Enhanced optical and electric manipulation of a quantumgas of KRb moleculesbyJacob P. Covey
B.S., University of Wisconsin-Madison, 2011
M.S., University of Colorado-Boulder, 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
2017This thesis entitled:
Enhanced optical and electric manipulation of a quantum gas of KRb molecules
written by Jacob P. Covey
has been approved for the Department of Physics

Prof. Jun Ye

Prof. Ana Maria Rey

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
Covey, Jacob P. (Ph.D., Physics)
Enhanced optical and electric manipulation of a quantum gas of KRb molecules
Thesis directed by Prof. Jun Ye

Polar molecules are an ideal platform for studying quantum information and quantum simulation due to their long-range dipolar interactions. However, they have many degrees of freedom
at disparate energy scales and thus are difficult to cool. Ultracold KRb molecules near quantum
degeneracy were first produced in 2008. Nevertheless, it was found that even when prepared in theabsolute lowest state chemical reactions can make the gas unstable. During my PhD we worked tomitigate these limitations by loading molecules into an optical lattice where the tunneling rates,
and thus the chemistry, can be exquisitely controlled. This setting allowed us to start using therotational degree of freedom as a pseudo-spin, and paved the way for studying models of quantum
magnetism, such as the t-J model and the XXZ model. Further, by allowing molecules of two“spin”-states to tunnel in the lattice, we were able to observe a continuous manifestion of the quantum Zeno effect, where increased mobility counterintuitively suppresses dissipation from inelastic
collisions. In a deep lattice we observed dipolar spin-exchange interactions, and we were able toelucidate their truly many-body nature. These two sets of experiments informed us that the filling
fraction of the molecules in the lattice was only ∼ 5 − 10%, and so we implemented a quantum
synthesis approach where atomic insulators were used to maximize the number of sites with one K
and one Rb, and then these “doublons” were converted to molecules with a filling of 30%. Despite
these successes, a number of tools such as high resolution detection and addressing as well as large,
stable electric fields were unavailable. Also during my PhD I led efforts to design, build, test, andimplement a new apparatus which provides access to these tools and more. We have successfully
produced ultracold molecules in this new apparatus, and we are now applying AC and DC electricfields with in vacuum electrodes. This apparatus will allow us to study quantum magnetism in alarge electric field, and to detect the dynamics of out-of-equilibrium many-body states.

Dedication

To Prof. Debbie Jin, my late advisor (jointly with Prof. Jun Ye). We all miss her deeply,
and JILA will never quite be the same.

v

Acknowledgements

I have had the pleasure of working with many incredible people over the past six years.
The KRb members I worked with are Amodsen Chotia, Brian Neyenhuis, Bo Yan, Bryce Gadway,
Steven Moses, Matt Miecnikowski, Zhengkun Fu, Luigi De Marco, Kyle Matsuda, Will Tobias, andGiacomo Valtolina. I also worked Kang-Kuen Ni briefly while she was a post-doc working on the
JILA eEDM experiment. Although I only overlapped with Amodsen Chotia for a few months, I
learned a lot about the KRb experiment from him, and he taught me a lot about experimental
physics in general. Brian Neyenhuis was the guiding force on the KRb experiment during my firstyear, and our initial experiments with molecules in optical lattices were done under his leadership.
The majority of my PhD was done with Steven Moses, Bo Yan, and Bryce Gadway. The fourof us were very productive together. Bo is the ultimate experimentalist. He can solve any problem
on the experiment, and he always found a way to keep it running well. Bryce has an encyclopedic
knowledge of physics, particularly AMO physics. He somehow knows every paper that every group
has published. Most of the success during his time on KRb came from his ideas. Steven and I
worked together for five years. He has an incredible balance of technical skills, physics skills, andproblem solving skills. The latter is incredibly valuable on an experiment that is both aging andextremely complicated. These capabilities combined with his dedication and hardworking mentality
have led to a lot of success. I enjoyed working with all three of these guys.
The new KRb team of Luigi De Marco, Kyle Matsuda, Will Tobias, and Giacomo Valtolina
is incredibly talented and ambitious. I have full confidence in their capabilities. They all joined
the experiment during the second half of the implementation of the new apparatus, and thus theyvi
all contributed significantly to its success. Accordingly, they are all well prepared to operate theexperiment without me. They breathed new life into the experiment as Steven graduated, Matt
and Zhengkun moved on, and Debbie was taken from us. The results that are now coming out of
the new apparatus were only possible because of them. I worked with Kyle and Will during theirfirst year as PhD students, and their ability to push the experiment forward while taking threeclasses has been very impressive. Luigi is a postdoc who did his PhD in physical chemistry studying
ultrafast molecular dynamics. He has adapted his skillset to AMO physics remarkably quickly, andhis general knowledge of physics and science has been impressive. He brings a bit more depth to thephilosophy and direction of the experiment. The newest member is the postdoc Giacomo, who is
an AMO expert and has a broad knowledge of ongoing AMO research. The skills and backgroundof Luigi and Giacomo complement each other well.
It has been a pleasure working with Jun Ye, Debbie Jin, and Eric Cornell. The collaborative
atmosphere between these three advisors is very constructive and works quite well. The passing of
Debbie Jin has been difficult for everyone, but no one was more affected than Jun and Eric. The
way they were able to go on afterwords and maintain their groups was incredible. Jun has been
an excellent advisor and mentor to me, and he has always been incredibly available and accessible.
This in itself is surprising given how busy he is, and he always does his best to help all of his groupmembers and support us in any way we need. Moreover, his optimism and positive attitude has
kept us going through the rough patches, and his constant support and belief in us has been very
helpful. Debbie Jin was also quite invested in her group members and she developed very close
relationships with many of us. Her friendliness and openness made her loss even harder to bear.
Eric’s support and advise as a co-advisor had been incredibly useful. He has helped us solve many
important problems over the years.
I would be remiss if I did not acknowledge Jun’s and Eric’s groups for support and advice over
the years. Most notably, Jun’s Sr experiments are always developing cool new toys and pioneering
the best way to do everything. We are indepted to them for help with our Raman cavity, highpower distribution and fibers for optical traps, and the new version of our experimental control

vii
program. We have benefitted immensely from discussions with Sara Cambell, Ross Hutson, Ed
Marti, Shimon Kolkowitz, Aki Goban, Toby Bothwell, and Wei Zhang from Sr; Ming-Guang Hu,
Rabin Paudel, Roman Chapurin, and Michael Van De Graaff from the new K apparatus; and Ben
Stuhl, Tim Langen, Hau Wu, Dave Reens on the OH Stark decellerator for invaluable advice on
AC and DC electric fields.
We have also had a very fruitful collaboration with Ana Maria Rey’s group. In particular we
worked with Michael Foss-Feig and Kaden Hazzard in the early years, and more recently Michael
Wall, Martin Garttner, Arghavan Safavi-Naini, and Bihui Zhu. They have all done an excellent job
being aware of what our experiment is capable of, and giving us space when we need to address
technical issues. We attempted the spin-exchange measurements after Kaden and Ana Mariasuggested that we would be able to observe them in the lattice even at our relatively low filling
fraction at that time.
The JILA instrument shop played a very significant role in much of the work in this thesis,
and I have spent a lot of time working with Tracy Keep, Kels Detra, Kim Hagen, Blaine Horner,
Todd Asnicar, and Hans Greene. Tracy made the primary contributions to the new apparatus, andwas very helpful along the entire journey to address every problem that emerged. A lot can change
in six years, and the JILA instrument shop during my PhD is an excellent example of that. Tracy
recently passed away after battling with cancer. Kels left JILA a few years ago. Blaine recently
retired. Nevertheless, the JILA instrument shops remains an incredible resource.
The JILA electronics shop and computing team have been invaluable to our experiment.
Terry Brown and Carl Sauer are always available to help us with any electrical or electronics
problem, particularly high voltage or AC circuits. Their expertise has played a significant rolein the electronics of the new apparatus, such as the servos for large electric fields and the rf andmicrowave coils for K and Rb spin flips. J. R. Raith in computing has been very helpful with allof our computing needs over the years.
On a more personal note, I am thankful to Dan Gresh (Cornell group, eEDM experiment,
JILA) for his friendship over the past six years, and for teaching me the art of powerlifting. I

viii
am also grateful to Rory Barton-Grimley (Aerospace Eng., CU-Boulder) for his friendship and forgetting me back into hockey. I have enjoyed playing with him and Carrie Weidner, Seth Caliga, and
Cam Straatsma (Anderson group, JILA) over the past several years. I would also like to thank myfamily for their constant support over the past six years, especially my fiancé Jennifer.

ix

Contents

Chapter
1 Introduction
1.1

1Quantum physics with ultracold matter . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Quantum simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Quantum information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.3

Many-body quantum systems out of equilibrium . . . . . . . . . . . . . . . .

3

1.2

Dipolar interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Topics in this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 Experimental background and overview
2.1

7

Creation of ultracold molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.1

Magneto-association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.2

Coherent optical transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

The first generation KRb machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3

Moving the experiment into the JILA X-Wing . . . . . . . . . . . . . . . . . . . . . . 22

2.4

Laser systems in the X-Wing lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5

2.4.1

MOT/probe lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.2

Raman lasers and cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.3

Optical trap/lattice lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Computer control software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

x
3 Quantum-state controlled chemical reactions and dipolar collisions

32

3.1

Quantum state control of the molecule . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2

Inducing the dipole moment in a lab frame with DC and AC electric fields . . . . . . 33

3.3

Role of quantum statistics on collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4

Reduced dimensions: quantum stereodynamics of chemical reactions in a 2D gas . . 39

3.5

Sticky collisions: 3-molecule collision channels . . . . . . . . . . . . . . . . . . . . . . 43

4 Suppression of chemical reactions in a 3D lattice
4.1

45

Optical lattice techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1

Optical dipole potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.2

Band structure and eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.3

Wannier functions for localized atoms . . . . . . . . . . . . . . . . . . . . . . 50

4.1.4

Calibrating the lattice depths . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.5

Superfluid to Mott insulator transition . . . . . . . . . . . . . . . . . . . . . . 52

4.1.6

Band mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2

Long-lived molecules in a 3D optical lattice . . . . . . . . . . . . . . . . . . . . . . . 55

4.3

The continuous quantum Zeno mechanism: stability from strong dissipation . . . . . 57

4.4

Anisotropic polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5

Hyperpolarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6

Polarizabilities at other wavelengths . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6.1

λ2 = 532 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6.2

λ3 = 755 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6.3

λ4 = 790 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Quantum magnetism with polar molecules in a 3D optical lattice

71

5.1

The dipolar spin-1/2 XXZ Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2

Spin-echo Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3

Removing pair-wise entanglement - WAHUHA NMR sequence . . . . . . . . . . . . . 75

xi
5.4

Many-body dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5

Universal dipolar interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6

N = 1 to N = 2 rotational transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.7

Pulse sequences with variable tipping angles . . . . . . . . . . . . . . . . . . . . . . . 83

5.8

Inferring the molecule filling fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6 A low entropy quantum gas of polar molecules in a 3D optical lattice

87

6.1

The need for low entropy - percolation theory . . . . . . . . . . . . . . . . . . . . . . 87

6.2

Many previous efforts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2.1

Species selective dipole traps . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2.2

‘Magic’ traps for matching the oscillation frequencies . . . . . . . . . . . . . . 92

6.2.3

Sympathetic cooling of KRb with Rb

6.2.4

Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

. . . . . . . . . . . . . . . . . . . . . . 93

6.3

Quantum synthesis with atomic insulators in the lattice . . . . . . . . . . . . . . . . 99

6.4

Experimental considerations for in situ imaging . . . . . . . . . . . . . . . . . . . . . 101

6.5

6.4.1

Calibrating the saturation intensity . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4.2

Measuring the imaging resolution and the pixel size . . . . . . . . . . . . . . 103

6.4.3

Tunneling during the probe time . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4.4

Molecule-specific systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Simulating the Rb Mott insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5.1

Calculating the atomic distributions at T = 0 . . . . . . . . . . . . . . . . . . 109

6.5.2

Convolving the atomic distributions with the imaging resolution . . . . . . . 109

6.5.3

Calculating the OD per atom . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.6

The atomic peak filling fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.7

Measuring the temperature of K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.8

The role of interspecies interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.9

Maximizing the doublon fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

xii
6.10 An alternative measurement of the doublon fraction . . . . . . . . . . . . . . . . . . 117
6.11 Magneto-association efficiency on lattice sites . . . . . . . . . . . . . . . . . . . . . . 121
6.12 Atomic tunneling - doublon dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.13 The role of higher bands in the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.14 Photoassociation of doublons in the lattice . . . . . . . . . . . . . . . . . . . . . . . . 134
7 The new apparatus - enhanced optical and electric manipulation of ultracold polar molecules136
7.1

Electrode design for large, versatile electric fields . . . . . . . . . . . . . . . . . . . . 138
7.1.1

Macor insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.1.2

Triple points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.2

The flexibility of the electrode system . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.3

Homogeneous electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.3.1

Effects from a hole in the ITO-coated plates . . . . . . . . . . . . . . . . . . . 146

7.3.2

The effect of an angle between the rods and/or plates . . . . . . . . . . . . . 149

7.3.3

Charging scenerios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.4

Arbitrary electric field gradients in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.5

Coupling AC fields onto the rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.6High resolution imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.7

7.8

7.6.1

Objective design and aberration theory . . . . . . . . . . . . . . . . . . . . . 160

7.6.2

The coil and optics mounts around the science cell . . . . . . . . . . . . . . . 161

7.6.3

Deflection of the science cell windows due to the differential pressure . . . . . 161

Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.7.1

Designing the gradient/quantization coils . . . . . . . . . . . . . . . . . . . . 165

7.7.2

Designing the bias coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

The chamber and vacuum considerations . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Designing, building, and testing the new apparatus
8.1

171

Reaching large electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

xiii

8.2

8.3

8.4

8.1.1

Macor design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.1.2

Connecting the electrodes to the ITO coatings . . . . . . . . . . . . . . . . . 173

8.1.3

Electrode surface smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.1.4

Conditioning the electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.1.5

Reaching large fields with the ITO-coated plates . . . . . . . . . . . . . . . . 183

8.1.6

Large fields with high intensity beams through the ITO . . . . . . . . . . . . 184

8.1.7

Effect of Titanium-Sublimation pumps on electrode insulators . . . . . . . . . 187

Electrode parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.2.1

Assembling the electrodes in the science cell . . . . . . . . . . . . . . . . . . . 188

8.2.2

Measuring the parallelism and straightness of the rods and plates . . . . . . . 192

AC electric field coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.3.1

Building the coupling capacitors . . . . . . . . . . . . . . . . . . . . . . . . . 194

8.3.2

Designing the decoupling inductors . . . . . . . . . . . . . . . . . . . . . . . . 197

8.3.3

Measuring the coupling into the vacuum chamber . . . . . . . . . . . . . . . . 199

8.3.4

Measuring the coupling along the transmission line . . . . . . . . . . . . . . . 200

Chamber assembly procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.4.1

Turning the corner in the small cube . . . . . . . . . . . . . . . . . . . . . . . 201

8.4.2

The reducer going from the large cube to the small cube . . . . . . . . . . . . 203

8.5

Aligning the electrode assembly to the cell windows

8.6

Reaching ultrahigh vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8.7. . . . . . . . . . . . . . . . . . 205

8.6.1

Eliminating virtual leaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8.6.2

Cleaning procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.6.3

Chamber baking and pump activation procedure . . . . . . . . . . . . . . . . 209

8.6.4

High-potting the ion pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Design of the ITO coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.7.1

Coating thickness requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8.7.2

Interaction of ITO coatings with alkali flux . . . . . . . . . . . . . . . . . . . 213

xiv
8.7.3
8.8

8.9

Pairing the ITO coating with an anti-reflection coating . . . . . . . . . . . . . 215

Optical qualities of ITO plates and cell windows . . . . . . . . . . . . . . . . . . . . 218
8.8.1

Anti-reflection coatings on the cell windows . . . . . . . . . . . . . . . . . . . 220

8.8.2

Optical flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Testing the objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.10 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.10.1 Testing the bias coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.10.2 Testing the gradient/quantization coils . . . . . . . . . . . . . . . . . . . . . . 228
8.11 Electric field voltage servos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.11.1 Stabilizing large voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

8.11.2 Realizing ppm voltage stability . . . . . . . . . . . . . . . . . . . . . . . . . . 232
9 Experimental procedure - making molecules in the new apparatus

235

9.1

Dual species MOT with grey molasses cooling of 40 K and 87 Rb . . . . . . . . . . . . 237

9.2

Magnetic transfer and plugged quadrupole evaporation . . . . . . . . . . . . . . . . . 240

9.3

Optical transfer into the science cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

9.4

9.3.1

Transfer beam alignment onto the plugged quadrupole trap . . . . . . . . . . 242

9.3.2

Lifetime and heating in the transfer beam . . . . . . . . . . . . . . . . . . . . 244

9.3.3

Loss of atoms during transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

9.3.4

Magnetization in the chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

9.3.5

Moving further in the magnetic trap . . . . . . . . . . . . . . . . . . . . . . . 251

9.3.6

The optical transfer efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

9.3.7

Heating during transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

9.3.8

An elliptical transfer beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Optical evaporation to a degenerate mixture . . . . . . . . . . . . . . . . . . . . . . . 257
9.4.1

Compressing the cloud in the transfer beam with a B-field gradient . . . . . . 258

9.4.2

Loss of K during evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

xv
9.4.3

Implementing the 790 nm beam for additional trapping of K . . . . . . . . . 260

9.4.4

Optimized trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

9.5State preparation, and producing molecules . . . . . . . . . . . . . . . . . . . . . . . 261

9.6

Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.6.1

Vertical lattice path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

10 New physics with the new apparatus - high resolution optical detection and large, stableelectric field

268

10.1 Large, stable electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
10.2 Driving rotational transitions - AC electric field coupling . . . . . . . . . . . . . . . . 270
10.3 “Magic” electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
10.4 Stabilizing reactive dipoles in 2D with large electric fields . . . . . . . . . . . . . . . 272
10.5 Towards evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
10.6 Spectroscopic layer selection using an axial electric field gradient . . . . . . . . . . . 274
10.7 High resolution in situ imaging of polar molecules . . . . . . . . . . . . . . . . . . . 274
10.8 Single-molecule addressing in an optical lattice . . . . . . . . . . . . . . . . . . . . . 275
10.9 Holographic projection of arbitrary potentials using a DMD . . . . . . . . . . . . . . 279
11 Outlook

284

11.1 Spin-orbital coupling with lattice-confined polar molecules . . . . . . . . . . . . . . . 284
11.2 Quantum magnetism in the XXZ Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 288
11.3 An approach for spin-selective quantum gas microscopy of polar molecules in an
optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
11.3.1 A quantum gas microscope for polar molecules in an optical lattice . . . . . . 292
11.3.2 Spin-resolved molecular microscopy . . . . . . . . . . . . . . . . . . . . . . . . 293
11.4 Spin impurity dynamics: spin-transport simulations . . . . . . . . . . . . . . . . . . 301
11.5 Beyond ITO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
11.6 Eliminating the optical transfer step . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

xvi
11.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Bibliography

307

Appendix
A New tapered amplifier mount design

327

B Cart and stage motion programs for the new apparatus

333

xvii

FiguresFigure2.1

Potential energy wells of the closed and open channel near a Feshbach resonance . . 10

2.2

Scattering length and energy of two atoms in the vicinity of a Feshbach resonance . 11

2.3

The energy of two atoms as a function of magnetic field . . . . . . . . . . . . . . . . 12

2.4

RF association of NaK Feshbach molecules from two different initial states . . . . . . 13

2.5

The STIRAP pathway for KRb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6

The old vacuum chamber in the first generation KRb experiment . . . . . . . . . . . 17

2.7

The IP trap in the old apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8

Charging problems in the first generation apparatus . . . . . . . . . . . . . . . . . . 20

2.9

Moving the lab into the X-wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.10 The bichromatic optical cavity used to lock the Raman lasers . . . . . . . . . . . . . 24
2.11 Locking to the optical cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.12 The Nufern fiber amplifier layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.13 The power calibration of the Nufern . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.14 The hollow-core photonic crystal fiber . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1

Induced dipole moments of KRb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2

Loss rate coefficient as a function of temperature . . . . . . . . . . . . . . . . . . . . 38

3.3

Decay of an incoherent mixture of rotational states . . . . . . . . . . . . . . . . . . . 38

3.4

Collision of two polar molecules in an electric field . . . . . . . . . . . . . . . . . . . 40

xviii
3.5

Decay coefficient of ultracold 40 K87 Rb molecules colliding in an electric field . . . . . 40

3.6

Chemical reactions in the 1D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1

An illustration of a red-detuned dipole trap . . . . . . . . . . . . . . . . . . . . . . . 47

4.2

Band structure calculations for various lattice depths . . . . . . . . . . . . . . . . . . 49

4.3

The superfluid to Mott insulator transition of Rb with K . . . . . . . . . . . . . . . 54

4.4Band mapping of K in the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5

Long-lived molecules in an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6

Experimental setup for studying the continuous quantum Zeno effect . . . . . . . . . 58

4.7

Extracting the filling fraction from the loss measurements . . . . . . . . . . . . . . . 59

4.8

The setup for the magic angle measurement experiment . . . . . . . . . . . . . . . . 60

4.9

The magic angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.10 The Ramsey time as a function of angle . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.11 The Differential polarizability at the magic angle for different lattice depths . . . . . 64
4.12 Reducing the harrmonic confinement using λ2 = 532 nm dipole traps . . . . . . . . . 65
4.13 The polarizabilities at λ3 = 755 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.14 The polarizabilities and lifetimes at λ4 = 790 nm . . . . . . . . . . . . . . . . . . . . 69
5.1

Schematic of spin exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2

The spin exchange experiment

5.3

The spin echo experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4

WAHUHA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5

Many-body nature of the dipolar interactions . . . . . . . . . . . . . . . . . . . . . . 78

5.6

Fitting the contrast decay with three frequencies vs. one frequency . . . . . . . . . . 80

5.7

Different excited states to use for spin-exchange interactions . . . . . . . . . . . . . . 80

5.8

Comparing contrast decay for | ↑i = |1, −1i and | ↑i = |1, 0i . . . . . . . . . . . . . . 82

5.9

The N = 1 to N = 2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.10 Pulse sequences with variable tipping angle . . . . . . . . . . . . . . . . . . . . . . . 85

xix
6.1

Low entropy lattice cartoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2

Degenerate K-Rb pairing in momentum space . . . . . . . . . . . . . . . . . . . . . . 91

6.3

Elastic and inelastic collision rates in 2D . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.4

Evaporative cooling of molecules in 1D lattice using an electric field gradient . . . . 97

6.5

Quantum synthesis of a low entropy molecular gas in the optical lattice . . . . . . . 100

6.6

Calibrating the saturation intensity of the probe beam . . . . . . . . . . . . . . . . . 102

6.7

Fourier optics analysis of the imaging system . . . . . . . . . . . . . . . . . . . . . . 105

6.8

The thermal hopping rate as a function of the probe time . . . . . . . . . . . . . . . 107

6.9

Simulating the Mott insulator shells . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.10 The filling fractions vs. their respective atom numbers . . . . . . . . . . . . . . . . . 112
6.11 The effect of K on the BEC and filling fraction of the intial Rb . . . . . . . . . . . . 116
6.12 High conversion efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.13 Inelastic collisions between K and Rb causes loss of doublons. . . . . . . . . . . . . . 118
6.14 An alternative method for determining the atomic filling fractions . . . . . . . . . . 120
6.15 Preparing doublons in the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.16 Characterizing the d-wave resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.17 K tunneling dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.18 K and Rb tunneling dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.19 The role of higher bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.20 Correlation between K and KRb in higher bands . . . . . . . . . . . . . . . . . . . . 133
6.21 Photoassociation of doublons in the lattice. . . . . . . . . . . . . . . . . . . . . . . . 135
7.1

The dipole moments of the bi-alkalis as a function of the electric field. . . . . . . . . 137

7.2

Overview of the new apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.3

The front macor holder in the science cell . . . . . . . . . . . . . . . . . . . . . . . . 141

7.4

Triple points at the end of the rod electrodes . . . . . . . . . . . . . . . . . . . . . . 143

xx
7.5

The electric field orientations that are possible with the electrode configuration of
the new apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.6

The energy shift of the |0, 0i to |1, 0i transition as a function of the electric field,
and the strength of J⊥ and Jz as a function of field . . . . . . . . . . . . . . . . . . . 145

7.7

Simulated electric field cuts along the axial and radial directions . . . . . . . . . . . 147

7.8

Simulating the field effects of a hole in the ITO plate . . . . . . . . . . . . . . . . . . 148

7.9

Electric field gradients induced by a wedge angle between the rods . . . . . . . . . . 149

7.10 Analysis of the field profile with an axial gradient . . . . . . . . . . . . . . . . . . . . 152
7.11 Analysis of the field profile with an axial gradient . . . . . . . . . . . . . . . . . . . . 154
7.12 The electrodes inside the new apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.13 The electrode layout inside the science chamber . . . . . . . . . . . . . . . . . . . . . 157
7.14 The optical path between the atoms/molecules to the objective . . . . . . . . . . . . 159
7.15 The coils and objective path around the science cell . . . . . . . . . . . . . . . . . . 162
7.16 Photographs of the coil mount and objective and dichroic mounts around the sciencecell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.17 Deflection of the vacuum windown due to the differential pressure. . . . . . . . . . 164

7.18 The full vacuum chamber of the new apparatus . . . . . . . . . . . . . . . . . . . . . 167
7.19 The back macor pieces which hold the electrodes in the science cell . . . . . . . . . . 169
8.1

The macor insulators which the electrodes in the science cell . . . . . . . . . . . . . . 172

8.2

Electropolished steel and tungsten rods . . . . . . . . . . . . . . . . . . . . . . . . . 174

8.3

An SEM image of mechanically polished tungsten rods . . . . . . . . . . . . . . . . . 175

8.4

SEM images of blemishes on the tungsten rods . . . . . . . . . . . . . . . . . . . . . 176

8.5

A schematic of the conditioning layout . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8.6

The voltage and current during a typical conditioning measurement . . . . . . . . . 179

8.7

Observation of blue flashing in the science cell during the conditioning measurement 181

8.8

A cartoon on the observation of triple-point-induced electron beam breakdown . . . 181

xxi
8.9

Damaged ITO-coated plates after conditioning . . . . . . . . . . . . . . . . . . . . . 183

8.10 The setup for ITO-coated plate conditioning with an incident high intensity beam . 185
8.11 The output mode of a Mach-Zehnder interferometer with an ITO-coated plate in one
arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.12 Activating the Titanium-sublimation pump, as seen from the small cube . . . . . . . 187
8.13 Rigs for accurately bending the electrodes . . . . . . . . . . . . . . . . . . . . . . . . 189
8.14 Tungsten rods bent to go into the science cell . . . . . . . . . . . . . . . . . . . . . . 190
8.15 The first assembly of the electrodes including the ITO-coated plates . . . . . . . . . 191
8.16 Measuring the angles between the rod and plate electrodes . . . . . . . . . . . . . . . 193
8.17 The capacitor design and construction for coupling AC signals onto the rods . . . . . 195
8.18 Designing the coupling capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.19 Assembly of the capacitors and inductors and their connection to eachother and thetransmission line to the science cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.20 The fully assembled large cube with all the inductors and capacitors connected . . . 198
8.21 A decoupling inductor and its resonances measured by its reflection as seen with a
network analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.22 A PCB designed to couple the AC signal into the transmission line . . . . . . . . . . 199
8.23 The AC coupling through the transmission line . . . . . . . . . . . . . . . . . . . . . 200
8.24 Connecting the electrode transmission line in the small cube . . . . . . . . . . . . . . 202
8.25 A solid model of the electrode transition which occurs in the small cube . . . . . . . 204
8.26 The reducer between the large cube and the small cube . . . . . . . . . . . . . . . . 205
8.27 Images of an ITO-coated plate before and after exposure to a high K flux . . . . . . 214
8.28 Baking a paired ITO-AR coating in air to various temperatures for one hour . . . . 216
8.29 An ITO/AR coating pair using SiO and an ITO thickness of 10 nm . . . . . . . . . . 217
8.30 The optical and electronic properties of the ITO/AR coating during a bake to 300 o C219
8.31 The AR coating on the cell windows before baking to the cell-assembly temperature 220
8.32 Fizeau interferometry of the cell windows . . . . . . . . . . . . . . . . . . . . . . . . 221

xxii
8.33 Testing the microscope objective by imaging an Air Force test target onto a profile
camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.34 Characterizing the resolution limit of the objective using a small pinhole . . . . . . . 224
8.35 Measuring the resistance and calibrating the magnetic field from the bias coils. . . 225

8.36 Thermal effects on the coils as a function of the applied current . . . . . . . . . . . . 226
8.37 Curvature of the z-component of the bias field in the axial and radial directions of
the coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8.38 Calibrating the field and field gradient from quantization/gradient coils . . . . . . . 229
8.39 A schematic of the servo circuit to stabilize the voltages after the high-voltage amplifiers231
8.40 The circuit for stabilitizing the voltages after the high-voltage amplifiers . . . . . . . 233
8.41 Measuring the voltage stability of the LM399 reference and the servo ouput . . . . . 234
9.1

Photographs of the process of moving the vacuum chambers . . . . . . . . . . . . . . 235

9.2

A render of a solid model of the entire new apparatus vacuum chamber . . . . . . . 236

9.3

UV LEDs for LIAD of K and Rb in the MOT chamber . . . . . . . . . . . . . . . . . 237

9.4

Bright and grey molasses laser detunings for both K and Rb . . . . . . . . . . . . . . 239

9.5

Aligning the cart coils with the chamber axis . . . . . . . . . . . . . . . . . . . . . . 241

9.6

Alignment of the very strong transfer beam onto the plugged quadrupole trap . . . . 243

9.7

Characterizing the heating of Rb in the transfer beam from pointing instability . . . 246

9.8

Transferring Rb atoms in the transfer beam using two different hyperfine states . . . 248

9.9Transferring Rb atoms past the bump . . . . . . . . . . . . . . . . . . . . . . . . . . 250

9.10 The optical layout of the transfer beam . . . . . . . . . . . . . . . . . . . . . . . . . 253
9.11 Heating during transfer in the transfer beam . . . . . . . . . . . . . . . . . . . . . . 256
9.12 Timing diagrams for the procedure of the experiment . . . . . . . . . . . . . . . . . . 259
9.13 Ultracold atoms and molecules in the science cell . . . . . . . . . . . . . . . . . . . . 262
9.14 The vertical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10.1 A schematic for applying electric fields to molecules in optical lattices . . . . . . . . 269

xxiii10.2 Applying AC electric fields and driving rotational transitions . . . . . . . . . . . . . 271
10.3 The magic electric field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

10.4 Simulations of atoms on sites of the lattice imaged with two different resolutions . . 276
10.5 A Schematic for combining the imaging and addressing beams . . . . . . . . . . . . . 278
10.6 The mounting scheme used for our DMD and its attached circuitry . . . . . . . . . . 280
10.7 Patterns on the DMD imaged onto a beam profile CCD . . . . . . . . . . . . . . . . 282
11.1 Spin-Orbital Coupling with polar molecules in an optical lattice . . . . . . . . . . . . 286
11.2 A quantum gas microscope for polar molecules . . . . . . . . . . . . . . . . . . . . . 290
11.3 Various steps required to realize a spin-resolved microscope, mapping | ↓i onto Rb
and | ↑i onto K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
11.4 Various steps required to realize a spin-resolved microscope, mapping | ↓i onto K
and | ↑i onto Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
11.5 TWA simulations of the imbalance of a 36 × 36 lattice system with z-excitations ina checkerboard pattern for three filling fractions and several checker sizes . . . . . . 303
A.1 An image of the tapered amplifier (TA) set-up in the Rb laser system . . . . . . . . 328
A.2 The brass base for holding the TA chip . . . . . . . . . . . . . . . . . . . . . . . . . . 329
A.3 The aluminum baseplate that mounts the TA fixture to the optical table or breadboard330
A.4 The plexiglass lid for the TA box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
A.5 The lens holder, which threads into the brass base . . . . . . . . . . . . . . . . . . . 332
B.1 The code for the stage motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
B.2 The motion profile of the stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
B.3 The code for the cart with the quadrupole trap . . . . . . . . . . . . . . . . . . . . . 336

Chapter 1

Introduction

In this chapter I provide a very brief context of the state-of-the-art in ultracold quantum
matter, and how dipolar interactions have become an important tool for many-body physics. I
then provide a discussion of the context of my thesis and the topics that it includes.

1.1

Quantum physics with ultracold matter
Since the demonstrations of laser cooling and optical molasses [159, 179, 210] and then evap-orative cooling to quantum degeneracy [8, 61], the field of quantum physics with ultracold matterhas burgeoned. In the intervening twenty years, a plethora of directions have been pursued, andthe capabilities of modern laboratories have been extended beyond anyone’s wildest dreams. While
the field has grown during this time, most research being pursued within the last decade can beplaced into one of the following catagories. Researchers are hoping to use ultracold quantum matterto learn more about real-life material systems, such as cuprate high-Tc superconductors; to create
new quantum systems that provide enhanced capabilities for measurement; and to create high fidelity quantum bits, or qubits, that can be wired into quantum circuits for quantum information,
computation, and communication.

1.1.1

Quantum simulation

The field of quantum simulation started with bulk, three-dimensional gases of bosonic or
fermionic atoms, where research was primarily centered on collective behavior [134, 240] such as2vortex formation [1, 154, 169], soliton dynamics [7], and the BCS-BEC (Bardeen-Cooper-Schrieffer
to Bose-Einsten Condensate) crossover [64, 101, 219, 288, 289]. From there scientists started reducing the dimensionality using optical lattices [93], which are typically constructed by retroreflecting a laser beam to create a standing wave. Three-dimensional lattices were the natural next
step [100, 138], and an enormous amount of quantum simulation work has been done with stronglyinteracting bosons and fermions in optical lattices [22]. The interactions between the atoms in such
experiments are tuned with a Feshbach resonance [43], which will be discussed in detail in Chapter
2.

1.1.2

Quantum information

While the interactions between neutral atoms can be very strong at short distances, they
originate from van der Waals forces and are thus very short range. Conversely, the goal of generating
entanglement between many qubits for quantum gate operations requires long-range interactions
that can be exquisitely controlled. Such interactions can be derived from electric monopoles,
magnetic dipoles, or electric dipoles. Electric monopoles are created using trapped ions, whichhave been the workhorse of quantum information experiments with cold atoms [155, 257]. Dipolarinteractions are the subject of the next section, and they can be generated from highly magnetic
atoms [5, 164], polar molecules, or highly excited Rydberg atoms [229].
Trapped ions and Rydberg atoms have been used for quantum gates [155, 229], and thegoal of both fields is to scale up the number of qubits for error correction and quantum registers.
Both approaches have advantages and disadvantages. For ions, scaling is very difficult because
they have such strongly repulsive interactions. Moreover, increasing the dimensionality from the
1D ion chains, which are used almost exclusively, to 2D or 3D crystals while maintaining singleion addressability and read-out has been prohibitively difficult, although recent attempts lookpromising [29, 224]. For Rydberg atoms the complexity is not in scaling up the system or changing
the dimensionality, but rather entangling larger systems and maintaining high gate fidelities.

3
1.1.3

Many-body quantum systems out of equilibrium

Before discussing dipolar interactions in more detail, I would like to describe a new area of
research that is becoming incredibly popular in recent years, which is non-equilibrium many-bodyquantum systems. As the control over neutral atoms continues to improve and as system sizes ofstrongly-interacting particles like ions continue to grow, the two approaches have begun to meet in
the middle. Thus, many experiments to date have excellent control of systems of 10-100 particles
with strong interactions. Simultaneously, topics like many-body localization [44, 125], quantum
thermalization [143], and correlation growth [140, 223] have begun to receive much more attention.
Ultracold quantum matter is an excellent template to study such physics because these systems
are inherently isolated quantum systems. I mention this because I think these directions are very
exciting, and I believe polar molecules will shed a lot more light on these subjects in the future, as
I mention later in the thesis.

1.2

Dipolar interactions
For many research directions, dipolar interactions would be ideal. Firstly, the interactions

between electrons in many real materials have a power-law scaling due to screening from the ionsforming the lattice, and thus a 1/R3 scaling is a reasonable approximation. Secondly, very shortrangle and very long-range interactions are difficult to work with, as I alluded to above. Therefore,
there is an enormous amount of open space in the interaction strength versus dimensionality parameter space which dipolar particles are particularly suited to explore. Thirdly, dipolar interaction
power laws are often the most challenging to simulate theoretically or numerically. Many-bodycorrelations are essential for describing the dynamics, and mean-field treatments are typically not
appropriate. From the experimental perspective, however, dipolar systems have been exceedingly
difficult to control, as discussed in this work. Nevertheless, important progress has been made.
The difficulty stems from the following facts: 1) highly magnetic atoms have been surprisingly
easy to manipulate, but their interactions are relatively weak and not tuneable, 2) Rydberg states

4
are too short-lived to realize dynamics, and 3) while polar molecules have strong interactions
that are long-lived, they are hard to control. Recent work has been done on Rydberg dressing,which allows Rydberg-state character to be mixed into ground-state atoms, thereby embuing themwith long-lived, strong dipolar interactions [87, 279]. I think this is a very promising direction.
Nevertheless, this direction is very new and many important questions are so far unanswered. Infact even highly magnetic atoms were largely unexplored when ground-state polar KRb molecules
were first made in 2008.

1.3

Topics in this thesis
In this thesis I will cover all the way from the first creation of ultracold ground-state polar

molecules to the present day where we are working towards building a quantum gas microscope ofpolar molecules in optical lattices for quantum simulation and quantum information. Prior to thework in this thesis the progress of ultracold polar molecules was limited by their complexity. The
molecular gases were too hot, the motion in the trap was not entirely controlled, their interaction
with the trapping light was poorly understood, and the molecules suffered fast loss from chemical
reactions. Further, large electric fields only enhanced the chemical loss in 3D gases. This was only
beginning to be controlled by the beginning of my thesis using 2D systems [63]. Moreover, there
was a quickly growing number of theoretical proposals for ultracold polar molecule experiments,
but many of them required new tools.
In Chapter 2 I will provide an experimental background and overview, where I describe thesetting in which polar molecules were first created. Then I will describe the physics of so-called
Feshbach molecules and how a Raman laser sequence can be used to convert Feshbach molecules tothe rovibronic ground state where they have large dipole moments. Then I will provide an overview
of the apparatus, specifically the first generation apparatus and the laser systems that we used inboth the old and new experiments.
In Chapter 3 I will outline the physics and chemistry of ultracold molecules that was observedin bulk gases in 2D or 3D systems. While I was not directly involved in these experiments, I will

5
describe them because they are important to set the stage for the work I was involved in with opticallattices, as well as future work on evaporative cooling and stabilizied dipoles in large electric fields.
Moreover, I coauthored a chapter for a book entitled “Low energy and low temperature molecular
scattering” in which we review all of these topics.
In Chapter 4 I will introduce molecules in optical lattices, and the stability they afford. I will
give an overview of the physics of optical lattices, band structure, superfluids, and Mott insulators.
Then, I will describe all the experiments we have performed to understand molecules in opticallattices and the effects of optical traps which are unique to polar molecules.
In Chapter 5 I will present our experiments studying dipolar spin-exchange interactions between molecules in two rotational states pinned in a deep optical lattice. These observations include
many-body dynamics, and constitute an important first step towards quantum magnetism with polar molecules.
In Chapter 6 I will describe our extended efforts to reduce the entropy, or temperature, of
our molecular sample. Ultimately, our success came using a quantum synthesis approach of atomic
insulators in the lattice that we convert to molecules with high fidelity. This approach has allowed us
to make the first low-entropy quantum gas of polar molecules, which could be considered a quantum
degenerate sample in a 3D optical lattice, and could allow for novel experiments on many-bodyout-of-equilibrium spin systems.
In Chapter 7 I will describe the new generation KRb experiment and all the design featuresthat will present enormous improvements to the old apparatus. Specifically, it is designed for large,
tunable electric fields generated by in-vacuum electrodes, as well as high resolution detection andaddressing. Such tools are necessary to take the next steps with capabilities described in the prior
two chapters: quantum magnetism and spin physics, and a low entropy system where exotic, truly
many-body dynamics and phases emerge.
In Chapter 8 I will report how the new apparatus was designed, built, and tested. There werean enormous number of technical limitations that must be overcome, and a huge number of tests
were required to successfully build the new apparatus machine to provide all the tools described in6the previous chapter. This chapter is entirely technical, but it is by far the longest chapter in thisthesis. I hope this fact illustrates just how complex the new apparatus is, and how exciting ourultimate success has been upon the completion of its implementation.
In Chapter 9 I will outline the procedure for producing ultracold atoms and molecules in
the new apparatus, and describe the conditions that we have achieved. These conditions are quite
comparible to the old chamber, despite a significantly more complicated experimental procedure.
In Chapter 10 I will present the novel experiments that we have performed in the new
apparatus, taking advantage of all the new tools for which it was designed. We were immediately
able to go beyond anything that we were capable of in the old apparatus, such as large electricfields, polarization-selectivity of rotational transitions, and high resolution imaging.
In Chapter 11 I will close with an outlook of where we have come and what the next steps
are for polar molecules, and the new apparatus in particular. The new apparatus opens the door
to many experiments which have been proposed over the years, and I will highlight the ideas thatare next on our list.

Chapter 2

Experimental background and overview

In this chapter I present an introduction to the field of ultracold polar molecules, which isless than a decade old. By extending the techniques that have been developed for ultracold atoms,
I describe how a molecular quantum gas can be produced. I then describe the first generation JILA
KRb apparatus, it’s move into the new X-wing of JILA in 2012, and the laser systems that havebeen in use on both the first and second generation KRb machines since the move.

2.1

Creation of ultracold molecules
The field of ultracold polar molecules has exploded over the past decade as researchers haveworked to extend the control offered in ultracold atom experiments producing atomic Bose-Einsteincondensates (BECs) [8, 26, 27, 61] and Degenerate Fermi gases (DFGs) [64, 68, 256] to the worldof molecules. Heteronuclear molecules can have intrinsic electric dipole moments, and they provide strong, long-range, and anisotropic interactions with precise tunability. A strong interest has
emerged in the scientific community to study systems with long-range interactions, as discussed in
the previous chapter. These systems are ideal candidates for the study of strongly correlated quantum phenomena, as well as for quantum simulation of lattice models relevant to some outstanding
problems in condensed-matter physics. The anisotropic nature of dipolar interactions provides powerful opportunities to control chemical reactions in the low energy regime, and dipolar interactions
can also give rise to novel forms of quantum matter such as Wigner crystallization [146], d-wave
superfluidity in optical lattices [151], fractional Chern insulators [276], and spin-orbit coupling [250].

8Efforts to produce stable gases of ultracold polar molecules started in the early 2000’s, andare documented in a recent review [36]. However, the production of ground-state polar molecules in
the quantum regime, where motional degrees of freedom must be described quantum mechanically,
was a challenging task and it took a major effort to achieve the first success in 2008 [190]. The key
was the combination of the use of a Feshbach resonance for magneto-association of bialkali atoms
and coherent optical state transfer via stimulated Raman adiabatic passage (STIRAP) [167, 216].
Feshbach resonances, where the energy of a colliding pair of free atoms matches the energy of abound state associated with a molecular potential (see Fig. 2.1), have been studied extensively in thecontext of controlling atomic interactions via magnetic fields, and they have become one of the mostpowerful tools for quantum gas experiments. Homonuclear, weakly-bound molecules associatedfrom free atoms using a Feshbach resonance (i.e. Feshbach molecules) were produced directlyfrom single-species ultracold atomic samples [69, 115, 135]. In particular, homonuclear Feshbach
molecules, even if composed of fermionic atoms, are themselves bosons, and their production enables
studies of a crossover from the BCS-type Cooper pairing of fermions into a BEC of weakly-boundmolecules [18, 101, 289]. In an optical lattice, such weakly-bound molecules can be effectively
protected: when the tunneling rate is smaller than the on-site reaction rate, molecular tunnelingis suppressed via the quantum Zeno effect [249]. Work on using the STIRAP technique to transfer
molecules to more deeply bound states was performed on both Rb2 [153, 270] and Cs2 molecules [57,59]. However, homonuclear molecules do not have an intrinsic electric dipole moment, and are thus
not useful for studying dipolar scattering or long-range dipole-dipole interactions.
Inspired by the work with weakly bound homonuclear molecules, many researchers started
producing heteronuclear dimers, involving various combinations of alkali atoms [43, 147]. However,the path to deeply bound, or ground-state, molecules was challenging. The earliest efforts for
producing cold polar molecules relied on photoassociation to optically couple free atoms to an
excited molecular electronic state, which then spontaneously decayed to many rovibrational statesin the ground electronic potential. In order to collect the population in the rovibrational groundstate, one can choose an excited state with good Franck-Condon overlap [48, 81] with both the9free atom state and the ground rovibrational state [136]. However, the efficiencies for both theexcitation from free atoms to an excited molecular state and the subsequent spontaneous decay
to a specific rovibrational state in the ground are very low. As a consequence, the phase-spacedensity (particle density in coordinate and momentum space) of such cold molecule samples istypically below ∼ 10−10 [119]. Several more recent experiments have managed to improve on thislimitation by using electronically excited states with stronger Franck-Condon coupling to deeply
bound states (see e.g. References [222, 242]), but they are not in the rovibrational ground state,
and the phase-space densities are still very far from reaching the quantum regime. For reference,the phase-space-densities routinely reached with atomic quantum gases are ≥ 1, and quantizedmotions, quantum statistics, and collective behavior become apparent.
Alternatively, the coherent transfer approach first adiabatically converts a pair of free atomsof different species into a highly vibrationally excited state in the ground electronic potential usinga Feshbach resonance, and then couples these Feshbach molecules to the rovibrational groundstate via an adiabatic Raman transfer process through an intermediate electronic excited state.
This approach effectively maintains coherence from the initial to final quantum state, and thus
introduces a minimal amount of entropy during the atom-to-molecule conversion process. The firstsuccess came in 2008 with the production of heteronuclear fermionic KRb molecules [190, 199].
Research on deeply bound heteronuclear molecules has exploded in recent years. Several groups
have now produced heteronuclear molecules in their deeply bound ground state, in species suchas bosonic RbCs [180, 252], fermionic NaK [203], and bosonic NaRb [106]. The fermionic KRb
molecules recently reached low entropies in an optical lattice [52, 183], to be described in Chapter
6.

2.1.1

Magneto-association

A pair of free atoms can be directly converted to a weakly bound molecule in the ground
potential when this bound molecular state (associated to a closed channel) becomes degenerate inenergy with the continuum state of two free atoms (associated to an open channel). This type of

10

Figure 2.1: Potential energy wells of the closed and open channel near a Feshbach resonance. Theenergy offset Ec can be adjusted via a differential Zeeman shift using a magnetic field to realize
resonant coupling of the two channels. Reproduced from Reference [43].

Feshbach resonance exists for essentially all atomic species with a magnetic moment, and can beused to tune the relative energy between the atomic and molecular state, and the two-channel model
for a Feshbach resonance is shown in Figure 2.1. Since the two channels typically have different
magnetic moments, the energy difference between them, Ec , can be adjusted with a magnetic field
to realize resonant coupling of the two channels [43]. As a result of this resonant coupling, thescattering length acquires a dependence on magnetic field, shown in Fig. 2.2a, and given bya(B) = abg 1 −∆,B − B0(2.1)

where abg is the background scattering length that parameterizes the interaction strength, and ∆is the magnetic field width of the resonance, which is the difference between B0 and the field that
corresponds to a = 0 (see Fig. 2.2a). Note that the energy scale plotted in Fig. 2.2b is normalizedby δµ, which is the difference in magnetic moments of the two channels. Typical widths in themost experimentally used resonances are ∼ 1 − 300 G, although some molecules (e.g. Cs2 [59, 115])
use a resonance width of several mG.
At the field B0 , the energy difference between the closed and open channels vanishes, andthe resultant scattering length of the two atoms diverges. Tuning B away from B0 on the side of

11

Figure 2.2: Scattering length and energy of two atoms in the vicinity of a Feshbach resonance. (a)shows the scattering length normalized to the background scattering as a function of field, while(b) shows the binding energy as a function of field. Note that this figure assumes abg > 0, which is
not always the case. Reproduced from Reference [43].

the positive scattering length leads to a molecular binding energy Eb , and it is given by
Eb = ~2 /2mR a2 ,(2.2)

where mR is the reduced mass of the atom pair and a is the scattering length of the atoms [43].
Typical binding energies in Feshbach molecule experiments are ∼ 0.1 − 1 MHz.
One way to produce Feshbach molecules is to sweep B across the resonance from the attractiveside (abg < 0) to the repulsive side (abg > 0) so that the pair of free atoms adiabatically enters
the closed molecular channel. Figure 2.3 illustrates this process schematically, and also shows thechange of the molecular binding energy as a function of B. The free atoms are confined in aharmonic trap, and the different entrance levels correspond to different harmonic oscillator statesin the trap. In the limit that the sweep is adiabatic and the harmonic oscillator states are wellresolved, only the lowest energy free atomic state couples to the bound channel. This process iswell described by a Landau-Zener coupling mechanism, and the probability that free atoms are
adiabatically converted to molecules is given by [147]:
PFbM = 1 − e−2πδLZ .

(2.3)

12

Figure 2.3: The energy of two atoms as a function of magnetic field. The harmonic oscillator levelsof the atoms in a harmonic trapping potential are shown above the resonance. The closed channel
energy is Eres (B). The lowest harmonic oscillator level couples to the Feshbach molecule, whoseenergy reduces below the resonance as Eb (B). Reproduced from Reference [147].

Here δLZ is the Landau-Zener parameter, which in a weak harmonic trap is [147]
√

ho
δLZ
=

where lho =6~ abg ∆,3π2φlhoḂ(2.4)

p~/mω̄ is the harmonic oscillator length of the molecule in the harmonic trap (withmean trap frequency ω̄), and Ḃ = dB/dt is the sweep rate across the resonance.
An alternative method to create Feshbach molecules is to couple the open and closed channels
with a radio frequency (rf) field. This technique is used in many experiments, and enables coherent
Rabi oscillations between free atoms and molecules [69, 193]. Fig. 2.4 shows a beautiful illustrationof this technique.

2.1.2

Coherent optical transfer

After producing weakly bound Feshbach molecules, a two-photon coherent population transfer technique, STImulated Raman Adiabatic Passage (STIRAP) [167, 216], is used to transfer
molecules to the rovibrational ground state. Three states are involved in the process: the initial
Feshbach molecule state |ii, an electronically excited state |ei (which is usually short-lived), andthe rovibrational ground state |gi. The excited state is chosen, after an exhaustive spectroscopic
search, to optimize the Franck-Condon factors with both |ii and |gi. The states are coupled with

13

Figure 2.4: RF association of NaK Feshbach molecules from two different initial states. In bothcases the binding energy can be seen as a separate peak that can be spectroscopically resolved fromthe free K atom peak. Reproduced from Reference [272].

14
two laser fields: the up leg, with Rabi frequency Ω1 , couples |ii and |ei; the down leg, with Rabi
frequency Ω2 , couples |gi and |ei. The transition dipole moments (including the Franck-Condon
factor) for both up and down legs are 0.5 to 1 ×10−2 atomic units [190]. The goal of STIRAP is to Ω1maintain a field-dressed dark state cos θ|ii + sin θ|gi, where θ = tan−1 Ω. The laser intensities
2
should adiabatically change from Ω2 /Ω1  1 to Ω1 /Ω2  1. In practice, the laser intensities areramped over a duration τ  1/Ω1,2 . Since the population is trapped in the dark state, population transfer takes place from |ii to |gi without populating the short-lived state |ei. The typical
STIRAP protocol for ultracold molecules is based on a dark resonance configuration, where boththe one-photon and the two-photon detunings are zero. The excited state population is therefore
adiabatically eliminated under these conditions, provided β  1/τ , where β is the relative laser
linewidth of the two-photon transition [277]. Thus, the two laser fields must be phase coherent
with each other during the time τ . For typical pulse times ∼ 10 µs, relative laser linewidths lessthan 1 kHz are required, which can be achieved by stabilizing the lasers to an optical frequency
comb [190, 199] or a high-finesse optical cavity [4, 96].
Once in the ground state, we can remove the unpaired atoms by sending in resonant light on
the K and Rb cycling transitions. Then, by reversing the order of the intensity ramps, the groundstate molecules are converted via STIRAP back to Feshbach molecules. The Feshbach molecules
are then dissociated and the resulting atoms are imaged on either the K or Rb cycling transition.
It is also possible to directly image the ground-state molecules, but the lack of optical cyclingtransitions limits the signal to noise ratio [264].
Figure 2.5 shows the states used for efficient production of ground-state KRb molecules. Bialkali molecules have orbital angular momentum Λ = 0 in their ground molecular potentials (e.g.
Σ states), and are thus described by Hund’s Case (b) [30]. The initial Feshbach state is the mostweakly bound vibrational level of the 3 Σ+ ground potential, while the rovibrational ground state isin the 1 Σ+ ground potential. Thus, the state |ei must contain an admixture of singlet and triplet
characters in order to allow reasonable transition strengths on both legs of the transfer. Such an
admixture results from significant spin-orbital coupling in the excited electronic state. Using this15

Figure 2.5: Two-photon coherent state transfer for KRb from the weakly bound Feshbach moleculestate |ii to the absolute molecular ground state |f i (ν=0 (vibration), N = 0 (rotation) of X1 Σ).
Reproduced from Reference [190].

16
scheme, we typically find a ∼ 90% one-way transfer efficiency from the initial Feshbach moleculestate to the ground state (since the coherent process is fully reversible, we have the same efficiency
from the ground state back to the Feshbach state). An important aspect to consider is that thedifference in energy between the initial and final states is ∼ 4000 K, which is enormous comparedto the temperature of the gas (∼ 100 nK). Therefore, one might wonder whether any of this extra
energy is deposited into the kinetic energy of molecules, which would heat the molecular gas outof the ultracold regime. However, since the transfer process is fully coherent, the energy difference
between the states is carried away by the photons, and the gas remains in the ∼ 100 nK regime.
However, because the Raman lasers have different wavelengths, the molecules do get a momentum
kick equal to the difference of the photon momenta [195]. When producing molecules directly inan optical lattice (to be discussed later), this process allows the molecules to occupy the groundband of the lattice with very high probability in typical conditions of our experiment, e.g. lattice
depth, etc. [58, 183, 227].

2.2

The first generation KRb machine
In this section I will describe the basics of the first generation KRb machine. More details

can be found primarily in the thesis by Josh Zirbel [284], but also Kang-Kuen Ni [189], Marcio de
Miranda [62], Brian Neyenhuis [187], and Steven Moses [181]. Therefore I will only describe what
I think is important to contrast with the second generation apparatus which I will be describing inlater chapters.
The first generation apparatus is based on a double chamber design with a long differential
pumping tube between, as shown in Fig. 2.6. The first generation chamber has K dispersers, which
are enriched to 5% 40 K (natural abundance is 0.012%), and Rb ampules which we cool with athermo-electric controller (TEC). In this chamber we load a dual MOT (Magneto-Optical Trap)of K and Rb from a background vapor of both species. The pressure is maintained by controlling
the temperatures of the K dispenser and Rb ampule, and we typical operated with a pressure∼ 10−10 mbar, which corresponded to a MOT fill time and lifetime of ∼ 5 seconds. The MOT sizes

17

Figure 2.6: The old vacuum chamber in the first generation KRb experiment. The vapor cell MOT
area is on the right, and seperated by a large differential pumping section and a gate valve is thescience cell side.

were typically 2 − 3 × 109 for Rb and 1 − 2 × 107 for K. Upon loading the MOT, we performed
a compressed MOT stage for both K and Rb. Then we did bright optical molasses (we now use
grey optical molasses for both species), and finally optical pumping to the stretched, magneticallytrappable states.
After loading the quadrupole trap we transfer atoms through the differential pumping tube
to the science cell by translating the anti-Helmholtz coils on a motion track. The transfer efficiency
is ∼ 50% for Rb and ∼ 20% for K. The temperature after transfer is roughly the same for both,but the temperature of K is much higher than Rb before transfer. The differential pumping tube
actually shaves down the K cloud and thus reduces the temperature. The tube actually contains a
strange “bump” region in the differential pumping tube that is the primary limitation to the transfer
efficiency (its original intention was to prevent the migration of alkali metal from the hotter MOT
region to the room temperature science region). This bump region is described in Josh Zirbel’s andKang-Kuen Ni’s thesis, so I will just say that it is an unnecessary complication which reduces thetransfer efficiency. The need for a new apparatus, to be described later, was initially recognized
before my arrival partially with the goal of removing this bump.
Once we reach the science cell, the atoms are loaded into a Ioffe-Pritchard (IP) trap (shownin Fig. 2.7) where Rb undergoes forced rf-knife evaporative cooling by driving the |F, mF i = |2, 2i
to |1, 1i transition (where F is the total atomic angular momentum and mF is its projection

18

Figure 2.7: The phenolic form that holds the Ioffe-Pritchard trap around the science cell of the oldchamber.

19
onto the magnetic field axis). K is sympathetically cooled by Rb during the evaporation. Notethat we continually remove Rb atoms that may appear in the |2, 1i state, which will undergo
inelastic collisions with K. This is done by using a so-called “|2, 1i-cleaner” which continuously
drives the |2, 1i to |1, 0i transition resonant with the bottom of the trap during the evaporation.
This evaporation starts with 2 − 3 × 108 Rb and 2 × 106 K atoms at a temperature of ∼ 300 − 400µK and ends with 4 × 106 Rb and 6 − 7 × 105 K atoms at a temperature of ∼ 1 µK. The trap
frequencies in the IP trap are ωz = 2π · 20 Hz and ωr = 2π · 156 Hz for Rb in |2, 2i.
From these conditions we loaded atoms into a crossed optical dipole trap of beams with 1/e2
radii of 40 × 200 µK. Typical powers before optical evaporation were ∼ 3 W in each beam, and thenthey lower over 2-3 seconds to evaporate Rb against gravity, again sympathetically cooling K. This
would yield typical conditions of 3 × 105 Rb and 3 × 105 K at 300 nK, just above Tc and TF , theBose-Einstein transition temperature and the Fermi temperature of Rb and K, respectively. Such
conditions yield the highest number of Feshbach molecules and ground state molecules, and wewere routinely able to get 3 − 4 × 104 at T = TF . However, for the later discussion of a low-entropy
quantum gas in an optical lattice we made deeply degenerate Bose-Fermi mixtures, and we could
get 2−3×105 Rb at 0.1Tc and 2−3×105 K at 0.3TF . This improvement in the phase-space-density
was obtained both by evaporating further in the optical trap and by increasing the aspect ratio of
the optical traps.
The glass cell on the old chamber was an uncoated pyrex cuvette that was fused to a glassto-metal seal. This assembly and fusing process was done at JILA by Hans Greene, but resulted invery large surface features such as ripples in the glass. Therefore, it was never possible to generate
precise optical potentials or to perform high resolution optical detection. Moreover, the lack of ananti-reflection coating for the optical lattice beams made the vertical lattice path significantly moredifficult to align in a way that minimized super-lattice effects (see Steven Moses’ thesis [181]). The
combination of this low quality glass cell and the limited optical access served as another source of
motivation for the new apparatus.
While electric fields were used only minimally in published results from this experiment, we20

Figure 2.8: Charging problems in the first generation apparatus. (a) shows the glass cell (blue) withelectrodes around it in a particular arrangement to generate a vertical field with a radial gradientfor evaporative cooling. (b) shows a rotational transition moving by 10’s of kHz after only 5 shots.
(c) shows how the transition frequency relaxes over 10’s of kHz over many hours. (d) shows howeven the STIRAP resonance is shifted by the electric field induced by the charges on the glass, andshows how this induced field decays to zero over a timescale of several hours.

21
have spent a lot of time working with them over the years. Electric fields were generated by placing
a small phenolic form between the inside of the IP trap (shown in Fig. 2.7), and the pyrex cell(shown in Fig. 2.8a). This phenonic form held indium-tin-oxide (ITO) coated glass plates, to whicha small piece of copper foil was glued with a conductive epoxy. Then a high-voltage cable wassoldered to the copper foil. The magnitude of the field was limited to 5 − 6 kV/cm by the nearby
grounded coils on the IP trap (although we attempted to go higher in the earlier years), from which
we were able to reach a dipole moment of 0.2 Debye [47, 63, 190].
The problem with electric fields that really limited us in the old chamber, however, wasenormous transients and instability. We believe this is due to patch charges accumulated the glass
between the electrodes, and even when the electric field was ostensibly zero, rotational transitions
and even the STIRAP lineshapes were completely shifted by the electric field from the residual
charges on the glass, as shown in Fig. 2.8b&c. Figure 2.8d shows how the field would relax on
a timescale of roughly 5 hours. This problem made it very difficult to do any rotational state
spectroscopy in an electric field, and even made evaporative cooling of molecules prohibitively
difficult. Figure 2.8 was also shown in Steven Moses’ thesis because this was an extremely important
problem which took a lot of our time in the early years.
Indeed, we tried many approaches to remove these charges, all of which were unsuccessful.
We tried baking the chamber and shining UV lamps on it; we tried “de-Gaussing” the chamber withhigh voltage electrodes that we switched at various frequencies and pointed in various directions;and we even removed and replaced the glass cell on the chamber! This last attempt was actuallya very large project, and required baking the chamber again to reach the typical vacuum lifetime.
This latter effort actually worked, and all the charges were removed...until we applied large fields
again and recharged the new pyrex cell, which was identical to the old cell. At this point we gave
up on using electric fields for the next several years, and these problems served as the primary
motivation for the second generation apparatus to be described later.

22

Figure 2.9: Moving the lab. (a) shows the old lab, S1B05. (b) shows moving the experiment downa half-flight of stairs. (c) shows the new lab, X1B20, before moving the optical table from the otherlab. Note that a second table is in place to the right, and this is the table where almost all of thelasers are situated.

2.3

Moving the experiment into the JILA X-Wing
At the end of 2012 and into the new year of 2013 we moved our experiment into the lab spacethat was prepared for us in the new X-wing of JILA. The two labs were only separated by 20-30
meters, but there was a half-flight level change between the two wings, which required a crane asshown in Fig. 2.9. We were very careful during the process, and we closed the gate valve between
the two halves of the chamber before the move. Nothing went wrong during the move, and as with
the OH experiment which had moved earlier to the other side of our new lab, our first paper inthe X-wing was in published in Nature. We were able to create a MOT within 1 week and a BEC
within two months despite changing several aspects of the system.

2.4

Laser systems in the X-Wing lab
Here I will give a brief overview of all the laser systems used in the X-wing lab since 2012,

which include the MOT/probe lasers, the Raman lasers, and the high power optical trapping andlattice lasers at 1064 nm. More recently we have added a MSquared tunable Ti:Sapphire laser,which I will discuss more during the chapter on the new apparatus.

23
2.4.1

MOT/probe lasers

While most of the experimental apparatus did not change as a result of the move, we tookthat opportunity to completely change the K and Rb trapping and probing lasers. The old systems
were based on homemade external cavity diode lasers (ECDLs) and Newport Vortex lasers, which
had become very unreliable and required frequent relocking. The new laser systems were build by
Steven Moses, and they are based on distributed Bragg reflector (DBR) lasers from Photodigm.
These lasers have high power (∼ 100 mW), and very large mode-hop-free tuning ranges of 10’s to100’s of GHz. Therefore, these lasers routinely stay locked for weeks. Eagleyard tapered amplifiers(TAs) are used to deliver enough power for the MOTs, which require ≈ 100 − 300 mW.
The repump lasers for both K and Rb are locked to an absorption cell using saturated absorption frequency-modulation (FM) spectroscopy. This is done using counter-propagating beams
of a few 100 µW whose phase is modulated with an electro-optic modulator (EOM) at 20 MHz.
The error signal is derived by demodulating the probe beam transmission with a phase-shifted rf
tone of the modulation frequency [21].
The trap lasers are locked to their respective repump lasers by beating them on a fast photodiode and locking to a frequency offset. This is done by mixing the ∼GHz beat notes with avoltage controlled oscillator (VCO), and in the case of Rb we divide the beat note frequency by 8
before mixing with a VCO. After mixing these frequencies down, they are converted to a voltage
with a F-V converter, which has an adjustable offset. While this is a very robust way to offset locka laser, we are constantly limited by the stability of the VCO, which has a large thermal coefficient.
Therefore, we are in the process of updating our offset lock approach to use direct digital synthesis(DDS)-generated frequencies instead of VCOs.

