Realizing quantum spin models with 7Li atoms in an
optical latticebyIvana Ljubomirova Dimitrova
B.A., Princeton University (2010)
Submitted to the Department of Physics
in Partial Fulfillment of the Requirements for the Degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2020
c Massachusetts Institute of Technology 2020. All rights reserved.
○

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
20 December 2019
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nergis Mavalvala
Associate Department Head

2

Realizing quantum spin models with 7 Li atoms in an opticallatticeby
Ivana Ljubomirova Dimitrova
Submitted to the Department of Physics
on 20 December 2019, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Quantum spin Hamiltonians are paradigmatic models, which display different kindsof quantum phase transitions, strongly-correlated and topological ground states, andvarious regimes of transport. Expanding their significance, many mappings exist
between quantum spin models and other systems in different areas of physics, mathematics, and beyond. Even though quantum spin models have been studied extensively, there are still many open questions. Simulating these Hamiltonians with thesystem of ultracold atoms in optical lattices provides a new perspective with the wide
tunability of parameters and the minimal coupling to the environment. The mapping involves using the Mott insulating state of ultracold atoms in optical lattices,where the energy of a second-order tunneling process (superexchange) maps to theparameters of a Heisenberg model.
This thesis provides a detailed roadmap for the design and building of such aquantum simulator with 7 Li atoms in optical lattices. Each step of the process is
described, together with the methods and techniques used for the building and thecharacterization of the physical system. A focus is placed on using the Mott insulator
as a starting point for spin physics experiments and, in particular, on the characterization and improvements of the mapping from a density sector description to a spin
sector description of the system.
Several schemes for implementing and studying spin systems are presented. In
particular, the feasibility of implementing the Heisenberg spin-1/2 and spin-1 models
in this system is described. The tilted lattice is presented as a tool for studying puresuperexchange-driven dynamics and for increasing their timescale by suppressing firstorder tunneling and the role of number defects. The first measurements and thetuning with this machine of superexchange-driven dynamics over a wide range in theanisotropic Heisenberg spin-1/2 models are presented.
Finally, the versatility of the BEC 5 machine is showcased by a study which does
not involve an optical lattice. It explores the realization of an exotic quantum phase,a supersolid, in a new way. After many years of building and improvements, theBEC 5 machine emerges as a repeatable and reliable quantum simulator which has
3

a clear scientific agenda of exploring many-body ground states and non-equilibrium
dynamics.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

4

To my family

5

6

Acknowledgments
I would like to thank first and foremost my advisor Wolfgang for always making
time, for listening to what I have to say, and for showing me how to find solutions toseemingly unsolvable problems both in terms of technical and scientific issues in lab
and in terms of interpersonal situations. I have learned so much in the many years of
my graduate work and I really appreciate Wolfgang’s approach to experimental workand his exceptional ability to extract the essential components of each problem until
a clear solution is revealed.
I would also like to thank my labmates Niki and Jesse with whom we have together
gone through the stages of building a BEC machine and developing it for fulfilling thepotential of Li 7 atoms for implementing spin Hamiltonians. Our different inclinations
and working styles, which have very often clashed, have now asymptoted to a wellbalanced mixture of pushing forward as fast as possible and ironing out details whichsometimes make the difference between an experiment succeeding or failing. Thelong discussions with Jesse over both various aspects of our work and also of life
and its meaning have contributed significantly to me enjoying my phd so much. The
laughs with Niki and our efforts to both understand the theoretical description andthe intuitive picture of various problems have been a fantastic way to learn. Even
though we were in the unusual situation of not having a senior grad student or a
postdoc, I think our team has managed to work together and produce now a working
quantum simulator.
I have also enjoyed working with the other people who have at one point or
another been part of the BEC 5 team. Michael and I built together the oven chamberand made the first MOT, which is one of my most memorable experiences at MIT.
Yichao was in BEC 5 at a time when we were desperate to realize a BEC and his
involvement has really helped us to move forward. Although Will and I have slightly
different areas of expertise, I think we have a similar working style and I have always
enjoyed debugging problems with him. I would also like to thank David, who waswith us during the initial stages of building the machine and always tried to make7

time and explain things.
I also want to thank all the people in the CUA with whom I have laughed andshared problems over the years. Sometimes it would just be giving me the right lensat the right time, and sometimes it would be sharing expertise or lasers in criticaltimes. I would like to thank the entire BEC 4 team, who have a similar research
agenda as ours and it has always been inspiring to talk to them. In particular, it has
been great to have long discussions with Colin about the news in the field and what
our place in it is. I would like to thank Timur and Lawrence, with whom we started
in the same year, for always encouraging me that phd life is good and for discussing
the latest setbacks or achievements. Even though it is impossible to list everyone that
I would like to thank, these people have significantly improved my phd experience:
Hiro, Georgious, Cody, Woo-Chang, Wenlan, Jinggang, Tout, Alan, Julie, Hyungmok,
Aviv, Christian, Wujie, Junru, Boris, Chris, Furkan, Pierre, Michael, Li, Alex, Eunice,
Yair, Matt, Thomas(es), Melih, Waseem, Ariel, Mark, Tarik, Julian, Peyman, Peter,
Sebastian, Zhenjie, Parth, Zoe, Carsten, Richard, Airlia, Cedric, Kristi, Dorian, Ian,
Zak.
Finally, I want to thanks my friends and family for listening to me and encouraging
me, for bringing joy and happiness to my life.

8

Contents
1 Introduction
1.1

19

Quantum simulation of spin models . . . . . . . . . . . . . . . . . . .

19

1.1.1

Platforms for simulating spin models . . . . . . . . . . . . . .

22

1.1.2

The 7 Li platform . . . . . . . . . . . . . . . . . . . . . . . . .

31

1.2

The history of BEC 5 . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

1.3

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2 The 7 Li machine
2.1

2.2

41

The road to a 7 Li Mott insulator . . . . . . . . . . . . . . . . . . . .

41

2.1.1

Oven and intermediate chambers . . . . . . . . . . . . . . . .

44

2.1.2

Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.1.3

MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

2.1.4

CMOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

2.1.5

Gray Molasses . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

2.1.6

Dark State Pumping . . . . . . . . . . . . . . . . . . . . . . .

63

2.1.7

Magnetic trap . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.1.8

ODT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

2.1.9

BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

2.1.10 Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

2.2.1

High power beam delivery . . . . . . . . . . . . . . . . . . . .

78

2.2.2

High field states and admixtures . . . . . . . . . . . . . . . . . 101

2.2.3

Bakeouts and the titanium coated viewport . . . . . . . . . . 107
93 Atoms in Optical Lattices
3.1

3.2

111

Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.1.1

Band structure and wavefunctions . . . . . . . . . . . . . . . . 112

3.1.2

Scattering in a lattice . . . . . . . . . . . . . . . . . . . . . . . 113

3.1.3

The Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . 115

Lattice Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.2.1

Interaction Spectroscopy . . . . . . . . . . . . . . . . . . . . . 120

3.2.2

Loading into a lattice . . . . . . . . . . . . . . . . . . . . . . . 123

3.2.3

Lifetime of the MI and heating processes . . . . . . . . . . . . 129

3.2.4

Three-body recombination . . . . . . . . . . . . . . . . . . . . 141

4 Spin Models and Dynamics
4.1

4.2

155

Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.1.1

Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4.1.2

Spin 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.1.3

Implementation of spin models with 7 Li . . . . . . . . . . . . . 175

4.1.4

Beyond spin chains: 2D and 3D physics . . . . . . . . . . . . . 181

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.2.1

What is the timescale for dynamics? . . . . . . . . . . . . . . 182

4.2.2

Dynamics in tilted lattices . . . . . . . . . . . . . . . . . . . . 184

4.2.3

Relaxation of non-equilibrium spin distributions . . . . . . . . 191

5 Case study: a tilted Mott insulator
5.1

197

Choice of tilt implementation . . . . . . . . . . . . . . . . . . . . . . 198
5.1.1

Magnetic field gradient . . . . . . . . . . . . . . . . . . . . . . 198

5.1.2

Optical tilt with an AC stark shift . . . . . . . . . . . . . . . 200

5.2

Characterization of the tilt beam . . . . . . . . . . . . . . . . . . . . 201

5.3

Suppression of the SF-MI transition . . . . . . . . . . . . . . . . . . . 207
5.3.1

Nonequilibrium density distributions . . . . . . . . . . . . . . 208

5.3.2

Minimum lattice depth . . . . . . . . . . . . . . . . . . . . . . 209

5.3.3

Maximum superexchange rate . . . . . . . . . . . . . . . . . . 213
10

5.3.4Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . 214

5.4

Effect of holes and doublons . . . . . . . . . . . . . . . . . . . . . . . 216

5.5

Measurement of the superexchange rate

5.6

Spin-1/2 phase diagram with a tilt . . . . . . . . . . . . . . . . . . . 221. . . . . . . . . . . . . . . . 217

6 Supersolids and Superradiance

223

6.1

Supersolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6.2

Two-beam Superradiance . . . . . . . . . . . . . . . . . . . . . . . . . 228
6.2.1

Rayleigh scattering . . . . . . . . . . . . . . . . . . . . . . . . 230

6.2.2

Between the 𝐷1 and the 𝐷2 lines . . . . . . . . . . . . . . . . 230

6.2.3

Threshold power . . . . . . . . . . . . . . . . . . . . . . . . . 232

7 Outlook

235

7.1

Transport in anisotropic spin-1/2 Heisenberg chains . . . . . . . . . . 236

7.2

Adiabatic state preparation . . . . . . . . . . . . . . . . . . . . . . . 237

7.3

Other directions and machine improvements . . . . . . . . . . . . . . 240

A Supplement to Chapter 3243

A.1 Mean-field Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . 243
B Supplement to Chapter 4

247

B.1 Mapping to spin-1/2 Heisenberg model . . . . . . . . . . . . . . . . . 247
B.2 Six-site Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
C Observation of two-beam collective scattering phenomena in a BoseEinstein condensate

253

D Interaction spectroscopy of a two-component Mott insulator

261

E Enhanced Superexchange in a Tilted Mott Insulator

267

11

12

List of Figures
1-1 Feshbach resonances in 7 Li. . . . . . . . . . . . . . . . . . . . . . . .

35

2-1 The 7 Li machine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2-2 Oven and intermediate chambers. . . . . . . . . . . . . . . . . . . . .

44

2-3 Microchannel nozzle. . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

2-4 Fluorescence of the atomic beam after the microchannel nozzle. . . .

46

2-5 Atomic beam shutter. . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2-6 Home-made 800 MHz EOM. . . . . . . . . . . . . . . . . . . . . . . .

51

2-7 EOM response as a function of drive frequency. . . . . . . . . . . . .

52

2-8 Li 7 level diagram at zero external magnetic field . . . . . . . . . . .

56

2-9 Number of BEC atoms as a function of MOT fluorescence . . . . . .

57

2-10 CMOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

2-11 CMOT absorption images.

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

59

2-12 Gray Molasses with Λ-enhanced cooling. . . . . . . . . . . . . . . . .

60

2-13 Evaporation transition in the magnetic trap. . . . . . . . . . . . . . .

65

2-14 Crossed ODT potential. . . . . . . . . . . . . . . . . . . . . . . . . .

68

2-15 Magnetic fields suitable for ODT evaporation. . . . . . . . . . . . . .

71

2-16 Cuts along the 3D lattice potential. . . . . . . . . . . . . . . . . . . .

73

2-17 Choice of lattice wavelength.

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

75

2-18 Thermal lensing after an optical isolator. . . . . . . . . . . . . . . . .

80

2-19 Measurement of the RIN in the Nufern fiber amplifier. . . . . . . . .

84

2-20 RIN as a function of output power. . . . . . . . . . . . . . . . . . . .

85

2-21 High power optical system. . . . . . . . . . . . . . . . . . . . . . . . .

87

13

2-22 Rotating the angle of linear polarization with a galvo. . . . . . . . . .

89

2-23 Galvo implementation and performance. . . . . . . . . . . . . . . . .

902-24 AOM operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

2-25 Polarization rotation in an AOM. . . . . . . . . . . . . . . . . . . . .

92

2-26 Moveable mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

2-27 Picomotor mirrors used for lattice, ODT, and plug alignment. . . . .

96

2-28 Decreasing alignment sensitivity.

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

97

2-29 Beam focusing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

2-30 Lattice depth calibration by AM modulation.

. . . . . . . . . . . . . 101

2-31 Energy level structure of 7 Li in a magnetic field.

. . . . . . . . . . . 103

2-32 Effective level diagram at high fields for each 𝑚𝐼 sublevel.
2-33 High field imaging frequencies.

. . . . . . 104. . . . . . . . . . . . . . . . . . . . . 104

2-34 Admixtures of 𝑚𝐽 = +1/2 states in the lowest two hyperfine states. . 106
2-35 Differential magnetic moment of the lowest two hyperfine states.

. . 106

2-36 Titanium coating on bottom bucket window. . . . . . . . . . . . . . . 108
2-37 Bakeouts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3-1 Energy bands for a particle in a sinusoidal potential for three lattice
depths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3-2 Wannier functions at 11 𝐸𝑅 . a) Linear scale. b) Logarithmic scale.

. 113

3-3 Hubbard parameters as a function of lattice depth. . . . . . . . . . . 116
3-4 Mean-field Bose-Hubbard phase diagram.

. . . . . . . . . . . . . . . 117

3-5 Wedding cake structure of a MI in a 3D harmonic potential . . . . . . 118
3-6 Variation of the Hubbard parameters across the cloud. . . . . . . . . 119
3-7 Interaction spectroscopy for two particles per site. . . . . . . . . . . . 121
3-8 Interaction spectrum.

. . . . . . . . . . . . . . . . . . . . . . . . . . 122

3-9 Controlling condensate fraction with a gradient. . . . . . . . . . . . . 124
3-10 ODT to Lattice transfer.

. . . . . . . . . . . . . . . . . . . . . . . . 125

3-11 Lattice ramp into the MI. . . . . . . . . . . . . . . . . . . . . . . . . 127
3-12 Controlling the number of atoms in the MI shells. . . . . . . . . . . . 12914

3-13 Lifetime of the MI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3-14 Band gaps in a lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3-15 Laser intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3-16 Lifetime as a function of (added) white noise. . . . . . . . . . . . . . 136
3-17 Raman transitions between the lattice arms. . . . . . . . . . . . . . . 138
3-18 Lifetime as a function of starting BEC number. . . . . . . . . . . . . 139
3-19 Lifetimes of different Mott shells. . . . . . . . . . . . . . . . . . . . . 141
3-20 Feshbach resonances in 7 Li, as measured by interaction spectroscopy . 143
3-21 Two-body loss coefficient for 7 Li. . . . . . . . . . . . . . . . . . . . . 146
3-22 Lifetime of triply occupied sites. . . . . . . . . . . . . . . . . . . . . . 148
3-23 Lifetime of doubly-occupied sites. . . . . . . . . . . . . . . . . . . . . 150
3-24 Three-body loss coefficient.

. . . . . . . . . . . . . . . . . . . . . . . 151

3-25 Lifetime of |𝑎𝑏⟩ doublons at 846.8377 Gauss. . . . . . . . . . . . . . . 152
4-1 Superexchange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4-2 Energy spectrum of the Heisenberg model in a double-well. . . . . . . 163
4-3 Energy spectrum of the Heisenberg model for 6 sites. . . . . . . . . . 164
4-4 First three excited states of the ferromagnet.

. . . . . . . . . . . . . 165

4-5 XY phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4-6 The antiferromagnet.

. . . . . . . . . . . . . . . . . . . . . . . . . . 168

4-7 Exactly-solved spin-1 Hamiltonians . . . . . . . . . . . . . . . . . . . 171
4-8 Valence bond states. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4-9 Phase diagram of the Heisenberg spin-1 chain for variable anisotropy

173

4-10 Phase diagram of the spin-1 Heisenberg model with interchain coupling 174
4-11 Feshbach resonances in 7 Li. . . . . . . . . . . . . . . . . . . . . . . . 176
4-12 Spin-1/2 phase diagram for the Feshbach resonances of 7 Li. . . . . . . 178
4-13 Spin 1 phase diagram for 7 Li around 𝑢 = 0. . . . . . . . . . . . . . . . 179
4-14 Wannier Stark functions compared to Wannier functions. . . . . . . . 188
4-15 Tunneling and superexchange with and without a tilt. . . . . . . . . . 189
4-16 Spin Spiral preparation. . . . . . . . . . . . . . . . . . . . . . . . . . 192

15

4-17 Spin spiral on 5 sites realizing a spiral with a wavelength of 8 sites. . 195
5-1 Superexchange matrix element as a function of tilt. . . . . . . . . . . 198
5-2 Calibration of the magnetic field gradient . . . . . . . . . . . . . . . . 199
5-3 Interaction spectroscopy in an 𝑛 = 2 MI in the presence of a magnetic
field gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5-4 Shape of the tilt beam at the focus.

. . . . . . . . . . . . . . . . . . 202

5-5 Lattice potential with and without a tilt. . . . . . . . . . . . . . . . . 202
5-6 Tilt calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5-7 Characterization of the strength and inhomogeneity of the tilt acrossthe cloud. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5-8 Choice of tilt strength. . . . . . . . . . . . . . . . . . . . . . . . . . . 206
5-9 Equilibrium density distributions in the MI . . . . . . . . . . . . . . . 208
5-10 Suppression of the transition to the superfluid.

. . . . . . . . . . . . 209

5-11 Fraction of atoms in doubly-occupies sites as a function of lattice depth.210
5-12 Mott insulator breakdown. . . . . . . . . . . . . . . . . . . . . . . . . 211
5-13 Phase diagram showing the superfluid and Mott insulating regions . . 212
5-14 Zener breakdown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5-15 Maximum superexchange rate in 7 Li. . . . . . . . . . . . . . . . . . . 214
5-16 Implementation of the tilt in 2D. . . . . . . . . . . . . . . . . . . . . 215
5-17 Superexchange across a doublon. . . . . . . . . . . . . . . . . . . . . 216
5-18 Relaxation of the spin spiral. . . . . . . . . . . . . . . . . . . . . . . . 218
5-19 Measurement of the superexchange rate in a tilted lattice. . . . . . . 219
5-20 Relaxation time of the spin spiral as a function of tilt . . . . . . . . . 220
5-21 Effect of a tilt on the spin-1/2 phase diagram. . . . . . . . . . . . . . 221
5-22 Effect of changing the value of the tilt on the spin-1/2 phase diagram. 222
6-1 Roton minimum in the excitation spectrum. . . . . . . . . . . . . . . 225
6-2 Supersolid from two counter propagating laser beams incident on an
elongated BEC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
6-3 Difference between one beam and two beams above threshold power . 229
16

6-4 Rayleigh scattering rate. . . . . . . . . . . . . . . . . . . . . . . . . . 230
6-5 Raman superradiance and optical pumping. . . . . . . . . . . . . . . 231
6-6 Threshold power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
A-1 Obtaining the mean-field Bose-Hubbard phase diagram.

. . . . . . . 245

B-1 Heisenberg spin-1/2 Hamiltonian on 6 sites. . . . . . . . . . . . . . . 250

17

18

Chapter 1
Introduction
This chapter provides the main motivation and an overview of the scientific agenda of
the BEC 5 lab, implementing a 7 Li in optical lattices quantum simulator. Then, thisplatform is compared to other platforms used to study quantum spin Hamiltonians.
Finally, a brief overview of the history of BEC 5 since we started building it is given.

1.1

Quantum simulation of spin models

Quantum lattice spin Hamiltonians are paradigmatic models which can be used to
study many physical phenomena. The two most widely known models are the Isingand the Heisenberg model. In the Ising model, each spin on the lattice can be in one oftwo states, while in the Heisenberg model, the spin can be in any superposition of thetwo states. These models have various extensions. For example, the total spin does
not have to be limited to 1/2 but higher values can be used, the dimensionality can bevaried, external fields can be added, the interactions can have different range, disordercan be added, etc. Due to this versatility, they exhibit rich phase diagrams hosting
phases, such as ferromagnets and anti-ferromagnets, counterflow superfluids, topological phases, valence-bond states, frustrated states, spin liquids, etc. In addition, thesemodels provide a way of studying quantum phase transitions between phases with different symmetries and spontaneous symmetry breaking. Beyond ground states, thedynamics and transport in different regimes of the spin Hamiltonians are topics of19

recent research and debate. Although these models are quite simple to write down,
in many cases they are quite difficult to solve.
The problem stems from the exponentially growing configuration space with thenumber of interacting particles. For example, if a particle can be in either of two states
and the system has N interacting particles, the Hilbert space is 2𝑁 . Therefore, solving
these models could take prohibitively long time on existing classical computers. The
idea, proposed by Richard Feynman, is to use another quantum system to simulate thesystem under study. Since the properties of a system are described by a Hamiltonian,
if the Hamiltonian of one system can be mapped to the Hamiltonian of anothersystem, then the energy levels, correlations, and dynamics can also be studied by thisother system. This has inspired the field of quantum simulation, which has grown to
include various quantum systems, such as ultracold atoms and molecules, solid statesystems, and photonic systems.
The main scientific agenda of the BEC 5 lab falls in this broad category. In
particular, ultracold atoms in optical lattices can be used to simulate various models,
relevant for condensed matter physics, statistics, and high-energy physics. The focus
of BEC 5 is on simulating quantum spin Hamiltonians. The spin is mapped onto theinternal atomic state and the periodic potential is implemented by the optical lattice.
The advantage of such a mapping is that the atomic system provides a much wider
tunability of the Hamiltonian parameters, more direct access to physical quantities,
such as energy levels and spin correlations, and better control over the noise sources.
Therefore, compared to condensed matter systems, the system of atoms in opticallattices can be used to study spin Hamiltonians in much more detail and to accessnew regimes.
The wide tunability of interactions and of the timescale for dynamics provide
access to novel ground states and to the non-equilibrium dynamics in different regimesof the Hamiltonians. Adiabatic state preparation protocols of many-body states canbe implemented, allowing the study of both the feasibility of these protocols andthe properties of the new states. Then, quantum phase transitions and quenches
between states with different symmetries can be studied in detail. Also, ultracold
20

atoms in optical lattices provide the unique perspective of a system isolated fromthe environment, lacking a bath. The unitary evolution of such systems can giveinsight about eigenstate thermalization and prethermal states. Another avenue of
exploration are the transport properties of quantum systems in different regimes. For
example, with ultracold atoms in optical lattices, the transport properties of purestates close to zero temperature can be compared to those of statistical mixtures andhigh temperature states.
However, one needs to show that the model quantum mechanical system does
indeed reproduce the properties of the system under study, i.e. a benchmark is needed.
This is necessary because the mapping between the two systems is established only
under certain conditions and assumptions. This can be due either to the inherent
properties of the experimental system or to technical issues, such as heating and loss
processes, which become important after some evolution time.
Very often, this benchmark is a 1D system. In the context of spin systems, 1D
spin chains have been studied extensively theoretically (either by analytical methods
like the Bethe ansatz, or by numerical simulations), while much less can be calculated in 2D and 3D for systems of more than a few sites. Having the benchmark, one
is now free to explore higher dimensions or more complex 1D systems which show
qualitatively new features but cannot be easily computed. From another perspective, many 1D systems are quite different from higher dimensional ones. Quantum
fluctuations are more pronounced in lower dimensions and in 1D transport is quite
different because one cannot go around particles in the chain. This could result indifferent phases in different dimensions, different behavior of spin-spin correlations,etc. Therefore, often the study of the properties of 1D systems is an intriguing research agenda of its own. The ability to easily access and compare 1D, 2D, and 3D isanother advantage of the atoms in optical lattices platform. In addition to quantum
problems, this can address some classical systems since sometimes a correspondence
between d-dimensional quantum systems and (d+1)-dimensional classical systems canbe established.
In addition to dimensionality, temperature can be used to benchmark systems.
21

Sometimes, theoretical results are more easily obtained in the high-temperature limit
or in the zero-temperature limit but it is difficult to determine the behavior in between. With ultracold atoms in optical lattices, temperature can be controlled too,
exploring the properties of the system in different regimes.

1.1.1

Platforms for simulating spin models

This section compares the types of spin models and the tunability of their parametersthat can be simulated with the system of alkali ultracold atoms in optical lattices tothe ones with other platforms, such as dipolar atoms, molecules, and Rydberg atoms
in optical lattices or tweezer arrays, trapped ions, solid state systems, and photonic
systems.
With alkali ultracold atoms in optical lattices, due to the short-range interactions,
Heisenberg Hamiltonians with nearest-neighbor interactions can be simulated:
𝐻 = 𝐽𝑧∑︁𝑧
𝑆𝑖𝑧 𝑆𝑖+1
+ 𝐽𝑥𝑦

𝑖

∑︁ (︀
𝑖

𝑦
𝑥
𝑆𝑖𝑥 𝑆𝑖+1+ 𝑆𝑖𝑦 𝑆𝑖+1

)︀

(1.1)

Here 𝑆𝑖𝛼 are spin matrices on site 𝑖 for 𝛼 = 𝑥, 𝑦, 𝑧. Effective external magnetic fields∑︁∑︁of the form ℎ𝑧
𝑆𝑖𝑧 and ℎ𝑥 𝑆𝑖𝑥 can be added.
𝑖𝑖With alkali atoms, the effective spin Hamiltonian is derived from the Hubbard

model, which has two parameters: the tunneling matrix element 𝑡 and on-site interaction 𝑈 . It exhibits a phase transition from a superfluid with delocalized particles
to a Mott insulator with particles localized on each site. It is in the Mott-insulating
regime that the mapping from the Hubbard model to the Heisenberg model is valid.
The effective spin-spin interactions come from superexchange: the process of virtual
hopping and hopping back of particles on neighboring sites. It is a second-ordertunneling process and scales as 𝑡2 /𝑈 .
The Hamiltonian in Eq. (1.1) is the anisotropic Heisenberg model for a system withtotal spin S. In the system of ultracold alkali bosons in an optical lattice, different
total spin S can be engineered by, for example, varying the on-site occupation [10].
If two hyperfine states with long lifetimes are used, |𝑎⟩ and |𝑏⟩, a spin-1/2 model can
22be implemented in a Mott insulator with one particle per site by mapping:|↓⟩ = |𝑎⟩|↑⟩ = |𝑏⟩

(1.2)

A spin-1 model is implemented in a Mott insulator with two particles per site bymapping:|+1⟩ = |𝑏𝑏⟩

1
|0⟩ = √ (|𝑎𝑏⟩ + |𝑏𝑎⟩)2|−1⟩ = |𝑎𝑎⟩

(1.3)

Typically, three-body loss limits the implementation of schemes with more than two
particles per site.
Since the interaction energies between the two atomic states can be different, there
could be three types of superexchange terms: 𝑡2 /𝑈𝑎𝑎 , 𝑡2 /𝑈𝑎𝑏 , and 𝑡2 /𝑈𝑏𝑏 , assuming
the tunneling 𝑡 is not spin dependent. The Heienberg parameters 𝐽𝑧 , and 𝐽𝑥𝑦 arefunctions of all three terms [49]. They can be tuned by the lattice depth, whichchanges the barrier between neighboring sites, thus varying the tunneling 𝑡 between
them, and by Feshbach resonances, in which an externally applied magnetic field tunes
the two-body interactions 𝑈 over a very wide range by bringing the free atom energy
in resonance with a molecular bound state energy. For two spin states, the lattice
depth and the magnetic field tune the Heisenberg parameters in a non-trivial way,
which is explained in Chapter 4. The tunability depends on the particular structureof the Feshbach resonances for a given alkali species. Note that there are other ways
to tune the Heisenberg parameters, for example, by spin-dependent lattices.
Since heating and loss processes are always a problem, faster dynamics are preferable. For lithium the superexchange timescale is in a range between several hundred
Hz to 2 kHz, while for heavier atoms, such as Rb (which is commonly used in cold
atom experiments) it is up to a few hundred Hz. The figure of merit is the ratio
23between the lifetime of the Mott insulator and the superexchange rate 𝐽𝑥𝑦 /~, which
we address in Chapter 3. One of the inherent drawbacks of this scheme is that mass,
entropy, and energy transport happen on the faster timescale of first-order tunneling𝑡 while the time for spin-spin interactions is on the slower timescale of second-ordertunneling 𝑡2 /𝑈 . We address some of the resulting issues by applying a tilt to thelattice described in detail in Chapter 5.
There are other platforms used for quantum simulation of various many-bodysystems. Here the platforms are grouped in the following way: (i) atoms in opticallattices: (ia) alkali atoms; (ib) dipolar atoms; (ii) molecules: typically bi-alkali but
a growing variety; (iii) Rydberg atoms; (iv) trapped ions; (v) solid state systems:
superconducting circuits, quantum dots, NV centers; (vi) photonic systems. They
differ in the types of Hamiltonians, which can be simulated and the tunability ofthe parameters. For example, long-range (anisotropic) interactions can be modeledin systems dipolar atoms, molecules, trapped ions through the Coulomb interaction,and photonic systems. In some of the platforms involving optical lattices, tunneling is
not required if the interactions are long-range. Next, a brief overview of the features ofthese platforms is given. Although there are candidates for universal quantum simulator or a universal quantum computer, in most cases, these platforms are best suitedfor only a subset of many-body systems. Here the focus is on the spin Hamiltonians
that can be modeled in these systems.
(ib) Dipolar atoms. Atoms with large magnetic dipole moments, such as Dy,
Er, Cr, are trending platforms for quantum simulation. Here, the atoms are pinned
down in the lattice, with a density of one atom per site, and the effective spin-spin
interaction originate in the dipole-dipole interaction between pairs of atoms. Since
this interaction is long-range and anisotropic, the Heisenberg XXZ model can berealized. It has long-range coefficients, which depend on the distance between theinteracting spins. For example, in [124], the following Hamiltonian was realized:𝐻=∑︁𝑖,𝑗̸=𝑖𝐽𝑖𝑗[︂

1𝑆𝑖𝑧 𝑆𝑗𝑧 −2

24(︀ 𝑥 𝑥)︀
𝑆𝑖 𝑆𝑗 + 𝑆𝑖𝑦 𝑆𝑗𝑦

]︂

(1.4)

where𝐽𝑖𝑗 = 𝑉𝑑𝑑

1 − 3 cos2 (𝜃𝑖𝑗 )
3
4𝜋𝑟𝑖𝑗

(1.5)

is the dipole-dipole interaction between atoms at sites 𝑖 and 𝑗, 𝜃𝑖𝑗 is the angle between
the dipolar axis (set by the external magnetic field) and the interparticle axis. Here𝑉𝑑𝑑 is the matrix element between the corresponding atomic states which encode thespins. Variations and extensions of this Hamiltonian are possible. For example, the∑︁quadratic Zeeman shift gives a term
𝛿𝑖 (𝑆𝑖𝑧 )2 . Note that this Heisenberg model is𝑖anisotropic but the anisotropy is not directly varied.

The anisotropic interaction of dipolar gases enables other interesting studies, forexample collapse of ultracold dipolar gases and the study of supersolids. Recent
reviews of the general platform of atoms in optical lattices can be found in [64, 25,
59, 16].
(ii) Molecules. Ultracold polar molecules are a promising candidate for realizing
a larger variety of many-body phases due to their large electric dipole moment, which
is typically larger than the one of dipolar atoms. For heteronuclear molecules, since
the rovibrational ground state has no dipole moment, an external electric field (DCor AC) can be applied to mix opposite-parity rotational states and induce a dipole
moment in the lab frame. For reviews on using molecules for quantum simulation,see ([34, 114, 23]). Here we give a brief summary.
When the molecules are pinned in a lattice (with one molecule per site), quantum
spin Hamiltonians with long range interactions can be engineered. In many schemes
using diatomic polar molecules, the spin is encoded in the rotational degrees of freedom and the interactions are tuned by external microwave fields (e.g. [18, 112, 61]).
The dipole-dipole interaction here comes from the dipole moment associated withthe transition between the lowest two rotational states |𝑁 = 0⟩ → |𝑁 = 1⟩. For

example, in an applied DC electric field, the two pseudo-spins can be chosen to be
the eigenstate superpositions between the |𝑁 = 0⟩ and |𝑁 = 1, 𝑁𝑧 = 0⟩ states. The

25

resulting Hamiltonians is the anisotropic spin-1/2 XXZ Heisenberg model [23]:𝐻=∑︁𝑖,𝑗̸=𝑖𝐽𝑖𝑗𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 +∑︁𝑖,𝑗̸=𝑖(︀)︀
𝐽𝑖𝑗𝑥𝑦 𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦(1.6)

where the 𝐽𝑧 and 𝐽𝑥𝑦 parameters are proportional to the dipole-dipole interaction
between the spin states 𝑉𝑑𝑑 (similarly to the case of dipolar atoms Eq. (1.5)). They
can be tuned by external fields. Variations of the model, as well as other types ofspin encoding resulting in different Hamiltonians can also be simulated with polar
molecules, e.g. the realizing fractional Chern insulators has been proposed [162].
At the moment, the challenges with this platform involve producing ultracold
polar molecules in the desired state and loading them in a lattice with density of
one molecule per site. Various ultracold molecules prepared from two atomic species
combined into Feshbach molecules and then transferred to the rovibrational groundstate via STIRAP (Stimulated Raman adiabatic passage) have been realized. Directlaser cooling has also been employed for molecules such as SrF, which have almostclosed electronic transitions, making it suitable for laser cooling. For optical trapping,
it is important to match the polarizabilities of the two rotational states at the lattice
wavelength. So far, the densities have been low and preparing a lattice with one
molecule per site remains challenging, so that fillings of about 30 % have been realized.
At larger densities, for some molecules, chemical reactions between the molecules
become important, which could limit the lattice loading process. An alternative
approach is to load the atomic gases into the lattice (or into arrays of individual
traps) first, realizing a filling of one atom of each species per site, and then create
the molecules in the lattice. Of course, this approach is suitable only for molecules
whose constituent atoms can be easily cooled.
Ultracold polar molecules are also used to explore quantum chemistry, quantum
computation, and precision measurement.
(iii) Rydberg atoms. These are atoms in a highly-excited electronic state𝑛 > 50, typically alkali atoms. The dipole-dipole interactions between two Rydberg
atoms are very strong: at a distance of a few 𝜇m the van der Waals interaction can26be ∼ 10-100 MHz. Compared to this, the radiative and blackbody-limited lifetimes(∼ 100𝜇s) are long and it is rather the motion of the atoms when in the untrapped Rydberg state that limits the timescale available for experiments [22]. Although keeping
the trap off during the Rydberg-Rydberg interaction is a common strategy, implementing magic traps (which trap both the ground and the Rydberg state) would bea significant improvement. One of the appeals of Rydberg atoms are the long-range
interactions, which allow the use of arrays of individual traps separated by a large
enough distance to allow single-site imaging and addressing without the need of very
high resolution.
Arrays of Rydberg atoms have realized the 2D Quantum Ising model [97], Isingtype quantum phase transition [22, 81], and quench dynamics in a 2D Ising model[67]. For example, the Hamiltonian realized in [138] is:𝐻=∑︁

𝐽𝑖𝑗𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 + ℎ𝑧

𝑖,𝑗

∑︁

𝑆𝑖𝑧 + ℎ𝑥

𝑖

∑︁

𝑆𝑖𝑥

(1.7)

𝑖

where 𝐽𝑖𝑗𝑧 = 𝑉𝑖𝑗 /2 for 𝑉𝑖𝑗 ∼ 𝐶6 /𝑅6 at the distances used in that experiment, and ℎ𝑥,𝑧

are external fields, realizing the transverse-field Ising model, which has a quantum
phase transition. For much shorter distances 1/𝑅3 interactions could also be engineered. The interaction can be tuned either by varying the distance between theatoms or by coupling them to different Rydberg states.
In addition, for simulating many-body systems, Rydberg atoms are a promisingquantum computing platform. Here gates are based on the Rydberg blockade, inwhich only one atom can be excited in a certain area, because of the interaction shift
between a pair of Rydberg atoms. This can be used either to directly encode a qubit
per atom (with |0⟩ = |𝑔⟩, the ground state, and |1⟩ = |𝑟⟩, the Rydberg state), or an
ensemble of N atoms with one Rydberg excitation can be used as a qubit, enhancing√
the coupling but also the laser power required by 𝑁 .
Recent reviews of this platform can be found in [156, 133, 132, 22, 117].
(iv) Trapped ions. Trapped ions have high-fidelity gate operations, but scaling
to larger systems is not straightforward. Typically, a chain of laser-cooled ions in27

the sub-mK range is used (e.g. in a Paul trap), where the position of the ions isdetermined by the balance between the repulsive Coulomb force and the trapping
potential. The ions are in the Lamb-Dicke regime, where they are confined to space
smaller than the optical wavelength. The distance between the ions is on the orderof a few 𝜇m, large enough for the fluorescence of each ion to be observed individually
without high-resolution objectives.
The spin is encoded in the internal degrees of freedom. The collective motion of 𝑁
ions in the trap has 3 𝑁 normal modes, with resolved frequency differences, which areused to engineer effective spin-spin interactions. These are vibrational modes which
can be excited, for example, with pairs of laser beams with the exact frequency andmomentum difference, which act on all ions in the chain. When the exciting fields
are state-dependent, coupling the internal state to the external collective motion,effective spin-spin interactions can be engineered. For example, the transverse-field
Ising model has been implemented [164]:
𝐻=∑︁

𝐽𝑖𝑗𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧 + ℎ𝑥

𝑖<𝑗

∑︁

𝑆𝑖𝑥

(1.8)

𝑖

where the coupling is:𝐽=𝐽0|𝑗 − 𝑖|𝛼(1.9)

where 𝐽0 depends on the Rabi frequency and the coupling to the motional modes.
The the effective interactions are long-range, determined by the exponent 𝛼 which inmany schemes is 𝛼 = 3. In the scenario in [164], it can in principle be tuned in therange 0 < 𝛼 < 3, but in practice 0.5 < 𝛼 < 1.8 due to technical issues.
Trapped ions offer long-range interactions with single-site imaging and the optionof single-site addressing. However, scaling to higher dimensions or larger 1D systems
is one of the main challenges. Trapped ions are also major candidates for quantum
information processing and quantum computing. For reviews of the trapped ion
platform, see [24, 129, 43].
(v) Solid state. Here we focus on superconducting circuits. Reviews on solidstate systems used for quantum simulation and computation can be found in [44,2814, 84, 90]. Although the main focus in this field is on implementing a (universal)
quantum computer, spin Hamiltonians can be implemented in this platform too, withfocus on spin-1/2.
Superconducting circuits use LC circuits with a Josephson junction, which acts
as a nonlinear inductor. This makes the excitation level spacing non-uniform, so thatthe qubits (which can be used as effective spins) can be encoded in the bottom twolevels. It is worth noting that in contrast to other systems considered here, the superconducting qubits are macroscopic in size and lithographically defined. Therefore,
the properties of the qubits can be tuned. Several types of qubits are used with different circuit geometry and elements. The coupling between these qubits (realizing
the effective spin-spin interactions) is implemented either by direct capacitive or inductive coupling (resulting in nearest-neighbor spin-spin interactions) or by couplingthem to microwave resonators (i.e. a resonator bus). For example, transmon qubits
are capacitively coupled realizing a spin flip-flop interaction [90]:(︀)︀𝐻𝑖𝑗𝑐𝑎𝑝 = 𝐽𝑖𝑗𝑥𝑦 𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦(1.10)

where 𝐽𝑖𝑗𝑥𝑦 depends on the coupling capacitance, or, in the case of a resonator bus, on
the frequency detuning between the qubits and the resonator. For flux qubits [90]:𝐻𝑖𝑗𝑖𝑛𝑑 = 𝐽𝑖𝑗𝑧 𝑆𝑖𝑧 𝑆𝑗𝑧(1.11)

where 𝐽𝑖𝑗𝑧 can be tuned by the magnetic flux applied to an additional inductive coupling element. In both cases, which qubits are connected can be used as a design
parameter, forming graphs with different connectivities.
Recent examples of spin Hamiltonians realized with superconducting circuits include the XYZ Heisenberg model, which can be implemented in a digital way (withsuccessive application of XY, YZ, and XZ coupling gates) with superconducting transmon qubits [134]. However, this digital approach becomes less applicable for strong
interactions, comparable to the inverse time required for each gate. The transverse
field Ising model (with nearest neighbor interactions) was simulated on 8x8x8 D-Wave29

processor [70, 89]. However, the spin-flip term 𝐽𝑥𝑦 would be difficult to implementwith flux qubits (see e.g. [84]). Although superconducting circuits offer the flexibility
to engineer different geometries and the potential for scaling up, in the case of spin
Hamiltonians, the tunability of the parameters is limited.
(vi) Photonic systems. Photonic systems can also be used for quantum simulation [13, 71]. The effective photon-photon interactions can be engineered by anonlinear medium, e.g. atoms or solid-state crystals. For example, atoms in cavities
can mediate strong effective photon-photon interactions. Coupled cavities can realize
photonic networks which can be engineered to simulate lattice physics. For reviewsof these platforms, see ([13, 71, 151, 48]).
In summary, each platforms has it advantages and drawbacks and can be adapted
to simulate a specific set of systems. Alkali atoms in optical lattices can simulate
Heisenberg models with nearest-neighbor interactions. Compared to the platforms
which offer long-range interactions, such as dipolar gases, molecules, and trapped
ions, the production of this system is less involved and the methods are more advanced, maybe on par with the trapped ions platform. In lithium, the tunability of
the Heisenberg parameters with Feshbach resonances is probably larger than whatmost other systems can offer, but it is specific to the atomic species. The ability to
implement spin-1 systems with alkali atoms in a Mott insulator with two atoms persite is also a feature which, so far, has been implemented only in trapped ion systems[142].
In terms of scalability, the systems of alkali atoms in optical lattices involve alarge number of atoms but single-site imaging and addressing is more involved. By
contrast, in the systems of trapped ions, superconducting circuits, or Rydberg arrays,
for which single-site addressing is relatively straightforward, increasing the number ofparticles is a challenge. Single-site resolution and addressing has been implemented
in atoms in optical lattices by using quantum gas microscopes and the technology ismaturing [95, 64]. However, this is more involved and limits the dimensionality to 2D,
although there are prospects for imaging individual planes in a 3D system. Also, more
recently, atom arrays have been realized (in 1D, 2D, 3D) with AODs (acousto-optic
30

deflectors) and SLMs (spatial light modulators) [22, 97, 19], which allow for both
single-site imaging and addressing, so far used for Rydberg atoms. Combinations of
optical lattices (with single-site imaging) and tunable arrays could further increase
the versatility of the systems of cold atoms and molecules in optical lattices.
Finally, there are numerous tools and techniques which have been implemented
in these systems and here we list some of them. In addition to 3D cubic lattices, inwhich the effective dimensionality for dynamics can easily be changed by varying thelattice depth along each axis, other types of lattice geometries can be implemented.
For example, hexagonal, triangular, and Kagome lattices have also been realized[103, 79]. Also, external fields allow for the engineering of additional terms to the
Hamiltonian. Coupling the two spin states with microwaves implements a tunable ℎ𝑥
field in the spin-1/2 Heisenberg model. The addition of an AC Stark shift gradientvia far-detuned optical beam can realize a tilt in the lattice potential. The additionof a magnetic field gradient can do the same. However, if the two atomic states
realizing the spin states have different magnetic moments, the magnetic field gradientcan separate the two spins in space (if they are allowed to move) or a spin pattern canbe formed (when they are pinned) due to the different Larmor precession frequencies
at different magnetic fields. Adding Raman beams to a tilted lattice can implement
laser-assisted tunneling realizing gauge fields with neutral atoms. Combinations ofthese elements can implement more complex Hamiltonians.

1.1.2

The 7 Li platform

A platform of 7 Li in optical lattices for the study of quantum spin models is particularly appealing for two reasons. First, lithium has relatively easily accessible Feshbachresonances which allow for the tunability of the interactions. Second, the light massof lithium makes the timescale for dynamics fast, which can be beneficial in studying
long-time dynamics in the heating and loss limited lifetimes of the system.
The properties and features of the 7 Li system can be understood by considering
the interactions between the particles at ultralow temperatures. In the following,a brief overview of the two-body elastic scattering is given and the implications for31

lithium atoms are presented.
Scattering length
At very low temperatures, the elastic scattering between two particles is describedby a single parameter: the scattering length 𝑎. Consider two particles with the same
mass in free space and at ultracold temperatures. Going to center of mass coordinates,
the problem reduces to finding solutions to the Schrodinger equation for the relative
motion between the two particles (for a review, see [41]):
(︂

)︂
p2𝑟
+ 𝑉 (r) Φk (r) = 𝐸𝑘 Φk (r)2𝑚𝑟(1.12)

where pr = (p1 − p2 )/2, r = r1 − r2 , 𝑚𝑟 = 𝑚/2, and 𝑉 (r) is the interaction potential.

Typically, one looks for solutions with 𝐸𝑘 = ~2 𝑘 2 /(2𝑚𝑟 ). For 𝑘 ≪ 1/𝑏, where 𝑏 is thecharacteristic range of the interaction potential, the energy is low enough, so that the
scattering is isotropic. At distances 𝑟 ≫ 𝑏, the wavefunction can be written as:
Φk (r) ∼ 𝑒𝑖kr + 𝑓 (𝑘)𝑒𝑖𝑘𝑟𝑟(1.13)

where the first term is the incident wave and the second is the scattered wave. For
spherically symmetric potentials, the problem simplifies and an expansion in partial
waves is performed. At ultralow temperatures, only the s-wave (𝑙 = 0) contributes,
in the following sense. For 𝑙 = 0 the interaction term in the Hamiltonian is just theinteraction potential, while for 𝑙 ̸= 0, there is also a centrifugal term ~2 𝑙(𝑙+1)/(2𝑚𝑟2 ),which forms a centrifugal barrier. If the energy of the particles is low enough thecentrifugal barrier results in just a reflection of the incident wave.
For an incident plane wave along 𝑧, the expansion in partial waves gives the32following form for the incident and scattered waves [41]:Φ𝑘,0 (r) = 𝑒

𝑖𝑘𝑧

∞
(︀
)︀
1 ∑︁
∼
(2𝑙 + 1)𝑃𝑙 (cos 𝜃) (−1)𝑙+1 𝑒−𝑖𝑘𝑟 + 𝑒𝑖𝑘𝑟
𝑖2𝑘𝑟 𝑙=0

∞
)︀
(︀
1 ∑︁
(2𝑙 + 1)𝑃𝑙 (cos 𝜃) (−1)𝑙+1 𝑒−𝑖𝑘𝑟 + 𝑒𝑖2𝛿𝑙 𝑒𝑖𝑘𝑟
Φ𝑘,𝑓 (r) ∼𝑖2𝑘𝑟 𝑙=0(1.14)(1.15)The solution of the problem amounts to finding the phase shift 𝛿𝑙 (𝑘) of the scatteredwave. The phase shift can be expressed in terms of a scattering length, which is usedas the parameter which describes interactions at ultralow temperatures:tan 𝛿0 (𝑘)𝑘→0𝑘(1.16)

𝑎 = − lim

For alkali atoms, the potential can be described by a truncated van der Waals potential[41]:

𝑉 (𝑟) =

which gives the scattering length:⎧⎪⎨− 𝐶66⎪⎩∞(︂

𝑎 = 0.478𝑎𝑐 1 − tan
1/4

where 𝑎𝑐 = (2𝑚𝑟 𝐶6 /~2 )𝑟 > 𝑟𝑐𝑟(1.17)

𝑟 < 𝑟𝑐

(︂

3𝜋
𝑎2𝑐
−
2𝑟𝑐
8

)︂)︂

(1.18). Often, for simplicity, a contact interaction is used:
𝑉 (r2 − r1 ) = 𝑔𝛿(r2 − r1 )(1.19)

where 𝑔 is the interaction strength. This expression is treated analytically by usinga pseudo-potential:
𝑉 (𝑟) = 𝑔𝛿(𝑟)

33

𝜕
𝑟
𝜕𝑟

(1.20)

which simplifies the results for the scattering length and the scattering cross section:
𝑔𝑚
4𝜋~2
8𝜋𝑎2
𝜎(𝑘) =
1 + (𝑘𝑎)2𝑎=(1.21)(1.22)The pseudo potential result is a good approximation for the scattering cross section and its variation with incident energy (determined by 𝑘) for repulsive interactions (𝑎 > 0), but not so much for attractive interactions (𝑎 < 0) [41]. The scattering
length of the |𝐹, 𝑚𝐹 ⟩ = |2, 2⟩ state for 7 Li is 𝑎 = − 27 𝑎0 at zero magnetic field andzero temperature [5]. The variation of the scattering cross section (and therefore of

the scattering length) with temperature (incident energy) is typically not considered
in cold atom experiments because the zero-field interactions are typically repulsive.
However, for a few cases including lithium, there is a zero-crossing of the scattering
length for increasing temperature, not captured by the pseudo-potential approximation. For lithium, this happens to be at around 8 mK [65, 41], which is a little bit
above the MOT temperature of 1-2 mK. Therefore, further laser cooling is required
before evaporative cooling, which relies on a large elastic scattering rates, can be used.
The atomic potentials are typically not known well enough to predict the scattering length. In cold atom experiments it is relatively straightforward to directly
measure the scattering length either spectroscopically (e.g. by photoassociation orby amplitude modulation in a Mott insulator) or by measuring the Thomas-Fermi
cloud size in a harmonic potential of known trap frequency. The scattering lengthcharacterizes the low-temperature properties of the particles. Particles at low enoughtemperatures which interact with the same scattering length, will behave in the sameway, even if the details of the interaction potentials are different.

34

Feshbach resonances
The scattering length can be tuned by varying the magnetic field near a Feshbachresonance, which to a good approximation is described by (for a review see [36]):(︂𝑎(𝐵) = 𝑎𝑏𝑔 1 −∆𝐵 − 𝐵0)︂(1.23)

where 𝐵0 is the position of the resonance and ∆ is the resonance width. Feshbachresonances originate from the proximity of molecular bound states of the atomic
hyperfine levels at a given magnetic field. When the energy of a pair of free atoms
equals the energy of the least bound state, a resonance in the two-body interactions
occurs. This resonance alters the atomic scattering length for pairwise interactions
but also leads to three-body recombination in which two atoms form a molecule andthe third atom gets the extra kinetic energy (this process is discussed in Section 3.2.4).
We use Feshbach resonances to produce a BEC (as explained in Chapter 2), to tune
the size of the Mott insulator plateaus, and to vary the interactions between thelowest two hyperfine states in the Mott insulator in order to realize different spin
Hamiltonians.
Scattering length (a 0)

400
aa
ab
bb

200
0
-200
-400
600

650

700

750

800

850

900

950

1000

Magnetic field (Gauss)

Figure 1-1: Feshbach resonances in 7 Li. Shown are the interactions in the lowest two hyperfine states at high field: |𝑎⟩ = |𝑚𝐼 = +3/2, 𝑚𝐽 = −1/2⟩ and |𝑏⟩ = |𝑚𝐼 = +1/2, 𝑚𝐽 = −1/2⟩.
The scattering lengths are based on hyperbolic fits to the interaction spectroscopy data from[11].

The Feshbach resonances in the lowest two hyperfine states are shown in Fig. 1-1.
A feature of this system is that for many magnetic fields, the interactions are negative.
This makes it possible to implement anti-ferromagnetic coupling with bosons, because35the signs of the Heisenberg parameters can be changed. Applying a tilt ∆ to thelattice with ∆ > 𝑈 , can also change the sign of the Heisenberg parameters [152, 45].
This allows for the realization of both the full ferromagnetic and anti-ferromagnetic
phase diagram of the Heisenberg Hamiltonian. This opens the possibility of adiabatic
state preparation of magnetically ordered states and the study of topological states,of phase transitions between states with different symmetries, and of non-equilibrium
spin dynamics in different Hamiltonian regimes.

1.2

The history of BEC 5

Building and developing the Li 7 machine went through many different stages andhurdles. We started from an almost-renovated room in the Fall of 2010. The initial
vacuum and laser cooling setup was complete in the Fall of 2011. The next few months
were spent on measuring the atomic flux at different stages of the experiment andoptimizing the Zeeman slower, which proved difficult. The resolution was to increase
the oven temperature to 100𝑜 higher than the design, wrap a guiding field on theflanges just before the slower coils in order to overcome the magnetic field of the ion
pump, and to improve the polarization of the Zeeman slower beam. Finally, we madethe first MOT in April 2012. This was one of the most magical moments in the lab
when things worked on the first try! Michael and I read the textbook description ofa MOT in the Foot book in the morning and had a MOT by the evening. It took
several months to make it stable.
The next problem was to get a 7 Li BEC. Initial attempts of RF evaporation failed
until we improved the laser cooling by implementing a Gray Molasses. During thistime, we had another big fallback, when we realized that the brass rods holding
the coils together were magnetic, so we had to redo much of the magnetic coils
setup. After some work on the IGBT H-bridge setup which switches the coils from
Helmholtz to anti-Helmholtz configuration, implementing either a constant bias field
or a quadrupole trap (used in the evaporation), we achieved a BEC in December 2013.
Although quite unstable initially, we slowly fixed various issues around the machine36and could produce a BEC at every run every day withing the next year. We also
started implementing a 3D optical lattice.
The main problem which haunted us for years was the short lifetime of the atoms
in the lattice. We looked at various heating sources and eliminated many of them,
improving the stability of the machine at the same time. Finally, we shortened theoutput fiber of our laser fiber amplifier (a 50 W Nufern NUAmp) which produces thelight for the lattice, eliminating Stimulated Brillouin scattering. This gave us long
lifetimes in December 2015 (∼ 1 s for 𝑛 = 1 MI). A lot of our efforts were hinderedby the unreliable fiber amplifier which needed to be sent for repair a few times.
Although we had observed these long lifetimes, it took several more months to makethe machine stable and the long lifetimes repeatable. In the summer of 2016 we had toface another major problem: the bottom bucket viewport was coated with Titanium
from the Titanium sublimation pump, which is needed to achieve ultra high vacuum.
A mistake in the initial vacuum design in 2011, lead to the Titanium filament being
too long. Although we could work around the ensuing thermal lensing in one of thelattice arms, the viewport became unusable in the summer of 2016 after we put muchhigher intensity in that lattice beam.
We diverted our efforts from lattice physics for the next year, looking into supersolids from Rayleigh scattering. This was a very exciting moment: this was our firstphysics project [46]! It was completed just before some of the main equipment failed:the borrowed TiSaph laser, instrumental for this project, had to be sent for repair,and one of the Zeeman slower MOSFETS, needed to make a MOT, failed.
In June-July 2017 we replaced the bottom bucket window, which required analmost complete rebuilt of the machine. We used this opportunity to float the table(by replacing the legs of the table with ones with controlled compressed air), andreplaced the Nufern with an ALS laser, which seemed to have better laser intensity
stability. Finally, in the Fall of 2017 we had long Mott insulator lifetimes! We also
abandoned the figure of merit we had used so far to measure these lifetimes: theremaining condensate fraction after ramping the lattice off. We instead began using
a more direct measurement, the lifetime of the number of 𝑛 = 1 or 𝑛 = 2 atoms in the37

Mott insulator for spin-1/2 or spin-1 physics respectively.
The next big setback was that the lattice depth we measured changed by morethan 50 % over the course of a few hours. This made it very time-consuming todiagnose. After a few months, we realized that the seed laser for the lattice light, theMephisto, was operating in a multi-mode fashion. A brief change in the temperatureof the laser crystal fixed the problem. Since this was the most stable and reliable
piece of the lattice setup and we had measured its lineshape in the past, we did notsuspect it until the very end. With that, in May 2018, we had a machine capable ofproducing stable, long-lived 7 Li Mott insulators and our studies of spin physics finallybegan!
We first looked into a technique: interaction spectroscopy [11], with which tomanipulate and detect lattice sites with different occupations in the Mott insulator.
This allows us to distinguish also between the different 𝑛 = 2 configurations (|𝑎𝑎⟩, |𝑎𝑏⟩,and |𝑏𝑏⟩). Our plan was to study spin-1 physics, because of the topological Haldane
phase, which we can potentially implement. However, in December 2018, we realizedthat the three-body loss which necessarily accompanies superexchange physics in
𝑛 = 2 could be a setback. We then turned to one particle per site and spin-1/2.
We first implemented a method for studying superexchange with several advantages[45]. By tilting the lattice, first-order tunneling is suppressed, leaving superexchange
to dominate the dynamics. This can lead to the suppression of the transition toa superfluid, thus speeding up the dynamics. We then measured superexchangedriven dynamics for the first time in our system by preparing a non-equilibrium
spin distribution. We studied the effect of the anisotropy 𝐽𝑧 /𝐽𝑥𝑦 the superexchangedriven transport, which we are currently finalizing. Next, the long-standing goal of
adiabatically preparing a many-body ground state (in this case the XY ferromagnet)
seems withing reach.

38

1.3

OutlineThis thesis is organized as follows: Chapter 2 describes the main aspects of building
the 7 Li machine. Chapter 3 looks into ultracold atoms in optical lattices and themethods for preparing and characterizing a Mott insulator with a long lifetime andfewer defects. Chapter 4 describes the spin Hamiltonians which can be implemented
in the system of 7 Li in an optical lattice. In Chapter 5, the experimental details
and outlook of using tilted lattices are presented. Chapter 6 showcases the versatilityof the BEC 5 machine by describing an experiment which does not involve opticallattices but which is also focused on exploring novel quantum phases. Chapter 7 gives
an outlook.

39

40

Chapter 2
The 7Li machineThis chapter describes each of the main steps and components that have been used in
the BEC 5 lab for producing a Mott insulator in an optical lattice, the starting point
for implementing spin Hamiltonians. It is divided in two parts. First, the sequence ofstages required for making a 7 Li Bose-Einstein condensate (BEC) and transferring it
to a 3D optical lattice is presented. Second, a few methods in designing and building
some aspects of the machine are detailed.

2.1

The road to a 7Li Mott insulator

The desired starting point of our experiments is a cold, low-entropy Mott insulator(MI). To prepare it, we adiabatically load a BEC from a crossed optical dipole trap
(ODT) into a 3D optical lattice. Starting with a BEC with little or no thermal fraction, the lowest momentum state of the ground band of the optical lattice can beloaded. Therefore, we first need to produce a 7 Li BEC. There are two main difficulties, inherent in the elastic scattering properties of lithium. First, the scattering
length has a temperature dependence and happens to have a zero-crossing aroundthe temperature of the Magneto-optical trap (MOT) of ∼ 1-2 mK. This means thatthe elastic scattering rate goes to zero at that point, preventing evaporative cooling,which relies on elastic collisions for thermalization. In the vicinity of the zero-crossing,evaporation can be performed, but it requires a very long time. The way to over41

come this issue is to use sub-Doppler cooling schemes, so that the zero-temperaturescattering length can be recovered. Second, the scattering length at low temperatures and zero magnetic field is negative 𝑎 = −27 𝑎0 for the highest-energy hyperfine

state |𝐹, 𝑚𝐹 ⟩ = |2, 2⟩, typically used for laser cooling [5]. This means that the BEC
is unstable against collapse unless very small atom numbers are used. The way to

overcome this is to use the Feshbach resonance of the lowest-energy hyperfine stateat large magnetic fields |𝑚𝐼 , 𝑚𝐽 ⟩ = |+3/2, −1/2⟩ and to bring the scattering length
to around 100 𝑎0 , where efficient evaporation can be performed.

When we started building this machine, only a few groups had produced 7 Li BECs:(i) the Hulet lab in Rice, where initially the evaporation took very long time sinceno sub-Doppler cooling methods were used [149]; (ii) the Salomon group in France,
where 6 Li has been used in the evaporative cooling of 7 Li due to the larger scatteringcross section of +38 𝑎0 between in the |𝐹, 𝑚𝐹 ⟩ = |1/2, −1/2⟩ state of 6 Li and the 7 Li
|1, −1⟩ state of 7 Li [140]; (iii) the Khaykovich group in Israel (where a BEC also in

the |𝑚𝐼 , 𝑚𝐽 ⟩ =|+1/2, −1/2⟩ (our 𝑏-state) has been all-optically produced [65]). Since

then, the group of our postdoc David Weld made a 7 Li BEC in UCSB and the groupof Jae-yoon Choi in Korea [88].
Our implementation of a 7 Li BEC has several defining features. First, in orderto overcome the effects of the zero-crossing of the scattering length, a sub-Doppler
cooling scheme is used. We implement the 𝐷1 -line Gray Molasses (GM), following
its first realization in 40 K [54] and then the Λ-enhanced version in 7 Li [63]. Until
recently, such schemes had not been implemented in lithium because of the very
closely spaced hyperfine levels in the 2𝑃3/2 state. The Gray Molasses scheme isimplemented using the 2𝑃1/2 state. Second, a hybrid approach is used for evaporation,in which the evaporation is performed almost to degeneracy in a magnetic trap in
the highest hyperfine Zeeman level |𝐹, 𝑚𝐹 ⟩ = |2, 2⟩ and then the final evaporationis performed in a crossed optical dipole trap (ODT) in the lowest-energy hypefinestate |𝑚𝐼 , 𝑚𝐽 ⟩ = |+3/2, −1/2⟩ at high fields. There are several considerations forthis approach. At low fields, the lowest hyperfine state |𝐹, 𝑚𝐹 ⟩ = |1, 1⟩ also has anegative scattering length 𝑎 = −5 𝑎0 but it is too small for evaporation. This state is42also not magnetically trappable, so an optical dipole trap has to be used. Although
all-optical production of 7 Li BECs is possible, smaller clouds are produced due tothe smaller mode volume of the trap. It also requires the use of very high power
lasers (> 100 W). Instead, it is straightforward to optically pump the atoms in the|𝐹, 𝑚𝐹 ⟩ = |2, 2⟩ after laser cooling. Just before degeneracy is reached, the atoms
are transferred to an ODT, where a Landau-Zener sweep is performed transferring

the atoms to the |𝐹, 𝑚𝐹 ⟩ = |1, 1⟩ state, which, as the field is increased, becomes the|𝑚𝐼 , 𝑚𝐽 ⟩ = |+3/2, −1/2⟩ state and a BEC is produced in that state. This approachhas the added advantage that the evaporation in the magnetic trap is fast since thestarting densities in the trap are large due to the large magnetic field gradients. We
reuse the magnetic field coils required to create the 800 Gauss Feshbach fields formagnetic trapping using an H-bridge and switching the current in one of the coils, sothat we can obtain a tight magnetic trap.

Figure 2-1: The 7 Li machine. From left to right: oven, differential pumping, Zeeman slower,
main chamber. The optical breadboards around the main chamber and the shim coils are
SOLIDWORKS
Educational Product. For Instructional Use Only.
also shown.

An overview of the machine is shown in Fig. 2-1. It has a single-chamber design
which requires no transport, but limits optical access. In the following, we outline
43

the steps required to realize a Mott insulator in an optical lattice.

2.1.1

Oven and intermediate chambers

The first part of the machine prepares the atomic beam. This is shown in Fig. 2-2.
We use an effusive oven, followed by a nozzle which selects only atoms with velocity
towards the main chamber. This is followed by a cup which was intended to collect
the atoms with large transverse velocities with respect to the atomic beam. Then
there are two differential pumping tubes which keep the pressure difference between
the oven chamber (higher pressure) and the main chamber to be at least two orders ofmagnitude. There is an atomic beam shutter which blocks the atoms after the initialloading so that they do not interfere with the experiments in the main chamber.
Finally, there are two gate valves, which separate the main chamber from the ovenand intermediate chamber (housing the shutter). We used two MDC metal valves(GV-1500M-P) instead of one because these are known to fail. An alternative would
have been a VAT valve (VAT Series 48), which is more expensive.
ion gauges
atomic beam
shuttercold cup
reducernozzlegate valveselbow
Zeeman slower
cup
turbo pumpdifferential pumping tubes
ion pumps (not shown)

Figure 2-2: Oven and intermediate chambers.

44

Atomic beam source
The atomic beam source is an effusive oven, typical for cold atom experiments.
Lithium rods from Sigma-Alrdich or ESPI Metals (99.9% pure from trace metals)
are heated up to 485 o C, resulting in a density of 3 × 1013 /cm3 [9]. The first aperturein the path of the atoms is a nozzle. We have used two types of nozzles: a circular

aperture and microchannel tubes. The circular aperture has a diameter of 6.5 mm andtogether with the aperture of the cold cup and the differential pumping tubes forms
the atomic beam which enters the Zeeman slower. The cold cup is a copper structure which is supposed to capture lithium atoms coming out of the nozzle with largetransverse velocities. However, with the large circular aperture nozzle, the lithium
atoms bounce off the cup and accumulate at the reducer right after the nozzle. This
caused a leak, as Lithium corroded the copper gasket.
The microchannel nozzle is composed of tubes with length to width ratio of 10. A
picture of it is shown in Fig. 2-3. This design allows for the atomic beam to be collimated already at the nozzle, significantly improving the lifetime of the oven, because
atoms with large transverse velocities are not lost. The current nozzle (installed in
February 2017).

Figure 2-3: Microchannel nozzle.

Viewports and atomic flux measurement
There are four viewports arranged in two pairs, one on each side of the atomic beam.
They can be used for monitoring and measuring the atomic flux by scanning the laser
frequency and measuring the absorption spectrum. The first place where the atomic
45

beam can be detected is right after the nozzle in front of the cold cup. Here the full
Doppler width of the atomic beam exiting the nozzle can be measured with a large
circular aperture used as a nozzle. The fluorescence of the atomic beam as seen fromthe closer of the two viewports is shown in Fig. 2-4. The flux of atoms here is about
1−5 × 1015 atoms/s. The second pair of viewports is in the intermediate chamber.

Here the beam has been collimated by the cold cup and the first differential pumping
tube. Finally, the flux in the main chamber can be measured and is about 1−5 × 1012
atoms/s, although the signal can be pretty weak.

Figure 2-4: Fluorescence of the atomic beam after the microchannel nozzle. The edge of the
viewport is seen on the left and the edge of the cold cup on the right.

Differential pumping
The oven and the main chamber are separated by two differential pumping stages.
These are tubes with small conductance so that a pressure difference is maintained
between the two chambers which each of them connects. Here the mean free path islarger than the tube diameter, resulting in the regime of molecular flow. Consider
the throughput 𝑄˙ defined as (see e.g. [101]):𝑑(𝑃 𝑉 )
𝑑𝑉
𝑄˙ =
=𝑃
= 𝑃𝑆
𝑑𝑡
𝑑𝑡

(2.1)

where 𝑃 is the pressure, 𝑉 is the volume and 𝑆 is the pump speed. A pipe restricts theflow rate and decreases the pumping speed. The pipe can be modeled as a resistive46

element in an electrical circuit with conductance:𝐶=𝑄˙
𝑃 𝑢 − 𝑃𝑑

(2.2)

where 𝑃𝑢 , 𝑃𝑑 are the upstream and the downstream pressures respectively. If thepressure of the first chamber is 𝑃1 and the pump speed in the second chamber is 𝑆2 ,the pressure in the second chamber is 𝑃2 = 𝑃1 𝐶/𝑆2 , so that the pressure difference
is:
𝑃1
𝐶
=
𝑃2
𝑆2

(2.3)

Here we have ignored the base pressure in the second chamber assuming it is much
lower than the gas leak from the first chamber. Therefore, if the conductance is madesmall, the pressure difference can be high. The conductance 𝐶 ∝ 𝐷3 /𝐿 where 𝐷 is theinside tube diameter and 𝐿 is the length. The differential pumping tubes we use have

(𝐷, 𝐿) = (0.180, 4.5) inches (the one closer to the oven) and (𝐷, 𝐿) = (0.375, 5) inches(the one closer to the main chamber). The angle of the first tube can be adjusted,following the design of Aviv Keshet [85] and the second tube is welded into a disk
welded perpendicularly to the flange.
Together with the slower, this gives 3 orders of magnitude difference in pressure
between the main chamber and the oven, when the oven is off, and 2 orders of magnitude when the oven is on. The pressure in the main chamber is about 2×10−11 Torrand in the oven the pressure is 1.2×10−8 Torr when the oven is off. When the oven ison, the pressure there rises to 1.1×10−7 Torr, while the pressure in the main chamber
remains roughly the same.
Ion pumps and ion gauges
To measure the pressure, two ion gauges are used, one in the oven chamber and one
in the intermediate chamber (see Fig. 2-2). They are the UHV-24 9715007 ThoriaIridium filament gauges. They are less accurate, but exhibit less outgassing comparedto Tungsten filaments. The ion pumps are 40 l/s from Varian/Agilent.

47

Atomic beam shuttera)b)Figure 2-5: Atomic beam shutter. The solenoid is on the left, with a red rubber tip to avoid
noises from scratching the flange. The bellows holding the shutter inside the chamber is on
the right. a) Shutter closed. b) Shutter open.

The atomic beam shutter is a metal plate which blocks the atomic beam, as shown
schematically in Fig. 2-1. The metal plate is attached by a rod to the top flange of aflexible bellows. A solenoid pushes the top flange of the bellows so that the shutter
plate moves in an out of the atomic beam. A picture is shown in Fig. 2-5.

