof a miniscule magnetic field, magnetic dipole moments exist and take a fixedvalue. Electric dipole moments, on the other hand, are usually an induced dipole
whose magnitude depends on the strength of the electric field. Let us take KRb
molecules as an example. The electric dipole moment, pointing from the K nucleus
to the Rb nucleus, comes from the fact that the two valence electrons in KRb
prefer to be near the K nucleus. Without an external electric field, we could not
fix the orientation of the molecular axis and the averaged electric dipole moment
is therefore zero.
KRb singlet (1 Σ) molecules have no electronic spin or orbital angular mo-

103
mentum. Their total angular momentum apart from the nuclear spin comes solely
from their rotational quantum number, N , which describes the rotation of themolecule about the molecular axis. The external electric field couples states of
opposite parity and thereby polarizes molecules along the field direction. For
singlet KRb molecules, these opposite parity states are rotational states. The
Hamiltonian in the basis of |N mN �, where mN is the projection of the rotational
state along the field direction, has two terms. The first term is the rotationalenergy without the presence of an electric field and the second is the Stark eﬀect:
�N mN |H|N � m�N �= B · N (N + 1) δN N � ,mN m�N
−d·ε·



�

(2N + 1)(2N � + 1)(−1)mN 

N
−mN

1

�



�



N  N 1 N 


0 m�N
0 0 0

where B is the rotational constant, d is the permanent electric dipole moment,and ε is the magnitude of the external electric field. The factor that modifies d · ε
is a geometric factor came from rotating from the molecular fixed-frame to thelaboratory frame. The geometric factor, in the three-j symbols form, ensures thatelectric field only mixes states with the same mN and that |N mN � and |N − mN �
are degenerate.
Since our molecules are in the ro-vibronic ground state, we are mainly interested in the Stark eﬀect of the rotational ground state, |0 0�. In principle,
this state will mix with all |N 0� states and therefore we have to diagonalize an
infinitely large matrix. However, the dominant contribution of the Stark eﬀect
comes from the |1 0� state, so in practice we only need to consider up to a finite
maximum rotational quantum number, Nmax . To evaluate what Nmax will be
suﬃcient, I calculate the saturated value for the electric dipole moment vs Nmax
(Fig. 8.3). For this calculation, the saturated dipole moment is considered at an

104electric field that is much larger than we can achieve experimentally, 100 times
the critical electric field (ε0 is defined when B/(d·ε) = 1. For singlet rovibrational
ground-state molecules, ε0 ≈ 4kV/cm). From Fig. 8.3, we see that the saturated
value stops growing for Nmax > 5 and is around 92% of the “permanent electric
dipole moment.” 1 Therefore, for the subsequent analysis, I consider mixing ofstates up to N = 5. 2
1.0
0.9
0.8

|def f |
d 0.7
0.6
0.5

1

2

3

4

5

6

7

8

Nmax
Figure 8.3: Calculation of the saturated electric dipole moment of the N = 0
state vs the maximum rotational state that is included√in the calculation. FordNmax = 1, the saturated value |def f | = dεEstark = d/ 3. As Nmax increases,
the saturated polarization approaches a value � 90%, but never reaches 100% at
100 ε0 . The dipole moment will reach 100% only in the limit of infinitely large
electric field. For further Stark spectroscopy analysis, we diagonalized the Stark
Hamiltonian with Nmax = 5.

The saturated dipole moment will reach the full “permanent electric dipole moment” onlyin the limit of ε → ∞ and N → ∞.
2
It is worth mentioning that for ε < 2ε0 , which is what we can achieve experimentally, thesaturated dipole moment is 10% lower if considering Nmax = 1 and 0.2% lower if considering
Nmax = 2 as comparing to considering Nmax = 5.
1

105
Figure 8.4 shows the measured Stark shifts (relative to the zero electric fieldcase) for the N = 0 and the N = 2 states of the v = 0 level of KRb. The splittingof N = 0 and N = 2 state at zero electric field measures 2 · (2 + 1) B. From thesplitting, we extracted a rotational constant of B = 1.1139(1) GHz. Finally, withthe measured rotational constant, we fit the N = 0 Stark shift data to the Stark
eﬀect calculated including the mixing of rotational states up to N = 5 and extractthat the permanent electric dipole moment is 0.566(17) D[15].3 The 3% error bar
is a systematic error from our estimated uncertainty in the external electric field(see Chapter 3.2). We have also measured the Stark shift for the N = 2 state.
This data is consistent with the dipole moment of 0.566 D that we extracted fromStark Shift (MHz)the N = 0 Stark shift.
5
0
-5
0

|mN|=0|mN|=1

N=2|mN|=2
N=0-10-20
X1! v=0

-30
-40

0.0

0.5

1.0

1.5

Electric Field (kV/cm)

2.0

Figure 8.4: Stark spectroscopy. The measured Stark shift of the rovibrational
electronic ground state of KRb is shown. From the measured Stark shift, and using
the rotational constant extracted from the measured energy diﬀerence between the
N = 0 and N = 2 levels at zero field, we found the v = 0 KRb permanent electric
dipole moment to be 0.566 D. [15]Since [15], we found a small error for the measurement of the separation of the E-field
plates. This changes our calculated electric field and hence our measured KRb dipole moment.
The updated dipole moment is 0.574(17) D. The E-field plates dimensions given in Chapter 3.2
are the updated values.
3

106
8.4

Population Transfer
Having determined detailed spectroscopy information of the ro-vibronic groundstate, we were ready to use STIRAP [102] to transfer KRb Feshbach molecules
into the absolute ground state (Fig. 8.1). As discussed in Chapter 6 and 7, thepopulation of |i� can be adiabatically transferred to |g� by an appropriate choicefor the time dependence of the coupling laser fields characterized by Rabi frequencies Ω1 (t) and Ω2 (t). The counter-intuitive STIRAP pulse sequence is to turn on
the coupling Ω2 first, and then ramp down the coupling Ω2 (by lowering the laser
intensity) while ramping up Ω1 (see Fig. 8.5(a)). The time when Ω2 and Ω1 areboth > 0 is when the transfer actually occurs. At the end of the STIRAP transfer,Ω1 can be turned oﬀ.
Fig. 8.5(b) shows STIRAP transfer of KRb from the weakly bound Feshbach molecule state to the absolute ground state. More specifically, the figure
shows the measured time evolution of the initial state (Feshbach molecule) population during the pulse sequence shown in Fig. 8.5(a). The disappearance of the
Feshbach molecules between the time of 15 to 47 µs corresponds to transfer into
the absolute ground state. To confirm this within the constraint that we can only
detect Feshbach molecules, we add a reverse STIRAP pulse sequence to bring
the deeply bound molecules back to Feshbach molecules for detection. For theparticular dataset in Fig. 8.5(b), the round-trip transfer has an eﬃciency of 67%.
Assuming equal eﬃciency each way, the one-way STIRAP transfer eﬃciency tothe absolute ground state is 81%. Our detection method is perhaps non-standard
in that it does not use direct detection of the ground-state molecules. However,this method works well because the STIRAP transfer is eﬃcient and because absorption imaging of the Feshbach molecule gas has good signal-to-noise, similar
to that typically obtained for ultracold atom gas imaging. Furthermore, this de-4

Number (10 )Intensity(arb.)

107

1.0
0.5
0.0

(a)

!2

!1

4
3
2
1
0

(b)
0

10

20

30

40

50

60

Time (µs)
Figure 8.5: (a) Intensity vs. time for the round-trip STIRAP pulse sequence. The
actual transfer step (ramping down the intensity of one laser field while ramping
up the other) takes 4µs. (b) Initial state (Feshbach molecules) population during
the STIRAP pulse sequence. The initial state population vanishes after beingtransferred to the absolute ground state in the first 4 µs period when both coupling
laser fields are on. To detect the ground-state molecules, a reversed STIRAP
transfer (the second 4 µs period when both coupling laser fields are on) is used
to bring these molecules back to Feshbach molecules. Assuming equal transfer
eﬃciency each way, the one-way STIRAP eﬃciency is 81% [104, 100].

108
tection is convenient in that it is completely compatible with our ultracold gas
apparatus. The reverse STIRAP step (which takes ∼ 4 µs) can be viewed simply
as an internal state manipulation that is part of the detection sequence for probing

4

Molecule Number (10 )the ro-vibronic ground-state molecules.

5
4
3
2
1
0
-1.0 -0.5 0.0 0.5 1.0! (MHz)-125319.703GHz! (MHz)Figure 8.6: STIRAP lineshape after round-trip STIRAP (the axis of molecule
number shown here has to be scaled up by 30% due to Feshbach molecule detection
eﬃciency (85%) and imaging OD saturation (90%) that were not accounted for).
We optimize STIRAP transfer by varying the two-photon detuning of the two
Raman lasers, δ. Here we achieved one-way transfer eﬃciency > 90%. The single
red point is measured after one-way STIRAP. The line is only for the guide of theeyes.

The data shown in figure 8.5 was taken with both the two-photon detuning
and the one-photon detuning on resonance (δ = 0 and ∆ = 0). We saw in Chapter
6 that maintaining ∆ = 0 is not crucial and detuned (∆ �= 0) STIRAP also works.
The usual one-photon linewidth is large (> 10 MHz) compared to other frequency
drift in the system, e.g. magnetic field slow drift. However, maintaining the two-

109
photon resonance δ = 0 is crucial. For δ �= 0, STIRAP sequence populates theexcited state and molecules are lost. Experimentally, we optimize the two-photon
detuning by taking a lineshape as shown in Fig. 8.6. The width of the lineshapedepends on the Rabi frequency of the two coupling lasers as well as the durationof the STIRAP transfer. The round-trip STIRAP lineshape in Fig. 8.6, which
has a linewidth of ∼ 700 kHz, was taken using a 4 µs one-way transfer pulse andcoupling laser fields with Rabi frequencies of a few MHz. This lineshape shows our
optimized results with one-way transfer eﬃciency > 90%, which corresponding to> 5.4·104 KRb polar molecules in their absolute ground state.
8.5

Trapped Molecular Temperature and Density
We have demonstrated that transferring molecular population to the abso-lute ground state using STIRAP is robust and eﬃcient. In addition, because of thecoherent and the eﬃcient nature of the transfer, there should be no heating during
the transfer process. However, there can remain a small momentum recoil of themolecules caused by the two-photon Raman process. This is minimized by having
the two laser beams for STIRAP be co-propagating. With this configuration for
our KRb transfer, the calculated momentum kick corresponds to an energy of only
kb ·13 nK. This kick is not random, so it is not heating. Fig. 8.7 shows a measurement of the molecular gas temperature using time-of-flight absorption imaging
for Feshbach molecules before and after round-trip STIRAP transfer. We do not
observe heating in the transfer process [104]. This is especially crucial because
we rely on the reversed STIRAP sequence as part of our detection scheme thatgives information about the number and the expansion energy (temperature) of
the absolute ground-state molecules.
For data taken with short hold times (∼ 100 µs) in the absolute groundstate (of course, the reverse STIRAP would provide a reverse recoil kick), we

110

Cloud
Cloud Size
Size (!m)( m)

8no STIRAP
after roundtrip

6

4

2

slope=5.1(3) !m/ms
slope=5.3(3) !m/ms

0

0

2

4

6

8

Expansion Time (ms)Figure 8.7: We can measure the expansion energy of Feshbach molecules before
and after a round-trip transfer to the absolute ground state. The hold time in theabsolute ground state was 20 µs. Within their uncertainty, the two measurements
give the same gas expansion velocity, and there is no observable heating during
the STIRAP transfer process.

saw that the STIRAP transfer process itself does not introduce heating of thegas. However, since Feshbach molecules and ro-vibronic ground-state molecules
have diﬀerent AC polarizibilities at our optical trap wavelength, they see slightly
diﬀerent optical trapping potentials. Therefore, we observe the excitation of adensity oscillation, or a breathe mode, that occurs at twice the trap frequency.
The oscillation of the ground-state molecule cloud is seen immediately after thetransfer (Fig. 8.8, measured in our crossed-beam dipole trap). The breathing
damps out in a timescale of 50 ms most likely due to anharmonicity of the trap
and the excitation energy then appears as a heating of the gas.
The AC polarizibility, α, can be directly extracted from the breathe frequency at a corresponding optical trap intensity. Another method to extract
α is to perturb the equilibrium position of the trap and then measure the fre-

111

Cloud Size (arb.)