2.4.2

Raman lasers and cavity

In 2012 Steven build a bichromatic high finesse optical cavity for locking the Raman lasers,
instead of the Ti:Sapphire comb that had been in use since 2007 or so. While the Ti:Sapphire

24

Figure 2.10: The bichromatic optical cavity used to lock the Raman lasers. The two laser wavelengths are 690 nm (blue) and 970 nm (red). As shown in this figure, an integer number ofwavelengths for both lasers is required to match the cavity length.

system allowed for enornous tunability of the Raman lasers, which was critical during the initial
spectroscopy period in 2007 and 2008, the stability of the Raman laser system was limited by thestability of the comb. During the first year of my PhD when we were locking the Raman lasers to
the comb, it would take hours or even the full day in order to get the comb to stay robustly locked.
The situation was further complicated by the fact that the comb was in a different room from theKRb experiment, with a long fiber strung between them.
We typically operate with 200 mW of power in the up leg and 20 mW in the down leg, whosetransition dipole moments are 0.005(2) ea0 and 0.012(3) ea0 , respectively [190]. Such powers areneeded to achieve Rabi frequencies of a few MHz. For the up leg at 968 nm we use an Eagleyard antireflection (AR)-coated 980 nm laser diode in a Littrow configuration (the AR coating gnenerally
shifts the wavelength blue by ∼ 10 nm). This is then amplified with an Eagleyard 970 nm TA.
The down leg is also an Eagleyard anti-reflection AR-coated diode laser operating at 689 nm in the
Littrow configuration, though it has been changed several times over the years and was a Sacher
Lasertechnik diode in the past. For the down leg we amplify the transmitted power through thecavity (∼ 500 µW) instead of the raw diode because it is much more stable. In the end we did notobserve an significant effect on the STIRAP efficiency compared to using the down leg before thecavity, but nevertheless we inject an OpNext slave laser (HL6750MG) with the cavity-transmitted
light, which is ∼ 100’s of µW.
The cavity has a finesse of ∼ 2 × 104 , and is comprised of a cylindrical piece of Zerodur with a25bore through the center (see Fig. 2.10). The mirror substrates are fused silica; one mirror if flat andthe other has a radius of curvature of 50 cm to ensure that the transverse modes are non-degenerate.
The coating was done by Advanced Thin Films in Boulder, and the cavity is temperature stabilized
to ∼ 10−2 o C, from which its resonances drift by roughly 100 kHz daily. The two-photon STIRAP
lineshape has a width of 500 kHz, so this drift can easily be corrected by optimizing the STIRAP
efficiency every few days.
The lasers are locked to the cavity using a standard PDH locking technique [71] to thetransverse electromagnetic TEM0,0 mode. The servo bandwidths are ∼ 2 MHz for both lasers. The
PDH error signal, cavity-ringdown spectroscopy, and the cavity alignment are shown in Fig. 2.11.
The frequency difference between the nearest TEM0,0 cavity mode and the desired laser frequency
is compensated using an AOM. Indeed, the STIRAP lineshape is scanned by adjusting the up leg
AOM which is in a double-pass configuration. Thus, the down leg AOM is configured so that
STIRAP operates on resonance with the intermediate state, as discussed earlier in this chapter,and the up leg double-pass AOM is adjusted to scan the two-photon resonance. This Raman laser
system has become incredibly robust, and is now one of the most reliable parts of the experiment.

2.4.3

Optical trap/lattice lasers

All the optical traps and lasers are at λ = 1064 nm, and while the laser systems havechanged many times over the years, we have been using a 50 W Nufern fiber amplifier since 2013
when we moved to the X-wing. This amplifier is seeded with a Mephisto NPRO (non-planar ringoscillator) solid-state laser, which has a linewidth of ∼ 10’s of kHz. The linewidth of the Nufern
after amplification is increased to ∼ 100’s of kHz, which is still quite good. Mephisto now makes
their own laser amplifier to accompany their NPRO which has an unequivocally lower noise profile
than the Nufern, and most optical lattice groups have moved away from Nufern lasers. However,
we have been using the Nufern since before the Mephisto option was popular or even available, andwe believe that we are not as sensitive to laser noise on our lattices as for example lithium quantumgas microscope experiments.

26

Figure 2.11: Locking to the optical cavity. (a) The PDH error signal for the up leg at 970 nm(black) as a function of the detuning from the cavity resonance. The cavity transmission is also
shown in red. (b) A cavity ring-down measurement with the up-leg, which is used to measure thefinesse and the free-spectral range of the cavity. (c) The PDH error signal for the down leg at 690
nm (black) as a function of the detuning from the cavity resonance. The cavity transmission is also
shown in red. Note that the transmission on resonance is very noisy. This measurement was takenwith the diode at the end of its life, and was subsequently replaced. (d) The alignment procedureof the cavity produces many beautiful Hermite-Gaussian spatial modes. These images show thetransverse electromagnetic TEM0,0 and the TEM1,0 modes.

27

Figure 2.12: The Nufern fiber amplifier layout. The main beam path “arms” are shown in thick
red, and diffraction from each AOM is shown in dotted red. Obstructions from other optics andother parts of the experiment are shown in dark grey. Note that an isolator has since been added
to the right arm.

All of our beams are derived from the Nufern, and so the power needs to be intelligently
distributed between many paths in a way that maximizes the power available at any stage of theexperiment. To accomplish this we use a layout as shown in Fig. 2.12, where the power follows
a path through several acousto-optic modulators (AOMs), and can be diverted to each fiber atwill using each AOM. Typical AOM diffraction and fiber coupling efficiencies are ∼ 80%, and wetypically operate at a total power of 40 W. Note that recently an isolator has been added to theright arm to reduce back-reflection.
While this layout style is inherently challanging to maintain since any change to an AOM willseverely affect the subsequent paths, we eventually converged on a set-up that is quite robust. We
use a photodiode to detect the power going to either arm, as controlled with the waveplate before
the polarizing beam splitter at the beginning of the path. The photodiodes are placed at the end ofeach arm, and are picked off from the beam before the water-cooled beam dumps using back-side
polished mirrors. These mirrors reflect ∼ 99.5% of the light, so looking at the transmitted portionis inherently noisy since a 0.1% change in the reflection can be a 100% change in the transmission.
Indeed, we see ∼ 30% fluctuations in the photodiode voltages over several days.
These photodiodes are very important for us because we see enormous polarization fluctuations coming out of the Nufern. We spent a large amount of time trying to understand and remove

28

Figure 2.13: The power calibration of the Nufern. (a) The power as a function of the chiller
temperature. (b) The power as a function of current for a chiller temperature of 23 o C.

these fluctuations, but they appear to be caused by the laser mode coming out of the fiber fromthe amplifier. To solve this problem, we use a Thorlabs motorized rotation mount with a λ/2
waveplate, which we feed back to using one of the photodiode voltages and regulate with a digital
servo. This works very well, although we occasionally have to change the servo setpoint slightly as
the power fraction to the photodiode drifts.
Another major complication from the Nufern is its inherent pointing instability. We see thatthe direction the beam is coming out of the fiber changes with time, and typically has a ≈ 100 µm
RMS deviation over a distance of ≈ 1 m, over timescales of 10’s of minutes to hours. We thinkthat this issue also comes from the mode of the light in the fiber, which changes slightly in time.
Therefore, this pointing fluctuation appears to be correlated with the polarization fluctuations. It
is particularly problematic for us given the length and complexity of our two arms. We tried touse a piezo-actuated mirror and a quadrant photodiode to servo this pointing drift, but our ability
to fully correct for this is limited by how close we can place the mirror to the Nufern fiber output.
Thus, in the end we have to re-optimize most of the path alignment every few weeks.
Some of these issues are potentially exacerbated by the fact that we were using the JILA
chilled water system to cool our Nufern, which operates at ≈ 17 o C. The output power of the Nufern

29

Figure 2.14: The hollow-core photonic crystal fiber. (a) The brass cooling block attached to thefiber connector, and the coupling optics for the fiber. (b) The profile of a focused beam far from thefiber output. Note the hexagonal shape caused by the core structure of the fiber. Any aberrations
wash out the corners.

is strongly dependent on the cooling temperature as shown in Fig. 2.13a, and the recommended
operating temperature is 23 o C. Accordingly, we switched from the JILA chilled water system to ahigh cooling power chiller which allows us to vary the temperature. With the laser operating at the
optimal temperature we can measure the power as a function of the input current, as in Fig. 2.13b.
Cooling the laser to its optimum temperature helps to mitigate the pointing and polarization drift,
though the difference has not been carefully quantified.
For beams where more than 3 W is delivered to the atoms we use hollow core photonic
crystal fibers from TraTech FiberOptics. We water cool the brass connector at the end of thejacket using a cooling block which also provides mechanical stress relief [196] (see Fig. 2.14a). The
cooling plates have holes for two thermisters; one for monitoring the temperature, and the otherfor an interlock on the corresponding AOM rf power. We typically optimize the fiber alignment by
pulsing the AOM with a 5% duty cycle. The temperature change is < 0.5 o C when well aligned
even with an input of 20 W. To couple light into and out of these fibers we use a plano-convex fusedsilica lens with focal length f = 20 mm, since the normal Thorlabs aspheres are made of KB-7 orother glasses with higher absorption. When the fiber output is focused with minimal aberrations,a slightly hexagonal shape is apparant even in the far field. The initial coupling of light into thesefibers is very difficult, and we’ve gotten pretty good at coupling light from a fiber alignment pen

30through the other side of the fiber without disturbing its mounting, or any optics between it andthe vacuum chamber.

2.5

Computer control software
Until very recently the computer control software for this experiment has never been updated

from the original mid-2000’s design. The control program was written in Labview in the early
2000’s, and I think Debbie Jin and Scott Papp were heavily involved in its original construction.
This program controls 64 digital output lines using a digital input/output (DIO) board, which ismade by Viewpoint Systems (64-DIO64PCI). The board’s internal clock sets the timing for theentire experimental sequence. There are also three digital to analog converter (DAC) boards madeby National Instruments (NI-PCI6733), which are synchronized with the DIO board. Two of the
DAC boards generate 16 analog outputs, and the other is used to generate the detailed timedependent ramps for optical traps. We call the former “static DACs” and the latter “dynamics
DACs”. The static DACs can perform linear ramps, but cannot be changed dynamically and cannot
do complicated waveforms.
In recent years we have run out of both digital and analog lines, and so we have begun to add
many FPGA and DDS boards that operate in parallel with the Labview architecture. These boards
are controlled with Matlab and Labview. Moreover, before we implemented the new apparatus
chamber, we attempted to update our DAC boards and the version of Labview used to control theexperiment. We run with Labview 7.1 which requires Windows XP. We bought new DAC boards(identical boards) to replace the old ones, but these also require a newer version of Labview andWindows. In the end we found it to be very complicated to update the version of Labview to run
on Windows 7 or Windows 10 while maintaining the original Labview program. Therefore, until
recently we have been operating with a very old computer control system.
We have become increasingly aware of problems with this system, as well as residual timingbased bugs in the Labview control program. Over the years many digital lines have begun to fail,and some of the DAC channels are also unreliable. Recently we learned that the DACs are unstable

31
at the 10’s of mV level such that ramps are jagged and non-monotonic. Therefore, once we createda degenerate K-Rb mixture and KRb molecules in the new apparatus, we decided to update ourcomputer control architecture.
We recently switched to an FPGA-based control software which is controlled with Python.
This program was recently designed by the Strontium group in Jun’s lab, and Ross Hutson inparticular. Both the anolog and digial controls are derived from FPGAs, such that eight are used
in total. Two boards with 64 digital lines are used to generate 128 digital lines, and six boards
with 8 DACs are used for 64 analog DAC lines.

Chapter 3

Quantum-state controlled chemical reactions and dipolar collisions

Immediately after ultracold polar KRb molecules were produced in the rovibrational groundstate, as well as in the lowest hyperfine state, it became clear that chemical reactions were responsible for the loss of molecules confined in a conventional far-off-resonance optical dipole trap [191, 198],
as the reaction 2KRb → K2 + Rb2 is exothermic. For understanding the nature of the dipolar collisions between molecules in the quantum regime, it is imperative to have a thorough understandingof the molecular states involved and the resulting quantum statistics of the colliding molecules. In
this chapter I will describe the experiments that led up to the beginning of my PhD research, which
is discussed in the next chapter. These discussions can also be found in our book chapter in “Lowenergy and low temperature molecular scattering” [53].

3.1

Quantum state control of the molecule
STIRAP transfers the molecules to the rovibrational ground state of the singlet (X 1 Σ+ )electronic ground potential. Although the electronic angular momentum is zero, there is hyperfine
structure arising from the nuclear magnetic moments of the constituent atoms. K has nuclear quantum number I K = 4, while Rb has nuclear quantum number I Rb = 3/2, so there are 36 hyperfine
states for each rotational state |N, mN i, where N is the principal rotational quantum number of
KRb and mN is its projection onto the quantization axis (which can be defined by either an electric
or a magnetic field applied in the lab). These states can be labeled as |N, mN , mK , mRb i, wheremK and mRb are the nuclear spin projection quantum number for K and Rb, respectively. Note

33
that since the ground electronic potential of KRb has both the orbital and spin angular momentum
equal to zero, the rotations of the molecular nuclei, N, are decoupled from the nuclear spin, I.
Therefore, |N, mN , mK , mRb i is an appropriate set of quantum numbers for such molecules [30]. In
the ultracold regime, the initial Feshbach association creates weakly-bound molecules in a single
quantum state since K and Rb atoms approach each other in the lowest partial wave, s (L=0),
where ~L is the quantized relative angular momentum. Thus, through a combination of angular momentum selection rules and the spectral resolution provided by STIRAP, the ground-state
molecules also occupy a single hyperfine quantum state, which for KRb is |0, 0, −4, 1/2i. The lowest
energy hyperfine state is |0, 0, −4, 3/2i [197]. To reach this absolute ground state and to also provide a general spectroscopic tool to coherently and efficiently transfer populations between different
hyperfine states, we have developed another coherent two-photon transfer technique, which is basedon the use of two microwave photons to couple from two different hyperfine states in |N = 0i to acommon hyperfine state in the |N = 1i manifold via electric dipole transitions. This two-photon
coupling is enabled by the fact that the |N = 1i state has non-negligible (a few percent) spin
mixing due to coupling between rotation and the nuclear electric quadrupole moment [6, 30, 197].
This microwave-based Raman transition allows us to transfer the molecular population to any hyperfine state in |N = 0i, including the absolute ground state |0, 0, −4, 3/2i. This technique is also
important for the creation of a 50:50 mixture of two hyperfine states in the ground state, or a 50:50
mixture of two rotational states for KRb molecules, which we use to study the role of quantum
statistics of identical molecules in the reaction process.

3.2

Inducing the dipole moment in a lab frame with DC and AC electricfieldsSince an electric dipole moment exists only between quantum states that have opposite

parity, individual rotational states |N, mN i have no dipole moment in the laboratory frame. In
other words, molecules are polarized along their internuclear axis, but since this axis changes rapidly
and randomly in the absence of an applied electric field, the dipole moment averages to zero in the34laboratory frame. When a DC electric field is applied, opposite-parity rotational states become
mixed, giving rise to a nonzero dipole moment in the lab frame. In other words, molecules becomealigned with the applied field. The Hamiltonian for the rotational states |N, mN i under an electricfield () has two contributions arising from the diagonal rotational energy and the state-mixing(off-diagonal) Stark effect:hN, mN |Ĥ|N 0 , m0N i = BN (N + 1)δN,N 0 δmN ,mN 0 − hN, mN |d̂ · |N 0 , m0N i.

(3.1)

Here B = ~2 /2I is the rotational constant for the state under consideration (I is the moment ofinertia, which is the reduced mass φ times the square of the internuclear separation), and the term−d̂ ·  describes the dipole-field coupling operator. If the electric field, of magnitude , is definedalong the z axis (indeed the electric field defines the quantization axis), the matrix elements of
Eq. 3.1 simplify to
hN, mN |Ĥ|N 0 , m0N i = BN (N + 1)δN,N 0 δmN ,mN 0 −pD (2N + 1)(2N 0 + 1)(−1)mN 

N
−mN

1

N0



 N

0 m0N
0

1
0

N0
0



 . (3.2)

Here D is the permanent, molecule frame dipole matrix element (0.574 Debye for KRb [190]) andthe terms in parentheses are Wigner 3-j symbols [75]. The first 3-j symbol is zero unless mN = mN 0 .
So the eigenstates in an electric field are|Ñ , mN i =X

cN 0 |N 0 , mN i,

(3.3)

N0

where |Ñ , mN i is the state that adiabatically connects to |N, mN i when the field is turned down to
zero. The dressed state energies EÑ ,mN can be determined by diagonalizing Eq. 3.2, and then the
induced dipole moment of state |Ñ , mN i is known through the quantity, −∂EÑ ,mN∂. The induced

dipole moments for the N = 0 and N = 1 states of KRb are displayed in Fig. 3.1. A field strengthof more than 40 kV/cm in the lab is required to reach 80% of the intrinsic dipole moment of themolecules. In general, the value of this critical field scales as B/D, and it can be much smaller forsome bialkali molecular species.

35

Figure 3.1: Induced dipole moments of KRb for the |0̃, 0i (blue), |1̃, 0i (orange), and |1̃, ±1i states(green), calculated using the lowest 20 rotational levels in Eq. 3.2.

36
Since successive rotational states have opposite parity, a more efficient option for realizing alab-frame dipole moment is to use a resonant rf or microwave field to directly couple two neighboringrotational states. When the N = 0 and N = 1 states are coherently mixed by applying a microwave√
pulse that creates a superposition (e.g. a π/2-pulse), we realize a dipole moment of magnitude D/ 3
in the rotating frame of the molecule. This is the technique used in References [110, 273] to realize
dipolar-interaction-mediated spin exchanges in a three-dimensional optical lattice, to be discussed
in Chapter 5.

3.3

Role of quantum statistics on collisions
Owing to the antisymmetrization requirement of the overall wavefunction for two identicalfermions, spin-polarized (i.e., the same internal state) 40 K87 Rb molecules collide via odd partial
waves. At ultralow temperatures, the lowest odd partial wave is the p-wave (L = 1). In contrast, distinguishable molecules (such as a hyperfine or a rotational mixture) can collide via s-wave(L = 0). Thus, when molecules are brought to the quantum regime, their interactions and reactions will be governed by the molecular quantum statistics, single partial-wave scattering, andquantum threshold laws at a vanishing collision energy. The first set of experiments that exploredmolecular reactions is reported in Reference [198], which studied the dependence of reaction rate
on temperature and quantum statistics without applying a laboratory electric field.
As a result of the exothermic exchange chemical reaction (2KRb → K2 + Rb2 ), the number
density n of trapped molecules decreases according todn= −βn2 − αn,
dt

(3.4)

where β is the two-body loss coefficient and α is the single-particle loss coefficient, which is set by
a combination of effects including heating of the gas in the trap, light scattering from the trapping
beams, and collisions with background particles. The origin of this heating is associated with theenhanced loss at the trap center where the density is the highest [198]. The remaining one-body loss
mechanism has a much longer time constant. The two-body loss rate β is found to be proportional

37
to T for spin-polarized samples, which is the expected dependence from the Bethe-Wigner thresholdlaws for p-wave collisions (Fig. 3.2). The lack of any non-monotonic behavior in the rate coefficientdependence on temperature is consistent with a simple model that assumes that after the molecules
tunnel through the p-wave barrier they chemically react with a near unity probability [127].
For an incoherent 50:50 mixture of two hyperfine states in the ground state (|0, 0, −4, 1/2i
and |0, 0, −4, 3/2i), s-wave collisions start to play a dominant role since the molecules from thetwo different states are distinguishable. As a result, the loss rate shows no measurable temperaturedependence and its magnitude is increased by ten- to hundredfold in comparison to the p-wave rate.
The measured rate agrees well with the predicted universal loss rate related to the two-body van der
Waals length. We note that the absolute ground state (|0, 0, −4, 3/2i) demonstrates the same loss
rate as other higher lying hyperfine states, which implies that the chemical reactions dominate overhyperfine state-changing collisions. Molecules prepared in an incoherent 50:50 mixture of rotational
states (half in |0, 0, −4, 1/2i and half in |1, −1, −4, 1/2i) have the largest loss rate observed at zero
electric field (see Fig. 3.3). Finally, when both atoms and molecules are prepared in their lowestenergy states, we have also observed a universal s-wave loss rate from the exothermic atom-exchange
reaction for K + KRb → K2 + Rb. For the endothermic process Rb + KRb → Rb2 + K, theobserved loss rate approaches zero within the measurement uncertainty [198]. However, if the Rb
density becomes too high, a three-body loss process involving two Rb atoms and a KRb will limit
the molecular lifetime, as discussed in Chapter 6.
When an electric field is applied, the molecules develop an electric dipole moment in thelaboratory frame and this modifies how they interact with each other. First, the isotropic centrifugal barrier arising from the p-wave becomes anisotropic. For L = 1, the projection onto thequantization axis (defined by the applied electric field), mL , dictates how the molecules collide:mL = 0 corresponds to a “head-to-tail” collision, which is attractive under the dipole-dipole interaction; while mL = ±1 corresponds to a “side-by-side” collision, which is repulsive under the
dipole-dipole interaction. This is shown schematically in Fig. 3.4. Furthermore, the dipole-dipole
interaction mixes states with different partial waves, such that the L = 1 partial wave is replaced

38

Figure 3.2: Loss rate coefficient as a function of temperature in a bulk gas for spin-polarized
40 K87 Rb molecules, showing a linear increase with temperature characteristic of the p-wave threshold behavior. For comparison, loss rate for a 50:50 mixture of hyperfine spin states is shown, where
the nature of s-wave collisions removes the temperature dependence for the rate constant while theloss magnitude is increased by a factor of 10-100. Reproduced from Reference [198].

Figure 3.3: Decay of an incoherent mixture of rotational states |0, 0, −4, 1/2i and |1, −1, −4, 1/2iof 40 K87 Rb molecules in a harmonic trap. The extracted loss rate is 9.0(4) × 10−10 cm3 s−1 ,which is almost 5 times higher than for the ground-state hyperfine mixture. Reproduced from
Reference [273].

39
by the lowest-energy channel with odd parity. Similarly, the L = 0 partial wave becomes thelowest-energy channel with even parity, which does not have a centrifugal barrier.
In a 3D optical dipole trap, we do not have control over the projection of the relative angularmomentum of the colliding particles on the quantization axis. Therefore, while for mL = ±1 thecentrifugal barrier is significantly increased with a reasonable electric field, the observed loss rateis dominated by the mL = 0 collision channel, which has a much lower barrier height for reactions(Fig. 3.4). Consequently, the measured loss rate increased sharply as a function of the electric field,
as shown in Fig. 3.5. In fact, β ∝ d6 (d is the induced dipole moment in the laboratory frame),
which agrees well with a quantum threshold model that takes into account the long-range dipolarinteraction [191].
Chemical reactions for ultralow temperature polar molecules confined in a 3D harmonic trap
thus depend strongly on the quantum statistics of the molecular gas, individual collision partial
waves, temperature, and the induced dipole moment in the lab frame. In the remainder of thischapter I present results on the control of the reaction rate when we modify the dimensionality ofthe optical traps in which the molecules are confined.

3.4

Reduced dimensions: quantum stereodynamics of chemical reactions in
a 2D gas
To suppress the large reaction rate of KRb molecules in an electric field, we must suppress thehead-to-tail collisions that lead to inelastic loss. Hence, we need to gain control over the projectionof the relative angular momentum of the colliding molecules on the quantization axis, mL . Thiscan be achieved by confining molecules in a two-dimensional optical trap, which can be realized byinterfering two counterpropagating optical beams to create a 1D lattice that consists of an array of
2D disk-shaped traps. To control the stereodynamics of the reaction (the orientation and relative
motion of the reactants), a DC electric field is applied along the z axis and perpendicular to the 2Dtraps. In such a configuration, the lossy head-to-tail collisions with mL = 0 are greatly suppressed,
and the repulsive side-by-side collisions with mL = ±1 stabilize the molecular gas [32, 63, 176].

40

Figure 3.4: Collision of two polar molecules in an electric field. (a) The p-wave barrier at zero
electric field is isotropic and does not depend on the orientation in which the molecules collide. (b)In an applied DC electric field, dipolar interactions reduce the centrifugal barrier for “head-to-tail”
collisions (blue arrow) and increase the centrifugal barrier for “side-by-side” collisions. Reproduced
from Reference [191].

Figure 3.5: Decay coefficient of ultracold 40 K87 Rb molecules colliding in an electric field as afunction of induced dipole moment in a bulk 3D gas. A clear onset of increased loss at 0.1 D isapparent. The inset shows the induced dipole moment as a function of electric field. Reproduced
from Reference [191].

41
For sufficiently strong optical confinement along z, the motion along z is fully quantized andcan be labeled with a quantum number νz (which is the band index, to be discussed in Chapter
4). At ultralow temperatures, the majority of molecules occupy the motional ground state νz = 0,
with only a small fraction of them in νz = 1. The angular momentum L in the 3D case is nolonger a good quantum number in 2D; instead, νz and M (the angular momentum projection
along z) become relevant in the antisymmetrization of the two-molecule wavefunction for fermionicKRb. For example, if two KRb molecules are prepared in the same internal state and have thesame νz , they must collide with odd M , which gives rise to a repulsive dipole-dipole interactionthat suppresses the chemical reaction. On the other hand, if the two molecules have different νz ,then the even M channel is accessible, and the reaction proceeds strongly. In the limit of a verylarge applied electric field and a very strong optical confinement along z, repulsive dipole-dipole
interactions dominate for both Bose and Fermi quantum statistics [32, 63, 176]. In this regime, thecharacteristic length scale associated with the dipole-dipole interaction is larger than the harmonic
oscillator length in z.
Stereo-chemical reactions in the quantum regime were explored in Ref. [63], where the reaction
rate as a function of the molecular temperature, the harmonic confinement along z, and the induced
dipole moment were systematically measured. Specifically, Fig. 3.6a depicts the three different
configurations that were studied in the experiment [63]: (1) molecules in different internal statesand same νz , (2) spin-polarized molecules with different νz , and (3) spin-polarized molecules withthe same νz . Collisions via channel 1 are s-wave and result in very fast loss. Channel 2 is also
undesirable, since it corresponds to unsuppressed head-to-tail collisions with even M . Channel 3
is the most ideal case, as it corresponds to side-by-side collisions with odd M . Modulation of thelattice depth (parametric heating to be discussed in Chapter 4) was used to promote molecules tohigher lattice bands, which sets the relative importance of channels 2 and 3. Figure 3.6b displays
measured loss curves for two different initial conditions. The black curve shows the loss for a casewhere about 75% of the molecules were initially in the lowest band. The red curve shows the loss
for a distribution where parametric heating promoted a large fraction of molecules to the second

42

(a)

(b)

(c)

Figure 3.6: 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 to stabilize themolecular gas in a 2D geometry since it corresponds to repulsive side-by-side collisions. The arrows next to the curves indicate whether the barrier increases or decreases when the electric field
increases. (b) Loss curves for two different initial band populations. The initial loss rate is faster
when a larger fraction of molecules occupy higher bands initially. (c) The measured 2D loss coefficients β2D for the three cases depicted in (a), with the same color coding. Reproduced from
Reference [63].

43
excited band. The decay curves were used to extract the loss rate coefficient β2D for the 3 channels(Fig. 3.6c). As expected, β2D is the smallest for channel 3. The suppression of the reaction rate via
odd M , in comparison to the head-to-tail collisions, was about a factor of 60 at an induced dipole
moment of 0.174 D (Fig. 3.6c).
This two-dimensional trap configuration can be ubiquitously applied to dipolar systems tocontrol collision processes by tuning the dipole orientation relative to the collision plane. Similar
experiments have recently been performed with Feshbach molecules of highly magnetic erbium
atoms confined in a 1D lattice with similar goals of removing the attractive part of the dipole-dipole
interaction [83]. New experiments at MIT with NaK molecules are also moving in the direction of
a 1D lattice to use strong repulsion to stabilize the system against inelastic collisions [290].

3.5

Sticky collisions: 3-molecule collision channels
Given that the KRb + KRb reaction is exothermic, there is a strong interest in the scientificcommunity in working with bialkali species that are predicted to have endothermic reactions [34,214, 128]. However, it was predicted that there could be substantial loss in three-body collision
channels [54, 84, 171, 172]. The simple intuitive picture invoked for these processes is the following:
two molecules can collide and temporarily form a reaction complex prior to separating. Whilethey are very near each other, a third molecule can come in and force the complex to a deeper
bound molecular state that results in inelastic loss of all three molecules. These processes could
be responsible for observed losses that are difficult to attribute to two-body loss mechanisms, but
rigorous experimental confirmation of these lossy three-body processes is work in progress.
The two-body nature of the collisions discussed in the previous section is strongly supportedby the observation that inelastic loss rates are ∼100 times larger with non-identical fermionsthan with spin-polarized fermions. Let us consider the case of three-body collisions of fermionicmolecules. If the fermions are spin polarized, three-body collisions will require the p-wave channel in the same way as two-body collisions. However, if the molecules are not spin polarized, thedifference between two-body and three-body could become experimentally distinguishable. For

44
example, suppose we restrict ourselves to only two unique internal states for the fermions. Then,collisions between two non-identical fermions can take place in the s-wave channel, and would not
require tunneling through a barrier. However, three-body collisions will necessarily still require ap-wave channel since two of the three fermions must be identical. In this situation, therefore, wewould not expect the three-body loss rate to be very different between identical and non-identicalfermions. This is in stark contrast to the scenario we have presented on the KRb loss, which
again points to two-body loss as the dominant loss process. The work at MIT with fermionic NaK
molecules [269, 290], which are ostensibly not reactive in the two-body channel, could shed lighton three-body loss.
With bosonic molecules the situation becomes significantly more complex. Since bosons cancollide in the s-wave channel, the loss rates can be extremely fast for both two-body and threebody collisions. Even bosonic molecules that are ostensibly chemically stable have been observed
to decay very quickly [180, 252], and experimental signal-to-noise ratios make it very difficult todifferentiate between two-body and three-body loss mechanisms. Just as for our reactive KRb
molecules, these experiments can benefit from confining the molecules in zero spatial dimension
achieved with a three-dimensional optical lattice to completely remove collisional losses. We thus
focus on the production, stabilization, and applications of ultracold polar molecules in 3D opticallattices in the next chapter.

Chapter 4

Suppression of chemical reactions in a 3D lattice

For many experiments based on the long-range interactions of polar molecules, contact between the molecules is unnecessary or even unwanted. For such experiments, a full 3D opticallattice can be used to pin molecules to individual sites where they will remain without colliding forthe duration of the experiment. While research on ultracold polar molecules in optical lattices has
started only recently [47, 58], promising initial experimental results have already been achieved, as
will be summarized in this Chapter. I will begin with a review of optical trapping, optical lattices,and lattice techniques.

4.1

Optical lattice techniques
Before describing how molecules can be stabilized in a deep optical lattice, I will describe thephysics of optical lattices and how they are created from optical dipole forces of off-resonant laser
beams. This discussion also provides a background on optical dipole traps.

4.1.1

Optical dipole potentials

The oscillator model for an optical dipole potential is given by [102]Vdip = −1
Re(α)I,
20 c

(4.1)

where 0 is the electric permittivity of vacuum, c is the speed of light, α is the polarizability of theatom at the wavelength of the laser, and I is the intensity of the laser beam. At the waist of a

46
Gaussian beam, this intensity is given by I0 = 2P0 /πωx,0 ωy,0 . The polarizability α as a function of
the laser frequency ω is given byα(ω) = 6π0 c3

Γ/ω02,
ω02 − ω 2 − i(ω 3 /ω02 )Γ

(4.2)

where ω0 is the resonance frequency of the atomic transition and Γ is the corresponding linewidth.
Atoms have many atomic transitions, so the full polarizability is the sum over all transitions withresonance frequencies ω0,i . While the real part of the polarizability describes the dipole force thatthe light exerts on the atoms, the imaginary part describes the scattering rate of the light by theatoms, which is given byΓsc = −

1
Im(α)I.
~0 c

(4.3)

When the laser is sufficiently detuned from any atomic transitions as is typically the case for
optical dipole traps, the rotating wave approximation can be made and the dipole potential andscattering rate can be approximated byVdip (r) ≈3πc2 Γ
I(r)2ω03 ∆(4.4)

andΓsc (r) ≈

3πc2
2ω03



Γ
∆

2
I(r),

(4.5)

respectively, where ∆ = ω − ω0 is the detuning from an atomic resonance. Again, this will be asum over all atomic transitions. For alkali atoms in red-detuned traps (i.e. when ∆ < 0, such aswith λ = 1064 nm) there are only two atomic transitions which need to be considered, which arenS to nP1/2 and nP3/2 , also known as D1 and D2.
For a Gaussian trapping beam, the intensity profile I(r) is given by
√

2
2
2P
2 x2 +yw (z) ,
I(r) =eπw2 (z)where P is the total power, w(z) = w0

(4.6)

p
1 + (z/zR )2 is the 1/e2 radius of the beam, and zR = πw2 /λis the Rayleigh length for a beam propagating in the z-direction. At the center of the beam (i.e.

47

Figure 4.1: An illustration of a cloud of atoms (blue) attracted to the waist of a red-detuned optical
dipole trap (grey).

the waist), the trapping potential as illustrated in Fig. 4.1 can be approximated as [102] 2  2 rz
V (r, z) ' −V0 1 − 2
−
,
(4.7)
w0
zR
q
√
where w0 = ωx,0 ωy,0 . The trap frequencies are given by ωxi = 4V0 /mwx2i ,0 where xi = x or yq2.
and ωz = 2V0 /mzR
4.1.2

Band structure and eigenfunctions

An optical lattice can be generated by retroreflecting a dipole trap beam upon itself to form
a interference pattern, or a standing wave of nodes and antinodes. Such an interference pattern
results in a periodic trapping potential, which is given by2

V (r, z) = Vlatt e−2r /wx,0 wy,0 sin(kz),(4.8)

where Vlatt is the depth of the lattice, which is maximal at an antinode for red-detuning, and k =2π/λ is the wavevector of the laser light. Near the center where r  w0 , this can be approximatedby
V (r, z) = −Vlattr2
1 − 2 2 sin2 (kz).
w0

(4.9)

It is convenient to describe the lattice depth in units of recoil energies Er = ~2 k 2 /2m. In asufficiently deep optical lattice, the confinement on an individual lattice site can be approximatedby a harmonic potential with trapping frequency
2
ωlatt
=

Vlatt ~2 k 4.
Er m2(4.10)

48
A particle in such a periodic potential V (r) has eigenstates which can be written as Blochwave functions (see e.g. Refs. [9, 145])
φnq (x) = eiqx/~ · unq (x),(4.11)

where
unq =

X n,q
cl ei2lkx ,(4.12)

lwith l an integer. The spectrum consists of energy bands separated by fobidden energy regions(gaps). Each energy eigenstate is described by two quantum numbers: n and q. q is the quasimomentum wich takes values between −k and k, and n is a discrete number that labels the bands.
The energy associated with a band n can be computed using the Schrödinger equation, where
the eigenstates are the Bloch wave functions unq :HB unq (x) = Eqn unq (x),(4.13)

where
HB =

1
(p̂ + q)2 + V (x).
2m

(4.14)

While in general the lattice potential could be written as a discrete Fourier series, in the case of an
optical lattice the actual potential is very well approximated by a sinusoid, and thus we can write1
V (x) = − Vlatt 2Cos(kx) + 2 .
4(4.15)For such a cosine potential the eigenvalues and eigenstates can be exactly written analytically interms of Mathieu functions [217]. Numerically, the Schrödinger equation can be written in termsof the coefficients cn,qlXHl,l0 cn,q
= Eqn cn,ql
l0 ,(4.16)lwhere Hl,l0 = (2l + q/~k)2 Er when l = l0 , −1/4 · Vlatt when |l − l0 | = 1, and zero otherwise [98].
The eigenstates and eigenenergies can be computed if this Hamiltonian is truncated formoderate positive and negative l. The corresponding cl ’s become sufficiently small after |l| ≤5 in order to accurately compute the energies of the lowest bands [98]. The eigenenergies Eqn

49

Figure 4.2: Band structure calculations for various lattice depths. The energies of the first several
bands (in units of Er ) are shown for several lattice depths as a function of the quasimomentum inthe first Brillouin zone. The dashed red line shows the lattice depth at which each is calculated.

50(in units of Er ) are shown for the first several bands in the first Brillouin zone in Fig. 4.2 forseveral lattice depths Vlatt (denoted in units of Er ). The ground-band tunneling matrix element
J 0 can be calculated using the eigenenergy of the ground band Eq0 , and is represented in thetight-binding approximation (nearest-neighbor tunneling only) where the energies are of the form
−2JCos(qa) [217] by






1
1
0
0
0
0
J =
=
max Eq − min Eq
E~k − E0 .
44
0(4.17)The tunneling matrix elements for excited bands J n can be calculated accordingly from Eqn in thelimit of a deep lattice (a more general definition will be given below). Moreover, extending theanalysis presented here to 3D is trivial in separable potentials, where the tunneling matrix elements
simply become Jin where i denotes x, y, z.
4.1.3

Wannier functions for localized atoms

With an understanding of how Bloch functions describe the motion of particles in periodic
potentials of arbitrary depth Vlatt , let us now consider specifically the case of deep lattices whereparticles are localized to individual lattice sites. In this limit an appropriate basis of eigenstates isgiven by Wannier functions, which are given by [98, 145]
1 X −iqxi /~ nwn (x − xi ) = √eφq (x),
L q(4.18)

where q is summed over the first Brillouin zone and N is the total number of lattice sites andxi is the position of the ith lattice site in units of the lattice spacing a. The extent to which aparticle is localizated on a lattice site is determined by how much overlap this Wannier function has
with a neighboring site. To quantify this, the rigorous way (valid even beyond the tight-binding
approximation) to calculate the tunneling matrix element J n is given by [98]Z
~2 ∂ 2n
J = wn (x − xi ) −+ V (x) wn (x − xj )dx
2m ∂x2for neighboring sites i and j. Moreover, the on-site interaction energy U is given byZ
4π~2 anU =|wn (r)|4 d3 r,m(4.19)(4.20)

51
where a is the scattering length between the two particles.
When the lattice is sufficiently deep a few very useful approximations can be made. The
Wannier functions can be approximated as Gaussians of RMS width a/(πs1/4 ), where a is thelattice period, and s = Vlatt /Er [209]. Moreover, the tunneling matrix element in the ground band
can be approximated by [90]√J 0 /Er = 1.39666s1.051 e−2.12104 s(4.21)and the on-site interaction energy in the ground band can be approximated by [90]
U 0 /Er = 5.97 a/λ s0.88 ,(4.22)

where λ is the wavelength of the light generating the lattice.

4.1.4

Calibrating the lattice depths

We use two main approaches to calibrate the depth of the lattice. The first is a parametricdrive of the lattice intensity (called “parametric heating”) which excites atoms from the n = 0 to
the n = 2 band of the lattice. In time of flight this results in a much larger cloud. Therefore, we canstudy the size of the could as a function of the modulation frequency to learn the resonant frequencyof this transition. The above discussion allows us to infer s = Vlatt /Er . This approach is very simple,
but inherently less accurate because the width of the modulation resonance is broadened by thewidth of the two bands, and is potentially shifted to lower frequencies by modulating the lattice
too hard and introducing more anharmonicity. Therefore, we typically perform the modulation at
a depth of around ≈ 20 Er for ≈ 2 ms with ≈ 5 − 10% modulation depth. This is the method used
to determine the anisotropic polarizability of molecules in a 1D lattice, which will be described ina following section.
A more precise method is to perform Kapitza-Dirac scattering of a BEC off of the latticepotential. By flashing on the lattice potential for τ < 10 µs (much shorter than any motionaltime in the lattice), the BEC is diabatically projected onto plane-wave components (components

52
in many bands), whose relative populations can be used to calculate the lattice depth. Specifically,
the population fraction in the ±2n~k component is approximately equal to Jn2 (sEr τ /2~), where
Jn is the ordinary Bessel function of order n [85, 181]. This approximation is only valid at shorttimes, and so we try to operate at a lattice depth sufficiently high to minimize the n = 0 population
within a few µs. This approach gives results consistent with parametric heating, and allows us tocalibrate the lattice depth to an uncertainty of ≈ 5%.

4.1.5

Superfluid to Mott insulator transition

The dynamics of ultracold bosons (such as our Rb atoms) in an external potential V(x) is
described by
Z
Ĥ =



Z
Z
~2 2
1 4πa~23††d xψ̂ (x) −∇ +V (x) ψ̂(x)+d xψ̂ (x)ψ̂ (x)ψ̂(x)ψ̂(x)−µ d3 xψ̂ † (x)ψ̂(x),
2m
2 m
3

†

(4.23)
where ψ̂ and ψ̂ † are bosonic field operators. In the case of a lattice potential they can be expanded
into the Wannier functions discussed above, asψ̂(x) =

Xâi w(x − xi ),(4.24)iwhere âi (â†i ) are bosonic annihilation (creation) operators on the site i. These operators obey the
canonical commutation relation [âi , â†j ] = δij .
This expansion allows us to rewrite the Hamiltonian in terms of the parameters discussed
above, J and U , and in the tight-binding limit the Hamiltonian simplifies to the so-called BoseHubbard HamiltonianĤ = −JX<i,j>â†i âj +Xi(i − µ)n̂i +

X1
i

2

U n̂i (n̂i − 1),(4.25)

where n̂i = â†i âi is the number operator that counts the number of bosons on the lattice sitei, i = Vext (xi ) is a site-by-site energy shift for example coming from an additional harmonic
confinement generated by the lattice beams and dipole trap beams Vext , and µ is the chemical
potential of the atoms in the trap. If Vext is too large compared to U , the system will prefer to put

53
more atoms on the sites in the middle rather than putting them further out where i is larger. The
third term describes this energy cost for multiple occupancy, as indicated by the presence of U .
This Hamiltonian exhibits a quantum phase transition from a superfluid to a Mott insulator
which was first observed in cold atoms in in 2002 [100]. The evidence of a superfluid is phasecoherence between all the sites, which gives rise to an interference pattern upon a sudden releaseof the atoms from the trap (see Fig. 4.3a). Once the Mott insulating phase is reached, the phasecoherence is lost, and there is no longer interference upon expansion. In fact, a superfluid canbe ramped into the Mott insulator phase and then back into the superfluid phase, and the phasecoherence returns. These tools will be discussed in Chapter 6 where we describe how we used themto determine if the the Rb Mott insulator is behaving properly. The phase transition should occur
for Rb in a lattice of wavelength λ = 1064 nm at ∼ 16 Er , which provides another way to diagnose
the system, particularly in the presence of K even for zero K-Rb intereaction [19]. This is shownin Fig. 4.3b-d by plotting the visibility (i.e. the number of atoms at the position of a diffraction
peak compared with the number of atoms half way between two adjacent diffraction peaks) of theinterference peaks versus the lattice depth.

4.1.6

Band mapping

For the experiments described in this thesis, the goal was to load the atoms and molecules
into the ground band of the optical lattice. However, due to the finite temperature of the atoms
and nonadiabaticity during the loading, we often have atoms or molecules in the first excited band.
Identifying the higher band fraction is particular important for fermions, such as 40 K and KRb,
where double occupancy in a spin-polarized sample can only occur through higher band excitations.
The standard way to detect the fraction of atoms in higher bands is called band mapping, by which
the different bands are mapped onto different positions upon expansion. Band mapping works bymapping each band into the quasimomentum corresponding to a Brillouin zone. To do this, it
is important that the lattice intensity is ramped off quickly compared to the motion of a latticesite yet still slowly enough to be adiabatic with respect to the band gap ∆. Band mapping is an

54

Figure 4.3: The superfluid to Mott insulator transition of Rb with K. (a) shows the disapearance
of interference beams in crossing the superfluid to Mott insulator phase transition for Rb alone.
The three images are 12, 17, and 22 Er , from left to right. Reproduced from Ref. [183]. (b) and(c) show the visibility of the intereference peaks as a function of lattice depth in the presence of K
for aK-Rb = 0, and −137 a0 , respectively. (d) shows the visibility in the case of Rb alone, whichcorresponds to images as in (a).

55

Figure 4.4: Band mapping of K in the lattice. The two cases show a lower excited band fraction in
i and a higher excited band fraction in ii. Reproduced from Reference [52].

incredibly useful tool, and has been described in a number places, such Refs. [62, 98, 181, 187].
We found the refocusing technique described in Ref. [184] to be particularly useful and provided the sharpest images which were easy to analyze. In this approach, the lattice intensity isramped to zero as discussed, but the underlying harmonic confinement (optical dipole traps) is left
on. After waiting for a quarter period in this harmonic potential, the initial momentum distribution
is perfectly mapped onto the position distribution. The resulting size of the first Brillouin zone
in position space is given by 2~k/(mωhc ), where ωhc is the oscillation frequency in the harmonic
confinement potential. The ground band is from x = −1~k/(mωhc ) to 1~k/(mωhc ), and the firstexcited band is from x = −2~k/(mωhc ) to −1~k/(mωhc ) and 1~k/(mωhc ) to 2~k/(mωhc ). This
analysis can be applied to two dimensions, as in Fig. 4.4, and can thus be generalized to estimate
the full 3D ground band fraction. I will come back to this topic in Chapter 6.

4.2

Long-lived molecules in a 3D optical lattice
I will now discuss molecules in an optical lattice, which can be obtained by either loading abulk gas of molecules into the lattice, or by loading K and Rb into the lattice and then making
molecules on sites that happen to have one (and only one) of each. Both approaches tend to yield
3−4×104 molecules in the lattice with a filling fraction of ∼ 5%. This is because given our trapping
conditions the entropy per particle in a Fermi gas of molecules at T = TF roughly matches 5%filling in the lattice, and a Bose-Fermi mixture of atoms at T = Tc , TF loaded into the lattice tend

56

Figure 4.5: Long lifetimes are experienced, after a fast initial loss attributed to dark molecules andresidual unpaired atoms. The inset shows that this long lifetime is independent of the electric field.
Reproduced from Reference [47].

to have ∼ 5% of sites with one of each in our specific case (this is not a generic argument; depends
on number of atoms, loading conditions, and external trapping conditions, etc.). Therefore, both
approaches were used in the early years of our work with optical lattices. However, by the next
chapter we were exclusively using the latter approach. The quantum synthesis idea described in
Chapter 6 is in fact based entirely on this latter idea, and the ability to put one K and Rb in everysite using atomic insulators.
Two years before the first production of ground-state polar molecules, researchers had alreadyprepared homonuclear Feshbach molecules in an optical lattice by magneto-association of pairs of
atoms contained in single sites of a bosonic Mott insulator of Rb [254]. They were then able toremove unpaired atoms, leaving only sites that were either empty or contained a Feshbach molecule.
In such an experiment, no collisions between molecules were allowed in a deep lattice and therefore
molecules could not undergo inelastic scattering via vibrational quenching from the Feshbach state.
The lifetime was measured to be 700 ms, which is significantly longer than that in a bulk gas wheremolecules can collide with atoms and with each other [254]. However, there is no fundamental limit
to this lifetime, and it was ultimately limited by inelastic light scattering from the trapping light.
More recently, we have demonstrated the creation of KRb Feshbach and ground-state molecules
in 3D optical lattices [47], and have shown that their lifetimes can exceed 20 seconds (see Fig. 4.5).

57
This lifetime is also limited by off-resonant light scattering from the optical trap, which is fairly far
detuned from molecular resonances. The lifetime of these molecules in the optical lattice was shown
to be significantly longer than the timescales that are relevant for the dominant interaction energies
between the molecules [112, 156], and so this offers an excellent starting point for the investigationof dipole-dipole interactions in an optical lattice and for the engineering of a lattice model with aspin-1/2 Hamiltonian.
As discussed above, applying an electric field in the laboratory frame can potentially reduce
the lifetime of the molecular gas by allowing the attractive dipolar interaction to reduce the pwave barrier. In the case of a deep 3D lattice and therefore the absence of tunneling, the shortrange collisional physics is fully suppressed, and hence the mechanism for reducing the lifetimeby an electric field is no longer valid. Indeed, we observed experimentally that the lifetime of themolecules in a deep 3D lattice remains the same with an electric field of ∼ 4 kV/cm as with nofield (see Fig. 4.5). This was very encouraging; we can now turn on the full-strength dipole-dipole
interactions with electric fields without having to worry about any extra loss our molecular sample
may experience [47].

4.3

The continuous quantum Zeno mechanism: stability from strong dissipation
I now consider molecules in a shallow lattice in which there is significant tunneling. The

largest measured two-body loss rate for a bulk gas of KRb at zero electric field was for the rotational
mixture discussed in Chapter 3. In this case, s-wave collisions are allowed and so the on-site loss
rate in a 3D lattice would be very large, even larger than both the tunneling rate and the energy
gap between the two lowest lattice bands. In this regime, however, the process for a molecule
to tunnel onto a lattice site that is already occupied by another molecule is greatly suppressedby the continuous quantum Zeno effect [249], where the effective loss rate of the whole systemcounterintuitively decreases as the on-site loss rate increases. In other words, as the loss rate of
two distinguishable molecules on a single lattice site increases, tunneling onto an occupied site is

58

Figure 4.6: (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 . Reproduced from Reference [282].

suppressed, as if the molecules knew they would undergo inelastic loss on that site.
The experimental setup for the work reported in Refs. [273, 282] is shown schematicallyin Fig. 4.6. About 104 ground-state KRb molecules were produced in a deep 3D lattice (along
all three directions). An incoherent mixture of | ↓i = |0, 0, −4, 1/2i and | ↑i = |1, −1, −4, 1/2i(|N, mN , mK , mRb i) was created by first applying a π/2 pulse between these two rotational states
and then waiting 50 ms for the superposition to decohere due to stray fields and the trapping light.
The two rotational states constitute a pseudo-spin-1/2 system, where the Hilbert space consistsof two orthogonal states (i.e. described by Pauli matrices). Then, the lattice depth along y was
reduced within 1 ms to allow tunneling with a rate Jy /h given by the tunneling matrix element
between neighboring sites J. This created a system of decoupled 1D tubes with a weak latticealong the tube axis (y). After holding the molecules in the lattice for a variable amount of time,
the number of molecules in the N = 0 state (| ↓i) was measured to determine the loss of thismixed-spin system.
There is no Pauli suppression preventing a | ↓i and a | ↑i molecule from occupying the sameJlattice site, and their on-site loss rate Γ0 is very large. In fact, Γ0  ~y . In this limit, tunnelinginduced on-site loss becomes a second-order perturbative process and the effective loss rate for the

59

Figure 4.7: (a) Typical loss curves for two different lattice depths along the y-direction, where thedata is fit to the solution of the rate equation (Eq. 4.27). (b) and (c) The dependence of κ on Γ0(b) and Jy ~ (c). Reproduced from Reference [273].

overall system becomesΓeff =

2Jy2
.
~2 Γ0

(4.26)

Since particles are lost via a two-body process [282],Ṅi (t) = −κNi (t)2,
Ni (0)(4.27)

where i = | ↑i or | ↓i, κ = 4qΓeff n↓ (0) is the loss rate coefficient, and Ni (0) is the initial number ofmolecules in state i. n↓ (0) is the initial filling fraction of | ↓i molecules, so 2n↓ (0) is the total filling
fraction, and q is the number of nearest neighbors. Measuring the dependence of κ on Jy and Γ0 isthus directly achievable in the experiment. An extra benefit is the determination of the molecule
filling fraction in the 3D lattice.
The transverse lattice depth V⊥ was typically 40 ER , while the lattice along y direction wasvaried between 5 and 16 ER . Here, ER = ~2 k 2 /2m is the lattice photon recoil energy for KRb.
To vary Γ0 only, the transverse lattice depths were changed. To vary Jy , the lattice depth along
the y direction was changed; however, to keep Γ0 fixed, the transverse lattice depths were alsochanged accordingly. Figure 4.7a shows a typical loss curve for Vy = 5ER and V⊥ = 25ER . The
data was fit to the solution of Eq. 4.27. Naı̈vely calculating the filling using Eq. 4.27 provided
an overestimate of the filling fraction arising from the modification of the single particle Wannier

60

Figure 4.8: (a) The geometry of the experiment with a 1D lattice and the quantization field set
by the magnetic field. (b) The energy level diagram of the relevant rotational states. (c) The
effect of a differential polarizability on the rotational state transition energy. Reproduced from
Reference [188].

orbitals due to interactions which is not taken into account in this equation. Since Γ0 is larger thanor on the same order of magnitude as the band gap, higher band populations must be included
to reach the correct value of Γ0 . For the experimental parameters, properly accounting for higher
bands can decrease Γ0 by about a factor of 5, which then increases Γeff by the same factor. The
curve in Fig. 4.7b clearly demonstrates that the loss rate coefficient κ and also the effective loss
rate Γeff decrease as Γ0 increases, while the data in Figure 4.7c shows that κ and Γeff have roughlya quadratic dependence on Jy /~. These scalings are in accordance with Eq. 4.26. These results
provide strong evidence for the continuous quantum Zeno effect, and at the same time allowed us
to determine that the lattice filling fraction was around 6%.

4.4

Anisotropic polarizability
The long lifetime for molecules in a deep lattice allows for investigations of the real part of themolecular AC polarizability in addition to its lossy imaginary component. This is important for us
to understand the molecular energy level shifts inside the lattice for the implementation of the latticespin model (see Chapter 5). The wavefunctions for many molecular states are spatially anisotropic,

61
and hence the corresponding polarizability depends on the angle between the polarization of thelattice light and the quantization field defined by an applied electric or magnetic field (see Fig. 4.8).
By tuning this angle carefully, KRb molecules are found to exhibit a ‘magic’ trapping conditionin which the polarizability of two rotational states (as used to define a spin-1/2 system) can be
adjusted to match with each other [150, 188] (see figure 4.9).
Such ‘magic’ conditions are reached in optical lattice clocks by adjusting the lattice lightwavelength and are routinely used to minimize the effects of the trapping light on the clock transition [165]. For molecules in this ‘magic’ angle lattice, the coherence time for a coherent superposition of the two rotational states is found to be orders of magnitude longer than otherwise (seeFig. 4.10), which is an important prerequisite for studying spin physics with polar molecules inan optical lattice. Note that a coherence time of roughly one second has been observed recently
between two hyperfine states of NaK in a single rotational state in an optical dipole trap [204].

4.5

Hyperpolarizability
In addition to an anisotropic polarizability, the molecules suffer from significant hyperpolariz-ability, which is a nonlinear optical property originating from a second-order electric susceptibilityof polar molecules. This second-order polarizability is what gives rise to second-harmonic generation in molecular samples. Essentially, the polarizability itself is a function of the beam intensity
such that the trap depth Vdip is given by
Vdip (I) = −1
α(I)I,
20 c(4.28)

where α(I) = α0 + α1 I with α0 as described in the previous section.
This hyperpolarizability is significant because the trap has an intensity curvature across thecloud. Therefore, even if the differential polarizability is zero at a certain lattice depth, the variationof the intensity across the cloud induces a finite differential shift. As shown in Fig. 4.8c, a nonmagic trap leads to a differential trapping potential between the two states of the form ∆V = ∆αI.
Conversely, the hyperpolarizability gives rise to a term that looks like ∆V = ∆α(I)I even if the62

Figure 4.9: The ac polarizability of KRb at 1064 nm for the |0, 0i (black squares), |1, 0i (bluecircles), |1, 1i (red inverted triangles), and |1, −1i (green triangles) states (|N, mN i) as a functionof the angle between the quantization field (B-field) and the polarization of the light field in a 1D
optical lattice. Reproduced from Reference [188].

Figure 4.10: The Ramsey coherence time is sharply peaked at the magic angle, and the inset shows
the Ramsey fringe as a function of dark time. Reproduced from Reference [188].

63
trap is magic at a given intensity. Since the α(I) is different between the two states, there is a net
polarizability difference across the trap. In practice we find that the total ac Stark shift across theentire cloud can be limited to only ≈ 1 kHz, which is  10 Hz per site at the center.
Figure 4.11 shows the differential ac Stark shift as a function of the lattice depth in unitsof Er . Their nonlinearity is evidence of the hyperpolarizability of the states involved. The inset
shows the differential polarizability of the |0, 0i to |1, −1i transition between 20 to 60 Er , and iszero when the lattice depth is about 40 Er in each beam. However, the differential light shift has
zero slope at 20 Er , so the effect of the intensity curvature across the cloud can be mitigated at thislattice depth. Note that the differential light shift of the transition to |1, 0i never has a vanishing
derivative.

4.6

Polarizabilities at other wavelengths
We have long been interested in using other wavelengths for trapping the molecules. There

are two primary reasons for this: 1) since the dipolar interaction scales as 1/R3 , reducing the latticespacing by a factor of 2 increases the interaction energy by a factor of 8! 2) it may be easier tomake a colder, denser, lower-entropy gas of molecules when starting from atomic gases trapped bywavelengths other than λ = 1064 nm. I will now discuss our efforts to use other wavelengths, andour measurements of the atomic and molecular polarizabilities at these wavelengths.

4.6.1

λ2 = 532 nm

A factor of eight increase in the interaction energy of the molecules in the lattice would put
us at J⊥ /2 ∼ 1 kHz (see Chapter 5), where we could even start to enter the blockade regimeof excitations with Rabi frequencies ∼ 1 kHz or less. Such interaction energies could make spin
diffusion and impurity dynamics much easier to observe. Moreover, realizing itinerant models suchas the t-J model based on the stability afforded by the continuous quantum Zeno mechanism would
be much more straightforward. Further, high power lasers are readily available at this wavelength,
and we used a Verdi by Coherent.

64

Figure 4.11: The differential light shift (with respect to the |0, 0i state is shown as a function of
the lattice depth. Green, black, and blue lines show calculated light shifts for the |1, 0i, |1, 1i, and|1, −1i rotational states, respectively. Red points are experimental data for the |1, −1i state. Inset:
An expanded view for the |1, −1i state is displayed. Reproduced from Reference [273].

65

Figure 4.12: Reducing the harrmonic confinement using λ2 = 532 nm dipole traps. (a) Increasing
the power of a λ2 = 532 nm dipole aligned onto the λ1 = 1064 nm dipole trap reduces the combined
trap frequency. (b) The addition of the λ2 beam increases the anharmonicity, and the descriptionof the trapping potential becomes more complicated. (c) The lifetime of ground state moleculesin the λ1 lattice, with and without the λ2 dipole beam on, whose 1/e2 radius is ∼ 50 µm. The
lifetime with it is ∼ 100× shorter. (d) The loss rate of molecules in the lattice with the λ2 beam
as a function of its intensity.

66
Additionally, since λ2 = 532 nm is blue-detuned for both K and Rb, a λ2 = 532 nm could
supplement the λ1 = 1064 beams already in use, and serve to decrease the harmonic confinement
since λ2 is anti-confining. The significance of this will become more clear in Chapter 6, but a flatter
trap with less harmonic confinement would allow a larger region in the lattice where molecules
could be created efficiently. This effect is shown in Fig. 4.12a, where increasing the λ2 beam
power decreases the trap frequency of the combined trap. However, the combined trap is now
a more complicated potential which is very sensitive to alignment. Figure 4.12b highlights thatthe addition of the λ2 beam substantially increases the damping rate, and depends strongly on the
power. The combined trap is very anharmonic and its properties are prone to enormous fluctuation.
Therefore, while this could potentially be useful, we ultimatley found it to be too difficult to use
as a part of the lattice loading.
Perhaps even more important is the lifetime of the molecules in a λ2 trap. To measure this,
we loaded ground state molecules into the λ1 lattice and turned on a λ2 beam. The results areshown in Fig. 4.12c&d. The λ2 1/e2 beam radius is ≈ 50 µm, and it reduces the lifetime by afactor of almost 100. The loss rate is proportional with the λ2 beam intensity, suggesting thatis entirely from Im(αKRb (532)) being very large. These observations corroborate calculations by
Svetlana Kotochigova, and the loss is rate is prohibitively high for most experiments we would
like to do. Thus, we never pursued 532 nm wavelength dipole traps or lattices further for either
anticonfinement or an enhanced dipolar interaction strength.

4.6.2

λ3 = 755 nm

Another intriguing wavelength on the blue-detuned side is λ3 = 755 nm, where K and Rb
would feel the same lattice depth in units of their respective Er . Since this lattice has a smaller
lattice constant, each lattice site is smaller, which increases the on-site interaction energy U , discussed above. Therefore, such a lattice could accommodate more atoms in the n = 1 Mott shell
before the onset of double occupancy at µ/V > 1. Moreover, the anti-harmonic confinement of thisbeam could (at least partially) cancel the harmonic confinement of the λ1 = 1064 nm dipole traps,

67

Figure 4.13: The polarizabilities at λ3 = 755 nm. (a) The slosh of K in the λ1 = 1064 trap froma ∼ 1 ms pulse of the λ3 = 755 nm trap. (b) The slosh of KRb from the same pulse from the λ3
beam. Note that the slosh of KRb has the opposite sign as the slosh of K. (c) The slosh of K, Rb,
and KRb all on the same plot induced from the λ3 pulse.

thereby further increasing the number of atoms in the n = 1 shell. The significance of the large
single occupancy Mott shell will become more clear in Chapter 6.
We studied this wavelength using a tunable Eagleyard ECDL and TA system operating
between ∼ 750 − 770 nm. Figures 4.13a&b show the slosh of K and KRb clouds, respectively, ina λ1 dipole trap after a ≈ 1 ms pulse of the λ3 beam. Note that the amplitude is ≈ 20× smaller
for KRb than it is for K, but also the sign is opposite. Therefore, while λ3 is blue detuned for
K and Rb, it is red detuned for KRb. Figure 4.13c shows the same measurement for K, Rb, andKRb all on the same plot. The ratio of the K and Rb amplitudes matches the difference in theirpolarizabilities at λ3 and their mass ratio.
The scenerio at λ3 is quite the opposite as λ1 where the polarizability is much larger and thesame sign for KRb as K and Rb. At λ3 the polarizability is opposite but also much smaller for
KRb. Moreover, this situation is nearly the same at 759 and 763 nm. Further, the lifetime of KRb
in the λ1 and λ3 trap is similar to what it is with just λ1 , which is ≈ 1 second. These observations
suggest that the polarizability for KRb may be shallowly crossing zero, and thus be blue-detunedon the blue of λ3 . Unfortunately, we were unable to confirm this. However, it is unclear how useful
this trap would be since the lattice depth would be very low at realistic beam intensities.
A large window of low Re(α) and Im(α) is quite uncommon for molecules, which have dense
spectra at most wavelengths. We have compared our results to polarizability calculations by Svet-

68
lana Kotochigova, which demonstrate a similar behavior. Nevertheless, a qualitiative comparisonis lacking to date, and a thorough understanding of this issue has not been reached. Yet, enoughis known that we may decide to investigate this further later on to take advantage of the longmolecular lifetime in optical traps at this wavelength.

4.6.3

λ4 = 790 nm

Wavelengths near the atomic resonances of K and Rb allow species selective manipulation,and λ4 = 790 provides a red-detuned trap for just K where the Rb polarizability crosses zero.
This tool could be very useful for tailoring the atomic gases to efficiently make molecules, and I will
return to this topic at the beginning of Chapter 6. Because of the different mass, trapping frequency,and quantum statistics of K and Rb, they tend to have different positions, sizes, and momentum
distributions at cold temperatures in weak traps. Therefore a trap that only manipulates K could
be very useful for enhancing the number of molecules that could be produced in the quantum
degenerate regime.
To study the effects of this trap we used a tunable Eagleyard ECDL and TA system operating
between ∼ 785 − 795 nm. Figure 4.14 shows the position of the K and Rb clouds as the λ4 = 790
nm beam is scanned across the λ1 = 1064 nm beam. Since the trap is species-selective, Rb hasvery little change while K moves substantially. The dispersive effect on the position of K can beseen in Fig. 4.14b as the λ4 beam is scanned for two different intensities. This capability allows us
to cancel the differential sag, and by increasing the power we can squeeze the large K Fermi gas sothat is is better spatially matched with the Rb BEC.
Although the absolute laser stability is not very important, we locked the lasers to a HighFinesse wavemeter, and carefully ensured single-mode laser operation. Fgure 4.14c shows the lifetimeof Feshbach molecules in a λ4 dipole trap for ≈ 60 mW with a 1/e2 beam radius of ≈ 100 µm.
The lifetime is incredibly short, ≈ 1 ms, which is perhaps not surprising given the proximity of the
wavelength to the enormous number of rovibronic excited states. The same measurement is carriedout with ground state molecules in Fig. 4.14d, and the lifetime is ≈ 100× longer. As discussed

69

Figure 4.14: The polarizabilities and lifetimes at λ4 = 790 nm. (a) The position of the K and Rb
clouds as the λ4 = 790 nm beam is scanned across the λ1 = 1064 nm beam. Since the trap is
species-selective, Rb has very little change while K moves substantially. (b) The dispersive effecton the position of K as the λ4 beam is scanned for two different intensities. (c) The lifetime of
Feshbach molecules in the combined λ1 /λ4 trap. This lifetime is prohibitively short. (d) The samemeasurement, but for ground state molecules. This lifetime is ∼ 100× longer, but still too shortfor most experiments.