2.1.2

Zeeman slower

Operation
Exiting the differential pumping tubes is a collimated atomic beam moving with velocity 𝑣𝑧 ≫ 𝑣𝑥 , 𝑣𝑦 , where the speed is a Maxwell-Boltzmann distribution as a functionof the temperature of the oven. In the Zeeman slowing stage, some of the atoms are
slowed down to or below the target velocity of 30 m/s, so that they can be captured
into a Magneto-optical trap (MOT). A beam propagating in a direction opposite theatomic beam and resonant with the Doppler-shifted atomic frequency is scattered by
the atoms. At each scattering event, the absorbed photon imparts a momentum −~𝑘,where 𝑘 is the wavevector of the light and the +𝑧 direction is chosen along the atomic

beam. Since the photon is emitted in a random direction, on average, the atomic
momentum is decreased by ~𝑘. As the atoms slow down, they go out of resonancewith the light. To remedy this, a guiding bias field is used which decreases according48

to the atomic velocity decrease as a function of 𝑧. The bias magnetic field induces a
Zeeman shift on the resonant atomic frequency, which compensates the Doppler shift.
Energy balance gives:
𝜔0 + 𝜇˜𝐵 = 𝑘𝑣 + 𝜔(2.4)

where 𝜇˜ = 𝜇𝐵 𝑔𝐹 𝑚𝐹 /~ = 2𝜋 × 1.4 MHz/Gauss for the |𝐹, 𝑚𝐹 ⟩ = |2, 2⟩ state, 𝜔 is thefrequency of the light, and 𝜔0 is the atomic resonant frequency. This gives the velocity
which is resonant with the light at each field along the Zeeman slower.
Setup∙ Magnetic field 𝐵(𝑧):𝐵(𝑧 = 0) = 600 Gauss
𝐵(𝑧 = 1 m) = −400 Gauss(2.5)

This constitutes a spin-flip Zeeman slower with a zero-crossing 𝐵 = 0.
∙ Light. (i) Frequency: In principle, the cycling transition on the 𝐷2 line is used:|𝐹 = 2, 𝑚𝐹 = 2⟩ ↔ |𝐹 ′ = 3, 𝑚′𝐹 = 3⟩. The level structure of 7 Li at zero bias
magnetic field is shown in Fig. 2-8. Since atoms can be optically pumped into
the |𝐹 = 1⟩ where they no longer interact with the slowing light, a repumping

beam targeting the |𝐹 = 1⟩ state is also used. Note that the excited state

hyperfine splittings are small (a few MHz), so that they cannot be resolved
with the lasers we use (which have a linewidth of around 1 MHz). The light isdetuned from the primary transitions 𝑓𝐹 2 and 𝑓𝐹 1 by:
𝑓2 = 𝑓𝐹 2 − 600 MHz
𝑓1 = 𝑓𝐹 1 − 600 MHz = 𝑓2 − 803.5 MHz(2.6)(ii) Intensity: 80 mW into a beam of 6.5 mm 1/𝑒2 radius. Note that the Zeeman
slower was initially operated at 40 mW but deposits on the slower window have49required the increase in power to ensure repeatable operation. Assuming 40 mW,
the intensity is 𝐼 = 60 mW/cm2 . Given the saturation intensity
𝐼sat =~Γ𝜔03= 2.55 mW/cm2
12𝜋𝑐2

(2.7)

where Γ = 2𝜋 × 5.9 MHz is the natural linewidth of the excited state, 𝜔0 is theatomic resonant frequency, the saturation parameter of the Zeeman slower beam
is:
𝑠=

𝐼
𝐼sat

= 24

(2.8)

Note that the intensity varies a little bit as a function of propagation distance.
The beam is almost focused onto the nozzle and is a bit clipped by the 1 inch
diameter Zeeman slower tube. In practice, the focusing is determined experimentally, by maximizing the initial loading rate into the MOT, which is predominantly determined by the cold atom flux.
(iii) Polarization. The light is circularly polarized so that the cycling transition
is used.
* 1 Guiding field. Wire is would around the vacuum chamber of the differentialpumping tubes just before the Zeeman slower. This guiding field is used todefine the quantization axis for the atomic beam entering the slower, because
the magnetic field of the ion pumps in the vicinity of the atomic beam is notaligned with the magnetic field of the Zeeman slower. For optical pumping, themagnetic field has to be parallel to the direction of propagation of the circularly
polarized laser beam.
* Compensation field. One of the shim coils (the one on the side of the chamber

towards the Zeeman slower) is used to cancel the fringing field of the final partof the Zeeman slower at the position of the MOT. We pick the value of this fieldby making the position of the atoms in the MOT same whether the Zeeman

1
In the following, we label the typically required components with a ∙ and the ones specific to
our setup with a *.

50

slower is on or off.
With this setup, the Zeeman slower can in principle slow atoms down by ∆𝑣,
which from Eq. 2.4 is:𝜇˜∆𝐵 = 𝑘∆𝑣
(2.9)

⇒ ∆𝑣 = 940 m/s

Assuming the atoms with final velocity of 𝑣𝑓 = 30 m/s or slower are captured in the
MOT, then atoms with initial velocity of 𝑣𝑖 = 970 m/s or slower can be captured in
the MOT. Note that the most probable velocity at the oven temperature 485o C is𝑣𝑟𝑚𝑠 = 1600 m/s, so that Zeeman slowing in our setup is inefficient. The detuning∆𝑓 = 𝑓 − 𝑓0 should be:
∆𝑓 =

1
(˜
𝜇𝐵𝑓 − 𝑘𝑣𝑓 ) = −605 MHz
2𝜋(2.10)

EOM
crystaladjustment screw

copper sheet

thermal connection
to the table

RF source

Figure 2-6: Home-made 800 MHz EOM. The crystal is secured inside the two parallel foldsof the copper sheet inside a plastic mount by the plastic screws. This avoids electricalconnection to the table.

About 5% of the light is resonant with the 𝐹 = 1 manifold. In the absence of
this light, the atoms get pumped into the 𝐹 = 1 state and no longer interact with51

the slowing light. The 803.5 MHz shift (the hyperfine splitting) is implemented witha home-built EOM (electro-optic modulator), design from [82]. The electro-optic
crystal is embedded in an inductor formed by a folded copper sheet. The crystal acts
as a capacitor, forming an LC circuit, which is driven by an RF source at the hyperfine
splitting 803.5 MHz. The resonant frequency of the circuit is controlled by adjusting
L with the help of a screw which can go in an out of the folded copper, changing
the effective inductance. When the driving frequency 𝜔𝑅𝐹 is at resonance with thecircuit, the light passing to the crystal acquires sidebands at 𝜔 ± 𝜔𝑅𝐹 . Since in theprocess, the system heats up, changing the resonant frequency of the LC circuit, we
use passive cooling to keep the temperature stable. The circuit is thermally connected

Sidebands strength (arb. un.)to the optical table which provides a cold thermal reservoir.

1000

800

600

400

200
795

800

805

810

815

Drive frequency (MHz)Figure 2-7: EOM response as a function of drive frequency. Plotted is the strength of the
sidebands. The dashed line indicates the hyperfine splitting.

Alignment
The initial alignment of the slower beam was performed when the atomic source was
not attached and instead a viewport was placed at the position of the nozzle. Then,the slower beam was turned on and aligned so that it passes through all apertures(slower, differential pumping, cold cup) with little or no clipping. The bellows between
the two Zeeman slower parts can be used to align these apertures so that there is lineof sight between the nozzle and the final Zeeman slower viewport (see Fig. 2-1 and
2-2). After this initial alignment, subsequent alignments are done by monitoring the52initial MOT loading rate by MOT fluorescence. The initial MOT loading rate isdominated by the flux of slow atoms and not so much by the performance of the
MOT. This is also used to tune the magnetic field, especially in the last Zeeman
slower coil, which determines the exit velocity. It is important that this exit velocity
is within the MOT capture range.
Spin-flip Slower
BEC 5 followed the history of the Ketterle group to use a spin-flip Zeeman slower,
which provides several advantages. To slow atoms with a given starting velocity acertain ∆𝐵 is required between the beginning and the end of the slower: Eq. 2.9. In
principle, this can be implemented by having 𝐵(0) = 0 and 𝐵(𝐿) = ∆𝐵. However,the large fringing field at the end of the slower affects the atoms in the MOT position. Alternatively, 𝐵(0) = ∆𝐵 and 𝐵(𝐿) = 0 can be used, but then the Doppler
shift 𝜔𝑑 = 𝑘𝑣 at the end of the slower will be too small (because 𝑣 is small). Thenthe slowing light will be close to resonance with the atoms in the MOT. A spin-flip
slower design has the 𝐵 = 0 point somewhere in the middle. The fringing field fromboth sides defines the quantization axis there. This technique also avoids having to
use large magnetic fields.
Cleaning the Zeeman slower viewport
A large fraction of the atoms in the atomic beam are not captured by the MOT
and are deposited on the Zeeman slower viewport. Since this deposition can block
the propagation of the slower light, active heating of that viewport or UV light isnecessary. We heat the flange to 180-200o C and sometimes use UV light at 495 nm
to enhance desorption.

53

2.1.3

MOT

Operation
The magneto-optical trap (MOT) traps and cools atoms slowed down by the Zeeman
slower. It is formed by a quadrupole magnetic field with a zero-crossing in the middleof 3 pairs of beams counter-propagating along 𝑥,𝑦, and 𝑧. This configuration ofbeams, without the magnetic field, is known as an optical molasses. It is describedby a damping force proportional to the velocity of the atoms: 𝐹 = − 𝛼𝑣, where 𝛼 isrelated to the spontaneous scattering rate (see e.g. [111]). For red-detuned light, an
atom moving along one pair of counter-propagating beams is more likely to absorb a
photon from the beam propagating in a direction opposite to its own. This is due tothe Doppler shift, which shifts the frequency of that beam in the frame of the atom
closer to resonance.
When a quadrupole magnetic field is added (from a pair of coils with currents
in opposite directions), the optical molasses is turned into a trap. The trapping is
not due to the confinement of the quadrupole field (it is typically quite weak), but it
rather makes the force position-dependent through the Zeeman effect: 𝐹 = −𝛼𝑣 −𝛽𝑧,where 𝛽 is related to the Zeeman shift (see e.g. [55]). Due to the different magnetic

moments of the ground state Zeeman sublevels, the resonant frequency of each statefor 𝜎 + and 𝜎 − light (right and left-circularly polarized light, respectively) is different
on each side of the 𝐵 = 0 point, the center of the trap. Is is possible to construct
a situation in which atoms moving away from the center of the trap become more
and more resonant with the light beam propagating in a direction opposite to theirs,
so that they are more likely to scatter from that beam and are effectively pushed
towards the trap center. In out system, the MOT has about 5×109 atoms at 1 mK
and a density of 1011 cm−3 .

54

Setup∙ Light. (i) Frequency. The MOT operates on the 𝐷2 line (Fig. 2-8) with thefollowing detunings:

𝑓2 = 𝑓𝐹 2 − 45 MHz
𝑓1 = 𝑓𝐹 1 − 35 MHz(2.11)(ii) Intensity. The two frequency components need about equal light levels. The
minimum and maximum powers used for the two frequency components are:
14 mW ≤ 𝑃𝐹 2 ≤ 20 mW
11 mW ≤ 𝑃𝐹 1 ≤ 16 mW(2.12)The beam size is 1 cm 1/𝑒2 radius, so that the saturation parameters are in theranges:3.5 ≤ 𝑠𝐹 2 ≤ 5
2.7 ≤ 𝑠𝐹 1 ≤ 4(2.13)(iii) Polarization. For the polarization of each beam defined with respect toits direction of propagation, each pair of counterpropagating MOT beams has
the same circular polarization. The choice of 𝜎 + vs 𝜎 − is determined by themagnetic field gradient. Since the gradient along the z-direction has the opposite
sign from the gradients in the horizontal plane, the polarization of the top andbottom beams is opposite to the polarizations of the side beams.
∙ Magnetic field gradient. Typically 30 Gauss/cm are used, but the performanceof the MOT is not very sensitive to this value.

55

MOT
GM

Figure 2-8: Li 7 level diagram at zero external magnetic field

Implementation
Since the MOT relies on the scattering force, the spatial profile of each pair of counterpropagating beams needs to be the same, so that no single beam “pushes” the atoms
away from the MOT center. In our machine, this is implemented by having a 2-to-6
fiber splitter so that each arm of the MOT comes from identical fibers and is thenpassed through identical optics. In fact, for the MOT operation it is not too importantthat all beams are the same, rather each pair of beams need to be balanced. The fiber
splitter has two inputs: for the 𝐹 = 2 and 𝐹 = 1 light, each of which is generated by
the same laser (Toptica TA Pro which combines a diode laser with a tapered amplifier)but passed through two different tapered amplifiers (Toptica BoosTA). The 800 MHz
frequency shift between the 𝐹 = 2 and 𝐹 = 1 light is implemented by a double-pass
400 MHz AOM (Intraaction). The fiber splitter itself relies on evanescent coupling
between the fibers and is very sensitive to temperature fluctuations and acoustic
noise. It is therefore kept inside the temperature-stabilized optical table enclosure,thermally connected to the table. The six outputs of the splitter are designed to havethe same output power, although in practice, some have lower output power thanothers. In fact, the fibers in the splitter are PM (polarization-maintaining) and slightdeviations from the correct incident polarization result in different splitting ratios.
For each pair of MOT beams we have selected those fibers which have similar output
56

powers. For alignment of the MOT beams, see Section 2.1.5.
BEC number vs MOT fluorescence
The number of atoms loaded in the MOT is monitored in real time by using a photodiode which collects the MOT fluorescence. For the same MOT light frequency,
the number of atoms in the MOT is proportional to the fluorescence. We use the
MOT loading time to vary the number of atoms in the MOT. We then record thefinal BEC number as a function of fluorescence. This is shown in Fig. 2-9. Beyond acertain critical point, loading more atoms does not increase the BEC number. Thistypically corresponds to 2-3 s of MOT loading. However, the BEC number stability
is improved by loading for a bit longer: 3-4 s. After enough MOT atoms have been
loaded, there are two other steps which limit the final BEC number: (i) the loading
into the ODT, where we load about 10% of the atoms limited by laser power, and (ii)three-body loss in the ODT.
MOT loading time (ms)
1

2

3

4

5

0.25

0.3

0.35

Number of BEC atoms (x10 -4 )

14
12
10
8
6
4
2
0
0

0.05

0.1

0.15

0.2

MOT fluorescence (V)Figure 2-9: Number of BEC atoms as a function of MOT fluorescence (shown here in unitsof the monitor photodiode voltage), controlled by the MOT loading time. MOT loading of
5 s typically results in MOT numbers ∼ 5 × 109 .

57

2.1.4CMOT

Operation
In many experiments, following the MOT step, there is a compressed-MOT (CMOT)
stage, in which the density is increased by increasing the magnetic field gradientand the temperature is decreased by tuning the MOT beam closer to resonance withdecreased intensity. In our case, increasing the magnetic field gradient did not significantly increase the density, so that we do not change the magnetic field gradientin this step. We only ramp the 𝐹 = 2 light closer to resonance and decrease the intensity of the 𝐹 = 1 light, as shown in Fig. 2-10. This pumps the atoms in the 𝐹 = 1
state, which is almost dark towards the end of the step, which decreases scattering.
Absorption images of the cloud after the CMOT stage are shown in Fig. 2-11. The
result is denser clouds with a small decrease in temperature. As in the case of opticalmolasses, the equilibrium temperature is determined by the balance of heating andcooling, where the heating comes from the random walk in momentum space causedby the scattering. The lowest temperature is achieved for detunings 𝛿 = − Γ/2. Bygoing closer to resonances during the CMOT stage, the temperature of the cloud isdecreased.
b)

-20
-25
20
-30
-35

Detuning (MHz)

-15

-40
15

-45
0

1

2

3

F=1

-32
10

Time (ms)

-34
-36

5
-38
0

4

-30

15

Detuning (MHz)-10

Individual beam power (mW)

F=2

25

Individual beam power (mW)

a)

0

1

2

3

4

-40

Time (ms)Figure 2-10: CMOT. Plotted are the powers and frequencies as a function of time for a) the𝐹 = 2 light and b) the 𝐹 = 1 light.

58

Setup∙ Light: The power of the 𝐹 = 1 light is swept from the MOT value to about
20-30 𝜇W. The detuning of the 𝐹 = 2 light is swept from the MOT value to avalue in the range of -8 to -12 MHz. Both ramps are linear over 4.5 ms.
∙ Magnetic field gradient: same as the MOT.
a)b)

Figure 2-11: CMOT absorption images images. a) With 𝐹 = 2 light, and b) with 𝐹 = 1.
The majority of the atoms after the CMOT step are in 𝐹 = 1.

2.1.5

Gray Molasses

Operation
Laser cooling has an inherent limit due to the recoil the atoms get from scattering.
This is known as the Doppler limit for a two-level atom [55]:𝑇𝐷 =~Γ= 140 𝜇𝐾 (for Li)
2𝑘𝐵(2.14)

Typical sub-Doppler cooling schemes on the 𝐷2 line (Sysyphus cooling, polarizationgradient cooling) do not work so well for Li, because of the narrow energy splittings of
the 𝑃3/2 state. Instead, 𝐷1 -line Gray Molasses (GM) cooling has become popular inrecent years. In addition, a Λ-enhanced cooling is implemented which decreases thetemperature further. The GM scheme is implemented on the |𝐹 = 2⟩ → |𝐹 ′ = 2⟩ tran-

sition (see Fig. 2-8). When a second laser frequency on the |𝐹 = 1⟩ → |𝐹 ′ = 2⟩ tran-

sition is added on two-photon resonance with the first, a VSCPT (velocity-selective

59

coherent population trapping)-like dark state is created [63]. Below we give a brief
description of these two effects.

a)

b)

|ei

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⌦1

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Figure 2-12: Gray Molasses with Λ-enhanced cooling. a) Typical GM cooling scheme inwhich the atoms in the dark state |𝐷⟩ are transferred by motional coupling to the brightstate |𝐵⟩, where they climb up the potential hill, decreasing their energy. In the brightstate, they scatter light and can be pumped back to |𝐷⟩. b) Λ scheme: if the 𝐹 = 1 and the
𝐹 = 2 frequencies are on two-photon resonance (and are blue-detuned with respect to theatomic transition), the temperature can be lowered. The Rabi frequency of the 𝐹 = 1 lightΩ1 is much smaller than the Rabi frequency of the 𝐹 = 2 light Ω2 . Figures adapted from[54, 63].

The Gray molasses is typically implemented on a 𝐽 → 𝐽 transition 𝐽 > 1 or on

a 𝐽 → 𝐽 − 1 transition with blue detuned light. This results in a state Ψ𝑁 𝐶 (𝑧)which is dark, i.e. not coupled to the light, which is a linear superposition of the
Zeeman sublevels. It is called “gray” because the dark state is not entirely dark. Theenergy of the dark state does not vary in space: Fig. 2-12a. The bright states haveenergy which varies as a function of space in a sinusoidal manner for a pair of beams
with the same polarization due to the AC Stark effect in the optical lattice formedby the counterpropagating beams. If the atom is in the bright state and moves up
the potential hill, it looses kinetic energy, just like in Sisyphus cooling. The key toachieving cooling is to have a situation in which an atom in the dark state is more60

likely to get coupled to the bright state at the potential minima (through motional
coupling), after which it scatters light and is pumped back to the dark state before
going over the potential maximum, see e.g.[54] and references therein.
The addition of repumping light implements a Λ scheme, in which there is alsoa dark state: Fig. 2-12b. It can be shown that when the 𝐹 = 1 and the 𝐹 = 2 lightare on two-photon resonance, there is additional cooling. This is similar in spirit
to VSCPT schemes, in which the atoms which have already been cooled beyond acertain critical velocity, no longer scatter light and cannot be heated by recoil. For atoy model description, see [63].
Setup∙ Light on the 𝐷1 line. (i) Frequency:
𝑓2 = 𝑓𝐹 2 + 30 MHz
𝑓1 = 𝑓𝐹 1 + 30 MHz(2.15)
Here the two-photon resonance is important: 𝑓1 − 𝑓2 = 803.5 MHz, the hyperfine splitting at zero field. Each frequency component is controlled with aseparate AOM. We observe that the temperature is significantly decreased if|𝑓1 − 𝑓2 | ≤ 1.5 MHz(2.16)

Then, subsequent evaporation in the magnetic trap can proceed as designed. In
our optimized sequence, when the relative detuning is larger than that, we do
not obtain a BEC.
(ii) Intensity:
10 mW ≤ 𝑃F2 ≤ 14 mW
0.3 mW ≤ 𝑃F1 ≤ 0.6 mW
61(2.17)

The beam size is 3.5 mm 1/𝑒2 radius, so that the saturation parameters are:20.4 ≤ 𝑠F2 ≤ 28.5
0.61 ≤ 𝑠F1 ≤ 1.22

(2.18)∙ No magnetic fields or gradients
GM and MOT Alignment
The light for the MOT and for the GM is produced in a similar way, except that
there is a separate diode laser, locked to the 𝐷1 line, which seeds the two BoosTAs.
After that, there are separate AOMs which act as switches for the MOT and the GM
light. Both the MOT and the GM light is coupled into the same 2-to-6 fiber splitterbut with opposite linear polarizations. This means that one of them is aligned withthe fast and the other one with the slow PM fiber axis. After the fiber, each of the 6arms is implemented by identical cages (Thorlabs) for extra stability. The two beams
need to have different sizes: large for the MOT and small for the GM. The two are
split with a PBS (polarization beam splitter) and passed through different expanding
telescopes after which they are recombined on another PBS. Since the final mirror
is the same for the MOT and the GM beams, the alignment of the two has to bedone at the same time. For more precision, each beam is passed through a cage iris
(Thorlabs CPA1), so that only the middle of each beam is used in alignment.
First, using the last two mirrors in the GM path, one of the GM beam in each of
the three axis is aligned to middle of the viewports using targets on the viewports.
This procedure does, in fact, result in the GM being overlapped with the center ofthe magnetic trap. For fine alignment, the atoms in the magnetic trap at maximum
gradient can be imaged along each GM axis and then the GM beam can be positioned
to the same spot. After that, the counter-propagating GM beams are overlapped with
the incident ones. Finally, each MOT beam is overlapped with the GM beam comingfrom the same fiber. This results in the atoms being in the same position in space,
avoiding extra heating when transitioning from one cooling stage to the next.
62After the polarizations of the beams are set, the next step is balancing the powers
in each pair of MOT and GM beams. It is important to also minimize and balance thepowers in the “wrong” polarization in each pair. For example, during the MOT phase,some of the MOT light propagates in the GM path but in the wrong polarization.

2.1.6

Dark State Pumping

Operation
In this stage, the atoms are transferred to a single hyperfine state: |𝐹, 𝑚𝐹 ⟩ = |2, 2⟩.
Having only the stretched-state has several advantages for magnetic trapping: (i)in the magnetic trap, not all states are trappable. The ones which are not trapped
have negative magnetic moments and, therefore, lower their energy by going to higher
fields (termed “high-field seekers”). This means that some fraction of the atoms arelost if they are transferred without optical pumping into the magnetic trap. (ii)Spin-changing collision between the trapped states could lead to loss. For example: |2, 0⟩ + |2, 0⟩ ↔ |2, 1⟩ + |2, −1⟩ and the |2, −1⟩ state is not trapped. If thegas is composed of only one of the stretched states |2, ±2⟩, it cannot decay by thismechanism.

When a small magnetic field is applied, so that the Zeeman sublevels are resolved,
the 𝐷1 line can be used to optically pump all atoms in the |2, 2⟩ state when circularly-

polarized 𝐹 = 2 and 𝐹 = 1 light is used. This state is then dark for 𝜎 + light (seeFig. 2-8).
Setup∙ Light on the 𝐷1 line. (i) Frequency:
𝑓2 = 𝑓𝐹 2 + 30 MHz
𝑓1 = 𝑓𝐹 1 + 30 MHz(2.19)

63

(ii) Intensity. Typical values are:
𝑃F2 = 5 mW
𝑃F1 = 8 mW

(2.20)The beam size is about 1.5 mm 1/𝑒2 radius and the pumping time is 300-600 𝜇s.
These values have a wide range: pump powers can be between 1-10 mW withcorresponding pump times between 1 ms and 200 𝜇s. The F = 1 light needs tobe about two times stronger than the F = 2 light. However, if too large pumplight intensities are used, the atoms can gain large recoil and start moving in
the direction of the pump beam.
∙ Magnetic field: a bias magnetic field along the axis of the propagation of thelight is used to define the quantization axis. It is about 5-6 Gauss.

2.1.7

Magnetic trap

Operation
In forced evaporation cooling, the hottest atoms are expelled from the trap and theremaining atoms thermalize to a lower temperature by elastic collisions. This resultsin higher phase-space densities. This process can go into a run-away regime which
can be very efficient and can result in the atoms condensing into a BEC.
We use a quadrupole magnetic trap which has a 𝐵 = 0 point (a “Majorana hole”),
where the atoms can be spin-flipped to untrapped states. Therefore, a plug beamis used: a blue-detuned beam implementing a repulsive barrier. It expels the atoms
from that region. An in-depth review can be found at [86].
Setup∙ RF sweep: evaporation is performed by spin-flipping atoms from the |𝐹, 𝑚𝐹 ⟩ = |2, 2⟩
state to the |𝐹, 𝑚𝐹 ⟩ = |1, 1⟩ state, as shown in Fig. 2-13. This state is notmagnetically-trappable. The RF frequency is swept from 1 GHz to 806 MHz.
64

600

Energy (MHz)

400
200
2,2
2,1
2,0
2,-1
2,-2
1,-1
1,0
1,1

0
-200
-400
-600
0

50

100

150

200

Magnetic field (Gauss)

Figure 2-13: Evaporation transition in the magnetic trap. A spin flip from |𝐹, 𝑚𝐹 ⟩ = |2, 2⟩
to |𝐹, 𝑚𝐹 ⟩ = |1, 1⟩ takes hot particles out of the magnetic trap.

∙ Plug. The beam is generated by a 532 nm single mode IPG GLR-20 laser. The
beam size is 20 𝜇m 1/𝑒2 radius and the power is 16 W.

* Magnetic field gradient sweep. During the evaporation, the density increases, sothat tree-body loss becomes important, limiting the density to about 𝑛 = 1013 cm−3 .

Therefore, the magnetic field gradient is decreased during evaporation.
* Shim coils. Due to stray magnetic fields (possibly from the steel chamber), the𝐵 = 0 point shifts in the process of decreasing the magnetic field gradient. Shimcoils in all three directions are used to compensate for the trap motion during
each step of the evaporation. In this way the plug beam stays aligned with the
𝐵 = 0 point and extra heating from the motion is eliminated.

We transfer into the magnetic trap by first rapidly turning on a small magnetic
field gradient which captures the atoms after dark state pumping. We then ramp up
the magnetic field gradient to 150 Gauss/cm in 100 ms to increase the starting density.
This also increases the temperature, but the high starting density is necessary for alarge collision rate to start the evaporation. The evaporation lasts for 2.5 s.

65

2.1.8ODT

Operation
While laser cooling relies on the scattering force, which is important for light frequencies close to atomic resonance, light trapping relies on the dipole force and is typically
implemented far from resonance. There, scattering is sufficiently suppressed by the
detuning 𝛿 and the dipole force provides a conservative potential. The scattering rate
scales as 1/𝛿 2 , while the dipole potential scales as 1/𝛿. The dipole potential is:
(︂
)︂
1
1
~Ω2
+
𝑈dip =
4
𝜔0 − 𝜔 𝜔0 + 𝜔
(︂
)︂
3𝜋𝑐2
1
1
= 3Γ
+
𝐼(𝑥, 𝑦, 𝑧) ≡ 𝑉dip 𝐼(𝑥, 𝑦, 𝑧)
2𝜔0𝜔0 − 𝜔 𝜔0 + 𝜔(2.21)

where Ω is the Rabi frequency, 𝜔 is the frequency of the light, 𝜔0 is the atomic
resonance frequency, and 𝐼(𝑥, 𝑦, 𝑧) is the spatial intensity distribution of the light.
Optical dipole trapping is necessary because the state used for evaporation, the|𝐹, 𝑚𝐹 ⟩ = |1, 1⟩, is not magnetically trappable, but is needed because of its Feshbachresonance and its stability against dipolar relaxation.
Setup∙ Light: 1064 nm light in two orthogonally polarized arms propagating at 45o anddetuned from each other by at 10 MHz. The beam size is 90 𝜇m 1/𝑒2 radiuswith 9 W of power each.
Gaussian beam
To describe the optical dipole potential, we must first look at a Gaussian beam.
Consider a single Gaussian beam propagating along the 𝑧-axis with polarization along
the 𝑥-axis. The electric field is:𝐸(𝑥, 𝑦, 𝑧) = 𝐸0

(︂

𝑤00
𝑤(𝑧)

)︂

2

2

2

2

2

𝑒−(𝑥 +𝑦 )/𝑤(𝑧) 𝑒−𝑖(𝑘𝑧+𝑘(𝑥 +𝑦 )/2𝑅𝑐 (𝑧)−𝑔(𝑧)) 𝑥ˆ

66(2.22)

where the spot size (1/𝑒2 radius) is 𝑤00 at the focus at 𝑧 = 0 and 𝑤(𝑧) as a functionof 𝑧:𝑤(𝑧) = 𝑤00

√︃

1+

(︂

𝑧𝜆
2
𝜋𝑤00

)︂2

(2.23)

where 𝜆 is the wavelength of the light. 𝑅𝑐 (𝑧) is the radius of curvature, and 𝑔(𝑧) is
the Gouy phase. The beams considered here have large enough radii compared tothe atomic cloud size, so that to a good approximation, we can evaluate eq.2.22 at𝑤(𝑧) = 𝑤00 . A measure of the distance over which a Gaussian beam diverges is theRayleigh range
𝑧𝑅 =

𝜋𝑤00
𝜆

(2.24)√The Rayleigh range is defined such that the beam size increases by a factor of 2√
at 𝑧𝑅 away from the focus: 𝑤(𝑧𝑅 ) = 2 𝑤00 . For example, for the lattices we use
𝑤00 = 125 𝜇𝑚 and the cloud is at most 𝑧𝑚𝑎𝑥 = 20 𝜇𝑚 in radius, so 𝑧𝑅 = 4.6 cm and(𝑧𝑚𝑎𝑥 /𝑧𝑅 ) = 4.3 × 10−4 . Therefore:(︂)︂𝑧𝑔(𝑧) = arctan= 9 × 10−4 ≈ 0
𝑧𝑅
(︁
𝑧𝑅 )︁
= 103 m ≈ ∞𝑅𝑐 (𝑧) = 𝑧 1 +𝑧(2.25)

Therefore, the intensity of a single Gaussian beam can be written as:22

2

𝐼(𝑥, 𝑦, 𝑧) = 𝐼0 𝑒−2(𝑥 +𝑦 )/𝑤00(2.26)

The atoms are located near the minimum and to a good approximation, the potential is harmonic there. For an axis perpendicular to the direction of propagation,
the Gaussian can be expanded in powers of 𝑥, so that the potential is:22

𝑉 (𝑥) = 𝑉0 𝑒−2𝑥 /𝑤00 ≈ −

2𝑉0 2
1
2 2
𝑥 ≡ − 𝑚𝜔𝑡𝑟
𝑥
2
𝑤00
2

(2.27)

where we have dropped the overall energy offset. We can identify the trap frequency

67

𝜔𝑡𝑟 :
𝜔𝑡𝑟 =

√︃

4𝑉0
2
𝑚𝑤00

(2.28)

2
where 𝑉0 = 𝑉𝑑𝑖𝑝 (2𝑃 )/(𝜋𝑤00) from Eq. (2.21), where 𝑃 is the power.

Crossed dipole trap potential
For two beams propagating at 45o with respect to each other, the trapping region in
the middle forms a dimple, ash shown in Fig. 2-14.

a)b)

Figure 2-14: Crossed ODT potential. The two beams are at 45o with respect to each other.
a) Top view. b) Side view

The trap frequencies along the main chamber axis can be calculated. Note thatthe 𝑧 axis is perpendicular to the plane of the ODT and is along gravity, while the 𝑥
and the 𝑦 axis are along the two symmetry axis of the crossed ODT trap. If the trap
frequencies of the two beams are the same and the angle between them is 2𝜃 = 45o ,the combined trap frequencies are:(𝜔𝑡𝑟,𝑥 , 𝜔𝑡𝑟,𝑦 , 𝜔𝑡𝑟,𝑧 ) =√2(sin 𝜃, cos 𝜃, 1) 𝜔𝑡𝑟(2.29)(𝜔𝑡𝑟,𝑥 , 𝜔𝑡𝑟,𝑦 , 𝜔𝑡𝑟,𝑧 )init = 2𝜋 × (0.432, 1.044, 1.130) kHz(2.30)(𝜔𝑡𝑟,𝑥 , 𝜔𝑡𝑟,𝑦 , 𝜔𝑡𝑟,𝑧 )final = 2𝜋 × (72, 174, 188) Hz(2.31)

where 𝜔𝑡𝑟, init is the value at the beginning of evaporation with 9 W in each beam and𝜔𝑡𝑟, final is the value at the end of evaporation with 250 mW in each beam.
68

Transfer to the ODTSince the mode volumes of the ODT and the magnetic trap are very different, thetransfer to the ODT is accompanied by a large atom number loss. In fact, at most
10 % of the atoms are transferred, limited by the total power in the ODT beams. We
have found that the transfer efficiency is higher if the ODT is kept on at full power
during the entire magnetic trap evaporation. The ODT is offset from the position of
the plug beam and the center of the magnetic trap, thus forming an effective dimple
potential. It is aligned to the saddle point of the Feshbach field which does notexactly coincide with the minimum of the quadrupole trap. When optimizing only
the transfer to the ODT we have found that the efficiency is the largest when the
ODT is placed about a beam radius (about 100 𝜇m) radially away from the center ofthe magnetic trap. It so happens that the Feshbach saddle point is about twice that
distance away. Although the transfer there is not as efficient, the Feshbach saddle
point still overlaps with the extent of the evaporated cloud in the magnetic trap. The
number of atoms transferred is sufficient to start ODT evaporation without having totransport the atoms. Surprisingly, when we had to transport the evaporated atoms
from the magnetic trap by a few millimeters to the ODT in the saddle of the Feshbach
field with the old coils, the plug was not moved along, but there was no significant
decrease in the number of atoms transferred to the ODT.
After the transfer to the ODT, the magnetic trap is switched off in the presence ofa small bias field which keeps the quantization axis defined. The atoms are then transferred to the |1, 1⟩ state by a Landau-Zener (LZ) sweep of the transition frequency,

making the transfer less sensitive to bias field drifts. After that, the remaining |2, 2⟩

atoms, leftover because of inefficiencies in the LZ sweep, are blown away by using

resonant 𝐹 = 2 light. Then, the atoms in the wings of the crossed ODT need to be
removed because they interfere with the ODT evaporation. This is done by applying amagnetic field gradient along the 𝑧 axis, which act as a force pushing the atoms in thewings away. Finally, the Feshbach field is turned on adiabatically and we evaporate
to a BEC.

69

2.1.9BEC

Operation
The goal of evaporative cooling is to increase the phase space density until the transition to a BEC is crossed. The phase space density expressed in terms of the thermal√︀de Broglie wavelength 𝜆𝑇 = 2𝜋~2 /(𝑚𝑘𝐵 𝑇 ) and the density 𝑛 is(2.32)

𝜌 = 𝑛𝜆3𝑇 ≈ 2.612

at the transition point. For a trapped cloud, the transition temperature is [125]:𝑇𝑐 ≈ 0.94~¯
𝜔𝑡𝑟 1/3
𝑁𝑘𝐵(2.33)

where the trap frequency is 𝜔¯ 𝑡𝑟 = (𝜔𝑡𝑟,𝑥 𝜔𝑡𝑟,𝑦 𝜔𝑡𝑟,𝑧 )1/3 . In the ODT, the trap frequencies
are given in Eq. (2.31). Then, for 𝑁 = 2 × 105 , the critical temperature is around
350 nK.

Using the Thomas-Fermi approximation (see e.g. [86]), we can obtain the extent
of the cloud, the Tomas-Fermi radius:
𝑅𝑖2 =

2𝜇
2
𝑚𝜔𝑡𝑟,𝑖(2.34)

in direction 𝑖, where 𝜇 is the chemical potential:
152/5
𝜇=
2

(︂

𝑁𝑎
𝑙𝐻𝑂

)︂

1/3

~¯
𝜔𝑡𝑟

(2.35)ODT evaporation
ODT evaporation involves an exponential decrease of the power 𝑃 in the two arms
with the same parameters:(𝑃0 − 𝑃𝑓 )𝑒−𝑡/𝜏 + 𝑃𝑓 − 𝑃0 𝑒−𝑡𝑓 /𝜏
𝑃 (𝑡) =
1 − 𝑒−𝑡𝑓 /𝜏

70(2.36)

where 𝑃0 = 9 W, 𝑃𝑓 = 250 mW, 𝑡𝑓 = 2000 ms, 𝜏 = 500 ms at a scattering length of
𝑎 = 100 𝑎0 . Here 𝑡𝑓 is the duration of the evaporation sweep and 𝜏 determines its
shape. The parameters can be optimized for a range of scattering lengths. If thescattering length is too large, three-body loss dominates, if the scattering length is
too small, experimental time is the limiting factor. Therefore, a conservative estimate
for the scattering lengths over which a BEC can be formed is:(2.37)

40 𝑎0 ≤ 𝑎 ≤ 300 𝑎0

Fig. 2-15 shows the regions of Feshbach fields which correspond to this range of scattering lengths 𝑎𝑎𝑎 and 𝑎𝑏𝑏 in the lowest two hyperfine states, labeled here as |𝑎⟩ and|𝑏⟩.

Scattering length (a0)

300

aaa
abb

200
100
0
-100
-200
-300

700

750

800

850

900

Magnetic field (Gauss)

Figure 2-15: Magnetic fields suitable for ODT evaporation. A BEC can be produced with
evaporation parameters similar to the ones in the text, either in the |𝑎⟩ state (red) or in the|𝑏⟩ state (blue) when the scattering lengths 𝑎𝑎𝑎 and 𝑎𝑏𝑏 are in the range of Eq. (2.37).

2.1.10Lattice

Operation
The physics of ultracold atoms in optical lattices is the subject of Chapter 3. Here
we detail the implementation and design choices.

71

Setup∙ Light: 1064 nm in three retro-reflected beams, orthogonally-polarized with eachother and detuned by at least 5 MHz from each other and from the ODT beams,
propagating along 𝑥, 𝑦, and 𝑧.
Intensity distribution
The electric field a pair of retroreflected beams propagating along the 𝑧 axis and withthe same linear polarization in the 𝑥 axis is:
2

2

2

2

2

2

𝐸(𝑥, 𝑦, 𝑧) = 𝐸0 𝑒−(𝑥 +𝑦 )/𝑤00 𝑒−𝑖𝑘𝑧 𝑥ˆ + 𝐸0 𝑒−2(𝑥 +𝑦 )/𝑤00 𝑒+𝑖𝑘𝑧 𝑥ˆ

(2.38)

so that the intensity is:
2

2

2

2

2

2

2

2

2

2

2

2

(2.39)𝐼(𝑥, 𝑦, 𝑧) = 4𝐼0 𝑒−2(𝑥 +𝑦 )/𝑤00 cos2 (𝑘𝑧)
For 3 pairs of lattices:𝐼(𝑥, 𝑦, 𝑧) =4𝐼0𝑧 𝑒−2(𝑥 +𝑦 )/𝑤0𝑧 cos2 (𝑘𝑧)
+4𝐼0𝑦 𝑒−2(𝑥 +𝑧 )/𝑤0𝑦 cos2 (𝑘𝑦)(2.40)+4𝐼0𝑥 𝑒−2(𝑦 +𝑧 )/𝑤0𝑥 cos2 (𝑘𝑥)2where 𝐼0𝑖 = 2𝑃/(𝜋𝑤0𝑖). The confinement along each axis comes from the beams propa-

gating in the perpendicular directions and the lattice potential comes from the pair ofretroreflected beams propagating along that axis. A 1D and 2D cuts of this potential
are plotted in Fig. 2-16.
The dipole potential on the atoms is then (from Eq. (2.21)):
3𝜋𝑐2
𝑉 (𝑥, 𝑦, 𝑧) = 3 Γ
2𝜔0

(︂

1
1
+
𝜔 − 𝜔0 𝜔 + 𝜔0

2
−2(𝑥2 +𝑧 2 )/𝑤0𝑦+ 4𝐼0𝑦 𝑒

)︂ [︂
2
2
2
4𝐼0𝑧 𝑒−2(𝑥 +𝑦 )/𝑤0𝑧 cos2 (𝑘𝑧)2
−2(𝑦 2 +𝑧 2 )/𝑤0𝑥2

cos (𝑘𝑦) + 4𝐼0𝑥 𝑒72]︂cos (𝑘𝑥)2(2.41)

a) 12

b)

10

I/I0

8
6
4
2
0
-150

-100

-50

0

50

100

150

m

Figure 2-16: Cuts along the 3D lattice potential. a) 1D for 𝑥 = 0 and 𝑦 = 0 and b) 2D for
𝑦 = 0 (zoomed-in).

The atoms are confined in a small region around (𝑥, 𝑦, 𝑧) = (0, 0, 0). To obtain theoverall trap frequency, the Gaussian can be expanded, just as in Eq. (2.28). Also, thesinusoidal potential on each lattice site can be approximated by a harmonic potential
and the trap frequency can be obtained by expanding the cosine term. For instance,along 𝑧 at 𝑥 ≈ 0 and 𝑦 ≈ 0:
𝑉 (𝑧) ≈ 𝑉0𝑧 cos(𝑘𝑥)2 ≈ 𝑉0𝑧 𝑘 2 𝑥2

(2.42)

so that the on-site trap frequency is
𝜔𝑠, 𝑧 =

√︂

2𝑉0𝑧 𝑘 2𝑚(2.43)

2
where now 𝑉0𝑧 = 4 (2𝑃𝑧 )/(𝜋𝑤0𝑧)𝑉𝑑𝑖𝑝 . The ground state is then a Gaussian:Ψ(𝑥) =

(︁ 𝑚𝜔

𝑠, 𝑧

𝜋~

)︁1/4

2

𝑒−𝑚𝜔𝑠, 𝑧 𝑧 /2~(2.44)The Gaussian is a fairly good approximation to the ground state wavefunction fordeep lattices. However, in a harmonic potential, the energy levels are equally spaced
and, while, as shown in Section 3.1.1, the energy levels in a lattice are not equally
spaced and that has some important implications.

73

Choice of lattice wavelength
As for the ODT, the lattice wavelength needs to be sufficiently far detuned from theatomic transition to avoid heating from spontaneous scattering. Here we look at thisin more detail. Both the scattering rate:
3𝜋𝑐2
𝑅=
2~𝜔03

(︂

𝜔
𝜔0

)︂3

2

Γ

(︂

1
1
+
𝜔0 − 𝜔 𝜔0 + 𝜔)︂2𝐼(𝑥, 𝑦, 𝑧)(2.45)and the strength of the dipole potential Eq. (2.21) need to be considered.
It is important to note that there is a difference between blue and red detuned
light. For red detuned light, the atoms are trapped at the maxima of the lightintensity and for blue detuned light, they are trapped at the minima. This changes
the scattering rate but not the rate of energy increase. This is detailed in [126]. Herea brief summary for the 1D case is provided, which can easily be generalized to 3D.
If the location of the atoms is set to be 𝑥 = 0, then the lattice potentials the atoms
feel can be expressed as:
(︂
)︂
1
2
𝑉red (𝑥) = − 𝑉0 cos (𝑘𝑥) ≈ −𝑉0 1 − (𝑘𝑥)2(2.46)

𝑉blue (𝑥) =𝑉0 sin2 (𝑘𝑥) ≈ 𝑉0 (𝑘𝑥)2(2.47)

2

where the deep lattice limit is used to expand around small 𝑘𝑥. The Lamb Dicke pa√︀rameter 𝜂 describes the ratio between the extent of the wavefunction 𝑙ℎ𝑜 = ~/(𝑚𝜔𝑠 )(approximated by the harmonic oscillator length in a deep lattice) and the lattice site𝑑 = 𝜆/2 = 𝜋/𝑘:𝑙ℎ𝑜
𝜂 ≡ 𝑘𝑙ℎ𝑜 = 𝜋
=
𝑑

(︂

𝐸𝑅
4𝑉

)︂1/4

(2.48)

where 𝜂 was expressed as a function of the recoil energy𝐸𝑅 =~2 𝑘 22𝑚(2.49)

which is the natural energy scale for a lattice. Therefore, to lowest order in the74Lamb-Dicke parameter, the scattering rates at the position of the atoms are:
𝑅red =𝑅max(2.50)

𝑅blue =𝑅max 𝜂 2

In the red-detuned case, this rate corresponds to scattering from the lowest lattice
band back to the lowest lattice band. In the blue-detuned case this process is suppressed and the lowest order process is scattering from the lowest band to the firstexcited band. There is an equivalent process at the same rate in the red-detuned
case, therefore, if we consider the energy increase per particle to be due to scattering
to higher bands, the red and blue detuned cases are the same. However, since in spin
physics experiments we are interested in the entropy and the correlations of particles in magnetically ordered states in the lowest band, any form of scattering can be

10 1

10 0

10 -1

400

600

800

1000

1200

120

350

100

300
250

80

200
60
150
40

100

20

50
400

Lattice wavelength (nm)Recoil (kHz)

b)

10 2

Power needed to get 40 E R (W)a)

Lifetime of a 40 E R lattice (s)detrimental to the spin-spin correlations.

600

800

1000

0
1200

Lattice wavelength (nm)Figure 2-17: Choice of lattice wavelength. a) Lattice lifetime limited by spontaneous scattering of a 40 𝐸𝑅 lattice with 1/𝑒2 beam radius of 125 𝜇m. Dashed lines correspond to theminimum requirement for a lifetime of 1 s (blue) and for a more desirable scattering-limited
lifetime of 10 s (black). b) Power required to obtain a 40 𝐸𝑅 lattice (blue) and the recoil
𝐸𝑅 /ℎ (red). The recoil increases (faster dynamics) but the power requirements also increase
for blue detuning.

Fig. 2-17a shows the scattering-limited lifetime 1/𝑅 (where 𝑅 is the scattering
rate at the position of the atoms) from the pair of Eq-s. (2.50) as a function of lattice
wavelength for a 40 𝐸𝑅 lattice, since it is the lattice depth in recoil units which sets
the timescales of dynamics. If a scattering-limited lifetime of 1 s can be tolerated,75

than any lattice wavelength which is at least ∼ 30 nm away from the resonancecan be chosen. However, a more realistic requirement is that the scattering-limited
lifetime be at least 10 s, especially if long-time dynamics are to be studied. This isthe black dashed line in Fig. 2-17a, limiting the choice of wavelength to 𝜆 > 834 nm
and 𝜆 < 637 nm.
The power required to make a 40 𝐸𝑅 lattice is plotted in Fig. 2-17b, as well as therecoil 𝐸𝑅 /ℎ in kHz. As we will see in Section 3.1.3, the timescale for dynamics at theMott insulator transition scales as the recoil. The higher the recoil, the faster the
dynamics. Fig. 2-17b shows that the recoil energy is larger for smaller wavelengths
but also the power required to make a lattice of a given depth in recoil units is large
there. We use 1064 nm due to the availability of lasers and fiber amplifiers at thatwavelength and would potentially consider 532 nm (frequency-doubled 1064 nm).
Beam size
There are two considerations for picking beam size: (i) The smaller the beam, thedeeper the lattice; (ii) The larger the beam, the more homogeneous the trap. A small
beam size means that in order to achieve a density of one or two atoms per site, theatom number needs to be decreased significantly. However, a deeper lattice allows
for more time for state preparation, because the deeper the lattice, the more isolated
from each other the sites are.
We pick a beam size of 125 𝜇m 1/𝑒2 radius, which corresponds to a maximum
depth of 35 𝐸𝑅 requiring 4.3 W of 1064 nm light. For a cloud size of 20 𝜇m (40 latticesites), this gives a variation of the lattice depth of 1.27% across the cloud.
Polarization
In the cubic lattice design, the three lattice arms do not interfere with each other.
This can be implemented by orthogonal linear polarizations. However, since many
components can introduce polarization rotation, the lattice beams are detuned byat least 5 MHz with respect to each other. Any interference between the different
arms of the lattice will beat at this frequency and average to zero for the atoms. For76example, dielectric mirrors have different S and P polarization reflection coefficients(which also vary as a function of the angle of the incident beam) so if the beampolarization does not exactly align with S and P polarizations of the mirror, a phaseshift between the S and P components of the incident light is introduced. Also, thepolarization properties of the vacuum viewports are not well characterized.
We use polarizing beam splitters suitable for high power applications (UVFS ionbeam sputtered PBSI-1064 from CVI with 𝑇𝑝 /𝑇𝑠 = 1000:1 with 1 MW/cm2 damage
threshold). Typically, polarizing beam splitters have better polarization extinction
for transmission than for reflection, while they offer lower power losses in reflection
than transmission. Also, zero-order waveplates have to be used in sensitive applications, because they offer smaller temperature sensitivity compared to multi-order
waveplates.
We note that we have used circular polarization for the lattices. We had evidence
that even for angled fibers, the tip still reflects some of the light, which could thenpropagate to the intensity stabilization photodiode. Therefore, a design in which
the retro is reflected out by a PBS in front of the fiber was used. The incident
beam passes through a PBS followed by a 𝜆/4-waveplate turning the polarization into
circular. When the retro passes through the same waveplate, its circular polarization
is turned into linear, crossed with the original linear polarization. At the cube, theretro is reflected out, never reaching the fiber tip. We note that we have so far seen
no difference in the lifetime of the lattice between the two designs (linear polarization
with retro hitting the fiber tip and circular polarization with the retro reflected out).
However, this might make a difference in the future.
The implementation of the 3D optical lattice completes the discussion of the stages
required to load a 7 Li BEC in an optical lattice. Once loaded, the atoms can be placedin the Mott insulating state with each atom localized on a single site of the lattice. We
will discuss the physics of the Mott insulator and its characterization in Chapter 3.

77

2.2

Methods

In this section, we focus on a few methods and techniques we have developed whenbuilding BEC 5. First, we describe the beam-delivery system from start to finish:from lasers to calibration of the final beams, since the lattices and the ODTs aremain ingredients of the machine. Then, we look into the atomic nature of the states
which are used as spin states once the atoms are placed in the Mott state in the opticallattice. Finally, we summarize the time sequence of baking the steel components ofthe machine in order to achieve ultrahigh vacuum (UHV).

2.2.1

High power beam delivery

Due to the large recoil energy of lithium as compared to other atoms, achieving thesame lattice depth, requires much more laser power. We use a 50 W fiber amplifier at1064 nm. With 1 mm beam radius, at these intensities many optical elements can exhibit thermal lensing or optical damage, so that the choice of optic material is crucial.
In the following, the elements of the beam delivery system are described, focusing on
the ones specific to high laser power. Also, the design choices and considerations foreach of the components are presented. First, we describe thermal lensing, since itis the direct consequence of high laser power. After that, we follow the path of thebeam, staring from the laser source and finishing with the atoms.
Thermal lensing
Thermal lensing is caused by absorption of the incident light from the material whichcauses local heating. The increase in the temperature can lead to changes in theoutgoing wavefront which resemble the effect of a lens. The textbook model of thermal
lensing is to consider a rod with radius 𝑟0 and length 𝐿 and index of refraction 𝑛0 .
There is uniform pumping light which is absorbed by the rod. The absorbed power
can be characterized by Beer’s law(︀)︀
𝑃𝑎𝑏𝑠 = 𝑃0 1 − 𝑒−𝛼𝐿78

(2.51)

where 𝛼 is the absorption coefficient. The uniform incident light results in a uniform
internal heating rate. The heat is transported to the surface of the rod, assumed to be
kept at a fixed (ambient) temperature. Then, the temperature decreases quadraticallywith distance from the center. It can be shown that the effect of this on an incident
laser beam is the same as a thin lens with focal length [92]:
𝐾𝜋𝑟02
𝑓=
𝑃

(︂

𝛼𝐶𝑇 𝐸 𝑟0 (𝑛0 − 1)1 𝑑𝑛
+ 𝛼𝐶𝑇 𝐸 𝐶𝑟,𝜑 𝑛30 +
2 𝑑𝑇
𝐿

)︂−1

(2.52)

where 𝑃 is the dissipated power, 𝐾 is the thermal conductivity, 𝐶𝑟,𝜑 are elasto-optic
coefficients, and 𝛼𝐶𝑇 𝐸 = (1/𝑥)𝑑𝑥/𝑑𝑇 is the coefficient of (linear) thermal expansion.
A strong thermal-lensing effect is characterized by a small effective focal length. Thisequation describes three mechanisms which lead to the formation of an effective lens.
(1) The index of refraction changes as a function of temperature. (2) The local
temperature gradient causes mechanical stress which result in changes of the index
of refraction (Photoelastic effect). (3) The temperature change causes the material
to expand (or contract). In the typical case encountered in experiments, the incident
beam is not uniform (it is Gaussian), which leads to modifications of this formula.
Also, in the case of lenses, the thermal lensing effect is to modify the initial curvatureso that focal length changes. Nevertheless, Eq. (2.52) describes the main ingredientsfor thermal lensing.

UVFS
NBK-7
TeO2
TGG

K
𝛼
o
(W/(m C))
(cm−1 )
1.38 [1]1 × 10−5 [2]1.114 [2]1 × 10−3 [4]
2 [53]
3.5 × 10−4 [27] 2
7.4 [27]𝑑𝑛/𝑑𝑇(10−6 /o C )11.9 [4]2.4 [4]< 5 (𝑛𝑒 , at 0.7𝜇m) [153]< 1 (𝑛𝑜 , at 0.7𝜇m) [153]
20 [27]1.31 × 10−3 [27]𝛼𝐶𝑇 𝐸
(10−6 /o C)
0.55 [4]7.1 [4]19.5 along (110) [3]
6.1 along (001) [3]
7 [163]

Table 2.1: Optical and thermal properties of common optical materials: 𝐾 is the thermal
conductivity, 𝛼 is the absorption coefficient, 𝑑𝑛/𝑑𝑇 is the thermo-optic coefficient, 𝛼𝐶𝑇 𝐸 isthe coefficient of thermal expansion. Note that the values are for a wavelength of 𝜆 = 1 𝜇m.
2Assuming 𝑑𝑛/𝑑𝑇 = 1 × 10−6 .

79The most commonly used types of glasses in our field are fused silica (UVFS,
i.e. the UV grade, brand names Suprasil 3001, Corning 7980) and the borosilicate
NBK−7. Their properties are summarized in Table 2.1. The main difference between
the two is that UVFS has two orders of magnitude lower absorption coefficient 𝛼,
which makes it the better choice for high power applications. We note that the dominant thermal lensing effect comes from (𝑑𝑛/𝑑𝑇 ), which is higher for UVFS but due
to the lower absorption, we observe no lensing for intensities up to 50-60 W/mm2
in UVFS lenses, while we do observe it for NBK-7 lenses. Note that the new lattice/MOT viewports are made out of Corning 7980 [1].

1/e 2 beam radius (mm)

wx
wy

1.8

1.6

1.4

1.2
0

5

10

15

20

25

30

35

Power (W)

Figure 2-18: Thermal lensing after an optical isolator (Thorlabs, IO-5-1064-VHP ). Plotted
are the horizontal and vertical beam sizes 𝑤𝑥 and 𝑤𝑦 respectively after a propagation distanceof 10-20 cm as a function of output power.

Table 2.1 also includes the most common AOM material (tellurium dioxide TeO2 )and the optical isolator crystal material TGG (terbium-gallium-garnet). At these
intensities, we typically observe little or no lensing in the AOMs, but we do observe
lensing in the isolators. Fig.2-18 shows a measurement of the size of a laser beam
after it has passed through a Thorlabs isolator (IO-5-1064-VHP). The addition of
an AOM does not change this behavior significantly. A characterization of thermal
lensing effects in similar setups can be found in [147]. Next, we follow the beam as itpropagates from the laser to the atoms.

80Seed and amplifier choice
The lattice and ODT light comes from a 500 mW Mephisto seed laser (from Coherent)
and a 50 W fiber amplifier. The Mephisto provides low noise power and has a narrow
linewidth (∼ 1 kHz) suitable for lattice physics. Originally, we used a Nufern NUAmp fiber amplifier which has been replaced by an AZUR ALS amplifier. The Nufern
has long output fiber which suffered from stimulated Brillouin scattering (which isaddressed next) and also switching electronics embedded in the drive circuits which
introduce intensity noise at 125 kHz and 250 kHz. These frequencies can drive transitions from the ground lattice band to the second excited band (see Section 3.2.3).
Stimulated Brillouin scattering (SBS) in fibers
One of the major issues we encountered with implementing a Mott insulator wasits poor lifetime, as detailed in Chapter 3, and a large portion of the problem wasresolved after the Nufern fiber was shortened. Therefore, here we give a brief overview
of the process of Stimulated Brillouin scattering (SBS) in fibers and present themeasurements of this process in the Nufern amplifier. At very high laser powers orfor very long fibers a stimulated process becomes important, in which a large fraction(sometimes even all) of the incident light is backscattered in a stimulated way. Thisis known as Brillouin scattering. At low powers, this process occurs spontaneously:
an incident photon is scattered by the fiber medium into a photon and an acoustic
phonon. Since the scattered photon has lower frequency than the incident one, it
is commonly referred to as the Stokes photon (from Stokes scattering). At higher
powers, this process becomes stimulated. In principle, the photon can be scatteredin any direction but due to the geometry of the fiber, the forward and backward
directions are preferred. Forward scattering should have a zero momentum transfer,
so that it should in principle not occur. However, the thermal excitations in thefiber are the eigenmodes of the cylindrical structure of the fiber rather than planewaves (which are assumed in the discussion of SBS), which makes forward scattering
possible. It is known as guided SBS and it was first studied in [143]. Here we focus

81

on backward scattering, the more dominant process.
The main cause of Brillouin scattering is electrostriction. This process occurs in all
dielectric materials: the presence of an electric field (light) causes a strain to develop
in the material, i.e a density change from ions moving in response to the applied
electric field, and a corresponding change in the index of refraction. The incident
photon interferes with the backscattered photon forming a moving optical latticewhich generates a moving density grating. When the intensity of the incident lightis high, the moving density grating affects the next photon and stimulates scattering
into the same backpropagating mode (Bragg scattering). Thus, for high intensities,
this process amplifies itself until, at some power, all the light is backscattered, orsome other process causes a breakdown.
In our case, SBS causes extra intensity noise in the lattice light. It is initiatedby thermally excited spontaneous photons. Coupled rate equations can be used todescribe the process (see e.g. [91]):𝑑𝑃𝑖= − 𝛾𝑚 ℒ(𝜔)𝑃𝑠 𝑃𝑖 − 𝛼𝑃𝑖𝑑𝑡𝑑𝑃𝑠=𝛾𝑚 ℒ(𝜔)𝑃𝑠 𝑃𝑖 − 𝛼𝑃𝑠𝑑𝑡(2.53)

where 𝑃𝑖 is the incident (pump) power, 𝑃𝑠 is the backscattered power. The absorption
coefficient 𝛼 accounts for loss in the fiber. This term can be ignored because for fiberson the order of a few meters, the attenuation is not significant. For instance, in 50 km
fibers, there is 10 % loss. Here 𝛾𝑚 is the gain coefficient for the m-th SBS mode.
Typically, there is only one dominant mode, so that 𝛾1 ≫ 𝛾𝑚′ , 𝑚′ = 2, 3, .... It is
given by [91]:𝛾=2(2𝜋)2 𝑛8 𝑝2𝑖𝑗
𝑔𝐵
= 3𝐴𝑐𝜆 𝜌𝐴𝜔𝑎𝑐 Γ(2.54)

where 𝑔𝐵 is the Brillouin gain coefficient, 𝐴 is the acousto-optic area, 𝑛 is the index
of refraction, 𝑝𝑖𝑗 is the respective electrostriction tensor, 𝜌 is the mean density of
the fiber material, 𝜆 is the wavelength of the incident light. The SBS gain has a82

Lorentzian spectral profile ℒ(𝜔) :ℒ(𝜔) =

Γ/2(𝜔 − 𝜔0 + 𝜔𝑎𝑐 )2 + (Γ/2)2(2.55)

where 𝜔0 is the frequency of the incident light and 𝜔𝑎𝑐 is the frequency of the phonon.
Typically, 𝜔𝑎𝑐 /(2𝜋) ∼ 11 GHz. With these definitions and with the approximations
𝛼 → 0 and that the pump light is not significantly depleted, we get:𝑑𝑃𝑖
= 0 → 𝑃𝑖 = 𝑃0
𝑑𝑧
𝑑𝑃𝑠
= 𝛾ℒ(𝜔)𝑃0 𝑃𝑠
𝑑𝑧(2.56)The second equation can be written in terms of the number of photons in the SBS
mode 𝑁 and the number of spontaneous photons 𝑁𝑠𝑝 = 1 + (𝑒~𝜔𝐵 /𝐾𝑇 − 1)−1 in thatmode [91]:𝑑𝑁=𝐺𝑃0 (𝑁 + 𝑁𝑠𝑝 )𝑑𝑧(︀)︀𝑁 (𝑧) =𝑁𝑠𝑝 𝑒𝐺𝑃0 (𝑧−𝐿) − 1(2.57)

where 𝐺 = 𝛾ℒ(𝜔) and we have used the boundary condition 𝑁 (𝐿) = 0. We cansee that in this model, thermally excited phonons seed the scattering and at higher
powers the number of phonons in one of the modes becomes dominant and leads tostimulated backscattering. The gain into that mode is exponential with fiber lengthand input power.
In our system, we measure the relative intensity noise (RIN) of the output light.
Over the frequency range DC to 1 MHz the noise increases with input power. We
interpret this to be due to SBS: when there is significant backscattering into a givenmode, the overall output power gets more noisy.
The noise spectrum is measured using the setup shown in Fig. 2-19. The twobackside-polished mirrors (Thorlabs BB1-E03P) attentuate the power. The time trace
is recorded on a photodiode (Thorlabs PDA10A) and then the Fourier transform is83

taken. The sensitivity of this photodiode to 1064 nm light is relatively low, so 100 mWof light are used, resulting in 9 V photodiode output for optimal performance. The
noisefloor is measured by turning the light off. The measured relative intensity noise(RIN) for a few input powers is shown in Fig. 2-19. To extract the RIN, two time
traces of total time 𝑇 need to be taken: an AC coupled one 𝑉𝐴𝐶 (for more resolution)
and a DC coupled one (for the average level 𝑉 ). The power spectral density 𝑆(𝑓 ) is
computed in the following way:
𝑉𝐴𝐶 (𝑡) − 𝑉
𝑉
⃒∫︁
⃒2
⃒
1 ⃒⃒ 𝑇𝑖𝜔𝑡 ⃒𝑆(𝑓 ) =2𝜋 ⃒𝜖(𝑡)𝑒 𝑑𝑡⃒
𝑇

(2.58)

𝜖(𝑡) =(2.59)

0so that the RIN can be expressed in dB/Hz as 10 log10 𝑆(𝑓 ).

-70
38W (39A)-80

36W (36A)
32W (33A)

RIN (dB/Hz)-90

29W (30A)
25W (27A)
2W (7A)noisefloor

-100
-110
-120
-130
-140
-150

50

100

150

200

250

300

350

400

450

500

550

600

Frequency (kHz)

Figure 2-19: Measurement of the RIN in the Nufern fiber amplifier. Shown are the measurement setup (top) and the RIN for several output powers.

84

This measurement can be used to get an estimate of the Brillouin gain coefficient𝑔𝐵 . Using Eq.(2.57), the total backscattered power 𝑃𝐵 can be modeled as having a
constant part and a fluctuating part:
𝑃𝐵 =

∑︁

𝑃𝑛 𝑒𝐺𝑃0 𝐿 𝑒−𝑖𝜔𝑛 𝑡 + 𝑃𝑐 𝑒𝐺𝑃0 𝐿(2.60)𝑛where 𝜔𝑛 is the frequency of fluctuations, and 𝐿 is the length of the fiber. We measure
the total output power:(2.61)

𝑃out (𝑡) = 𝑃0 − 𝑃𝐵 (𝑡)from which the RIN can be extracted:
𝜖=𝑃out (𝑡) − 𝑃out
𝑃out

(2.62)

where 𝑃out = 𝑃0 − 𝑃𝑐 𝑒𝐺𝑃0 𝐿 is the average output power and assuming that the in-put power is constant for the small backscattered powers. Fig. 2-19 shows that in

the region between 150 and 600 kHz the spectrum is flat, so just a single frequency
component 𝜔𝑛 can be used for the estimate:⃒⃒⃒ −𝑃𝑛 𝑒𝐺𝑃0 𝐿 ⃒2⃒
𝑆𝑓 ∝ ⃒⃒
𝑃0 − 𝑃𝑐 𝑒𝐺𝑃0 𝐿 ⃒(2.63)-80-90

RIN (dB/Hz)

-100
-110
-120
-130
-140

0

10

20

Output Power (W)30

40

Figure 2-20: RIN as a function of output power. The values are taken as the average overa frequency range 150-600 kHz, ignoring the peaks. The solid line is a fit of the form ofEq. (2.64).

85

This is shown in Fig. 2-20. Below 25 W, the RIN is constant, which is probably
just the intensity noise in the seed (Mephisto) light. Adding this in quadrature to the
SBS noise to get the RIN:

RIN(P0 ) = 20 log10

√︃

C2 +

(︂

A
B − e−GLP0)︂2(2.64)

where all constants of proportionality have been absorbed in the coefficients 𝐶 (which
accounts for any other noise source), 𝐴 (which accounts for the fluctuating part of
the SBS power), and 𝐵 (which accounts for the constant part of the SBS power).
Using this as a fitting function to the measured RIN in the range 150 kHz to 600 kHz,
results in the following fit parameters:
𝐶 = 2.4 (±0.1) × 10−7
𝐴 = 3.1 (±4.9) × 10−13
𝐵 = 1.8 (±3.5) × 10−8
𝐺0 = 𝐺 × 𝐿 = 0.46 (±0.05) 1/Watt(2.65)

If the fiber length over which SBS mostly happens is assumed to be 𝐿 = 1.5 m (1 moutside of the box and another 0.5 m inside) and the core of the fiber is 𝑑𝑐 = 7.5 𝜇m
in diameter, the Brillouin gain coefficient can be estimated to be:𝑔𝐵 =𝐺0 𝜋𝑑2𝑐= 1.35 × 10−11 m/Watt
𝐿 4(2.66)

We compare this to typical Brillouin gain coefficients 𝑔𝐵 in silica fibers which are on
the order of 1.9 to 5 × 10−11 m/W (see e.g. [119]). The errors in our estimate

are: (i) the exact fiber length is not known; (ii) the effective mode diameter could be
different from the actual core diameter; (iii) we have only a few points in our fit.
We note that this analysis is valid only for small 𝑃𝐵 , before depletion of thepump light becomes important. This is where all fiber amplifiers operate, because
the output power is maximized. When the Nufern output fiber length was decreased86to 20 cm, the noise spectrum was dominated by the noise from the seed laser.
Layout

PBS 2

AOM 1

AOM 3

PBS 3
AOM 2

Fiber amplifier

PBS 5

ODT 2

ODT 1

AOM 4

LATTICE 2

galvo

AOM 5

LATTICE 3

isolator

beam
dump

PBS 4

LATTICE 1

PBS 1

Figure 2-21: High power optical system. PBS 1 regulates the power in the setup. The galvo
sets the ratio between the power in the ODT and the power in the lattice. PBS 3 sets theratio between the power in the two ODT arms. PBS 4 sets the ratio between the power inlattice 1 and the total power in lattices 2 and 3. PBS 5 sets the ratio between the powers
in lattice 2 and lattice 3. Each of the five beams is fibercoupled. Since the lattices are
retroreflected, there are isolators in front of each fiber to prevent the retroreflected beamfrom propagating to the fiber amplifier.

