A quantum gas of polar molecules in an optical lattice
by
Steven A. Moses
B.S., University of Michigan, 2010
M.S., University of Colorado, 2013

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
2016This thesis entitled:
A quantum gas of polar molecules in an optical lattice
written by Steven A. Moses
has been approved for the Department of Physics

Jun YeDeborah S. Jin

Date

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

iii
Moses, Steven A. (Ph.D., Physics)
A quantum gas of polar molecules in an optical lattice
Thesis directed by Prof. Jun Ye

Ultracold polar molecules, because of their long-range, spatially anisotropic interactions, area new quantum system in which to study novel many-body phenomena. In our lab, we have produced the first quantum gas of 40 K87 Rb polar molecules. These molecules were found to undergo
exothermic chemical reactions, and this led to interesting studies of chemistry near absolute zero.
By creating the molecules at individual sites of a 3D optical lattice, we completely suppress thesechemical reactions, and the polar molecule gas becomes stable and lives for tens of seconds. This
thesis documents our efforts to explore coherent, many-body phenomena resulting from long-range
dipolar interactions in the lattice. By encoding a spin-1/2 system in the rotational states of themolecules, we were able to realize spin-exchange interactions based on a spin Hamiltonian, which isone of the first steps in studying quantum magnetism with polar molecules. While this study was
the first realization of such coherent dipolar interactions with polar molecules in a lattice, its fullpotential was limited by the low lattice filling fractions. Using our ability to exquisitely control the
initial atomic gas mixture, we loaded a Mott insulator of Rb and a band insulator of K into thelattice. This quantum synthesis approach led to significantly higher molecular filling fractions andrepresents the first fully connected system of polar molecules in an optical lattice. This low-entropy
quantum gas of polar molecules opens the door to interesting quantum simulations, which should
be attainable in the next generation of the experiment.

Note: This work was jointly supervised by Prof. Deborah Jin.

Dedication

To my family, for their constant love and support.

v

Acknowledgements

My interest in physics started pretty early in life. I had several great professors in college
who were very influential in my decision to go to graduate school. In particular, I would like tothank Jean Krisch, Ctirad Uher, Georg Raithel, and Aaron Leanhardt. It was while I was an REU
summer student with James Thompson at JILA in 2008 that I developed my interest in atomic
physics, and I realized that Boulder was a very special place. I liked it so much that I came back
to JILA for grad school!
My time at JILA has been an adventure, and there are many people I would like to thank formaking my time here a great experience. First, I am grateful for having the opportunity to workwith Jun and Debbie. They have been incredible leaders. Debbie’s patience and positive attitude
are incredible, and among the many things that I’ve learned from her are how to ask the rightquestions and how to break hard problems into the simplest, most important pieces. Jun has anincredible passion for physics, and his infectious optimism consistently kept my spirits high, evenwhen it seemed like the experimental progress had come to a screeching halt. I also want to thankEric Cornell, who gave us many useful suggestions over the years. I’m continually amazed by his
intuition for experimental physics and his ability to find creative solutions to problems that might
initially sound crazy, but almost always work!
When I first came to the lab, I worked with Brian Neyenhuis, Amodsen Chotia, and Marciode Miranda, who taught me the nuts and bolts of the experiment. I also overlapped briefly with
Dajun Wang and Silke Ospelkaus, and I have kept in contact with them from time to time asthey’re setting up their own polar molecule experiments. After my first year, I met Kang-KuenviNi, and since then I’ve had many fruitful physics discussions with her. Around the same time, Bo
Yan joined the project as a postdoc and Jake Covey joined as a grad student. A year later, Bryce
Gadway arrived. Bo had an amazing work ethic and could fix anything. Bryce had an encyclopedic
knowledge of atomic physics, was an expert at Mathematica, and could always get interesting dataout of the experiment. Jake spearheaded the design and construction of the new experiment, andI’ve learned a lot about designing and building experiments from working with him. I’m grateful forall of their hard work and dedication, which were instrumental for many of the results presented inthis thesis. More recently, I’ve had the pleasure of working with Matt Miecnikowksi and Zhengkun
Fu. I trust the experiment is in good hands with Jake and Matt, as they start to get the new
system working.
Mike Martin, Matt Swallows, and Jan Hall were very helpful when I was setting up the cavity
for the Raman lasers. The staff in the machine shop and electronics shop (especially Hans Green,
Carl Sauer, and Terry Brown) were incredibly helpful at various points of the experiment, andthe computing department (especially J.R. Raith) helped us with any computing issues we had. I
would also like to thank Sara Campbell, Ben and Ruth Bloom, Xibo Zhang, Craig Benko, Bryce
Bjork, Brian Lester, Adam Kaufman, Ross Hutson, Toby Bothwell, Wei Zhang, Matt Grau, Will
Cairncross, Ed Marti, Shimon Kolkowitz, Tim Langen, Hao Wu, Dave Reens, Alejandra Collopy,
Matt Hummon, Mark Yeo, Phil Makotyn, Dan Lobser, Yoav Sagi, Tara Drake, Ming-Guang Hu,
Roman Chapurin, Cathy Klauss, Xin Xie, Kevin Cossel, Dan Gresh, Sebastian Blatt, Tom Allison,
Arman Cingoz, Brian Sawyer, Travis Nicholson, Oliver Heckl, and all of the other members of theJin, Ye, and Cornell groups with whom I’ve overlapped.
I had the pleasure of working and interacting with many great theoretical physicists at JILA
and beyond, including Ana Maria Rey, John Bohn, Murray Holland, Paul Julienne, Svetlana Kotochigova, Michael Wall, Kaden Hazzard, Martin Gärttner, Arghavan Safavi, Bihui Zhu, Johannes
Schachenmayer, Goulven Quéméner, Michael Foss-Feig, Stefan Natu, and Norman Yao. Thanks
for being patient and answering my questions, however silly they may have been.
Rabin Paudel, Adam Reed, Bryce Bjork, and Karl Mayer are great friends, with whom I’ve

vii
had many great discussions and enjoyed and brewed many beers. Going into the mountains helped
me to maintain a healthy work-life balance. When I moved to Boulder, I never imagined that I
would one day be running up mountains, biking centuries, and skiing double blacks. But I had agreat group of people to do stuff with, and we constantly pushed each other to be our best and doincreasingly harder things. In particular, I would like to thank Karl, Adam, Bryce, Carlos Pinilla,Paul Ackerman, Dan Palken, Adam Keith, Phoebe Tengdin, Brad Pelz, Dhruv Kedar, Scott Miller,
and all of the other people who I’ve adventured in the outdoors with.
Finally, I would like to thank my family, who have constantly supported me throughout theyears. Their sacrifices during the early years of my education helped pave the way to where I am
today.

Contents

Chapter
1 Introduction
1.1

1Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Apparatus and techniques

4
6

2.1

The vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

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

7

2.2.1

MOT lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2.2

Raman lasers and cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3

Optical trap/lattice lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3

Electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1

2.4

Electric-field problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Optical lattice techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1

Band structure and eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2

Dependence on beam parameters . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3

Avoiding superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.4

Calibrating the lattice depths . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.5

Band mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.6

Generating the lattice beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

ix
3 Chemical reactions and the Zeno effect

38

3.1

Brief summary of molecule production . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2

Rotational states and driving rotational transitions . . . . . . . . . . . . . . . . . . . 41

3.3

Chemical reactions in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4

Suppressing chemical reactions in a 1D lattice . . . . . . . . . . . . . . . . . . . . . . 45

3.5

Shutting off chemical reactions in a 3D lattice . . . . . . . . . . . . . . . . . . . . . . 47

3.6

Long-lived Feshbach molecules and conversion of preformed pairs . . . . . . . . . . . 50

3.7

Quantum Zeno effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.8

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Observation of spin exchange

59

4.1

Deriving the dipolar Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2

Anisotropic polarizability and differential light shifts . . . . . . . . . . . . . . . . . . 71

4.3

Simple Ramsey sequence and some technical issues . . . . . . . . . . . . . . . . . . . 74

4.4

4.5

4.6

4.3.1

Effect of the inhomogeneous light shift . . . . . . . . . . . . . . . . . . . . . . 77

4.3.2

Imaging both spin states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.3

Creating phase stable microwaves . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.4

Experimentally measured Ramsey decays . . . . . . . . . . . . . . . . . . . . 82

Initial experiments in the {|0, 0i, |1, −1i} manifold . . . . . . . . . . . . . . . . . . . 82
4.4.1

WAHUHA pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4.2

Robustness to imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Second set of experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5.1

Contrast decay for the |0, 0i → |1, 0i transition . . . . . . . . . . . . . . . . . 88

4.5.2

Fitting to multiple frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.5.3

Theory comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Future ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6.1

Simple simulation of spin transport . . . . . . . . . . . . . . . . . . . . . . . . 93

x
4.6.2

Realizing additional terms in the Hamiltonian . . . . . . . . . . . . . . . . . . 95

5 Increasing the filling fraction
5.1

5.2

5.3

97

Target distributions: Rb Mott insulator and K band insulator . . . . . . . . . . . . . 101
5.1.1

Rb Mott insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.1.2

K band insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.1.3

Challenges of making a dual insulator . . . . . . . . . . . . . . . . . . . . . . 106

Top imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.1

Calibrating the saturation intensity . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2.2

Fitting the atomic distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Measuring the filling of the individual atomic gases . . . . . . . . . . . . . . . . . . . 117
5.3.1

Measuring the filling of the Rb Mott insulator. . . . . . . . . . . . . . . . . 117

5.3.2

Measuring the filling of the K band insulator . . . . . . . . . . . . . . . . . . 120

5.3.3

Using inelastic loss to measure the K filling . . . . . . . . . . . . . . . . . . . 122

5.4

Studying the dual insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.5

Putting all of the ingredients together . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.6

Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Studying doublons in the lattice

133

6.1

Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2

Effect of the d-wave resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3

Detrimental effects due to tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7 Conclusions and future work

143

xi
Bibliography

147

Appendix
A Deriving the dipolar Hamiltonian

159

xii

TablesTable
2.1

Polarizabilities and lattice depths of K, Rb, and KRb at 1064 nm. . . . . . . . . . . 33

4.1

Coupling constants of the dipolar Hamiltonian . . . . . . . . . . . . . . . . . . . . . 68

xiii

FiguresFigure2.1

Rb laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

K laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Energy levels and timing diagram for STIRAP . . . . . . . . . . . . . . . . . . . . . 12

2.4

Comparison of wavelength tuning of Littrow and Littman configurations for an Eagleyard 980 nm laser diode9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5

Optics layout for the Raman lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6

Pictures and diagrams of the Raman laser cavity . . . . . . . . . . . . . . . . . . . . 18

2.7

Scheme for bridging the frequency difference between the cavity modes and the
STIRAP transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.8

Measuring the free spectral range of the cavity . . . . . . . . . . . . . . . . . . . . . 20

2.9

Ringdown measurements to measure the cavity finesse . . . . . . . . . . . . . . . . . 21

2.10 Beatnote of the down leg master laser with the Sr red MOT laser at 689 nm . . . . . 22
2.11 Electric-field problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.12 Band structure in the lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.13 Bloch and Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.14 Calculation of J and U for an isotropic 3D lattice . . . . . . . . . . . . . . . . . . . . 29
2.15 Illustration of how we get superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.16 Band mapping in the harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.17 Different configurations for the optical trap and lattice beams . . . . . . . . . . . . . 37

xiv
3.1

Anisotropy of dipolar collisions in an applied DC electric field . . . . . . . . . . . . . 39

3.2

Chemical reactions in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3

Loss rate for a mixture of |0, 0, −4, 1/2i and |1, −1, −4, 1/2i molecules . . . . . . . . 44

3.4

Chemical reactions in the 1D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5

Lifetime of ground-state molecules in the 3D lattice . . . . . . . . . . . . . . . . . . . 49

3.6

Going from 2D to 3D lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.7

Lifetime of Feshbach molecules after removing all unpaired atoms . . . . . . . . . . . 52

3.8

Experimental setup for studying the continuous quantum Zeno effect . . . . . . . . . 54

3.9

Lattice parameters for the Zeno experiment . . . . . . . . . . . . . . . . . . . . . . . 56

3.10 Extracting the filling fraction from the loss measurements . . . . . . . . . . . . . . . 56
4.1

Schematic of spin exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2

Induced dipole moments of the N = 0 and N = 1 states . . . . . . . . . . . . . . . . 64

4.3

Electric field dependence of Jz and J⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4

Diagram showing the anisotropic couplings in the lattice . . . . . . . . . . . . . . . . 70

4.5

Polarizability of the N = 0 and N = 1 states at 1064 nm . . . . . . . . . . . . . . . . 72

4.6

Differential light shifts vs. lattice depth . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.7

Calculation of how exchange is suppressed by site-to-site energy shifts . . . . . . . . 76

4.8

Timing diagram for Ramsey and spin echo sequences and typical Ramsey fringe . . . 76

4.9

Kinetics imaging for ground-state molecules . . . . . . . . . . . . . . . . . . . . . . . 80

4.10 Setup used to generate phase stable microwaves for driving rotational transitions . . 80
4.11 Typical Ramsey contrast decay for |0, 0i → |1, −1i and |0, 0i → |1, 0i transitions . . 81
4.12 Contrast decay for the {|0, 0i, |1, −1i} manifold . . . . . . . . . . . . . . . . . . . . . 84
4.13 WAHUHA sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.14 A variant of the WAHUHA sequence where the inner π pulse is about ŷ instead of x̂

87

4.15 Looking at the effect of imperfect π pulses and less magic traps on the contrast decay 87
4.16 Comparing contrast decay for | ↑i = |1, −1i and | ↑i = |1, 0i . . . . . . . . . . . . . . 90

xv
4.17 Comparing one- and three-frequency fits . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.18 Theory comparsion for many-body dynamics . . . . . . . . . . . . . . . . . . . . . . 92
4.19 Simulating spin transport in a randomly filled 2D array . . . . . . . . . . . . . . . . 94
5.1

Ramping on and off the lattice causes very little heating for K, but significant heating
for ground-state molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2

Image of a Feshbach molecule cloud with a hole in the center due to the initial Rb
density being too high . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3

Quantum synthesis for realizing high molecule fillings in the lattice . . . . . . . . . . 101

5.4

Images showing the disappearance of the interference pattern in momentum space
as we increase the lattice depth beyond the superfluid-to-Mott insulator transition . 103

5.5

Calculating densities of Fermi gases in the lattice in the limit of zero tunneling . . . 107

5.6

Comparing the densities of degenerate Bose and Fermi gases in a harmonic trap . . . 109

5.7

Issues with the side imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.8

Problems caused by the ITO plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.9

New setup for top imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.10 Typical calibration of I/Isat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.11 Calculated distributions of the Mott insulator for typical trap parameters . . . . . . 119
5.12 Peak filling of the Rb Mott insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.13 Peak filling of K and K temperature vs. K number . . . . . . . . . . . . . . . . . . . 121
5.14 Band mapping of the K gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.15 Hole in the K distribution after undergoing inelastic loss with Rb . . . . . . . . . . . 123
5.16 Using loss induced from spin-changing collisions to measure the K filling fraction . . 124
5.17 The dependence of the Mott insulator filling on interspecies interactions . . . . . . . 126
5.18 The Rb filling vs. interspecies interactions for different K temperatures . . . . . . . . 127
5.19 Conversion efficiency of making Feshbach molecules vs. Rb number and in situ imagesof ground-state molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xvi
5.20 The final molecule filling fraction for different lattice loading procedures . . . . . . . 129
6.1

Experimental scheme for studying the conversion of preformed pairs . . . . . . . . . 135

6.2

Effect of the d-wave resonance on molecule production. . . . . . . . . . . . . . . . . . 137

6.3

Probability to be adiabatic with respect to the s-wave and d-wave Feshbach resonances in the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.4

Conversion of doublons vs. interaction strength and lattice depth . . . . . . . . . . . 141

7.1

Pictures of KRb generation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Chapter 1

Introduction

One of the major thrusts of modern atomic, molecular, and optical (AMO) physics researchis to use systems of cold atoms or molecules to create novel quantum systems that don’t exist innature. The scientific insights gained from studying these systems are useful on their own, but
often can be applied to other areas of physics, such as condensed matter physics [1, 2]. AMO
systems can be probed and manipulated in a straightforward and clean way. The quantum control
of AMO systems has seen remarkable progress in the last few decades, and now experiments cancontrol nearly all degrees of freedom, both internal and external, of the particles. This makes
AMO systems ideal for simulating other quantum systems that are difficult to study otherwise[3, 4], and for making very precise measurements [5, 6, 7, 8]. The immense progress in controlling
AMO systems really took off with the development of laser cooling [9, 10] and evaporative cooling[11], which led to the first realization of degenerate gases of bosons [12, 13, 14] and fermions [15].
More recently, quantum gas microscopes for bosons [16, 17] and fermions [18, 19, 20, 21] havebeen realized that can probe individual atoms in an optical lattice. Many other platforms, suchas trapped ions [22] and superconducting qubits [23], are quickly building up the infrastructure to
perform complex quantum simulations and computations.
Systems of cold neutral atoms are usually very dilute gases and normally weakly interacting.
The strongly interacting regime is highly nontrivial and relevant to many interesting phenomena
in condensed matter physics. There have been many recent experiments looking at the stronglyinteracting regime [24, 25, 26, 27, 28]. For alkali atoms, the main tool used to control the interaction

2strength is the Fano-Feshbach resonance, which is reviewed nicely in Ref. [29]. Even with this tunability, the particles are still effectively pointlike and interact via isotropic contact interactions thatare well parametrized by a quantity called the scattering length a. This doesn’t mean that neutral
atoms cannot be used to study interesting phenomena. For example, the Fermi-Hubbard model,
which is a leading candidate for the underlying mechanism of high-Tc superconductivity [30, 31],
can be realized with cold atoms in optical lattices [32, 33]; however, one of the main issues withusing cold atoms to study the Hubbard model is that the energy scales are weak and the timescales
for observing interesting dynamics are relatively long [34, 35, 36]. Systems with long-range interactions open up the possibility of stronger interactions (and correspondingly faster timescales). In
addition, systems with long-range interactions could give insight into different physics questions,such as how thermalization and localization work in an isolated quantum system with both disorderand long-range interactions [37], and how correlations spread in a many-body system with longrange interactions [38, 39] (which is relevant for understanding how quickly quantum information
can propagate through a system [40]).
There are many ways to realize long-range interactions in the laboratory. Some examples
include coupling to phonon modes in trapped ion systems [41], placing atoms in optical cavities [42],using highly excited Rydberg atoms [43], or integrating cold atoms with nanophotonic structures[44]. This thesis discusses long-range interactions that arise from dipole-dipole interactions, which
originate from interacting electric or magnetic dipoles. For typical electric and magnetic dipole
strengths, electric dipoles are stronger by a factor of ∼ 1/α2 , where α is the fine-structure constant.
Highly magnetic atoms (such as Cr, Er, and Dy) are relatively easy to cool, and both degenerate
Bose and Fermi gases of highly magnetic atoms have been produced [45, 46, 47, 48, 49, 50]. In
contrast, polar molecules usually have stronger interactions but are more difficult to manipulate
and cool [51, 52, 53, 54, 55, 56, 57].
There has been a lot of experimental work trying to directly cool polar molecules. Somecommon techniques for cooling molecules starting from room temperature include buffer gas cooling[58], Stark deceleration [59, 60, 61], and making a magneto-optical trap (MOT) of polar molecules3[56, 57]. In the case of the MOT, only certain diatomic molecules can be cooled [62]. There aresome exotic techniques being employed to cool polyatomic molecules using Sisyphus cooling [63] ora centrifuge decelerator [64]. An alternative approach, which is the method we employ, is to takevery cold atoms, in our case 40 K and 87 Rb, first create weakly bound molecules using a Feshbachresonance [65, 66], and then optically transfer the weakly bound molecules to the rovibrational
ground state using Stimulated Raman Adiabatic Passage (STIRAP) [51]. This technique produces
molecules with a phase space density that is several orders of magnitude higher than the othermethods but restricts the possible molecules that can be produced to those molecules comprised of
atoms that can be easily laser cooled and for which Feshbach resonances exist [51, 52, 53, 54, 55].
Practically speaking, this usually means bialkali mixtures, although some groups are using an
alkaline earth atom in place of one of the alkalis. An alternative option to the Feshbach association
is to use STIRAP or photoassociation to first transfer pairs of atoms on individual sites of an
optical lattice into weakly bound molecules, and this has worked reasonably well in specific cases,
particularly Sr2 [67, 68]. However, for most of the physics that we want to study, the actual
atomic species involved are not so important, as our main requirement is that the molecules havea sufficiently large dipole moment. Due mainly to our group’s work over the past ten years, acommon theme throughout this thesis is that most of the hard work is done in cooling the atoms,
and the molecule production is the final step that usually works quite well (there are some caveats,
which are discussed in Chapters 5 and 6).
After producing the molecules [51], it was found that KRb suffers from inelastic loss due tochemical reactions [69, 70]. In the next few years, work focused on using optical lattices to suppress
this chemical loss [71, 72]. After stabilizing the gas in the 3D lattice, we embarked on a series ofexperiments looking at coherent manifestations of the dipole-dipole interaction. There have been
many proposals for realizing lattice Hamiltonians with multiple rotational states [73, 74, 75, 76, 77,
78, 79, 80, 81, 82]. In particular, Ref. [83] predicted we should be able to see signatures of dipolarinteractions even at relatively high entropies and low lattice fillings. By mapping the two lowest
rotational states to a spin-1/2 system we realized an XY Hamiltonian, which is one of the first4
ingredients necessary for studying quantum magnetism with polar molecules [84, 85]. Models ofquantum magnetism, such as the XY model, are some of the simplest yet ubiquitous models thatare computationally hard to handle and are impossible to simulate on a classical computer evenfor as few as 30 spins [38]. There are a host of other interesting phenomena that polar molecule
experiments could potentially study, such as Wigner crystallization [86], d-wave superfluidity in
optical lattices [87], and novel forms of spin-orbit coupling [88]. Open-shell polar molecules could
potentially be used to engineer topologically ordered states that could enable topologically protected
quantum memories [73, 89].
An important question is whether our experiments are actually doing quantum simulation.
After all, the results of our spin-exchange work [84, 85] could be simulated reasonably well using
a cluster expansion [85]. If our experiment could achieve better imaging resolution and we could
measure spin correlations between individual lattice sites, we should be in a position to measuresomething that would be classically intractable to simulate. However, even to simulate the global
quench experiment, the theory had to be refined and improved in order to converge quickly enough[85]. For our system of polar molecules, the theory and experiment build off of one another in a
symbiotic way. The validity of the improved theoretical methods is checked by comparison withthe experimental results. The theory can then be used to make further predictions. It is in thissense that our experiment is pushing the field forward and hopefully leading to bigger and better
results. In general, most experiments claiming to be doing quantum simulations are still in thebenchmarking stage, but benchmarking the results of small systems with theory is very important
before attempting to study something really nontrivial.

1.1

Contents
In Chapter 2, I describe the apparatus and the upgrades and changes made since I’ve been at

JILA, as well as some relevant experimental techniques. In Chapter 3, I discuss chemical reactions
in KRb and how we are able to control them. As an application of the chemical reactions, we studied
the continuous Zeno mechanism in a rotational mixture [90]. In Chapters 4 and 5, I present the5main results of this thesis. In Chapter 4, I discuss our work in realizing a spin-1/2 system with polar
molecules and then engineering one of the basic ingredients of quantum magnetism, spin exchangeinteractions. We observed spin exchange between molecules pinned in a deep 3D optical lattice,even though the lattice fillings were quite dilute. A systematic study was undertaken that studied
the density dependence, dependence on choice of excited state, and implementing more complex
pulse sequences that disentangled isolated pairs. In Chapter 5, I document a detailed study of themolecular filling fractions we can obtain in the 3D optical lattice. By loading a Mott insulatorof bosons and a band insulator of fermions into the optical lattice, we optimized the number oflattice sites that have one atom of each species. Combined with efficient magnetoassociation andoptical state transfer, this led to significantly higher molecular filling fractions. The achieved filling
fractions of ∼ 25% should enable future studies of transport and entanglement propagation ina many-body system with long-range dipole-dipole interactions. In Chapter 6, I present some
additional studies related to the molecule association in the 3D lattice. In particular, we observed
tunneling dynamics that were strongly modified by strong attractive interspecies interactions, aswell as effects due to a d-wave Feshbach resonance that’s about 1 G above the s-wave resonance
usually used to make molecules. In Chapter 7, I conclude and give an outlook to future work.
The work described in this thesis is published in several papers, and I am an author on severalof them. In particular, Chapter 3 discusses work presented in Refs. [72, 90], Chapter 4 discusses
work presented in Refs. [84, 85, 91], Chapter 5 is based on Ref. [92], and Chapter 6 follows Ref. [93].

Chapter 2

Apparatus and techniquesThis chapter describes the ultracold molecule apparatus, with a particular emphasis on thecomponents that were built or modified during my time at JILA. Most of my time here was spent
doing experiments with the first generation apparatus, which was started by Josh Zirbel and KangKuen Ni in 2003. Except for when we broke vacuum in 2012 to change the science cell in an attempt
to solve the electric-field charging problem, the vacuum chamber itself wasn’t really modified andthere were no significant vacuum issues. However, many other things on the experiment (laser
systems, electronics, etc.) have been modified. A few years ago we started thinking about building
a new generation of the experiment to address many technical limitations of the first generation
experiment. This effort was spearheaded by Jake Covey, and in October 2015, we switched to thesecond generation apparatus.

2.1

The vacuum system
Similar to many of the other quantum gas machines at JILA, our vacuum chamber consistsof two parts separated by a gate valve: a higher pressure MOT chamber and a lower pressurescience chamber. The vacuum lifetimes are a few seconds in the MOT chamber and a few hundred
seconds in the science chamber, implying the pressure is about two orders of magnitude lower inthe science chamber. In contrast, many other experiments have a single chamber utilizing an ovenand an atomic beam fed from a Zeeman slower. In our experiment, atoms are transferred between
the two chambers via a cart with coils that provide a strong quadrupole field (these are the same

7
coils that produce the quadrupole field for the MOT). There are two ion pumps, one for the MOT
chamber and another for the science chamber, and there is also a titanium sublimation pump near
the science cell. The gate valve between the two sides allows one to break vacuum on one side
without affecting the other side.

2.2

Laser systems
Many lasers are required to cool and trap two atomic species, and then create molecules. The

lasers can be divided into three categories: cooling lasers which we use for cooling and imaging K
and Rb, Raman lasers for the STIRAP transfer to the ground-state, and the optical trap/latticelasers.

2.2.1

MOT lasers

When I first came to the lab in 2010, the laser systems for trapping and cooling Rb and K
were less robust than we would desire. Many of the lasers were old and the beam height was about
6 inches, which meant that beams had to be realigned quite often. The trapping light was derived
from external cavity diode lasers (ECDLs), some of which weren’t AR coated, and a slave laser wasused for both K and Rb since the power of the master lasers wasn’t sufficient to inject a tapered
amplifier (TA). The K slave laser was actually a 780 nm diode that was cooled to -40◦ C. Also, thelasers were on the same table as the main experiment, so there were a lot of issues due to vibrations
from the cart and shutters.
To make the system more robust, I decided to rebuild the cooling diode lasers from scratch.
We decided to replace the ECDLs with distributed Bragg reflector (DBR) lasers from Photodigm.
The K laser diodes have a maximum power of 60 mW, while the Rb laser diodes have a maximum
power of 120 mW, so no slave lasers are required. Because the feedback is provided by a distributed
Bragg grating integrated on the diode, there is no external cavity, and the mode-hop-free tuning
range is significantly higher than that of an ECDL, typically > 50 GHz. The downside is that the
linewidth is about 1 MHz, which is larger than an ECDL but adequate for cooling alkali atoms.

8
Because there is no external cavity, the laser is incredibly sensitive to optical feedback, and we need
two optical isolators to provide at least 60 dB of isolation. The beam height is roughly 2 inches
and the system rarely needs realignment.
The repump lasers for both K and Rb are locked to an absorption cell using frequencymodulation (FM) spectroscopy. A beam of a few hundred µW is phase modulated at around 20
MHz with an electro-optic modulator (EOM), and then split into a strong pump beam and weakprobe beam. We then do standard saturated absorption spectroscopy. The error signal is obtained
from demodulating the probe beam transmission with a phase shifted copy of the same tone used
to drive the EOM. This technique is described in Ref. [94] and is similar to Pound-Drever-Hall(PDH) locking, which will be discussed later.
The schematic of the Rb laser system is shown in Fig. 2.1. We lock the repump laser to the
F = 1 to F 0 = 1/2 crossover. This is a fairly weak transition, and we heat the Rb vapor cell to
get a better signal-to-noise ratio (SNR). For both K and Rb, the trap lasers are frequency locked
to the repump lasers using a frequency-to-voltage (F-V) converter. We chose this option becauseof the large servo bandwidth that would be necessary to phase lock the DBR lasers to each other.
The frequency offset is controlled by an RF frequency derived from a voltage controlled oscillator(VCO). The F-V converter is set up so that a frequency of around 30 MHz corresponds to 0 V (theuseful range of the device is about 0-70 MHz). We also find it useful to put a 50 MHz low passfilter on the input to the F-V converter.
For K, we lock the repump laser to a 39 K absorption line since the absorption cell contains
only the natural abundance of 40 K, which is ∼ 0.01%, whereas it has about 93% 39 K. Specifically,
we lock to the F = 2 to F 0 = all feature, since the hyperfine structure is not resolved in 39 K. To shift
the laser to the right frequency for 40 K, we use a 345 MHz acousto-optic modulator (AOM). In theold experiment, we derived two additional beams from the repump laser, the high-field imaging lightand the |9/2, −5/2i blast light. The high-field imaging, which is on the |9/2, −9/2i → |11/2, −11/2i
cycling transition, is about 770 MHz lower in frequency at 550 G than at zero magnetic field, so
it can be accessed from the repump laser. For the experiments described in Chapters 5 and 6, we9Figure 2.1: Rb laser system. AOM 1 shifts the frequency for the repump light used during opticalpumping and high-field imaging, when it pumps atoms from the F = 1 manifold to the F = 2
manifold. AOM 2 undoes the frequency shift of AOM 1 and allows us to reduce the power of therepump light going to the MOT, which is useful during the CMOT and molasses stages. AOM 3
gives the frequency shift and switches the light for the Rb probe light, optical pumping, fluorescencepulse, and high-field imaging. To adjust the laser frequency for these different processes, thebeatnote frequency between the trap and repump lasers is changed. AOM 4 is used to servo thetotal MOT intensity (which is mostly trap light). All beamsplitters are polarizing unless noted
otherwise. The AOM orders are listed next to the beams after the AOM. All focal lengths are
given in mm.