70
in Chapter 6, such short lifetimes for the Feshbach molecules render this wavelength effectively
unusable.

Chapter 5

Quantum magnetism with polar molecules in a 3D optical latticeWith chemical reactions under control and the molecular gas stabilized in the 3D optical lattice, we have recently begun to study a conservative manifestation of the dipole-dipole interaction,
where the molecules never come in contact with each other. This is a new experimental regime
where the internal degrees of freedom (spins) of molecules interact strongly, but the motional degrees of freedom are largely decoupled from the system dynamics. Certainly this is not the case forcold, ground-state alkali atoms in a lattice whose interactions are primarily short-range, although
recently experiments with highly magnetic atoms and Rydberg atoms have demonstrated similar
long-range interactions [10, 16].

5.1

The dipolar spin-1/2 XXZ Hamiltonian
References [14, 94] proposed that polar molecules can be used to study quantum magnetism,and dipolar interactions should be observable even at low lattice fillings and high entropies [112].
Specifically, by defining a spin-1/2 degree of freedom using the lowest two rotational states (| ↓i ≡ |0, 0i and | ↑i ≡ |1, −1i or | ↑i ≡ |1, 0i), the molecules can undergo energy-conserving spin
exchanges mediated by dipolar interactions between nearby lattice sites, as depicted in Fig. 5.1.
The full spin-1/2 Hamiltonian that describes this exchange process (J⊥ ) as well electric fielddependent processes (Jz and W ) is [94, 263, 273]:
1X
Ĥ =Vdd (ri − rj )
2
i6=j






J⊥  + −
− +
z
z z
z
,
Ŝi Ŝj + Ŝi Ŝj + Jz Ŝi Ŝj + W Ŝi nj + Ŝj ni
2

(5.1)

72

Figure 5.1: Spin exchange between molecules in N = 0 (↓) and N = 1 (↑). Because of the long-range
interactions, the exchange can occur over distances of many sites in the lattice.

Figure 5.2: (a) and (b) Spin exchange between molecules in N = 0 (↓) and N = 1 (↑). Because of
the long-range interactions, the exchange can occur in principle over distances of many sites in thelattice. (c) The geometrical factor associated with the distance and angle between each moleculeand the center molecule (green). Reproduced from Reference [273]73
where Vdd (ri −rj ) =1−3 cos2 θij, which is a geometrical factor pertaining to dipole-dipole interactions,3
rij

ni is the population on site i, and Ŝ + and Ŝ − are spin-1/2 raising and lowering operators. Notethat there is another term which depends only on the densities and not on the spins [263]. The
kd2

coupling constant J⊥ = 4π ↓↑3 , where d↓↑ is the transition dipole moment between the two spin
0 alat√states | ↓i and | ↑i (see Reference [263]) (at zero DC field, d↓↑ = D/ 3), 0 is the permittivity offree space, alat is the lattice spacing, and subscripts i and j index lattice sites.
It is constructive to understand the four prefactors and how they depend on the electric field.
As stated above, J⊥ ∼ d2↓↑ , where again d↓↑ = h↓ |dˆ0 | ↑i = h↑ |dˆ0 | ↓i. The second term has Jz ∼(d↑ − d↓ )2 , where d↓ = h↓ |dˆ0 | ↓i and d↑ = h↑ |dˆ0 | ↑i [263]. The third term has W ∼ d2↑ − d2↓ /2,
and the fourth term (∼ ni nj ) has V ∼ d2↑ + d2↓ /4 [263]. Note that k = 2 for | ↑i = |1, 0i andk = −1 for | ↑i = |1, −1i, which are set by the matrix elements coupling to | ↓i ≡ |0, 0i. With
molecules distributed in a 3D lattice, the dominant frequency (energy) component present in the
Hamiltonian is |J⊥ /(2h)|, where h is Planck’s constant. The experiments of References [110, 273]
were performed at zero DC electric field, so Jz = W = 0 [263], and thus the remainder of thischapter focuses only the the spin-exchange term.

5.2

Spin-echo Ramsey spectroscopy
The dynamics were probed with a spin-echo Ramsey spectroscopy sequence, which is de-scribed below and schematically depicted in Fig. 5.3a. One additional complication in the experiment is that there is a site-to-site differential energy shift due to a residual inhomogeneous lightshift arising from the anisotropic polarizability of the molecules, caused by a difference in how
the two rotational states couple to the electronic excited states. The use of a ‘magic’ angle (seeFig. 4.9) for the lattice polarization with respect to the quantization axis reduces the inhomogeneity by more than a factor of 10, but a residual effect remains [188]. A spin echo technique isan effective approach to mitigate the effects of single-particle dephasing from this energy spread
across the molecular cloud. Initially, all of the molecules are | ↓i. The first π/2 pulse excites every√
molecule to a coherent superposition of (| ↑i + | ↓i)/ 2. After a free evolution time T /2, an echo

74

Figure 5.3: (a) Spin echo setup and timing diagram. (b) Typical Ramsey fringes at short time
(green) and long time (orange). (c) Contrast vs. time showing the decay and oscillations forintermediate density. The inset shows that the contrast decay for short times is concave down. (d)Contrast decay for two different particle densities. The oscillation frequency is basically the samefor the two datasets but the coherence time is shorter for higher density. (e) The data is fit toC(T ) = Ae−T /τ + B cos2 (πf T ). Compiling all of the measured coherence times shows τ ∝ 1/N .
Reproduced with permission from Reference [273]75
pulse is applied to reverse the single-particle dephasing from the inhomogeneous light shift. After
another free evolution time of T /2, a final π/2 pulse is applied with a phase φ relative to the first
pulse. This rotates the Bloch vector about the axis n̂ = cos φ ŷ + sin φ x̂. We measure the number of molecules in | ↓i by using STIRAP to transfer the ground-state molecules back to weakly
bound Feshbach molecules, followed by absorption imaging of the constituent atoms. By varying
the angle φ we can obtain a Ramsey fringe contrast for a given free evolution time T , as shown inFig. 5.3b, which we fit to N2tot 1 + C cos(φ + φ0 ) , and describes oscillation with amplitude givenby the contrast C around 50% the total number. Here, Ntot is the total molecule number and C
is the fringe contrast (0 ≤ C ≤ 1). The contrast determines the amount of spin coherence left inthe system after a time T , and it usually decays as a function of T due to residual single-particledephasing and many-molecule interaction effects. Some typical contrast decay curves are shownin Figure 5.3c and d. The most striking feature of these curves is the oscillation superimposed onan overall decay. We attribute both the oscillations and the overall decay to dipolar interactions.
Dilute lattice fillings and long-range interactions lead to a spread of interaction energies, whichcause dephasing and loss of contrast, and the interaction energy spectrum has the strongest contribution from the nearest-neighbor interaction (of frequency f = J⊥ /2h), which we fit empiricallyto (1 − A)e−T /t + Acos2 (πf T ), where A ∼ 0.1 [110, 273].

5.3

Removing pair-wise entanglement - WAHUHA NMR sequence
Figure 5.4a shows how the spin echo sequence can be extended with a multipulse sequencewhich removes pair-wise entanglement. Such multipulse sequences are examples of dynamical decoupling, which is widely used in nuclear magnetic resonance (NMR) [265] and quantum information
processing [170] to remove dephasing and extend coherence times. This sequence in particular removes dephasing due to two-particle dipolar interactions by swapping the eigenstates of the dipolarinteraction Hamiltonian for two isolated particles to allow for subsequent rephasing. This pulse
sequence is named WAHUHA after its authors - Waugh, Huber, and Haeberlen [265].
This procedure is straightforward to understand by considering two particles initially prepared

76

Figure 5.4: The WAHUHA pulse sequence. (a) shows the pulse sequence used to remove pair-wiseentanglement. The first, middle, and last pulse are the same as the spin echo sequence describedabove, and the extra pulses serve as an echo sequence in the two-molecule basis. (b) shows thecontrast decay vs. time for Ramsey (purple), spin echo (red), and WAHUHA (black). The inset
shows the difference between the spin echo and the WAHUHA for each time. Reproduced from
Reference [273].

77
in | ↓↓i. Then an intial (π/2)y pulse transfers them to
O 1
 1

1
√ | ↓i + | ↑i
√ | ↓i + | ↑i = | ↓↓i + | ↑↑i + | ↓↑i + | ↑↓i .
2
2
2

(5.2)

Because of the spin-exchange term, | ↓↑i and | ↑↓i are not eigenstates of the Hamiltonian. However,√
the three triplet states | ↓↓i, | ↑↑i, and (| ↓↑i+| ↑↓i)/ 2 are eigenstates with eigenenergies 0, 0, andJ⊥ /2, respectively. Note that a single (π/2)x pulse can swap the states | ↓↓i+ | ↑↑i and | ↓↑i+ | ↑↓i,and can thus act as an effective spin echo for the two-particle wavefunction.
During the first free evolution time of duration T /8 as shown in Fig. 5.4a, the eigenstates | ↓↓i√and | ↑↑i accumulate no phase, whereas (| ↓↑i + | ↑↓i)/ 2 acquires a phase of e−i(J⊥ /~)T /16 , which
entangles the state. After this evolution time we apply a (−π/2)x pulse to swap the contributions
from | ↓↑i + | ↑↓i and | ↓↓i + | ↑↑i, and swapping the accrued phases. After another evolution
time of T /4, the (π/2)x pulse swaps the phases again. This state then evolves freely for anothertime T /8, after which both | ↓↑i + | ↑↓i and | ↓↓i + | ↑↑i have accumulated the same total phase,e−i(J⊥ /~)T /8 , and the state is no longer entangled as a result. Hence the oscillations due to pairwise
dipole-dipole interactions is cancelled.
Figure 5.4b shows the decay of the Ramsey contrast for three different pulse sequences. In
purple is the simple two-pulse Ramsey sequence without an echo. The coherence time is ≈ 1 ms,
which matches the ≈ 1 kHz energy shift across the cloud discussed in Chapter 4. This is limitedby the anisotropic polarizability, hyperpolarizability, and any residual magnetic or electric field
noise. In red is shown the three-pulse spin echo sequence, as discussed in the previous section. The
black curve shows the seven-pulse WAHUHA sequence, and the pair-wise oscillations have clearly
vanished. To elucidate this effect further, the inset shows the difference between the spin-echo and
WAHUHA contrasts at each time. We attribute the finite decay time of the coherence that remains
after pairwise entanglement removal to many-body (beyond two-body) effects, as well as residual
decoherence mechanisms.

78

Figure 5.5: The many-body nature of the dipolar interactions. (a) shows the simulations of thespin-exchange dynamics allowing only nearest-neighbor interactions (blue, purple), and the full
simulation with the filling fraction as a fitting parameter. (b) shows nearest and next-nearest
neighbors (blue, purple), with the same simulation in black. Reproduced from Reference [110].

5.4

Many-body dynamics
While this NMR sequence allows us to remove pairwise entanglement, extenting it to larger

clusters becomes highly nontrivial. Therefore, to assess the many-body nature of our interactions
we must turn to more indirect methods of analysis. The first method is by comparing the dynamics
observed in our experiment to a theoretical simulation carried out in the group of Ana Maria Rey.
The dotted black lines in Fig. 5.5 (and the lines in Fig. 5.7) are a many-body simulation where theonly fitting parameter is the filling fraction. This simulation technique is described in Ref. [110],
and represents a step forward in the simulation of complex, many-body systems. Further, thistheory is instructive for understanding the experiment.
Specifically, only nearest-neighbor, or nearest-neighbor and next-nearest-neighbor interactions can be considered, as shown in Fig. 5.5a and b, respectively. Again, the only fitting parameteris the filling fraction. The blue and purple lines show these simulations for a few different filling
fractions that describe well either the short time or long time part of the contrast decay. However,
all of these curves are in qualitative disagreement with the observation and the full theory (black
dotted line), which illustrates that beyond-neighbor interactions are required, and the system is a

79
truly long-range, many-body system.
Another approach for studying the many-body nature of our data is with the simple, exponential fit discussed above: (1−A)e−T /t +Acos2 (πf T ). Looking at the interaction map in Fig. 5.2c,√
it is clear that there are three dominant interactions with relative frequencies f , f / 2, and f /2.
Therefore, we can consider instead a fit that includes these three frequencies:√C(T ) = (1 − A)e−T /t + A cos2 (πf T ) + cos2 (π(f / 2)T ) + cos2 (π(f /2)T ) .

(5.3)

Note that we force the amplitude of the three oscillation terms to be identical. While this equation
has more terms, it does not have any more fitting parameters, which are still just the frequency f ,the coherence time t, and the amplitude A.
Figure 5.6a shows a contrast decay dataset, and it is fit with the one-frequency function(green) and the three-frequency function (blue). The three-frequency fit is clearly better even byeye, but we quantify this in Fig. 5.6b by considering the reduced χ2 of the two fits. We study thisfit quality as a function of frequency, and we find that the three-frequency fit is clearly better near
the correct frequency (J⊥,|1,0i /2h = 100 Hz), and that the three-body fit is best very close to thecorrect frequency at 100 Hz. This suggests that every molecule is interacting strongly with manyof its neighbors, and thus the system is necessarily described by many-body physics.

5.5

Universal dipolar interactions
Another way to change the interaction strength is to couple |0, 0i to a different state in the

N = 1 manifold, as J⊥ is twice as large for the {|0, 0i, |1, 0i} transition. To test this, the same
experiment described above was repeated with | ↑i = |1, 0i [110]. Figure 5.8a shows typical contrast
decay curves (coherence time is given by 1/e = 0.37) for ∼ 104 molecules for both choices of | ↑i,
and Figure 5.8b compares the two curves when time is rescaled by a factor of 1/2 for the |1, −1idata. The contrast clearly decays faster for | ↑i = |1, 0i for which the interactions are stronger,
and the fact that the two curves nearly collapse onto each other when time is rescaled by exactly
the same ratio of the interaction energy highlights that dipole-dipole interactions give rise to the80

Figure 5.6: Fitting the contrast decay with three frequencies vs. one frequency for the |0, 0i to|1, 0i transition. (a) shows the fitting with one frequency in green, and three frequencies in√ blue.
There is only one fitting parameter f in both cases, but the other frequencies used are f / 2 andf /2. (b) shows the reduced χ2 associated with both fits vs. f . The three frequency fit is clearly
better. Reproduced from Reference [110].

Figure 5.7: Different excited states to use for spin-exchange interactions. The strength of the spinexchange term varies by a factor of two between the two projections of the excited rotational state|1, −1i and |1, 0i. Reproduced from Reference [110].

81
observed dynamics. The solid curves are based on theoretical simulations obtained from a cluster
expansion [110]. A new approach to the cluster expansion technique called a “moving-average”
cluster expansion [110] was developed in order to explain our data, and it exemplifies the needfor new development of theory tools to help understand the complex many-body quantum systems
that are now found in laboratories, such as trapped ion chains, magnetic atoms, and atoms in
optical lattices. This theory comparison also informed us that the molecular filling fraction in thelattice was low, at about 5% for the data in Fig. 5.8, which is consistent with the earlier estimation
based on the quantum Zeno effect. This has motivated work to increase the filling fraction in thelattice [183], to be discussed in Chapter 6.

5.6

N = 1 to N = 2 rotational transitions
Another way to tune the strength of the dipolar interaction is to use the N = 1 to N = 2

transition instead of N = 0 to N = 1. The rotational transitions, and hence the spin exchange
process, is an electric dipole transition. Therefore, only states of opposite parity can be coupled.
The next transition on the lattice is N = 1 to N = 2, for which the dipole moment is ∼ 0.1 − 0.2
Debye rather than 0.567 Debye. The exchange energies of this spin qubit are J⊥,N =1−2 /2 = 10 − 20
Hz, and almost all signatures of the spin exchange process should be undetectable. This could
serve as a useful benchmark to show that the oscillations and many-body dynamics we observe are
not somehow a technical artifact of the experiment, and would supplement other evidence like thedensity-dependence, many-body theory, and the WAHUHA sequence.
As discussed already in the previous section, reducing the interaction strength increases thecoherence time since many-body entanglement is playing less of a role. Thus, for the N = 1 toN = 2 transition the coherence time should be significantly longer than the |1, −1i case. Figure 5.9
shows spectroscopy of the N = 1 to N = 2 transition, where the exact frequencies and coupling
powers to the five projections are unknown. We measure the transition frequency to be 4.4558 GHz,
which is consistent with the rotational constant. We were never able to use this transition for spinexchange dynamics, however, because we could not find any magic angle to minimize the differential

82

(a)

(b)

Figure 5.8: (a) Typical contrast decay curves for | ↑i = |1, −1i (top) and | ↑i = |1, 0i (bottom) forroughly 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. Note that the data for |1, −1i is rescaled by a factor of two shorter in (b), sodata out to 70 ms is included. The lines are theory [110]. Reproduced from Reference [110].

Figure 5.9: The N = 1 to N = 2 transition. Spectroscopy of this transition to an unknown
projection gives the frequency needed to reach nearly a π-pulse with a given power. The time ofthe pulse was similarly scanned to reach τ Ω ∼ π.

83
polarizability. We worked on this only briefly, but the magic angle is extremely important for thelong coherence times observed with the N = 0 to N = 1 transitions.

5.7

Pulse sequences with variable tipping angles
All the pulse sequences shown so far have had an area of either π/2 or π, but a lot of

information could potentially be gleaned from using other tipping angles [168]. We can write~i · S~j = Ŝix Ŝjx + Ŝ y Ŝ y + Ŝiz Ŝjz .
S
i j

(5.4)

~i · S~j ≈In the mean-field picture, the spin operators can be replaced by ensemble averages, and so S
hSi2 at short times, provided the initial state can be represented in the Dicke manifold. Then, thespin exchange term can be rewritten Ŝix Ŝjx + Ŝiy Ŝjy ≈ hSi2 − Ŝiz Ŝjz . Further, the projection operators
can also be replaced with ensemble averages, and so Ŝiz is replaced by the average magnetizationhS z i.
The first pulse of our Ramsey sequence of area θ then changes the average magnetization
by hS z i = cos(θ). Now the spin-exchange term can be thought of in the mean-field picture as aspin interacting with the effective magnetic field due to all the other spins. In this picture thespins accrue a phase shift of ∆φ ∼ cos(θ)∆τ , such that the Ramsey fringe evolves in time asRe(e−icos(θ)τ ). This trend is shown in Fig. 5.10a for θ = π/16 and 15π/16. Figure 5.10b shows
the contrast of the final Ramsey fringe as a function of the hold time, and Fig. 5.10c shows theintial slope from Fig. 5.10a as a function of tipping angle (or pulse area) between zero and π. The
corresponding calculations are shown in Fig. 5.10d-f, and they show excellent qualitiative agreementwith the data.
However, as seen in comparing Fig. 5.10a and d, the ampitude of the data is far too large.
In fact, we found that developing good quantitative agreement was incredibly difficult, and we
attributed this to the non-magicness of the trap. Due to the anisotropic polarizability [188] andhyperpolarizability [273] discussed in the previous chapter, there is a differential light shift acrossthe cloud. Thus, the tipping angle varies across the cloud, and the spins start to become off-

84
resonant compared to J⊥ /2 towards the outside of the cloud. Thus the phase evolution is highly
nontrivial to understand, and phase evolution can be observed even at a tipping angle of π/2 (seeFig. 5.3b) where it should not happen. Many-body effects would certainly cause our experimental
observations to differ from the simple mean-field approximation, but we were unable to demonstrate
a quantitative agreement between our data and the many-body theory described in the sections
above.
This differential light shift that dephases individual spins can be included into the HamilP
tonian, and should be written as 1/2 i hi Ŝiz , which looks like a magnetic field. The strength ofhi varies from site to site in accordance with the harmonic confinement. While this effect canbe mitigated with the spin echo pulse, imperfections become particularly acute at small tipping
angles. Therefore, we were never able to learn much about the many-body dynamics of our system
by varying the tipping angle, and a better approach to combat the hyperpolarizability would beneeded to proceed with these experiments. In the future we would like to make the lattice very flatand have molecules occupy only the central region.

5.8

Inferring the molecule filling fraction
The curves in Fig. 5.3d display a clear density dependence, which is a signature of an in-teraction effect. The density was varied by loading the same initial distribution of molecules andthen holding them in the lattice for a variable amount of time to allow for single-particle loss fromlight scattering. In Ref. [47], we acquired a full understanding of the inelastic light scattering fromthe trapping light, and so we can use the trapping light to remove molecules or drive them todark states at a well understood rate, which is very long compared to the Ramsey spectroscopy
sequence. Thus, the density ρ is proportional to the number of molecules we detect [47]. We expect
the coherence time τ to scale with the molecule number N as τ ∝ 1/N , since
τ∝

1
R̄3
1
∝
∝
,
hEint i
J⊥
J⊥ ρ

(5.5)

85

Figure 5.10: Pulse sequences with variable tipping angle. (a) shows the phase of the Ramsey fringe
as a function of the dark time when the tipping angle is π/16 and 15π/16. (b) shows the contrastof these Ramsey fringes. (c) shows the slope of the intial phase change (e.g. short times in (a)) for
different tipping angles. (d)-(f) show simulations of the plots in (a)-(c). Note that the amplitude
in (d) is significantly smaller than the measured amplitude in (a).

86
where R̄ is the average interparticle spacing and the density ρ = R̄−3 . For our loading scheme,ρ ∝ N . Figure 5.3e shows the coherence time clearly scales as 1/N .
As stated in several places so far in this thesis, the filling fraction in the lattice has beenmeasured to be ∼ 5−10% by several methods, including the decay time of the contrast. To proceedwith quantum magnetism or out-of-equilibrium many-body dynamics using dipolar interactions
such as the spin-exchange or Ising interactions, it will be important to increase this filling fraction.
We have tried many times over the past five or six years to reduce the temperature and entropyof our molecular sample by many different methods, and we succeeded in doing so within the last
few years using a quantum synthesis approach with atomic insulators in the optical lattice. These
efforts are the subject of Chapter 6.

Chapter 6

A low entropy quantum gas of polar molecules in a 3D optical lattice

The first successful creation of KRb ground state molecules in 2008 reached close to quantum
degeneracy, with a temperature of 1.3TF . However, despite an intense effort to further cool themolecular gas, the lowest temperature for the KRb gas reached in a harmonic trap is T /TF = 1.
No other groups have produced colder, denser gases to date. As discussed at the end of Chapter
5, the best filling fraction we had previously reached in the optical lattice is 5 − 10%. I will begin
this chapter by discussing why it is important to increase the filling fraction, and what new andexciting experiments await a low entropy molecular sample. Then I will discuss many of our efforts
to reduce the entropy, with the ultimate success coming from a quatum synthesis approach of
atomic insulators in the lattice. This successful approach will then be the topic for the remainderof the chapter.

6.1

The need for low entropy - percolation theory
Low temperatures and entropies are required for most of the exciting future directions to be

pursued with polar molecules. For bulk gases, one would expect qualititatively new phenomena
when the temperature is on the order of the dipolar interaction energy. For KRb at moderatedensities of n = 1013 cm−3 , the dipolar interaction has an energy scale that corresponds to on
average 5 − 10 nK. Therefore it will be important to reach temperatures well below 100 nK.
In the lattice, on the other hand, the desire for low entropy comes from a need for a high filling
fraction. Nearly any quantum simulation or quantum information research requires a well defined

88
initial state, where the location of each particle is known with high fidelity. Indeed, realizing exotic
quantum magnetic phases of matter in the XXZ model would likely require near-unity filling.
Moreover, since the enery scale between neighboring sites is J⊥ /2 = 100 Hz, most dynamical
experiments centered around the spin-exchange mechanism will require a dense network of nearest
neighbor interactions. While we have demonstrated conclusively that the interactions we observe
are many-body, the interaction range is too small to support significant interaction between evennext-nearest-neighbor sites. Additionally, the hyperpolarizability discussed in Chapter 4 causes anenergy offset between distant sites which adds a detuning their interactions.
It is constructive to understand what filling fraction is actually needed for the onset of dynamics in an out-of-equilibrium spin system. To do this we invoke the idea of a percolation threshold,which is the formation of long-range connectivity in a random system. Below the threshold alarge connected component of the system does not exist; while above it, there exists a connectedcomponent with a size on the order of the system size. The notion of connectivity is based on
the competition between the energy scales discussed above. Since the dipolar interaction has finite
range, inhomogeneity or finite coherence time will set the length scale in the system. The hyperpolarizability causes a kHz-level energy shift across the cloud and limits the Ramsey coherencetime to 1 ms without an echo pulse. Therefore, even next-nearest-neighbor interactions at ∼ 10 Hz
energy scales are starting to set the length scale of the interaction. Accordingly, percolation occurs
when every molecule has a nearest neighbor with which it can interact.
The percolation threshold for an infinite cubic lattice of nearest-neighbor interactions can becalculated analytically, and is given in Ref. [241] to be fPT = 0.3. Since our system has finite sizeand our interactions are slightly longer than nearest neighor, we expect the percolation thresholdin our system to be < 0.3. Therefore, this number serves as a benchmark for what we would liketo achieve in our experiment, and we have been trying since 2010 or so to create a lower entropy
system.

89

Figure 6.1: Low entropy lattice cartoon. This shows a high filling fraction of molecules in thelattice, and illustrates how they are all connected in a percolating network. Reproduced from
Reference [183].

90

6.2

Many previous efforts
There are several challanges to efficiently make Feshbach molecules (FbM) in a bulk 3D gas.

The first is good spatial overlap between the two clouds. There are two reasons why this could bepoor. As the clouds become very cold and the optical trap becomes very weak, the difference in mass
2 − 1/ω 2 ) [200],
causes the two clouds to separate. The differential sag is given by ∆ysag = −g(1/ωRb
K

where ω is the oscillation frequency in the trap. In our optical traps at λ1 = 1064 nm, thepolarizability of K and Rb are similar to within 80%, and so their oscillation frequencies arepdifferent by ≈ mRb /mK . In typical conditions of a degenerate Bose-Fermi mixture of K and Rb,
this differential sag could be ≈ 3 − 5 µm.
The other problem with their spatial overlap also arrises from reaching quantum degeneracy.
The fermions have enormous Fermi pressure due to the Pauli exclusion principle. Typical sizes foreven the coldest degenerate Fermi gases of NK ≈ 105 are σK ≈ 50 µm. Conversely, the bosons
condense to a very small cloud described by the Thomas-Fermi (TF) approximation, and typicalconditions are NRb ≈ 105 where σRb ≈ 10 µm. In this regime the differential sag becomes similar
to the size of the Rb cloud. Therefore, not only are their sizes very poorly matched, their positions
are significantly different. The latter problem cannot be solved easily with a magnetic field gradient
which generates a differential force because of the magnetic moments of the states we use.
In addition to these problems, the high density of the BEC introduces another problem withthree-body loss. If even 50% of the Rb is converted to a FbM, the remaining 50% will quickly cause
three-body interactions between one FbM and two Rb atoms. The lifetime of FbM with atoms isseveral ms, which makes it difficult to do STIRAP and remove the unpaired atoms quickly enough.
The timescale of three-body loss increases as n3Rb , so dense BECs pose significant challanges.
The next challange is the poor momentum space overlap. The BEC has macroscopic population of the lowest momentum state of the trap by definition, but the DFG stacks up K atoms
in momentum states with one per level, as required by the Pauli exclusion principle. Therefore,
similar to coordinate space, the momentum space distribution of K is much larger than Rb. It

91

Figure 6.2: Degenerate K-Rb pairing in momentum space. (a) The ideal case where each Rb (red)picks the next lowest available K (blue). In our experiment this typically occurs when NRb  NK ,but then we suffer strong three-body loss. (b) The more realistic case where each Rb picks a K at
random through the momentum distribution, and is actually biased to the top by the weight of the
Fermi surface. In our experiment this typically occurs when NRb  NK .

is constructive to think about how this pairing looks in momentum space, and a few possibilities
are shown in Fig. 6.2. The ideal case is when each Rb takes the lowest K available, such that the
momentum distribution of the molecules is the same as K, and the FbM T /TF is similar to that ofK. This would allow us to make a DFG of molecules in bulk, but this is certainly not what actually
happens.
Realistically, each Rb will pair with any K atom in the distribution, but it will be biasedby the Fermi surface which carries the most weight because of the k 2 dk factor in the momentum
integral. Therefore, the FbMs actually inherit the temperature of K (or slightly higher) rather thanits T /TF . Moreover, since NFbM ≈ 0.2NK , the T /TF immediately jumps up by a factor of ≈ 3.
Concomitantly, even starting with TK = 0.3TFK , we have never been able to create FbM colder than
TFbM = 1.0TFKRb . Nevertheless, we have pursued virtually every possible route to increasing thephase-space-density, and I will now discuss several of these efforts.

6.2.1

Species selective dipole traps

Despite the complications with momentum space overlap, we spent a lot of time trying to
improve the coordinate space overlap. Since the K cloud is bigger and higher in the trap than Rb, aspecies-selective trap which only affects K could be very useful. In order to make the polarizabilityof the trap equal to zero for Rb, we need the contributions from the P3/2 (called D2) at 780 nm
and P1/2 (called D1) at 795 nm to cancel each other. This is done at a wavelength of 790 nm,

92
though it is sensitive to the polarization since vector and tensor light shifts are significant so close
to resonance.
As discussed in Chapter 4, this trap can be used to move the K cloud relative to Rb (seeFig. 4.14), and it can be used to compress K. We had to be careful, however, to limit the heatingof K and Rb since the scattering rates are high for both and the clouds are initially very cold.
Nevertheless, this beam was actually very useful since it allowed us to efficiently make FbM at very
low temperatures in very weak traps, and we used this approach to make ≈ 10 − 20 × 104 FbM at
T = 70 nK (T /TF = 1). This is the coldest molecular sample ever created as far as I am aware.
The caveat is that this beam kills FbM in 1 ms and ground state molecules (GSM) in 100
ms, as also shown in Fig. 4.14. Therefore, as soon as FbM are created, we had to turn the 790
nm beam off suddenly. In doing so, the FbM are out of equilibrium since the trap was changed
diabatically, and thus they slosh and breathe in the trap with large amplitudes. Further, since
the lifelime of FbM is only ≈ 15 ms even in ideal circumstances, the FbM must be converted
to GSM immediately. At these temperatures molecules cannot collide with s-wave collisions andhence cannot thermalize. Consequently, the initial improvement in the phase-space-density cannot
be utilized, and we ultimately moved on to other approaches.

6.2.2‘Magic’ traps for matching the oscillation frequencies

A related idea which we also pursued is a ‘magic’ wavelength red-detuned trap, where thepolarizability ratio matches the mass ratio in such a way that the trap frequencies are equivalent.
Specifically, we want αK = mK · αRb /mRb . This occurs for K-Rb at 807-808 nm, depending on thepolarization. Such a trap was used extensively in the work of Ref. [200] to remove the differential
gravitational sag. However, as with 790 nm, it is impornant to minimize heating due to off-resonant
light scattering.
Though we did not study the effects of this beam too thoroughly, we learned that the FbM
and GSM also suffered from short lifetimes. Thus, it was difficult to take advantage of this trap,
and typically we found that the benefit from reducing sag was not offset by the heating of the93atomic clouds and the fast loss of the molecular samples. However, if we were to load atoms into
the lattice with the aid of the 808 nm beam, and then turned it off before making molecules, thisbeam could be very useful. Indeed, we may return to the idea of a magic trap later, but we will
always have to be careful to avoid the detrimental effects of the atomic heating.

6.2.3

Sympathetic cooling of KRb with Rb

With limited success in creating a degenerate bulk gas of molecules in 3D simply throughhigh fidelity magneto-association, we now turn to methods by which the molecules could be cooled.
Perhaps the most obvious approach to cooling a gas of molecules is to sympathetically cool itwith an atomic species, such as Rb. When we make molecules there are many leftover Rb atoms.
By evaporatively cooling the remaining Rb and allowing it to thermalize with KRb, it could bepossible to further cool KRb into quantum degeneracy. However, this requires a suitable elastic
cross-section for Rb-KRb collisions, and a sufficiently small inelastic collision cross-section.
While KRb was calculated to be unstable (exothermic) with K and a lifetime of only several
ms was observed [198], KRb is expected to be stable (endothermic) with Rb [198]. Despite thesecalculations the lifetime of KRb with Rb was measured to be only 60 ms, and the Rb density wasquite low, nRb = 0.6 × 1012 cm−3 [198]. To reach even this lifetime, however, it is essential totransfer KRb to its lowest energy hyperfine state, while Rb was similarly in its lowest hyperfine
state |1, 1i. The short lifetime was attributed to a large fraction of dark molecules which originatedfrom FbM that underwent vibrational-quenching collisions with Rb. If this was the case, thensympathetic cooling with Rb would be very difficult since we cannot spend less time in the FbM
step.
Nevertheless, we set out to test the hypothesis that large numbers of KRb molecules in darkstates are created during the FbM step by collisions with Rb. We did this by using a lattice toprevent collisions between Rb and FbM. We loaded K and Rb into the lattice, and converted sites
with one of each into a FbM with some fidelity. Then the FbM were converted to GSM with ∼ 90%
fidelity. The unpaired K was then removed with a blast pulse of resonant light. This light would

94
also dissociate any weakly-bound dark molecules (such as d-wave FbM, to be discussed later), andthe resulting K would then be removed. Then, the lattice should only contain sites with GSM,
a small number of deeply-bound dark molecules, and Rb. Next, the GSM are transferred to thelowest energy hyperfine state with ≈ 95% fidelity. Finally, the lattice can be ramped down, andthe lifetime of GSM with Rb can be probed in a cleaner system.
Upon doing this experiment we found that the lifetime is τ = 60 − 100 ms, which is similar
to the original measurement without the lattice. We can verify that it is indeed essential for KRb
to be in the hyperfine ground state, and the lifetime is τ < 10 ms if we leave it in its original state.
These results suggest that Rb is indeed the cause for the loss. Reference [198] found a very weak
dependence on the atomic density, which is inconsistent with two-body Rb-KRb collisions, and isalso inconsistent with three-body Rb-Rb-KRb collisions.
We believe that this loss is due to the sticky collisions discussed in Chapter 4. A Rb atom
collides with a KRb molecule and forms a collision complex where they spend a relatively long timenear each other (long compared to the collision rate). Although they would not undergo chemistry,
they spend enough time together for a second Rb to hit their complex. The second Rb drives the
Rb-KRb complex to a deeply bound state, and carries away the binding energy as kinetic energy.
Because this process does not require all three particles to collide at the same time, its dependence
on density is very shallow. Thus, sympathetic cooling of KRb with Rb does not appear to be an
option for cooling the molecules.

6.2.4

Evaporative cooling

The only remaining option to cool the molecules is with direct evaporative cooling. As
discussed in Chapter 3, ultracold fermionic spin-polarized molecules cannot collide in the s-wave
channel, as is often used for thermalization during evaporation. Moreover, KRb molecules undergo
large inelastic losses due to chemistry, and evaporation will require a large ratio of elastic to inelastic
collisions. Therefore, upon first inspection this approach seems very unlikely to succeed. In fact,
the solution will be to apply a large electric field.

95

Figure 6.3: Elastic and inelastic collision rates in 2D. (a) The 2D elastic (red) and inelastic, or reactive (yellow/green/blue) collision rates as a function of the dipole moment and the axial trapping
frequency which confines in 2D, for a collision energy of 500 nK. Reproduced from Reference [215].
(b) The same elastic and reactive collision rates as a function of collision energy, specifically forν = 23 kHz and d = 0.2 D. These conditions are appropriate for our efforts with the first generation
apparatus. (c) The elastic and reactive collision rates as a function of collision energy for ν = 100
kHz and d = 0.4 D. These conditions are appropriate for what we anticipate doing in the new
apparatus.

The dipolar interaction mixes the partial waves, which allows thermalization of identicalfermions even at ultralow temperatures. The dipole-dipole interactions prevent the elastic crosssection between identical fermions from vanishing at T = 0 because the corresponding Wigner
threshold law gives a finite energy-independent elastic cross section [5]. This feature has beenused in evaporation of highly magnetic atoms, enabling very efficient evaporation of spin-polarized
fermionic samples down to T < 0.2TF [5, 163]. The dipolar interactions for molecules are muchstronger than for highly magnetic atoms, and therefore we expect this approach also to work forus.
As discussed in Chapter 3, molecules can be stabilized in 2D using an electric field that isoriented perpendicular to the plane, such that their collisions are aligned only with the repulsive
part of the anisotropic dipolar interaction. This suppresses chemical reactions, and we were able
to observe lifetimes of ≈ 1 s at d = 0.2 D [63]. For evaporation, an important figure of merit is
the ratio of elastic collisions which can thermalize the sample, to inelatic collisions which lead tochemistry and loss. Figure 6.3a shows both the elastic collision rate and the reactive collision rate
as a function of both the dipole moment and the axial trapping frequency in the 1D lattice. A larger

96
dipole moment increases the strength of the repulsive dipolar interactions, further stabilizing themolecules. On the other hand, as the axial trap frequency is reduced, the ability for the molecules
to collide head-to-tail increases, and thus the reactive loss increases. Therefore, the electric fieldneeds to be very large and the axial trap frequency needs to be very high for successful evporation.
For the experiments done in 2011 and 2012 in the first generation chamber, we could only
reach a field of 4 kV/cm, which corresponds to 0.2 D. Moreover, at the time we could only reachan axial trap frequency of ν = 23 kHz. Figure 6.3b considers this exact case as a function of thecollision energy, e.g. temperature. At the time we were starting at a temperature of T ≈ 500 nK,
where the ratio of elastic to inelastic collisions was roughly 100. This ratio is marginally sufficient tosuccessfully evaporate and increase the phase-space-density [283], and so we attemped evaporation.
Figure 6.4 summaries our evaporation efforts. We used ITO-coated plates to generate anelectric field in the vertical direction, and the plates had a small hole in the middle to accomodatea vertical optical lattice (see Fig. 6.4a). The coating on the plates was split in half across thecenter such that four unique voltages can be applied. A small bias can be added to the two plate
sections on the right, which creates an electric field gradient in the radial (horizontal) direction.
This gradient tilts the individual 2D traps, and spills the hottest molecules as depicted in Fig. 6.4b.
To characterize the evaporation, we plot the slope of a line fitted to data on a log(N ) versus
log(T ) plot, which we label as log(N )/log(T ) in Fig. 6.4c. With no electric field to stabilize thedipoles, reactive loss causes anti-evaporation since chemistry preferentially occurs in the coldest,
highest-density part of the cloud. Therefore, log(N )/log(T ) < 0 since the cloud gets hotter as theloss occurs. At about 2 kV/cm the electric field is sufficiently high to stabilize the dipoles enoughto prevent anti-evaporation, and thus log(N )/log(T ) crosses zero. Now let us consider what slope
sets the criteria for evaporation that increases the phase-space-density of the molecules.
√
The Fermi temperature in a 2D gas is given by TF = N ~ωr /kB , where ωr is the radial trap frequency [73]. To increase the phase-space-density we want to reduce T /TF , and thus√
we want T to decrease faster than N ~ωr /kB . Specifically we want to look at log(T ) versus
log(TF ) = 1/2log(N ) + log(~ωr /kB ). Therefore, in order to increase the phase-space-density we97

Figure 6.4: Evaporative cooling of molecules in 1D lattice using an electric field gradient. (a) Split
plate electrodes are placed around the vacuum cell, and a 1D lattice propagates through a small
hole between them. The voltages can be biased to one side to create a radial electric field gradientof a vertical electric field. (b) All the molecules on each layer of the 1D lattice are aligned withthe electric field. The radial gradient spills the layers and tips them in one direction such that thehottest molecules can be selectively removed. (c) The efficiency of the evaporative cooling vs. theelectric field strength. Below ∼ 2 kV/cm, there is anti-evaporation due to the inelastic chemicalloss, which happens selectively with the coldest molecules. Above ∼ 2 kV/cm there is cooling, andthe phase-space density is increasing above 4 kV/cm. The inset zooms in on this region, and theevaporation efficiency is actually becoming quite good up to the limit of the electric field we canreach in the old apparatus.

98
require log(N )/log(T ) < 2. Figure 6.4c shows that the phase-space-density is indeed increasing at
fields above ≈ 4.5 kV/cm. However, we were never able to evaporate enough to make a degenerate
gas with T /TF < 0.5.
The procedure for initializing this experiment was very detrimental to its success for thefollowing reason. We created molecules in a bulk 3D gas, and then ramped up the 1D lattice andturned on the electric field. As a result, any non-adiabaticity during the ramp would heat themolecules, which cannot thermalize until the electric field is applied. Moreover, since the cloud is
initially in a 3D trap, the 1D lattice slices the cloud into ≈ 10 − 15 pancakes [63]. The number ofatoms in the center pancakes is only ≈ 4 − 5000, coming from the initial cloud of ≈ 3 × 104 . This
fact immediately increases T /TF by a factor of three or so. Moreover, the density increases when
we go to 2D, and so there is adiabatic heating of the molecules. While adiabatic heating does notaffect T /TF , higher temperatures cause the ratio of elastic to inelastic collisions to decrease.
Additionally, TF changes in going from 3D to 2D. In 3D TF3D = ~ω̄/kB · (6N )1/3 [218], whilep√in 2D TF2D = N ~ωr /kB . ω̄ = ωr2 ωz is the geometric mean trap frequency, where ωr ≈ 2π × 40
Hz and ωz ≈ 2π × 250 Hz. Note that ωz does not enter into TF2D , and therefore T /TF2D takes
another hit to an even higher value since ωz > ωr . The different scaling with number is in the favorof 2D, but both scalings are so shallow that the number change going from 3D to 2D dominates
over the change in scaling. As a result of all of these factors we typically start evaporation with
T /TF2D = 4 even though we started in 3D with T /TF3D = 1. The best evaporation we have observed
allowed us to get back to T /TF2D ≈ 1 − 2.
During all of these studies we also had enormous problems with charging of the glass cell andthe resulting electric field instability, as discussed in Chapter 2. The new apparatus is designed
to reach much larger electric fields, and since the electrodes are in the chamber, we anticipate
excellent field stability. This will be discussed more in Chapter 7. The ability to reach 20 − 30
kV/cm will give us dipole moments of d = 0.4 D. Moreover, we expect to be able to reach muchlarger lattice depths, providing axial trapping frequencies of ν = 100 kHz. These conditions areshown in Fig. 6.3c, and the ratio of elastic to reactive collisions will be roughly 400, which is ≈ 4×

99
better than the conditions used in the old apparatus. Moreover, as the molecules get colder thisratio increases very rapidly, even to 3000 at 10 nK.
Another important improvement can be made in the new apparatus by starting in a 2D(or 2D-ish) trap from the beginning. This could be done by loading atoms into the lattice andperforming atom evaporation to reach degeneracy in 2D before making molecules on each pancake.
Alternatively, the 3D trap aspect ratio could be increased such that the cloud in 3D would load
into only a few pancakes. While the prospects for evaporation in the new apparatus are very
exciting, this list of problems with our efforts in the old apparatus has made evaporative coolingof molecules to quantum degeneracy prohibitively difficult in the old apparatus. We will return tothis experiment in the new apparatus, as discussed in Chapter 10.

6.3

Quantum synthesis with atomic insulators in the lattice
Thus, it gradually became more attractive to create KRb molecules directly in a 3D op-

tical lattice and optimize the filling fraction to realize a low-entropy system. A natural way toaccomplish this task is to take advantage of the precise experimental control that is available formanipulating the initial atomic quantum gas mixture in the 3D lattice. Specifically, we need toprepare low-entropy states of both atomic species and then utilize the already familiar techniquesof coherent association and state transfer for efficient molecule production at individual lattice sites(see Fig. 6.5). The combination of efficient magnetoassociation of preformed pairs [47] and coherent optical state transfer via STIRAP means the second step should work well; however, creating
a low-entropy state for both species with the optimal density of one particle each per site is very
challenging. This idea was first proposed in 2003 [56] and later specifically for the KRb system [82].
The creation of Rb2 Feshbach molecules in a 3D lattice followed this basic idea, where the molecules
were produced out of the region of a Mott insulator that has two atoms per site [261]. However,the KRb system faces a much bigger challenge due to the fact that we must address two different
species with different masses and different quantum statistics.
The initial experimental target is to produce spatially overlapped atomic distributions that

100

Figure 6.5: Quantum synthesis of a low entropy molecular gas in the optical lattice. The 40 K Fermi
gas needed to obtain a band insulator is much larger than a Rb BEC which gives one atom persite, and the goal is to efficiently make molecules in the center where there is a 40 K atom and a Rb
atom at each site. Reproduced from Reference [183].

101
consist of a Mott insulator (MI) of Rb bosons [100] and a spin-polarized band insulator of 40 K
fermions [232]. The occupancy of the MI depends on the ratio of the chemical potential µ to onsite interaction energy U , and can be higher than 1 per site if there are too many Rb atoms or if theunderlying harmonic confinement of the lattice is too high. The filling fraction of fermions in the
optical lattice depends on an interplay between initial temperature, lattice tunneling, and external
harmonic confinement. For our experiment, we need a sufficiently tight harmonic confinement andlarge 40 K atom numbers to achieve a filling approaching unity in the center of the lattice. The
main challenge of the experiment is the opposite requirements between the filling of the Fermi gas
and the number of Rb atoms that can be accommodated in the n = 1 Mott insulator. This requires
a careful compromise to achieve experimental optimization.

6.4

Experimental considerations for in situ imaging
In order to measure the filling fractions of K and Rb by in situ detection in the lattice, there

are a number of crucial experimental systematic checks that must be performed. Therefore, before
we go on to learn the atom numbers of K and Rb that are required to reach nearly unity peakoccupation for both species, we must calibrate the inaging system and remove imaging systematics
that could significantly impact the interpretation of the in situ measurements.

6.4.1

Calibrating the saturation intensity

For absorption imaging it is very important to know the intensity of the probe beam since thescattering rate, and thus the OD, will depend on it. The pictures of our cloud are generated using
three images: the first uses resonant light to probe the atoms which in turn cast a shadow onto thecamera. This image is called Ishadow . The second image contains the probe pulse, but (typically
500 ms) after the atoms are released from the trap. This is called Ilight . Finally an image without
the probe beam at all is taken to subtract the background light. This is called Idark . Therefore, we
use If = Ishadow − Idark and Ii = Ilight − Idark . These calculations are done for every pixel, and a

102

Figure 6.6: Calibrating the saturation intensity of the probe beam. (a) The number of counts on
the camera per pixel per µs for the K probe beam as a function of the peak OD of the K cloudthat is imaged. (b) The number of counts on the camera per pixel per µs for the Rb probe beam
as a function of the peak OD of the Rb cloud that is imaged.

simple estimate of the OD isODmeas = ln

Ii
.
If

(6.1)

However, if any photons cannot be absorbed by the atoms, which we call “bad light”, then themeasurable OD will be limited accordingly. For all of the measurements reported in this chapter,
light with linear polarization is used to drive a σ + transition. Therefore, the absorption crosssection is reduced by a factor of two. The maximum OD that can be measured is called ODsat ,and this is typically measured to be between three and five by simply looking at the cut-off OD ofvery dense BECs where the OD is roughly 10. We account for this effect with a modified OD given
by [268]
1 − e−ODsat
ODmod = ln −ODmeas.
e− eODsat(6.2)

The intensity of the probe beam plays a significant role, and so the actual OD must accountfor the intensity, through [268]ODactual = ODmod + (1 − e−ODmod )

I
Isat

.

(6.3)

In general the saturation intensity Isat of the probe beam is not known, so this equation can be used
to calibrate the intensity. This is done by measuring ODmod (using ODmeas and ODsat as described

103
above) as a function of the intensity, as shown for K and Rb in Fig. 6.6. Then, this equation isrearranged to give I as a function of ODmod , with ODactual and Isat as fitting parameters.

6.4.2

Measuring the imaging resolution and the pixel size

An extremely important parameter for looking at small Mott insulators in the lattice is theresolution, as it artificially inflates the apparant size of small objects. We focus the imaging systemby imaging the smallest object we can make reliably (typically a few hundred atoms at a few nK),
and then we scan the position of the camera to minimize the size and maximize the optical depth(OD). We measure the resolution at the optimal focus position by measuring the size of the object.
We can then vary the size of the BEC loaded into the lattice, and compare the measured size tocalculations of the Thomas-Fermi (TF) radius (i.e. when kinetic energy is neglected) corresponding
to the appropriate trap frequencies and atom number calculated from the measurements. The size at
which the calculation starts to significantly deviate from the measurement indicates the resolution
limit. For typical BECs of Rb, RTF ≈ 10 µm. Due to the limitations of all the coils in the IP trap
around the cell and the cell’s poor optical quality, our imaging resolution was limited to ≈ 3 µm.
Therefore, the resolution limit was already playing a significant role for all but the largest BECs
that we studied.
In is constructive to use Fourier optics to calculate the effect that finite resolution has on themeasured size of the cloud. Consider the cloud to be a Gaussian function g of width σ. The third
row of Fig. 6.7 shows the function g for four different values of σ. In generating the image, theobject plane is mapped with the first lens to its Fourier transform, but is first convolved with theresolution (aperture size) of the lens. This function is given by an Airy function, which is simply a
Bessel function, and is labeled as f . Thus, the image plane is given by F{f ? g}, where F denotes
the Fourier transform, and ? denotes the convolution. This can be rewritten, however, using theconvolution theorem as
F{f ? g} = F{f } · F{g}.

(6.4)

The Fourier transform of the Gaussian cloud g is another Gaussian, where the size is inverted.

104
The Fourier transform of the lens Airy function f is given by a disk, or aperture of the lens. The
product of these two Fourier transforms is shown the first row of Fig. 6.7. Note that when σ is verysmall its Fourier transform is large, and then it is significantly limited by the aperture of the lens.
The second lens of the imaging system images the image plane from the first lens on thecamera by taking its Fourier transform. The second row of Fig. 6.7 shows the image on the camera
after the second lens. Note that when σ is very small the object on the camera is again small, but
limited by the resolution, and the ripples of the Airy function are clearly visible. When σ is verylarge, the cloud is completely unaffected by the imaging system. This can be seen by comparing to
the third row, with shows the cloud function g. This analysis was done for a magnification M = 1,
but it can easily be generalized to an arbitrary M by simply multiplying the width in the second
row by M .
To understand how the imaged size σimage compares to the cloud size σ, we can fit the image
function in the second row to a Gaussian. In plotting σimage versus σ, we can generate a roughestimate of how the resolution affects the fitted image size. The resolution function f is given byrmin , which is the radius where the Airy function first crosses zero. To convert this to a Gaussian
σmin , we must first multiply by ≈ 0.75 to get the 1/e2 value (fitting the Airy function to a Gaussian),and then divide by 2 to get the 1/e value. Thus the resolution in the first column rmin = 2.5 µm
becomes σmin = 1 µm. Such analysis suggests thatq2
σimage = σ 2 + σmin

(6.5)

is an excellent estimate for most of the range, though it becomes unreliable when σ ≈ σmin . As
described above, σ can be calculated from the TF approximation, and this method can be used tomeasure σmin .
Another effect that was very important in order to avoid artificially inflating the cloud size
was the optical repump beam. As discussed in Chapter 3, we use Rb in the |1, 1i state when
we make molecules, yet we must image the atoms in the |2, 2i state in order to drive the cyclingtransition to the |3, 3i0 state. There are two ways to do this: either a repump beam that drives

105

Figure 6.7: Fourier optics analysis of the imaging system. The first row shows the convolution of
the object g with the resolution of the lens f . Then, the Fourier transform is taken to show thetransfer function of the lens to the image plane. Using the convolution theorem this can also bethought of as the Fourier transfer of the cloud times the Fourier transform of the lens. Thus, asmall cloud becomes very large in the image plane, and a very large cloud becomes small. The
Fourier transform of the lens is a circular aperture, which limits the size in the image plane. Fromleft to right the cloud size σ is increasing, and thus the limitation of the aperture is decreasing.
The next row shows the inverse Fourier transform of the first row, which is done by the second(or so-called eyepiece lens), and this is what the object looks like on the camera. Note the fringes
on the object when the cloud is very small, which serve to form an Airy disk pattern. The third
row shows the object function g, which can be compared to the second column. Clearly the size islimited by the resolution for the objects in the left two columns.

106|1, 1i to |2, 2i0 down to |2, 2i can be used for a few µs before the image from the probe beam isacquired, or a microwave pulse can be used to transfer the atoms to |2, 2i without light. We foundit to be very important that we use the latter approach, as the former method scatters several
photons which have a probability of causing atoms to tunnel due to the heating, which in turnmakes the cloud appear larger.
The size in the object plane corresponding to one pixel (referred to as the pixel size) canbe estimated simply from the resolution and the magnification of the imaging system. However, amore accurate measurement will be needed to correctly count the number of atoms and the size ofthe cloud. For side imaging paths this can easily be done by measuring the vertical position during
expansion. The cloud falls ballistically, and so the position in pixels can be fit to a quadratic polynomial and compared to 1/2 gt2 . For vertical paths, as we used for all of our filling measurementswith (relatively) high resolution, this technique does not work. The alternative we used was to lookat the expansion of a superfluid from the lattice. The position of the interference peaks relative tothe center could be fit to 2~k × t, where t is the expansion time.

6.4.3

Tunneling during the probe time

Typically in situ imaging is done with short probe times so that there is no atomic motion
during the probe. This often requires a large intensity I ≈ Isat simply to get enough light on thecamera. We find empirically that probe times longer than ≈ 10 µs cause the cloud to appear larger.
We can quantify this a bit more by consider the heating rate from the probe beam, and the thermal
tunneling rate that it induces.
The heating rates are calculated using the photon scattering rates given by:
Γsc =

Γ

(I/Isat ),2 1 + 4(∆/Γ)2 + (I/Isat )(6.6)

where Γ is the linewidth of the electronic excited state, and ∆ is the detuning from resonance ofthe excited state. The heating rate can be calculated from this scattering rate via Ṫ = 2/3 ×(2 × Er )/kB × Γsc , where Er is the recoil energy [102]. For a probe beam of intensity I = Isat

107

Figure 6.8: The thermal hopping rate as a function of the probe time. The integral under the curve
gives a hopping probability of ≈ 15% in 4 µs.

and detuning ∆ = 0, the scattering rate is Γsc ∼ 107 s−1 , which corresponds to a heating rate of∼2 µK/µs for Rb in the |2, 2i state (∼4 µK/µs for K in |9/2, −9/2i).
Since typical optical lattice depths are ∼ 100’s of µK, it seems very possible that atoms could
undergo significant thermal hopping during the probe time. We can quantify this by approximating
the lattice as a harmonic trap and defining a thermal hopping rate Γh [266]Z ∞Γh ≈ Γsc−E/kB Tep√dE /(kB T ) = Γsc πerfc( Vlatt /kB T ).


(6.7)

Vlatt

An upper bound to this hopping rate is twice the oscillation frequency, which is the frequency of
hopping attempts.
Assuming the lattice depth is sRb = 50, the trap oscillation frequency is ωRb ≈ 20 kHz.
Further, assume the initial temperature (kinetic energy) in the lattice is slightly higher than thetemperature of the initial BEC, so Ti = 200 nK. The above equation can be used to estimate thethermal hopping rate for this case, and gives ≈ 10−6 s−1 , while the coherent tunneling rate is ≈ 0.1
s−1 . Therefore, at this temperature thermal hopping is insignificant, but it will drastically increaseas the atoms are heated during the probe pulse, and the band index in the lattice starts to becomelarge. The thermal hopping rate as a function of the probe duration is shown in Fig. 6.8, and it

108
is calculated using the heating rate from the probe beam given above. The area under this curve
gives a rough estimate of the tunneling probability during the probe time. For these parameters
this probability is ≈ 15% in 4 µs. This is sufficiently low to avoid significantly perturbing the sizeof the cloud, but it illustrates why a short probe pulse is important.

6.4.4

Molecule-specific systematics

There are also effects that arise from having two species and two atoms per site which areimportant to consider. Specifically, it is important that the presence of one species does not affectthe counting of the other species. This could happen through photoassociation loss during imaging,or by inelastic spin changing collisions, to be discussed later in this chapter. These effects will beconsidered in much more detail in Chapter 11, but here I will focus on the following point: once themolecules are dissociated into doublons (i.e. sites with one K and one Rb), we should be able tomeasure the same number and filling fraction of both species. If we cannot do this, then it will bedifficult to trust any measurements of the filling fraction of molecular clouds. References [52, 183]
focus on this point, and we were able to demonstrate that we do indeed measure the same filling
fraction and atom number of both species. Moreover, we demonstrated that we can image thespecies in either order, which further strengthens our confidence in our filling measurements of K,
Rb, and KRb.

6.5

Simulating the Rb Mott insulators
In addition to the excellent understanding of how to probe atoms in situ with high fidelity,

it is important to develop a theoretical understanding of what we expect the atomic distributions
to be in our lattice. To do this we return to the discussion in Chapter 4 of the superfluid to Mott
insulator transition. The simulations we perform are at T = 0, which is appropriate for all of ourdata. However, finite temperature corrections were considered in the group of Ana Maria Rey [228],
and are used later in the discussion of the doublon fraction.

109
6.5.1

Calculating the atomic distributions at T = 0

For a perfect, T = 0 Rb MI, we can calculate the distribution without tunneling, which isbased on the analysis of Ref. [65]. This allows us to numerically find the relationship between
the chemical potential µ0 and the particle number N , and the local chemical potential is given byµ(i, j, k) = µ0 − V (i, j, k), where (i, j, k) are the site indices for (x, y, z) and V is the harmonic
confinement. In the absense of tunneling, the occupancy n on site (i, j, k) must satisfy(n − 1) <µ(i, j, k)
≤ n,
U

(6.8)

where U is again the on site interaction between two atoms. The green staircase in Fig. 6.10a isexactly this calculation, given the uncertainty in the harmonic confinement, which is ωr = 2π×38(2)Hz and the aspect ratio of the trap is A = 6.4(1).
Figure 6.9a shows how the atoms are distributed in the lattice as a result of such a simulation.
The number of atoms integrated through the vertical direction is shown as a function of the radial
coordinates. Figure 6.9b shows the peak filling in the center of the distribution as a function of theatom number. Note that the staircase behavior is washed out because the distribution is convolved
with the imaging resolution, to be discussed next. Figure 6.9c shows a cross-section cut throughthe simulation in Fig. 6.9a, where the different colors show cross sections at varying distances fromthe center. A Gaussian fit to the center cross section is shown in Fig. 6.9d, and demonstrates what
the distribution looks like when fit with a Gaussian function.

6.5.2

Convolving the atomic distributions with the imaging resolution

We compare these simulations to experimental data by convolving them with a Gaussian filter
which has RMS width of 4.5(5) lattice sites, in accordance with our resolution. We also accountfor the pixelation due to the finite imaging magnification by mapping an area of 6 × 6 lattice sites
onto one pixel. This gives us a convolved, pixelated 2D distribution that we then fit with a 2D TF
surface to extract fRb . This is shown as the orange band in Fig. 6.10a. The width of the bandaccounts for uncertainties in the trap, the resolution, and the pixel size. We found it more rigorous

110

Figure 6.9: Simulating the Mott insulator shells. (a) The number of atoms intregrating through thevertical direction. The axes denote lattice sites, and the Mott shells are clearly apparant. (b) The
peak filling in such simulations as a function of the total atom number. Excellent agreement withexperimental data can be seen in later figures. (c) A cross-section cut through the simulation in (a).
The different colors show cross sections at varying distances from the center. (d) A Gaussian fit thecenter cross section shows what the distribution looks like when convolved with the experimental
imaging resolution.

111
to compare the pixelated and convolved theory to the data as opposed to trying to deconvolve thedata and reverse-pixelate it. Obviously a lot of information is missing, so this approach requires
an assumption of a TF distribution anyway.

6.5.3

Calculating the OD per atom

In order to compare the calculations of the number of atoms per site with the measured
ODs it is important to understand the OD per atom. The OD per area is given by τ = n × σ0 .
n = N/A is the 2D area of the cloud integrated through the z-direction. To consider a single atom,
the number is N = 1, and the area is the size of the atomic wavefunction on the lattice site. Basedon the discussion in Chapter 4, for the typical value of s = 25 Er and a = λ/2 = 532 nm, thewavefunction size γ = 75 nm in the ground band. However, during imaging there is significant
heating at a rate ≈ 1 µK/µs, so the size will quickly get much bigger. The upper limit is the size ofthe lattice site λ/2 = 532 nm, so we can assume an atom size of γ ≈ 100 − 200 nm. The absoprtion
cross section for circular polarization is given by σ0 = 3λ2 /2π, but we use linear polarization which
reduces this cross section by a factor of two. Combining these factors together provides an estimateof τ ≈ 0.5 − 1 /atom.

6.6

The atomic peak filling fractions
In Reference [183], we first studied the filling of the atomic gases separately and determinedthat we need to work with a very small MI (fewer than 5000 Rb atoms) and a large Fermi gas(more than 105 40 K atoms) to achieve fillings approaching one atom of each species per site forthe given external confinement potential (see Fig. 6.10). For the Fermi gas we reached the band
insulator limit where the filling is saturated beyond 105 40 K atoms where the Fermi gas is muchlarger than the Bose gas (see Fig. 6.5).
A band insulator at T = 0 is characterized as an incompressible many-body state of spinpolarized fermions in the lattice where every site has a fermion, and thus tunneling is suppressedby the Pauli exclusion principle [138]. While T = 0 is an appropriate approximation for Rb, the112

Figure 6.10: The filling fractions vs. their respective atom numbers. (a) The peak filling (filling in
the center of the atomic distribution) of Rb as a function of the Rb number. The green lines show
the zero temperature theory for filling, and the orange band shows the same theory predictions
under a finite imaging resolution realized in the experiment. (b) The peak filling in the lattice(blue circles, left vertical axis) and temperature of 40 K prior to loading the lattice (red circles, rightvertical axis) as function of 40 K number. The temperature is normalized to the Fermi temperatureof the 40 K gas in the optical harmonic trap. Reproduced from Reference [183].

113
Fermi gas is relatively hot and the band insulator is thus riddled with thermal holes. Thus, thefilling fraction saturates at only ≈ 80 − 90% even for the band insulator.

6.7

Measuring the temperature of K
It is useful to understand what the temperature of the K gas is before loading into lattice,as this will ultimately determine the filling fraction that we can reach. In general it is difficult
to accurately measure the temperature of a degenerate Fermi gas (DFG) because its profile inexpansion deviates from a Gaussian at low temperatures. Therefore, a poly-log fitting function
with a fugacity must be used [218]. Alternatively, the cloud in expansion can be truncated toremove atoms within ≈ 50% of the maximum OD, and then the wings can be fit to a Gaussian
to obtain the temperature, since the deviation of a Fermi-Dirac distribution from a Gaussian ismaximized at the center. Then the number can be obtained by fitting the entire cloud, from which
TF = ~ω̄/kB · (6N )1/3 can be calculated [218]. ω̄ is the geometric mean trap frequency.
While we used both approaches and found that they give consistent results, we preferred touse a third approach in which we use Rb as a thermometer of the K cloud. To do this we holdthe Rb BEC and the K DFG in the trap with finite inter-species interaction. Within 100’s of ms
the BEC will melt, and equilibrate with the hotter K gas. This temperature can be measured very
accurately, and we wait until the BEC fraction becomes < 20%. This gives the T for K, and N for
K can be measured by fitting the entire K gas, thus giving T /TF .
Figure 6.10b shows the temperature of K as well as the peak filling as a function of the
K number. The trend in T /TF is actually very strongly anticorrelated with the trend in fK .
We understand this phenomena by considering that when the trap is very weak at the end of
evaporation, the Rb BEC fails to cool the DFG further. Nevertheless, TF ∼ N 1/3 , and thus T /TF
continues to go down for increasing NK until it saturates at ≈ 2 × 105 at a value of T /TF = 0.3.
The entropy per particle in this Fermi gas can be calculated and compared with the entropy
per particle that corresponds to 80 − 90% filling in the lattice. The latter is calculated using
S/N = −kB /f · [f ln(f ) + (1 − f )ln(1 − f )] [183], and gives 0.6kB . Using the finite temperature

114
calculations for the K and Rb gases loaded into the lattice in Ref. [228], we find that our intial T /TF
is in good agreement with filling fraction in the lattice. This suggests that our loading procedureof ramping up the lattices with a smooth, s-shaped function in 200 ms is sufficiently adiabatic.
Further cooling of the DFG would require more efficient thermalization at the bottom of thetrap, which is very challanging once the Rb BEC becomes quite pure. An alternative which weare considering for the new experimental apparatus is to have K efficiently cool itself at the end of
evaporation by trasnferring some of the atoms to |9/2, −7/2i so that s-wave collisions can be used
for K to thermalize with itself, and then we can evaporate K as well as Rb by selectively removing
the hottest K atoms using a combination of gravity and a magnetic field gradient. We have nevertried this in the old apparatus, but we intend to return this idea in the new apparatus in an effort
to reach > 90% filling of K in the lattice.