After the laser source has been optimized, the light needs to be distributed to
the ODT and the lattice arms. Fig. 2-21 shows the laser path for high-power beam
delivery. Note that there is an isolator after the amplifier. The ALS laser has abuilt-in isolator, while the Nufern requires an external one. Since there is thermal
lensing in the isolator, when one has to be used, we run the full power through theisolator and then collimate the beam, i.e. the thermal lens at the highest power of
the amplifier is included as part of the setup. Then a PBS and a half-waveplate areused to attenuate the power for alignment.
The first design choice we make is pass each beam through an optical fiber. This87

has two advantages. (i) The beam profile after the fiber is almost a pure Gaussian, as
the only optics after the fiber are polarizers, mirrors, and focusing lenses. This leads
to clean ODT and lattice beam profiles. (ii) The physical setup can be separated inspace, minimizing the free-propagation distance and enabling a more modular andeasier to maintain setup.
The second design choice is whether to “chain” the AOMs, i.e. to reuse the 0thorder of the AOMs. While this seems to recycle power, the beam profile of the 0thorder is poor and does not lead to good fiber-coupling. Also, the power of each beam
becomes harder to control, because it depends on the level of the previous AOMs in
the chain. This increases the intensity noise of each subsequent arm. Finally, some
AOMs are sensitive to the input polarization and after passing through an AOM thepolarization may not be as well-controlled as if there were a PBS in front of the AOM.
Therefore, we split the power with polarizing-cubes (PBS) and waveplates into fivearms. Next we describe how this is implemented.
Galvo
Since one 50 W fiber amplifier does not provide enough power for all five arms at fullpower, but the lattice and the ODT are never on at full power at the same time inour sequence, we dynamically switch between the ODT and lattice paths during thesequence. This is done by rotating a half-waveplate, implemented with a modified
commercial galvanometer (GVS series from Thorlabs) with a home-built waveplate
mount. Alternatively, a commercial motorized rotation stage (Thorlabs DDR25) canbe used.
The principle of operation is illustrated in Fig. 2-22. A half-waveplate is attached
to a rod which is rotated by an angle ± 𝜃. The beam is positioned such that it isalways inside the waveplate. In general, a half-waveplate has two axis, known as thefast (F) and the slow (S) one. When a half-waveplate is placed between two parallel

polarizers, the first one for setting the initial polarization and the second one fordetecting the angle of the outgoing polarization, the polarization rotation from the
waveplate is turned into intensity variation. When either the fast or the slow axis is
88

a)

b)

Ideal transmission through PBS (%)

F
100
100

80

99
-22.5

-20

60
40
20
0
-100

-50

0

50

(degrees)

Figure 2-22: Rotating the angle of linear polarization with a galvo. a) A half-waveplate is
attached to a rod which is rotated by an angle ± 𝜃 (blue, green). The beam (red circle) is
positioned such that it is always inside the waveplate. b) Transmission through a polarizing
beam splitter positioned after the galvo as a function of the waveplate angle. The orange
line represents the range of operation 𝜃 = ± 20o . The inset is a zoomed-in version showingthat within this range not all the power can be switched from one path to the other.

aligned with the axis of the polarizers, the output has maximum intensity. Therefore,
if the half-waveplate is rotated by 45o , the output intensity goes between 0 and 100 %.
The Thorlabs galvos have a rotation rangle from 0 to 40o , so that it is not possible toswitch the entire power from the ODT to the lattice path. This is shown in Fig. 2-22,
where the polarization in front of the galvo is set (with another half-waveplate) sothat when the galvo is horizontal (𝜃 = 0), the transmission is 50 %.
In the sequence, the galvo rotates the waveplate in such a way that its motion
does not limit the ODT or lattice intensity stabilization. The best time to do theswitch is after ODT evaporation when the power in the ODTs is already low. At thattime, the lifetime of the BEC in the ODT is very long so that there is no heating
during the hold time required for the galvo to switch. The waveplate in front of thegalvo allows for the leftover power in the ODTs after the galvo has switched to bemodified. It is set so that there is 15-20% more than the final ODT power, so thatthe performance of the intensity stabilization system is not compromised.
The gavlo is the GVS series from Thorlabs (e.g. GVS011). We replace the mirror
with a threaded waveplate mount, see Fig. 2-23a, implemented on a 3D printer from
Formlabs 3 . There are two threads: one for mounting a 1-inch waveplate with a 13

https://formlabs.com/

89

b)

a)

Figure 2-23: Galvo implementation and performance. a) Picture of the galvo with thehome-built waveplate mount. b) Switching on, hold time, and switching off time traces.
The ramps are 100 ms each. Pink: drive voltage; Yellow: galvo position (this is the outputof the internal galvo position sensor). Note that the voltage scales are arbitrary.

inch ring from Thorlabs, the other for attaching an adjustable balance (a metal nut),
which allows for adjusting the waveplate mount so that it is horizontal when the galvo
PID is not engaged.
Fig. 2-23b shows the switching on and off of the galvo from 𝜃 = −20o , corresponding

to all power in the ODT, to 𝜃 = +20o , corresponding to all power in the lattice. The

switching can be done in 50-100 ms. We use the commercial driver which comes with
the GSV011 system, although the feedback parameters need to be tuned manually.
The procedure is a variant of the one described in a Thorlabs manual, which they
provide upon request, with the square-wave drive replaced by using a sinusoidal 1 Hz
drive. This is due to the larger moment of inertia of the home-built galvo comparedto the small mirror in the original system.
AOMs
Once the light is split into five arms, the intensity and frequency of each of themneeds to be controlled. This is done with acousto-optic modulators (AOMs). We use
TeO2 AOMs from Intraaction (80 MHz) and from Gooch&Housego (110 MHz). The
following briefly describes the principles of operation of the AOMs and then summarizes our observations about their performance, since those will be important for the90performance of the optical lattice and the Mott insulator, described in Chapter 3.
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vRF<latexit sha1_base64="ilkGYYetP4dXKqQ129jmaliD7RM=">AAAB7XicbVDLSsNAFL3xWeur6tLNYBFclUSLj11REJdV7APaUCbTSTt2kgkzk0IJ/Qc3LhRx6/+482+cpEHUeuDC4Zx7ufceL+JMadv+tBYWl5ZXVgtrxfWNza3t0s5uU4lYEtogggvZ9rCinIW0oZnmtB1JigOP05Y3ukr91phKxUR4rycRdQM8CJnPCNZGao57yd31tFcq2xU7A5onTk7KkKPeK310+4LEAQ014VipjmNH2k2w1IxwOi12Y0UjTEZ4QDuGhjigyk2ya6fo0Ch95AtpKtQoU39OJDhQahJ4pjPAeqj+eqn4n9eJtX/uJiyMYk1DMlvkxxxpgdLXUZ9JSjSfGIKJZOZWRIZYYqJNQMUshIsUp98vz5PmccU5qVRvq+XaZR5HAfbhAI7AgTOowQ3UoQEEHuARnuHFEtaT9Wq9zVoXrHxmD37Bev8CqXCPUw==</latexit>

2✓B<latexit sha1_base64="iSc+rEJnIoSFa/2+PWqb/C9FxyU=">AAAB8HicbVDLSsNAFJ3UV62vqks3g0VwVZJafOxK3bisYB/ShjKZTtqhk0mYuRFK6Fe4caGIWz/HnX/jJA2i1gMXDufcy733eJHgGmz70yqsrK6tbxQ3S1vbO7t75f2Djg5jRVmbhiJUPY9oJrhkbeAgWC9SjASeYF1vep363QemNA/lHcwi5gZkLLnPKQEj3dcGMGFAhs1huWJX7Qx4mTg5qaAcrWH5YzAKaRwwCVQQrfuOHYGbEAWcCjYvDWLNIkKnZMz6hkoSMO0m2cFzfGKUEfZDZUoCztSfEwkJtJ4FnukMCEz0Xy8V//P6MfiXbsJlFAOTdLHIjwWGEKff4xFXjIKYGUKo4uZWTCdEEQomo1IWwlWK8++Xl0mnVnXOqvXbeqXRzOMooiN0jE6Rgy5QA92gFmojigL0iJ7Ri6WsJ+vVelu0Fqx85hD9gvX+BXKekFE=</latexit>

I0

I 1

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Figure 2-24: AOM operation. When the incident light is at the Bragg angle 𝜃𝐵 , maximum
efficiency is achieved in the 1st order.

When an RF frequency is applied to a transducer at one end of an acousto-optic
crystal, an acoustic wave propagates within the crystal causing regions of compression
and changing the index of refraction at a period given by the drive frequency. An
incident optical beam is defracted because of the periodic modulation of the index ofrefraction. When the incident beam is at the Bragg angle, see Fig. 2-24, maximum
efficiency in the first diffracted order is achieved:
𝜃𝐵 =

𝜆
𝑓𝑅𝐹

𝑣𝑅𝐹

(2.67)

where 𝜆 is the wavelength of the incident light, 𝑓𝑅𝐹 is the frequency of the drive
signal, and 𝑣𝑅𝐹 is the velocity of propagation of acoustic waves in the crystal. For
TeO2 , 𝑣𝑅𝐹 = 4.2 mm/𝜇s, so that 𝜃𝐵 = 1.16o for 𝑓𝑅𝐹 = 80 MHz, and 𝜆 = 1064 nm. The
separation between the orders is at 2𝜃. If the incident beam has 1/𝑒2 radius of
𝑤00 = 1 mm, then clear separation between the two orders can be achieved when the
distance between their centers is about 4 to 5 𝑤00 , which corresponds to 10 to 12 cm.
When 𝑓𝑅𝐹 = 110 MHz, this distance is shorter: 7 to 9 cm.
By varying the power of the RF drive, the power in the diffracted order canbe tuned. This is used for intensity-stabilizing the ODT and lattice beams. The
propagation time of the acoustic wave in the crystal leads to a delay time, which91

limits the stabilization bandwidth (to about 50-100 kHz). On the other hand, if
the delay time can be accounted for, in open-loop mode, the bandwidth of the AOMs
allows fast switch on and off times less than 1 𝜇s. Here we summarize our observationsof the properties of these AOMs.
(1) The +1 order, which is at frequency 𝑓0 + 𝑓𝑅𝐹 , contains a little bit of the -1
order frequency 𝑓0 − 𝑓𝑅𝐹 (and vice versa). This is possibly due to a reflection of theincident RF wave from the edge of the crystal. Although crystals with slanted cuts
are used, some of this effect persists (see Section 3.2.3).

PBS 1<latexit sha1_base64="/cnPJi5Yv/mnys3iSKVEsOEUdvU=">AAAB+nicbVDLSsNAFJ3UV62vVJduBovgqiS1+NgV3bisYGuhCWUymbRDJ5MwM1FKzKe4caGIW7/EnX/jJA2i1gMDh3Pu4d45XsyoVJb1aVSWlldW16rrtY3Nre0ds77bl1EiMOnhiEVi4CFJGOWkp6hiZBALgkKPkVtvepn7t3dESBrxGzWLiRuiMacBxUhpaWTWnUAgnDpMR3yUpa1sZDasplUALhK7JA1QojsyPxw/wklIuMIMSTm0rVi5KRKKYkaympNIEiM8RWMy1JSjkEg3LU7P4KFWfBhEQj+uYKH+TKQolHIWenoyRGoi/3q5+J83TFRw5qaUx4kiHM8XBQmDKoJ5D9CngmDFZpogLKi+FeIJ0l0o3VatKOE8x8n3lxdJv9W0j5vt63ajc1HWUQX74AAcARucgg64Al3QAxjcg0fwDF6MB+PJeDXe5qMVo8zsgV8w3r8Apc+UZA==</latexit>

2<latexit sha1_base64="/cnPJi5Yv/mnys3iSKVEsOEUdvU=">AAAB+nicbVDLSsNAFJ3UV62vVJduBovgqiS1+NgV3bisYGuhCWUymbRDJ5MwM1FKzKe4caGIW7/EnX/jJA2i1gMDh3Pu4d45XsyoVJb1aVSWlldW16rrtY3Nre0ds77bl1EiMOnhiEVi4CFJGOWkp6hiZBALgkKPkVtvepn7t3dESBrxGzWLiRuiMacBxUhpaWTWnUAgnDpMR3yUpa1sZDasplUALhK7JA1QojsyPxw/wklIuMIMSTm0rVi5KRKKYkaympNIEiM8RWMy1JSjkEg3LU7P4KFWfBhEQj+uYKH+TKQolHIWenoyRGoi/3q5+J83TFRw5qaUx4kiHM8XBQmDKoJ5D9CngmDFZpogLKi+FeIJ0l0o3VatKOE8x8n3lxdJv9W0j5vt63ajc1HWUQX74AAcARucgg64Al3QAxjcg0fwDF6MB+PJeDXe5qMVo8zsgV8w3r8Apc+UZA==</latexit>

PBS 2

2 4
<latexit sha1_base64="2+8xv88acXRTlk2MZDdiOZy4TaQ=">AAAB+nicbVDLSsNAFJ3UV62vVJduBovgqiRafOyKblxWsA9oQplMJu3QySTMTJQS8yluXCji1i9x5984SYOo9cDA4Zx7uHeOFzMqlWV9GpWl5ZXVtep6bWNza3vHrO/2ZJQITLo4YpEYeEgSRjnpKqoYGcSCoNBjpO9Nr3K/f0eEpBG/VbOYuCEacxpQjJSWRmbdCQTCqcN0xEdZ2spGZsNqWgXgIrFL0gAlOiPzw/EjnISEK8yQlEPbipWbIqEoZiSrOYkkMcJTNCZDTTkKiXTT4vQMHmrFh0Ek9OMKFurPRIpCKWehpydDpCbyr5eL/3nDRAXnbkp5nCjC8XxRkDCoIpj3AH0qCFZspgnCgupbIZ4g3YXSbdWKEi5ynH5/eZH0jpv2SbN102q0L8s6qmAfHIAjYIMz0AbXoAO6AIN78AiewYvxYDwZr8bbfLRilJk98AvG+xeo2ZRm</latexit>

Photodiode 3

AOM
0
-1

Photodiode 2

b)

2

1.5

1

0.5

0

0

50

100

150

Relative power in 80MHz peak (dBu)a)

Percent rejected power (at PBS 2)

Photodiode 1

-55

-60
1st order
0th order

-65

-70

0

Linear polarization angle (deg.)

20

40

60

80

Efficiency (%)

Figure 2-25: Polarization rotation in an AOM. The measurement setup is shown on the
top. a) Percent rejected power measured at photodiode 1 as a function of the angle of theincident linear polarization. The rejected power has been minimized with the polarization
optics after the AOM. b) Relative power if the peak in the noise intensity spectrum at 𝑓𝑅𝐹
as a function of RF drive power, measured here as efficiency in the 1st order.

(2) If the incident linear polarization is not aligned with the crystal axes, theoutput polarization becomes elliptical and less pure. When a polarization sensitiveelement, such as a PBS, is placed in the light beam, this results in intensity modulation. We characterize this effect by using the setup shown in Fig. 2-25. A polarizing
92

beam splitter (PBS 1) sets the input linear polarization, which is then rotated by the
first half-waveplate. After the AOM, PBS 2 detects the state of the output polarization. The rejected power is measured at photodiode 1 as a function of the angleof the incident polatization. For each angle of the incident polarization, we optimize
the angles of the two waveplates after the AOM in order to minimize the rejected
power at PBS 2. The result is shown in Fig. 2-25a. There are two points at which
the rejected power is minimized, corresponding to horizontal and vertical polarization
with respect to the AOM. We interpret this variation of the percent rejected power
to signify that the polarization state of the outgoing light is less pure. For chained
AOMs this would lead to a chain-reaction: the more the output polarization of thefirst AOM varies, the more the output polarization of the second AOM varies and soon.
(3) The intensity spectrum of the 1st and 0th orders of the AOM contain a peak at
𝑓𝑅𝐹 when measured after PBS. In the setup shown in Fig. 2-25, we use photodiodes 2
and 3 together with a PicoScope FFT spectrum analyzer. Note that the amplitude ofthis peak is independent of the propagation distance after the AOM, i.e. it is not theresult of interference between the Gaussian beams of the two orders. The dependenceof the amplitude of the peak on AOM efficiency (varied by varying the RF power) is
shown in Fig. 2-25b. The higher the efficiency, the smaller the effect in the first order.
Note that this effect could be due to rescattering in the crystal or to polarization
rotation at 𝑓𝑅𝐹 , which then is measured as intensity modulation. However, we observe
little dependence of the amplitude of the peak on incident polarization angle.
(4) As a function of RF power, there is beam steering. This affects the fiber coupling efficiency even when the fibers are watercooled. Depending on the application,
this effect can be decreased by fiber-coupling at lower or higher RF power. Since wekeep the laser beams on for a long time at full power but also require that intensity
stability during the initial stages after turning on the beams is not compromised, we
optimize the fiber coupling efficiency at intermediate RF powers.

93Fibers and fibercoupling
After each beam has passed through an AOM, providing individual control, the lightneeds to be delivered to the atoms. At this point, the transverse intensity distributionof the beam (the beam shape) is not a pure Gaussian, i.e. it is not the TEM00 mode,due to lensing in an isolator or in other optics in the laser amplifier head or due tosmall distortions in the AOM crystal. Therefore, the next design choice we make is topass the beam through a cleanup fiber. This also makes the initial part of the setup
independent from the final beam delivery in terms of alignment and spatial location.
We pass each of the ODT and lattice arms through optical fibers. We use thelarge mode area polarization maintaining photonic crystal fibers from NKT Photonics LMA-PM-15. These photonic crystal fibers use a microfabricated region which
improves the single-mode operation. They are not hollow core and are not photonic
bandgap fibers, but rely on a modified index of refraction principle which enables
them to be single mode over a large wavelength range. For fibercoupling we use amolded aspheric lens from Thorlabs (C280-TME-1064, material ECO-550) 18.4 mm.
Although this material shown in principle exhibits thermal lensing, the short focal
length decreases the sensitivity to the addition of a second (thermally formed) lenswith a larger focal length.
Just before each of the three lattice fibers, we place an optical isolator (Thorlabs
IO-5-1064-VHP). This provides additional attenuation of the lattice retro beams.
Since the isolators exhibit thermal lensing (Fig. 2-18), the isolators have to be placed
before the fibers. Then, when the lattice arms are switched on at high power, thermal
lensing in these isolators would result in the power after the fibers decreasing slowly(over several milliseconds), which the intensity stabilization circuit should be able
to eliminate. Note that significant thermal lensing in the isolators is observed forincident powers above 10 W and each lattice arm requires at most 5 W (or 10 W if
the retro is included), so significant thermal lensing is not expected in these isolators
anyway. However, since the beam shape is important for preparing and interpreting
Mott insulators in the lattice, we aim to minimize the number of optical components

94

after the fibers which could distort the beam in any way. Therefore, the only opticalelements after the fiber are: a waveplate and a PBS, which set the polarization, twoalignment mirrors and a focusing UVFS lens, none of which have been observed to
exhibit thermal lensing.
Moveable mirrors
After the light beams have been prepared, they need to reach the atoms. The BEC
5 machine uses a single-chamber design, which does not require transport, but which
compromises optical access. This means that the MOT and the lattice beams sharethe same viewports. In order to avoid dichroics in the lattice beam paths, the MOT
mirrors are moved out of the way during the magnetic trap evaporation stage. Thisis done by using translation stages which are moved by air pressure pistons, shownin Fig. 2-26. Borrowing from the design by Aviv Keshet in BEC 2 [85], standardtranslation stages (Newport 433) are modified so that the pneumatic actuators from
Bimba can be mounted.

Figure 2-26: Moveable mirrors

The MOT position of the stages is set by the standard stage stopper (down left on
Fig. 2-26). The final position when the air is on is determined by the balance between
the air pressure in the piston and the spring in the stage. The air pressure is set suchthat the stages exactly reach the end of their travel range. Too much pressure could
lead to a sudden stop at the end, missaligning the mirrors. Note that after repeateduse, the pistons need to be cleaned to ensure they move all the way out. Since themirror mount uses a single screw to attach it to the post by a Thorlabs C-clamp,95

an additional plastic piece is added, securing the angle of the mirror. Also, it isimportant to use small orifices in the supply of the air to allow for smooth operation.
The orifices limit the flow so that there is no interference between the airflow of the
air tubes going to the different mirrors. The actuators use standard pressurized air
provided by MIT facilities, since too much air is used in a cycle and a pressurized
cylinder would not suffice. The stages can move to their final position within 1-2 s.
In the machine sequence, the air pressure is switched on during the magnetic trap
evaporation, leaving about 5 s for the stages to move. The mirrors have been tested
with repeated cycling resulting in no detectable missalignment of the MOT and Gray
molasses beams.
Picomotors and beam alignment
After the beams are prepared, they needs to be aligned with the atoms. Since alignment of the lattice, ODT, and the plug is very sensitive, we use motorized mirrors.
This also helps to keep the temperature in the enclosures constant, because alignment
does not require opening the enclosures. We use the New Focus 8821 series mirror
mounts, shown in Fig. 2-27. They use piezo actuators resulting in angular resolutionof 0.7 𝜇rad. These mirrors display hysteresis proportional to the step size. Therefore, they cannot be used for absolute positioning. Instead, an alignment procedure,
described below, has to be followed.

Figure 2-27: Picomotor mirrors used for lattice, ODT, and plug alignment.

Typically, the last mirror before the focusing lens is used for alignment, so thatangular displacements of the mirror lead to linear displacements of the beam focus.
However, this is too sensitive to align the plug and a different strategy has to be used.
96

The plug has the smallest beam size of around 20 𝜇m 1/𝑒2 radius, compared to 90 𝜇m
for the ODTs and 125 𝜇m for the lattices. For plug alignment, the picomotor mirror isplaced before an expanding telescope, so that large displacements of the mirror leadto small displacements of the beam position in the focus. This makes the alignment
easier and more precise.

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Figure 2-28: Decreasing alignment sensitivity. As used in the plug beam path, the angulardisplacement Δ𝜃 of the first mirror (left) results in linear displacement Δ𝑥𝐹 of the beam in
the focus of the last lens. The blue lines indicate the beam path and the dashed green lines
are lines parallel to the corresponding light rays through the center of the lenses, used todetermine the refraction through the lenses.

Fig. 2-28 illustrates this idea. To displace the focus of a beam by ∆𝑥𝐹 in the focal
plane of a lens with focal length 𝑓𝐹 , the incident collimated beam needs to propagate
at an angle 𝜃3 with respect to the optical axis. The displacement is:∆𝑥𝐹 = 𝑓𝐹 tan 𝜃3(2.68)

If a mirror produces the angular displacement 𝜃3 , the rotation of the mirror is directly
proportional to the displacement of the focus. The proportionality constant is the
focal length of the lens 𝑓𝐹 . If 𝑓𝐹 is large, small angular displacements 𝜃3 result inlarge ∆𝑥𝐹 . This works well for the lattice and ODT beams, but aligning the plug
beam in this way requires too small rotation of the mirror. Instead, the mirror before97

an expanding telescope is used, so that:tan 𝜃3 =𝑓1∆𝑥2=
tan 𝜃1
𝑓2
𝑓2(2.69)

as defined in Fig. 2-28. Finally, the angle 𝜃1 = 2∆𝜃, so that for tan 𝜃1 ≈ 𝜃1 :∆𝑥𝐹 = 2𝑓𝐹

𝑓1
∆𝜃
𝑓2

(2.70)

If the telescope is expanding 𝑓1 < 𝑓2 , large angular displacements ∆𝜃 result in small
position changes ∆𝑥𝐹 . In practice, the mirror also introduces linear and not justangular displacement, but this is a small effect.
Focusing, alignment, and calibration
Here we describe the final step of beam delivery: placing a beam of known shape andpower at the position of the atoms and calibrating the optical dipole potential which
the atoms experience. A rough alignment is performed by imaging the atoms froman axis along the beam we would like to align. Then, the beam is imaged along thesame axis, so that it can be placed where the atoms are. Since the resonant light used
for imaging the atoms (671 nm) and the ODT and lattice light (1064 nm) have quite
different frequencies, there is chromatic aberration in the optics used to form theimaging system. Nevertheless, this alignment is sufficient to place the beam so thatthe atoms feel its dipole potential, after which a more precise alignment procedure isused.
The next step is to position the focus of the beam at the position of the atoms.
This is done by maximizing its trap frequency. For each lattice beam (with blocked
retro), the trap frequency is measured by displacing the beam from the BEC. Thisis done by adiabatically loading into the combined potential of the crossed ODT and
the displaced lattice beam and then suddenly turning off the ODT beams. This leaves
the atoms a little bit away from the center of the lattice beam, so that center of massoscillations can be subsequently observed. Note that since we are interested in the98harmonic part of the Gaussian beam, the displacement needs to be small. For the
ODT beams, this procedure can be done by displacing the cloud by other means, e.g.
by pulsing on and off a gradient or by using the same setup of a crossed ODT anda displaced lattice arm (with blocked retro) and turning off one of the ODT beams
and the lattice arm. The center of mass oscillations can be observed in short time offlight, which is still sensitive to the initial position but which enhances the amplitudeof oscillation. A sample of this measurement is shown in Fig. 2-29. To vary the focus,
the fiber collimating lens, which is mounted on a cage z-stage (Thorlabs SM1Z), isCenter of mass position ( m)used rather than the focusing lens because it is more sensitive.
460

440

420

400
0

5

10

15

Time (ms)Figure 2-29: Beam focusing. Shown is the position of the atoms as a function of hold timein the beam potential. The solid line is a fit of the form 𝑓 (𝑡) = 𝑎 sin(2𝜋𝑓 𝑡 + 𝜑) + 𝑐. To focus,
the frequency 𝑓 is maximized.

After the beam is focused onto the atoms, it needs to be aligned. The alignmentof the ODT and the lattice beams is different and we describe both below. Since
the simulation of spin Hamiltonians with lithium in an optical lattice require theuse of the Feshbach resonances at fields between 600 Gauss and 1000 Gauss, the fullHelmholtz potential of the coils needs to be considered. In this configuration, there isa strong bias field in the 𝑧 direction and it has a saddle point in the middle. In order
to avoid magnetic field gradients, the atoms need to be placed at the saddle point.
This is done by aligning the crossed-ODT trap with this point.
In principle, the BEC can be produced anywhere around this point, as long as
the gradients are not too big to spill the atoms out of the cloud. After that, the finealignment of the ODTs to the saddle point can be performed by imaging the position99

of the BEC in each of the two arms from the top (imaging the 𝑥𝑦 plane). When oneof the ODT beams is adiabatically ramped to zero, the BEC moves, so that it is in
the minimum of the potential of the ODT and the magnetic field, i.e. the cloud moves
closer to the saddle point. Using the position of the BEC in each of the two ODTs,
the position of the saddle point can be triangulated. Finally, we can test if there is a
gradient on the atoms by observing their time of flight position. If they stay at thesame place, we have located the saddle point.
The intros of the lattice beams (retros blocked) can be aligned by “kicking”: bypulsing on and off the lattice beams faster than the trap frequency and observing
the position of the cloud in time of flight. The position of the atoms in TOF before
the kick can be marked on a camera. The imaging axis has to be chosen so that
motion perpendicular to the direction of propagation of the beam can be detected.
Longer times of flight lead to better sensitivity. The lattice beam is then moved(with a picomotor mirror) until the cloud does not move when the kick is applied.
This results in the atoms being in the middle of the lattice beam. The retro is thenoverlapped with the into by backcoupling the retro into the fiber. Note that theretro has a cat-eye configuration, i.e. after the collimating lens, there is another lens
which focuses the beam onto the retro mirror. This makes the setup less sensitive tovibrations and misalignment. In fact, although the intros have to be aligned once or
twice a week, the cat-eye retros stay aligned for much loner, on the order of weeks.
Finally, the lattice depth needs to be calibrated. This is done by amplitude
modulation, which couples the ground band to the second excited band. The BEC
is loaded into a single-arm lattice at typically 35 𝐸𝑅 , whose intensity is periodicallymodulated and the frequency of the modulation is varied. When the modulation
frequency is equal to the bandgap (at zero quasimomentum) the BEC is heated up.
This can be detected by ramping the lattice to zero and observing the cloud density
distribution in time of flight. The number of BEC atoms or the condensate fraction
can be plotted as a function of the modulation frequency. The modulation depth
is about 3 %. The modulation time is adjusted so that there is a detectable BEC
depletion at the resonance. One such measurement is shown in Fig. 2-30b. The
100a) 600a)
Number of BEC atoms

Frequency (kHz)

12000

500
400
300
200
100

q=0
q=2 /a L

0

10

20

30

8000

4000

40

450

Lattice depth (ER)

475

500

525

550

Modulation frequency (kHz)

Figure 2-30: Lattice depth calibration by AM modulation. a) The difference in energy between the ground band and the second excited band for all quasimomenta. We
use the 𝑞 = 0 value. b) Number of BEC atoms after modulation. The dip corresponds
22
to the resonance frequency, here at 35 𝐸𝑅 . The solid line is 𝑎(1 − 𝑏 𝑒−(𝑥−𝑥0 ) /2𝑤0 ) with𝑎 = 1.18(2)×104 , 𝑏 = 0.65(6)), 𝑤0 = 5.4(6), 𝑥0 = 499.7(6).

calculated band structure is then used to to determine the depth in 𝐸𝑅 , shown in
Fig. 2-30a. Now the lattice is ready to use!

2.2.2High field states and admixtures

The mapping to spin Hamiltonians typically requires the use of two long-lived nondegenerate states, so that the atoms in each state can be individually addressed andimaged. In the alkalis, typically, two of the ground hyperfine states are chosen. In
this section, we describe the hyperfine structure of 7 Li at high field, and the choice ofspin states. We calculate the imaging frequencies that need to be used and then look
into the nature of these states.
The Hamiltonian
Because of the Feshbach resonances of 7 Li in the region between 500 and 950 Gauss,
the atomic structure at these fields needs to be considered. The laser cooling is
performed at fields around zero Gauss, but evaporation in the ODT and all thelattice studies are done at high field. The Hamiltonian describing an alkali atom inan extrenally applied magnetic field 𝐵 is:
𝐻 = 𝐴𝐻𝐹 𝑆 I · J + 𝜇
˜B (gJ J · B + gI I · B)
101

(2.71)

were we use units of MHz, so that 2𝐴𝐻𝐹 𝑆 = 803.5040866 MHz [20] is the hyperfine
splitting at 𝐵 = 0, J = L + S is the electronic angular momentum and I is the nuclear
angular momentum (which for lithium is 3/2), the Bohr magneton is 𝜇˜𝐵 = 𝜇𝐵 /~ =1.399625 MHz/Gauss, 𝑔𝐼 = −0.001182213 [12] and the Lande g-factor 𝑔𝐽 is:𝑔𝐽 = 𝑔𝐿

𝐽(𝐽 + 1) + 𝐿(𝐿 + 1) − 𝑆(𝑆 + 1)
𝐽(𝐽 + 1) − 𝐿(𝐿 + 1) + 𝑆(𝑆 + 1)
+ 𝑔𝑆
2𝐽(𝐽 + 1)
2𝐽(𝐽 + 1)(2.72)

where 𝑔𝑆 = 2.00231930 and 𝑔𝐿 = 1. The Hamiltonian Eq. (2.71) can be directly diagonalized. We pick a basis: |𝐼, 𝐽, 𝑚𝐼 , 𝑚𝐽 ⟩. 𝐼 and 𝐽 are always good quantum

number while 𝑚𝐼 and 𝑚𝐽 are good quantum numbers only at high magnetic field.
The Hamiltonian couples states with the same 𝑚𝐹 = 𝑚𝐼 + 𝑚𝐽 . Therefore, only thetwo “stretched” states are not coupled: |𝐹, 𝑚𝐹 ⟩ = |2, 2⟩ and |𝐹, 𝑚𝐹 ⟩ = |2, −2⟩ corresponding to |𝑚𝐼 , 𝑚𝐽 ⟩ = | 23 , 21 ⟩ and |𝑚𝐼 , 𝑚𝐽 ⟩ = | − 32 , − 21 ⟩ respectively, where 𝐹 = 𝐼 + 𝐽.

The matrix elements are:𝜇𝐵
⟨𝑚𝐼 , 𝑚𝐽 |𝐻|𝑚𝐼 , 𝑚𝐽 ⟩ = 𝐴𝐻𝐹 𝑆 𝑚𝑖 𝑚𝑗 +𝐵(𝑔𝐽 𝑚𝐽 + 𝑔𝐼 𝑚𝐼 )
~
√︀
√︀
1
⟨𝑚𝐼 +1, 𝑚𝐽 −1|𝐻|𝑚𝐼 , 𝑚𝐽 ⟩ = 𝐴𝐻𝐹 𝑆 𝐽(𝐽 + 1) − 𝑚𝐽 (𝑚𝐽 − 1) 𝐼(𝐼 + 1) − 𝑚𝐼 (𝑚𝐼 + 1)
2
√︀
√︀
1
⟨𝑚𝐼 −1, 𝑚𝐽 +1|𝐻|𝑚𝐼 , 𝑚𝐽 ⟩ = 𝐴𝐻𝐹 𝑆 𝐽(𝐽 + 1) − 𝑚𝐽 (𝑚𝐽 + 1) 𝐼(𝐼 + 1) − 𝑚𝐼 (𝑚𝐼 − 1)
2
The resulting matrix is block-diagonal.
Eigenenergies and high field imaging frequencies
The energy level structure for the ground state 2𝑆1/2 is shown in Fig. 2-31a. Notethat the same diagram can be obtained with the Breit-Rabi formula. In the range ofFeshbach fields, the Zeeman levels have already separated into manifolds according
to their 𝑚𝐽 angular momentum. The Feshbach resonances which we use are between
the states in the lowest manifold. For simplicity, we will label them |𝑎⟩, |𝑏⟩, |𝑐⟩, and|𝑑⟩. In order to avoid dipolar relaxation, the lowest two states |𝑎⟩ and |𝑏⟩ can bechosen as spin states. Next, we calculate the resonant frequencies which are required
to image them.
102

a)

b) 500

2𝑆1/2

1500

2𝑃1/2
𝑚𝐽 = + 12

Energy (MHz)

Energy (MHz)

1000
500
0
-500

0

𝑚𝐽 = − 12

-1000
-1500

-500
0

200

400

600

800

1000

0

200

Magnetic field B (Gauss)

Energy (MHz)2000

𝑚𝐽 = + 23

1000𝑚𝐽 = + 120-1000

𝑚𝐽 = − 12

-2350

-2000

-2360

+3/2
+1/2
-1/2
-3/2

-2370
-2380

-3000

842

0

844

200

30

d)

2𝑃3/2

846

𝑚𝐽 = − 32

848

400

600

800

600

800

1000

2𝑃3/2

20

F=0
10
F=1
F=2

0

F=3

-10

MHz

c)

400

Magnetic field B (Gauss)

-20
-30

1000

Magnetic field B (Gauss)

0

2

4

6

Magnetic field B (Gauss)

8

10

Figure 2-31: Energy level structure of 7 Li in a magnetic field. a) 2𝑆1/2 . The states arelabeled by |𝑚𝐼 , 𝑚𝐽 ⟩. b) 2𝑃1/2 . The labels of the states are the same as in a). c) 2𝑃3/2 .
Here each 𝑚𝐽 manifold is further divided into four states labeled by their 𝑚𝐼 . The inset
shows this for 𝑚𝐽 = − 23 . d) 2𝑃3/2 at low fields.

To compute the imaging frequencies, the excited state manifolds need to be diagonalized as well. Here 𝐴𝐻𝐹 𝑆 = 45.914 MHz for the 2𝑃1/2 and 𝐴𝐻𝐹 𝑆 = − 3.055 MHz

for the 2𝑃3/2 [12]. The the energy level structure is shown in Fig. 2-31b-c. Notethat in the 2𝑃3/2 state, the hyperfine-structure constant is negative, resulting in the“inverted” level structure at low field, as show in Fig. 2-31d.
As in the ground state, the states are now split into manifolds with the same𝑚𝐽 and each manifold has four 𝑚𝐼 sublevels. Since electric dipole transitions (usedin imaging) only couple states with the same 𝑚𝐼 , i.e. they cannot flip the nuclear
spin, we can consider an effective level diagram which only accounts for 𝑚𝐽 . Thislevel diagram applies to each 𝑚𝐼 sublevel and is shown in Fig. 2-32. For imaging, the𝜎 − transition on the 𝐷2 line can be used because it requires the smallest frequency
detuning with respect to the zero field 𝐷2 line. In practice, we use linearly polarized103light perpendicular to the external magnetic field, so that both the 𝜎 − and the 𝜎 +transition are allowed. However, since they are separated by more than 2 GHz, thecoupling to the |𝑚′𝐽 = +1/2⟩ state is negligibly small.

Imaging frequency difference (MHz)Figure 2-32: Effective level diagram at high fields for each 𝑚𝐼 sublevel. The levels are
separated based on their 𝑚𝐽 angular momentum. The possible electric dipole transitions
are shown out of the ground state. For imaging, the 𝜎 − transition is used.

0
a
b
c
d

-200
-400
-600
-800
-1000
-1200
-1400
600

650

700

750

800

850

900

950

Magnetic field B (Gauss)

Figure 2-33: High field imaging frequencies. The frequency difference between the zero-field𝐷2 line and the high field 𝜎 − transition are shown for the four lowest hyperfine states. We
use the |𝑎⟩ and |𝑏⟩ states as spin states and the |𝑐⟩ state as an auxiliary state.

The frequencies required for imaging the 𝑚𝐽 = −1/2 levels in the ground state areshown in Fig. 2-33 with respect to the 𝐷2 line at zero field. The high field imaging
system is realized by an offset lock from the 𝐷2 line lock, since the frequency difference

between the two transitions is several hundreds of MHz. Since the imaging frequencies are relatively far detuned from each other (∼ 200 MHz), we can consider state104

preparation and detection schemes which involve imaging only one of the states without affecting the others. The probability of absorption for atoms in one state, if lightresonant with one of the other states is used, is suppressed by a factor 1/(1+(2𝛿/Γ)2 ),where 𝛿 is the detuning. For example, for atoms in state |𝑎⟩, if light resonant withthe |𝑏⟩ state is used 200 MHz away, the probability of absorption is suppressed to 1 %of its resonant value. If, on the other hand, light resonant with the |𝑑⟩ state is used

at 650 MHz, the suppression goes down to 0.1 %. Therefore, imaging the |𝑎⟩ and the|𝑏⟩ states separately is possible. Moreover, state preparation schemes involving “blow-

ing away” one of the species are also feasible, although probably the largest possible
detuning has to be chosen. One such scheme involves preparing a pure 𝑛 = 2 Mott
insulator plateaus without the surrounding 𝑛 = 1 shell. The atoms could be preparedin the Mott insulator in the |𝑎⟩ state and then the 𝑛 = 1 atoms could be sequentiallytransferred to the |𝑑⟩ state by Landau Zener sweeps (which is allowed, as we will seein the next section), after which resonant light could blow them away.
Eigenstates and RF transitions
So far we have assumed that the states are purely the |𝑚𝐼 , 𝑚𝐽 ⟩ states. However,

although this is true in the limit of very high magnetic fields, at the fields we areusing, there are admixtures which need to be taken into account. In fact, it is these
admixtures that allow us to drive the |𝑎⟩ to |𝑏⟩ transition with RF fields. Given the
diagonalization procedure above, states with the same 𝑚𝐹 = 𝑚𝐼 + 𝑚𝐽 are coupled,leading to admixtures of 𝑚𝐽 = +1/2 states in the lowest two hyperfine states. The|𝑎⟩ and the |𝑏⟩ state can be written in the following way:
⃒ 3 1⟩⃒ 1 1⟩⃒⃒|𝑎⟩ =𝐶𝑎 ⃒ + , −+ 𝛿𝑎 ⃒ + , +2 2
2 2⃒ 1 1⟩⃒ 1 1⟩⃒⃒|𝑏⟩ =𝐶𝑏 ⃒ + , −+ 𝛿𝑏 ⃒ − , +2 2
2 2(2.73)

which are normalized so that ⟨𝑎|𝑎⟩ = 1 and ⟨𝑏|𝑏⟩ = 1.

The values of 𝛿𝑎 and 𝛿𝑏 are shown in Fig. 2-34. From Eq. (2.73), the Rabi frequency

for coupling |𝑎⟩ and |𝑏⟩ is proportional to 𝛿𝑎 , so that high field spin flips require large
105a) 0.24
0.2
0.18
0.16
0.14
0.12
0.1
600

6
a
b

Probability | |2 (%)

Amplitude admixture

b)

a
b

0.22

650

700

750

800

850

900

5
4
3
2
1
600

950

650

700

Magnetic field B (Gauss)

750

800

850

900

950

Magnetic field B (Gauss)

Figure 2-34: Admixtures of 𝑚𝐽 = +1/2 states in the lowest two hyperfine states. Plottedare 𝛿𝑎 and 𝛿𝑏 from Eq. (2.73).
70- a (kHz/Gauss)

a
b

-1.3

-0.95

-1.35

50

-0.9

-1.4
600

700

750

800

800

Gauss

900

/ B

(MHz/Gauss)

-1.25

60

-1

b

40
30
600

650

700

850

Magnetic field (Gauss)

900

950

Figure 2-35: Differential magnetic moment of the lowest two hyperfine states. Inset: theindividual magnetic moments.

RF power which is proportional to 𝛿𝑎2 . Indeed, to have a Rabi frequency of 5 kHz, we
need a 100 Watt amplifier. This also means that the high field imaging transitions
are not cycling at the few percent level.
Finally, the different magnetic moments of the spin states need to be considered.
This difference can play a role when manipulating the atomic cloud with magnetic
field gradients. The magnetic moment is the derivative of the energy as a functionof magnetic field, i.e. the slope. This is shown in Fig. 2-35. The magnetic moments(inset) approach the Bohr magneton value of 1.4 MHz/Gauss at high fields. The
differential magnetic moment 𝜇𝑏 − 𝜇𝑎 approaches zero as the magnetic field goes toinfinity. The small differential magnetic moment implies that schemes in which the106two spin states are separated spatially by a magnetic field gradient, e.g. for observing
transport and spin mixing, are not suitable for lithium. On the other hand, this offers
better control for creating spin patterns, such as a helix within the cloud. This will
be discussed in Section 4.2.3.

2.2.3

Bakeouts and the titanium coated viewport

Building BEC 5 has involved several bakeouts and, for completeness, here we include
some time graphs of the temperature and pressure. Note that the initial Main chamberbakeout and the subsequent oven bakeout were done in 2011. Here we include themain chamber bakeout after swapping the Titanium-coated bottom bucket window
in the summer of 2017. The oven bakeout is from the winter of 2017 after a leakcoming from lithium accumulating on an unheated flange: at the reducer connecting
the oven to the six-way cross containing the cold cup, shown in Fig. 2-2.
Air molecules remaining in the main chamber can decrease the lifetime of theatomic samples we prepare. Therefore, ultra-high vacuum (UHV) is required, i.e.
pressures below 10−10 Torr. Air is pumped out of the vacuum chamber in several
stages: roughing pump, turbo-molecular pump, ion-pump. However, this can achieve
vacuum of around 10−9 to10−10 Torr, when the molecules adsorbed to the walls of thesteel chamber limit the pressure. The next step involves raising the temperature of
the steel chamber so that the molecules desorb and are pumped out by the ion pumps.
Finally, a layer of titanium is typically sprayed so that any remaining molecules stick
to it.
The problem with BEC 5’s main chamber design is that the Titanium filaments
had direct line of sight with the bottom bucket window so that it too got coated, as
shown in Fig. 2-36. The titanium layer heated up when high-intensity IR light for thelattice passed through it, creating a strong thermal lens. Although we could operatea lattice with a retro going through that viewport for some time, finally, when aneven larger intensity of the IR beam was applied, the coating was destroyed, making
the viewport opaque at that spot. We have since replaced the viewport.
Fig. 2-37 shows how the pressure in the chambers changes during the bakeout as107

Titanium filamentt linedirecht
of sigBottom bucket
window

Figure 2-36: Titanium coating on bottom bucket window. The red arrow shows the directline of sight of the filaments to the bottom bucket window.

a function of temperature. Although different components are heated up to different
temperatures and each one is monitored individually, here we plot an estimate of
the average temperature using these measurements. The pressure in each chamber ismonitored by a Thoria-Iridium ion gauge. After the temperature is first increased,
the pressure spikes, indicating the release of adsorbed molecules. After that, it slowly
plateaus. When the temperature is decreased back to room temperature, the pressure
also decreasing, signifying that the rate of outgasing decreases with temperature. Theoven and intermediate chambers included a second bake at higher temperature. Notethat the final pressure is much lower at the end of the bake. For the oven, the pressure
rises again when the oven is turned on. The final pressure in the main chamber isreached after firing the Titanium sublimation pump.
In this chapter, the road to placing ultracold atoms in an optical lattice has beenpresented. The operation and specific features of each step have been described.
After that, the design choices and methods for assembling the high power setup forthe lattice and ODT beams, which is at the heart of the experiment, have beenpresented. In the next few chapters, the behavior of the atomic system in an opticallattice and the different spin models which can be implemented with this machine
will be considered. Finally, two studies will be described, one with and one without
108

103

a)

10-4

Oven chamber

10-6

102

10-7

Temperature (C)Pressure (Torr)

10-5

Ion pumps ON
Second bakeOven ON
Final for oven ON
Final for oven OFF

10-8

0

1

2

3

4

5

6

7

8

9

10

101
11

Days
103

b)

10-4

Int. chamber

10-6

102

10-7

10

Temperature (C)Pressure (Torr)

10-5

Ion pumps ON
Second bakeOven ON
Final for oven ON
Final for oven OFF

-8

0

1

2

3

4

5

6

7

8

9

10

101
11

Days
10-5

c)

103
TiSub degas
Ion Pump ON

10-7
10-8

102

10-9

Main chamber

10-10
10-11

Temperature (C)Pressure (Torr)

10-6

0

1

2

3

4

5

6

7

8

101

Days

Figure 2-37: Bakeouts. Plotted are the pressures (black) and the temperatures (orange)
during the bakes. These are two separate bakes: (i) oven a) and intermediate chamber b)and (ii) main chamber c).

109

a lattice, which both aim at investigating novel quantum phases.

110Chapter 3
Atoms in Optical Lattices
The scientific agenda of BEC 5 is to study quantum lattice spin models, which canbe simulated by atoms in optical lattices by mapping the Hamiltonians of the twosystems. Interacting atoms in an optical lattice are described by the Hubbard model
which has a quantum phase transition from a superfluid (SF) to a Mott insulator(MI). In the MI, the atoms are confined to individual sites and it is in this state thatthe mapping to spin models is realized. In this chapter, the theoretical background of
the Hubbard model and the SF-MI transition is presented. Then, the specific featuresand difficulties associated with implementing and diagnosing a MI are discussed. The
focus is placed on ways of preparing and characterizing the MI as a starting point ofspin experiments. Here, atoms in a single state are considered, except when a second
state is needed for diagnostics, while Chapter 4 describes the spin models when asecond state is added.

3.1

Theoretical description

When a single particle is placed in a periodic potential, the energy structure is modified and is characterized by the presence of separate energy bands. When interacting
particles are placed in a lattice, the interplay between the individual particle kinetic
energy and the interaction energy between two particles on the same site is capturedby the Hubbard model. This chapter describes this behavior.
111

3.1.1

Band structure and wavefunctions

A single particle in an optical lattice is described by the Hamiltonian:
𝐻=−

~2 2
∇ + 𝑉 cos2 (𝑘𝐿 𝑥)
2𝑚

(3.1)

where 𝑘𝐿 = 2𝜋/𝜆 is the lattice wavevector. The Schrödinger equation(3.2)

𝐻Ψ𝑛,𝑞 = ℰ𝑛 (𝑞)Ψ𝑛,𝑞

can then be solved (for an introduction to periodic structures see [165] and for amore cold-atom perspective see [62]), revealing that the eigenenergies ℰ𝑛 (𝑞) form
separate bands, indexed here by 𝑛. They are plotted in Fig.3-1 for several values ofthe depth of the potential 𝑉0 = 𝑉 /𝐸𝑅 , where 𝐸𝑅 = ~2 𝑘𝐿2 /2𝑚 is the recoil energy. For
increasing lattice depth, the gaps between the bands increase and the widths of the
bands decrease. When atoms are loaded adiabatically into an optical lattice startingfrom a BEC in the ground band of the trapping potential (e.g. the optical dipole
trap), they occupy the lowest band only of the lattice potential at quasi-momentum
𝑞 = 0.
V0 =0

V0 =15

V0 =35

30

25

35
25

Energy (ER)

20

30
20

25

15

20

15
10

15

10

10

5

5

0
-1

0

1

0
-1

5

0

1

0
-1

0

1

1st Brillouin zone (qa L/2 )Figure 3-1: Energy bands for a particle in a sinusoidal potential for three lattice depths.

For a single particle in a periodic potential, the eigenstates are Bloch wavefunctions112

Ψ𝑛,𝑞 (x) in energy band 𝑛 and quasimomentum 𝑞:(3.3)

Ψ𝑛,𝑞 (x) = 𝑒𝑖𝑞𝑥 𝑢𝑛,𝑞 (x)where 𝑢𝑛,𝑞 (x) = 𝑢𝑛,𝑞 (x + aL ) are periodic functions in space. The Bloch wavefunctions are delocalized in space. They can be expanded into Wannier functions, which
are localized on each site, and form an orthogonal basis. The Wannier function forsite 𝑖 is:

1 ∑︁ 𝑖𝑞𝑥𝑖
𝑤𝑛 (x − xi ) = √𝑒 Ψ𝑛,𝑞 (x)
𝒩 𝑞

(3.4)

with normalization constant 𝒩 . The Wannier function at 11 𝐸𝑅 centered around𝑥 = 0 is plotted in Fig. 3-2. The Wannier functions are more suited to describing

particles in the Mott insulating state, where particles on a lattice become localized
due to the strong interactions between them compared to their kinetic energy. This
corresponds to a delocalization in quasimomentum space.
a)

10 6

6

b)

10

10

5
10

0

|w(x)|2|w(x)|2

4
3

10 -10

2
1
0
-5

-4

-3

-2

-1

0

1

2

3

4

10 -20-55Position ( /2)

-4

-3

-2

-1

0

1

2

3

4

5

Position ( /2)

Figure 3-2: Wannier functions at 11 𝐸𝑅 . a) Linear scale. b) Logarithmic scale.

3.1.2

Scattering in a lattice

The discussion so far involved non-interacting particles. However, the MI phase relies
on interactions. The free space scattering properties could in principle be alteredwhen the atoms are placed in a confining potential. Typically, one compares thedifferent length scales which characterize the interactions and the confining potential.
113

For example, a tight harmonic confining potential could be applied along one axis
(e.g. along 𝑧), characterized by the harmonic oscillator length. For 7 Li in a lattice of
𝑉 = 35 𝐸𝑅 , it is:

√︂

~
=
𝑚𝑟 𝜔

√︃

2~
≈ 100 nm
(3.5)(2𝑚𝑉 𝑘𝐿2 )1/2
√︀
where we have used 𝑚𝑟 = 𝑚/2 (the reduced mass) and 𝜔 = 2𝑉 𝑘𝐿2 /𝑚 from Eq. (2.43)
𝑙ℎ𝑜 =(the on-site trap frequency). This could be compared to the van der Waals radius,
the effective radius of closest approach between two particles with van der Waals
interactions. For lithium, it is 𝑟𝑣𝑑𝑊 ∼ 180 pm, so that 𝑟𝑣𝑑𝑊 ≪ 𝑙ℎ𝑜 . Therefore, theBusch model [30] can be used to describe the scattering. In this model, two interacting
cold atoms confined in a harmonic potential are described by the 1D Schrödinger
equation with pseudo potential interaction.
The strength of the interactions for cold enough temperatures, can be described by
the scattering length 𝑎 (Section 1.1.2). When the harmonic oscillator length becomescomparable to the scattering length, a resonance occurs in the following sense. Using
the pseuso potential
𝑉 (𝑟) = 𝑔𝛿(𝑟)

𝜕
𝑟
𝜕𝑟

(3.6)

the effective interaction strength in the confining direction can be expressed as [121]:
𝑔1𝐷 =

2~2 𝑎
1
𝑚𝑟 𝑙⊥ (1 − 𝐶𝑎/𝑙⊥ )(3.7)

where 𝑎 is the free-space scattering length, 𝐶 ≈ 1.46. There is a resonance when𝑎 = 𝑙⊥ /𝐶. In the resonance regime a Tonks-Girardeau gas can be realized [123], inwhich the strongly-interacting bosons acquire fermionic properties, because they candecrease their energy by not occupying the same spatial position. For lithium in a
𝑉 = 35 𝐸𝑅 lattice:𝑙⊥ /𝐶 =

√︃

2~
1
= 1275 𝑎0
2 1/2(2𝑚𝑉 𝑘𝐿 ) 1.46

(3.8)

Since in our experiments, typically the scattering length is kept below 400 𝑎0 , thisregime is never entered.

114

Finally, for large scattering lengths (on the order of hundreds of 𝑎0 ), the on-site
interactions can be modified because of the large repulsion/attraction between the twoparticles. For example, for repulsive interactions (𝑎 > 0), the on-site interaction energy
𝑈 (Eq. (3.10)) is decreased because the spatial overlap between the two wavefunctions
is decreased from the repulsion. This effect is sometimes described by the admixtureof higher bands to the wavefunction and has been observed with cold atoms in opticallattices [159, 107].
In summary, to a very good approximation, for the scattering lengths used in our
experiments, the interactions can be described by the free space scattering length 𝑎.

3.1.3

The Bose-Hubbard model

The Bose-Hubbard model describes well the properties of atoms in optical lattices.
This section presents the model and the phase transition between a superfluid and a
Mott insulating state. The Hamiltonian is:
𝐻𝐵𝐻 = −𝑡∑︁ †∑︁ †𝑈 ∑︁ † †𝑎𝑖 𝑎𝑗 +
𝑎𝑖 𝑎𝑖 𝑎𝑖 𝑎𝑖 − 𝜇
𝑎𝑖 𝑎𝑖
2 𝑖
𝑖

(3.9)

⟨𝑖,𝑗⟩

where 𝑡 is the tunneling matrix element between neighboring sites (the sum ⟨𝑖, 𝑗⟩ isover nearest neighbors), 𝑈 is the on-site interaction, and 𝜇 is the chemical potential.
Hubbard parameters
Using the Wannier functions 𝑤(𝑥), the two parameters can be expressed as [78]:
(︂
)︂
~2 𝑑2
𝑡 = 𝑑𝑥 𝑤 (𝑥 − 𝑥𝑖 ) −− 𝑉𝐿 (𝑥) 𝑤(𝑥 − 𝑥𝑗 )2𝑚 𝑑𝑥2∫︁
4𝜋~2 𝑎𝑈=
𝑑3 x|𝑤(x)|4𝑚∫︁

*(3.10)

where 𝑉𝐿 (𝑥) is the lattice potential. In the limit of a deep lattice: 𝑉0 = 𝑉 /𝐸𝑅 ≫ 1

the tight-binding approximation can be used and the Hubbard parameters can be

115

a)

b)

2.2

2

20

UTB/U

10

1.6

U
UTB

5

1.6

t (kHz)

1.8

1.8

15

0
0

20

40

0

1.4

Lattice depth (E R)

1.4

t
t TB

10 0

t TB/tU (kHz)220

40

Lattice depth (E R)

1.2

1.2
1

1
0

5

10

15

20

25

30

35

0

5

10

Lattice depth (E R)

15

20

25

30

35

Lattice depth (E R)

Figure 3-3: Hubbard parameters as a function of lattice depth. The main plots show theratio between the tight-binding expressions (Eq. (3.11)) and the exact values (Eq. (3.10)) fora) the on-site interactions 𝑈 and b) the tunneling 𝑡. The insets show the actual values as afunction of lattice depth. Here only the lowest band of the lattice is considered.

expressed as [26]:√4
3/4
𝑡𝑇 𝐵 = √ 𝐸𝑅 𝑉0 𝑒−2 𝑉0𝜋√︂8
𝑎𝑘𝐿 𝐸𝑅 (𝑉0𝑥 𝑉0𝑦 𝑉0𝑧 )1/4
𝑈𝑇 𝐵 =𝜋(3.11)

with 𝑉0 in recoil units. The expression for 𝑈 can be obtained by using the harmonic
oscillator wavefunctions (Eq. (2.44)) instead of the Wannier functions. The expression
for 𝑡 is valid for the lowest band and is obtained by equating 𝑡 with the bandwidthof the lowest band of the 1D Mathieu equation. In the tight-binding approximation,
the energy of the lowest band is ℰ = ℰ0 + 2𝑡 cos(𝑞𝑥 𝑥) and the width is 4𝑡. The analytic

expressions Eq. (3.11) are fairly good approximations to Eq. (3.10) for a deep lattice.
A comparison between the two is plotted in Fig.3-3. For depths below 5 𝐸𝑅 the two
begin to deviate.
Superfluid to Mott insulator transition
In the Bose-Hubbard model, there is a transition from a superfluid state with delocalized particles to a Mott insulating state in which the particles are localized oneach site. To illustrate this transition, here a mean-field approach is used, following[155, 144]. The goal is to transform the tunneling term in such a way as to obtain116

an effective Hamiltonian which does not couple different sites. Using the superfluidorder parameter 𝜓 = ⟨𝑎𝑖 ⟩ = ⟨𝑎†𝑖 ⟩, which is assumed to be real, the tunneling term canbe transformed:(︁)︁𝑎†𝑖 𝑎𝑗 → ⟨𝑎†𝑖 ⟩𝑎𝑗 + 𝑎†𝑖 ⟨𝑎𝑗 ⟩ − ⟨𝑎†𝑖 ⟩⟨𝑎𝑗 ⟩ = 𝜓 𝑎†𝑖 + 𝑎𝑗 − 𝜓 2(3.12)

After inserting this into Eq.(3.9) and summing over nearest neighbors 𝑧 = 2𝑑 for a𝑑-dimensional system, the effective Hamiltonian on site 𝑖 becomes:(︁)︁˜ 𝑖 = −𝑧𝑡 𝑎† + 𝑎𝑖 𝜓 + 𝑧𝑡𝜓 2 + 𝑈 𝑛𝑖 (𝑛𝑖 − 1) − 𝜇𝑛𝑖𝐻𝑖2(3.13)

where the number of atoms on a site is 𝑛𝑖 = 𝑎†𝑖 𝑎𝑖 . This Hamiltonian can be diagonalized by truncating the space to some maximum 𝑛𝑖 , which is much larger than the
typical occupation per site. Then, the lowest eigenenergy is minimized with respect
to the superfluid order parameter 𝜓. The phase diagram can be obtained by plotting 𝜓𝑚𝑖𝑛 . When 𝜓 = 0, the system is a MI, otherwise it is a SF, which is shown in
Fig. 3-4a. For more details, see Appendix A.
a)

4

b)

n=4
3

/U

n=3

SF

2

n=2
1

n=1
0

0

0.05

0.1

0.15

0.2

0.25

0.3

zt/U

Figure 3-4: Mean-field Bose-Hubbard phase diagram. a) Plotted is the 𝜓 which minimizes
the ground-state energy with 𝜓 = 0 corresponding to the Mott state. b) Phase boundaries
using Eq. (3.14). The dashed line indicates the critical point for 𝑛 = 1 using Eq. (3.15).

The phase boundaries can be found by using second-order perturbation theory
for 𝑈 ≫ 𝑡 and treating the tunneling term in Eq.(3.13) as a perturbation. The eigen-states of the unperturbed Hamiltonian are Fock states with a fixed number of atoms
117

(0)

|𝑛⟩ with energy ℰ𝑛 . Since the first-order corrections to the energy vanish because
(︀
)︀
(2)
⟨𝑛| 𝑎† + 𝑎 |𝑛⟩ = 0, the energy has to be expanded to second order ℰ𝑛 . When the to(0)(2)

tal energy ℰ𝑛 = ℰ𝑛 +ℰ𝑛 is minimized with respect to the superfluid order parameter𝜓, the phase boundaries are obtained (Appendix A):
𝜇± =1 √︀ 2
1[𝑈 (2𝑛 − 1) − 𝑧𝑡] ±𝑈 − 2𝑈 (2𝑛 + 1) 𝑧𝑡 + (𝑧𝑡)222(3.14)

They are plotted in Fig. 3-4b. The SF-MI transition is typically defined as the point
where the upper and lower phase boundaries of the MI meet 𝜇+ = 𝜇− at:
(︂

𝑈
𝑧𝑡

)︂

= (2𝑛 + 1) +𝑐√︀
(2𝑛 + 1) − 1(3.15)

so that for 𝑛 = 1, the well-known result (𝑈/𝑡)𝑐 = 𝑧 5.8 is obtained. Higher order
calculations, numerical Monte-Carlo, and DMRG methods show that the meanfield treatment is a good approximation in 2D and 3D, however in 1D quantum
fluctuations have a significant effect. In 1D the prediction for the phase transition is (𝑈/𝑡)𝑐 = 3.128 = 𝑧 1.564 [42]. Quantum Monte Carlo simulations in 2D give(𝑈/𝑡)𝑐 = 16.7392 = 𝑧 4.1848 [33] and in 3D: (𝑈/𝑡)𝑐 = 29.34 = 𝑧 4.8900 [32].
Density distributions in a harmonic trapa)b)

Figure 3-5: Wedding cake structure of a MI in a 3D harmonic potential. a) At a lattice
depth of 35 𝐸𝑅 , deep in the MI, and b) at 17 𝐸𝑅 where some of the plateaus are melted.
Here the scattering length is 100 𝑎0 and the total number of atoms is 100,000.

118

In most cold-atom experiments, a trapping potential is used, which results in avariation of the chemical potential across the cloud. Therefore, in experiments, a cutof the phase diagram along a constant (to within a few percent) 𝑡/𝑈 is implemented.
Typically, a local chemical potential 𝜇(r) is defined in terms of the global chemical
potential 𝜇, which determines the total atom number, and the trap potential 𝑉𝑡 (r):(3.16)

𝜇(r) = 𝜇 − 𝑉𝑡 (r)This results in a “wedding cake” structure, in which MI plateaus alternate with small
superfluid regions. A sample is shown in Fig. 3-5 for two lattice depths. In a deep
lattice of 35 𝐸𝑅 , all three MI shells are formed, while in a shallower lattice of 17 𝐸𝑅 ,the 𝑛 = 3 MI shells has not formed yet and the system is superfluid in the middle.
14
tunneling t
interactions U

12superexchange t 2 /U

Percent %

10
8
6
4
2
0

0

5

10

15

20

25

30

35

Lattice depth (E R)

Figure 3-6: Variation of the Hubbard parameters across the cloud. For reference, the variation of the superexchange matrix element 𝑡2 /𝑈 is also shown. Here the lattice depth isassumed to vary by 1.27% across the cloud (see main text).

Note that due to the trap, the lattice depth varies across the cloud, resulting ina variation of the Hubbard parameters as well. As described in Chapter 2, in oursystem the trapping potential is due to the lattice beams, which are chosen to have
125 𝜇m 1/𝑒2 radius. In this potential, for a cloud size of 20 𝜇m (40 lattice sites),
there is a 1.27% variation of the lattice depth across the cloud. The variation of the
Hubbard parameters are plotted in Fig. 3-6 for a 3D lattice. When only one or two of
the lattice arms are varied (for 1D or 2D dynamics), the variation of the interactionenergy 𝑈 is smaller.
119

3.2

Lattice Methods

The Hubbard model describes the physics of the density sector: how the number ofatoms per site in a given band changes as a function of lattice depth and interactions.
One of the main assumptions in the mapping to the Heisenberg model is that thedensity distribution is homogeneous, e.g. a MI with one atom per site. However, ina real system, the trap leads to a wedding cake structure, there are number defects(such as holes and doublons in the 𝑛 = 1 MI), and some particles can be excited tohigher bands, leaving the Hilbert space of the Heisenberg model and also heating thesystem. In this section, we describe the methods we have used to characterize theMI in order to eliminate imperfections or decrease their importance. This involves
two steps: first, the preparation of the MI should not cause excitations and second,heating and loss processes in the MI should be avoided. Note that our system does
not have single-site imaging resolution, therefore, bulk techniques need to be used.

3.2.1

Interaction Spectroscopy

Since throughout this section, we will refer to a technique that we have developed,
Interaction spectroscopy, we describe its principle first. For details, see [11], reprinted
in Appendix D, here we summarize the main points. Interaction spectroscopy enables
the manipulation of individual Mott insulator shells based on their on-site occupation. In the spirit of spectroscopy, the lowest two hyperfine states of 7 Li at highmagnetic field are labeled here as |𝑎⟩ and |𝑏⟩. For exactly one atom per site, thetransition frequency 𝜔𝑎𝑏 is determined by the atomic structure at a given magnetic
field. When a second atom is added, the interaction between the two atoms shifts
the energy levels. This shift, also known as clock shift [31], is large enough in 7 Li in
the region of magnetic fields around the Feshbach resonances to be easily detectable
spectroscopically. For example, if there are two 𝑏 atoms on a site, i.e. the |𝑏𝑏⟩ state,
and we would like to flip exactly one atom to 𝑎, i.e. to obtain the |𝑎𝑏⟩ state:(︂)︂1|𝑎𝑏⟩ ≡ √ |𝑎𝑏⟩ + |𝑏𝑎⟩2
120

(3.17)

the transition frequency is:𝜔𝑏𝑏→𝑎𝑏 = 𝜔𝑎𝑏 + (𝑈𝑎𝑏 − 𝑈𝑏𝑏 ) /~ ≡ 𝜔𝑎𝑏 + ∆1 /~(3.18)

Similarly, if we start with the |𝑎𝑏⟩ state, the transition frequency to the |𝑎𝑎⟩ state is:𝜔𝑎𝑏→𝑎𝑎 = 𝜔𝑎𝑏 + (𝑈𝑎𝑎 − 𝑈𝑎𝑏 ) /~ ≡ 𝜔𝑎𝑏 + ∆2 /~(3.19)

This technique relies on the three scattering lengths 𝑎𝑎𝑎 , 𝑎𝑎𝑏 , 𝑎𝑏𝑏 (and hence on theinteractions 𝑈𝑎𝑎 , 𝑈𝑎𝑏 , 𝑈𝑏𝑏 ) being different from each other. Fig. 3-7 illustrates theconcept. If the the interaction shifts ∆1 and ∆2 are the same, we cannot spin flip√
only one of the two atoms. Instead, each atom will be in a superposition (|𝑎⟩+|𝑏⟩)/ 2,
resulting in:
(︂
)︂(︂
)︂
(︂
)︂
1
1
1
1
|𝑎⟩ + |𝑏⟩ |𝑎⟩ + |𝑏⟩ = |𝑎𝑎⟩ +|𝑎𝑏⟩ + |𝑏𝑎⟩ + |𝑏𝑏⟩
|Ψ⟩ =
2
2
2
2

(3.20)

This shows that the |𝑎𝑎⟩ state will also be populated. However, if the interaction

shifts are different, we can flip exactly one of the two atoms and prepare the |𝑎𝑏⟩state without populating the |𝑎𝑎⟩ state, whose energy is detuned from the driving
frequency 𝜔𝑏𝑏→𝑎𝑏 .

|aai

|aai

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1(|ai + |bi) (|ai + |bi)2
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1
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Figure 3-7: Interaction spectroscopy for two particles per site. When all three interactions
are the same 𝑈𝑎𝑎 = 𝑈𝑎𝑏 = 𝑈𝑏𝑏 , the energy differences Δ1 = Δ2 (left). Therefore, it impossible
to flip only one atom on a doubly-occupied site. When Δ1 ̸= Δ2 , a state with exactly one
atom in each state |𝑎𝑏⟩ can be prepared (right).