10wanted to be able to image K at any magnetic field within 10-15 G of the Feshbach resonance.
For this purpose, the original high-field imaging was insufficient since the frequency could only betuned by a few MHz without requiring realignment. To solve this problem, we implemented anoffset lock with a wider tuning range by switching to a VCO with larger bandwidth (part number
ZX95-1300-S+ from Mini Circuits, which has a range 400-1300 MHz). With this expanded range
for the offset lock, we were then able to derive the high-field imaging light from the trap laser.
At one point, we used 41 K instead of 40 K. For this, we offset locked both DBR lasers to a master
laser, which was an ECDL (this is the “third” laser in Fig. 2.2). In general this is the most flexible
approach, and even though it requires one more laser, fewer AOMs (just 2 as opposed to 4 or 6)
would be required.

2.2.2

Raman lasers and cavity

Stimulated Raman adiabatic passage (STIRAP) is used to transfer weakly bound Feshbach
molecules to the ground state [95]. It’s a Raman process, and the two colors we need (689 and 968nm) are accessible with modern laser diode technology. The relevant energy levels are sketched inFig. 2.3a. We start in the Feshbach state |f i and couple through a lossy excited state |ei (withdecay rate γ) that has decent Franck-Condon factors with both the Feshbach state and the groundstate |gi. We apply two laser fields: the up leg (with Rabi frequency Ωu ), which couples |f i and|ei, and the down leg (with Rabi frequency Ωd ), which couples |gi and |ei. The goal of STIRAP uis to coherently maintain the dark state cos θ|f i + sin θ|gi, where θ = tan−1 ΩΩd . There are two
relevant detunings, the single photon detuning ∆ and the two-photon detuning δ. We normallyoperate with ∆ = δ = 0, but it’s more imperative that δ be zero for efficient transfer. If we choosea pulse sequence where initially Ωd /Ωu  1 and then adiabatically change the intensities so thatat the end Ωu /Ωd  1 (Fig. 2.3b), we can transfer most of the population from |f i to |gi without
populating the lossy state |ei. Experimentally, we define a pulse time τ as the time duration during
which the intensity ramping occurs. For more details about STIRAP, see the thesis of Kang-Kuen
Ni [96].

11

Figure 2.2: A schematic of the K laser system. Similar to the Rb system, there is a trap and repump
laser, as well as a tapered amplifier. AOM 1 is at 345 MHz and is used to bridge the frequency
difference between the 39 K and 40 K transitions (we lock the laser to one of the 39 K transitions).
AOM 2 is for the optical pumping repump. We also use this AOM to control the trap/repump
power ratio for the MOT light. AOM 3 is for the probe and optical pumping beam. AOM 4 is forstabilizing the total power of the MOT. AOMs 5 and 6 provide the switch and frequency shift for the
K |9/2, −5/2i → |11/2, −5/2i blast light and the K high-field imaging |9/2, −9/2i → |11/2, −11/2i.
All beamsplitters are polarizing unless noted otherwise. The third laser was an ECDL locked to
one of the 39 K transitions, which we used as a reference when trapping and cooling 41 K.

12

(a)

(b)

|e>

Field Strength

Δ

γ

Ωu

Ωu

Ωd

Ωd
|f>

δ
τ

time

|g>

Figure 2.3: (a) The energy levels relevant for STIRAP are the Feshbach molecules |f i, the lossy
excited state |ei (with decay rate γ) and the ground state |gi. Usually we set the single-photon
detuning ∆ = 0 and we scan the two-photon detuning δ by adjusting the up leg double-pass AOM,
driven at frequency fDP . (b) Timing diagram for STIRAP. Initially Ωd  Ωu . We then ramp
Ωd to 0 as we ramp Ωu to its maximum value. Here a linear ramp in field strength is shown. In
practice, we usually ramp the intensities linearly. The pulse time τ is the duration of the ramps.
To STIRAP back to Feshbach molecules, we reverse the order of the ramps (start with Ωd at zero,Ωu at the maximum value, and then ramp Ωd to the maximum value and Ωu to zero).

13
To maintain the dark state and transfer a significant fraction of the population, it’s necessary
for the two laser fields to be phase stable during the pulse time τ . More precisely, the transfer
22efficiency scales as e−π γ/(Ω0 τ )−ητ /2 , where η is the relative linewidth, τ is the pulse time, andqΩ0 = Ω2u,max + Ω2d,max [97]. For efficient STIRAP, we require ητ  1 and γ/(Ω20 τ )  1. For us,γ = 2π × 6.7 MHz [96], Ω0 is typically 2π × 6 MHz, and τ is usually 6-10 µs. Plugging these in,
the first factor is ∼ e−0.05 = 0.95, and the relative linewidth should be less than 1 kHz in order tonot be limited by the laser linewidth. In the work of Ref. [92], the one-way transfer efficiency wasroutinely 0.9-0.95 for cold, small samples.
To get a sense of the optical powers required for STIRAP, it’s necessary to know both thetransition dipole moments and the sizes of the beams. The beams are focused to roughly the same
size as the optical trap beams (200 × 40 µm in the radial and vertical directions, respectively) to
give roughly homogeneous coupling over the cloud while still getting a reasonably large intensity.
The transition dipole moments are 0.005(2) ea0 for the up leg and 0.012(3) ea0 for the down leg [96].
These are substantially weaker than typical cycling transitions in alkali atoms (∼ 1 ea0 ). We wantto achieve Rabi frequencies of a few MHz, which means we need powers in the range 20-100 mW. As
a result, the power from the master lasers is insufficient, so we need some additional amplification.
For the up leg, we use a laser diode operating at 968 nm. Currently, this is an Eagleyard AR-coated
980 nm laser diode in a Littrow configuration (see Fig. 2.4 for a discussion of how these lasers don’t
perform well in the Littman configuration) along with an Eagleyard 970 nm TA. For the down leg,
we use an AR-coated laser diode operating at 689 nm (again in the Littrow configuration), alongwith a slave laser (HL6750MG), which is optically injected.
When I first came to the lab, we were stabilizing both of the Raman lasers to a Ti-sapphirefrequency comb [98]. While the comb is useful for the initial spectroscopy, it can be a bit cumbersome on a daily basis. We stabilized both degrees of freedom of the comb, the repetition rate frep
and the carrier offset frequency fceo . To stabilize frep , we locked one comb tooth to Jan Hall’s 1064
nm stable laser, which was stabilized to an iodine cell. To stabilize fceo , we frequency doubled thepart of the comb around 1064 nm and beat it with the part of the comb at 532 nm. Thus, we needed14Power (mW)

980 laser diode range
60 Serial number AD-18597
55
50
45
40
35
red: Littman
30
blue: Littrow (~107 mA)
25
20
15
10
best performance5
around 943 nm
0
900
920
940
960
980

1000

Wavelength (nm)Figure 2.4: We could never get the Eagleyard laser diodes to work well in the Littman configuration.
Here we compare the performance of an Eagleyard 980 nm AR-coated diode in the Littrow and
Littman configurations. The power in the Littman configuration is lower (which is to be expected),
but the wavelength tuning range is smaller and the peak wavelength is shorter, such that the laser
doesn’t work well at our desired wavelength of 968 nm.

15
an octave spanning comb from 532 to 1064 nm and also required sufficient optical spectrum at 690
and 970 nm to obtain strong beatnotes for locking the Raman lasers. In principle, by phase locking
both lasers to the comb, the two lasers should be perfectly phase-locked to each other; however,the comb itself had a linewidth of about 1 kHz. Furthermore, the comb was located in a different
room from the main experiment, with ∼ 25-meter fibers sending the light from the lasers to the
comb and similar length BNC cables transmitting the error signal back to our room. We weren’t
canceling the phase noise induced by acoustic vibrations in the fibers, which could broaden each of
the lasers, and the long cables meant a long time delay and correspondingly low servo bandwidth.
This may have led to a mediocre relative linewidth. We would definitely see fluctuations in thetransfer efficiency from day to day depending on how well the lasers were locked, and some days
we spent the entire day getting the comb and lasers locked.
To make the experiment more reliable, we decided to replace the comb with a high finesse
cavity. This system is much more robust and usually it takes less than 5 minutes every day to get thelasers locked well. Most other groups making ultracold polar molecules have followed our example
and are using cavities for locking their Raman lasers. The cavity is comprised of a cylindrical pieceof Zerodur (a glass-ceramic with very small coefficient of thermal expansion) with a bore throughthe center, and fused silica substrates optically contacted on the ends. One of the mirrors is flat andthe other has a 50 cm radius of curvature, which ensures the transverse modes are not degenerate.
A highly reflective coating at both 690 and 970 nm gives finesses of several 104 , which are measured
via ringdown (Fig. 2.9). The coating was done by Advanced Thin Films, a company in Boulder.
The cavity is housed inside of an aluminum box and pumped down to about 10−6 torr (see Fig. 2.6).
We passively stabilize the temperature of this box to ∼ 10 mK, which gives day to day frequency
drifts of roughly 100 kHz (inferred from scanning the two-photon detuning of STIRAP), which isless than the typical two-photon width of STIRAP of 500 kHz. We scan the STIRAP lineshaperoughly once a day and rarely see any degradation in the transfer throughout the day due to thefrequency shifting. When we moved the lab from JILA’s S-wing to the new X-wing at the end of
2012, there seemed to be a small leak in the cavity chamber. Upon pumping down again, the cavity

16
seems to have slightly changed length or moved, since the resonant frequencies shifted by about 10
MHz.
We lock the lasers using standard PDH locking techniques [99, 100] to the TEM0,0 modes. In
both cases we fiber couple the lasers to the cavity platform to have a stable alignment and to getbetter mode matching into the TEM0,0 modes. To achieve the highest possible servo bandwidth,
we minimize the length of the cables from the current controller to the laser and from the servo tothe current controller. We achieve servo bandwidths of 2-3 MHz for the down leg laser and 1.5-2
MHz for the up leg laser (the lower servo bandwidth in this case seems to arise from the laser
diode itself since we could achieve higher servo bandwidths with different 970 nm laser diodes).
This servo bandwidth is much larger than we achieved when locking the lasers to the comb and iscomparable to the Rabi frequencies Ωu,max and Ωd,max . For the down leg, we observed that the
STIRAP efficiency was very sensitive to the servo gain. To avoid deleterious effects of the relaxation
bumps of the laser (which occur at the servo bandwidth), we optically inject the 690 slave laser
with the transmitted light from the cavity. Good spatial mode matching between the transmitted
beam and the slave laser beam is important in order to be able to reliably inject the slave laser
with less than 200 µW. In order to use the transmitted beam, it’s very important to have an opticalisolator after the cavity. Otherwise, a small amount of light from the slave laser is coupled into thecavity the wrong way and makes it all the way to the photodiode (PD 3 in Fig. 2.5). This lightinterferes with the normal reflection off of the cavity and writes noise on the error signal. For theup leg, we inject the TA with the unfiltered master light, as we didn’t observe a similar sensitivity
to the servo gain.
In general, there will be a frequency difference between the TEM0,0 cavity modes and thetransitions for STIRAP. The difference could be as large as νFSR /2, where the free spectral range
νFSR = c/2L is the spacing between longitudinal modes of the cavity (c is the speed of light and L
is the cavity length). We heat the chamber to stabilize the temperature, but since it’s ULE glass
we don’t get a large tuning range with temperature. To compensate for this frequency difference,
we use two AOMs for both lasers (see Fig. 2.7). For the up leg, the STIRAP transition is about 200

17

Figure 2.5: Layout for the Raman lasers. Both lasers are fiber coupled to the cavity platform.
The frequency difference between the up leg STIRAP transition and the cavity mode is bridged
by a double-pass AOM (AO2) and a single-pass AOM (AO1) of opposite orders, so that the total
difference is fSP + 2fDP . For the down leg, two single-pass AOMs of opposite orders bridge the verysmall (less than 10 MHz) frequency difference between the cavity mode and the STIRAP line. We
scan the two-photon detuning δ by changing fDP . PD1 is for the master-slave beatnote. PD2 is for
PDH locking for the up leg, and PD3 is for PDH locking for the down leg. On the cavity platform,
the two colors are combined/separated using dichroic mirrors. The 690 slave laser is injected bysending the transmitted light from the cavity through the isolator the “wrong” way.

18

Figure 2.6: The cavity spacer is a cylindrical piece of zerodur, which is supported on top of anotherpiece of ULE glass by four small oring pieces. This entire assembly sits inside of a vacuum chamberthat is pumped down to ∼ 10−6 torr and temperature stabilized to ∼ 10 mK. The picture in
the lower right shows how the chamber is currently installed in the lab. There are two layers oftemperature control and a lot of insulation. The drawings are courtesy of Hans Green and the
instrument shop.

19

Cavity
fDP

fDP
Laser

f1
Laser
Cavityf2
STIRAPfSP
STIRAP

Figure 2.7: Left: Scheme for bridging the frequency difference between the cavity and up leg
STIRAP line. Increasing the frequency of the double-pass AOM fDP reduces the laser frequency.
In Fig. 2.5, AOM 1 is driven at fSP and AOM 2 is driven at fDP . Right: Similar diagram for the
down leg. The STIRAP transition is within 10 MHz of the cavity mode. AOM 3 is driven at f1
and AOM 4 is driven at f2 .

950

1.29939(7) GHz

900
850
-60

-40 -20020Mode number (arb.)

620
Freq. (309.xxx THz)

Freq. (434.xxx THz)

1000

Measure FSR at 690 nm

40

Measure FSR at 970 nm

600

20

1.29956(8) GHz

580
560
540

0

20
40
Mode number (arb.)

Figure 2.8: Position of many TEM0,0 modes, using our wavemeter to measure the frequency (to 8digits). The measurements were taken on the same day to minimize any drifts of the wavemeter
calibration. Fitting lines, we extract νFSR , which is around 1.3 GHz.

MHz lower than the closest cavity mode, and this difference can easily be bridged by double-pass
and single-pass AOMs of opposite order. By scanning the RF frequency fDP of the double-pass
AOM, we scan the two-photon detuning δ. For the down leg, the difference between the cavity
mode and the STIRAP line is less than 10 MHz, so we use two single-pass AOMs of opposite order,
one that deflects the transmitted light from the cavity to the slave laser and the other that switches
the light to the main experiment.
To check the cavity is behaving as expected, I measured the FSR, the finesse, and the transmission on resonance. To measure the FSR of the cavity, I locked the laser to many different cavity
modes and recorded the frequency using our wavemeter. Although our wavemeter is only accurate
to about 100 MHz and only reads to the 10 MHz place (8 digits), on a given day the relative
difference between two nearby frequencies should be known to ∼ 10 MHz. Fig. 2.8 shows thesemeasurements for both colors, and the resulting νFSR = 1.3 GHz implies the cavity length is 11.5cm.
A typical way to measure the finesse F of the cavity is to do a ringdown measurement, inwhich the incident light to the cavity is instantaneously shut off, which can be achieved by switching
off an AOM. The light from the cavity will leave, but not all at once, since photons make roughly
F roundtrips before exiting the cavity. The power leaving the cavity decays exponentially with aCavity ringdown up leg

21

(c)

120

0.8
τ = 5.68(1) µs

0.6
0.4
0.2
0.0

Cavity ringdown down leg

2.5

100

PD signal (V)PD signal (V)

1.0

(b)
PD signal (mV)(a)

τ = 3.87(1) µs

80
60
40
20

0

5

10
15
Time (µs)

20

0

5

10
15
Time (µs)

20w=29.1(6) kHz

2.0
1.5
1.0
0.5
0.0
-200

-100
0
100
Detuning (kHz)200

Figure 2.9: Ringdown measurements at 968 nm (a) and 689 nm (b). From the time constants we
extract finesses of 4.6 × 104 and 3.2 × 104 at 968 and 689 nm, respectively. (c) The transmissionof the up leg when it was locked to the comb. The FWHM κ can be used as an independent4
measurement of the finesse, using F = νFSR
κ . The result is F = 4.5 × 10 , which is slightly lower
than the finesse extracted from the ringdown measurement.

time constant given byτ=FL→ F = 2πτ νFSR .
πc

(2.1)

This is shown in Fig. 2.9a for the up leg and Fig. 2.9b for the down leg. Alternatively, when thelasers were locked to the comb, I scanned the up leg across the cavity resonance (Fig. 2.9c). Thenby measuring the transmission spectrum, we can obtain the cavity linewidth κ through the equation
F = νFSR /κ. The two methods agree reasonably well, with the finesse extracted from the linewidth
measurement slightly lower than the finesse extracted from the ringdown measurement.
Since we want to optically inject the down leg slave laser with the transmitted light from thecavity, we care about how much light is transmitted on resonance. I measured Pt /Pin ≈ 0.25 forthe down leg, which means we need to send more than 1 mW to the cavity in order to get enoughtransmitted light to inject the slave laser. The transmission for the up leg is about 0.35.
Going back to the condition stated earlier that the relative linewidth should be less than 1
kHz for efficient STIRAP transfer, we’re interested in measuring the linewidth of our lasers. The
most direct way to do this is to build two ostensibly identical systems and then beat them against
each other. This is not really an option for us since we don’t have the resources to build anothercomplete cavity setup. A more indirect method is to use delayed self-heterodyne interferometry[101, 102], in which the laser is separated into two parts, one of which has a long delay (achieved

22

PD signal (mV)

2.5

Beat with Sr 689 nm laser

2.0
1.5

211(11) Hz

1.0
0.5
0.0
-600 -400 -2000

200 400 600

Relative detuning (Hz)Figure 2.10: The average of 10 single sweeps on the spectrum analyzer with resolution bandwidth
30 Hz. Each time the beat frequency was subtracted. The fit is a Lorentzian.

by using a long optical fiber) and then creating a beatnote between the two paths. We only havea 1 km fiber available, which is not really sufficient to reliably measure sub-kHz linewidths.
Conveniently, the down leg transition is within 100 GHz of the Sr red MOT transition. So
we decided to lock our laser to the nearest cavity mode to the Sr red MOT transition and beat ourlaser against the 689 nm Sr laser in the neighboring lab, which should have a linewidth  100 Hz.
Because the lasers are not referenced to each other, the beatnote frequency drifted over time, but
averaging 10 single sweeps on the RF spectrum analyzer, we measure a relative linewidth of around
200 Hz (see Fig. 2.10). Since our two Raman lasers are locked to the same cavity, their relative
linewidth should be at a similar level or even smaller, which is adequate for performing efficient
STIRAP.

2.2.3

Optical trap/lattice lasers

For most of the work described in this thesis, the optical trap (OT) and lattice lasers are
at 1064 nm. This is a convenient wavelength since high-power, low-noise lasers are commerciallyavailable, and the molecules have a long lifetime when exposed to 1064 nm light (1064 nm photons
have a lower energy than necessary to excite to the bottom of the excited molecular potential [72]).

23
We used several different types of 1064 nm lasers for the OT and lattice during my time at JILA,
including a Verdi-IR, an IPG fiber laser, and a Nufern fiber amplifier seeded by a Mephisto NPRO.
This latter combination was chosen because the Mephisto has a relatively narrow linewidth (∼ 1
kHz) and the Nufern fiber amplifier has low noise and a high gain (50 W for a 1-15 mW input).
Towards the end of the generation 1 apparatus, we were using the Verdi-IR for just one of the OT
beams. This is because the power of the Verdi-IR had deteriorated and so we couldn’t reliably get
all the power we needed for both OT beams, which is more than 1 W per beam at the beginningof the OT evaporation.
At one point the Mephisto laser that seeded our Nufern fiber amplifier became multimode,and this affected the lattice depths. We noticed this problem when we switched to using the Verdi
for the seed laser, and all of the lattice depths increased, even though the power and polarizationof the lattice beams didn’t change. When using the Mephisto, a large fraction of the light was at adifferent frequency, and presumably the phase of the two lattices was different. Either way, there
was likely some kind of superlattice and a large fraction of the light was not interfering, leading toa smaller lattice depth.

2.3Electric fields
We want to apply large DC electric fields to polarize the molecules in the lab frame andinduce dipole-dipole interactions. As will be discussed in Chapter 4, applying a DC electric fieldrealizes Ising-type interactions between the two lowest rotational levels. We need to apply large
fields (more than 10 kV/cm) to begin to saturate the dipole moment (see Fig. 4.2). Ideally, wewant very flat fields without having to sacrifice optical access. To do this, we use electrodes withtransparent indium-tin-oxide (ITO) coatings.

2.3.1

Electric-field problems

There were several limitations with the electric-field control in the old experiment. The layout
and geometry of the electrodes was very similar to that described in the thesis of Kang-Kuen Ni

24[96]. In short, the plates are arranged in a boxlike structure, with plates usually only on the top
and bottom, similar to the setups depicted in Fig. 2.11a. In 2012, we were trying to use an electricfield gradient to tilt the 2D traps formed from a vertically oriented 1D optical lattice. After doing
this for a while, we learned that there was a severe charging problem occurring in the glass walls
forming our cell, where an electric field opposite in direction to our applied field would build up
over time. We determined this charging field Ech was in the opposite direction of the applied fieldby looking at the shift in the dark resonance with small applied electric fields of both polarities, anddetermining which direction of applied field would bring the transition closer to the zero field value(this was the same polarity used to get to the charged condition). The presence of Ech manifested
in two ways: (1) a shift in the two-photon detuning of STIRAP, and (2) a shift in the microwave
transition frequency between the two lowest rotational states.
Since the ground-state is a strong-field seeking state, if we don’t change the frequency of thedown leg laser, the presence of an electric field means that in order for the STIRAP transfer tobe on resonance, we have to adjust the two-photon detuning by decreasing the frequency of theup leg laser (by increasing fDP ). When we were looking for charging we always did STIRAP withthe applied field off, so the electric field present during STIRAP is the charging field Ech . We
observed that running the experiment even just a few times (where the electric field was on for
500 ms or less out of a ∼ 60 second total duration) would cause the STIRAP lineshape to shift by
more than the two-photon linewidth (Fig. 2.11b). The maximum shifts we observed were 1-2 MHz,
which imply Ech = 500 − 1000 V/cm. The charging problem was much more pronounced at higher
applied fields. There was some hope that the field would be more stable at lower applied fields.
To test this, we drove a microwave transition between the N = 0 and N = 1 states at a modest
field of about 1 kV/cm. We saw that over a few hours, the frequency shifted, and the width of thefeature became broader (Fig. 2.11c), which implies some uncontrolled charging field even at these
low applied fields. From the magnitude of the shift, we inferred a shift of a few V/cm (out of 1
kV/cm) over the 1.5 hour experiment. We tried many things to get rid of this charging, such as
applying blue light and heating the cell, but these didn’t work. The charging on the cell would

25
relax over the time scale of ∼ 5 hours (Fig. 2.11d).

2.4

Optical lattice techniques
Optical lattices have become a common tool in AMO physics. On a very naı̈ve level, they’reanalogous to the lattice structure found in materials. One notable difference is that we don’t usually
have to worry about phonons in our system. The ability to tune the lattice strength and geometry
and the ability to add disorder are the main features that make optical lattices so useful for AMO
experiments, as most of these things cannot be controlled easily in solids. For us, optical lattices
serve a very practical purpose, which is to shield the molecules from chemical loss (see Chapter 3).
In this section, I briefly review the relevant aspects of lattice physics for our experiment.

2.4.1

Band structure and eigenfunctions

The eigenstates for particles in a periodic potential have the same periodicity as the lattice.
Instead of the free-particle quadratic dispersion, only certain energy bands are allowed, and atoms
occupy one of the allowed energy bands. The curvature of these bands determines the effective
mass of the particles in the lattice. The calculation of the band structure is presented in manyother places, including several theses [103, 104, 105]. Fig. 2.12a-c shows the dispersion for a few
2 2

klattice depths. The lattice depth U0 is expressed in units of the recoil energy ER = ~2m
by adimensionless number s = U0 /ER . In our case, we have a simple cubic lattice and k = 2π/λ = π/a,where a is the lattice spacing.
(n)

The eigenstates, which are the Bloch wavefunctions φq (x), are given byiqx/~ (n)φ(n)
uq (x),
q (x) = e(2.2)

where q is the quasimomentum (restricted to −π/a to π/a for the first Brillouin zone), n is the(n)band index, and uq (x) is a function with period a [103]. The Bloch functions are a convenientbasis for shallow lattices, since they are delocalized over the entire lattice, but in deep lattices the

26

(a)

(c)

(b)
+V

+V

+(V+δV)-V-V-(V+δV)(d)Figure 2.11: (a) Different configurations for applying electric-field gradients. A cross-section of
the glass cell is indicated by the blue rectangle. We tried a configuration where one of the plates
was tilted (left), and another configuration where the plates were flat but the electrical connection
between the two sides was broken to allow independent voltages on the two sides (right). (b)The STIRAP lineshape would shift even though there was no applied field during STIRAP. Afterrunning the experiment for 5 shots the transition shifted by more than the two-photon linewidth(from the black curve to the red curve). (c) The microwave transition between the (molecular)|0, 0i and |1, 0i states also shifted. (d) The charging would dissipate in several hours, which wemeasured by periodically measuring the STIRAP resonance.

27

(a)

(b)

(c)

(d)

Figure 2.12: Dispersion relations for lattices of depths 2ER (a), 8ER (b), and 20ER (c). (d) The
transition frequency between bands 0 and 2 (f0→2 = E0→2 /h) at q = 0 vs. lattice depth. Thisfrequency is measured by parametric heating.

28
Wannier functions are a more useful description. The Wannier function centered at site xj is [103]:
1 X −iqxj /~ (n)wn (x − xj ) = √eφq (x),
N q(2.3)

where q is summed over the first Brillouin zone, and N is a normalization factor. A decent approximation of the ground-band Wannier function in the limit of a deep lattice is a Gaussian ofrms width a/(πs1/4 ) [106]. Fig. 2.13 compares the densities of the Bloch functions and Wannierfunctions.
There are several relevant energy scales for atoms in the lattice, the tunneling J, the on-site
interaction U when there are two atoms on the same site, and the bandgap ∆. We usually operate
in a regime where J, U  ∆ and the initial atomic temperatures are sufficiently low, so we usuallyconsider only the lowest band.
J and U can be calculated from the Wannier functions by the following equations [103]:
Z
J=



~2 ∂ 2wn (x − xi ) −+ V (x) wn (x − xi+1 )dx,
2m ∂x24π~2 aU=mZ|wj (r)|4 d3 r.

(2.4)

(2.5)

It was recently shown how dipole-dipole interactions modify these quantities [107].
J can be calculated directly from the band structure calculation, since J is 1/4 of the bandwidth of the ground band. An approximate functional form is given in Ref. [104] as√

J/ER = 1.39666 s1.051 e−2.12104 s .

(2.6)

An approximate formula for U is given in Ref. [108] as
U/ER = 5.97(a/λ)s0.88 ,(2.7)

where a is the scattering length and λ is the lattice wavelength. Fig. 2.14 shows J for both Rb and
K and URb-Rb at 1064 nm.

29

(a)

(b)

(0)

Figure 2.13: (a) Density |φq (x)|2 for Bloch functions in the lowest band, for q = 0 and q = 0.75~kand a lattice depth of 8 ER . (b) Density |w(x)|2 and |w(x − a)|2 for different lattice depths. For
deeper lattices, the Wannier function is more confined to one particular lattice site and the overlap
with neighboring sites goes down, which leads to less tunneling.

1200

Energy/h (Hz)

1000
800
600

URb-Rb /h

400

JK /h

200

JRb /h

0
0

5

10
15
20
Lattice Depth (ER Rb )25

30

Rb , calculated for an isotropic 3D lattice
Figure 2.14: JRb , JK , and URb-Rb vs. the Rb lattice depth ER
at 1064 nm, and using Eqs. 2.6 and 2.7. The tunneling for K takes into account that sK = 0.4sRb
at 1064 nm.

30
2.4.2

Dependence on beam parameters

We have a Gaussian beam of power P and beam waists Wx and Wy that propagates along
the z direction. For simplicity, I assume that the retroreflection is the same size. To account for
the intensity imbalance between the incoming and retroreflected beams (due to reflections and loss
within optics), there is an attenuation factor f for the retroreflected beam intensity (f < 1), which√
is a factor of f in the electric field amplitude. The peak intensity of one beam is
I0 =

2P
.
πWx Wy(2.8)

The corresponding trap depth is U0 = αI0 , where α is the polarizability. The full dependence of U
is
p
2
2
2
2
U (x, y, z) = −U0 |eikz + f e−ikz |2 e−2x /(Wx ) e−2y /(Wy )


p
2
2
2
2
= −U0 1 + f + f (e2ikz + e−2ikz ) e−2x /(Wx ) e−2y /(Wy )


p
p
2
2
2
2
= −U0 1 + f − 2 f + 2 f (1 + cos(2kz)) e−2x /(Wx ) e−2y /(Wy )


p
p
2
2
2
2
= −U0 (1 − f )2 + 4 f cos2 (kz) e−2x /(Wx ) e−2y /(Wy ) .
The trap frequencies are ωi =s
ωx,y =
and

q

(2.9)
(2.10)(2.11)(2.12)

1 ∂2U
m ∂x2i |x=y=z=0 , and the results are that√√ s
16U0 (1 + f )2
1+ f
16U0
=
,
2
2
4mWx,y
2
mWx,y

r √
r
8 f k 2 U0
8k 2 U0
1/4
=f
.
ωz =
m
m

(2.13)(2.14)

Finally, the lattice depth s = U0 /ER is given bys= mω 2
z
2
~k

=

p
8m
U
f.
0
~2 k 2(2.15)A relevant issue is whether attenuation in the beam path leads to unwanted harmonic confinement
ωx,y from the running-wave component. Using Eqs. 2.13-2.15, it can be shown that ωr increases√fby a factor of 1+
when increasing the power to get the same lattice depth as when f = 1. An
2f 1/4

31
example here clarifies this statement. If f were 1, we would have some lattice strength s0 withharmonic confinement ω0 . Using the same incident power, if f = 0.5, we would have lattice depth√1s0 / 2 and harmonic confinement 21 + 2√
ω0 = 0.85 ω0 . If we increase the power by a factor2√
of 2, the lattice depth increases to s0 , and the harmonic confinement becomes 1.015 ω0 . Thus
we see that even for this fairly large attenuation the additional harmonic confinement (1.5%) of
the running wave is almost negligible. In the experiment, f is usually 0.6-0.8, so we don’t have toworry about this.