6.8

The role of interspecies interactions
In order to preserve the filling of the Rb MI in the presence of such a large 40 K cloud, it

is imperative to turn the interspecies interactions off by loading the lattice at aK-Rb = 0. This isshown in Fig. 6.11. The top shows the filling fraction of Rb in the lattice normalized to what it
is without K as a function of the scattering length. The background scattering length is shown as
the vertical dotted line, so that the data on the left of the line is above the resonance. The bottom
shows the BEC fraction with K prior to loading the lattice also vs. the scattering length. The redband shows the BEC fraction of Rb alone.
Note the clear correlation between the two curves, which suggests that the interaction between
K and Rb deteriorates the BEC and thus has deleterious effects on the Rb filling fraction. We
attribute this heating of the BEC to the large heat load of the K Fermi gas which is at a relatively
high temperature of T = 0.3TF . Once Rb forms a BEC, it thermalizes with K very poorly, and the
K number is ∼ 50× the Rb number for this data. Thus, enabling interactions before loading thelattice melts the BEC and distorts the filling for a trivial reason. While we have studied the effects
of K on the superfluid to Mott insulator transition of Rb as shown in Chapter 4, we do not have115
a definitive understanding of the role of interactions [19], even though they are expected to play asignificant role in the lattice loading [82, 228]. Thus, we work at zero K-Rb scattering length.
However, the magnetoassociation proceeds by ramping the magnetic field from high to low
values across the Feshbach resonance, while the location of aK-Rb = 0 is below the resonance (seeFig. 6.12a). Hence, before we can perform magnetoassociation we need to first ramp the magnetic
field from where aK-Rb = 0 is located to above the resonance. To avoid populating higher bands
when we do this, we first transferred 40 K to the |F, mF i = |9/2, −7/2i state to avoid the Feshbachresonance. This procedure is depicted in Fig. 6.12a. We then ramped the B field above theresonance, and transfer the 40 K atoms back to the |9/2, −9/2i state. Finally we converted the
K-Rb pairs into Feshbach molecules on each site and we studied the fraction of Rb atoms that wereconverted to molecules (Rb is the minority species).

6.9

Maximizing the doublon fraction
In the limit of a small Rb atom number, we converted more than 50% of the Rb to FbM(Fig. 6.12b). We then produced ground-state molecules and removed all unpaired atoms from thelattice. The final number of ground-state molecules can be determined by reversing the transfer
process and then performing the normal absorption imaging on atoms. Typical in situ images areshown in Fig. 6.12c. In the case of high conversion, we achieved molecule fillings of at least 25%,
which is significantly higher than our previous work [110, 273]. With some quick experimental
improvements we should be able to increase the filling even further.
This filling fraction is at the percolation threshold, where every molecule would be connected
to every other molecule in the entire lattice [183], and it marks the first time that the polar
molecules have entered the quantum degenerate regime, with an entropy per particle of 2.2kB(kB is Boltzmann’s constant). Studying the spin dynamics described in the previous section inthis higher-filled lattice will be the subject of future work. We note that a similar procedure has
recently been pursued in the RbCs experiment in Innsbruck, where a Rb superfluid was moved tooverlap a Cs Mott insulator before increasing the lattice depth to produce a dual Mott insulator.

116

Figure 6.11: The effect of K on the BEC and filling fraction of the intial Rb. The top shows thefilling fraction of Rb in the lattice with normalized to what it is without K as a function of thescattering length. The background scattering length is shown as the vertical dotted line, so thatthe data on the left of the line is above the resonance. The bottom shows the BEC fraction with
K prior to loading the lattice also vs. the scattering length. The red band shows the BEC fractionof Rb alone. Note the clear correlation between the two curves. Reproduced from Reference [183].

117
This technique has lead to a gas of RbCs Feshbach molecules with 30% filling [221], but has not
yet been combined with STIRAP to make ground-state molecules.

6.10

An alternative measurement of the doublon fraction
The previous studies suggest that ≈ 50 − 60% of the Rb atoms are on a doublon site, but

the actual conversion should only be limited by the filling of K, which is 80 − 90%. Therefore it isunclear why the FbM filling, or equivalently the conversion from Rb to FbM, is low. To address
this question we have developed an alternative method for measuring the doublon fraction. Insteadof associating doublons into FbM, we can turn on strong inelastic loss which causes doublons todecay in a few ms. This is done by flipping the Rb spin from |1, 1i to |2, 2i, which is h × 8.1 GHz
higher in energy at 550 G. Since Rb and K (|9/2, −9/2i) are stretched on opposite sides, angularmomentum conservation does not help, and energy conservation is the only law which makes the
Feshbach-resonant mixture stable since |1, 1i and |9/2, −9/2i are the lowest energy states. Onceh × 8.1 GHz of internal energy is added by flipping Rb to |2, 2i, the system decays very quickly,
with β = 6 × 10−12 cm3 s−1 [52].
By looking at the number of atoms lost from this inelastic decay, we can learn the number ofdoublons. Figure 6.13 shows how these measurements are made. Fig. 6.13a shows a Mott insulatorof Rb in the lattice with NRb = 2−3×104 , and Fig. 6.13b shows a band insulator of K in the latticewith NK ≈ 2 × 105 . Once the inelastic loss is initiated, all the doublons are lost and a hole appears
in the center of the K cloud which is the same size as the Rb cloud, as shown in Fig. 6.13c. Thismeasurement also serves to verify that the two clouds are well overlaps; an obvious but non-trivial
requirement.
Such images contain a wealth of information, most of which can be gleaned by counting the
Rb number before and after the inelastic loss as a function of time. To describe this idea, let us
first assume the filling of Rb is fRb ≤ 1. Then, the fraction of Rb in a doublon will be equal to thefilling fraction of K. In Fig. 6.14a we consider this case, and once we turn on the inelastic loss we seethat ≈ 80% of the Rb is lost within τ = 2 ms (Vlatt = 25 ErRb ). This suggests that 80% of the Rb

118

Figure 6.12: (a) The K-Rb scattering length as a function of the magnetic field. Initially the
K atoms are prepared in the resonant state |9/2, −9/2i. We spin flip the K atoms to the nonresonant |9/2, −7/2i state, before we sweep the magnetic field to above the Feshbach resonance,
at which time we drive the K atoms back to the |9/2, −9/2i state. We can then sweep the field tobelow the resonance to make Feshbach molecules. (b) The number of Feshbach molecules produced
normalized to the initial Rb number versus the number of Rb atoms. Theory band corresponds tocalculations based on the zero temperature curve in Fig. 6.10a. (c) Images of clouds of ground-state
molecules in the optical lattice for high and low filling fraction. Reproduced from Reference [183].

Figure 6.13: Inelastic collisions between K and Rb causes loss of doublons. (a) An in situ imageof a Mott insulator of ∼ 2 × 104 Rb atoms in an optical lattice. (b) An in situ image of a band
insulator of K in an optical lattice. (c) An image of K after the inelastic loss of doublons. The holein the middle represents the location of the Rb atoms and hence the doublons. Reproduced from
Reference [183].

119
were doublons and 20% were alone. Therefore, this measurement corroborates the measurement offK discussed above by in situ imaging of the K.
The residual decay on the longer timescale is also very interesting though we did not study
it thoroughly. Since the K cloud is larger than the Rb cloud, the Rb atoms that remain after the
doublon decay are alone inside a shell of K atoms. For Vlatt = 25 ErRb , the lattice depth is only
0 = 90 Hz. Therefore, the K tunnels around quickly and
10 ErK , and so the K tunneling rate is JK

can tunnel onto sites with an immobile Rb, which are then lost quickly before the K can tunnel
again. While this setting sounds as though it could give rise to the continuous quantum Zenomechanism which would prevent the K from tunneling onto sites occupied with Rb (similarly towhat we observed with a spin mixture of molecules as discussed in Chapter 4), the on-site inelastic
loss rate is too low for the Zeno regime.
Let us now consider the case when fRb > 1, where sites with two Rb atoms and one K atom
will be common. The loss on these sites will still be dominated by pairwise loss of a K and a Rb.
Indeed, three-body loss of Rb does not become significant until we reach four Rb per site [35].
Therefore, we expect the inelastic loss of Rb to be less than fK . This is shown in Fig. 6.14b, where
the loss is ≈ 50%. This trend can be measured for all Rb numbers, and is shown in Fig. 6.14c.
Notice that the data indicates a corner where the fraction lost drops below fK , and this corner
exactly matches the onset of significant double occupancy of Rb.
This technique tells us both the filling fraction of K and the number of Rb which corresponds
to the onset of double occupancies, and both of which are in excellent agreement with the in situ
measurements discussed above. The curves in Fig. 6.14a&b are dashed because they are simply
double exponential fits. The curve in Fig. 6.14c, however, is a finite temperature simulation of the
Rb distribution. The width is due to the uncertainty in the harmonic trapping frequency and theuncertainty in the trap aspect ratio, and the fitting parameter is fK . Note that the data starts todeviate from theory when fRb ≈ 4 at NRb ≈ 105 because three-body loss of Rb becomes significant
and serves to overestimate the fraction lost.
This doublon decay technique confirms that indeed 80% of the Rb atoms are in doublons,120

Figure 6.14: An alternative method for determining the atomic filling fractions. (a) and (b) show
the loss of Rb from inelastic collision with K as a function of time for low and intermediate number,respectively. (c) shows the fraction lost in ∼ 2 ms as a function of the intial Rb number. Reproduced
from Reference [52].

121
and so when fRb = 1, the doublon filling fraction should be 0.8. Yet, we only convert ≈ 50%of the Rb to Feshbach molecules. This discrepency is very important to understand, and could
originate from two sources. First, since the tunneling rate of K is high and there is an enormous
number of unpaired K atoms, it is possible that some K atoms are tunneling onto sites with FbM
and then quickly undergoing inelastic decay. The other possibility is that the efficiency with whicha doublon can be converted to a FbM is less than unity. We believe the latter effect to play amore significant role for these particular experiments, and so we must devise a way to measure themagneto-association efficiency of doublons in the optical lattice (however, the former problem could
be solved using a colder K cloud which requires less excess K atoms to enter the band insulating
regime).

6.11

Magneto-association efficiency on lattice sites
Even if every site has one 40 K and one Rb (i.e. a doublon), a highly filled lattice of molecules

can only be achieved if the conversion of doublons to Feshbach molecules is very efficient. We
already know that the STIRAP efficiency of converting Feshbach molecules to the ground state isabout 90%. So we set out to develop another experimental technique to allow us to investigate thisFeshbach molecule conversion efficiency.
Once the KRb molecules are produced in the ground state, unpaired atoms can be removed
with resonant light. In principle, this atom-removal process does not affect the molecules. However,the fact that a large number of 40 K atoms is required and that KRb molecules can chemically reactwith 40 K if they encounter each other at short range does lead to a loss of a small fraction ofground state molecules. Nevertheless, after the free atoms are removed, the 3D optical lattice
now has sites that are either empty or contain one ground-state molecule. The STIRAP process
can then be reversed, which leads to sites that are either empty or contain a Fesbach molecule.
Then, the field is swept back above the Feshbach resonance to dissociate the molecules, leading toa scenario where lattice sites are either empty or contain a Bose-Fermi atomic pair, referred to as
a doublon. This process is outlined in Fig. 6.15, and the preparation of this clean initial condition

122
can be used to study the conversion efficiency of doublons into Feshbach molecules.
We have identified effects from a narrow, higher order d -wave Feshbach resonance near thebroad s-wave resonance we use for molecule association (shown in Fig. 6.16a). In the doublon
preparation step, the speed of the dissociation sweep from below to above the resonance is extremely
important. Figure 6.16b shows the consequences of the d -wave resonance. The dissociation starts
in the green channel at (1). Then, the rate of the magnetic field ramp determines whether themolecule converts to doublons in (2) or (3). The case of (2) is when the d -wave resonance is crossed
diabatically, and the FbM are connected to a doublon in the ground band of the optical lattice.
However, when the ramp is too slow and the d -wave resonance is crossed adiabatically, the FbM
connect to (3) where there is a motional excitation in the doublon, and a weighted superpositionof K and Rb is in the first excited band in the optical lattice.
Then, the Feshbach association measurement is conducted by sweeping from above the resonance back to the other side of the s-wave resonance, and this is done very quickly (≈ 10 − 20G/ms). For ground-band doublons in (2), the fast sweep is diabatic with respect to the d -wave
resonance, and they are converted to s-wave FbM. For the excited-band doublons in (3) the fast
ramp rate back below the resonance is also diabatic with respect to the d -wave resonance and the
doublons are not connected to any FbM, but rather a doublon in the ground band.
The results of this measurement, where the ramp rate from below to above is varied andthe ramp rate from above back to below is very fast, are shown in Fig. 6.16c. This measurement
is fitted with a Landau-Zener description P = e−A/|Ḃ| , where P is the probability of creating ans-wave FbM, Ḃ is the rate of the magnetic field ramp, and√4 3ωHO |abg ∆d |A=,
LHO
where ωHO is the harmonic oscillator trapping frequency on a lattice site, LHO =(6.9)
p~/(µωHO ) is the

harmonic oscillator length with the doublon reduced mass µ, abg = −187(5)a0 is the backgroundscattering length of the s-wave resonance, and ∆d is the width of the d -wave resonance [52]. Thisdata provides the most accurate measurement of the width of this resonance, |∆d | = 9.3(7) mG [52].

123

Preparation

Measurement

Bhold
Ḃ

res

Bs
STIRAPK rf pulse

Doublon Dynamics

t

τ

0

JK

0

JK1

UK-Rb

Figure 6.15: The experimental setup for producing K-Rb doublons in an optical lattice. The firststep is the preparation, where a pure sample of molecules in the optical lattice are turned into
sites with one K and one Rb using the STIRAP sequence. We then let these doublons evolvein the lattice for a variable time at a variable scattering length. During the evolution, both K
and Rb can tunnel (potentially together), but for short evolution times only K will tunnel since itfeels a weaker lattice due its smaller mass. After the evolution period, we associate doublons thathave not fallen apart back into Feshbach molecules, which can be spectroscopically differentiatedin the measurement from unpaired K atoms that have separated from their doublon Rb partner.
Reproduced from Reference [52].

124
In Fig. 6.16d we ramp slowly to various fields to identify to position of the resonance as 547.47(1)G. This is the most accurate determination of its position.
Historically, the ramp rate we have always used in the lattice is 3.4 G/ms (0.34 mT/ms),and thus this narrow resonance was causing our Feshbach molecule creation efficiency to be 70%
since we first produced molecules in an optical lattice [47]. When we actually make FbM we sweep
from above to below, and so ≈ 30% of the FbM molecules produced in our experiment have always
been d -wave FbM. These FbM will stay dark during the subsequent STIRAP sequence because
their initial rotational angular momentum is L = 2~. As a result of this, of the doublons which
account for 80% of the Rb, only 50 − 60% are converted to s-wave FbM, which is consistent with
our measurements.
By simply ramping across the resonance sufficiently fast so that we avoid making d -wave
molecules adiabatically, we are able to solve this problem. With this mechanism understood, we
anticipate 100% conversion efficiency for all future experiments. This improvement alone should
allow us to reach GSM fillings of fGSM > 0.5. I want to emphasize that since the d -wave resonanceis only 9 mG wide, it would play no role in molecule formation in a harmonic dipole trap. However,since the on-site density in an optical lattice is so high, the timescales for keeping the sweep diabaticwith respect to this resonance are much faster. We believe this to be the only investigation to dateof narrow heteronuclear Feshbach resonances in an optical lattice. Note, however, that similar
resonances have been explored in Cs2 , and navigation through the state manifold was carried out
in Ref. [58].

6.12

Atomic tunneling - doublon dynamics
We have also investigated tunneling dynamics and interaction effects between the bosonicand fermionic atoms composing the doublons. Tunneling dynamics of doublons in the lattice canalso affect molecule production. In the quantum synthesis approach described above, achieving ahigh lattice filling for molecules requires not only the preparation of a large fraction of lattice sitesthat have doublons, but also that these doublons are not lost due to tunneling and/or collsions

125

Figure 6.16: Characterizing the d-wave resonance. (a) shows the scattering length as a function ofmagnetic field, for both the s-wave and d-wave resonance. (b) shows the adiabatic mapping of thebound and unbound states below the resonances to their corresponding harmonic oscillator statesin the lattice above the resonance. (c) shows the probability of crossing the d -wave resonancediabatically (i.e. not making a d-wave molecule) as a function of the ramp speed. (d) shows
the same efficiency as a function of the field in order to identify its center. Reproduced from
Reference [52].

126

Figure 6.17: K tunneling dynamics. (a) The survival probability for doublons in a lattice for 1 ms
as a function of lattice depth, shown for several intra-doublon interaction strengths. (b) The same
survival probability as a function of scattering length for several lattice depths. The solid lines
represent single doublon calculations, while the dotted lines show a lattice with ∼ 15% doublon
filling, where the doublons can interact and enhance their survival. Reproduced from Reference [52].

prior to conversion to molecules. At λlatt = 1064 nm, K feels a lattice depth that, in units ofrecoil energy, is 2.6 times weaker than for Rb due to differences in atomic mass and polarizability.
Consequently, K tunnels faster than Rb.
Figure 6.17 illustrates doublon dynamics due to the interplay between tunneling and interactions, which we control by varying the lattice depth and interspecies scattering length aK-Rb
through Bhold . We note that for aK-Rb > −850a0 , the B sweep crosses the d-wave resonance witha Ḃ that varies from 5 to 19 G/ms, and so the data in Fig. 6.17 has been multiplied by a factor
that increases the doublon fraction to account for the finite Ḃ when crossing the d-wave resonance.
Figure 6.17a shows the effect of the lattice depth for τ = 1 ms at three different valuesof Bhold , corresponding to different values of aK-Rb . This timescale is relevant for both molecule
production and K tunneling dynamics. We observe that the remaining doublon fraction is highly
sensitive to the lattice depth for weak interspecies interactions, e.g. aK-Rb = −220a0 , with a lower
doublon fraction for shallower lattices that exhibit higher tunneling rates. For stronger interactions
the dependence on lattice depth becomes less significant and almost disappears in the stronglyinteracting regime, e.g aK-Rb = −1900a0 . Similar behavior is observed if we fix the lattice depth
but vary the interspecies interaction, as shown in Fig. 6.17b.

127
The data in Fig. 6.17 clearly shows evidence of decay of doublons due to tunneling that is
affected by both the lattice depth and interspecies interactions. We can model doublon dynamics
with the following Hamiltonian: [52]0H = −JRb

X †
X η †
X η
X
U0
ai aj −
JK ci,η cj,η +UKRb n0Rb,i nηK,i + RbRb
n0Rb,i (n0Rb,i − 1),2hi,jiη,hi,jii,η

(6.10)iwhere η = 0 (1) denote the ground (first) excited band in the lattice. The first and second terms
are the kinetic energy of the K and Rb atoms, respectively. Here, ai a†i is the bosonic annihilation(creation) operator for a Rb atom at the lattie site i in the ground band, and ci c†i is the fermionicannihilation (creation) operator for a K atom at lattice site i and band η. We use hi, ji to indicate
nearest-neighbor hopping between sites i and j with matrix element Jαη , where α = K or Rb. We
assume η = 0 for Rb throughout, and only K in the excited band in considered. The third termη
. The last term is thedescribes the inter-species on-site interactions with matrix element UKRb
0
, with n0Rb,ion-site intra-species interaction between ground-band Rb atoms with strength URbRb

as the occupation of site i.
The solid lines in Fig. 6.17 show the calculations based on this Hamiltonian, where we have0 = 0. We start with a single doublon, evolve the K for a holdneglected Rb tunneling by setting JRb

time τ , and then extract the doublon fraction from the probability that the K atom remains on thesame site as the Rb atom. In this treatment we ignore the role of the magnetic-field sweeps. Cal2
0 /U 0[52],
culations for a single doublon (solid lines), where the initial decay scales as 1 − 12 JK
KRbagree well with the data, except at doublon fractions below ∼ 30%, where the disagreement arises
from the finite probability in the experiment that a K atom finds a different Rb partner.
The doublon fraction where this starts to matter depends on the doublon filling fraction, andcould present another way to measure the FbM or GSM filling fractions. Simulating a Gaussian
distribution of doublons with 10% peak filling, which is appropriate for this data, accounts for thiseffect. This data is shown as the dashed lines, and the good agreement of these calculations with thedata shows that tunneling of K, which is suppressed in deeper lattices, is the dominant mechanism
for the reduction of the doublon fraction at short (∼ 1 ms) times. The on-site interaction with Rb

128
suppresses the K tunneling when the interaction energy becomes larger than the width of the K
Bloch band [52, 113].
Moreover, if sufficient tunneling time is allowed, we can even observe effects of doublondoublon interactions. In Fig. 6.18 we present data taken for τ up to 40 ms in order to look for theeffects of Rb tunneling. Measurements of the remaining doublon fraction are shown for two lattice
depths (10Er and 15Er ) and two values of aK-Rb (−910a0 and −1900a0 ). The doublon fraction
has been normalized by the measured value for τ = 1 ms in order to remove the effect of theshorter-time dynamics discussed above. Similar to the shorter-time dynamics, we see a reductionin the doublon fraction at long hold times which is suppressed by a deeper lattice and stronger
interspecies interactions.
Modeling these dynamics is theoretically challanging, and the lines in Fig. 6.18 are exponential
fits that are intended only as guides to the eye. Compared to doublons composed of identicalbosons [271] or fermions in two spin states [244], the heteronuclear system has the additional
complexities of two particle masses, two tunneling rates, and two relevant interaction energies. For
example, for large aK-Rb the interspecies interactions will strongly suppress Rb tunneling from adoublon to a neighboring empty site. Similarly, tunneling of a doublon to an empty site is a slow
0 J 0 /U 00second-order process at the rate Jpair = 2JRbK
KRb due to the energy gap of UKRb , as is shownin the inset.
However, Rb tunneling between two neighboring doublons, which creates a triplon (RbRb-K) on one site and lone K on the other, may occur on a faster timescale due to a much0
smaller energy gap URbRb, which is smaller than the K tunneling bandwidth. While the theoreticaldescription is complicated, we observe that the time scale of the doublon decay roughly matches
1/(2πJpair ). Therefore, we believe that the loss of doublons on longer timescales comes fromneighboring doublons, and the pair-hopping of next-nearest-neighbor doublons is the limiting step
at long times when nearest-neighbor doublons have already interacted. This type of system could
be very interesting for the study of quasi-crystallization and many-body localization in an opticallattice [52, 92, 233, 238].

129

Figure 6.18: K and Rb tunneling dynamics. The survival probability for doublons being held in thelattice for a variable time. Several combinations of lattice depth and interaction strength are shown,and the date is fit to simple exponentials. Note that the doublon fraction has been renormalized
to the corresponding number in Fig. 6.17 to highlight the additional loss upon waiting longer
times, which is due to Rb. The inset shows processes by which this additional loss can occur, anddemonstrate a qualitative agreement with our data. Reproduced from Reference [52].

130

6.13

The role of higher bands in the lattice
When studying doublon dynamics measured for two different initial atom conditions, we finddirect evidence for excited-band molecules. We compare results for our usual molecule preparation
using atomic insulators to a case where we start with a hotter initial atom gas mixture at atemperature above Tc or TF (i.e. thermal gases). Using the band mapping technique discussed in
Chapter 4, we measure the initial population of K in the ground band and the first excited band,as shown in Fig. 6.19b. For this data we found that 11(2)% of the K atoms occupy the first excited
band for the colder intial gas (i), and 31(6)% of the K atoms occupy the first excited band for thesignificantly hotter intial gas (ii).
When looking at the doublon dynamics for these two cases (Fig. 6.19a) we observe a lower
doublon fraction for the hotter initial gas for Vlatt ≤ 25 ErRb . These data are taken for 16.8
G/ms sweeps, τ = 1 ms, and aK-Rb = −220a0 . The lower doublon fraction can be explained
0 /h and J 1 /h are 89.3 and 1110
by excited-band K atoms, which have a high tunneling rate (JKKHz, respectively, for Vlatt = 25 ErRb ). The presence of excited-band K atoms suggests that the B
sweeps for magneto-association (and dissociation) couple excited-band K atoms (plus a ground-band
Rb atom) to excited-band Feshbach molecules. Moreover, the data suggests that the conversion
efficiency for the excited-band Feshbach molecules is still high for Vlatt ≤ 25 ErRb . This can be seenby the observed difference (roughly 20%) in the doublon fraction, which matches the difference inexcited-band fraction of K between the two cases (as shown in Fig. 6.19a).
Since we prepare the doublons directly from the dissociated GSM, these results further indicate that a polar molecule sample prepared from a fintie-temperature gas can contain a small
fraction of molecules in an excited motional state in the lattice. We also observe that a Rb excitedband population of 31(5)% after loading the thermal gas in the lattice; however, even in the excited
band, the off-resonant Rb tunneling is slow compared to the 1-ms timescale of these measurements.
The green dashed curve in Fig. 6.19a shows the theoretical results for a K excited-band fraction of
24%. For comparison, the red solid curve, which is the same as the red curve in Fig. 6.17a, includes

131

Figure 6.19: The role of higher bands. (a) The doublon survival probability for a 1 ms hold vs.
lattice depth with aK-Rb = −220 a0 . The Case i (red) data has a very low fraction of K atoms in
excited bands prior to making molecules in the lattice. (b) The corresponding band mapping of theinitial K, where a larger excited band fraction is shown in Case ii (green). The green data in (a)is clearly lower, demonstrating a strong correlation between initial K in higher bands, molecules inhigher bands, and dissociated molecules in higher bands leading to K in higher bands. The green
dotted curve is the same as the red, except that the tunning rate of ∼ 20% (extracted from thedata in (b) Case ii) of the doublons is the higher band value instead of the ground band value.
Reproduced from Reference [52].

132
no excited-band population.
The estimated excited band fraction ignores the effects of harmonic confinement on tunneling,which are more significant for the hotter initial atom gas, where the resulting molecular cloud is also
larger. For the hotter initial atom gas, the green dashed curve overlaps the data at the shallower
lattice depths, but deviates from the measured doublon fraction at larger lattice depths (the excited
band fraction of the initial K gas is independent of lattice depth). This may be expected since in thelimit of a very deep lattice and a fully adiabatic magneto-association sweep, one expects that only
the heavier atom (Rb) in excited bands (plus a ground-band K atom) will couple to excited-band
Feshbach molecules.
Further evidence for the correlation between excited-band atoms in the initial K gas andexcited-band molecules created from it is shown in Fig. 6.20, where the measured KRb excitedband fraction is plotted versus the excited-band fraction of the K gases from which it was createdfor several temperatures of the initial atomic gases. Sample images of the band-mapped K andKRb clouds are shown for very low and very high initial temperatures. This trend corroborates
all the indirect measurements based on doublon dynamics. However, the slope of the fitting line
cannot be explained by the above reasoning based on the harmonic oscillator levels for excited K
and Rb atoms in the lattice. Nevertheless, it is clear that the temperature of the initial atomic
gases is highly correlated with excited-band fraction of the GSM and FbM they create.
It is also possible that even if all the FbM are in the ground band, the two-photon STIRAP
sequence couples them to a GSM in an excited band. Typical STIRAP linewidths are ∼ 200 kHz,
which is significantly larger than the band spacing of ∼ 20 kHz, and so the band structure is fully
unresolved. To understand this possibility we must consider the Lamb-Dicke factors for the lattice
depth typically used to make molecules, which is Vlatt = 25 ErRb = 40 ErKRb . We define the wave
vectors of the two Raman beams as ~ku and ~kd for the up leg and the down leg, respectively. The~~

momentum transfer scales as ∼ ei(ku −kd )·~r .
~ this momentum transfer is k~ξ = (~ku −~kd )·ξ.
~ Moreover, theFor excited bands in the direction ξ,
lattice spacing is λ/2 = a = 532 nm. For the parameters typically used for STIRAP, we find that our

133

Figure 6.20: Correlation between K and KRb in higher bands. The fraction of K atoms in higher
bands initially loaded into the lattice is correlated with the number of molecules in higher bands
in the lattice that were synthesized from these atoms. Sample band mapping images are shown for
K and KRb for both low and high excited band fractions.

134
1/4

KRb /E KRb [228].
experiment is well within the Lamb-Dicke regime kξ a/sKRb  1, where sKRb = Vlattr

Therefore, the total population in the first excited band due to STIRAP is ∼ 1% [228]. This is forbeams co-propagating at a 45 degree angle with respect to the x and y lattice axes and a 90 degree
angle with respect to the z lattice axis. In the new apparatus (discussed in Chapter 7) the Raman
beam will be co-propagating with a 0 degree angle with respect to the z lattice axia and 90 degrees
with respect to the x and y lattice axes. Nevertheless, this probability is still be negligibly small.

6.14

Photoassociation of doublons in the lattice
Another question we addressed with doublons is whether the FbM step could possibly becircumvented by efficiently photo-associating doublons to a bound state in the electronically excitedpotential, which could be coupled to the ground state. If it were possible to do this with high fidelity,
we could even imagine doing STIRAP straight from a doublon to a GSM. To address this question
we study the effects of the up leg Raman beam on doublons in the lattice. Figure 6.21a shows
the normal up leg transfer from FbM to the excited intermediate state of the STIRAP transition,
which is dark to atomic detection. The pulse shown is ≈ 10 µs, which is consistent with the Rabi
frequency of that transition and the transition dipole matrix element discussed in Chapter 2.
Figure 6.21b shows the same curve except with doublons instead of FbM. While the widthis similar, the center frequency has shifted by ≈ 400 kHz, which is consistent with the binding
energy of the FbM at the field that we typically use. The pulse time in Fig. 6.21b is τ ≈ 1 ms,
which indicates that the Rabi frequency is much lower. Figure 6.21c shows the fraction of doublons
lost as a function of time, and it shows that the characteristic pulse time is τ ≈ 1 ms. However,this timescale is approaching the tunneling rate of K in the lattice, and so it is difficult to gain aquantitative understanding about the Rabi frequency of the photo-association pulse.
Nevertheless, it is parhaps not surprising that the Rabi frequency would be so low since theFranck-Condon overlap of the doublon center-of-mass wavefunction and the intermediate STIRAP
state is very poor. The molecular state of the FbM is very highly excited in the vibrational
degree of freedom, so the wavefunction that describes the internuclear separation has many nodes.

135

Figure 6.21: Photoassociation of doublons in the lattice. (a) The lineshape of the up leg pulse on
Feshbach molecules, taking them to a dark state which is not detected. (b) The up leg pulse on
doublons (not associated to Feshbach molecules). The pulse times are very different between thetwo, reflecting the very different Franck-Condon Factors. Moreover, the frequency has shifted by
400 kHz, which matches the binding energy of the Feshbach molecules at this field. (c) the doublon
photoassociation loss as a function of the up leg pulse time. Note that the time in (a) is ∼ µs. The
loss timescale here is incredibly long, suggesting that the Rabi frequency is incredibly small. Also
note that the loss timescale matches the K tunneling rate in the lattice.

Conversely, if both K and Rb are in the ground band of the lattice their motional wavefunctions
have no nodes. Even though doublons adiabatically connect to FbM, their wavefunctions are quite
different in size and shape.
Although the strong confinement of the lattice alters the properties of the Feshbach resonance and can lead to so-called confinement induced resonances [109], the Franck-Condon overlap
between such doublons and bound molecular states in the excited molecular potential is very small.
Therefore, direct photo-association as a first step of a Raman transition to GSM is not practical.
It is worth pointing out, however, that the situation may be much more favorable in microtraps,
called tweezers, rather than an optical lattice because typical oscillation frequencies are ∼MHz
rather than ∼ 10’s of kHz. This question is now being explored at Harvard [126, 161] and is quickly
attracting more interest.

Chapter 7

The new apparatus - enhanced optical and electric manipulation of ultracold
polar molecules

Ultracold polar molecules have long been heralded as an excellent toolbox for probing manybody long-range interacting systems [36, 175] as well as quantum information [66]. Recently, dipolar spin-exchange interactions and many-body dynamics have been observed with fermionic KRb
molecules in an optical lattice [110, 273] as described in Chapter 5, and low-entropy samples ina lattice have been realized for KRb ground-state molecules [52, 183] (described in Chapter 6)
and RbCs weakly-bound Feshbach molecules [221]. Future work towards realizing the XXZ modelof quantum magnetism will require a large DC electric field to control the strength of the Isinginteraction [95, 112, 263]. Moreover, studying many-body non-equilibrium dynamics will require
high resolution in situ detection and addressing [267].
Several groups are working to implement these tools in experiments with polar molecules [89,103], although there are few published results to date. Further, there are no published results to ourknowledge that have yet demonstrated ultracold polar molecules in such an advanced apparatus.
The challanges of implementing such an apparatus include the already complex nature of ultracold
polar molecules which often require large magnetic fields to reach a Feshbach resonance [43, 285],
combined with the seemingly conflicting requirements of large, versatile electric fields and highresolution optical detection and addressing.
In terms of the fields required to nearly saturate the dipole moments of bi-alkalis, KRb isparticulatly demanding. For KRb d = 0.566 Debye [190], but another figure of merit is B/d, where

137

Figure 7.1: The dipole moments of the bi-alkalis as a function of the electric field. The top plot shows
reactive species, and the bottom plot shows nonreactive species. Reproduced from Reference [215].

138
B is the rotational constant, which describes the characteristic electric field. For KRb this valueis ≈ 4 kV/cm, while for e.g. NaK it is ≈ 2 kV/cm [203, 262], which is thus easier to polarize.
Concomitantly, to reach > 80% of d for KRb, a field of ≈ 20 − 30 kV/cm is required. Most of
the other molecules being pursued have larger d and smaller B/d, so they can reach larger dipole
moments at smaller fields [215]. This is true for NaRb, NaK, RbCs, KCs, and NaCs, as shown in
Fig. 7.1.
In this chapter, I describe our approach to combining all the tools together that are needed
for large electric fields with high resolution detection and addressing. I begin by describing theelectrode system, which allows for large, stable, homogeneous electric fields that simultaneously
allow arbitrary gradients in two dimensions (2D). I go on to describe how we can couple AC
microwave frequencies onto the electrodes to drive rotational transitions with high polarization
fidelity. I then describe the high resolution imaging system.

7.1

Electrode design for large, versatile electric fields
While there has been an enormous amount of work in the cold molecule community withlarge electric fields [20, 23, 259], designs that are compatible with ultracold atom technology andhigh resolution optical control remain unexplored. In order to reach such large electric fields ina manner than allows both homogeneous fields as well as arbitrary gradients while still providing
excellent optical access, our design is based on four tungsten rods between two Indium Tin Oxide(ITO) coated fused silica plates. This electrode assembly is located inside a fused silica octagonalcell (radius r = 20 mm, height h = 35 mm) whose windows are anti-reflection coated on bothsides. The cell is surrounded by hollow-core water-cooled coils for the large magnetic fields (550
G) required to access the K-Rb feshbach resonance [285] (see Fig. 7.2a). Figs. 7.2b&c show theelectrodes in greater detail, with and without the cell. Below the cell is a high resolution microscope
objective, as shown in Fig. 7.2d.
Transparant plate electrodes are required to allow high resolution imaging as well as the
propagation of optical trapping and atom/molecule addressing beams in the vertical direction. The

139

Figure 7.2: The new apparatus KRb machine (a) shows the cell with electrodes inside, and issurrounded by the coils needed for atom manipulation and Feshbach association. (b) shows the cellalone, and the electrodes and macor holders can be seen extending further into the chamber. (c)shows a close-up render of a solid model of the electrodes and the macor insulators. (d) shows asection view of a solid model of the entire chamber and coils. The high resolution objective below
the cell is also apparent.

140
thickness of the ITO coating is ≈ 10 µm, which has an optical absorption at a wavelength of
λ = 1064 nm of ≈ 2% and a coating resistance of ≈ 1 kΩ/, where  means “square” and denotes
the entire surface area of the coating. Tungsten was chosen for the rods because it is very hardand has a high work function, which are important for reaching large electric fields, and because it
is minimally magnetic. Moreover, its coefficient of thermal expansion is very small, which reduces
thermal expansion problems associated with baking the vacuum chamber. The electrode assembly isheld together using macor insulating pieces, which were carefully designed to maximize the surface
path between adjacent electrodes, and electrodes with nominally opposite sign.

7.1.1

Macor insulators

A major challange with this design is the macor insulators that hold the electrodes together.
It is very difficult to machine such small intracate pieces and assemble the electrodes without themcracking or chipping. Figure 7.3 shows a solid model of the front macor electrode holder whichgently rests against the front side of the science cell. The JILA instrument shop did an incredible
job developing the expertise to create such pieces with a high success rate. This work was doneby the late machinist Tracy Keep, and former machinist Kels Detra. None of the current JILA
machinists have attempted these parts, and so certainly some expertise has been lost as a result of
the untimely passing of Tracy after battling with cancer.
We anticipate reaching fields of 20 − 30 kV/cm, and all the macor insulators were designed tomaximize the surface path length between adjacent electrodes. While the bulk dielectric strengthof macor (500 − 1000 kV/cm) is much larger than any field we could apply, pathways on the surface
that allow current to flow develop at much lower fields. Indeed, this phenomena will actuallylimit the fields we can reach, and a general rule of thumb is that the surface path length shouldbe ≈ 10× the distance between the electrodes [245]. Note that our electrode design requires us
to place the electrodes in the region of the largest field. Other high-field experiments like Stark
decellerators typically try to avoid this, and often have metal rods that come together from distantsupports [245]. Nevertheless, even Stark decellerators are often limited by this type of breakdown.

141

Figure 7.3: The front macor holder in the science cell. The hole diameter for the rods is 1 mm,
and the entire diameter is ≈ 15 mm. The groove on the front side holds the ITO-coated plates.

142
7.1.2

Triple points

Another important consideration while designing electrodes for large electric fields is triple
points, which are defined as the junction of conductor, dielectric, and vacuum. Since the electric field
inside a perfect conductor is identically zero, there is a discontinuity at the surface of the electrodes(see Fig. 7.7c), and the addition of a dielectric material can add substantial complications. These
triple points serve as electron sources, and are generally regarded as the location where flashoveris initiated in the insulators [137]. They are also a vulnerable location from which radio-frequency(rf) breakdown is triggered [137].
This issue will be discussed more in Chapter 8, but the most accute area for triple pointinduced breakdown on our electrodes is at the end of the rods inside the macor sleeves. These
problems are illustrated in Fig. 7.4, and two cases are considered. If the macor sleeve is longer
than the rod as shown in Fig. 7.4a, then the triple-point angle θ is very small as shown in Fig. 7.4c.
However, if the rod extends beyond the macor sleeve as in Fig. 7.4b, then the triple-point angle isθ ≈ 90 degrees. The latter case is much better for minimizing the electron density in the dielectricmaterial, and thus reduces the propensity to generate an electron beam from the electrode or
facilitate rf breakdown. This hypothesis has been verified with measurements in the new apparatus,
and will be discussed in Chapter 8.

7.2

The flexibility of the electrode system
Such an electrode configuration offers many useful electric field orientations, as depicted

in Fig. 7.5. Figure 7.5a shows how a homogeneous electric field in the vertical direction can begenerated, and will next be discussed in detail. The direction of this field is highly adjustable, as
shown in Figs. 7.5b&c (although the field is less uniform when oriented at an angle), which is usefulfor tuning the dipole angle relative to the collision angle (see e.g. conceptually similar experimentswith highly magnetic atoms [83, 164]). Further, Figs. 7.5d&e show how arbitrary gradients canbe applied in the plane perpedicular to the long direction of the electrodes. Lastly, a quadrupole

143

Figure 7.4: Triple points at the end of the rod electrodes. Blue is the metal conductor (electrode),
green is the dielectric (macor insulator), and orange is vacuum. (a) The macor sleeve extending
beyond the end of the rod electrode. (b) The macor sleeve ending before the conical taper of therod electrode. (c) A zoom in of the triple point when the macor exceeds the length of the rod. The
triple point angle is very small. (d) A zoom in of the triple angle when the macor sleeve is shorted
than the rod. The triple point angle is ≈ 90 degrees.

144
configuration can be used, as shown in Fig. 7.5f. Any combination of these six orientations is also
possible, yielding enormous flexibility.

7.3

Homogeneous electric fields
As stated above, homogeneous electric fields are important for realizing the XXZ model ofquantum magnetism. This is manifest upon consideration of the strength of the spin-exchange termJ⊥ /2 (see e.g. Ref. [263]), which for KRb is 100 Hz between neighboring sites in a lattice of spacing
λ/2 = 532 nm [110, 273]. In order for spin-exchange processes to be resonant between distant sites,
the energy offset between them must be ∼ 10’s of Hz. Any electric field gradient or curvature will
limit the length scale of these interactions. A figure of merit specificly from recent KRb experiments
in an optical lattice is the energy spread across the cloud due to residual differential AC Stark shifts
from the optical traps (discussed in Chapter 5) which were limited to ∼ 1 kHz [188, 273].
Several groups are pursuing an electrode system of just four rods, similarly aligned in aquadrupole fashion. However, this configuration is incapable of providing purely homogeneous
fields. On the other hand, perfectly parallel plates will provide a homogeneous field, but are notversatile and cannot provide arbitrary gradients. The configuration presented in this work canallow homogeneous fields that are simultaneously versatile, but the relative voltages on the rods
and plates must be chosen very carefully such that the curvature from the rods does not dominate
the flat field contribution from the plates. The separation between the plates (whose thickness is 1mm) is 6 mm, and the vertical seperation between the rods (whose diameter is 1 mm) is 3 mm. The
horizontal seperation between the rods is 5 mm. Finite Element Analysis (FEA) using COMSOLMultiphysics was performed to study the electric field distribution for arbitrary voltages on all six
electrodes.
It is constructive to understand what electric field homogeneity is needed for all the experiments that we would like to do. The 0th order requirement is that the electric field gradient orcurvature is small compared to the optical dipole force of the trap, such that applying a field does
not spill molecules out of the optical trap. A higher order requirement that the new apparatus was

145

Figure 7.5: The electric field orientations that are possible with the electrode configuration of the
new apparatus. (a) shows how large, homogeneous fields in the vertical direction can be generated.
(b) and (c) show how this field can be tilted to an arbitrary angle in the 2D plane. Note that thefield homogeneity is reduced in these orientations. (d) and (e) show how arbitrary gradients canbe applied in any direction in the 2D plane. (f) shows a quadrupole field which can be used to trap
molecules in 2D.

Figure 7.6: The energy shift of the |0, 0i to |1, 0i transition as a function of the electric field (a),and the strength of J⊥ and Jz as a function of field (b). Adapted from Reference [112].

146
designed to meet is an energy shift between sites of less than J⊥ . The energy shift of the |0, 0i to|1, 0i transition as a function of the electric field is shown in Fig. 7.6a. For a cloud of ≈ 100 µm(ntot ≈ 200 sites) the total shift across the cloud ntot J⊥ ≈ 40 kHz, for which we require a gradientof E 0 < 20 V/cm2 . At a field of E = 20 kV/cm this level of homogeneity corresponds to 1/105
across the cloud.
Figures 7.7a&c show cuts of the magnitude of the electric field when the voltages on theplates are ±10 kV and the rods are varied around ±4.225 kV, for which the electric field is 32
kV/cm. Figure 7.7a shows the horizontal cut, and the cat ears immediately between the rods areapparent. Thus, the field is not homogeneous over the entire range, but it is infinitely flat over aroughly ±1 mm region around the center. Figure 7.7c shows the vertical cut, and a similar behavior
is evident around the geometric center.
It is constructive to look at the flatness of the field near the center as a function of the voltage
ratio between the rods and the plates. Figures 7.7b&d show a zoom of the axial and radial cuts,
respectively, where the rod voltage is varied around ±4.225 kV while holding the plates at ±10
kV. The different colors show the different values of the rod voltages, and it is clear that field in
the center has curvature either up or down depending on whether the rod voltage is too high or
low, respectively. The purple curves show the optimal field where the field is flat over ≈ 1 mm oneither side of the center. Note that the curvature is opposite in the radial and axial directions, as~ ·E~ = 0.
is required by Gauss’s law ∇7.3.1

Effects from a hole in the ITO-coated plates

We can also use these simulations to understand how sensitive this maximally homogeneous
regime is to imperfections. The first case we consider is a hole in the ITO plate. Such a configuration
is interesting since we could have alternatively used metal plates, which would require a hole forthe vertical lattice, Raman beams, and a probe beam. Moreover, as discussed in Chapter 8, we
initially had a lot of trouble pairing the ITO coating with an AR coating. Lastly, the ITO coating
could be damaged in a very localized region. This could happen if a lot of atoms or other dirt

147

Figure 7.7: Simulated electric field cuts along the axial and radial directions. (a) A cut throughthe origin along the axial direction (through the plates) for the plates at ±10 kV and the rods at±4.225 kV. (b) A cut through the origin along the radial direction (between the plates) for theplates at ±10 kV and the rods at ±4.225 kV. (c) A zoom in of the axial cut for sevaral values of
the rod voltage with the plates at ±10 kV. (d) A zoom in of the radial cut for sevaral values of therod voltage with the plates at ±10 kV. Reproduced from Ref. [50].

148

Figure 7.8: Simulating the field effects of a hole in the ITO plate. (a) The field profile through theorigin and along the symmetry axis of the electrodes. The shallow dip in the middle is caused bya hole in the plates. (b) The gradient of the field 500 µm from center of the hole as a function of
the size of the hole.

land on a small region, or if the vertical lattice beam goes through a dirty region and locally burns
something off.
Figure 7.8 shows the case of a hole centered on the electrode assembly and centered on themolecules. Figure 7.8a shows a cut through the origin of the electrodes and along the symmetryaxis. The dip in the middle is due to the hole in plates. Note that the ears at the ends of the flat
field region are an artifact of a difference between the lengths of the plates versus the rods. To
quantify the effect of this hole we consider the gradient of the field slightly off axis of the center ofthe hole, by 500 µm. We can look at this gradient as a function of the size of the hole, as shown in
Fig. 7.8b.
The scaling is rather unfavorable, and even a hole of radius R = 500 µm causes a gradient of≈ 150 V/cm2 at a field of ≈ 30 kV/cm. Note that the energy gradient due to the variation in the
DC Stark shift is comparable to the optical dipole force in the radial direction of typical optical
dipole traps, and so such gradients will severely perturb or even spill the molecules out of an optical
trap. A hole of this size is required to safely propagate typical laser beams, and so this scheme is
not practical. Note, however, that if the hole is very small or far away from the molecules, theseeffects are mitigated enormously. Therefore, we expect the effects of localized damage or patch

149

Figure 7.9: Electric field gradients induced by a wedge angle between the rods. (a) The geometryof the simulation, where many wedge angles were simulated. (b) The gradient of the field at theorigin as a function of the wedge angle.

charges to be negligible.

7.3.2

The effect of an angle between the rods and/or plates

To understand whether it is possible to reach 10−5 -level field homogeneity it is also important
to consider the parallelism of the plates and rods, which can be roughly estimated without the useof FEA. If the spacing between two rods or plates are d and d0 on the two ends separated by r,then the gradient is roughly E 0 = (∆V /d − ∆V /d0 )/r = ∆V /d · ∆d/(d · r) = E∆d/(d · r), where∆d = d − d0 . Note that ∆d/r is also the wedge angle θ. For E = 30 kV/cm, d = 6 mm, and r = 20mm, limiting the gradient to E 0 = 20 V/cm2 requires ∆d ≈ 10 µm. To confirm this, Fig. 7.9 showsthe effect of a wedge angle between the rods and plots the gradient as a function of the angle.
Figure 7.9b shows that reaching a gradient of 20 V/cm2 requires a wedge angle less then
θ = 0.02 degrees, which corresponds to ∆d = 7 µm. While this is not impossible, it will certainly
be difficult to realize in practice. However, the situation becomes much more favorable at lower
fields. Firstly, the gradient scales as E, so we win linearly there. More importantly, however, theslope of the energy shift decreases as the dipole moment decreases. Therefore, at a field of E ≈ 10

150
kV/cm we expect this level of homogeneity to be possible. Figure 7.6b shows the strength of J⊥
and Jz as a function of the electric field. This shows that Jz is actually maximized at around thisvalue [112], so there will likely not be any reason to try to do quantum magnetism in larger fields.

7.3.3

Charging scenerios

Chapters 2 and 6 described the severe problems in the old apparatus with the build-up oftransient charges on the glass cell. Hence, it is very important that such issues do not plague ourwork with electric fields in the new apparatus. To estimate the significance of these effects weconsider putting potentials of ±1 kV on various dielectric materials in the chamber, such as theglass cell or the macor insulators. We then look at the field profile when all the electrodes are
grounded. We simulated the effects of potentials in three places: 1) the edges of ITO-coated plates,or the edges of the windows where they are sealed to the quartz frame of the cell, specifically theedges of the side and top and bottom windows which are nearest to the ITO-coated plates; 2) the
end window and its edges; and 3) the macor holders, which are the surfaces nearest the molecules.
First note that the electrode assembly acts as an effective Faraday cage, which blocks electricfields from distant objects. In case 1), the potential applied to the edges of the ITO-coated plates
or the edges of the windows gives rise to a field at the center of the electrodes of ≈ 1 V/cm, andis flat over roughly the separation of the rods, which is 4 mm. Note that the shielding from theelectrodes is the best along this radial axis. In the case of 2) where potential is put on the end
window, the symmetry of the electrodes along this access do not allow us to shim any stray fields.
Fortunately, the end window is far enough away that the electric field from this potential has fallen
to also ≈ 1 V/cm at the center of the science cell. The gradient of this field is also very small, ≈ 1
V/cm2 . Case 3), where potentials are placed on the macor insulators, is by far the most concerning
scenerio. These potentials give rise to fields at the center of ≈ 20 V/cm, but this of course depends
on the potential. ±1 kV is perhaps unrealistically high, but hopefully this will give us an idea ofwhat to expect in the worst case. While this field is larger than the other cases, it is still fairly
uniform at the center. Nevertheless, we will ultimately be limited by such effects, as well as patch

151
charge build-up on the ITO-coated plates, as discussed above.

7.4

Arbitrary electric field gradients in 2D
While a homogeneous field is ideal for studying spin models with polar molecules, gradientsare essential for pursuing other directions. For instance, consider an array of 2D gases in a 1D opticallattice which is propagating in the vertical direction (through the ITO-coated plates). Applying
the field along the vertical direction stabilizes the molecules through the quantum stereodynamics
of their collisions [63]. Then, a gradient in the horizontal direction can be used to tilt the 2Dtraps for direct evaporative cooling of the molecular gas [283], which is a promising route to create
the first bulk quantum degenerate gas of polar molecules as discussed in Chapter 6. Further, avertical gradient can be used to create a unique DC Stark shift of a rotational transition for each
2D subsystem of the 1D lattice. This can be used to spectroscopically select a single layer, as isdone routinely using the Zeeman shift for atoms [235], which is important for high resolution insitu detection.
We can start to add a gradient of the electric field in any arbitrary direction in 2D. This isdone by biasing some of the rods away from the voltage ratio between the rods and plates of 0.4225
that is nominally used for the homogeneous field described above. Figures 7.10 and 7.11 show thegradient of the field as a function of the voltage bias δ in the horizontal (radial) and vertical (axial)
directions, respectively. Typical gradients required in the horizontal direction for evaporation giveenergy shifts similar to the radial trap frequency, which are ≈ 100 V/cm2 for KRb at large electricfields [283]. Figure 7.10c shows that such a gradient can be generated with δH = 8 V, which iseasily controllable with typical voltage stability.
For selection of a single layer in the 1D lattice, energy shifts of ≈ 10 kHz/site have been used
for ultracold atoms [235]. For KRb at large electric fields (> 30 kV/cm), the typical energy shifts
are ≈ 100 kHz/(V/cm) (see Fig. 7.6a). Figure 7.11b&d shows the sensitivity to a bias of the rodsin the vertical direction, which suggests that a bias of δV ≈ 200 V is sufficient to give a 10 kHz
energy shift (2 kV/cm2 needed to separate adjacent layers of spacing λ/2 = 532 nm by 0.1 V/cm).

152

Figure 7.10: Analysis of the field profile with a radial gradient. (a) The potential color map of theelectrodes when the plates are at ±10 kV and the rods are at Vr = 4.225 kV with some a bias δHalong the horizontal (radial) direction. The lines are equipotentials, and the color map is within±10 kV. (b) The electric field at the center as a function of the horizontal position. The colors
represent different values of δH between 0 and 100 V. The inset shows a zoomed out view where
the peaks on the edges are the locations of the rods, and the voltage on the left rod is larger thanthe voltage on the right by 2δH . (c) The gradient of the electric field in the radial direction as afunction of δH . (d) The field variation between the center and 0.5 mm in the vertical direction as
a function of δH . The inset shows the field variation in the vertical direction as a function of theposition between 0 and 1 mm. The different colors denote the different values of δH between 0 and
100 V. Reporoduced from Reference [50].

153
Moreover, in order to select individual layers of the 1D lattice with high fidelity, the field strength
will have to be stable enough such that field variation at the layer during the pulse is much smaller
than the field difference between layers (0.1 V/cm). We have thus designed high voltage servos
for electrodes that are expected to operate at the several parts per million level, giving ≈ 10 mV
stability and precision. This corresponds to fluctuations of the order 0.1 V/cm.
In fact, layer selection becomes easier at lower electric fields where the absolute field fluctuation is pushed down. However, by operating at a lower electric field, the dipole moment isstill in the linear regime, and can easily be tuned between 0 − 200 kHz/(V/cm). Thus, the field
gradient will need to be even larger to reach the same energy shift between layers. Fortunately,
the electrode geometry allows for very large vertical gradients, and the plot shown in Fig. 7.11d
continues up to ∼ 15 kV/cm2 . For concreteness, consider working at 5 kV/cm where the energy
shift is ≈ 50 kHz/(V/cm). An energy shift of 10 kHz between layers can be reached with ≈ 10
kV/cm2 gradients, induced by δV ≈ 1 kV. Note that the bias δV is becoming similar to the voltages
nominally applied to the rods for the homogeneous field at 5 kV/cm (i.e. ≈ 4.225/6 kV). Herethe field difference between adjacent sites is still 0.1 V/cm, but the absolute field stability is 10
mV/cm.
~ ·E~ = 0. We
Figure 7.11c shows the curvature in the radial direction due to Gauss’s law ∇require the field variation across one layer to be well within the Rabi frequency Ω ≈ 10 kHz, andmuch smaller than the field difference between adjacent layers. Even for the largest bias δV = 1 kV
where the field difference between adjacent layers is 1 V/cm, the electric field variation in the radial
direction near the center is smaller than the numerical error,  1 V/cm. Note that it is important
to work in the maximally flat field configuration Vr /Vp = 0.4225, and that it is important that thecloud be within ≈ 200 µm from the center in the horizontal direction especially. Additionally, it isimportant the bias δ is divided evenly between each pair of rods ±δ/2, as in Figs. 7.10a and 7.11a.
This is essential for maintaining the homogeneity of the field and minimizing the curvature along
the direction opposite the applied gradient.

154

Figure 7.11: Analysis of the field profile with an axial gradient. (a) The potential color map of theelectrodes when the plates are at ±10 kV and the rods are at Vr = 4.225 kV with some a bias δValong the vertical (axial) direction. The lines are equipotentials, and the color map is within ±10
kV. (b) The electric field around ±2.5 mm of the center in the vertical direction. The different
colors denote different values of δV between 0 and 1000 V. (c) The electric field as a function of thehorizontal position between the center and 1 mm. The different colors denote the same range of δV
between 0 and 1000 V. The inset shows a zoom around 0.5 mm. (d) The gradient of the electricfield in the axial direction as a function of δV . Reporoduced from Reference [50].

155

7.5

Coupling AC fields onto the rods
We also designed our apparatus for coupling AC microwave frequencies onto the rods, which

we can use to directly drive rotational transitions. The rods are seperated by several mm’s, which
is significantly smaller than the wavelength of the microwave field in the frequency range of 2 − 8GHz. Therefore, the microwave field can be thought of as a homogeneous AC electric field on therods. We have the ability to change the relative angle between the AC and DC electric fields,
which gives us the ability to control the polarization of the microwave field and thus the angularmomentum of the rotational transitions.
Figure 7.12 shows the elements of the electrode system that are required to couple microwaves
onto the rods. A bias tee of high voltage inductors and capacitors are used to independantly
couple in DC and AC voltages, respectively [248]. The schematic is shown in Fig. 7.12a, and thearrangement of these elements is shown in b&c. The inductors are simply two loops of StainlessSteel 316 (SS316) wire, and the inductance was chosen to maximize the transmission of microwave
signals of frequency 2 − 8 GHz. The capacitors are shown in c, and the top capacitor plates arecomprised of SS316 pillars. The disks at the bottom of the pillars sit on a macor base, whose
thickness of 1 mm serves as the dielectric material of the capacitor. Below the macor is anothersuch disk which continues below to a four-rod low power feedthrough.
This feedthrough is not impedence matched with a coaxial input, but it allows us to change
from outside the chamber how the microwaves are coupled to the four rods, giving us control ofpolarization. It is imperative that this bias tee is placed inside the vacuum chamber because theDC electric field across the capacitor plates will be up to 100 kV/mm when 10 kV are applied to
the rods, and this field is much larger than the dielectric breakdown of air. We measure that theinsertion loss going from a coaxial cable below the chamber through the capacitors is ≈ 5 dB. This
will be discussed more in Chapter 8.
The capacitance of the coupling capacitors were chosen based on an estimate of the capacitance between the rods in the science cell, which is < 1 pF. The capacitance of the coupling

156

Figure 7.12: The electrodes inside the new apparatus. (a) shows a schematic of the bias tee andhow AC fields are coupled into the chamber. (b) shows the bias tee in the large cube inside thevacuum chamber. (c) shows the home-made capacitors used to couple the AC microwave fields
onto the rod electrodes in the science cell. (d) shows the transmission line from the large cube into
the science cell.

157

Figure 7.13: The electrode layout inside the science chamber. The large cube that houses the
feedthrough for the AC and DC voltages is connected to the pumping region of the chamber, butalso the electrode path to the science cell. The cube contains four inductors and four capacitors
to decouple the AC and DC signals. Six rods go from the large cube all the way to the science
cell, where two of the rods connect to ITO-coated plates. The four-channel feedthrough connects
high voltage through the inductors that ultimately go to the rods in the science cell. The twochannel feedthrough connects high voltage ultimately to the plates in the science cell. There are
no inductors because the plates are not connected to capacitors or high frequency feedthroughs.
We opted to use a four-pin low power feedthrough for coupling in the microwaves so that we cancontrol outside the chamber which rods are connected.

158
capacitors can thus be thought of as an AC voltage divider, and since the impedence of a capacitor
is ZC = 1/(iωC), we want the capacitance of the coupling capacitor to be larger than that of
the rods. Thus the capacitance was designed to be ≈ 4 pF. Figure 7.12d shows the transmission
line from the feedthroughs in the large cube into the science cell. This is also shown in Fig. 7.13.
We measure ≈ 20 dB transmission loss of microwaves from the coupling capacitor into the rod
electrodes in the science cell. This will be discussed more in Chapter 8. Despite these losses, we
expect the coupling of the microwave electric field to the molecules to be similar or larger than the
broadband microwave horn that we used in the past because the rods are so much closer and there
are no longer any grounded magnetic field coils between the molecules and source.

7.6

High resolution imaging system
The rods are arranged within the plates in a rectangular way in order to accomodate an opticalnumerical aperture (NA) in the vertical direction as high as 0.75. In practice, the imaging resolution
will be limited by the objective as well as the optical quality of the intermediate surfaces. The RMS
surface flatness before assembly and baking over the entire aperture of the large, vertically-oriented
windows of the cell is λ/5 at λ = 780 nm. The ITO-coated plates are expected to have better
flatness, but slightly worse surface figure. The eletrode configuration of the rods and plates was
carefully oriented to be as parallel to the large cell windows as possible. Any angle will likely limit
the ultimate NA that can be achieved, and an imaging system that corrects the coma aberations
induced by this angle would likely be needed for high NA objectives. The working distance of theobjective is 22 mm, and the optical path between the atoms/molecules and the objective is shownin Fig. 7.14a. The design of the objective is shown in Fig. 7.14b.
The microscope objective shown in Fig. 7.2d is designed to have NA = 0.53, correcting forthe spherical aberation of the ITO-coated plate and the cell window (whose thickness is 6.4 mm).
The coma induced by the tilted ITO-coated plates is not corrected, but we expect its effect to berelatively small for a 0.1 − 0.2 degree angle at this NA. The diffraction-limited Rayleigh spot size is
Rmin = 880 nm, and such spot sizes have been reached in testing. The ITO-coated plate is paired

159

Figure 7.14: (a) The optical path between the atoms/molecules to the objective. The thicknessesof the substrates are indicated in green. (b) The design of the objective, accounting for all the
obtical elements shown in (a). The optical elements are outlined in purple, including the fusedsilica substrates. Courtesy of Special Optics.

160
with a broadband anti-reflection coating from λ = 700 − 1100 nm, which is designed to minimize
reflection at 1064 nm in order to accomodate a vertical optical lattice beam. The cell window iscoated over a similar bandwidth, and also optimized at 1064 nm. In order to reflect the verticallattice beam, which propagates from above, before going through the objective, a UVFS dichroic
filter plate of 2 mm thickness is placed below the cell. Figure 7.2d shows this filter, which can be
adjusted with a 2D kinematic mount. The microscope objective is inside the mount for this filter,and can be independently adjusted with a five-axis stage.

7.6.1

Objective design and aberration theory

It is constructive to consider the mathematical structure of the primary (called Seidel) aberrations, where the wave aberration is given by the Zernike polynomials [25, 105]
1
1
φ(4) = − Bρ4 − Cy0 ρ2 cos2 θ − Dy02 ρ2 + Ey03 ρcosθ + F y0 ρ3 cosθ,
4
2

(7.1)

where ρ is the radial coordinate in cylindrical coordinates, θ is the polar angle, and y0 is an offset
from the center of the imaging path defined by the lenses. These terms are specific aberrations:the first term B is spherical aberration, the second term C is astigmatism, the third term D iscurvature of field, the fourth term E is distortion, and the last term F is coma [25].
The substrates between the objective and the object plane cause spherical aberrations, whichmust be corrected in the objective. This is done using a miniscus lens as the first lens in the five-optic
objective lens system, which can be seen in Figs. 7.2d, 7.14b, and 7.15. Since spherical aberration
scales as ρ4 an objective design where the first lens is a miniscus lens can be used for corrections.
The final lens pair in Fig. 7.14b is a doublet. As discussed above another important term in theaberration error is coma, and a possible source is the relative angle between the cell windows andthe ITO-coated plates. The cos θ scaling of this term suggests that a wedged compensation plate
could be placed after the objective. However, since it also scales as ρ3 this wedged plate would
likely need a polynomial cylindrically-symmetric taper in addition to the wedge.
The objective is designed only for λ = 767 and λ = 780 nm, and thus substantial chromatic161
aberrations are expected for significantly different wavelengths. This will be important for the highresolution addressing beams and holographic digital micromirror device (DMD) projection on to
the atoms discussed in Chapter 10. The wavelengths that we intend to use for this are λ = 690 nm
and λ = 1064 nm. We expect that the effects of shorter wavelength (smaller Rmin ) and chromaticaberration (larger Rmin ) roughly cancel for λ = 690 nm, allowing us to reach a Gaussian 1/e2 radius
for an addressing beam of ≈ 750 nm. λ = 1064 nm is both further away from λ = 780 nm, and has
a longer wavelength. Accordingly we expect the minimum waist of an addressing beam with thiswavelength to be roughly 1 µm, which is still sufficiently small to allow addressing at the two-sitelevel in the lattice.

7.6.2

The coil and optics mounts around the science cell

The coils and objective path around the science cell are shown in Fig. 7.15. The objective isshown below the science cell, and around it is the holder for the dichroic plate that goes between
the science cell and the objective. This dichroic filter is used to retroreflect the vertical latticebeam, while the Raman beams and probe beams propagate through the objective (to be discussed
more in Chapter 9). The objective is on a five-axis stage, and the dichroic is separately on a twoaxis kinematic mount. Photos of how this looks in the lab are shown in Fig. 7.16. The objective
and dichroic holders are made of ultem, and most of the other components are made of aluminum(which is anodized). The black aluminum plate has a slit cut in it to prevent eddy currents whenthe vertically-oriented fields switch quickly. The coil mount is made of G-11 glass/epoxy phenolic,and then painted black with a low-outgassing paint. The legs for the coil mount and the objective
and dichroic mounts are titanium in an efffort to reduce magnetization of nearby materials.

7.6.3

Deflection of the science cell windows due to the differential pressure

It is also constructive to consider the deflection of the windows on the vacuum chamber
due to the differential pressure. Continuum mechanics provides Kirchhoff-Love plate theory, which
describes the deflection of a material (i.e. Fused Silica) due to a differential load with various types

162

Figure 7.15: The coils and objective path around the science cell. The objective is shown below
the science cell, and around is the holder for the dichroic plate that goes between the science celland the objective. The objective is on a five-axis stage, and the dichroic is separately on a two-axis
kinematic mount.