5.2
5.0
4.8
4.6
4.4

f = 590(10)Hz
0

2

4

6

8

10

20

40

60

80

100

Time (ms)Figure 8.8: The molecular cloud exhibits a breathe motion as a function of time after the STIRAP internal state transfer from Feshbach molecules to the ro-vibronic
ground state. Molecules in these two internal states have diﬀerent AC polarizibilities, and therefore see an optical trapping potential with diﬀerent strengths. The
breathe motion damps out on a time scale of 50 ms. (This was measured in ourcrossed-beam optical trap.)

radial position /µm

10 (a)
0−10
10 (b)
0

−10

0

2

4
6
time/ms

8

10

Figure 8.9: Molecule cloud center-of-mass motion after perturbing optical dipole
trap by shutting it oﬀ for 0.5 ms [104]. (a) Feshbach molecules slosh frequency isνF eshbach = 190(3) Hz and (b) Ro-vibronic ground-state molecule slosh frequency
is νX 1 Σ = 175(5) Hz. (This was measured in our single-beam optical trap.)

112
quency of the resulting cloud slosh. Fig. 8.9(a) and (b) (measured in a singlebeam optical trap) are slosh measurements for Feshbach molecules and ro-vibronic
ground-state molecules, respectively, at the same trap intensity. The opticaltrap here uses a multi-mode 1090 nm laser. The measured slosh frequencies areνF eshbach = 190(3) Hz for the Feshbach molecules and νX 1 Σ = 175(5) Hz for thero-vibronic ground-state molecules. The AC polarizibility ratio αX 1 Σ /αF eshbach =22νX
1 Σ /νF eshbach = 0.85(5), where the Feshbach molecules have an AC polarizibil-

ity αF eshbach = 5.7 · 10−5 (MHz/W/cm2 ) that is the sum of the K and the Rb
polarizibilities [107].
Combining information of the number (detection eﬃciencies due to STIRAP
transfer eﬃciency and Feshbach molecule detection eﬃciency have to be taken into
account), the temperature, and the trap frequencies of ro-vibronic ground-state
molecules, we can calculate their peak density in the optical dipole trap and thenormalized temperature T /TF , where TF = �ω̄(6N )1/3 , ω̄ being the geometrickbmean of the trap frequency, is the Fermi temperature of the gas. Typically, thepeak density of the gas is 1012 cm−3 for a temperature of a few hundreds of
nanoKelvins. The coldest gas of absolute ground-state molecules we achieved was
at 160 nK with T /TF = 1.4, and was created with STIRAP from the coldest
Feshbach molecule gas we could produce. This T /TF achieved is only a factorof 2 away in temperature and a factor of 15 away in phase-space density fromquantum degeneracy.
The realization of a high phase-space-density gas of ro-vibronic ground-statepolar molecules opens new and exciting prospects for studying electric-field controlled dipolar interactions in a quantum gas, and modeling exotic many-body
physics Hamiltonians. At the same time, from a practical point of view, we havejust begun to learn how to work with and manipulate these molecules. These
singlet ground-state molecules do not have an electronic magnetic moment since

113
their total electronic spin is zero. (They do however have a much weaker nuclear
magnetic moment.) In our first experiments realizing STIRAP transfer to singlet
ground-state molecules [15], we used an atom trap that had a magnetic potential providing additional confinement along the weak direction of the single-beam
optical trap. This magnetic confinement was significant for atoms and for Feshbach molecules, but did not aﬀect the ro-vibronic ground-state molecules. This
resulted in rapid axial expansion of the ground-state molecular cloud after the
STIRAP transfer. In addition, there could be extra potential energy given to theground-state molecules if the magnetic and the optical trap centers were not exactly overlapped. Eventually, we replaced the single-beam optical dipole trap by acrossed-beam dipole trap (described in Chapter 3.1) to increase the confinement,
particular in the weak direction of the original optical trap. On the flip side, one
advantage of a small, almost negligible magnetic moment of the molecules is that
we can ramp the magnetic field in any time scale and do not have to worry that
it aﬀects the molecules.
In the next chapters, I will present experiments concerning manipulation ofthis tiny nuclear magnetic moment and studying the dependence of collisions ofro-vibronic ground-state molecules in a single hyperfine state.

Chapter 9
Identifying and Manipulating Molecular Hyperfine States

In the last chapter, we discussed the manipulation of KRb molecules’ rotational, vibrational, and electronic state quantum numbers. To control the complete internal states of a molecule, we also need to address its hyperfine state.
Full control of all the internal state quantum numbers is a prerequisite for further
experiments with a quantum gas of polar molecules where collisions and interactions depend on the quantum statistics. The discussion in this chapter follows ourwork in [108].
Unlike the triplet molecules (3 Σ, Chapter 7) that have a total electronic
spin S = 1 and have hyperfine splittings on the order of rotational splittings (∼GHz), singlet molecules (1 Σ) have S = 0 and a hyperfine structure that comesfrom their nuclear spins (with energy splitting < MHz). Their hyperfine splittings→− →−come from two contributions, with the nuclear Zeeman eﬀect ( B · I ) dominating−→ −→over the spin-spin coupling (IK · IRb ) at magnetic fields > 20 G. In this regime,
we can label hyperfine states by the individual nuclear spin projections of the K
and the Rb nuclei, |mK , mRb �. Because K has a nuclear spin IK = 4 and Rb has
IRb = 32 , there are a total of 36 hyperfine states (Fig. 9.1(a)) for the ro-vibronic
ground state. The hyperfine energy splitting due to ∆mK = ±1 is ∼ h · 130 kHz
and for ∆mRb = ±1 the splitting is ∼ h · 760 kHz at 546 G. These small energy
splittings are a lot more diﬃcult to resolve compared to the hyperfine structure

115of triplet molecules. However, their energy scales are still about 10 - 100 times
larger than the thermal energy of our gas at a few hundreds of nanoKelvins.

11/2

(a)

mK = 43

mF = mK+mRb

7/2

2
1

3/2

0
-1

-1

0

mR

b=

-3/

2

-1/
mR

b=

b=

-11/2
-2
1.5

2

1/2

3/2

mR

-7/2

b=

-4

mR

-3/2

-2
-3

1

2

Number (104)

(b)

1.0

0.5

0.0
-2
more deeply bound

-1

0

1

2

E/h (MHz)

Figure 9.1: Hyperfine structure of ro-vibronic ground-state KRb molecules at 546
G. (a) There are 36 total hyperfine states that can be labelled by their nuclear
spin projections |mK , mRb �. The lowest hyperfine state is | − 4, 3/2�, circled inred. The energy splitting between states with ∆mK = ±1 is ∼ h · 130 kHz, while
the splitting for ∆mRb = ±1 is ∼ h·760 kHz. (b) Observed hyperfine states in our
STIRAP transfer. The blue circles correspond to data taken using a single circularpolarization two-photon STIRAP and the green triangles correspond to data takenusing mixed polarization. From the selection rules and the energy splitting among
features, we can assign the observed final states as having corresponding quantum
numbers of | − 4, 1/2�, | − 3, −1/2�, and | − 4, −3/2�. The lines through the data
are only guides to the eyes.

116
9.1

Identify Hyperfine States
We first identified what hyperfine state we populated in our STIRAP trans-

fer to ro-vibronic ground-state molecules. The keys to identifying the specifichyperfine state(s) populated after STIRAP transfer were to understand the selection rules and to spectroscopically resolve diﬀerent hyperfine states by narrowing
down the STIRAP linewidth. In our setup, the Raman lasers for the two-photon
STIRAP transition are co-propagating along the magnetic-field direction (see Fig.
3.1) Using a quarter waveplate, we can vary polarization of the two Raman laser
beams to both have the same pure circular polarization or to both have a mixtureof σ + and σ − . Conservation of angular momentum tells us that the two-photontransition only allows the change of total angular momentum, ∆ mF = 0, 2, or−2. Since the Feshbach molecules has mF = −7/2 (see Chapter 4.1), the final
molecule state after transfer must have mF = −3/2, -7/2, or -11/2.
Our ability to spectroscopically resolve diﬀerent hyperfine states depends on
the linewidth of the STIRAP transfer, which is in turn set by the STIRAP pulse
duration. To resolve diﬀerent hyperfine states, we narrowed down the linewidth
as compared to the one in Fig. 8.6 by applying a longer STIRAP pulse (15µs). We observed a total of three diﬀerent hyperfine states (Fig. 9.1(b)) when
we iterated through three diﬀerent ro-hyperfine states of the electronic excitedlevel (Table 5.3). Two of the final hyperfine states were accessed through thetwo-photon transition where both photons had the same polarization (blue circles
in Fig. 9.1(b)), while one of the final hyperfine states was accessed only whenboth photons had a mixture of polarizations (green triangles in Fig. 9.1(b)).
From selection rules and the relative energies, we identified the three states as|mK = −4, mRb = 1/2�, | − 3, −1/2�, and | − 4, −3/2�.
Out of the three intermediate excited states in Table 5.3, the one with the117transition frequency of 309602.753 GHz only couples to | − 4, 1/2�. The intrinsic
selection rule of this intermediate state guarantees that we always populate asingle hyperfine state without having to worry about polarization and to narrow
STIRAP lineshapes.
9.2

Manipulate Molecular Hyperfine States
Using STIRAP, we can populate ro-vibronic ground-state molecules in asingle hyperfine state. However, because our starting Feshbach molecules havemF = −7/2, using a two-photon transition in our setup will not allow us todirectly populate molecules in the lowest hyperfine state, | − 4, 3/2�, which has
mF = −5/2. Reaching the lowest hyperfine state of the ro-vibronic ground-state
has turned out to be important for studying collisional stability of these molecules.
9.2.1

State Transfer to the Lowest Hyperfine StateTo reach the lowest hyperfine state (| − 4, 3/2�), we can in principle startby populating molecules in | − 4, 1/2� and then drive a nuclear spin transition.
However, this nuclear magnetic transition (mRb = 1/2 to mRb = 3/2) couples only
weakly to external magnetic field. Moreover, this transition is nearly degeneratewith the spin flip of mRb = 1/2 to mRb = −1/2, where the energy diﬀerence isonly about 20 Hz out of 760 kHz. Therefore, the straightforward single nuclear
spin flip does not work for us.
To manipulate nuclear spins, we came up with a general scheme that relies
on electronic dipole transitions involving the rotational excited state N = 1 [108].
Since the transition relies on a large electric dipole moment, this method requires
polar molecules with a permanent electric dipole moment.
Before we drive transitions from N = 0 to N = 1, our theorist friend Goulven Quéméner first calculated the N = 1 hyperfine structure at 545.9 G. This

118
calculation included states of rotational projection mN = 0, 1, and -1 onto themagnetic field axis (which usually would thought to be degenerate in the presenceof zero electric field) and the hyperfine states, using the relevant ab initio parameters from [109]. Together, there are a total 3 × 36 = 108 states. The splittingsof states diﬀering only in mN are about a few hundreds of kHz. These splittings
come from interactions of the nuclear electric quadrupole moments of the 40 K and87

Rb nuclei and the electric-field gradients created from molecule’s electron cloud.