2.4.3

Avoiding superlattices

Another thing we worry about is unwanted reflected beams interfering with the primary lattice
beams and creating a superlattice. Fig. 2.15 shows the setup from the old chamber. There are
four reflected beams that can hurt us, two from the incoming beam and two from the retroreflected
beam. At normal incidence, each would be roughly 4% in intensity. This is problematic since the√interference scales as Iin Ir , which means each of these 4% reflections would produce a latticethat’s 0.2 times the strength of the initial lattice (if there were perfect overlap of the beams). Thissuggests we can’t propagate the beam normal to the cell. In principle, the superlattice should bestable if there’s no relative motion between the cell and the retroreflecting mirror, but there would
likely be a phase difference between the two lattices. This is bad, and we want to avoid it.
If we have beams with I0 and 0.04I0 with the same waist, and we want the four possible
reflections (Fig. 2.15) to give a lattice less than 0.01 times the strength of the regular lattice, thebeams would need to be separated by about 2.5 beam waists. We typically use a 250 µm beam forthe vertical lattice and the cell is 1 cm tall, so θ would need to be larger than about 4◦ to satisfy
the above condition. Most recently the lattice beams were a little larger, so we chose 6◦ to be safe.

2.4.4

Calibrating the lattice depths

We regularly use two different methods to measure the lattice depth, parametric heating and
Kapitza-Dirac scattering. Parametric heating, which consists of amplitude modulating the lattice

32

Figure 2.15: We get reflections off of the cell surfaces. The ones that could hurt us by interferingwith the primary lattice beams are shown in light red (there are 4 total). For uncoated glass at
normal incidence, each of these would give ∼ 4% reflection. We tilt the beam at an angle θ withrespect to normal. The atoms sit at the center, and we require the separation between the reflectedbeams and the desired lattice beams to be at least 2-3 beam waists to keep the superlattice depth
below 1% of the primary lattice depth.

Species
RbK
KRbα1064 /h (10−5 MHz/(W/cm2 ))
3.2
2.8
5.5

i )
Relative lattice depth (ER
1
0.40
2.48

33

Table 2.1: Polarizabilities and lattice depths of K, Rb, and KRb at 1064 nm.

beams and driving transitions that change the band index by 2, is discussed at length in Ref. [105].
Starting with atoms in the ground band, we measure f0→2 , corresponding to the energy difference
between the ground band and the second excited band (see Fig. 2.12d). In Kapitza-Dirac scattering,
the lattice is turned on for a short time τ , which is typically less than 10 µs (which is faster thanthe lattice oscillation frequency). This diffracts the BEC into plane-wave components [109, 110].
Measuring the populations in the plane-wave components, one can extract the lattice depth. For
short, strong pulses, the motion of the atoms during the pulse duration can be neglected, and thepopulation in the ±2n~k component is approximately Jn2 sE2~R τ , where Jn is the ordinary Bessel
function of order n [109]. This approximation is only valid for short times. We find it’s relatively
simple to find the shortest τ that minimizes the population of the 0~k component. This occurs
when the argument of J0 is about 2.405, which leads to the condition sτ ≈ 377 µs for Rb (solving
the time-dependent Schrödinger equation exactly, this is true to within 2% for s ≥ 30). We findthat parametric heating and Kapitza-Dirac scattering give consistent answers for the lattice depth.
Unless otherwise noted, the uncertainty in the lattice depth calibration is about 5%.
We measure the lattice depth for Rb and use our knowledge of the different mass and polarizability of K at 1064 nm to calculate the lattice depth for K. At 1064 nm, αRb /αK = 1.15, which
implies ωK /ωRb = 1.37 [105]. The recoil energy is higher for K by a factor of mRb /mK . Putting
all of these factors together, sK = 0.4 sRb , so Rb can be well localized in the Mott insulating phase
while K is still mobile. We do the same procedure for ground-state molecules; there, the conversion
KRb αKRb
is sKRb /sRb = mm
≈ 2.48. Note that it was by measuring parametric heating resonances
Rb αRbthat we measured the polarizability of molecules in both the N = 0 and N = 1 states (see Fig. 4.5)
[91]. The results are summarized in Table 2.1.

34
2.4.5

Band mapping

Due to both finite temperature and nonadiabaticity in loading the lattices, we can haveatoms and molecules populated in higher lattice bands. The standard way to detect particles inhigher bands is via band mapping, whereby the lattice intensity is ramped to zero (instead of being
instantaneously shut off) on a timescale fast compared to motion on a lattice site but slow enoughto be adiabatic with respect to the band separation. This technique is described well in other places[98, 103, 105]. Here I only discuss a variant of the technique that we’ve recently implemented. We
use a refocusing technique [111], where the lattices are ramped down in the same way, but theunderlying harmonic trap is not turned off. Waiting a quarter trap period (of the harmonic trap),
there is a one-to-one correspondence between the position of the atoms, given by the distributionfunction n(x, t), and the initial momentum, given by the distribution function Π(p, t):n(x, T /4) = Π(p, 0),(2.16)

where p = mωx. Note that in this case we are making a measurement in position space, whereas
in normal band mapping, one makes a measurement in momentum space. In this case, the size ofthe first Brillouin zone is 2~k/(mωhc ), where ωhc is the harmonic trap frequency. Ordinary band
mapping requires a sufficiently long expansion such that the cloud is in momentum space, but the
refocusing technique requires a shorter expansion time, such that the cloud is smaller and the OD
and SNR are higher. When imaging from the top, we can fit the entire 2D distribution to thefollowing functional form (similar to that in Ref. [98]):

35

 


  
y − yc − ∆yy − yc + ∆yx − xc − ∆xx − xc + ∆ x
K + A erf
− erf
× erf
− erf
wx
wx
wy
wy
 


  
y − yc + ∆ y
y − yc + 2∆yx − xc − ∆xx − xc + ∆ x+ B1 erf
− erf
× erf
− erf
wx
wx
wy
wy
 


  
y − yc − 2∆yy − yc − ∆yx − xc − ∆xx − xc + ∆ x+ B2 erf
− erf
× erf
− erf
wx
wx
wy
wy
 


  
y − yc − ∆yy − yc + ∆ yx − xc + ∆ x
x − xc + 2∆x+ C1 erf
− erf
× erf
− erf
wx
wx
wy
wy
 


  
y − yc − ∆ yy − yc + ∆ yx − xc − 2∆xx − xc − ∆x
− erf
× erf
− erf
,
+ C2 erf
wx
wx
wy
wy
(2.17)
which includes an imaging offset K and a contribution from the ground band and 4 additional
regions corresponding to higher bands in x and y (see Fig. 2.16a). Since the probability of being
excited along any given direction is small (usually < 0.1), multiply excited states will be ignored.
wx and wy take into account imaging resolution and momentum smearing from not being perfectly
adiabatic at the band edge (the momentum smearing seems to be the dominant effect). The centerof the distribution is (xc , yc ) and the width of the Brillouin zone in x (y) is 2∆x (2∆y ). Ideally,
these should be the same, but a different trap frequency along the two directions might cause slightdifferences in ∆x and ∆y . We assume the higher-band population in z is the same as in x and y
and thus multiply the higher-band fraction measured for the x and y directions by 1.5 to account(B1 +B2 +C1 +C2 )for those atoms excited along z. Thus, the fraction of atoms in higher bands is 1.5
2A+B1 +B2 +C1 +C2 .

2.4.6

Generating the lattice beams

The optical layout for the lattice beams changed since the original 3D lattice work [72].
Initially we were using the OT beams as the horizontal lattice beams. We would do the evaporationin the OT, load the vertical lattice, ramp the OTs to 0, open shutters to unblock the retroreflections,and then finally ramp on the horizontal lattices (see Fig. 2.17a). Eventually we wanted to be able
to ramp the lattices independently of the dipole trap, so we made the polarization of the lattice
beams orthogonal to the OT beams, sent in the lattice beams from the opposite side of the cell as

36

(a)

(b)

B2

y

C1

A

C2

(c)

3

2

1

B1
x

0

Figure 2.16: (a) A “map” that defines the different regions for band mapping when imaging suchthat excitations along two directions can be observed (see Eq. 2.17). (b) and (c) Typical imagesof band-mapped K from the experiment, using the refocusing technique, with lower (b) and higher(c) higher-band fraction. The OD is given by the color scale on the far right. These images comefrom Ref. [93].

Intensity (a.u.)

(a)

(b)

5
4

37
OT
lattice

3
2

OT
Horizontal lattice
Vertical lattice
1

0
0

1

2
3
Time (seconds)

4

Figure 2.17: (a) Timing diagram for the initial lattice loading scheme, as used in Ref. [72]. We
would ramp up the vertical lattice, ramp down the OTs, open the shutters, and then ramp up
the horizontal lattice beams. (b) The most recent setup for the horizontal lattice beams and OT
beams. The OT and lattice beams have orthogonal linear polarizations and the OT and latticeincoming beam propagate in opposite directions.

the OT beams, and then used a polarizing beamsplitter on the far side (on the incoming side for
the OT) to separate the lattice beam from the optical trap (see Fig. 2.17b).
The typical OT beam size is 200 µm in the radial direction and 40 µm in the vertical direction.
At one point, we made them even more oblate to reduce the radial harmonic confinement. Afterswitching to separate horizontal lattice beams, the lattice beams were initially round (with a waistof 125-250 µm), and then oblate, with sizes ∼ 500 × 100 µm in the radial and vertical directions,
respectively. We wanted to make the horizontal lattice beams elliptical to better match the size ofthe cloud and have a lower harmonic confinement along the radial direction. The vertical lattice
was round with a 200-300 µm waist, and most recently it was around 280 µm at the position of theatoms.

Chapter 3

Chemical reactions and the Zeno effect

One of the first observations made with ultracold KRb molecules in a 3D harmonic trap wasthat the molecules had a significantly shorter lifetime in the optical trap than the constituent K
and Rb atoms [69]. Furthermore, upon applying a DC electric field, which is the simplest way to
induce dipolar interactions between the molecules, the lifetime became much shorter [70]. Thisis a problem because it correlated strong interactions with strong inelastic losses, which makes it
difficult to study dipolar interactions in a controlled way. Since the molecules were prepared in thelowest internal energy state with the thermal energy less than all other energy scales, no other statesof the molecule were accessible. Therefore, it was concluded that exothermic chemical reactions
are responsible for the loss, as the energy of K2 +Rb2 is about 10 cm−1 lower than the energy of 2
KRb, and the reaction 2KRb→K2 +Rb2 is barrierless [112]. The observations of chemical reactions
in KRb signal a new epoch in the study of ultracold chemical reactions.
Fermionic 40 K87 Rb molecules prepared in the same internal quantum state collide in thepartial-wave channel with L = 1 (p wave), where ~L is the quantized relative angular momentum.
The projection of this angular momentum onto the quantization axis, mL , determines the orientation in which the molecules collide: mL = 0 corresponds to an attractive head-to-tail collision,
while mL = ±1 corresponds to a repulsive side-by-side collision. Distinguishable molecules cancollide via s-wave collisions (L=0). In an applied DC electric field, the dipole-dipole interaction
mixes states with different L, so the L = 0 channel becomes the lowest-energy adiabatic channel
with even L, and the L = 1 channel becomes the lowest-energy adiabatic channel with odd L. This

39

Figure 3.1: Cartoon illustrating the anisotropy of collisions in an applied DC electric field. The
p-wave centrifugal barrier increases for repulsive side-by-side collisions (red arrow) and decreases
for attractive head-to-tail collisions (blue arrow). Figure reproduced from Ref. [70].

odd-L adiabatic channel has a centrifugal barrier. A simple model for the chemical reactions is thatthe molecules tunnel through the p-wave centrifugal barrier and then chemically react with unit
probability (if the molecules are not in the lowest hyperfine state, another possibility is hyperfinestate changing collisions) [69]. In an electric field, the centrifugal barrier decreases for mL = 0(head-to-tail) collisions but increases for mL = ±1 (side-by-side) collisions [70] (see Fig. 3.1). Thissuggests a natural strategy to stabilize the molecular gas in a strong electric field, which is to prevent the molecules from colliding head-to-tail. This can be accomplished by trapping the molecules
in a 1D optical lattice, with the electric field perpendicular to the quasi-2D traps. In this geometry,the loss rate can be reduced by almost two orders of magnitude [71]. To completely shut off thechemical reactions, we can create the molecules in a 3D lattice. In the full 3D lattice, the molecules
live for tens of seconds, independent of the electric field strength [72].
In the first part of this chapter, I briefly summarize the molecule production and chemical
reactions in the harmonic trap and in optical lattices. Since a lot of this has been covered in othertheses from our group, particularly those of Kang-Kuen Ni [96], Marcio de Miranda [98], and Brian
Neyenhuis [105], I focus on the most important aspects. In the second part of the chapter, I describe
our studies of the continuous Zeno effect. This occurs in a regime where the chemical reaction rateis the largest energy scale in the lattice, which occurs for chemical reactions on a single latticesite between distinguishable molecules in a rotational mixture (half in N = 0 and half in N = 1,40
where N is the molecule’s principal rotational quantum number). The result of the Zeno effect isthat, counterintuitively, the loss rate decreases as the chemical reaction rate increases. Much ofthis chapter follows Refs. [72] and [90], of which I am a coauthor.

3.1

Brief summary of molecule production
The experiment starts with magneto-optical traps (MOTs) of both 87 Rb and 40 K. Afterloading the MOTs for about 10 seconds, we do a compressed MOT stage, which gets the MOT to
the correct position for loading into a magnetic quadrupole trap. We then have a molasses stage(where the magnetic field is completely shut off), which does sub-Doppler cooling. Finally, we use
optical pumping to put the atoms in the magnetically trappable stretched states |F, mF i = |2, 2i for
Rb and |9/2, 9/2i for K. We then load the atoms into the quadrupole trap and transfer the atoms
to the science cell by a pair of moving coils. The evaporative cooling is done in a Ioffe-Pritchard(IP) trap by using ∼ 6.8 GHz microwaves to transfer hot Rb atoms from the |2, 2i state to the|1, 1i state, which are not trapped. The K gas is sympathetically cooled by the Rb and remains
thermalized with the Rb throughout the evaporation.
The s-wave Feshbach resonance that we use to create weakly bound Feshbach molecules
occurs at a field of 546.6 Gauss between K in the |9/2, −9/2i state and Rb in the |1, 1i state[113]. These are both strong-field-seeking states so we can’t trap them in a magnetic trap. Instead,
we cool the atoms in the IP trap to about 1-2 µK and then load the atoms into a crossed-beam
optical dipole trap at 1064 nm, which can trap all spin states. At this point, we ramp on a roughly
homogeneous bias magnetic field to about 30 Gauss. About 300 ms after transferring into thedipole trap, we use an adiabatic rapid passage (ARP) to transfer K from the |9/2, 9/2i state to the|9/2, −9/2i state and use a microwave ARP at around 6.9 GHz to transfer Rb to the |1, 1i state.
We then do further evaporation by reducing the intensity of the dipole trap beam over several
seconds. After 3-4 seconds we produce a degenerate Fermi gas of K and a BEC of Rb. For optimal
molecule production in the harmonic trap, we usually work with cold thermal gases (T /Tc > 1 forRb) to maximize the phase-space overlap of the two species. For optimal molecule production in41
the lattice, we use very small BECs and large, degenerate Fermi gases (see Chapter 5).
We ramp the magnetic field across the Feshbach resonance from high field to low field toadiabatically convert pairs of atoms to weakly bound Feshbach molecules (molecules in the leastbound vibrational state). In a harmonic trap this is not very efficient, as we can only convert 1020% of the minority species, but in the 3D lattice it is quite efficient (see Section 3.6 and Chapter
6). We use STIRAP to transfer the Feshbach molecules to the absolute ground state [51, 114].
When the molecules are in the ground state, we apply resonant light at 767 and 780 nm to remove
the K and Rb atoms that didn’t make molecules. This is necessary since the atoms can chemicallyreact with the molecules. To image the molecules, we use STIRAP to bring the molecules backto the Feshbach molecule state and then dissociate the Feshbach molecules, either with light on
the K cycling transition or by sweeping the magnetic field above the resonance. Either way, wethen image K or Rb atoms on a cycling transition, but to image Rb we first have to do anothermicrowave ARP to put the atoms in the |2, 2i state.

3.2

Rotational states and driving rotational transitions
For 1 Σ molecules such as KRb, the rotational degree of freedom is instrumental in get-ting dipolar interactions, as successive rotational states have opposite parity. Transitions between
rotational states are electric dipole transitions. Furthermore, the rotationally excited states are
long-lived and we can easily couple to them using microwaves. As will be discussed in the next
chapter, we can encode spin in the rotational degree of freedom and realize an XY coupling between
molecules in the lattice. For studying chemical reactions, driving transitions to different rotational
states is useful because it allows us to control the distinguishability of the molecules and to create
molecules in the absolute hyperfine ground state. The energies of the excited rotational states are
EN = hBN (N +1), where B = 1.113950 GHz is the rotational contstant [96]; however, higher-orderterms break the degeneracy between different hyperfine states and different projections along thequantization axis. This section summarizes how we drive rotational transitions in the molecules
and follows Ref. [115].

42
Although the ground-state molecules are a singlet (X 1 Σ+ ), there is still hyperfine structurearising from the nuclear magnetic moments of the K and Rb atoms. With nuclear spin quantum
numbers I K = 4 and I Rb = 3/2 for K and Rb, respectively, there are 36 hyperfine states foreach rotational state |N, mN i, where N is the principal rotational quantum number and mN is
its projection onto the quantization axis. Thus, the states can be labeled by |N, mN , mK , mRb i,
where mK is the nuclear spin projection for K, and mRb is the nuclear spin projection for Rb. mKand mRb are good quantum numbers for magnetic fields larger than about 20 Gauss, where thelinear Zeeman energy shift for each of the nuclear spins is much larger than the mutual coupling of
the nuclear spins. At 550 Gauss, the splitting between states with ∆mRb = ±1 is about h × 760
kHz, while the splitting between states with ∆mK = ±1 is about h × 130 kHz. In addition,
there is a coupling between the nuclear electric quadrupole moments of the 40 K and 87 Rb nuclei
and the electric-field gradients created by the molecules’s electron cloud [115, 116]. This coupling
breaks the degeneracy of states with the same N but different mN , and mixes states in the N = 1
manifold that have a constant sum mN + mRb or mN + mK . This leads to eigenstates of the form
|1, 0, mK , mRb ± 1i + δ|1, ±1, mK , mRb i or |1, 0, mK ± 1, mRb i + δ|1, ±1, mK , mRb i and allows us todrive transitions that ultimately put the molecules in the absolute ground state |0, 0, −4, 3/2i.
From selection rules and spectroscopic resolution, STIRAP populates a single hyperfine stateof the ground rovibrational state, |0, 0, −4, 1/2i [96]. We then use a microwave signal to drive
transitions to states in the N = 1 manifold. The microwave horn we use doesn’t give us very muchcontrol over the microwave polarization, and we find we get much better coupling for σ ± transitions
than for π transitions. To transfer molecules to the hyperfine ground state, we drive two microwave
transitions, one of which changes mRb . Specifically, we drive |0, 0, −4, 1/2i → |1, 0, −4, 3/2i +δ|1, 1 − 4, 1/2i → |0, 0, −4, 3/2i (note that these are σ + and π transitions, respectively). For
this particular intermediate state, |δ|2 ≈ 0.1. Because |δ|2 < 1, for fixed microwave power, theRabi frequency for a hyperfine-state changing transition is less than for a transition that doesn’tchange hyperfine state. For the spin-exchange work described in the next chapter, we drive the|0, 0, −4, 1/2i → |1, 0, −4, 1/2i and |0, 0, −4, 1/2i → |1, −1, −4, 1/2i transitions.

43

3.3

Chemical reactions in 3D
Chemical reactions of ultracold KRb molecules in the 3D harmonic trap depend on whichstate(s) the molecules are in, the temperature, and the applied electric field. There are three
relevant cases: (1) indistinguishable molecules (spin-polarized), (2) a hyperfine mixture with bothstates in the N = 0 manifold, and (3) a hyperfine mixture with one state in the N = 0 manifold
and one state in the N = 1 manifold. In the first two cases, the temperature and electric-fielddependences were studied. The third case is relevant to the quantum Zeno work, where we havean incoherent mixture of N = 0 and N = 1 molecules.
The chemical reactions cause the number density of trapped molecules, n, to decrease according todn= −βn2 − αn,
dt

(3.1)

where β is the two-body loss coefficient and α reflects a decrease in the density due to an increasein temperature [69]. Fig. 3.2a shows how β depends on temperature for cases 1 and 2. For spinpolarized cases, the loss rate β ∝ T , which is expected from the Bethe-Wigner threshold laws
for p-wave collisions of indistinguishable fermions. For the hyperfine mixture, the molecules are
distinguishable, s-wave collisions are allowed, and the loss has no significant temperature dependence. Another important conclusion from this data is that hyperfine changing collisions don’t
play a large role, since the loss rates are the same within experimental error for the |0, 0, −4, 1/2i
state (for which hyperfine-state changing collisions are allowed in principle) and the |0, 0, −4, 3/2i
state (for which there cannot be hyperfine-state changing collisions, as this is the hyperfine groundstate). Fig. 3.2b shows that β ∝ d6 (d is the induced dipole moment in the lab frame), consistent
with a quantum threshold model [70]. Fig. 3.3 shows the loss of molecules in a mixture of N = 0
and N = 1 states, from which we extract β = 9.0(4) × 10−10 cm3 s−1 , which is almost 5 times
higher than for the N = 0 mixture [84]. To create an incoherent spin mixture, we applied a π/2pulse on the |0, 0, −4, 1/2i → |1, −1, −4, 1/2i transition and waited 50 ms for any superpositions
to decohere.

44

(a)

(b)

Figure 3.2: Chemical reactions in the 3D harmonic trap. (a) Chemical reaction rates at zero electricfield for two different hyperfine states and a mixture. The loss rate β is proportional to temperature
for the spin-polarized case and is independent of temperature for the mixture. Figure reproduced
from Ref. [69]. (b) The loss rate vs. induced dipole moment for a spin-polarized sample. β ∝ d6 ,which is consistent with a quantum threshold calculation [70]. Figure reproduced from Ref. [70].

Figure 3.3: Decay of an incoherent mixture of rotational states |0, 0, −4, 1/2i and |1, −1, −4, 1/2i.
The extracted loss rate, 9.0(4) × 10−10 cm3 s−1 , is almost 5 times higher than for the mixture of
two N = 0 hyperfine states (|0, 0, −4, 1/2i and |0, 0, −4, 3/2i). Figure reproduced from Ref. [84].

45
In addition to reacting with themselves, the molecules can also react with the K and Rb
atoms (K is worse for molecules in the ground hyperfine state, but both are bad for molecules in
excited hyperfine states [96]). Thus, it’s very important to remove all remaining atoms after making
molecules, especially in cases where we convert a small fraction of one or both of the atomic species(this will be important for K in Chapter 5).

3.4

Suppressing chemical reactions in a 1D lattice
Because of the asymmetry between head-to-tail and side-by-side collisions, a natural strategy

for stabilizing the gas against chemical reactions is to suppress the lossy head-to-tail collisions [71].
In particular, the lifetime should be enhanced in a 1D lattice geometry with the electric field oriented
perpendicular to the pancake-shaped lattice sites. Here, the molecules are created in a verticallyoriented lattice (z direction) and an electric field is also applied along the z direction. The lattice
has a strong optical confinement with trap frequency ωz . For the applied electric field strengths,
molecular temperatures, and harmonic confinements of Ref. [71], the experiments are performed inquasi-2D, and the combination of sufficiently tight optical confinement and Fermi statistics makes
molecules preferentially collide in the side-by-side orientation. Chemical reactions are suppressed by
the repulsive dipole-dipole interactions. In the quasi-2D geometry, L is not a good quantum numberq~
to describe how molecules approach each other for intermolecular separations  aho (aho = mωzis the harmonic oscillator length along z), and instead the relative motion is described by mL . A
quantum number ν labels which harmonic oscillator motional state (lattice band) a molecule is in.
This quantum number acts as a new internal degree of freedom for molecules as they approach eachother from long range. Fermionic molecules in the same internal state and same ν must collidewith odd mL (side-by-side), whereas fermions in the same internal state but different ν collide witheven mL (head-to-tail).
Again, there are three separate cases to consider (see Fig. 3.4a): (1) distinguishable molecules,(2) indistinguishable molecules in different bands, and (3) indistinguishable molecules in the same
band. Similar to the 3D case, channel 1 leads to s-wave losses and should be avoided. Channel

46

(a)

(b)

(c)

Figure 3.4: Chemical reactions in the 1D lattice. (a) The barrier for collision channel 1 (green,
distinguishable molecules), 2 (red, indistinguishable molecules in different bands), and 3 (black,indistinguishable molecules in the same band). Channel 3 is the most desirable since it corresponds
to repulsive side-by-side collisions. The arrows next to the curves indicate whether the barrier
increases or decreases as the electric field increases. (b) Loss rates for two different initial band
populations. The initial loss is faster when the initial higher band fraction is higher. (c) 2D decaycoefficients β2D for the three cases depicted in (a). Figures reproduced from Ref. [71].

47
2 corresponds to head-to-tail collisions (mL = 0), which are also undesirable. Channel 3 is thedesirable scenario, and corresponds to mL = ±1 (side-by-side collisions). In an applied electricfield, dipole-dipole interactions cause the barrier for head-to-tail collisions to go down and thebarrier for side-by-side collisions to go up. To control the fraction of molecules in excited bands
and thus the relative importance of channels 2 and 3, parametric heating was used to promotemolecules to higher bands. Fig. 3.4b shows the loss for two different initial conditions: for colder
molecules, where about 75% of the molecules in the lowest band, and another case where parametricheating was used to heat the molecules to the second excited band. Fig. 3.4c shows the loss rate
coefficient, β2D , vs. induced dipole moment for the three cases depicted in panel a and clearlyshows the loss rate is significantly less for channel 3. At an induced dipole moment of 0.174 D, thesuppression is about a factor of 60.
It’s worth noting that for very large applied electric fields and very strong optical confinements, repulsive dipole-dipole interactions can suppress chemical reactions, regardless of quantum
statistics [71, 117, 118]. In this regime, the ratio of elastic to inelastic collisions should be favorable
for evaporatively cooling the molecules [119], similar to pioneering work in dipolar Fermi gases ofmagnetic atoms [50]. However, Ref. [120] shows that for d > 0.3 D, the ratio of elastic to reactive
collisions only really starts to significantly increase (beyond 100) for ωz /(2π) > 100 kHz, which isa significantly higher trap frequency than we have used so far.

3.5

Shutting off chemical reactions in a 3D lattice
The success in suppressing reactions in a 1D lattice motivated us to add lattices along thehorizontal directions (x and y) to create a 3D lattice [72]. Again, the strategy is to load the atoms
into the lattice and then make molecules. In the 3D lattice, the molecules enjoy a long lifetime,
as long as they are all in the lowest band, which for spin-polarized fermions trivially means there
can’t be more than one molecule per site. Furthermore, the lifetime is independent of electric field,
and is actually limited by single-photon scattering from the lattice beams (the 1064 nm light isonly about 30 nm detuned from the bottom of the excited molecular potential).

48
Fig. 3.5 shows a typical decay of ground-state molecules in a deep 3D lattice. After a fast
initial loss, which could be due to collisions with dark molecules (those that didn’t make it to theground state), we observe a simple exponential decay of the molecules, with a 1/e lifetime of 16(2)seconds. These dark molecules could be d-wave molecules produced during the magnetoassociationin the lattice (see Chapter 6). The inset compares the lifetime at zero electric field with thatfor polarized molecules with an induced dipole moment of 0.17D, and we see no difference within
experimental error.
To understand the loss mechanism, we first studied the crossover from vertically oriented
tubes to an isotropic 3D lattice. Fig. 3.6 shows how the lifetime depends on the lattice depth in
the vertical direction. For these experiments the horizontal lattices were 56 ER . For just verticallyoriented tubes (no lattice along z), the lifetime is about 1 second and decreases to about 0.1
seconds when the molecules are polarized to 0.17 D. Adding even a shallow lattice along z, thelifetime increases significantly (5 seconds for sz = 12 and 20 seconds for sz = 20), and even at 5
ER the lifetime doesn’t significantly change in an applied electric field. Pauli blocking should leadto a rapid suppression of loss as the lattice is increased along z, but would do so more rapidly than
we observe. This, combined with the insensitivity of the lifetime at 5 ER to the induced dipole
moment, suggest that an incoherent process is responsible for the loss in this regime of strong
horizontal and weak vertical lattices. A model that describes the loss well is that there is someheating rate that promotes molecules to higher bands, and then these higher band molecules cantunnel much faster and collide with other molecules at a rate much higher than the promotion rate.
A heating rate ∼ 1 − 2 ER /s matches the data well (red and blue curves in Fig. 3.6).
The lifetime reaches a maximum value of about 25(5) seconds for sz = 34 (point b in Fig. 3.6)
and then it starts to decrease. At large lattice depths, the lifetime becomes limited by off-resonant
scattering from the lattice laser. The rich internal structure of the molecules makes it likely thatupon absorbing a photon, the molecule won’t return to the ground state. The lifetime depends
on the total light intensity, which we verified by adding a running wave beam (points c and d in
Fig. 3.6). Measuring this scattering rate allows us to measure the imaginary part of the polariz-

49

Figure 3.5: The lifetime of ground-state molecules in the 3D lattice, with sx = sy = 56 and sz = 70
(the uncertainty in the lattice depths is at most 10%). After a fast initial loss of some of themolecules, the remaining molecules enjoy a long lifetime of 16.3 ± 1.5 seconds. The inset shows thatthe lifetime doesn’t depend on electric field (these measurements were performed in an isotropic 50
ER lattice). Figure reproduced from Ref. [72].

Figure 3.6: The lattice depth along the horizontal directions is fixed at 56 ER , and the lattice depth
along the vertical direction is gradually increased. This allows us to study the lifetime as we gofrom a 2D lattice to tubes with a small corrugation along the third direction to a full 3D lattice.
For large z lattice depths the lifetime becomes limited by off-resonant scattering. We can vary theintensity both by changing the z lattice strength and by adding an additional running wave beamthat increases the total intensity but doesn’t increase the lattice depth. Figure reproduced from
Ref. [72].

50
ability at 1064 nm to be h × 2.052(9) × 10−12 MHz/(W/cm2 ).
The achieved lifetimes of ∼ 10 − 20 seconds are sufficient to realize a stable dipolar gas. As
we will see in the next chapter, the exchange timescales for molecules in the lattice are as fast as
10 ms, so our lifetime provides ample opportunities to study interesting physics.