Figure 7.16: Photographs of the coil mount and objective and dichroic mounts around the sciencecell. The objective is on a five-axis mount that is inside and separate from the tube of the two-axis
kinematic mount which holds the dichroic filter. This filter is necessary to reflect the vertical latticebeam before the objective.

163
of supports. Using this analysis we can write the deflection as a function of the distance from thecenter of the window [255]w(r) =

2q
a2 − r2 ,0
64 · D(7.2)

where q is the differential pressure 105 Pa, a ≈ 20 mm is the radius of the window, and D0 =2h3 E/(3(1 − ν)2 ) is the flexural rigidity, where 2h is the plate thickness, E is the elastic modulus,and ν is Poisson’s ratio. An interesting note from this equation is that spherical aberrations also
scale as r4 , so vacuum deflections particularly exacerbate the spherical aberration which are alreadypresent from the vacuum window.
Figure 7.17 shows a finite element simulation of the deflection amount, and it suggests thatthe maximum deflection is wmax ≈ 0.2 µm, which is order λ/5. This level is similar to the flatness
due to the science cell assembly procedure and baking, and analysis of the flatness will be discussed
in Chapter 8. Note that wmax depends strongly on a, and so larger windows will be more affectedby this issue unless h is increased accordingly. However, even if h is increased the spherical abberation it causes will become more significant, and the objective design will inevitably become more
complicated in order to compensate.

7.7

Magnetic fields
The coils around the science cell are shown in Figs. 7.2, 7.15, 7.16, and 7.18. The bias coils are

16 turns (4×4) of hollow-core square coil where water flows through every turn. Inside the bias coils
and concentric with them are the gradient/quantization coils, which are a 3 × 9 array of magnetic
wire. These coils are not actively cooled, but their proximity to the bias coils provides excellent
passive cooling. These coils have an H-bridge which allows us to change the current direction of
one coil with respect to the other, and switch them from an anti-Helmholtz quadrupole field to a
Helmholtz field which could be used for fast magnetic field sweeps, providing a quantization fieldin the vertical direction, or shimming out the vertical component of the stray field during opticalmolasses or Raman side band cooling in a quantum gas microscope (to be discussed in Chapter

164

Figure 7.17: Deflection of the vacuum windown due to the differential pressure. A finite element
simulation of the deflection of window, where the color scale shows the deflection as described by
the color bar.

165
11).

7.7.1

Designing the gradient/quantization coils

To model the field from the coils when the current is flowing the same direction in both, we
assume the coils are oriented in a perfect Helmholtz geometry where their radius is equal to theirseparation (R = d). In our case R > d, but we can use this approximation to desribe the field by 3/24B=µ0 · N · I/R,
5

(7.3)

where N = 27 is the number of turns, I is the current, and R = 4.2 cm is the average radius of thecoils. Taking the derivative with respect to I and plugging in the numbers gives 6 G/A, which isin excellent agreement with the measurements shown in Chapter 8.
When the H-bridge is switched the coils have current flowing through them in the opposite
direction. In this case the gradient of the field in the axial direction z near the geometric center isgiven by3
d/2
B z = 2 · µ0 · N · I · R 2
5/2 ,2(d/2)2 + R2

(7.4)

where the factor of two in front is because this gradient is from each coil. Again, taking the derivativewith respect to I gives the gradient as a function of current, which is calculated to be ≈ 2 G/cm/A.
Measurements in Chapter 8 show that this value is 2.4(1) G/cm/A, so the calculation is not perfect.
This could be caused by a slight error in the estimates of R and d, for which average values are used.
The way this equation scales with R makes it particularly sensitive to miscalculations. Neverthelesssuch a simple equation can be used for reasonable qualitative agreement.

7.7.2

Designing the bias coils

The bias coils were painstakingly designed such that the average separation between the 16
turns of the two coils d and the average radius of each coil R are exactly the same. Moreover, thetwo coils were wound in the opposite way such that the gradients from the winding inperfections ofeach coil would cancel each other out. The careful consideration to detail in the design of these coils

166
is what gives rise to the incredible flatness measured in Chapter 8. The equation for the magnetic
field from these coils is the same as the equation above for the quantization coils, and the field per
current for the bias coils is calculated to be 2.55 G/A, which is in excellent agreement with themeasurements presented in Chapter 8.
The K-Rb Feshbach resonance is at 547 G, which can be reached with ≈ 220 A. At 1005
G is a narrow Feshback resonance for Rb, which could be useful to measure double occupancy of
Rb by making Rb2 Feshbach molecules. This field can reached with ≈ 400 A. Another importantfield is 1260 G, at which the excited rotational states |1, 1i, |1, 0i, and |1, −1i are degenerate [250](discussed in Chapter 11). This field can be reached at ≈ 480 A, which is still within reach of ourhigh-current power supply. The coils in the old apparatus had 12 turns, and roughly 330 A wererequired to reach the Feshbach resonance. Moreover, the old coils were not in the perfect Helmholtz
configuration, and so they had a larger curvature across the cloud.

7.8

The chamber and vacuum considerations
The full vacuum chamber of the new apparatus is shown in Fig. 7.18. Similar to the oldapparatus, a moving quadrupole trap carries atoms from the MOT chamber (not shown) to thequadrupole evaporation region. A gate valve and a flexible bellows separates the two regions, asin the old apparatus. The evaporation region is near the small cube, attached to which is a nonevaporable getter (NEG) pump (SAES CapaciTorr D 400-2) which nominally has a pumping speedof 400 L/s. After plugged quadrupole evaporation (discussed in Chapter 9) in the region near thesmall cube, the K and Rb clouds are optically transferred into the science cell. Behind the large
cube where the high voltage feedthroughs are shown there is a 55 L/s ion pump and titanium
sublimation pump.
More details of the chamber will be described in Chapters 8 and 9, and here I will present
only the details necessary to analyze the gas conductance through the chamber. The pressure P in
the science cell is given by P = Q/S, where Q is the gas load and S is the effective pumping speed.
The gas load is generally complicated to calculate, but it scales with the surface area SA and the167

Figure 7.18: The full vacuum chamber of the new apparatus. Similar to the old apparatus, amoving quadrupole trap carries atoms from the MOT chamber (not shown) to the quadrupole
evaporation region. A gate valve and a flexible bellows separate the two regions, as in the oldapparatus. The evaporation region is near the small cube, attached to which is a non-evaporable
getter pump. After plugged quadrupole evaporation in the region near the small cube, the K and
Rb clouds are optically transferred into the science cell. Behind the large cube where the highvoltage feedthroughs are shown there is a ion pump and titanium sublimation pump.

168
outgassing rate R as Q ∼ SA · R. The effective pumping rate S is given in terms of the pumping
speed from the ion pump and Titanium-sublimation (Ti-sub) pump Sp , and the conductance [117]
1
1
1
=
+
S
C
Sp

(7.5)

such that they add in parallel.
The conductance through a tube in the molecular flow limit is given by C = 78D3 /L, where
D is the tube diameter and L is the tube length [117]. Adding material in the chamber such as
electrodes and macor insulators reduces the vacuum quality by both increasing the surface area anddecreasing the effective tube diameter, and thus we must be very careful to correctly account forit all. Given all these factors the pressure in the system is given by [117]P = SA · R ·


1
L
+
.
Sp 78D3(7.6)

While it is difficult to calculate the outgassing rate, we can assume that all materials outgas thesame amount (Rnew = Rold ) and simply compare the surface area of the new apparatus to thatof the old apparatus, whose pressure we know. The outgassing rates of macor and tungsten are
actually similar to glass and steel, so this is a reasonable approximation.
We can now compare the pressure in the new chamber to that in the old chamber, and thisratio is given by
Lnew
1
3
Pnew
SAnew Sp,new + 78Dnew
=
.
L
1
old
Pold
SAold S
+
3
p,old

(7.7)

78DoldEstimating the effective diameter of the new tube out of the science cell Dnew requires detailed
knowledge of the macor pieces in the tube. The most important of which is the back macor piece,
as shown in Fig. 7.19. This piece has undergone many revisions to remove as much material as
possible because it plugs the access of the science cell to the connecting tube. Shown in grey are
the macor sleeves. We found that as a result of the assembly process the rods bow out and only
need to be supported from outside. This solution makes the machining much simpler since thesleeves are very thin and weak. The groove holds the ITO-coated plate, and the hole behind thegroove is where the rods come through to contact the ITO coating on the plates.

169

Figure 7.19: The back macor pieces which hold the electrodes in the science cell. For vacuum
conductance reasons we want to keep this piece as small and thin as possible. Shown in grey are
the macor sleeves. We found that as a result of the assembly process the rods bow out and only
need to be supported from outside. This solution makes the machining much simpler since thesleeves are very thin and weak. The groove holds the ITO-coated plate, and the hole behind thegroove is where the rods come through to contact the ITO coating on the plates.

For the old chamber we estimate Lold = 76 mm and Dold = 25 mm, while for the newchamber we estimate Lnew = 100 mm and Dnew = 25 mm. Note that these numbers are actuallysimilar between the two cells. Given these numbers, we can estimate Pnew /Pold as a function ofSAnew /SAold . Based on the geometry of all the electrodes and macor insulators, we estimate
SAnew /SAold ≈ 4 − 5. The pumping speed of the ion pump in the old chamber was Sp,old = 35L/s,
and as expected Pnew /Pold is minimized when Sp,new is large. Nevertheless, Pnew /Pold saturates for
Sp,new ≈ 50 L/s, and the value depends on SAnew /SAold . Pnew /Pold = 3 for SAnew /SAold = 3, and
5 for SAnew /SAold = 5. Note that if Dnew = 20 mm instead, these numbers scale up by ≈ 1.7.
Since Pold  10−11 mbar, it is difficult to measure with a pressure gauge, so we measure the1/e lifetime of atoms in a conservative magnetic trap τold ≈ 300 seconds. Therefore, since P ∼ τ −1
and we expect Pnew /Pold = 5, then the lifetime in the science cell should be τnew ≈ 60 seconds.
I will discuss this more in Chapter 8 and 9, but since we only use an optical dipole trap in thescience cell, the lifetime is limited by light scattering and heating and thus we cannot measure thevacuum-limited lifetime. Nevertheless, we have measured τnew > 30(5) seconds in the science cell,which is in good agreement with these rough estimates. As stated above, we perform evaporative

170
cooling in a magnetic trap outside of the science cell. This trap also has a loss mechanism (discussed
in Chapter 9), but we use it to measure the lifetime in the small cube outside the science cell to beτcube > 150(10) seconds. Note that we added the NEG pump which has a pumping speed of ≈ 400
L/s, and this was not included in the above analysis. We added this to be as safe as possible sincepressure problems are notoriously time-consuming to solve.

Chapter 8

Designing, building, and testing the new apparatus

With a promising set of ideas in place for the design of the new apparatus, it is time to discuss
the building and testing of the new chamber, coils, and imaging systems for the new apparatus.
We begin with the assembly and testing of the electrodes and their ability to reach large electricfields.

8.1

Reaching large electric fields
Testing of the electrodes was done in many test configurations in small chambers pumped

with turbo-molecular pumps. While these tests all had promising outcomes, the performance of theelectrodes in the fully assembled new apparatus chamber is ultimately what matters. In order todo these tests, however, the full chamber (or at least the science side) must be assembled. Chapter
9 will describe in more detail how the chamber is in two halfs: the science half and the MOT half.
They are connected with a differential pumping tube, a flexible bellows, and a gate vacuum whichallows the two vacuum regions to be separated.

8.1.1

Macor design

To build the full electrode assembly we must start with the design and fabrication of themacor insulating pieces. As was discussed in Chapter 7, we want the surface path length between
adjacent electrodes to be ≈ 10× their separation [118, 245]. Figure 8.1 shows some of the macor
pieces used to hold the electrodes. The one in the right image is the front holder in the science

172

Figure 8.1: The macor insulators which the electrodes in the science cell. The smallest piece whichgoes in the science cell is compared to a dime for reference.

cell, which gently presses against the window at the end of the cell. Note that its diameter is ≈ 15mm, or roughly the size of a dime. The insulators on the left hold the electrodes from the small
cube to the science cell. They are covered in blue glue because the machining was performed witha slab of macor glued to a block of aluminum. This glue is one of many contaminants which must
be carefully removed during the cleaning process, to be discussed later.
There are six rods that go into the science cell: the four rods between the ITO-coated plates
and the two rods which contact the ITO coatings on the two plates. These six rods are arranged asa perfect hexapole down the transmission line from the feedthroughs to the science cell. The small
piece in the left image is an old version of the back electrode in the science cell, which resides near
the entrance to the tube that is connected to the science cell. Even this amount of material would
have significantly reduced the vacuum conductance out of the science cell and therefore limited thepressure achievable in the science cell.
Note the slots on the bottom macor piece in the left image. This is the piece that holds theelectrodes in the flange of the science cell. These slots are essential to indicate the entire electrodeassembly in the cell relative to the windows. The relative angle between the ITO-coated plates andthe cell top and bottom windows was minimized by rotating the electrode assembly in these slots,

173
and the finally fixing the orientation. The holes for the electrodes on this piece have long sleevesbecause of the orientation of the slots with respect to the rods, which requires one of the slots tobe adjacent to one of the rods that connects to the ITO-coated plates. These rods have the largest
magnitude potential, and thus it particularly important to maximize its path length to ground.

8.1.2

Connecting the electrodes to the ITO coatings

The two rods on the top and bottom of the hexapole guide connect to the ITO coatings on
the fused silica plates. This is done by facing off the rod through half of its 1 mm diameter, leaving
a flat section 1 mm wide and typically 3-4 mm long. These rods are made of tungsten, which is
hard to cut or grind. The facing of these rods as well as any other cutting of tungsten rod describedthroughout was done with a water-cooled diamond-blade grinder.
The plates sit in grooves on the two macor pieces that hold all the electrodes together in thescience cell, which are shown in Figs. 7.3 and 7.19. The ITO coating on the plates extends to ≈ 1
mm from the ends along the long axis, and so the part of the plate that fits into the grooves on themacor holders is uncoated. This region is required to hold the plates during their ITO/AR coatingprocess. Figures 7.3 and 7.19 also show a hole in the back macor piece of diameter 1 mm throughwhich the rod extends to connect to the plate. The center of this hole is lined up with the bottom
surface of the plate, such that when the rod is faced off half way through, it will contact the plate.
No epoxy or conductive glue is used, so once the rod is faced off, it is angled in slightly such thatthe connection to the plate is maintained by the resulting cantilever spring force.
To ensure that the plate is centered on the macor holders and the rod connector, a small
slot is grinded ≈ 1 mm into the end of the plate with the diamond-blade grinder. The rod thenfits into this slot, and the faced-off last several mm extend into the ITO-coated region and contact
the coating. This approach also makes the connection between the rod and the ITO more robust.
We measure Ω-level resistance between the rod and the ITO coating, which is negligibly small
compared the ≈ 1 kΩ resistance of the ITO coating itself. This connection can be seen by looking
carefully at Fig. 8.15.

174

Figure 8.2: Electropolished steel and tungsten rods. The lower rod is SS316, and the upper rod istungsten. This image was taken with an optical microscope.

8.1.3

Electrode surface smoothness

Stark decelerators are typically designed using the least magnetic stainless steel, SS316. The
smoothness of the surfaces of their electrodes is very important, and is often the limiting factor
on the current drawn during steady state operation. SS316 can be electropolished nicely in a 10%
NaOH solution, which we did for all the steel parts of the electrode system. However, the electrodes
between the small cube and the science cell are tungsten to mitigate any magnetization problems,
which cannot be easily electropolished to my knowledge. In fact we sent some tungsten rods to beelectropolished by a company which claimed to be able to electropolish tungsten, and they cameback with crators and more surface roughness than they had originally. Images of electropolishedSS316 and tungsten can be seen in Fig. 8.2. We expect that the challenge with tungsten is thatthe rod is a sintered material, which is likely the cause of the cratoring upon electropolishing.
After exhausting the possibilities of electropolishing tungsten, we turned to the idea of mechanical polishing. We found that a buffing wheel and buffing compound work very well to polishtungsten. Buffing compound is an oil-based clay, called a vehicle, in which are embebbed microabrasives. We found that a grey compound works very well as a coarse polish, and then a red
ccompound works very well as a fine polish. This procedure was so effective that we could make

175

Figure 8.3: An SEM image of mechanically polished tungsten rods. (a) The end of the rod, withthe scale bar indicating 300 µm, and (b) the side of the rod with the scale bar indicating 20 µm.

our tungsten rods smoother through mechanical polishing than electropolished SS316 [118]. We
measured the surface quality using a Scanning Electron Microscope (SEM), and such images areshown in Fig. 8.3. The long, axial ridges seen in Fig. 8.3b are expected to be due to the mechanicsof the buffing wheel, which was mostly spinning against that axis.
Unfortunatley, the mechanical polishing approach has a huge downfall of which we did not
become aware until we prepared a vacuum system and attempted to reach ultrahigh vacuum (<10−11 mbar). The cleaning procedure was unable to completely remove this clay, which persisted in
the vacuum system with a relatively high outgassing rate. The outgassing was low enough that theoperation with the turbo-molecular pump during the vacuum pump-down procedure was typical,
and pressures in the 10−8 − 10−9 range could easily be reached. In fact, at the same time that wewere working on this, Noah Fitch in Heather Lewandowski’s group was building their travelingwave stark decelerator, which is made of tantalum electrodes. They similarly had a difficult timeelectropolishing tantalum, so they decided to electropolish and mechanically polish their electrodes
with the same clays that we used. Their vacuum system is very different than ours, and they were
ultimately able to reach sub-10−10 mbar in their large, open chamber using several gigantic turbo
pumps.

176

Figure 8.4: SEM images of blemishes on the tungsten rods, with and without being mechanicallypolished. White objects are dielectric materials, since they reflect electrons.

As discussed in Chapter 7, our vacuum system needs to reach sub-10−11 mbar pressures,and the pumping speeds and conductances are limited. Therefore, the additional gas load fromthe buffing compound is unacceptable. Indeed, we found that when we baked the chamber, thepressure was higher after the bake than before, indicating that the clay had been activated anddispersed at high temperature. The pressure levels at this stage required an ion pump to reach,and thus the dispersal of this clay got into the ion pump and rendered it unusable. It took us
several tests to learn that the problem was indeed coming from the clay, and several ions pumps
were contaminated in the process since the pressure would saturate the range of a turbo pump. In
the end this was a rather expensive and frustrating process.
We were never able to find smoking gun evidence that the clay was causing the problem,
merely a 100% correlation during the vacuum tests. The only way we could imagine detecting clay
residue on the tungsten rods was with the SEM. Dielectric materials appear white on a SEM because
they reflect electrons, unlike conductors like tungsten. Therefore, we suspected that perhaps we
could find white clumps near points of elevated surface topography like bumbs or ridges. Figure 8.4
shows such images for polished and unpolished rods. We certainly found more white clumps on thepolished rod, but we also found some white-ish sections on the unpolished rod. Hence, I would say
this measurement was suggestive and perhaps even circumstantial evidence, but still not definitive
proof.

177

Figure 8.5: A schematic of the conditioning layout. The 0.7 gain op amp circuit was used to ensurethat the voltage to the ampifier never exceeded ±10 kV.

Since both electropolishing and mechanical polishing with buffing compound have now been
ruled out, we decided to test how important it actually is that the tungsten rods are polished.
Stark decelerators often reach E ≈ 100 kV/cm, which is well beyond the fields we wish to reach.
Moreover, Stark decelerators are designed to have relatively small fields at the insulators such thatthey are not limited by surface currents on the insulators, as discussed in Chapter 7. However, ourmacor insulators are in the positions with the largest electric fields, and thus they will limit theultimate field we can reach. Therefore, it is possible that our system would never be limited by thesmoothness of the electrodes.

8.1.4

Conditioning the electrodes

Therefore, instead of mechanical polishing with buffing compound we used three consecutively
finer sandpapers of grits 2400, 4000, and 12000. These were primarily used on the ends of the rods
and the faced-off sections where contact the ITO-coated plates. We performed many tests with justtwo rods, both smooth and faced off at the end. We found that the limitation of rod smoothness
only comes from small burrs and spikes on the faced-off rods, but the ultrafine sand paper is actuallyquite helpful to mitigate these problems.
Conditioning of electrodes is a violent process by which small spikes and other featureson the surface of the rods ablate each other by shooting a high-energy electron beam. These178
features serve as nucleation sites like a lightning rod, and eject or receive an electron beam. We
performed a conditioning procedure, similar to what is used in Stark decelerators [80, 118, 245].
This conditioning data was taken using molybdenum plates instead of ITO-coated glass, for reasons
to be discussed in a later section. Conditioning is done by progressively increasing the voltages
to the electrodes, and measuring the current response as a function of time. A schematic of thesystem is shown in Fig. 8.5. We performed most conditioning measurements with the same voltage
on the top plate and top rods such that only two voltages were used ±Vtest .
To limit the current that can be drawn during the electron beam ablation, we put 100 MΩ
resistors in series with the rods, and thus Ilim = Vtest /R = 100 µA for Vtest = 10 kV. However,this is still a lot of current. An interesting feature of our apparatus is that the electrodes are quite
visible inside the glass cell, as opposed to a Stark decelerator. Moreover, in a dark room the human
eye is nearly a single photon detector in the blue wavelength range, and thus we can literally see
every electron emitted by the electrodes during conditioning. Accordingly, this measurement isboth really exciting and really terrifying.
Figure 8.6 shows typical data from a conditioning measurement. The upper plot shows ±Vtest
as a function of the time during conditioning. Note that progressively more time is spent at eachvoltage since it takes longer for the transient current response to stabilize. As discussed abve,
the smoothness of the tungsten rods (besides the faced-off connections to the plates) was never asource of large current and never limited the voltages we could reach. Rather, the developmentof small surface paths on the macor insulators in the science cell and throughout the transmission
line to the feedthroughs has always been the dominant breakdown mechanism. The total surfae
area of the electrodes in our system is incredibly small compared to a Stark decelerator. Therefore,
each individual current spike, or electron emission, could be detected. Instead of drawing a finite
current that changed slowly, we often drew very little current ≈ 1 − 2 µA, and had discrete spikesof ≈ 5 − 10 µA beyond the background, whose frequency would vary.
The lower plot shows the time in seconds between these discrete spikes. For Vtest < 4 kV these
spikes were very infrequent, and spaced by > 60 seconds. As the voltage increased, however, the179

Figure 8.6: The voltage and current during a typical conditioning measurement. Such small spikes
are typically 5 − 10 µA.

180
spike frequency increased until it saturated at 10 − 15 seconds/spike. This is a very encouranging
response, and the field reached in this test was 32 kV/cm. We typically only apply the electric fieldfor ≈ 1 second, so it is unlikely that any damage could occur at these timescales even at larger
voltages. However, it would be beneficial to understand where on the electrode system the spikes
are occurring. Any spikes inside the science cell would be visible by eye, and so we can carefully
look for flashing during these conditioning measurements.
Figure 8.7 shows the observation of some blue flashing which was observed inside the pyrex
test cell that was used for these particular conditioning measurements. The location of the emission
is on the left edge of the front macor holder which is gently pressed against the science cell’s end
window. Thus, the breakdown mechanism involves the macor rather than the surface smoothness,
as anticipated. While conditioning is a self-healing process that smoothes the electrodes, thedevelopment of surface currents in the insulators is a run-away problem that will continue tolower the voltages that can reached in the experiment. Indeed, during the conditioning of a Stark
decelerator it is common to go to voltages a few kV above the operatoring voltages such that
operation is even more stable. However, when destructive processes occur in the insulators thisis not an option. Therefore, it is important that we do not consistently operate at such voltages
where this breakdown occurs in the final assembly.
Another important limitation of having the insulators in the region of maximal electric fieldis the presence of triple points, as discussed in Chapter 7, which can lead to rf breakdown and thedevelopment of an electron beam. These issues are particularly acute in a small glass cell whereother dielectric surfaces are relatively near the triple points. The front macor holder used in thistest had macor sleeves which extend well beyond the end of the rod electrodes, thus leading to θ
being small as discussed in Chapter 7. The generation of an electron beam was clearly apparent
during some of our measurements at high voltages. This beam only originated from the rods withnegative potentials, as expected, and the beam was clearly directed towards the end window.
The evidence of this beam was the development of a circular deposit of macor powder on theend window. The generation of this instability was quasi-periodic, where every several seconds a

181

Figure 8.7: Observation of blue flashing in the science cell during the conditioning measurement.
The blue emission is localized to part of the front macor holder which is gently pressed against thescience cell’s end window.

Figure 8.8: A cartoon on the observation of triple-point-induced electron beam breakdown. The
circles on the left are macor powder dust on the end window, and the orange lines or arrows show
the electron beam. The presence of the macor insulator (green) beyond the electrode (blue rod)has an enormous bearing on our observations.

182
small burst of macor powder would hit the end window followed by a large burst. We believe thistimescale corresponds to the growth of an rf breakdown instability near the triple point. Figure 8.8
shows a cartoon of what we observed. The circles on the left are macor powder dust on the end
window, and the orange lines or arrows show the electron beam. The presence of the macor insulator(green) beyond the electrode (blue rod) has an enormous bearing on our observations. It appears
that the solid angle subtended by the electron beam is influenced by the presence of the macor, andthe macor dust depositing on the end window is caused by the electron beam penetrating the macor
sleeve. Note that one of the negative rods had this problem while the other did not, which suggeststhat it could be exacerbated by a particular rounding shape of the end of the rod. Nevertheless,this problem has been entirely solved simply by having the macor sleeve not reach the end of therod, as discussed in Chapter 7.
While these macor-related problems must be taken seriously, there is a lot of reason to
be optimistic. First, the triple point issue has a simple solution which immediately corrected
the problem. Second, as discussed in Chapter 7 the homogeneous electric field occurs when therod voltages are at 42.25% of the plate voltages, and thus will never exceed ±4.225 kV. The
surface currents on the front macor insulator developed at ≈ ±7 kV, and thus are negligible in
the homogeneous field configuration. Third, we operate these electric fields with a very small duty
cycle. Unlike Stark decelerators that use large electric fields to produce cold molecules, we operate
an ultracold atom experiment where molecular manipulation with large electric fields only happens
for roughly 1 second every 30 seconds. Fourth, the onset of the macor-related instabilities takes
several seconds. Therefore, even if they were limitations, they could potentially be circumvented
simply by apply the field for short times. Fifth, as discusssed in Chapter 7, Jz is actually maximized
at the relatively low field of 10 kV/cm, and so such large fields are necessary only for a limited
number of experiments, such as evaporative cooling of molecules.

183

Figure 8.9: Damaged ITO-coated plates after conditioning. Notice that the damage on the positiveplate is much worse than the negative plate.

8.1.5

Reaching large fields with the ITO-coated plates

The above conditioning measurements were done with molybdenum plates rather than ITOcoated fused silica plates, and thus we must repeat some of these measurements using ITO-coatedglass plates instead. One particular test we did was just with two ITO-coated plates, and the
faced-off rods that connect to them as discussed above. This test was done without using theultrafine sandpaper mentioned above, and elucidates the need for such a procedure. The faced-off
rods have sharp edges and small burrs which ablate away throughout the conditioning process as
expected. The proximity of the ITO-coated plates, however, makes this ablation process particularly
damaging. The negative faced-off rod shoots electrons at the positive faced-off rod, but they subtend
some angle around the faced-off rod and hit the plate. Figure 8.9 shows the damage to the ITO
coatings that result, and it is clearly much worse on the plate which had the positive voltage. Theedge of the ITO coating and connection slot are apparant at the bottom of the plates.
The dark moat on the positive plate around the contact point of the rod shows how thecoating was locally destroyed, and the ITO at the contact point is completely electrically isolated
from the rest of the coating. This is obviously a huge problem as the ability to control the plate
voltages is lost and they simply float to whatever potential. Moreover, there is some evidence of184
damage in one of the corners at the far end of the plates. This is due to the makeshift way theplates were held where there was no support at the far end, and the plates got too close together.
Nevertheless, these measurements show the propensity for damage from such fragile coatings insuch a violent atmosphere.
Our solutions to this problem are two-fold. First, the ultrafine sandpaper enormously improved the smoothness of the faced-off sections of the rods and mitigated the ablation when applying
large fields. However, it is still not quite smooth enough to entirely avoid damaging the ITO-coating.
Second, the final assembly was done first without any plates present, and the conditioning was done
as such. Then, once the conditioning had been performed and the faced-off rods were exquisitely
smooth, the vacuum system was opened and the ITO-plates were inserted. This approach allowed
us to reach fields of ≈ 40 kV/cm without the plates with minimal current draw and no dischargeor spiking in the science cell, and then ≈ 30 kV/cm when the plates were added before the onsetof any spiking in the science cell.

8.1.6

Large fields with high intensity beams through the ITO

I will now discuss additional conditioning measurements using ITO-coated plates where highintensity beams are incident on the coating during the conditioning measurement. It is possiblethat the photons from a high intensity beam could facilitate the emission of electrons from theconductive coating, in accordance with the photo-electric effect. Fortunately, the work functionof ITO is ≈ 4.2 − 4.8 eV depending on the coating method and thickness, which is quite high.
Nevertheless, the energy of the work function corresponds to an optical photon of wavelength of
280 nm, within a factor of 5-6 of our lattice wavelength of λ = 1064 nm.
We performed the conditioning measurements with a high intensity beam focused on the
ITO-coatings, as shown in Fig. 8.10. The highest intensity we anticipate using in the experiment is
a 5 W beam focused to a 1/e2 radius of 50 µm. Therefore, to match this intensity we used a 1 W
beam focused to 20 µm. The Rayleigh length of such a small beam is sub-mm, however, and so weused a translating mount for the focusing lens to ensure that we could have the highest intensity

185

Figure 8.10: The setup for ITO-coated plate conditioning with an incident high intensity beam.
The focusing lens could be translated to ensure that the highest intensity was hitting the ITO
coating.

186

Figure 8.11: The output mode of a Mach-Zehnder interferometer with an ITO-coated plate in
one arm. This mode was measured for low power and high power for several conditions of the
ITO-coated plate, such as before and after baking to a high temperature.

incident on the ITO-coatings. During the conditioning measurements there was no effect of the
focused beam on the current drawn, and no discharging was observed even up to potentials of ±9
kV, or 30 kV/cm. This was an excellent result for the operation of large electric fields with anincident high-intensity beam.
Another question, however, if what effect the high intensity has on the optical properties of
the ITO-coating. Since ITO is a semiconducting material, it has finite absorption near the band gap
in the IR range, and the absorption at λ = 1064 nm is proportional to the coating thickness. Whilean AR coating paired with the ITO can alter the reflection and transmission properties defined byRe(n(λ)) where n is the index of refraction, the absorption caused by Im(n(λ)) is an unalterable
property of the material. As discussed later, the absorption of our ITO coatings is ≈ 2%, which
could cause heating that in turn changes the transmission or absorption, or even causes thermal
lensing.
To study this we put an ITO coated plate in one arm of a Mach-Zehnder (MZ) interferometer.
The output fringe on the optical mode of the recombined beam was then monitored on a beamprofile camera, as shown in Fig. 8.11. We looked at this mode for both low intensity and highintensity under several circumstances. As discussed later, we have observed that optical propertiesof the ITO coating change irreversibly upon baking, and so we tested an unbaked plate as well as

187

Figure 8.12: Activating the Titanium-sublimation pump, as seen from the small cube. These images
are from the small cube where the electrodes are turning the corner and heading to the science cell,which is to the left. The blackbody radiation from the filaments can be seen with the room lighton or off.

plates that had been baked to various temperatures. For all of these cases, however, we saw nodiscernible change in the optical mode of the MZ interferometer output even after minutes of hold
time.

8.1.7

Effect of Titanium-Sublimation pumps on electrode insulators

Another potential compatibility issue in this apparatus is high voltage and ultrahigh vacuum.
In Chapter 7 I discussed the vacuum conductance and outgassing limitation of the electrodes andinsulators in the vacuum, and I will come back to this issue in the next section. However, we must
consider the effects of the titanium-sublimation pump, which operates by sputtering titanium onto
the chamber walls. Titanium is a gettering material, which means that it serves as a pump andoften can be used to reach pumping speeds of ≈ 1000 L/s when sputtered onto a large enoughsurface area.
This type of pumping technology is rarely used in a system with nearby high voltage electrodes, and an obvious issue is whether a titanium layer could develop on any of the macor insulators. Even a nm-thick layer would cause enormous problems with surface currents developing
between the electrodes. Therefore, we must place the Ti-sub pump in a position which has minimal

188
contact with any electrode or insulator. The titanium is sputtered onto an area determined by aline-of-sight from the Ti-sub pump filaments. Therefore, we placed the pump in its own arm on atee behind the large cube which houses the feedthroughs. The pump is activated by heating thefilaments to a temperature at which titanium has a large vapor pressure, at which it is red hot.
Figure 8.12 shows a view from the small cube back into the large cube where the Ti-sub pumpis behind the electrodes in an arm that is oriented vertically. The blackbody radiation from thefilaments can be seen with the room lights on or off.
To test the effect on the electrodes of Ti sputtering in the chamber, we compared a conditioning test before and after the first activation of the Ti-sub filaments. Fortunately, we did not observe
any difference in the current drawn or the nature of any current spikes. Therefore, we sufficiently
shielded the Ti-sub filaments from any electrodes in the large cube area. Yet, its pumping effects
are still apparent as we have been able to reach excellent pressures in this chamber (< 10−11 Torr),
to be discussed later in this chapter.

8.2

Electrode parallelism
Now I describe the design and testing of all the electrodes, and the procedure used to assemble

everything between the feedthroughs and the science cell. As was discussed in Chapter 7, thehomogeneity of the field that can ultimately be reached is limited by how parallel all the plate androd electrodes are in the science cell. I will begin by discussing how the electrodes in the sciencecell were assembled and what angle we were able to reach.

8.2.1

Assembling the electrodes in the science cell

The entire electrode assembly inside the science cell was assembled on its own, and theninserted into the cell like a ship in a bottle. Therefore, the electrode assembly had to be quite rigid
and robust, and the cell was only used to indicate the assembly relative to the chamber. I will
begin by discussing how this was accomplished.
As the hexapole configuration of the rods approaches the science cell, all six rods are angled

189

Figure 8.13: Rigs for accurately bending the electrodes. The left image shows a solid model of the
rig used to bend the electrodes near the science cell, and the right image shows both rigs with abent rod between the dowel pins.

together to reach a smaller separation for larger electric fields. However, the ability to do thisrequired that all six rods were bent in exactly the same way, and so a bending rig was needed to do
this accurately. Another rig was used to taper the rods from the large cube down to the small cube,
to be discussed later. These rigs are shown in Fig. 8.13, and are simply dowel pins protuding froma phenolic block. The positions of the pins were chosen to move the rod by the desired amount,and the angle was chosen based on the chamber or cell tapering around the rods.
Figre 8.14 shows the six tungsten rods that are bent to go into the science cell. The top fourrods go between the ITO-coated plates, and the bottom two will be faced off to contact the plates.
Note that the angle with which the bottom two rods is faced off must exactly match the plane of
the bend of the rod. These six rods are rounded at the tip using a polishing wheel. This image wasbefore any sanding or cleaning of the rods.
While tungsten is very hard, it is also a brittle material. Roughly one third of the times wetried to bend the tungsten rods in this way to go into the science cell, they splintered at the bend
like a piece of wood. The fracture was always obvious, but we looked at all the rods using an optical
microscope to ensure that no microfractures were present. We were unable to find a correlation
between the fracturing and anything else about the rods, but we assume it must be related to how
the internal domains and microlayers are aligned relative to the bend angle.
Once the rods were bend, a fine angling procedure using graphing paper was employed to190

Figure 8.14: Tungsten rods bent to go into the science cell. The top four rods go between the
ITO-coated plates, and the bottom two will be faced off to contact the plates. This image wastaken after the rod tips were rounded, but before any cleaning.

191

Figure 8.15: The first assembly of the electrodes including the ITO-coated plates. The left image
is with the ITO-coated plates, and the macor piece to the right of the bend is replaced with analuminum mounting block with six holes arranged in a hexapole pattern. Note how the faced-off
rod is contacted with the ITO-coated plate. The right image shows the assembly using all themacor pieces, but the plates have not yet been included.

ensure that the sections of the rod on either side of the bend were parallel. Moreover, the rods wereplaced on a flat surface such as a super-polished granite block to ensure that they lay on a plane,
and that there were no angles perpendicular to the plane defined by the bend. This procedure was
useful, but a final angle adjustment was done with the electrodes in their assembly.
With the ability to bend the rods we were able to assemble the electrodes to be inserted into
the science cell. Figure 8.15 shows the first assembly of all the electrodes together, including the
ITO-coated plates. The left image shows a metal rig just after the rods bend up to a larger radius
which was used occasional instead of the macor piece so that small tweeks to the rod angles could
be made without inducing too much stress in the fragile macor pieces. The length of the rods was
ultimately chosen such that the connection in the cube before the science cell gently pushed theelectrode assembly against the end window of the science cell. This will be discussed more in alater section.

192
8.2.2

Measuring the parallelism and straightness of the rods and plates

One of the first things we wanted to test after assembling the electrodes is how parellel the rods
and plates are to one another. The angle between the plates could be measured interferometrically,
but this approach would be difficult to extend to the rods. Therefore, we used a dial indicator ona super-polished granite block to measure these angles, as shown in Fig. 8.16. The indicator was
swept along the rods and plates, and the change of height was measured over their length. This
allowed us to measure angles as well as bowing.
We found that the macor pieces which hold the electrodes inside the science cell had toexactly match the diameter of the rods and the thickness of the plates to minimize the angles. Any
tolerance caused a much larger angle. While this may be obvious, it made the assembly much moredifficult because the variation in diameter between the different plates and rods was important.
Therefore we had to choose four rods that had the same exact diameter and similarly two plateswith identical thickness, at the level of 1/10, 000 inch or 2.54 µm. This was also challanging on themachining side, since the rods and plates are already very small, and macor is incredibly fragile
and brittle.
Nevertheless, in doing so we were able to get the two plates parallel to within ≈ 5 µm relative
separation over their length. We were able to reach similar levels with the four rods, though slightbowing due to stress was observed. However, when we put everything together with the rods
contacting the plates we found that these numbers went up to ≈ 10 µm for the rods and plates.
This is still quite the accomplishment, and puts us within the flatness goals discussed in Chapter 7,
potentially allowing for investigations of quantum magnetism in a large DC field. It is important
to note, however, that we don’t know how much this was altered by the baking procedure of thechamber, and the moving of the new chamber (see Chapter 9). Since tungsten and macor bothhave very low coefficients of thermal expansion and tungsten is very hard, we expect that thesenumbers have changed only slightly.

193

Figure 8.16: Measuring the angles between the rod and plate electrodes. A dial indicator on asuper-polished granite block was swept along the length of the rods and plates to measure theirrelative angles.

194

8.3

AC electric field coupling
Now I will go on to discuss the assembly of the electrodes from the feedthroughs to the large

cube. This region includes the coupling capacitors and decoupling inductors in the bias tee. This
discussion will include measurements of the AC coupling at various parts of the transmission linefrom the feedthrough to the science cell.

8.3.1

Building the coupling capacitors

The design of the coupling capacitors was described in Chapter 7, so I will now discuss how
they were built. Figure 8.17 and 8.18 show the four capacitor pillars mounted on the bottom
flange in the large cube. A four-pin low-power feedthrough is below this flange, and the four macor
standoffs from the flange contact the four pins on the vacuum side of the feedthrough, eminatingfrom the hole shown in the middle (see Fig. 8.18). The top of the standoffs then go to steel disks, on
which the macor piece sits. This macor piece serves as the dielectric material between the capacitor
plates. These steel disks serve as the bottom capacitor plates, and the top capacitor plates are thefour pillars shown. The holes near the top of the plates are where the inductors from the highvoltage feedthroughs will connect, and ultimately go down the transmission line to the science cell.
Even though the four-pin feedthrough is not designed for coupling AC and its impedence isnowhere near 50 Ω, we chose this design so that we could control oustide the vacuum chamber
how the AC signal is connected to the four rods. This will ultimately allow us to control thepolarization of the microwave field inside the electrodes, and thus the angular momentum of therotational transitions we can drive. As discussed in Chapter 7, the capacitor area and gap werechosen such that this capacitance is ≈ 10× larger than the capacitance between the rods in thescience cell to make an effective AC voltage divider that will optimally couple the majority of the
AC power onto the rods in the science cell.
Figure 8.19 shows how the pillars for the capacitors are connected with the inductors (right)
and into the transmission line which goes to the science cell (up) in the large cube. These images

195

Figure 8.17: The capacitor design and construction for coupling AC signals onto the rods. The left
image shows the capacitor design on the bottom flange of the large cube, attached to which are
low-power feedthrough to couple in the AC from outside the chamber. The right image shows thecapacitor assembly inside the large cube.

Figure 8.18: Designing the coupling capacitors.

196

Figure 8.19: Assembly of the capacitors and inductors and their connection to eachother and thetransmission line to the science cell. These images show a time sequence left to right of eachsuccessive electrode being connected.

197
show a time sequence left to right of each successive electrode being connected. To the left isa two-pin high voltage feedthrough (Kurt J Lesker EFT1223093, Copper, rated to 30 A and 12kV). These feedthroughs ultimately connect with the ITO-coated plates and do not go throughan inductor or connect to a capacitor because no AC is applied to the plates. To the right isa four-pin high voltage feedthrough (otherwise identical), and an inductor is connected to each.
These inductors connect to the pillars which are the top capacitor plates. Another rod then goesup to the transmission line to the science cell. Needless to say, the large cube assembly was very
challanging to build.
Figure 8.20 shows other views of the fully assembled large cube with all the inductors andcapacitors connected. The top flange in the left image is where an inline valve will be placed andused for connecting the turbo pump during vacuum preparation. The right image is from thelocation of the ion pump. The vertical flange on this tee is where the arm for the Ti-sub is housed.
Past the macor insulating piece in the right image the rods go to the small cube, and then turnright to go to the science cell.

8.3.2

Designing the decoupling inductors

The design constraints for the decoupling inductors are less stringent since the long highvoltage cables to the high voltage amplifiers are very poor AC transmission lines and are wellshielded with braided grounding wires. The goal is to avoid having a resonance in the coil over thefrequency range of interest, which is between 2.2 GHz and 8 GHz corresponding to the N = 0 to
N = 1 transition between 0 kV/cm and 30 kV/cm, respectively.
Figure 8.21 shows a typical coil, which is made by winding 1 mm diameter steel wire arounda large screw. Various combinations of screw diameter, pitch, and number of turns were explored.
The best inductor we observed is shown here. Its reflection as a function of frequency was measured
using a network analyzer which goes up to 9 GHz. A large dip in the reflection shows a coil resonance
where transmission is maximized, which we want to avoid. This scan shows that there are no dips
between 2 and 8 GHz. The marker is at 2.1 GHz, and the two small dips to the left are below our

198

Figure 8.20: The fully assembled large cube with all the inductors and capacitors connected. The
top flange in the left image is where an inline valve will be placed and used for connecting theturbo pump doing vacuum preparation. The right image is from the location of the ion pump. The
vertical flange on this tee is where the arm for the Ti-sub is housed.

Figure 8.21: A decoupling inductor and its resonances measured by its reflection as seen with a
network analyzer.

199

Figure 8.22: A PCB designed to couple the AC signal into the transmission line. The left image
shows the PCB whose four holes go around the four feedthroughs, and the right image shows how
it all looks when connected together.

frequency range. Most of the other inductors we tried had dips of ≈ −20 dB between 3 and 6 GHz.

8.3.3

Measuring the coupling into the vacuum chamber

We couple AC signals into the large cube through the four-pin feedthrough whose pins connect
to the four bottom capacitor plates inside the large cube. We use a custom, yet very simple, printed
circuit board (PCB) below the large cube which takes an SMA bulkhead and breaks it into fourholes with the spacing of the four feedthroughs. This PCB and its connection to the feedthroughs
are shown in Fig. 8.22. Two connections are AC ground, and two are AC signal. By simplingchanging the orientation of this PCB with respect to the feedthroughs we can control whether theAC electric field oscillates in the horizontal plane or vertical plane in the science cell.
The first thing we wanted to measure is how effectively the AC signal is coupling onto thefeedthroughs and through the capacitors in the large cube. These measurements were made with
the 9 GHz network analyzer and a fast oscilloscope probe which operates up to 10’s of GHz and canbe touched to the electrodes with minimal perturbation to the boundary conditions of the system.
We found that the insertion loss in going from the 50 Ω coaxial SMA bulkhead to the four-pin
feedthroughs is ≈ 5 dB. The coupling loss through the capacitor depends on the frequency, but
is typically 10 − 20 dB. The left and middle images of Fig. 8.23 show in blue the transmission

200

Figure 8.23: The AC coupling through the transmission line. The left image shows the transmission(blue) through the feedthrough rods, but before the capacitors. The middle image shows thetransmission (blue) through the capacitors, and on the top of the steel pillars. The right image
shows the transmission (blue) to the electrodes in the science cell.

measured before and after the capacitors, respectively. These images show that low frequencies
around 2-3 GHz couple very well through the capacitors, but the coupling falls off at around 6 − 7
GHz.

8.3.4

Measuring the coupling along the transmission line

The transmission from the top of the capacitors all the way to the science cell can be seen bycomparing the middle and right images in Fig. 8.23. This was tested by glass-blowing a small hole in
the pyrex text cell shown in Fig. 8.7, and placing the AC scope probe right against the pair of rods
on one side. The PCB is oriented to drive the AC field in the vertical direction, but we confirmedthat the results are virtually the same when it is being driven in the horizontal direction. The
transmission is generally very good, with a relative dip of at most 10 dB in at certain frequencies.
This is very promising, and suggests that the AC voltage divider is working very well.
In total we lose ≈ 25 dB on average in going from the coaxial SMA bulkhead all the way tothe science cell. Our microwave amplifier allows us to put ≈ 35 dBm onto the SMA bulkhead, andso we should be able to get ≈ 10 dBm to the electrodes in the science cell. In the old apparatus, weused a large broadband horn to couple microwaves to the molecules, and the horn’s transmission
was severely blocked and perturbed by the IP coils around the old science cell. The polarization

201
from the horn is roughly linear, but was completely scrambled upon going through all the coils.
We used to use ≈ 10 dBm to drive the |0, 0i − |1, ±1i transition and ≈ 33 dBm for |0, 0i − |1, 0i.
However, it is difficult to estimate the microwave intensity within the volume of the cloud.
Since the IP coils severely altered the boundary conditions for the field, it is likely that the coupling
into the coils was attenuated by > 10 dB. Moreover, the power from the coils subtends a large solid
angle, and so the power within the volume of the cloud is also likely attenuated by a similar factor,> 10 dB. The electrodes in the science cell, on the other hand, are spaced by less than a wavelength
at these microwave frequencies, and thus the entire power of the microwave field that makes it thatfar down the transmission line will couple directly to the molecules. With these considerations inmind we anticipate that we will be able to drive rotational transitions with Rabi frequencies similar
to if not better than in the old apparatus.

8.4

Chamber assembly procedure
I have discussed how the electrodes are assembled within the science cell and within the large

cube. It is now time to discuss the sections in between, such as going around the corner in thesmall cube and the reducer between the small and large cubes.

8.4.1

Turning the corner in the small cube

Figure 8.24 shows the small cube and how the electrodes are connected inside. In the left image, the science cell is above. In the middle image the science cell is to the left and the feedthroughs
are to the right. The right image is taken from the MOT side of the chamber, and cold atoms comealong this direction first in a magnetic quadrupole trap and then into the science cell in a optical
dipole trap. As discussed above, all the electrodes are already assembled inside both the sciencecell and the large cube.
This small cube serves several purposes: 1) it connects the science cell electrodes to the large
cube electrodes, 2) it allows cold atoms to pass in from the MOT side of the chamber such thatthe quadrupole magnetic trap can move over the small cube, and 3) it allows us to change from the202

Figure 8.24: Connecting the electrode transmission line in the small cube. In the left image, thescience cell is above. In the middle image the science cell is to the left and the feedthroughs areto the right. The right image is taken from the MOT side of the chamber, and cold atoms comealong this direction first in a magnetic quadrupole trap and then into the science cell in a optical
dipole trap.

203
tungsten in the science cell to the SS316 used in the large cube. This is shown in Figs. 8.24 and 8.25.
The connection was made using right-angle pieces of SS316, with three different lengths on eitherside of the angle (see the middle image of Fig. 8.24). These right angle pieces are connected to therods on the science cell side and the large cube side using pieces of SS316 hypodermic tubing whichis slid over both rods to be joined, and then crimped on both sides of the junction. Such pieces oftubing are shown over the rods in the left image of Fig. 8.24.
This procedure was highly nontrivial, and several custom tools were needed to successfully
crimp the rods in place. The SS316 rods on the large cube side are rigidly secure. However, as
discussed above, the rod positions in the science cell are not rigidly connected to anything. Thus,
the goal is to gently push the two tungsten rods that connect to the plates, which will in turn movethe entire electrode assembly such that the front macor piece is gently contacting the end window.
Then, the other four rods can be positioned such that they all extend the same amount past themacor sleeve on the front macor piece. The role of the hypodermic tubing and the right-angle
pieces is to secure all six of these electrodes in place. This worked very well, but again was noteasy to do.

8.4.2

The reducer going from the large cube to the small cube

The remaining connection to discuss is shown in Figs. 8.25 and 8.26, and is the reducer
between the large cube and the small cube. Between the large cube and the reducer, and between
the reducer and the small cube are double-sided blank flanges which were modified to hold themacor holders on either side of the reducer. The rod sections between these two macor holders
were made with the larger rod-bending rig discussed above. The length of these rods was chosen
to extend a specific distance on either side of the reducer.
On both sides hypodermic tubing was used to connect it to the next rod section: the rightangle piece on the small cube side, and a rod to the top of the capacitor pillars on the large cubeside. The large side of the reducer is shown in the right image of Fig. 8.26, and this shows how thehexapole guide in the large cube is reduced down to the hexapole guide in the small cube. Once

204

Figure 8.25: A solid model of the electrode transition which occurs in the small cube. This figure
shows how the connections are done, and how we go from tungsten in the science cell to SS316 in
the large cube.

205

Figure 8.26: The reducer between the large cube and the small cube.

this reducer was assembled, it was first connected to the large cube. Then the small cube andfinally the science cell were connected to it. The NEG pump discussed in Chapter 7 is connectedto the small cube opposite the reducer.

8.5

Aligning the electrode assembly to the cell windows
Another important question arising during the assembly of the small cube part of the trans-mission line is how small the angle can be between the ITO-coated plates and the top and bottom
windows of the glass cell. This will affect the amount of coma aberrations in the high resolution
imaging system, and will ultimately inform the resolution which we can reach without correction
optics.
During the connection phase with the right-angle pieces and the hypodermic tubing, theslots in the macor insulating piece on the flange of the science cell were left loose. This allowed
the electrode assembly inside the science cell to be rotated as the connections were made. The
angle between the ITO-coated plate and the cell windows was measured using a beam of λ = 1064
propagating through the cell in the vertical direction. This allowed us to look at the reflections

206
from both substrates over a long distance from the cell, from which we could measure the relative
angle.
As the electrode assembly was rotated in the cell, this angle inevitably crossed zero degrees,
however it was very difficult to prevent it from rotating as the connections were made and thefasteners locked the slotted macor insulator in place. In the end we were able to reach < 0.1
degrees, but not much better than this. The coma caused by this angle is sufficiently small to allow
nearly diffraction-limited performance with an NA = 0.53, but will likely become more of an issue
as we try to reach NA = 0.65 in the future.
Another point of concern, however, is how much this relative angle may have changed during
the extensive baking phase and the final moving of the chamber. Recently we measured this angle
again to be ≈ 0.2 degrees. This is starting to become more of an issue for our NA = 0.53 imaging
system, but it is still sufficiently small to be relatively unimportant at this stage in the developmentof the experimental capabilities.

8.6

Reaching ultrahigh vacuum
There were many challanges that needed to be overcome before we ultimately reached excel-

lent pressures in the new apparatus chamber. Not least of which was the contamination issues withthe buffing compound. Therefore it was imperitive that none of the vacuum components we wereusing even contacted this compound. However, there were many other issues which were essential
to reach ultrahigh vacuum.

8.6.1

Eliminating virtual leaks

Virtual leaks inside a vacuum system are a very well known problem, and so it was relatively
straightforward for us to avoid them sufficiently well. Of course every vacuum chamber has virtual
leaks, so the goal is really to make them small enough to be insignificant at the desired pressure.
There are four main sources of virtual leaks in this system with trapped volumes that need to bevented: 1) inside the hypodermic tubing sleeves, 2) inside the holes tapped into the custom flanges

207
which hold the macor insulating pieces, 3) inside the macor standoff below the bottom capacitor
plates, and 4) inside the copper rods of the high voltage feedthroughs in which the inductors areinserted.
These four sources of trapped volume are vented with small holes in their sides. The hypodermic tubing sections were cut to length with a dremel, which was then used to make a small slitin the side towards the middle of the length. Then the hypodermic tubing sections were mounted
such that the slit was also within the trapped volume between the two adjacent rods (typically
only a few mm apart). The custom flanges were tapped through in all cases where they mate withthe macor insulating pieces using a fastener. This approach allows us to avoid a trapped volume
at the bottom of the screws. The macor standoffs below the bottom capacitor plates have a holein their sides so that there is no trapped volumes. The same is true for the copper rods on the
feedthroughs. These were nontrivial to drill, especially using isopropanol as a cutting fluid sincevacuum feedthroughs cannot be cleaned easily.
There are several remaining sources of trapped gas, such as in every thread. Such sources
can be quite long since their length is the integrated helix of the thread pitch, and they empty
themselves very slowly since their width is only the slop between the mating threads. This type ofleak is ubiquitous, however, and seems largely insignificant even at the level of < 10−11 mbar. The
different coefficients of thermal expansion between the mating materials (i.e. steel and macor) may
help during the baking since the treads of the two materials move relative to each other.
The other major source of virtual leaks in this chamber is between adjacent surfaces, such as
the macor holders and the rods within the sleeves, or between the macor holders and the custom
steel flanges to which they are connected. Similarly, there are large surface areas between the disksof the capacitor plates and the macor piece which serves as the dielectric material. In most ofthese places additional sleeve length of macor is used to maximize the surface path length between
electrodes, which again demonstrates the competing nature of high voltage and high vacuum.
While these trapped volumes have been insignificant, it is likely that the baking process has helped
significantly to slightly rub these surfaces against each other and open small gaps from which the208trapped volumes can escape.

8.6.2

Cleaning procedures

Due to the large number of surfaces and materials in the chamber and the vacuum requirements of < 10−11 mbar, we developed a specialized procedure for preparing the vacuum chamber.
Some of these steps originated from our unsuccessful efforts to remove the buffing compound fromthe electrodes, and other steps were helpful to breakdown the wax or paint used in the machining
procedure for macor. It is not clear how important all of these steps are when there is no buffing
compound in use. For example, dichloromethane is excellent at removing tough greases and evendissolves clays.
I will begin by listing the procedure for the metal parts in the chamber, which includes all
the chamber components, screws for electrode-related assembly, and all the electrodes:• 10 minutes in the ultrasonic with Bio T Max citrus solution (highest setting)• Rinse with water and Simple Green degreasing detergent (all-purpose cleaner)• 10 minute soak in dichloromethane (just a rinse for chamber parts too large to soak)• 10 minutes in the ultrasonic in Alconox detergent and water (highest setting)• Rinse in deionized water• 10 minutes in the ultrasonic with acetone (highest setting)• 10 minutes in the ultrasonic with isopropanol (highest setting)• 10 minutes in the ultrasonic with methanal (highest setting)• 2-3 hour air bake at 250 − 300 o C for the chamber parts (electrodes not baked to avoid
oxidation, which would affect the conditioning and electron discharging)
The baking procedure for the macor pieces is similar but a few of the steps are different.
Since macor is very fragile we used the ultrasonic on its lowest setting. The macor procedure is:209• Soak in Bio T citrus solution• Rinse with water and Simple Green degreasing detergent (all-purpose cleaner)• 10 minute soak in dichloromethane• Air bake over night at 600 − 700 o C• 10 minutes in the ultrasonic in Alconox detergent and water (lowest setting)• Rinse in deionized water• 10 minutes in the ultrasonic with acetone (lowest setting)• 10 minutes in the ultrasonic with isopropanol (lowest setting)• 10 minutes in the ultrasonic with methanol (lowest setting)• Air bake over night at 600 − 700 o C
Note that for all the ultrasonic sequences with macor and with the metals, the heater in themachine was on and temperature of the solution reached ≈ 70 o C. At these temperatures acetone,
isopropanol, and methanol had very high vapor pressures and were even on the verge of boiling.

8.6.3

Chamber baking and pump activation procedure

The procedure for baking the chamber and activating all the pumps is also quite complicated,partially because of the complexity of the electrode system inside the chamber. The bakes were
performed with a turbo pump attached to the inline valve on the big cube. Note that much of thebaking of the MOT chamber and the science side were done separately. This was possible becauseof the differential pumping tube and the gate valve in between them. The baking procedure for the
MOT chamber is much less involved as that chamber is rather simple. The only complexity of the
MOT chamber bake was the activation of the dispensers for K and Rb during the bake.
In total the science chamber was baked for more than three weeks, though the temperature
was changed a few times during this span. The pressure was measured using the ion pump and a210nude ion gauge near the ion pump. We baked near 200 o C for almost a week, and then near 300 o
for more than 10 days. We then went down to 200 o C for a few days, and then down to just over
100 o C for a few more days. The final bake at ≈ 100 o C was used for the activation of the NEG
pump and the Ti-sub pump.
The activation procedure for the NEG pump and the Ti-sub was rather complicated since
the goal was to progressively clean the dirt from one and the other in an alternating fashion. The
procedure was as follows:• Turn off the ion pump (the ion pump was mostly off during the previous baking, but
occasionally it would be on, especially at the lower temperature during the final bake)• Activate the NEG pump for 45 minutes total as initial cleaning (not continuously on because
gas load is too high, but up to 33 minutes continuous)• Wait a few days for the baking and pumping to remove the extra gas load from the NEG
pump• Activate the NEG pump again for a total of 40 minutes• While the NEG pump is on, clean the Ti-sub pump filaments by running 25 A througheach of them in turn for 120 seconds each
• Then, while the NEG is still on but after the Ti-sub pumps turn off, flash the ion pump on
five times to clean any dirt off its electrodes• Then, right before turning off the NEG pump, activate each of the Ti-sub filaments in turnto 33 A for 10 seconds each
• Finally turn off the NEG (everything is off now, except the turbo pump), and wait for thepressure to fall below 10−8 Torr as measured on the turbo pump
• Decrease the bake temperature from 130 to 100 o C, as a first step in returning to room
temperature

211• Next activate the Ti-sub pump filaments to 48 A for 45 seconds each
• Wait one day for the pressure to stabilize, still holding at 100 o C• Flash the ion pump on five times to clean any dirt off its electrodes• Cool the chamber down to room temperature with everything off except the turbo pump
• Turn on the ion pump permanantly when the chamber has reached nearly room temperature
The pressure read on the ion pump after this entire procedure and after closing the inline valve tothe turbo pump was typically 1.3 × 10−11 Torr (1000 mbar = 760 Torr), which is higher than we
wanted.

8.6.4

High-potting the ion pump

We also found that the current (pressure) reading on the ion pump was unstable, and frequently had spikes to several 10−11 Torr that were random and unphysically fast. Moreover, we
could cross reference the pressure reading with the nude ion gauge, which did not show any of these
spikes. To understand this it is important to consider the current measured by the ion pump. The
current is converted to pressure based on the assumption that its only contribution is from theionization of atoms and molecules that move into the pump. This is called the discharge current.
However, current could also be drawn by so-called secondary electron currents where electrons
can be ejected from the cathode plates. This current increases when the ion energy increases, i.e.
the operating voltage of the pump electrodes increases. Another current source is leakage currents,
where current can flow between the pump anode and cathode. This can happen when coatings
develop on the ceramic elements in the pump, and thus behave like a resistance between highvoltage and ground. Yet another possibility is field emission currents, which are caused by large
electric field gradients. The most common cause of such currents is spiky burrs developing on theelectrodes, which emit electrons in a similar fashion to the conditioning discussion above.

212
Such irregular current spikes on the ion pump suggests problems from these latter two mechanisms. These problems are common in ion pumps as they age, but we believe that it happened to
us during the pump activation procedure described above. Even though the ion pump was off for
most of the sequence, some of the initial activation of the NEG pump was done with the ion pumpon, during which it observed gas loads as high as ≈ 10−5 Torr. The lifetimes of ion pumps are
typically given in pressure-hours units, and operating at such high pressures even for short times
really cuts into the lifetime of good operation of the pump. Note that the pumping speed of theion pump may not be affected at all by these issues, but we want to be able to use the pump as anindication of the pressure. The nude ion gauge requires heating to activiate, and thus it artificiallyinflates the pressure and cannot be used constantly.
In order to remove all the deposits onto the ion pump electrodes we used a so-called hi-potting
procedure, which is built into the Gamma Vacuum pump controller for Gamma Vacuum ion pumps.
The principle of hi-potting is identical to conditioning, where the current at a given voltage canbe minimized by conditioning the electrodes at higher voltages. The maximum operating voltageof the ion pump is 7.5 kV, and so hi-potting is done by increasing the voltage to 10.5 kV fortens of seconds before going back down. We used this procedure a few times over several months.
Moreover, we operate the ion pump at 5.5 kV to limit the leakage and field emission currents. By
using these tools the current drawn by the ion pump decreased enormously, and the pressure it
calculates is limited by its current resolution to be P  10−11 Torr.

8.7

Design of the ITO coatings
I now discuss the ITO coatings and their designs. We have always used ITO-coated plates

that went around the cell in the old apparatus experiment. The coatings were ≈ 40 nm thick, andthere was substantial absorption and reflection as a result. However, the old cell was uncoated andalso had ≈ 5% reflection.

213
8.7.1

Coating thickness requirements

There are two factors that determine the thickness of ITO that we need on the plates in thescience cell. The first is the resistance of the coating, which we model as being in series with thecapacitance between the plates. We want to be able to ramp on the electric field in ∼ 100’s of mssuch that it is adiabatic with respect to the optical trapping frequencies. Therefore, we want the
RC time constant to be at a similar level. Since τ = RC and the capacitance between the plates
is C ≈ 0.1 pF, the resistance of the coating for which τ = 100 ms is R = 1012 Ω. In this regard
the resistance of the transmission line is not a concern, and so we typically put 100 MΩ resisters
in series with the plates simply to reduce the current, as discussed above.
The other requirement is that the coating be at least ≈ 100 atomic layers thick such that itis robust to microscopic sources of damage such as alkali atoms or dust which could be burned byan incident laser beam. An atomic layer is roughly 0.1 nm thick for ITO [258], and so a coating
thickness of 10 nm is sufficiently thick. The coatings that we used in the old apparatus were 40 − 50
nm thick, but since the absorption increases with thickness we would prefer to use a thinner coating.
Moreover, it becomes more difficult to reduce the reflection by pairing ITO with an AR coatingwhen the ITO coating is too thick. Therefore, we chose to use a coating thickness of 10 nm, andthe resistance across the plate that results is ≈ 800 Ω.

8.7.2

Interaction of ITO coatings with alkali flux

Another important issue with ITO coatings is their known reactivity with alkalis [60]. We
could have tried to use a Ni thin film coating instead, but the reflection and absorption would bemuch higher than ITO. To estimate the reactivity and lifetime we expect with an ITO coating
given the atomic flux in the science cell (≈ 105 atoms/second), we can measure the lifetime of atest coating in high K vapor pressure environments. Before and after images of such a test plate
are shown in Fig. 8.27. These measurements were done by preparing a vacuum chamber that has
a K source in it which is not isotopically enriched. We measure the pressure in the chamber as a

214

Figure 8.27: Images of an ITO-coated plate before and after exposure to a high K flux.

function of the current through the K dispenser, which controls its temperature. We can also verify
the presence of K spectroscopically by perfoming an absorpotion measurement when the pressure
is high enough. We can also heat the vacuum system to help control the pressure of K, which has
a very low vapor pressure at room temperature and thus deposits on the walls.
By measuring the pressure P and the temperature T of the vacuum chamber we can calculate
the flux Ṅ of atoms onto the ITO-coated plate, which has an area A. This can be derived from theideal gas law, for which we getṄ =P · Ap
3kB T /mK ,
kB T

(8.1)

where mK is the mass of K and kB is Boltzmann’s constant. The latter factor is simply the velocity
v through one the axes of the plate that comes from turning the volume in the ideal gas law intoA · x and taking the deriviate of both sides, giving ẋ = v and Ṅ . We measure the lifetime of
the coating in two cases: P = 10−7 Torr and T = 150 o C, and P = 10−8 Torr and T = 23 o C.
The lifetime in these two cases is τ = 10(2) and > 200 hours, respectively. We expect that thereaction has a strong temperature dependence, so operating at room temperature helps our casesignificantly.
We can now extend this analysis to the flux we expect from cold atoms being dropped onto
the plate every roughly 30 seconds. The above calculation for the conditions of these tests gives aflux of Ṅ ∼ 1012 atoms/second over the area of the plate. Given that we will drop ∼ 106 atoms
every 30 seconds the flux through whatever area of the plate the atoms fall on to (they will expand
over ≈ 5 mm2 after a resonant probe pulse or while very hot at the beginning of optical evaporation)

215
is Ṅ = 5 × 104 atoms/second. This flux is a factor 107 − 108 smaller than the measurements, andso we expect the lifetime to be larger than the measured lifetime by a similar factor. Therefore, weestimate that the lifetime of our ITO plates will be τ ∼ 108 − 1010 hours if we ran the experiment24-7. This is a very long time, and so we are confident that ITO should work in our system. In
other experiments where a Zeeman slower constantly shoots a beam of hot alkali atoms into thechamber containing ITO, the issue of atomic flux onto the ITO coatings becomes a bit more serious.