121

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122

y (kHz)ermine the differential
nput to a many-bodyAtom Number (a.u.)

this technique to mean=1
n=2
of two confined atoms
in by preparing an n=
847.0 G
erfine state in a 1064847.4 G
us and Bose-Einstein
847.7 G
cribed elsewhere [13].
848.1 G
-2
0
2
-4
-6
rf Detuning (kHz)e lattice depth is 35ER
848.3 G
848.5 G
plateau, approximately
849.5 G
= 1 shell containing a849.9 Grf drive for 2.9 ms and
850.9 G!
her hyperfine state as a
!ab + 1 /~861.0 G
quency corresponding
868.1 Gbserve a peak coming
871.1 Gbserve a second peak
874.6 G
Figure 3-8: Interaction spectrum. Top: Number of 𝑎-atoms as a function of central Landaud by the differencein877.7 G
Zener sweep frequency relative to the pure transition frequency 2𝜋𝜔𝑎𝑏 . Bottom: schematicstate (|aa⟩ or representation|bb⟩) and of each peak of the spectrum. The left peak corresponds to the |𝑏𝑏⟩ → |𝑎𝑏⟩881.5 Gand the right peak corresponds to the |𝑏⟩ → |𝑎⟩ transition. Spectrum from [11].
orresponds to transitiona π pulse
-80
-60
-40
-20
0
20
40
60
8
ize the signal), Fig.
which
rf detuning(kHz)insulator with
3-8 illustrates the advantages of this technique.
In a Mott
that of the n= 1 sites,
several MI shells of different site occupations, we can detect the number of atoms ineraction blockade [14]FIG. 2. Fits to representative spectra of the transition between
each shell by picking the RF drive frequency and measuring the number of transferredrgies in the three states
and a, taken between the two bb Feshbach resonances. The spectr
atoms. For example, by picking the RF frequency to be the pure transition frequency
lectively between |aa⟩
have been plotted so that the Zeeman-shifted peaks of the n=
𝜔
,
only
atoms
in
𝑛
= 1 will be addressed.
the RFfrequency
is chosen
to be 𝜔𝑏𝑏→𝑎𝑏,𝑎𝑏transitions
overlap.IfThe
inset
shows
an example
spectrumin whic
n we probe theatoms
only atoms
sites will
be isaddressed.
When ofthefourtransition
frequencies forandthe the fit is a sum
point
an averagemeasurementst that is resonant
onlyin 𝑛 = 2eachdifferent
shells are sufficiently
separated, Landau-Zenersweepsdrive frequencyof two Gaussians.
The crossingof ofthethefrequenciesof the two peakd atom per n=2 site.
= 0. the number of transferredcorresponds
to Ufrequency,
performed overeach transitionab − Ubb making
orresponds tocan
thebeZeecorrespond to the number of atoms in each MI shell divided by the occupation
f the applied atoms
magnetic
7
ar gfactor for
Li 𝑛.areThis technique
numbercan beused forstate
preparation and detection.
It
has beenused
to bothdrive
singlon-to-doublon
conversion
in th
ft of the n=allows
2 peak,the number lowest
of atoms inHubbard
each MI shell
to be monitored
which
can
band,
and theseparately,
resonant
frequencyof this a direct measure
ofdiagnosing
be used for
the MIhas
lifetime,
we will describe
below.
an 𝑛 =interaction
2 MI
process
beenasassociatedwith
theForon-siteenerg
Using this technique,inofbothand attractive[17]
plateau, the number
atoms repulsivein each of the[16]three possible
configurations
on asingle-componensite,bias fields fromwhich
systems.
employthe spin-orderedsame technique to ma
i.e. |𝑎𝑎⟩, |𝑎𝑏⟩, |𝑏𝑏⟩,bosoniccan be measured.
ThisHerecan beweusefulin detecting
7
littings Ubb −phases
Uab inandon-siteintraspecies
interactions
Liwithacross a broa
the spin-1theHeisenberg
model,
discussed in Chapter
4. Finally, ofa stateorks equally well for
range ofcanmagnetic
the just
lattice depth b
only a given site occupation
be prepared.fields.
The 𝑛 =We
1 MImodulate
can be preparedso long as the system
50(a)

Uab - U aa
U-U

by decreasing the atom number while monitoring the remaining atoms in 𝑛 = 2. A
system of only 𝑛 = 2 sites can be prepared by decreasing the atom number so that
there are no sites with 𝑛 > 2 and then transferring the atoms in 𝑛 = 1 sites to anotherstate, where they can be blown away with resonant light. Care has to be taken thatthe two states have sufficiently different resonant frequencies, so that the atoms in
𝑛 = 2 are not affected by the light.

3.2.2

Loading into a lattice

The first step of realizing Heisenberg models with ultracold atoms in optical lattices,is preparing a pure MI. Typically, this involves starting with a BEC and transferring
it into an optical lattice, rather than trapping hot atoms in the lattice and thencooling, since efficient cooling techniques for this scenario are lacking. By startingwith a pure BEC in the lowest state in an optical dipole trap and then adiabaticallyturning on the lattice potential, the lowest state in the lattice can be loaded (the 𝑞 = 0
quasimomentum of the ground band). After that, with an appropriate lattice ramp,a MI with no excitations can be prepared. In principle, in an infinite-size system, thesuperfluid and the MI states in a lattice cannot be connected adiabatically. However,in a finite system of size 𝐿, the gap ∼ 1/𝐿 can be used to enable this preparation.

In this section, we describe the methods we have used to prepare a pure BEC and tominimize excitations when loading the MI.
Preparation of a pure BEC
First, we need to ensure that the BEC has no thermal fraction. One way to do thisis to “spill out” the thermal atoms from the trapping potential. There are several
approaches to do that. The simplest one is to control the trap depth with the final
power of the ODT beams at the end of evaporation. By lowering the power justenough, only the condensed component can be kept trapped. However, this presents achallenge in laser intensity stability control because the ODT uses a large dynamical
range of powers. The power needs to be as large as possible initially to load the123maximum number of atoms from the magnetic trap. The final ODT power has to bequite low to complete the evaporation. In order to completely spill out the thermal
component, the power required is so low that the signal-to-noise ratio in the interlock
setpoint and in the photodiode signal are quite small. This causes some intensity
instability which excites dipole oscillation modes of the BEC in the trap. Ideally,
setpoint-dependent gain feedback should be used.

Condensate fraction

1
0.8
0.6
0.4
0.2
0

6

7

8

9

10

Evaporation gradient (Gauss/cm)

Figure 3-9: Controlling condensate fraction with a gradient. As the gradient is increased,
the thermal atoms leave the trap, leaving a pure BEC within our detection limitations.

Another approach is to use a magnetic field gradient to modify the trapping potential. As illustrated in Fig. 3-9, when a linear potential is superimposed with aGaussian dipole trap, the number of atoms filling the trap can be controlled byvarying the power of the gradient. This can be done after evaporation is complete.
However, we have found better results (higher final condensate fraction without exciting dipole oscillations of the BEC in the trap) when we use the gradient during
evaporation. We turn the gradient on at the beginning of ODT evaporation and keep
it constant during the evaporation. Towards the end of evaporation, the ODT depth
is lowered beyond a certain point, so that even the lowest temperature thermal atoms
leave the trap. The final condensate fraction as a function of the gradient is shown in
Fig. 3-9. We note that we cannot detect condensate fraction larger than ∼ 90 % due
to limitations in the image fitting procedure.

124

0.25

1

b)

9

35

8
0.8

0.6

0.1

0.4

0.05

30

7

ODT power (W)0.15

Lattice Power (ER)ODT power (W)

0.2

25

6
5

20

4

15

3

10

2

0.2

5

1
0
0

50

100

150

Lattice Power (ER)

a)

0
200

0

0
0

Time (ms)

500

1000

1500

2000

Time (ms)Figure 3-10: ODT to Lattice transfer. a) Power in each ODT beam (orange) and lattice
depth of each lattice arm (blue) as a function of time a) during the transfer and b) from thestage of a thermal cloud to the MI stage. The shaded area is the transfer stage from a).

Transfer from the ODT to the Lattice
Since the confining potential in the lattice is provided by the lattice beams, theODT is completely switched off when the lattice is turned on, in order to avoid
extra confinement. Therefore, the BEC needs to be transferred from the ODT to
the lattice without creating excitations. The most likely form of excitations in thisprocess are dipole oscillations in the trap. There can also be three-body loss fromthe compression of the cloud in the lattice sites. In order to avoid the former, we
perform a slow transfer between the ODT and lattice potentials by using a 200 ms
linear ramp, with the smallest ODT trap frequency being 72 Hz. Note that although
it is important to make sure the trap centers overlap, this long transfer also makes thesystem less sensitive to slight missalignments. This ramp is shown in Fig. 3-10a. In
order to avoid three-body loss from compressing the BEC in the lattice during the longtransfer time, the lattice is ramped up to a small depth of 1 𝐸𝑅 . For completeness,
the full sequence of the light powers from the stage of a thermal cloud to the MIstage is shown in Fig. 3-10b, including ODT evaporation, transfer, and lattice ramp
up. Next we discuss the ramp into the MI.

125

Lattice ramp into a MI
In order to enter the MI regime, the interactions 𝑈 need to be larger than the tunneling 𝑡. In our system, this can be achieved in two ways: by increasing the interactions
via a Feshbach resonance, or by increasing the lattice depth. When the lattice depth
is increased, the tunneling decreases (exponentially) because the tunnel barrier between the lattice site increases, decreasing the overlap between the wavefunctions onneighboring lattice sites, while the interactions between particles on the same site increase, because the wavefunctions become more localized (increasing the local density
on each site). We follow the typical route for cold atom experiments: we increase thelattice depth. This has the advantage that the tunneling timescale decreases, which isalso the timescale for transporting heat and entropy by the superfluid shell surrounding the MI or by number defects in the MI. A deep lattice in the MI with negligible
tunneling timescale realizes a system of isolated sites, which is an ideal starting point
for state preparation of spin physics experiments.
Next we discuss the ramp of the lattice depth. There are several considerations:(i) the ramp should not be faster than the bandgap, causing excitations to higher
bands or particle-hole excitations in the MI; (ii) density redistribution should be
avoided. The first set of conditions can be easily satisfied by slower ramps andthe second can be implemented by measuring the initial and final cloud sizes andmatching them by either changing the trapping potential or adjusting the interaction
strength. Numerical simulations show that the dominant source of excess energy
during the lattice ramp is density redistribution and that an exponential ramp leads
to the smallest energy added [47]. We have tried to use other types of ramps andhave more or less verified this claim. A rational behind this is that the ramp needs tobe slow enough during the initial stages, when the system is still superfluid and has
low-lying excitations. Once the density is fixed in the MI, the ramp should be fast inorder to avoid other, slower, heating processes.
We experimentally measure how fast the ramp should be by picking an exponential

126ramp of the form:

𝑒𝑡/𝜏 + 1
𝑉 (𝑡) = 𝑉𝑖 + (𝑉𝑓 − 𝑉𝑖 ) 𝑇 /𝜏𝑒−1

(3.21)

for initial depth 𝑉𝑖 and final depth 𝑉𝑓 for a total time 𝑇 . In order to avoid excitations
to higher bands, the ramp speed needs to be smaller than the bandgap. The adiabaticity criterion typically used in cold atom experiments is that for a Bloch state in
the 𝑛-th band Ψ𝑛,𝑞 (𝑥) [21]:∫︁Ψ*𝑛,𝑞 (𝑥)𝑑𝐸𝑛 (𝑞) − 𝐸1 (𝑞)Ψ1,𝑞 (𝑥)𝑑𝑥 ≪𝑑𝑡~(3.22)For 𝑞 away from the band edge, this simplifies to [21]:
𝑑
𝑑𝑡

(︂

𝑉 (𝑡)
𝐸𝑅

)︂

√
≪ 32 2𝐸𝑅 /~(3.23)

for 𝑉 ≪ 𝐸𝑅 , which gives a lower bound, since for larger lattice depths, the bandgaps

are larger. For our exponential ramp, we typically fix 𝜏 = 𝑇 /2.5 and vary 𝑇 .
Eq. (3.23) limits 𝑇 ≫ 10 𝜇s. A similar analysis can be done now taking into accountparticle-hole excitations in the MI, where the effective bandgap is the interaction energy 𝑈 . Since 𝑈/ℎ is a few tens of kHz for the scattering lengths we are using, thisis a factor of about 10 smaller than the bandgap to higher bands, so that the latticea)b)

Lattice depth (E R)

30
25
20
15
10
5
0
0

20

40

60

80

Remaining condensate fraction

ramp should be slower than about 100 𝜇s.

Time (ms)

0.6

0.5

0.4

0.3
0

20

40

60

80

100

time (ms)Figure 3-11: Lattice ramp into the MI. A symmetric ramp into and out of a 3D 33 𝐸𝑅 lattice
is used shown in a). The remaining condensate fraction after transferring back to the ODTis shown in b). The depth is varied in all three lattice arms in the same way.

127

We have measured the effect of varying 𝑇 on the remaining condensate fraction.
Fig. 3-11 shows the remaining condensate fraction after a symmetric lattice ramp intoand out of a MI at 𝑉𝑓 = 33 𝐸𝑅 , measured after transferring the cloud back to the ODT.
Note that there is no hold time after reaching the maximum lattice depth. Very slow
ramps lead to heating and very long ramps become sensitive to other noise processes.
We typically operate with intermediate ramps between 20 and 40 ms.
Eliminating higher on-site occupations
In order to simulate spin Hamiltonians, it is important to have control over the numberof atoms in each MI shell. For example, to realize a spin-1/2 system, an 𝑛 = 1 MI
is needed, to realize a spin-1 system, an 𝑛 = 2 MI is need, and to study three-bodyloss, an 𝑛 = 3 MI is needed. In the simplest possible form, this means maximizing thecorresponding MI shell and eliminating all higher-occupation MI shells. In principle,
lower-occupation MI shells could be transferred to another hyperfine state and blown
out with resonant light. In the present discussion, we limit ourselves to eliminating
higher occupations. This can be done by simply loading fewer atoms into the lattice.
First, a way of controlling the BEC number reliably is needed. Using a gradientto cut into the BEC number, similarly to spilling off the thermal fraction, is simple,
but experimentally it leads to heating and atom number fluctuations. Instead, we
decrease the atom number before ODT evaporation by relying on three-body loss.
The thermal cloud is kept in a deep ODT for a certain amount of time at a certain(large) scattering length. After that, the same evaporation procedure is performed.
This results in a BEC with a different atom number. Fig. 3-12a shows the numberof final BEC atoms as a function of the scattering length at which the thermal cloud
was being held.
Next, the number of the MI shell we are interested in with on-site occupation 𝑛 is
maximized. This is done by picking the starting BEC number which completely fills
that shell. After loading into a deep MI, the number of atoms in the 𝑛 + 1 MI shellis measured as a function of initial BEC number using interaction spectroscopy. For
example, if we want to maximize the number of atoms in the an 𝑛 = 1 MI plateau,128we count the number of atoms in an 𝑛 = 2 and pick a starting BEC number which
results in no 𝑛 = 2 atoms. One such measurement is shown in Fig. 3-12b. Finally, thescattering length at which the lattice is transferred from the ODT can be varied, sothat the number of atoms in the MI shell we are interested in is maximized. This
procedure is described in more detail in [45], reprinted in Appendix E. This completes
the loading process. Using atom number control and interaction spectroscopy, we canload large MI plateaus with atoms predominantly in sites of occupation 𝑛 = 1, 2, or
3, which we can choose.
10 4

b)

14
12
10
8
6
4
2
0
100

5

Number of atoms in n=2

Number of BEC atoms

a)

200

300

4
3
2
1
0
10 3

400

Scattering length at B set (a0)

10 3

10 4

Initial BEC number

Figure 3-12: Controlling the number of atoms in the MI shells. a) The BEC number iscontrolled by holding the atoms prior to evaporation for 1 s at a given scattering length.
Plotted is the BEC number after evaporation as a function of the scattering length at which
we hold. b) Loading the lattice at 150 𝑎0 . The BEC number prior to loading into the latticedetermines the number of atoms in each MI shell. Plotted here is the number of atoms in
𝑛 = 2.

3.2.3

Lifetime of the MI and heating processes

After preparation of the MI, an important question is how long the initial distribution
remains the same. Without a microscope, it is more challenging to characterize thissystem directly. In this section, we describe several approaches for characterizing the
lifetime of the MI and the sources of noise we have considered and measured.

129

There and back again
The simplest way to characterize the lifetime of the MI is to ramp into a MI, hold fora variable amount of time, and ramp out in a symmetric way. The remaining atom
number, condensate fraction, and BEC number once the atoms are back in the ODT
can be treated as a signature of the loss and heating processes which occur during thelattice ramp and in the MI. A variant of this is to not go all the way back to the ODTbut to measure the “visibility” of the interference pattern of the atoms in a shallow
lattice. A BEC released from a shallow lattice displays the typical interference patternin time-of-flight. In our system, both procedures give similar results. Fig. 3-13 shows
one such lifetime measurement (in a 3D 33 𝐸𝑅 lattice) performed by transferring backto the ODT (from May 2015). The lifetime of the remaining BEC number is about
90 ms, quite small. The small lifetime has been a main problem for the machine, but
we have significantly improved it, as we describe below.

Number of BEC atoms

10 4
1.2
1
0.8
0.6
0.4
0.2
0
0

50

100

150

200

250

time (ms)Figure 3-13: Lifetime of the MI in a 3D 33 𝐸𝑅 lattice measured by ramping back to the
ODT. In this figure the fitted exponential lifetime (no offset) is 𝜏 = 91.43 ± 16 ms (fromMay 2015).

When considering the processes which could lead to a decreased MI lifetime, adistinction needs to be made between heating and loss. Using the procedure above,the signature of loss processes is a decreased total atom number as a function ofhold time in the MI and that of heating processes is a decreased condensate fraction.
Typically, in a deep MI, loss processes are due to three- or two-body interactions ona site which change the internal state of the atoms, or to atoms being transferred130

to untrapped lattice bands. Heating processes remove atoms from the lowest bandand transfer them to higher bands or create excitations at the interaction energy 𝑈
but the atoms do not leave the trap. Here heating due to light-scattering is ignored
because it is very small in the far-detuned lattices we are using. Except in a few
distinct cases, we have typically observed that the atoms do not leave the lattice
but the condensate fraction after ramping back to the ODT decreases within tens tohundreds of ms. We interpret this to signify that there is a heating process in the
MI. In the following, several models of heating in a lattice are presented. After that,a few experimental measures of noise processes are described.
Single-particle model
Using a single-particle model, the effects of lattice amplitude noise and phase (orfrequency) noise can be taken into account. The deep lattice potential is modeled by
a Harmonic oscillator. In this context, amplitude noise is a periodic modulation of
the trap frequency 𝜔𝑡𝑟 [57, 136]:
1
2
𝑉 (𝑥) = 𝑚𝜔𝑡𝑟
(1 + 𝜖 sin(2𝜔𝑡))𝑥22(3.24)

where 𝜖 ≪ 1, and 𝜔 is the modulation frequency. The 𝜖 term is treated as a perturbation which couples bands 𝑚 and 𝑛 of the trap with a matrix element:⟨𝑚|𝑥2 |𝑛⟩ ∝ 𝛿𝑛+2,𝑛(3.25)

Therefore, amplitude noise can bring atoms from the ground band to the 2nd excited
band in the lattice. In the harmonic oscillator approximation, the correspondingtransition rate can be calculated [57, 136]:
𝑅0→2 =

𝜋𝜔 2
𝑆(2𝜔)
8

(3.26)

where 𝑆(𝜔) is the single-sided power spectrum of the noise (see Eq. (2.59)). Transitions to higher bands lead to heating in the MI. Therefore, in this model, amplitude131

noise in the laser light lead to a lifetime of 𝜏 = 1/𝑅0→2 .
The heating rate from phase noise can be calculated in a similar way: [57, 136]:12(𝑥 + 𝜖0 sin(2𝜔𝑡))2
𝑉 (𝑥) = 𝑚𝜔𝑡𝑟
2

(3.27)

where now the position is being modulated at a frequency 𝜔. The matrix element is:⟨𝑚|𝑥|𝑛⟩ ∝ 𝛿𝑛+1,𝑛(3.28)

so that this type of noise transfers atoms from the ground band to the 1st excited
band. The transition rate is:𝑅0→1 =𝜋𝑚𝜔 3 𝑆(𝜔)2~(3.29)

Eq.-s (3.26) and (3.29) can be used as the starting point for searching for noise
sources. First, the relevant frequency ranges for both amplitude and phase noise
need to be calculated for the the lattice band structure. The band gaps over the 1st
Brillouin zone are shown in Fig. 3-14. In the MI (typically 𝑉 > 15 𝐸𝑅 ), laser intensity
noise in the band 250-500 kHz and phase noise in the band 200-300 kHz can leadto heating. Transitions to even higher bands, eventually leading to atoms loss, arealso possible, for example, when the bandgaps overlap and the noise power at thesefrequencies is large (for a given hold time).
The amplitude modulation transition rate is proportional to the frequency squared.
For a flat noise spectrum, this means that lighter atoms suffer more from intensity
noise due to the larger transition frequencies. However, noise spectra are almost
never flat and it depends on the particular laser sources and drive electronics which
frequency bands have more noise. AOM-based stabilization schemes suffer from atime delay in the crystal, which limits how much gain can be applied above ∼ 50 kHz,making them unsuitable for eliminating intensity noise for a lattice of lithium atoms.

Alternatively, EOMs can be used, but thermal lensing in the crystal limits their performance for high power lattices, such as the one for lithium. Although there are132a) 550

b)

350

20

500

300

350
300
250

10

250

10

200
150
5
100

200
(0 2)
(2 4)

150
100
0

5

10

15

20

25

30

(0 1)
(1 2)
(2 3)

50
5
35

Bandgap (ER)15

Bandgap (kHz)400

Bandgap (ER)Bandgap (kHz)

450

0
0

Lattice depth (E R)

5

10

15

20

25

30

35

Lattice depth (E R)

Figure 3-14: Band gaps in a lattice. The energy differences for all quasimomenta are plottedfor bands a) coupled by amplitude modulation and b) coupled by phase modulation. For
reference, the bandgaps at 𝑞 = 0 are denoted by thick black lines.

ways to (partially) overcome these limitations, it is preferable to start with a lownoise laser source. The relative intensity noise of the Mephisto seed is reasonably
low, however Stimulated Brillouin scattering in the fiber amplifier (Section 2.2.1) significantly increased the noise level. After that was eliminated, the dominant sourceof intensity noise were the AOM driver electronics, which have been redesigned. If
a RIN of -130 dB/Hz is assumed (roughly the Mephisto level), the lifetime in thismodel should be ∼ 16 s. Since we measure a lifetime almost two orders of magnitude

smaller, either another noise source needs to be found or another model. We consider
both below.
The phase modulation transition rate is proportional to frequency cubed, making
this type of noise particularly important for light atoms. The lattice frequency is determined by the Mephisto seed, which has a linewidth of ∼ 1 kHz, which is negligible.

Therefore, vibrations of the retro mirror are the dominant source of phase noise. In
fact, as we will see below, there is some evidence that this type of noise was indeed
limiting after SBS was eliminated.
Interacting system
A single particle estimate predicts lifetimes much larger than what we observe. Therefore, interactions need to be considered. If particles are excited to higher bands, they
can interact with particles in the lower bands, causing heating. In particular, the133interactions between particles in the ground band and in the 1st excited band could
be quite strong. Although it is possible to count the number of particles in higher
bands through bandmapping, we have observed that almost no atoms populate higher
bands. Note that our bandmapping method has a few major limitations. First, interactions probably have to be switched off so that they do not affect the observed
density distribution. Second, and more importantly, our imaging is not done alonga lattice arm, but in the horizontal plane at a 45o angle with respect to the 𝑥 and𝑦 lattice arms. This makes the Brillouin zones overlap, which could make it difficult
to detect atoms in higher bands. Nevertheless, we can assume that at most a small
fraction of atoms (less than 10%) are in higher bands. In terms of explaining theshort MI lifetime, this leaves two options: either a small number of particles in higher
bands can cause a lot of heating in the Mott insulator, or a different type of excitation
is present.
The lowest-lying excitations in a MI with 𝑛 particles per site are pairs of 𝑛 ± 1

occupations, e.g. a pair of a hole and a doublon in an 𝑛 = 1 MI. These excitations areat the interaction frequency 𝑈/ℎ ∼ 1−30 kHz. This case has been studied theoreticallyin [127], where white-noise variation of the lattice depth ⟨𝛿𝑉 (𝑡)𝛿𝑉 (𝑡′ )⟩ = 𝑆0 𝛿(𝑡 − 𝑡′ ) in

the Bose-Hubbard model leads to the average energy increase per particle in the MI:⟨𝐸˙ 𝑀 𝐼 ⟩
=𝑆0
𝑁

(︂

1 𝜕𝑡
1 𝜕𝑈
−
𝑡 𝜕𝑉
𝑈 𝜕𝑉

)︂2

𝑧𝑡2 𝑈 (𝑛 + 1)

(3.30)

where 𝑧 is the number of nearest neighbors. An experimental measure of this process
could be the creation of doubly-occupied sites in an 𝑛 = 1 MI as a function of holdtime.
In summary, for lithium atoms in optical lattices, noise in the frequency bands
1-30 kHz and 100-500 kHz (for lattice depths up to 35 𝐸𝑅 ) can lead to heating. The
theoretical models presented here have served as a guidance, predicting the effects
of different types of noise. However, since often there are many noise sources ofdifferent origin, we have relied on experimental explorations of the dependence of theMI lifetime on different parameters. Below, a few examples of our characterization

134

are presented.
Laser intensity noise
We focus on laser intensity noise and illustrate two ways of characterizing it. First,
the total number of atoms and the BEC number after a hold time at a given lattice
depth can be measured. Second, adding a controlled amount of noise can reveal theimportance of intensity noise for the MI lifetime.
10 4

a)

7

4

6

BEC number

Total number of atoms

10 4

b)

5
4
3
2

3
2
1

1
0

0

5

10

15

20

0

25

0

Lattice depth (E R)

5

10

15

20

25

Lattice depth (E R)

Figure 3-15: Laser intensity noise. a) Total atom number and b) BEC number after a 40ms
hold time at each lattice depth.

A measurement of the the remaining total atom number and the remaining BEC
number after loading in a 3D lattice and holding for 40 ms at each lattice depth isshown in Fig. 3-15. The measurement is performed after ramping back to the ODT.
The atom number drops significantly after holding at around 12 𝐸𝑅 . The intensity
spectrum of the lattice light reveals a peak at 250 kHz. Coincidentally, this is thefrequency of a switching power supply in the Nufern fiber amplifier electronics boards
and also the frequency of an AC-DC converter in the AOM drive electronics. Fig. 314a shows that at around 12 𝐸𝑅 , intensity noise at this frequency can couple the bands
0 → 2 and then 2 → 4, so that laser intensity noise at that frequency can eventually
drive atoms out of the lattice, as seen in Fig. 3-15a.

When the atoms are kept at lattice depths above 12 𝐸𝑅 , the atom number “recovers”, probably because the time spent at 12 𝐸𝑅 is small in that case. However,the BEC number never goes back to the original value, showing that either there is135another loss process at higher lattice depths or that even a small number of atoms
in higher bands can lead to a lot of heating. Subsequently, the main source of the
250 kHz peak (the AC-DC converter) was eliminated, decreasing the magnitude ofthe effect.
If uncontrolled, laser intensity noise leads to heating and loss. However, controlled
amplitude modulation of the lattice can be used to deliberately drive particles outof the ground band. In fact, we use this to calibrate the lattice depth (Chapter 2).
Intensity modulation can also be used as a tool to clean up particles which are alreadyin excited lattice bands. Due to the different bandgaps, particles in different bands
can be targeted separately [15]. For example, one could target the 2 → 4 transition
which could lead to atoms leaving the lattice, while avoiding the 0 → 2 transition. As

shown in Fig. 3-14a, this can be done only at lattice depths below 12 𝐸𝑅 or above
27 𝐸𝑅 .

Heating rate (s-1 )

102

101

Superfluid, (Mephisto)
Superfluid (Nufern)
Mott insulator (Nufern)

-140

-130

-120

-110

-100

-90

Power Spectral Density (dB/Hz)

Figure 3-16: Lifetime as a function of (added) white noise. Plotted is the heating rate (1/𝜏 )as a function of the power spectral density 10 log10 (𝑆(𝑓 )) for three different setups. The
superfluid is a 1-arm 10 𝐸𝑅 lattice and the Mott insulator is a 3D 33 𝐸𝑅 lattice.

Similarly, adding intensity modulation in a controlled way can show if the system
is limited by this type of noise. We add white noise to either a superfluid in a
lattice or to a MI and measure the remaining condensate fraction after the atoms aretransferred back to the ODT as a function of hold time, during which the noise isadded. The heating rate, defined as 1/𝜏 , where 𝜏 is the measured lifetime, is shown in
Fig. 3-16. When a single arm of the lattice at 10 𝐸𝑅 is used, the system is superfluid
136and the heating rate increases even when a small amount of noise is added. When thesystem is in the MI, the heating rate begins to increase only after the power spectral
density rises above -110 dB/Hz. This suggests that in the superfluid phase, intensity
noise is likely the dominant source of noise, while in the MI in a deep lattice it isprobably not. Nevertheless, because reaching a deep lattice requires passing througha superfluid phase, decreasing the intensity noise could improve the starting entropy
in the MI. Note that in the superfluid phase, the addition of the Nufern amplifier does
not change the measured heating rate, indicating that the amplifier does not add anoise source which limits the lifetime at the levels of intensity noise used.
Raman transitions between lattice arms
In addition to intensity and phase noise in the lattice, undesired interference between
different lattice arms could transfer atoms to higher bands in the lattice. In our
setup, AOMs are used to shift the frequencies of the different lattice and ODT arms
and the frequency differences are chosen to be large compared to all relevant energy
scales. Often, both the positive and the negative AOM diffraction orders are used,typically to reuse frequency sources. This gives a separation of 2 𝑓𝑅𝐹 between thearms. However, light at the frequency of the -1st order is found in the beams of the+1st order and vice versa. This leads to interference between the arms with smallbeat frequencies, which could drive Raman transitions between the lattice bands.
To measure this effect, we tune the frequency of one of the lattice AOMs, 𝑓𝐿1 ,with respect to the other two 𝑓𝐿2 , 𝑓𝐿3 . For
𝑓𝐿1 = 𝑓0 + 𝑓𝑅𝐹 + ∆𝑓
𝑓𝐿2 = 𝑓0 − 𝑓𝑅𝐹
𝑓𝐿3 = 𝑓0 + 𝑓𝑅𝐹(3.31)∆𝑓 is varied. We hold for 100 ms in a 3D lattice at 15 𝐸𝑅 , ramp back the lattice
and transfer the atoms to the ODT and measure the remaining condensate fraction.
Fig. 3-17a shows the final condensate fraction around 𝑓𝐿1 = 𝑓𝐿3 , i.e. ∆𝑓 = 0. There is137

no condensate left in a region of ± 1 MHz around the resonance and the condensate
fraction reaches its nominal value when |𝑓𝐿1 − 𝑓𝐿3 | ≥ 3 MHz. Fig. 3-17b shows the fi-

nal condensate fraction around ∆𝑓 = −2 𝑓𝑅𝐹 , i.e. when one lattice arm uses a positive
and the other a negative AOM order. There are several distinct peaks, corresponding
to Raman transitions to higher bands, the most prominent being |0⟩ → |1⟩.
a)

b)
0.4

Condensate fraction

Condensate fraction

0.4

0.3

0.2

0.1

0.3

0.2

0.1

0

0
-1

0

1

Detuning

2

3

-1

4

0

Detuningf (MHz)

1

2

3

4

f - 2f RF (MHz)Figure 3-17: Raman transitions between the lattice arms. Plotted is the remaining condensate fraction after 100 ms hold time in a 15 𝐸𝑅 3D lattice. There are resonances when the
AOM frequencies of two of the lattice arms are the same, either when they use a) the sameand b) the opposite diffraction order. The black line corresponds to the transition |0⟩ → |1⟩
and the grey is for |0⟩ → |2⟩ and |0⟩ → |3⟩.

Therefore, in order to avoid driving Raman transitions to higher bands, the AOM
frequencies need to be spaced by at least 5 MHz between all five beams (lattice and
ODT), where we consider each arm to have a frequency spectrum:𝐴(𝑓0 +𝑓𝑅𝐹 ) + 𝜖𝐴(𝑓0 −𝑓𝑅𝐹 )(3.32)

where 𝐴𝑓 is the amplitude of the corresponding frequency component 𝑓 and 𝜖 ≪ 1. In

principle, if controlled, interference between the lattice arms could be used to engineer
more exotic lattice geometries [141]. However, since in our setup we do not stabilize
the relative phase between the lattice arms, interference between the arms needs to
be avoided.

138

MI lifetime after improvements
After eliminating SBS in the Nufern and cleaning up the intensity spectra of the AOM
driver electronics (by eliminating ground loops, noisy amplifiers, etc), we measure thelifetime of the MI as a function of atom number. This is shown in Fig. 3-18a. The
lifetimes are measured by ramping in and out of the MI. As the initial BEC number
is decreased, the lifetime increases. The extracted lifetimes after exponential fits
are shown in Fig. 3-18b for the remaining BEC number, condensate fraction, andtotal atom number. There are two observations. First, the lifetimes extracted bymeasuring the remaining BEC number and condensate fraction have increased by afactor of 10, as compared to our first measurements above. In fact, a lifetime of 1 s
is quite acceptable for studying superexchange physics with ∼ 1 kHz superexchange
rate.

10

3.5

4

32 10 3

BEC number

3

29 10 318 10 3

2.5

12 10 3
8.8 10 32

6.6 10 3

1.5

b)Atom number
Condensate fraction
BEC number

6

Lifetime (s)

a)

2
1
0.5

1
0.5
0

0

500

1000

0.5

1500

1

1.5

2

Initial BEC number

Hold time (ms)

2.5

3
10

4

Figure 3-18: Lifetime as a function of starting BEC number. The measurement is performedby ramping into a 3D 33 𝐸𝑅 lattice, holding for a variable amount of time, then ramping
back and transferring to the ODT at 125 𝑎0 in the 𝑎-state. a) BEC number as a function ofhold time. b) Extracted lifetime from an exponential fit. Note that the 𝑛 = 2 MI shell forms
for initial BEC numbers above around 2 × 104 . From Nov 2015.

Second, the lifetime of the total atom number is around a factor of 5 larger. In fact,
although the overall lifetime has increased, the lifetime measured by using the total
atom number has so far always been larger than the lifetime measured by using theremaining BEC number. Up to this point, we interpreted this difference to suggestthat the MI atoms are transferred either to higher bands or to particle-hole pairs,
which thermalize when the cloud is transferred back to the ODT. However, another139interpretation is that there is a cold MI core surrounded by a bath which heats
up. Ramping out of the MI and into the ODT thermalizes all excitations, which
could produce misleading results. Therefore, we have resorted to another method forcharacterizing the MI lifetime.
A different characterization method
Here we measure the lifetime of the number of atoms in each MI shell, using interaction
spectroscopy, without ramping back to a superfluid. We believe this is a bettercharacterization of the MI, as it is a more direct measurement. Note that in thissection, the lifetimes are measured after three more improvements: (i) the titaniumcoated bucket window was swapped, (ii) the Nufern was replaced with an ALS fiber
amplifier, and (iii) the table was floated. Eliminating thermal lensing could havedecreased density redistribution during lattice loading. Swapping the fiber amplifier
could have eliminated intensity noise peaks, although it could have introduced new
ones. Floating the table could have decreased phase noise.
First, interaction spectroscopy is used to prepare a MI with exactly known maximum number of atoms per site. Fig. 3-18a shows the lifetimes of three MIs: with up
to 𝑛 = 1, up to 𝑛 = 2, and up to 𝑛 = 3 atoms per site in a 3D lattice at 20 𝐸𝑅 at 111 𝑎0
in the 𝑏 state. The atom number loss in the 𝑛 = 3 MI is dominated by three-bodyloss, the lifetime of the 𝑛 = 1 MI is extremely long, and the lifetime of the 𝑛 = 2 MI isprobably dominated by three-body related loss. Note that for technical reasons (e.g.
Feshbach coils heating up, AOMs and fibers heating up), we cannot keep the lattice
beams on at full power for more than 6-8 s.
Next, the lifetime of an 𝑛 = 2 MI is characterized at the same parameters. The
lifetime of only the atoms which are in doubly occupied sites is measured by performing a Landau Zener sweep so that all |𝑏𝑏⟩ sites turn into |𝑎𝑏⟩ and we measure

the number of 𝑎 atoms. Similarly, only the number of atoms in singly occupied sites
is measured by performing a Landau Zener sweep |𝑏⟩ → |𝑎⟩. Finally, the total atom
number is also measured. This gives the lifetimes in Fig. 3-19b. The inset shows that

if the three curves are rescaled by the initial number, they collapse on top of each
140

10 4
n=1 MI
n=2 MI
n=3 MI

b)

10 4

14
12

Atom number

Atom number

15

Atom number

a)

10

5

10

2

n=2 only
n=1 only
Total number

1
0
0

1

2

Time (s)

8
6
4
2

0
0

2

4

0

6

Time (s)

0

0.5

1

1.5

2

Time (s)Figure 3-19: Lifetimes of different Mott shells. The measurements are done in a 3D 20 𝐸𝑅
lattice of 𝑏 atoms at 111 𝑎0 . a) For Mott insulators with different maximum occupation
number. The fitted exponential lifetimes for the 𝑛 = 1, 𝑛 = 2, and 𝑛 = 3 MI are respectively
12 s, 4.1 s, and 2.4 s. b) Lifetime of an 𝑛 = 2 MI. The lifetime of the two shells and of thetotal atom number are measured individually. The inset shows that if the three curves arerescaled by the initial number, they collapse on top of each other. From Nov 2018.

other. This suggests that there is a loss process. This could be three-body related
loss from admixtures of pairs of triplons and singlons on top of the 𝑛 = 2 MI, or fromtwo-body loss in the |𝑏⟩ state. Both of these are discussed more in the next section.

Since we have measured that an 𝑛 = 1 MI (with no higher occupation) has a lifetimeof more than 10 s, we could hypothesize that the loss of atoms in 𝑛 = 1 sites could be
due to the loss of 𝑛 = 2 atoms in the middle of the cloud, leading to density redistribution from the surrounding 𝑛 = 1 sites. Note that the tunneling rate at 20 𝐸𝑅 is~/𝑡 = 2.5 ms.
Although more data is needed to characterize the particular loss processes atthis scattering length, this example shows how powerful the method of interaction
spectroscopy is in terms of preparing and characterizing Mott insulators. Next, we
use this method to study the loss from three-body recombination in 𝑛 = 3 and 𝑛 = 2
Mott insulators.

3.2.4

Three-body recombination

So far we have focused mostly on processes which lead to heating in the MI. In thissection, we focus on processes which lead to atom loss. The main source of loss in Mott

141

insulators of the lowest two hyperfine states |𝑎⟩ and |𝑏⟩ is three-body recombination.
This process involves the collision of three particles leading to two of them forming

a bound state (a dimer). Since typically the bound state has energy different fromthe energy of the free particles, the dimer and the third particle acquire extra kinetic
energy, leading to their loss from the trap. The two energies are equal at the points of
Feshbach resonances. Therefore, this process is more prominent close to a Feshbachresonance, where the energy of the bound molecular state comes close to the energyof the free atoms. In practice, three-body recombination results in all three particles
being lost from the trap. In addition, the bound state is typically not trapped at allby the optical dipole potential used for the individual atoms.
Note that this process is important not only for three particles per site but also
for an 𝑛 = 2 MI at lower lattice depths, where tunneling is non-negligible, admixing
triplon-singlon pairs to the wavefunction. It is in this regime that superexchangephysics can be realized by mapping the system to a spin-1 Heisenberg model (Chapter 4). In fact, both the admixture of triplon-singlon pairs and the superexchange
matrix element are proportional to 𝑡2 . This makes the study of three-body recombination at the Feshbach fields of interest a necessary step in realizing spin Hamiltonians
with an 𝑛 = 2 MI.
Three-body loss can be described by a rate equation:𝑑𝑁= −Γ3 𝑁𝑑𝑡

(3.33)

where Γ3 is the three-body loss rate, which can further be expressed in terms of a loss
coefficient 𝒦3 and the particle density 𝑛:∫︀𝑛(r)3 𝑑r
Γ3 ∝ 𝒦3 ∫︀= 𝒦3 ⟨𝑛2 ⟩𝑛(r)𝑑r

(3.34)

The coefficient of proportionality depends on the state of the particles. From dimensional analysis, the three-body loss coefficient of this process should scale as:
𝒦3 ∼

~ 4
𝑎
𝑚

142

(3.35)

of two Gaussians. The crossing of the frequencies of the two peaks
per n= 2 site.
corresponds to Uab − Ubb = 0.
nds to the Zeeplied magnetic
tor for 7 Li arehas been used to drive singlon-to-doublon conversion in thee n= 2 peak, wherelowest
𝑎 is the Hubbard
scattering length.
band, and the resonant frequency of thisct measure of Inprocess
been
associatedwiththe occurs
on-siteforinteractionenergyaddition,has
there
is anotherprocess
whichthree particles
on a site
this technique,in bothrepulsive [16] and attractive [17] single-component
results in the formation of three-body bound states, known as Efimov
ds from which but whichbosonic systems. Here we employ the same technique to map
states.
Efimov in the 1970s [50], such states
been observed with
Ubb − Uab andthePredicted
on-sitebyintraspecies
interactions of 7 Li haveacrossa broadually well for ultracold
atoms
[94]
and
becomerelevantforthescattering
properties
near
a Feshbachrange of magnetic fields. We modulate the lattice depthbyas the system resonance. In the region of resonant two-body interactions |𝑎| → ±∞, bound Efimov
trimers exist for both positive interactions (when a dimer bound state exists) and

Scattering length(units of a0)

Interaction energy (kHz)the differential(a)aa
50interactions (when there is no dimer state). As a functionUabfornegativeof- Uincreasinga many-body
Ubb - U abrate, which at scattering |𝑎|, the energies of the Efimov states are separated by a universal ratio (for0
optical lattice reviews, see [116, 154]). This process modifies the total three-body loss coefficient 𝒦3rtz, depending at the positions of the Efimov resonances.
-50
e measure themagnetic field400(b)by the stability
aaa
5
3
aab
200
10 at 10 G)
abb
two hyperfine0
teraction ener-200
or bars are too
is technique is
-400
650
700
750
800
850
900
950
ristically large
Feshbach field (G)wide separation
ces permits the Figure 3-20:
resonancesinteractionsin 7 Li, as measured
by interaction
spectroscopy in [11].
FIG.Feshbach
3. (a) RelativeUab − Uaa and Ubb − Uab plottedfrom [11].
signal size and Figureasadapted
a function of magnetic field, measured via rf interaction spectroscopy in a 35ER × 35ER × 35ER optical lattice. (b) Scattering

ful and precise, another tech- processes
and this valueon theproximity
Feshbachfield,aremeasuredusingdepends
lattice AM
andshown toin aunitsof theresonance
Bohr ra-whereraction energy the two-bodyshown
as a diverges.
bold dashedis theresonances
ab scattering
length,dius a0 . Also
scattering
lengthThe line
Feshbachof 7 Liare shownmine the latticeobtained using simultaneous hyperbolic fits to the rf 7spectroscopy
in Fig. 3-20, as measured by interaction spectroscopy in [11]. For Li, the three-bodyeach hyperfine
and lattice AM data sets. The fits are shown as solid black lines.
loss
coefficient𝒦3 haslines
beenindicate
measureda near-degenerate
thermal cloud for thedulation (AM)
Dotted vertical
theforpositionof a resonance.
Therefore,
is theof interactions
𝒦3 which determines
importantlengths ofit theaa valueand bbplotted ashow
a functionof three-bodymagnetic loss

lowest two hyperfine states: 𝒦𝑎𝑎𝑎 and 𝒦𝑏𝑏𝑏 [128, 66]. These coefficients have been

033612-2
found to be universal as a function of scattering length near the 𝑎𝑎-resonance at 737
Gauss and the broad 𝑏𝑏-resonance at 895 Gauss. This means that the loss coefficient
𝒦𝑎𝑎𝑎 = 𝒦𝑏𝑏𝑏 when 𝑎𝑎𝑎 = 𝑎𝑏𝑏 . The effect of Efimov resonances has been observed as143

local maxima or minima in the otherwise monotonic curve 𝒦3 (𝑎) as a function of
scattering length 𝑎. In principle, there are two other coefficients 𝒦𝑎𝑎𝑏 and 𝒦𝑎𝑏𝑏 whichhave not been measured. Also, the narrow 𝑏𝑏-resonance at 845 Gauss has not been
explored. In fact, some of the spin Hamiltonians which can be implemented with
7

Li, have interesting points close to this resonance. In particular, there is a point

where a symmetry-protected topological state, the Haldane phase, can be realized,
as discussed in Chapter 4. Therefore, measuring the three-body loss coefficients is an
important part of characterizing a MI for its use as a starting point for spin physics.
Having particles in a 3D lattice is an ideal platform to measure these coefficients
because the exact number of particles on a site can be controlled and the on-site
densities can be calculated with the Wannier functions.
Theoretical description
Here we summarize the description of the three-body loss rate for two and three atoms
per site in a MI. We compare the three-body driven loss rate to the superexchange
matrix element for an 𝑛 = 2 MI, revealing the importance of measuring all four 𝒦3
coefficients.

Following [77], the exact form of Eq. (3.34) for three bosons on a site can be derived
by considering a rate equation for the loss of particles per site:∫︁𝑑𝑁
𝒦3 ∑︁ †3 3=−
⟨𝑎𝑖 𝑎𝑖 ⟩ |𝑤(𝑥)|6 𝑑3 (𝑥)
𝑑𝑡
6 𝑖
∫︁
𝒦3 ∑︁
=−
⟨𝑛(𝑛 − 1)(𝑛 − 2)⟩ |𝑤(𝑥)|6 𝑑3 (𝑥)6 𝑖∫︁𝑁= − 𝒦3 |𝑤(𝑥)|6 𝑑3 (𝑥)3(3.36)

where 𝑤(⃗𝑟) is the Wannier function on a site1 . For the sum over sites, we assume
that all 𝑁 particles are in 𝑛 = 3 sites, so that there are 𝑀 = 𝑁/3 sites. This gives us
1Here the authors of [77] define 𝐾3 to be the recombination event rate, which is related to theloss rate 𝒦3 = 3𝐾3 /6 as detailed in [52]. The factor of 3 comes from the fact that all three particles
are lost and 1/6 from the 𝑁 3 /6 triplons in a fixed volume.

144

the loss rate:
Γ3 = 𝒦3(︂∫︁

6 3|𝑤(⃗𝑟)| 𝑑 ⃗𝑟

)︂

1
3

(3.37)For comparison, for two atoms per site and two-body loss coefficient 𝒦2 , the loss

rate is:

∫︁
𝑑𝑁
𝑁
†2 2
= −𝒦2 ⟨𝑎 𝑎 ⟩ |𝑤(𝑥)|4 𝑑3 𝑥
𝑑𝑡2∫︁
Γ2 =𝒦2 |𝑤(𝑥)|4 𝑑3 𝑥

(3.38)(3.39)

This coefficient has been calculated for two atoms in the |𝑏⟩ state in 7 Li [66] and isshown in Fig. 3-21. In the region of magnetic fields relevant for spin physics (close to
the 𝑏𝑏 Feshbach resonances) 𝒦2 it is at most 1 × 10−15 cm3 /s at 848.7 Gauss but istypically on the order of 1 × 10−16 cm3 /s. An upper bound can be calculated for theloss rate (lower bound for the two-body limited lifetime) by considering a deep latticewhere the densities are largest. For a 3D lattice with depths (35,35,35) 𝐸𝑅 this gives alifetime of at least 6 s (for 𝒦2max = 1 × 10−15 cm3 /s), which is quite large. Accidentally,

the Haldane point (Section 4.1.3) is at 847.5 Gauss where 𝒦2 = 2×10−16 cm3 /s, givinga lifetime of 30 s in a deep lattice. Therefore, we can assume that two-body loss isnever limiting, except at the two resonance points plotted in Fig. 3-21. However, atthese points, the scattering lengths are either too small or too large, so we do notplan on using these magnetic fields for realizing spin Hamiltonians.
We next consider the effect of triplon-singlon admixtures on the lifetime of an𝑛 = 2 MI. This admixture can lead to three-body loss but it is suppressed by the energy difference of the interaction energy 𝑈 . Nevertheless, the superexchange process,required for realizing spin physics with an 𝑛 = 2 MI, occurs via the same intermediate
state: a triplon-singlon pair. The probability of tripton admixture is:𝑃3 = 𝑧(︂

6𝑡2𝑥 6𝑡2𝑦 6𝑡2𝑧
+ 2 + 2
𝑈2
𝑈𝑈)︂

(3.40)

where 𝑧 is the number of nearest neighbors. The factor of 6 comes from the matrix
element, which in the Fock basis for two sites |𝑛1 , 𝑛2 ⟩ with atom numbers 𝑛1 and 𝑛2
145

Fig. 1. (a) The broad Feshbach resonances of the |m F = 1⟩ (dashed red line) and the |m F = 0⟩ (solid blue line) states, centered at 738.2 G and 893.7 G,
respectively. (b) The two Feshbach resonances of the |m F = 0⟩ state.

Fig. 2. A coupled-channels calculation of dipolar relaxation rate coeﬃcients as a function of magnetic field for the |m F = 0⟩ state.

Figure 3-21: Two-body loss coefficient for 7 Li. The red dashed lines indicate the positions of
the two𝑏𝑏 Feshbachresonances.
Theinteraction
shadedpotentials
regions
to |𝑎|<40
𝑎0Feshbachand |𝑎|a coupled-channels
calculationby using recent[19].correspond
Except for two peaks
whichsignify
res-> 400
onances, the loss rate coeﬃcients are extremely small, ∼ 3 orders of magnitude smaller than the corresponding measured
and aretobeavoided.
Figureadaptedfrom[66].
rate coeﬃcients, if the experimental losses were treated as purely two-body related. For example, at 880 G the two-bodyis:loss coeﬃcient is ∼ 5 × 10−17 cm3 /s which yields, given that the atom density is ∼ 1012 cm−3 , a life time of ∼ 20 000 s.
As a comparison, the life time in our dipole trap due to vacuum is less than 100 s. As a result, we exclude two-body losses
from the analysis described below and determine that the loss processes in the region of interest are related to three-bodyrecombination. Note that while we find this mechanism to be negligible for 7 Li atoms, it can be important for heavier alkali
atoms. For instance, 133 Cs experiences large dipolar losses caused by the second-order spin–orbit interaction [20].

√⟨3, 1|𝑎†1 𝑎2 |2, 2⟩ = 6

2.2. Experimental measurement of three-body recombination loss

(3.41)

According to a dimensional analysis, taking the scattering length as the only parameter describing the interaction, one
The three-bodydriven loss rate of 𝑛 = 2 sites in a 1D system is then:finds that the three-body recombination loss rate coeﬃcient (K 3 ) scales as h̄a4 /m [21]. Above this general scaling, taking

into account the fact that the scattering length has to be supplemented by a three-body parameter to describe the interaction of three bosons, universal theory predicts log-periodic oscillations of K 3 due to the presence of Efimov trimer states.
For positive scattering lengths the oscillations are caused by destructive interference conditions between two possible decay
pathways at certain values of a [4,22]. For negative scattering lengths the loss rate coeﬃcient exhibits a resonance enhance-

Γ2, rec =Γ3 𝑃3
)︂
(︂∫︁
(︂ 2
)︂
1
6𝑡𝑥 6𝑡2𝑦 6𝑡2𝑧
6 3
Γ2, rec =𝒦3|𝑤(⃗𝑟)| 𝑑 ⃗𝑟2
+ 2 + 22
𝑈2
𝑈𝑈
)︂ (︂ 2
(︂∫︁
)︂
𝑡2𝑦
𝑡𝑥
𝑡2𝑧
|𝑤(⃗𝑟)|6 𝑑3⃗𝑟
Γ2, rec =𝒦3++
6𝑈2 𝑈2 𝑈2

(3.42)

where the factor of 1/2 replaces the factor of 1/3 in Eq. (3.37) in the sum over sites,
because now the total number of sites is 𝑁/2, again, assuming all 𝑁 atoms are in
𝑛 = 2 sites.
For spin physics, the superexchange rate for two atoms per site needs to be compared to this loss rate. Let’s assume, as we have been doing so far, that all spins arein the same state. The superexchange matrix element is then in 1D:
𝐽=

12𝑡2
𝑈

(3.43)

for 𝑡 = 𝑡𝑧 ≫ 𝑡𝑥 , 𝑡𝑦 . The factor 12 is the same as in Eq. (3.40): a factor of 2 from thenumber of nearest neighbors and a factor of 6 from the matrix element for two atoms
146per site. The ratio of the superexchange rate to the three-body loss rate in an 𝑛 = 2
system is then:
𝐽/~
2𝑈
(︀∫︀
)︀
=
Γ2, rec~𝒦3 |𝑤(⃗𝑟)|6 𝑑3⃗𝑟

(3.44)∫︀
Since 𝑈 ∝ 𝑎 |𝑤(⃗𝑟)|4 𝑑3⃗𝑟, in a deep lattice the ratio of the integrals over Wannierfunctions is almost constant with lattice depth. Then the ratio 𝐽/(~Γ2, rec ) cannot
be tuned by the lattice depth and is determined by the ratio 𝑎/𝒦3 , which can bevaried by the magnetic field. The choice of magnetic field is dictated by the choiceof the Heisenberg Hamiltonian parameters (see Chapter 4). Therefore, whether somespin Hamiltonians can be realized with 7 Li solely depends on the value of the threebody loss rate coefficient 𝒦3 . Since for the two spin states, these coefficients are not

all known in the magnetic fields of interest for us, experimental measurements are
required.
Experimental measurement of 𝒦3

We then set out to measure the 𝒦3 coefficients for the region of Feshbach fields
between the 𝑏𝑏 resonances where the Haldane phase can be realized. Although wehave not completed this work, here we present the initial measurements. We perform
two measurements: (1) we measure the lifetime of triply-occupied sites. Then, using
Eq. (3.37), 𝒦3 can be extracted. (2) We measure the lifetime of doubly-occupied sites

in a system with no higher occupation. If three-body loss from triplon admixture
is the main loss process, 𝒦3 can be extracted using Eq. (3.42). Here we focus on

measuring 𝒦𝑏𝑏𝑏 in order to benchmark our methods. At the end of the section, we
present one data point suggesting what the ratio 𝒦𝑎𝑏𝑏 /𝒦𝑎𝑎𝑏 is.

First, to measure the lifetime of the atoms in |𝑏𝑏𝑏⟩ sites, a MI in the 𝑏-state isprepared at a field which maximizes the size of the 𝑛 = 3 shell. The field of 867 Gauss ischosen, optimized experimentally. Then, interaction spectroscopy is used to transfer
all atoms in 𝑛 = 2 and 𝑛 = 1 sites to the 𝑎 state and measure the lifetime of the 𝑏atoms. We label this configuration (a,aa,bbb). In this way, the three-body loss rate
can be precisely measured because only atoms which started in 𝑛 = 3 sites are imaged.
147

In this way, the process in which triply-occupied sites in the middle are replenishedfrom atoms in the wings is excluded, because the atoms in the wings are in the 𝑎state which is not imaged. The field can then be changed to the desired value while
keeping all lattices at 35 𝐸𝑅 . A sample loss curve and the three-body loss coefficientare shown in Fig.3-22. This reproduces the results in [145], obtained by using athermal cloud of 𝑏 atoms close to degeneracy. The three-body loss below 850 Gauss
becomes prohibitively large for our preparation sequence. The range of magnetic
fields over which this measurement can be performed is limited by the preparation
time. If the three-body loss rate is faster than the preparation time, this method
fails.
a) 10

10

3

8

Nb

6

4

2

0
0

10

20

30

40

50

60

Time (ms)Magnetic field (Gauss)
850

b)

855

860

865

870

875

c)

10

-26

5
4

30

K3 (cm6/s)

Lifetime of n=3 (ms)

40

20

10

3
2
1
0

0
0

50

100

150

200

0

50

100

150

200

Scattering length (a0)Scattering length (a0)Figure 3-22: Lifetime of triply occupied sites. The lifetime of 𝑏 atoms is measured in an(a,aa,bbb) system. a) A sample loss curve at a field of 854 Gauss. The black curve is a fit
𝑓 (𝑡) = 𝐴 exp −𝑡/𝜏 + 𝐶 with 𝜏 = 8.107 ± 0.8250, 𝐴 = 7258 ± 303, 𝐶 = 1932 ± 230. b) Lifetime
𝜏 as a function of scattering length. c) Three body loss coefficient 𝒦3 .

To extend the range of fields for which 𝒦3 is measured, the loss from triplon
148

admixture to an 𝑛 = 2 MI can be used. If the three-body loss rate is large, thenthe lifetime of 𝑛 = 2 sites is dominated by the loss associated with the admixture of
triplons. From Eq. (3.42), it follows that for deep lattices
𝑡2
Γ2, rec ∝ 𝒦3 2𝑎(3.45)The Wannier function integrals for the density and for the interactions 𝑈 cancel, up
to a numerical prefactor which is constant with lattice depth. This is particularlytransparent when the Wannier functions are approximated as Gaussians. Therefore,
we expect that measuring the three-body loss coefficient by using the lifetime of the
doubly-occupied sites in an 𝑛 = 2 MI is appropriate when Γ2, rec is large: for large
tunneling rates, small scattering lengths, or at magnetic fields where 𝒦3 is large.

In the measurement, an 𝑛 = 2 MI is used and the number of atoms in doubly-

occupied sites is maximized by picking the MI loading field, similarly to the caseabove. The 𝑛 = 1 atoms are then transferred to the 𝑎 state, realizing an (a,bb)
system. The lifetime of the 𝑏 atoms is then directly measured. An exponential fit
with an offset is used to extract the lifetime. The lifetime is measured as a function oflattice depth in 1D, while the other two lattices are kept constant at 35 𝐸𝑅 . One such
lifetime is shown in Fig.3-23a. This is done at a point where the Haldane phase is
expected to be. The lifetime does indeed scale as 1/𝑡2 , but only for a range of lattice
depths. At very low depths, tunneling plays a role and triplon admixture is no longer
the main mechanism responsible for the formation of triply occupied sites. For verylarge lattice depths, other loss processes become dominant. Beyond about 3 s the loss
rate does not depend on lattice depth anymore. This experiment is done at several
magnetic fields between the 𝑏𝑏 resonances, where only 𝑎𝑏𝑏 varies significantly. Thisis shown in Fig.3-23b for positive values of 𝑎𝑏𝑏 and in Fig.3-23c for negative values.
For 𝑎𝑏𝑏 positive, the 𝒦𝑏𝑏𝑏 coefficient has been measured (Fig. 3-22), so that we plot a

shaded region to indicate the predicted lifetime. While for small positive scattering
lengths, the loss rate of 𝑛 = 2 atoms is consistent with the three-body limited rate,
for large positive scattering lengths, the lifetimes are much shorter, indicating that a149

Lifetime of n=2 atoms (sec)a) 102
100

10-2

10-4

0

5

10

15

20

25

30

35

Lattice depth (E R)

c)

101

100

10-1

10

-2

10

-3

12.4
18.2
20.5
36.6
111.6

0

5

10

15

20

25

30

Lifetime of doublons (sec)Lifetime of doublons (sec)b)

35

Lattice depth V z (ER)

101

100

10-1

10

-2

10

-3

-25.1
-58.1
-96.2

0

5

10

15

20

25

30

35

Lattice depth V z (ER)Figure 3-23: Lifetime of doubly-occupied sites. The lifetime of 𝑏 atoms is measured in an(𝑎, 𝑏𝑏) system. This is done at several magnetic fields where only 𝑎𝑏𝑏 varies significantly.
a) 𝑎𝑏𝑏 = − 96.2 𝑎0 , the Haldane point in spin-1. The solid line is a one-parameter fit to
Eq. (3.42). In the regime where the solid line fits the data (i.e. the loss rate scales as 𝑡2 ), we
believe the lifetime is limited by triplon admixture, as shown in the cartoon. b) 𝑎𝑏𝑏 > 0. The
shaded areas are the expected variation of the lifetime with lattice depth given the measured
𝒦𝑏𝑏𝑏 . c) 𝑎𝑏𝑏 < 0. Here we do not have a measured 𝒦𝑏𝑏𝑏 .

different loss process dominates.
To elucidate this point, the loss rate Γ2, rec and the 𝒦3 coefficient can be extracted

as a function of scattering length 𝑎𝑏𝑏 . This can be done using both the 𝑛 = 3 and 𝑛 = 2
lifetime measurements. In the case of 𝑛 = 2, only the lattice depths where the lifetime
scales as 1/𝑡2 are used. This is shown in Fig. 3-24. In the region of scattering lengths
where there is data from both 𝑛 = 2 and 𝑛 = 3 sites, the loss rates and 𝒦3 coefficients

extracted from the two measurements agree. At the minimum of the three-body loss

coefficient (around 𝑎𝑏𝑏 = 100 𝑎0 ), the measured lifetime of 𝑛 = 2 sites is smaller thanthe one predicted from the loss rate of 𝑛 = 3 sites. Note that at this point, the 𝑛 = 2
lifetime does not scale at all like 1/𝑡2 (for 𝑎𝑏𝑏 = 116 𝑎0 in Fig. 3-23b). This can be150interpreted in the following way. A constant background loss rate can be assumed,which does not vary with magnetic field but comes from other sources. This could
be technical noise (laser intensity noise or magnetic field noise) at a frequency on theorder of 𝑈/~ of 1−30 kHz, which leads to the creation of 𝑛 ± 1 type of excitations in
the MI, followed by three-body loss. Then, this limits the 𝑛 = 2 lifetime when thethree-body loss rate falls below about 1 to 5/s. Then the points at negative 𝑎𝑏𝑏 (forwhich we have no 𝑛 = 3 data) are a measurement of the loss rate Γ2, rec and the threebody loss coefficient 𝒦3 . In the region of the Haldane point 𝑎𝑏𝑏 ≈ −100 𝑎0 , although
three-body loss from 𝒦𝑏𝑏𝑏 is likely to limit the lifetime, it is not too severe, i.e. there

are probably tens of superexchange times before three-body loss takes over.

102
101

847.5

Magnetic field (Gauss)

848.5 850

855 860

870

875

878

from n=3 sites
from n=2 sites

847.5

b)

Magnetic field (Gauss)

848.5 850

855 860

870

875

878

10-25

K3 (cm6/s)(1/sec)

10

3

2,rec

a) 10

4

10-26

100
10

-1

-100

-50

0

50

100

Scattering length (a0)

150

10

200

-27

-100

-50

0

50

100

Scattering length (a0)

150

200

Figure 3-24: Three-body loss coefficient. The results from the lifetime measurements ofatoms in 𝑛 = 3 and 𝑛 = 2 sites are compared. a) Γ2, rec at 11 𝐸𝑅 . b) 𝒦𝑏𝑏𝑏 .

To complete the story, all four loss coefficients 𝒦𝑎𝑎𝑎 , 𝒦𝑎𝑏𝑏 , 𝒦𝑎𝑎𝑏 and 𝒦𝑏𝑏𝑏 need

to be measured. However, since this region is dominated by the 𝑏𝑏-resonances, we
believe that the largest of the four coefficients will be 𝒦𝑏𝑏𝑏 and 𝒦𝑎𝑏𝑏 , since they involvethe formation of a 𝑏𝑏 molecule. Bose enhancement also suggests that 𝒦𝑏𝑏𝑏 should belarger than 𝒦𝑎𝑏𝑏 . Here we present a lifetime measurement using both states 𝑎 and 𝑏,

which suggests that 𝒦𝑎𝑎𝑏 is much smaller than 𝒦𝑎𝑏𝑏 . The measurement is performedat one particular field, where the Haldane phase is expected to be. Starting withan 𝑛 = 2 MI in the 𝑏 state and using Interaction spectroscopy, a system of (c,ab)
is prepared, where |𝑐⟩ is the next hypefine level. The lifetimes of the 𝑎 and the 𝑏atoms are measured separately and are shown in Fig.3-25a. If three-body loss from151

triplon admixture is the only loss mechanism, then the loss rates of 𝑎 and 𝑏 atoms
are respectively:
𝑡2
𝑡2
+
𝒦
𝑎𝑏𝑏
2
2
𝑈𝑎𝑎
𝑈𝑏𝑏
𝑡2
𝑡2
Γ𝑏 ∝ 𝒦𝑎𝑎𝑏 2 + 2𝒦𝑎𝑏𝑏 2
𝑈𝑎𝑎
𝑈𝑏𝑏Γ𝑎 ∝ 2𝒦𝑎𝑎𝑏(3.46)

Here we have used Eq. (3.42) and the fact that the number of 𝑎 and 𝑏 atoms are thesame. If we define the ratio 𝑟 = Γ𝑏 /Γ𝑎 = 𝜏𝑎 /𝜏𝑏 , then:
𝒦𝑎𝑏𝑏
𝑈 2 2𝑟 − 1
= 𝑏𝑏
2 2−𝑟
𝒦𝑎𝑎𝑏
𝑈𝑎𝑎

(3.47)

which we plot in Fig.3-25b. If we average the first 5 points (9 < 𝑉𝑧 < 20 𝐸𝑅 ), assumingthat other loss processes dominate for larger lattice depths (corresponding to longer
measurement times), we get 𝒦𝑎𝑏𝑏 /𝒦𝑎𝑎𝑏 ≈ 10. Of course, this is only one data point
and probably a different approach has to be used to verify these results.
b)

2

104

1.5
1

103

0.5
10

20

102

10

K abb /K aab

Lifetime (sec)

a

/ b

a)

1

a-atomsb-atoms

10

15

20

25

30

15

20

35

30

35

200

10

150
100
50

0

0

8 10 12 14 16 18

Lattice Depth (ER)

25

250

Lattice Depth (ER)

20

25

30

35

Lattice Depth (ER)Figure 3-25: Lifetime of 𝑎𝑏 doublons at 846.8377 Gauss. a) Lifetimes of 𝑎 and 𝑏 atoms in
the |𝑎𝑏⟩ sites. b) Ratio of the 𝑎 and 𝑏 lifetimes (top) and the corresponding ratio of thecoefficients 𝒦𝑎𝑏𝑏 /𝒦𝑎𝑎𝑏 . Note that at this magnetic field 𝑎𝑎𝑎 = −58 𝑎0 and 𝑎𝑏𝑏 = −185 𝑎0 .

In general, if there is more than one state, then the superexchange rate and theloss rates will be the sums of all processes. If one of the loss coefficients is muchsmaller than the others, then we could in principle bias the system so that this is themore frequent event. This is due to the fact that the loss rate is the product of the loss2
coefficient 𝒦𝑛𝑛𝑚 and the corresponding admixture probability 𝑡2 /𝑈𝑛𝑚
(here the indices152

correspond to the different combinations of the two states). By applying a linear tilt
across the lattice (as in Chapter 5), we can modify this admixture probability andresonantly enhance the one which corresponds to the lowest loss rate. This will
enhance the total loss rate but will change the ratio of superexchange rate to loss rate
for the total system.
This chapter has summarized the main results of the theoretical description ofultracold atoms in optical lattices using the Bose-Hubbard model and has presenteddifferent ways of loading and characterizing Mott insulators. Starting with a 90 ms
lifetime, we have improved the setup and our characterization strategy, resulting inseveral-second lifetimes of 𝑛 = 1 MIs. We can point to a few reasons for the small
lifetime initially: three-body loss from the presence of higher MI shells with 𝑛 > 2,intensity noise from SBS in the Nufern, general noise in the drive electronics (e.g.
ground loops, noisy amplifiers, large signal to noise in the setpoint, etc), phase noisefrom vibrations in the retro mirror, intensity and potentially frequency noise in the
Nufern fiber amplifier, density redistribution due to thermal lensing. We have shown
the advantages of using the number of atoms in each MI shell as a measure of the lifetime and purity of the system, instead of the ramp-in and ramp-out method. We havealso identified and characterized three-body loss in 𝑛 = 2 MIs from triplon admixtures
to the wavefunction in a range of scattering lengths. Although this study requires
further work, it points to an inherent loss mechanism in superexchange schemes for
implementing spin-1 Heisenberg Hamiltonians with a MI with two atoms per site. In
the next chapter, the spin models that can be studied with 7 Li in an optical lattice
will be described.

153

154

Chapter 4Spin Models and Dynamics
Spin models can be realized with ultracold atoms in optical lattices by mapping
particle hyperfine states to spin states. The mapping is valid in the Mott insulator
regime in an optical lattice, where particles are localized on individual lattice sites.
The effective spin-spin interactions rely on a second-order tunneling process, known
as superexchange, requiring non-zero tunneling between lattice sites on experimental
timescales. This is due to the very short-range van der Waals interactions in alkali
atoms, which, for all practical purposes, can be considered contact interactions. In
an optical lattice, this means that, to a good approximation, particles interact only ifthey are on the same site. Superexchange involves the virtual hopping and hoppingback of a particle in the Mott insulator, realizing the coupling between neighboring
lattice sites.

Figure 4-1: Superexchange. It is a the second-order tunneling processes in the Mott insulator
at a rate proportional to 𝑡2 /𝑈 .