3.6

Long-lived Feshbach molecules and conversion of preformed pairs
Because Feshbach molecules are so weakly bound, it’s difficult to cleanly remove all of theatoms that didn’t get converted. This is especially problematic for K, where the light that wewould use to remove the unpaired K atoms would kill the Feshbach molecules as well. Instead, wejust “hide” the K in the |9/2, −7/2i state so we don’t image them, but the atoms are still trapped.
Furthermore, we typically convert only 10-20% of the minority species to Feshbach molecules, so thenumber of remaining atoms is significantly larger than the number of Feshbach molecules. These
atoms can then collide with the Feshbach molecules and limit their lifetime. To verify this, we canuse our ability to create ground-state molecules to more cleanly remove the unpaired atoms. If wethen STIRAP the ground-state molecules back to the Feshbach state, we observe lifetimes ∼ 150
ms in the dipole trap, significantly longer than the lifetime when “hiding” the atoms, which is lessthan 10 ms [114].
If we do the same procedure in the lattice, we find that the Feshbach molecules can live forup to 10 seconds (see Fig. 3.7). Varying the magnetic field after making molecules we can vary
the size and binding energy of the Feshbach molecules. Similar to the ground-state molecules, wefind that the lifetime is limited by off-resonant scattering. To explain the dependence on binding
energy, we approximate the Feshbach molecule wavefunction |f i as
|f i ≈√
Z|Ci +√

1 − Z|Oi(3.2)

where |Ci is the closed-channel molecular wavefunction and |Oi is the open-channel wave function
for free atoms. Z depends on the magnetic field and is 1 for B  B0 (deeply bound molecules)
and 0 for B  B0 (free atoms), where B0 is the location of the Feshbach resonance. The total

51
scattering rate is ZΓmol + (1 − Z)Γatoms , where Γmol and Γatoms can be extracted from the lifetimes
in the limiting cases, and Z can be computed from a simple coupled-channel theory [29, 121]. The
red fit curve in Fig. 3.7 has no free parameters and agrees well with the data.
By dissociating these long-lived Feshbach molecules to atoms, we can study the conversion
of preformed pairs (lattice sites that have one K and one Rb atom). This is important because it
allows us to understand how efficiently we can convert lattice sites with one K and one Rb atom
to a Feshbach molecule. To do this we prepare a clean lattice of Feshbach molecules in the same
way as described above. We then dissociate the Feshbach molecules by ramping the magnetic field
above the resonance. At this point, we should have a lattice consisting of only preformed pairs. By
then ramping B to below the resonance to recreate molecules we can study how well we can convert
these preformed pairs to Feshbach molecules. To measure the number of molecules, we apply an
RF pulse which transfers free K atoms in the |9/2, −9/2i state to the |9/2, −7/2i state, but leaves
Feshbach molecules unaffected. This renders the unpaired K atoms invisible during the subsequentmolecule detection. We then dissociate the Feshbach molecules once again by ramping B to above
the resonance, and measure the number of K atoms with spin-selective resonant absorption imaging.
We determine the conversion efficiency by dividing this molecule number by the total number of
K atoms measured when we do not apply the RF pulse. In Ref. [72], we found a conversion of
87 ± 13%. In Chapter 6, we investigate this issue in much further detail and confirm that in theideal case of deep lattices and fast magnetic-field sweeps the conversion is near unity [93].

3.7

Quantum Zeno effect
As we’ve seen in the previous sections, the chemical reaction rate can be tuned over several

orders of magnitude by changing the distinguishability of the molecular gas, the induced dipole
moment, and the temperature. The highest two-body loss rate we observed is for the rotational
mixture. In this case, the molecules are distinguishable, and there is no Pauli blocking and they
can occupy the same site and chemically react. This onsite loss rate is very large and can be madeto be larger than both the tunneling rate and the energy gap between different lattice bands. In

52

Figure 3.7: Using our ability to remove all unpaired atoms after making ground-state molecules, thelifetime of Feshbach molecules can be long. The lifetime depends on magnetic field because of thedifferent admixture of open and closed channel contributions to the wave function, which dependon magnetic field. Above the resonance we have confinement-induced molecules, which don’t exist
in the absence of the lattice. Figure reproduced from Ref. [72].

53
this regime, tunneling onto an already occupied site is suppressed due to the continuous quantum
Zeno effect, and as a result, the effective loss rate of the entire system decreases as the onsite loss
rate increases. Some previous observations of the Zeno effect in AMO systems were reported inRefs. [122, 123]. It’s worth noting here that the loss rates in the 3D lattice work [72], where we
prepared a spin-polarized sample, were not large enough to be in the Zeno regime (at least forground-band collisions). This section closely follows Ref. [90], of which I am an author.
Similar to the rotational mixture described above, we work with the |0, 0, −4, 1/2i and|1, −1, −4, 1/2i states, which we map onto a spin-1/2 system (|0, 0i ≡ | ↓i and |1, −1i ≡ | ↑i).
To see how the Zeno suppression comes about, consider just two lattice sites and take the initialstate to be | ↑, ↓i (one particle in each well). The states that lead to loss are |0, ↑↓i and | ↑↓, 0i.
To account for the loss we add an imaginary term −i~Γ0 /2 to the energies of |0, ↑↓i and | ↑↓, 0i(these two states also have a real energy shift ∆E, which we assume to be small compared to ~Γ0[123, 124]). The system can be described by the following Hamiltonian, in the basis | ↑, ↓i, | ↓, ↑i,|0, ↑↓i, and | ↑↓, 0i:| ↑, ↓i | ↓, ↑i

 0


 0
H=

 −J


−J

0

|0, ↑↓i| ↑↓, 0i

−J

−J

0

J

J

J

0
∆E − i~Γ
2

0

J

0

0
∆E − i~Γ
2






.





(3.3)

Using second-order perturbation theory, the states | ↑, ↓i and | ↓, ↑i acquire an imaginary part to2

J. This implies that thetheir energy, which in the limit that ~Γ0  J, ∆E, is proportional to ~Γ0
2

states | ↑, ↓i and | ↓, ↑i decay at a rate proportional to ~J2 Γ0 .
A schematic of the experimental setup is shown in Fig. 3.8. We used our standard preparation
scheme to create ∼ 104 ground-state molecules in a deep 3D lattice (along all three directions).
Next, we applied a π/2 pulse on the |0, 0, −4, 1/2i → |1, −1, −4, 1/2i transition and waited 50 ms
to create an incoherent mixture. Then, we reduced the lattice depth along y within 1 ms to allow

54

Figure 3.8: (a) Experimental setup for studying the continuous quantum Zeno effect. The molecules
are held in a 3D lattice, with strong lattices along x and z and a weak lattice along y. This realizes
a system of decoupled 1D tubes with a weak corrugation along the tubes. (b) The tunneling energy
along the tube direction is J and the onsite loss rate is Γ0 . Figure reproduced from Ref. [90].

tunneling along that direction (at a rate J/h), which realizes a system of decoupled tubes withweak corrugation along the tube. We held the molecules in the lattice for a variable hold time andthen measured the number of | ↓i molecules.
When an N = 0 and N = 1 molecule encounter each other, there is no Pauli suppression
preventing them from occupying the same site and their onsite loss rate due to chemical reactions
Γ0 is very large. In fact, we can have Γ0 > ω0  J~ , where ~ω0 is the band separation and J is thetunneling energy along the y direction. The loss is this regime is governed by an effective loss rate
Γeff =

2J 2
.
~2 Γ0

(3.4)

The loss is two-body and the equation for the loss of either spin state isṄi (t) = −κNi (t)2,
Ni (0)(3.5)(2)

where i = | ↑i or | ↓i, Ni (0) is the initial number of molecules in state i, and κ = 4qΓeff g↓↑ n↓ (0) (themaster equation is needed to get the correct numerical prefactors). In the experiment we measure
the population of molecules in | ↓i, and n↓ (0) is the initial density, or filling fraction, of molecules in
the | ↓i state, so that the total initial filling fraction is 2n↓ (0). q is the number of nearest neighbors,(2)and g↓↑ =−→−→hnˆi nˆj −4Si ·Sj ihnˆi 2 iis the correlation function for the two spin states on neighboring sites i

55→−
and j (n̂i is the number operator on site i and Si is the spin-1/2 vector operator). Since we have
(2)

(2)

an incoherent mixture, we assume g↓↑ = 1, but if our system were correlated such that g↓↑ 6= 1,
we should be able to tell this by looking at the loss and comparing to an uncorrelated case. One
goal of the experiment is to measure κ for different values of J and Γ0 and verify the expected
dependences from Eq. 3.4. Another goal is to get an idea of the filling fraction.
For our parameters, we likely occupy ∼ 1000 − 2000 tubes with ∼ 6 molecules/tube. Ultimately the small number of molecules per tube slows the loss down quite a bit for long holdtimes, and to account for this we only fit the loss until we lose 25% of the initial number. The
typical transverse lattice strength is 40 ER , while the lattice along the weak direction was varied
between 5 and 16 ER . We can adjust the lattice depths to tune over a wide range of Γ0 and J
values (see Fig. 3.9). To change J, we primarily adjust the lattice depth along the y direction.
However, decreasing J by reducing the lattice depth along y causes Γ0 to go down slightly since theonsite wavefunction size becomes slightly larger. So to reduce J but keep Γ0 fixed, we reduce sy
and slightly increase sx and sz . To change Γ0 we need to change the size of the Wannier function,
which we accomplish by changing the transverse lattice depths. Considering only the lowest band,Γ0 is given byΓ0 = β(3D)Z|w(r)|4 d3 r,(3.6)

where w(r) is the (ground-band) Wannier function, and the rate coefficient β (3D) was measured in
the harmonic trap to be 9.0(4) × 10−10 cm3 s−1 (see Fig. 3.3).
Fig. 3.10a shows a typical loss curve. Here sy = 5 and sx = sz = 25. The fit is the solutionof Eq. 3.5. If we naı̈vely calculate the filling using Eq. 3.5, we get an answer that is much too high,something in the range of 30%, whereas we know that the filling is more like 5-10%. This is because
we have neglected the effects of higher bands. Higher bands play a very important role because
Γ0 is on the same order of magnitude as the band gap. An admixture of higher bands causes
the molecules to be less localized (since the onsite density is much lower for molecules in higher

56

(a)

(b)

Figure 3.9: (a) Tunneling rate along the tube direction (J/~) vs. lattice depth Vy along the tube
direction. In the experiment Vy was varied between 5 and 16 ER . (b) Onsite loss rate Γ0 (considering
only the lowest band) vs. Vx,z (Vx = Vz ) and Vy = 5ER , calculated using Eqs. 3.6 and 2.7.

Figure 3.10: (a) Typical loss curve, with a fit to the rate equation (Eq. 3.5). (b) and (c) Comparisonof the theory, considering both a single band (green) and multiple bands (blue). In both cases, therate equation was used. The only difference is that in the case of multiple bands a renormalizedgΓ
eff , which accounts for higher bands, is used. The multi-band theory fits the data with a much
lower filling fraction than a single-band theory. Figure reproduced from Ref. [90].

57
bands) and hence the integral over W is lower, which causes Γ0 to be lower. This causes Γeff toincrease and reduces the filling fraction extracted from a measured value of κ. For our parameters,
properly accounting for higher bands can increase Γeff by about a factor of 5. Fig. 3.10b shows thedependence of κ on Γ0 , and clearly shows the Zeno suppression. Fig. 3.10c shows that κ depends
roughly quadratically on J/~. Panels b and c show that lattice fillings around 6% lead to the best
agreement with the experiment.

3.8

Conclusion
This chapter described our efforts to understand, control, and finally shut off chemical reac-

tions in KRb. It’s widely viewed that chemical reactions are a disadvantage for studying quantum
many-body physics with a system of molecules; however, we have shown that creating the moleculesin the lattice completely avoids this problem. Our studies of the quantum Zeno effect show an interesting application of chemical reactions, which could be used to measure correlations between
particles without relying on amazing imaging resolution. These experiments also informed us thatthe lattice fillings were quite low, less than 10%. Most of the rest of this thesis will focus on cases
where the molecules are created and confined in a deep 3D lattice, so the chemical reactions areirrelevant and the lifetimes are long.
As an alternative way to avoid chemical reactions in ultracold molecular gases, a number ofexperiments are currently underway to produce gases of ultracold non-reactive molecules. However,it’s not completely clear that other processes cannot occur in these “nonreactive” molecules thatwould limit the lifetime. For example, there are predictions for a three-body loss process whereby
two nonreactive molecules form a transient complex which collides with a third molecule, leading
to the loss of all three [125]; however, in reactive molecules like KRb, the chemical reactions makethese loss processes unobservable [125].
Molecules in the lowest rotational state suffer from chemical reactions, and the situation only
gets worse as the dipole moment is increased. However, collisions between molecules in mixed
rotational states can be tuned via an applied electric field. In particular, Ref. [126] shows that the

58
ratio of elastic to quenching (reactive plus inelastic) collisions can be tuned by several orders ofmagnitude by using an electric field to make a second excited colliding channel cross the thresholdof the incident channel. The specific cases considered were |1̃, 0i + |1̃, 0i with either |0̃, 0i + |2̃, 0i
or |0̃, 0i + |2̃, ±1i, where |Ñ , mN i is the dressed state which asymptotically approaches |N, mN i in
the limit of zero electric field. These specific resonances occur around 12 kV/cm, which should beaccessible in the second generation of our experiment.

Chapter 4

Observation of spin exchange

From the point of view of pursuing many-body physics mediated by long-range interactions,chemical reactions are a nuisance. Previous observations of effects caused by dipolar interactions inultracold molecules involved inelastic collisions or chemical reactions [69, 70, 71]. In this chapter,I discuss a manifestation of coherent dipole-dipole interactions, where the molecules never actually
come in contact with each other. It was proposed in Refs. [74, 82] that the molecules can behave
as quantum magnets, and even at low lattice fillings, it should be possible to observe dipolarinteractions [83]. In this scheme, we encode a spin-1/2 degree of freedom in two rotational statesof the molecules, and the molecules can swap their spin orientation via a flip-flop process that canoccur in principle over arbitrary distances. More specifically, the N = 0 and N = 1 rotational states
have opposite parity, and by driving electric dipole transitions between them with microwaves, werealize an XY coupling, as depicted in Fig. 4.1. We probe these spin-exchange interactions using
Ramsey spectroscopy [84]. This experiment can also be viewed as a global quench, as the initial
π/2 pulse in our Ramsey sequence instantaneously turns on the interactions.

Figure 4.1: Spin exchange between molecules in N = 0, denoted by ↓ and N = 1, denoted by↑. To emphasize the role that long-range interactions play, the exchange depicted here is between
next-nearest neighbors.

60
There has been a lot of interest recently in quantum magnetism based on effective spin1/2 systems. One of the appeals of quantum magnetic models is that although they appear very
simple, they are actually very hard to solve with classical computers [34, 38]. In addition, quantum
magnetic models are relevant to many classes of materials, including antiferromagnets, spin glasses,
spin nematics, and unconventional superconductors [34]. Any experimental system with spinspin couplings can engineer quantum magnetic models, including neutral atoms in optical lattices[35, 36, 46, 127], trapped ions [38, 39], NV centers [128], and Rydberg atoms [129]. Much of the workin this field was pioneered with neutral atoms in optical lattices. However, the contact interactions
in cold atoms require wavefunction overlap, and effective spin-spin interactions are usually mediatedby tunneling in a process called superexchange [130]. The resulting interactions are weak and limited
to nearest neighbors, and very low temperatures are required for the interactions to manifest.
Furthermore, the energy scale decreases exponentially as the lattice depth increases. In contrast,
dipolar interactions give rise to spin-spin interactions that don’t require tunneling, which enables
coherent spin dynamics to persist in systems with high entropies and low lattice filling fractions,and in very deep lattices where the molecules are completely pinned. It’s also worth noting that thedipolar interactions present in our system are different from the exchange interactions in electrons inmetals. There, an effective spin-spin interaction arises from the combination of the spin-independent
Coulomb interaction and the fermionic exchange symmetry of electrons, which is responsible forlifting the degeneracy between the singlet and triplet states.
This chapter is organized as follows. First, I describe how the molecules acquire a dipole moment and then give the expression for the dipolar Hamiltonian. Next, I discuss how the anisotropic
polarizability of polar molecules affects the observation of dipole-dipole interactions. Then I describe how we used Ramsey spectroscopy (with a spin echo) to see dipolar effects in the lattice, anddescribe the various pieces of evidence that support that the dynamics of our system is governedby an XY Hamiltonian. Finally, I give an outlook of future experiments that could be done, which
motivates the next chapter on increasing the lattice filling fraction. The discussion in this chapter
closely follows Refs. [84] and [85], of which I am an author.

61

4.1

Deriving the dipolar Hamiltonian
An electric dipole moment only exists between quantum states of opposite parity. For polar

molecules, the rotational states |N, mN i are states of definite parity, and thus molecules in the
N = 0 state have no dipole moment in the lab frame at zero electric field. In an electric field,
the eigenstates become dressed states that are a superposition of many different rotational levels,and this gives a nonzero dipole moment in the lab frame. However, it takes a very large DC field
to fully mix the rotational states and saturate the dipole moment. A field of more than 10 times
the critical field EC = B/d, where B is the rotational constant and d is the dipole moment, isrequired to get to 80% of the saturated dipole moment (see Fig. 4.2). For KRb, EC ∼ 4 kV/cm.
Given the maximum DC field we could apply in the first generation of the experiment, as wellas the instabilities described in Chapter 2, applying a DC field to study coherent dipole-dipole
interactions is problematic. An alternative mechanism for getting dipolar interactions is to apply
an AC electric field, which directly couples, on-resonance, two opposite parity rotational states.
This is the technique discussed in this chapter for realizing strong dipole-dipole interactions in thelattice.
The classical dipole-dipole interaction between two dipoles di and dj is
Vdd =di · dj − 3(di · r̂ij )(dj · r̂ij )
,
3
rij

(4.1)

where rij is the distance between the dipoles and generally the dipoles are aligned along somequantization axis. For the rest of this section, I consider the case of molecules in a 3D square
optical lattice with lattice spacing alat . The sites are labelled by i and j, and rij is in units of thelattice spacing.
The exchange process depicted in Fig. 4.1 is described by the Hamiltonian
1X
Vdd (ri − rj )
H=
2
i6=j

where Vdd (ri − rj ) =J⊥  + −− +Ŝi Ŝj + Ŝi Ŝj
,
2

(4.2)

1−3 cos2 θij, Sˆi+ and Sˆi− are spin-1/2 raising and lowering operators on site i,3
rij

and J⊥ = kd2↓↑ /(4π0 a3lat ) (k is a constant and d↓↑ is the transition dipole moment between the62
two states of our spin-1/2 system). In this section, I outline how to get Eq. 4.2 from Eq. 4.1. The
derivations closely follow Ref. [34].
To understand how the dipoles can behave like quantum magnets, we would like to find thequantum analog of Eq. 4.1. To do this, it’s useful to understand that the dipolar interaction is
the result of the contraction of two rank-two tensors. One of these tensors acts on the molecules’internal states via the dipole operators, and the other acts on the orbital motion of two molecules(this is responsible for the angular dependence of Eq. 4.1) [34]. As a result, Eq. 4.1 can be expressed
as a sum of terms that allow q units of rotational angular momentum to be transferred to orbital
angular momentum [34]. Appendix A shows that an equivalent way to write Eq. 4.1 is:
√

q=2

6 X

2
Vdd = − 3(−1)q C−q(θ, φ)Tq2 (di , dj ),rij q=−2

(4.3)

where
r
Cqk (θ, φ) =4π
Ykq (θ, φ),
2k + 1(4.4)

and the Ykq (θ, φ)’s are spherical harmonics [131]. The T ’s are irreducible tensor operators:
ˆ±
2
T±2
= dˆ±
i dj .
ˆ± ˆ0dˆ0i dˆ±j + di dj
√
.
2

(4.6)

ˆ−
ˆ− ˆ+
ˆ0 ˆ0
dˆ+
i dj + di dj + 2di dj
√
.
6

(4.7)

2
T±1
=

T02 =

(4.5)

Only the q = 0 term is relevant for this chapter, since the other terms don’t conserve energy (seesection 4.6.2 for more details). The q = 0 term of Eq. 4.3 is
q=0
Vdd

=

1 − 3 cos2 θij
3rij

dˆ0i dˆ0j += Vdd (ri − rj ) dˆ0i dˆ0j +ˆ− ˆ− ˆ+dˆ+i dj + di dj

!
(4.8)

2

ˆ− ˆ− ˆ+dˆ+i dj + di dj
2

!
.

(4.9)

ˆ are spin-1/2 operators (dˆ± are analogous to the spin-1/2 raising and lowering operators,
The d’sand dˆ0 is equivalent to dˆz ). In the basis of rotational states |N, mN i, the matrix elements of the

63dipole operators dˆp , with p = 0 corresponding to dˆ0 , and p = ±1 corresponding to dˆ± are



0
0
1 N  N 1 N p NhN, mN |dˆp |N 0 , m0N i = D (2N + 1)(2N 0 + 1)(−1)mN .
0−mN p mN
0 0 0
(4.10)
Here, D is the full dipole matrix element (which is 0.574 D for KRb), and the terms in parentheses
are 3-j symbols.
The Hamiltonian for the rotational states |N, mN i consists of the rotational energy and the
Stark effect.
hN, mN |Ĥ|N 0 , m0N i = BN (N + 1)δN,N 0 δmN ,mN 0 − hN, mN |d · |N 0 , m0N i,(4.11)

where B is the rotational constant and the coupling of the dipole with an external electric field is −d · . We take the electric field, of magnitude , to be along the z direction; thus, the second
term of Eq. 4.11 involves the p = 0 component of Eq. 4.10.
hN, mN |Ĥ|N 0 , m0N i = BN (N + 1)δN,N 0 δmN ,mN 0 − hN, mN |dˆ0 |N 0 , m0N i.

(4.12)The p = 0 term of Eq. 4.10 preserves the value of mN (if mN 6= mN 0 , then the first 3-j symbol iszero). Thus, the eigenstates in an electric field can be expressed as |Ñ , mN i, which are the states
which adiabatically connect to the states |N, mN i at zero electric field.
|Ñ , mN i =X

cN 0 |N 0 , mN i.

(4.13)

N0

By diagonalizing Eq. 4.12, we obtain the dressed state energies EÑ ,mN and the coefficients cN 0 .
The induced dipole moment of state |Ñ , mN i is then −∂EÑ ,mN∂. Fig. 4.2 shows the induced dipole

moments for the |0̃, 0i, |1̃, 0i, and |1̃, ±1i states vs. applied DC field.
For our mapping to a spin-1/2 system, we choose | ↓i ≡ |0̃, 0i and | ↑i ≡ |1̃, 0i or |˜↑i ≡ |1̃, ±1i(following the notation of Ref. [34]). The induced dipole moments are:d↓ ≡ h0̃, 0|dˆ0 |0̃, 0i.

(4.14)d↑ ≡ h1̃, 0|dˆ0 |1̃, 0i.

(4.15)d˜↑ ≡ h1̃, ±1|dˆ0 |1̃, ±1i.

(4.16)

64

Figure 4.2: Induced dipole moments of the |0̃, 0i (blue), |1̃, 0i (orange), and |1̃, ±1i states (green).
At a field of around 19 kV/cm, the dipole moment of the |1̃, 0i state changes sign. The lowest 20
rotational states were included in the calculation, which introduces an error of less than 1% in thelimit of infinite electric field. The first generation of the experiment could access electric fields up
to about 6 kV/cm.

65
For completeness, let’s work out the expression for d↓ . To do this, we use Eq. 4.13 to write|0̃, 0i =X

cN |N, 0i.

(4.17)

 X
Xc∗N 0 hN 0 , 0| dˆ0cN |N, 0i =c∗N 0 cN hN 0 , 0|dˆ0 |N, 0i.

(4.18)

N

Then
d↓ =

X
N0

N

N,N 0

For states with mN = 0, the two 3-j symbols in Eq. 4.10 are the same, and


s
s
0
N
1
N
N
N0
0


N
+ (−1)N δN,N 0 −1.

 = (−1) δN,N 0 +1
0
(2N + 1)(2N + 1)
(2N + 1)(2N 0 + 1)
0 0 0
(4.19)
Then the sum in Eq. 4.18 becomes∞Xc∗N +1 cN hN + 1, 0|dˆ0 |N, 0i,(4.20)N +1
c∗N +1 cN p.
(2N + 1)(2N + 3)
N =0

(4.21)d↓ = 2

N =0

which by Eqs. 4.10 and 4.19 becomesd↓ = 2D∞X

We are also interested in the transition dipole moments between | ↓i and | ↑i or |˜↑i. Since|0̃, 0i and |1̃, 0i are coupled by dˆ0 , we haved↓↑ ≡ h1̃, 0|dˆ0 |0̃, 0i = h0̃, 0|dˆ0 |1̃, 0i.
Again, it’s useful to write out the full expression for this. If we let |0̃, 0i =P|1̃, 0i = N bN |N, 0i, then
d↓↑ =

∞ 
X

P

N aN |N, 0i anda∗N bN +1 hN, 0|dˆ0 |N + 1, 0i + aN +1 b∗N hN + 1, 0|dˆ0 |N, 0i

N =0
∞
X

= D

(4.22)N +1.
(a∗N bN +1 + aN +1 b∗N ) p
(2N
+
1)(2N
+
3)
N =0

(4.23)(4.24)

In contrast, |0̃, 0i and |1̃, ±1i are coupled by dˆ± .
d↓˜↑ ≡ h1̃, ±1|dˆ± |0̃, 0i = −h0̃, 0|dˆ∓ |1̃, ±1i,(4.25)

66
where the minus sign in the last equality comes from the (−1)mN in Eq. 4.10.
With all of these definitions, we’re now ready to proceed with turning Eq. 4.8 into somethingthat looks like Eq. 4.2. First, let’s consider the {|0̃, 0i, |1̃, 0i} manifold. We project Eq. 4.8 onto thebasis {| ↑i ↑j i, | ↑i ↓j i, | ↓i ↑j i, | ↓i ↓j i} to get the Hamiltonian Ĥij for two molecules on sites i and j.
Since the dipole-dipole interaction is a pairwise interaction, we can then sum over all pairs i, j to
get the full Hamiltonian for the system.
Ĥij =X

Cαβ |αihβ|,(4.26)α,βq=0|βi, and |αi, |βi ∈ {| ↑i ↑j i, | ↑i ↓j i, | ↓i ↑j i, | ↓i ↓j i}. For example,
where Cαβ = hα|Vdd
q=0| ↑i ↑j i = Vdd (ri − rj )h↑i ↑j |dˆ0i dˆ0j | ↑i ↑j i = Vdd (ri − rj )d2↑ ,h↑i ↑j |Vdd

(4.27)

since h↑i ↑j |dˆ0i dˆ0j | ↑i ↑j i = h↑i |dˆ0i | ↑i ih↑j |dˆ0j | ↑j i = d2↑ . Note that only the dˆ0 terms of Eq. 4.8
contribute. Similarly,
q=0| ↓i ↑j i = Vdd (ri − rj )h↑i ↓j |dˆ0i dˆ0j | ↓i ↑j i = Vdd (ri − rj )d2↓↑ ,h↑i ↓j |Vdd

(4.28)

since h↑i ↓j |dˆ0i dˆ0j | ↓i ↑j i = h↑i |dˆ0i | ↓i ih↓j |dˆ0j | ↑j i = d2↓↑ by Eq. 4.22. The result is a 4 × 4 matrix:| ↑↑i | ↑↓i | ↓↑i | ↓↓i




2
 d↑



 0
Ĥij = Vdd (ri − rj ) 

 0


0

0

0

d↓ d↑

d2↓↑

d2↓↑

d↓ d↑

0

0

0 


0 
.

0 


2
d↓

(4.29)For the {|0̃, 0i, |1̃, ±1i} manifold, everything is the same except d↑ is replaced by d˜↑ , and theflip-flop matrix elements are different.
q=0
h˜↑i ↓j |Vdd
| ↓i ↑˜j i = Vdd (ri − rj )h˜↑i ↓j | 

ˆ−
ˆ− ˆ+
dˆ+
i dj + di dj2 | ↓i ˜↑j i = − Vdd (ri − rj ) d2˜ .
↓↑2(4.30)

67
To understand the origin of the factor of −1/2, consider |˜↑i = |1̃, 1i. Then we have
1 ˜
ˆ−
ˆ− ˆ+
˜
h↑ ↓j |dˆ+i dj + di dj | ↓i ↑j i =
2 i
=
=
=

1 ˜
1 ˜
ˆ− ˆ+
ˆ−
˜
˜
h↑ ↓j |dˆ+i dj | ↓i ↑j i + h↑i ↓j |di dj | ↓i ↑j i
2 i
2
1 ˜ ˆ+
1 ˜ ˆ−
ˆ+ ˜˜h↑i |di | ↓i ih↓j |dˆ−j |↑i i + h↑i |di | ↓i ih↓j |dj |↑i i
2
2
1(d ˜ )(−d↓˜↑ ) + 0
2 ↓↑−d2↓˜↑.
2(4.31)(4.32)(4.33)(4.34)

The answer is the same for the |1̃, −1i state. In this case, the 4 × 4 matrix is


2
0
 d˜↑


 0 d↓ d˜↑Ĥij = Vdd (ri − rj ) 2 0 −d↓↑˜

2

0
0

0
−d2 ˜

↓↑

2

d↓ d˜↑
0


0 


0 
.