This mechanism provides coupling between nuclear hyperfine states and rotationand can serve as a means to manipulate hyperfine states!

(a)E|0, mI �|1, mI �| − 1, mI �

σ|0, mI �(b)

E |0, m ± 1� + δ| ± 1, m �
I
I
N =1
N =1
−

σ

π
+

N =0

σ±

π

∆m

= ±1

|0, mI ± 1�|0, mI �

N =0

Figure 9.2: A general scheme of hyperfine state manipulation. (a) Electric dipole
transitions between N = 0 and N = 1, which leave the nuclear spin unchanged.
(b) Strong nuclear electric quadrupole interactions mix rotations and hyperfine
states. Eigenstates considering this interaction become |0, mI ± 1� + δ| ± 1, mI �,
with |δ|2 � 1. This allows an eﬀectively spin changing transition from N = 0
to N = 1. With an additional non-spin-flip transition from N = 1 to N = 0,
we can eﬀectively manipulate hyperfine states within the ro-vibronic ground-state
manifold.
To leading order, an applied microwave field (∼ 2 GHz) can drive σ + , σ − ,and π electric-dipole transitions from |N = 0, mN = 0� to |1, 1�, |1, −1�, and|1, 0� respectively while leaving the nuclear spins unchanged. We drive these transitions using an amplified microwave source (∼ 5W) that is delivered by a waveg-

main state|1 0 -4 3/2�
|1 0 -3 1/2�admixed state|1 1 -4 1/2�
|1 1 -4 1/2�|δth |2
0.045
0.0085|δexp |2
≈ 0.1
≈ 0.0064

119

Table 9.1: Mixing of diﬀerent hyperfine states in the rotationally excited stateN = 1 at B = 545.9 G. We compare the theoretically calculated admixture |δth |2
to the experimentally measured admixture |δexp |2 .

uide (31-cm long and 20.5×10.4 cm2 in cross-section area). The relative strength ofthese transitions allows us to calibrate the strength of individual polarization component in our microwave field. Now, the nuclear electric quadrupole interactions
for N = 1 mix states of diﬀerent hyperfine characters with mN + mK + mRb being
the conserved quantity. This mixing results in eigenstates (written in the uncoupled basis) that have the form of |N, mN , mK , mRb ± 1� + δ|N, mN ± 1, mK , mRb �(or |N, mN , mK ± 1, mRb � + δ|N, mN ± 1, mK , mRb �), where |δ|2 � 1 characterizes the small admixtured state. The small admixture allows us to eﬀectively havea nuclear spin flip when driving between rotational states. Figure 9.3 show the
Rabi floppings between N = 0 and N = 1 hyperfine states. Since the electronic
dipole transition only talks to the non-spin-flip part, all transitions here have thepolarization of σ + . For a given microwave power, we can obtain relative Rabi
frequencies that characterize the admixtured fractions for Fig. 9.3(b) and (c).
The admixture measurement is summarized in Table 9.1.
With a combination of one spin flip transition from N = 0 to N = 1
and then a non-spin-flip transition from N = 1 back to N = 0, we can eﬀectively
drive population between two hyperfine states within the ro-vibronic ground-state
manifold. We demonstrated this general scheme of manipulation by populating
molecules in the lowest hyperfine states of the ro-vibronic ground-state (Fig. 9.4).
We start with molecules in | − 4, 1/2� of the ro-vibronic ground state directly
after a STIRAP transfer. We then use a π-pulse to drive a spin-flip transition to

120

1.0

a)

0.5
0

0

50

100

150

fraction

1.0

200
b)

0.5
0

0

100

200

300

1.0

400
c)

0.5
0

0

100

200

time / µs

300

400

Figure 9.3: Rabi oscillations on a (a) hyperfine preserving microwave transition |0, 0, −4, 1/2� → |1, 1, −4, 1/2� (b) Rb nucleus hyperfine changing transition|0, 0, −4, 1/2� → |1, 0, −4, 3/2� + δ|1, 1, −4, 1/2� (c) K nucleus hyperfine changing
transition |0, 0, −4, 1/2� → |1, 0, −3, 1/2�+δ|1, 1, −4, 1/2�. Note the diﬀerent time
axis in panel (a). The microwave power was reduced by a factor of 4 for the datain (a) resulting in an eﬀective decrease of the Rabi frequency by a factor of 2.

|1, 0, −4, 3/2� + δ|1, 1, −4, 1/2�. The admixture here was measured to be about
10%. A single π-pulse eﬃciency is typically around 90-95%. Then we perform
another π-pulse to drive the non-spin-flip transition to |0, 0, −4, 3/2�, and thereby
populate the lowest energy state!
9.2.2

More Hyperfine Spectroscopy

The manipulation scheme introduced in the last section is general and canbe applied to move population anywhere within the ro-vibronic ground-state manifold. Here we present a summary, Table 9.2, of microwave hyperfine state spectroscopy where we map out all transitions connecting to the initial |0, 0, −4, 1/2�state. One exception was the transition to |1, −1, −3, 1/2�, which we were unable
to measure because the transition is weak and lies very close to two other strong

E |0, −4, 3/2� + δ|1, −4, 1/2�π(2.227 009 GHz)|0, −4, 3/2�

121

N =1

σ+
(2.227 765 GHz)|0, −4, 1/2� N = 0

Figure 9.4: The specific scheme we used to transfer molecules from |0, 0, −4, 1/2�to the lowest hyperfine state |0, 0, −4, 3/2� at 546 G.

microwave transitions.
Using all our measured transition frequencies, G. Quéméner had used rootmean-square fit to extract rotational constant B = 1.113950(5) GHz and theelectric quadrupole term coeﬃcients eqQK = 0.45(6) MHz and eqQRb = −1.41(4)MHz. The electric quadrupole moment terms are 10-30% diﬀerent from the abinitio calculations in [109].
The general scheme of manipulation of hyperfine states allowed us to populate molecules in a single hyperfine state and the lowest energy state. This is thefirst time that we have the exquisite control of all internal quantum states of ultracold molecules, and this is extremely crucial for the collisional studies presented
in the next chapter. Moreover, many proposals recently have been exploiting possibilities of using microwave transitions that couples rotational states for shielding
inelastic collisions [110] and studying exotic many-body physics [39, 40, 111]. Ourdetailed hyperfine spectroscopy gave a realistic picture of the rich structure of a“single” rotational line that needs to be taken into consideration for such proposals.

122State istate jexp. (kHz)theory (kHz)abs. diﬀ.

|0 0 -4 1/2�|0 0 -4 1/2�|0 0 -4 1/2�|0 0 -4 1/2�|0 0 -4 1/2�|0 0 -4 1/2�|0 0 -4 1/2�|0 0 -4 1/2�

|1 1 -4 1/2�
|1 0 -4 1/2�
|1 -1 -4 1/2�
|1 0 -4 3/2�|1 -1 -4 3/2�
|1 0 -3 1/2�
|1 1 -4 -1/2�
|1 0 -4 -1/2�

2227 837(5)2228 125(5)2227 774(7)2227 009(2)2227 133(20)2228 237(10)2228 588(5)2228 804(1)2227 835
2228 1192227 776
2227 008
2227 128
2228 225
2228 593
2228 805

2
6
2
1
5
12
5
1

2227 765(10)2227 761

4|0 0 -3 1/2�

|1 0 -3 3/2�2228 109(16)2228 091

18|0 0 -4 3/2�

|1 0 -4 3/2�Table 9.2: Spectrum of rotational transitions from hyperfine state |i� within N = 0
to hyperfine state |j� within the N = 1 manifold. We compare the experimentallymeasured transition frequencies to the theoretical calculation.

Chapter 10
Ultracold Chemistry and Inelastic Collisions

In the last two chapters, we have demonstrated a complete control over
the KRb molecular internal-state quantum numbers, which allows us to probeinelastic collisions one quantum state at a time. Moreover, the possibility of
putting molecules into the lowest electronic, vibrational, rotational, and hyperfine
energy state, raises an immediate question that is “can these molecules still belost from the trap?” If they can, “what determines the loss rate?” Since molecules
are already in the lowest energy state, the only way for them to suﬀer inelastic
collisions, which release energy and cause loss from the trap is through chemical
reactions. Here the exiting particles after the collisions are diﬀerent chemical
species. In Table 10.1, we consider the binding energy of various types of moleculesthat could be made from diﬀerent combinations of 40 K and 87 Rb atoms. Theseenergy estimates allow us to assess whether a specific two-body reaction process
is allowed (exothermic) or not (endothermic) in the ultracold gas. In the sections
below, we will refer back to Table 10.1 to test each “entrant” channel for possible
reactions, and show that exothermic reactions take place at universal rates thatare determined solely by long-range interactions and the quantum statistics ofcollisions.

Molecules
87
Rb2
40 87
K Rb
40
K2
K2 Rb
KRb2

v = 0 binding energy (D0 )3965.8(4) cm−1
4180.417 cm−14405.389(4) cm−1
∼13000 cm−1 ?
∼12000 cm−1 ?

reference
[112]
[15]
[113]

124

Table 10.1: Summary of the relevant molecular energetics involved in possible
chemical reactions. The binding energies are given with respect to the thresholdenergy for free atoms in the absence of a magnetic field. The 87 Rb2 and 40 K2
binding energies include isotope shifts from the data in the respective references.
The trimer binding energies are unknown and we give here crude estimates as asum of the binding energies of the three relevant pairs. For example for K2 Rb, welist twice the binding energy of KRb plus the binding energy of K2 . Calculations
are needed and will undoubtedly change the values given here.