8.7.3

Pairing the ITO coating with an anti-reflection coating

Since 2011 all the research we have done has been in at least a vertical lattice, and much ofit has been in a full 3D lattice. We expect this trend to continue, and several experiments suchas layer selection and evaporative cooling rely heavily on a lattice in the vertical direction. Since
the vertical beam will go through many surfaces at small (or zero) angles, we must be mindfulof the standing wave contributions from reflections off of all the surfaces. Since there is no welldefined phase relationship between all of these reflections and the main reflection from the dichroic
high reflection (HR) plate after the cell, the reflections cause an unstable superlattice which causes
enormous heating and destroys BECs very quickly. See Steven Moses’ thesis for a more thoroughdiscussion of this effect and a calculation of how the superlattice strength scales with the angle ofthe beam.
Therefore, we want to pair the ITO coating with an AR coating to minimize the reflection of
the ITO-coated plate, particularly at λ = 1064 nm. Since the ITO layer must be the outer layer,
we will put an AR coating on the substrate and then put the ITO coating onto the AR coating.
The ITO coatings and AR coatings were developed together by the same company, VacuLayer
Corporation in Ontario, Canada. The AR coating was designed specifically for the ≈ 10 nm
thickness of ITO as discussed above.
The first AR coating to be paired with ITO was made of MgF2 , and we found that it
underwent chemical reactions with ITO when baked in air or vacuum. Figure 8.28 shows that the
optical properties through the plate changed significantly when the baking temperature of coated216

Figure 8.28: Baking a paired ITO-AR coating in air to various temperatures for one hour. The
plots from left to right show the reflection, absorption, and resistance, respectively.

substrates approached 300 o C, even when only baking for one hour. Moreover, the resistance of thecoating increased by 1000× at 300 o C. These observations suggest that the ITO coating is getting
destroyed during the bake; however, it could be either that the ITO coating is unstable due to how
thin it is, or that the ITO-MgF2 interface is unstable.
To address this question we performed a test where we baked two substrates with just an
ITO coating, where one was 10 nm thick and the other was 150 nm thick. We found that the ITO
coating was stable for both thicknesses when baked in vacuum, even up to 300 o C for many days.
Therefore the problem must be coming from the ITO-MgF2 interface, and we suspect that thefollowng chemical reaction between the oxygen in ITO and the AR coating is being activated at≈ 200 o C: O+MgF2 → MgO+F2 . Upon researching this issue, we found that similar problems havebeen observed where atomic oxygen in the atmosphere degrades MgF2 AR coatings on spacecrafts
via this reaction pathway [239].
Therefore, we clearly need to use a different AR coating, and we next turned to SiOx coatings,
where x = 1−2. VacuLayer prepared such an ITO-AR pairing for us where the ITO coating is ≈ 10
nm thick, and the measured optical properties before any baking are shown in Fig. 8.29. Using
this combination they were able to minimize the reflection at λ = 1064 nm to ≈ 0.2%, and thecorresponding transmission was 98%. Both of these numbers are excellent, and they suggest thatthe absorption is < 2%. Note that the other side of the plate has an AR coating which is different

217

Figure 8.29: An ITO/AR coating pair using SiO and an ITO thickness of 10 nm. The two different
colors of the reflection data are two different coating runs.

218
than the coating paired with ITO.
Now we need to bake these substrates as before to see if the coatings survive. We baked the
ITO/AR-coated plates to 300 o C in vacuum for up to 6 days, and the results are shown in Fig. 8.30.
We can see that while the optical and electrical properties of the coatings change slightly, there are
no dramatic changes indicating damage to either coating. In fact, the resistance of the ITO coating
actually decreases by ≈ 20% within a few days and then appears to saturate. This suggests that the
ITO coating changes irreversibly as the bake progresses. We suspect that the chemical composition
is changing, and our first guess would be that the indium is evaporating slightly during the bake.
Since ITO is a semiconductor material even minute changes to its chemical composition can change
its band gap, which would affect the index of reflection at λ = 1064 nm and its electrical resistance.
It is important to note that the changes in these quantities appear to saturate within six daysor so. This is corroborated by the observation that after all the final baking proceedures discussed
above, these properties are still the same as the values after the six-day test. The reflection has
increased from ≈ 0.5% to ≈ 2%, and the absorption has increased from ≈ 2% to ≈ 5%. Bothof these changes will have deleterious effects on the vertical lattice path, and it would be highly
advantageous to try to avoid them. While the change to Re(n) could be accounted for using the
AR coating if it was understood well enough, there is nothing we can do about the changes inIm(n). Fortunately, the tests of high intensity on the coated substrates discussed above were also
performed for plates which were baked to high temperatures for extended periods. This shows thatthe increased absorption and reflection still allows us to operate the vertical lattice as desired, but
we need to be even more aware of super-lattice effects since the reflections are stronger.

8.8

Optical qualities of ITO plates and cell windows
The substrates for the ITO-coated plates and the cell windows were produced by LatticeElectro Optics in California. The glass cell was assembled by Precision Glassblowing Inc. here
in Colorado. Their procedure is based on mechaning the quartz skeleton which holds all thesubstrates, and then frit-contacting the substrates and the quartz tube while pulling vacuum at219

Figure 8.30: The optical and electronic properties of the ITO/AR coating during a bake to 300 o C.
The four plots show (clockwise from top left) the transmission through the substrate, reflection off
the ITO/AR coating, absorption of the substrate and coatings, and the electrical resistance acrossthe length of the ITO coating.

220

Figure 8.31: The AR coating on the cell windows before baking to the cell-assembly temperature.
The blue curve is before baking, and the orange is after baking.

high temperature. This type of contact uses glass powder (called frit) between the quartz frame andthe substrate as a glue when the powder melts at ≈ 1000 o C. The other viable contacting techniqueis called optical contacting, and requires that the contacting surfaces are polished to better than
λ/10, at which point Van der Waals forces cause the two surfaces to fuse together. This type offusing can be done at only a few hundred o C, but is much more expensive since super-polishedsubstrates are required.

8.8.1

Anti-reflection coatings on the cell windows

The frit-based fusing technique requires that the entire cell be heated to ≈ 1000 o C, and so
the AR coatings must be designed to survive such temperatures. Most thin film companies cannot
make such coatings, and only two companies in the US have worked with Precision Glassblowing.
One such company is Lattice Electro Optics, the company that provided the substrates for the cell
windows and ITO-coated plates. The initial plan for our cell was that they would do the coating,but they backed out in the middle of the job, and so Precision Glassblowing had to turn to theother thin coating provider.
Figure 8.31 shows a photospectrometer scan of the first version of the AR coating before andafter baking to the cell-assembly temperature. The blue curve is before baking, and the orange is221

Figure 8.32: Fizeau interferometry of the cell windows.

after baking. Clearly the coating irreversibly shifts blue by ≈ 150 nm as a result of the bake. The
coating is broadband from ≈ 750 − 1100 nm, but it is designed to be optimized at λ = 1064 nm.
To correct for the blue-shift which results from the baking, the coating manufacturer had to shift
the coating red by this same amount. After doing so the coating beautifully matched the desired
wavelength range. Note that the transmission at λ = 1064 nm is > 99%.

8.8.2

Optical flatness

For high resolution imaging and addressing, as well as for low-wavefront-error optical lattice
beams it is important that the optical substrates of the cell windows and the ITO-coated plates
have excellent surface flatness and figure. The flatness refers to bowing, twisting, or wedging due
to differential stress or an inhomogeneity in the polishing procedure. For example, the differential
pressure between the two sides of the window will cause a r4 flexure as discussed in Chapter 7.
The surface figure refers to how microscopically smooth the surface is, and this of course depends
enormously on the polishing procedure.
These quantities can be measured with a Fizeau interferometer which analyzes the interference between a pair of beams, where one of the arms goes through the substrate of interest. The
resulting Newton rings can be analyzed to infer the detail of the substrate. The results are shownin Fig. 8.32, where a qualitiative difference in the type of abberations are apparent. The left image

222(window 1) shows a clear wedge across one axis of the window, which causes coma abberations.
The right image (window 2) shows a roughly cylindrically symmetric Newton ring resulting frombowing, which causes distortion (i.e. variation of the magnification of the imaging system acrossthe image plane) and spherical aberration.
The traces in these two figures are cuts through the interference pattern across the substrate,
and they can be used to quantify the surface quality. The larger feature across the substrate is ameasure of the surface flatness, which is λ/5 for both windows at λ = 780 nm. The RMS amplitudeof the smaller oscillations of this envelop is a measure of the surface figure, which is ≈ λ/10 for
both. The quality of the substates for the ITO/AR coatings was measured to be ≈ λ/5. These
measurements were not repeated with the chamber under vacuum or after baking, but we expect
the surface flatness to be λ/5 − λ/4 for all the substrates after all baking. Such values are typicaleven for quantum gas microscope experiments where the imaging system has NA= 0.65 [266], andan optical system with an optical path difference (OPD) of λ/4 is considered a perfect system froma geometrical optics point of view, and this criteria is called the Rayleigh limit [105].

8.9

Testing the objective
We tested the resolution of the objective with two techniques. The first test used an Air

Force test target, which is a glass substrate with sets of three lines progressively smaller in a spiralpattern. A portion of the test target is shown in Fig. 8.33, and the smallest line sets that canbe resolved are in the middle of the image. The target is imaged onto a profile camera using amagnification large enough that sizes limited by the resolution of the objective (rmin ) are magnified
to occupy many pixels on the camera. The line set of spacing 770 nm cannot quite be resolved, but
the 880 nm line set can be resolved easily. Therefore, the resolution inferred from this measurementis ≈ 850 nm, which is similar to the expectation based on NA = 0.53.
A more rigorous and useful measurement of the resolution can be done by illuminated apinhole much smaller than the resolution, and placing it in the object plane. In this limit theimage of the pinhole is equivalent to the image of a single atom (point-source scatter), and this

223

Figure 8.33: Testing the microscope objective by imaging an Air Force test target onto a profile
camera. The smallest line spacing that can be resolved is 880 nm, suggesting that the resolution is≈ 850 nm.

224

Figure 8.34: Characterizing the resolution limit of the objective using a small pinhole. The left
plot measures the depth of focus by translating the pinhole relative to the object plane. The rightimage measures the effect of tilt of the 2 mm thick plate between the pinhole and the objective.

approach allows us to measure the point spread function of the objective. We used a pinhole ofdiameter D = 500 nm, and some characterizations of the resolution are shown in Fig. 8.34. The
measurement is performed in the same way as the test target, where a second lens is used to image
the pinhole onto a camera with sufficiently high magnification to avoid pixelation. Note that bothof these tests are done with glass pieces between the object plane and the objective which matchthe thickness of the elements used in the new apparatus.
The plot on the left shows the depth of focus, which is measured by translating the pinhole
along the imaging axis. The size in the image plane gets larger as the object is moved out of focus,
and the fit is to a Gaussian Rayleigh range function. The depth of focus is defined when the object√
becomes larger by 2, which is ≈ 3 − 5 µm, consistent with what we expect for this NA. The plot
on the right is looking at the effect from tilting along the x-axis a 2 mm-thick glass plate between
the object plane and the objective. This test is designed to simulate the relative angle between the
ITO-coated plates and the cell window. A large tilt angle causes significant coma aberrations, as
shown in the inset images on the camera. This data suggests that the measured tilt angle of 0.2
degrees should be negligible.

225

Figure 8.35: Measuring the resistance and calibrating the magnetic field from the bias coils. The
left plot measures the resistance of the coils, and the right plot shows the magnetic field measured
with a Gauss meter as a function of the current. The measured data points and fit are shownin black, and the calculated points and theoretical fit are shown in red. The two are in excellent
agreement.

8.10

Magnetic fields
Now I will discuss all the coils that go around the science cell. Measurements of magnetic

fields were performed with a Gauss meter that was calibrated to zero field using a µ-metal shield.

8.10.1

Testing the bias coils

Figure 8.35 shows measurements of the voltage and magnetic field as a function of the current
in the coils. The left plot measures the resistance of the coils, and the right plot shows the magnetic
field measured with a Gauss meter as a function of the current. The measured data points and fit
are shown in black, and the calculated points and theoretical fit described in Chapter 7 are shownin red. The two are in excellent agreement. The K-Rb Feshbach resonance is at 547 G, which canbe reached with ≈ 220 A. At 1005 G is a narrow Feshback resonance for Rb, which could be useful
to measure double occupancy of Rb by making Rb2 Feshbach molecules. This field can be reachedwith ≈ 400 A. Another important field is ≈ 1260 G, at which the excited rotational states |1, 1i,|1, 0i, and |1, −1i are degenerate [250]. This field can be reached at ≈ 480 G, which is still withinreach of our high-current power supply.

226

Figure 8.36: Thermal effects on the coils as a function of the applied current. The left plot shows
the steady state temperatures measured on the water inlet to the coil (red), on the top turns of
the 4x4 turn coil (black), and on the water outlet (green). The right plot shows the transientsmeasured in the same places when 220 A is stepped on, and then later stepped off.

We also measured the heating as a result of going to such high currents. Figure 8.36 shows
the steady state operating temperature measured in a few places as a function of the current. The
coils are hollow core and have water flowing through them, so each of the 16 turns should be atroughly the same temperature. The temperature of the coils enters a linear regime at above ≈ 250
A, where the steady-state temperature increases by ≈ 5 o C for every 50 A. This trend suggeststhat at even 500 A the coil temperature would be < 40 o C, which is acceptable for hold times ofseveral seconds.
We can also look at the transients to see how long it takes to reach the steady-state temperatures. This is shown in the right plot of Fig. 8.36 for the same three temperature probe locations.
This shows that for 220 A (550 G, where the K-Rb resonance is), the heating and cooling occurs ona τ ≈ 10 sec timescale. This timescale will be roughly the same at any current, only the amplitude
will vary. Typically we only need to apply large fields for a few seconds, and so this heating in
negligible even to 500 A. Such thermal cycling of several seconds per 30 − 40 seconds is certainly
acceptable, but mechanical fatique will eventually set in, and the glue between the coils will haveto accomodate slow mechanical drift over several years.

227

Figure 8.37: Curvature of the z-component of the bias field in the axial and radial directions of thecoils. Measurements were made with a Gauss meter oriented along the z-direction mounted on a
3D translation stage.

228
It is important to consider the homogeneity of the field to identify how much field curvatureexists at the center, and to estimate how sensitive we will be to exactly how the coils are oriented
around the science cell. Figure 8.37 shows the curvature of the z-component of the field in theaxial and radial directions of the coils as measured with a Gauss meter. The field is fit to a quartic
polynomial, where the only fitting parameters are the quartic term amplitude and the offset. Theseestimates suggest that the field variation over the size of the cloud (100 µm) in the center is ≈ 6
mG in both directions at 250 G, which is roughly 2 parts in 105 . The figure of merit is the spreadin the K transition |9/2, −9/2i to |9/2, −7/2i at 550 G, where the field variation over the cloud
would be ≈ 10 mG. The sensitivity of this transition is ≈ 40 kHz/G, and thus this field variation
will broaden the transition to 400 Hz. This is a smaller effect then the field noise due to the servo
quality, and is also smaller than the Rabi frequency of the K 80 MHz rf coil, which is ≈ 10 kHz.

8.10.2

Testing the gradient/quantization coils

We can perform similar tests of the small, magnet wire coils inside and concentric with thebias coils. These coils will be used for quantization coils, fast magnetic field sweeping coils, andalso for gradient coils. The direction of the current in one coil can be switched relative to theother coil using an H-bridge, which allows us to change them from a Helmholtz configuration toan anti-Helmholtz configuration. The field from the coils in the Helmholtz orientation and thegradient from the coils in the anti-Helmholtz configuration as a function of current are shown in
Fig. 8.38.
The field in Helmholtz configuration is in excellent agreement with the predicted value, giving
6 G/A. We can apply up to ≈ 10 A for a few seconds, allowing us to apply fields of aroung 60
G. These fields can be switched very quickly, allowing for fast magnetic field sweeps across the
Feshbach resonance. We configure the current directly such that this field adds with the field fromthe bias coils. Then we can initialize the system above the resonance, where both the bias coils andthe fast Helmholtz coils are on. We can set the bias coils to the desired field below the resonance,
and then when we quickly turn the current in the small Helmholtz coils to zero, we sweep across the229

Figure 8.38: Calibrating the field and field gradient from quantization/gradient coils. The left plotshows the field as a function of current with the coils in the Helmholtz configuration. The rightplot shows the field gradient as a function of current with the coils in anti-Helmholtz configuration.

230
Feshbach resonance. The high-field bias coils have a more stable servo and their field profile is more
uniform. These factors are only important below the resonance when we perform rf manipulationof K, and in this way the small Helmholtz coils are off for such manipulation.
The calibration for the coils in the anti-Helmholtz configuration gives 2.4 G/cm/A, and thus
we can reach 25 G/cm with these coils, which is sufficient to levitate atoms against gravity. We
use these coils for spin-dependent detection to measure the relative spin populations in the clouds
of atoms. We can also use this quadrupole field to compress the clouds of atoms in the science cellthat are spread out along a single laser beam. We optically transfer atoms into the science cell in
the focus of one beam, and this gradient from the quadrupole field allows us to compress the cloudin order to enhance the subsequent loading into a crossed dipole trap. This will be discussed in
Chapter 9.

8.11

Electric field voltage servos
Any high-fidelity manipulation of rotational states in a large electric field will require excellent

electric field stability. If the Rabi frequency of the rotational transitions is Ω ≈ 10 kHz, we need
the field fluctuation during the pulse to result in frequency shifts < 1 kHz. As discussed in Chapter
7 the shifts of the |0, 0i to |1, 0i transition are ≈ 100 − 200 kHz/(V/cm), and thus we need the field
to be stable to ∆E ≈ 10 mV/cm. At an electric field of E ≈ 10 kV/cm, the required stability is∆E/E = 10−6 . Accordingly we want to be able to stabilitize the voltages applied to the electrodes
at the part-per-million (ppm) level.

8.11.1

Stabilizing large voltages

The design, construction, and testing of the servos to stabilize the voltages from the highvoltage amplifiers was done by Terry Brown of the JILA instrument shop and Jamie Shaw, an
undergraduate writing his honors thesis in our group. More information on these servos can befound in the thesis of Jamie Shaw [234]. A schematic is shown in Fig. 8.39. The input voltage to be
amplified is from a digital-to-analog converter (DAC), and it is derived from a field programmable

231

Figure 8.39: A schematic of the servo circuit to stabilize the voltages after the high-voltage amplifiers. Reproduced from Reference [234].

gate array (FPGA) connected to a computer. Moreover, the DAC uses a temperature-stabilitizedvoltage reference such as an LM399. The requirement for the stability of this DAC voltage is as
stringent as the stability of the high voltage.
This analog voltage needs to be compared with the voltage after amplification, and we dothis by using an ultrastable voltage divider. The voltage divider was designed carefully to minimize
thermo-couples (connections between different metals), but a larger concern is the stability of theresisters. We used ultrastable Vishay resistors (Z-foil type) with a temperature coefficient of 0.05
ppm/o C. Since we want to reach ppm stability, we must be careful of the temperature stability of
the resistors. Thermal effects are generally slow, but since the voltage divider actually dissipatescurrent we must be aware of the heat load on the resistors during measurement. While this is an
important factor, we believe that this is not a major limitation at the few ppm level.
The divided voltage is summed in with the input voltage, and the sum goes into an error
amplifier and the loop filter, and is then summed with a feed forward voltage. The output of that
goes to the high voltage amplifiers. The grounding of all of these elements is critical, as shown in
Fig. 8.39. The voltage dividers and servo circuits are shown in Fig. 8.40. Below the voltage divider
is a compensator plate which is used to improve the servo bandwidth. It does this by couplingsignals back into the error amplifier board connected to the voltage divider that would have been

232
lost due to capacitance between the resistors and the nearby metal box [234].

8.11.2

Realizing ppm voltage stability

In order to characterize the quality of the servo we can begin by looking at the stability of
the DAC voltage reference, which is shown in Fig. 8.41a. These measurements were performed
with a 6-digit digital multi-meter (DMM), and the measurement was broken into two ranges to get
the best data from the DMM: 1 V and 10 V. The black and red data shows the stability of the
DAC voltage as a function of the voltage. Their fractional stability improves enormously as thevoltage approaches the LM399 reference voltage of ≈ 7 V. Both of which are reasonably close to
the noisefloor, which is shown as the black line.
We performed an open loop measurement of the stability of the high voltage amplifiers. Thiswas done by amplifying the DAC voltages (1000×), and then dividing it by 1000 with the voltage
divider. The blue and green data show this divided voltage, and suggest that the high voltage
amplifier is ≈ 20 ppm worse at most voltages. The amplified voltage follows the DAC voltages,
suggesting that its noise is not strongly dependent on the voltage.
Now we can turn on the servo loop and see how effectively it removes the noise from the highvoltage amplifier. The loop filter error signal is analyzed on a fast Fourier transform (FFT) as afunction of the frequency in Fig. 8.41b. The noise floor of the FFT is shown in blue, and the redshows the error. This suggests that the servo bandwidth is 10’s of kHz. Therefore, we expect tohave several-ppm-level stability at ±7 kV potentials up to frequencies of around ≈ 10 kHz. Suchvalues should enable high-fidelity rotational state control in large electric fields, which can be used
for quantum magnetism and non-equilibrium many-body dynamics.

233

Figure 8.40: The circuit for stabilitizing the voltages after the high-voltage amplifiers. (a) The
voltage divider to compare the voltage after the amplifiers to the input voltage. (b) The Servo box,with the voltage divider on the top and the loop filter servo on the bottom. Note the compensation
plate below the voltage divider. This was painstakingly adjusted to optimize the servo performanceand bandwidth. Reproduced from Reference [234].

234

Figure 8.41: Measuring the voltage stability of the LM399 reference and the servo ouput. (a)The voltage stability versus the voltage. This measurement was broken into two ranges to get thebest data from the DMM: 1 V and 10 V. (b) The error signal from the servo loop as a functionof frequency on an FFT machine. This curve shows that the servo bandwidth is several kHz.
Reproduced from Reference [234].

Chapter 9

Experimental procedure - making molecules in the new apparatus

Most of the laser systems for the experiment are the same between the old apparatus and the
new apparatus, so to change to the new apparatus, we moved the old vacuum chamber off of themain optical table for the experiment, and on to a smaller optical table for testing and building.
The new chamber was then moved from this table to the position on the main experiment table
where the old chamber was previously. The moving of the chambers was done with a mobile
jacking lift platform which was outfitted with a large aluminum breadboard. The two sides of theflexible bellows were braced on both chambers when moving them. Figure 9.1 shows the entire
chamber-moving process.
Now I go on to discuss the experimental procedure for creating gases of ultracold atoms andmolecules in this new apparatus. A view of the full vacuum system is shown in Fig. 9.2, and thedifferent regions for atom manipulation and trapping are indicated.

Figure 9.1: Photographs of the process of moving the vacuum chambers. (a) Moving the oldchamber off of the main experiment table. (b) Moving the new chamber onto the main experimenttable. (c) Installing the new chamber in the position that the old chamber was previously.

236

Figure 9.2: A render of a solid model of the entire new apparatus vacuum chamber. The threechambers used to prepare ultracold atoms are molecules are identified. The large cube where thehigh voltage feedthroughs couple in large DC voltages are evident, and the high resolution objective
can be seen under the science cell.

237

Figure 9.3: UV LEDs for LIAD of K and Rb in the MOT chamber. The quadrupole coils are shownon the right, and the LEDs are shown in blue. The purple cloud indicate that the LEDs facilitate
the desorption of atoms from the walls of the pyrex chamber.

9.1

Dual species MOT with grey molasses cooling of 40 K and 87 Rb
The first section of the chamber is the vapor MOT cell, where we load a bichromatic MOT

for both K and Rb from a background vapor. The K dispensers are enriched to 14% aboundanceof 40 K, above the natural aboundance of 0.012%. The pressure in this regime is maintained to givea 1/e MOT fill time and magnetic trap lifetime of roughly 2.5 seconds. The MOT beams have a
1/e2 radius of 1.5 cm and power of ∼ 200 mW, and typical MOT sizes are NRb = 2 × 109 and
NK = 5 × 107 . The quadrupole gradient used during the MOT is ≈ 15 G/cm on the strong axis.
In the new apparatus we started using laser-induced atomic desorption (LIAD) to reduce
the MOT loading time and enhance the MOT size. LIAD is induced by UV light which facilitates
desorption of atoms from glass surfaces [174]. While the efficacy of LIAD depends on many factors
such as the atomic species, the type of material in the chamber to which the atoms are adsorbed,
and the temperature of the chamber, it works very well in our system. We use a UV diode centeredat ≈ 260 nm near the pyrex MOT chamber as shown in Fig. 9.3. The MOT chamber is heated to≈ 40 − 50 o C to prevent K deposits from developing on the glass and obstructing the MOT beams.
We turn the LEDs on to 0.5 A for one second at the beginning of the MOT loading phase, and indoing so we were able to save ≈ 10 sec on the experimental cycle time.

238
Following the MOT loading, we use a compressed MOT stage (CMOT) [210], and then
optical molasses [159] for both K and Rb. For Rb, bright molasses allows us to reach a minimum
temperature of 40 µK with small clouds, and 200 µK for maximum density, limited by photon
rescatter. For K, bright molasses on the D2 (P3/2 ) transition at λ = 766.7 nm is even less efficientdue to the limited hyperfine splitting of the electronic excited state [179], and we reach ≈ 250 µK.
In the new apparatus we have added grey molasses for both K and Rb to circumvent these issues
and cool both to 10’s of µK or less.
For grey molasses of K we use the D1 (P1/2 ) at λ = 770 nm instead [33, 79], where thehyperfine structure in the electronic excited state is better resolved. As shown in Fig. 9.4, thegrey molasses operates as a Raman transition between the two hyperfine ground states with ablue detuning in the electronic excited state. Typical detunings for both species are ∆B ≈ 20
MHz, which are similar to the bright molasses detunings ∆R . However, we find it best to ramp
∆B for both K and Rb closer to resonance during the grey molasses time. The scheme for Rb issimilar, except the D2 (P3/2 ) can still be used since the hyperfine splitting is well resolved in theelectronically excited state.
Since the grey molasses relies on coherent Raman transitions between the hyperfine groundstates (see Fig. 9.4), phase coherence between the two lasers is necessary. Therefore, for K and
Rb we use EOM phase modulators with sidebands at 1.29 GHz and 6.84 GHz, respectively. Both
EOMs are from New Focus, and we drive them with ≈ 35 dBm, which gives ≈ 7 − 8% of the total
power in each sideband. Note that only one sideband is useful for each species, and that it happens
to be the opposite sideband for each due to the inversion of the hyperfine structure of 40 K. This
phase coherence allows us to observe electromagnetical induced transparence (EIT) between the
two laser fields as a function of the EOM frequency.
For both K and Rb we find that the overall molasses number and temperature is optimized
when we do a few ms of bright molasses prior to grey molasses. We find that ≈ 8 ms of grey
molasses is optimal, during which we ramp the intensity and power of the molasses beams after the
first few ms. During these ramps we decrease the power and the detuning. This approach allows

239

Figure 9.4: Bright and grey molasses laser detunings for both K and Rb. (a) The level structureof K, including fine and hyperfine structure. Note the inverted hyperfine structure for 40 K. The
bright molasses is red-detuned as shown with the red beams. The dashed beam is the repump.
Both have a detuning from their respective electronically excited hyperfine state of ∆R . The grey
molasses is blue-detuned, and operated by a Raman condition with detuning ∆B , as shown with theblue arrows. For K the bright molasses uses the D2 state and the grey molasses uses the D1 state.
(b) The level structure of Rb, including hyperfine structure. Note that only the D2 electronically
excited fine structure is shown. The bright molasses is red-detuned as shown with the red beams.
The dashed beam is the repump. Both have a detuning from their respective electronically excited
hyperfine state of ∆R . The grey molasses is blue-detuned, and operated by a Raman condition
with detuning ∆B , as shown with the blue arrows. For Rb the bright and grey molasses both use
the D2 state. Note that the level stuctures in (a) and (b) are not necessarily to scale.

240
us to reach 60(8) µK for K and 7(2) µK for Rb at the end of the molasses stage. For the Rb grey
molasses there is a ≈ 50% loss of atoms, presumably because the capture velocity is comparable tothe temperature during the bright molasses that precedes it. Nevertheless, the phase space density
is significantly higher with the grey molasses for both species.
After molasses we optically pump Rb to |F, mF i = |2, 2i (F is the hyperfine quantum number
and mF its projection onto the magnetic field) and K to |9/2, 9/2i, which are low magnetic fieldseeking states, both with a magnetic moment of 1µB . Then we load the atoms to a quadrupole trapwith gradient on the strong axis of ≈ 40 G/cm. To characterize the efficiency of all these steps,
we ruitinely find that the number loaded into the quadrupole trap after CMOT, molasses, andoptical pumping is 80% of the initial number in the MOT. The temperature after loading into thequadrupole trap with grey molasses is ≈ 100 µK for Rb, and ≈ 220 µK for K. The corresponding
temperature with D2 bright molasses cooling of K instead is 700 µK. Note that there is considerable
adiabatic and non-adiabatic heating of both species during the loading of the quadrupole trap frommolasses.

9.2

Magnetic transfer and plugged quadrupole evaporation
Once the atoms are loaded into the quadrupole trap, which is composed of anti-Helmholtz

coils on a motion track, we move the clouds through the vacuum chamber to the evaporation region
using a gradient of ≈ 100 G/cm, as depicted in Fig. 9.2. The coils were aligned to the chamber
axis using axial imaging through the differential pumping tube, as shown in Fig. 9.5. We perform
forced microwave evaporation in the quadrupole trap driving Rb from the magnetically trapped|2, 2i state to the anti-trapped |1, 1i state with radiation from a microwave horn at ≈ 6.85 GHz.
We sympathetically cool K with Rb [285].
The starting temperature before evaporation is ≈ 200 µK for both species, although this isdifficult to measure. We find that the initial number of atoms is improved using grey molasses
by 6× for K, and by 2× for Rb. While the number of K atoms in the quadrupole trap before
magnetic transfer is similar with or without grey molasses, the lower temperature with it allows

241

Figure 9.5: Aligning the cart coils with the chamber axis. The three images here show the cloudin the magnetic coils at two different positions with two different alignments. The left image
shows the cloud well centered on the differential pumping tube. The middle image shows the cloudsignificantly off-center from the tube. The right image shows the same alignment as the middle,
but a further position where the cloud is beginning to crash into the side of the chamber.

for a substantially higher transfer efficiency through the chamber, where a hotter cloud is severely
clipped by the aperture of the differential pumping tube in the vacuum chamber. The pressurein the quadrupole region of the vacuum chamber is orders of magnitude lower because of thisdifferential pumping, and is expected to  10−11 mBar. The lifetime in the quadrupole trap
depends on the temperature because of Majorana spin flips at the zero-field center of the trap [28],but was measured to be > 150 seconds.
While evaporation to 10’s of µK with Rb in a quadrupole trap before Majorana loss becomessignificant is routine [160, 235], the smaller mass of K and the larger number of magnetic states(larger F) causes significant loss at much higher temperatures. Therefore, we use a blue-detuned
optical plug beam that repels the atoms from the magnetic field zero crossing [61, 72, 114]. We use
a plug with wavelength λ = 750 nm, which is ∼ 16 nm blue-detuned of the K D2 line, and ∼ 30 nm
blue of the Rb D2 line. This laser is derived from a M2 SOLSTIS, which is a Titanium-Saphhire
tunable laser from ≈ 740 nm to ≈ 950 nm capable of delivering ≈ 5 W. We use a 1/e2 beam radiusof 50 µm with a power of ≈ 1 W, giving a repulsive barrier height of 300 µK for K and 200 µK for
Rb. We evaporate to a temperature of 16(2) µK in 13 seconds, yielding NRb = 2.0(3) × 107 and
NK = 5.0(4) × 106 .
Evaporation of two species using sympathetic cooling often requires careful removal of coolant
atoms in the wrong spin state, such as Rb in |2, 1i [284]. In the old apparatus we used a so-called242“|2, 1i”-cleaner in the IP trap as discussed in Chapter 2. The application of such a cleaner using
a second rf tone is more complicated in the quadrupole trap where both |2, 1i and |2, 2i become
degenerate at the bottom of the trap, although the plug beam ostensibly prevents atoms fromspending appreciable time in this region. We spent some time working with a |2, 1i-cleaner since
we anticipated that it would be important, but in the end we found that the Rb purity in |2, 2i
is excellent throughout evaporation, and we never needed the |2, 1i-cleaner in the new apparatus.
This is because we choose the gradient of the quadrupole trap after optical pumping to be smallenough that we cannot efficiently hold |2, 1i atoms, and thus they never appear in appreciable
quantity in the quadrupole trap. The K number changes by a factor of ≈ 3 − 4 throughout theevaporation, but this is expected as a result of three-body loss, collisions with Rb atoms in |1, 1i asthey leave the system, off-resonant light-scattering from the plug beam, and other loss mechanisms(a similar factor of K was lost in the old apparatus).

9.3

Optical transfer into the science cell
After evaporation in the magnetic trap we ramp down the quadrupole trap and the plug

beam in 600 ms while ramping up a single optical dipole trap beam with wavelength λ = 1064
nm. The waist is 46(3) µK, and the final power is 7 − 8 W, such that the trap depth is ≈ 500 µK
for both K and Rb. We align the dipole trap beam (called the transfer beam) onto the plugged
quadrupole trap in order to maximize the loading efficiency, but the relative positions of the plugand transfer beam depend on the relative power.

9.3.1

Transfer beam alignment onto the plugged quadrupole trap

In the limit that the trap depth of the transfer beam is much larger than the barrier height ofthe plug (we were briefly using a vary large trap depth of the transfer beam), the two together givea net attractive potential. This potential holds all the atoms at the zero-field location (where theplug is aligned), and so there is substantial loss when the transfer beam is at this position. Sucha dip is apparent for Rb in Fig. 9.6 where the transfer beam position is scanned in the horizontal

243

Figure 9.6: Alignment of the very strong transfer beam onto the plugged quadrupole trap. (a)Scanning the horizontal position of the transfer beam with the vertical roughly centered. (b)Scanning the vertical position with the horizontal roughly centered. “Above” and “below” indicate
the relative position of the transfer beam focus with respect to the quadrupole trap center.

244
and vertical direction. The vertical profile is asymmetric because of gravity, and thus we set thetransfer beam to be centered in the horizontal and below in the vertical, which corresponds to thebest points in Fig. 9.6b.

9.3.2

Lifetime and heating in the transfer beam

Typically we operate with the transfer beam depth roughly the same as the plug barrierheight. In this case they roughly cancel each other out when perfectly overlapped, and the loss rateof either species at this temperature is small compared to the time scale that both beams are on
together. Therefore, we typically do not see these dips when both beams are perfectly overlappedin scanning the transfer beam (as in Fig. 9.6) during typical operation. We find typically 50 − 60%
loading efficiency for both K and Rb. The lifetime in the transfer beam is typically ≈ 10 seconds,
which is limited by pointing noise of the transfer beam. The transverse motion of the beam drives
excitations to excited motional states in the trap which causes significant heating which leads toloss. The lifetime was initially ≈ 1 second, and it took some dedicated effort to understand thesource of the heating and reduce the pointing noise.
Typical transverse frequencies are f ≈ 1 kHz, which is a ubiquitous frequency for acoustic
noise of small, metal objects such as mirror mounts and posts. Moreover, since the separation
between the radial states is not well resolved at these temperatures (1 kHz ∼ 50 nK), atoms caneasily be excited multiple times which gives rise to very large heating rates. To characterize theseeffects we can drive excitations in the radial degree of freedom using a piezo-actuated mirror in
the beam path to measure the v = 0 to v = 1 transition, and amplitude modulation (parametricheating) to drive the v = 0 to v = 2 transition. Figures 9.7a&b shows the resonances of these twomodulations at a slightly lower trap depth, and the width gives an estimate of the anharmonicity
sampled by the atoms at such temperatures.
The lifetime of Rb in the transfer beam as a function of power is shown in Fig. 9.7c, andthere are obvious resonances at which the lifetime is particularly short. The lifetime at very hightrap depth starts to decrease because of three-body loss. We use a quadrant photodiode and an

245
FFT machine to characterize the pointing noise of the transfer beam, and a typical measurement√from the initial set-up is shown in Fig. 9.7d where the amplitude of the noise is plotted in mV/ Hz
as a function of the frequency. In fact, the trap depths in Fig. 9.7c where the lifetime is very shorthave corresponding transverse trapping frequencies exactly matching the frequencies of the largest
spikes in Fig. 9.7d.
To solve this problem we had to rebuild almost the entire beam set-up for the transfer beam.
Since the beam has to be too large for one-inch optics, we are required to use a beam height of two
inches above the breadboard and optical transfer stage. We use only stainless steel optics holders
and 1 inch pedastals for all optics, so there were not many potential upgrades available with theoptics. Most of the optics were on a 1/2 inch thich aluminum breadboard held above the table bylong 2 inch diameter pillars. We found this mounting scheme to be particularly resonant at mostacoustic frequencies, and since the breadboard is undamped the vibrations would persist for longtimes. Therefore, most of the optics are now on a honeycomb breadboard that is 4.3 inches thick,under which are two 1-inch thick stainless steel slabs. We use thin sheets of highly-damping Viton
between the table and the first steel slab, between the two steel slabs, and between the upper steelslab and the breadboard. All of these elements are bolted to the table through the breadboard
with long screws, where there are clearance holes through everything.
The motion stage used to be mounted alone on four 2-inch diameter pedastals, but we reduced
the pedastal height by two inches and we use two 1-inch thick steel slabs here as well, again with
Viton between each layer. The stage is bolted to the pedastals through clearance holes in the steelslabs (see Fig. 9.10). As a result of these improvements to the optical surfaces for the transfer beam
optics we now have a lifetime of ≈ 10 seconds and we do not observe any heating during this time.
We are satisfied with this result, and as I will discuss later we have measured lifetimes of ≈ 30
seconds in this beam in the science cell. Nevertheless, it is still likely the case that the lifetime issignificantly less at specific trap depths whose resonant trapping frequencies exactly match somepointing noise, although the predominant peaks that remain are ≈ 100 − 200 Hz, which is well
below any relevant trap depth.

246

Figure 9.7: Characterizing the heating of Rb in the transfer beam from pointing instability. (a)Pointing modulation using a piezo mirror, providing a dip at the trap frequency. (b) Amplitude
modulation creates a dip at roughly twice the trap frequency (depending on the role of anharmonicity). The trap depth is ≈ 150 µK for this data, which is lower that we typcally use. (c) The
lifetime in the transfer beam as a function of the beam power as measured by the atom number
after holding two seconds. (d) A FFT scan of the quadrant photodiode at the focus of the transfer
beam. The large peaks match the frequencies of the traps that have large dips in the lifetime.

247
9.3.3

Loss of atoms during transfer

With this problem solved we then go on to translate the focus of this optical trap beam byusing an Aerotech magnetic-bearing translation stage (ANT130-110-L-PLUS) holding two mirrors
which are between the focusing lens and the vacuum chamber. Since the lens is not on the stage andwe do not transfer BECs, we are less sensitive to the smoothness of the transfer process, and thus
we do not need to put the stage on a superpolished granite slab. We align the beam incident to thestage and the mirrors on the stage such that the beam translates by less than 10 µm between thetwo ends of the travel range (see discussion below on changes to the mirror mounts on the stage).
The beam goes into the chamber through the end window of the science cell, and its focus carries
the atoms ≈ 17 cm from the quadrupole evaporation region into the science cell (see Fig. 9.2).
Initially we ran into enormous problems with the optical transfer that took quite a while
to understand. We found that in transferring Rb, the number of atoms dropped sharply after
transferring 2 cm. We were eventually able to determine that this sharp loss was a magnetic effectsince it was significantly less drastic for atoms in |1, 1i instead of |2, 2i, as shown in Fig. 9.8a. Partof the reason it was difficult to identify this problem is that every measurement we could makelooked perfectly normal at this position of 2 cm where we lost all the atoms. Figure 9.8b-d showthat the lifetime and the transverse trapping frequency is exactly the same at this position as it isin the initial position of the quadrupole trap. Note that the trap lifetime here is relatively shortbecause we were using a very deep trap in an effort to combat this sharp loss.
These results are perhaps not surprising since a sharp magnetic field gradient would affectthe axial trap frequency, which is ≈ 5 Hz, much more than the transverse frequencies. Moreover,the loss of atoms is commensurate with the reduction in the trap depth, and so the temperatureof the atoms that remain is such that the ratio of the effective trap depth to the temperature issimilar. Therefore, we need other ways to diagnose this problem, which are shown in Fig. 9.9.
The first test we would like to do is confirm that the problem is coming from some feature thatis specifically at the position of 2 cm. Other possibilities which made it difficult to identify the248

Figure 9.8: Transferring Rb atoms in the transfer beam using two different hyperfine states. (a)The number of Rb atoms as a function of the transfer distance for the |2, 2i state and the |1, 1i
state, showing that the sharp loss at 2 cm is a magnetic effect. (b) The lifetimes of Rb in |2, 2i at
0 cm and 2 cm, which are almost the same. (c) Parametic measurement of the trap frequency at 0cm. (d) Parametric measurement of the trap frequency at 2 cm.

249
problem are that the motion profile of the stage (to be discussed later) could cause a lot of loss fromtoo much acceleration, and it happens to be at 2 cm; or that the transfer beam could be clippingon something in the science cell when the focus is between 0 and 2 cm.
To address this question, we can use the oscillate function of the stage to move the stage
back and forth several mm at a certain frequency, thus giving a total distance we can vary without
moving vary far. This is shown in Fig. 9.9a, where the loss as a function of transfer distance fromthe oscillation is much more shallow then when moving linearly in the direction of the science cell.
A more direct measurement is shown in Fig. 9.9b, where we can first move backwards 2 cm, andthen we see that the sharp loss occurs after 4 cm, which is exactly in the same position. Thus, it
is now very clear that there is a sharp yet localized magnetic field gradient at 2 cm.

9.3.4

Magnetization in the chamber

There is nothing we can do to mitigate this problem, even when using a very deep transfer
beam. The gradient is just too strong to overcome with an optical trap whose axial frequency is
5 Hz. Moreover, K is lighter and thus more easily affected by such perturbations. It is also very
difficult to use less magnetic states of K, and we are limited by the stable combinations of K and
Rb, both of which are stretched in the F = 9/2 manifold for K. We tried to use several coils oriented
along the transfer direction to dynamically shim this gradient in such a way to cancel it out, but
it is just too sharp and too localized, such that the profile from any coil cannot compensate itadequitely.
It is important to understand what this sharp magnetic field gradient is from. As was
discussed in Chapter 8, the small cube in which this plugged-quadrupole evaporation is performed
houses the electrodes where they turn 90 degrees to connect the science cell with the large cube
which holds the feedthroughs. Additionally, the small cube is where we transfer from the tungsten
rods in the science cell to the SS316 that continues all the way to the feedthroughs. The tungsten
rods are connected to the SS316 rods using SS316 hypodermic tubing which is crimped onto bothrods, and a slit is cut into the gap between the two rods to avoid virtual leaks. These junctions

250

Figure 9.9: Transferring Rb atoms past the bump. (a) The oscillate function of the stage can beused to move the stage a large total distance (red) without it moving more than several mm fromthe initial position. This shows that the loss is occurring at a specific position (black) rather than aspecific distance. (b) We can also initially move backwards by 2 cm, and then move forwards withthe same profile (red). This shows that the sharp loss occurs 2 cm further than the normal profile(black), which suggests again that the loss occurs at a specific position. (c) Moving the quadrupole
trap further, we see loss when we stop at exactly the position of the magnetic field bump. (d) The
horizontal (clack) and vertical (red) position of the cloud as we move past the bump position. Thisshows that there is a sharp change of the field at the same position where we lose atoms from thetransfer beam.

251
between the right-angle SS316 sections (see Chapter 8) and the tungsten rods are centered very
nearly at 2 cm, and our best guess is that they cause the sharp magnetic field gradient.
It is well known that mechanical stress can induce magnetic domain formation in otherwise
minimally-magnetic materials like SS316. The SS316 rods have been rounded, and the crimped andmachined SS316 hypodermic tubing sections certainly have substantial mechanical stress. Further,
the gradient field from the quadrupole field causes fields of ≈ 100 G at this location, which are
sufficiently high to eventually magnetize a magnetically susceptible material. We tried to de-Gauss
the magnetic domains by reversing the field direction, and by leaving the magnetic field on when
we move the coils back to the MOT chamber. Typically we turn the current off before we returnthe coils to the MOT chamber. None of these approaches were suscessful in completely removing
the magnetization, but the axial dipole force in the transfer beam is very small and thus opticaltransfer is very unforgiving.

9.3.5

Moving further in the magnetic trap

Other groups have dealt with magnetization issues during transfer (magnetic rare-earth atoms
are especially susceptible to this issue). Our solution was to use the large magnetic field gradientfrom the quadrupole trap to completely swamp the local, sharp gradient from the magnetizedregion. There are two options: either we move the quadrupole trap and optical trap together overthe first several cm in a magnetic-optical hybrid trap [160, 235], or we initially transfer further
in the magnetic trap and evaporate several cm past the center of the small cube. Since the plug
beam is already aligned through the science cell, we decided to pursue the second option and all we
had to change was the focus position of the plug by several cm (the cart coils had to be modified
slightly to accomodate obstacles which arose within the additional several cm, but this was quite
straightforward).
Therefore, we can now magnetically transfer up to 5 cm further than the center of the small
cube, and by transferring to variable positions within this 5 cm, we can see if there is a significant
change in the magnetic field. Any change in the background field will move the position of the252
zero-field region of the quadrupole trap and in turn move the cloud. This is shown in Fig. 9.9c&d,
where the number is adversely affected by transferring to exactly the location of this sharp magnetic
field bump. Transferring past it with sufficient speed appears to mitigate any loss, and the heatingthat results is minimal. Figure 9.9d shows how the cloud position changes near the location the
sharp gradient, and this measurement corroborates all the measurements with the optical transfer.
Therefore, we solved this problem by transferring ≈ 4 cm further in the cart quadrupole coils andperforming evaporative cooling offset from the center of the cube. We load the transfer beam fromthis position as discussed above.

9.3.6

The optical transfer efficiency

The transfer efficiency is now ruitinely ≈ 90% for both species to go the remaining ≈ 12 cm.
The transfer profile is a s-curve to maintain constant jerk, and reaches a maximum accelaration of≈ 480 mm/s2 . The motion profile of the stage was carefully chosen based on the axial trap frequencyof the transfer beam [42, 107], which is ≈ 6 Hz for Rb and ≈ 8 Hz for K, though it is slightly nonadiabatic [49] and some heating is observed. The acceleration, velocity, and position during the
motion of the stage is shown in Fig. 9.12a. We find that the temperature after transfer is slightly
higher than before transfer, and we find that the difference depends on the intial temperature,suggesting that the efficiency is limited by the trap depth when the intial cloud is too hot. This
will be discussed more in the next section. The clouds are typically 20 µK for both K and Rb after
transfer to the science cell.
Figure 9.10 shows how we mount the optics for the transfer beam, and the two mirrors on
the stage are mounted together at exactly a right angle using a custom aluminum frame mounted
onto the stage. These mirrors are not adjustable, so it is quite important that the incoming beam
is parallel to the axis of translation of the stage. We do this using a pinhole on the stage and thefirst two mirrors after the fiber, while the intermediate lenses are removed. The springs on typicalkinematic mounts can cause substantial pointing noise resonances, and these resonances can be
excited by the acceleration (and any minimal jerk) of the transfer beam during its motion. We

253

Figure 9.10: The optical layout of the transfer beams. All the optics are stainless steel, except theright-angle mirror mount on the stage which is aluminum.

254
find that using the corner-cube-style aluminum block instead of kinematic mirror mounts slightly
reduces the heating of the clouds during transfer. Note that we observe a finite jerk at the very
beginning of the motion profile using the digital scope program for the stage. We can vary theparameters for the stage feedback and motion, but they have been observed to not significantly
affect the transfer efficiency or transfer heating.
Further, the alignment of the lenses is critical to mitigate any aberrations on the beam.
This is particularly important since the axial profile of the beam does the trapping against theacceleration of the motion profile, and thus any astigmatism or coma will adversely affect thetransfer efficiency. We align the lenses in the 6.7× telescope and the final focusing lens (the three
optics before the stage) one at a time. All three are achromatic doublets. With the beam fromthe prior two mirrors perfectly aligned through the pinhole on the stage, we place the second lensf = 200 mm in the mirror mount, and position the lens such that it is still aligned to the pinhole
which should correspond to the exact center of the lens. We can then replace the lens with a mirror
to reflect the beam, and we align the kinematic mount to retroreflect the beam back into the fiber.
We may have to interate this process a few times. Then we do the same procedure with the firstlens f = 30 mm. Now the beam on the pinhole has been magnified by 6.7×. Finally we repeat
this procedure with the third lens f = 750 nm, which is the focusing lens onto the atoms. This
procedure produces a perfectly round beam over a large range around the focus.

9.3.7

Heating during transfer

To characterize the amount of heating during the optical transfer we move the stage anappropriate distance to carry the atoms into the science cell, and then we move it back to theposition of the magnetic quadrupole trap. This round-trip approach allows us to remove any
systematics from different imaging systems in the two different regions of the chamber. Figure 9.11
shows the measured temperature and atom number with round-trip transfer and with no transfer
at all. The hold time in the transfer beam is the same for both. This measurement is carried out
for two different powers of the transfer beam, corresponding to trap depths of 180(10) and 250(10)

255
µK.
We control the temperature of the atoms loaded into the transfer beam by varying theevaporation cut in the plugged quadrupole magnetic trap. The abscissa of these plot is the cut
frequency in the magnetic trap, and it is proportional to the temperature. As a point of reference,a cut frequency of 6838 MHz corresponds to a temperature of ≈ 10 µK in the magnetic trap. The
difference in temperature after loading into the transfer beam comes from the adiabatic compression
due to the higher geometric-mean trap frequency in the transfer beam, and also the different powerlaw of the linear magnetic quadrupole trap versus the harmonic optical dipole trap.
Figure 9.11 shows that the round-trip transfer efficiency is excellent for all cases. However,
the temperature after round-trip transfer is always the same regardless of the initial temperature. In
all cases the cloud temperature saturates at ≈ 1/10th of the trap depth in the transfer beam. Thisis a common η-parameter for optical traps and optical evaporation. This measurement suggeststhat there is a very large amount of heating due to the transfer. In fact, we find that virtually all of
this heating comes from the very first ≈ 1 mm of transfer, where clearly something non-adiabaticis happening to the cloud at the beginning of the motion profile.
We attribute this to the fact that we are using a magnetic mechanical-bearing stage, which are
known to have more friction than air-bearing stages. Indeed, upon monitoring the motion profileof the stage we see that the position error, velocity error, and driver current error all have a spikeright as the motion profile initiates. This is needed to overcome the static friction of the system.
By varying the servo parameters of the stage we were able to reduce this spike by a factor of two
or so, but we were never able to remove it. Such changes had no observable effect on the heating
during transfer, and it seems that these spikes need to be completely removed in order to reduce
this heating. We have recently switched from this magnetic mechanical-bearing stage (Aerotech
ANT130-110-L-PLUS) to an air-bearing stage (Aerotech ABL10100-LT), which is presently being
investigated.

256

Figure 9.11: Heating during transfer in the transfer beam. (a) The temperature of Rb with (black)and without (red) round-trip transfer in the transfer beam with a trap depth of 180 µK as a functionof the evaporation cut in the magnetic trap, which changes the temperature of the cloud. (b) The
same measurement except with a trap depth of 250 µK. (c) The number of atoms with transfer
normalized to the number without as a function of the evaporation cut in the magnetic trap, fora trap depth of 180 µK. (d) The same measurement except with a trap depth of 250 µK in thetransfer beam.

257
9.3.8

An elliptical transfer beam

We recently changed the beam shape of the transfer beam from round with waist ≈ 45 µm toan elliptical beam of sizes ≈ 130 µm in the horizontal direction and ≈ 30 µm in the vertical direction.
This serves to increase the axial trap frequency by ∼ 30% and decrease the axial size similarly. We
thought this would improve the evaporation efficiency, particularly of K, in the subsequent crossed
optical dipole trap, and we thought that this geometry could improve the loading efficiency into thecrossed dipole trap. We found that the loading efficiency and the evaporation efficiency was very
similar between the round and elliptical transfer beams. The loading efficiency into the transfer
beam and the transfer efficiency to the science cell is similar between the two, but the transfer inthe elliptical beam may actually have slightly less heating with the same efficiency.

9.4

Optical evaporation to a degenerate mixture
Once the atoms reach the science cell, we load them into a crossed optical dipole trap com-posed of 1) a horizontal optical dipole trap (H OT) with 1/e2 radius of 145(10) µm in the transfer
direction and 30(3) µm in the vertical direction, and 2) a vertical beam (V OT) propagating throughthe ITO-coated plate which is round with radius 100(10) µm. The powers of these two beams are
6.5(5) W and 4.0(5) W, respectively. Figure 9.12b shows the timing diagrams throughout the experiment for K and Rb. The next step is to optically evaporate in the crossed optical dipole trapby reducing the power. We load to the crossed optical dipole trap in 200 ms, while simultaneously
reducing the transfer beam power to ∼ 30% of the initial value. The loading efficiency to this trap is∼ 60% for both K and Rb. Typical conditions in the crossed dipole trap are NRb = 9.0(5)×106 and
NK = 1.5(3) × 106 . Many ramping and loading procedures have been used, and most trajectories
give essentially the final conditions after evaporation, to be discussed later.

258
9.4.1

Compressing the cloud in the transfer beam with a B-field gradient

We find that the loading efficiency from the transfer beam into the crossed dipole trap isenhanced by using a magnetic field gradient of ≈ 25 G/cm on the strong axis, whose center is ≈ 1
mm above the cloud. This serves to compress the cloud along the axial direction of the transfer
beam, reducing the size by ≈ 30%. However, this effect is less for the elliptical transfer beam
because the axial trap frequency is higher and thus the axial size is smaller. After the cloud reaches
the science cell we ramp up this gradient in 200 ms, and then load the crossed dipole trap in 200
ms. We ramp the gradient to zero quickly during the subsequent evaporative cooling in the crossed
dipole trap. We have also performed rf-knife pre-evaporation of Rb in the transfer beam plus
magnetic field gradient (which we call the squeezed transfer beam) by driving the |2, 2i to |1, 1i
in a decreasing frequency sweep that cuts into the cloud from its edge along the axial direction.
This evaporation allows us to cool the cloud in the squeezed trap from ≈ 25 µk to < 10 µK whilelosing a factor of roughly 2 in Rb and almost none in K. This technique yields lower numbers in
the crossed dipole trap, but the initial temperature is lower, and thus we end up with slightly more
K at the end of evaporation. This method was considered in an efort to perserve our K number,which has been a major limitation, as discussed next.

9.4.2

Loss of K during evaporation

Again, we sympathetically cool K as Rb is lost first from the trap due to its larger mass andsimilar polarizability. A major problem, however, is that at higher temperatures when the trap isvery deep the effect of gravity is negligible. Therefore, since the polarizability of K is lower than thatof Rb by ≈ 20% at λ = 1064 nm, we lose both K and Rb during evaporation. Once the trap depth
is sufficiently low the effect of gravity becomes significant and thus the trap depth of Rb begins
to drop faster than for K. This is desired since we want to evaporate Rb while sympatheticallycooling K. Moreover, the crossover value of the trap depth between these two regimes depends
on the vertical waist of the horizontal beam (the waist opposing gravity) since this dipole force

259

Figure 9.12: Timing diagrams for the procedure of the experiment. (a) The motion profile of stage,
showing the position (top), velocity (middle), and acceleration (bottom) commands of the stage
as a function of time. Analysis of the position and velocity errors during the motion has beenvery useful, but is not shown here. (b) A timing diagram for the experiment, showing MOT beam
powers, magnetic field strengths, and optical trap powers.

260
directly counters the linear potential from gravity. For wz = 45 µm the crossover occurs at T ≈ 2µK at which the trap depth is V ≈ 20 µK. For wz = 30 µm the crossover occurs at an even lower
trap depth. Since we start with T = 20(2) µK, we lose ≈ 80% of the initial K before we reach thecrossover into efficient sympathetic cooling after which we lose another factor of two or so.

9.4.3

Implementing the 790 nm beam for additional trapping of K

We have tried many approaches to combat this issue. One of which is to load a colder cloud inthe crossed dipole trap by pre-evaporative cooling using the rf-knife of Rb in the squeezed transfer
beam. While this helped us load a colder cloud into the crossed dipole trap, it is difficult to get it
cold enough in the transfer beam such that we can efficiently load into the crossed dipole trap witha trap depth sufficiently low that gravity causes the trap depth of K to be larger than Rb, which
we calculate to be ≈ 30 µK for our trap geometry.
To this end we turned to species-dependent traps that helped us hold K in the trap more
effectively than Rb. We use λ = 790 nm which was discussed in Chapter 4 and Chapter 5, andcan be used to trap K only. The caveat here is that the heating rate for Rb is ≈ 1 µK/s for a
power of ≈ 50 mW in a beam the same size as the horizontal dipole trap. This heating should bemanageble for Rb during the first third of evaporation, which is only ≈ 500 ms. After this gravity
will cause the trap depth for K to be larger than for Rb, and thus we do not need the 790 nm beam
after the trap depth decreases from ≈ 80 µK down to ≈ 30 µK. However, we have never seen any
significant improvement from using this beam, and the heating of Rb is much higher than expected.
We expect that the issue could be due to the overlap between the 1064 and 790 nm beams.

9.4.4

Optimized trajectory

Ultimately, we do not rely on the 790 nm beam or the squeeze coils to enhance the loadingor evaporation since neither approach significantly enhanced the final conditions. The optimized
trajectory using the transfer beam, H OT, and V OT is shown in Fig. 9.12b. After evaporation we
end up with NRb = 1.0(2) × 105 Rb and NK = 1.0(2) × 105 at T = 300(30) nK. These conditions

261
are comparable to the conditions in the old apparatus, and we believe we can still make further
improvements. Figure 9.13a shows time of flight expansion of a quasi-pure BEC of 5 × 104 atoms.
Figure 9.13b shows a degenerate Fermi gas of 5 × 104 K atoms with temperature 0.5TF . We
measure the temperature by fitting the wings of the density profile in expansion to a Gaussian, andextracting the temperature from that fit [64], as discussed in Chapter 6. Fitting the entire cloud
to a Fermi-Dirac distribution [64] gives a consistent temperature.

9.5

State preparation, and producing molecules
During the optical evaporation we transfer K and Rb from |9/2, 9/2i and |2, 2i to the

Feshbach-resonant states |9/2, −9/2i and |1, 1i [285]. This is done at 26 G using Adiabatic Rapid
Passages (ARPs) by sweeping the frequencies. For Rb, we sweep the frequency from 6.898492 to
6.898992 GHz in 2 ms with a power to the antenna of 40(2) dBm. For K, we sweep the frequency
from 10.2 MHz to 8.5 MHz in 20 ms with a power to the rf coil of 35(2) dBm. We demonstrate
excellent control of the population in all 10 spin states for K in Fig. 9.13c by during Stern-Gerloch
spin seperation during expansion [225].
We initially had a difficult time coupling rf and microwaves into the science cell, and weeventually learned that the electrodes are partially the reason for this. As discussed in Chapter
7, they act as an effective Faraday cage for electric fields, and thus they attenuate the rf andmicrowave signals. In order to have a good enough coupling between the rf and microwave coils
and the atoms in order to reach Rabi frequencies of Ω ≈ 10 kHz, we must use > 30 dBm at 10 and
80 MHz and > 40 dBm at 6.8 and 8 GHz.
The disadvantage of using large rf fields is that they can couple onto electronics, such as
the servo for the magnetic field, and rectify to DC or frequencies below the servo bandwidth of≈ 1 kHz. To address this issue we used a lot of filtering on the Hall probe for the magnetic field
servo, and on the servo input. We found that the grounding scheme of the B-field coils and theirservos was a very important part of removing this coupling. We also needed to shield many of theelectronics to minimize their exposure to the rf field. In the end we were able to do this sufficiently

262

Figure 9.13: Ultracold atoms and molecules in the science cell. (a) shows a series of expansion
images of a Rb BEC of 5 × 104 atoms. (b) shows an image of 5 × 104 K in expansion, whosecross-section fits to a Fermi-Dirac distribution give a temperature of 0.5(1)TF . (c) shows state
preparation of K at 30 G, indicating excellent control of the populations of individual spin statesin the F = 9/2 manifold. Shown from left to right are mF = 9/2, −1/2, −3/2, −5/2, −7/2, and−9/2. The gradient does not allow for separation of the +mF states. (d) shows an image of < 104
Feshbach molecules in expansion.

263
well that the rf has negligible effects on the magnetic field, but it took us some time to learn how
to do this successfully.
Once both species are in the Feshbach resonant states we sweep the magnetic field up to560 G in 500 ms towards the end of evaporation such that the atoms are above the Feshbachresonance as the final temperature is reached. Then, by sweeping the field from above to below theresonance we can associate K and Rb into Feshbach molecules [52, 183, 285]. Feshbach molecules
can be detected by probing with resonant light of either species (Fig. 9.13d shows < 104 Feshbach
molecules imaged using K). The final field used after the Feshbach sweep embues the molecules witha binding energy of ∼ 100 kHz, and so resonant with either atom dissociates the molecules [285].
However, they can be differentiated from free K atoms by transferring free K to the |9/2, −7/2istate. The binding energy can be sufficiently large compared to the Rabi frequency of this π-pulse
in order to not affect the Feshbach molecules. This pulse is at 80 MHz, and the time of the π-pulse
is typically ∼ 100 µs.
To make ground-state molecules, all unpaired atoms can immediately be removed with resonant light as soon as the molecules are transferred to the ground state, and therefore no statetransfer manipulation is needed. The STIRAP pulse time is typically ∼ 10 µs. The STIRAP beams
propagate down along the vertical direction. It is important that they are along the quantization(B-field) axis, since their polarization is important [190]. In both the old apparatus and now thehighest molecular yield from a bulk gas was found using atomic temperatures near TC and TF of∼ 100 − 200 nK. In such cases, we are able to produce ≈ 1 × 104 ground state molecules with atemperature of ≈ 2 − 3 TF , as shown in Fig. 9.13d. Such conditions are a factor of three worse
than in the old apparatus [190], and our work on this issue is ongoing. We believe the new motion
stage for the transfer beam will help us with this problem.

9.6

Optical lattices
The horizontal lattice beams for the 3D optical lattice come through the diagonal windowsof the octagonal cell, and they are the only beams which use these paths. This allows us to keep

264
the horizontal lattice beams simple. They are retro-reflected with a λ/4, lens, and mirror after
the incident beam exits the science cell. The lens collimates the beam after the incident pass, andre-focuses the retro-reflection back to the same place in the center of the science cell. We alignthe retro-reflection using a kinematic mount for the mirror, and optimize the power that goes backthrough the optical fiber.
We choose the lattice beam size to be ωz = 40(5) µm in the vertical direction and ωr = 160(10)µm in the horizontal direction. We maintain an angle of ≈ 3o between the horizontal lattice beams
and the axes normal to the windows of the science cell through which they pass. This is to avoid
superlattice effects due to the ≈ 1% reflection per surface from the cell windows, as discussed in
Chapter 4. We calibrate the lattice depths using parametric heating and Kapitza-Dirac scattering,as discussed in Chapter 4. We again find that the two techniques give consistent results.
We put a λ/2 on the incident path of the lattice such that we can tune the polarization of thelattice beams. This is important to achieve the magic polarization angle discussed in Chapter 4. The
DC electric field will nominally be oriented in the vertical direction, which is also the direction of
the magnetic field. Thus, vertical polarization will give a 0 degree angle, and horizontal polarization
will give a 90 degree angle. However, since these fields are oriented along the vertical direction,
the vertical lattice will always be at a 90 degree angle, and thus we will not be able to reach themagic angle of ∼ 50 degrees with this beam. For this reason the “magic electric field” discussed in
Chapter 10 will be very important [150].