0 


d2↓

(4.35)

z , Sˆ+ , and Sˆ− = (Sˆ+ )† . These
Now we want to write this in terms of spin operators Sˆi,j
i,j
i,j
i,j

operators have the commutation relations [Sˆiz , Sˆj± ] = ±δij Sˆi± and can be represented as 4 × 4
matrices. A lot of algebra can be used to rewrite Eqs. 4.29 and 4.35 asJ⊥  + −
− +
z
z
z z
Ŝi Ŝj + Ŝi Ŝj + W (1i Ŝj + Ŝi 1j ) + V 1i 1j ,Ĥij = Vdd (ri − rj ) Jz Ŝi Ŝj +2(4.36)

where the 1’s are identity matrices, and the coupling constants are given in Table 4.1. The final
step is to replace the identity matrices with the molecule density (1k → n̂k , where n̂k = 1 if there isa molecule on site k, and 0 if there’s not), and then sum over all lattice sites i and j. The simplest
way to do this is to sum over all i and j with i 6= j and then multiply by 1/2 to account for double
counting. Finally, this gives usJ⊥  + −
1X
− +
z z
z
z
Vdd (ri − rj ) Jz Ŝi Ŝj +Ŝi Ŝj + Ŝi Ŝj + W (n̂i Ŝj + Ŝi n̂j ) + V n̂i n̂j .
Ĥ =
2
2

(4.37)

i6=j

Note that to get the coupling constants in SI units requires dividing by 4π0 a3lat , where 0 is thevacuum permittivity.
The first term, proportional to Jz , is the Ising term, which comes about because parallel andantiparallel spin configurations have different energies. At zero electric field, this term is zero since68
Parameter

Coupling {|0̃, 0i, |1̃, 0i}

Coupling {|0̃, 0i, |1̃, ±1i}(d↑ − d↓ )2(d˜↑ − d↓ )2(d2↑ − d2↓ )/2(d2˜↑ − d2↓ )/2

Jz

2d2↓↑

J⊥
W

(d↑ + d↓ )2 /4V−d2↓˜↑(d˜↑ + d↓ )2 /4

Table 4.1: Coupling constants of the dipolar Hamiltonian, Eq. 4.37. To get the values in SI units,divide by 4π0 a3lat .

both induced dipole moments are zero. The second term, proportional to J⊥ , can be rewritten as
J⊥ (Ŝix Ŝjx + Ŝiy Ŝjy ) and hence is referred to as the XY term. Physically, it gives rise to the flip-flopprocess depicted in Fig. 4.1. Fig. 4.3 plots Jz and J⊥ vs. applied electric field (for KRb in a 532
nm lattice spacing), and shows that we actually get the largest interactions, corresponding to an√
induced dipole moment of D/ 3, at zero electric field with a pure XY coupling (Jz = 0, J⊥ 6= 0).
The XY term gets monotonically weaker as the electric field is increased. Second, any value of Jz /J⊥
can be obtained in principle by tuning the electric field; however, it’s not possible to independently
control the strength of the total coupling. Third, Jz has a maximum around 12 kV/cm.
The different signs for J⊥ for the {|0̃, 0i, |1̃, 0i} and {|0̃, 0i, |1̃, ±1i} manifolds imply antiferromagnetic and ferromagnetic interactions, respectively. The term proportional to W is thedensity-spin interaction, and in the limit of unit filling, it’s a constant of motion and can be ignored. The term proportional to V is the density-density interaction and is a constant for molecules
pinned in a deep lattice. The experiments described in this chapter are performed at zero electricfield, so Jz = W = V = 0, and the Hamiltonian simplifies to a long-range XY Hamiltonian:H=

1X
Vdd (ri − rj )
2
i6=j




J⊥  + −Ŝi Ŝj + Ŝi− Ŝj+.
2

(4.38)

One additional complication in our experiment is that there is a site-to-site differential energy shift
due to a residual inhomogeneous light shift arising from the molecules’ anisotropic polarizability.
This will be discussed more in the next section but it gives rise to an additional term in the

69
Hamiltonianz
i hi Ŝi .

P

These site-to-site energy shifts suppress the exchange when |hi − hj | >J⊥ Vdd (ri − rj ). Thus, the Hamiltonian we will study in this chapter is:
1X
H=
Vdd (ri − rj )
2



i6=j

 X
J⊥  + −− +Ŝi Ŝj + Ŝi Ŝj+hi Ŝiz .
2(4.39)i

Since we’re working at zero electric field all the tildes on the states will be dropped, and we will focus
our attention on spin-1/2 systems formed from the {|0, 0i, |1, −1i} and {|0, 0i, |1, 0i} manifolds.
The rate at which an isolated pair of molecules would swap from | ↑i ↓j i to | ↓i ↑j i (theexchange rate), is Jh⊥ Vdd (ri − rj ). In the experiments of Refs. [84, 85], we measured the contrast ofa Ramsey fringe, which depends on hSx i and hSy i. These expectation values oscillate at a frequencyof J2h⊥ Vdd (ri − rj ) [132]. This is because the populations in a spin-1/2 system undergoing coherent
evolution (such as Rabi flopping) oscillate twice as fast as the wavefunction. For example, consider
Rabi flopping at frequency Ω. If the initial state is | ↓i, then the state after a time t isψ(t) = cos

Ωt
2




| ↓i + sin

Ωt
2


| ↑i.

(4.40)

After a 2π pulse, ψ(2π/Ω) = −| ↓i, and |h↓ |ψ(2π/Ω)i|2 = 1, so the state has a minus sign but the
population is the same. Thus, the population is periodic with a period of 2π but the state itself(which is what hSx i depends on) is periodic with a period of 4π.
The Vdd factor gives the dependence on the distance between molecules and their orientation
relative to the quantization axis, which for the experiments in this chapter is set by a magnetic fieldin the horizontal plane (see Fig. 4.4a). The maximum value of J2h⊥ Vdd (ri − rj ) is about 52 Hz for the{|0, 0i, |1, −1i} manifold [84] and 104 Hz for the {|0, 0i, |1, 0i} manifold [85] (assuming a 532 nm
lattice spacing). Hence, this is the dominant frequency component observed in the contrast decay.
Fig. 4.4b shows the calculated spectrum of the contrast decay for the {|0, 0i, |1, −1i} manifold,which shows three pronounced oscillation frequencies on top of a broad, structured background [85].
Certain couplings are more pronounced: in particular, the head-to-tail orientation has strength -1(with respect to the maximum possible coupling), and two couplings in the horizontal plane have√strengths of 1/2 and 1/ 2.

70

Figure 4.3: Magnitude of the coupling constants for the {|0̃, 0i, |1̃, 0i} manifold (solid) and{|0̃, 0i, |1̃, ±1i} manifolds (dashed), Jz /h (red) and J⊥ /h (blue), for a 532 nm lattice spacing. For
these choices of states, we get the largest couplings at zero electric field. The lowest 20 rotational
states were included in the calculation.

(a)

(b)

Figure 4.4: (a) Couplings between the molecule in green and its neighbors, normalized to thestrongest coupling, which occurs for nearest√ neighbors along ẑ. There are three relatively wellpronounced couplings of magnitude 1, 1/ 2, and 0.5. Figure reproduced from Ref. [84]. (b)Calculated interaction spectrum for the {|0, 0i, |1, −1i} manifold. Pronounced interactions
occur√
around 52 Hz, 37 Hz, and 26 Hz, corresponding to the interactions of magnitude 1, 1/ 2, and 0.5
in (a). Figure reproduced from from Ref. [85].

71

4.2

Anisotropic polarizability and differential light shifts
As I discussed in the last section, it takes a very large DC electric field to substantiallypolarize the molecules. Coupling the rotational states with an AC electric field is much more
selective than with a DC field. Furthermore, for our choice of spin-1/2 system, we actually get thestrongest interactions at zero electric field with just the spin-exchange terms. However, there wereseveral issues we needed to address before we could actually see the spin-exchange interactions in
the experiment. In particular, the anisotropic polarizability of the molecules creates a spread intransition frequencies between | ↑i and | ↓i across the cloud and leads to a rapid dephasing whenprobing the rotational coherence [91].
A well-established technique in precision spectroscopy is to match the shape of the trapping
potentials for the two states of interest. For neutral atoms, it’s usually possible to find somewavelength of light for which the polarizabilities of the two states are equal. A well-known example isthe “magic wavelength” optical lattice for optical lattice clocks [133]. The polarizability of diatomic
polar molecules is more complicated than atoms because the molecules can be polarized about
different axes (in particular, parallel and perpendicular to the internuclear axis) [91]. For molecularstates where the electron distribution is not spherically symmetric, the polarizability parallel andperpendicular to the molecular quantization axis can be significantly different. Although it would
be possible, in principle, to find a magic wavelength for molecules, the large number of rovibrational
states in the excited molecular potential make it difficult to find a suitable wavelength where therate of off-resonant light scattering is tolerable [91]. However, there exist magic values of the electricfield (for KRb this is about 12 kV/cm [134]), as well as magic angles [91], where the angle between
the light polarization and the quantization axis is tuned to match the polarizabilities.
Ultimately we chose to find the magic angle, since a magic angle should exist even at zero
electric field. Fig. 4.5 shows measurements of the real part of the polarizability at 1064 nm forthe |0, 0i state, as well as for the three N = 1 states [91]. For the experiments discussed in thischapter, we use both the {|0, 0i, |1, 0i} manifolds and the {|0, 0i, |1, −1i} manifolds. The crossings

72

Figure 4.5: Real part of the polarizability at 1064 nm for the |0, 0i (black), |1, 0i (blue), |1, 1i(red), and |1, −1i (green) states. This was measured in a 1D vertical lattice with peak intensityof 2.3 kW/cm2 , corresponding to a lattice depth of about 92 ER for the |0, 0i molecules. Magic
angles exist for the combinations {|0, 0i, |1, 0i} and {|0, 0i, |1, −1i}, and are indicated by the circles.
Figure reproduced from Ref. [91].

73
for both states are indicated by the circles. Since the polarizabilities of the |0, 0i and |1, 1i states
never cross, we chose to not use the {|0, 0i, |1, 1i} manifold. In addition to depending on the angle
between the light polarization and the quantization axis, the polarizability also depends on lightintensity. The energies of the three N = 1 states can be found by diagonalizing the following 3 × 3
Hamiltonian:|1, 0i|1, −1i|1, 1i−α12 I−α13 I −α11 I + 1





H =  −α12 I.
−α22 I + 2−α23 I−α13 I−α23 I−α33 I + 3

(4.41)

The coefficients αij depend on the polarizabilities parallel to (αk ) and perpendicular to (α⊥ ) thequantization axis and on the angle θ between the quantization axis and the light polarization.
α11 =3αk + 2α⊥αk + 4α⊥sin2 θ +cos2 θ.
55

αk + 4α⊥2αk + 3α⊥sin2 θ +cos2 θ.
55√ αk − α⊥α12 = −α13 = 2
sin θ cos θ.
51
α23 = (α⊥ − αk ) sin2 θ.
5α22 = α33 =(4.42)(4.43)(4.44)(4.45)

By doing a global fit to the data shown in Fig. 4.5, we extracted αk /h = 10.0(3)×10−5 MHz/(W/cm2 )and αk /h = 3.3(1) × 10−5 MHz/(W/cm2 ).
A difference in polarizability between the two states also leads to a difference in the transition
frequency compared to the bare transition frequency at zero light intensity. This differential lightshift is plotted in Fig. 4.6a for light polarizations of {45◦ , −45◦ , 45◦ } with respect to the magnetic
field B̂ (the quantization axis) along the x̂, ŷ, and ẑ directions (coordinate system defined inFig. 4.4a). This is a convenient choice given that the horizontal lattice beams intersect the cell atroughly 45◦ , so if they’re horizontally polarized then the polarizations already make an angle of 45◦
with respect to the magnetic field. A global differential light shift does not really cause a problem,

74
since a global uniform shift could be accounted for by simply shifting the microwave frequency.
Instead, the problem arises because the lattice is formed from Gaussian laser beams, such that themolecules have a spread of positions, leading to a spread of differential light shifts.
We see that the differential light shift is quite small for the {|0, 0i, |1, −1i} manifold for lattice
depths below 40 ER ; thus, we chose to do the initial experiments with this state. When using the|1, 0i state, we had to put half waveplates before and after the cell for the horizontal lattices to get
the linear polarizations correct (since for the |1, 0i state, the magic angle is at about 52◦ [91]).
Another consequence of this inhomogeneous light shift is that it gives a site-to-site energy
shift. In Ref. [84], we estimated the average site-to-site shift to be about 6 Hz (for nearest neighbors)by finding the standard deviation of differential light shifts across the cloud and dividing it by thecharacteristic size of the cloud. This means that the exchange interaction is suppressed over a few
sites (see Fig. 4.7).

4.3

Simple Ramsey sequence and some technical issues
Ramsey spectroscopy is a powerful tool for studying decoherence in quantum systems. We’reinterested in using it to probe dipolar interactions in our experiment. In its simplest implementation, Ramsey spectroscopy consists of two π/2 pulses separated by some free evolution time T ,as shown in Fig. 4.8a. We typically use microwave pulses that are resonant with the rotationaltransition. The experiment begins with all of the molecules in | ↓i, which is the south pole of the
Bloch sphere. The first π/2 pulse rotates the Bloch vector about the ŷ axis from the south pole to√the equator, which puts every molecule in a coherent superposition (| ↑i + | ↓i)/ 2. After a freeevolution time T , we apply another π/2 pulse with a phase φ relative to the first pulse, which rotates
the Bloch vector about the axis n̂ = cos φ ŷ + sin φ x̂. We then measure the number of molecules in| ↓i by doing STIRAP from the ground-state back to weakly bound Feshbach molecules, followedby our normal imaging of Feshbach molecules. By scanning φ at a fixed T , we obtain a fringe asshown in Fig. 4.8c. We fit the fringe to N2tot 1 + C cos(φ + φ0 ) , where Ntot is the total number ofmolecules and C is the contrast (0 ≤ C ≤ 1). φ0 should be equal to π if the microwave frequency

75

(a)

(b)

Figure 4.6: (a) The differential light shift for the three N = 1 states relative to the N = 0 state,
for lattice polarizations of {45◦ , −45◦ , 45◦ } with respect to the magnetic field B̂ along the x̂, ŷ, andẑ directions. For this beam configuration, the |1, −1i has the smallest differential light shift. This
motivated us to use the {|0, 0i, |1, −1i} manifold as our spin-1/2 system for the initial experiments
in Ref. [84]. Note that the lattice depth is expressed in units of the recoil energy for the |0, 0i state.
Figure reproduced from Ref. [84]. (b) Measured frequency shifts vs. lattice strength for different
polarizations of the vertical lattice for the |0, 0i → |1, 0i transition. The angle of a half-waveplate
before the cell was scanned, so the total range here corresponds to scanning the polarization by 16◦ .
Here we could achieve even smaller differential light shifts, which led to longer Ramsey coherence
times (see Fig. 4.11b).

76

Figure 4.7: Because of the site-to-site energy offset, the exchange interaction is suppressed formolecules separated by more than one lattice site. Similar to off-resonant Rabi flopping, if the bare2
exchange rate is J and the detuning is δ, then the amplitude of the actual exchange is J 2J+δ2 . The
plot shows this amplitude vs. distance between molecules, assuming J = Jmax /|∆r|3 and δ = k|∆r|,
where k is the shift per site, and ∆r is the distance between two molecules (in lattice sites). Twocases are plotted: Jmax = 52 Hz and δ = 6 Hz/site (blue), relevant for the work in Ref. [84] andJmax = 104 Hz and δ = 2 Hz/site (orange), relevant for the work in Ref. [85], where we used the{|0, 0i, |1, 0i} manifold. The 2 Hz is an estimate based on the fact that the coherence times couldbe ∼ 3 times longer for the {|0, 0i, |1, 0i} manifold.

(a)

(c)

(b)

Figure 4.8: Timing diagrams and Bloch vector representation for a simple Ramsey sequence (a)and a spin echo sequence (b). Figure reproduced from Ref. [84]. (c) Typical Ramsey fringe using
the DDS to program the phase offset of the final pulse. After the initial π/2 pulse, we wait a time
T . We then apply a phase-shifted π/2 pulse and measure the number of molecules in | ↓i ≡ |0, 0i
vs. the phase of the final pulse.

77
is exactly on resonance. Measuring the contrast of this fringe then determines the amount of spin
coherence left in the system after time T . The contrast can be less than 1 for a variety of reasons,
including single-particle dephasing and interaction effects.
Before getting to the experimental results, it’s useful to discuss a few technical issues. First,
we would like to understand the role of the inhomogeneous light shift described in the previous
section. This will lead us to constraints on the pulse strengths that we should use. A quick
calculation using the expected inhomogeneous light shifts tells us the coherence times we should
expect for the simple two-pulse Ramsey sequence. Finally, I discuss the system we use to generate
microwave pulses with programmable phase delay and discuss the prospects for imaging both spinstates.

4.3.1

Effect of the inhomogeneous light shift

Several constraints determine the Rabi frequency Ω we should use (the time for a π pulse isπΩ ). For these experiments, we used square pulses, but it’s worth noting that many groups use pulse

shaping to mitigate some of the following issues. First, there is a finite energy splitting between
the three different N = 1 substates, and we want to isolate one of them as our | ↑i of a spin-1/2
system. Let’s call the smallest splitting ∆. Clearly Ω must be less than ∆ so we avoid populatingother states. However, we need the π pulse to be broad enough spectrally in order to overcome any
differential light shifts in the sample, which cause an energy spread σ and set a lower limit on the
Rabi frequency.
The relationship between the energy spread σ and the Ramsey coherence time τc depends
on the shape of the distribution of frequency deviations δ, f (δ). Let’s assume that the frequency
22

deviations have a Gaussian distribution, so f (δ) = e−δ /(2σ ) . Then taking the Fourier transform,
2 2

2

2

F (t) = σe−σ t /2 = σe−t /(2τc ) , with the Ramsey coherence time τc = σ1 . Another way to get therelationship between σ and τc is to actually calculate the Ramsey fringe as a function of time. To
do this, all we need to do is to calculate the probability that each particle is in | ↓i or | ↑i at theend of the sequence as a function of the dark time T and the phase φ of the final pulse, and then

78
average over all of the particles. The Rabi frequency is Ω. For a particle with frequency deviationqδi , there is an effective Rabi frequency Ωeff,i = Ω2 + δi2 , and thus the pulses are not exactly π/2pulses. As a function of the phase φ of the final pulse,π Ωeff,iΩ2
1 + cos(δi T + φ) ,P (T, φ) = 2 sin2
4 ΩΩeff,i(4.46)

where P = 0(1) corresponds to | ↓i(| ↑i) and the first two factors account for the tipping angle notbeing exactly π/2. The contrast is the amplitude of the average of all of these oscillations, which is
 2



Ω
2 π Ωeff,iC(T, φ) =sin
1 + cos(δi T + φ) .
(4.47)4 ΩΩ2eff,iGoing through this prescription, one finds that the contrast has a Gaussian decay with τc ≈ σ1
(assuming Ω  σ). If f (δ) were Lorentzian, then the decay would be approximately exponential.
Thus, we require1 Ω  ∆,
τc

(4.48)

where all frequencies are angular frequencies. This tells us there’s some minimum tolerable Ramsey
coherence time; otherwise, we wouldn’t be able to satisfy both inequalities in Eq. 4.48. For the|1, −1i state, ∆ ≈ 2π × 60 kHz, and the typical energy spreads are a few hundred Hz, calculated
by converting the size of the cloud to a distribution of intensities, and then to a distribution of
frequencies. This leads to Ramsey coherence times a little less than 1 ms. In practice, Rabi
frequencies around a few kHz work well and allow us to achieve π pulse fidelities > 99%. For mostof the work in Ref. [84], we use π pulses of ∼ 180 µs duration, corresponding to a Rabi frequencyof ∼ 2π × 2.6 kHz. Experimentally, we examined the robustness of the experiment to having alarger inhomogeneous light shift (which corresponds to having a shorter Ramsey coherence time)
and to imperfect pulses, and we found the dynamics to be rather insensitive to such imperfections(Section 4.4.2).

4.3.2

Imaging both spin states

To read out the spin evolution after the Ramsey sequence, we need the excitation fraction,N↑ /Ntot or N↓ /Ntot , where Ntot = N↑ + N↓ is the total number. Normally, we measure the number

79
of | ↓i molecules, N↓ , at the end of the sequence. Assuming that the total molecule number is stable,
this gives a measurement of the excitation fraction. Normalization via the measurement of the total
molecule number would allow us to remove technical noise, but requires that we image both spinstates. To achieve this, we STIRAP back the N = 0 molecules, image the resulting atoms, do a πpulse to transfer the N = 1 molecules to N = 0, and then repeat STIRAP back and imaging. The
only problem with this is that during the first STIRAP pulse, the down leg laser is on for about
5 µs, and this is enough to kill about 20-25% of the molecules in the N = 1 state (see Fig. 4.9),
presumably by exciting molecules in the N = 1 state to an excited electronic and rotational state.
The fraction η of N = 1 molecules that remain can be calibrated, as shown in Fig. 4.9b. However,
we never consistently saw a significant improvement in the SNR of the Ramsey fringes using thistechnique and thus we ended up not using it. This could be because the calibration itself requires
many shots to get a small uncertainty and any drifts in laser power could change η. In the future,
we may be able to use a more spectrally selective STIRAP transfer (with improved coherence) toachieve this goal.

4.3.3

Creating phase stable microwaves

It’s desirable to have precise and fast control of the microwave phase, especially for more
complex pulse sequences. For the simple Ramsey and spin echo sequences, we only need twophases, since all we care about is the phase of the final pulse relative to the initial pulse(s). To
accomplish this, we initially used two Agilent signal generators (which were both referenced to thesame 10 MHz oscillator), and changed the phase of one of them by hand. In this scheme, one of
the generators was used for the first pulse(s) and the other was used for the final pulse. There may
have been a more efficient way to do this, but ultimately we decided to pursue other options. To domore complicated sequences such as the WAHUHA sequence (section 4.4.1), we need four distinct
phases and we need to be able to switch between them relatively quickly (less than 1 ms).
Initially we tried an analog solution, which was an analog phase shifter from Mini Circuits(part JSPHS-2484+). A control voltage shifts the phase, but also affects the amplitude. To account

80

(a)

(b)

Figure 4.9: Imaging both spin states. We first do STIRAP back and image the molecules that werein | ↓i = |0, 0i. Next, we apply a π pulse, which transfers | ↑i to | ↓i. Then we STIRAP back andimage a second time. The only problem with this is that there’s some loss of the N = 1 molecules
during the imaging of the N = 0 molecules. (a) The loss of | ↑i = |1, −1i molecules vs. pulse timeof the down leg laser. (b) Starting with all molecules in |0, 0i, we perform a Rabi experiment. This
allows us to normalize this loss by fitting both components to sine functions. For this dataset, thefraction of |1, −1i molecules lost is 21(8)%. More data would be required to reduce the error.

Figure 4.10: System used to generate microwaves with arbitrary phase control. An Agilent E8257D
signal generator, which is referenced to a 10 MHz Wenzel oscillator, provides a 2 GHz tone. Halfof the 2 GHz signal is sent to the DDS, where it is frequency divided by 2 and then used as the
DDS clock; the other half is used as the local oscillator for the mixer. We program an AD9858
to put out an RF frequency of around 228 MHz. This goes to the IF port of the mixer. The
output is ∼ 2.228 GHz, and we send this through a bandpass filter to get rid of the carrier andother unwanted frequency components before amplifying the signal and sending it to the horn. The
mixing process maps the phase of the IF (which we can control with 14 bits of resolution) directly
onto the phase of the microwave signal.

81

(a)

(b)

1.2

1.0
τ = 0.69(3) ms

0.8

Contrast

Contrast

1.0

0.6
0.4

0.6
0.4
0.2

0.2
0.0

τ = 2.2(2) ms

0.8

0.0

0.5
1.0
1.5
Dark time (ms)

0.0

0

1

2
3
4
Dark time (ms)5

6

Figure 4.11: Typical Ramsey contrast decay for |0, 0i → |1, −1i (a) and |0, 0i → |1, 0i transitions(b). We could achieve smaller inhomogeneous light shifts for the |0, 0i → |1, 0i transition, resulting
2
2
in a longer Ramsey coherence time. The fitting function is e−t /(2τ ) , which is the expected form if
the energy spread has a Gaussian distribution.

for the amplitude variation as the phase changed, we used different pulse times depending on thephase we wanted. Ultimately, this was not flexible and efficient, so we decided a digital solution was
better. Since it’s hard to find an inexpensive DDS operating at 2.3 GHz (for this the clock would
need to be more than 4.5 GHz), we decided to use a lower frequency DDS (at around 230 MHz)and mix it with a 2 GHz local oscillator to get the required frequency. We use an AD9858 from
Analog devices, which has a maximum clock frequency of 2 GHz. The device can be programmedwith four phase-frequency profiles, and it’s feasible to switch between these profiles in  1 µs using
two digital inputs. We use a 2 GHz signal from an Agilent synthesizer (E8257D) as both the local
oscillator and also to clock the DDS (it’s divided by 2 in the DDS so we end up with a 1 GHz
clock). The mixer maps the phase of the RF directly onto the phase of the microwaves. After themixer, we use a bandpass filter to filter out unwanted frequency components. A splitter, amplifier,and horn complete the setup, which is shown schematically in Fig. 4.10.

82
4.3.4

Experimentally measured Ramsey decays

Experimentally measured Ramsey decay curves are shown in Fig. 4.11. The data for the|0, 0i → |1, −1i transition (a) was taken in a 40 ER deep lattice with the same lattice polarization
configuration as in Fig. 4.6a. We find a Ramsey coherence time of around 700 µs. For the |0, 0i →|1, 0i transition (b), we can achieve a smaller differential AC Stark shift (Fig. 4.6b), and as a result
the coherence time is longer (more than 2 ms). For these experiments the lattice depth was typically
20 ER , although we could also achieve similar coherence times in deeper lattices. However, all ofthese coherence times are far too short to observe dipolar interactions, and we need to incorporatea spin echo to mitigate the single-particle dephasing.

4.4

Initial experiments in the {|0, 0i, |1, −1i} manifold√
For the |0, 0i → |1, −1i transition, the transition dipole moment d↓↑ is 0.98×D/ 3, where thefactor of 0.98 reflects an estimated 2% admixture of another hyperfine state. This gives |J⊥ /(2h)| =52 Hz, which implies that the dynamics occur on the tens of ms timescale. As we saw in the previous
section, the Ramsey coherence time is limited by the inhomogenous light shift to less than 1 ms.
This inhomogeneous light shift causes the Bloch vector for each molecule to precess at a slightly
different rate. However, this rate is determined only by where the molecule sits and in principle
it doesn’t change throughout the experiment. By applying a spin echo pulse halfway through thefree evolution (Fig. 4.8b) we can reverse this precession so that after a total time T , every molecule
has no net precession from the inhomogeneous light shift. However, the spin echo has no impact
on the dipolar interactions, and ideally any remaining dephasing should be due to interactions.
Typical contrast decay curves with the spin echo pulse are shown in Fig. 4.12. One of themost striking features of these curves is the oscillations on top of an overall decay. We attributeboth the decay and the oscillations to dipolar interactions. We fit the contrast decay curves to an
empirical function
C(T ) = Ae−T /τ + B cos2 (πf T )(4.49)

83
to extract both a coherence time τ and oscillation frequency f . Low lattice fillings and the longrange interactions in the lattice both give rise to a spread of interaction energies, which leadto dephasing and loss of contrast. At very dilute fillings, the interaction energy spectrum canhave a strong contribution from the strongest nearest-neighbor interaction (the vertically oriented
interaction in Fig. 4.4). The oscillations can then arise from the beating of this particular frequency
with the contribution from molecules with much smaller energy shifts.
One of the signatures of an interaction effect is the dependence on density. We reduce thedensity without changing the overall shape of the molecular distribution by holding the moleculesin the lattice for a few seconds to allow molecules to be removed by single-particle loss [72]. The
reduced density is proportional to the number of molecules remaining. The two curves shown in
Fig. 4.12a show that the oscillation frequency is basically the same for different densities (also see
Fig. 4.12c), but the coherence time becomes shorter for higher densities. We expect τ ∝ 1/N
because
τ∝

1
R̄3
1
∝
∝
,
hEint iJ⊥
J⊥ N

(4.50)

where R̄ is the average interparticle spacing and the density ρ = R̄−3 ∝ N , given the way we reducethe density. Fig. 4.12b shows the coherence time vs. N for many datasets and clearly shows a 1/N
dependence.

4.4.1

WAHUHA pulse sequence

The spin-echo sequence, as well as more complicated pulse sequences, can be used for dynamical decoupling, which can remove dephasing and extend coherence times [135, 136, 137, 138].
One particular pulse sequence, named WAHUHA after its inventors, disentangles the dipole-dipole
interactions of two particles [139]. The timing diagram is shown in Fig. 4.13a. The WAHUHA
sequence swaps between eigenstates of the dipolar Hamiltonian, allowing for subsequent rephasing
after a total dark time T . When we apply the multi-pulse sequence, the oscillations in the contrast
are suppressed and the data fits well to a simple exponential decay (Fig. 4.13b). The inset shows
the difference between the black points (WAHUHA sequence) and red points (spin echo), which

84

(a)

(b)

(c)

Figure 4.12: (a) Two contrast decay curves corresponding to different initial densities. The fit curves
are Eq. 4.49. (b) All of the experimentally measured coherence times for a spin echo experimenton the |0, 0i → |1, −1i transition. The fit is C + A/N , with A and C fit parameters. Figures
reproduced from Ref. [84]. (c) Looking at typical datasets corresponding to different densities, we
see the frequency obtained from fitting to Eq. 4.49 doesn’t significantly change with density.

85
clearly shows that the oscillations are suppressed with the multi-pulse sequence.

Figure 4.13: WAHUHA sequence: (a) The pulse sequence used for WAHUHA. The pulses about±x̂ have a relative phase of ±90◦ with respect to the initial pulse. (b) The contrast decay for a
two-pulse Ramsey sequence (purple, same data as in Fig. 4.11a), spin echo (red), and WAHUHA(black). The inset (green points and curve) shows the difference between the red and black curves
and highlights that the multi-pulse sequence largely suppresses the oscillations in the red data.
(c) Number of nearest neighbors in the lattice vs. filling fraction. For fillings < 0.05, most of themolecules have 0 or 1 nearest neighbors. Figure reproduced from Ref. [84].

To understand how WAHUHA works, we examine the wavefunction for two molecules after
each of the pulses. Initially, both molecules are in | ↓i. The π/2 pulse puts each of them in asuperposition of | ↑i and | ↓i.
111
ψ(0) = √ (| ↑i + | ↓i) ⊗ √ (| ↑i + | ↓i) = (| ↑↑i + | ↓↓i + | ↑↓i + | ↓↑i).
2
2
2(4.51)The states | ↑↑i and | ↓↓i don’t interact via the XY Hamiltonian, while | ↑↓i and | ↓↑i do. The
three triplet states | ↑↑i, | ↓↓i, and √12 (| ↑↓i + | ↓↑i) are eigenstates with energy 0, 0, and J⊥ /2
respectively. A (π/2)x pulse converts between |N i = √12 (| ↑↑i+| ↓↓i) and |Ii = √12 (| ↑↓i+| ↓↑i), so
it basically acts as an echo between the noninteracting and interacting states. Thus we can rewrite
Eq. 4.51 as
1
ψ(0) = √ (|N i + |Ii).
2(4.52)

After the first T /8 free evolution, |N i acquires no phase, while |Ii acquires a phase e−iJ⊥ T /(16~) .