As illustrated in Fig. 4-1, this process has a matrix element proportional to 𝑡2 /𝑈 ,155

where 𝑡 and 𝑈 are the Hubbard parameters. In the Mott insulator, this process
corresponds to a second order perturbation (in 𝑡/𝑈 ) to the energy. It is the coherent admixture to the wavefunction of states with higher and lower occupancy 𝑛 ± 1,

where 𝑛 is number of atoms per site in the unperturbed Fock basis. In realizing

spin models with atoms in optical lattices, the superexchange rate needs to be fasterthan heating and loss processes. Therefore, in a given system, the maximum possiblesuperexchange rate should be used. It can be varied experimentally by changing thelattice depth or the interactions. The choice of interactions is typically dictated by the
choice of spin model parameters. Therefore, the lattice depth, which mostly affects
the tunneling 𝑡 as we have seen in Chapter 3, is used to vary the superexchange rate.
In a shallow lattice, the system is still superfluid and the mapping to spin models is
not valid, because the particles are not localized on individual sites. In a very deep
lattice, tunneling is too small, hence the superexchange rate can be small compared tothe heating and loss processes in the lattice. Therefore, typically, spin physics experiments are performed at lattice depths a little bit above the superfluid-MI transition,
where particles are localized but the superexchange rate is still relatively large.
The advantages of using lithium for studying superexchange-mediated spin physics,
compared to the other alkali, are that lithium is light and has Feshbach resonances.
The small mass makes the recoil energy large: 𝐸𝑅 = ~2 𝑘 2 /(2𝑚) = ℎ × 25 kHz, where
𝑘 is the lattice light wavevector. As we have seen in Chapter 3, in the tight-binding

limit, the tunneling 𝑡 is proportional to 𝐸𝑅 for a given lattice depth (Eq. (3.11)).
Therefore, for different atomic species at the same lattice depth (𝑉0 in recoil units),
larger recoil energy means faster dynamics. For experiments to be meaningful, thetimescales of spin dynamics need to be at least an order of magnitude smaller thanthe heating and atom-loss timescales. However, larger recoil energy also means that
more laser power is required to implement a lattice with the same depth 𝑉0 in recoil
units. As we have seen in Chapter 2, although this can lead to some experimental
complications, the implementation of a high laser power lattice is experimentally feasible. Finally, the Feshbach resonances allow the tuning of the interactions. In 7 Li,
the Feshbach resonances between the lowest two hyperfine states |𝑎⟩ and |𝑏⟩ at fields
156

of around 800 Gauss allow the tuning of all three interaction energies 𝑈𝑎𝑎 , 𝑈𝑎𝑏 , 𝑈𝑏𝑏
and, therefore, of the parameters of the spin Hamiltonians. Although they cannot be
tuned independently, it is possible to find many points of interest where the scattering
lengths at a given magnetic field correspond to the spin parameters of various spinmodels.
In this chapter, we consider the spin models which can be realized with 7 Li in an
optical lattice: the spin-1/2 and spin-1 Heisenberg models. We detail the mappingfrom the two-component Bose-Hubbard model to the Heisenberg model, discuss thephase diagrams and the advantages and limitations of our platform for studying thesemodels. In addition to ground states, we discuss the spin dynamics which can bestudied with our platform.

4.1

Spin Models

The spin models which can be realized with the lowest two hyperfine states in 7 Li
depend on the site occupation in the Mott insulator. With one atom per site, thetwo-component Bose-Hubbard model maps to a spin-1/2 Heisenberg model and with
two atoms per site, it maps to a spin-1 Heisenberg model. In principle, this can beextended to higher occupations [10] but for three or more particles per cite, threebody recombination is limiting for large interactions.

4.1.1

Spin 1/2

Mapping from the Hubbard model
Here we discuss the mapping from the two-component Hubbard model to the spin1/2 Heisenberg model. This mapping can be done by a modified Schrieffer–Wolff
transformation [105] or by a more direct second-order perturbation theory approach[96]. The phase diagram has been investigated in [49]. Here we give a brief overview
of the mapping and explore the implications for studying superexchange-mediatedspin physics with 7 Li.

157

The mapping to a Heisenberg model starts with the two-component Bose-Hubbard
model:)︁∑︁∑︁ (︁ ††𝑛↑𝑖 𝑛↓𝑖𝐻 =−𝑡𝑎↑𝑖 𝑎↑𝑗 + 𝑎↓𝑖 𝑎↓𝑗 + 𝑈↑↓
𝑖

⟨𝑖,𝑗⟩

𝑈↑↑ ∑︁
𝑈↓↓ ∑︁
+𝑛↑𝑖 (𝑛↑𝑖 − 1) +𝑛↓𝑖 (𝑛↓𝑖 − 1)
2 𝑖
2 𝑖

(4.1)

where 𝑎†𝜎𝑖 and 𝑎𝜎𝑖 are the creation and annihilation operators of spin 𝜎 on site 𝑖.
The Hubbard parameters 𝑡 and 𝑈 are assumed to be independent of the lattice site,
i.e. they are uniform across the lattice. This is a fairly good approximation, except
for a small variation of the lattice depth (1%) due to the harmonic confinement (seeSection 3.1.3). In our system, 𝑡 is independent of spin, while 𝑈 is spin-dependent.
The goal of the mapping is to obtain a Heisenberg Hamiltonian of the form:
𝐻 = 𝐽𝑧∑︁

𝑆𝑖𝑧 𝑆𝑗𝑧 + 𝐽𝑥𝑦

⟨𝑖,𝑗⟩

∑︁ (︀
)︀
𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦

(4.2)

⟨𝑖,𝑗⟩

where the 𝐽𝑧 terms are the diagonal matrix elements (the energies) and the 𝐽𝑥𝑦 terms
are off-diagonal matrix elements (spin flip-flop terms which flip spins on neighboringsites). To construct the effective spin Hamiltonian, the tunneling term in Eq. (4.1) istreated as a perturbation in the limit 𝑡 ≪ 𝑈 , following the approach in [96]. The spin
operators on site 𝑖 are defined as:

1
𝑆𝑖𝑧 = (𝑛𝑖↑ − 𝑛𝑖↓ )21𝑥
𝑆𝑖 = (𝑎†𝑖↑ 𝑎𝑖↓ + 𝑎†𝑖↓ 𝑎𝑖↑ )2𝑖𝑦𝑆𝑖 = − (𝑎†𝑖↑ 𝑎𝑖↓ − 𝑎†𝑖↓ 𝑎𝑖↑ )
2

(4.3)
(4.4)
(4.5)

From the definition of the spin raising and lowering operators 𝑆 ± = 𝑆 𝑥 ±𝑖𝑆 𝑦 , thesecond term in Eq. (4.2) can be expressed as:
𝐽𝑥𝑦

∑︁ (︀
)︀
)︀ 𝐽𝑥𝑦 ∑︁ (︀ + −𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 =𝑆𝑖 𝑆𝑗 + 𝑆𝑖− 𝑆𝑗+
2
⟨𝑖,𝑗⟩

⟨𝑖,𝑗⟩

158

(4.6)

elucidating the fact that this term causes spins on neighboring sites to flip. The
effective Hamiltonian is found to be (for details, see Appendix B):
𝐻 = 𝐽𝑧∑︁

𝑆𝑖𝑧 𝑆𝑗𝑧 + 𝐽𝑥𝑦

⟨𝑖,𝑗⟩

∑︁ (︀
⟨𝑖,𝑗⟩

∑︁
)︀
𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 + ℎ𝑧
𝑆𝑖𝑧 + ℰ0

(4.7)

𝑖Here ℎ𝑧 is a longitudinal field and ℰ0 is an overall energy offset. The parameters of

this Hamiltonian, to second order in perturbation theory, are:
4𝑡2
4𝑡2
4𝑡2
−
−
𝑈↑↓ 𝑈↑↑ 𝑈↑↑
4𝑡2
𝐽𝑥𝑦 = −
𝑈↑↓
2
2𝑡
2𝑡2
ℎ𝑧 =
−
𝑈↓↓ 𝑈↑↑
4𝑡2
4𝑡2
4𝑡2ℰ0 = −−−𝑈↑↑ 𝑈↑↓ 𝑈↓↓

𝐽𝑧 =

(4.8)
(4.9)
(4.10)(4.11)The relative strength of 𝐽𝑧 and 𝐽𝑥𝑦 can be tuned by a magnetic field, varying therelative strength and sign of the three interactions energies 𝑈↑↑ , 𝑈↑↓ , 𝑈↓↓ . The overall
strength of the parameters can be tuned by the lattice depth, varying 𝑡. The point ofzero energy can be defined to be ℰ0 = 0. The longitudinal magnetic field is typically∑︀ 𝑧ignored in cold atom experiments, because the total magnetization 𝑆 𝑧 =
𝑆𝑖 =
𝑖
∑︀
(𝑛𝑖↑ − 𝑛𝑖↓ ) is fixed, so that this field does not play a role, except in a few cases𝑖we describe below. Therefore, when ℎ𝑧 and ℰ0 are ignored, the pure anisotropic

Heisenberg Hamiltonian Eq. (4.2) is recovered.

We have observed the effect of ℎ𝑧 in a few cases. Since ℎ𝑧 is a differential energy
between the |↑⟩ and the |↓⟩ spins, it can bias the system to have a higher or lower
number of |↑⟩ spins relative to the number of |↓⟩ spins. In experiments, the two spinstates are mapped onto atomic hyperfine states, which cannot be transformed from
one to the other unless a coupling field is applied. When experiments are performedin the spin 𝑧 basis, the ℎ𝑧 field does not play a role in the absence of a coupling field.
If a coupling field is applied, ℎ𝑧 shifts the resonant frequency accordingly. Whenexperiments are performed in the 𝑥𝑦 spin basis, ℎ𝑧 shifts the precession frequency. In
159

a homogeneous system, this shift is not detectable. In a trapped system, the trapping
potential makes the Hubbard parameters vary across the lattice because the lattice
depth varies. Then, the effective magnetic field ℎ𝑧 will also vary, which can lead todephasing of the spin components in the 𝑥𝑦 plane between lattice sites with different
lattice depth.
Jordan-Wigner transformation
It is instructive to transform this Hamiltonian back to the density sector by a JordanWigner transformation. This transformation is typically performed in order to simplify numerical simulations of spin-1/2 chains. It also provides a different and sometimes more intuitive perspective of the spin phases and spin transport. In this transformation, the spin model is mapped to a spinless fermion model with either one orzero fermions per site:|↑⟩ → |1⟩
(4.12)|↓⟩ → |0⟩In this mapping, care has to be taken to fix the phase of the fermionic operators oneach site 𝑓𝑗 so that they obey the fermionic anticommutation relations [40]:𝑆𝑗𝑧 =𝑓𝑗† 𝑓𝑗 −1
2

𝑆𝑗+ =𝑓𝑗† 𝑒𝑖𝜋𝑙<𝑗 𝑛𝑙∑︀

𝑆𝑗− =𝑓𝑗 𝑒−𝑖𝜋∑︀(4.13)

𝑙<𝑗 𝑛𝑙where 𝑛𝑙 = 𝑓𝑙† 𝑓𝑙 is the number of fermions on site 𝑙. Then, the XXZ Heisenberg modelmaps to [40]:)︁∑︁∑︁𝐽𝑥𝑦 ∑︁ (︁ ††𝐻𝐽𝑊 = −
𝑓𝑖 𝑓𝑗 + 𝑓𝑗 𝑓𝑖 − 𝐽𝑧𝑛𝑖 𝑛𝑗 + 𝐽𝑧
𝑛𝑖
2
𝑖
⟨𝑖,𝑗⟩

⟨𝑖,𝑗⟩

160

(4.14)

Then, the 𝐽𝑥𝑦 term is interpreted as the kinetic energy, or the tunneling, and the 𝐽𝑧
term as the interactions between fermions on neighboring sites. Although initiallyintroduced for 1D systems, the Jordan-Wigner transformation can be extended tohigher dimensions.
Ground states and phase diagram
To determine the phase diagram of the Heisenberg spin-1/2 model, a variational
approach can be used, as in [49]. For reasons outlined above, in homogeneous systems
of ultracold atoms, Eq. (4.2) (with no external fields) can be used as a spin model andin a trapped system, it serves as a good approximation. We take four trial states andevaluate their energies for a double-well system. Since these states are product statesover all sites with the same wavefunction on each site, they are not the many-bodyground states, but this approach gives an estimate of the character of the groundstate as a function of the anisotropy 𝐽𝑧 /𝐽𝑥𝑦 :Z Ferromagnet | ↑↑⟩ : 𝐸↑↑ =

𝐽𝑧
4

𝐽𝑧
Z Anti-ferromagnet | ↑↓⟩ : 𝐸↑↓ = −4 )︂ (︂(︂)︂|↑⟩ + | ↓⟩|↑⟩ + | ↓⟩𝐽𝑥𝑦√√
XY Ferromagnet | →→⟩ =: 𝐸→→ =
4
2
2
(︂
)︂ (︂
)︂
|↑⟩ + | ↓⟩|↑⟩ − | ↓⟩𝐽𝑥𝑦√√
XY Anti-ferromagnet | →←⟩ =: 𝐸→← = −4
2
2
Therefore, we can distinguish between two cases depending on the sign of 𝐽𝑥𝑦 , which
for 7 Li depends only on the sign of the scattering length between the two hyperfine
states 𝑎𝑎𝑏 . The corresponding ground states are listed in Table 4.1.
𝐽𝑧
>1
𝐽𝑥𝑦
𝐽𝑧
−1 < 𝐽𝑥𝑦 < 1
𝐽𝑧
< −1
𝐽𝑥𝑦

𝐽𝑥𝑦 > 0Z antiferromagnet
XY antiferromagnet
Z ferromagnet

𝐽𝑥𝑦 < 0Z ferromagnet
XY ferromagnet
Z antiferromagnet

Table 4.1: Ground states for the anisotropic spin-1/2 Heisenberg model.

161

This approach can serve as a good starting point in studying spin physics. In
real systems, more effects need to be accounted for. Depending on dimensionality,
system size, presence of defects, the properties of the spin systems can change. In
the following, we focus on 1D systems, which are easier to calculate numerically andhave faster dynamics in cold atom experiments. In 1D the SF-MI transition is at
lower lattice depths (see Section 3.1.3), so that experiments can be performed in the
MI at lower lattice depths, hence with faster superexchange 𝑡2 /𝑈 . On the one hand,by comparing experimental 1D results to numerical simulations, our system can bebenchmarked as a quantum simulator. On the otherhand, the unique properties ofcold atom systems (pure states, low temperatures, tunability, isolation from environment) can give a new perspective on Heisenberg spin chain studies and can inspire
further research.
Heisenberg spin-1/2 chains: 1D physics
The spin-1/2 Heisenberg chains are integrable and have long-range correlations. In
integrable models, the number of conserved quantities is the same as (or larger than)
the number of degrees of freedom. The spectrum of the Hamiltonian and the spin-spincorrelations can be computed in many cases analytically by the Bethe ansatz. It has
several variants, of which the algebraic Bethe ansatz is conceptually simplest. The
main idea is that the Hamiltonian of the integrable system can be constructed via atransfer matrix, which satisfies the Yang-Baxter relation. Then, the eigenstates andeigenvalues of the transfer matrix can be computed. For a review of this method see[148].
For small systems, we can directly diagonalize the Heisenberg Hamiltonian Eq. (4.2).
We will do this for two and for six sites, because some aspects of the model are not162transparent for two sites. We start with two sites, where the eigenstates are:|1⟩ = |↑↑⟩
|2⟩ = |↓↓⟩

1
|3⟩ = √ (|↑↓⟩ + |↓↑⟩)
2
1|4⟩ = √ (|↑↓⟩ − |↓↑⟩)
2(4.15)

These are the typical triplet (|1⟩, |2⟩, |3⟩) and singlet (|4⟩) states. Their energies
are plotted as a function of 𝐽𝑧 /𝐽𝑥𝑦 in Fig.4-2. The Z ferromagnet is the degen-erate states |↑↑⟩ and |↓↓⟩. We can associate the XY ferromagnet with the state√√(| ↑↓⟩ + | ↓↑⟩) / 2 and the XY antiferromagnet with the state (| ↑↓⟩ − | ↓↑⟩) / 2,

which are ground states when 𝐽𝑥𝑦 < 0 and 𝐽𝑥𝑦 > 0 respectively. Then, as in thevariational approach, the phase transition from the Z ferromagnet to the XY antiferromagnet is at 𝐽𝑧 /𝐽𝑥𝑦 = − 1 (Fig. 4-2a) and to the XY ferromagnet at 𝐽𝑧 /𝐽𝑥𝑦 = + 1(Fig. 4-2b). The classical Z antiferromagnet, the Neel state, in the double-well pictureis the degenerate states |↑↓⟩ and |↓↑⟩, which are not eigenstates of the Hamiltonian,but superpositions of the eigenstates. To see see how the Z antiferromagnet comes tobe the ground state, more than two sites are needed.
Jxy>0

1
0.5
0
-0.5
-1
-1.5
-3

Jxy<0

b) 1.5
Energy (in units of Jxy )Energy (in units of Jxy )a) 1.5

1
0.5
0
-0.5
-1

-2

-1

0

1

2

-1.5
-3

3

J z/J xy

-2

-1

0

1

2

3

J z/J xy

Figure 4-2: Energy spectrum of the Heisenberg model in a double-well. a) Antiferromagnetic
coupling 𝐽𝑥𝑦 > 0; b) Ferromagnetic coupling 𝐽𝑥𝑦 < 0.

Next, we directly diagonalize the Hamiltonian Eq. (4.2) for six sites in the basis|𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 ⟩ where 𝜎𝑖 = ±1/2 is the spin on site 𝑖. Note that the Hamiltonian is163block diagonal (see Appendix B), so that manifolds with different total magnetization∑︀∑︀(𝑛𝑖↑ − 𝑛𝑖↓ ) are not coupled. Put another way, 𝑆 𝑧 commutes with the𝑆 𝑧 = 𝑆𝑖𝑧 =𝑖𝑖

Hamiltonian. The energy spectrum is shown in Fig. 4-3. The energy levels with totalmagnetization 𝑆 𝑧 = 0 are shown in green and the ones with 𝑆 𝑧 = 3 are shown in red.
The ground states for different values of the anisotropy are in either of these two
manifolds. The eigenstates can be expressed as:
|Ψ⟩ =

𝐿
∑︁
𝑗

(4.16)

𝐶𝑗 |𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 ⟩𝑗where 𝐿 depends on the value of 𝑆 𝑧 . For example, for |𝑆 𝑧 | = 3, 𝐿 = 1 and for |𝑆 𝑧 | = 0,
𝐿 = 20.
a)

Jxy > 08
Z-FM

XY-AFM

b)
Z-AFM

Z-AFM

XY-FM

Z-FM

6

Energy (in units of Jxy )Energy (in units of Jxy )

6
4
2
0
-2
-4
-6
-8
-2

Jxy < 0

8

4
2
0
-2
-4
-6

-1

0

1

2

3

4

-8
-5

5

J z/J xy

-4

-3

-2

-1

0

1

2

J z/J xy

Figure 4-3: Energy spectrum of the Heisenberg model for 6 sites. The energy levels withtotal magnetization 𝑆 𝑧 = 0 are shown in green and the ones with 𝑆 𝑧 = 3 are shown in red.
The ground states are indicated on the top. a) Antiferromagnetic coupling (𝐽𝑥𝑦 > 0). b)
Ferromagnetic coupling (𝐽𝑥𝑦 < 0).

The 𝐽𝑥𝑦 > 0 and 𝐽𝑥𝑦 < 0 cases are similar for the same sign of 𝐽𝑧 except that symmetric states are replaced by antisymmetric states except for the 𝑆 𝑧 = ±3 manifolds,

which each contains only the fully spin-polarized state which is a product state. Notethat for 7 Li, 𝐽𝑥𝑦 > 0 (antiferromagnetic coupling) but a ferromagnetic coupling canbe realized by a tilt, switching the sign of 𝐽𝑥𝑦 (Chapter 5). We focus our discussion
on the ferromagnetic case 𝐽𝑥𝑦 < 0 for simplicity. As in the double-well system, at
𝐽𝑧 /𝐽𝑥𝑦 = 1, there is a phase transition from a Z ferromagnet to an XY ferromagnet.
At this transition, the ground state changes nature: from one with total magneti164zation 𝑆 𝑧 = ±3 to one with 𝑆 𝑧 = 0. At the other transition point, 𝐽𝑧 /𝐽𝑥𝑦 = − 1, theground state does not switch magnetization manifolds, signaling that the transition
between the XY phase and the Z antiferromagnet has a different nature. For large
and negative 𝐽𝑧 /𝐽𝑥𝑦 , the lowest two states become almost degenerate. In the limitof infinite chain and large and negative anisotropy, the ground state of the system
is expected to be the two classical Neel states. We discuss each of the three regionsnext.
E = 0.73 J xy

E = 0.77 J xy
0.4

0.4

0.2

E = 0.86 J xy
0.6
0.5
3

0.5

2

0.6

1

0.6

0

0.3

0.4
0.3

-0.2

0.2

0.2

-0.4

0.1

0.1

-0.6

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0

0.4
0.4

0.3
0.3

| 3|2

| 2|2

| 1|2

0.3
0.2

0.2

0.2

0.1

0.1

0.1

0

0

0

Figure 4-4: First three excited states of the ferromagnet. Shown are the coefficients 𝐶𝑗
from Eq. (4.16). Here 𝐽𝑧 /𝐽𝑥𝑦 = 2 and 𝐽𝑥𝑦 < 0. The two colors indicate the two degeneratestates for each energy level. The first and the second excited state have total magnetization
𝑆 𝑧 = ±2 and the 3rd excited state has 𝑆 𝑧 = ±1. The 4th excited state (not plotted) has
𝑆 𝑧 = ±0. The energy of each state is indicated on the top.

In the Z ferromagnet region 𝐽𝑧 /𝐽𝑥𝑦 > 1, the ground state is two-fold degenerate
and the states are |↑↑↑↑↑↑⟩ and |↓↓↓↓↓↓⟩. It is the classical ground state with total

spin ± 𝑁/2 for 𝑁 particles on the chain. The Z-ferromagnetic phase has magnon
excitations and is gapped. The first three excited states are shown in Fig. 4-4 and are
all two-fold degenerate, just as the ground state. The first two excited states havetotal magnetization 𝑆 𝑧 = ± 2, i.e. they have one flipped spin compared to the groundstate. The states with more weight have a single domain wall (i.e. a spin is flipped
at the edges of the system). The next excited state has two flipped spins and so on.
At 𝐽𝑧 /𝐽𝑥𝑦 = 1, the nature of the ground state changes character and for |𝐽𝑧 /𝐽𝑥𝑦 | < 1

it includes all states with zero total magnetization 𝑆 𝑧 = 0. This is the XY ferromagnet
165

Jz/Jxy = 1, Jxy < 0

Jz/Jxy = 0, Jxy < 0

0.3

0.3

0.3
0

0.4

0

0.4

0

0.4

0.1

0

0

0

0.2

0.2

0.15

0.15

0.15

0.1

| 0|2

0.2

| 0|2

0.1

0.1

0.05

0.05

0.05

0

0

0

Jz/Jxy = 0, Jxy > 0

Jz/Jxy = 1, Jxy > 0
0.4

0.2

0.2

0.2

0

0

0.4

0

0.4

0

0

-0.4

-0.4

-0.4

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0.15

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0.1

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0.2

Jz/Jxy = -1, Jxy > 0| 0|2

Jz/Jxy = -1, Jxy < 0

0.1

0.1

0.05

0.05

0.05

0

0

0

Figure 4-5: XY phase. Shown are the coefficients 𝐶𝑗 from Eq. (4.16). Here |𝐽𝑧 /𝐽𝑥𝑦 | < 1. For
𝐽𝑥𝑦 < 0, it is a ferromagnet (top two rows) and for 𝐽𝑥𝑦 > 0, it is an anti-ferromagnet (bottom
two rows). The ground state is a superposition of all states with total magnetization 𝑆 𝑧 = 0.

and the ground state as a function of anisotropy is shown in Fig. 4-5. For comparison,
this figure also shows the ground state for antiferromagnetic coupling, illustrating thatthe difference between the two is just the symmetry of the state. It is an easy-plane
state: a superposition of all states with zero z-component of the total magnetization,
so that the spin vector points in all directions in the 𝑥𝑦 plane.
Focusing on the ferromagnetic case, there are two notable points: 𝐽𝑧 = 0 and
𝐽𝑧 = 𝐽𝑥𝑦 (technically the phase-transition point). At 𝐽𝑧 = 0, the XX Heisenberg modelis realized. In the language of spinless fermions (Jordan-Wigner transformation),

166

this is the non-interacting point. Transport is expected to be ballistic here. At theisotropic point 𝐽𝑧 = 𝐽𝑥𝑦 , the system is symmetric in all directions of space. This is ahighly degenerate point.
As 𝐽𝑧 /𝐽𝑥𝑦 is varied between +1 and −1, the weights of the different states with𝑆 𝑧 = 0 change. The amplitudes of two of these states, the two Neel states | ↑↓↑↓↑↓⟩

and | ↓↑↓↑↓↑⟩, grow continuously and for 𝐽𝑧 /𝐽𝑥𝑦 < − 1, they dominate. At the otherend of the XY phase, at 𝐽𝑧 /𝐽𝑥𝑦 = 1, the ground state is equally likely to be in anyof the 𝑆 𝑧 = 0 basis states. This means that this state has maximum entanglement(maximum information). It can be interpreted as a spin-squeezed state, i.e. thevariance of 𝑆 𝑥𝑦 is maximum (it goes all around the Bloch sphere) and the variance of𝑆 𝑧 is minimum. Therefore, it could potentially be useful for metrology (for a review
on spin squeezing and its uses for metrology, see e.g. [104]).
In the infinite chain, the XY phase is gapless, however, in a finite system a gap
opens up, which is on the order of 𝐽𝑥𝑦 /𝑀 , where 𝑀 is the number of spins in the
chain. In the XY phase, it is maximum at the 𝐽𝑧 = 0 point and goes to zero at theisotropic point, even for a system of 6 sites considered here. Therefore, adiabatic
state preparation schemes work better at the 𝐽𝑧 = 0 point. In an infinite system, thedispersion relation is linear at the XX point, while it gradually becomes quadratic at
the isotropic point.
For 𝐽𝑧 /𝐽𝑥𝑦 < − 1, the system is expected to be in the Z antiferromagnetic state.

The ground state and the first excited state are plotted in Fig. 4-6. The groundstate is still in the 𝑆 𝑧 = 0 manifold but now the weight of the Neel states is muchlarger. For example, for 𝐽𝑧 /𝐽𝑥𝑦 = −5, the case plotted in the figure, the probability

is 87 % and 94 % for the ground and the first excited state respectively. Therefore, toa good approximation, the ground state is the |+⟩ = | ↑↓↑↓↑↓⟩ + | ↓↑↓↑↓↑⟩ state andthe first excited state is |−⟩ = | ↑↓↑↓↑↓⟩ − | ↓↑↓↑↓↑⟩. For large and negative 𝐽𝑧 /𝐽𝑥𝑦 ,the gap between the two decreases. The gap also decreases with system size. For asystem with 𝑀 sites the gap between the two scales exponentially with system size
∆01 ∝ exp(−𝑐𝑀 ), where 𝑐 is constant with size [113]. The second excited state, as in

the Z ferromagnetic case, has one flipped spin and is doubly degenerate, as plotted167

E = 0 J xy

E = 0.13 J xy
0.5

0.4

1

0

0.6

0

0.2
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|

|

|

|

|

|

|

|

|

|

|

|

|

|

|

|

|

|

|

0

| 2|2

0.4
0.3
0.2
0.1
0

Figure 4-6: The antiferromagnet. Shown are the coefficients 𝐶𝑗 from Eq. (4.16). Here𝐽𝑧 /𝐽𝑥𝑦 = −5 and 𝐽𝑥𝑦 < 0. The ground state and the first excited states are plotted in the
top two rows. The probability of being in either of the two Neel states is 87 % and 94 % for
the two states respectively. The energy gap here is 0.129 |𝐽𝑥𝑦 |. The second excited state isplotted in the bottom two rows.

in the bottom two rows of Fig. 4-6.

4.1.2

Spin 1

Mapping from Hubbard model
A similar approach can be taken to derive the effective Hamiltonian for a system of
two particles per site. Then, two-component Bose-Hubbard model maps to a spin-1

168

Heisenberg model. The basis states on each site are now:|+1⟩ = | ↑↑⟩1|0⟩ = √ (| ↑↓⟩ + | ↓↑⟩)
2|−1⟩ = | ↑↑⟩1|𝑠⟩ = √ (| ↑↓⟩ − | ↓↑⟩)
2

(4.17)

where the singlet state |𝑠⟩ is not coupled to the triplet manifold (|+1⟩, |0⟩, |−1⟩). The

singlet state is antisymmetric, while the triplet states are symmetric and therefore
the two cannot be connected adiabatically. The effective Hamiltonian in this case is:
𝐻=𝐽

∑︁
⟨𝑖,𝑗⟩

S𝑖 · S𝑗 + 𝑢∑︁(𝑆𝑖𝑧 )2 + 𝛿 2 𝐻1

(4.18)𝑖where:
2𝑡2
2𝑡2+
𝑈↑↑ 𝑈↓↓1𝑢 = (𝑈↑↑ + 𝑈↓↓ ) − 𝑈↑↓2

𝐽=(4.19)

and at the region of interest (around 𝑢 = 0), 𝛿 ≈ 0.1, so the last term can be ignored.
The 𝐽 term is the typical Heisenberg term. When 𝑢/𝐽 ≫ 1, the ground state is a

product state with the two spins on each site in the |0⟩ state. We call this state theSpin Mott state. When 𝑢/𝐽 ≪ −1 is large and negative, the system is a Z antifer-romagnet or a Z ferromagnet for 𝐽 > 0 and 𝐽 < 0 respectively. Around the isotropic
point 𝑢 = 0, there is a symmetry-protected topological phase for antiferromagnetic
coupling 𝐽 > 0 in 1D, which we discuss next.
Heisenberg spin-1 chains
As in the case of spin-1/2, 1D systems are more amenable to numerical and theoretical
exploration. However, in an antiferromagnetic spin-1 chain, there is a symmetry-

169

protected topological (SPT) phase, the Haldane phase, absent in the spin-1/2 case.
We focus the rest of the discussion on the antiferromagnetic case 𝐽 > 0 and notethat there is a region in the 7 Li Feshbach diagram for which the effective spin-1
Hamiltonian is in the Haldane phase regime.
The SPT phase is named after Haldane, who in a series of 1983 papers [69, 68]
realized that there is a difference between half-integer and integer AFM spin chains.
This came as a surprise to the community because the behavior of spin chains with
spin-1/2 (which could be solved analytically by Bethe ansatz) was similar to theclassical limit behavior of 𝑆 → ∞ chains and it was assumed that spin chains withany spin in between would behave the same way. Haldane’s original papers did notgive a rigorous proof but later analytical, numerical, and experimental work (forreviews see [131, 6, 8, 158]), have shown for the zero-temperature ground state that:𝑆 = 𝑛 (integer spin)

𝑆 = 𝑛2 (half integer spin)

Excitation gap

Continuous spectrum (spin waves) forsmall 𝑞∆ℰ ∝ 2𝐽𝑆𝑒−𝜋𝑆ℰ(𝑞) = 𝜋2 𝐽| sin(𝑞𝑑)|decay exponentiallydecay by a power law

spin-spin correlations ⟨𝑆(𝑥)𝑆(𝑥 + 𝑑)⟩spin-spin correlations ⟨𝑆(𝑥)𝑆(𝑥 + 𝑑)⟩

the ground state has no long-range or-

the ground state has quasi-Neel long-der (but hidden string order)

range order

We note that his discussion applies to the antiferromagnetic coupling case. In 1D
quantum fluctuations play an important role for Hamiltonians with antiferromagnetic coupling 𝐽 > 0. By contrast, for ferromagnetic coupling 𝐽 < 0, the ground state
has all 𝑁 particles with spin 𝑆 aligned, acting as one big spin 𝑁 𝑆. In fact, as we haveseen with the 6-sites toy model in the discussion for spin-1/2, the classical ferromagnetic ground state is the true quantum mechanical ground state, while the classicalantiferromagnetic state (the Neel state | ↑↓↑↓ ...⟩) is not.

The spin-1 Heisenberg Hamiltonian has been solved analytically only for a set of170

special cases. In fact, the Hamiltonian which has been considered is:
𝐻=

∑︁ (︀
⟨𝑖,𝑗⟩

52

cos 𝜃 (S𝑖 · S𝑗 ) + sin 𝜃 (S𝑖 · S𝑗 )2)︀(4.20)

2 Haldane Quantum Spin Chainstrimerized
phase

Sutherland (θ = π/4 )

ferromagneticphase

AKLT (β = 1 /3 )Haldane (θ = 0 )Haldane phase

dimerized phase
Takhtajan, Babujan(θ = - π/4 )Barber, Batchelor (θ = - π/2 )Fig. 1. Angular representation of the phase diagram of the S = 1 bilinear biquadratic hamiltonian. (Adapted from Ref. [16].)

Figure 4-7: Exactly-solved Hamiltonians of the form of Eq. (4.20) as a function of 𝜃. Here𝛽 = sin 𝜃/ cos 𝜃. This diagram illustrates that the Haldane phase is of the same class as the
In this representation, the exact solutions discussed above correspond to θ = π/4
AKLT phase and the two (Sutherland),can be adiabaticallyconnected.
from et[131].
θ = −π/4 (Takhtajan,
Babujian), θ = Figure
arctg(1/3) (Affleckal.),θ = −π/2 (Barber, Batchelor) and the Haldane point is for θ = 0. In this diagram
the Haldane phase should occur in the region −π/4 < θ < π/4; the points θ = −π/4
and θ = π/4 at which the gap vanishes, should be critical points separating the
Haldane phase from, respectively, the dimerized and trimerized AF phases [16]. The
Haldane phase at β = 0 can be intuitively interpreted as being similar to the VBS
phase, which is the exact ground state for β = tg θ = 1/3. In this frame, spin 1 ateach lattice site is obtained by symmetrization of two S = 1/2 variables at the samesite. The VBS ground state is realized as a symmetrical linear superposition of states
where one of the two S = 1/2 variables at each site is paired in a singlet state (valencebond) with one of the two S = 1/2 variables at the neighboring site. This picture
clearly reveals the difference between integer spin chains, which may have a VBS
ground state of unbroken symmetry with a gap, and half-integer spin chains, which
are necessarily partially dimerized. In this latter case, if the translational symmetry2
is restored by quantum fluctuations leading to a unique resonating-valence-bond𝑖𝑗 [18].
𝑖𝑗(RVB) state [17], the gap vanishes
The VBS⟨𝑖,𝑗⟩state of the S = 1 AF chain is characterized by the development ofstring correlations which reflect a hidden topological long-range order [19,20]. The
string correlations have been extensively studied for the general hamiltonian, Eq. (6),Fig.4-7a shows the exactly-solved points as a function of 𝜃 and the corresponding
phases. The Haldane phase (𝜃 = 0) is expected to be of the same class as the AKLT
phase, so we next discuss the AKLT phase. It is the exact solution of:
)︂
∑︁ (︂
1
𝐻=
(S · S ) + (S · S )3(4.21)

given by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [7, 8]. The ground state isa valence-bond state (VBS). Here, each spin-1 (on each site) is represented as asuperposition of two spin-1/2 particles (similar to our implementation with a MIwith two atoms per site), which are in a triplet state. Each of the two spin-1/2
particles is in a singlet state with one of the spin-1/2 particles on the neighboringsites, as illustrated in Fig.4-8a. In the case of a finite chain, this leaves two uncoupled
spin-1/2 particles on each end of the chain. Since each end can be in |𝑠 = ± 1/2⟩,
this leaves a four-fold degeneracy. These spin-1/2 particles are similar to Majorana

fermions, because they are correlated through the chain, so that this state can be of171interest
quantum
can be offorinterestfor computing.
quantum computing.
a)
b)
b)

-1
+1

+1

0

-1

+1

-1

+1

0

0

+1

0

0
-1
-1

-1

0

0

+1

-1

0

0

0

+1

0

0

-1

-1
-1

+1
+1
0

0

-1

00
+1

-1-1
-1

0 0

+1
+1

+1

-1

00

+1+1

-1-1

+1
-1
+1
00a
+1 states00-1 a) a)+1Pictorial
Figure 1-7:
4-8: Valence
Valence bondbondstates.
of 0theasof-1Oneach -1
site,representation
the00spin-100particle
a +1
superposition+1 particle
0 is spin-1
superpositionof twostate.
spin-1/2
in a triplet sites,
state eachon thesamesite andin singlet
spin-1/2 in a triplet
On neighboringof thespin-1/2
particles(onestatesfrom withthethe
Hiddenstringorderstate.
in theThis
HaldaneThe ground
state
is a
left neighboringand one fromsite.
the b)
right)are ina singletleavesphase.
two uncoupledspins
at the0superpositionofconfigurations
withalternating
|+1⟩
and
|+1⟩
sites
with
any
numberof|0⟩0edges. Since each end canb)
Hidden00 leaves-1-1+10degeneracy.
-1-1
00
+1
0 be in0 |s = ±001/2i, this+1 a four-fold
0
+1
sites
between.
Here
we show
threeThesuchgroundconfigurations.
stringinorderin theHaldanephase.
state is a superposition of configurations
with alternating |+1i and |+1i sites with any number of |0i sites in between.

-1-1

0

0

+1

00

0

00

The Haldane phase is also a VBS but it has hidden string order (for a discussion
The Haldane phase is also a VBS but it has hidden string order (for a discussion seesee [83]). A pictorial representation is shown in Fig.4-8b. The state is a linear[23]). A pictorial representation is shown in Fig.1-7b. The state is linear superposition
superposition of all configurations with zero magnetization in which sites with spinof all configurations (with zero magnetization) in which sites with spin |+1i alternate|+1⟩ alternate with sites with spin |−1⟩ with any number of sites with spin |0⟩ inwith sites with spin | 1i with any number of sites with spin |0i in between. The
between. The state has no long-range order in the sense that if one were to pick astate has no long-range order in the sense that if one were to pick a site in the chain,site in the chain, one cannot predict what the spin would be further down the chain
one cannot predict what the spin would be further down the chain because of the
because of the uncontrolled number of spin |0⟩ sites. Mathematically, this hidden
uncontrolled number of spin |0i sites. Mathematically, this hidden order is capturedorder is captured by the correlator:
by the correlator:(︁ ∑︀𝑘−1 𝛼 )︁𝑆𝑙 ⌘(4.22)𝒪↵𝛼 = lim ⟨Ψ𝑔 | − 𝑆↵𝑗𝛼⇣ 𝑒ik𝑖𝑘Pk𝑙=𝑗+1
𝑆↵𝑘𝛼 |Ψ𝑔 ⟩1
Sl↵|𝑗−𝑘|→∞
l=j+1
O = lim h g | Sj e
Sk | g i(1.16)|j k|!1

where 𝛼 = 𝑥, 𝑦, 𝑧. The exponent keeps track of the number of |0⟩ sites between sites
where ↵ = x, y, z. The exponent keeps track of the number of |0i sites between sites(𝑖, 𝑗): it is (−1)n𝑛 , where 𝑛 is the number of non-zero spins between sites (𝑖, 𝑗). The(i, j): it is ( 1) , where n is the number of non-zero spins between sites (i, j). The
correlator is non-zero in the Haldane phase. This a necessary but not sufficientcorrelator is non-zero in the Haldane phase but this a necessary but not suﬃcientcondition for a state to be the Haldane phase.
condition for the Haldane phase.
The Hamiltonian which describes the majority of real materials which realize the
Although the Haldane gap in spin-1 chains has been experimentally observed insome materials, typically, D/J ⌧ 1 [35]. The control of the parameters with ultracold
12
172

00

diagonalization of finite-size systems. The numerical data are analyzed using conformal field theory, the level
spectroscopy, phenomenological renormalization-group, and finite-size scaling method. We thus present aquantitatively reliable ground-state phase diagram of this model. The ground states of this model contain the
Haldane phase, large-D phase, Néel phase, two XY phases, and the ferromagnetic phase. There are four
different types of transitions between these phases: the Brezinskii-Kosterlitz-Thouless-type transitions, theGaussian-type transitions, the Ising-type transitions, and the first-order transitions. The locations of these
critical lines are accurately determined.

spin-1 Heisenberg chain
is: number!s": 75.10.Jm, 75.40.Mg, 75.40.Cx
DOI: 10.1103/PhysRevB.67.104401
PACS∑︁phases,∑︁(︀ 𝑥 the)︀ phase,the∑︁large-D𝑧 phase,
two XY𝑧
𝑥 ferromagnetic
4,5
𝐻 = 𝐽and𝑆𝑖 𝑆phase.
+ 𝑆𝑖𝑦various
𝑆𝑗𝑦 +types
𝐷 of (𝑆𝑖𝑧 )2𝑧 the Néelthese
𝑗 + 𝐽𝑥𝑦Between𝑆𝑖 𝑆𝑗phases,

I. INTRODUCTION(4.23)

phase
mensional antiferromagnetic spin chains have been⟨𝑖,𝑗⟩transitions take place.
⟨𝑖,𝑗⟩There is a gapful phase to gapful𝑖
phase transition !Gaussian transition" between the Haldaneof recent investigations by numerous groups. A
phase and large-D phase, gapful-gapless Berezinskiitiferromagnetic Heisenberg chain is known to!BKT" transitions
between
XY 4-9.
phase Note that the ferroess ground state for half where
integer spin.
Inparticu𝐷 ≡ 𝑢 above. Kosterlitz-Thouless
The phase diagram
is shown
in theFig.
and the Haldane or large-D phase, an Ising transition bect solution is available for S!1/2 chains.1 In contween the Néel phase and Haldane phase, a first-order traneger spin,2 there is a gap magnetic
between the firstexcitedandanti-ferromagnetic
sides differ in the 𝐷 → 0 region. The ferromagnetic
sition between the ferromagnetic phase and the large-D
e ground state. This state is destroyed by various
phase or XY phase. The character of the transition between
rturbations such as single-ion-type
exside hasanisotropy,
an XY-ferromagnet
in thespin-1/2
case)although
and theantiferromagnetic side
the large-D(likeand Néelphase
is still unclear
it issotropy, and bond alternation.3– 6 In this context,
likely to be the first-order transition.
XXZ chain with single ion
has been
hasanisotropy
a Haldanephase. Our
Forphase
large
𝐷, thesystem is
Spin
Mott phase, described
diagramis summarized
in in
Fig.the1. ForJ z #0,many authors from the early stage of the study oftheHaldane-large-Dtransition
line
of
is
shown
by
the
!.
p problem. In the present work, we present theabove, which is known
in thecondensed
as thethe large-D phase. For
For large
D, theground statematterbecomes literaturea large-D phase,
phase diagram of this model analyzing the exact
Haldanephase
appearsunderthelarge-Dphase.
With
thetion data by various methods
including the re7
Li, 𝐽𝑧7 =𝐽𝑥𝑦onsoconthat decrease
we realize
theground
red statelinebecomesin thethe
figure.
of D, theNéel phase. The
loped level spectroscopy method
basedline with the symbol " represents the Haldane-Néel transid theory and renormalization group. These methysis allow us to obtain an accurate phase diagram
D/Jxy
the numerical data for relatively small size sys<latexit sha1_base64="QDmiFFistfUPtsgfI3m8Lz/+828=">AAAB73icbVBNSwMxEJ31s9avqkcvwSJ4qrsq6LGoB/FUwX5Au5Rsmm1Dk+yaZMVl6Z/w4kERr/4db/4b03YP2vpg4PHeDDPzgpgzbVz321lYXFpeWS2sFdc3Nre2Szu7DR0litA6iXikWgHWlDNJ64YZTluxolgEnDaD4dXYbz5SpVkk700aU1/gvmQhI9hYqXV9fNvNntJRt1R2K+4EaJ54OSlDjlq39NXpRSQRVBrCsdZtz42Nn2FlGOF0VOwkmsaYDHGfti2VWFDtZ5N7R+jQKj0URsqWNGii/p7IsNA6FYHtFNgM9Kw3Fv/z2okJL/yMyTgxVJLpojDhyERo/DzqMUWJ4aklmChmb0VkgBUmxkZUtCF4sy/Pk8ZJxTutuHdn5eplHkcB9uEAjsCDc6jCDdSgDgQ4PMMrvDkPzovz7nxMWxecfGYP/sD5/AHQBo/R</latexit>

er is organized as follows. In the next section, themiltonian is defined and the obtained phase diaesented. The numerical exact diagonalization remethods of analysis are explained in Sec. III. The
n is devoted to a summary and discussion.
HAMILTONIAN AND GROUND-STATE PHASEDIAGRAMJz /Jxy<latexit sha1_base64="v+Ekk3Lj4mphQM4jomuB90AQQ9c=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU01U0GPRi/RUwX5gG8Jmu2mXbjZhdyPG0H/hxYMiXv033vw3btsctPXBwOO9GWbm+TFnStv2t1VYWl5ZXSuulzY2t7Z3yrt7LRUlktAmiXgkOz5WlDNBm5ppTjuxpDj0OW37o+uJ336gUrFI3Ok0pm6IB4IFjGBtpPu693RS97LHdOyVK3bVngItEicnFcjR8MpfvX5EkpAKTThWquvYsXYzLDUjnI5LvUTRGJMRHtCuoQKHVLnZ9OIxOjJKHwWRNCU0mqq/JzIcKpWGvukMsR6qeW8i/ud1Ex1cuhkTcaKpILNFQcKRjtDkfdRnkhLNU0MwkczcisgQS0y0CalkQnDmX14krdOqc1a1b88rtas8jiIcwCEcgwMXUIMbaEATCAh4hld4s5T1Yr1bH7PWgpXP7MMfWJ8/cxKQxA==</latexit>

miltonian is given byNxyz"S ly S l"1" "J z S zl S l"1$J ! S xl S l"1%"D# S z2l ,

l!1

!1"

s a spin-1 operator. The parameter D represents
gle-ion anisotropy. The periodic boundary condiFIG. 1. The phase diagram of S!1 XXZ chains with uniaxial
Figure 4-9:
Phaseof theanisotropy.
Heisenberg
anisotropy 𝐽𝑧 /𝐽𝑥𝑦
med unless specifically mentioned.
In whatfol- diagram
single-ion-type
The solidspin-1
lines andchain
symbolswith
are thevariable
tran7
et J!1 to fix the energy as
scale.
The
ground-statesitionlines.
The
dotted
line
shows
the
curve
J
!$1/2
!
D
!
expectedin Eq. (4.23). Figure adapted from [35]. With Li,z the red dashed line 𝐽𝑧 = 𝐽𝑥𝑦 can be
am of this model consists of the Haldane phase,from the perturbation calculation for large negative D.

realized.

003/67!10"/104401!7"/$20.00

67 104401-1©2003 The American Physical Society

A Hamiltonian which better describes the 7 Li system has been considered in [158].
Here, a term 𝐽 ′ is added which represents interchain coupling:
𝐻=𝐽

∑︁
⟨𝑖,𝑗⟩

S𝑖 · S𝑗 + 𝑢∑︁𝑖(𝑆𝑖𝑧 )2 + 𝐽 ′

∑︁
[𝑖,𝑘]

S𝑖 · S𝑘(4.24)

where ⟨𝑖, 𝑗⟩ denotes nearest-neightbors along the chains and [𝑖, 𝑘] denotes nearestneighbors between neighboring chains. The phase diagram is shown in Fig.4-10.

The control of the Hamiltonian parameters which ultracold atoms offer can reveal
173

Characterizing the Haldane phase in quasi-one-dimensional spin-1 Heisenberg antiferromagnets

9

Néel

0.020

0.012

0.008

NDMAP, Honda et al. [23]NENP, Zaliznyak et al. [28]NENP@2.5GPa, Zaliznyak et al. [28]PbNi2V2O8, Zheludev et al. [30]Ising AFM

J’J’/J

0.016

Present work (constant D)
Present work (constant J’)
AgVP2S6, Takigawa et al. [22]SrNi2V2O8, Zheludev et al. [30]
Y2BaNiO5, Xu et al. [35][Ni(HF2)(3-Clpy)4]BF4, Manson et al. [57]

XY AFM

0.004

Haldane
0.000
-0.5

0.0

QPM
1.0

0.5

1.5

2.0

D

u/J

5. Phase diagramin theD–J 0 planewith phase boundaries
as indicatedthe dotted black
Figure 4-10: PhaseFig.
diagramof thespin-1Heisenbergmodelwith byinterchain
coupling 𝐽 ′ aslines. The borders of the Haldane and QPM phases are obtained as polynomial fits to the present
in Eq. (4.24). Here work
QPM
standsforforquantum
paramagnet
isdetermined
the Spin
Mott (large-D)
and represent
guidesthe eye. Data points
for the present andwork areby QMC
simulations at constant D and J 0 (solid red circles and blue squares, respectively). Several Haldane
phase. Figure from compounds[158]. are plotted as large crosshatched symbols using estimates for D and J 0 from theindicated sources. Adapted from K. Wierschem and P. Sengupta, Phys. Rev. Lett. 112, 247203(2014).

the different phase transitions and the properties of the Haldane phase. Althoughcompound TlCuCl3 .36 By contrast, the isostructural compounds PbNi2 V2 O8 andthe gap of the Haldane
hasbeenlieexperimentallyobserved
inmaking
some materials,
SrNi2 V2 O8phase
are foundto alreadyvery near to a quantum
critical point,them prime candidates for the study of pressure-induced quantum criticality.

In additionpressure,
substitution
represents
anothermechanism
for most materials 𝐷/𝐽≫ 1,to sothatchemical
the groundstate
is thelarge-Dphase and theby which the e↵ective Hamiltonian parameters of magnetic compounds can be

modified.beThisreachedis already [131].
somewhat The
apparent
in the Heisenberg
isostructural compounds
Haldane phase cannot
spin-1
Hamiltonian
has
2+
2+
PbNi2 V2 O8 and SrNi2 V2 O8 , where the di↵ering influence of the Pb

and Srleads SrNi2 V
be closer
to the Ising
AFM phasethan PbNi2across
V2 O8 . In-the Haldane
2 O8 tobeen realized withions
ultracoldions
andalthoughquenchdynamics

deed, SrNi2 V2 O8 was initially thought to magnetically order below TN = 7K basedon experiments on powder samples,30 whereas recent results on polycrystalline31
phase have been observed,
the ground state remains elusive [38, 39, 142]. As notedand single crystals32 have established its low temperature magnetic behavior to beconsistent with a non-magnetic Haldane ground state.
in [39], the HaldaneThe
phaseisofnot
connected
the Spin Mott or theprospect
chemicalfine-tuning isadiabaticallyeven more exciting tofor molecule-based

antiferromagnetic phase. The authors propose to add a term to the Hamiltonian of∑︀
the form 𝐻pert = − ℎ 𝑖 (−1)𝑖 𝑆𝑖𝑧 (a staggered magnetic field). This term breaks thecorresponding symmetries, so that in the presence of this term, an adiabatic ramp
ramp 𝑢 → 0 can be implemented, after which the perturbing term can be switched

off adiabatically, realizing the Haldane phase. Other schemes with ultracold bosons
in optical lattices could also realize a Haldane phase, for example, in a zig-zag latticewith rapid modulation of the magnetic field [161].
Indeed, the overlap between the Spin Mott phase (large-D phase) and the Haldane
phase has been studied for chains of several lengths [160]. Both phases have total
𝑆 𝑧 = 0 and are gaped, but the gap closes at the phase transition between them. Also,

one is topologically trivial and the other one is not. Therefore, it is expected that the174overlap between these two states will be vanishing in the limit of large chains. The
study [160] shows that as a function of 𝑢, the overlap quickly drops to 0 for chainswith 102 atoms, but for chains with up to 10 atoms, it remains non-zero even for|𝑢/𝐽| ≈ 5.

So far we have described the Hamiltonians which the two-component Hubbard

model with one or two atoms per site maps to. Next, we will explore to what degree
the parameters of these Hamiltonians can be varied with the Feshbach resonances of7

Li and discuss the feasibility of simulating Heisenberg spin models with 7 Li atoms.

4.1.3
7

Implementation of spin models with 7 Li

Li atoms in an optical lattice is a promising platform for simulating Heisenberg

models because the parameters of the Hamiltonians can be widely tuned. This is dueto the set of Feshbach resonances between the lowest two hyperfine states at magnetic
fields between 600 and 1000 Gauss. We have detailed the atomic nature of these statesin Section 2.2.2. Here we will refer to them as |𝑎⟩ and |𝑏⟩. We have mapped out the

Feshbach resonances using interaction spectroscopy and amplitude modulation (seeChapter 3 and [11], reprinted in Appendix D). They are shown in Fig.4-11. Although
the three interactions cannot be varied independently, by picking different magnetic
fields, many interesting magnetic phases can be realized with both spin-1/2 and spin1. There are several considerations which can determine what range of Heisenberg
parameters can be accessed with this system.
First, note that in a large part of the Feshbach diagram, one or more of the scattering lengths is negative. For a BEC, this would mean that it would collapse unlessvery small atom numbers are used [60]. However, a MI with attractive interactions
is metastable [108]. An attractive single-species MI is the highest excited state andits decay mechanisms are the same as the excitation mechanisms in a repulsive MI.
Therefore, from the perspective of the MI, we need to consider only the absolute valueof the scattering length. From the perspective of the spin models, the negative interactions flip the sign of some of the Heisenberg parameters, enabling the preparationof antiferromagnetic Heisenberg models with bosons.
175and precise
other techion energy
the latticeh hyperfine
tion (AM)Interaction e

0

-50
400

Scattering length(units of a0)

ical latticedependingeasure thegnetic field
he stability
at 103 G)o hyperfine
ction enerars are toochnique iscally large
separation
permits theal size and

(b)

aaa

200

aab
abb

0

-200
-400

650

700

750

800

850

900

950

Feshbach field (G)7 Li. Figure from [11].
4-11: Feshbachresonances
FIG. 3. Figure(a) Relativeinteractions
Uabin −
Uaa and Ubb − Uab plotted
as a function of magnetic field, measured via rf interaction specTo form ain
MIawithexperimen× experimentally
35ER × 35ERattainableoptical lattice
lattice.depth(b) and
Scattering
troscopy
35EatR an
lengthsof timescale
the aa andinteractions
plottedscattering
as a functiontallyfeasiblefor bbdynamics,
the minimum
lengthofhasmagnetic
to be larger
field,
measuredusing
lattice
AMthreeand body
shown
units ofthe Bohrra-thethanabout
40 𝑎0 . On
the otherhand,lossinbecomes
limiting
duringshown as a bold dashed line is the ab scattering length,
dius a0 . Also
preparation
phase at around 400 𝑎0 . Although the MI preparation can be done at one
obtained using simultaneous hyperbolic fits to the rf spectroscopy
scattering length and the experiments at another (see Chapter 5), we have avoided
and lattice AM data sets. The fits are shown as solid black lines.
performing
experiments
400 𝑎thethere
band admixtures could
0 because
Dotted verticallinesabove
indicate
positionof higher
a resonance.

play a role, changing the Hubbard parameters. Although this effect can be accountedfor (see for example [108]), if the scattering lengths of the two species are very differ033612-2
ent, then they will not simultaneously be in an equilibrium density distribution. We
have so far ignored this, relying on there being enough residual curvature from thetrap to keep mass transport suppressed (see Chapter 5 for tilted lattices). Also, neara Feshbach resonance in the 7 Li system, only one of the scattering lengths goes above
400 𝑎0 , while the others remain ∼ 75 𝑎0 . In order to keep the ratios of the scattering

lengths within a factor of 10, we have restricted the experimentally useful regions in
the Feshbach diagram to points where all three scattering lengths are in the region
40 a0 < |a| < 400 a0(4.25)

Next, we look for special points in the Feshbach diagram. First, there is no
point at which all three interactions are the same 𝑈↑↑ = 𝑈↑↓ = 𝑈↓↓ , although they are
all predicted to asymptote to about −40 𝑎0 at fields above the second 𝑏𝑏-resonance176

(about 𝐵 > 950 Gauss). Second, for magnetic fields between the two 𝑏𝑏 resonances,
there are two points at which 𝑈↓↓ = ± 𝑈↑↑ for the SF-MI transition occurs at the samelattice depth for 𝑎 and 𝑏 atoms. Also, in the same region of Feshbach fields, there isa point where 𝑈↓↓ = 𝑈↑↓ , so that there is a tunneling resonance between |𝑏⟩ and |𝑎𝑏⟩sites. Although these points might be interesting from the perspective of the density
sector, we next focus on the points relevant for the spin schemes.
Given these considerations, the region between 820 and 950 Gauss is suitable
for realizing spin physics. In this region, there is one scattering length which varies
significantly (𝑎𝑏𝑏 ), while the other two stay almost constant and in the region of
Eq. (4.25). As we discuss in the next section, in this region, a large variety of magnetic
phases can be realized.
Finally, in the discussion so far, only the interactions 𝑈 have been considered

to be spin-dependent, while the tunneling 𝑡 is the same for both spin states. One
way to realize spin-dependent tunneling with alkali atoms in an optical lattice is toimplement a spin-dependent lattice. For example, in rubidium this can be done bychoosing a lattice wavelength between the 𝐷1 and 𝐷2 lines so that the difference
between the vector polarizabilities of the two Zeeman levels used as pseudo-spins canbe utilized by using light with different polarizations [106, 109, 137]. If two separate
beams are used for the different polarization components, spin-dependent tunnelingcan be implemented. However, in lithium, the small fine structure splitting of 10 GHz
between the 𝐷1 and 𝐷2 lines leads to large scattering rates and loss from spontaneousemission.
Spin-1/2 with 7 Li
We first consider the case of simulating the spin-1/2 Heisenberg model with a MIwith one atom per site. The Hamiltonian regimes, characterized by their groundstates, which can be realized are shown in Fig. 4-12a. Fig. 4-12b shows the spin1/2 Heisenberg parameters 𝐽𝑧 and 𝐽𝑥𝑦 at a lattice depth of 11 𝐸𝑅 as a function ofmagnetic field. At this depth, the system is always in the MI regime in the range ofscattering lengths of Eq. (4.25). The optimal depth can be tailored for each field. The
177

400

Scattering length (a0)

a)

Z FM
Z AFM
XY AFM

200
0

-200
-400
1

b)

820

840

860

880

900

920

940

J/ħ (kHz)

0.5
0

-0.5
-1

c)

Jz
Jxy

820

840

860

820

840

860

880

900

920

940

880

900

920

940

Jz/Jxy

2
0

-2

Gauss

Figure 4-12: Spin-1/2 phase diagram for the Feshbach resonances of 7 Li. a) Groundstates. The blacked-out regions correspond to scattering lengths outside the range
40 𝑎0 < |𝑎| < 400 𝑎0 . b) 𝐽𝑧 and 𝐽𝑥𝑦 at a lattice depth of 11 𝐸𝑅 . c) Anisotropy 𝐽𝑧 /𝐽𝑥𝑦 .

blacked-out regions correspond to scattering lengths outside the range of Eq. (4.25).
This condition can be relaxed a little to lower scattering lengths if the lattice depth
is raised accordingly and to higher scattering lengths if higher band admixture isaccounted for and no density redistribution occurs (e.g. by a tilt). Since 𝑎𝑎𝑏 < 0
in this region of magnetic fields, 𝐽𝑥𝑦 > 0 from Eq. (4.11), realizing antiferromagnetic
couplings. The sign of 𝐽𝑥𝑦 can be flipped by applying a tilt ∆ > 𝑈𝑎𝑏 (see Section 5.6
for the corresponding phase diagram). On the other hand, 𝐽𝑧 has a resonance at
𝑈𝑏𝑏 = 0 and switches sign.
Fig. 4-12c shows the ratio 𝐽𝑧 /𝐽𝑥𝑦 as a function of field. This ratio determines the
behavior of the system: energy spectrum, ground states, transport properties, etc.
The range of parameters we can access covers most types of behavior, which allows us
to explore very different systems. Notable points which can be realized in this system178

are: (i) 𝐽𝑧 = 0, where the system maps to non-interacting fermions (Jordan-Wignertransformation); (ii) 𝐽𝑧 = 𝐽𝑥𝑦 , the isotropic point; 𝐽𝑧 = − 𝐽𝑥𝑦 , where 𝑈↑↑ = 𝑈↓↓ andcan be considered the ferromagnetic counterpart of the isotropic point. Nevertheless,the Hamiltonian at these two points is different.
Spin-1 with 7 Li

In 7 Li, we can realize the Haldane phase in a small region around the point 𝑢 = 0, i.e.
when:
(4.26)

2𝑈↑↓ = 𝑈↑↑ + 𝑈↓↓

We call this the degeneracy point. Away from that point for 𝐽 > 0, the system ispredicted to be in an antiferromagnetic state (𝑢 < 0) and a Spin Mott state (𝑢 > 0).
As in Fig. 4-10, we assume that the system is in the Haldane phase when
− 0.5 <

𝑢<1𝐽

(4.27)

Fig. 4-13a shows this phase diagram for a (35,35,11) 𝐸𝑅 lattice and Fig. 4-13b shows
how the width of the region in which the system is in the Haldane phase varies as afunction of lattice depth. Since the width is on the order of tens of mG around 11 𝐸𝑅 ,field stability needs to be improved in our system.

3

b)
Z AFM

HaldaneSpin Mott

Haldane phase width (mG)

a)

2

u/J

1
0
-1

400

300

200

100

-2
0
-20

-10

0

10

20

30

6

40

8

10

12

14

16

18

20

Lattice depth in 1D (E R)

Magnetic field (mG)

Figure 4-13: Spin 1 phase diagram for 7 Li around 𝑢 = 0. a) At 11 𝐸𝑅 . Here zero fieldis the point where 𝑢 = 0, which is at 847.5091 Gauss. b) Magnetic field region over which−0.5 < 𝑢/𝐽 < 1 and the system is in the Haldane phase, plotted as a function of lattice depth𝑉𝑧 for a (35,35,𝑉𝑧 )𝐸𝑅 lattice. The dashed line is 𝑉𝑧 = 11 𝐸𝑅 , the case in a).

179

The Spin Mott phase can easily be prepared by starting in a Z ferromagnet andLandau-Zener sweeping over the transition |𝑎𝑎⟩ → |𝑎𝑏⟩ on each site. This can bedone in a deep lattice, so that the sites are isolated and in the region of magnetic
fields where the interactions are different and interaction spectroscopy can be used.
Once the system is in the Spin Mott phase, the magnetic field can be ramped to
the Haldane phase. However, given the discussion above, adiabatic state preparation
would require the addition of terms to the Hamiltonian.
Since superexchange inevitably involves a higher on-site occupation as part ofthe virtual tunneling, loss processes from having two or three particles on a site forspin-1/2 and spin-1 respectively have to be considered. This type of loss is inherent
to the superexchange process. For the lowest two hyperfine states |𝑎⟩ and |𝑏⟩, thetwo-body spin relaxation of two-particles per site of the form |𝑎𝑎⟩ and |𝑎𝑏⟩ is excluded
by |𝑎⟩ being the lowest state. Given the calculated dipolar relaxation coefficient for

two 𝑏-particles, dipolar loss from |𝑏𝑏⟩ sites can mostly be ignored (see Section 3.2.4).
Therefore, superexchange in the spin-1/2 system does not induce loss. In the spin-1

system, admixtures of triply occupied sites in the 𝑛 = 2 MI does lead to loss, as shownin Section 3.2.4. Depending on the values of the four loss coefficients, 𝒦𝑎𝑎𝑎 , 𝒦𝑎𝑎𝑏 , 𝒦𝑎𝑏𝑏 ,and 𝒦𝑏𝑏𝑏 , at the fields of interest, some spin-1 schemes with 7 Li could not be possible
or could be limited to short observation times.

Although some of the results above are more general, we have been focusing on
the 1D case (spin chains), in which the spins are allowed to move only along one direction. The implementation of such a system with an optical lattice is straightforward.
Typically, a 3D lattice is used in order to enter the MI regime. Then, only one of thelattice arms is lowered to allow dynamics along that direction, while the other two are
kept large and constant. More specifically, if we want superexchange dynamics along𝑧, then 𝑡2𝑧 /𝑈 ≫ (𝑡2𝑥 /𝑈, 𝑡2𝑦 /𝑈 ). This realizes a quasi-1D geometry with many tubes, i.e.

many independent 1D realizations of the same chain, which we typically average over.
Due to the trap, these chains are not exactly the same: the ones towards the edgesof the cloud are shorter and have lower lattice depth. The lattice beam geometry canbe designed to minimize this effect by, for example, using elliptical beams.
1804.1.4

Beyond spin chains: 2D and 3D physics

With a cubic optical lattice, it is fairly easy to effectively switch on and off the dynamics along a certain axis by changing the lattice depth. It is therefore straightforward
to create 1D, 2D, or 3D systems. In many cases, dimensionality does not change
the physics qualitatively but only quantitatively. For example, for ultracold atoms
in optical lattices, higher dimensions typically mean slower dynamics. This is dueto the fact that the SF-MI transition is at higher lattice depths in 2D and 3D (seeSection 3.1.3). Also, again from an experimental perspective, it is easier to add features to 1D systems than to 3D systems, just because in many cases identical copies
are needed along each lattice axis. Such features include magnetic or AC-Stark shift
gradients, superlattices, trap-compensation beams, etc.
However, a change in dimensionality could result in a qualitative change of thesystem behavior. For example, quantum fluctuations are much more pronounced
in 1D systems and, therefore, interest towards lower-dimensional physics has been
𝛼high. Similarly, the decay of spin-spin correlations ⟨𝑆𝑖𝛼 𝑆𝑖+𝑑⟩, where 𝛼 = 𝑥, 𝑦, 𝑧 andthe distance between the spins is 𝑑, could depend on dimensionality. For example,
the Haldane phase is predicted to exist only in 1D. As shown in Fig. 4-10, when coupling between the chains is added, transitioning to a 2D system, the phase disappears.
Spin transport can also be different: while diffusion is observed in 1D, superdiffusion
is observed in 2D [73]. Another illustration of the increased importance of quantum
fluctuations in low-dimensional systems is provided by the Mermin-Wagner theorem. It states that in systems with continuous symmetry, there is no phase withspontaneously-broken symmetry (no long-range order) at any finite temperature for𝑑 ≤ 2.

Finally, 1D spin systems have been a model system in the condensed matter com-munity for the last decades and most numerical studies are only possible in 1D. This
makes 1D spin chains a useful benchmark for quantum simulations, for an example
of such a study see [29].

181

4.2

Dynamics

We have so far discussed ground states of the Heisenberg Hamiltonian. However, thedynamics in the different regimes of the Hamiltonian are also different. Sometimes,
the nature of the dynamics also depends on the type of initial state used (e.g. purestate vs thermal state), so it can be non-universal. The dependence of the dynamics
on the Hamiltonian anisotropy 𝐽𝑧 /𝐽𝑥𝑦 in the spin-1/2 chain is the subject of our mostrecent research (paper in preparation). Here we will focus on two other aspects. Firstwe will discuss whether the superexchange timescale is ℎ/𝐽 or ~/𝐽. Then we will look
into the dynamics when a uniform energy offset is applied, motivated by our studyof superexchange in a tilted lattice [45] (addressed in Chapter 5).

4.2.1

What is the timescale for dynamics?

In this section we compare the timescales for coherent Rabi oscillations in a doublewell and for coherent ballistic expansion in a 1D lattice. We also show how and when
the tunneling dynamics with matrix element 𝑡 of a single particle in a lattice map
to the superexchange dynamics with matrix element 𝐽 = 𝑡2 /𝑈 of a single spin |↑⟩in a lattice filled with spin |↓⟩ particles. This is in the spirit of the Jordan-Wignertransformation.

Let’s first consider a single particle in a double-well. We label the two wells left|𝐿⟩ and right |𝑅⟩. In this basis, the Hamiltonian is:
𝐻 = − 𝑡 (𝑎†𝐿 𝑎𝑅 + 𝑎†𝑅 𝑎𝐿 ) =
⎡
⎤
0
𝑡
⎦
=−⎣
𝑡
0

(4.28)The eigenvalues are 𝜆± = ±𝑡 and the eigenstates are:1|±⟩ = √ (|𝐿⟩ ± |𝑅⟩)2

(4.29)

If the particle starts in the left well |Ψ(𝜏 = 0)⟩ = |𝐿⟩ = √12 (|+⟩ + |−⟩), the evolution
182

under the Schrödinger equation is:1]︀
1 [︀|Ψ(𝜏 )⟩ = √ 𝑒−𝑖𝑡𝜏 /~ |+⟩ + 𝑒𝑖𝑡𝜏 /~ |−⟩2]︀
1 [︀ 𝑖𝑡𝜏 /~= (𝑒+ 𝑒−𝑖𝑡𝜏 /~ )|𝐿⟩ + (𝑒𝑖𝑡𝜏 /~ − 𝑒−𝑖𝑡𝜏 /~ )|𝑅⟩
2 (︂ )︂
(︂ )︂
𝑡𝜏
𝑡𝜏|𝐿⟩ + sin
|𝑅⟩
= cos
~
~

(4.30)The evolution of the probability is then:2

𝑃 (𝜏 ) = ⟨Ψ(𝜏 )|Ψ(𝜏 )⟩ = cos

(︂

𝑡𝜏
~

)︂

− sin

2

(︂

𝑡𝜏
~

)︂

= cos

(︂

2𝑡𝜏
~

)︂

(4.31)The particle undergoes Rabi oscillations with frequency 𝜔𝑅 =2𝑡/~. The time to tunnel
to the next site, which is half the period 𝑇 = 2𝜋/𝜔𝑅 , can be considered as the timescalefor dynamics in this system:
𝑇double−well =

ℎ
𝜋~
=
4𝑡
2𝑡

(4.32)

We now consider a 1D chain with one particle. The dispersion relation can beused to obtain the maximum group velocity. The tight-binding approximation forthe dispersion relation can be used, which as we have seen in Chapter 3 is a good
approximation for the dispersion relation of a particle in the sinusoidal potential ofan optical lattice for lattice depths above a few recoil. The dispersion relation in thetight-binding limit in a 1D periodic potential is:ℰ(𝑞) = −2𝑡 cos(𝑞𝑎𝐿 )(4.33)

so that the width of the band is 4𝑡. The velocity is
𝑣=
1

𝑑ℰ
𝑑(~𝑞)

In this chapter, we denote time with 𝜏 and the tunneling matrix element with 𝑡.