10.1

Atom-Molecule Collisions
We first study two-body collisions of atoms and molecules. As a prerequisite,

we assume and will verify in section 10.3 that inelastic collisions of our spinpolarized fermionic molecules occur only at a relatively low rates.
To prepare the atom-molecule mixture, we control the atom density by
selectively removing or heating up unpaired atoms after the initial molecule creation.1 For these experiments, we typically work with an atom number about 5
to 15 times larger than the molecule number. All atoms and molecules are prepared in their lowest energy states at 545.9 G. Specifically, K atoms are in their|F = 9/2, mF = −9/2� state, Rb in |F = 1, mF = 1�, and KRb in | − 4, 3/2 >.
The procedure of removing selective amounts of K is the following. We partially transfer
the population of K atoms from |F = 9/2, mF = −9/2� to |9/2, −5/2� using two successive rf
pulses. We then clean out K atoms in the |9/2, −9/2� state using resonant light as describedin Chapter 4.1.2. Once they are heated out of the trap, we again use rf pulses to spin flip K|9/2, −5/2� atoms back down to |9/2, −9/2�. To selectively vary the density of Rb, we perform
a complete spin flip from |1, 1� to |2, 2� and then turn on resonant light for a duration between
1 µs to 10 µs. The resonant light both removes Rb atoms and heats up the remaining gas. Afterthe resonant light pulse, Rb atoms are transferred back to the lowest hyperfine state |1, 1�. We
then apply an additional resonant light pulse to remove any residual Rb in |2, 2�, thus ensuringthat all Rb atoms are in the |1, 1� state.
1125Here F is the total atomic spin and mF is the spin projection. We performed
two separate experiments, one with a combination of K and KRb and the second
with a combination of Rb and KRb. In both cases, the background of atoms in
the other, undesired, atomic species was less than 1000 atoms, corresponding to adensity below 5×109 cm−3 . Since both the atoms and the molecules are preparedin their lowest energy states, trap loss due to inelastic spin-changing collisions is
not possible. However, loss from chemical reactions for the case of K + KRb isenergetically allowed since K + KRb → K2 + Rb is exothermic, while Rb + KRb→ Rb2 + K is endothermic (Table 10.1).

KRb Number (104)

1.2

KRb+K
KRb+Rb

1.0
0.8
0.6

nRb = 0.61x1012 cm-3,!=60(8)ms

0.4
0.2
0.0

nK = 0.77x1012 cm-3 ,!=7(2)ms

0

20

40

60

80

100

Time (ms)Figure 10.1: Sample decay curves for the molecular number when subject to collisions with K atoms (circles) and with Rb atoms (triangles). All species areprepared in their lowest energy internal states. The atom numbers are 5 to 15
times higher than the molecule numbers. The reduced initial molecule number for
KRb+Rb is due to the collisional quenching of the initial weakly bound KRb Feshbach molecules with Rb [24] before the two-photon Raman process that transfers
the molecules down to the ro-vibronic ground state.

126
For each experiment, we measured the time-dependence of the trappedmolecule population. Typical molecular loss curves are shown in Fig. 10.1. To
extract the inelastic collision rate, we assume that the atom number density isconstant. This is approximately true because the number of atoms is much larger
than the number of molecules. The trapped molecular number should then decayasdNmolecule = −β · Nmolecule · natom ,dt(10.1)

where Nmolecule is the number of molecules, natom is the atomic density, and β
is the inelastic rate coeﬃcient. We can then extract β via an exponential fit,e−β·natom ·t . As can be seen in Fig. 10.1, in general we find that the molecules arelost from the trap at a much faster rate when K atoms are present than when asimilar density of Rb is present.
To see if the molecule loss is due to atom-molecule collisions, we measure
the dependence of the loss rate on the atom gas density. For the case of KRb +K, we see a clear linear dependence on the atomic density and extract an inelastic
collision rate coeﬃcient of 1.7(3) · 10−10 cm3 /s. In contrast, for KRb + Rb collisions, the density dependence of the loss rate is much less clear. Nevertheless,
we again fit the dependence as linear and obtain a suppressed rate coeﬃcient of0.13(4) · 10−10 cm3 /s. A possible mechanism for a loss rate that is independentof the atom density would be collisions of ground-state KRb molecules with undetected molecules in high lying vibrationally excited states. These contaminantmolecules could be produced by inelastic collisions of Rb atoms with our weakly
bound KRb Feshbach molecules [24] before the two-photon Raman process that
produces ground-state molecules. The fact that we observe a smaller initial number of ground-state molecules when Rb atoms are present is consistent with thispicture. The linear fit to β vs nRb provides an upper limit for the KRb + Rb

Molecule Decay Rate (1/s)

127

200
KRb+K
KRb+Rb150
β=1.7(3)x10-10 cm3/s

100
50
0
0.0

0.2

0.4

0.6

0.8

1.0

Atom Density (1012 cm-3)
Figure 10.2: Dependence of the molecule loss rate on the atomic gas density
for collisions of atoms and molecules in their lowest energy internal states. We
observe strong molecule loss with β = 1.7(3) · 10−10 cm3 /s for KRb+K collisions
and suppressed loss of KRb for KRb+Rb collisions. This suggests that the KRb
decay mechanism is due to the exothermic chemical reaction K + KRb → K2
+ Rb. The corresponding reaction for Rb +KRb, Rb + KRb → Rb2 + K, isendothermic and is not energetically possible at ultralow temperatures.

inelastic rate coeﬃcient that is one order of magnitude smaller than what wemeasure for KRb + K. This is consistent with KRb + K collisional loss due to thechemical reaction KRb + K → K2 + Rb (at ultracold temperatures!) In the caseof KRb + Rb, there is no two-body chemical reaction pathway and the inelastic
rate for atom-molecule collisions is strongly suppressed.
We note that we only observe this suppressed loss rate when both KRb and
Rb are prepared in their lowest energy internal states at 545.9 G. We have measured an inelastic rate coeﬃcient of the order 10−10 cm3 /s when KRb is prepared| !− 4, 1/2�
decay
rate
KRbKRbDecay
Rate(Hz)
(1/s)

128
-10

3

=1.82(5)x10 -10
cm cm/s 3/s100 |6>"=1.82(5)x10

80
60
40
20
0
0.0

0.2

0.4

0.6

12
Rb Density (1012 cm-3)

3

Rb avg density (10 /cm )Figure 10.3: KRb+Rb collisions when KRb molecules are prepared at an excited
hyperfine state, | − 4, 1/2�, and Rb atoms are prepared in their lowest energy state|1, 1�. We observe a strong molecular loss rate of 1.82(5)·10−10 cm3 /s.

in the higher energy | − 4, 1/2� hyperfine state (Fig. 10.3) and/or Rb is preparedin its |2, 2� hyperfine state. Therefore, for practical reasons, removing Rb atoms
is important for creating a long-lived sample of ground-state KRb molecules.
10.2

Universal Loss RateTo understand the loss rates, we use a model based on the formalism ofmulti-channel quantum defect theory (MQDT) [114, 89] to estimate their values.
In this model, our collision process is characterized by a van der Waals intermolecular potential, −C6 /R6 , where C6 is the van der Waals dispersion coeﬃcient.
When the collisional partners come within the van der Waals length of each other,inelastic collisions happen with some probability which we assume to be unity for129now. The van der Waals length is defined ascos(π/4) Γ(3/4)
lvdw =
2
Γ(5/4)

�

2µC6
�2

�1/4

,

(10.2)

where µ is the reduced mass of the system and Γ stands for the Gamma function.
The inelastic loss rate is
βinelastic =

�

4π�
µ

�

Im(a),(10.3)

where Im(a) is the imaginary part of the scattering length. In this model wherethe van der Waals length is assumed to be the characteristic length, we can replaceIm(a) with lvdw , and predict a universal loss rate that is solely determined by thelong-range part of the potential.
Now, we estimate the universal inelastic collision rates for KRb + K (due tochemical reactions) from the van der Waals length characterizing the long-range
part of the potential between the collision partners. The van der Waals lengths are
directly calculated from the corresponding C6 coeﬃcients which are calculated by
S. Kotochigova. For KRb + K, C6 is 7019 atomic unit (a.u., 1 a.u. is EHartree · a60 ,where EHartree = 4.36 · 10−18 J and a0 = 0.53 · 10−10 m.), which gives a predicted
reaction rate of 1.1 · 10−10 cm3 /s. The rate agrees well with the experimental
measurement, which supports the universal loss model where long-range quantum
scattering and short-range unity loss characterize the inelastic process.
10.3

Molecule-Molecule Collisions
We have learned so far that atom-molecule collisions lead to rapid inelastic

or chemically reactive loss of trapped KRb molecules. Therefore, to have a longlived sample of molecules and to study intrinsic collisions of molecules, we remove
all atoms using the procedure described in Chapter 4.1.2. Right away, we observed
molecular lifetimes on the order of 300 ms. However, the observed lifetime did

130
not depend on the molecular density but rather depended on the intensity of ourcrossed-beam dipole trap (corresponding roughly to a power law with exponent 2to 3). Since we were using a broad linewidth (∼ 1 nm) multi-mode 1090 nm laser,
we suspected that the two laser beams might give rise to molecule loss througha multi-photon processes. We then exchanged the dipole trap laser for a single
frequency 1064 nm laser and oﬀset the frequencies of the two crossed beams by
130 MHz. This resulted in a longer molecule lifetime and enabled our study of
intrinsic molecular collisions.
Molecule-molecule collisions can lead to a few possible decay mechanisms.
From a purely energetic argument based on the energetics summarized in Table 10.1, the molecule-molecule collisions have three possible exothermic chemical
reactions, namely KRb + KRb→K2 + Rb2 , KRb + KRb→K2 Rb + Rb, and KRb+ KRb→KRb2 + K. All of these reactions require breaking and making molecular bonds. If the KRb molecules are prepared in an excited hyperfine state, anadditional inelastic scattering mechanism is spin relaxation to a lower hyperfine
state.
The quantum statistics of the molecules plays an essential role for collisionsof molecules at a temperature of a few hundreds of nanoKelvin. At these ultralow
temperatures, collisions with large-impact parameters and correspondingly large
centrifugal barriers are frozen out and the collisions are typically dominated by asingle partial wave, with relative angular momentum quantum number L=0 (swave) or L=1 (p-wave). Our KRb molecules are fermions and therefore the collisional wave function has to be antisymmetric with respect to particle exchange.
For spin-polarized molecules, where they are all prepared in exactly the same internal quantum state, p-wave is the lowest allowed collision channel. The heightof the centrifugal barrier for L=1 KRb molecule-molecule collisions is kB · 24 µK.2

131This barrier height is an order of magnitude higher than kB T , where T is thetranslational temperature of the molecular gas. Thus, collisions of spin-polarizedmolecules are expected to proceed predominately by tunneling through the p-wave
barrier. If two molecules make it through the barrier to short range, chemical reactions or hyperfine state-changing can take place, leading to a loss of the entrant
channel population.
10.3.1p-wave Collisions

The quantum nature of the collisions can be seen in the temperature dependence of loss rates. The Bethe–Wigner threshold law [115] predicts that thep-wave inelastic/reactive collision rate should be linear in temperature (∝ T ). To
look for this behavior, we first prepared spin-polarized molecules in the single hyperfine state |−4, 1/2� for various T ranging from 200 to 900 nK. The temperatureis measured from the expansion energy of the molecular gas after releasing it fromthe optical trap. For each initial temperature, we observed the time-dependent
molecular loss and extracted a two-body loss rate β (which is 2 times the collisional event rate ) by fitting the measured decay of the molecular gas density n
vs. time t (Fig. 10.4) to the solution of the following diﬀerential equation,dn= −βn2 − αn.
dt(10.4)