86
The first −(π/2)x pulse swaps this phase to |N i (there’s also an additional factor of −i). Thus,
the state after the second pulse is

ψ

T+
8




−i 
= √ e−iJ⊥ T /(16~) |N i + |Ii .
2(4.53)

Immediately before the third pulse, the state is

ψ

3T −
8




−i 
= √ e−iJ⊥ T /(16~) |N i + |Iie−iJ⊥ T /(8~) .
2(4.54)The (π/2)x swaps the phases again so

ψ

3T +
8




1 
= √ e−iJ⊥ T /(8~) |N i + e−iJ⊥ T /(16~) |Ii .
2(4.55)

Finally in the evolution from 3T /8 to T /2, |Ii picks up another phase e−iJ⊥ T /(16~) . This meansthat the state after T /2 is 1T= √ e−iJ⊥ T /(8~) (|Ii + |N i) = e−iJ⊥ T /(8~) ψ(0).
ψ
2
2(4.56)

Thus apart from an overall phase, the state after half of the sequence is identical to the initialstate. The pulses about x̂ in the second half have the same effect, and the center π pulse is stillnecessary for mitigating single-particle dephasing. It’s worth noting that the WAHUHA sequence
has no effect on dipole-dipole interactions beyond those that come from isolated pairs. As shown in
Fig. 4.13c, for very dilute lattice fillings (< 5%), most of the molecules have at most one neighbor;as a result, the multi-pulse sequence can have a large effect.
It’s quite important that the echo pulse is a rotation about x̂. Fig. 4.14 shows what happens
when the echo pulse is about ŷ instead of x̂. In this case, the entanglement doesn’t reverse at the endof the sequence, and the contrast decay is qualitatively very different from that in Fig. 4.13b. With
the echo pulse about x̂, the state immediately before the readout pulse is −e−iJ⊥ T /(4~) ψ(0); however,


i−|Ii
when the echo pulse is about ŷ it would be e−iJ⊥ T /(4~) |N√, which cannot be expressed as a2
trivial multiple of the initial state.

87

Figure 4.14: Same as WAHUHA, except the echo pulse in the middle is about ŷ instead of x̂. The
contrast decay is qualitatively very different from that in Fig. 4.13b, as it decreases sharply at shorttimes and then decays on a longer timescale.

(a)

(b)

Figure 4.15: (a) Varying the duration of the echo pulse by up to 6% (so that it’s 0.94-1.06 π),
we see no significant effect on the coherence time or oscillation frequency. (b) Contrast decay for
different lattice depths, which varies the spread of energies across the cloud. We don’t observe any
significant differences over the range 20-40 ER . Figure reproduced from Ref. [84].

88
4.4.2

Robustness to imperfections

One natural question is how sensitive we are to various imperfections in the experiment.
Earlier I stressed that we could achieve > 99% π-pulse fidelities. But is this actually necessary?
To answer this question, we introduced pulse errors of up to 6%. As shown in Fig. 4.15a, we don’t
see any significant change in the contrast decay when the echo pulses aren’t perfect. However, if
we were to use a different initial tipping angle (other than π/2) then these pulse errors are moreof an issue. We also looked at the effect of doing the experiment at different lattice depths (again
using the same lattice polarizations as in Fig. 4.6a). This changes the magicness of the trap andthe Ramsey coherence time. Fig. 4.15b shows that the oscillation frequency and coherence timeare very similar over the range 20-40 ER .

4.5

Second set of experiments
As was discussed in Section 4.1, J⊥ is twice as large for the {|0, 0i, |1, 0i} manifold. This

gives us an easy way to change the strength of the interactions. In particular, the faster dynamics
were useful to see that there were multiple frequency components present in the contrast decay.
In addition, the observed dynamics for the faster interactions are basically identical to the slower
interactions (except that time is rescaled), and this strongly suggests that processes other thaninteraction effects are negligible in the system’s dynamics. Furthermore, the theoretical tools to
describe the experiment had to be improved in order to explain our data, which exemplifies how
the theory and experiment build off of one another in our system. The discussion in this section
closely follows Ref. [85], of which I am an author.

4.5.1

Contrast decay for the |0, 0i → |1, 0i transition

The main experimental change when working with the |1, 0i state is that we needed to add
some additional waveplates to the horizontal lattice beams to get close to the magic angle (around52 degrees for each beam). After doing this, we were able to achieve Ramsey coherence times longer

89
than 2 ms (Fig. 4.11b). We repeated the spin echo experiments following the same protocol outlined
earlier. Typical contrast decay curves for ∼ 104 molecules for both choices of | ↑i are shown in
Fig. 4.16a. Larger couplings are observed for | ↑i = |1, 0i and the faster spin dynamics makes theexperiment less sensitive to technical limitations. Fig. 4.16b compares the contrast decay for thetwo choices of | ↑i when time is rescaled by a factor of two for the |1, −1i data. This highlights thatthe dipole-dipole interactions are responsible for the observed dynamics and that other possible
contributions play a negligible role in the decay.

4.5.2

Fitting to multiple frequencies

For the |1, −1i data, we fit to the sum of a single frequency oscillation and an exponential
decay (Eq. 4.49). The low sampling rate makes it difficult to fit to more frequencies. Furthermore,as shown in Fig. 4.7, for the |1, −1i state the exchange is suppressed even at molecular spacings of
1.5-2 alat , which could suppress some of the strong couplings on the diagonal. However, looking at
Fig. 4.4, there should be other pronounced frequency components in the data. Since the timescales
are faster for the |1, 0i state, we tried fitting the contrast decay to three frequencies, with thefunctional form√C(T ) = Ae−T /τ + B1 cos2 (πf T ) + B2 cos2 (πf T / 2) + B3 cos2 (πf T /2).

(4.57)

These three frequencies are the three dominant interactions as shown in Fig. 4.4a, and are predicted
to give rise to observable frequency components in the contrast decay. The spacing between the
couplings gets increasingly smaller at lower frequencies, and this is responsible for the overall decay.
We see better agreement with the three frequency fit (Fig. 4.17). We see that the reduced χ2 iscloser to 1 with the three frequency fit, with the lowest χ2 occurring for a frequency around 108
Hz, which is close to the expected value of 104 Hz from J⊥ .

90

(a)

(b)

Figure 4.16: (a) Typical contrast decay curves for | ↑i = |1, −1i (top) and | ↑i = |1, 0i (bottom)
for roughly the same density (∼ 1.2 × 104 molecules). The coherence time is clearly shorter for the|1, 0i data, which is expected due to the stronger interactions. (b) Taking the two datasets from(a) and scaling the time axis for the |1, −1i data by a factor of two, we see that the curves collapse
onto each other reasonably well, which highlights that dipolar interactions are responsible for theobserved dynamics. Figures reproduced from Ref. [85].

Figure 4.17: (a) Fitting a |1, 0i contrast decay curve to Eq. 4.49 (1 frequency, dashed green) andEq. 4.57 (3 frequencies, solid blue). (b) For this dataset, fitting to 3 frequencies yields a reduced
χ2 closer to 1. For the 3 frequency fit, the best fit occurs for a frequency around 108 Hz, in good
agreement with the expected value from J⊥ . Figure reproduced from Ref. [85].

91
4.5.3

Theory comparison

Our simple fitting functions, Eqs. 4.49 and 4.57, do a good job at extracting the coherencetime and telling us about the dominant energies in the interaction spectrum. However, our theory
colleagues can fit the data to a more sophisticated model. The basic idea, illustrated in Fig. 4.18a, is
to use a cluster expansion, whereby the system of ∼ 104 molecules is broken up into small clustersof ∼ 10 molecules, and the dynamics within the clusters is computed exactly. The theoreticaladvance in Ref. [85] was to develop a “moving-average cluster expansion” (MACE), which is able
to converge much faster than a discrete cluster expansion. The MACE builds an optimal cluster foreach molecule i by finding the set of largest couplings Vij to it. The expectation values hSix i and hSiy ipare computed exactly for each cluster, and the global Ramsey fringe contrast C = 2 hSix i2 + hSiy i2
is calculated by summing these expectation values over all molecules. Details of the convergence of
the algorithm are given in Ref. [85].
The theory was able to estimate the filling fraction by matching to the data in Fig. 4.16, andthe result is that the filling is about 5% for that data. The molecular distribution was assumed
to be a shell, given our expectation that the initial Rb density is high in the center and there
will be many multiply occupied sites that do not produce molecules (this will be discussed morein depth in the next chapter). However, the theoretical decay curves are largely independent of
the exact geometry in which the molecules are created. Another issue that the theory-experimentcomparison can address is the role of long-range interactions in the experiment. In particular, it’sinteresting to see if the experimental data can be described accurately by only considering nearest
neighbor interactions (or nearest and next-nearest neighbor interactions). As shown in Fig. 4.18b,c,
the curves that only consider short-range interactions don’t agree well with the experimental data,
which says that long-range interactions are needed to explain the dynamics of our system.

92

(a)

(b)

(c)

Figure 4.18: (a) Schematics of the discrete cluster expansion (DCE) and the moving-average cluster
expansion (MACE). (b) and (c) Theoretically calculated contrast decay truncating the interactions
at nearest-neighbor (b) and nearest and next-nearest neighbor (c). In both cases, the black points
and dashed curve are the data and theory shown in Fig. 4.16 for | ↑i = |1, 0i, and the purple
and light blue solid curves are the predicted contrast decay curves for different filling fractions andtruncated interactions. The disagreement of the short-ranged calculations with the data supports
the claim that long-range interactions are present in the experiment. Figures reproduced from
Ref. [85].

93

4.6

Future ideas
Our initial observations of spin-exchange are just the first step in studying quantum mag-netism with polar molecules. The results of the theory comparison tell us that although we can see
effects of long-range interactions in the experiment, the lattice fillings are low (less than 10% evenfor the highest densities). This is consistent with other methods for determining the filling, such as
the losses in tubes [90] and direct imaging [84]. We would really like to increase the lattice filling sothat we can study more complex dynamics, such as the buildup of entanglement and correlations,and the propagation of spin excitations. In the next chapter, we show that by tailoring the initialatomic distributions, we can achieve much higher lattice fillings.

4.6.1

Simple simulation of spin transport

To motivate why higher lattice fillings are important, it’s useful to think about future experiments we would like to do. One example is to study spin transport. Here, the idea would be tocreate a small number of | ↑i molecules in a sea of | ↓i molecules. The molecules are pinned in thelattice and don’t move, but the rotational excitations can move from site to site. Experimentally,
we would want to see how these excitations propagate at different lattice fillings and electric fields,
as well as in the presence of disorder, both from the finite filling fraction and from intentionallycreated disorder in the potential that the spin excitations see. This is intimately linked to theidea of many-body localization, where strong disorder could potentially localize these excitations[37]. This closely parallels work in trapped ions looking at the speed limit of the propagation ofcorrelations in 1D chains of trapped ions [38, 39]. In our system, we could study these effects in 2
or 3 dimensions.
A naı̈ve expectation is that the excitations will propagate faster for higher lattice fillings. To
test this, I did a simple 2D simulation on a 15 × 15 grid. The center molecule is always | ↑i and therest are | ↓i, distributed randomly with filling f . I then compute the coupling matrix, calculate thewavefunction after some time T , and compute how far on average the excitation has moved. Note

94

(a)

2

5

1

10
5

(c)

10

15

15

0

distance moved

4
2
50
time (ms)

100

40
20
0

0

2
distance moved

4

10

15

0

6
4
2
0

(f)
# of occurences

(e)

0

5

(d)

6

0

0.5

10

0.5

15

distance moved

1

1.5

5

# of occurences

(b)

0

50
time (ms)

100

0

2
4
distance moved

6

40
20
0

Figure 4.19: Simulation of spin transport. Dynamics in a single 2D plane (for example, the xy
plane of Fig. 4.4a) is considered for the coupling strengths for the {|0, 0i, |1, −1i} manifold. (a)Initial distribution with f = 0.3. The colorbar represents excitation: 0 is an empty site, 1 is | ↓i,and 2 is | ↑i. (b) Distribution after evolving for 50 ms. Note the change in colorbar. There is a
finite probability of finding the excitation on any site. (c) Distance the excitation moves (in latticesites), R̄ (Eq. 4.58), vs. evolution time T for f = 1 (blue), f = 0.5 (red), f = 0.3 (green), f = 0.15
(magenta), and f = 0.05 (black). The open circles are averages of 50 realizations (except for f = 1
where the dynamics is always the √same), and the solid red curve is a fit to the f = 0.5 data,assuming diffusive transport (R̄ ∝ T ). (d) Same as (c) except there is a site-to-site energy shift
3 Hz × ((i − 8)2 + (j − 8)2 ) (i and j label the lattice sites along the two directions). (e) Histogramof R̄ after 50 ms for f = 0.05. The simulation was repeated 500 times. (f) Same as (e) but for
f = 0.5. Here there is a clear displacement from the center.

95
that since this is the one excitation subspace, this requires diagonalizing at most a 152 ×152 matrix,
which is computationally easy. Fig. 4.19a shows a typical initial distribution for a 30% filling, whilepanel b shows the distribution after 50 ms. After 50 ms, the excitation has some probability to beon any site. The expected distance R̄ (in lattice sites) is given byR̄ =X

pi ri ,

(4.58)iwhere pi is the probability that the particle is on site i and ri is the distance from the center site(in lattice sites). Panel c shows R̄ vs. T for different filling fractions, and panel d is the same
but adding a site-to-site energy shift. Higher lattice fillings clearly lead to faster propagation. For
fillings less than unity, the transport is not exactly diffusive (the red solid curve in panel c shows a√
fit of the red data to R̄ ∝ T ) and there appears to be a saturation at longer times, which could
be due to the finite system size. In the new experiment, we plan to have an imaging resolution ofaround 1 µm, which is about 2 lattice sites. We would want the excitation to propagate at least
this far in order to detect that it moved.
For very dilute fillings the actual distribution is very different from one realization to the next,
with some realizations having two particles close to each other (which will exchange quickly) andother realizations having only very small couplings. In the simulation this manifests as a very large
standard deviation of the expected distance the excitation moved, which suggests that it would
be difficult to get good statistics in the experiment for small lattice fillings. This is highlighted inpanels e and f, which show histograms of the expected distance after 50 ms for both f = 0.05 (e)and f = 0.5 (f).

4.6.2

Realizing additional terms in the Hamiltonian

In addition to achieving higher lattice fillings, it would also be interesting to realize additionalterms in the dipolar Hamiltonian (Eq. 4.3). As was shown earlier, the full dipolar Hamiltonian has
more terms than just those in Eq. 4.8. In this chapter, I have so far focused on the q = 0 term.
The q = ±2 terms are particularly interesting because they can be used to realize a novel form of

96
spin-orbit coupling [88]. Most realizations of spin-orbit coupling in cold atoms can be understood asa single-particle effect, with the coupling usually arising from Raman beams that couple hyperfine
states, which generally means that spontaneous emission and heating are problematic. In contrast,
the spin-orbit coupling described in Ref. [88] arises from interactions between the molecules and inprinciple shouldn’t have any issues with spontaneous emission or heating.
The q = ±2 terms look like
q=±2
Vdd
=


−3  ˆ+ ˆ+ −2iφij
ˆ− dˆ− e2iφij sin2 θ ,
d
d
e
+
d
ij
i
j
i
j
3
2rij

(4.59)and correspond to processes that transfer between rotational angular momentum and orbital angular momentum. This method of coupling the spin degree of freedom to the particles’ orbital motion
is similar to recent experiments in magnetic atoms [140, 141]. Because of the sin2 θ term, Ref. [88]proposes to do the experiment in a single 2D layer, with θ = π/2. The interactions occur withthree states of the molecules and can drive the following process:|1, −1i + |0, 0i → |0, 0i + |1, 1i.

(4.60)As shown in Ref. [88], the strength of the coupling is 3J⊥ , so in order for this to work, the energy
difference between the |1, 1i and |1, −1i states must be less than this. The easiest way to bringthese states to degeneracy is to apply a large magnetic field of about 1260 G [88]. The second
generation experiment will have the capability of reaching such large magnetic fields.

Chapter 5

Increasing the filling fraction

The last chapter described our observation of spin-exchange interactions for molecules pinned
in a deep lattice; however, the lattice filling was quite dilute (less than 10%). This is consistent
with the filling extracted from the measured loss rate of molecules confined in tubes and from direct
imaging of the spatial extent of the cloud [84, 90]. The reason we could observe spin exchange atsuch low fillings is that there is a strong decoupling between spin and motion that makes the spin
entropy the relevant quantity [83, 85]. Since we can easily initialize all of the molecules in the same
quantum state, the spin entropy can be very low. This chapter addresses the question of why thefilling was so low and presents a strategy that directly leads to a much higher lattice filling in our
experiment.
Experimental imperfections, such as finite temperature, limit the filling fraction of cold atomic
gases when they are loaded into optical lattices. Atomic Fermi gas experiments achieve fillings
close to, but not exactly, unity [21, 142]. Exactly what filling fraction is required for polar molecule
experiments depends on the experiment one wants to do. As shown in the simulations at the endof the previous chapter, the propagation of rotational excitations can occur at any density, but
occurs much faster for higher fillings. Many other interesting proposals don’t require unit filling.
Fillings in the range 0.25-0.5 are sufficient for studying many-body localization [37], and the spinorbit coupling discussed in the previous chapter can work even at fillings of ∼ 0.1 [88]. However,even when an experiment is possible at low fillings, the SNR will almost surely be better at higher
fillings.

98
One notion of sufficiently high filling is that the system is well connected, which means there
is a reasonably efficient way to propagate information between any two parts of the system. Anotherway of saying this is that there has been a percolation in the system, which is defined as the fillingat which an infinite cluster develops in an infinite system [143]. For an infinite 3D cubic latticewith nearest neighbor interactions, the percolation threshold occurs at a filling of around 0.3 [143],
so this gives us a rough target. In our system, the exact critical filling will be different, and likely
less than this, since we have long-range interactions and a finite system size.
There are two main approaches one might adopt to achieve higher fillings. The first is to try
to directly cool the molecules and then load them adiabatically into the lattice, while the second
is to start with two species of cold atom gases, optimize their loading into the lattice, and thenmake molecules, as proposed in Refs. [144, 145]. This second option has been demonstrated for
Rb2 Feshbach molecules, where the molecules were created out of a Mott insulator with two atoms
per site [146].
The first option doesn’t work very well for KRb because it’s very difficult to cool the moleculesin the presence of chemical reactions. There are proposals to evaporatively cool reactive polar
molecules by using an electric-field gradient to remove hot molecules and the dipolar interactions torethermalize [119]. Using universal dipolar scattering to evaporatively cool spin-polarized fermions
was demonstrated beautifully for magnetic atoms, but there the ratio of good to bad collisions
was very favorable [50]. For KRb, very large electric fields and harmonic confinements would benecessary in order to reach a favorable regime of good to bad collisions [119, 120]. In addition, we’vealso seen evidence that the molecular gas is heated a lot when loading and unloading the lattice.
Doing the same experiment for K, we saw almost no heating (Fig. 5.1). This suggests problems intrying to adiabatically load molecules into the lattice. More experimental work would be required
to determine the origin of this problem. In 2011-2012, we spent a lot of time trying to create colder
molecules in the harmonic trap by using a species selective dipole trap around 790 nm, which
affected K but not Rb and was useful for optimizing the overlap between the two species. However,ultimately the poor fractional conversion of the atom gases into Feshbach molecules limited the99
temperature of the molecular gas to T /TF ∼ 1.
The second option benefits from the high conversion of preformed atomic pairs to Feshbach
molecules, and the subsequent efficient optical transfer to the ground state. As shown in Ref. [72],the conversion of sites with one K and one Rb atom to molecules is nearly unity (87 ± 13%). Thisissue will be addressed further in Chapter 6, where we show that in the case of deep lattices and fastmagnetic-field ramps, the conversion efficiency seems to be exactly 100%. Previous measurements
of inelastic loss rates of Feshbach molecules with K and Rb atoms suggest that having multiple Rb
atoms on a lattice site is detrimental to molecule production on that site [65, 69]. This motivatedus to maximize the number of lattice sites that have one K and one Rb atom. The ideal states of
the initial atomic gases are a bosonic Mott insulator (MI) with one atom per site and a fermionicband-insulator where the filling approaches unity at the center. If the temperature of the K gas
is not sufficiently low, the number of K atoms will need to be much larger than the number of
Rb atoms, so that we avoid multiple occupancies of Rb and so that the K density approaches one
particle per site in the center of the lattice. Even in the harmonic trap, we see that if the Rb
density is too high, this negatively affects the molecule production (Fig. 5.2).
Many of our previous attempts to increase the molecule filling looked at the final result asa function of many parameters (atom number, interspecies interactions, lattice depth, harmonic
confinement, etc.) without seeing a significant dependence on these parameters. Since the molecule
production involves several steps, we decided to systematically study each step individually before
putting them all together. This ultimately led to a better understanding of the dependence of themolecular density on the parameters listed above and motivated a strategy to increase the densityof molecules.
In Section 1, I give a brief theoretical discussion of the bosonic Mott insulator and fermionicband insulator, and then discuss the challenges of creating a dual insulator. In Section 2, I describechanges to the imaging system that were instrumental to the success of this study. In Section 3,
I present measurements of the fillings of the individual atomic gases. In Section 4, I describe theeffect of the insulators on each other and show it’s optimal to turn off the interspecies interactions

100

(a)

(b)

160

50

120
100
80
60
40

before lattice ramp: 198(5) nK
after lattice ramp: 202(9) nK

20
0

0

4

8 12 16
Time (ms)20

24radial size (µm)

radial size (µm)

140

40
30
20

before lattice ramp: 155(6) nK
after lattice ramp: 331(12) nK

10
0

0

2

4
6
8
Time (ms)1012Figure 5.1: (a) Ramping up and down the vertical lattice in 200 ms and holding for 50 ms (total
time 450 ms), the K gas is not significantly heated.
We measure the temperature T by fitting theqradial size during a TOF expansion to σr = σ02 + kBmT t2 , where σ0 is the in situ size and m is
the particle mass. (b) Repeating the same experiment for ground-state molecules, we see a lot ofheating, which could mean we didn’t load the lattice adiabatically. The exact origin of this problem
is not very well understood.

Figure 5.2: We see a hole in the center of the Feshbach molecule distribution when the Rb density
is too high. This image was taken in the optical dipole trap.

101

Figure 5.3: Quantum synthesis for realizing high molecule fillings in the lattice. Left: We load K(blue) and Rb (red) atoms into a 3D optical lattice, with many more K atoms. In the center of thelattice, we have a MI of Rb and band-insulator of K, each with a filling near unity. Right: Sites
with one Rb and one K produce molecules with high probability, whereas sites with multiple Rb
atoms or only a single atomic species don’t produce molecules. Figure reproduced from Ref. [92].

by loading the lattice at aKRb = 0. In Section 5, I describe experiments that studied the conversion
efficiency of Feshbach molecules vs. Rb number and show the final results of the improvement of
the molecule filling fraction. In the best case, we achieve molecule fillings > 25%. In Section 6,
I conclude and give an outlook for future work. Most of the discussion of this chapter follows
Ref. [92], of which I am the lead author.

5.1

Target distributions: Rb Mott insulator and K band insulator
Our strategy for realizing higher molecule fillings in the lattice relies on the precise experi-mental control that’s available for manipulating the initial atomic quantum gas mixture in the 3D
lattice. Specifically, we need to prepare low entropy states of both species and combine this withefficient molecule production at individual lattice sites. The combination of efficient magnetoassociation of preformed pairs [72] and efficient optical state transfer via STIRAP means the second stepshould work well; however, creating a low entropy state of both species with the optimal density of
one particle per site is very challenging.
The basic scheme is illustrated in Fig. 5.3. By loading a nearly pure Rb BEC into a deep
3D lattice, we achieve a MI, where repulsive interactions between the Rb atoms drive a transition

102
to a state that has an integer number of particles per site. The initial BEC density should be
sufficiently low that we end up with one atom per site. For spin-polarized K atoms, Pauli blocking
prevents any site from having more than one K atom, assuming that all atoms are loaded into thelowest band. The initial K gas needs to be cold in order to load most of the atoms into the lowest
band. The optimum case is a spin-polarized K band insulator with a filling ∼ 1 atom per site in
the center, which given our relatively high initial temperature, requires starting with a relatively
large number of K atoms. In addition, we require that the Rb MI is well spatially overlapped withthe center of the much larger K distribution and that the high filling of each species is preservedin the presence of the other.

5.1.1

Rb Mott insulator

Ultracold bosons in an optical lattice are well described by the Bose-Hubbard Hamiltonian[32, 103]:
H = −JXhi,jiâi † aˆj +

XU
i

2

n̂i (n̂i − 1) +X
(V (ri ) − µ)n̂i ,

(5.1)

i

where âi † and âi are bosonic creation and annihilation operators, and n̂i is the number operator
for site i. The first term represents tunneling at a rate J between adjacent lattice sites (the sum isover neighboring sites i and j), the second term is the onsite interaction energy (which is repulsive
for Rb), and the third term represents the external harmonic confinement. The chemical potentialµ sets the particle number.
In shallow lattices (U/J  1), the atoms are delocalized over the entire lattice and there is awell-defined phase on each lattice site [147]. In this superfluid phase, the number of atoms on any
site is uncertain and is given by a Poisson distribution. In momentum space, the superfluid displays
an interference pattern with well-resolved peaks spaced by twice the lattice recoil momentum 2~k(Fig. 5.4). For deep lattices (U/J  1), the energy cost U of having multiple occupancies suppresses
number fluctuations. The result is a MI with Fock states on each lattice site. In the MI, themacroscopic phase on each lattice site has a maximum uncertainty and the interference pattern in103

Figure 5.4: The superfluid-to-Mott insulator transition. The lattice depth is 12, 17, and 22 ER
from left to right. The white bar shows twice the lattice recoil energy. The optical depth (OD) isdisplayed by the color scale. Figure reproduced from Ref. [92].

momentum space disappears (Fig. 5.4). The superfluid-Mott insulator transition was first observedin 2002 [147], and seen under a quantum gas microscope in 2010 [148, 149].
To gain an intuitive understanding, it’s useful to consider the case of zero temperature andzero tunneling. In this case, the density of the MI depends on the ratio of the chemical potential
to the interaction energy, µ/U . Because of the additional harmonic confinement V (r), the local
chemical potential µ − V (r) determines the occupancy of a lattice site. Specifically, the occupancy
of site (i, j, k) is α if
(α − 1) <µ − V (i, j, k)
≤ α.
U

(5.2)

This condition allows for a self-consistent computation of the number of atoms for a given chemical
potential, N (µ) [150]. In the thermodynamic limit, µ has a form very similar to the Thomas-Fermi(TF) result for a harmonic trap. It’s useful to calculate this limit in order to understand howµ depends on N and the harmonic trapping frequency. The calculation is outlined in Ref. [103].
Assuming a spherically symmetric distribution, the density n(r) (in particles per lattice site) is
given byn(r) =µ − 21 mω 2 r2
,
U

(5.3)

where ω is the underlying harmonic confinement of the lattice beams. The extent of the cloud,104
2
Rmax , occurs when µ − 12 mω 2 Rmax
= 0, which gives Rmax =q

2µ(to get this in number of latticemω 2sites, divide by alat ). Then N is fixed by
Z Rmax /alat3n(r)d r =

N=

0

0

µ − 12 mω 2 r2
4πr2 dr.
U(5.4)

N 2/5 ω 6/5 .

(5.5)

Z Rmax /alat

Doing the integral and solving for µ we obtainµ=

15 U a3lat m3/2
29/2 π!2/5For different harmonic confinements along the three lattice directions, like we have in our experiment, ω is replaced by the geometric mean harmonic confinement ω̄. What’s important about thisexpression is how µ scales with N and ω̄: µ ∝ N 2/5 ω̄ 6/5 . To have at most one atom per site, we
require µ ≤ U . As shown in Fig. 2.14, a typical value of U is h × 1 kHz (for Rb in a 1064 nm
lattice). For the harmonic confinements needed to overlap the species and achieve the required
densities, the maximum N we can have and still satisfy µ ≤ U is a few thousand. But in general,
we want to have as small of a harmonic confinement as possible in order to maximize the numberof atoms N in the n = 1 MI.

5.1.2

K band insulator

The fermionic analog to the bosonic superfluid and MI is the metal and fermionic MI, respectively. In the fermionic MI, strong repulsive interactions (onsite interaction U ) between two
spin components lead to a state with one particle per site. The fermionic nature of the particles
combined with repulsive onsite interactions provides an ideal experimental setting to study fundamental questions related to strongly correlated electron systems [151]. The fermionic MI was
first observed in 2008, with experimental signatures including a suppression of doubly occupied
sites, a reduction of the compressibility, and the appearance of a gap in the excitation spectrum[151]. More recently, the fermionic Mott insulator was observed with imaging at the single-particle
level [142]. If the chemical potential increases further (µ > U ), the state transitions to a band
insulator, where there is one particle of each spin state per site [21, 152]. In our experiment, we105
use spin-polarized fermionic 40 K atoms in the |9/2, −9/2i state, so if the atoms are in the lowest
band, there can be at most one atom per site. In this case, the state with one atom per site is aspin-polarized band-insulator.
The filling of fermions in optical lattices is more complicated to calculate than for bosons, anddepends on an interplay between temperature, tunneling, and harmonic confinement. In contrast
to the case of bosons, low temperature alone is not sufficient to guarantee unit filling, but we do
require sufficiently low T /TF . A few rough arguments tell us that large particle numbers and strong
harmonic traps are going to be necessary to achieve a density ∼ 1 particle per lattice site. Ref. [152]gives the following condition for a band insulator:mω̄ 2 λ2
8



3N
4π

2/3
 12J,(5.6)

where J is the tunneling energy, λ is the lattice wavelength, and ω̄ = (ωr2 ωz )1/3 is the geometricmean harmonic confinement. The left hand side of Eq. 5.6 is the Fermi energy for noninteracting
fermions in a deep 3D lattice [153], while the right side is the total tunneling bandwidth. Examining
K tells us that N 2/3 (ω /(2π))2  106 Hz2 , which we roughly satisfy forthis condition at 10ERr

ωr = 2π × 50 Hz and 105 atoms.
Another simple way to think about this is to calculate the partition function on each lattice
site, assuming no tunneling. This calculation is outlined in Ref. [151] for a two-component gas.
Although tunneling definitely plays a role in our experiment, the conclusions of this simple model
give intuition that is useful for optimizing the K conditions. Each lattice site can have at most one
particle, so the partition function for each site isZi = 1 + Ze−βi .