183(4.34)

The maximum group velocity is:
]︂
2𝑡𝑎𝐿
2𝑡𝑎𝐿
sin(𝑞𝑎𝐿 ) =𝑣𝑚𝑎𝑥 = max~~[︂(4.35)The time to travel one site is 𝑇 = 𝑎𝐿 /𝑣𝑚𝑎𝑥 :~2𝑡(4.36)

𝑇double−well=𝜋
𝑇1D−chain(4.37)

𝑇1D−chain =so thatComparing the kinetic term in the Hubbard model and in the Heisenberg model:
𝐾Bose−Hubbard = − 𝑡∑︁ †(𝑎𝑖 𝑎𝑗 + 𝑎†𝑗 𝑎𝑖 )⟨𝑖,𝑗⟩

𝐽𝑥𝑦 ∑︁ + −(𝑆𝑖 𝑆𝑗 + 𝑆𝑗+ 𝑆𝑖− )𝐾Heisenberg =2⟨𝑖,𝑗⟩reveals that we can map 2𝑡 → 𝐽𝑥𝑦 . In a 1D chain, mass transport happens at a rate2𝑡/~ and spin transport happens at a rate 𝐽𝑥𝑦 /~, so that the superexchange time is:𝑇1D−chain, J =~𝐽𝑥𝑦(4.38)

In other words, if the dynamics are driven by superexchange and the lattice depth isvaried, then when the time is scaled by 𝑇1D−chain, J , the dynamics will be the same.

4.2.2

Dynamics in tilted lattices

In Chapter 5 and in [45], we demonstrate the advantages of adding a constant energy
offset (tilt) to 1D lattices for studying superexchange-mediated spin dynamics. Here
we outline the regime in which these experiments are to be carried out. We compare
the dynamics of a single particle in a tilted lattice to the dynamics of a single spin|↑⟩ in a (perfect) chain of spin |↓⟩ particles in a tilted lattice.
184

The spin Hamiltonians discussed in this chapter can be realized with atoms in
optical lattices in the MI regime, where first-order tunneling at a rate 𝑡/~ is suppressedby the interactions since 𝑡 < 𝑈 . Then, second order tunneling, superexchange, at a
rate of 𝐽/~ drives the dynamics, enabling the study of spin physics. However, intrapped systems with defects (or at non-zero temperature), the mapping is only anapproximation. For example, number defects (sites with occupations of 𝑛 ± 1 where 𝑛is the MI filling) can still propagate at the tunneling rate 𝑡/~. Moreover, in a trappedsystem, superfluid shells form around the MI plateaus, enabling the propagation ofentropy from them into the MI also at rates on the order of 𝑡/~. Since this rate ismuch faster than the superexchange rate 𝐽/~, transport of entropy and mass could
interfere with the study of spin Hamiltonians. A tilt of the lattice potential could
suppress first order tunneling by turning linear transport into Bloch oscillations. If
the tilt per site is larger than the tunneling matrix element, the Bloch oscillations are
confined to a single site, so that transport through first-order tunneling is inhibited.
However, superexchange is not necessarily suppressed, unless the tilts are too large.
For example, in [45] we use tilts on the order of 𝑈 in the MI regime. These effects ofthe tilt are explored in this section.
Finally, if the tilt is large enough so that higher bands are coupled, the dynamics
are different: particles can tunnel to the next band. This is the Zener breakdown
regime, analogous to the Zener breakdown in diodes and to the diabatic LandauZener crossings between two coupled levels.
Bloch oscillations
From the perspective of single particles, the tilt is nothing more than a constantforce 𝐹 . The dynamics of a single particle can then be described by the following
Hamiltonian (for reviews, see [75, 72]):~2 𝑑2+ 𝑉0 cos2 (𝑘𝐿 𝑥) + 𝐹 𝑥
𝐻(𝑥) = −
2
2𝑚 𝑑𝑥

185

(4.39)

From a quantum mechanical perspective, we want to solve for the time evolution of a
wavepacket (e.g. a Gaussian) in this potential. The solutions to Eq. (4.39) for 𝐹 = 0
are Bloch waves Ψ𝑛,𝑘 (x), as we have seen in Chapter 3. We now construct spatiallylocalized wavepackets from the Bloch waves:𝜑(𝑥, 𝜏 ) =∞ ∫︁ 𝜋/𝑑∑︁

𝑑𝑘 𝑔𝑛 (𝑘, 𝜏 )Ψ𝑛,𝑘 (x)(4.40)−𝜋/𝑑𝑛=1

where 𝑔𝑛 (𝑘, 𝜏 ) is the momentum distribution. This is then substituted in the Hamiltoian eq.(4.39) to find the time evolution of the momentum-space wavefunctions
𝑔𝑛 (𝑘, 𝜏 ), which can be simplified by considering only a single band 𝑛. It can beshown that the evolution of |𝑔(𝑘, 𝜏 )|2 is governed by a transport equation [75, 80]:
𝐹 𝑑𝑑|𝑔(𝑘, 𝜏 )|2 =|𝑔(𝑘, 𝜏 )|2𝑑𝜏~ 𝑑𝑘(4.41)

with solution of the form:(︂𝐹𝜏
|𝑔(𝑘, 𝜏 )| = 𝐺 𝑘 −~2)︂(4.42)

This suggests that the evolution of the center of the wavepacket in momentum spaceis governed by a classical equation of motion:𝑑(~𝑘) = −𝐹𝑑𝜏(4.43)

𝐹𝜏
~(4.44)

with solution:𝑘(𝜏 ) = 𝑘0 −Following this semi-classical picture, the group velocity of the wavepacket is thederivative of the dispersion relation ℰ(𝑘) = − ℰ1 cos(𝑘𝑑), where 𝑑 is the lattice spacing
and 2ℰ1 is the band width, and for simplicity we have taken the 𝐹 = 0 dispersion

relation. Going back to the Hubbard model, the width of the ground band is 4𝑡, so186

that ℰ(𝑘) = − 2𝑡 cos(𝑘𝑑). The group velocity is then:
1 𝑑
ℰ(𝑘)
~ 𝑑𝑘
2𝑡𝑑
=−
sin(𝑘(𝜏 )𝑑)~2𝑡𝑑
sin(𝑘0 𝑑 − 𝜔𝐵 𝜏 )=−~𝑣𝑔 (𝜏 ) =(4.45)

where we have defined the Bloch frequency:
𝜔𝐵 =

𝐹𝑑
~

(4.46)The time evolution in position space is also periodic:𝑥(𝜏 ) =

∫︁𝑣𝑔 (𝜏 )𝑑𝜏 = 𝑥(0) −2𝑡cos(𝑘0 𝑑 − 𝜔𝐵 𝜏 )𝐹(4.47)We have thus shown that the motion of a particle in a periodic potential under aconstant force is periodic. This motion is known as Bloch oscillations. The amplitudeof the oscillation depends on the strength of the force. The peak-to-peak amplitude
is 𝐴𝑝𝑝 = 4𝑡/𝐹 , which can be expressed in terms of the energy offset per site (the tilt)∆ = 𝐹 𝑑, so that
𝐴𝑝𝑝 =

4𝑡
𝑑
∆

(4.48)

If ∆ > 4𝑡, the Bloch oscillations are confined to one lattice site. This is, in fact, theregime we want to be in. In [45] we have used ∆ = 1.65 𝑈 and in 1D 𝑡/𝑈 ≈ 0.2 atmost in the presence of the tilt, so that 𝐴 = 0.48 𝑑.

It is much easier to observe Bloch oscillations in systems of ultracold atoms in
optical lattices than in real materials, where the period is on the order of THz.
Oscillations in both momentum space [21] and position space have been observed withultracold atoms. In position space, when the initial density distribution is a single
Wannier function localized on one site, the dynamics are a “breathing” mode: thewavefunction expands and relocalizes [130]. When the initial distribution is extended,
the center of mass does periodic oscillations [58].
187

Wannier-Stark states
Another way to describe this system is by the eigenstates of the Hamiltonian Eq. (4.39)
in the tight-binding approximation. Without the tilt, these are the Wannier statesfrom Section 3.1.1. With a tilt, the eigenstates are the Wannier-Stark states Φ(𝑥𝑗 )centered around site 𝑗, which can be expressed in terms of the Wannier states 𝑤(𝑥𝑖 )centered around site 𝑖 (here we have assumed the lowest band) [72]:Φ(𝑥𝑗 ) =

∑︁
𝑖

𝒥𝑖−𝑗

(︂

2𝑡
∆

)︂

(4.49)𝑤(𝑥𝑖 )

where 𝒥𝑛 (𝑧) is a Bessel function of the first kind. These functions are localized on asingle site when the argument of the Bessel function is small 𝑡 ≪ ∆, the result from

Eq. (4.48).

2500
Wannier
Wannier-Stark

2000

(x)

1500
1000
500
0
-3

-2

-1

0

1

2

3

Lattice site ( /2)

Figure 4-14: Wannier Stark functions compared to Wannier functions. Both are plotted at
11 𝐸𝑅 and for the Wannier Stark states Δ = 1.65 𝑈 . Here Δ > 0 so that the lattice sites on
the right have higher energy.

Tunneling and superexchange dynamics in a tilted lattice
Next, we look at the tunneling and superexchange dynamics with and without a tilt.
The evolution of a single particle in a non-tilted lattice starting from the particle
localized on a single site (i.e. initial state Ψ(0)⟩ = |0...010...0⟩) in an infinite chain is188

given by:
𝑖𝐻𝜏 /2~|Ψ(𝜏 )⟩ =𝑒
where 𝐻 = − 𝑡

∑︀

⟨𝑖,𝑗⟩

|Ψ(0)⟩ =∞∑︁
(𝑖𝜏 /2~)𝑛𝑛=0𝑛!

𝐻 𝑛 |Ψ(0)⟩(4.50)𝑎†𝑖 𝑎𝑗 . The odd and even powers of 𝐻 𝑛 can be computed separately,

giving the occupation probability on a site 𝑖 after a time 𝜏 is [93]:
⃒ (︂
)︂⃒2
⃒
⃒
2𝑡𝜏
⃒
𝑃𝑖 (𝜏 ) = ⃒⃒𝒥𝑖
~ ⃒(4.51)

where we have picked the particle to be at 𝑖 = 0 initially and 𝒥𝑛 (𝑧) is a Bessel functionof the first kind. The dynamics are plotted in Fig. 4-15a, showing the coherent ballistic
expansion of the wavefront. Such dynamics have been observed in [130]. This systemmaps to the system of a single |↑⟩ spin in a chain of |↓⟩ spins when the tunnelingmatrix element 𝑡 is replaced by the superexchange matrix element 𝐽 = 2𝑡2 /𝑈 . Thishas been observed in [56].

a)

b)

c)

(i)
(ii)
(iii)

Figure 4-15: Tunneling and superexchange with and without a tilt. The time units are 𝑡𝜏 /~.
a) (i) Time evolution of a single particle initially localized on a single site, Eq. (4.51). The
dynamics are the same for spin transport via superexchange for a spin | ↑ ⟩ in a chain of| ↑ ⟩ spins if (ii) 2 𝑡 → 𝐽 = 4𝑡2 /𝑈 (no tilt) and (iii) 2 𝑡 → 𝐽(Δ) = 2𝑡2 [1/(𝑈 − Δ) + 1/(𝑈 + Δ)](tilt Δ). b) Bloch oscillations, Eq. (4.52) for a small tilt Δ/4𝑡 = 0.125 and c) for a large tilt
Δ/4𝑡 = 2.06.

189However, when the lattice is tilted, a single particle in an empty lattice behavesdifferently from a single spin | ↑ ⟩ in chain of spin | ↓ ⟩. The evolution of a single particle
in a tilted lattice is described by Bloch oscillations with occupation probability [72]:⃒ (︂(︁ 𝜔 𝜏 )︁)︂⃒⃒2
⃒
4𝑡
𝐵
⃒
sin
𝑃𝑖 (𝜏 ) = ⃒⃒𝒥𝑖⃒∆
2

(4.52)In the limit ∆ → 0, this reduces to Eq.(4.51). Fig. 4-15b shows the “breathing” modeof the dynamics for small tilts ∆ < 4𝑡, when the Bloch oscillation amplitude 𝐴𝑝𝑝

in Eq. (4.48) extends over several lattice sites. When the tilt is large, the Blochoscillations are confined to the same site, Fig. 4-15c. Therefore, for ∆ > 4𝑡, firstorder tunneling is suppressed. This is clear also from an energy argument. The widthof the lowest band is 4𝑡. When the energy difference between neighboring sites ismuch larger than this width, there is either not enough energy to tunnel or there is
no state to dissipate the energy into, thus suppressing tunneling both up and down
the hill. However, second order tunneling is still allowed in a tilted lattice but at
a modified rate (Chapter ]refchapter5). Therefore, the evolution of the single spin| ↑ ⟩ in a spin | ↓ ⟩ chain in a tilted lattice is the same with and without the tilt (iftunneling resonances are avoided).
Zener breakdown
The results so far were derived in the single-band approximation. We now turn tothe probability that the wavepacket will transition to the next band. As we have seen
so far, in momentum space, the particle’s momentum increases linearly, i.e. it cycles
through the first energy band of the lattice. The bandgap is the smallest at the edgesof the first Brillouin zone, so the probability that the particle will tunnel to the next
band is largest there. We can view this in terms of an avoided crossing and computethe Landau-Zener transition probability. An approximate result can be derived in
the limit of a weak lattice by considering the free-particle wavefunctions and energy
levels and treat the lattice as a perturbation. This results in the following transition

190

probability [75]:

where
𝜁=

𝑃 = 𝑒−2𝜁

(4.53)𝜋 2 (𝐸𝑏𝑔 /𝐸𝑅 )2
16 ∆/𝐸𝑅

(4.54)

This result remains valid for deeper lattices, only the prefactor in Eq. (4.54) changes.

4.2.3

Relaxation of non-equilibrium spin distributions

In order to measure the dynamics of spin Hamiltonians, we need a way of preparing
non-equilibrium spin distributions in the Mott insulator. One way to do this is to
use the different magnetic moments of the lowest two hyperfine states and preparea spin spiral, as in [17, 73]. In a 1D spin spiral, the total magnetization along thechain rotates in the XZ or in the XY plane, hence the term spin helix might be more
appropriate. After preparation of this pattern in a deep lattice, the lattice depth canbe lowered to enable the dynamics. Since the spin spiral is a non-equilibrium distribution, it relaxes. The relaxation rate can be used as a measure of the superexchange
rate and of the main transport mechanism of a given Hamiltonian. In the following,
we assume that the MI has one atom per site, realizing a spin-1/2 Heisenberg model.
In this section, we describe the spin spiral and its preparation as a tool for measuringtransport properties of spin systems. The relaxation dynamics in a tilted lattice arepresented in Section 5.5 and the variation of the relaxation properties as a function of
spiral wavevector and anisotropy 𝐽𝑧 /𝐽𝑥𝑦 are the subject of our most recent project.
Spin Spiral preparation
As illustrated in Fig.4-16, the starting point is a 1D chain of atoms in an 𝑛 = 1 MI in
the |↑⟩ state. A 𝜋/2 pulse is applied so that, on each site, the atom is in:1|Ψ(𝑡 = 0)⟩ = √ (| ↑⟩ + | ↓⟩)2

191(4.55)

<latexit sha1_base64="eqf3wGSZ1av0NaGSCoBvrLCsrAY=">AAAB/HicbVBNS8NAEJ34WetXtEcvwSJ4KkkV9Fj04rGC/YAmlM120y7dbMLuRggh/hUvHhTx6g/x5r9xk+agrQ8GHu/NMDPPjxmVyra/jbX1jc2t7dpOfXdv/+DQPDruyygRmPRwxCIx9JEkjHLSU1QxMowFQaHPyMCf3xb+4JEISSP+oNKYeCGachpQjJSWxmbDDQTCmRvTPGvn9RJjs2m37BLWKnEq0oQK3bH55U4inISEK8yQlCPHjpWXIaEoZiSvu4kkMcJzNCUjTTkKifSy8vjcOtPKxAoioYsrq1R/T2QolDINfd0ZIjWTy14h/ueNEhVcexnlcaIIx4tFQcIsFVlFEtaECoIVSzVBWFB9q4VnSKehdF5FCM7yy6uk3245Fy37/rLZuaniqMEJnMI5OHAFHbiDLvQAQwrP8ApvxpPxYrwbH4vWNaOaacAfGJ8/6c6S9g==</latexit>⇡
2

Magnetic field gradient<latexit sha1_base64="eqf3wGSZ1av0NaGSCoBvrLCsrAY=">AAAB/HicbVBNS8NAEJ34WetXtEcvwSJ4KkkV9Fj04rGC/YAmlM120y7dbMLuRggh/hUvHhTx6g/x5r9xk+agrQ8GHu/NMDPPjxmVyra/jbX1jc2t7dpOfXdv/+DQPDruyygRmPRwxCIx9JEkjHLSU1QxMowFQaHPyMCf3xb+4JEISSP+oNKYeCGachpQjJSWxmbDDQTCmRvTPGvn9RJjs2m37BLWKnEq0oQK3bH55U4inISEK8yQlCPHjpWXIaEoZiSvu4kkMcJzNCUjTTkKifSy8vjcOtPKxAoioYsrq1R/T2QolDINfd0ZIjWTy14h/ueNEhVcexnlcaIIx4tFQcIsFVlFEtaECoIVSzVBWFB9q4VnSKehdF5FCM7yy6uk3245Fy37/rLZuaniqMEJnMI5OHAFHbiDLvQAQwrP8ApvxpPxYrwbH4vWNaOaacAfGJ8/6c6S9g==</latexit>

⇡
2

time

A. B. Bardon, S. Beattie, C. Luciuk, W. Cairncross, D. Fine, N. S. Cheng, G. J. A. Edge, E. Taylor, S. Zhang, S. Trotzky, and J. H.

344, 722(2014).
Figure 4-16:Thywissen,
Spin. Science
Spiralpreparation.
The spins are initialized in the |↑⟩ state. Then, a 𝜋/2
S. Hild, T. Fukuhara, P. Schauß, J. Zeiher, M. Knap, E. Demler, I. Bloch, and C. Gross. Phys. Rev. Lett. 113, 147205 (2014).
pulse transfers them to the 𝑥𝑦 plane on the Bloch sphere where they precess at a frequency
which depends on the total magnetic field. A magnetic field gradient is added along thechain, causing the spins to precess at different rates. This creates a spin spiral in the XY
plane. If the second 𝜋/2 pulse is applied after the magnetic field gradient is switched off, aspin spiral in the XZ plane is prepared.

Then, a magnetic field gradient 𝐵 ′ is applied so that the total field along the chain is
𝐵(𝑧) = 𝐵0 + 𝐵 ′ 𝑧, where 𝐵0 is the bias field we use in the Feshbach resonances region.
The evolution of each atom is described by:)︀
1 (︀|Ψ(𝑡, 𝑧)⟩ = √ 𝑒−𝑖𝜇↑ 𝐵(𝑧)𝑡/~ | ↑⟩ + 𝑒−𝑖𝜇↓ 𝐵(𝑧)𝑡/~ | ↓⟩
2
(︀
)︀
1
|Ψ(𝑡, 𝑧)⟩ = √ 𝑒−𝑖𝜇↑ 𝐵(𝑧)𝑡/~ | ↑⟩ + 𝑒−𝑖(𝜇↓ −𝜇↑ )𝐵(𝑧)𝑡/~ | ↓⟩2

(4.56)

If the free evolution time is 𝑇 , the differential magnetic moment ∆𝜇 = 𝜇↓ − 𝜇↑ (cal-culated for this region of magnetic fields in Section 2.2.2), the differential phase shiftis:∆Φ(𝑇, 𝑧) =∆𝜇(𝐵0 + 𝐵 ′ 𝑧)𝑡~(4.57)

We can split this in two parts ∆Φ(𝑇, 𝑧) = 𝜑0 (𝑇 ) + ∆𝜑(𝑇, 𝑧), where the total magnetic
field contributes only an overall phase shift 𝜑0 = ∆𝜇𝐵𝑇 /~. The relative phase between
the |↑⟩ and the |↑⟩ components varies along the chain:∆𝜇𝐵 ′∆𝜑(𝑇, 𝑧) =
𝑧𝑇
~

192

(4.58)

This is a spiral in the XY plane. A spiral in the XZ plane can be created by applying
another 𝜋/2 pulse after the magnetic field gradient has been switched off:[︀]︀1|Ψ𝑓 ⟩ = 𝑒−𝑖𝜇↑ 𝐵(𝑧)𝑡/~ | ↑⟩(1 + 𝑒𝑖ΔΦ(𝑧,𝑡) ) + | ↓⟩(1 − 𝑒𝑖ΔΦ(𝑧,𝑡) )2(4.59)

After that the free evolution of the spiral is measured as a function of hold time. Boththe XY and the XZ spirals are measured by imaging the probability of being in either
the |↑⟩ or the |↓⟩ state. In the case of the XY spiral, a second 𝜋/2 pulse is applied
before imaging in order to transfer the atoms back in the 𝑧 basis. The probabilitiesare:1[1 + cos(∆𝜑(𝑧, 𝑡) + 𝜑0 )]2
1
𝑃↓ = [1 − cos(∆𝜑(𝑧, 𝑡) + 𝜑0 )]2𝑃↑ =(4.60)

If, for example, only the atoms in the |↑⟩ state are imaged, a sinusoidal variation of

the density along the tubes is measured. Note that although both the XY and the
XZ spirals can decay by transport by superexchange, the XY spiral is also sensitive
to dephasing. For example, in our system, we realize many 1D tubes which weaverage when imaging. Due to the variation of the lattice depth due to the trapping
potential, the effective magnetic field ℎ𝑧 also varies (Section 4.1.1), which leads toslightly different time evolution of the XY spirals in each tube. When the spin
distributions of the different tubes are averaged, this is leads to dephasing.
The wavevector of this distribution is determined by the phase shift ∆𝜑 = 𝑄𝑧 in
Eq. (4.58):
𝑄=∆𝜇𝐵 ′ 𝑇~

(4.61)

Therefore, by varying the evolution time 𝑇 at a given magnetic field gradient 𝐵 ′ , thewavelength of the spin spiral can be controlled. In principle, spirals of any wavelength
can be prepared. However, if the spiral wavelength becomes larger than the size of thecloud 𝐿, finite-size effects dominate. In Fourier space, the 𝑄 vector is Fourier-limitedby the inverse cloud size 2𝜋/𝐿. Since the relaxation of the spin spirals is limited to193

Q-vectors of at least 2𝜋/𝐿, the relaxation of spirals with smaller Q-vectors cannot
explore all relevant decay mechanisms and cannot point to the underlying transport
properties of the spin Hamiltonian. In the other extreme, we are limited by our
resolution, so we cannot image spirals with wavelengths smaller what about 5 latticesites. However, if an echo pulse sequence is implemented during the measurement
stage after the spiral is prepared, the evolution of small wavelength spirals can be
explored as well. As in [17], the spin spiral can be prepared in the 𝑥𝑦 plane and a𝜋 pulse (an echo) can be used halfway through the evolution time, with the rest of
the measurement protocol as describe above. If there are no decay mechanisms, after
the final 𝜋/2 pulse used to rotate the spins to the 𝑧 basis, all the spins will be in thesame state and if there are decay mechanisms, they will not.
We now consider the effect of noise in 𝐵 ′ and 𝐵, which can be seen from Eq. (4.60).
If the magnetic field gradient 𝐵 ′ fluctuates during the evolution time, then the∫︀ 𝑇
wavevector is determined by the integral 𝐵𝑇′ = 0 𝐵 ′ (𝑡)𝑑𝑡. The effect of low frequency noise in the overall magnetic field (which results in the overall magnetic field
being different for different realizations of the experiment) is to shift the position
along the cloud where 𝜋/2 pulses are resonant, i.e. where we have assumed 𝑧 = 0 in
the above analysis. This just changes the overall phase of the fringes. Since for ourcurrent magnetic field stability, this does not allow us to directly average images withthe same spiral wavelength 𝜆 = 2𝜋/𝑄. Therefore, using spiral wavelengths smaller
than the cloud size also enables us to extract the spiral contrast from a single shot,because the phase of the density distribution can be fitted, so that this fluctuatingphase is just a fit parameter.
Spin spiral relaxation
We first note that the spin spiral is a non-equilibrium spin distribution which is nota single spin wave. Focusing on the Z ferromagnet, as in Section 4.1.1, the first bandof excited states are different superpositions of all states with one flipped spin, thenext one have two flipped spins and so on. These states are spin waves and since thetotal 𝑧 magnetization 𝑆 𝑧 is a good quantum number because 𝑆 𝑧 commutes with the194

Hamiltonian, each spin wave is an eigenstate of 𝑆 𝑧 . The spin pattern we create is asuperposition of states with a different number of slipped spins. As an example, let us
consider 5 sites with spin as illustrated in Fig. 4-17, realizing a spiral with wavelengthof 8 sites.

Figure 4-17: Spin spiral on 5 sites realizing a spiral with a wavelength of 8 sites.

We can represent the spin on each site as a superposition of spin | ↑⟩ and spin | ↓⟩:|Ψ𝑠𝑝𝑖𝑟𝑎𝑙 ⟩ = | ↑⟩ (0.92| ↑⟩ + 0.38| ↓⟩) (0.71| ↑⟩ + 0.71| ↓⟩) (0.38| ↑⟩ + 0.92| ↓⟩) | ↓⟩= 0.25| ↑↑↑↑↓⟩ + 0.6| ↑↑↓↑↑⟩ + 0.25| ↑↓↓↓↓⟩++ 0.6| ↑↑↑↓↓⟩ + 0.25| ↑↑↓↑↓⟩ + 0.25| ↑↓↑↓↓⟩+(4.62)+ 0.1| ↑↓↑↑↓⟩ + 0.1| ↑↓↓↑↓⟩

This state contains states with a different number of flipped spins (relative to the Z
ferromagnet), so it is a superposition of different eigenstates.
The relaxation of the spin spiral monitors the relaxation of a single spatial frequency. By varying the Q-vector a range of these frequencies can be measured and thecorresponding transport behavior can be extracted. The Z component of the spiralcan decay only through spin transport, i.e. via superexchange at a rate proportional
to |𝐽𝑥𝑦 |/~. The prefactor depends on the Heisenberg anisotropy 𝐽𝑧 /𝐽𝑥𝑦 , because thetransport regimes are different in the different regions of the Hamiltonian parameter
space. For example, at the 𝐽𝑧 = 0 point, which maps to non-interacting fermions, thetransport is expected to be ballistic. As 𝐽𝑧 is increased, the transport is expected toslow down. The XY components of the spin spiral can decay also through dephasing.
The decay rates of the XZ and XY spirals as a function of 𝐽𝑧 /𝐽𝑥𝑦 are the subject ofour next project.
Variations of this method include preparing the spiral as a perturbation, i.e. by not
195using the full 𝜋/2 pulse, but some other angle. Also, varying the temperature of thesystem could reveal more features of the dynamics. Other types of non-equilibrium
spin distributions can also be prepared. For example, the cloud can be separated
into a |↑⟩ side and a |↓⟩ side and the mixing between the two can be measured (as in[157, 118]).

In this chapter, we have looked into the main research direction of BEC 5: spin
physics. We have described the implementation of both spin-1/2 and spin-1 Heisenberg models with 7 Li atoms in an optical lattice and have looked at the specificadvantages and drawbacks of this approach. Our platform provides wide tunabilityof the Heisenberg parameters, enabling us to create different ground states and to also
study various transport regimes. These can be studied by measuring the relaxationof a spin spiral. In the next chapter we look into a method which can further enhance
our ability to study spin physics by suppressing first order tunneling (which is responsible for entropy and mass transport) and not superexchange. Finally, it remains tobe seen how low our entropy is and if ground-state physics are within reach.

196

Chapter 5
Case study: a tilted Mott insulator
In [45] we have demonstrates several advantages and new features of adding a uniform
potential energy offset, a tilt, to a Mott Insulator (MI). In this chapter we add details
and focus on the implications of using the tilt to study spin physics with 7 Li.
A disadvantage of using superexchange to obtain effective spin-spin interactions isthat spin dynamics happen at a rate proportional to 𝑡2 /(~𝑈 ) which is much slower thatfirst-order tunneling at 𝑡/~. Although in the MI first-order tunneling is suppressedby interactions, number defects in the MI and the proximity to the superfluid shellin a trapped cloud, where entropy can accumulate, can cause entropy and heat to
be transported by first-order tunneling. By using a controlled potential-energy offset,
first order tunneling is suppressed (in the sense explained in Section 4.2.2) and secondorder tunneling is still allowed but at a modified rate:
4𝑡2
𝐽(∆) =𝑈(︂1
1+1 − ∆/𝑈
1 + ∆/𝑈)︂

(5.1)

The prefactor is valid for a single-species 𝑛 = 1 MI. This function if plotted in Fig. 5-1.
For ∆ = 1.4 𝑈 , the superexchange matrix element is the same with and without thetilt. Therefore, by using tilts ∆ ∼ 1.4 𝑈 , the rate of superexchange is not significantly
altered by the tilt and the advantages of studying suprexchange in a tilted lattice canbe explored.

197

J( )/J(0)

10 1

10 0

10 -1

0

0.5

1

1.5

2

2.5

3

/U

Figure 5-1: Superexchange matrix element as a function of tilt. Plotted is 𝐽(Δ) from
Eq. (5.1) relative to 𝐽 at zero tilt. We focus on the range of tilts in the shaded region forstudying spin physics.

5.1

Choice of tilt implementation

We have used two physical implementations of the tilt: (i) a magnetic field gradient;(ii) an optical AC Stark shift gradient. A magnetic field gradient can be used for spinstates with the same magnetic moments. Although it provides a more homogeneous
tilt, a magnetic field gradient has slower turn on and off times. Due to the high power
demands of lithium, we have used an AC Stark shift to explore the effects of the tilt
on superexchange.

5.1.1

Magnetic field gradient

Here we describe the characterization of the magnetic field gradient in our system1 .
The magnetic field gradient is calibrated by measuring the acceleration it imparts
on a BEC. The gradient is switched on while the BEC is held in the ODT. Afterthe ODT is switched off, the position of the atoms is recorded as a function of time,
as show in Fig.5-2a. The acceleration is plotted in Fig.5-2b as a function of thecurrent in the coils. For reference, the resulting gradient in Gauss/cm and the tilt
in kHz/site are also shown. With a 150-Amp power supply the maximum tilt is 3.4
kHz/site and with a larger power supply of 500 Amps, the tilt can be at most 11.3
1

For the magnetic field gradient we use the pair of double-layer coils which are furthest from theatoms. The rest of the coils are used to create the Feshbach field.

198

kHz/site. This tilt is equal to the on-site interaction for a lattice depth of 11 𝐸𝑅 andscattering length of 100 𝑎0 in 1D and of 200 𝑎0 in 3D. The inset in Fig.5-2b shows azoom into the zero crossing of the acceleration. At zero applied gradient, the residual
effective magnetic field gradient due to the steel chamber and the welding causes
an acceleration of −2𝑔, measured at 869.3174 Gauss. This measurement includes
gravity with acceleration +𝑔, so that the acceleration due to the residual magnetic

field gradient is −3𝑔 (pointing up).
Magnetic field gradient (Gauss/cm)

b)

400

0

300

20

30

40
3

Acceleration (g)

100
0
-100
-2.1
-1.8
-1.5
1.7
0.6
5.4

-200
-300

2.5
25
2

20
15
10
5

1.5

2

1

0
-2

0.5
0

5

10

-400
0

2

4

6

8

10

0

TOF (ms)

15

Gradient (Amps)

0
50

100

Tilt per site (kHz/site)

30

200

Position ( m)1035

Acceleration (g)

a)

0
150

Magnetic field gradient (Amps)

Figure 5-2: Calibration of the magnetic field gradient formed with the outermost coil pairin anti-Helmholtz configuration. a) Center of mass position of the BEC cloud after release
Fun gradients.
figures The legend shows the fitted acceleration in units of 𝑔.
in several magnetic field
Wednesday, September 30, 2015
1:44 AMb) The acceleration as a function of current in the coils.
Fff

2.5 G/cm
11 G/cm
23 G/cm
36 G/cm
45 G/cm

Figure 5-3: Interaction spectroscopy in an 𝑛 = 2 MI in the presence of a magnetic field
gradient. The left peak corresponds to the: |𝑎⟩ → |𝑏⟩ transition in n = 1 and the right peakcorresponds to the |𝑎𝑎⟩ → |𝑎𝑏⟩ transition in n = 2.

We also note that this set of coils produces a small bias field in the anti-Helmholtz
199configuration. To measure it, we use interaction spectroscopy (Section 3.2.1) in an𝑛 = 2 MI at different values of the applied magnetic field gradient. Starting with a MI
in state 𝑎, we do spin-flips to state 𝑏 at different frequencies and measure the atom
number in 𝑏. Fig. 5-3 shows that the resonance lines for both 𝑛 = 1 and 𝑛 = 2 shift andbroaden in the presence of larger tilts. The broadening is expected and is due to thefact that in a magnetic field gradient, different portions of the cloud are at different
magnetic fields. Atoms in the top and bottom parts of the cloud are resonant forslightly smaller or slightly larger frequencies as compared to atoms in the center ofthe cloud. The shift must be due to an additional bias field which this pair of coils
introduces.

5.1.2

Optical tilt with an AC stark shift

We implement an AC Stark shift potential gradient by a 1064 nm Gaussian beam
offset by about a beam radius from the atoms. The total potential which the atoms
feel is the sum of the potential of the lattice beams (which are also responsible forthe overall trapping) and that of the tilt beam. For dynamics in 1D along the tilt,
the potential is:
𝑈 (𝑧) = 𝑈𝐿 (𝑧) + 𝑈𝑇 (𝑧)
(︁
)︁
2
2
2
2
where: 𝑈𝐿 (𝑧) = − 𝑉𝑧 cos2 (𝑘𝐿 𝑧) + 𝑉𝑥 𝑒−2𝑧 /𝜎𝑥,𝐿 + 𝑉𝑦 𝑒−2𝑧 /𝜎𝑦,𝐿
2

(5.2)

2

𝑈𝑇 (𝑧) = −𝑉𝑇 𝑒−2(𝑧+Δ𝑧) /𝜎𝑧,𝑇(︂)︂
3𝜋𝑐2 2𝑃
1
1
and 𝑉 =
Γ+
2𝜔03 𝜋𝜎𝑇2 𝜔0 − 𝜔 𝜔0 + 𝜔

The lattice potential 𝑈𝐿 is the sum of the three orthogonal lattice beams and thetilt potential 𝑈𝑇 is the AC stark shift of the beam offset by ∆𝑧 from the center ofthe lattice beams. The maximum gradient of the intensity of a Gaussian beam is at
𝜎𝑧,𝑇 /2, which is half of the 1/𝑒2 radius of the beam. As we will see in the next section,
the non-Gaussian components of the beams become important for the fine alignmentand calibration of the tilt.
200

5.2

Characterization of the tilt beam

The value of the tilt across the cloud is determined by the intensity distribution in
the wings of the tilt beam (around the point of maximum tilt 𝜎𝑧,𝑇 /2). Although this
distribution is expected to be Gaussian, experimental imperfections lead to deviations,requiring experimental calibration of the parameters of the system. This includes the
position of the beam with respect to the cloud, the value and the inhomogeneity ofthe tilt across the cloud and their calibration. In this section, we present the detailsof this characterization.
DesignThe beam is prepared in a similar way to the lattice and ODT beams, as described in
Section, 2.2.1. In short, the light source is a 50 W ALS 1064 nm laser amplifier witha Mephisto seed. The power of the beam is controlled with an AOM, after which
the beam passes through a large mode area polarization-maintaining fiber LMA-PM15 from NKT-Photonics with a watercooled input section. This allows us to use
about 20 W. We pick a beam size of 73 𝜇m so that we can get tilts per site of up to
30 kHz/site. This is equal to the on-site interaction in an 11 𝐸𝑅 1D lattice at 265 𝑎0
and a 35 𝐸𝑅 1D lattice at 190 𝑎0 .
Beam shape
Although a single-mode fiber suppresses modes other than the TEM00 mode (Gaussian
spatial distribution of the intensity), the photonic crystal fiber core has a hexagonal
shape. This affects only the wings of the beam, but since we would like to utilize
the intensity gradient in the wings, we need to characterize them. Fig. 5-4a shows
the intensity distribution at the focus of the tilt beam and the hexagonal shape canbe clearly seen. Since the lattice beams suffer from the same effect, we need toexperimentally optimize the parameters of the combined tilt and lattice system.
To calibrate this profile with the atoms, we measure the AC stark shift gradientby "kicking" the atoms with the tilt beam, i.e. we pulse the tilt beam on and off
201

-250

-200

-150

-100

-50

0

50

a) -150b) 1-150-100

0.8-100

Cloud position (a.u.)

Z-stage position

Measured
Gaussian

0.6 -50

-50
0

-150

400
300

-100

200
-100

0

-50

100

Tilt beam position ( m)

0

0

0.4
50

0.2

100
150
25

50

50

100

100
150

0 150
-100

0

-100

100

0

-50

10

0

50

z ( m)20

100

150

30

-100

Figure 5-4: Shape of the tilt beam at the focus. a) Measured intensity profile. b) Intensity
profile extracted from the kicking measurement as compared to a Gaussian profile. Inset:
measured position of the cloud in 3 ms time of flight after kicking as a function of the position
25of the tilt beam. The main plot is obtained by integrating the shape in the inset.
3

35

15

5

1

0
-100

0

25
20
15
10

100

0.5
0
-20

Individual potentials (E R)

1.5

Total potential (ER)

Tilt (kHz/site)

10

70

5

-10

0

10

0
-20

20

-10

40

10

0
-150

20

70

30

60

Total potential (ER)

13
12
11
10

Individual potentials (E R)

35

25
20
15
10
5

9
0

z ( /2)

-100

-100

-50

10

20

0
-20

0

50

100

150

z ( m)

15

-10

0

20

16

8
-20

5

30

z ( /2)14

Tilt (kHz/site)

0Lattice
Tilt

50
40
30
20
10

-10

0

z ( /2)

10

20

0
-150

-100

-50

0

50

100

150

z ( m)

Figure 5-5: Lattice potential with and without a tilt. Shown are the magnitude of the tilt
per site (left), the total potential for 40 lattice sites (middle) and the individual Gaussian
potentials of the tilt and lattice beams (not including the interference). The lattice has
a depth of 𝑉𝑧 = 12 𝐸𝑅 and has 1/𝑒2 radius of 125 𝜇m. a) No tilt. b) With a tilt of 8 W
positioned half a beam waist away (at the theoretical point of maximum tilt).

faster than the trap frequency. The initial position of the atoms is determined bytheir equilibrium position in a dipole trap formed by the three lattice beams with
their retros blocked. We vary the position of the tilt beam and measure the position

202

15
10

50

10

z ( /2)b)LatticeTilt

x-direction:153 m 0 m60
y-direction:139 m 0 m

30

2.5
2

20

1/e 2 -diameter

20

a)

25

of the cloud after 3 ms time of flight. We convert this to an effective beamshape by
integrating the measured cloud position as a function of position of the tilt beam.
This is shown in Fig.5-4b.
Given the measured beam profile of the tilt beam, we construct the expected total
potential including the potential of the lattice beams. Although the lattice beams
are prepared in a similar way and have similar deviations from a perfect Gaussian,we ignore that for this calculation. Fig.5-5 shows the tilt per site across a MI of 40
sites as well as the total and the individual potentials with and without a tilt. Notethe large inhomogeneity of the tilt when the tilt beam is positioned at the point of
maximum gradient 𝜎𝑧,𝑇 /2.
Calibration
The tilt beam is calibrated by ramping down the lattice depth in the direction along
the tilt to 12 𝐸𝑅 and then adiabatically turning the tilt on. When the tilt crosses∆ = 𝑈 there is an increase in the number of doublons, as shown in Fig. 5-6a. This
reproduces the results in [146, 110], where a transition between a paramagnet (a statewith one atom per site) to an antiferromagnet (a state with alternating empty anddoubly occupied sites) was observed. We associate the tilt value corresponding to
the middle of this transition with 𝑈 . More quantitatively, we fit a phenomenological
function of the form (solid line in Fig. 5-6a):𝑓 (𝑥) = 𝐴 tanh(︂𝑥 − 𝑥0
𝑤

)︂

+𝑐

(5.3)

and identify the center 𝑥0 of the transition with ∆ = 𝑈 .
Displacement from the lattice
Depending on the relative position between the center of the lattice and the center of
the tilt beam, both the value of the tilt for a given beam power and the inhomogeneity,caused by the curvature of the tilt and the lattice beams, can vary. For a perfect
Gaussian, the maximum slope is at ±𝜎/2, so we expect that the tilt is maximized
203

Tilt (kHz/site)
40
35

30

b) 6000

25

5000

20

25

15

20
15

10

10
5
5

Tilt (kHz/site)

Lattice (ER)30

Number of atoms in n=2

a)

5

10

15

6

8

20

25

4000
3000
2000
1000

0

0

0
0

5

10

15

20

25

0

2

4

10

12

14

Tilt beam power (Watts)

Time (ms)Figure 5-6: Tilt calibration. a) After the lattice arm along the tilt is lowered to 12 𝐸𝑅 , thetilt beam is adiabatically turned on. b) The number of atoms in 𝑛 = 2 sites is measured asa function of the final power in the tilt beam. The solid line is a fit function of the form in
Eq. 5.3.

when the tilt beam is positioned ±𝜎/2 away. This is the case illustrated in Fig. 5-

5. The inhomogeneity across the cloud also depends on both the relative position
and the relative strength of the lattice and the tilt beams. Since we found that thetilt beam deviates from a perfect Gaussian, we use the calibration procedure above
to experimentally measure the tilt strength (in units of kHz/site per Watt of beam
power) and the inhomogeneity as a function of tilt beam displacement.
The calculated tilt and inhomogeneity for a system of 40 lattice sites (a cloud withdiameter of about 20𝜇m) are compared to the measured ones in Fig. 5-7. Given thecalculated total potential in Fig. 5-5, the inhomogeneity can be calculated by defining
it as:
𝐼=∆max − ∆min∆max + ∆min(5.4)

over the extent of the cloud. Note that this potential includes the measured beam
shape of the tilt beam and perfect Gaussian beams for the three lattice arms. The
inhomogeneity is measured by using the calibration technique and taking the widthof the fitted curve of Eq. (5.3). It is defined as:
𝐼𝑚 = 2.2

204

𝑤
𝑥0

(5.5)

which is the range over which 𝑓 (𝑥) goes from 10 % to 90 % of its asymptotic values.
a)

b)

1.6

1

0.8

1.2

Inhomogeneity I

Average tilt (kHz/Watt)

1.4

1
0.8
0.6
0.4

0.6

0.4

0.2

0.2
0
-80

0
-80 -60 -40 -20

0

20

40

60

80

-60

-40

Displacement ( m)

c)

d)

1.1

0

20

40

60

80

40

60

80

0.6
0.5

1

Inhomogeneity

Tilt strength (kHz/Watt)-20

Displacement ( m)

0.9
0.8

0.4
0.3
0.2

0.7
0.1
0.6
-80

-60

-40

-20

0

20

40

60

0
-80

80

Displacement ( m)

-60

-40

-20

0

20

Displacement ( m)Figure 5-7: Characterization of the strength and homogeneity of the tilt across the cloud.
The lattice depth is 12 𝐸𝑅 . a)-b) Calculated average tilt per Watt and inhomogeneity I(Eq. (5.4)) of the total potential as a function of tilt beam displacement. c)-d) Measured tilt
strength 𝑈/𝑥0 and inhomogeneity 𝐼𝑚 (Eq. (5.5)).

The plots in Fig. 5-7 are not symmetric around zero due to the asymmetric tilt
beam profile measured above. While the expected positions of maximum tilt andminimum inhomogeneity agree qualitatively with the measurements, the quantitative
differences are probably due to imperfections in the lattice beams, similar to the ones
measured for the tilt beam. For more details, see the Supplementary information in
paper [45], Appendix E.
Alignment
In the measurements shown above, the beam was focused to the position of theatoms. We do this by measuring the dipole oscillation frequency after the BEC
has been suddenly loaded into the tilt beam, positioned slightly off-center in the 𝑧
205direction. We move the focus of the tilt beam until this frequency is maximized. To
a good approximation, if the displacement is small compared to the beam size in that
direction, the potential is harmonic and characterized by a trap frequency 𝑤𝑡𝑟𝑎𝑝 =√︁24𝑉𝑇 /(𝑚𝜎𝑧,𝑇). After focusing, we position the tilt beam so that the inhomogeneity

is minimized (Fig. 5-7d). In the measurements shown above, we record the positionof the atoms when loaded adiabatically into the tilt beam on a camera and use thisas a reference.
Choice of tilt strength
In order to suppress tunneling, the tilt has to be ∆ > 4𝑡 (Section 4.2.2) but in order
to not suppress superexchange, it has to be not more than about 2𝑈 (Fig. 5-1). As
explained above, to avoid higher order resonances ∆ = 𝑈/𝑚, the tilt should be alittle bit larger 𝑈 . How much larger than 𝑈 is determined by the tilt inhomogeneity.
Since the inhomogeneity varies with the power in the tilt beam, which in tern isdetermined by 𝑈 , which we vary in the experiment, the tilt inhomogeneity will vary.
In order to make sure that no ∆ = 𝑈 resonances exist in the extent of the cloud,we experimentally determine what the ratio ∆/𝑈 should be in order to avoid theformation of doubly-occupied sites at ∆ = 𝑈 .

40

30

35

b)

25

15

20
15

10

10
5
5

Tilt (kHz/site)

Lattice (ER)

30
20

4000
5 ER

3500

25

Number of atoms in n=2

a)

6 ER
8 ER
9 ER

3000

10 E R
12 E R2500

14 E R
1.65

2000

/U

1500
1000
500

0

0

0
0

5

10

15

20

25

0.5

1

1.5

2

2.5

3

/U

Time (ms)Figure 5-8: Choice of tilt strength. Here 𝑎 = 55 𝑎0 . a) After the lattice arm along the tilt
is lowered to 12 𝐸𝑅 , the tilt beam is adiabatically decreased from Δ ≫ 𝑈 . b) The numberof atoms in 𝑛 = 2 sites is measured as a function of the final power in the tilt beam several
lattice depths 𝑉𝑧 .

206

We use a procedure similar to the calibration protocol but we now start from alarge tilt and adiabatically decrease it until we see the formation of doublons. This isshown in Fig. 5-8. We pick a tilt of ∆/𝑈 = 1.65 in order to avoid having a resonance∆ = 𝑈 within the cloud.
Now that the tilt beam is aligned and calibrated, we can study its effects on an𝑛 = 1 MI. As we have illustrated in [45], there are four main advantages of studying
superexchange physics in the presence of a tilt. First, since the tilt suppresses firstorder tunneling by turning linear transport into Bloch oscillations with amplitude
smaller than the lattice site (for ∆ > 4𝑡), nonequilibrium density distributions can befrozen in. We use this to load large Mott insulators by loading the lattice at a large
scattering length. Second, in the presence of the tilt, the transition to a superfluid,which occurs when the lattice depth is lowered starting from the MI, is suppressed.
This means that spin physics experiments can be performed at lower lattice depths,
resulting in faster dynamics. Typically, superexchange experiments are performedin the MI but at a lattice depth a little bit above the SF-MI transition so that theatoms are localized on individual sites but the tunneling timescales are still fast.
Third, number defects, such as holes and doublons in an 𝑛 = 1 MI, are frozen in andact as reflective barriers, which allows pure spin physics to be studied. Fourth, the tilt
can be used to tune the Heisenberg parameters. We have illustrated these features in[45] and in the following we give more details focusing on the implications for studying
spin physics with 7 Li.

5.3

Suppression of the SF-MI transition

In the presence of a tilt, the superfluid transition can be suppressed when approachedfrom the Mott phase. In this section we describe this effect and the benefits forstudying superexchange-driven spin models at lower lattice depths.

207

Number of atoms per site

50 a0
100 a0200 a0300 a0

3
2.5
2
1.5
1
0.5
0
-30

-20

-10

0

10

20

30

Lattice site ( /2)

Figure 5-9: Equilibrium density distributions in the MI as a function of scattering length
for a fixed total number of 50,000 atoms. Plotted is a cut along the middle of the cloud at
35 𝐸𝑅 .

5.3.1

Nonequilibrium density distributions

Due to the small mass of lithium, high laser power is required to achieve lattice depths
large enough so that the atoms are deep in the MI. A system deep in the MI is usefulfor preparation and detection. Therefore, the lattice beams need to be relatively
small: 125 𝜇m 1/𝑒2 radius. This results in large harmonic confinement due solely
to the curvature of the lattice beams, so that it is difficult to have a large numberof atoms in a given MI plateaus. The equilibrium density distributions in such a
potential are plotted in Fig.5-9 for 50,000 atoms for several scattering lengths in a
3D lattice at 35 𝐸𝑅 . When the scattering lengths is small, MI plateaus with higher
occupation are populated, while when the scattering length is large, more atoms are
in 𝑛 = 1. If large MI plateaus are required, the scattering length can be increased
just before the lattice is loaded. In the experiments in [45], we load 45,000 atoms at
300 𝑎0 resulting in spin chains of 40 sites.
For experiments, typically lower lattice depths and smaller scattering lengths are
required. Although the initial Mott loading can still be performed at a large scattering length to maximize the number of atoms with a given occupation, when thelattice depth is lowered and the scattering length decreased, density redistribution is
expected due to the non-equilibrium initial density distribution. The tilt can be used
as a method of preserving the initial density distributions. Note that even without
208

an externally applied tilt, the curvature of the lattice beams provides a non-negligible
energy offset, which freezes first-order tunneling dynamics at the edges of large clouds.

5.3.2

Minimum lattice depth

In the experiments described in [45], we observe that we can suppress the transition
to a superfluid by starting from a MI with a tilt. The experiments are performed
in 1D tubes and the lattice is loaded at a large scattering length of 300 𝑎0 . When
all three lattice arms are at 35 𝐸𝑅 , deep in the MI, the tilt beam is turned on fasterthan the tunneling time ~/𝑡. Then, the lattice depth along the tilt is decreased
adiabatically and after a short evolution time of 10 ms, the lattice is rampred backup, so that doubly-occupied sites can be detected with interaction spectroscopy. Notethat the lattice ramp up is on the order of a few tunneling times, so that there islocal but not global equilibrium. This means that the state locally is not projected
onto the Fock basis (which a sudden ramp up would do) but if there has been density
redistribution during the hold time (atoms accumulating close to the center of thelattice due to the nonequilibrium initial density distribution), it would be detected as
doublon formation. The fraction ∆/𝑈 = 1.65 is kept constant constant and the lattice
depth and the scattering length are varied. We use the appearance of doubly-occupied
sites as an indicator of the breakdown of the Mott insulator.
=0= 1.65U

Fraction of atoms in n=2

0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
5

10

15

20

25

30

35

Lattice depth (E R)

Figure 5-10: Suppression of the transition to the superfluid. Plotted is the number of atoms
in n = 2 sites at 50 𝑎0 with and without a tilt. The dashed lines indicate the lattice depths
below which doublons appear (Eq. 5.6). Figure adapted from [45].

209Fig. 5-10 compares the data with and without a tilt for a scattering length of
50 𝑎0 . In both cases, below a certain lattice depth, the number of atoms in 𝑛 = 2 sites
sharply increases. The main observation here is that with a tilt the lattice depth
can be lowered further than without a tilt before doublons begin to appear in thesystem. Since, as we show below, the main mechanism responsible for the breakdownof the Mott plateaus in the case without the tilt is the transition to the superfluid,
it is in this sense that this transition is suppressed in the presence of the tilt. This
enables the use of lower lattice depths for superexchange experiments, increasing thetimescale for dynamics. Note that the superexchange matrix element scales with
𝑡2 and tunneling increases exponentially when lowering the lattice depth. For moredetails, see [45].
=1.65U

0.3
0.25
0.2
0.15
0.1
0.05
0

0

10

20

30

=0

b)

50
75
100
125
150

Fraction of atoms in n=2

Fraction of atoms in n=2

a)

0.3

0.2

0.1

0

40

Lattice depth (ER)

50
75
100
125
150

0.4

0

10

20

30

Lattice depth (ER)Figure 5-11: Fraction of atoms in doubly-occupies sites as a function of lattice depth a) witha tilt of Δ/𝑈 = 1.65 and b) without a tilt.

Next, we analyze the breakdown mechanisms for the case with and without a tilt.
Fig. 5-11a-b shows the data for all scattering lengths with a tilt and without a tilt
respectively. In order to analyze the system, we extract a critical lattice depth 𝑉𝑐𝑟
below which doublons appear. We fit exponentials to part of the data: down to thelattice depths where the fraction of atoms in 𝑛 = 2 sites begins to decrease or plateau.
The critical lattice depth is defined as the point at which the fraction of atoms in210doubly-occupied sites is 3 % above the noisefloor:
𝑓 (𝑥) = 𝐴𝑒−𝑥/𝜏 + 𝐶
𝑓 (𝑉𝑐𝑟 ) ≡ 𝛼 = 0.03 + 𝐶)︂(︂1(𝛼 − 𝐶)𝑉𝑐𝑟 = −𝜏 log
𝐴

(5.6)

The critical lattice depth is plotted in Fig. 5-12a and the ratio (𝑡/𝑈 ) evaluated at
𝑉𝑐𝑟 is plotted in Fig. 5-12b. With a tilt, the data is consistent with there being a
critical lattice depth, below which the MI breaks down detected as increased numberof doublons. Without a tilt, the data indicates that there is a critical (𝑡/𝑈 ), so thatthe critical lattice depth varies as a function of scattering length.
a)14=1.65U
=0
SFMI

12

b)

0.25
=1.65U=00.2

8(t/U)c

Vcr

10

6

0.15
0.1

4
0.05

2
0

50

100

0

150

Scattering length (a0)

50

100

150

Scattering length (a0)Figure 5-12: Mott insulator breakdown. a) Critical lattice depth 𝑉𝑐𝑟 above which doublons
appear (Eq. (5.6)). b) Critical 𝑡/𝑈 (calculated at 𝑉𝑐𝑟 ).

We first focus on the case without a tilt (blue points). The data is consistent witha critical (𝑡/𝑈 ). This is indeed what we would expect since the transition between
the superfluid and the Mott insulator phases happens at a critical ratio 𝑡/𝑈 (Section 3.1.3). In fact, the critical ratio that we measure by detecting the appearanceof doubly-occupied sites is, averaging the values in Fig. 5-12b, (𝑡/𝑈 )𝑐 ≈ 0.07. Notethat in 1D, (𝑡/𝑈 )𝑐 ≈ 0.3. The difference can be rationalized by looking at the phase

diagram. Fig. 5-13 shows the mean field phase diagram and a numerically obtainedphase diagram in 1D using DMRG. In particular, the point (𝑡/𝑈 ) which is the maximum extent of the MI and is typically associated with the SF-MI transition seems
211

a poor indicator of the size of the MI plateaus in a trap (where there is a range ofchemical potentials). Since our critical (𝑡/𝑈 ) is defined in terms of the appearance of
doublons, this measurement is much more sensitive to the shape of the MI lobes. We
will use our measurement of (𝑡/𝑈 )𝑐 as the maximum (𝑡/𝑈 ) suitable for superexchange
S. Ejima experiments
et al.
in a MI.

b)2

1.5 MI
ρ=2µ/U/U

neighbouring
2 sites, U > 0 denotes the strengtha) latticeof the on-site Coulomb repulsion, Vc parameterises thecurvature of the quadratic confining potential, and rc =(L + 1)/2 denotes the central position of the chain. In thefollowing, we set1.5
U = 1n=2
as our energy unit, unless statedotherwise.

tc ~ 0.180

SFConstrained Bose-Hubbard
model.
In general, the1
1SF
Bose-Hubbard model cannot be solved analytically. In
the low-density limit, the model reduces to the Bose gas
with δ-potential interaction which was solved by Lieb
0.5Then=1
0.5 MIand Liniger [16].
fact that three or more bosons
tc ~ 0.305
may occupy the same site forms the major obstacle on
ρ=1
the way to an exact solution. Since multiple occupancies
also pose technical
0 problems in numerical approaches,
0
0
0.1
0.2
0.3
0.4
0
0.1
0.15 by the0.2
the Bose-Hubbard model
is0.05usually approximatedt/U
constraint that there is a maximalzt/U
number of bosons
per site, 0 ! nb ! N − 1 . This constrained Bose-Hubbard
Fig. 1: Phase diagram of the one-dimensional constrained Bose5-13:
Phase perdiagramshowingand Mott insulating regions. a) Frommodel hasFigure
N degreesof freedomsite so thatit canthe
be superfluid
Hubbard model (nb ! 5) from DMRG with “superfluid” (SF)written inmean
terms ofspinasvariables
withinS Section= (N − 1)/2.
Theb) andfield
derived3.1.3;From
DMRG.
from The[51].symbols
The confine
blue line
Mott insulatingFigure(MI) regions.
thecase N = 2 is trivial because the hard-core Bose-Hubbard regions with a finite Mott gap, ∆ > 0, extrapolated from ∆(L)
indicates the critical 𝑡/𝑈 we have measured.
model has no interaction term. It reduces to a model for L ! 128. The position of the Mott tips has been obtainedfor free spinless fermions whose properties are known from the Tomonaga-Luttinger parameter.
exactly [17]. The Bose-Hubbard model is recovered in
In the case with the tilt, the data is consistent with a critical lattice depth 𝑉𝑐𝑟 ≈the limit N → ∞ . In general, however, the SU (N )-Bethe
Ansatz equations do not solve the constrained Bose- less than half filling,
6 𝐸𝑅 , [18,19].
mostly independent of the scattering length (Fig. 5-12a). At even lower lattice
Hubbard model+In our work, we study the restricted Bose-Hubbard∆(L)out= µof(L)µ− (L), Fig. 5-14 shows
depths (4 𝐸 ), we see that the atoms are acceleratedthe− cloud.
model with N = 6, i.e., 𝑅nb ! 5. Our results are represen+tative for the original Bose-Hubbard model (1) because(L) = E0(L, N + 1) − E0(L, N ),(2)
the position of the atoms as a function of latticeµdepth.
We interpret the breakdown
multiple lattice occupancies are strongly suppressed in the−µ (L) = E0(L, N ) − E0(L, N − 1),parameter regions of interest to us, U/t > 2 and fillingsinthepresence
of
the
tilt
to
be
a
Zener
breakdown,
so that atoms tunnel to the next
ρ = N/L < nb .
where E0(L, N ) is the ground-state energy for L sites
band of the lattice. Note that at 6 𝐸𝑅 thewidth of the first excited band is 1.7 𝐸 .
Numerical algorithm.
We adopt the DMRG and N particles. In the thermodynamical limit, N, L𝑅→
andbreakdownρ = N/L integer,
the Mottgap is insulator
finite for the
method [12] Inas summary,our numerical
toolinfor
whilethethenocalculatilt case∞ theof theis Mott
welltion of ground-state properties for constrained Bose insulator, ∆ = limN,L→∞ ∆(L) > 0, so that the system
when𝑡/𝑈
we gofrom
thecase“superfluid”
systems [13,14].
For theproperties,
employ becomes
described
by spectrala transition
to awesuperfluid
aboveincompressible
a critical ratio, intheof thephase to the Mott insulating phase.
the dynamical DMRG (DDMRG) [15].
TheforMotttransitionlinesisinatheµ-U ground-statephase
The considered
are large
enough to mechanism,
permittilt therelattices
is anotherbreakdown
which
there
criticallattice depth
reliable extrapolations to the thermodynamic limit for diagram have been previously determined by various
analytical and numericalstrong-coupling
the physicalquantities ofof
interest
to us. We
keep upto interpretation
independent
scattering
length.
One
is thatmethods,
at smalle.g.,
latticedepths,m = 2000 density-matrix eigenstates, so that the discarded expansions [20,21], variational cluster approach [22],. bandwidth become
QMC [23,24],
and DMRG[13,14].
In fig. tunneling
1 we show theweight is sincealways thesmallerthan 1 ×and10−10bandgap
thecomparable,
there
is Zenerto
We checked our algorithm for nb = 1 against the exact phase diagram for the first Mott lobe (ρ = 1) and theresult [17].
The exact
ground-statein the thermosecond Mott lobe
= 2)veryas obtained
from ourdepths,
DMRG
higherbands.
This isenergy
supportedby the observation
that(ρ forsmall latticedynamical limit and the extrapolated ground-state energy calculations with system sizes up to L = 128.
from DMRG
to four-digit
accuracy.
The overall
shape ofof
thethisMottsetlobes
with previous
the agree
atoms
leave thetrap. Finally, the main
conclusion
ofagrees
experimentsisresults. Here, we provide accurate data for the second
Ground-state phase diagram.
At integer filling ρ = Mott lobe, and the values for the critical interaction
212 for the first two Mott lobes which we obtain
N/L, the Bose-Hubbard model in one dimension describes strengtha Mott transition between the “superfluid” phase, char- from the Tomonaga-Luttinger parameter. At the tip ofacterised by a divergence of the momentum distribution each Mott lobe, the model is in the universality class ofat momentum k = 0 [10], and a Mott insulating phase, the XY spin model so that there is a Kosterlitz-ThoulessPosition of the cloud (px)

270

265

260

255
0

10

20

30

Lattice depth (E R)

Figure 5-14: Zener breakdown. Plotted is the position of the cloud along the tilt axis as afunction of lattice depth. The atoms are accelerated out of the cloud around 4 𝐸𝑅 .

that with a tilt, the lattice depth can be lowered further than without a tilt, enabling
faster dynamics. Next we explore how much the superexchange rate can be increased
by going to lower lattice depths.

5.3.3

Maximum superexchange rate

In practice, for 7 Li the three scattering lengths (𝑎𝑎𝑎 , 𝑎𝑎𝑏 , 𝑎𝑏𝑏 ) are very different, so the
maximum superexchange rate in the presence of a tilt is determined by all three inthe following way. First, the rate of transport dynamics is determined by |𝐽𝑥𝑦 |, whichin the presence of a tilt is:

⃒
(︂
)︂⃒
⃒
⃒
1
1
1
2
⃒
|𝐽𝑥𝑦 | = ⃒⃒−4𝑡+
2 (𝑈𝑎𝑏 − ∆) (𝑈𝑎𝑏 + ∆) ⃒(5.7)

The interactions 𝑈 are determined by 𝑎𝑎𝑏 and by the lattice depth. The tunneling 𝑡
is also given by the lattice depth, which is determined through the criterion of thephase transition by the smaller of the two scattering lengths {|𝑎𝑎𝑎 |, |𝑎𝑏𝑏 |}min , so thatboth components are in the MI regime. The tilt is determined by the maximum of

the three scattering lengths, so that all resonances ∆ = 𝑈/𝑚 (with 𝑚 = 1, 2, 3, ...) areavoided. In principle, if the tilt inhomogeneity is smaller, the tilt can be fine-tuned tobe between the resonances for up to some maximum value of 𝑚. Working in 1D andassuming that the minimum lattice depth with the tilt is 𝑉𝑐𝑟 = 6.3 𝐸𝑅 (the average

213

𝑉𝑐𝑟 from Fig. 5-12a) and without the tilt is the depth corresponding to 𝑡/𝑈 = 0.07
(the average (𝑡/𝑈 )𝑐 from Fig. 5-12b), and the tilt is ∆ = 1.65{|𝑈𝑎𝑎 |, |𝑈𝑎𝑏 |, |𝑈𝑏𝑏 |}max ,=0=1.65U3b)|Jxy|/hbar (kHz)

2.5
2

c)

1.5
1
0.5
0
820

830

840

850

860

870

880

400
a min
a max

300
200
100

14

Lattice depth (E R)a)scattering length (a0)

we can plot 𝐽𝑥𝑦, max as a function of magnetic field: Fig. 5-15.

=0=1.65U

12
10
8
6
820

830

840

Gauss

850

860

870

880

Gauss

Figure 5-15: Maximum superexchange rate in 7 Li. a) 𝐽𝑥𝑦, 𝑚𝑎𝑥 with and without a tilt
Δ = 1.65 max (|𝑈𝑎𝑎 |, |𝑈𝑎𝑏 |, |𝑈𝑏𝑏 |) at the minimum depth 𝑉𝑐𝑟 . Plotted for reference are: b)
the minimum and maximum scattering lengths and c) the lattice depths at which 𝐽𝑥𝑦, max
is calculated. The ranges where 𝑎min < 40 𝑎0 and 𝑎max > 400 𝑎0 are excluded.

When 𝑎max = 𝑎𝑎𝑏 , the term in brackets in Eq. (5.7) is kept as large as possible.
When 𝑎max ̸= 𝑎𝑎𝑏 , then ∆/𝑈𝑎𝑏 can become much larger than 2, significantly slowing
down the dynamics. Due to the particular Feshbach resonances in lithium, the mini-

mum lattice depth without the tilt is always larger than the minimum lattice depthwith the tilt, resulting in larger superexchange rates with the tilt in the range ofmagnetic fields relevant for implementic spin Hamiltonians.

5.3.4

Higher dimensions

The tilt can be implemented in 2D and 3D by offsetting the beam along the otherone or two axis. As discussed in [45], since the motion along x, y, and z is separable,
we expect to be able to reach the same minimum lattice depth of 6 𝐸𝑅 with thetilt as in 1D. Since the MI transition is at much higher depths, the increase in the
superexchange rate is expected to be much larger in higher dimensions. For example,
we can use Eq. (5.7), at a scattering length of 100 𝑎0 , and assume that minimum
lattice depth with the tilt is 6.3 𝐸𝑅 and without the tilt it is the one corresponding214

to the SF-MI transition. Table 5.1 lists the superexchange rates ~/|𝐽𝑥𝑦 |.
∆/𝑈
0
0
1.65
0
1.65

(𝑡/𝑈 )𝑐 (no tilt)
𝑉0
𝑉Δ
𝐽0 /~
𝐽Δ /~
𝐽Δ /𝐽01D
0.07
8.3 𝐸𝑅
6.3 𝐸𝑅
1.3 kHz
2.2 kHz
1.7

2D
0.06
10.2 𝐸𝑅6.3 𝐸𝑅698 Hz3.6 kHz
5.2

3D
0.034
13.3 𝐸𝑅6.3 𝐸𝑅
203 Hz6.0 kHz30

Table 5.1: Maximum superexchange rates in 1D, 2D, and 3D with and without a tilt for
100 𝑎0 . Here we assume that the tilt allows the lattice depth to be lowered to 6.3 𝐸𝑅 in 2D
and 3D as well. In the case without the tilt, we use the measured (𝑡/𝑈 )𝑐 = 0.07 for 1D andthe numerically calculated literature values for (𝑡/𝑈 )𝑐 for 2D and 3D, listed in Section 3.1.3.

zx

Figure 5-16: Implementation of the tilt in 2D. Illustration of the competing second orderprocesses in 2D when the tilt is along a diagonal of the lattice.

Implementing a given tilt by offsetting a Gaussian beam requires more power inhigher dimensions. In addition, if the beam is positioned along the diagonal, there
are lattice sites on equipotential lines. This is illustrated in Fig.5-16 in 2D. Tunnelingover two sites at the same potential through an intermediate off-resonant state cantake place, which is on the same timescale as superexchange. For example, if one ofthese sites on equipotential lines is a hole, it can propagate through a second-orderprocess on a timescale comparable to ~/𝐽. To suppress these processes, the tilt beam
can be offset slightly from the diagonal.

215

5.4

Effect of holes and doublons

In an 𝑛 = 1 MI the lowest energy number defects are holes and doublons, which,ignoring Bose enhancement factors, propagate at the tunneling rate 𝑡/~. When a tilt
is added, these defects become pinned in the same sense that first order tunneling gets
suppressed. In [45], we have illustrated the effects of the tilt on holes and doublons.
We have shown that a pinned hole acts as a reflective barrier, while a pinned doublon
also acts as barrier but with a reflectivity which depends on the sign and magnitudeof the tilt. Here we discuss this effect.
We consider the case of a single |↑⟩ atom next to a |↓↓⟩ doublon resulting in the|↑⟩ atom becoming part of the doublon for all 𝑈 being equal. As shown in Fig.5-17,
the superexchange matrix element is:

2𝑡2
2𝑡2
𝐽𝑑 = −+∆
2𝑈 + ∆(5.8)

This is the case illustrated in [45].
35

30

U+

25

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20

35
15

35

30

U +2

10

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2

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5

25

20
0
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0

20
4

2

6

8

15

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10

JT+ =

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25
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2t2

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0

2

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6

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4

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5

0
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6

Figure 5-17: Superexchange across a doublon.