Here, the first term accounts for number loss and the measured β can be comparedto theoretical predictions. The second term accounts for density change due toheating of the trapped gas during the measurement. Within a single measurement,
we observe an increase in temperature that is at most 30%. In the analysis foreach data set, we fit the measured temperature to a linear heating and obtain
The height of the p-wave barrier is determined by the molecule-molecule long-range poten2
C6tial, namely � L(L+1)−R
6 , for L = 1 and C6 = 16130 a.u.
2µR2
2

132
c
a constant slope c. In Eq. 10.4, we then use α = 32 T +c
, where T is the initialttemperature. The time evolution of the molecular density based on Eq. 10.4 isn(t) =

c n0 T 3/2√√.
(c t + T )(−2n0 βT 3/2 + c c t + T + 2n0 βT c t + T )(10.5)

At our lowest temperature of 250 nK, the heating was 7(1) nK/s and β = 3.3(7) ·10−12 cm3 /s (Fig. 10.4). The measured dependence of β vs. T is summarized inFig. 10.5 (closed circles). Here, we fit the data to a power law β(T ) ∝ T L andfind that L = 1.1(2), which agrees with the predicted p-wave threshold law. This
demonstrates that indistinguishable 40 K87 Rb molecules at ultralow temperaturescollide via tunneling through a p-wave barrier followed by an inelastic collision in
the short range. A linear fit to the data (L = 1) yields a slope of the decay rate

Molecule Density (1012 cm-3)

coeﬃcient of 1.2(3) · 10−5 cm3 /s/Kelvin.
0.4
E

0.3
R

0.2

T=250 nK!= 3.3(7)x10-12 cm3/s

0.1

0.0
00

2

4

6

8

Time (s)Figure 10.4: A sample molecular density decay vs time for inelastic collisions
between indistinguishable fermionic molecules in the ro-vibronic ground-state of
40 87
K Rb. Here the molecules are prepared in a single hyperfine state, | − 4, 1/2�,and the molecular density decays slowly with a rate coeﬃcient of 3.3(7) · 10−12
cm3 /s at T = 250 nK.

Decay Coefficient β (10-12 cm3/s)

133

200
100
15
10

|-4, 1/2>|-4, 3/2>
50/50 mixture
QT
MQDT

5
0
0.0

0.2

0.4

0.6

0.8

1.0

Temperature (μK)
Figure 10.5: Inelastic collision rate coeﬃcient vs temperature for fermionicmolecules in the ro-vibronic ground state of 40 K87 Rb. For the lower decay coeﬃcients, the collision rate coeﬃcients were measured for molecules prepared in either| − 4, 1/2� (closed circles) or the lowest energy state | − 4, 3/2� (open triangles).
We observe the loss rate increases linearly with temperature for spin-polarizedmolecules, which verifies that the dominant collision channel is p-wave. A linear
fit (solid line) to the | − 4, 1/2� data yields the temperature-dependent loss rate
to be 1.2(3)·10−5 cm3 /s/Kelvin. For the | − 4, 3/2� case, where the collisional loss
can only be due to chemically reactive scattering, the loss rate is similar. The dotted and dashed lines are theoretical predictions from the QT model and MQDT(describe in the text), respectively. In contrast, when the molecular sample isprepared in a mixture of two hyperfine spin states, | − 4, 1/2� and | − 4, 3/2� (filledsquares), s-wave collisions dominate. Here we observe a temperature-independent
decay rate that is 10 – 100 times higher than for the spin polarized case.

We repeated this measurement for molecules in the lowest hyperfine state| − 4, 3/2� (open triangles in Fig. 10.5). The data again show β ∝ T with aslope of 1.1(3) · 10−5 cm3 /s/Kelvin, similar to that measured for molecules in
the | − 4, 1/2� state. However, in the case of | − 4, 3/2� molecules, hyperfine
state-changing collisions are no longer possible and the only possible loss channels

134
are the three chemical reactions discussed earlier. Thus, we find that the rate ofchemical reactions is determined by the p-wave angular momentum barrier and thechemical reaction barrier must be below the collision energy. This suggests thatthese reactions are barrierless and can thus occur freely at ultralow temperatures.
However, we cannot distinguish among the three possible exothermic mechanisms
at this time. Meanwhile, the fact that the same loss rate is observed for both|−4, 1/2� and |−4, 3/2� state molecules suggests that chemical reactions dominate
the loss in these ground-state molecular collisions.
To understand the loss rates, we again use the MQDT model introduced in
section 10.2 to estimate the rates where the collision process is characterized by
combining the p-wave barrier and a van der Waals inter-molecular potential. The
MQDT formalism for p-wave collision is provided by Z. Idziaszek and P. Julienne[116], where they derived the loss rate analytically and gave β = (11.48 · Im(a))3(kB T /h), where Im(a) can be replaced by lvdw in Eq. 10.2. Based on C6 =16130
a.u. for KRb-KRb (calculated by S. Kotochigova), β = 0.8(1)·10−5 cm3 /s/Kelvin,which is close to the experimental result.
In addition, we also use a simple quantum threshold (QT) model developedby Quéméner and Bohn to estimate the rate [117]. (This same model will be usedin the next chapter when describing inelastic dipolar collisions.) The inelastic
collisional cross section for indistinguishable particles (a factor of 2 bigger thanthe distinguishable particle case) is given quantum mechanically by
in
σL,m
=
L

�2 π in 2|T|,µEc L,mL(10.6)

inwhere Ec is the collisional energy and TL,mis the inelastic transition matrix. In
L

the QT model, the maximum loss rate is determined by setting the |T |2 matrix
to unity for the collisional energy Ec at the height of the p-wave barrier V (i.e.
chemical reaction proceeds with unit probability once the molecules pass through135
the barrier.) When the collisional energy is below the height of the p-wave barrier,the loss is suppressed and the rate scales linearly with collisional energy (temperainture) by the Bethe-Wigner threshold law. In this regime, |TL,m|2 = (Ec /V )L+1/2 .
LSince the p-wave barrier height is directly related to the attractive van der Waals
potential (characterized by C6 ), the QT model gives an upper limit for the rate
coeﬃcient to beπ
4� 17 3 3 �1/4
3 µ C6kB T.
�10(10.7)For KRb, β = 1.5(1) · 10−5 cm3 /s/Kelvin, which again agrees well with our experimental measurement (Fig. 10.5).
These simple theories, which both provide reasonable agreement with ourmolecule-molecule collisional loss measurements, suggest that chemical reaction
rates are strongly influenced by the long-range interactions. This opens intriguing
control possibilities as the long-range interaction can be controlled by selecting
quantum states and tuning collision energies.
10.3.2s-wave Collisions

Reaction rates should be dramatically diﬀerent if molecules are preparedin a mixture of diﬀerent hyperfine states as s-wave scattering is now allowed.
We measured the inelastic collision rates for ro-vibronic ground-state moleculesthat were prepared in a roughly 50-50 incoherent mixture of the two hyperfine
spin states | − 4, 3/2� and | − 4, 1/2�. The time-dependent number density of
trapped molecules was measured for both spin states. We observed the same loss
rate for both states, consistent with loss due to collisions between distinguishable
molecules in diﬀerent spin states. The rate coeﬃcient is determined to be 1.9(4) ·10−10 cm3 /s, independent of temperature, as shown in closed squares in Fig. 10.5.
In comparison to our measurements for p-wave collisions between spin-polarized

136
molecules, the s-wave collision rate between molecules in diﬀerent hyperfine statesis 10 - 100 times larger for a similar temperature range.
As in the atom-molecule s-wave collisions, molecule-molecule s-wave inelastic collision rates can be estimated using the MQDT model. Here, the relevant
length scale is determined by the inter-molecular van der Waals potential without
any angular momentum barrier [85]. We assume that when the molecules approach each other within the van der Waals length, chemical reactions take place
and remove these entrant molecules with a near-unity probability. The universal
loss rate coeﬃcient predicts a β value of 0.8 · 10−10 cm3 /s, which is about halfof the experimentally observed value. This diﬀerence suggests that short-range
physics may play some role in the loss dynamics. An enhancement in the rate
coeﬃcient (up to the energy-dependent unitarity limit, which is 4 · 10−10 cm3 /s fora gas at 400 nK) is possible if there is a partial reflection of the colliding species
back into the entrance channel. The reflected amplitude interferes with the incoming amplitude and can either increase or decrease the rate coeﬃcient from its“universal” value. Additional theory and experiment are needed to explore thispossibility [116].
Together the results presented here allow us to draw the conclusion that
we have observed barrierless chemical reactions in the short range, with the rates
determined completely by long-range scattering dynamics dictated by quantum
statistics, angular momentum barriers, and threshold laws. In particular, thesestudies show that chemical reactions can proceed with high rates in the ultracold
regime, and that, furthermore, it will be possible to experimentally control thereaction rates because they depend primarily on long-range interactions. And
most importantly, with our understanding of the intrinsic collisional molecular
lifetime, we can now turn on an applied electric field and begin to study electric
dipole-dipole collisions.

Chapter 11
Electric Dipole-Dipole Collisions

In the last chapter, we investigated density- and temperature-dependent
lifetime of our fermionic ro-vibronic ground-state molecules, which is on the orderof a second. Finally, we are ready to turn on an electric field to manipulatea unique property of polar molecules – their large and tunable electric dipole
moment. After working with alkali atoms and weakly bound Feshbach molecules
for many years, where their magnetic moment is about ∼ 1 µb and their electric
dipole moment is essentially zero, it took some getting used to molecules with
an electric dipole moment and essentially zero magnetic dipole moment. In thischapter, I will start by introducing technical things we learned while working withelectric dipoles and then discuss some of the exciting first results of electric-field
controlled dipolar collisions.
11.1

Working with Electric Dipoles
The induced dipole moment of our polar molecules is directly related to theapplied external electric field. Figure 11.1 shows the induced dipole moment vselectric field from the permanent electric dipole moment (0.574 Debye) and therotational constant (h·1.1139 GHz) we measured previously (Chapter 8.3). We
typically work with an electric field up to ∼ 5.0 kV/cm (corresponding to a dipole
moment up to ∼ 0.21 Debye.)138

Dipole Moment (Debye)

0.25
0.20
0.15
0.4

0.10

0.3
0.2

0.05

0.1
0.0
0

0.00
0

1

2

5

3

10

4

15

20

5

25

30

6

E-field (kV/cm)
Figure 11.1: Calculated KRb induced dipole moment vs electric field for thepermanent dipole moment of 0.574 Debye and the rotational constant of h·1.1139
GHz that we measured[15]. A larger range of dipole moment vs electric field isshown in the inset.