(5.7)

The energy offset i is due to the harmonic confinement ( = 0 at the center of the lattice),β = 1/(kB T ) is the inverse temperature, and the fugacity is Z = eβµ . µ is the chemical potential,
which sets the number of particles. The probability that a site is occupied is Ze−βi /Zi , so theZpeak filling is the probability that the central site is occupied, which is Z+1. To find µ, we require106that the total number of particles is N :N=

X Ze−βi
i

Zi

.

(5.8)

Fig. 5.5 shows results for small systems. We see that keeping the particle number fixed,
the peak filling fraction increases when we reduce the temperature or increase the harmonic confinement. Increasing the number of atoms at fixed temperature and harmonic confinement also
increases the peak filling.

5.1.3

Challenges of making a dual insulator

One of the main challenges of producing a dual Bose-Fermi insulator with one atom of eachspecies per site is that there is generally a large mismatch in the size and spatial overlap of thetwo species, which comes from the different masses, polarizabilities, and quantum statistics. In
this section we show that we need large harmonic confinement frequencies to support both species
against gravity with a small differential sag and to achieve sufficiently high densities for K. We also
briefly discuss challenges of evaporatively cooling the mixture to very cold temperatures. For theexperiments described in this chapter, all optical potentials are at λ=1064 nm.
A lot of intuition about the constraints on the K and Rb distributions can be obtained fromconsidering the densities in the optical trap. The distributions will change when loading the latices;
however, in order to achieve a density of ∼ 1 particle per lattice site (6.6 × 1012 cm−3 ), we require acomparable density in the optical trap. The distribution of a sufficiently large BEC in a harmonic
trap is given by a TF function, where the density has an inverted parabolic distribution and thecloud radius along an axis with harmonic confinement ωi is
4/5

Ri = aho (15N aRb,Rb )1/5 ,where aho =

q

~
mωi

(5.9)

is the harmonic oscillator length and aRb,Rb = 100 a0 is the background scat-tering length [154]. For our typical trap frequencies for Rb of ωr (ωz ) = 2π × 25 (180) Hz and 105
atoms, Rx = Ry ≈ 17 µm, and Rz ≈ 2.3 µm, which are quite small. For a degenerate Fermi gas,

107

Occupancy

1.0
0.8

250 Hz, 50 nK, N=103 , f0 =0.62

0.6
0.4
0.2
0.0
-20

(b) 1.0
Occupancy

(a)

0.8
0.6
0.4
0.2
0.0
-20

-10
0
10
20
Position (lattice sites)

300 Hz, 50 nK, N=103 , f0 =0.79

-10
0
10
20
Position (lattice sites)��������

Occupancy

1.0
0.8

250 Hz, 30 nK, N=103 , f0 =0.85

0.6
0.4
0.2
0.0
-20

(d) 1.0 250 Hz, 50 nK, N=2 103, f0 =0.83
Occupancy

(c)

-10
0
10
20
Position (lattice sites)

0.8
0.6
0.4
0.2

0.0
-20

-10
0
10
20
Position (lattice sites)

Figure 5.5: Calculating the density of spin-polarized fermions in the lattice in the zero tunneling
limit for different temperatures and harmonic confinements. Here the trap is spherically symmetricand only a radial cut is shown. Comparing (a) and (b) we see that higher harmonic confinements
give higher peak fillings. Comparing (a) and (c) we see that lower temperatures give higher peakfillings. Comparing (a) and (d) we see that more atoms give higher peak fillings.

108
the density is given by the Fermi-Dirac distribution:n(ρ) = −

kB mT
2π~2

3/2

Li3/2 (−Z exp(−mωr2 ρ2 /(2kB T ))),

(5.10)

where Li3/2 is a polylog function, Z is the fugacity, and ρ2 = x2 + y 2 + λ2 z 2 is a scaled coordinate,
where λ = ωz /ωr characterizes the asymmetry of the trap [155]. The fugacity is related to T /TF
by Li3 (−Z) = 6(T−1[155]. Fig. 5.6a compares the density distributions of a pure BEC and/TF )3a Fermi gas at T /TF = 0.25 (each with 105 atoms), and shows that the size of the Fermi gas issignificantly larger. When optimizing the lattice loading to get one atom of each species per site,
we would need to use a much smaller number of Rb atoms and the Rb cloud would be even smaller.
Fig. 5.6b shows the peak density of K in the harmonic trap for different harmonic confinements,
atom numbers, and temperatures. Like the simple model in the last section, there are several ways
we can achieve high densities of K: increase the number of atoms, increase the trap frequencies,
and use colder gases. Since we can only increase the number so much (perhaps to 3 × 105 ), and thescaling of density with harmonic confinement is much stronger, we choose to use relatively large
harmonic confinement frequencies. For Rb, we’re fighting the opposite battle. The typical BEC
densities are very high, and as such, we have to go to small atom numbers.
We next briefly discuss the vertical overlap. Since the size of the K cloud is much larger
than the BEC, we can tolerate a vertical offset of a few µm. K and Rb have different masses andpolarizabilities at 1064 nm (see Table 2.1), so the two species have different trap frequencies (ωK =
1.37 ωRb ). Consequently, there will be a differential gravitational sag ∆z = g ω21 − ω21≈z,Rb0.27 g, where ωz is the vertical trap frequency.
2
ωz,Rb

z,K

Clearly, ∆z becomes smaller as the vertical trap

frequencies increase. For our normal optical trap frequencies of ωr (ωz ) = 2π × (25) 180 Hz for Rb,
the differential sag is about 2 µm, which is sufficiently small.
We want both gases to be as cold as possible. For Rb, this means a nearly pure BEC and for
K it means a deeply degenerate Fermi gas. However, the way we do the evaporative cooling limits
the temperature of K. In the optical trap, we evaporate the gases by lowering the intensity of the
OT beams. However, Rb is heavier and has a smaller trap frequency, so it falls out of the trap first.

109

(a)

(b)

Figure 5.6: Comparing the densities of degenerate Bose and Fermi gases in a harmonic trap. (a)Comparison of a density cut along the radial direction for 105 atoms for a K gas at T /TF = 0.25
(blue) and a Rb BEC (orange) with a harmonic confinement ωr (ωz ) = 2π × 51 (290) Hz for K (thefrequencies for Rb are 1.37 times less). This is the expected harmonic confinement of the opticaltrap plus lattice beams in the experiments of Ref. [92]. The density of each species is normalized
to its peak density ρ0 . (b) Density of Fermi gases of T = 0.25 TF (blue) and T = 0.5 TF (red) ina 3D harmonic trap vs. atom number. The harmonic confinement is ωr (ωz ) = 2π × 51(290) Hz(solid) and 2π × 32(180) Hz (dashed). The density corresponding to one particle per lattice site isdenoted by the dashed black line.

110This means we do most of the cutting on Rb and we lose very few K atoms. We rely on collisions
between K and Rb to thermalize the K. Once Rb condenses the K is no longer efficiently cooled
since the BEC is much smaller than the Fermi gas. As a result, we observe that as the temperaturedrops, the K and Rb clouds are not in thermal equilibrium, with the K hotter. Experimentally we
see that we can only reach temperatures of T /TF ≈ 0.3 (see Fig. 5.13).
To summarize, there is a huge compromise between the harmonic confinement and the densities of the atomic species. If we use a weak harmonic trap, the n = 1 MI will be larger but the
K filling will be lower. If we use a much stronger trap, the size of the n = 1 MI will be very small
but the density of K will be high. Ultimately we choose a harmonic confinement such that the Kfilling fraction saturates around 105 atoms (see Section 5.3), and we have to live with the small Rb
MIs (less than 5000 atoms).

5.2

Top imaging
Normally we would image the atoms from the side, using a probe beam that propagatedthrough the entire chamber. This imaging direction is at 45◦ with respect to the principal axes
defined by the OT and horizontal lattice beams. As a result, we were able to measure one radial size(σr ) and the vertical size (σz ). However, the imaging resolution was about 5 µm (Airy disk radius),
which was sufficient for imaging in time-of-flight (TOF), but caused problems for imaging in situ,
especially for very small clouds. As discussed in the previous section, the typical cloud radius of a
BEC in the vertical direction is only a few µm, which is less than the imaging resolution. Measuring
the filling of the atomic gases requires knowing the size along all three directions. Since σz is lessthan the imaging resolution, we get no information about that direction. Thus, we were extracting
all of the information from σr . Since the filling fraction scales like 1/σr3 , any uncertainty in σr
contributes a larger fractional uncertainty to the measured filling fraction.
The bigger problem was that the system was unstable. The filling of the MI would changeover time (over the course of a few hours) even if the lattice alignment didn’t seem to change.
Looking at the in situ distribution, we would sometimes see the cloud split into two parts (see111

Figure 5.7: An image showing how the vertical lattice would split the cloud in two.

Fig. 5.7). To better understand the origin of this problem, and to see the cloud size along bothradial directions, we decided to image the gas along the vertical direction. Once we set up thevertical imaging, we quickly understood the origin of these problems. It turns out that the ITO
plates were probably vibrating when we ramped to high magnetic field. Because the plates weremechanically coupled to the coil assembly, this could have been due to mechanical forces arisingfrom the changing current in the coils. This manifested as terrible fringes in the images caused bya relative motion of the probe beam between the shadow and light frames, which were separated by∼ 500 ms (see Fig. 5.8a). We determined that the magnetic-field ramps were problematic because
if we waited at high field for a few seconds before imaging, the amplitude of the fringes was lower.
The vibrations would also affect the vertical lattice, since the lattice beam also propagated throughthe ITO plates. This was likely what caused the cloud to sometimes split in half. Since we didn’t
need to apply electric fields, we decided to remove the ITO-coated glass from the region that thebeams propagated through (we couldn’t remove the whole plate since it was necessary to hold themicrowave coil assembly together). This made the OD images much better (Fig. 5.8b) and thefilling fraction of the atomic gases became much more robust, at least as far as we could tell fromour detection. This also improved the stability of the vertical lattice.
A schematic of the top imaging setup can be seen in Fig. 5.9. For the objective, we used asingle achromatic doublet from Edmund optics, with a focal length of 125 mm. In a test setup,
we used this objective to image a 500 nm pinhole, which acts as a point source. This allowed us

112

Figure 5.8: (a) OD image before we removed the ITO. The fringes between the shadow and lightframes have an amplitude of about 1 OD. (b) Similar OD cloud after we removed the ITO plates.
Removing the ITO made a huge difference, since the two clouds have a similar OD, and the SNR
is much higher in (b).

113
to measure the imaging resolution to be 6.0(3) µm, which translates into a Gaussian rms widthof 2.5(1) µm, or 4.7(2) lattice sites. The objective lens is not particularly well AR-coated (thereflection is 1-2% at 1064 nm). To avoid reflections that could create a superlattice, we decided tohave separate beam paths for the lattice and probe beams. The probe beam propagates roughly
vertically, while the lattice beam propagates at roughly 6◦ from vertical. The vertical lattice is not
focused at the position of the atoms but rather at the retroreflecting mirror, which is a half mirrorthat sits above the objective lens without obstructing it.

5.2.1

Calibrating the saturation intensity

For reliably measuring the filling fractions of the atomic gases, it’s necessary to be able toaccurately measure the OD. In the past, we usually imaged dilute gases after a TOF expansion.
For studying the filling fraction of the atomic insulators, we needed to image dense gases in situ.
This introduces several systematic effects in the absorption imaging that need to be understoodand characterized. Here I discuss the relevant aspects for our experiment, following the discussion
from Rob Wild’s PhD thesis [156].
We take three images, a shadow frame with the atoms, a light frame with the probe beam
but no atoms, and a dark frame with no probe beam. These images are 2D arrays of “counts” on
the CCD camera that we call Ishadow , Ilight , and Idark , respectively. The measured optical depth(OD) is then
ODmeas = ln

Ilight − Idark
Ishadow − Idark.

(5.11)

There are two main effects that give corrections to Eq. 5.11. The first is that there is a maximum
OD we can measure (ODsat ) due to “bad” light, which can have the wrong polarization or frequency
to be absorbed or could be light that scatters around the atoms. This “bad” light isn’t absorbed
and limits the contrast between the shadow and light frames. Ultimately, the dynamical range ofthe digital CCD camera also limits the maximum OD one can measure (a camera with N bits canmeasure an OD up to ln(2N )) [156]. Applying the correction from ODsat gives a modified OD given

114

Figure 5.9: New setup for top imaging. The lattice beam is focused at a half mirror, which
retroreflects the beam. The lens for the imaging beam sits below the half mirror. A PBS on theprobe beam after the final mirror before the cell allows us to align the light polarization with thequantization axis, which is important for achieving a large ODsat . The position of the ITO plates
is denoted by the black dashed lines above and below the glass cell. The region of glass on the ITO
plates around where the probe beam and lattice beam go through was removed.

115
by [156]:

ODmod = ln

1 − e−ODsat
e−ODmeas − e−ODsat

.

(5.12)

There is also a correction due to imaging at finite intensity, which is a result of the probe beamsaturating the transition. This gives the actual OD [156].
ODact = ODmod + (1 − e−ODmod )

I
Isat

.

(5.13)

Especially when doing absorption imaging of small clouds, we need to use short probe pulse durations so the atoms don’t move significantly during the probe pulse. We find ∼ 10 µs is sufficiently
short for our purposes. However, these short probe pulses are usually done at relatively high intensities. We can measure ODsat by directly measuring very dense clouds, and we usually findODsat ∼ 3.5 − 4. To calibrate the intensity of our probe beam, we follow the procedure outlined inRef. [156], where we image the same thermal cloud at different intensities and then fit to Eq. 5.13
(see Fig. 5.10).

5.2.2

Fitting the atomic distributions

When imaging from the top, we measure the two radial sides and compute the verticalsize according to σz = σr /A, where A is the aspect ratio. Using the side imaging, we measured
A = 6.4(1) for large thermal clouds in the combined potential of the OT and a 24 ER 3D lattice.
We assume that A is the same for small clouds in the lattice.
We fit the absorption images (after correcting the OD pixel by pixel) to obtain σx and σy
(σr =

√

σx σy ) and then fit to either a 2D Gaussian surface or a 2D TF surface to extract the peakdensity ρ0 . The Gaussian isnGauss (x, y) =

√

2

2

2

2

2πσz ρ0 e−x /(2σx ) e−y /(2σy ) .

(5.14)and the peak filling isfGauss =N (λ/2)3
N A(λ/2)3=.
(2π)3/2 σx σy σz(2πσx σy )3/2

(5.15)

3

Ilight (10 counts/pixel/µs)

116

3.5
3.0

Isat=1.1(1)e3 counts/pixel/µs
OD=1.8(1)

2.5
2.0
1.5
1.0
0.5
0.0
0.4

0.8

1.2
OD

1.6

2.0

Figure 5.10: Typical calibration of I/Isat , following the procedure outlined in Ref. [156]. We image
the same cloud with different intensities. The actual OD is the intersection of the curve with the xaxis, while the steepness of the curves tells us Isat in counts/pixel/µs. As a measure of the intensity,
we average the number of counts in the light frame around the location of the center of the atom
cloud.

117
The 3D TF distribution is
x
Rx

2



2



n3D
TF (x, y, z) = ρ0

1−

y
Ry

2



2


−

z
Rz

2 !



2 !


−

.

(5.16)

Then integrating along z,
n2D
TF (x, y) = ρ0rwhere zmax = Rz

1−



x
Rx

2

−

Z zmax
1−
−zmax



y
Ry

2

x
Rx

−

y
Ry

−

z
Rz

dz,

(5.17). Doing the integral yields

4ρ0 Rzn2D
TF (x, y) =
3


1−

x
Rx

2


−

y
Ry

2 !
.

(5.18)

Integrating along x and y gives
ZZ
N=

n2D
TF (x, y) dx dy =8π
ρ0 Rx Ry Rz ,15

(5.19)

which can be inverted to givefTF =15N (λ/2)3
15N A(λ/2)3=.
8πRx Ry Rz
8π(Rx Ry )3/2

(5.20)

f
In both cases I used the fact that ρ0 = (λ/2)
3 (λ is the lattice wavelength), and f represents thepeak filling.

5.3

Measuring the filling of the individual atomic gases
In this section, we characterize the density of the individual atomic gases. This is an impor-tant step as fillings of ∼ 1 particle per lattice site are necessary for our quantum synthesis scheme
to be successful. We find good agreement with the conclusions from Section 1, namely that we needsmall number Rb clouds and large number K clouds to achieve densities ∼ 1 particle per latticesite for each species at the trap center.

5.3.1

Measuring the filling of the Rb Mott insulator

Looking back at Eq. 5.5, we see that for our typical trap frequencies, the number of atoms in
an n = 1 MI will be less than 5000. Given the aspect ratio of our trap, this implies a radial size of

118
about 10 µm, which is slightly less than twice the imaging resolution. For these small clouds, we
begin to worry about the effect of the imaging resolution on the determination of the size of thecloud, and consequently on the determination of the filling fraction. Normally what one would do
is to measure the point spread function of the imaging system and then deconvolve the images withthis function. In our case, it’s difficult for us to make such a small object (since a BEC of 5000
atoms is already very small and the size scales weakly with number), and also the magnificationof the imaging system is such that the pixel size is not so different from the imaging resolution(this was chosen mainly due to space constraints). Instead, our approach is to model what wewould expect the n = 1 MI to look like, given the finite imaging resolution and pixelation. We
then fit the expected distribution the same way as we would fit the experimental data to obtainthe filling fraction vs. number. We then compare the model’s predicted filling fractions with thefilling fractions measured in the experiment. Even though they’re not useful for making molecules,
we can also produce large MIs in the experiment, which should be much less sensitive to the effects
of imaging resolution and pixelation, and we can benchmark the model by its agreement with theselarge clouds.
To be more specific, we calculate the expected distribution at T = 0 with no tunneling basedon Ref. [150], using the expected trap frequencies for the combined OT and 3D lattice potential.
The simulation begins by choosing a value of µ/U and then distributing the particles according to
Eq. 5.2. Summing over the occupation of each site allows us to numerically find the relationshipbetween µ and N . To compare with the experiment, we sum the number of atoms along the z
direction (following the experimental geometry where the probe beam integrates along z), andconvolve the resulting 2D distribution with a Gaussian filter of rms width 4.5(5) lattice sites toaccount for the finite imaging resolution. To account for pixelation, every 6 × 6 group of latticesites is mapped to one pixel (in the experiment, the pixel size is 3.2 µm ≈ 6alat ). In the limit oflarge µ/U , the density approaches a TF distribution; thus we fit both the experimental data andthe distributions from the model to a TF distribution.
The distribution of a perfect MI of roughly 2.5 × 103 atoms with ωr = 2π × 38 Hz and

119

(a)

(b)

(c)

Figure 5.11: (a) 2D distribution of a perfect MI of ∼ 2.5 × 103 atoms after integrating along thez direction. The colorbar represents the number of atoms. (b) Same distribution after convolving
with a Gaussian filter of rms width 4 lattice sites. (c) Same distribution after binning every 6 × 6
sites into one pixel. This is what we would expect to see in the experimental images.

120
A = 6.4 is shown in Fig. 5.11a. Applying the Gaussian filter gives the distribution shown in panelb, and binning the distribution into pixels finally leads to the distribution shown in panel c. Thisis our best estimate of what the experimental distribution should look like. We then fit the model
distribution to a 2D TF surface (Eq. 5.18) and extract fTF according to Eq. 5.20, which is shownby the orange band in Fig. 5.12. The band reflects the experimental uncertainties in the trap
frequencies (ωr = 2π × (38 ± 2) Hz for Rb and ωz = (6.4 ± 0.1) ωr ) and imaging resolution. For thenumber of atoms, we use the number extracted from the fit, not the sum of the array (althoughthese agree to within a few percent). Note that we do not subtract off the imaging resolution in
any way. This leads to predicted fillings less than 1 for very small MIs. From the model, we canalso compute the peak occupancy in the lattice, which is just µ/U promoted to the next largest
integer. This is shown by the green curves in Fig. 5.12.
Experimentally, we fit the in situ images to a TF distribution and calculate fTF according to
Eq. 5.20. Fig. 5.12 shows the measured fillings over a range of Rb numbers of 2 × 103 to ∼ 105 . We
find excellent agreement with the T = 0 model for large clouds, as expected because the imaging
resolution and pixelation play less of a role; additionally, the production of larger BECs is much
more robust. For smaller clouds, the agreement is also quite good, and we verify that the n = 1
MI occurs for small Rb numbers, less than about 5000 atoms.

5.3.2

Measuring the filling of the K band insulator

Rb lattice (which is only 9E K ),Fig. 5.13 shows the filling fraction for K measured for a 23ER
R

extracted from a Gaussian fit to the in situ distribution. Note that the distribution of cold fermionsin the lattice is not necessarily given by Eq. 5.10. We choose a Gaussian distribution since it matches
the observed density distribution well. We observe that the filling saturates to around 0.8 for K
numbers > 105 . At this point, the gas appears to be incompressible, since the peak filling no longer
increases as we increase the number, which is indicative of the formation of a band insulator in thecenter of the lattice. We also see that the atoms nicely fill up the lowest Brillouin zone (Fig. 5.14),
which we verify by taking band mapping images. We attribute the filling being less than 1 to finite

121

Peak filling

5
4
3
2
1
0

1

2

5 10 20 50 100
Rb number (103 )5 10 20 50 100
b number (103 )

1

0.7

0.8

0.6

0.6

0.5

0.4

0.4

0.2

0.3

0

0

10
20
30
K number (104 )T /TF

Peak filling

Figure 5.12: Peak filling of the Rb Mott insulator. For more than 5 × 103 atoms, the agreementwith the T = 0 model is quite good. The orange band is a calculation of the filling extracted fromT = 0 model, and the width reflects the uncertainty in the trap parameters. The green is thepeak occupancy (restricted to be an integer) for the same distributions. Figure reproduced from
Ref. [92].

0.2

Figure 5.13: Peak filling of K vs. K number (blue points). For NK > 105 , the filling no longer
significantly changes as we increase the number of atoms, which suggests the formation of an
incompressible band insulator. The filling is less than 1, which we attribute to finite temperature.
The red points show the measured T /TF of the K gas prior to loading the lattice. TF is calculated
from the number of atoms and trap frequencies and T is measured by measuring the temperatureof a Rb thermal gas that’s equilibrated with the K. Figure reproduced from Ref. [92].

122

Figure 5.14: Left: Image of the K gas after 11.5 ms TOF, where the lattices were ramped off slowly
to do band mapping. The colorbar shows the optical depth (OD). Right: A cut along the whiteline showing the distribution vs. quasimomentum (averaged along the other direction). This showsthat most of the atoms are in the lowest band. Figure reproduced from Ref. [92].

temperature and tunneling, but experimental errors in the imaging procedure could also play a role.
As discussed earlier, the temperature of the K gas is limited to T /TF ≈ 0.3 by the poor efficiencyof the evaporative cooling once Rb condenses (see the red points in Fig. 5.13).

5.3.3

Using inelastic loss to measure the K filling

We can also measure the K filling fraction using an inelastic loss process. This work is
described in Ref. [93]. The states normally used for making molecules are the |1, 1i state for Rb
and the |9/2, −9/2i state for K. These are the lowest energy states for both species and the mixture
is stable. If we transfer Rb to the |2, 2i state (which is ∼ 8 GHz higher in energy at 550 G), themixture is no longer stable and undergoes spin-changing collisions, which leads to loss of both
species. At a collision energy of 1 µK, the calculated inelastic collision rate is β = 6 × 1012 cm3 /s
Rb lattice. Fig. 5.15 shows an in[121], which corresponds to a loss timescale of ∼ 2 ms in a 25 ER

situ image of the K gas after transferring Rb to the |2, 2i state. The hole in the center occurs where
the K and Rb overlap and shows that the species are well overlapped.
Fig. 5.16a and b show typical loss curves for Rb after a 2.1 ms RF sweep that transfers Rb
to the |2, 2i state. The data show loss on two different timescales: a fast loss in a few ms, which we

123

Figure 5.15: After transferring Rb to the |2, 2i state, K and Rb undergo inelastic spin-changing
collisions, leading to the loss of both species. This creates a hole in the center of the K distribution.
Figure reproduced from Ref. [92].

attribute to loss on sites that initially have a K and Rb atom, followed by a slower decay, which we
attribute to tunneling of atoms followed by inelastic loss once they get on the same site. The dashedlines are a fit to the sum of two exponential decays with different time constants. We can extract
the fraction of Rb that is lost on the shorter timescale from these fits. Fig. 5.16c plots this fraction
lost on the shorter timescale vs. Rb number. The solid curve shows the expected loss for a MI withtemperature T /JRb = 15 and total radial harmonic confinement ωr = 2π × 33 Hz. The shadedband reflects a 10% uncertainty in the trap frequency and 30% uncertainty in the temperature. For
small Rb numbers, we expect to have only one Rb per site, and the fraction lost should be equal
to the K filling fraction (assuming no double occupancy of K). From the fit, we extract a K filling
fraction of 77(2)%, which is in good agreement with the direct measurements shown in Fig. 5.13.
For higher Rb numbers, multiple occupancies cause a reduction of the fractional loss, assumingthat one Rb and one K are lost per inelastic collision.

5.4

Studying the dual insulator
In the last section, we showed that both atomic species can have fillings near unity. However,

the optimal atom number for the two species is very different: less than 5 × 103 for Rb but morethan 105 for K. A natural concern is that the K will act as a heat bath and melt the Rb MI. There
have been a few recent experiments looking at the effect of a bath of fermions on the superfluid-

124

a

6
5
4

0.7

3
2
1

Rb number (103 )

c

0.8

0
30

b

25
20

Fraction lost

Rb number (103 )

7

0.6

0.5

0.4

15
10

0.3

5
0

0

20

40

60

Hold time (ms)

80

103

5⨯103

104

5⨯104

105

Rb number

Figure 5.16: An alternative way to measure the filling fraction of K is to look at the loss of Rb
induced by spin-changing collisions, in the limit of a small number of Rb atoms. The fits in (a) and(b) are double exponential fits, and the points in (c) are either the ratio of the coefficients of these
two exponential fits (blue diamonds) or the fraction of atoms remaining after 8 ms (red circles).
Figure reproduced from Ref. [93].

125Mott insulator transition [157, 158]; however, these experiments focused on the visibility of theinterference pattern, the lattice depth at which the coherence disappears, and the sharpness of thetransition. We’re more interested in the filling fraction of the Rb MI in the presence of a large bathof K.
When loading the dual insulator at low temperatures, one might expect that attractive interactions will enhance the overlap of the two species [145, 159]. However, for the relatively hot K
temperatures in our system, we see that nonzero interspecies interactions detrimentally affect theRb filling fraction (Fig. 5.17). However, at aKRb = 0, fRb is unaffected by the presence of K. We
attribute the degradation in the filling when aKRb 6= 0 to the fact that the BEC fraction is significantly less than 1. To directly verify that heating from the K is playing a large role, we measured
the dependence of filling on interaction strength for two K temperatures. The fRb dependence on
aKRb is more sensitive when the K gas is hotter (Fig. 5.18). However, at aKRb = 0, fRb is the samefor the two cases. This highlights that the K and Rb distributions are basically independent at
aKRb = 0.

5.5

Putting all of the ingredients together
The first step of molecule production is magnetoassociation, where we adiabatically sweep

the magnetic field across the Feshbach resonance at B0 = 546.6 G from high to low field. As
shown in the previous section, we want to load the lattice at aKRb = 0 to preserve the filling of
the MI. Loading the lattice at aKRb = 0, we would then first need to jump the magnetic field
above the resonance so that we can initiate the magnetoassociation process. This jump should bediabatic to avoid promoting atoms to higher bands [160]. This effect will be discussed further in
the next chapter. The high atomic densities on isolated lattice sites make it difficult to sweep thefield fast enough. To circumvent this issue, we use an RF pulse to transfer K to the |9/2, −7/2istate, which doesn’t experience the 546.6 G resonance. After ramping B to 563 G, we transfer
the K back to the |9/2, −9/2i state and then proceed with the magnetoassociation. The final
sweep goes from 563 G to 545.6 G in 5 ms. We use STIRAP to transfer the Feshbach molecules

126fwith K /fRb

1
0.8
0.6
0.4
0.2

BEC fraction

0
1
0.8
0.6
0.4
0.2
0

−300−200−100 0

100 200

Scattering length (a0 )Figure 5.17: Top: fRb vs. aKRb . The filling is normalized to fRb with Rb alone. For points below
the resonance, the scattering length was ramped from around −120 a0 to the final value between
140 ms and 40 ms prior to loading the lattices. For points above the resonance, the evaporation
was done above the resonance and the scattering length was ramped from around −220 a0 to thefinal value with the same timing. Away from aKRb = 0, the filling degrades. Bottom: Initial BEC
fraction vs. aKRb , measured after holding in the OT for 150 ms. Away from aKRb = 0, the BEC
fraction is lower, likely from equilibrating with the hot K gas. The BEC fraction with Rb alone isdenoted by the red shaded region. Figure reproduced from Ref. [92].

127

Figure 5.18: We looked at how the filling of Rb vs. interspecies interactions depends on the temperature of K. Here, the hot condition corresponds to T /TF ≈ 0.7 while the cold condition corresponds
to T /TF ≈ 0.3 − 0.4. In the hotter case, fRb more strongly depends on aKRb , but at aKRb = 0
the filling is the same for both cases, which highlights that the densities of the two species are
independent at aKRb = 0. Only data with (6 − 10) × 103 Rb atoms are shown.