This expression will change when the interactions 𝑈 are not all equal or if the
doublon is different. We consider the most general case when all three interaction
strengths are different. The propagation of an |↑⟩ spin across a |↓↓⟩, |↑↓⟩, and |↑↑⟩

216

doublon are respectively:
2𝑡2
2𝑡2
+
(𝑈↑↓ − 𝑈↓↓ ) − ∆ 2𝑈↑↓ + ∆
𝑡2
𝑡2
𝐽↑↓ =
+
−∆ (𝑈↑↑ + 𝑈↑↓ ) + ∆
4𝑡2
3𝑡2
𝐽↑↑ =
+
−∆ 2𝑈↑↑ + ∆
𝐽↓↓ =

(5.9)
(5.10)(5.11)As a function of ∆, these expressions have resonances at different points, so thatin a general two-species MI with doublons, the rates at which the single |↑⟩ atoms

propagate across the doublons is strongly dependent on the exact value of the tilt.
It is therefore best to avoid doublons all together, which can be done by erring on
the side of using smaller atom numbers. By decreasing the atom number, the chemical potential is decreased, making doublons less probable than holes. Nevertheless,doublons can appear due to noise processes, limiting the lifetime of the MI.

5.5

Measurement of the superexchange rate

To show the tunability of the Heisenberg parameters with tilt, we measure the superexchange rate in a spin-1/2 system, described in Section 4.1.1. We implement the
XXZ Heisenberg model:
𝐻 = 𝐽𝑧∑︁

𝑆𝑖𝑧 𝑆𝑗𝑧 + 𝐽𝑥𝑦

⟨𝑖,𝑗⟩

∑︁ (︀
)︀
𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦(5.12)⟨𝑖,𝑗⟩with a MI with one atom per site. In order to measure dynamics, we create a nonequilibrium spin distribution: a spin spiral (Section 4.2.3) and observe the decay of
its contrast. We measure the density distribution of the number of atoms in one if thetwo spin states, which has a sinusoidal variation along the 1D chains. We then fit for
the amplitude of the sinusoid, the contrast, and record how it decays as a function oftime. Note that the spin spiral is created along the same lattice axis as the tilt andthe lattice depth only along this axis is decreased, realizing 1D dynamics. We have217

detailed these measurements in [45], here we provide additional data sets and moredetails.
6
7
8
9
10
11
12
13
14
15
16
17

1

Contrast (C/C 0)

0.9
0.8
0.7
0.6
0.5

b)

1.1
1

Contrast (C/C 0)a) 1.1

0.9
0.8
0.7
0.6
0.5
0.4

0.4

0.3

0.3
0

200

400

600

0

800

5

10

15

20

25

Time (t/tSE)

Time (ms)Figure 5-18: Relaxation of the spin spiral. Plotted is the contrast obtained when imaging
the probability of being in state |↑⟩. The contrast is rescaled to the initial contrast 𝐶0 . a)
Relaxation at different lattice depths, indicated in the legend in recoil units. Time in ms. b)
Time rescaled by 𝑡𝑠𝑒 = ~/|𝐽𝑥𝑦 |. Here the decay curves collapse, showing that the relaxation
is driven by superexchange. Here Δ = 1.65 𝑈 and 𝐽𝑧 /𝐽𝑥𝑦 = 0.25.

We create the spiral in the XZ plane, so that the z-component of the magnetization can only decay through transport, because spin-flips between the two hyperfine
states can only happen if a microwave field is applied, which we do not apply for thismeasurement. Therefore, measuring the timescale for the decay of the spin spiral isa measure of the rate of transport. We confirm that the decay is driven by superexchange by varying the lattice depth and collapsing the decay curves by rescaling thetime by the superexchange time 𝑡𝑆𝐸 = ~/|𝐽𝑥𝑦 |, Eq. (5.7). This is shown in Fig. 5-18 fora range of lattice depths. In this case, the spiral decays over about 7 superexchange
times.
We also plot the lifetime of the spin spiral at each lattice depth, extracted byfitting an exponential with an offset to the data. This is shown in Fig. 5-19a for
two tilts ∆/𝑈 = 1.65 and ∆/𝑈 = 1.35. For the smaller tilt, the dynamics are fasterbut at low lattice depths another process dominates, which is probably due to theappearance of doublons at ∆ = 𝑈 due to tilt beam inhomogeneity (Fig. 5-8b). The
solid line is a fit of the form 𝐴~/|𝐽𝑥𝑦 |. Note that the anisotropy 𝐽𝑧 /𝐽𝑥𝑦 varies asa function of tilt, as shown in Fig. 5-19b for this magnetic field. The fact that the218coefficient 𝐴, characterizing how many superexchange times are required for the spin
spiral to decay, is different for the two anisotropies can be explained by the different
transport properties at different Hamiltonian anisotropies 𝐽𝑧 /𝐽𝑥𝑦 .
a)b)= 1.65 U ab2
1.5

= 1.35 U ab

103

10

0.5

Jz/JxyLifetime (ms)

1
2

101

10

0
-0.5
-1

0

-1.5
10

-1

5

10

15

-2

20

0

Lattice depth (E R)

0.5

1

1.5

2

2.5

/Uab

Figure 5-19: Measurement of the superexchange rate in a tilted lattice. a) Relaxation timeof the spin spiral as a function of lattice depth for two tilts Δ/𝑈 = 1.65 and Δ/𝑈 = 1.35. The
solid lines are functions of the form: 𝐴 ~/|𝐽𝑥𝑦 | with 𝐴 = 7.0 and 𝐴 = 5.5 respectively. The
magnetic field is 848.1014 Gauss so that 𝐽𝑧 /𝐽𝑥𝑦 = 0.25 and 𝐽𝑧 /𝐽𝑥𝑦 = 0.03 respectively. The
scattering lengths are (𝑎𝑎𝑎 , 𝑎𝑎𝑏 , 𝑎𝑏𝑏 ) = (−58, −80, −65) 𝑎0 . b) Variation of the anisotropy
𝐽𝑧 /𝐽𝑥𝑦 as a function of tilt at 848.1014 Gauss. The dashed and dotted lines correspond to
the resonances Δ = |𝑈𝑎𝑎 | and Δ = |𝑈𝑏𝑏 | respectively. The solid black lines are the two tilts
from a).

In fact, the variation of the relaxation time constant as a function of anisotropy
is the subject of our most recent project. We have shown that different types oftransport are realized for different regimes of the Hamiltonian. Here, we would liketo show that the tilt can be used to tune the Heisenberg parameters. Since varying
the tilt, varies the anisotropy 𝐽𝑧 /𝐽𝑥𝑦 :
(︂
)︂
4𝑡2 1
1
1
𝐽𝑧 (∆) =+𝑈𝑎𝑏 2 1 − ∆/𝑈𝑎𝑏 1 + ∆/𝑈𝑎𝑏
(︂
)︂
4𝑡2 1
1
1
+
+
𝑈𝑎𝑎 2 1 − ∆/𝑈𝑎𝑎 1 + ∆/𝑈𝑎𝑎
(︂
)︂
1
1
4𝑡2 1
+
+
𝑈𝑏𝑏 2 1 − ∆/𝑈𝑏𝑏 1 + ∆/𝑈𝑏𝑏(5.13)in order to single out the effect of the variation of the tilt only on the superexchange
time ~/|𝐽𝑥𝑦 |, we pick a special point where 𝑎𝑎𝑎 = − 𝑎𝑏𝑏 so that the last two terms
cancel: 𝐽𝑧 = − 𝐽𝑥𝑦 . Then, it is only the value of |𝐽𝑥𝑦 | which drives the dynamics as219

the tilt is varied at this magnetic field.
Fig. 5-20 shows the relaxation times for two spiral wavevectors. Although we do
expect the relaxation time to depend on wavevector, the two wavelengths used here
are too similar. In fact, since this data was taken, we have improved the resolution
and by using polarization-rotation imaging instead of absorption imaging, and we are
now able to vary the spiral wavelength by a factor of 5. Here, as a function of tilt, weobserve the expected variation of the lifetime. Note that for tilts close to resonances∆ = 𝑈/𝑚, plotted as dashed and dash-dotted lines, the lifetimes no longer fall on the
curves 𝐴~/|𝐽𝑥𝑦 |.
=11.2 m=10.45 m

Lifetime (ms)

102

101

0

0.5

1

1.5

2

Tilt ( /U )Figure 5-20: Relaxation time of the spin spiral as a function of tilt for two spiral wavevectors.
The lattice depth is 12 𝐸𝑅 . The magnetic field is 857.0052 Gauss so that 𝐽𝑧 /𝐽𝑥𝑦 = −1 andis constant as a function of Δ. The solid lines are functions of the form: 𝐴 ~/|𝐽𝑥𝑦 | with
𝐴 = 6.5 and 𝐴 = 6.6 for 𝜆 = 11.2 𝜇m and 𝜆 = 10.45 𝜇m respectively. The dotted parts
are the same curves over regions where resonances Δ = 𝑈/𝑚 occur. These are plotted for
Δ = 𝑈𝑎𝑏 and Δ = 𝑈𝑎𝑎 = 𝑈𝑏𝑏 in dashed black and grey lines respectively and in dash-dottedlines for Δ = 𝑈𝑎𝑏 /2 and Δ = 𝑈𝑎𝑎 /2.

In addition to showcasing the tilted lattice as a method for studying spin models,these two experiments serve as a benchmark that spin Hamiltonians can be realized
in the system we have built. In fact, this is the first time superexchange mediateddynamics have been measured in our system. Moreover, we measure superexchange
rates varying over two orders of magnitude and we show that we can control this ratewith a tilt over an order of magnitude. As we have shown that the tilt can be used
to tune the parameters of the Heisenberg model, next we explore how this affects thephases which can be accessed with 7 Li in the region of Feshbach resonances between
220the lowest two hyperfine states.

5.6

Spin-1/2 phase diagram with a tilt

In Section 4.1.3 we presented the phase diagram of the spin-1/2 Heisenberg modelfor 7 Li. We now explore how the tilt changes this. Fig. 5-21 compares the no tiltcase to the case of a tilt ∆ = 1.65{|𝑈𝑎𝑎 |, |𝑈𝑎𝑏 |, |𝑈𝑏𝑏 |}max . This tilt flips the sign of

the Heisenberg parameters, so that the XY phase is now antiferromagnetic. It alsochanges the ratio 𝐽𝑧 /𝐽𝑥𝑦 , as shown in Fig. 5-21b. In the majority of the magnetic
field regions, this results in the Z phases becoming XY and vice versa. 𝐽𝑥𝑦 is positive
for no tilt and negative for the 1.65 U tilt. Since in the entire (usable) range of the
Feshbach diagram with no tilt 𝐽𝑥𝑦 is positive, even a system with many Feshbachresonances, such as ours, can benefit from using the tilt as a tool to enlarge the

200
Δ=0
150
100
50
0
-50
-100
-150
-200
200
150 Δ = 1.65|𝑈𝑚𝑎𝑥 |
100
50
0
-50
-100
-150
-200
810
820
830
840

b)

2
=0
=1.65 |Umax|

1.5
1
0.5

Z FM
Z AFM
XY AFMXY FM

Jz/Jxyscattering length (a0)a)

scattering length (a0)available Hamiltonian regimes.

0
-0.5
-1
-1.5

850

860

870

-2
810

880

820

830

840

850

860

870

880

Gauss

Gauss

Figure 5-21: Effect of a tilt on the spin-1/2 phase diagram. a) 7 Li phase diagram without
and with a tilt. The red, purple, and blue lines correspond to the scattering lengths 𝑎𝑎𝑎 , 𝑎𝑎𝑏 ,and 𝑎𝑏𝑏 respectively. b) Anisotropy 𝐽𝑧 /𝐽𝑥𝑦 . 𝐽𝑥𝑦 is positive for no tilt and negative for the
1.65 U tilt.

When the tilt is varied in the range ∆/|𝑈 |max = 1.2 to 2, the phase diagram does

not change significantly. It is rather the speed of superexchange which changes overabout an order of magnitude. This is shown in Fig. 5-22. It is true that for a given
magnetic field, varying the tilt changes the Heisenberg parameters, so different behavior is expected but it is typically within the same phase.
221

200
Δ = 1.20
150
100
50
0
-50
-100
-150
-200
200
150 Δ = 2.0|𝑈𝑚𝑎𝑥 |
100
50
0
-50
-100
-150
-200
810
820
830

b)

2
=1.2 |Umax|

1.5=2.0 |Umax|

1
0.5

Z FM
Z AFM
XY FM

Jz/Jxyscattering length (a0)scattering length (a0)

a)

0
-0.5
-1
-1.5

840

850

860

870

-2
810

880

820

830

840

850

860

870

880

Gauss

Gauss

Figure 5-22: Effect of changing the value of the tilt on the spin-1/2 phase diagram. a) 7 Liphase diagram. The red, purple, and blue lines correspond to the scattering lengths 𝑎𝑎𝑎 , 𝑎𝑎𝑏 ,and 𝑎𝑏𝑏 respectively. b) Anisotropy 𝐽𝑧 /𝐽𝑥𝑦 .

In this chapter, we have explored the effect of a controlled potential energy offset
on a Mott insulator. The main advantages of the tilt are to increase the speed ofsuperexchange by allowing the MI plateaus to be preserved to lower lattice depths,
to freeze number defects, and to vary the Heisenberg parameters. An important
application for our lab is that the tilt can be used to increase the fidelity of adiabatic
state preparation protocols. But also, since the tilt freezes holes and doublons, it
can be used to implement schemes with disorder in the following way. If a superfluid
is loaded semi-adiabatically into a 3D lattice so that a (small) fraction of the sites
are holes and doublons, the tilt can freeze this density distribution. Then, by spinflips, a second spin component can be added. If the lattice depth is now decreased,spin transport via superexchange is allowed in this potential with random holes anddoublons, which in addition can have tilt-variable reflectivity.

222

Chapter 6
Supersolids and Superradiance
Even though the BEC 5 machine was built to study spin models with optical lattices, it is versatile enough to explore other ways of forming novel quantum phases.
This chapter focuses on the supersolid, a new form of matter, long sought-after incondensed matter physics, but recently demonstrated in systems of ultracold atoms.
Here we review the supersolids formed in ultracold atom systems and put in context
our take on it: to form one by superradiance in a BEC. Our work is described indetail in [46] reprinted in Appendix C.

6.1

Supersolids

At first glance, a supersolid is an impossible form of matter because it is both a solidand a fluid at the same time. However, bosonic atoms can be in a supersolid state
if a BEC develops a density modulation with a spontaneously chosen phase. In fact,
the defining features of a supersolid are the breaking of two continuous symmetries:(i) translational invariance, characteristic for crystals with periodic structure and(ii) phase invariance, characteristic for superfluid systems. The energy spectrumof the system is often characterized by a roton minimum at the wavevector of themodulation. In other words, the excitation spectrum of the superfluid contains anenergy state with a density modulation at a wavevector 𝑞, which, as a function ofsome parameter, becomes lower and lower in energy, until, at the phase transition, it
223becomes energetically more favorable for this state to form because its energy goes
below the energy of the original ground state. Although this is a very generic behavior,
we now look at a few physical systems in which supersolids have been realized.
The early efforts to look for a supersolid phase were focused on superfluid helium4, which has such a roton minimum (see e.g. [98]). Here the roton minimum inthe dispersion relation is attributed to a peak in the pair correlation function. Notethat the term “roton” is a historical artifact and does not correspond to rotation in
the system. Despite some controversy about the observation of a supersolid in thissystem, the current consensus is that the supersolid was in fact not observed [87].
In a typical BEC of atoms with contact interactions, the excitation spectrum
does not have a roton minimum. However, for dipolar gases confined in 2D pancakeswith strong dipolar interactions, such a minimum develops. Physically, it is theresult of the angular dependence of the dipole interaction, which in this geometryis manifested as a momentum dependence of the excitations, as discussed in [135].
When the dipoles are confined to a 2D plane and oriented perpendicular to it, theyinteract repulsively. However, when the 2D character of the confinement potential
is relaxed and the interactions become more 3D-like, the interaction energy can be
decreased. For small excitation wavevectors (𝑞 < 1/𝐿, where 𝐿 is the size of thecondensate perpendicular to the 2D planes), the interaction is 2D-like, while for large
momenta, 𝑞 > 1/𝐿, it has a 3D character. This means that at some momentum
𝑞 ∼ 1/𝐿, a minimum in the excitation spectrum can develop for sufficiently strong

dipole interactions. The system can be described by the Gross-Pitaevskii equation
with both contact and dipolar interactions. It can then be linearized around theground state for no confinement in the 𝑥𝑦-plane and with a Thomas-Fermi profile in
the 𝑧-direction. The lowest branch of the excitation spectrum looks like [135]:(︂ 2 2 )︂2~𝑞~2 𝑞 2 (2𝛽 − 1)(5 + 2𝛽)ℰ (𝑞) =
+ ~2 𝜔 2 +𝜇2𝑚
2𝑚 3(1 + 𝛽)(2 + 𝛽)
2

(6.1)

where 𝛽 = 𝑔/𝑔𝑑 parameterizes the ratio between contact interactions (with strength𝑔) and dipolar interactions (with thrength 𝑔𝑑 ), 𝜔 is the trap frequency of the strongly224

3.03

3.03

2.5

(a)

2.5

(b)

2.02

2.02

1.5

ε0
hω 1.5

1.01

1.01

0.5

0.5

00 0
0

week ending
27 JUNE 2003

PHYSICA L R EVIEW LET T ERS

5

0.5
0.5

1
1.0

1.5
1.5

2
2.0

00
0

0.5

1.0

1.5

2.0

ql0

ql0

Figurein the"=excitation
The figure
adaptedfrom"q# for (a) % ! 1=2,"=h!
!6-1:!Roton
343; minimum(b) % ! 0:53,
h!
! ! 46spectrum.
(upper curve)
and %is !
0:47, "=h!
! [135],! 54
which describes the case of a dipolar gas. Dashed line: The top branch has 𝛽 = 0.53 andrves show the numericalresults, and the dotted curves the result of Eq. (9).
the lower branch has 𝛽 = 0.47 from Eq. (6.1). The solid lines are numerical simulations.
The roton minimum is formed in the lower branch around 𝑞𝐿 ∼ 1.

2
2
2
zero for the maximum density n 0
0 "z# ! n 0 "1 $ z =L #, with n 0 ! "="g % g d # being theconfined
direction 𝑧 (perpendicular
to the pancakes),
𝜇 is the2 chemical
potential.
maximum density,and L !and"2"=m!
At higher densities,
excitations#1=2 the TF
size.
icinity of this minimum
Linearizing
Eq. the(1)excitation
around thegroundstate asolution
When thebecome
second term is negative,𝛽 < 1/2,
spectrumdevelops
roton
sate collapses. Below we find the
0 "z# we obtain the Bogoliubov– de Gennes (BdG) equaminimum.andThis
We note that
thisareanalyticalexpression
isn, excitation spectrum,
theis plottedtionsinforFig.
the6-1.
excitations.
Thosecharacterized
by thenization and instability.
momentumof the supersolids
in-plane havefree been
motionand inbyDyan
valid in the region aroundthe roton. qRecently,
observed
ate of dipolar particles harmoniinteger quantum number (j '0) related to the motion
systems in a metastable regime [150, 28, 37].
irection of the dipoles (z) andin the z direction. The excitation wave functions take theway of inducing
a minimum is through atom-photon interactions.
ections (! ! fx; yg).Another
The dynamform fsuch("z# exp"iq!#, where f( ! u (v, and u; v are theve function "r; t#
in waythis toinfinite
Bogoliubov
fu;laservg functions.
thea BdG
equationsreadOne
think about
it is that theexcitationThen
induces
dipole-dipole
interd by the time-dependent Gross# intensities
action between the atoms for high enoughand"large$atomic densities. For
h! 2d2(see [7] and references therein),
0
$f$ !
(3)
f% )Hkin f% ;$ 2 % q2 %example, in a BEC with contact interactions
2m
dz in an elongated
0 trap along the 𝑧 axism
" % !2 z2 % gj and"r; light
t#j2 beam incident along the 𝑦 axis, the Bogoliubov excitation spectrum for2"
wavevector𝑞𝑧 = 𝑞 gets modified by the $f
total
effectivewhich
Hkin f$atom-atom% Hint *f$interaction
+;
(4)
% !
dr0 Vd "r $ r0 #j "r0 ; t#j2 "r; t#;includes the coupling to a light field [120]:(1)

where Hkin is the sum of kinetic energy operators, and(︂ 2 2 )︂2(︂ 2 2 )︂ [︂ 2]︂2 2
~H𝑞 *f + ! 2"g~%𝑞g#f 4~𝑎𝛼𝑘22ment frequency, m is the particleℰ (𝑞) =
+ 𝜋𝑛0 𝑅 dint $$ "z#2 +0 "z# 2 𝐼 𝑄(𝑞, 𝑅)2𝑚2𝑚𝑚𝑅Z2𝜋𝜖0 𝑐oment. The wave function "r; t#
2

1

(6.2)

(5)
dz0 f$ "z0 # 0 "z0 #l number of particles. The third$1
where 𝑛0 is the peak density, 𝑅 is the radial width of the condensate,
𝐼 is the intensity,de (rhs) of Eq. (1) corresponds to$ z0 j#:𝛼(𝛿) isforces,
the atomic
2𝜋/𝜆 is the light wavevector. The term inange (van der Waals)
and polarizability, 𝑘, =exp"$qjz
n field of the dipole-dipole
inter- the effective atom-atom interactions: the first term is the contact
brackets describes
For each j we get the excitation energy $j as a function ofnstant for the short-range interq. Welength
will bemostlyinterested
thesecond
lowestone
excitation
with scattering
𝑎 inthis geometry
andinthecomes"1 $ 3cos2 !#=r3interactions
is the potential
branch $0 "q# for which the confined motion is not excited
eraction, with ! being the angle
in the limit q ! 0.
d the direction of the dipoles (z).
225 in the rhs of Eq. (5) originates fromThe second term
ave function is independent of
the nonlocal character of the dipole-dipole interaction
e ! and can be written as
and gives rise to the momentum dependence of an effec" is the chemical potential.
$ "3=2#gd q

0 "z#from the coupling to the light field. The momentum dependence is captured by the𝑄(𝑞, 𝑅) which is the dimensionless integral of the dipole-dipole interaction over theatomic density (so it is geometry-dependent) and depends on the difference 𝑞 2 − 𝑘 2 .

If the second term is negative, a minimum in the excitation spectrum can develop

around 𝑞 ∼ 𝑘, i.e. for momenta close to the wavevector of the incident light. One

potential issue in this scheme is superradiant Rayleigh scattering, which can heatup the cloud. Its relative importance can be suppressed by placing the system ina cavity, which increases the atom-photon interaction or by the choice of geometry
and laser polarization, or by significantly increasing the atomic density compared totypical BEC densities.
Although such roton minima in the excitation spectra of BECs in cavities havebeen observed [115], the breaking of gauge invariance is also required for a supersolid.
This means that the phase of the density modulations needs to be spontaneouslychosen when measured. Put another way, the ground state is highly degenerateand contains configurations with all possible phases. In the cavity experiments, thecavity defines the phase of the modulation since the cavity mirrors define a boundary
condition for the light in the cavity. This issue has been resolved by placing the atoms
in two cavities at an angle with each other. This additional degree of freedom enables
the continuous symmetry breaking [99].
Another way to realize a system which breaks both continuous translational symmetry and gauge invariance is to introduce spin-orbit coupling between two atomic
hyperfine states, as realized in BEC 2 [100].
The scheme which we implemented was originally proposed in [122]. It also uses
a light field to introduce an additional term in the lowest branch of the excitation
spectrum. The setup uses two counter-propagating laser beams, which do not interfere and are detuned from atomic resonance, incident along the long axis of anelongated BEC. To obtain an analytical solution for the lowest branch of the excitation spectrum, similarly to the treatments above, the Gross Pitaevskii equation in
1D is linearized around the ground state. Here the density is assumed to be spatially√homogeneous of length 𝐿, so that the ground state is Ψ0 = 1/ 𝐿, and the light field
226

(0)

can be treated as plane waves 𝐸1,2 = 𝐸𝑒𝑖𝑘𝑒𝑓 𝑓 𝑥 , where the effective wavevector takes
into account the light-atom interaction [122]:2𝜋𝑘eff =
𝜆

√︂
𝛼
1+ 𝑛
𝜖0

(6.3)

Here 𝜆 is the wavelength of the incident light, 𝑛 is the density, and 𝛼 is the polarizability. This results in the following dispersion relation [122]:]︃(︂ 2 2 )︂2 (︂ 2 2 )︂ [︃
2 2
~
𝑞
16𝛼
𝑘
𝑛
1
~
𝑞
ℰ 2 (𝑞) =2𝑛𝑎 −𝐼1,2 2
+
2
2𝑚
2𝑚
𝑐𝜖20
𝑞 − 4𝑘𝑒𝑓𝑓(6.4)

where and 𝐼1 and 𝐼2 are the light intensities in each beam, which are assumed to be
the same. This equation has form similar to the one of Eq. (6.2). Here the spectrum
develops a roton minimum at 𝑞 = 2𝑘𝑒𝑓 𝑓 above a critical intensity:
𝐼𝑐𝑟𝑖𝑡 =

~2 𝑞 2 2𝜋𝑐𝜖202𝑚 𝛼2 𝑘𝑛𝐿

(6.5)

For larger densities and longer clouds, the critical intensity is lower.
This is illustrated in Fig. 6-2. Two counter-propagating laser beams, which do
not interfere and are detuned from atomic resonance, are incident upon an elongated
BEC. In our setup, the two beams are detuned by 𝜔1 − 𝜔2 = 180 MHz with respect toeach other, which is higher than the relevant frequencies, so that they can be treatedas non-interfering for all practical purposes. They are detuned by 1-20 GHz fromatomic resonance. Each one triggers superradiant Rayleigh scattering preferentiallyin the backward direction. We describe superradiance in the next section. For eachbeam, the incident and backscattered beams interfere to form a moving optical lattice,corresponding to a moving density modulations. When the two beams act independently, there are two moving optical lattices and two moving density modulations.
Above a critical power, the two beams act together, i.e. the lowest-energy state isa stationary optical lattice and a stationary density modulation. In this regime, thetwo moving lattices get phase-locked by the atoms and are stationary with respect
to the atomic density modulation. This constitutes a supersolid because the atomic
227

a)

b)

!1

!2

c)
!1

!1

!1

d)
!1

!2

!1

!2

!1

!2

!2

Figure 6-2: Supersolid from two counter propagating laser beams incident on an elongated
BEC. a) Setup. b) Superradiance from a single beam. Shown are the light fields only:
the interference between the incident and the backscattered beam form a moving opticallattice. This corresponds to a moving density modulation (not shown). c) Superradiancefrom two independent beams. d) Supersolid: stationary density modulation (also shown)
and a stationary phase-locked lattice.

superfluid develops a density modulation with a spontaneously chosen phase.

6.2

Two-beam Superradiance

Each beam acting individually in this regime would trigger a collective scattering
response from the atoms: Superradiance, as studied in a BEC [76]. As the atoms
Rayleigh scatter from the incident beam, they gain momentum. In the strongly
elongated geometry, the photon is scattered preferentially along the axis of the condensate, opposite to the direction of the incident photon. This process is analogous
to the Stimulated Brillouin scattering in fibers, described in Section 2.2.1. The interference between the recoiling atoms and the BEC atoms forms a moving density
modulation. This modulation acts as a Bragg grating and further stimulates more
atoms to scatter into that mode. The backscatted light has slightly lower frequency
228

OBSERVATION OF TWO-BEAM COLLECTIVE(a) 1 ms

8 (b) 1 ms
6
4

the incident light, forming a moving optical lattice. This process can be described2

by rate equations, assuming a quasiparticle description, for the number of atoms in0

the superradiant mode. However, for large laser powers, this description is no longer(c) 50 s

adequate and rather a description in terms of mode softening and roton formation in
the excitation spectrum is relevant. Indeed, a roton minimum forms in the excitation

10 4

b)

1 beam2 beams(e) 3 s

8

-4

10

10

a)

Number of atoms in one satellite peak

spectrum above a threshold power, as described above.

Atom number in one peakthan the incident one due to the imparted recoil on the atoms. It interferes with

00.2(d) 50 s
6

1 beam2 beams

4
2
0
0

1

(f) 3 s
4

6

2
4
2

100 m

0
0

0
0

1

2

3

4

5

5

Power in a sin

6Time ( R t)

FIG. 2. Observation of superradiant light scat

Figure 6-3: Difference between one beam and two beams above threshold power. a) Number
regimes. Shown are momentum distributions af
of atoms in a single satellite peak as a function of time. b) Time of flight image. Figuresflight for three different pump times and differen
adapted from [46]7

18.62 GHz red of the Li D2 line. For each timecase with pump propagating from the left (top pi
compared to the case of two balanced beams fro
In our experiment, we see that sudden turn on (a quench)of circles).
a single The
pumpbeam of collective light(bottom,strength
is characterized
number of atoms wi
results in a different response as compared to a sudden power
turn onof two beamsby(inthethe(right peak). Solid lines are a guide to the eye. The
geometry of Fig. 6-2). The response of the BEC can be observedin timeof flight,
at the powers
indicatedbywhenthe dashed vertical line
the atoms with recoil momentum can be counted separately from the BEC atoms,.

because they form satellite peaks to the main cloud. Thisis shown
in Fig.
The atoms are lostalready
present.
The6-3.
recoilingat ±~𝑘a rateis L,
eitherforbyone
collisions
or because thtime evolution of the number of atoms with momentum
plottedandthe condensate volume.

for two pump beams. While for a single beam, we see andTheincreasein the rate
numberofresultingequation
describes boand the initiallyexponential
outcoupled atoms, until the BEC density has dropped significantly
and thecollective gain for the casebeam [7],scattering decreases, for two beams, we see decaying oscillations. We interpret thisdN1
dt

to show that in the two-beam case a new regime is entered (potentially= Ref (Na 1supersolid).
+ 1) − LN1 = Ref + (RN0
However, it forms only transiently due to Rayleigh scattering to other modes.

229

For weak pump beams and negligible s
one would expect, at least in the perturbative
addition of a second counterpropagating pum
trigger superradiant scattering into the opp
However, due to bosonic stimulation, the rainto the end-fire modes are proportional to6.2.1

Rayleigh scattering

Since Rayleigh scattering is important for this experiment, we measure the lifetimeof a BEC in the lowest hyperfine state |𝑎⟩, limited by the Rayleigh scattering rate,

which, ignoring the counterrotating term and interference effects is:
3𝜋𝑐2 2
𝑅𝑠𝑐 =
Γ𝐼
2~𝜔03

(︂

1
1
1
2
+
2
3 (𝜔 − 𝜔𝐷1 )
3 (𝜔 − 𝜔𝐷2 )2

)︂

(6.6)

Here a single pump beam is used, which is expanded so that the lines are not powerbroadened significantly. Fig. 6-4 shows the measured BEC lifetime as a function ofdetuning from the 𝐷2 line. Note that there are two minima, corresponding to the 𝐷1
and 𝐷2 line resonances, which are split by 10 GHz. The measured lifetime is fit fairly
well by 1/𝑅𝑠𝑐 , serving as a calibration for our system.

BEC lifetime (ms)

1.5

1

0.5

0
-15

-10

-5

0

5

Detuning from the D 2 line (GHz)

Figure 6-4: Rayleigh scattering rate. Plotted is the BEC lifetime as a function of frequencyof the pump light. The solid line is a fit of the form: 𝐴/(1/(𝜔 − 𝜔𝐷1 )2 + 2/(𝜔 − 𝜔𝐷2 )2 ) with
𝐴 = 119.6.

6.2.2

Between the 𝐷1 and the 𝐷2 lines

Here we describe an interesting observation. Note that the data presented in [46] isfor laser frequencies either red or blue detuned with respect to both the 𝐷1 and the𝐷2 line. If the pump frequency is tuned between the two lines, many of the atoms
get pumped into the next Zeeman sublevel, the |𝑏⟩ state. On the one hand, this canbe interpreted as optical pumping, since the transitions are not entirely cycling. On
230

the other, it could also signify Raman superradiance, as observed in [139].
The pump lasers are linearly polarized and drive a 𝜋 transition from the |𝑎⟩ state.

As explained in Section 2.2.2, at the magnetic fields we use, the two lowest hyperfine
states |𝑎⟩ and |𝑏⟩ contain an admixture of a state with 𝑚𝑗 = +1/2, so that in the|𝑚𝐼 , 𝑚𝐽 ⟩ basis they are:⃒ 3 1⟩⃒ 1 1⟩⃒⃒|𝑎⟩ =𝐶𝑎 ⃒ + , −+ 𝛿𝑎 ⃒ + , +2 2
2 2⃒ 1 1⟩⃒ 1 1⟩⃒⃒|𝑏⟩ =𝐶𝑏 ⃒ + , −+ 𝛿𝑏 ⃒ − , +
2 2
2 2

(6.7)
(6.8)

where 𝛿𝑎 ∼ 0.15, and 𝛿𝑏 ∼ 0.2. Since electric-dipole transitions do not couple stateswith different nuclear magnetic moment, the |𝑎⟩ state is coupled to the |𝑏⟩ state by

the 𝛿𝑎 part of the |𝑎⟩ state. Therefore, spontaneous emission from the excited state
can populate the |𝑏⟩ state.

b)1
BetweenBlue
Red

Fraction Nb/Na

0.8
0.6
0.4
0.2
0

0

0.5

1

1.5

2

2.5

3

10 4

Maximum number of b atoms

a)

6

4

2

0
-15

-10

-5

0

5

Detuning from the D 2 line (GHz)Time (ms)Figure 6-5: Raman superradiance and optical pumping. a) The number of atoms in |𝑏⟩
relative to the number of atoms in |𝑎⟩ for a single pump beam. Three different detunings
are used: between the lines (Δ𝐷2 = −6.5 MHz, Δ𝐷1 = 2.9 MHz), blue of both lines (Δ𝐷2 =5.0 MHz, Δ𝐷1 = 14.5 MHz), and red of both lines (Δ𝐷2 = −13.2 MHz, Δ𝐷1 = −3.8 MHz).
The pumping light is 𝜋-polarized. b) The maximum number of atoms pumped into the |𝑏⟩state. The inset shows a time-of-flight image of atoms in the |𝑏⟩ state for pump frequency
between the 𝐷1 and 𝐷2 lines.

By using a single pump beam targeting the |𝑎⟩ state, the number of atoms in the |𝑏⟩state can be measured as a function of pump time. Note that the frequency difference
between the resonant transitions for the |𝑎⟩ and the |𝑏⟩ state is around 200 MHz. Thisis shown in Fig. 6-5a for three detunings of the pump beam with respect to the 𝐷1 and231

the 𝐷2 lines. The number of atoms pumped into the |𝑏⟩ state is significant only when

the pump frequency is between the 𝐷1 and 𝐷2 lines. Fig. 6-5b shows the maximum
number of atoms measured in the |𝑏⟩ state for the time period used in Fig. 6-5a as

a function of the detuning of the pump light. Note that in between the 𝐷1 and the𝐷2 lines this number is notably larger. Also shown is a time of flight image taken in
the |𝑏⟩ state for pump frequencies between the two lines. It is interesting to see thatboth the main cloud and a cloud with momentum ~𝑘 are visible. Typical Raman
superradiance would transfer atoms to the other state always with recoil momentum.
Instead, in the situation where the transitions are not entirely cycling, between the𝐷1 and 𝐷2 lines we apparently observe a combination between optical pumping andRaman superradiance. Note that when both pump beams are on, there are little
or no atoms in the |𝑏⟩ state for all pump powers. In principle, it is possible thata stationary coherent grating involving a superposition between the |𝑎⟩ and the |𝑏⟩states could form, but we do not observe it here.

6.2.3

Threshold power

One way to shed more light on the new phase formed when the cloud is pumped by
two beams in the transient regime is to measure how the threshold power scales withdetuning. In [46], we presented a single dataset, and here we combine all datasets.
From Eq. (6.5): 𝐼𝑐𝑟𝑖𝑡 ∝ 𝛼−2 ∝ 𝛿 2 , therefore, the process is driven by a critical

Rayleigh scattering rate, which scales the same way with detuning: Eq. (6.6). Ignoring
interference effects, the threshold power can be expressed as:
𝑃𝑡ℎ,𝑞𝑢𝑎𝑑 = 𝐴

(︂

1
2
+
2
(𝜔 − 𝜔𝐷1 )(𝜔 − 𝜔𝐷2 )2

)︂−1

(6.9)

We should compare this to a scenario in which the threshold power is driven by the
AC Stark shift of the applied beams. Then, the threshold power would scale linearlywith detuning:
𝑃𝑡ℎ,𝑙𝑖𝑛 = 𝐴

(︂

1
2
+
|(𝜔 − 𝜔𝐷1 )| |(𝜔 − 𝜔𝐷2 )|
232

)︂−1

(6.10)

b)
Threshold power (a.u.)

Threshold power (a.u.)

a)

10 0

10 -1
-20

-10

0

10

20

10 0

10 -1
-20

Detuning from D 2 line (MHz)

-10

0

10

20

Detuning from D 2 line (MHz)Figure 6-6: Threshold power. a) Three datasets marked by three different colors. The circles
use the critical power in beam 1 and the number of atoms in the right satellite peak andthe triangles use the critical power in beam 2 and the number of atoms in the right satellite
peak. The two beams are balanced. b) Different fits: red: quadratic (Eq. (6.9)), green:
linear (Eq. (6.9)), both are weighted fits.

We next average over all datasets we have taken by scaling them to the mean
threshold power in each dataset. We believe the variations between the three datasets
are due to different starting BEC numbers, resulting in different densities, which from
Eq. (6.5) change the critical power 𝐼𝑐𝑟𝑖𝑡 ∼ 1/𝑛, where 𝑛 is the density of the cloud. The

different datasets are shown with different colors in Fig. 6-6a. We then try both fits
(Eq. (6.9) and Eq. (6.10)) to all the data, Fig. 6-6b. The quadratic fit seems better,
confirming that the critical power for the formation of the new phase is driven by acritical Rayleigh scattering rate.
We have observed a new regime in Superradiant Rayleigh scattering, in which
the emergent phase is predicted to be a supersolid. There is a marked differencein the response of the system when pumped with a single beam and when pumped
with two balanced beams. The latter shows an oscillatory behavior in the numberof atoms in the satellite peaks after a quench, compared to an exponential increasein the former. However, due to the large Rayleigh scattering rate into modes otherthan the backscattered modes, this new phase forms only transiently. Although ourobservations indicate the formation of a phase with a density modulation, we could
not unambiguously establish supersolid behavior. We have seen that the threshold
power for this phase is consistent with a critical Rayleigh scattering rate, as predicted233

theoretically. The lifetime of the new phase can be increased by increasing the density,
thus lowering the threshold power.
It would be interesting to measure the backscattered light as a means to detect
if the density modulation in the BEC is stationary and perhaps if its phase is spontaneously chosen at each realization. By interfering the backscattered light and thepump light and measuring the frequency difference, we could detect if the two are
frequency shifted (in the case of Rayleigh scattering and single-beam superradiance)or they have the same frequency (in the case of a stationary density modulation).
Finally, the excitation spectrum of a supersolid is also of interest. Due to thetwo broken symmetries, two different excitation modes (the corresponding Goldstone
modes) are expected. A phonon mode, characteristic of solids, is expected due to thebroken continuous translational symmetry and a phase mode, related to the broken
gauge symmetry. The excitation spectrum of the system we have studied has been
explored numerically in [122], predicting three excitation branches. The appearanceof the first gap at momentum 𝑞 = 𝑘eff from Eq. (6.3) is due to the formation of an
optical lattice potential above threshold power. A second gap opens up at momentum
𝑞 = 2𝑘eff . Above that momentum, a phononic excitation branch forms. Recently, theexcitation spectrum of a supersolid in a dipolar gas has been measured and two
different excitation modes detected [150].

234

Chapter 7
Outlook
We have laid out the features of the BEC 5 machine and the advantages of the7

Li platform for exploring novel quantum phases. The ability to tune between non-interacting and strongly interacting systems of particles of different states together
with the fast timescale for dynamics, open up a wide range of systems that can bestudied both with and without a lattice. We have developed a straightforward routeto a BEC with 7 Li atoms with a fast cycle time of 10 s, which has enabled us to bettercharacterize the Mott insulator as a starting point of spin physics experiments. Spin
Hamiltonians can be studied by a mapping in the the MI to the spin sector, whichassumes a homogeneous density. In order to improve the accuracy of this mapping,
we have explored new methods and approaches.
Creating a MI with a long lifetime is particularly challenging for lithium due to
its light mass. We have identified the main noise sources and decreased their effect.
Due to the large bandgaps in the optical lattice, traditional noise-canceling schemes
are not appropriate, therefore decreasing the noise of the laser source is important.
Similarly, the system is particularly sensitive to phase noise, so that floating the tableis probably necessary. The three-body loss coefficients are higher as compared to thosefor heavier atoms, so that having good control over the number of atoms per site in
the MI is essential. More importantly, we have moved on from the traditional methodof characterizing a MI (without single-site resolution) by ramping in and ramping out,
because this method does not distinguish between heating in the MI and heating in235

the rest of the system. We have found that using interaction spectroscopy to measure
the lifetime of individual MI shells is a more direct way to characterize a MI. Further,
we have developed the tilted lattice as a tool for bringing the MI system closer to thespin system we wish to simulate. In a tilted lattice superexchange-driven dynamics
stand out, because first-order tunneling is suppressed as is the effects of number
defects and the surrounding superfluid.
With this, we believe that we have characterized the main issues involved in the
mapping from the density sector description of the system with the Hubbard model tothe spin sector description with a Heisenberg model. By creating a non-equilibrium
spin distribution and studying its relaxation in the anisotropic Heisneberg spin-1/2
model, we have directly measured superexchange-mediated physics over two ordersof magnitude in the coupling parameters, making our first steps in studying spinHamiltonians. We are now in a position to realize spin-1/2 and spin-1 models and to
study nonequilibrium dynamics, adiabatic state preparation protocols, ground states,and phase transitions. Here we discuss several directions in which the 7 Li platform isprepared to go.

7.1

Transport in anisotropic spin-1/2 Heisenberg chains

The direct continuation of this work is to explore transport in the spin-1/2 Heisenberg
model. Indeed, this has been the first goal of BEC 5 since its conception, after BEC
4’s study on transport with 87 Rb [157]. We have measured different decay rates ofthe spin spiral as a function of the anisotropy 𝐽𝑧 /𝐽𝑥𝑦 . Different transport regimes
emerge, from ballistic at 𝐽𝑧 = 0 to super-diffusive at 0.5 < 𝐽𝑧 /𝐽𝑥𝑦 < 1, to diffusive at
the isotropic point 𝐽𝑧 = 𝐽𝑥𝑦 , and to sub-diffusive at 𝐽𝑧 /𝐽𝑥𝑦 > 1, where 𝐽𝑥𝑦 > 0.
To some degree, these properties can be understood if we consider the JordanWigner transformation, introduced in Section 4.1.1. There, the spin system is transformed into a system of spinless fermions on a lattice by mapping |↑⟩ → |1⟩ and|↓⟩ → |0⟩. The fermions tunnel with a matrix element 𝐽𝑥𝑦 and have nearest-neighborinteractions 𝐽𝑧 . At 𝐽𝑧 = 0 point, the system maps to noninteracting fermions. There236

fore, the ballistic transport at that point is expected. As the interactions are increased,
the transport slows down. More notably, at the isotropic point 𝐽𝑧 = 𝐽𝑥𝑦 , we measure
diffusive transport, just as in a previous study of transverse spiral relaxation with87

Rb [74]. Moreover, when 𝐽𝑧 changes sign, the transport is different. It is ballistic

almost all the way to 𝐽𝑧 /𝐽𝑥𝑦 = 1 and then becomes diffusive at large negative 𝐽𝑧 , but
it does not seem to show sub-diffusive behavior, which was the case for 𝐽𝑧 > 0.
Transport in the Heisenberg model is still an active research topic. The focus so
far has been on the high-temperature limit. For example, recent numerical studies[102] show that for a system consisting of two halves, each in a different thermal
state, transport is ballistic for 0 ≤ 𝐽𝑧 /𝐽𝑥𝑦 < 1, super-diffusive at the isotropic point

𝐽𝑧 /𝐽𝑥𝑦 = 1, and diffusive at 𝐽𝑧 /𝐽𝑥𝑦 > 1. What our measurements contribute to thediscussion is the realization of a different regime: a pure state in an isolated system,
which shows a different transport behavior. Extensions of our study could include
preparing different initial states, rather than the spin spiral, and studying the effectof holes. It is possible that holes become more important for larger anisotropies(stronger effective fermion interactions), or that a spiral with a smaller tilt angle in
the XZ plane could lead to different behavior. Finally, it would be interesting tostudy transport in higher dimensions. This can be done either by using the same spin
spiral we have created here and decreasing one or both of the other two lattice arms
to enable transport in 2D or 3D respectively, or by preparing a spiral along the 2D or
3D diagonal. These are two different systems which could potentially show different
behavior.

7.2

Adiabatic state preparation

Realizing the ground states of the model Hamiltonians is one of the prime goals ofquantum simulation. However, so far noise and heating have prevented their observation. Conceptually, there are two routes to prepare the ground states: either by direct
cooling, or by adiabatic state preparation. The latter relies on the ability to adiabatically transform one Hamiltonian with the system in its ground state to another237

Hamiltonian, whose ground state we are interested in. The two ground states canthen be connected by varying a parameter. There are two main caveats. First, thistype of state preparation is possible only if the two ground states have the same symmetries. Put another way, there have to be no level crossings between the two groundstates. Second, the tuning parameters needs to be varied slowly with respect to theband gap to the first excited state. The band gap is typically minimal at avoided
crossings. This gap sets the timescale over which the protocol is to be implemented
and it needs to be within the experimental limits.
Often, if the first condition is not satisfied, terms can be added to the Hamiltonian,in order to create an adiabatic pathway. For example, in spin-1/2 Heisenberg models,ferromagnets and antiferromagnets cannot be adiabatically connected. However, withthe addition of a magnetic field gradient, they can be linked. One way to see this,is by considering as an initial state the Z ferromagnet and the desired final state an
Z Neel state. In a similar way to creating the spin spiral, after an initial 𝜋/2 pulse,a magnetic field gradient can be applied for long enough that after a final 𝜋/2 pulse
the accumulated phase difference results in a Z Neel state.
Similarly, a way to prepare an XY-ferromagnet starting from a Z-ferromagnet
in the spin-1/2 Heisenberg chain, is to add strong ℎ𝑥 and ℎ𝑧 fields which can thenbe adiabatically turned off. This can be experimentally realized by applying an
RF coupling between the two hyperfine states, initially with a large Rabi frequency(realizing the ℎ𝑥 field) and a large detuning (realizing the large ℎ𝑧 field). The adiabatic
protocol consists of first adiabatically decreating the detuning to zero, i.e. performing
half of a Landau-Zener sweep. In a double-well description, this leaves the system ina superposition between the three triplet states. After that the Rabi frequency has tobe adiabatically brought to zero. It is in this stage that the bandgap is the smallest
and is non-zero only because the system is finite. In fact, the XY phase has no gap in
the inifinite-size system. As an initial probe, the adiabatic sweep into the XY phasecan be reversed and the remaining magnetization can be measured. A magnetic field
gradient of varying strength can be adiabatically turned on and off while the system
is in the XY phase and the results compared to the case of isolated lattice sites (deep238

lattice). Since the XY ferromagnet should be insensitive to small perturbations of
this kind, while for separate lattice sites, a spin spiral pattern should emerge, thecomparison between the two could suggest if the phase we have prepared is indeed
the XY phase.
This protocol has been considered by the Daley group. By calculating the fidelityof the final state with and without a lattice tilt, they have shown that the fidelity isdecreased by mobile holes but this effect can partially be compensated by freezing in
the holes with a tilt [45], reprinted in Appendix E. Therefore, a promising direction for
the BEC 5 machine is to pursue adiabatic state preparation schemes in the spin-1/2
system. Calculations show that for chains of 15-30 sites, the gaps are large enoughfor the protocols to be realized on experimentally feasible timescales (under 100 ms).
We note that the Z-ferromagnet is the natural state to start with, since this is the
single-species 𝑛 = 1 Mott insulator, whose initial entropy is limited only by our Mott
insulator preparation, which we have studied in detail. Moreover, this state has a gapof size 𝑈 , which is large not a function of system size.
Similar approach can be taken for preparing ground states in spin-1. There, bydefinition, the Haldane phase cannot be prepared adiabatically from a topologicallytrivial phase, like the Spin Mott or the Z antiferromagnet. This holds true for an
infinite chain where the gaps of both states close at the phase transition point. In fact,
the overlap between the Z ferromagnet and the Haldane phase has been calculated
for several lattice chains [160] and it drops significantly for systems larger than 10
sites. When this system was realized in a chain of ions, the authors saw dynamics
when they directly drive the transition [142]. They suggested adding a term to the∑︀
Hamiltonian of the form 𝐻 ′ = −ℎ 𝑖 (−1)𝑖 𝑆𝑖𝑧 , which can lead to an adiabatic pathway[38]. Finally, the Haldane phase occurs in the 1D case, while it disappears in higher
dimensions. Therefore, varying the interchain coupling can be of significant interest.
In our systems we need to first measure all three-body loss coefficients at the Haldane
point and then design a Hamiltonian which adiabatically connects to the Haldane
phase, in a way similar to adding 𝐻 ′ .
Although each of these ground states is interesting on their own right, for example
239

the XY ferromagnet can be though of as a counterflow superfluid [96], their realization
open up the possibility to study different phase transitions quenches between them.
In fact, just by changing the magnetic field, we can drive various types of transitions
between states of different symmetries. For example, we can directly drive the transition between a Z ferromagnet and a Z antiferromagnet in spin-1/2, or between the
Haldane phase and the Z antiferromagnet in spin-1.

7.3

Other directions and machine improvements

In addition to Heisenberg spin systems, the 7 Li platform with its fast dynamics andtunable interactions is suitable for both simulating other types of systems or for studying other quantum phenomena in general. For example, another goal for BEC 5 isto study non-equilibrium dynamics. Cold atoms in optical lattices provide a unique
opportunity to study a isolated systems which are not coupled to a bath, by contrast
to what is typically encountered in condensed matter systems. Therefore, the dynamics and relaxation of pure quantum states can be studied with ultracold atoms.
One avenue of exploration is how isolated quantum systems thermalize. In classicalsystems non-equilibrium states thermalize, i.e. relax to a state with thermal distribution. For isolated many-body quantum systems with nonintegrable Hamiltonians, theeigenstate thermalization hypothesis suggests that nonequilibrium quantum systems
following unitary time evolution also relax. It is a topic of recent research under what
conditions this relaxation happens and when a prethermal long-lived state is realized.
In this context, dissipation could also be added in a controlled way, exploring a different set of quantum systems. Finally, gauge fields with controllable interactions canbe engineered for applications in condensed matter and high energy physics.
Machine improvements can also lead to new capabilities. For example, a microscope can be added along realizing single site resolution. For that, the top viewport
would probably have to be replaced by one with flatness and thickness matched tothe microscope design. Also, shaping of the lattice potential could improve someexperiments. For example, we are currently interested in 1D experiments because of240

the faster dynamics and larger many-body gaps for adiabatic state preparation. However, with a symmetric 3D cubic lattice, we average over many isolated realizations of
the same system: many tubes, which also have different lengths. A smaller numberof tubes can be engineered by shaping the lattice beams, providing less averaging
and more access to individual system realizations. Also, adding blue-detuned beams
could shape the overall trapping potential, leading to more homogeneous systems.
This could also be done by using digital micromirror devices (DMD) or spatial light
modulators (SLM). However the high laser power required to make a 1064 nm opticallattice for lithium often limits the types of optical devices that can be conveniently
used for optical potential shaping.
After many years of design and improvements, the 7 Li quantum simulation platform that we have implemented now works reliably and repeatably and is ready to
study a wide range of interesting quantum systems.

241

242

Appendix A
Supplement to Chapter 3
A.1

Mean-field Bose-Hubbard model

Here we derive in details the phase diagram of the Bose-Hubbard Hamiltonian, discussed in Chapter 3, following [155, 144]. The transformation of the Hamiltonianis similar in spirit to the Bogoliubov transformation used in describing a condensed√
superfluid gas, for example, (𝑎 → 𝑎𝑖 + 𝑁 ). Here we define the superfluid order√parameter 𝜓 = ⟨𝑎𝑖 ⟩ = ⟨𝑎†𝑖 ⟩ = 𝑁 , which is assumed to be real. The transformationis then:(︁)︁𝑎†𝑖 𝑎𝑗 → ⟨𝑎†𝑖 ⟩𝑎𝑗 + 𝑎†𝑖 ⟨𝑎𝑗 ⟩ − ⟨𝑎†𝑖 ⟩⟨𝑎𝑗 ⟩ = 𝜓 𝑎†𝑖 + 𝑎𝑗 − 𝜓 2(A.1)

Which gives the Hamiltonian
˜ = −𝑡
𝐻

∑︁ †
∑︁
∑︁ †
𝑈 ∑︁ † †(𝑎𝑖 + 𝑎𝑗 )𝜓 + 𝑡
𝜓2 +𝑎𝑖 𝑎𝑖 𝑎𝑖 𝑎𝑖 − 𝜇
𝑎𝑖 𝑎𝑖
2 𝑖
𝑖
⟨𝑖,𝑗⟩

⟨𝑖,𝑗⟩

We look at the first term:
−𝑡

∑︁
⟨𝑖,𝑗⟩

(𝑎†𝑖 + 𝑎𝑗 )𝜓 = −𝑡

∑︁
⟨𝑖,𝑗⟩

(𝑎†𝑖 )𝜓 − 𝑡

∑︁
∑︁ †
∑︁
(𝑎𝑗 )𝜓 = −𝑡𝑧𝑎𝑖 𝜓 − 𝑡
𝑧𝑎𝑖 𝜓
⟨𝑖,𝑗⟩

243

𝑖

𝑖

(A.2)

and similarly for the second therm:𝑡∑︁

𝜓2 = 𝑡

∑︁

𝑧𝜓 2

𝑖

⟨𝑖,𝑗⟩

For the interaction term, we simply use the commutation relation for the raising andlowering operators [𝑎, 𝑎† ] = 1 to express 𝑎† 𝑎† 𝑎𝑎 = 𝑎† (𝑎𝑎† − 1)𝑎 = 𝑛(𝑛 − 1) for 𝑛 =𝑎† 𝑎.

We obtain a Hamiltonian which is valid on each site of the lattice and does not couple
to other sites. This significantly simplifies the analysis.
(︁
)︁
𝑈
†
˜
𝐻𝑖 = −𝑧𝑡 𝑎𝑖 + 𝑎𝑖 𝜓 + 𝑧𝑡𝜓 2 + 𝑛𝑖 (𝑛𝑖 − 1) − 𝜇𝑛𝑖
2

(A.3)

Now we can write this Hamiltonian in the Fock state basis with |𝑛⟩ atoms per site,𝑛 = 1, 2, 3, .... We set 𝜇¯ = 𝜇/𝑈 and 𝑡¯=𝑧𝑡/𝑈 :⎡√(𝑡¯𝜓 2 − 𝜇)
− 2𝑡¯𝜓0⎢ √√⎢⎢ − 2𝑡¯𝜓 (𝑡¯𝜓 2 + 1 − 2𝜇)− 3𝑡¯𝜓
⎢√⎢⎢0− 3𝑡¯𝜓(𝑡¯𝜓 2 + 3 − 3𝜇)
⎣
..
..
..
.
.
.

...

⎤

⎥
⎥
... ⎥
⎥
⎥
... ⎥
⎦
..
.

(A.4)

We truncate this at 𝑛 = 10 and diagonalize it. We then minimize the lowest energy
eigenstate with respect to the superfluid parameter 𝜓. This is plotted in Fig. A-1
for two values of 𝑚𝑢.
¯ The regions in the phase diagram where 𝜓𝑚𝑖𝑛 = 0 are the MI
regions.
We now use second-order perturbation theory to obtain the phase boundaries. We
˜ (Eq.(A.3)) into:
separate the Hamiltonian 𝐻
˜ = 𝐻 (0) + 𝑉
𝐻
𝐻 ( 0) =𝑈𝑛(𝑛 − 1) − 𝜇𝑛 + 𝑧𝑡𝜓 22 (︁)︁𝑉 = −𝑧𝑡 𝑎†𝑖 + 𝑎𝑖 𝜓

(A.5)

The eigenstates of the unperturbed Hamiltonian are Fock states |𝑛⟩ with 𝑛 atoms per
244

a)

0.7
/U=1, zt/U=0.1/U=0.5, zt/U=0.1

0.6
0.5

E0

0.4
0.3
0.2
0.1
0
0

0.5

1

1.5

2

2.5

3

3.5

Figure A-1: Obtaining the mean-field Bose-Hubbard phase diagram. Plotted is the energy
˜ (Eq.(A.3)) as a function of 𝜓. The minimum of the red curveof the lowest eigenstate of 𝐻
is at 𝜓 = 0, signifying a MI phase.

site with eigenenergies:ℰ𝑛(0) = ⟨𝑛|𝐻 (0) |𝑛⟩ =𝑈𝑛(𝑛 − 1) − 𝜇𝑛 + 𝑧𝑡𝜓 2
2

(A.6)

To second order in 𝜓, the corrections are:ℰ𝑛(2) =

∑︁ |⟨𝑛|𝑉 |𝑘⟩|2(0)(2)
𝑘̸=𝑛 ℰ𝑛 − ℰ𝑛√√
| − 𝑧𝑡𝜓(⟨𝑛|𝑘 + 1⟩ 𝑘 + 1 + ⟨𝑛|𝑘 − 1⟩ 𝑘)|2=𝑈𝑛(𝑛 − 1) − 𝜇𝑛 + 𝑧𝑡𝜓 2 − 𝑈2 𝑘(𝑘 − 1) + 𝜇𝑘 − 𝑧𝑡𝜓 2
𝑘̸=𝑛 2√√∑︁|𝛿𝑘+1,𝑛 𝑛 + 𝛿𝑘−1,𝑛 𝑛|22= (𝑧𝑡𝜓)𝑈𝑛(𝑛 − 1) − 𝜇𝑛 + 𝑧𝑡𝜓 2 − 𝑈2 𝑘(𝑘 − 1) + 𝜇𝑘 − 𝑧𝑡𝜓 2
𝑘̸=𝑛 2
)︂
(︂
𝑛
𝑛+1
2
= (𝑧𝑡𝜓)+(𝑛 − 1)𝑈 − 𝜇 𝜇 − 𝑛𝑈
∑︁

(0)

(A.7)

(2)

The energy is then ℰ𝑛 = ℰ𝑛 + ℰ𝑛 :
[︂
(︂
)︂]︂
𝑛
𝑛+1
𝑈
2
2
+
ℰ𝑛 = 𝑛(𝑛 − 1) − 𝜇𝑛 + 𝜓 𝑧𝑡 + (𝑧𝑡)2(𝑛 − 1)𝑈 − 𝜇 𝜇 − 𝑛𝑈

(A.8)

Since this expression is a parabola with respect to 𝜓, then ℰ𝑛 (𝜓 = 0) is the groundstate when the term is square brackets is positive, otherwise ℰ𝑛 (𝜓 = 0) is the highest

excited state. Therefore, the sign of the term in square brackets determines the phaseboundary between a MI with 𝜓 = 0 and a superfluid with 𝜓 ̸= 0. From this we can245get the phase boundaries:
)︂
𝑛+1
𝑛
+
=0
𝑧𝑡 + (𝑧𝑡)(𝑛 − 1)𝑈 − 𝜇 𝜇 − 𝑛𝑈
1
1 √︀ 2𝜇± = [𝑈 (2𝑛 − 1) − 𝑧𝑡] ±𝑈 − 2𝑈 (2𝑛 + 1) 𝑧𝑡 + (𝑧𝑡)2
2
2
2

(︂

(A.9)

The SF-MI transition is typically defined as the point where the upper and lower
phase boundaries of the MI meet 𝜇+ = 𝜇− at:
(︂

𝑈
𝑧𝑡

)︂

= (2𝑛 + 1) +
𝑐

246

√︀
(2𝑛 + 1) − 1(A.10)

Appendix B
Supplement to Chapter 4
The different spin models which can be studied with 7 Li are discussed in Chapter 4.
Here, the mapping to the spin-1/2 Heisenberg model is presented for a double-welland some details of the six-site system are given.

B.1

Mapping to spin-1/2 Heisenberg model

The goal is to map the two-component Hubbard model:)︁∑︁ (︁ †
∑︁†𝐻 =−𝑡𝑎↑𝑖 𝑎↑𝑗 + 𝑎↓𝑖 𝑎↓𝑗 + 𝑈↑↓𝑛↑𝑖 𝑛↓𝑖𝑖⟨𝑖,𝑗⟩𝑈↑↑ ∑︁
𝑈↓↓ ∑︁
+𝑛↑𝑖 (𝑛↑𝑖 − 1) +𝑛↓𝑖 (𝑛↓𝑖 − 1)
2 𝑖
2 𝑖

(B.1)

to the Heisenberg spin-1/2 model:
𝐻 = 𝐽𝑧∑︁

𝑆𝑖𝑧 𝑆𝑗𝑧 + 𝐽𝑥𝑦

⟨𝑖,𝑗⟩

∑︁ (︀
)︀
𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦

(B.2)

⟨𝑖,𝑗⟩

Here we consider a double well with one particle per site. Following [96], the tunnelingterm in Eq. (B.1) is treated as a perturbation:)︁(︁
𝑉 = −𝑡 𝑎†1 𝑎2 + 𝑎†2 𝑎1
247

(B.3)

The four Fock states {|↑↑⟩, |↑↓⟩, |↓↑⟩, |↓↓⟩} can be used as basis states, where |𝛼𝛽⟩ istaken to represent that there is a spin 𝛼 on site 1 and spin 𝛽 on site 2 in the double

well. These states are the eigenstates (or linear combinations of eigenstates) of the
Hamiltonian Eq. (B.2) in the limit 𝑡 = 0. The matrix elements of the Bose-Hubbard
Hamiltonian Eq. (B.1) can be calculated to second order in perturbation theory for𝑡 ≪ 𝑈 in this basis:
𝐻𝛼𝛽 =∑︁ |𝑉𝛼𝑘 𝑉𝑘𝛽 |2(B.4)

𝐸𝛼 − 𝐸𝑘𝑘where 𝑉𝛼𝛽 = ⟨𝛼|𝑉 |𝛽⟩ and |𝛼⟩ and |𝛽⟩ are two of the basis states. Note that in theunperturbed case 𝑡 = 0 the basis states have the same energy ℰ = 0. Including the
perturbation, the matrix elements are:
⎡

−4𝑡2
⎢ 𝑈↑↑

⎢
⎢ 0
⟨𝐻𝐵𝐻 ⟩ = ⎢
⎢
⎢ 0
⎣
0

0

0

−2𝑡2

−2𝑡2

−2𝑡2
𝑈↑↓

−2𝑡2
𝑈↑↓

0

0

𝑈↑↓

0

𝑈↑↓

⎤

⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎦

(B.5)

−4𝑡2
𝑈↓↓

Next, the matrix elements of the Heisenberg Hamiltonian Eq. (B.2) can be computed:
⎡

1
𝐽
⎢4 𝑧

⎢
⎢ 0
⟨𝐻𝐻 ⟩ = ⎢
⎢
⎢ 0
⎣
0

0

0

− 14 𝐽𝑧

1
𝐽
2 𝑥𝑦

1
𝐽
2 𝑥𝑦

− 41 𝐽𝑧

0

0

0

⎤

⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎦
1
𝐽
4 𝑧

(B.6)

where the spin matrices 𝑆𝑖𝛼 are defined as:
1
𝑆𝑖𝑧 = (𝑛𝑖↑ − 𝑛𝑖↓ )21𝑥
𝑆𝑖 = (𝑎†𝑖↑ 𝑎𝑖↓ + 𝑎†𝑖↓ 𝑎𝑖↑ )2𝑖𝑦𝑆𝑖 = − (𝑎†𝑖↑ 𝑎𝑖↓ − 𝑎†𝑖↓ 𝑎𝑖↑ )2
so that 𝑆𝑧 | ↑⟩ = 1/2.
248

(B.7)
(B.8)
(B.9)

We now match ⟨𝐻𝐵𝐻 ⟩ to ⟨𝐻𝐻 ⟩ element by element. However, the resulting systemof equations for 𝐽𝑧 and 𝐽𝑥𝑦 does not have a solution, so that more terms need to be
added to Eq. (B.2). We add both a longitudinal field term and an overall energy
offset:𝐻 ′ = 𝐽𝑧∑︁𝑆𝑖𝑧 𝑆𝑗𝑧 + 𝐽𝑥𝑦

⟨𝑖,𝑗⟩

1
𝐽 + ℎ𝑧 + ℰ0
⎢4 𝑧

⎢
⎢
′
⟨𝐻𝐻 ⟩ = ⎢⎢⎢⎣0

(B.10)

𝑖⟨𝑖,𝑗⟩

so that:
⎡

∑︁
∑︁ (︀
)︀
𝑆𝑖𝑧 + ℰ0
𝑆𝑖𝑥 𝑆𝑗𝑥 + 𝑆𝑖𝑦 𝑆𝑗𝑦 + ℎ𝑧

0

0

0

− 41 𝐽𝑧 + ℰ0

1
𝐽
2 𝑥𝑦

0
0

0

1
𝐽
2 𝑥𝑦

− 14 𝐽𝑧 + ℰ0

0

0

0

1
𝐽 − ℎ𝑧 + ℰ0
4 𝑧

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦

(B.11)′
Now, ⟨𝐻𝐵𝐻 ⟩ can be matched to ⟨𝐻𝐻⟩ element by element, resulting in the following

expressions for the parameters:

4𝑡2
4𝑡2
4𝑡2
−
−
𝑈↑↓ 𝑈↑↑ 𝑈↑↑
4𝑡2
𝐽𝑥𝑦 = −
𝑈↑↓
2
2𝑡
2𝑡2
ℎ𝑧 =
−
𝑈↓↓ 𝑈↑↑
4𝑡2
4𝑡2
4𝑡2ℰ0 = −−−𝑈↑↑ 𝑈↑↓ 𝑈↓↓
𝐽𝑧 =

B.2

(B.12)(B.13)(B.14)(B.15)

Six-site Hamiltonian

We diagonalize the six-site spin-1/2 Heisenberg Hamiltonian Eq. (B.2). It is block
diagonal in the basis |𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 ⟩, where 𝜎𝑖 is the spin on site 𝑖 which is |↑⟩ or∑︀ 𝑧|↓⟩. Each block corresponds to a value of the total 𝑧 magnetization 𝑆 𝑧 =
𝑆𝑖 =
𝑖
∑︀
(𝑛𝑖↑ − 𝑛𝑖↓ ), which commutes with the Hamiltonian.
𝑖

We use the following basis states which are arranged with progressively lower

magnetization. Shown are only states 1 to 42, the states 43-64 are the same as 22 to
1 but progressively higher negative magnetization (i.e. |↑⟩ ↔ |↓⟩. We note the XY
249

Figure B-1: Heisenberg spin-1/2 Hamiltonian on 6 sites for 𝐽𝑥𝑦 = 1 = 𝐽𝑧 .

phase consists of states with 𝑆 𝑧 = 0, which in this basis are the states between 23-42.
The matrix elements are plotted in Fig.B-1.