11.1.1

E-field Gradients

The electric field at our molecules is provided by a pair of parallel voltage
plates (see Fig. 3.2). However, due to the machining precision of a spacer/holder
in between the plates plus gluing the two plates to the holder, the plates may not
be perfectly parallel. We first suspected this when we saw a shift in the molecular
cloud equilibrium position when we turned on the electric field to increasing values(see Fig. 11.2). This shift occurred in the weak direction of a single-beam opticaltrap (perpendicular to the field direction). We then estimated the electric-field
gradient that could arise from a slight tilt between two plates. First, to a good
approximation within the range of our applied electric field, we consider electricfield mixing of only N = 0 and N = 1 rotational states. The induced dipole

cloud position (pixels)

139
180 position = 116(1) + 3.6(2)E

2

160
140
120
100

0

1

2

3

4

Electric field (kV/cm)

Figure 11.2: An electric-field gradient pushes the molecular cloud in the weak
direction of the optical trap confinement. Each camera pixel is 5.4 µm in oursystem. For this data, the electric field is turned on for 40 ms. For a given
duration, the cloud position shift is proportional to the gradient force which inturns is proportional to the square of the applied electric field. We fitted the shift
to the quadratic term of the electric field that is 20 µm/(kV/cm)2 and backed out
the amount of electric-field gradient.

moment, �d�, isd�d� ≈ √
3�(ε/ε0 )2,
3 + (ε/ε0 )2(11.1)

where d is the permanent electric dipole moment, ε is the applied electric field,and ε0 = 3.85 kV/cm is the critical electric field (see Chapter 8.3 for definition)
for KRb. The force on the molecules from the electric field is
F =−

d �
dε
∂�d� ∂εdε/ε0dε(�d�·�ε) = −�d�· −ε≈ −2· √ �·
∝ ε2 (11.2)
2
dx
dx
∂ε ∂x
dx
3 3 + (ε/ε0 )The shift of the cloud position (Fig. 11.2) was due to the electric-field gradient(countered by an optical confinement in this weak direction which we neglectedhere). From Eq. 11.1, we can calculate the gradient we measured in Fig. 11.2
to be 1.8 V/cm2 per 1 kV/cm applied. The gradient could be explained by theseparation of the E-field plates varying by ∼ 25 µm over 1 cm. This is a very

140
small fraction of the 1.35 cm average separation. From the experimental point ofview, this shows that the eﬀect of the molecular electric dipole moment is huge!
11.1.2

Fringing Fields and a Non-symmetric Ground Plane

We overcame the electric-field gradient problem by replacing the single beam
optical dipole trap with a crossed-beam dipole trap. This increased the opticaltrap confinement in the weak direction by a factor of 10 in trap frequency andreduced the eﬀect of the field gradient in this direction. (This was done for all
the data presented in Chapter 9 and 10). However, we encountered yet anothergradient problem in the direction parallel to the electric field (Fig. 11.3) (also thedirection of gravity) where the optical confinement is the strongest. Since any tiltof the field plates would only cause a gradient in the horizontal plane, we needed
to investigate other possible causes.

Figure 11.3: Molecule leaking out of the optical trap due to a strong electric field
gradient at an applied field 3 kV/cm.

We considered the eﬀect of the electric-field plates being surrounded byground planes and also the fringing field from the finite-size of the plates. In our
first attempt to apply an electric field, we had the top plate grounded and applieda high voltage to the bottom plate. We did this because the top plate was close
to a grounded microwave coil (Fig. 3.2) and we were concerned about dielectric
breakdown. This asymmetry in the applied voltage combined with the existenceof nearby grounded magnetic coils give rise to a vertical E-field gradient. The

141
4.0 cm

(a)

1 kV

2.6

1.3 cm

0V
0V

(b)
0.16
cm

(C)

!

!

!

5c

m

0V

1 kV

!

zoom in on
the color scale

0V
0V
0.5 kV

Figure 11.4: Finite-element simulation of the electric-field for our E-field plates
surrounded by nearby grounded planes. (a) We ground the top E-field plate andapply 1 kV at the bottom plate. The plates are surrounded by a grounded 3-D box
that is 4 times bigger than the space spanned by the plates and their separation.
The field gradient is ∼ -40 V/cm2 at the geometric center. (b) Add an additional
ground plane that is only 0.16 cm away from the top plate. The field gradient is
the same as in (a) but the eﬀect of fringe fields moves closer to the center. (c)When splitting the voltage diﬀerence between the top and the bottom plates, thefield gradient is reduced to 10 V/cm2 and changes sign.

-0.5kV142
electric field will be stronger near the bottom plate and weaker near the top plate
because field lines can go to grounded planes other than our grounded E-field
top plate. To estimate the magnitude of the gradient, we performed a finiteelement calculation using a program called ANSYS. To illustrate the importanceof each eﬀect – finite size, fringing field, position of the grounded plane, and thenon-symmetric applied voltage, we calculated the field by adding one eﬀect at atime (we ignored the glass cell between the plates here). Our simulation was thefollowing: 1. We first considered grounding the top plate, and applying 1kV at
the bottom plate. In the model, the pair of E-field plates are surrounded by agrounded 3-D box that is four times the size of the space spanned by the area
and the separation of the plates. At the geometric center between the plates, thefield gradient is about −40 V/cm2 while the E-field ∼ 750 V/cm (Fig. 11.4(a)).
Experimentally we observed a gradient that is about a factor 2 larger than thesimulation based on the crude geometry. 2. Adding a grounded plane to simulate
the eﬀect of the nearby microwave coil, the gradient at the center did not change,
but the fringing fields cause the region of uniform field strength to shrink farther
from the edge of the plates. Therefore, if the cloud is oﬀ-centered in the horizontal
plane, the gradient can be larger and pointed in a direction other than the verticaldirection. (Fig. 11.4(b)) 3. When split the applied voltage diﬀerence between thetwo plates (top plate at -0.5 kV and bottom plate at 0.5 kV), the sign of thevertical gradient changed and the magnitude is reduced to <10 V/cm2 . (Fig.
11.4(c)) This is obviously an improvement which we have experimentally verified
as well. However, to get the lowest possible field gradient, we need to further
experimentally vary the voltage ratio of the two plates due to the non-symmetricground planes, which comes from surrounding metal coils in the experiments.

143
11.2

Inelastic Dipolar Collisions
With the two electric-field gradient problems solved, we can turn up theelectric field in our experiment without perturbing cloud position. Previously, we
have established that at zero applied electric field (zero eﬀective dipole moment)
our indistinguishable fermionic KRb collide via p-wave followed by a near-unity
probability of chemical reactions in the short range (Chapter 10.3.1). This collision
is universal in the sense that the collision rate is solely determined by the longC6
range potential, � L(L+1)−R
6 , which in our case sets the p-wave barrier height.
2µR2
2

By tuning the external electric field, we can adjust the eﬀective dipole moment,3�d�, of the molecules (Fig. 11.1), which introduces an additional −C
term fromR3

the electric dipole-dipole interaction in our gas. The C3 is given by



�
�
�
2 L   L 2 L  �d�2
 L
C3 = 2 · (−1)mL (2L + 1)(2L� + 1) 
.


4π�0
−mL 0 m�L
0 0 0(11.3)

This term shows that dipole-dipole interactions 1. are long-range and anisotropic
among mL s and 2. mix diﬀerent partial waves of the same parity.
The anisotropy nature of the dipole-dipole interaction directly modifies thelong-range potential according to the mL s (shown in Fig. 11.5 and 11.6). At zeroeﬀective dipole moment, the p-wave barrier for mL = 0 and mL = ±1 channels aredegenerate (dashed line in Fig. 11.5). As the eﬀective dipole moment is turned
on, e.g. 0.1 D (Fig. 11.5), the p-wave barrier is raised for the mL = ±1 “sideby-side” repulsive collision channels and is lowered for the mL = 0 “head-to-tail”
attractive channel. Figure 11.6 shows the anisotropy of the barrier heights vsdipole moment. At a large eﬀective dipole moment (> 0.15 D), the mL = ±1
p-wave barrier is again lowered, which is due to mixing with higher partial waves,e.g. L = 3, 5, 7, ... As seen in the QT model (developed by Quéméner and Bohn

144

40

Energy�kb �ΜK�30

mL = ±1 (0.1 D)

20
10
0
�10
0

mL = 0 (0.1 D)

200 400 600 800 1000
Inter-molecular
Separation�a(a00�)
Internuclear
Separation

Figure 11.5: Adiabatic energy curves of the long-range part of KRb-KRb potential(shown for p-wave, L = 1). When no electric field is applied, the p-wave barriers
for angular momentum projection mL = 0, 1, −1 are degenerate (central dashedline). When an electric field is applied and the dipole moment of the molecules
is turned on, e.g. 0.1 Debye, the barriers for the mL = ±1 channels are raised(upper branch) and the barrier for the mL = 0 channel is lowered (lower branch).

Barrier Height �ΜK�

80
60
40

mL = ±120mL = 00
0.00

0.05 0.10 0.15 0.20
Dipole Moment �Debye�

0.25

Figure 11.6: Barrier height vs eﬀective dipole moment, �d�. The solid lines arecalculated using the adiabatic energies considering partial wave mixings up to
L = 7. The mL = ±1 barrier heights rise as the eﬀective dipole moment increases.
However, when d> 0.15 D, the barrier height reduces due to mixings of higher
partial waves. The mL = 0 barrier height reduces as the dipole moment increases.
The dashed lines are the diabatic curves without partial wave mixing.

145
in [117]) in Chapter 10.3.1, the inelastic collision rate is directly determined by
the collisional barrier. The consequence of dipolar interaction in our gas, which
is a thermal average of all collisional channels, is that the inelastic collision isdominated by the lowering of the p-wave barrier due to the attractive mL = 0
channel.
11.2.1

Anisotropy of Losses

Experimentally, we measured the inelastic loss rate of molecules as a function of the eﬀective dipole moment (Fig. 11.7(a)). At small dipole moments,
we observed a constant inelastic collision rate that is mainly determined by theattractive van der Waals potential. At large dipole moments, the inelastic loss
rate grows with �d�p where p was measured to be 6.1(0.4) (fitting to an additionaloﬀset, c + A · �d�p , p = 6.7(0.5)). The large power-law dependent came from thesteep lowering of the barrier height due to the attractive dipolar interaction (themL = 0 channel, Fig. 11.7(b)). This can be seen in the analytical formulas of the
QT model, which predicts the inelastic rate for the attractive channel (mL = 0) toC3C6
grow as �d�6 in the limit of R
3 � R6 [117]. For the repulsive channel (mL = ±1)
C3
C6
in the limit of R
3 � R6 , but before mixings with higher partial waves become

important (i.e. in the diabatic limit), the inelastic rate decreases as �d�−6 [117].
Our experimental data (Fig. 11.7(a)) shows qualitative agreement with the
QT model (Fig. 11.7(b)). However, the quantitative agreement can be made if we
introduce two fitting parameters, pb and C6 , to adjust the QT predicted loss rate.
“pb” is the probability or an overall scaling factor of the loss at the short-range
and C6 determines the zero E-field barrier height, hence the molecular loss rate
at low dipole moment. For the best fit to our data, we found pb= 0.34(12) andC6 = 31000(16000) a.u.

146Β�T �10�5 cm3 �s�Kelvin�!/T (10-5 cm3/s/Kelvin)!/T (10-5 cm3/s/Kelvin)

100
(a)

10

Fit to experimental data gives
short-range chemical reaction
probability (pb) = 0.34 andC6= 31400 a.u.

1
0.01

100.0

0.1

Dipole Moment (Debye)
(b)

50.0

10.0

QT model using C6=16130 a.u.
and pb =1.