128
to the rovibrational ground-state. For these experiments, the typical one-way STIRAP efficiency
is 89 ± 4%. Once the molecules are in the ground state, we remove the unpaired K atoms withresonant light on the |9/2, −9/2i → |11/2, −11/2i cycling transition and we remove the Rb atoms
with a series of adiabatic rapid passages to the |2, 2i state followed by pulses of resonant light on
the |2, 2i → |3, 3i cycling transition. To detect the ground-state molecules, we STIRAP them backto Feshbach molecules, dissociate the Feshbach molecules by ramping B back to 563 G in 1 ms, andfinally image the resulting K atoms. We can also image the molecules by measuring the number ofdissociated Rb, and the numbers agree within experimental uncertainty.
Since Rb is by far the minority species, we first looked at the number of Feshbach molecules
produced vs. Rb number, since the best we can do is to convert all of the Rb to molecules. Thisis shown in Fig. 5.19a. Note that we only convert a very small fraction of the K to molecules (lessthan 5%). This makes the measurement challenging, since we can’t completely remove the unpaired
K atoms without also killing the Feshbach molecules. Rather, we just hide them in the |9/2, −7/2istate by using an RF pulse that can discriminate atoms from Feshbach molecules. The atoms in
the |9/2, −7/2i state are invisible to the imaging (since the imaging light is 80 MHz detuned).
K , the K atoms are still mobile and can collide withSince we typically operate in lattices ∼ 10 ER

the Feshbach molecules. Also, the RF pulse is only ∼ 99% efficient, and the small background of
K atoms in the |9/2, −9/2i state is not so much less than the number of molecules we create.
To directly see a positive effect of the spin flips, we did two experiments. In the first, we
loaded the lattice at aKRb = 0 and compared the final filling of ground-state molecules with andwithout doing the spin flips. We observe that the filling is about 60% higher when we do thespin flips. In the second experiment, we compared the case of loading the lattice at aKRb = 0
and then doing the spin flips against a case where we loaded the lattice above the resonance (ataround aKRb = −220 a0 ). This comparison is shown in Fig. 5.20 (note that the filling fraction of
the ground-state molecules is plotted), where we see that for small Rb numbers there is a clear win
to loading the lattice at aKRb = 0 and doing the spin flips.
Rb lattice for 40
We then imaged the ground-state molecules in situ, after holding in a 25ER

129

(a)

(b)

(c)

Figure 5.19: (a) Conversion efficiency of making Feshbach molecules vs. Rb number (Rb is theminority species). In the limit of small Rb number, the conversion is around 50%. The shadedband shows the predicted conversion, which is the product of the measured K filling fraction(0.80 ± 0.05), the calculated fraction of Rb atoms on singly occupied sites (from our simple model),and the conversion efficiency of preformed pairs reported in Ref. [72] (0.87 ± 0.13). (b) and (c) In
situ images of the ground-state molecules after holding in the lattice for 40 ms, for high conversion(b, average of three images) and low conversion (c, average of seven images). The high conversion
case corresponds to a filling of around 25% (see text for details). The scale bar is 40 µm. Figure
reproduced from Ref. [92].

0.30
0.25

filling

0.20
0.15
0.10
loading lattice at a=0
loading lattice above resonance

0.05
0.00

0

1

2
3
3
Rb (10 )4

Figure 5.20: Filling of ground-state molecules vs. Rb number for two different lattice loading
schemes. For small Rb numbers, loading the lattice at aKRb = 0 definitely leads to higher groundstate molecule fillings.

130
ms (Fig. 5.19b and c). Since the molecules are created within the confines of the Rb MI, we fit
the images to a TF distribution. For the case of low conversion (c), the filling fraction is very low
and there is a hole in the center of the molecular distribution. This is consistent with the central
lattice sites containing multiple Rb atoms and not making molecules. This condition corresponds
to starting with about 2.5 × 104 Rb atoms. When starting with about 2500 Rb atoms (b), we
observe a much denser molecular sample. A TF fit yields 7.9(5) × 102 molecules with a TF radiusof 12.0(2) µm.
There are two ways we can extract the filling. One is to compute the filling directly fromthe TF fit. Doing this gives fmol = 0.27(2). The other is to determine the filling by comparing
the TF width of the molecular cloud to that of our simulated T = 0 Rb distribution and assume auniform conversion of Rb into molecules. The molecules are best described by a distribution that
corresponds to an initial Rb number of 3.2(4) × 103 . Taking the ratio of the measured number ofmolecules to this number, we extract fmol = 0.25(4), which is consistent with the result from direct
fitting. From the product of the measured fRb , NKRb* /NRb , and the STIRAP efficiency, one might
expect fmol = 0.35. One possible explanation for the lower measured filling is molecular loss causedby the K atom removal.
Rb
As shown in Table 2.1, the different polarizability and mass of KRb means that a 25ER
KRb . In such a deep lattice, the molecule tunneling rate is negligible. In
lattice corresponds to 62ER

this case, we can estimate the entropy per molecule from the filling fraction, with some assumption
about the shape of the distribution. As shown in Ref. [161], the entropy per particle in a uniform
lattice with average filling f is

S
−kB
=
f ln(f ) + (1 − f ) ln(1 − f ) .
N
f(5.21)

Our quantum synthesis approach likely leads to a molecular distribution that is much more homogeneous than if we were to adiabatically load a Fermi gas of molecules into the lattice. The K Fermi
gas is roughly homogeneous within the confines of the Rb MI, and the entropy per particle in thisregion is roughly that of a uniformly filled lattice with f = 80%, which is about 0.6 kB . Applying

131
the same argument to the molecules with fmol = 25%, we find the entropy per molecule is about
2.2 kB . For comparision, the entropy associated with the lowest temperatures we have achieved in
the harmonic trap (T /TF ≈ 1) is about 6 kB per molecule.

5.6

Conclusion and outlook
This chapter presented a detailed investigation of the molecule production in the 3D opticallattice. It was very important to diagnose each step in the molecule creation process in orderto realize a significant increase in the final molecule filling fraction. In particular, it was very
important to have a clear understanding of the appropriate sizes the Bose and Fermi gases shouldhave in order to have roughly unit filling in the center of the lattice. Turning off the Bose-Fermi
interactions when loading the lattice was also important. Finally, flipping K to a noninteracting
spin state when crossing the resonance the wrong way was also a crucial step in maximizing thefilling.
At the achieved filling fractions of ∼ 25% the system should be reasonably well-connected,
given our simple argument based on percolation theory. Additionally, given the results of thesimulation in Section 4.6, the propagation of excitations at 25% filling should be reasonably fast.
The number of molecules created at higher fillings is quite low, but is appropriately sized forquantum gas microscopy [16, 17]. There is no fundamental limit to our dual insulator approach,and I foresee filling fractions > 50% should be attainable in the new experiment. Two of the biggest
issues that will likely need to be addressed in order to increase the filling further are that the K is
too hot and the n = 1 MIs are small.
Colder K will help in many ways. We already see direct evidence of the K temperature on
the Rb filling in the data of Fig. 5.18. Reducing the temperature at constant atom number andharmonic confinement leads to higher filling fractions. Having colder K should allow us to reduce
the harmonic confinement and achieve larger n = 1 MIs. The temperatures we achieved in Ref. [92]seem to be limited by the evaporative cooling, and to get K colder while still having pure BECs,
the evaporation strategy may need to be modified. One idea would be to transfer some of the K

132
to the |9/2, −7/2i state towards the end of the evaporation and allow the two spin components
to thermalize with each other instead of with Rb (however, only one of the spin components is
actually useful for making molecules).
In principle, there’s not a huge problem with having to use such small Rb clouds, but producing them reliably in the presence of a huge K cloud is challenging. One way to make the Rb
MI bigger is to have less harmonic confinement; however, to keep the K filling constant we would
require colder K. Another option is to use a shorter wavelength lattice. The onsite interactionenergy U is proportional to 1/λ3L , where λL is the lattice wavelength (the dipolar interactions also
get stronger by 1/λ3L ). This means that the chemical potential µ can be larger while still satisfyingµ/U < 1. The challenge is to find a wavelength that’s favorable for K, Rb, and the ground-state
molecules. For example, 532 nm would be nice, since high-power lasers are readily available, but
we measured the lifetime of ground-state molecules to be quite short at that wavelength. Anotherchoice we considered is 755 nm. At that wavelength, the size of the n = 1 MI can be more thanan order of magnitude larger than at 1064 nm, and K and Rb feel the same lattice depth in unitsof their respective recoil energies. Experimentally, we observed that the lifetime of ground-state
molecules is quite long at this wavelength; however, the real part of the polarizability is small andopposite in sign to the Rb and K (so the beam would be red-detuned for the molecules while being
blue-detuned for both K and Rb), which is not so ideal. More work will be required in the future
to determine whether shorter wavelength lattices offer advantages in molecule production.

Chapter 6

Studying doublons in the lattice

In the previous chapter, we devised a strategy to optimize the loading of the initial quantumgas mixture in the lattice. This led to a significant increase in the molecule filling fraction; however,
the actual molecule production from preformed pairs was not examined that closely. In this chapter,
we investigate the molecule production more thoroughly, to try to understand any effects that might
limit molecule production. In particular, we want to isolate any effects that would cause preformed
pairs to not make molecules. To do this, we leverage our capabilities in producing ground-state
molecules and prepare a distribution where lattice sites are either empty or occupied by a doublon
composed of an interacting K and Rb atom. This well-defined initial state allows us to address
limitations in the molecule production process by measuring the efficiency at which these doublons
are converted back to molecules, after varying the atomic tunneling rates, interspecies interactions,and adiabaticity of magnetic-field sweeps through a higher partial-wave Feshbach resonance.
There are two main ways that the doublons can fail to make molecules: the doublon makes
something other than an s-wave molecule, or one of the atoms (especially the lighter K atom)tunnels off of the site before making a molecule. A d-wave resonance about 1 Gauss above the swave resonance can play a detrimental role in molecule production in the lattice. If we ramp acrossthe d-wave resonance too slowly, it’s very likely that we will make d-wave molecules (when sweepingfrom high to low field) or atoms with an excitation in the relative coordinate (when sweeping from
low to high field). Because of the high onsite density in the lattice, the magnetic-field sweep ratenecessary to be diabatic with respect to the d-wave resonance is not so different than our typical134
sweep rate for making molecules. We show that in the limit of deep lattices and fast magnetic-fieldsweeps, we get near unity conversion of preformed pairs. We see that strong Bose-Fermi interactions
hold the doublons together, even for weak lattices where the K would otherwise be very mobile; forweaker onsite interactions, we see evidence for K tunneling.
In Section 1, I discuss the preparation and measurement of doublons in the lattice. In Section
2, I describe our investigation of the role of the d-wave resonance on molecule production in thelattice. In Section 3, I discuss the interplay between tunneling and interactions. This chapter isbased on Ref. [93], of which I am a coauthor.

6.1

Experimental setup
The experimental scheme for creating and studying a lattice of doublons is shown in Fig. 6.1.

After making ground-state molecules in the lattice, we can cleanly remove the unpaired atoms,
STIRAP back to Feshbach molecules, and then dissociate the Feshbach molecules by ramping themagnetic field above the resonance to a field Bhold . This realizes a lattice where each site is either
empty or occupied by a doublon comprised of a K and Rb atom. We then let the system evolvefor some time τ . To measure the doublon fraction after the evolution, we measure the conversion
of preformed pairs to molecules, following the same protocol described in Section 3.6.

6.2

Effect of the d-wave resonanceIn our normal magnetoassociation protocol, we start far above the resonance (by usuallymore than 10 Gauss), and then ramp below the s-wave resonance in a few ms. In doing this, wecross a narrow d-wave resonance that is located about 1 G above the 3 G-wide s-wave resonance.
A schematic of the two resonances is shown in Fig. 6.2a. Normally, the d-wave resonance wouldn’t
play a role in the molecule production because it’s so narrow; however, in the lattice, it can playa larger role since the onsite density is orders of magnitude higher than in a regular optical trap.
Crossing the d-wave resonance too slowly from high to low B field will produce d-wave molecules(Fig. 6.2b), which are not coupled to the ground state by the subsequent STIRAP pulses. Crossing

135

Figure 6.1: We start with a mixture of K (blue balls), Rb (red balls) and doublons (pair groupedwith gray background) in a 3D lattice. We sweep the magnetic field across the s-wave Feshbachresonance at 546.6 G to create Feshbach molecules. These molecules are then transferred to thero-vibrational ground state via STIRAP. We then remove the unpaired atoms with resonant light,
reverse the STIRAP process to recreate Feshbach molecules, and then sweep the magnetic field
above the resonance to dissociate the molecules and create a lattice of doublons. After holding fora time τ , we measure the remaining doublon fraction by converting to Feshbach molecules again.
To discriminate between Feshbach molecules and free K atoms, we use an RF pulse to transfer
unpaired K atoms to the dark |9/2, −7/2i state (blue ball with dashed black edge). The bottom
panel shows the possible dynamics during the hold time. Doublons have an energy shift due to the00onsite interaction energy UK-Rb. The K tunneling rates in the ground and first excited band are JK
1and JK , respectively. Since Rb experiences a deeper lattice, the Rb tunneling rate is much slower
and is irrelevant for the experiments discussed in this chapter. Figure reproduced from Ref. [93].

136
the d-wave resonance diabatically has no impact and is the desirable scenario. We seek to determine
how fast we need to ramp the magnetic field across the d-wave resonance in order to be diabatic.
First, it is useful to understand the possible states we can have in the lattice system. These
are shown in Fig. 6.2b. For simplicity, the states are illustrated for a harmonic potential where both
species have the same trap frequency ω. There are five relevant states: s-wave molecules 1 , atoms
in the ground band above both resonances 2 , atoms with an excitation in the relative coordinate
3 , free atoms below the resonance 4 , and d-wave molecules 5 . The ideal trajectory for makings-wave molecules is to follow the lower solid arrow from 2 to 1 . Being adiabatic with respect tothe d-wave resonance would correspond to going from 1 to 3 (if ramping from low to high field)
or 2 to 5 (if ramping from high to low field).
We investigate the d-wave resonance by varying the rate of the sweep, Ḃ, that creates doublons, as well as Bhold . For these experiments, the lattice depth is 30 − 35ER (unless otherwise
noted, the lattice depths are in units of the Rb recoil energy). We then measure the molecule
conversion after τ = 1 ms using a 16.8 G/ms sweep for the final association. Fig. 6.2c shows themeasured conversion vs. Ḃ after sweeping from 545.6 G to 562.4 G. For the largest Ḃ, the conversion is near unity. This high conversion fraction is crucial for creating molecules in the lattice athigh filling.
Using a Landau-Zener formalism [162], we can fit this data to extract the width of the dwave resonance. The probability to cross the resonance diabatically (and proceed to create s-wave
Feshbach molecules) is Pdia = exp(−A/|Ḃ|), where A depends on the on-site densities and on theparameters of the Feshbach resonance. We approximate the sites of the deep optical lattice asharmonic oscillators [162], so√
4 3ωHO |abg ∆d |
A=
,
LHO

where ωHO is the harmonic trap frequency for the two atoms’ relative motion, LHO =(6.1)
p~/(µωHO )is the corresponding harmonic oscillator length, µ is the reduced mass, ∆d is the width of thed-wave resonance, and abg is the background scattering length. We ignore the coupling between

6

2

ν=2

1
s- waved- wave

Energy (ℏωrel )Scattering length aK- Rb (103 a0 )

137

0
-1
-2

4
ν=1

3
4

2
ν=0

2

0
1

a

-3
546.0

546.5

547.0

547.5

548.0

546.0

Magnetic field (G)

547.0

547.5

548.0

1.0
Conversion efficiency

Conversion efficiency

546.5

Magnetic field (G)

1.0

0.8

0.6

0.4

0.2

b

5

-2

c
0

5

10

Ramp speed (G/ms)

15

0.8

0.6

0.4

0.2

d
547.42547.44547.46547.48547.50Magnetic field (G)Figure 6.2: (a) The s-wave Feshbach resonance is used to create molecules. About 1 Gauss above
the s-wave resonance there is a narrow d-wave resonance. (b) Pair states for K and Rb in a 1D
lattice (we treat the 3D lattice as separable along the three directions). See the text for details.
(c) Measurement of molecule conversion efficiency at 35 ER (circles) and 30 ER (diamonds), after
ramping from 545.6 G to 562.4 G and varying Ḃ. The data for 30 ER is exponentiated by (35/30)3/4
to account for the expected dependence of Pdia on lattice depth. The solid curve shows a fit to a
Landau-Zener avoided crossing, which yields |∆d | = 9.3(7) mG. (d) The magnetic field at which
the resonance occurs is determined by sweeping upwards to various fields at 1.8 G/ms and thendownwards at 18 G/ms. By fitting to an error function, we determine the resonance is at 547.47(1)G in a 35 ER lattice. Figure adapted from Ref. [93].

138
the center of mass and relative coordinates and take ωHO =q
2 + m ω 2 )/(m(mRb ωK
K Rb
Rb + mK ).

The scattering length depends on B in the following way:a(B) = abg

∆s
∆d
1−
−
res
B − BsB − Bdres


.

(6.2)

Since ∆d  ∆s , this can be simplified near the d-wave resonance toa(B) = abg∆s∆d
1 − res−res
Bd − Bs
B − Bdres


,

(6.3)

which can be approximated asa(B) ≈ a0bg



∆0d
1−
B − Bdres


,

(6.4)



∆d
s
where a0bg = abg 1 − B res∆−Band ∆0d = 1−∆s /(Bres −B res ) . We fit the data from Fig. 6.2c to
res
d

s

d

s

exp(−A/|Ḃ|), extract A = 1.10(7) G/ms, and then get ∆d = a0bg ∆0d /abg . The results are ∆0d =−2.0(2) mG and ∆d = −9.3(7) mG. For the s-wave resonance, A is about 300 times larger. The
probability to be adiabatic with respect to both resonances is shown in Fig. 6.3.
Fig. 6.2d shows the measured conversion vs. Bhold for a slow 1.8 G/ms sweep. From Fig. 6.2c,
we see that this sweep rate corresponds to about a 50% probability to be diabatic. For Bhold below
the d-wave resonance, we observe conversions near unity, but for Bhold above the d-wave resonance,
we observe conversions ∼ 50%. By fitting the data to an error function, we extract the position of
the resonance to be 547.47(1) G, in agreement with previous experiments that looked at atom loss
[163, 164].
The precise determination of the width of the d-wave resonance enables us to determine its
significance in molecule creation. In the work described in the previous chapter, we used Ḃ = 3.5
G/ms for the molecule association, which gives a ∼ 70% probability to be diabatic with respect tothe d-wave resonance. This suggests we may have created a significant fraction of d-wave moleculesthat are dark to our detection, which may have limited the filling fractions we achieved. Future
work will be necessary to confirm this hypothesis.

139

Figure 6.3: Probability to be adiabatic with respect to the s-wave (dashed blue) and d-wave (solidred) resonances in the lattice, using Eq. 6.1 with ωHO = 2π × 30.4 kHz (which corresponds to 35
ER ). For any experimentally accessible Ḃ, we are adiabatic with respect to the s-wave resonance.
For Ḃ > 20 G/ms, the effect of the d-wave resonance is negligible. The timescales are different by
about a factor of 300 between the two resonances.

140

6.3

Detrimental effects due to tunneling
In order to acheive high lattice fillings, we require not only a large fraction of sites that containdoublons, but also that these doublons are not lost due to tunneling or collisions before converting
to molecules. By varying both the lattice depth and Bhold , we can explore the interplay between
tunneling and interactions. In our system, sK = 0.4 sRb (recall s = Ulat /ER ), so K tunnels fasterthan Rb. Fig. 6.4 shows the measured doublon fraction vs. lattice depth for different interaction
strengths (τ = 1 ms). For these experiments, the ramp time was fixed, so Ḃ depends on Bhold andranges from 5 to 19 G/ms. The data are multiplied by a factor to account for these ramps not
being perfectly diabatic with respect to the d-wave resonance.
We see that for deep lattices, the conversion is near unity for all interaction strengths, which
is just trivial localization of the doublons (even resonant tunneling would be slow). For shallower
lattices, the conversion is still high for very strong interactions (aKRb = −1900 a0 ), because the0
suppresses tunneling to empty sites, which are very far off-resonant.
strong onsite interaction UK-Rb

This modification of tunneling from interactions has been seen in other experiments, which studied
the stability of repulsively interacting atom pairs [165] and the modification of tunneling of fermionsdue to interactions with bosons [166]. In our case, the interactions are attractive, but the signdoesn’t really matter, as tunneling is off-resonant for either sign of the onsite interactions. For
weaker interactions, the conversion decreases for shallower lattices, since the interactions are notstrong enough to hold the doublons together.
The solid curves in Fig. 6.4 show the expected evolution of a single doublon in the lattice,which generally describes the data well. Here, only K tunneling is considered. For small doublon
fractions, the solid curves start to disagree with the data, because there is a finite probability that
the K will find another Rb partner. The dashed lines account for this by simulating the dynamics
with a peak filling of 10%.
The relevance of this effect to molecule production is a little misleading, since this data wastaken with a relatively dilute lattice (typically around 10% of the sites are initially occupied by141

Doublon fraction

1.0
0.8
0.6
0.4

aK-Rb =-220a0
aK-Rb =-410a0

0.2

aK-Rb =-1900a0
0.0

10

15

20

25

30

Lattice depth Vlatt (ER )Figure 6.4: The remaining doublon fraction after a 1 ms hold in the lattice vs. lattice depth fordifferent interaction strengths. For strong attractive interactions the conversion is high for any
lattice depth, which is due to the doublons being held together by the strong onsite interactions.
For weaker interactions, the conversion decreases for shallower lattices, which we attribute to thelighter K atoms tunneling. The solid curves show the expected evolution of a single doublon,
while the dashed curves simulate the dynamics with a peak filling of 10%, which accounts for the
possibility that the K will find another Rb atom after it tunnels. Figure reproduced from Ref. [93].

142
a doublon), and so the K have many sites to which they can tunnel. When we actually makemolecules, the fillings of the atomic gases are much higher, and the K atoms have many fewer sites
to which they can tunnel. However, the most important result of this study is that the conversion
of preformed pairs is near unity, in the limit of fast magnetic-field sweeps and deep lattices.

6.4

Conclusions
In this chapter, we showed the important roles that tunneling, interspecies interactions, andmagnetic-field sweep rates play in the conversion of doublons to molecules via magnetoassociation.
Most importantly, we verified that we can get near unity conversion of preformed pairs, assuming
we use sufficiently deep optical lattices and sufficiently fast magnetic-field sweeps. While we focused
specifically on the 40 K87 Rb mixture, our conclusions should be applicable to other heteronuclear
systems, where narrow Feshbach resonances might exist in the vicinity of wider ones used for
producing Feshbach molecules, and where the two species might have different mobilities in thelattice.

Chapter 7

Conclusions and future work

Over the last six years, the field of ultracold polar molecules has made immense strides.
Our group has learned how to control and shut off the chemical reactions that make the bulk gas
unstable, and that were initially viewed as a huge obstacle to studying quantum many-body physics
with long-range dipolar interactions. With the chemical reactions under control, we learned how
to induce and probe coherent long-range interactions between the molecules, which should leadto many interesting future experiments using polar molecules to study quantum magnetic models.
More recently, our work in increasing the filling fraction shows that it’s possible to use many of
the tools of modern AMO physics to assemble heteronuclear molecules from the ground up. Other
groups around the world have also succeeding in producing ground-state molecules [52, 53, 54, 55],
and this will hopefully enable the field to progress even faster.
However, despite all of the successes of our experiment, there is much room for improvement.
For example, our quantum synthesis scheme was able to produce molecules at ∼ 25% filling, but
future improvements based on the considerations discussed in Chapter 5 will be needed to increase
the filling further and make the production of such dense samples more robust. The dream is thatmaking dense samples of ultracold molecules in optical lattices would become routine, much likehow the production of BECs of alkali atoms has become much more robust over the last 20 years.
To do this, the importance of improving the technical capabilities of the experiments cannot be
stressed enough. The dense molecular clouds described in Chapter 5 are very small, and better
imaging resolution will be required to probe and manipulate them more precisely. Being able to144
apply large, stable electric fields will make even richer physics accessible.
To this end, we recently installed a second generation KRb apparatus. The salient features
are outlined in Fig. 7.1. The two main technical improvements are better electric field control andbetter imaging resolution. The glass cell has much more optical access and every window is nicely
AR coated. The electric-field plates are now inside the vacuum chamber, with a combined ITO-AR
coating such that there is no dielectric between the electrodes and the molecules. The plates willallow us to apply flat DC fields, while the four rods will enable us to apply field gradients bothvertically and radially. A high numerical aperture (NA) objective will give us resolution around∼ 1 µm, which is about 5 times better than in the first generation machine, and in principle we
could get to ∼ 700 nm with a higher NA objective. The ultimate goal would be to create a quantumgas microscope that can observe the dynamics of polar molecules at the single lattice site level.
Most likely, the first set of experiments in the new apparatus will highlight its improved
technical capabilities. By applying a vertical electric-field gradient, it should be possible to perform
manipulations of a single 2D layer of the lattice. This is because the molecules in each layer would
experience a different electric field and hence different Stark shift, so the N = 0 → N = 1 transition
would be different in frequency in each layer and we could use microwaves to spectroscopicallymanipulate only molecules in a desired layer. This could be useful as a simple implementation of
the excitation propagation experiment outlined in Section 4.6. If we can flip just a single layer
to | ↑i, then we could study the propagation of these spin excitations to other layers. The same
manipulation that created the initial distribution could be used to read out the final distribution.
We would either look for fewer | ↑i molecules in the starting layer or shift the microwave frequency
to detect | ↑i molecules in other layers.
With the improved imaging capabilities, the experiments should be able to more precisely
measure the spin dynamics. I think the trend of the experiment and theory building off of one
another will continue. Similar to the work in Ref. [85] where the cluster expansion was greatly
improved, as the experiment provides us means for more precisely probing and manipulating themolecules, we should be in a position to help validate other theoretical tools and methods. The

145

(a)

(b)

1 inch (25.4 mm)

(c)

(d)

(e)

Figure 7.1: (a) A picture of the new science cell. There are two flat plates for applying DC fields
and four rods for applying gradients. The entire structure is inside of a glass cell made by Precision
Glassblowing and is held together by macor pieces. (b) Sketch of the science cell with the high-NA
objective below it. (c) Sketch of the entire chamber. Similar to the first generation experiment,
the MOT cell and science cell are separated by ∼ 1 m, with a gate valve in between. The maindifference is that in the new chamber there is another cell before the science cell. The evaporative
cooling will be done in this cell in a plugged quadrupole trap and then the atoms will be opticallytransported to the science cell. (d) Sketch of the electrode structure. (e) In-vacuum inductors andcapacitors separate AC and DC on the rods and should enable us to couple microwaves onto therods to drive rotational transitions.

146
future of polar molecule experiments looks bright, and I’m confident that the field will continue toprogress in understanding and manipulating these interesting quantum many-body systems.

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Appendix A

Deriving the dipolar Hamiltonian

Here I show how to connect the classical and quantum expressions for the dipole operator,
which are useful in understanding the Hamiltonian presented in Chapter 4. The derivations in thissection come mainly from Refs. [34, 167].
Recall the classical dipole-dipole interaction
Vdd =d1 · d2 − 3(d1 · r̂12 )(d2 · r̂12 )
.
3
r12

(A.1)

To get the quantum analog of this expression, we need to replace the classical dipole d with anˆ It turns out that the familiar angular dependence (1 − 3 cos2 θ) from Eq. 4.1 represents
operator d.
just one term in the full expression for the dipole-dipole interaction, and to get the full angulardependence it is necessary to decompose the dipole operator into irreducible tensor components.
This decomposition yields a sum of terms corresponding to q units of angular momentum being
exchanged between rotational and orbital angular momentum [34].
The result is that
√

q=2

6 X
2
Vdd = − 3(−1)q C−q(θ, φ)Tq2 (d1 , d2 ),r12 q=−2

(A.2)

where
r
Cqk (θ, φ) =4π
Ykq (θ, φ)
2k + 1(A.3)

and Ykq (θ, φ) is a normal spherical harmonic. The T ’s are irreducible tensor operators [131]:

ˆ±
2
T±2
= dˆ±
1 d2 ,

(A.4)

160
ˆ0
+ dˆ±
dˆ01 dˆ±
2√
1 d2
,
2
dˆ+ dˆ− + dˆ−dˆ+ + 2dˆ01 dˆ02
1
√2
T02 = 1 2
.
6
2
T±1
=

(A.5)

(A.6)

From now on, I will omit the hats to denote operators since every d will be understood to be an
operator. We will also need the following relationships between the Cartesian and tensor forms:
d± = ∓

dx ± idy√,2d0 = dz .

(A.7)
(A.8)

To make things simple, we’ll put one dipole at the origin and the other at the point (x, y, z),with the electric field in the z direction. Then r12 = r, and the polar angle θ and azimuthal
angle φ are defined in the usual way (z = r cos θ, x = r sin θ cos φ, and y = r sin θ sin φ, wherepr = x2 + y 2 + z 2 ).
For completeness, the C’s are given by√6
sin2 θe±2iφ ,4√62C±1 = ∓sin θ cos θe±iφ ,2
1
C02 = (3 cos2 θ − 1).
2
2
C±2
=

(A.9)
(A.10)(A.11)

Then Eq. A.2 becomes√ √
66
sin θ cos θ iφ 0 − − 0++ 0−2iφ − −√Vdd = − 3sin2 θ(e2iφ d+d1 d2 )+(e (d1 d2 +d1 d2 )−e−iφ (d01 d+
1 d2 +e2 +d1 d2 ))+
r
4
2 2

3 cos2 θ − 1 + −− +0 0(d1 d2 + d1 d2 + 2d1 d2 ) . (A.12)
12
After using Eqs. A.7 and A.8 and simplifying, we obtain
6 1
Vdd = − 3sin2 θ (d1x d2x − d1y d2y ) cos(2φ) + (d1y d2x + d1x d2y ) sin(2φ)
r 4 3 cos2 θ − 1sin θ cos θ+(d1z d2x +d1x d2z ) cos φ+(d1z d2y +d1y d2z ) sin φ +2d1z d2z −d1x d2x −d1y d2y .
2
12(A.13)

161
Substituting sin φ = √ 2y

x +y 2

6
Vdd = − 3r

, cos φ = √ 2x

x +y 22

2, sin(2φ) = x22xy, and cos(2φ) = xx2 −y
we get
+y 2
+y 2

 2
x − y2xyxz(d1x d2x − d1y d2y ) + 2 (d1y d2x + d1x d2y ) + 2 (d1z d2x + d1x d2z )
2
4r
2r
2r

2
yz
3 cos θ − 1+ 2 (d1z d2y + d1x d2y ) +(2d1z d2z − d1x d2x − d1y d2y ) . (A.14)
2r
12


3x2
3y 2
3z 2
3xy
1
Vdd = 3 d1x d2x (1 − 2 ) + d1y d2y (1 − 2 ) + d1z d2z (1 − 2 ) − 2 (d1y d2x + d1x d2y )
r
r
r
r
r

3xz
3yz
− 2 (d1z d2x + d1x d2z ) − 2 (d1z d2y + d1x d2y ) . (A.15)
r
r




1
3
Vdd = 3 d1x d2x + d1y d2y + d1z d2z − 5 d1x d2x x2 + d1y d2y y 2 + d1z d2z z 2 + xy(d1y d2x + d1x d2y )rr+ xz(d1z d2x + d1x d2z ) + yz(d1z d2y + d1x d2y ) . (A.16)
Vdd =d1 · d2(d1 · r̂)(d2 · r̂)
−3
.
r3
r3

(A.17)