9.6.1

Vertical lattice path

The vertical lattice path is significantly more complicated since it has to propagate throughall the optical substrates in the science cell. As a result of the baking process the reflection of
the ITO-coated plates is a few percent, and thus we cannot directly form a lattice with normal
incidence. Moreover, since the dichroic below the science cell to reflect λ = 1064 nm cannot beangled by more than a few degrees, we were forced to use a more complicated beam path for thevertical lattice. This path is shown in Fig. 9.14, and it is based on two beams with a ≈ 10 degrees

265with respect to the cell.
The beam that hits the cloud before reflecting off of the dichroic (red) is the vertical opticaltrap beam discussed above. This beam is already part of the crossed optical trap, and to turn on
the vertical lattice we ramp on the beam which is aligned to the cloud after a reflection from thedichroic (green). We use a waist of 100 µm for both of these beams, but they can be independently
changed. The lattice spacing of the optical lattice scales as (λ/2)/ cos(θ), and for θ = 10 degrees,θ
0 cos(θ), and thus
the lattice spacing is 540 nm. The lattice oscillation frequency ωlattscales as ωlattthe oscillation frequency from this configuration is 98% of what it would be with a retroreflectedlattice.
This lattice is much easier to align than we initially expected. Both beams are essentiallythe same, except the position of their focus is different. We can independently align them onto thecloud. Both beams can be aligned such that either the incoming or the reflected beam hits thecloud. We can identify which is which by reducing the transfer beam to allow the atoms to slide
down the vertical beam. By observing whether the atoms slide down sloping left or right we are
immediately able to identify whether the beam is hitting the atoms before or after bouncing off thedichroic. Figure 9.14b shows horizontal images of the coherent diffraction (see Chapter 4) of a Rb
BEC off of the vertical lattice potential when it is pulsed on for a few µs.
An issue of immense important for spectroscopically selecting a single layer in the verticallattice is that each antinode moves during the procedure by an amount which is small comparedto their separation ≈ 0.5 µm. In order to guarentee the phase stability between the two beams weinterfere them both on a beam splitter. As shown in Fig. 9.14, the lattice is formed by only oneof the beams reflecting off of the dichroic and thus we are particularly sensitive to its drift. If weinterfere the two beams after they both reflect off the dichroic then the drift of the dichroic would
be common mode. Therefore, we interfere the incoming vertical optical trap beam before the cellwith the reflection of the vertical lattice beam after the cell and the dichroic.
This allows us to maintain a constant phase of the lattice when only one beam goes throughthe dichroic. We use a ring piezo actuator on the holder of the fiber for the vertical lattice so that

266

Figure 9.14: The vertical lattice. (a) Scheme for forming the vertical lattice using a reflection fromthe dichroic below the science cell. All the surfaces in blue are on substrates in the cell such as its
vacuum windows and the ITO-coated plates inside the cell. The hashed blue surface is the dichroic
below the science cell which reflects λ = 1064 nm. The vertical optical dipole trap hits the cloudat a ≈ 10 degree angle before hitting the dichroic (red). The vertical lattice beam has a similarangle, but it hits the dichroic first (green) and interferes with the incoming beam after reflecting
off of the dichroic. (b) Horizontal imaging of the diffraction of a Rb BEC off of the vertical lattice.
The images from left to right show the lattice pulsed on for successively longer times, from 1 µs
to 8 µs, with 8 ms expansion time. Notice that the central component is depleted and re-emerges
during this time. Note that a tilt angle of a few degrees has been removed, which is due to an angle
between the horizontal imaging system and the vertical direction.

267
we can change its phase independently. We use the interference fringe on a photodiode to feed backto this piezo. All other optics in the set up of these two beams are stainless steel mounts whichwere arranged in the most stable way possible in an effort to ensure that the drift of the relative
phase is predominantly caused by the dichroic.

Chapter 10

New physics with the new apparatus - high resolution optical detection andlarge, stable electric field

With the excellent conditions of KRb molecules in the new apparatus reported in the lastchapter, we are ready to start exploiting its novel capabilities. Again, these capabilities include
large, versatile electric fields; AC electric field polarization control; and high resolution in situ
detection and addressing. I will now discuss our work in the new apparatus which uses these tools.
Note that at the time of this writing our work on some of these subjects is very preliminary andlimited. Nevertheless, this chapter is intended to describe current research directions, and directions
we are preparing to investigate imminently.

10.1

Large, stable electric fields
We can begin to apply large electric fields to molecules confined in a bulk 3D gas or a 1D latticeeither along the vertical or horizontal directions. We measure the electric field by performing Stark
spectroscopy [106, 180, 190, 203, 252], in which we measure the energy of a rotational transition as
a function of the voltages on the electrodes. See Chapter 3 for a discussion of DC Stark shifts. For
these measurements we keep the electrodes in the homogeneous configuration discussed in Chapter
7. The frequency of the N = 0 to N = 1 rotational transition varies between 2.23 GHz at E = 0
to ≈ 8 GHz at E = 30 kV/cm, and thus the rod electrodes need to have sufficient AC couplingover this entire range (see Chapter 8). A schematic of the electrode geometry with the molecules
in optical lattices is shown in Fig. 10.1.

269

Figure 10.1: A schematic for applying electric fields to molecules in optical lattices. (a) A verticallattice propagates through the ITO-coated plates. (b) The horizontal lattices propagate at a 45oangle with respect to the symmetry axis of the electrodes. (c) The E = 0 level structure of theN = 0 and N = 1 rotational states. (d) The geometry of the DC electric field and electric field of
the lattice beam from one of the horizontal lattices. The angle between the DC electric field andlattice electric field θ is tunable between 0 and π/2.

270

10.2

Driving rotational transitions - AC electric field coupling
As discussed in Chapter 7 the rods were designed with the capability of applying microwave-

frequency AC electric fields. The orientation of this electric field can be controlled outside thevacuum chamber, and thus by changing the relative orientation between the AC driving field
and the quantization axis (either the DC electric field or the magnetic field) we can choose thepolarization of the microwave field. This allows us to selectively drive rotational transitions of a
specific relative anglular momentum; either a π transition when EDC k EAC , or σ + and σ − when
EDC ⊥ EAC . Previous work in the old apparatus did not have any polarization control, and thus it
was not possible to selectively drive these transitions. Only energy selectivity allowed us to choose
the state to which we coupled in the old apparatus.
Therefore, we expect the orientation of EAC to affect the excited rotational states to which weKRb
can couple (see Fig. 10.2). The ground state is |N, mN , mK
I , mI i = |0, 0, −4, 1/2i, where mI andmRb
I are the nuclear spin projections of K and Rb onto the magnetic field, respectively. Therefore
when we drive π transitions with EDC k B k EAC , the accessable excited states are |1, 0, −4, 1/2i,
Rb|1, 0, −3, −1/2i, |1, −1, −3, 1/2i, |1, −1, −4, 3/2i, |1, 1, −4, −1/2i such that mtot ≡ mN + mK
I + mI

remains the same (mtot = −7/2). When EDC k B ⊥ EAC , both the σ + and σ − transitions areavailable, so there are many options such that mtot = −9/2 or −5/2.

10.3“Magic” electric fields
The polarizability as a function of the angle between the quantization axis and the polariza-tion axis discussed in Chapter 4 was in the absence of an electric field (see Fig. 4.9). However, whena large electric field is applied it polarizes the molecules and the dependence on the polarization
angle changes [150]. At E = 0 the polarizability of the |1, 0i state is higher than |0, 0i for 0 degrees
and lower for 90 degrees. Conversely, when the electric field is sufficiently large this trend reverses.
At a special field Emagic the differential polarizability (difference in polarizability of the two states)
is zero for all angles, as shown in Fig. 10.3.

271

Figure 10.2: Applying AC electric fields and driving rotational transitions. (a) The DC electricfield (purple arrow) and the AC electric field (red arrow) are parallel, driving π transitions with∆mtot
I +∆mN = 0. (b) The DC electric field (purple arrow) and the AC electric field (red arrow) are
perpendicular, driving σ + and σ − transitions with ∆mtot
I + ∆mN = ±1. (c) Some of the rotational
tot
and hyperfine transitions that can be driven with ∆mtot
I + ∆mN = 0. Either ∆mI = ∆mN = 0,or ∆mtot
I = −∆mN . (d) Some of the rotational and hyperfine transitions that can be driven withtot+
∆mN = ±1. Either ∆mtot∆mtot
I = 0 and ∆mN = ±1, or ∆mI = ±1 and ∆mN = 0. Note thatI(c) and (d) are place holders and do not represent data. Data taking is currently underway. The
difference between orange and purple is supposed to represent the fidelity of the AC electric fieldpolarization.

e = 0 and 1 states (here called
Figure 10.3: The magic electric field. (a) The polarizability of the Ne~J) as a function of the electric field when the trap polarization ~ is parallel to the electric field E.
e(b) The polarizability of the N = 0 and 1 states as a function of the electric field when the trap
~ Reproduced from Ref. [150].
polarization ~ is perpendicular to the electric field E.

272
For KRb this occurs at 10 kV/cm [150], which is coincidently similar to the field at whichJz is maximized [112]. As stated in the previous chapter this “magic electric field” is particularly
important because the vertical lattice is oriented along the electric and magnetic field direction,and thus the relative polarization angle will always be 90 degrees (as in Fig. 10.3b). Accordingly,
the magic electric field will be the only way to reach zero differential polarizability.
We study the dependence of the differential polarizability on electric field by loading molecules
into a 1D lattice along the horizontal axis such that we can tune the polarization angle relative
to the field between 0 and 90 degrees (see Fig. 10.1b&d). We measure the polarizability of |e0, 0i
and |e1, 0i in two ways, such as in Ref. [188] (the tilde denotes the basis of rotational states in
the presence of a large electric field where eigenstates are no longer pure states. See Chapter 3).
The first method is to measure the lattice depth by performing parametric heating, as discussed
in Chapter 4. The second method is to use the transition frequency in the lattice compared to thetransition frequency in free space (expansion).

10.4

Stabilizing reactive dipoles in 2D with large electric fields
Figure 6.3a showed how the elastic and inelastic (reactive) collision channels varied as afunction of the lattice depth in a 1D lattice and the electric field along the lattice axis. Specifically,
when both the electric field and the axial confinement from the vertical lattice are very strong,
the repulsion between molecules is maximized and the ratio of elastic to inelastic collisions is veryhigh. In such a regime we expect that it may be possible to observe Wigner crystallization at
sufficiently low temperatures, where the potenial energy from the dipolar repulsion dominates the
kinetic energy and the molecules develop an ordered spatial distribution. This case was considered
in Reference [13], and this scenerio would allow us to study spin models without pinning themolecules in a deep 3D lattice. The dipole moment of KRb is marginal, and may preclude thisobservation. Nevertheless it should be experimentally viable for more polar molecules like NaK,
RbCs, and NaRb.
With the ability to reach large electric fields in hand, we can now enter the regime of stabilized273
dipoles in 2D by using the vertical 1D lattice. Previous work in Ref. [63] was only able to reachd = 0.2 Debye and νz = 23 kHz, where the ratio of elastic to inelastic collisions is only ≈ 200 at
T = 500 nK [283]. Under these conditions the lifetime was measured to be ≈ 1 sec [63], which
is similar to the case of a 3D gas with no electric field [191]. However, with the ability in handto reach d = 0.4 Debye and νz = 50 − 100 kHz at T = 200 nK, the ratio of elastic to inelastic
collisions can be > 1000 [283]. This is expected to lead to signficantly longer lifetimes. In thissetting it is possible to perform spin physics between | ↓i = |0, 0i and | ↑i = |1, 0i without a full3D lattice, where strongly repulsive dipolar interactions stabilize the system against the s-wave
collisions discussed in Chapter 3.

10.5

Towards evaporative coolingWith such strong dipolar repulsion between molecules in a 2D system, we can return to ourprevious efforts on evaporative cooling as discussed in Chapter 6. We tilt the 2D traps in the radial
direction of the vertical 1D lattice using a horizontal gradient of the electric field as discussed in
Chapter 7. Gradients of ≈ 100 V/cm2 are sufficiently high to begin spilling the molecules. We workin the homogeneous field case where Vrod /Vplate = 0.4225, except that we bias the rod voltages in
the horizontal direction as discussed in Chapter 7. The large ratio of elastic to inelastic collisions
will significantly improve the thermalization time relative to the chemical reaction time, leading tomuch more efficient evaporation than in our previous efforts discussed in Chapter 6.
Also as discussed in Chapter 6, we previously started from evaporation with T /TF = 3 − 4 inthe 2D pancakes. Our current stratagy in the new apparatus will allow us to begin at T ≈ TF , which
combined with the improved elastic to inelastic collision ratio should enable efficient evaporation toT < 0.5TF . The improvement in the initial PSD in the pancakes comes from two signficant changes.
The first comes from a change in the aspect ratio of the initial optical dipole trap. Previous thewaist of the beams in the vertical direction was ≈ 45 µm, but now we use ≈ 25 µm (discussed in√
Chapter 9). This reduces the number of pancakes occupied by ≈ 2×, which increases TF by 2.
We are also preparing another H OT beam with a vertical waist of ≈ 5 − 10 µm to compress the274
cloud into even fewer pancakes. The other improvement will come from cooling the atoms in the
1D lattice prior to making molecules. Previously, we loaded molecules in a 3D gas into the 2D
pancakes. Now we plan to cool the atoms in the 2D pancakes, and makes molecules in 2D from
sufficiently cold atomic clouds. If necessary, atomic evaporation in 2D can be done using a radial
magnetic field gradient [266]. The coils around the science cell discussed in Chapters 7 and 8 aredesigned for this purpose. As a result of these improvements we anticipate the initial conditions
for molecular evaporation to be T ≈ TF with N > 104 molecules per pancake.

10.6

Spectroscopic layer selection using an axial electric field gradientIn order to perform high resolution in situ imaging of the molecules we need to be able tocreate a single 2D subsystem. Further, in order to access bilayer exciton physics [111, 166] we need
to be to uniquely control individual layers. In order to do this we must spectroscopically select asingle layer in the vertical 1D lattice using an electric field gradient along the vertical direction as
discussed in Chapter 7. We work in the homogeneous field case where Vrod /Vplate = 0.4225, exceptthat we bias the rod voltages in the vertical direction as discussed in Chapter 7. Note that thephase stability of the vertical lattice as discussed in Chapter 9 is crutial.

10.7

High resolution in situ imaging of polar molecules
While ground-state molecules can be imaged directly [264], such imaging results in a poor

signal-to-noise ratio due to the lack of closed cycling transitions. However, it became clear thatmolecules can be detected in the same way they are created: by reversing the STIRAP process
and the magneto-association process of the Feshbach molecule, a ground-state polar molecule canbe converted to a pair of free atoms with ≈90–95% fidelity [190, 285], which is limited by thepresence of other states that make the three-level lambda-system picture only an approximation [3,260] (for further background see e.g. [53, 182]). It is important to point out, however, that the
STIRAP process is state-selective, and energy and angular momentum selection rules allow all
excited rotational states to remain decoupled from the STIRAP sequence.

275
Our initial work on high resolution in situ imaging of polar molecules is based on absorption
imaging. Many group have used high resolution in situ absorption imaging of 2D atomic samples
with Rmin ≈ 1 µm for many species. This has been done both in a lattice and in a bulk 2D gas
for both bosons and fermions. Species including Cs [88], 87 Rb [67], 40 K [78], and 6 Li [121, 177]
have been used. A nice overview of the set-up and characterization of such imaging systems canbe found in Refs. [46, 122, 124]. As discussed in Chapter 6, we can detect molecules by imaging K
or Rb, and high resolution absorption imaging of both has been demonstrated.
The new KRb apparatus has a microscope objective for high resolution in situ detection ofpolar molecules. Initial experiments will be done using absorption imaging [88], but future directions
will be based on Raman sideband cooling (RSC) of K and Rb for operating a dual quantum gas
microscope. Details of the new apparatus and its optical capabilities are outlined if Ref. [50].
Future work towards a quantum gas microscope will use an objective with numerical aperture(NA) of 0.6 − 0.65, which corresponds to an optical resolution at λ = 780 nm of Rmin = 790 − 730
nm (Reyleigh criteria). However, the current objective has an NA of 0.53, with a resolution of 900
nm. We use an Andor electron-multipying CCD (EMCCD) for detection. We currently image witha magnification of M ≈ 24 such that Rmin is mapped onto several pixels.

10.8

Single-molecule addressing in an optical lattice
Sending a beam through the objective backwards allows us to focus a beam onto the atoms,

which can impart an AC Stark shift onto that region of the array. For molecules most wavelengths
have deleterious effects since their photon scattering rates are high due to the enormous number ofrovibronic excited states. We have tried many wavelengths for KRb molecules (see Chapter 4), andfound 532, 790, and 808 nm to all have prohibitively large scattering rates. Moreover, we foundthat the polarizability is small and crosses zero at ∼ 750 nm. Fortuitously, we know that λ = 1064
nm has a low scattering rate [47] while still providing a large polarizability [188]. Therefore, wewould like to use a 1064 nm beam to selectively Stark shift specially tailored regions of the cloud.
An objective with resolution of Rmin = 900 (750) nm at λ = 780 nm can also be used to focus a276

Figure 10.4: Simulations of atoms on sites of the lattice imaged with two different resolutions.
(a) Atoms imaged with a resolution of Rmin = 900 nm. The orange circle indicates the region
illuminated with a relatively large addressing beam. (b) The atoms that remain in one spin state
after using the addressing beam imaged with resolution of Rmin = 900 nm. (c) The atoms in theother spin state imaged with resolution of Rmin = 900 nm. (d) Atoms imaged with a resolutionof Rmin = 750 nm. The light-blue circle indicates the region illuminated with a relatively large
addressing beam. (e) The atoms that remain in one spin state after using the addressing beam
imaged with resolution of Rmin = 750 nm. (f) The atoms in the other spin state imaged withresolution of Rmin = 750 nm. Reproduced from Reference [51].

277
Gaussian beam of wavelength λ = 1064 nm to a 1/e2 radius of 860 (700) nm.
We now consider what AC Stark shifts result from an addressing beam of these sizes, andwe determine what powers will be required to reach a desirable Stark shift. To uniquely address
the molecules within the addressing beam, we require that the AC Stark shift be large comparedto the Rabi frequency, which is typically ∼ 10 kHz for such state preparation. Therefore, for abeam of 1/e2 radius 860 (750) nm, a power of 10 (7) µW is needed for a 50 kHz differential AC
Stark shift of the |0, 0i − |1, 0i transition at the center of the beam. To create a larger impurity
region as shown in Fig. 10.4, we want an addressing beam with radius of i.e. 2 µm, for which 50µW is needed. Note that similar powers are needed for the |1, −1i. All of these powers are easily
manageable. Also, note that the polarization of the addressing beam is linear and perpendicular
to the quantization axis, and the AC Stark shift was calculated accordingly [188].
We could also use λ = 690 nm for the addressing beam for single-molecule addressing. Thisis the wavelength of the down-leg STIRAP transition, and so by tuning nearby we can achieve
large Stark shifts from the strong electronically-excited state. Moreover, since this wavelength issignificantly shorter than 1064 nm, we will be able to focus it more tightly, even to the level of asingle site. However, much less is known about the polarizability at this wavlength than at 1064
nm, and so some exploration would be required to identify a regime where the Stark shift is large
enough and the scattering rate is small enough.
Another exciting possibility for creating an initial condition that will lead to novel out-ofequilibrium many-body dynamics, is to have the left half as | ↓i and the right half as | ↑i, with ahard domain wall in between, similar to recent work with atoms tunneling in a lattice [44]. While
this state could be created using AC Stark shifts from an addressing beam, an alternative approach
may be to use an electric field gradient across the layer using the electrodes in the science cell of the
new KRb apparatus, as outlined in Ref. [50]. Operationally, this approach would be similar to thetechnique of selecting a single layer as outlined above and in Ref. [50], except that the frequency
would be swept in order to flip the spins of all molecules on one side of the domain wall, as in an
Adiabatic Rapid Passage pulse, or ARP. Such ARPs are very robust, and any fluctuation in the278

Figure 10.5: A Schematic for combining the imaging and addressing beams. The imaging wavelengths are 767 and 780 nm, and the addressing wavelengths are 690 and 1064 nm. These can beseparated using dichroic filters.

279
field would only change the location of the domain wall.

10.9

Holographic projection of arbitrary potentials using a DMD
We can go beyond simply using a single tightly-focused beam, and introduce an array of

spots imaged onto the molecules by holographically projecting a spatial light modulator (SLM),
such as a digital micro-mirror device (DMD). These techniques have been used to create arbitrary
potentials by several groups [44, 173, 212]. In our case we can create arbitrary, random patterns
with λ = 1064 nm which create a disordered landscape as a setting for investigating many-body
localization [274]. The scattering rate at λ = 690 nm may be too high to use it for this purpose, butsince 690 nm is blue-detuned for K and Rb this technique could be used to project an anti-confining
potential to reduce the harmonic confinement in the 1064 nm lattice, and thus accomodate a larger
N = 1 shell, as discussed in Chapters 4 and 6.
Figure 10.5 shows how the imaging path, single-spin addressing path, and the DMD path
are all combined together using dichroic filters. We use a mirror with a piezo actuator to position
the addressing beam or DMD onto the cloud. The DMD is the Texas Instruments Digital Light
Processing (DLP) LightCrafter 6500 model, with DLPC900 DMD digital controller and 1080p
resolution (1920 × 1080 array). The micromirrors deflect at angles of ±12o , such that ‘on’ and ‘off’
are only ±24o . The mirrors rotate diagonally, and thus the DMD array must be mounted at a 45oangle in order for all the beams to stay on the plane of the table. This is shown in Figure 10.6.
Ideally, the DMD is implemented in a holographic fashion to create an arbitrary beam shapewhile simultaneously cancelling aberrations [202, 286]. Accordingly, we would like to locally control
both the amplitude and the phase simultaneously. This has been successfully demonstrated in thegroup of Markus Greiner by placing the DMD in the Fourier plane [286]. We employ a similar
technique, as shown in Fig. 10.5. An optimization and aberration-correction procedure is outlined
in Ref. [286]. However, this technique is rather involved since the Fourier transform pattern on the
DMD must have the correct amplitude and phase distribution.
Our initial goal is to control the spatial amplitude profile, and so we begin by using the DMD

280

Figure 10.6: The mounting scheme used for our DMD and its attached circuitry. Even though thearray of micromirrors is not square, we must rotate it by 45o degrees such that the ‘on’ and ‘off’
components both stay on the plane of the table.

281
to generate arbitrary patterns and waveforms. The easiest way to do this is to place the DMD in
the image plane, and map the DMD pattern onto the atoms with a de-magnifying telescope. In
this way, the required feature size of the pattern on the DMD can be determined by the desiredfeature size on the atoms divided by the de-magnification factor. Arbitiray images can be generated
in Matlab, and uploaded to the DMD as a bitmap (BMP) file in a look-up table (LUT). Thus,
arbitrary images from the internet or elsewhere can easily be loaded to the DMD. We operate the
DMD at the blazing angle to maximize the available power [286].
Figure 10.7 shows patterns uploaded to the DMD, and subsequently imaged onto a beamprofile CCD with a magnification of 1/6.7. There are three effects in these images which are
immediately apparent. First, the intensity profile of the illuminated regions is not perfectly homogeneous. This effect can be corrected as discussed in the following paragraph. Second, the images
are tilted at an angle of a few degrees. This is partially due to the beam incident on the DMD,
which has an angle of a few degrees with respect to the table. This effect is the reason that theflat-top disk in Fig. 10.7h is not perfectly round. Further, since the DMD is rotated at 45o the
profile CCD must also be oriented at this angle. The tilt angle of the images is also partially caused
by a relative angle between the DMD and CCD. Third, there are high frequency fringes across theilluminated regions. These are caused by the infinitely sharp edges on the DMD which generatesinc profiles in the image plane. Essentially, this is a resolution limitation where the achromaticlens that collimates the k-vectors from the DMD has a finite NA. Concomitantly, the lens aperturemust be considered in the Fourier transform of the DMD pattern (for more discussion on Fourier
optics, see Chapters 6 and 7).
This latter problem can be addressed by using a higher power lens immediately after the
DMD, but also by using a grey-scale on the DMD to provide edges of finite width. This can be doneby choosing a magnfication such that n2 micromirrors map onto the diffraction-limited resolution
2
area Rminof the objective in the plane of the atoms. Therefore, the number of micromirrors within

this region which are ‘on’ can be varied, giving a grey-scale with resolution n2 . When the value ofn is chosen optimally, the edges can still be sufficiently flat for most applications while the flat-top

282

Figure 10.7: Patterns on the DMD imaged onto a beam profile CCD. (a)-(d) Patterns uploaded to
the DMD. The arrays in these images are 1920 × 1080 to match the size of the DMD array. (e)-(h)The DMD imaged onto a beam profile CCD using a magnification of 1/6.7. The homogeneity of
the light field is set by the size of the beam illuminating the DMD. Therefore, ideally the beam ismuch larger than the DMD array.

283
is much more uniform.
However, for certain applications the sharpness of the edge is the most important figure of
merit. Specifically, when using the light to impart an AC Stark shift for spatially-resolved hyperfine(rotational) transitions for atoms (molecules), the presence of light at that lattice site is binary.
The exact intensity is typically not crutial if the Stark shift is large enough to sufficiently spectrally
resolve ‘light’ and ‘dark’ sites. For such cases, the goal is rather to have a well defined edge between‘light’ and ‘dark’ on neighboring sites. Figures 10.7b, d, f, and h show such a sharp edge as a line
and a circle.

Chapter 11

OutlookWith all the tools in place in the new apparatus, and many of them demonstrated in theprevious chapter, I now turn to the future of this experiment and the next set of experiments that
we have in mind. Most of these ideas have been alluded to throughout this thesis, and they include
spin-orbital coupling with lattice-confined polar molecules, realizing the full spin-1/2 Hamiltonian,and demonstrating a spin-resolved quantum gas microscope for polar molecules. The future ofultracold polar molecules is very bright, and there are of course many proposals that I will not
mention. Finally, I will discuss ideas to go beyond ITO coatings in the future in search of a morerobust solution for transparent conductive coatings, and I will mention prospects for avoiding theoptical transfer step in the experimental procedure. Then I will conclude this thesis.

11.1

Spin-orbital coupling with lattice-confined polar molecules
Spin-orbital coupling has been realized in many alkali atomic systems [2, 160, 178], and typ-ically it is induced using Raman beams to couple momentum states with internal hyperfine states.
However, this generally results in large spontaneous emission and heating rates [41]. Noteable
exceptions include alkaline earth optical lattice clocks, where ultranarrow clock transitions can beused to eliminate heating and elucidate many-body effects [148]. Ultracold polar molecules canalso be used to realize spin-orbital coupling directly through the spin-exchange interaction, andtherefore without coupling to electronic excited states and thus no spontaneous emission. Furthermore, the presence of long-range dipolar interactions provide a rich, novel many-body nature to285
the spin-orbital coupling [250].
This novel spin-orbital coupling mechanism arises from additional terms in the dipolar spin1/2 Hamiltonian which was discussed in Chapters 5 and 10. The splitting between the |1, −1i, |1, 0i,
and |1, 1i states eliminates these terms, but the generalized dipolar interaction between molecules
in an optical lattice can be written as [181, 250]
√ q=2
6 X
2
Vdd = − 3(−1)q C−q(θ, φ)Tq2 (di , dj ),rij q=−2(11.1)

wherer
Cqk (θ, φ) =4π
Ykq (θ, φ)
2k + 1(11.2)

where Ypq is the spherical harmonic. The Tq is an irreducible tensor operator [181, 250]:
2
ˆ±
= dˆ±
T±2
i dj ,±±00dˆ dˆ + dˆ dˆ ,(11.4)


+ ˆ−
− ˆ+
0 ˆ0
ˆ
ˆ
ˆ
di dj + di dj + 2di dj ,(11.5)

1
2
T±1
=√

2

1
T02 = √

6

(11.3)i j

ijwhere dˆi is the spin-1/2 dipole operator on site i. The final term gives rise to the spin-1/2 Hamiltonian in Chapters 5 and 10.
To realize the spin-orbital interactions inherent in these terms, we must work in a regime
where the mN = ±1 projections of N = 1 are degenerate. This can be accomplished with AC or DC
stark shifts that move the three uniquely, but the most straightforward approach is to use magnetic
fields which Zeeman shift the nuclear degree of freedom. Accordingly, the states become resonant at
1260 G [250], and thus the q = ±2 term can be exploited. This term corresponds to a spin-exchange
process which does not conserve the angular momentum of the system: |1, −1i|0, 0i → |0, 0i|1, 1i,
as shown in Fig. 11.1a. The Hamiltonian that governs these exchange processes has the form
q=±2
Vdd
=



−3 ˆ+ ˆ+ −2iφij− ˆ− 2iφijˆ+ di dj esin2 θij .
3 di dj e
2rij(11.6)

286

Figure 11.1: Spin-Orbital Coupling with polar molecules in an optical lattice. (a) The scheme using
several states for generating spi-exchange processes that do not conserve angular momentum. (b)The desity profile of spins ranging between | ↓i = −1 and | ↑i = 1 for positions x and y in the 2D
plane in units of lattice sites. (c) The same plot as in (b) except with filling fraction of 10%, whichshows that the chiron behavoir is still apparant. Reproduced from Reference [250].

287
Since this process does not conserve the angular momentum of an individual pair of molecules,
a net circulation in the spin current develops. The e±2iφij factors give rise to a chiral excitation
with d -wave symmetry, called a “chiron” (see Fig. 11.1b&c), which has a nontrivial Berry phaseof 2π [250]. Because of the sin2 θij factor it is most straightforward to perform this experimentin a single plane perpendicular to the quantization axis such that θ = π/2. The initialization ofthis experiment is very similar to the spin-impurity dynamics experiments discussed in Chapter 10.
After an arbitrary spin impurity region has been created in the |1, −1i state, the magnetic field canquickly be ramped to 1260 G, which is within reach of the bias coils in the new apparatus. Then,the spin-exchange process can proceed by both |1, −1i|0, 0i → |0, 0i|1, −1i and |1, −1i|0, 0i →|0, 0i|1, 1i. The latter gives rise to the development of chirons.
The observation of such phenomena should be relatively straightforward since the chiron isa topological excitation and thus robust to system imperfections, and it is even expected to persist
to low filling fractions [250]. In order to prevent spin-exchange processes from occurring before
the excited spin projections are degenerate, we can leave the spin addressing beam on after the
localized spin excitation. The AC stark shift from the addressing beam is very large comparedto J⊥ , and thus any dynamics will be off-resonantly suppressed until the beam is turned off. We
anticipate that our tools of high resolution in situ detection and nearly-single-spin addressing in an
optical lattice combined with a bias field of 1260 G will allow us to observe spin-orbital coupling,which will be especially clear given our current filling fraction of 30% and our aniticapted filling
fraction of 50% after a few improvements.
These tools could be extended to the observation of emergent Weyl fermionic excitations [251].
Weyl fermions are massless chiral particles, such as the chirons that arise from the dipolar spinorbital couplings. To observe Weyl excitations we need a weakly broken time-reserveal symmetry,such as a large magnetic field (already present). This opens intriguing prospects of observing
chiral anomaly, non-local electrodynamics, and non-Anderson disorder-driven transitions [251]. The
Weyl character of the excitations could be probed with a Ramsey protocol, but requires pulses ofinterfering Raman beams [251]. Such a procedure should give rise to the observation of Weyl nodes

288
through in situ detection [251].

11.2

Quantum magnetism in the XXZ Hamiltonian
The ability to apply large, stable electric fields opens the door to possible experiments withboth the Ising term and the spin-exchange term i.e. the XXZ Hamiltonian, whose relative strengths
depend on the field as shown in Fig. 7.6. The ability to tune J⊥ and Jz allows us to access anenormous wealth of intriguing phenomena. For example, when J⊥ = Jz , dynamics are governedby the Heisenberg Hamiltonian, which can be used to study critical points and phase transitionsof magnetic systems. It is expected that there is a phase transition between an XY paramagnetic
spin fluid when J⊥ dominates (small electric field) and a Ne’el-ordered antiferromagnet when Jz
dominates (large electric field) [132]. Thus, at sufficiently large fields we may expect spin-exchange
oscillations to diminish and give way to the proliferation of spin correlations, which could be observed with high resolution in situ imaging. However, the phase diagram depends on dimensionality
and is in general poorly understood beyond 1D where the system is highly frustrated. Our ability
to tune the dimensionality of the system offers even more information, particularly since the dipolarinteractions are anisotropic.
Moreover, it is possible to study critical phenomena by quenching the system across thecontinuous quantum phase transition. However, this requires the system to be near the groundstate, and to have nearly 100% filling. By jumping the electric field from J⊥  Jz to Jz  J⊥at a rate large compared to J⊥ , Jz , we can access the Kibble-Zurek mechansim which arises fromspontaneously breaking a global symmetry [144, 287]. Here we would expect to see the formationof topological defects such as domain structures, whose size scales with the ramp rate across thetransition in accordance with the Kibble-Zurek scaling [133]. The density scale (inverse length scale)of such domains obeys a power law in the quench rate of the electric field across the transition, andthe power exponent is given in terms of the critical exponents of the transition.

289

11.3

An approach for spin-selective quantum gas microscopy of polar molecules
in an optical lattice
In situ single-molecule detection is quickly becoming a reality both in optical lattices [50, 183]and optical tweezers [126, 161]. Therefore, it is now especially timely to consider exactly how
quantum gas microscopy of polar molecules might work, and how two spin states can be uniquely
detected. The latter question is of particular importance for molecules due to the limited fidelity ofthe molecule creation process, which results in the occupation of a given site being a priori unknown.
Hence, the ability to detect the presence or absence of a molecule on a given site as well as its spinwould be incredibly powerful. A few groups have implemented advanced techniques to separate
atomic spins using magnetic field gradients such that their position describes their spin [24, 213],
but these approaches are not readily compatable with ultracold molecule experiments.
Figure 11.2 illustrates schematically how a quantum gas microscope for polar molecules might
operate. The spin degree of freedom in ultracold molecules is encoded in the molecular rotational
state, and Fig. 11.2a shows the first few rotational states of KRb. The states are denoted by|N, mN i, where N is the rotational quantum number, and mN is it’s projection onto the quantization axis (e.g. electric and magnetic fields). Any two states may be chosen to represent a spin–1/2
system, or qubit, and are thus represented by | ↓i and | ↑i. The particular choice of rotational statesoffers a means of controlling interactions. For example, in KRb, the interaction energy between
the two spin states varies by a factor of two whether | ↑i is chosen to be |1, −1i or |1, 0i [110]. The
excited rotational state projection can be chosen using their energy splitting, or potentially withpolarization control [50].
While the motives for studying ground-state polar molecules in optical lattices are numerous,
we focus on the enormous number of unanswered questions regarding energy transfer and nonequilibrium dynamics of excitations. For example, relatively little is known about many-body
localization [125], and exactly when excitations fail to propagate through a system. This is of course
not due to a lack of effort; there has been an immense amount of activity in this field, including

290

Figure 11.2: A quantum gas microscope for polar molecules. (a) The rotational energy levels of
KRb. (b) A single plane of an optical lattice sitting between in-vacuum electrodes. Site addressing
is realized by focusing a laser using the same objective used for imaging. (c) Illustration of spin
propagation following a local excitation. Reproduced from Reference [51].

291
neutral atoms in optical lattices [44, 149, 233], and trapped ions in a spin chain [140, 223], evenleading to the recent observation of time cyrstals [45, 275, 280]. However, an important parameterfor understanding such phenomena is the ratio of the scale of the power law interaction α tothe dimensionality of the system β [274], and since neutral atoms have only short-range contact
interactions and spin-chains of trapped ions have only been studied in one dimension (1D), a large
range of values for α/β remains unexplored.
Figure 11.2b shows a cartoon of how studies of excitation propagation can be carried out
with polar molecules in an optical lattice. A two dimensional (2D) plane of molecules rests between
in-vacuum electrodes [50] which are used to tune intermolecular dipole-dipole interactions. These
interactions are characterized by an energy scale J⊥ , and fall off with distance as r−3 . The green
laser beam, incident from the objective of the quantum gas microscope, indicates how a beam canbe tightly-focused onto nearly a single site in the optical lattice, so that the resulting AC Stark
shift [188] from this beam can be used to control the molecules’ rotational state independently ofthe other molecules. Figure 11.2c shows a zoom-in of the plane of molecules, illustrating how alocalized excitation region can diffuse throughout the array.
We also note that optical tweezers, which have the advantage of being loaded directly froma magneto-optical trap, provide a new avenue for single-molecule control and addressability. In
particular, the ability to non-destructively scatter many photons from atoms in tweezers allows
them to be imaged, and dynamically rearranged in order to make zero-entropy arrays [16, 77].
This technique is quickly being expanded to molecules, where tweezer arrays of two types of atoms
can be dynamically rearranged and merged to make an array of molecules in tweezers [126, 161].
Therefore, spin-resolved microscopy is perhaps even more important for tweezers, where singleparticle imaging and control is often accompanied by high fidelity read-out and even quantum gate
operations [130].
Here we begin with a look at how molecules are detected, and how molecular dissociation
to free atoms via a Feshbach resonance has become an essential tool. We then go on to discuss
how these detection techniques can be extended to fluorescence detection as is done in quantum

292
gas microscopes. Further, we present a general technique by which two molecular rotational “spin”
states can be uniquely detected, giving simultaneously site-resolved and spin-resolved detection
with high fidelity. We then present simulations of mobile spin impurities, and describe how theinvestigation of such dynamics would look in the new KRb apparatus [50]. Furthermore, while weuse KRb as an example throughout, we emphasize that these ideas and methods are rather generaland can be applied directly to many other molecular species.

11.3.1A quantum gas microscope for polar molecules in an optical lattice

Using the ability to dissociate molecules into free atoms, we can now consider the operationof a quantum gas microscope for molecules based on this principle. Formation and dissociation ofground-state molecules in an optical lattice is well understood [47, 52, 183], though high resolution
detection has yet to be demonstrated. Nevertheless, it has recently been demonstrated that the
K and Rb atoms from dissociated KRb molecules can both be imaged in situ, yielding the same
number of atoms [52, 183]. This is an important step towards building a quantum gas microscopeof polar molecules since it confirms that either species can be used as a proxy for molecules, andperhaps more importantly, that one species can be blasted with resonant light for removal without
deleterious effects on the other species.
It is well known that light-assisted collisions during fluorescence imaging induce pairwise
losses on sites of an optical lattice [11, 235]. Therefore, it is important to remove one of the two
atoms from the molecule prior to fluourescence imaging. One atomic species can be removed witha fast blast pulse of a single beam, which is also how many quantum gas microscope experiments
remove one atomic spin state [40, 97, 267]. In fact, removing a unique atomic species is much easierthan removing a unique atomic spin since the frequency difference will be on the order of several
THz instead of MHz or GHz. Indeed, K and Rb have the most similar atomic transitions to their
excited P3/2 (D2) states of all the alkalis, at λ = 767 nm and 780 nm, respectively. Li, Na, and Cs,
on the other hand, have D2 transitions at λ = 671 nm, 589 nm, and 852 nm, respectively.
With the ability to remove one atomic species at will, molecules in the lattice can be mapped

293
onto either atomic species, making a quantum gas microscope of KRb molecules operationallyidentical to a quantum gas microscope of either K or Rb, both of which have been demonstrated [11,38, 74, 108, 235]. Moreover, recent demonstrations of Raman sideband cooling (RSC) of bothRb [208] and K [38] have used lattice depths that are prohibitively shallow for the optical molasses
techniques employed in the intial observations with Rb [11, 235]. All alkali (and many other) species
are compatable with these techniques, and thus quantum gas microscopy can be readily extended
to virtually any heteronuclear molecule whose constituent atoms can be laser cooled.

11.3.2Spin-resolved molecular microscopy

The proposed ability to detect molecules with single-site and spin resolution is based on a
mapping protocol during the return STIRAP sequence where the two rotational states are uniquely
mapped onto the two atomic species. For example, in a KRb experiment, all of the | ↓i molecules
can be mapped onto a Rb atom while all of the | ↑i molecules can be mapped onto a K atom. Then,a quantum gas microscope can be operated on both species, and a dichroic filter can be used toseparate the two wavelengths, providing an image of just Rb and just K that are in correspondancewith images of | ↓i molecules and | ↑i molecules, respectively.
If the two wavelengths are significantly different, as in e.g. NaCs, the two microscopes should
be completely decoupled. However, in case like KRb where the two wavelengths are seperated
by only 13 nm, greater care must be taken to ensure the fidelity of the mapping procedure. For
example, Raman sideband cooling (RSC) of one species may affect the other. Nevertheless, withadequate attention to detail, both species can be imaged simultaneously.
Before describing the mapping protocol in detail, we note that there is some ambiguity inwhich molecular state is mapped onto which atomic species. Either option is possible in general,
but for a given molecular species one choice may prove to be more advantageous than the other.
In what follows, we focus on the case in which | ↓i is mapped onto Rb, and discuss the alternative
mapping procedure afterwards.
Figures 11.3 shows a particular mapping procedure for realizing spin-resolved microscopy.

294
Each step is discussed in detail in the next several subsections.

11.3.2.1

Step I: Dissociating | ↓i molecules

The first step in the mapping protocol is to convert | ↓i molecules to a pair of K and Rb
on the same site, i.e. a doublon [52]. This is done by reversing the STIRAP sequence, which
uniquely couples the | ↓i state to Feshbach molecules, and then sweeping the magnetic field acrossthe Feshbach resonance in order to dissociate the Feshbach molecules. The fidelity of this process
is ∼ 90 − 95% [52, 190], which is limited by the STIRAP efficiency.
This first step assumes that the | ↑i molecules are completely unaffected by the STIRAP
pulse. While it is true that this pulse will not couple | ↑i molecules to the Feshbach state, therequirement that it will not couple them to any other state, which would remain dark, is more
stringent. To understand whether this happens, we must look carefully at the full quantum state of
the molecules, including the nuclear hyperfine degrees of freedom. We work in an uncoupled basis
Rb
K
Rb|N, mN , mK
I , mI i, where mI and mI are the nuclear spin projections of K and Rb, respectively.

This basis is appropriate for molecule experiments where the Feshbach resonance is at a sufficiently
large magnetic field (such as 546.7 G for KRb), where the molecular hyperfine states have sufficiently
large splitting (i.e. cI Rb ·I K  gIX µB I X ·B, where c is the scalar nuclear spin-spin coupling coupling
constant and X is either K or Rb), and are therefore not significantly mixed [197].
The nuclear spins of 40 K and 87 Rb are I K = 4 and I Rb = 3/2, respectively, which gives a
total of 36 hyperfine states for each molecular rotational state. The ground state that is populatedwith STIRAP is |0, 0, −4, 1/2i, which is not the lowest energy hyperfine state [197], and has a total
Rb = −7/2. Note that the total electronic angularnuclear angular momentum of mI = mK
I + mI

momentum J = 0 in the X 1 Σ+ electronic ground potential, so mI = mF , where F refers to the total
angular momentum of the molecule. The hyperfine states of K and Rb that are Feshbach resonant
are |F, mF i = |9/2, −9/2i and |1, 1i, respectively, such that the total angular momentum of thesestates is thus also mtot = −7/2. Therefore, the net angular momentum change in the STIRAP
pulse sequence is zero, and this is accomplished using pulses with the same circular polarization.

295

i
ii
iii
iv
v

1,1
9/2,-9/2

2,2
2,2
1,1

1,1
9/2,-9/2
9/2,-9/2

Figure 11.3: Various steps required to realize a spin-resolved microscope, mapping | ↓i onto Rb
and | ↑i onto K. Reproduced from Reference [51].

296
The hyperfine states can be exploited to protect the | ↑i molecules during the STIRAP
process. Hyperfine and rotational degrees of freedom are coupled by interactions between electricquadrupole moments of the atomic nuclei and the electric-field gradient caused by the electrons
near the nuclei [197]. Therefore, the electric dipole transitions between rotational states can also
drive hyperfine-state changing transitions. Moreover, this coupling lifts the degeneracy between
the three mF projections of the excited rotational state even at zero magnetic field, as shown in
Fig. 11.2a and discussed in previous chapters.
Additionally, an electric field can be used to provide even greater selectivity. If STIRAP isdone in a DC electric field of a few kV/cm, the DC Stark shift of the | ↓i and | ↑i molecules will bedifferent, providing an even larger separation between the | ↑i state and the energy of the photonthat couples | ↓i to the electronically excited intermediate state. Moreover, the Stark structure in
the electronic excited states differs from the ground state, which further reduces the coupling. A
combination of these two selection rule tricks are likely to preserve the | ↑i molecules with highfidelity during the STIRAP sequence, though the specific requirements may vary between molecularspecies.

11.3.2.2

Step II: Removing an atomic species

With an understanding of how | ↑i molecules can be preserved with high fidelity during thefirst step, we now describe the rest of the mapping protocol. Following dissociation, one of thespecies must be removed such that each doublon becomes a site with a single atom. This can bedone with a single blast pulse of resonant light, preferentially normal to the plane of atoms. Indeed,such pulses are routinely used in quantum gas microscope experiments, with typical blast times
between 10 µs and 1 ms depending on the species and lattice depth, and such pulses have been
demonstrated to preserve the other atomic spin with very high fidelity [31, 39, 206]. As statedearlier, removal is even simpler with two atomic species rather than two atomic spins, since thewavelengths of their atomic transitions are far more separated. Moreover, such pulses can have anegligible effect on molecules [52, 190], and thus the | ↑i molecules remain unaffected.

297
At this point, a choice must be made and either K or Rb can be removed with a resonant
blast pulse, and whichever remains will be subsequently mapped to | ↑i molecules. We consider thecase in which K is removed first (see Fig. 11.3), and we discuss the alternate situation below (seeFig. 11.4). The removal of K can easily be accomplished by driving the |9/2, −9/2i → |11/2, −11/2i0
cycling transition which leaves behind Rb atoms in the |1, 1i state.
Once the first atomic species is removed, we need to flip the unpaired atoms to a unique spinstate such that they can be differentiated from the atoms that appear in the subsequent steps. For
Rb, this transition is from |1, 1i to |2, 2i at 8.05 GHz. The removal and spin changing is shown in
Fig. 11.3 in going from ii to iii, where the free atoms are shown as going to a lighter color after
transfer to a different spin state.
To quantify the fidelity of atom removal while preserving the other atomic species or atom
spin, we calculate the heating rates of both species due to the removal pulses. The heating rates
are calculated using the photon scattering rates given by:
Γsc =

Γ

(I/Isat ),2 1 + 4(∆/Γ)2 + (I/Isat )(11.7)

where Γ is the linewidth of the electronic excited state, I is the intensity of the removal beam, Isat is
the saturation intensity of the atomic transition, and ∆ is the detuning from resonance of the excitedstate. The heating rate can be calculated from this scattering rate via Ṫ = 2/3×(2×Erec )/kB ×Γsc ,where Erec is the recoil energy.
For a removal beam of intensity I = Isat and detuning ∆ = 0, the scattering rate is Γsc ∼107 s−1 , which corresponds to a heating rate of ∼2 µK/µs for Rb in the |2, 2i → |3, 3i0 states(∼4 µK/µs for K in |9/2, −9/2i → |11/2, −11/2i0 ). For the Rb |1, 1i → |2, 2i0 transition the
detuning is about 8 GHz at 546 G, and the scattering rate is therefore only Γ|1,1i ∼ 3 s−1 . However,after scattering one photon, the atom can fall from |2, 2i0 to |2, 2i, and thus we require the removal
pulse to be short compared to 1/Γ|1,1i→|2,2i0 ∼ 100 ms, which is trivial. The situation is similar for
K|9/2, −7/2i − |11/2, −9/2i0 , but the detuning is only 70 MHz, for which Γ|9/2,−7/2i→|11/2,9/2i0 =3.6×104 s−1 . While K removal in 1/Γ|9/2,−7/2i→|11/2,9/2i0 ∼ 30 µs may be challenging, an alternative

298
for K is to use instead |7/2, −7/2i to protect the atom, which is separated by ∼ 2.75 GHz at 550
G in the ground state rather than 80 MHz [39].

11.3.2.3

Step III: Dissociating | ↑i molecules

Next we convert the | ↑i molecules back to a doublon, first by applying a rotational-statechanging microwave pulse to transfer | ↑i → | ↓i. Then the same reverse-STIRAP and magnetodissociation sequence converts the | ↑i molecules to doublons. This is shown in steps iii to iv ofFig. 11.3.
The next step is to remove the other atomic species, which makes clear why the atomic statetransfer was necessary in the previous step. Since we are considering the case where Rb remains
from the previous step, the Rb atoms in the doublons must be removed. The Rb that remain fromthe previous step are in the |2, 2i state, and a global microwave pulse is needed to transfer themto |1, 1i and the Rb in the doublons to |2, 2i so that they may be removed on the |2, 2i → |3, 3i0transition. However, this spin combination of Rb in |2, 2i and K in |9/2, −9/2i is unstable withrespect to hyperfine-state-changing collisions and so these doublons are unstable. Typical inelastic
loss times in a lattice are ∼ms [52], and so the Rb removal time should be short compared withthis lifetime in order to not severely limit the fidelity of the state mapping process of the remaining
K. Since Rb is relatively heavy this timescale may be challenging, and previous experiments haveused a 5 ms removal pulse [267]. Furthermore, a repump beam cannot be used instead because
we need to preserve the state of the other Rb. This step demonstrates why one mapping may bebetter than the other.

11.3.2.4

Step IV: Imaging

After the three steps described above, the sites that were | ↓i molecules are Rb in |1, 1i andthe sites that were | ↑i molecules are K in |9/2, −9/2i. While K is in its stretched cycling state,
Rb is not in the state with its closed cycling transition. However, the subsequent optical molasses
or RSC is operated at zero magnetic field where the spin states of K are mixed, and the Rb atoms

299
will be optically pumped to their stretched states with the addition of a repump beam.

11.3.2.5

Alternate mapping scheme and discussion

Above we considered the case where | ↓i is mapped onto Rb and | ↑i is mapped onto K. We
could have, however, chosen to blast away Rb atoms in step ii as opposed to K atoms, as shownin Fig. 11.4. To do so, one could change Rb atoms from |1, 1i to |2, 2i in order to remove Rb on
the |2, 2i → |3, 3i0 cycling transition. In this process, we will have created an unstable doublon as
described above, requiring a fast blast pulse. In this scheme, however, Rb can be removed with arepump beam in addition to a beam on the cycling transition, since there are no other Rb atoms topreserve. This would limit the time spent in the |2, 2i state, resulting in higher fidelity preservationof K.
The K atoms would then be transferred to the |9/2, −7/2i state or the |7/2, −7/2i state
to differentiate them from the K in later steps. This approach results in a mapping of | ↓i to K
atoms. The remaining | ↑i molecules are then converted to | ↓i and dissociated. Now K is in the|9/2, −9/2i which can immediately be blasted away on a cycling transition to leave behind Rb in
the |1, 1i state. Imaging then proceeds as described above.
Given that this mapping protocol does not necessarily result in the production of unstabledoublons, mapping | ↓i onto K and | ↑i onto Rb results in a higher overall fidelity.
One factor that may play a role is photo-association during the removal pulses. Quantumgas microscopes operate with parity detection, because every pair of atoms is associated into theelectronically excited molecular potential during the roughly second-long integration time. However, the probability of this happening during the ∼ms removal pulse is low for the following threereasons. 1) The Franck-Condon Factors that describe the overlap of the vibrational wavefunctions
which couple a doublon on a lattice site to an electronically-excited molecular state are small (asdiscussed in Chapter 6), such that Rabi frequencies of these transitions are on the order of Hz.
2) The molasses cooling or RSC during imaging in quantum gas microscopes is red-detuned fromthe atomic resonances, which can couple to molecular states of finite binding energy. Resonant

300

i
ii
iii
iv
v

1,1
9/2,-9/2

9/2,-7/2

9/2,-7/2

9/2,-7/2

1,1
9/2,-9/2

1,1

Figure 11.4: Various steps required to realize a spin-resolved microscope, mapping | ↓i onto K and| ↑i onto Rb. Reproduced from Reference [51].

301
pulses for removal only couple to the continuum of the excited molecular potential, thereby further
reducing the matrix elements. 3) Photo-association loss of two identical atoms into a homonuclear
molecule is different than the case of two separate atoms associating into a heteronuclear molecule.
Crucially, the molecular potentials associated with an excited electron are non-degenerate in theheteronuclear case, thereby reducing the probability of excitation for the herteronuclear case compared to homonuclear.

11.4

Spin impurity dynamics: spin-transport simulations
We now present some simulations of the dynamics of a spin impurity through the array. The

goal here is to motivate the need for microscopic detection with spin resolution. The Hamiltonianthat describes this exchange process is again (see Chapter 5) [95, 263, 273]:
Ĥ = J

N X
X

||rl − rk ||−3 σ̂lx σ̂kx + σ̂ly σ̂ky ,(11.8)

k=1 l<kwhere σ̂ are Pauli matrices, and the positions r of the molecules in the lattice are randomly selectedin accordance with the molecular filling fraction. Disorder in such a model manifests itself in two
ways: either a disordered potential landscape can be added to the lattice using the DMD discussed
in the previous chapter; or since the lattice filling fraction is less than unity, there is a natural
disorder in the arrangement of the molecules and hence in the ||rl − rk ||−3 geometrical prefactors
coupling the molecules. Concomitantly, the lattice can be thought of as random in such a model.
The initial state will be prepared by driving all molecules to the −z (or −x) state and thenselecting an excitation region of the lattice in which the molecules are prepared in +z (or +x). The
initial ordering will eventually disappear with dynamics, and the main objective is to see how thedisorder induced by the random filling affects this memory loss. The main observable we can use
for quantifying the initial order memory is imbalance I defined as [51]Iˆα (t) = N −1

NX
hσ̂kα (0)iσ̂kα (t),(11.9)

k=1

where α can either be along z or x. This quantity will be equal to one initially and will eventually

302
decay through the dynamics. Moreover, we can also consider the scaled total spin [51]Ŝα = N

−1N
X

σ̂kα (t).

(11.10)k=1

Since the exact simulation of spins in a 2D lattice becomes intractable as the number of molecules
is increased, the simulations presented here are based on an approximation method known as the
Discrete Truncated Wigner Approximation (DTWA). Finally, while dynamics in the x-basis occur
within times of a few J and memory loss is quite pronounced, such a system is highly out-ofequilibrium since ≈ N/2 excitations in the z-eigenbasis are present. Therefore, for the following
analysis we limit our simulations to the z-basis where the excitation number is conserved and thesystem is much closer to the global ground state.
Our simulations suggest that the perimeter of the interface between | ↓i and | ↑i in the zeigenbasis must be quite long in order to see significant decay of Iˆ within experimentally relevent
timescales of several J (see Fig. 11.5). Therefore, a single circular impurity region of 10’s of sites
has a very small boundary perimeter and hence spin diffusion is quite slow and limited even forunity filling. We consider the case of a checkerboard pattern (which can be generated with a DMD),
where the size can be as small as 2 × 2 sites. In this case we observe significant decay of Iˆ withina few J even with filling fractions of ∼ 10%. The cross-over between localization and diffusion
appears to occur with checkerboard sizes of roughly 8 × 8, but this depends on the filling fraction.
Figure 11.5 shows 12.5%, 25%, and 50% filling for several checker sizes. These numerical predictions
provide strong motivation for the experimental capability of microscopic, spin-resolved detection
since substantial diffusion occurs only for microscopic areas.
The investigation of excitations on the x-plane is related to the notion of a so-called BEC
phase of rotational excitations that is predicted to appear in 2D above a critical filling fraction [152].
This BEC manifests as a divergent T2 coherence time of the rotational transition even in the presence of inhomogeneous broadening. The dynamical evolution of a dense gas of rotational excitations
shows regimes of nonergodicity, characterisitic of many-body localization and localization-protected
quantum order [152]. The spins on a lattice can be mapped to hard-core bosons, with | ↓i rep-

303

Figure 11.5: TWA simulations of the imbalance of a 36 × 36 lattice system with z-excitations
in a checkerboard pattern for three filling fractions and several checker sizes. The legend of thecheckerboard size n × n is shown on the right, and the filling fraction (randomly filled) is shown in
the plots. They are: (a) 12.5%, (b) 25%, and (c) 50% filling. Reproduced from Reference [51].

resenting an empty site and | ↑i representing an occupied site. This gives rise to the hard-core
Bose-Hubbard model, with the interesting feature of intersite hopping that scales with 1/r3 . ThisP ik·R
leads to the dispersion relation for the bosons k = Ja3 i e|R3 |i for which δk ≈ |k| at small waveivectors [152]. This relativistic dispersion has striking consequences for the thermodynamic phase
diagram of the model [152].

11.5

Beyond ITO
In the last section of this thesis I will discuss possible alternatives to ITO coatings, whichwere shown in Chapter 8 to have a number of limitations for high voltage applications as well as
alkali adsorption. While the longevity of this coating is not known at the time of this writing,
I would except that it will cause most of the problems experienced in the next several years.
Therefore it is worth pursuing other options, although the problems with large electric fields andthe destructive nature of high voltage conditioning will likely be general problems that plague any
thin film technology.
As I mentioned, other options include metal coatings such as nickel or copper, but thesehave far too much absorption and reflection. Another option could be thin films of carbon nanotubes [201], which are being explored in Stuttgart for thermal vapor cells of Rydberg atoms [211].
Similarly, graphene thin films are an intriguing option. It is possible that such coats will have lower
absorption and reflection. Further it is possible, even likely, that they will be more stable in the304
presence of alkali vapor. However, the surface figure may be worse in carbon nanotube thin film
coatings since they are essentially a spaghetti of carbon nanotubes, and so surface features may
exceed λ/10 ≈ 80 nm at which point they may affect very high resolution imaging systems.
Moreover, it is unclear whether these coatings will have better performance in high voltage
applications than ITO. The coating may be more chemically robust, but if it is less smooth it will
be more susceptible to electron emission. Further, the work function for carbon nanotube thinfilms is slightly lower than for ITO [162]. In any case, many tests would need to be performed in asimilar fashion to the tests we did with ITO.
Other intriguing options include a much thicker ITO coating than we are using, such as 100
nm, and then small regions of thickness ≈ 10 nm where the beams pass through the coating. A
nickel coating could also be used in such a way, or graphene thin films. As was discussed in Chapter
10, the lattice beam is at a ≈ 10 degree angle, and so the regions of thinner coating would be offset
from the cloud by a few mm. Higher absorption and reflection will not severely affect the highresolution imaging system, and it will simply mean that the integration time must be a few percent
longer to achieve the same number of photons collected on the camera.

11.6

Eliminating the optical transfer step
Chapter 9 illustrated the enormous number of complications we encountered using an optical

dipole trap with translating focus to carry the atoms through the vacuum chamber from the evaporation region to the science cell. The primary reason we use this transfer is because we have thehigh-NA microscope objective below the science cell which obstructs the access of the moving coils
carrying the atoms from the MOT chamber to the evaporation chamber. If this microscope objective were absent, we could simply use the moving coils to carry the atoms from the MOT chamber tothe science chamber directly without the need for an intermediate chamber for evaporative cooling.
It turns out that if the mount for the objective and the dichroic filter above it (see Chapter
9) are modified, we could potentially use the cart coils to go all the way to the science chamber. We
would just need to change the separation of the moving coils by ≈ 2 cm, which is a change of only

305
< 10%. With the transfer beam eliminated, we can load directly from the magnetic quadrupole
trap after evaporation to the crossed optical dipole trap. Thus, we never need to go from a round(ish) trap to a very elongated trap, and then back to a round trap. For now we are pushing forwardwith the transfer beam approach described in Chapter 9 (now using an air-bearing stage), but themodifications necessary to remove the optical transfer stage are being pursued in parallel, and may
be implemented at a later date.

11.7

Conclusions
Before 2008 the future of ultracold polar molecules was shrouded by their complexity andslow progress. The molecular gases were too hot, the motion in the trap was not entirely controlled,
their interaction with the trapping light was poorly understood, and the molecules suffered fast loss
from chemical reactions. Further, large electric fields only enhanced the chemical loss in 3D gases.
This was only beginning to be controlled using 2D systems [63]. Moreover, there was a quickly
growing number of theoretical proposals for ultracold polar molecule experiments, but many ofthem required the tools which were developed during the time of my thesis. My PhD work has
contributed to bringing us from thermal (T > TF ) samples of ultracold polar KRb molecules in 2D
or 3D gases all the way to a spin-exchange-interacting degenerate gas of molecules in an opticallattice where advanced AC and DC electric field tools and high resolution imaging and addressing
tools are beginning to be used to create a quantum gas microscope of polar molecules, in which
every lattice site can be resolved and addressed.
More generally, the field of ultracold polar molecules has changed enormously in the past six
years. Since 2014, there are now many ultracold molecule experiments in the world. These otherexperiments are moving very quickly, but they all have more to learn about their molecular species.
We felt that it was time to implement a new generation of our apparatus which was sufficiently
complex to go beyond the capabilities of any other experiment. Our design was intended to include
every tool we wanted, and in the end we were able to successfully combine all of these tools together
into the same apparatus. We hope that our new apparatus will serve as a standard for what is306
possible with ultracold polar molecules as we continue to pursue novel directions in the coming
years.

Bibliography[1] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle. Observation of vortex lattices
in bose-einstein condensates. Science, 292(5516):476–479, 2001.
[2] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch. Realizationof the hofstadter hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett.,
111:185301, 2013.
[3] K. Aikawa, D. Akamatsu, M. Hayashi, K. Oasa, J. Kobayashi, P. Naidon, T. Kishimoto,
M. Ueda, and S. Inouye. Coherent transfer of photoassociated molecules into the rovibrational
ground state. Phys. Rev. Lett., 105:203001, Nov 2010.
[4] K Aikawa, D Akamatsu, J Kobayashi, M Ueda, T Kishimoto, and S Inouye. Toward the
production of quantum degenerate bosonic polar molecules, 41 k 87 rb. New Journal of
Physics, 11(5):055035, 2009.
[5] K. Aikawa, A. Frisch, M. Mark, S. Baier, R. Grimm, and F. Ferlaino. Reaching Fermi
Degeneracy via Universal Dipolar Scattering. Phys. Rev. Lett., 112:010404, 2014.
[6] J. Aldegunde, Hong Ran, and Jeremy M. Hutson. Manipulating ultracold polar moleculeswith microwave radiation: The influence of hyperfine structure. Phys. Rev. A, 80:043410,
2009.
[7] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A.
Cornell. Watching dark solitons decay into vortex rings in a bose-einstein condensate. Phys.
Rev. Lett., 86:2926–2929, 2001.
[8] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor. Science, 269(5221):198–201,
1995.
[9] N. W. Ashcroft and N. D. Mermin. Solid state physics. Saunders College Publishing, Fort
Worth, TX.
[10] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller,
and F. Ferlaino. Extended bose-hubbard models with ultracold magnetic atoms. Science,
352(6282):201–205, 2016.
[11] W. S. Bakr, J. I. Gillen, A. Peng, S. Folling, and M. Greiner. A quantum gas microscope fordetecting single atoms in a Hubbard-regime optical lattice. Nature, 462:74–77, 2009.