5.0

total
mL=0

1.0
0.5

mL=1 and mL=-1
0.010

0.015 0.020

0.030

0.050 0.070 0.100

Dipole Moment �Debye�

0.150 0.200

Dipole Moment (Debye)
Figure 11.7: Inelastic dipolar collisions. (a) experimental observation of inelastic
losses as the function of eﬀective dipole moment. The fit is based on the QT loss
rate model that uses p-wave barrier height from the adiabatic energies with an
overall scaling factor of 0.34 and a van der Waals dispersion coeﬃcient C6 = 31400
a.u. (b) Predicted loss rate from the QT model based on C6 = 16130 a.u. (C6was calculated by S. Kotochigova) and pb = 1. The QT model is a “classical”
calculation tailored to describe quantum threshold laws. Contributions of mL = 0
and mL = ±1 respectively are shown, where the losses we observe at high dipole
moments are predominately from the attractive mL = 0 channel.

147
11.2.2Anisotropic Heating

Along with the strong inelastic loss of molecules with an increasing dipole
moment, we observed strong corresponding heating for the trapped gas (Fig.
11.9). Here, we consider the heating of the trapped gas due to the so-called“anti-evaporation” process from inelastic collisions. The main inelastic loss in ourgas is two-body collisions (ṅ = −β · n2 ), where the losses of molecules occur mostfrequently at the densest part of the cloud. Since our gas is in a harmonic trap,
the highest density region coincides with the coldest part of the gas (Fig. 11.8(a)).
Therefore, when molecules in this region are removed, the average temperature of
the remaining gas increases (Fig. 11.8(b)). However, there is a competing coolingeﬀect due to β ∝ T for p-wave collisions. To understand the overall heating of
the gas, we will consider both the cooling and the heating eﬀects from p-wave
inelastic collisions.

T
(a)

(b)

Figure 11.8: Anti-evaporation of a trapped gas due to inelastic collisions. (a)inelastic collisions occur most frequently at the densest part of the gas whichcoincides with the coldest part of the gas. (b) when molecules in the coldest partare lost, the average temperature of the remaining gas increases.

148(10-5 cm3/s/Kelvin)

HeatingṪ /T02 /n0

10

1
β/T0
12

0.1

0.01
0.01

0.1

Dipole Moment (Debye)
Figure 11.9: Strong heating in our trapped gas when an E-field is turned on. Thisis due to “anti-evaporation” where the cold molecules from the densest part of
the gas are preferentially removed. The line to the data is the inelastic dipolar
loss (β/T0 ) we measured in Fig. 11.7(a) scaled by 1/12, which confirms that theheating comes directly from inelastic collisions.
0.7

Temperature (µK)

0.6

Tx
Tz

0.5
0.4
0.3
0.2
Tx

Tz

0.1
E-field

0.0
0.00

0.05

0.10

0.15

0.20

Dipole Moment (Debye)

0.25

Figure 11.10: Individual temperatures, along (Tz ) and perpendicular (Tx ) to theapplied E-field direction. We observed heating in Tx and cooling in Tz (for small
dipole moment). The anisotropy of heating can be explained by the dominate
dipolar collisions in our gas being the attractive channel.

149
Here I will only give a brief sketch of the analysis as it is still ongoing during
the time when this thesis is written. The temperature evolution of the gas follows
directly from the inelastic collision events. (Temperature changes due to elastic
dipolar collisions are ignored here.) For each collisional event, two molecules are
removed from the trap with an energy change including their kinetic energy andpotential energy. The energy change is then re-distributed among the remaining
molecules in the trap. Furthermore, the rate of the temperature change is the
product of the temperature change per collision and the inelastic collision rate,
which depends on the collisional energy (temperature). Our model based on thisanalysis shows that the initial heating rate of the gas is directly related to theinelastic loss rate β by an overall factor of 1/12. Plotting Tβ0 /12 where Tβ0 was
fitted from the loss measurement in Fig. 11.7(a), we see that the overall heating
is explained well by the model sketched above (Fig. 11.9).
Beside an overall heating due to the inelastic collisions, we also observed ananisotropy in the heating along (Tz ) and perpendicular (Tx ) to the applied E-field
direction (along z). In Fig. 11.10, we measured the molecular gas temperature Tz
and Tx as a function of the eﬀective dipole moment after holding the gas in thetrap for 100 ms while the E-field is on. This measurement shows that Tz is mostly
cooled during the 100 ms period, while Tx is heated. At large eﬀective dipole
moment, both directions are heated. This eﬀect makes sense because we expect
cooling in Tz and heating in Tx for the attractive dipolar collisions, which are
the dominant collisions in our 3D gas. However, at high eﬀective dipole moment,
we observed a rise in the temperatures for both directions. This could be due tostrong elastic dipolar collisions that transfer energy from one direction to another.

150
11.3

Elastic Dipolar Collisions
The presence of an E-field not only turns on strong inelastic collisions, butalso turns on long-range elastic dipolar collisions. The elastic collisions due to thelong-range dipole-dipole interaction is independent of the short-range elastic crosssection, which for fermionic molecules is expected to be small and suppressed (asis the case of fermionic K atoms [118]). However, the long-range elastic collisioncross section could still be large. In Chapter 1.3.1 (Fig. 1.1), I have summarized
the result of universal elastic dipolar collisions from Bohn et al. [36] (a similar
result was also obtained in [119]), where the elastic collisional cross section in thethreshold regime (applicable to our experiments) is directly proportional to �d�4 .
This result was derived using the Born approximation, where the elastic collisionalcross section written in a generic T -matrix form is
σ el =

2
�
π lD
mL 2
|TL,L
�| .
2
k L,L� ,m(11.4)

L

Here lD is the dipole length as defined in Chapter 1.3 and k is the wavenumber ofthe relative motion of the dipoles. An additional factor of 2 needs to be multiplied
for collisions of indistinguishable particles. The T -matrix labelled by angularmL
momentum channels is an integral of both the angular part (CL,L� ) and the radial

part (ΓL,L� ) of the wavefunction.
mL
mL
TL,L
� = −k · CL,L� · ΓL,L� ,

where
mL
mL
CL,L
� = (−1)

and

�




(2L + 1)(2L� + 1) 
Γ

L,L�

=









L

2

−mL

4
,
L(L+1)
4,3(L+1)(L+2)(11.5)

�



�



L  L 2 L 


0 m�L
0 0 0

if L = L��

if L = L + 2.

(11.6)(11.7)

151
mL
From the explicit expression of CL,L
� , it is clear that dipolar collisions only mix

states of the same parity. Therefore, Bohn et al. have derived the elastic collisionalcross section for the even and the odd partial waves separately, which depend on
the quantum statistics of the collisions.
In our case, the indistinguishable fermionic molecules collide via odd partial
4

�d�
2
waves, and σ el = 6.7 · 10−8 Debye
4 cm . This collisional cross section is quite large.

For a comparison, Rb elastic collision cross section with a background scattering
length ∼ 100 a0 is 8πa20 = 7 · 10−12 cm2 , which is equivalent to the elastic dipolarcross section of fermionic polar molecules with a dipole moment of 0.1 Debye.
Since the total cross section is a sum of many angular momentum channels, we
can directly compute the contribution of each partial-wave channel using the Born
approximation. The contribution of L = 1, mL = 0 is ∼ 60 %, L = 1, mL = ±1
is ∼ 15 % each, and higher partial waves is only ∼ 10% of the total cross section.
Using the theoretical predicted elastic dipolar collision cross section andthe measured inelastic collision rates, we can calculate the ratio of the elastic toinelastic dipolar collision rate (βelastic = �σ el · v�/βinelastic ) as a function of theeﬀective dipole moment. From Fig. 11.11, we can see that the highest ratio isslightly above 1 around 0.13 D, which suggests that further evaporative coolingusing dipolar collisions is diﬃcult without suppressing the inelastic collisions.
11.4

Cross-dimensional Rethermalization
We attempt to measure the elastic dipolar collision cross section in the ul-

tracold gas of ground-state KRb by using the method of cross-dimensional rethermalization [120]. The molecules are confined in a pancake-shaped optical dipole
trap with an aspect ratio of ∼7. Typical trap frequencies for the optical dipole
trap used are fx ≈ fy ≈ 40 Hz and fz ≈ 280 Hz. For these measurements, weheat the gas in one direction, Tz (or Tx ), by modulating the trap at twice the

152

1

10

Βelastic�Βinelastic

1
10�1
10�2
T=100 nK
T=200 nK
T=500 nK

10�3
10�4
0.00

0.05
0.10
0.15
Dipole Moment �Debye�

0.20

Figure 11.11: The ratio of the elastic dipolar collision rate βelastic (calculated basedon [36]) to the inelastic dipolar collision rate βinelastic (measured) vs the eﬀective
dipole moment for KRb. The elastic collision rate scales as T 1/2 and the inelastic
collision rate scales as T . The ratio is the highest (> 1) around 0.13 D.

frequency 2fz (or 2fx ) to excite a breathing mode and watch the gas come to athermal equilibrium due to energy transfer from one direction to another. The
rethermalization time scale (τreth ) can be related to the elastic collisional crosssection by an expression τreth = (n · σ el · v)/α, where n is the density, v is themean velocity of the gas, and α accounts for the geometric factor of the numberof collisions required for rethermalization. However, in this measurement inelastic
collisions that cause anisotropic heating could mimic the rethermalization eﬀect
of elastic collisions. This is because both types of collisions aﬀect the thermal
anisotropy of the gas.
Figure 11.12 shows data from our cross-dimensional rethermalization measurements. For the data in the first column, we heat the gas in the z direction. We
observe the temperatures in two directions (Tx , Tz ) approach each other rapidly

153
as a function of the dipole moment. However, the temperatures eventually crosseach other. For the data in the second column, we heat the gas in the x and y
directions simultaneously. However, the temperature in the x direction and thez direction never appear to “rethermalize.” This can be explained mostly by theinelastic anisotropic heating model as briefly sketched out in section 11.2.2, which
reenforces our observation of the anisotropic nature of inelastic collisions. Furthermore, our preliminary attempt to fit the data based on the model shows good
agreement (Fig. 11.12) if we include the elastic collision term (�d�4 as described in
the last section 11.3) to describe the temperature behavior of the rethermalization
data.
Although the data analysis for the dipolar collisions is still on-going, I
have presented here our measurements that show both the long-range and theanisotropic nature of dipole-dipole interactions.

Tx
Ty

0.6
0.4
0.2
0.0
0
1.0

1

2

0.064 Debye

0.4
0.2
1

2

Time (s)0.8

30.084 Debye

0.4
0.2

1.0

1

2

Time (s)32Time (s)340.064 Debye

0.6
0.4
0.2
1

2

Time (s)

0.8

3

4

0.084 Debye

0.6
0.4

0.0
0

4

1.0

0.158 Debye

0.6
0.4
0.2

change time axis
0.5

1.0

Time (s)

1

2

1.5

3

4

Time (s)0.158 Debye

0.8

Tz<Tx(µK)

Tz>Tx (µK)

1

0.2

0.8

0.0
0.0

0.2

0.0
0
1.0

4

0.6

0.0
0

0.4

0.8

0.6

0.0
0
1.0

0 Debye

0.6

0.0
0
1.0

4

Tz<Tx(µK)

Tz>Tx(µK)

3Time (s)0.8

Tz>Tx (µK)

0.8

0 DebyeTz<Tx (µK)

0.8

154

1.0

Tz<Tx (µK)

Tz > Tx (µK)

1.0

0.6
0.4
0.2
0.0
0.0

change time axis
0.5

1.0

1.5

Time (s)Figure 11.12: Sample data for cross-dimensional rethermalization. For initial
Tz > Tx , we observed Tx and Tz approach each other with a decreasing timescale as a function of dipole moment. However, the temperatures did not cometo a thermal equilibrium but rather crossed each other. For initial Tx > Tz , thetwo temperatures never rethermalize and even diverge. This is mainly due toanisotropic heating for attractive dipolar collisions in our 3D gas.