308[12] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and
M. Greiner. Probing the Superfluid–to–Mott Insulator Transition at the Single-Atom Level.
Science, 329(5991):547–550, 2010.
[13] M. A. Baranov, H. Fehrmann, and M. Lewenstein. Wigner crystallization in rapidly rotating
2d dipolar fermi gases. Phys. Rev. Lett., 100:200402, 2008.
[14] R. Barnett, D. Petrov, M. Lukin, and E. Demler. Quantum Magnetism with Multicomponent
Dipolar Molecules in an Optical Lattice. Phys. Rev. Lett., 96:190401, 2006.
[15] J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle, G. Gabrielse, Y. V. Gurevich, P. W.
Hess, N. R. Hutzler, E. Kirilov, I. Kozyryev, B. R. O’Leary, C. D. Panda, M. F. Parsons,
E. S. Petrik, B. Spaun, A. C. Vutha, and A. D. West. Order of Magnitude Smaller Limit on
the Electric Dipole Moment of the Electron. Science, 343(6168):269–272, 2014.
[16] Daniel Barredo, Sylvain de Léséleuc, Vincent Lienhard, Thierry Lahaye, and Antoine
Browaeys. An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays. Science, 354(6315):1021–1023, 2016.
[17] J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. Demille. Magnetooptical trapping of a diatomic molecule. Nature, 512:286–289, 2014.
[18] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. Hecker Denschlag, and
R. Grimm. Crossover from a Molecular Bose-Einstein Condensate to a Degenerate Fermi
Gas. Phys. Rev. Lett., 92:120401, 2004.
[19] Th. Best, S. Will, U. Schneider, L. Hackermüller, D. van Oosten, I. Bloch, and D.-S. Lühmann.
Role of interactions in 87 Rb−40 K bose-fermi mixtures in a 3d optical lattice. Phys. Rev. Lett.,
102:030408, 2009.
[20] Hendrick L. Bethlem, Giel Berden, and Gerard Meijer. Decelerating neutral dipolar molecules.
Phys. Rev. Lett., 83:1558–1561, Aug 1999.
[21] Gary C. Bjorklund. Frequency-modulation spectroscopy: a new method for measuring weakabsorptions and dispersions. Opt. Lett., 5(1):15–17, 1980.
[22] I. Bloch, J. Dalibard, and S. Nascimbene. Quantum simulation with ultracold quantum gases.
Nature Phys., 8:267–276, 2012.
[23] J. R. Bochinski, Eric R. Hudson, H. J. Lewandowski, Gerard Meijer, and Jun Ye. Phase
space manipulation of cold free radical oh molecules. Phys. Rev. Lett., 91:243001, Dec 2003.
[24] Martin Boll, Timon A. Hilker, Guillaume Salomon, Ahmed Omran, Jacopo Nespolo, Lode
Pollet, Immanuel Bloch, and Christian Gross. Spin- and density-resolved microscopy of
antiferromagnetic correlations in fermi-hubbard chains. Science, 353(6305):1257–1260, 2016.
[25] M. Born and E. Wolf. Principles of optics. Cambridge University Press, UK, 1999.
[26] C. C. Bradley, C. A. Sackett, and R. G. Hulet. Bose-Einstein Condensation of Lithium:
Observation of Limited Condensate Number. Phys. Rev. Lett., 78:985–989, 1997.

309[27] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein
Condensation in an Atomic Gas with Attractive Interactions. Phys. Rev. Lett., 75:1687–
1690, 1995.
[28] D. M. Brink and C. V. Sukumar. Majorana spin-flip transitions in a magnetic trap. Phys.
Rev. A, 74:035401, Sep 2006.
[29] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. Joseph Wang, J. K. Freericks, H. Uys, M. J.
Biercuk, and J. J. Bollinger. Engineered two-dimensional ising interactions in a trapped-ion
quantum simulator with hundreds of spins. Nature, 484:489–492, 2012.
[30] J. M. Brown and A. Carrington. Rotational spectroscopy of diatomic molecules. Cambridge
University, Cambridge, UK, 2003.
[31] P. T. Brown, D. Mitra, E. Guardado-Sanchez, P. Schauß, S. S. Kondov, E. Khatami, T. Paiva,
N. Trivedi, D. A. Huse, and W. Bakr. Observation of canted antiferromagnetism with ultracold fermions in an optical lattice. arXiv:1612.07746, 2016.
[32] H. P. Büchler, E. Demler, M. Lukin, A. Micheli, N. Prokof’ev, G. Pupillo, and P. Zoller.
Strongly Correlated 2D Quantum Phases with Cold Polar Molecules: Controlling the Shapeof the Interaction Potential. Phys. Rev. Lett., 98:060404, 2007.
[33] A. Burchianti, G. Valtolina, J. A. Seman, E. Pace, M. De Pas, M. Inguscio, M. Zaccanti, and
G. Roati. Efficient all-optical production of large 6 Li quantum gases using D1 gray-molasses
cooling. Phys. Rev. A, 90:043408, 2014.
[34] J. N. Byrd, J. A. Montgomery, and R. Côté. Long-range forces between polar alkali-metal
diatoms aligned by external electric fields. Phys. Rev. A, 86:032711, 2012.
[35] Gretchen K. Campbell, Jongchul Mun, Micah Boyd, Patrick Medley, Aaron E. Leanhardt,
Luis G. Marcassa, David E. Pritchard, and Wolfgang Ketterle. Imaging the mott insulator
shells by using atomic clock shifts. Science, 313(5787):649–652, 2006.
[36] L. D. Carr, D. DeMille, R. V. Krems, and J. Ye. Cold and ultracold molecules: science,technology and applications. New J. Phys., 11(5):055049, 2009.
[37] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauss, T. Fukuhara, C. Gross,
I. Bloch, C. Kollath, and S. Kuhr. Light-cone-like spreading of correlations in a quantum
many-body system. Nature, 481:484–487, 2012.
[38] L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V. V. Ramasesh, W. S. Bakr, T. Lompe,and M. W. Zwierlein. Quantum-Gas Microscope for Fermionic Atoms. Phys. Rev. Lett.,
114:193001, 2015.
[39] Lawrence W. Cheuk, Matthew A. Nichols, Katherine R. Lawrence, Melih Okan, Hao Zhang,
Ehsan Khatami, Nandini Trivedi, Thereza Paiva, Marcos Rigol, and Martin W. Zwierlein.
Observation of spatial charge and spin correlations in the 2d fermi-hubbard model. Science,
353(6305):1260–1264, 2016.
[40] Lawrence W. Cheuk, Matthew A. Nichols, Katherine R. Lawrence, Melih Okan, Hao Zhang,and Martin W. Zwierlein. Observation of 2d fermionic mott insulators of 40 K with single-site
resolution. Phys. Rev. Lett., 116:235301, Jun 2016.

310[41] Lawrence W. Cheuk, Ariel T. Sommer, Zoran Hadzibabic, Tarik Yefsah, Waseem S. Bakr,
and Martin W. Zwierlein. Spin-injection spectroscopy of a spin-orbit coupled fermi gas. Phys.
Rev. Lett., 109:095302, 2012.
[42] A. P. Chikkatur, Y. Shin, A. E. Leanhardt, D. Kielpinski, E. Tsikata, T. L. Gustavson, D. E.
Pritchard, and W. Ketterle. A continuous source of bose-einstein condensed atoms. Science,
2002.
[43] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga. Feshbach resonances in ultracold gases.
Rev. Mod. Phys., 82:1225–1286, 2010.
[44] Jae-yoon Choi, Sebastian Hild, Johannes Zeiher, Peter Schauß, Antonio Rubio-Abadal, Tarik
Yefsah, Vedika Khemani, David A. Huse, Immanuel Bloch, and Christian Gross. Exploring
the many-body localization transition in two dimensions. Science, 352(6293):1547–1552, 2016.
[45] S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya,
V. Khemani, C. von Keyserlingk, N. Y. Yao, E. Demler, and M. D. Lukin. Observation of
discrete time-crystalline order in a disordered dipolar many-body system. arXiv:1609.08684,
2016.
[46] L Chomaz, L Corman, T Yefsah, R Desbuquois, and J Dalibard. Absorption imaging
of a quasi-two-dimensional gas: a multiple scattering analysis. New Journal of Physics,
14(5):055001, 2012.
[47] A. Chotia, B. Neyenhuis, S. A. Moses, B. Yan, J. P. Covey, M. Foss-Feig, A. M. Rey, D. S. Jin,
and J. Ye. Long-Lived Dipolar Molecules and Feshbach Molecules in a 3D Optical Lattice.
Phys. Rev. Lett., 108:080405, 2012.
[48] E. U. Condon. Nuclear Motions Associated with Electron Transitions in Diatomic Molecules.
Phys. Rev., 32:858–872, 1928.
[49] A. Couvert, T. Kawalec, G. Reinaudi, and D. Gury-Odelin. Optimal transport of ultracold
atoms in the non-adiabatic regime. EPL (Europhysics Letters), 83(1):13001, 2008.
[50] J. P. Covey, L. De Marco, K. Matsuda, W. G. Tobias, S. A. Moses, M. T. Miecnikowski,
G. Valtolina, D. S. Jin, and J. Ye. A new apparatus for enhanced optical and electric controlof ultracold krb molecules. In preparation, 2017.
[51] J. P. Covey, L. De Marco, G. Valtolina, and J. Ye. An approach to spin-resolved quantumgas microscopy of polar molecules in an optical lattice. In preparation, 2017.
[52] J. P. Covey, S. A. Moses, M. Garttner, A. Safavi-Naini, M. T. Miecnkowski, Z. Fu,
J. Schachenmayer, P. S. Julienne, A. M. Rey, D. S. Jin, and J. Ye. Doublon dynamics
and polar molecule production in an optical lattice. Nat. Commun., 7:11279, 2016.
[53] J. P. Covey, S. A. Moses, J. Ye, and D. S. Jin. Controlling a quantum gas of polar molecules
in an optical lattice. Royal Society of Chemistry book chapter: “Low temperature and low
energy molecular scattering”, 2017.
[54] J. F. E. Croft and J. L. Bohn. Long-lived complexes and chaos in ultracold molecular collisions. Phys. Rev. A, 89:012714, 2014.

311[55] T. D. Cumby, R. A. Shewmon, M.-G. Hu, J. D. Perreault, and D. S. Jin. Feshbach-molecule
formation in a Bose-Fermi mixture. Phys. Rev. A, 87:012703, 2013.
[56] B. Damski, L. Santos, E. Tiemann, M. Lewenstein, S. Kotochigova, P. Julienne, and P. Zoller.
Creation of a Dipolar Superfluid in Optical Lattices. Phys. Rev. Lett., 90:110401, 2003.
[57] J. G. Danzl, E. Haller, M. Gustavsson, M. J. Mark, R. Hart, N. Bouloufa, O. Dulieu,
H. Ritsch, and H.-C. Nägerl. Quantum Gas of Deeply Bound Ground State Molecules. Science,
321(5892):1062–1066, 2008.
[58] J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, R. Hart, J. Aldegunde, J. M. Hutson,and H.-C. Nägerl. An ultracold high-density sample of rovibronic ground-state molecules inan optical lattice. Nat. Phys., 6, 2010.
[59] J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, R. Hart, A. Liem, Z. Holger, and H.-C.
Nägerl. Deeply bound ultracold molecules in an optical lattice. New J. Phys., 11(5):055036,
2009.
[60] R. Daschner, R. Ritter, H. Kübler, N. Frühauf, E. Kurz, R. Löw, and T. Pfau. Fabricationand characterization of an electrically contacted vapor cell. Opt. Lett., 37(12):2271–2273,
2012.
[61] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn,and W. Ketterle. Bose-Einstein Condensation in a Gas of Sodium Atoms. Phys. Rev. Lett.,
75:3969–3973, 1995.
[62] M. H. G. de Miranda. Control of dipolar collisions in the quantum regime. PhD thesis,
University of Colorado, Boulder, 2010.
[63] M. H. G. de Miranda, A. Chotia, B. Neyenhuis, D. Wang, G. Quéméner, S. Ospelkaus, J. L.
Bohn, J. Ye, and D. S. Jin. Controlling the quantum stereodynamics of ultracold bimolecular
reactions. Nat. Phys., 7(6):502–507, 2011.
[64] B. DeMarco and D. S. Jin. Onset of Fermi Degeneracy in a Trapped Atomic Gas. Science,
285(5434):1703–1706, 1999.
[65] B. DeMarco, C. Lannert, S. Vishveshwara, and T.-C. Wei. Structure and stability of mottinsulator shells of bosons trapped in an optical lattice. Phys. Rev. A, 71:063601, 2005.
[66] D. DeMille. Quantum computation with trapped polar molecules. Phys. Rev. Lett., 88:067901,
Jan 2002.
[67] R. Desbuquois, L. Chomaz, T. Yefsah, J. Leonard, J. Beugnon, C. Weitenberg, and J. Dalibard. Superfluid behaviour of a two-dimensional Bose gas. Nature Physics, 8:645–648, 2012.
[68] K. Dieckmann, C. A. Stan, S. Gupta, Z. Hadzibabic, C. H. Schunck, and W. Ketterle. Decay ofan Ultracold Fermionic Lithium Gas near a Feshbach Resonance. Phys. Rev. Lett., 89:203201,
2002.
[69] E. A. Donley, N. R. Claussen, S. T. Thompson, and Wieman C. E. Atom-Molecule Coherence
in a Bose-Einstein Condensate. Nature, 417:529, 2002.

312[70] W. Dowd, R. J. Roy, R. K. Shrestha, A. Petrov, C. Makrides, S. Kotochigova, and S. Gupta.
Magnetic field dependent interactions in an ultracold Li-Yb(3 P2 ) mixture. New J. Phys.,
17(5):055007, 2015.
[71] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, andH. Ward. Laser phase and frequency stabilization using an optical resonator. Applied Physics
B, 31(2):97–105, 1983.
[72] R. Dubessy, K. Merloti, L. Longchambon, P.-E. Pottie, T. Liennard, A. Perrin, V. Lorent,and H. Perrin. Rubidium-87 bose-einstein condensate in an optically plugged quadrupole
trap. Phys. Rev. A, 85:013643, 2012.
[73] P. Dyke, K. Fenech, T. Peppler, M. G. Lingham, S. Hoinka, W. Zhang, S.-G. Peng, B. Mulkerin, H. Hu, X.-J. Liu, and C. J. Vale. Criteria for two-dimensional kinematics in an interacting
fermi gas. Phys. Rev. A, 93:011603, Jan 2016.
[74] G. J. A. Edge, R. Anderson, D. Jervis, D. C. McKay, R. Day, S. Trotzky, and J. H. Thywissen.
Imaging and addressing of individual fermionic atoms in an optical lattice. Phys. Rev. A,
92:063406, 2015.
[75] A. R. Edmonds. Angular momentum in quantum mechanics, 2nd edition.
University Press, Princeton, USA, 1960.

Princeton[76] M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross, L. Mazza, M. C.
Bañuls, L. Pollet, I. Bloch, and S. Kuhr. Observation of correlated particle-hole pairs andstring order in low-dimensional mott insulators. Science, 334(6053):200–203, 2011.
[77] Manuel Endres, Hannes Bernien, Alexander Keesling, Harry Levine, Eric R. Anschuetz,
Alexandre Krajenbrink, Crystal Senko, Vladan Vuletic, Markus Greiner, and Mikhail D.
Lukin. Atom-by-atom assembly of defect-free one-dimensional cold atom arrays. Science,
2016.
[78] M. Feld, B. Frohlich, E. Vogt, M. Koschorreck, and M. Kohl. Observation of a pairing
pseudogap in a two-dimensional Ferm gas. Nature, 480:75–78, 2011.
[79] D. Rio Fernandes, F. Sievers, N. Kretzschmar, S. Wu, C. Salomon, and F. Chevy. Subdoppler laser cooling of fermionic 40 k atoms in three-dimensional gray optical molasses.
EPL (Europhysics Letters), 100(6):63001, 2012.
[80] N. Fitch. Traveling-wave stark-decelerated molecular beams for cold-collision experiments.
PhD thesis, University of Colorado, Boulder, 2013.
[81] J. Franck and E. G. Dymond. Elementary processes of photochemical reactions. Trans.
Faraday Soc., 21:536–542, 1926.
[82] J. K. Freericks, M. M. Maśka, Anzi Hu, Thomas M. Hanna, C. J. Williams, P. S. Julienne,
and R. Lemański. Improving the efficiency of ultracold dipolar molecule formation by firstloading onto an optical lattice. Phys. Rev. A, 81:011605, 2010.
[83] A. Frisch, M. Mark, K. Aikawa, S. Baier, R. Grimm, A. Petrov, S. Kotochigova, G. Quéméner,
M. Lepers, O. Dulieu, and F. Ferlaino. Ultracold Dipolar Molecules Composed of Strongly
Magnetic Atoms. Phys. Rev. Lett., 115:203201, 2015.

313[84] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C. Makrides, A. Petrov, and S. Kotochigova. Quantum chaos in ultracold collisions of gas-phase erbium atoms. Nature, 507:475–
479, 2014.
[85] B. R. Gadway. Bose gases in tailored optical and atomic lattices. PhD thesis, Stony Brook
University, 2012.
[86] J. P. Gaebler, J. T. Stewart, J. L. Bohn, and D. S. Jin. p-Wave Feshbach Molecules. Phys.
Rev. Lett., 98:200403, 2007.
[87] C. Gaul, B. J. DeSalvo, J. A. Aman, F. B. Dunning, T. C. Killian, and T. Pohl. Resonant
rydberg dressing of alkaline-earth atoms via electromagnetically induced transparency. Phys.
Rev. Lett., 116:243001, 2016.
[88] N. Gemelke, X. Zhang, C.-L. Hung, and C. Chin. In situ observation of incompressible
Mott-insulating domains in ultracold atomic gases. Nature, 460:995–998, 2009.
[89] M W Gempel, T Hartmann, T A Schulze, K K Voges, A Zenesini, and S Ospelkaus. Versatileelectric fields for the manipulation of ultracold nak molecules. New Journal of Physics,
18(4):045017, 2016.
[90] Fabrice Gerbier, Artur Widera, Simon Fölling, Olaf Mandel, Tatjana Gericke, and Immanuel
Bloch. Interference pattern and visibility of a mott insulator. Phys. Rev. A, 72:053606, 2005.
[91] J. Goldwin, S. Inouye, M. L. Olsen, B. Newman, B. D. DePaola, and D. S. Jin. Measurementof the interaction strength in a Bose-Fermi mixture with 87 Rb and 40 K. Phys. Rev. A,
70:021601, 2004.
[92] Sarang Gopalakrishnan, Ivar Martin, and Eugene A. Demler. Quantum quasicrystals of spinorbit-coupled dipolar bosons. Phys. Rev. Lett., 111:185304, 2013.
[93] A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer,
A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle. Realization of boseeinstein condensates in lower dimensions. Phys. Rev. Lett., 87:130402, 2001.
[94] A. V. Gorshkov, S. R. Manmana, G. Chen, J. Ye, E. Demler, M. D. Lukin, and A. M. Rey.
Tunable Superfluidity and Quantum Magnetism with Ultracold Polar Molecules. Phys. Rev.
Lett., 107:115301, 2011.
[95] A. V. Gorshkov, S. R. Manmana, G. Chen, J. Ye, E. Demler, M. D. Lukin, and A. M. Rey.
Tunable superfluidity and quantum magnetism with ultracold polar molecules. Phys. Rev.
Lett., 107:115301, 2011.
[96] P D Gregory, P K Molony, M P Köppinger, A Kumar, Z Ji, B Lu, A L Marchant, and S L
Cornish. A simple, versatile laser system for the creation of ultracold ground state molecules.
New J. Phys., 17(5):055006, 2015.
[97] Daniel Greif, Maxwell F. Parsons, Anton Mazurenko, Christie S. Chiu, Sebastian Blatt,Florian Huber, Geoffrey Ji, and Markus Greiner. Site-resolved imaging of a fermionic mottinsulator. Science, 351(6276):953–957, 2016.

314[98] M. Greiner. Ultracold quantum gases in three-dimensional optical lattice potentials. PhD
thesis, Ludwig-Maximilians-Universität München, 2003.
[99] M. Greiner and S. Folling. Condensed-matter physics: Optical lattices. Nature, 453:736–738,
2008.
[100] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch. Quantum phase transition
from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature, 415(6867):39–44,
2002.
[101] M. Greiner, C. A. Regal, and D. S. Jin. Emergence of a molecular Bose-Einstein condensate
from a Fermi gas. Nature, 426:537–540, 2003.
[102] Rudolf Grimm, Matthias Weidemller, and Yurii B. Ovchinnikov. Optical Dipole Traps for
Neutral Atoms. volume 42 of Advances In Atomic, Molecular, and Optical Physics, pages 95– 170. 2000.
[103] M. Gröbner, P. Weinmann, F. Meinert, K. Lauber, E. Kirilov, and H.-C. Nägerl. A newquantum gas apparatus for ultracold mixtures of K and Cs and KCs ground-state molecules.
J. Mod. Opt., 0(0):1–11, 2016.
[104] T. Grünzweig, A. Hilliard, M. McGovern, and M. F. Andersen. Near-deterministic preparationof a single atom in an optical microtrap. Nature Physics, 6:951–954, 2010.
[105] R. Guenther. Modern optics. John Wiley and Sons Inc., USA, 1990.
[106] M. Guo, B. Zhu, B. Lu, X. Ye, F. Wang, R. Vexiau, N. Bouloufa-Maafa, G. Quéméner,
O. Dulieu, and D. Wang. Creation of an ultracold gas of ground-state dipolar 23 Na87 Rb
molecules. Phys. Rev. Lett., 116:205303, 2016.
[107] T. L. Gustavson, A. P. Chikkatur, A. E. Leanhardt, A. Görlitz, S. Gupta, D. E. Pritchard,
and W. Ketterle. Transport of bose-einstein condensates with optical tweezers. Phys. Rev.
Lett., 88:020401, Dec 2001.
[108] E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G. D. Bruce, and S. Kuhr.
Single-atom imaging of fermions in a quantum-gas microscope. Nat. Phys., 11:738–742, 2015.
[109] Elmar Haller, Manfred J. Mark, Russell Hart, Johann G. Danzl, Lukas Reichsöllner, Vladimir
Melezhik, Peter Schmelcher, and Hanns-Christoph Nägerl. Confinement-induced resonances
in low-dimensional quantum systems. Phys. Rev. Lett., 104:153203, 2010.
[110] K. R. A. Hazzard, B. Gadway, M. Foss-Feig, B. Yan, S. A. Moses, J. P. Covey, N. Y. Yao,
M. D. Lukin, J. Ye, D. S. Jin, and A. M. Rey. Many-Body Dynamics of Dipolar Molecules
in an Optical Lattice. Phys. Rev. Lett., 113:195302, 2014.
[111] K. R. A. Hazzard, A. V. Gorshkov, and A. M. Rey. Spectroscopy of dipolar fermions inlayered two-dimensional and three-dimensional lattices. Phys. Rev. A, 84:033608, 2011.
[112] K. R. A. Hazzard, S. R. Manmana, M. Foss-Feig, and A. M. Rey. Far-from-Equilibrium
Quantum Magnetism with Ultracold Polar Molecules. Phys. Rev. Lett., 110:075301, 2013.

315[113] J. Heinze, S. Götze, J. S. Krauser, B. Hundt, N. Fläschner, D.-S. Lühmann, C. Becker,
and K. Sengstock. Multiband spectroscopy of ultracold fermions: Observation of reduced
tunneling in attractive bose-fermi mixtures. Phys. Rev. Lett., 107:135303, 2011.
[114] Myoung-Sun Heo, Jae-yoon Choi, and Yong-il Shin. Fast production of large 23 Na boseeinstein condensates in an optically plugged magnetic quadrupole trap. Phys. Rev. A,
83:013622, 2011.
[115] J. Herbig, T. Kraemer, M. Mark, T. Weber, C. Chin, H.-C. Nägerl, and R. Grimm. Preparation of a Pure Molecular Quantum Gas. Science, 301(5639):1510–1513, 2003.
[116] T. A. Hilker, G. Salomon, F. Grusdt, A. Omran, M. Boll, E. Demler, I. Bloch, and C. Gross.
Revealing hidden antiferromagnetic correlations in doped Hubbard chains via string correlators. arXiv:1702.00642v1, 2017.
[117] D. J. Hucknall and A. Morris. Vacuum technology: Calculations in chemistry. Royal Societyof Chemistry, 2003.
[118] E. R. Hudson. Experiments on cold molecules produced via stark deceleration. PhD thesis,
University of Colorado, Boulder, 2006.
[119] E. R. Hudson, N. B. Gilfoy, S. Kotochigova, Jeremy M. Sage, and D. DeMille. Inelastic
collisions of ultracold heteronuclear molecules in an optical trap. Phys. Rev. Lett., 100:203201,
2008.
[120] J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds.
Improved measurement of the shape of the electron. Nature, 460:995–998, 2009.
[121] K. Hueck, N. Luick, L. Sobirey, J. Siegl, T. Lompe, and H Moritz. Two-Dimensional Homogeneous Fermi Gases. arXiv:1704.06315v2, 2017.
[122] Klaus Hueck, Niclas Luick, Lennart Sobirey, Jonas Siegl, Thomas Lompe, Henning Moritz,
Logan W. Clark, and Cheng Chin. Calibrating high intensity absorption imaging of ultracold
atoms. Opt. Express, 25(8):8670–8679, 2017.
[123] Matthew T. Hummon, Mark Yeo, Benjamin K. Stuhl, Alejandra L. Collopy, Yong Xia, and
Jun Ye. 2d magneto-optical trapping of diatomic molecules. Phys. Rev. Lett., 110:143001,
Apr 2013.
[124] Chen-Lung Hung, Xibo Zhang, Li-Chung Ha, Shih-Kuang Tung, Nathan Gemelke, and Cheng
Chin. Extracting densitydensity correlations from in situ images of atomic quantum gases.
New Journal of Physics, 13(7):075019, 2011.
[125] David A. Huse, Rahul Nandkishore, and Vadim Oganesyan. Phenomenology of fully manybody-localized systems. Phys. Rev. B, 90:174202, Nov 2014.
[126] N. R. Hutzler, L. R. Liu, Y. Yu, and K.-K. Ni. Eliminating light shifts in single-atom opticaltraps. arXiv:1605.09422v1, 2016.
[127] Z. Idziaszek and P. S. Julienne. Universal rate constants for reactive collisions of ultracold
molecules. Phys. Rev. Lett., 104:113202, 2010.

316[128] P. S. Żuchowski and J. M. Hutson. Reactions of ultracold alkali-metal dimers. Phys. Rev. A,
81:060703, 2010.
[129] S. Inouye, J. Goldwin, M. L. Olsen, C. Ticknor, J. L. Bohn, and D. S. Jin. Observation of
Heteronuclear Feshbach Resonances in a Mixture of Bosons and Fermions. Phys. Rev. Lett.,
93:183201, 2004.
[130] L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker,
and M. Saffman. Demonstration of a neutral atom controlled-not quantum gate. Phys. Rev.
Lett., 104:010503, Jan 2010.
[131] R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner. Measuring
entanglement entropy in a quantum many-body system. Nature, 528:77–83, 2015.
[132] R. Jafari and A. Langari. Phase diagram of the one-dimensional s = 12 xxz model with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions. Phys.
Rev. B, 76:014412, 2007.
[133] D. Jaschke, K. Maeda, J. D. Whalen, M. L. Wall, and L. D. Carr. Critical phenomena andkibble-zurek scaling in the long-range quantum ising chain. New Journal of Physics, In press,
2017.
[134] D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Collective
excitations of a bose-einstein condensate in a dilute gas. Phys. Rev. Lett., 77:420–423, 1996.
[135] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag,
and R. Grimm. Bose-Einstein Condensation of Molecules. Science, 2003.
[136] K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne. Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering. Rev. Mod. Phys., 78:483–535, 2006.
[137] Nicholas M. Jordan, Y. Y. Lau, David M. French, R. M. Gilgenbach, and P. Pengvanich. Electric field and electron orbits near a triple point. Journal of Applied Physics, 102(3):033301,
2007.
[138] R. Jördens, N. Strohmaier, K. Günter, H. Moritz, and T. Esslinger. A mott insulator offermionic atoms in an optical lattice. Nature, 455:204–207, 2008.
[139] P. S. Julienne, T. M. Hanna, and Z. Idziaszek. Universal ultracold collision rates for polar
molecules of two alkali-metal atoms. Phys. Chem. Chem. Phys., 13:19114–19124, 2011.
[140] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos. Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature,
511:202–205, 2014.
[141] A. M. Kaufman, B. J. Lester, and C. A. Regal. Cooling a single atom in an optical tweezer
to its quantum ground state. Phys. Rev. X, 2:041014, Nov 2012.
[142] A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig, K. R. A. Hazzard,
A. M. Rey, and C. A. Regal. Two-particle quantum interference in tunnel-coupled opticaltweezers. Science, 345(6194):306–309, 2014.

317[143] Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko,
Philipp M. Preiss, and Markus Greiner. Quantum thermalization through entanglement
in an isolated many-body system. Science, 353(6301):794–800, 2016.
[144] T W B Kibble. Topology of cosmic domains and strings. Journal of Physics A: Mathematicaland General, 9(8):1387, 1976.
[145] C. Kittel. Quantum theory of solids. John Wiley and Sons, New York, 1963.
[146] M. Knap, E. Berg, M. Ganahl, and E. Demler. Clustered Wigner-crystal phases of cold polar
molecules in arrays of one-dimensional tubes. Phys. Rev. B, 86:064501, 2012.
[147] T. Köhler, K. Góral, and P. S. Julienne. Production of cold molecules via magneticallytunable Feshbach resonances. Rev. Mod. Phys., 78:1311–1361, 2006.
[148] S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, A. P. Marti, G. E. andKoller, X. Zhang,
A. M. Rey, and J. Ye. Spin-orbit-coupled fermions in an optical lattice clock. Nature, 542:66–
70, 2017.
[149] S. S. Kondov, W. R. McGehee, J. J. Zirbel, and B. DeMarco. Three-dimensional anderson
localization of ultracold matter. Science, 334(6052):66–68, 2011.
[150] Svetlana Kotochigova and David DeMille. Electric-field-dependent dynamic polarizability
and state-insensitive conditions for optical trapping of diatomic polar molecules. Phys. Rev.
A, 82:063421, 2010.
[151] K. A. Kuns, A. M. Rey, and A. V. Gorshkov. d-wave superfluidity in optical lattices ofultracold polar molecules. Phys. Rev. A, 84:063639, 2011.
[152] M. P. Kwasigroch and N. R. Cooper. Bose-einstein condensation and many-body localization of rotational excitations of polar molecules following a microwave pulse. Phys. Rev. A,
90:021605, 2014.
[153] F. Lang, K. Winkler, C. Strauss, R. Grimm, and J. Hecker Denschlag. Ultracold triplet
molecules in the rovibrational ground state. Phys. Rev. Lett., 101:133005, 2008.
[154] A. E. Leanhardt, A. Görlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard, and
W. Ketterle. Imprinting vortices in a bose-einstein condensate using topological phases. Phys.
Rev. Lett., 89:190403, 2002.
[155] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland. Quantum dynamics of single trapped
ions. Rev. Mod. Phys., 75:281–324, 2003.
[156] M. Lemeshko, R. V. Krems, and H. Weimer. Nonadiabatic Preparation of Spin Crystals with
Ultracold Polar Molecules. Phys. Rev. Lett., 109:035301, 2012.
[157] B. J. Lester, A. M. Kaufman, and C. A. Regal. Raman cooling imaging: Detecting single
atoms near their ground state of motion. Phys. Rev. A, 90:011804, Jul 2014.
[158] Brian J. Lester, Niclas Luick, Adam M. Kaufman, Collin M. Reynolds, and Cindy A. Regal.
Rapid production of uniformly filled arrays of neutral atoms. Phys. Rev. Lett., 115:073003,
Aug 2015.

318[159] Paul D. Lett, Richard N. Watts, Christoph I. Westbrook, William D. Phillips, Phillip L.
Gould, and Harold J. Metcalf. Observation of atoms laser cooled below the doppler limit.
Phys. Rev. Lett., 61:169–172, 1988.
[160] Y. Lin, R. L. Compton, K. Jimenez-Gercia, J. V. Porto, and I. B. Spielman. Synthetic
magnetic fields for ultracold neutral atoms. Nature, 462:628–632, 2009.
[161] L. R. Liu, J. T. Zhang, Y. Yu, N. R. Hutzler, T. Rosenband, and K.-K. Ni. Ultracold
molecular assembly. arXiv:1605.09422v1, 2017.
[162] Peng Liu, Qin Sun, Feng Zhu, Kai Liu, Kaili Jiang, Liang Liu, Qunqing Li, and Shoushan Fan.
Measuring the work function of carbon nanotubes with thermionic method. Nano Letters,
8(2):647–651, 2008.
[163] M. Lu, N. Q. Burdick, and B. L. Lev. Quantum Degenerate Dipolar Fermi Gas. Phys. Rev.
Lett., 108:215301, 2012.
[164] Mingwu Lu, Nathaniel Q. Burdick, Seo Ho Youn, and Benjamin L. Lev. Strongly dipolarbose-einstein condensate of dysprosium. Phys. Rev. Lett., 107:190401, 2011.
[165] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt. Optical atomic clocks. Rev.
Mod. Phys., 87:637–701, 2015.
[166] Roman M. Lutchyn, Enrico Rossi, and S. Das Sarma. Spontaneous interlayer superfluidity
in bilayer systems of cold polar molecules. Phys. Rev. A, 82:061604, 2010.
[167] P. Marte, P. Zoller, and J. L. Hall. Coherent atomic mirrors and beam splitters by adiabatic
passage in multilevel systems. Phys. Rev. A, 44:R4118–R4121, 1991.
[168] M. J. Martin, M. Bishof, M. D. Swallows, X. Zhang, C. Benko, J. von Stecher, A. V. Gorshkov,
A. M. Rey, and Jun Ye. A quantum many-body spin system in an optical lattice clock. Science,
341(6146):632–636, 2013.
[169] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell.
Vortices in a bose-einstein condensate. Phys. Rev. Lett., 83:2498–2501, 1999.
[170] P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski,
D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin. Roomtemperature quantum bit memory exceeding one second. Science, 336(6086):1283–1286, 2012.
[171] M. Mayle, G. Quéméner, B. P. Ruzic, and J. L. Bohn. Scattering of ultracold molecules in
the highly resonant regime. Phys. Rev. A, 87:012709, 2013.
[172] M. Mayle, B. P. Ruzic, and J. L. Bohn. Statistical aspects of ultracold resonant scattering.
Phys. Rev. A, 85:062712, 2012.
[173] A. Mazurenko, C. S. Chiu, G. Ji, M. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt,
E. Demler, D. Greif, and M. Greiner. Experimental realization of a long-range antiferromagnet
in the Hubbard model with ultracold atoms. arXiv:1612.08436, 2016.
[174] M. Meucci, E. Mariotti, P. Bicchi, C. Marinelli, and L. Moi. Light-induced atom desorption.
EPL (Europhysics Letters), 25(9):639, 1994.

319[175] A. Micheli, G. K. Brennen, and P. Zoller. A toolbox for lattice-spin models with polar
molecules. Nat. Phys., 2:341–347, 2006.
[176] A. Micheli, G. Pupillo, H. P. Büchler, and P. Zoller. Cold polar molecules in two-dimensional
traps: Tailoring interactions with external fields for novel quantum phases. Phys. Rev. A,
76:043604, 2007.
[177] Debayan Mitra, Peter T. Brown, Peter Schauß, Stanimir S. Kondov, and Waseem S. Bakr.
Phase separation and pair condensation in a spin-imbalanced 2d fermi gas. Phys. Rev. Lett.,
117:093601, 2016.
[178] Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William Cody Burton, and Wolfgang Ketterle. Realizing the harper hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett., 111:185302, 2013.
[179] G. Modugno, C. Benkő, P. Hannaford, G. Roati, and M. Inguscio. Sub-doppler laser coolingof fermionic 40 K atoms. Phys. Rev. A, 60:R3373–R3376, Nov 1999.
[180] P. K. Molony, P. D. Gregory, Z. Ji, B. Lu, M. P. Köppinger, C. R. Le Sueur, C. L. Blackley, J. M. Hutson, and S. L. Cornish. Creation of Ultracold 87 Rb133 Cs Molecules in the
Rovibrational Ground State. Phys. Rev. Lett., 113:255301, 2014.
[181] S. A. Moses. A quantum gas of polar molecules in an optical lattice. PhD thesis, Universityof Colorado, Boulder, 2016.
[182] S. A. Moses, J. P. Covey, M. T. Miecnikowski, D. S. Jin, and J. Ye. New Frontiers for quantum
gases of polar molecules. Nat. Phys., 13:13–20, 2017.
[183] S. A. Moses, J. P. Covey, M. T. Miecnikowski, B. Yan, B. Gadway, J. Ye, and D. S. Jin.
Creation of a low-entropy quantum gas of polar molecules in an optical lattice. Science,350(6261):659–662, 2015.
[184] P. A. Murthy, D. Kedar, T. Lompe, M. Neidig, M. G. Ries, A. N. Wenz, G. Zürn, and
S. Jochim. Matter-wave fourier optics with a strongly interacting two-dimensional fermi gas.
Phys. Rev. A, 90:043611, 2014.
[185] K. D. Nelson, X. Li, and D. S. Weiss. Imaging single atoms in a three-dimensional array.
Nature Physics, 3:556–560, 2007.
[186] D. J. Nesbitt. Toward State-to-State Dynamics in Ultracold Collisions: Lessons from HighResolution Spectroscopy of Weakly Bound Molecular Complexes. Chem. Rev., 112(9):5062–
5072, 2012.
[187] B. Neyenhuis. Ultracold polar krb molecules in optical lattices. PhD thesis, University ofColorado, Boulder, 2012.
[188] B. Neyenhuis, B. Yan, S. A. Moses, J. P. Covey, A. Chotia, A. Petrov, S. Kotochigova, J. Ye,
and D. S. Jin. Anisotropic Polarizability of Ultracold Polar 40 K87 Rb Molecules. Phys. Rev.
Lett., 109:230403, 2012.
[189] K.-K. Ni. A quantum gas of polar molecules. PhD thesis, University of Colorado, Boulder,
2009.

320
[190] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye. A High Phase-Space-Density Gas of Polar
Molecules. Science, 322(5899):231–235, 2008.
[191] K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda, J. L.
Bohn, J. Ye, and D. S. Jin. Dipolar collisions of polar molecules in the quantum regime.
Nature, 464:1324–1328, 2010.
[192] F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Béguin, A. Vernier, T. Lahaye, and
A. Browaeys. Single-atom trapping in holographic 2d arrays of microtraps with arbitrarygeometries. Phys. Rev. X, 4:021034, May 2014.
[193] M. L. Olsen, J. D. Perreault, T. D. Cumby, and D. S. Jin. Coherent atom-molecule oscillations
in a Bose-Fermi mixture. Phys. Rev. A, 80:030701, 2009.
[194] A. Omran, M. Boll, T. A. Hilker, K. Kleinlein, G. Salomon, I. Bloch, and C. Gross. Microscopic Observation of Pauli Blocking in Degenerate Fermionic Lattice Gases. Phys. Rev.
Lett., 115:263001, 2015.
[195] C. H. R. Ooi. Laser cooling of molecules by zero-velocity selection and single spontaneousemission. Phys. Rev. A, 82:053408, 2010.
[196] N. D. Oppong. Towards a degenerate fermi gas of strontium-87 in a 3d optical lattice. MS
thesis, University of Colorado, Boulder, 2015.
[197] S. Ospelkaus, K.-K. Ni, G. Quéméner, B. Neyenhuis, D. Wang, M. H. G. de Miranda, J. L.
Bohn, J. Ye, and D. S. Jin. Controlling the Hyperfine State of Rovibronic Ground-State Polar
Molecules. Phys. Rev. Lett., 104:030402, 2010.
[198] S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyenhuis, G. Quéméner, P. S.
Julienne, J. L. Bohn, D. S. Jin, and J. Ye. Quantum-State Controlled Chemical Reactions of
Ultracold Potassium-Rubidium Molecules. Science, 327(5967):853–857, 2010.
[199] S. Ospelkaus, A. Pe’er, K.-K. Ni, J.J. Zirbel, B. Neyenhuis, S. Kotochigova, P.S. Julienne,
J. Ye, and D. S. Jin. Efficient state transfer in an ultracold dense gas of heteronuclear
molecules. Nat. Phys., 4:622–626, 2008.
[200] S. Ospelkaus-Schwarzer. Quantum degenerate fermi-bose mixtures of 40 k and 87 rb in 3d
optical lattices. PhD thesis, Universität Hamburg, 2006.
[201] A.K. Pal, R.K. Roy, S.K. Mandal, S. Gupta, and B. Deb. Electrodeposited carbon nanotube
thin films. Thin Solid Films, 476(2):288 – 294, 2005.
[202] Alexander T. Papageorge, Alicia J. Kollár, and Benjamin L. Lev. Coupling to modes of anear-confocal optical resonator using a digital light modulator. Opt. Express, 24(11):11447–
11457, 2016.
[203] J. W. Park, S. A. Will, and M. W. Zwierlein. Ultracold Dipolar Gas of Fermionic 23 Na40 K
Molecules in Their Absolute Ground State. Phys. Rev. Lett., 114:205302, 2015.
[204] J. W. Park, Z. Y. Yan, H. Loh, S. A. Will, and M. W. Zwierlein. Second-scale nuclear spin
coherence time of trapped ultracold 23 Na40 K. arXiv:1606.04184v1, 2016.

321[205] M. F. Parsons, F. Huber, A. Mazurenko, C. S. Chiu, W. Setiawan, K. Wooley-Brown, S. Blatt,and M. Greiner. Site-Resolved Imaging of Fermionic 6 Li in an Optical Lattice. Phys. Rev.
Lett., 114:213002, 2015.
[206] Maxwell F. Parsons, Anton Mazurenko, Christie S. Chiu, Geoffrey Ji, Daniel Greif, andMarkus Greiner. Site-resolved measurement of the spin-correlation function in the fermihubbard model. Science, 353(6305):1253–1256, 2016.
[207] B. Pasquiou, A. Bayerle, S. M. Tzanova, S. Stellmer, J. Szczepkowski, M. Parigger, R. Grimm,
and F. Schreck. Quantum degenerate mixtures of strontium and rubidium atoms. Phys. Rev.
A, 88:023601, 2013.
[208] Y. S. Patil, S. Chakram, L. M. Aycock, and M. Vengalattore. Nondestructive imaging of an
ultracold lattice gas. Phys. Rev. A, 90:033422, Sep 2014.
[209] P. Pedri, L. Pitaevskii, S. Stringari, C. Fort, S. Burger, F. S. Cataliotti, P. Maddaloni,
F. Minardi, and M. Inguscio. Expansion of a coherent array of bose-einstein condensates.
Phys. Rev. Lett., 87:220401, 2001.
[210] Wolfgang Petrich, Michael H. Anderson, Jason R. Ensher, and Eric A. Cornell. Behavior ofatoms in a compressed magneto-optical trap. J. Opt. Soc. Am. B, 11(8):1332–1335, 1994.
[211] Tilman Pfau. Private Communication.
[212] Philipp M. Preiss, Ruichao Ma, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Philip
Zupancic, Yoav Lahini, Rajibul Islam, and Markus Greiner. Strongly correlated quantum
walks in optical lattices. Science, 347(6227):1229–1233, 2015.
[213] Philipp M. Preiss, Ruichao Ma, M. Eric Tai, Jonathan Simon, and Markus Greiner. Quantumgas microscopy with spin, atom-number, and multilayer readout. Phys. Rev. A, 91:041602,
2015.
[214] G. Quéméner and J. L. Bohn. Dynamics of ultracold molecules in confined geometry andelectric field. Phys. Rev. A, 83:012705, 2011.
[215] G. Quéméner and P. S. Julienne.
112(9):4949–5011, 2012.

Ultracold Molecules under Control!

Chem. Rev.,[216] R. Unanyan and M. Fleischhauer and B.W. Shore and K. Bergmann. Robust creation andphase-sensitive probing of superposition states via stimulated Raman adiabatic passage (STIRAP) with degenerate dark states. Opt. Comm., 155(13):144 – 154, 1998.
[217] C. A. Regal. Ultracold bosonic atoms in optical lattices. PhD thesis, University of Maryland,College Park, 2004.
[218] C. A. Regal. Experimental realization of bcs-bec crossover physics with a fermi gas of atoms.
PhD thesis, University of Colorado, Boulder, 2006.
[219] C. A. Regal, M. Greiner, and D. S. Jin. Observation of Resonance Condensation of Fermionic
Atom Pairs. Phys. Rev. Lett., 92:040403, 2004.
[220] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin. Tuning p-Wave Interactions in an
Ultracold Fermi Gas of Atoms. Phys. Rev. Lett., 90:053201, 2003.

322[221] L. Reichsöllner, A. Schindewolf, T. Takekoshi, R. Grimm, and H.-C. Nägerl. Quantum
engineering of a low-entropy gas of heteronuclear bosonic molecules in an optical lattice.
arXiv:1607.06536v1, 2016.
[222] G. Reinaudi, C. B. Osborn, M. McDonald, S. Kotochigova, and T. Zelevinsky. Optical
Production of Stable Ultracold 88 Sr2 Molecules. Phys. Rev. Lett., 109:115303, 2012.
[223] P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V.
Gorshkov, and C. Monroe. Non-local propagations of correlations in quantum systems withlong-range interactions. Nature, 511:198–201, 2014.
[224] Philip Richerme. Two-dimensional ion crystals in radio-frequency traps for quantum simulation. Phys. Rev. A, 94:032320, 2016.
[225] T. M. Roach, H. Abele, M. G. Boshier, H. L. Grossman, K. P. Zetie, and E. A. Hinds.
Realization of a magnetic mirror for cold atoms. Phys. Rev. Lett., 75:629–632, Jul 1995.
[226] T. Rom, T. Best, O. Mandel, A. Widera, M. Greiner, T. W. Hänsch, and I. Bloch. State
Selective Production of Molecules in Optical Lattices. Phys. Rev. Lett., 93:073002, 2004.
[227] A. Safavi-Naini, M. L. Wall, and A. M. Rey. Role of interspecies interactions in the preparationof a low-entropy gas of polar molecules in a lattice. Phys. Rev. A, 92:063416, 2015.
[228] A. Safavi-Naini, M. L. Wall, and A. M. Rey. Role of interspecies interactions in the preparationof a low-entropy gas of polar molecules in a lattice. Phys. Rev. A, 92:063416, 2015.
[229] M. Saffman, T. G. Walker, and K. Mølmer. Quantum information with rydberg atoms. Rev.
Mod. Phys., 82:2313–2363, 2010.
[230] P. Schauß, J. Zeiher, T. Fukuhara, S. Hild, M. Cheneau, T. Macrı̀, T. Pohl, I. Bloch, andC. Gross. Crystallization in ising quantum magnets. Science, 347(6229):1455–1458, 2015.
[231] M Schlosser, J Kruse, C Gierl, S Teichmann, S Tichelmann, and G Birkl. Fast transport, atom
sample splitting and single-atom qubit supply in two-dimensional arrays of optical microtraps.
New Journal of Physics, 14(12):123034, 2012.
[232] U. Schneider, L. Hackermller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch,and A. Rosch. Metallic and Insulating Phases of Repulsively Interacting Fermions in a 3D
Optical Lattice. Science, 322(5907):1520–1525, 2008.
[233] Michael Schreiber, Sean S. Hodgman, Pranjal Bordia, Henrik P. Lüschen, Mark H. Fischer, Ronen Vosk, Ehud Altman, Ulrich Schneider, and Immanuel Bloch. Observation of
many-body localization of interacting fermions in a quasirandom optical lattice. Science,
349(6250):842–845, 2015.
[234] J. C. Shaw. External electric fields: A new tool for controlling ultracold polar molecules. BS
thesis, University of Colorado, Boulder, 2015.
[235] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr. Single-atomresolved fluorescence imaging of an atomic Mott insulator. Nature, 467:68–72, 2010.
[236] E. S. Shuman, J. F. Barry, D. R. Glenn, and D. DeMille. Radiative Force from Optical
Cycling on a Diatomic Molecule. Phys. Rev. Lett., 103:223001, 2009.

323[237] J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner. Quantum simulationof antiferromagnetic spin chains in an optical lattice. Nature, 472:307–312, 2011.
[238] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse,
and C. Monroe. Many-body localization in a quantum simulator with programmable random
disorder. Nat. Phys., 12:907–911, 2016.
[239] D. Speckman, D. Marvin, J. Matossian, N. Ianno, and W. Stuckey. Atomic oxygen testing of
mgf2 coatings. IEEE conference proceedings.
[240] D. M. Stamper-Kurn, H.-J. Miesner, S. Inouye, M. R. Andrews, and W. Ketterle. Collisionlessand hydrodynamic excitations of a bose-einstein condensate. Phys. Rev. Lett., 81:500–503,
1998.
[241] D. Stauffer and A. Aharon. Introduction to percolation theory. Taylor and Francis, London,
revised ed. 2, 1994.
[242] S. Stellmer, B. Pasquiou, R. Grimm, and F. Schreck. Creation of Ultracold Sr2 Molecules inthe Electronic Ground State. Phys. Rev. Lett., 109:115302, 2012.
[243] E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard.
Continuous and Pulsed Quantum Zeno Effect. Phys. Rev. Lett., 97:260402, 2006.
[244] Niels Strohmaier, Daniel Greif, Robert Jördens, Leticia Tarruell, Henning Moritz, Tilman
Esslinger, Rajdeep Sensarma, David Pekker, Ehud Altman, and Eugene Demler. Observationof elastic doublon decay in the fermi-hubbard model. Phys. Rev. Lett., 104:080401, 2010.
[245] B. K. Stuhl. Ultracold molecules for the masses: evaporative cooling and magneto-opticaltrapping. PhD thesis, University of Colorado, Boulder, 2012.
[246] B. K. Stuhl, M. T. Hummon, M. Yeo, G. Quemener, J. L. Bohn, and J. Ye. Evaporative
cooling of the dipolar hydroxyl radical. Nature, 492:396–400.
[247] B. K. Stuhl, B. C. Sawyer, D. Wang, and J. Ye. Magneto-optical Trap for Polar Molecules.
Phys. Rev. Lett., 101:243002, 2008.
[248] B. K. Stuhl, M. Yeo, M. T. Hummon, and J. Ye. Electric-field-induced inelastic collisions
between magnetically trapped hydroxyl radicals. Mol. Phys., 111(12-13):1798–1804, 2013.
[249] N. Syassen, D. M. Bauer, M. Lettner, T. Volz, D. Dietze, J. J. Garcı́a-Ripoll, J. I. Cirac,
G. Rempe, and S. Dürr. Strong Dissipation Inhibits Losses and Induces Correlations in Cold
Molecular Gases. Science, 320(5881):1329–1331, 2008.
[250] S. V. Syzranov, M. L. Wall, V. Gurarie, and A. M. Rey. Spin–orbital dynamics in a systemof polar molecules. Nat. Commun., 5:5391, 2014.
[251] S. V. Syzranov, M. L. Wall, B. Zhu, V. Gurarie, and A. M. Rey. Emergent Weyl excitations
in systems of polar particles. Nat. Commun., 7:13543, 2016.
[252] T. Takekoshi, L. Reichsöllner, A. Schindewolf, J. M. Hutson, C. R. Le Sueur, O. Dulieu,
F. Ferlaino, R. Grimm, and H.-C. Nägerl. Ultracold Dense Samples of Dipolar RbCs Moleculesin the Rovibrational and Hyperfine Ground State. Phys. Rev. Lett., 113:205301, 2014.

324[253] M. R. Tarbutt and T. C. Steimle. Modeling magneto-optical trapping of CaF molecules.
Phys. Rev. A, 92:053401, 2015.
[254] G. Thalhammer, K. Winkler, F. Lang, S. Schmid, R. Grimm, and J. Hecker Denschlag.
Long-Lived Feshbach Molecules in a Three-Dimensional Optical Lattice. Phys. Rev. Lett.,
96:050402, 2006.
[255] S. Timoshenko and S. Woinowsky-Krieger. Theory of plates and shells. McGraw-Hill, New
York, 1959.
[256] A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet.
Observation of Fermi Pressure in a Gas of Trapped Atoms. Science, 291(5513):2570–2572,
2001.
[257] Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe,
and D. J. Wineland. Deterministic entanglement of two trapped ions. Phys. Rev. Lett.,
81:3631–3634, 1998.
[258] S. Umadevi, V. Ganesh, and Sheela Berchmans. Liquid crystal (lc) monolayer on indium tinoxide (ito): structural and electrochemical characterization. RSC Adv., 4:16409–16417, 2014.
[259] Sebastiaan Y. T. van de Meerakker, Nicolas Vanhaecke, Hendrick L. Bethlem, and Gerard
Meijer. Transverse stability in a stark decelerator. Phys. Rev. A, 73:023401, Feb 2006.
[260] N. V. Vitanov, A. A. Rangelov, B. W. Shore, and W. Bergmann. Stimulated Raman adiabatic
passage in physics, chemistry and beyond. arXiv:1605.00224v2, 2016.
[261] T. Volz, N. Syassen, D. M. Bauer, E. Hansis, S. Dürr, and G. Rempe. Preparation of aquantum state with one molecule at each site of an optical lattice. Nature Phys., 2:692–695,
2006.
[262] M. L. Wall, Erman Bekaroglu, and Lincoln D. Carr. Molecular hubbard hamiltonian: Field
regimes and molecular species. Phys. Rev. A, 88:023605, 2013.
[263] M. L. Wall, K. R. A. Hazzard, and A. M. Rey. Quantum Magnetism with
Ultracold Molecules. In From Atomic to Mesoscale, chapter 1, pages 3–37. World Scientific,
2015.
[264] D. Wang, B. Neyenhuis, M. H. G. de Miranda, K.-K. Ni, S. Ospelkaus, D. S. Jin, and J. Ye.
Direct absorption imaging of ultracold polar molecules. Phys. Rev. A, 81:061404, 2010.
[265] J. S. Waugh, L. M. Huber, and U. Haeberlen. Approach to high-resolution nmr in solids.
Phys. Rev. Lett., 20:180–182, 1968.
[266] C. Weitenberg. Single-atom resolved imaging and manipulation in an atomic mott insulator.
PhD thesis, Ludwig-Maximilians-Universität München, 2011.
[267] C. Weitenberg, M Endres, J. F. Sherson, M. Cheneau, P. Schauss, T. Fukahara, I. Bloch, andS. Kuhr. Single-spin addressing in an atomic Mott insulator. Nature, 471:319–324, 2011.
[268] R. J. Wild. Contact measurements on a strongly interacting bose gas. PhD thesis, Universityof Colorado, Boulder, 2012.

325[269] S. A. Will, J. W. Park, Z. Y. Yan, H. Loh, and M. W. Zwierlein. Coherent Microwave Control
of Ultracold 23 Na40 K Molecules. Phys. Rev. Lett, 116, 2016.
[270] K. Winkler, F. Lang, G. Thalhammer, P. von der Straten, R. Grimm, and J. Hecker Denschlag. Coherent optical transfer of Feshbach molecules to a lower vibrational state. Phys.
Rev. Lett., 98:043201, 2007.
[271] K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hecker Denschlag, A. J. Daley, A. Kantian, H. P. Büchler, and P. Zoller. Repulsively bound atom pairs in an optical lattice. Nature,
441:853–846, 2006.
[272] C.-H. Wu, J. W. Park, P. Ahmadi, S. Will, and M. W. Zwierlein. Ultracold fermionic feshbach
molecules of 23 Na40 K. Phys. Rev. Lett., 109:085301, 2012.
[273] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin, andJ. Ye. Observation of dipolar spin-exchange interactions with lattice-confined polar molecules.
Nature, 501(7468):521–525, 2013.
[274] N. Y. Yao, C. R. Laumann, S. Gopalakrishnan, M. Knap, M. Müller, E. A. Demler, andM. D. Lukin. Many-body localization in dipolar systems. Phys. Rev. Lett., 113:243002, Dec
2014.
[275] N. Y. Yao, A. C. Potter, I.-D. Potirniche, and A. Vishwanath. Discrete time crystals: Rigidity,
criticality, and realizations. Phys. Rev. Lett., 118:030401, 2017.
[276] N.Y. Yao, A.V. Gorshkov, C. R. Laumann, A. M. Läuchli, J. Ye, and M. D. Lukin. Realizing
Fractional Chern Insulators in Dipolar Spin Systems. Phys. Rev. Lett., 110:185302, 2013.
[277] L. P. Yatsenko, B. W. Shore, and K. Bergmann. Detrimental consequences of small rapid
laser fluctuations on stimulated Raman adiabatic passage. 89, 2014.
[278] M. Yeo, M. T. Hummon, A. L. Collopy, B. Yan, B. Hemmerling, E. Chae, J. M. Doyle, and
Jun Ye. Rotational state microwave mixing for laser cooling of complex diatomic molecules.
Phys. Rev. Lett., 114:223003, 2015.
[279] J. Zeiher, R. van Bijnen, P. Schauß, S. Hild, J.-Y. Choi, T. Pohl, I. Bloch, and C. Gross.
Many-body interferometry of a rydberg-dressed spin lattice. Nature Physics, 12:1095–1099,
2016.
[280] J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano, I.-D. Potirniche,
A. C. Potter, A. Vishwananth, N. Y. Yao, and C. Monroe. Observation of a Discrete Time
Crystal. arXiv:1609.08684, 2016.
[281] V. Zhelyazkova, A. Cournol, T. E. Wall, A. Matsushima, J. J. Hudson, E. A. Hinds, M. R.
Tarbutt, and B. E. Sauer. Laser cooling and slowing of caf molecules. Phys. Rev. A, 89:053416,
May 2014.
[282] B. Zhu, B. Gadway, M. Foss-Feig, J. Schachenmayer, M. L. Wall, K. R. A. Hazzard, B. Yan,
S. A. Moses, J. P. Covey, D. S. Jin, J. Ye, M. Holland, and A. M. Rey. Suppressing the Loss of
Ultracold Molecules Via the Continuous Quantum Zeno Effect. Phys. Rev. Lett., 112:070404,
2014.

326
[283] B. Zhu, G. Quéméner, A. M. Rey, and M. J. Holland. Evaporative cooling of reactive polar
molecules confined in a two-dimensional geometry. Phys. Rev. A, 88:063405, 2013.
[284] J. J. Zirbel. Ultracold fermionic feshbach molecules. PhD thesis, University of Colorado,
Boulder, 2008.
[285] J. J. Zirbel, K.-K. Ni, S. Ospelkaus, J. P. D’Incao, C. E. Wieman, J. Ye, and D. S. Jin.
Collisional Stability of Fermionic Feshbach Molecules. Phys. Rev. Lett., 100:143201, 2008.
[286] Philip Zupancic, Philipp M. Preiss, Ruichao Ma, Alexander Lukin, M. Eric Tai, Matthew
Rispoli, Rajibul Islam, and Markus Greiner. Ultra-precise holographic beam shaping for
microscopic quantum control. Opt. Express, 24(13):13881–13893, 2016.
[287] W. H. Zurek. Cosmological experiments in superfluid helium? Nature, 317:505–508, 1985.
[288] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle. Vortices
and superfluidity in a strongly interacting fermi gas. Nature, 435:1047–1051, 2005.
[289] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic,
and W. Ketterle. Observation of Bose-Einstein Condensation of Molecules. Phys. Rev. Lett.,
91:250401, 2003.
[290] Martin Zwierlein. Private Communication.

Appendix A

New tapered amplifier mount design

328

Figure A.1: An image of the tapered amplifier (TA) set-up in the Rb laser system. All the electricalconnections come in the side of the aluminum base (anodized) via a DB9 jack. The brass base inshown inside the plexiglass lid, and it contains the TA chip. The plexiglass lid has windows on eachside an angle of approximately 5 degrees to avoid reflections.

329

.690

Diameter 0.2" hole bored all the way to opposite
Hole of diameter 0.2".
thread to create 0.1" deep slot.
Threaded to 0.800"#2-56 Tapped Hole
36 thread. A.360

Radius of
1/16"
.170
.350

.375

.050
.110 A

1.200
SECTION A-A.260
4 x Clearance holefor 8 32 screw.

.125
.225

Threaded to0.800" 36 thread.

#6-40 Tapped Hole0.09" diameter hole to depthof 0.25". Position not important.

1.200 .750

UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONALANGULAR: MACHBEND
TWO PLACE DECIMAL
THREE PLACE DECIMAL
INTERPRET GEOMETRICTOLERANCING PER:PROPRIETARY AND CONFIDENTIAL
THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

5

MATERIAL

NAME

DATE

DRAWN

TITLE:CHECKED
ENG APPR.

Amplifier Chip Holder

MFG APPR.
Q.A.
COMMENTS:SIZE DWG. NO.

Brass

NEXT ASSYUSED ON
APPLICATION

4FINISH

REV

A

None
3

SHEET 1 OF 1

SCALE: 2:1

DO NOT SCALE DRAWING

2

1

Figure A.2: The brass base for holding the TA chip. This base is mounted to the aluminum
baseplate (next image) with nylon screws, with a Peltier Thermo-Electric Cooler (TEC) in between.
The height of the brass base and the aluminum baseplate are designed to combine with a TEC togive a beam height of 2 inches.

330

2 x Counterbore clearance hole for
1/4 20 screw. Counterbore to depth 1".
4 x 8-32 Tapped Holeto depth of 0.5 inch.
2 x 3/8 hole to depth 1.10".
3/8 Hole
to depthof 2.8".

.850
.500

.813

.191
.625

.781

.094

1.063

.469

.375
2.875

2.625

.750

1.125
1.500

.125

1.438

.438
1.563

.250

4 x 8-32 Tapped Holeto depth of 0.5 inch.
2 x 4-40 Tapped Holeto depth of 0.5 inch.

2.875
3.125
UNLESS OTHERWISE SPECIFIED:PROPRIETARY AND CONFIDENTIAL
THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

5

NAME

DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONALANGULAR: MACHBEND
TWO PLACE DECIMAL
THREE PLACE DECIMALDRAWNINTERPRET GEOMETRICTOLERANCING PER:

Q.A.

DATE

TITLE:CHECKED
ENG APPR.

Aluminum Baseplate

MFG APPR.

COMMENTS:

MATERIAL

SIZE DWG. NO.

Aluminum
FINISHNEXT ASSYUSED ON
APPLICATION4SHEET 1 OF 1

SCALE: 1:1

DO NOT SCALE DRAWING

3

REV

A

None

2

1

Figure A.3: The aluminum baseplate that mounts the TA fixture to the optical table or breadboard.
The connections come through this baseplate in a DB9-style connector in the side of the baseplate.
The cables then run through the baseplate and come up the holes near the brass TA chip base.

331

3.125
2 x 0.535 40 Tapped holes
2.000
.850
.250

1.563

.125

A
.750
.063
2.625
.750

2.500 2.875

A.063

SECTION A-A.125
4 x Clearance hole for 8 32 screw.

UNLESS OTHERWISE SPECIFIED:PROPRIETARY AND CONFIDENTIAL
THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

5

NAME

DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONALANGULAR: MACHBEND
TWO PLACE DECIMAL
THREE PLACE DECIMALDRAWNINTERPRET GEOMETRICTOLERANCING PER:

Q.A.

DATE

TITLE:CHECKED
ENG APPR.

Plexiglass Lid

MFG APPR.

COMMENTS:

MATERIAL

SIZE DWG. NO.

Plexiglass
FINISHNEXT ASSYUSED ON
APPLICATION4SHEET 1 OF 1

SCALE: 1:2

DO NOT SCALE DRAWING

3

REV

A

None

2

1

Figure A.4: The plexiglass lid for the TA box. This lid eliminates convection air currents between
the TA base and the surrounding environment. There are tapped half-inch holes on the sides tomount windows. The holes are typically tapped at an angle of approximately 5 degrees to eliminate
reflections back into the TA chip.

332

Outside of cylinder to betapped with 0.800-36 tap

0.130

0.800
0.420
0.250

B

B
SECTION B-B
2 x 0.1" diameterclearance holes
0.290

Large hole to beclearance of diameter 0.42"
Small hole to betapped with M9-0.5
metric tap

Note: tolerance of 0.80"-36 thread
is not critical as long as all elementswith this threading are compatible
UNLESS OTHERWISE SPECIFIED:PROPRIETARY AND CONFIDENTIAL
THE INFORMATION CONTAINED IN THISDRAWING IS THE SOLE PROPERTY OF<INSERT COMPANY NAME HERE>. ANY
REPRODUCTION IN PART OR AS A WHOLEWITHOUT THE WRITTEN PERMISSION OF<INSERT COMPANY NAME HERE> IS
PROHIBITED.

5

NAME

DIMENSIONS ARE IN INCHESTOLERANCES:
FRACTIONALANGULAR: MACHBEND
TWO PLACE DECIMAL
THREE PLACE DECIMALDRAWNINTERPRET GEOMETRICTOLERANCING PER:

Q.A.

DATE

TITLE:CHECKED
ENG APPR.

9 mm lens holder

MFG APPR.

COMMENTS:

MATERIAL

SIZE DWG. NO.

Brass
FINISHNEXT ASSYUSED ON
APPLICATION

4

REV

A

None

3

SHEET 1 OF 1

SCALE: 2:1

DO NOT SCALE DRAWING

2

1

Figure A.5: The lens holder, which threads into the brass base. The outer threads on this cylidricalelement mate with the threads in the bore of the brass base shown above. The lens holders arelocked in place using a nylon-tipped set screw in a tapped hole on the top of the brass base (notshown).

Appendix B

Cart and stage motion programs for the new apparatus

334

Figure B.1: The code for the stage motion. Motion forward (back) is triggered with a TTL goinghigh (low). Note that the oscillate command has been used to dither the motion at a specified
frequency.

335

Figure B.2: The motion profile of the stage. The three curves from top to bottom are the positioncommand, velocity command, and acceleration command. The motion shown is from the initial
position to the final position. The profile during the reverse motion is symmetrical.

336

Figure B.3: The code for the cart with the quadrupole trap. Motion forward (back) is triggeredwith a TTL going high (low). Note that some extra commands are included and commented out.