Chapter 12Conclusions and Future Directions

12.1

Conclusion
In this thesis, I have presented the story of the experimental realization ofa high phase-space-density gas of 40 K87 Rb polar molecules and some initial investigations of ultracold molecular chemical reactions and dipolar collisions. Theseultracold polar molecules at a few hundreds of nanoKelvins and at 1.4 times theirFermi temperature are created with two coherent transfer steps from an ultracold
gas mixture of 40 K and 87 Rb. The first step is to transfer atoms into extremely
weakly bound molecules (molecular size of ∼ 250 a0 ) near a Fano-Feshbach resonance. The second step is to transfer these weakly bound molecules into theirro-vibronic ground state (molecular size of ∼ 8 a0 ) with a coherent two-photon
Raman technique. The first step was demonstrated in J. Zirbel’s thesis to beabout 25% eﬃcient [24], while the second step demonstrated in this thesis is 90%
eﬃcient. We verified that the ro-vibronic ground-state molecules are polar bymeasuring their permanent electric dipole moment spectroscopically.
Once molecules were prepared in the ro-vibronic ground state, we implemented a scheme to manipulate their molecular internal hyperfine state and explored various scenarios of ultracold collisions. We observed ultracold chemical
reactions between KRb and K. Once all the atoms were removed, we observed
fermionic 40 K87 Rb lifetime on the order of a second. This is limited by molecule-156
molecule p-wave collisions where molecules tunnel through a 24 µK p-wave angular momentum barrier followed by a near-unity probability of chemical reactions. Furthermore, we began to explore the polar nature of the molecules incollisions through their large and tunable electric dipole moment. For large eﬀective dipole moments, we observed enhanced inelastic collision rates that scale as
d6 and anisotropic heating of the gas.
12.2

Future WorkAn immediate goal in the future is to evaporate our near quantum degen-erate gas further and begin to realize some of the many exciting proposals formolecular quantum gases. To evaporate our gas of spin-polarized fermionic polar
molecules, one can turn on strong elastic dipolar collisions by applying an electricfield. However, our work so far suggests that strong inelastic collisions also happen at the same time as the elastic collisions, and these cause significant loss andheating of the gas. Therefore, to reach quantum degeneracy, inelastic collisions
need to be suppressed. One way is to suppress the dominant inelastic channel(the attractive dipole-dipole interactions) by going to reduced dimensions [110].
Figure 12.1 shows the ratio of the elastic to inelastic dipolar collision rate when
the attractive collision channel is completely suppressed. This ratio is favorable
for further evaporative cooling of KRb when the eﬀective dipole moment is increased. Given the advances we have already made and all that we have learned,
I am optimistic that the future of a quantum gas of polar molecules is bright andexciting!

157

102

Βelastic�Βinelastic

101
1
10�1
10�2
10�3
10�4
0.00

T=200 nK
T=200 nK, suppress mL=0 collisions

0.05
0.10
0.15
Dipole Moment �Debye�

0.20

Figure 12.1: The ratio of the elastic dipolar collision rate (calculated based on [36])to the inelastic dipolar collision rate (measured) vs the eﬀective dipole momentfor KRb at 200 nK. The dashed line shows the ratio when the mL = 0 collisional
channel is completely suppressed. In this case, the ratio increases dramaticallywith increasing dipole moment, which makes dipolar collisions favorable for further
evaporative cooling of our KRb gas.

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Appendix A
Sample Mathematica CodeThroughout the thesis, I have presented a few calculations, for examples,
the KRb Stark shifts and the KRb-KRb long-range adiabatic potentials. Those
calculations were based on a very simple code in Mathematica that were setup withthe help from D. Meiser and T. Rosenband. In addition to the calculations shownin this thesis, the same code have been adapted to calculate diﬀerent Hamiltonians
that use the basis set of angular momentum states |J, mJ �, e.g. Zeeman energies
of ground-state and excited-state K and Rb, etc. I found the code very useful
and would like to share it here. I will take the KRb Stark shift calculation as an
example. The parts labeled “list of J and mJ values to be evaluated” and “define
H in reduced units” are the main parts that can be adapted for any Hamiltonianof interest.

169

In[1]:=

Clear@B, mu, d, r, Efield, E0, EfD;
Off@ClebschGordan::"phy" D
Off@ ClebschGordan::"tri"D
Needs@"NumericalCalculus`"D
Needs@"NonlinearRegression`"D
Needs@"LinearRegression`"D
Needs@"ErrorBarPlots`"D

constants and units
In[8]:=

In[9]:=

hbar = 1.054571628 10-34 H*J s*L; amu = 1.660538782 10-27 H*kg*L;c = 299 792 458H*m s-1*L; kb = 1.3806504 10-23 H*J K-1*L; e = 1.602176462 10-19
H87 µ 40L
H*C*L; a0 = 0.5291772083 10-10 H*m*L; mKRb =amu;
H87 + 40L
SetOptions@Plot, BaseStyle -> 8FontFamily -> "Arial", FontSize Ø 18<,
PlotStyle Ø 8Thick<, TicksStyle Ø Directive@Black, 10D, Frame Ø TrueD;
SetOptions@ListPlot, BaseStyle -> 8FontFamily -> "Times", FontSize Ø 18<,
PlotStyle Ø 8Thick<, TicksStyle Ø Directive@Black, 10D, Frame Ø TrueD;

list of J and mJ values to be evaluatedIn[11]:=

mjlist@j_D := Table@8j, mj<, 8mj, 0, j<D;
jlist@Jmax_D := Table@j, 8j, Jmax, 0, -1<D;
JmJ@Jmax_D := Flatten@mjlist êü jlist@JmaxD, 1D

In[14]:=

JmJ@5DOut[14]=885, 0<, 85, 1<, 85, 2<, 85, 3<, 85, 4<, 85, 5<, 84, 0<, 84, 1<, 84, 2<, 84, 3<,
84, 4<, 83, 0<, 83, 1<, 83, 2<, 83, 3<, 82, 0<, 82, 1<, 82, 2<, 81, 0<, 81, 1<, 80, 0<<In[15]:=

Jmax = 5;H*define how many Js I want to take into consideration*L

define H in reduced units
In[16]:=

In[17]:=

H*write hamiltonian in the energy unit of B,
write Efield is in the unit of critical field=Bêd*L

hamiltonian2@8jprime_, mjprime_<, 8j_, mj_<D :=j * Hj + 1L * KroneckerDelta@j, jprimeD * KroneckerDelta@mj, mjprimeD -Ef * H2 j + 1L H2 jprime + 1L H-1Lmj ThreeJSymbol@8j, -mj<, 81, 0<, 8jprime, mjprime<D *ThreeJSymbol@8j, 0<, 81, 0<, 8jprime, 0<D;
HH2 = Outer@hamiltonian2@Ò1, Ò2D &, JmJ@JmaxD, JmJ@JmaxD, 1D;
ev = Eigenvalues@HH2D
In[20]:=

Out[20]=H* figure out ordering of eigenvalues at non-zero Efield *L
evOrder = Ordering@ev ê. Ef Ø 1 êê ND
evo = evPevOrderT;

816, 11, 17, 7, 12, 18, 4, 8, 13, 19, 2, 5, 9, 14, 20, 1, 3, 6, 10, 15, 21<

Printed by Mathematica for Students

170

v=0 of singlet groundIn[22]:=

singlet = :d Ø 0.22591382365886692 e a0, B Ø

In[23]:=

E0 =In[24]:=

StarkPlot = PlotB

H*dipole moment and rotational constant*LBd6.6836
6

109 hbar 2 p>;

ê. singlet; H*define critical electric field for singlet vibrational ground state*LB
evo ê. 9Ef Ø IEff 105 M ë E0= ê. singlet,
109 hbar 2 p
8Eff, 0, 30<, PlotLabel Ø "Stark effect of v=0",
FrameLabel Ø 8"Electric Field in HkVêcmL", "Energy HGHzL"<F

Stark effect of v=0
Energy HGHzL

30
20Out[24]=1000

5
10 15 20 25
Electric Field in HkVêcmL

30

Printed by Mathematica for Students

171

Dipole Moment (Debye) vs Electric Field (kV/cm)
In[26]:=

DvEPlot = PlotB-D@Hevo@@1DDL 0.574, EfD ê. Ef Ø 105

ê. singlet,
E0
8Eff, 0, 5<, PlotLabel Ø "induced dipole moment of v=0", FrameLabel ØEff

8"Electric FieldHkVêcmL", "Dipole Moment HDebyeL"<, PlotRange Ø 880, 5<, 80, 0.22<<F

Dipole Moment HDebyeL

induced dipole moment of v=0
0.20
0.15

Out[26]=

0.10
0.05
0.00
0

1

2
3
4
Electric FieldHkVêcmL

Printed by Mathematica for Students

5Appendix B
Tapered Amplifier Design

One of my very first projects as a graduate student was to design a tapered
amplifier (TA) mount for Eagleyard TA chips that come in a C-mount. With thehelp from Josh Zirbel, Scott Papp, and Carl Wieman, our design has now become
the standard JILA design. The major diﬀerence between previous designs andour design is the use of clamping to make the electric connections to the tapered
amplifier, rather than soldering. The design of the overall geometry and theflexure mount that is used for fine adjustment of the collimation is similar to
Brian DeMarco’s design [121].

Figure B.1: Overview of the TA mechanical design. The red arrow shows the
direction of the current from an electric wire through the screw hole to the clamp,
then through the lead of the TA chip to the body of the chip, and finally to thewire that is connected on the side of the TA holder.

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should be bigger to avoid
possible clipping of the lightlayers of Kapton
tape for insulation

Figure B.2: Detailed dimensions (in inches unless otherwise indicated) of the Cmount pocket. To isolate the lead (either current sourcing or sinking) of the TAchip from the holder, we place 1-2 layers of Kapton on the holder where the leadsits. In addition, a thin piece of indium is sandwiched between the C-mountpocket and the C-mount of the TA chip for a good thermal contact that allows
heat to flow away from the chip to the whole copper holder piece.

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Figure B.3: Detailed dimensions of the TA holder in inches. The “arm”-likestructure is to prevent dust falling onto the TA chip and the lens holder touching
the chip. This part can be removed.

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Figure B.4: Detailed dimensions of the clamp. A current source is attached to theclamp through a metal lug that is screwed onto the center screw hole. The clamp
itself is screwed onto the TA holder using nylon screws.

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Figure B.5: Detailed dimensions of the lens holder that is used to collimate laser
beams going into and out of the TA chip.

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Figure B.6: Lens adapter for Thorlabs Geltech aspheric lens.

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Figure B.7: Detailed dimensions of the flexure mount that is used for fine tuningof the collimation lens position relatively to the TA chip.