250

|23⟩ =| ↑↑↑↓↓↓⟩|24⟩ =| ↑↑↓↑↓↓⟩|8⟩ =| ↓↓↑↑↑↑⟩|9⟩ =| ↓↑↓↑↑↑⟩|10⟩ =| ↓↑↑↓↑↑⟩|11⟩ =| ↓↑↑↑↓↑⟩|2⟩ =| ↑↑↑↑↑↓⟩|3⟩ =| ↑↑↑↑↓↑⟩|1⟩ =| ↑↑↑↑↑↑⟩|4⟩ =| ↑↑↑↓↑↑⟩|5⟩ =| ↑↑↓↑↑↑⟩|6⟩ =| ↑↓↑↑↑↑⟩|7⟩ =| ↓↑↑↑↑↑⟩|12⟩ =| ↓↑↑↑↑↓⟩|13⟩ =| ↑↓↓↑↑↑⟩|14⟩ =| ↑↓↑↓↑↑⟩|15⟩ =| ↑↓↑↑↓↑⟩|16⟩ =| ↑↓↑↑↑↓⟩|17⟩ =| ↑↑↓↓↑↑⟩|18⟩ =| ↑↑↓↑↓↑⟩|19⟩ =| ↑↑↓↑↑↓⟩|20⟩ =| ↑↑↑↓↓↑⟩|21⟩ =| ↑↑↑↓↑↓⟩|22⟩ =| ↑↑↑↑↓↓⟩|25⟩ =| ↑↑↓↓↑↓⟩|26⟩ =| ↑↑↓↓↓↑⟩|27⟩ =| ↑↓↑↑↓↓⟩|28⟩ =| ↑↓↑↓↑↓⟩|29⟩ =| ↑↓↑↓↓↑⟩|30⟩ =| ↑↓↓↑↑↓⟩|31⟩ =| ↑↓↓↑↓↑⟩|32⟩ =| ↑↓↓↓↑↑⟩|33⟩ =| ↓↑↑↑↓↓⟩|34⟩ =| ↓↑↑↓↑↓⟩|35⟩ =| ↓↑↑↓↓↑⟩|36⟩ =| ↓↑↓↑↑↓⟩|37⟩ =| ↓↑↓↑↓↑⟩|38⟩ =| ↓↑↓↓↑↑⟩|39⟩ =| ↓↓↑↑↑↓⟩|40⟩ =| ↓↓↑↑↓↑⟩|41⟩ =| ↓↓↑↓↑↑⟩|42⟩ =| ↓↓↓↑↑↑⟩

251

252

Appendix C
Observation of two-beam collective
scattering phenomena in a
Bose-Einstein condensate
This appendix contains a reprint of the following paper [46]:I. Dimitrova, W. Lunden, J. Amato-Grill, N. Jepsen, Y. Yu, M. Messer, T. Rigaldo,
G. Puentes, D. Weld, and W. Ketterle, Observation of two-beam collective scattering
phenomena in a bose-einstein condensate, Phys. Rev. A 96, 051603 (2017).

253

RAPID COMMUNICATIONSPHYSICAL REVIEW A 96, 051603(R) (2017)

Observation of two-beam collective scattering phenomena in a Bose-Einstein condensate
Ivana Dimitrova,1 William Lunden,1 Jesse Amato-Grill,1,2 Niklas Jepsen,1 Yichao Yu,1,* Michael Messer,1,†
Thomas Rigaldo,1,‡ Graciana Puentes,1,§ David Weld,1,¶ and Wolfgang Ketterle1
1

Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA(Received 7 September 2017; published 22 November 2017)

Different regimes of collective light scattering are observed when an elongated Bose-Einstein condensate is
pumped by two noninterfering beams counterpropagating along its long axis. In the limit of small Rayleighscattering rates, the presence of a second pump beam suppresses superradiance, whereas at large Rayleighscattering rates it lowers the effective threshold power for collective light scattering. In the latter regime, thequench dynamics of the two-beam system are oscillatory, compared to monotonic in the single-beam case.
In addition, the dependence on power, detuning, and atom number is explored. The observed features of thetwo-beam system qualitatively agree with the recent theoretical prediction of a supersolid crystalline phase oflight and matter at large Rayleigh scattering rates.
DOI: 10.1103/PhysRevA.96.051603

Collective scattering of light can dramatically enhance
single-particle scattering and can lead to qualitatively new
phenomena. Since the pioneering work of Dicke [1], it has
been observed in various systems, including thermal atomic
and molecular gases [2–6], degenerate Bose and Fermi gases[7,8], atomic systems in cavities [9,10], solid state systems[11,12], and astrophysical systems [13]. Collective scattering
occurs when it is impossible to determine which particle
scattered a photon. As a result, correlations develop between
the particles which enhance subsequent scattering, leading
to superradiance. Due to their long coherence time, BoseEinstein condensates are well suited for studying superradiance [7]. Furthermore, the coherence between the atoms,
which is responsible for the superradiant scattering, can bedirectly observed from the momentum distributions and allows
a detailed study of superradiance without detection of theemitted light. Different regimes and geometries have been
explored by many experimental [6,14–24] and theoreticalstudies (see Refs. [25–30] and references therein). In elongatedclouds the superradiant gain of the optical mode along thecondensate axis, known as the “end-fire mode,” is largest, andone can use a single-mode approximation. If the pump laser
beam also propagates along this axis, all relevant modes ofboth the light and the atoms are aligned, allowing for a simple
one-dimensional description.

*

Present address: Department of Physics, Harvard University,
Cambridge, Massachusetts 02138, USA.
†
Present address: Institute for Quantum Electronics, ETH Zurich,
8093 Zurich, Switzerland.
‡
Present address: Université Pierre et Marie Curie (Paris 6), 4 place
Jussieu, 75 252 Paris Cedex 05, France.
§
Present address: (1) Departamento de Física, FCEyN, UBA,
Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina;(2) Instituto de Física de Buenos Aires, UBA CONICET, Pabellón 1,
Ciudad Universitaria, 1428 Buenos Aires, Argentina.
¶
Present address: Department of Physics, University of California,
Santa Barbara, California 93106, USA.
2469-9926/2017/96(5)/051603(6)Motivated by recent theoretical work [31], we have studied
the effect of illuminating a Bose-Einstein condensate by two
counterpropagating beams at the same intensity. Interference
between the two beams can be suppressed by orthogonal
polarization or rendered irrelevant by a large frequency offset.
Remarkably, we never find a regime in which the two beams
act independently, i.e., in which each of them independently
induces superradiant scattering into its corresponding directionof propagation. When the Rayleigh scattering rate into theend-fire modes is smaller than the recoil frequency ωR =h̄k 2 /2m, where m is the atomic mass and k is the wave numberof the pump light, superradiance is suppressed compared tothe single-beam case, which is readily explained by bosonicstimulation. By contrast, at high Rayleigh scattering rates,
the threshold for collective light scattering is lowered in thepresence of a counterpropagating beam. The two regimes arecharacterized by different microscopic mechanisms.
At small Rayleigh scattering rates, each scattered photon
creates a quasiparticle in the form of a recoiling atom, ora phonon, with momentum equal to the difference between
the momenta of the incoming and the scattered photons.
Each phonon mode is a moving density modulation whichis amplified by subsequent collective light scattering. In thegeometry of Fig. 1, a single pump beam triggers collective
light scattering along the condensate axis, resulting in an
enhanced number of atoms in the 2h̄k phonon mode and enhanced backscattered light. An additional counterpropagating
beam can create quasiparticles with opposite momentum orannihilate the quasiparticles created by the first beam.
When the Rayleigh scattering rate into the end-fire modes
is comparable to or larger than the recoil frequency, two or
several photons can be scattered quasisimultaneously. This
creates the possibility of new collective dynamics to emerge,
which are no longer governed by individual quasiparticle
resonances. The analysis of Ref. [31] shows that above a
critical pump power, corresponding to a Rayleigh scattering
rate on the order of ωR , the two-beam system develops atype of roton instability towards a periodic modulation andundergoes a phase transition to a crystalline phase. This
phase is a stationary situation, where an atomic density

051603-1©2017 American Physical Society

RAPID COMMUNICATIONSIVANA DIMITROVA et al.

PHYSICAL REVIEW A 96, 051603(R) (2017)

FIG. 1. Experimental geometry. An elongated Bose-Einstein condensate is exposed to two counterpropagating pump beams detuned
by 160 MHz from each other and by 1–20 GHz from the atomic
resonance at 671 nm. Collective scattering into the end-fire modes
creates two backscattered beams and a modulated atomic density
distribution. If this distribution is stationary, ω1 = ω1 and ω2 = ω2 .

modulation collectively creates two backscattered beams. Eachof them interferes with its corresponding pump beam so that
two standing waves are formed. In a self-consistent way, astationary density modulation is stabilized by the stationary
optical lattice potential, which, in contrast to usual opticallattices, consists of two different standing waves which are
created and therefore phase locked by the atoms. In contrast
to the single-beam case, in which the backscattered beam isfrequency shifted by 4ωR with respect to the incident beam
due to the 2h̄k of recoil momentum imparted to the atoms,in the crystal phase both backscattered beams have the same
frequency as their corresponding pump beams because theatomic density distribution is stationary. This establishes a newsupersolid form of matter by spontaneous crystallization oflight and atoms. For a sudden turn on of the pump beams, there
should be an oscillatory behavior around the new equilibriumphase, in contrast to the exponential growth and eventual gain
depletion of single-beam superradiance, as in Ref. [7].
In this Rapid Communication, we characterize the regimesof low and high Rayleigh scattering rates by determining
the threshold power for collective light scattering and bymonitoring the time evolution of the atomic momentum
distribution both for a single beam and for two beams. We
observe qualitatively different behaviors, including oscillatory
dynamics for two pump beams at high Rayleigh scattering
rates. Our observations are consistent with the predictions of
the crystal phase using the one-dimensional (1D) theoreticalmodel [31]. However, the experimental system is three dimensional (3D) and its lifetime is severely limited by Rayleighscattering into free space. Therefore, the crystal phase canonly form transiently and cannot reach equilibrium.
Experiments are performed with a new 7 Li machine which
produces Bose-Einstein condensates of typically 4 × 105
atoms in 10 s. Atoms from an effusive oven are laser cooledwith a spin-flip Zeeman slower and 5 × 109 atoms are captured
in a 3D magneto-optical trap operated on the D2 line. Aftera compression step, further sub-Doppler cooling is performed
using gray molasses on the D1 line, as in Ref. [32], which
reduces the temperature to 90 μK. Dark-state optical pumping
on the D1 line prepares the atoms in the magnetically trappable|F = 2,mF = 2 ground state. They are then transferred to aquadrupole magnetic trap with a repulsive “plug” optical beam

used to inhibit losses from the center of the trap [33]. During
rf evaporation the atomic density is kept at 1 × 1013 cm−3 bygradually opening up the magnetic trap to prevent strong losses
due to three-body recombination. The negative scattering
length of the |F = 2,mF = 2 state prevents the formation oflarge stable condensates [34]. The evaporation is terminatedjust before degeneracy is reached and the atoms are transferred
to a 1064-nm optical dipole trap. They are spin flipped to thelowest hyperfine |F = 1,mF = 1 state, which has a Feshbachresonance at 737 G [35]. The scattering length is tuned toabout 125aB , where aB is the Bohr radius, and the atoms areevaporated to degeneracy.
For the current experiment, a magnetic field is chosen to
realize a scattering length of 15aB to avoid strong scattering
between atoms in different momentum states. We obtained acloud of typical size of about 20 μm × 120 μm by releasing
it from a crossed optical dipole trap and letting it expand into asingle-beam dipole trap. A tunable Ti:sapphire laser generates
671-nm light detuned by 1–20 GHz from atomic resonance.
The two pump beams have e−2 radii of 140 μm and propagate
along the long axis of the condensate. Interference between
them is suppressed on experimentally relevant time scales byoffsetting them in frequency by 160 MHz using two acoustooptic modulators. This frequency offset is large enough toeliminate Raman coupling between momentum sidebands,ω  ωR , and small enough so that the recoil momenta ofthe two beams are indistinguishable, h̄k  h/L. Both beams
have the same polarization and drive a π transition. Rectangular pump pulses are applied, after which the trap is suddenly
switched off and momentum distributions are recorded after
ballistic expansion. Momentum distributions are then characterized by the number of atoms in the satellite peaks, which
are separated from the main cloud by recoil momentum.
The onset of collective light scattering is studied bymeasuring the number of atoms in the 2h̄k peak in time offlight as a function of laser power (Fig. 2). All three plotson the right-hand side show that there is an effective criticalpower for the onset of superradiance. For long pump times[Figs. 2(a)–2(d)], the thresholds are lower, corresponding tosmall Rayleigh scattering rates. For high pump powers (large
Rayleigh scattering rates), the dynamics occurs already on
short time scales and the critical powers are lower for two
beams than for a single beam [Figs. 2(e) and 2(f)]. By contrast,
at low Rayleigh scattering rates, collective light scattering is
suppressed in the presence of two beams. This is best seen in
Figs. 2(c) and 2(d): At the same powers, at which the system exhibits strong superradiance with a single pump beam, there are
no recoiling atoms visible in the presence of two pump beams.
For Rayleigh scattering rates smaller than the recoil
frequency [Figs. 2(a)–2(d)], a quasiparticle picture can be used
to describe the onset of superradiance (for discussions, see,e.g., Refs. [7,24,25,28]). Recoiling quasiparticles are createdby Rayleigh scattering into the end-fire mode, which occurs at
a rate of Ref = RN0 f , where R is the total Rayleigh scattering
rate per atom, N0 is the number of atoms in the condensate,
and f is the effective solid angle for scattering into theend-fire mode, approximately given by λ2 /D 2 , where λ is
the wavelength of the scattered light and D is the diameter ofthe condensate. This rate is enhanced via bosonic stimulation
by a factor N1 + 1 with N1 atoms with recoil momentum 2h̄k051603-2

RAPID COMMUNICATIONSOBSERVATION OF TWO-BEAM COLLECTIVE SCATTERING . . .
(a) 1 ms

8 (b) 1 ms(a)

10-4 Atom number in one peak

6
4
2

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10

(e) 3 sAtom number in one peak0(c) 50 s

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PHYSICAL REVIEW A 96, 051603(R) (2017)

0

0.2

0.4

0.6

(d) 50 s
6
1 beam2 beams

4
2
0
0

1

2

3

1 beam2 beams8(b)

N0 =(2.24 0.22) 10

5N =(1.82 0.20) 105
0

2
6
4

1

2
0

0
4 (c)

P/P0 =2.3
P/P =1.7
0

3

P/P0 =1.2

4 (d)

2

1

1

0

2

4

/ 0 =1.0
/

3

2

0

(f) 3 s

3

0
6 0

0

=0.57

/ 0 =0.33

2

4

6

4

Pump time (units of Rt)

2

FIG. 3. Time dynamics of collective light scattering for two
beams in the large Rayleigh scattering rate regime. Each beam is
at power P which is above the effective threshold power P0 at blue
detuning δ from the 7 Li D2 line. (a) Comparison of the single-beampump to the two-beam pump. In both cases, the single-beam poweris the same and P /P0 = 1.2, where P0 = 2.8 mW at δ = 9.9 GHz.
(b) For different initial condensate numbers (P /P0 = 1.8, where
P0 = 8.6 mW at δ = 17 GHz). (c) For different pump powers [P0 andδ are the same as in (a)]. (d) For different detunings but at constantRayleigh scattering rate, which was measured for a single beam ineach case to be 1.7 × 104 s−1 , which corresponds to P /P0 = 2.3.
Here, δ0 = 9.9 GHz. Solid lines are a guide to the eye.

0
0

5

10

15

Power in a single beam (mW)

FIG. 2. Observation of superradiant light scattering in different
regimes. Shown are momentum distributions after 2-ms time offlight for three different pump times and different pump powers at
18.62 GHz red of the 7 Li D2 line. For each time, the single-beamcase with pump propagating from the left (top picture, triangles) is
compared to the case of two balanced beams from opposite sides(bottom, circles). The strength of collective light scattering vs laser
power is characterized by the number of atoms with momentum 2h̄k(right peak). Solid lines are a guide to the eye. The images are taken
at the powers indicated by the dashed vertical lines.

already present. The recoiling atoms are lost from the system
at a rate L, either by collisions or because they move out of
the condensate volume.
The resulting rate equation describes both the threshold
and the initially exponential gain for the case of a single pumpbeam [7],dN1= Ref (N1 + 1) − LN1 = Ref + (RN0 f − L)N1 . (1)dt
For weak pump beams and negligible source depletion,
one would expect, at least in the perturbative regime, that theaddition of a second counterpropagating pump beam would
trigger superradiant scattering into the opposite direction.
However, due to bosonic stimulation, the rates of scattering
into the end-fire modes are proportional to the number ofatoms in the initial and the final states. For the case of
two counterpropagating beams which can transfer equal but
opposite momenta, the stimulated scattering rates, which areresponsible for superradiance, cancel in the rate equation
dN1= R1 N0 f (N1 + 1) − R2 N1 f (N0 + 1) − LN1
dt(2)

if the single-particle scattering rates are equal (R1 = R2 ). The
remaining terms simply reflect spontaneous scattering and loss
L as described above. A similar equation can be written forthe −2h̄k atoms.

Complete suppression of superradiance for the two-beamcase is observed for pump times on the order of tens of recoil
times ωR−1 = 2.5 μs [e.g., 50-μs case in Figs. 2(c) and 2(d)].
For even smaller pump powers and therefore longer pumptimes, as in the 1-ms data, the suppression is incomplete. Thisis possibly due to effects of decoherence, or defocusing of
the recoiling atoms by atom-atom interactions, which could
begin to have an effect after long pump times. For detailed
experimental studies of the behavior of the single-beam system
at low Rayleigh scattering rates, see Refs. [7,15,16,18–21].
When the Rayleigh scattering rate into the end-fire modes
becomes on the order of the recoil frequency ωR , thequasiparticle picture can no longer be used. In this regime
the system displays the opposite behavior compared to thelow scattering rate regime. The presence of the second beam
lowers the apparent threshold power for nonzero momentum
atoms to appear in time of flight [see Fig. 2(f)]. In addition,
the time dynamics of the system differs qualitatively forthe single-beam and the two-beam cases (Fig. 3). When thetwo beams are suddenly turned on, we observe temporaloscillations of the number of atoms with nonzero momentum.
This was predicted by Ref. [31] as oscillations around an
equilibrium crystal phase. By contrast, with a single pumpbeam the number of recoil atoms grows continuously until the
Bose-Einstein condensate gets depleted and the gain decreases,
as shown in Fig. 3(a), in agreement with the predictions of
Eq. (1). The frequency of the oscillations is on the orderof the recoil frequency ωR . The amplitude decays with time
due to the loss of atoms to Rayleigh scattering into modes

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PHYSICAL REVIEW A 96, 051603(R) (2017)

(a)

100 m

P = 4.37 mW

8Atom Number(b)Threshold power (mW)P = 2.33 mW6

P = 7.86 mW

104
4
2
0
0

5

10

Power (mW)

4
2
0

-20

-10

0

10

Detuning from D2 line (GHz)
FIG. 4. (a) Time-of-flight images at three different powers at 3-μs
pump time, 2 ms of ballistic expansion, and at 7 GHz blue of theD2 line. The power noted is the power in each beam. (b) Effective
threshold power for the onset of collective light scattering with twopump beams. Each point is obtained from a two-line fit to the numberof atoms in the 2h̄k peak after 3-μs pump time as a function of power,
as shown in the inset. The error bars are one standard deviation of thethreshold fit. The blue line is a quadratic fit for the threshold power vseffective detuning: P0 = Aδ̃ 2 with A being a single free parameter.

other than the end-fire modes. We observe distinct nonzero
momentum peaks in time of flight with two beams for about
12 μs at an inverse Rayleigh scattering rate of about 30 μs.
The amplitude and period of the oscillations depend strongly
on the initial number of condensate atoms [Fig. 3(b)]. Thisis characteristic of collective light scattering effects, e.g.,
in collective spontaneous emission the peak intensity of theemitted light scales with N 2 , where N is the number ofscatterers [3]. Sample oscillation curves for different initialcondensate numbers as well as different loss rates have been
numerically calculated in Ref. [36] for a 1D two-beam system.
The frequency and the amplitude of the oscillations increase
with power in the pump beams, as shown in Fig. 3(c). When
the Rayleigh scattering rate is kept constant and the detuning
is varied, oscillations with a larger amplitude are observed at
higher detunings, as evidenced in Fig. 3(d). This is consistent
with the optical dipole potential rather than Rayleigh scattering
governing the dynamics, as in the model used to describe thecrystal phase in Ref. [31]. A constant Rayleigh scattering
rate R  requires the power to be scaled with detuning δ
as P  ∝ R  δ 2 , so that the ac Stark shift and, therefore, thepotential depth ∝P  /δ ∝ δ [37].
In addition to the different time dynamics, the single-beamand the two-beam cases also differ in the shape of the observed
momentum peaks [Fig. 4(a)]. For the short times of flightused in this experiment, the observed density distributions
within each momentum peak still reflect the in-trap density
distribution. The shape of the ±2h̄k peaks follows closely theelongated shape of the condensate, while the shape of the 2h̄kpeak for a single beam is shorter and more rounded. This canbe explained by the inhomogeneous intensity distribution of

the backscattered end-fire beam along the condensate. The
power of the end-fire mode is largest where the pump beam
enters the cloud, increasing the generation of recoiling atoms
via Bragg scattering. In contrast, the combined potential of
the two stationary optical lattices is predicted to be almost
homogeneous along the condensate [31]. At even higher
powers [see the rightmost column in Fig. 4(a)], there isa backward peak of atoms with −2h̄k momentum in the
one-beam case. It is the result of the second-order processof rescattering of a backscattered photon. This is the KapitzaDirac regime, commonly associated with pump times smaller
than the recoil time [38]. We also observe momentum peaks
at 4h̄k due to higher-order superradiance as studied earlier [7].
In all cases, the two-beam geometry shows patterns distinctly
different from those obtained by adding up the peaks for thetwo one-beam geometries, indicative of a different mechanismof collective light scattering.
We have also studied the detuning dependence of theeffective threshold power P0 for collective light scattering[Fig. 4(b)]. Due to the fine structure splitting in the excitedstate (10 GHz), the effective detuning δ̃ for a pump beam at
frequency ω is given by δ̃ −1 = 2/(ω − ωD2 ) + 1/(ω − ωD1 ),where ωD1 and ωD2 are the frequencies of the D1 and D2 lines
and the coefficients reflect the relative strength of the dipole
matrix elements for π -polarized light. The observed thresholds
are consistent with P0 ∝ δ̃ 2 , i.e., the onset requires a critical
Rayleigh scattering rate. This agrees with the threshold power
predicted for the crystal phase in Ref. [31]. However, due toscatter in the data, the experimental results would also agree
almost equally well with a linear fit P0 ∝ δ̃, i.e., the onset isdriven by a critical ac Stark shift. Note that in between theD1 and D2 lines optical pumping to other hyperfine states
limits the gain into the end-fire modes. For the single-beam
pump, this leads to Raman superradiance, as observed inRefs. [39,40].
We have so far emphasized the qualitative differences
between the one- and two-beam cases. Those cases can beconnected by using two beams with imbalanced intensities.
In the large Rayleigh scattering limit, we observe that the
number of atoms in both satellite peaks initially goes throughan oscillation, but eventually the stronger beam wins over:With pump time, one momentum peak grows in number and theother one decreases. Reference [36] has investigated the phase
diagram of the imbalanced system, showing that there is asimilar instability to self-organization for all values of the beam
asymmetry. Further studies are needed to explore this regime.
In this work, we have observed a self-organization effect of
atoms coupled by light. Related effects have been observedwith cold atoms in optical cavities (see Ref. [41] for areview). Superradiant light scattering into a single-cavity mode
gives rise to only two distinct density modulations, i.e., acheckerboard phase which breaks Z2 symmetry. The system
studied here is in free space and the density modulation whichforms breaks a continuous U (1) symmetry, which constitutes
a supersolid phase. In fact, a system with two cavities also
realizes a supersolid with the spontaneous breaking of acontinuous symmetry [42]. The present system is conceptuallysimpler, but the study of the crystal phase is currently limitedby the large Rayleigh scattering rates into free space relative
to the scattering into the end-fire modes.

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PHYSICAL REVIEW A 96, 051603(R) (2017)

In conclusion, we have studied collective light scattering ofan elongated Bose-Einstein condensate when pumped with
one or two noninterfering beams. In the regime of small
Rayleigh scattering rates compared to the recoil frequency, aquasiparticle picture of the scattering explains the suppressionof superradiance in the presence of two beams. In the regimeof large Rayleigh scattering rates, the behavior of the twobeam system is qualitatively different and consistent with
the incipient formation of the predicted crystal phase. The
superradiant gain and, as a consequence, the lifetime of thecrystal phase could be increased by increasing the atomic
density or decreasing the cloud diameter D while keeping theFresnel number F ∝ D 2 /L0 λ on the order of one, where L0
is the length of the condensate (see Ref. [2] for a discussion).
This would allow the study of a supersolid formed by collective
light scattering, which features a spontaneously chosen phaseof the atomic density distribution and the emergent opticallattice. In particular, it would be interesting to confirm thepredictions that the backscattered light is not recoil shifted
and that the phase of the density modulation is spontaneouslychosen.

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We would like to thank H. Ritsch, S. Ostermann, and F.
Piazza for fruitful discussions and G. Siviloglou for help andadvice during the initial stages of building the 7 Li machine.
We acknowledge support from the NSF through the Centerfor Ultracold Atoms and from Award No. 1506369, from
ARO-MURI Non-equilibrium Many-body Dynamics (Grant
No. W911NF-14-1-0003), from AFOSR-MURI Quantum
Phases of Matter (Grant No. FA9550-14-1-0035), from ONR
DURIP (N00014-16-1-3141), and from an ARO seedling
grant. J.A.G. acknowledges support by the National Science Foundation Graduate Research Fellowship under Grant
No. DGE 1144152.

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RAPID COMMUNICATIONSIVANA DIMITROVA et al.

PHYSICAL REVIEW A 96, 051603(R) (2017)for our experimental conditions, the single-beam case does not
display oscillations. Numerical 1D simulations typically predict
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Appendix DInteraction spectroscopy of a
two-component Mott insulator
This appendix contains a reprint of the following paper [11]:J. Amato-Grill, N. Jepsen, I. Dimitrova, W. Lunden, and W. Ketterle, Interaction
spectroscopy of a two-component mott insulator, Phys. Rev. A 99, 033612 (2019).

261

PHYSICAL REVIEW A 99, 033612 (2019)
Editors’ Suggestion

Interaction spectroscopy of a two-component Mott insulator
Jesse Amato-Grill,1,2 Niklas Jepsen,1 Ivana Dimitrova,1 William Lunden,1 and Wolfgang Ketterle1
1

Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, and Department of Physics, Massachusetts Institute ofTechnology, Cambridge, Massachusetts 02139, USA
2
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA(Received 19 September 2018; published 18 March 2019)
We prepare and study a two-component Mott insulator of bosonic atoms with two particles per site. The
mapping of this system to a magnetic spin model and the subsequent study of its quantum phases require a
detailed knowledge of the interaction strengths of the two components. In this work we use radio-frequency
transitions and an on-site interaction blockade for precise empirical determination of the interaction strengthsof different combinations of hyperfine states on a single lattice site. We create a map of the interactions of thetwo lowest hyperfine states of 7 Li as a function of magnetic field, including measurements of several Feshbachresonances with exceptional sensitivity, and we identify promising regions for the realization of magnetic spinmodels.
DOI: 10.1103/PhysRevA.99.033612
I. INTRODUCTION

Ultracold atoms in optical lattices, described by a Hubbard
Hamiltonian, are a uniquely accessible platform for the studyof quantum magnetism. A dual-component Mott insulator
with n atoms per site, in which each of the components stands
in for a magnetic spin, implements a spin-n/2 Heisenberg
model with nearest-neighbor interactions [1]. With first-ordertunneling suppressed by on-site interactions, only exchanges
between sites that preserve the overall density distribution are
possible. This superexchange of particles mediates effective
spin-spin interactions [2], and in analogy to the spin system,
the ground state will be determined by ratios of the on-site
intra- and interspecies interactions, which can be varied bymeans of a state-dependent optical lattice [3] or by using
Feshbach resonances [4]. The model with a single particle persite, corresponding to spin 1/2, has been studied extensively
in many regimes; recent successes include observation of
three- and two-dimensional Néel ordering in fermions [5,6],
measurements of spin correlations in two-dimensional spinimbalanced systems [7], and spin-charge correlations in thepresence of hole doping [8].
In this work we focus on a two-component spin-1 bosonicmodel implemented using the two lowest hyperfine states of7
Li (hereafter a and b) in a cubic optical lattice. Integer-spin
models remain largely unexplored using cold-atom systems
and yet they are predicted to exhibit many interesting behaviors arising from both magnetic ordering and beyondmean-field effects due to topological considerations [9–11].
With two particles per site, the three√ states of our model, | +1 = |aa, |0 = (|ab + |ba)/ 2, and | − 1 = |bb, forma spin triplet manifold on each lattice site.1 As in the
spin-1/2 system, the nearest-neighbor interactions arise fromsuperexchange. Because of Feshbach resonances, the three1
Hereafter, for compactness, we will refer to these three states as|aa, |ab, and |bb, using the Fock basis notation.

2469-9926/2019/99(3)/033612(5)possible configurations have on-site energies that are a function of the applied magnetic field B, so one can tune theconfigurational energies of different distributions of spins in
the lattice (Fig. 1). Quantum phase transitions are expected inregimes where the nearest-neighbor interactions compete withon-site interactions. Determining the relevant regions of B foran exploration of the spin-1 Hamiltonian therefore requires
precise knowledge of the relative strength of the interactions
among the hyperfine states.
II. EXPERIMENTRadio-frequency spectroscopy has been used in the past tomeasure site occupancy in a single-species Mott insulator of

|bb
1
√
(|ab
2

ωbb→ab+ |ba )|aa

ωaa→abB
FIG. 1. The three different combinations of two hyperfine states
on a lattice site have interaction energies Uaa , Uab , and Ubb which canbe tuned via Feshbach resonances. At a given field B, the splittings
will in general be unequal, giving rise to an interaction blockade: The
two possible transitions can be individually addressed by an rf drivewith frequency ω = ωZeeman + U/h̄.

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PHYSICAL REVIEW A 99, 033612 (2019)

Magnetic Field

Atom Number (a.u.)

n=1

n=2

847.0 G
847.4 G
847.7 G
-2
0
-4
rf Detuning (kHz)-6

848.1 G2

848.3 G
848.5 G849.5 G
849.9 G
850.9 G
861.0 G
868.1 G
871.1 G
874.6 G
877.7 G
881.5 G

-80

-60

-40

-20

0

20

40

60

80

rf detuning (kHz)FIG. 2. Fits to representative spectra of the transition between b
and a, taken between the two bb Feshbach resonances. The spectra
have been plotted so that the Zeeman-shifted peaks of the n = 1
transitions overlap. The inset shows an example spectrum in which
each point is an average of four measurements and the fit is a sumof two Gaussians. The crossing of the frequencies of the two peaks
corresponds to Uab − Ubb = 0.

Interaction energy (kHz)has been used to drive singlon-to-doublon conversion in thelowest Hubbard band, and the resonant frequency of thisprocess has been associated with the on-site interaction energy
in both repulsive [16] and attractive [17] single-component
bosonic systems. Here we employ the same technique to map
the on-site intraspecies interactions of 7 Li across a broad
range of magnetic fields. We modulate the lattice depth by

50

(a)

Uab - U aa
Ubb - U ab

0

-50
400

Scattering length(units of a0)

ultracold bosons [12]. Here we extend this technique to measure the differential interaction energy of two confined atoms
as a function of magnetic field. We begin by preparing an n =2 Mott insulator of 7 Li in a single hyperfine state in a 1064nm cubic optical lattice [our apparatus and Bose-Einsteincondensate (BEC) production are described elsewhere [13].
The number of atoms is 1 × 105 and the lattice depth is 35ER
in each dimension. The central n = 2 plateau, approximately
3 × 104 sites, is surrounded by an n = 1 shell containing asimilar number of sites. We pulse the rf drive for 2.9 ms andmonitor the number of atoms in the other hyperfine state as afunction of drive frequency. At the frequency corresponding
to the transition of a bare atom, we observe a peak comingfrom the atoms on n = 1 sites. We observe a second peakfrom the n = 2 atoms, which is shifted by the difference ininteraction energy between the initial state (|aa or |bb) andthe final state (|ab). The pulse length corresponds to a π pulse
for the n = 2 sites (so√that we maximize the signal), whichhave a Rabi frequency 2 greater than that of the n = 1 sites,
due to bosonic enhancement. The interaction blockade [14]that arises from unequal interaction energies in the three statesmeans that one may drive the system selectively between |aaand |ab (or |bb and |ab). Thus when we probe the atoms
absorptively after an rf pulse with light that is resonant only
for a (or b), we measure a single flipped atom per n = 2 site.
The frequency of the n = 1 peak corresponds to the Zeeman shift and thus to the magnitude of the applied magnetic
field (the hyperfine constant and nuclear g factor for 7 Li aretaken from [15]). The frequency shift of the n = 2 peak,which may be positive or negative, is a direct measure of
the differential two-body interactions. Using this technique,we obtain rf spectra at many selected bias fields from which
we derive the two-body interaction splittings Ubb − Uab andUab − Uaa (Fig. 2). The technique works equally well for
attractive and repulsive interactions, so long as the systemremains in the Mott insulating state.
The precision to which we must determine the differential
interaction energies on a site as an input to a many-body
physics model is fixed by the superexchange rate, which at
Mott insulator depths for 7 Li in a 1064-nm optical lattice
ranges from hundreds of hertz to several kilohertz, dependingon lattice depth and dimensionality. Here we measure thedifferential interaction energies as a function of magnetic field
to a precision of about 100 Hz, limited only by the stabilityof our magnetic field (better than one part in 105 at 103 G)
and the difference in magnetic moments of the two hyperfine
states (approximately 33 kHz/G). As these interaction energies range over many tens of kilohertz, the error bars are too
small to see on a full scale plot [Fig. 3(a)]. This technique isparticularly well suited to atoms with characteristically large
interaction energies, such as Li, because the wide separation
between singlon and doublon spin-flip resonances permits theuse of high Rabi frequencies, which maximizes signal size anddecreases sensitivity to magnetic-field noise.
While rf spectroscopy in a lattice is a powerful and precise
tool for characterizing differential interactions, another technique is necessary to measure the absolute interaction energy(i.e., Uaa or Ubb ), for these interactions determine the lattice
depth for the transition to the Mott insulator in each hyperfine
state. In previous studies, lattice amplitude modulation (AM)

(b)

aaa

200

aab
abb

0

-200
-400
650

700

750

800

850

900

950

Feshbach field (G)FIG. 3. (a) Relative interactions Uab − Uaa and Ubb − Uab plotted
as a function of magnetic field, measured via rf interaction spectroscopy in a 35ER × 35ER × 35ER optical lattice. (b) Scattering
lengths of the aa and bb interactions plotted as a function of magnetic
field, measured using lattice AM and shown in units of the Bohr radius a0 . Also shown as a bold dashed line is the ab scattering length,
obtained using simultaneous hyperbolic fits to the rf spectroscopy
and lattice AM data sets. The fits are shown as solid black lines.
Dotted vertical lines indicate the position of a resonance.

033612-2

INTERACTION SPECTROSCOPY OF A TWO-COMPONENT …

PHYSICAL REVIEW A 99, 033612 (2019)

TABLE I. Parameters of the Feshbach resonances in the two lowest hyperfine states of 7 Li, determined with a simultaneous fit to rf
interaction and lattice AM spectroscopy data (except where specified otherwise). The reported errors in our measurements are 1σ statisticaluncertainties in the fit parameters. The AM data slightly bias the resonance positions derived from combined fits towards higher magnetic
fields, possibly due to unaccounted-for systematics. As the rf data are influenced only by two-body on-site effects rather than by many-body
physics, the resonance positions are likely captured more accurately by the rf data alone.
abg/a0 (G)Bres (G)(rf data only)

Bres (G)(combined fit)Bres (G)(previous works)aa− 25.8(1.2)− 135.9(6.9)

737.58(10)738.29(15)738.2(2)a737.8(2)b736.97(7)c

ab
bb
bb

− 29.8(1.3)− 23.0(1.4)− 23.0(1.4)− 90.5(4.0)− 14.9(0.9)− 172.7(10.0)

794.64(07)845.42(01)893.34(12)794.59(12)845.45(02)893.84(18)Channel

844.9(8)a893.7(4)aaReference [19].
Reference [20].
cReference [21].
b30% peak to peak along the shallow dimension of a 35ER ×
35ER × 20ER optical lattice and measure the entropy added
to the system by adiabatically ramping back to the BEC from
the Mott insulator and measuring the recondensed fraction.
While we must remain in the Mott insulator in regions ofmagnetic field where the scattering length is negative in orderto prevent the collapse of the atomic cloud, data taken above
but close to the transition for either attractive or repulsive
interactions display a bias towards higher frequency [16].
While this systematic bias prevents the collection of data forvery small scattering lengths, limiting the accuracy of ourdetermination of the background scattering length in eachchannel, the locations of the resonances themselves are notsignificantly affected.
We perform a simultaneous fit to both the rf differential
interaction spectroscopy data and the lattice AM spectroscopy
data in order to determine the interspecies interaction energy
Uab and to extract parameters of the Feshbach resonances wedetect [see Fig. 3(b)]. We take the form of the scattering lengthin each state to be a hyperbola (i)
as = abg 1 −(1)(i)
i B − Bres
with a background scattering length abg , width (i) , and res(i)onance location Bres
. In order to provide a useful comparison
with existing literature, we calculate the scattering lengths

1
4π h̄2 1=(2)|ψ (r)|4 d 3 r,asm U
where U is the on-site interaction energy we measure andψ (r) is the calculated Wannier wave function on a site,
given calibration of the lattice depth by interband parametric
excitation. This approximation for ψ systematically biases thescattering lengths towards lower values because interactions
modify the actual two-particle wave function by admixing
higher bands, but √the correction is not significant as long as|as |/aHO  V 1/4 / 2π , where aHO is the harmonic-oscillator
length on a site and V is the lattice depth in recoil units[18]. For the moderate scattering lengths considered here,
we remain more than one order of magnitude below thisthreshold. The complete Feshbach resonance spectrum for thetwo lowest hyperfine states of 7 Li is plotted in [Fig. 3(b)] andthe parameters of the resonances can be found in Table I.
III. RESULTS

We find a single resonance in the lowest hyperfine state,whose position is in good agreement with the most recent measurements made using rf spectroscopy of molecular binding energies [19], although previous measurements
made using the modification of in-trap condensate size dueto the mean-field energy of interactions have reported theresonance at slightly lower magnetic fields [20–22]. We alsofind a single resonance between the a and b states, previously
unmeasured, whose parameters are particularly relevant forstudies of two-component systems. Of interest is also the
double resonance in the b state, studied previously using threebody atom loss [23,24] and also rf spectroscopy of molecularbinding energies [19]. Compared to previous techniques, rf
interaction spectroscopy allows for the exploration of theinterspecies resonance. The method is as precise as rf spectroscopy of molecular binding energies, but does not require
knowledge of the molecular potential in order to extract the
scattering lengths.
The region between the two bb resonances provides anopportunity to vary the interaction energy of a |bb site withrespect to that of an |aa or |ab while keeping the latter two
relatively constant. We even find a point, near 849 G, where
2Uab = Uaa + Ubb(3)

so that the transitions from |aa to |ab and |ab to |bb occur
at the same frequency. The interaction blockade vanishes andit is possible to rotate around the entire spin-1 Bloch sphere.
This point is characterized by Rabi oscillations with twice
the amplitude(i.e., full contrast on all doubly occupied sites)√but 2 lower Rabi frequency than when the two transition
frequencies are different and the rf drive is resonant with only
one of them (Fig. 4). Moreover, for two sites connected bytunneling, the degeneracy condition (3) also means that thesuperexchange process corresponding to | + 1L | − 1R ⇒|0L |0R becomes resonant. This degeneracy implements a
special point in the spin-1 Heisenberg model: It is the point
where the spin-spin interactions are isotropic and where the033612-3

PHYSICAL REVIEW A 99, 033612 (2019)

(a)

2
1
0

(b)

2
1
0
0

1

2
Time (ms)Atoms/site detected in state a

JESSE AMATO-GRILL et al.

starting point for adiabatic state preparation, in analogy to
the band insulator in fermions [3]. Full diagnostics of doubly
occupied sites can be realized using transitions to a third hyperfine state, by selectively measuring the number of doublonsin different configurations of hyperfine states. We therefore
expect this method to prove useful in studies of stronglyinteracting multicomponent cold-atom systems, especiallywhen the relative interaction strengths can be modified with
Feshbach resonances or with a spin-dependent lattice.
IV. CONCLUSION

3

FIG. 4. Rabi oscillations of doublons (a) exactly at and (b) far
away from the magnetic field at which the interactions are degenerate, as in (3). The measured atom number is normalized to the total
number of n = 2 sites. The system is initially prepared in state b
and we detect the total number of atoms in state a after applying
a resonant rf drive. Decaying sinusoidal fits determine the Rabi
frequencies to be √1.68(4) and 2.30(1) kHz, respectively, consistent
with the expected 2 ratio in Rabi frequency between resonant threelevel and two-level systems. The time constant for the decoherenceaway from degeneracy is 10.2 ms. The oscillations in (a) seem todecay faster because these data were taken at the closest magnetic
field to the degeneracy point which is permitted by the present
resolution limit of our magnetic-field set point, so we ultimately
observed beating between two nearly equal but off-resonant Rabi
frequencies.

We have developed a technique to precisely determine
the relative on-site interaction energies of different configurations of two-component bosons in an optical lattice and havedemonstrated that technique using 7 Li over a broad range ofmagnetic fields. We have improved the precision parametersof three previously observed Feshbach resonances and report
the observation of an interspecies Feshbach resonance in
7
Li. The identification of a magnetic field which provides
for a degeneracy of differential interactions paves the way
for future investigation of a spin-1 bosonic system, which
is predicted to include spin-ordered phases such as an XY
antiferromagnet, a Z paramagnet, and topologically protected
phases such as the Haldane phase.
ACKNOWLEDGMENTS

on-site anisotropy, which biases the system towards local
pairing, completely vanishes.
As much as rf interaction spectroscopy in a lattice enablesprecise measurements of scattering lengths, it is also a tool
for state preparation and diagnostics. Starting with an n = 2
Mott insulator in a single hyperfine state, one can prepare thefully paired state |ab on every doubly occupied lattice siteby means of a π pulse or a Landau-Zener sweep. Figure 4(b)demonstrates coherent preparation of this fully paired state,or spin Mott state, which has a large gap and is a promising

We would like to thank Junru Li and Wenlan Chenfor critical readings of the manuscript. We acknowledge
support from the NSF through the Center for Ultracold
Atoms and Award No. 1506369, from ARO-MURI NonEquilibrium Many-Body Dynamics (Grant No. W911NF14-1-0003), from AFOSR-MURI Quantum Phases of Matter (Grant No. FA9550-14-1- 0035), from ONR (Grant No.
N00014-17-1-2253), and from a Vannevar-Bush Faculty Fellowship. J.A-G. acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant
No. DGE 1144152.

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033612-5

Appendix E
Enhanced Superexchange in a Tilted
Mott Insulator
This appendix contains a reprint of the following paper [45]:Ivana Dimitrova, Niklas Jepsen, Anton Buyskikh, Araceli Venegas-Gomez, Jesse
Amato-Grill, Andrew Daley, and Wolfgang Ketterle. Enhanced superexchange in atilted Mott insulator, arXiv:1908.09870 (2019)267

Enhanced Superexchange in a Tilted Mott Insulator
Ivana Dimitrova,1 Niklas Jepsen,1 Anton Buyskikh,2 Araceli Venegas-Gomez,2Jesse Amato-Grill,1, 3 Andrew Daley,2 and Wolfgang Ketterle1
1
Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms,
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
3
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA(Dated: December 19, 2019)

In an optical lattice entropy and mass transport by first-order tunneling is much faster than spintransport via superexchange. Here we show that adding a constant force (tilt) suppresses first-ordertunneling, but not spin transport, realizing new features for spin Hamiltonians. Suppression of thesuperfluid transition can stabilize larger systems with faster spin dynamics. For the first time ina many-body spin system, we vary superexchange rates by over a factor of 100 and tune spin-spin
interactions via the tilt. In a tilted lattice, defects are immobile and pure spin dynamics can bestudied.

The importance of spin systems goes far beyond quantum magnetism. Many problems in physics can bemapped onto spin systems. Famous examples are the
Jordan-Wigner transformation between spin chains andlattice fermions, and the mapping of neural networks
to Ising models. The study of spin Hamiltonians has
provided major insights into phase transitions and nonequilibrium physics. Therefore, the properties of wellcontrolled spin systems are explored using various platforms [1].
In the field of ultracold atoms, such Hamiltonians are
realized by a mapping from the Hubbard model in the
Mott insulating (MI) state to Heisenberg models witheffective spin-spin coupling given by a second order tunneling process (superexchange) [2, 3]. Although immense
progress has been made towards the realization of spinordered ground states [4–7], a major challenge is to reachlow spin temperatures. A promising route is adiabatic
state preparation [8], but in a trapped system a higher
entropy region surrounds a low-entropy MI core, whose
ultimate temperature and lifetime is limited in most casesby mass or energy transport. A fundamental limitationof superexchange-driven schemes is that the lattice depth
controls both mass transport (occuring at the tunnelingrate t/~) and the effective spin dynamics (at t2 /(~U ),where U is the on-site interaction). Schemes isolating
the MI by shaping the trapping potential have been proposed [9–13].
Here we use a controlled potential energy offset between neighboring sites (a tilt) to decouple spin transport
from density dynamics in the MI regime. Tilted lattices
have been used before to suppress tunneling (in spin-orbit
coupling schemes with laser-assisted tunneling [14–17]),or to implement spin models using resonant tunnelingbetween sites with different occupations [18–21]. Energy
offsets have been used in double-well potentials to modify
superexchange rates [22], between sublattices to suppress
first-order tunneling and to observe magnetization decay
via superexchange [23].

X
X

FIG. 1. In a tilted lattice with energy offset per site ∆, tunneling at t/~ is suppressed, while superexchange at J(∆)/~ isstill allowed. This enables the slower superexchange processes
to dominate the dynamics even in systems with defects.

The implications of using an off-resonant tilt for studying spin physics fall into four categories: (i) A tailoreddensity distribution can be chosen which is frozen-in by
the tilt. (ii) The tilt suppresses the transition to a superfluid (SF). We use these two features to stabilize larger
MI plateaus at lower lattice depths. (iii) The sign andmagnitude of the superexchange interaction can be tuned
with the tilt which allows access to a larger range of magnetic phases. (iv) In a tilted MI with n atoms per site,
number defects (n ± 1) are localized. This turns t-J models [24] into spin models with static impurities and allows
the study of pure spin dynamics.
In a tilted lattice, the energy difference between lattice sites prevents first-order tunneling. More precisely,
the dynamics of a single particle are Bloch oscillations[25, 26] and if the tilt per site ∆ is larger than the bandwidth, their amplitude is smaller than a lattice site. In
contrast, swapping particles incurs no energy cost, preserving superexchange (Fig. 1), but with a modified matrix element. For n = 1 it is [22]:


4t2 1
1
1
J(∆) =+(1)U 2 1 − ∆/U
1 + ∆/U
where tunneling resonances at ∆ = U/m (m = 1, 2, 3...)[21] should be avoided. We implement the tilt with an
AC Stark shift gradient from a far-detuned 1064 nm laser
beam, offset by a beam radius from the sample. We loada 7 Li Bose-Einstein condensate [27] into a 3D 1064 nm

2
a)
0.3

Critical depth Vc (ER)Fraction of atoms in n=2=0= 1.65U

0.2

10
5
0

0.1

0

Coherently admixed doublon fraction

100

200

300

Scattering length (a 0)

10

b)

20
Fraction of atoms in n=2

optical lattice in the MI regime. Although the tilt canbe applied in any direction, here we use a tilt only along
one axis of the lattice and study 1D dynamics (see [28]).
(i) Preparing large non-equilibrium MI plateaus. In
most optical lattice experiments, the number of atoms(and therefore the signal-to-noise ratio of measurements)
is not determined by the number of available atoms fromthe cooling cycle, but by the available laser power (andtherefore beam size) for the optical lattice. This determines the harmonic confinement potential at each lattice
depth. The equilibrium size of a MI plateau with n atoms
per site is determined by the balance between the local
chemical potential µ ≈ n U and the harmonic trapping
potential. Its radius r ∝ µ1/2 , so the total atom number
N ∝ U 3/2 , where U is controlled by the scattering lengtha via a Feshbach resonance [29]. We find that the n = 1
MI plateau loaded at a = 300 a0 has an order of magnitude more atoms than the one loaded at a = 50 a0 (seeFig. S3 in [28]). We initialize the experiment by loading
45,000 atoms at a = 300 a0 at a lattice depth V = 35 ER
in a pure n = 1 MI with diameter of 40-45 sites and thenfreeze in this distribution by applying a tilt with a 300 µs
linear ramp, much faster than ~/t = 28 ms. This allows
the decoupling of MI state preparation from further spin
experiments, which could be carried out at very different
scattering lengths and lattice depths.
(ii) Increasing the speed of superexchange. The speedof superexchange is proportional to t2 and therefore increases dramatically at lower lattice depths. Due to competing heating and loss processes, most experiments on
spin physics are carried out at lattice depths only slightly
above the SF-MI transition. The melting of the Mott
plateaus at the transition can be suppressed by a tilt
and spin Hamiltonians can be studied at lattice depths
even below the phase transition. Next, we experimentally
determine how much the lattice depth can be lowered.
We associate the breakdown of the initial MI plateau
with the appearance of doublons (two atoms per site[30]), which are measured by interaction spectroscopy.
We transfer only atoms in n = 2 sites to another hyperfine state by using the interaction-shifted transition frequency, so that atoms in n = 1 sites are not affected [29].
After loading, we decrease the scattering length, lower
the lattice depth Vz along the direction of the tilt while
keeping Vx = Vy = 35 ER , and hold for 10 ms. We detectdoublons by ramping Vz back to 35 ER on a timescale∼ ~/t but slower than ~/U , so that there is local (butnot global) equilibrium. The fraction of atoms on n = 2
sites at a = 50 a0 is shown in Fig. 2a. Below a critical lattice depth Vc , a sharp increase in the number of doublons
is observed. Without the tilt Vc = 11.7 ER , while with atilt of ∆ = 1.65 U , Vc = 7.3 ER , implying an increase in
the superexhchange rate from Eq. (1) by a factor of 5 at
the critical depth in the tilted lattice.
To generalize this result, we repeat the measurement
at several scattering lengths (inset of Fig. 2a). All Vc are

30

0.3

0.15

slow ramp-up
fast ramp-up

0.2
0.1

0.1

0

10

20

30

Lattice depth (E R)

0.05

0
10

20

Lattice depth (E R)

30

FIG. 2. Stabilization of large Mott plateaus at small lattice
depths. a) Fraction of atoms in doubly occupied sites as afunction of lattice depth Vz measured with (gold) and without (blue) a tilt at a = 50 a0 . Inset: Critical lattice depth
Vc (dashed lines in main plot) below which the fraction of
atoms in n = 2 is more than 3% above the noise floor. Two
initial density distributions are used: (i) nonequilibrium: ann = 1 plateau prepared at a = 300 a0 (circles) and (ii) equilibrium: an n = 1 plateau prepared at the final scattering length(triangles). The dotted line corresponds to the SF-MI transition. b) Virtual and real doublon populations at a = 50 a0
and ∆ = 1.65 U . The fraction of coherently admixed doublonsis the difference between the doublons from the fast and theslow ramp, shown in the inset. Solid line: probability of doublon admixture. The shaded region accounts for tilt inhomogeneity. The dashed lines corresponds to Vc . The negativevalues of the coherent doublon fraction are an artefact in the
breakdown regime.

above the threshold for the SF-MI transition because of
the spatial shrinking of the equilibrium Mott plateaus ina harmonic trap [31, 32]. Without the tilt, Vc is determined by the proximity to the SF-MI transition and thebreakdown of plateaus is driven by first-order tunnelingin the single-band approximation. Note that global density redistribution is not responsible for the breakdown inthis measurement, as indicated by the fact that when thelattice is loaded at the final scattering length, so that little or no density redistribution is expected, we see similar
Vc (triangles in Fig. 2a).
With the tilt, this melting can be suppressed and we
observe that Vc is decreased, resulting in faster spin dynamics. However, we observe that we cannot stabilize

3

hi,ji

hi,ji

where hi, ji denote nearest neighbors, Siα are spin matrices, and Jz and Jxy are the superexchange parameters (see [28]). Similarly to [36, 37] (our preparation is
described in [28]), we create a spin pattern and study
its relaxation. Using π/2 pulses and a pulsed magnetica)102
Contrast

Lifetime (ms)

103

101

1
0.8
0.6
0.4
0

20

10

Time (ħ/Jxy)

10

0

5

10

15

20

Lattice depth (E R)b) 102Lifetime (ms)the Mott plateaus by tilts for lattice depths smaller than
Vc ≈ 6.3 ER which we interpret as a breakdown of thesingle-band approximation. We find Vc to have only aweak dependence on tilt for the range of tilts used (0.3
to 0.9 ER ). Note that at this lattice depth, the bandgap
is 3 ER , and the width of the first excited band is 1.6 ER .
At a somewhat lower lattice depth of 4 ER , we observethat atoms are accelerated out of the lattice, a clear signfor the breakdown of single-band physics. In cubic 2Dand 3D lattices, the motion separates in x, y, and z andthe effective breakdown of the single-band approximation
should be independent of dimension. Assuming that thelattice can be lowered to 6.3 ER in 3D, then at a = 100 a0
where the SF-MI transition is at Vc = 13.3 ER , superexchange can be 50 times faster at ∆/U = 1.4, where J/~
in Eq. (1) is the same as for ∆ = 0.
The tilt suppresses the real population of doublons, responsible for the breakdown of MI plateaus,
but not the virtual ones (coherent doublon admixtures), responsible for superexchange. In leading order in perturbation theory in t, the n = 1 MI groundstate has doublon-hole admixtures with probability
P = 2t2 /(U − ∆)2 + 2t2 /(U + ∆)2 . These admixtures aretaken into account by a unitary transformation whichleads to the effective spin Hamiltonian acting on the unperturbed states [3, 33]. As perturbation theory breaks
down when t and U become comparable, the distinction
between real and virtual doublons is blurred. Virtual
doublons have been detected without a tilt in [34, 35].
We measure the number of coherently admixed doublonhole pairs as the difference between all doublons (measured with a lattice ramp-up faster than ~/U , projecting
the wavefunction onto Fock states) and the real doublons(incoherent doublons, measured with a slow, locally adiabatic ramp-up as in Fig. 2a). Fig. 2b shows that the presence of the tilt does not inhibit this coherent admixture,but only modifies its probability. At Vc perturbation theory breaks down.
(iii) Tuning the Heisenberg parameters with a tilt.
Tilts comparable to U tune the strength and sign of thesuperexchange interactions (Eq. (1)). This effect has so
far only been observed for two particles in a double-well[22]. Here we demonstrate it for the first time in a manybody system by measuring the relaxation dynamics of anonequilibrium state in a spin chain.
A spin-1/2 Heisenberg model [2] is implemented using
the lowest two hyperfine states of 7 Li in a high magnetic
field
X
X

H = JzSiz Sjz + Jxy
Six Sjx + Siy Sjy
(2)

101
0

0.5

1

1.5

2

Tilt ( /U )FIG. 3. Relaxation of a non-equilibrium spin pattern by superexchange, controlled by the tilt. a) Lifetime as a function of lattice depth Vz along the tilt for ∆ = 1.65 U↑↓ . Inset:
decay of the contrast of the |↑i state for 6 ER ≤ Vz ≤ 17 ER .
The lifetimes are obtained from exponential fits with an offset. b) Lifetime as a function of applied tilt at Vz = 12 ER .
The solid lines in both subfigures are A ~ /Jxy (∆), with one fit
parameter: a) A = 7.54 ± 0.31 b) A = 6.54 ± 0.34. The dottedline indicates the region where the single-band approximation of the Hubbard model breaks down due to resonances of∆ = U↑↑ , U↑↓ , U↓↓ .

gradient, a spiral spin pattern is created resulting in asinusoidal (cosinusoidal) variation of the z (x) projectionof the magnetization, which is a superposition of manyspin waves (magnons), and is therefore not an eigenstate.
The spiral has a pitch of 11.5 µm, and about two periods
fit within the cloud. We measure the relaxation of the
spiral by imaging the decaying contrast of the real-spacedensity distribution of |↑i atoms on a CCD camera (with
4 µm resolution) in the presence of a tilt ∆ = 1.65 U .
We first show that the tilt does not inhibit superexchange. To simplify the interpretation, we pick a magnetic field of 848.1014 Gauss at which Jz = 0 and the
dynamics are solely determined by Jxy = J from Eq. (1)with U = U↑↓ . The inset in Fig. 3a shows the decay of thecontrast at several lattice depths, which collapse onto asingle curve when the time is rescaled by ~/Jxy . This confirms, over a range of more than two orders of magnitude(0.015 kHz < Jxy /~ < 2.68 kHz), that the spin relaxation
is driven by superexchange. We note that the contrast decays to a long-lived offset, which is larger at higher temperatures probably due to the presence of holes. Also, theoffset in the spin-density modulation is smaller than the4observed offset in the contrast since our imaging methodenhances small contrasts. The dependence of the relaxation time and the offset on parameters of the system,
such as the anisotropy of the Heisenberg model, the pitchof the spiral and temperature will be addressed in a future study.
We now demonstrate the modification of the superexchange rate with tilt. In general, changing the strength of
the tilt also changes the ratio Jz /Jxy , which determines
the nature of the dynamics and the ground state. For example, when ∆ > U , the sign of the Heisenberg parameters can be flipped (see Eqs. S6-S7 in [28]), making it possible to go between ferromagnetic and antiferromagnetic
coupling. Here we pick a magnetic field of 857.0052 Gaussat which U↑↑ = − U↓↓ so that Jz /Jxy = − 1 is constant as
a function of tilt. Then, varying the tilt only changes
the speed of the dynamics and not the nature of the
Hamiltonian. Fig. 3b shows that the relaxation times
can be tuned by the tilt by an order of magnitude(0.067 kHz < Jxy /~ < 0.605 kHz).
(iv) Freezing in defects. A direct consequence of theabsence of first-order tunneling in a tilted MI is that
defects, which normally propagate at a rate ∼ t/~, arefrozen in. Here we illustrate the different effects of mobile
and immobile holes and doublons on the spin transportof a single |↑i atom in a chain of |↓i atoms. We numerically simulate the evolution of the two-component
Bose-Hubbard model (see [28]) for three inital states after tunneling is suddenly switched on. When there are no
defects (Fig. 4a,d), the dynamics are the same with andwithout the tilt. The time evolution of spin |↑i shows coherent ballistic expansion of the wavefront with a characteristic checkerboard pattern [38], akin to the dynamics
of a single particle in a non-tilted lattice [26]. The effectof mobile holes (Fig. 4b) is to displace the particles without impeding the overall dynamics significantly, which
was also observed for antiferromagnetic chains [39]. Somecoherent oscillations appear blurred and are restored by
the tilt. In the tilted case, the holes act as domain walls,
confining the dynamics to a shorter chain (Fig. 4e).
The effect of doublons is more subtle. Compared toholes, the presence of |↓↓i doublons enables the formation of an |↑↓i doublon, so that the |↑i spin can propagate at t/~. Note that due to Bose enhancement, an|↑↓i doublon quickly turns into a |↓↓i doublon. Fig. 4cshows that the |↑i spin is localized near the original position by collisions with |↓↓i doublons, which we suspectis due to destructive interference of all paths. With thetilt, doublons are pinned and act as reflective barriers.
The superexchange rate for spin |↑i to become part of adoublon is J2 = 2t2 [−2/∆ + 2/(2U↑↓ + ∆)], which is different for ±∆ and leads to the left-right asymmetry in
Fig. 4f. The effects of fixed and mobile defects in higher
dimensions will be somewhat different, but overall, mobile defects can have a significant effect on spin dynamics,
while immobile defects act, to a good approximation, as

a)

b)

c)

d)

e)

f)

FIG. 4. Effect of holes and doublons on the superexchange
dynamics of an |↑i spin in a chain of |↓i spins. Plotted is theprobability distribution of the |↑i spin as a function of time forthree initial states without (top row) and with (bottom row)
the tilt: no defects (left), two holes (middle), two doublons(right). Here ∆ = 1.25 U , U is spin independent, and the parameters are chosen so that the superexchange rate J/~ is thesame as in the case with no tilt.

domain walls or static impurities. This has implications
not only for dynamics, but also for adiabatic state preparation where the tilt prevents defects from increasing thefinal entropy (see Fig. S5 in [28]).
The implementation of tilts for heavier atoms, shouldbe less demanding since similar tilts (in units of recoil
energy) require lower laser power. Magnetic tilts are also
possible if the two spin states have the same magnetic
moment. Separation of spin and mass transport could
also be achieved with random offsets implemented withbichromatic lattices or laser speckle, as in the studies ofAnderson localization [40, 41].
We have introduced tilted lattices as a new tool withpractical and fundamental applications. On the practical side, we have shown that it can lead to an order ofmagnitude larger systems with spin-spin couplings which
are an order of magnitude faster. On the fundamentalside, the tilt can change not only the speed of superexchange, but also the anisotropy of Heisenberg models. It
also turns t-J models with mobile holes into spin systems with pinned impurities. This can be used to create lattices with disorder, similar in spirit to disorder in
species-dependent lattices created by pinning the second
species [42], and to study mixed-dimensional transport in2D systems with tilt along one axis [43]. The separation
between spin and density dynamics should be useful for
future quench experiments and for improving the fidelityof adiabatic preparation of magnetically-ordered groundstates.
We acknowledge support from the NSF through the5Center for Ultracold Atoms and Award No. 1506369,
from ARO-MURI Non- Equilibrium Many-Body Dynamics (Grant No. W911NF- 14-1-0003), from AFOSRMURI Quantum Phases of Matter (Grant No. FA955014-1- 0035), from ONR (Grant No. N00014-17-1-2253),
and from a Vannevar-Bush Faculty Fellowship. Part ofthis work was done at the Aspen Center for Physics,
which is supported by National Science Foundation grant
PHY-1607611. We acknowledge support from the EPSRC Programme Grant DesOEQ(EP/P009565/1), andby the EOARD via AFOSR grant number FA955018-1-0064. Results were obtained using the EPSRC
funded ARCHIE-WeSt High Performance Computer(EP/K000586/1).

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Supplemental material
REALIZATION OF TILTED POTENTIALSThe tilt is implemented by an AC Stark shift gradientacross the cloud. We use a 1064 nm beam, far off detuned
from the 671 nm D-lines of lithium so that the two hyperfine states used as spin states feel the same potential gradient. Using an optical beam makes it possible to switch
the tilt on and off suddenly (within 1 µs) with an acoustooptic modulator and to change the axis along which thetilt is applied. An alternative approach is to use a magnetic field gradient, which provides for better alignment
stability and homogeneity of the tilt across the sample.
However, the use of a magnetic field gradient can make it
more difficult to implement large tilts, due to geometrical and power constraints (for lithium 150 Gauss/cm areneeded for a tilt of 10 kHz/site), and also to make thetilts the same for all states used as pseudo-spins, due todifferential magnetic moments.
The generalization to 2D and 3D tilts is straightforward except that care has to be taken to avoid resonant
second-order tunneling (see also [44]). These processes
arise when nearest-neighbor sites have the same potential energy offset. For example, when the tilt is alongthe (1,1) direction in a 2 D lattice, all sites connected bya line parallel to (1,-1) have the same potential energy
and a two-step tunneling process can take place, competing with superexchange. This issue can be resolved
by missaligning the tilt from the (1,1) direction.

CALIBRATION OF THE TILTThe tilt beam is a Gaussian beam with 73 µm 1/e2
radius. After preparing an n = 1 MI in a 3D lattice of
35 ER , 1D dynamics are realized by lowering the latticealong the tilt direction to 12 ER and keeping the othertwo lattices at 35 ER . The calibration is done by thenadiabatically ramping up the tilt across the ∆ = U resonance. As a result, doublons are formed on every othersite. This reproduces the experiments in [19, 20]. Fig. S1shows the number of atoms in n = 2 sites as a function oftilt power. We fit a phenomenological function
f (x) = A tanhx − x0
w



+c

(S1)

and identify the center x0 of the transition with ∆ = U .
For slower ramp speeds we see a second resonance at half
the power, which we interpret as ∆ = U/2.

7
a)

6000

10

Tilt (kHz/site)
15

20

12

= U (Watts)

5

13

25

11

Number of atoms in n=2

5000

Tilt at which

10

4000
3000
2000

8
7
-80

1000
0

2

4

6

8

10

12

Tilt beam power (Watts)

b)

14

-60

-40

-20

0

20

40

60

-60

-40

-20

0

20

40

60

Displacement ( m)60
50

FIG. S1. Calibration of the tilted potential. Plotted are thenumber of atoms in the n = 2 MI shell (with lattice depths(Vx , Vy , Vz ) = (35, 35, 12) ER ) after an adiabatic ramp of thetilt as a function of the final strength of the tilt beam in
Watts. After identifying the center of the transition with U,
we obtain the power of the tilt in unites of kHz/site (top axis).

Inhomogeneity (%)

0

9

40
30
20
10

ALIGNMENT AND INHOMOGENEITY

0-80

We repeat the calibration measurement as a functionof the distance between the MI and the center of the tilt
beam. The center x0 and width 2.2 w/x0 of the fitted
function f (x) from Eq. (S1) are shown in Fig. S2. For aperfect Gaussian beam
2

2g(z) = A e−2(z−z0 ) /σ0

Displacement ( m)FIG. S2. Strength and inhomogeneity of the tilt (at lattice
depths (Vx , Vy , Vz ) = (35, 35, 12) ER ). The extracted a) center
point x0 and b) width 2.2 w/x0 as a function of the displacement of the tilt beam from the atoms.

(S2)

we expect that the tilt is maximized when the beamis z0 = ±σ0 /2 = ± 36.5 µm away from the cloud. However, due to imperfections in the beam, we find it atz0 = + 36.3 µm and z0 = − 49.8 µm. see Fig. S2.
The potential across the cloud is determined by thesum of the AC-Stark shifts of the tilt beam combinedwith the lattice beams. Since they are all Gaussian, theircurvatures lead to an inhomogeneity of the tilt per site
across the cloud. In fact, for our geometry, the majorityof the inhomogeneity comes from the lattice beams, eachwith a 1/e2 radius of 125 µm. In principle, for a given
tilt beam power, it is possible to find a displacement of
the tilt beam which leads to a cancellation of the curvature of the lattice and the curvature of the tilt beam at
the position of the cloud. We characterize the inhomogeneity by the width of the region over which doublons
form when we perform the calibration measurement in
Fig. S1. We define the width as the region over which thefit function f (x) goes from 10 % to 90 % of the asymptotic values. We note that the point of minimum width
does not exactly coincide with the point of maximum
tilt, as shown in Fig. S2, and that for lower powers of thetilt beam, the inhomogeneity increases, which is probably due to the partial cancellation of the lattice and tilt
beam curvatures. The inhomogeneity can be decreasedby using larger beams, which is an option for heavier
atoms for which the laser power is not so limiting as forlithium. For our experiments, we use a displacement of
z0 = + 45 µm to minimize the inhomogeneity of the tilt.
CHOICE OF TILT VALUE

For most of the experiments, we pick a tilt per site of∆ = 1.65 U . This choice avoids resonant tunneling andformation of doublons at ∆ = U/m for tunneling m sites
away. The most prominent such resonance is at ∆ = U
and tunneling of up to 5 sites away has been observed[21]. Due to the inhomogeneity of the tilt across thecloud of 10-15 %, we pick a ∆ > U to avoid any resonances
within the cloud. The scaling of the superexchange ratewith tilt x = ∆/U is


1
1
1
h(x) =+(S3)
2 1+x 1−x

8√For x > 1, the sign of h(x) is flipped. For 1 < x < 2, the√magnitude of superexchange is increased. For x > 2
the superexchange rate decreases, for example to 50 % at
x = 1.75, implying that the most useful range of applicability of the tilt is between 1 < x < 2 and any points x < 1that avoid resonances.

π/2 pulse rotates the spiral into the xz plane, after which
the magnetic field gradient is turned off. This results in aspiral with wavelength λs = h/(∆µB 0 T ) = 11.5 µm where
T is the evolution time.
SPIRAL IMAGE ANALYSIS

atoms (a.u.)

LOADING LARGE MI PLATEAUS
1.2

Number of atoms in n=2

4

103
50 a 0100 a0300 a0

0.6

3

0.4

2

0.2

1
0
103

1

0.8

200 a0

Average number of spin |5104

Initial BEC number

105

FIG. S3. Size of n = 1 MI plateaus. Plotted is the number ofatoms in the n = 2 MI shell as a function of initial BEC number (with condensate fraction of more than 95 %). This is ameasure of the size of the n = 1 MI plateau at each scattering length given the trapping potential, which in this case isdetermined solely by the curvature of the lattice beams. The
dashed lines indicate the maximum number of BEC atoms
which fit in the n = 1 MI shell at a scattering length of 50 a0
and of 300 a0 .

To determine the maximum atom number for the n = 1
MI shell, we probe the formation of the n = 2 MI shellby loading successively more atoms at each scattering
length and measuring the number of doubly-occupied
sites: Fig. S3. The smaller the scattering length, thesmaller the number of atoms that fill the n = 1 MI plateau
for a harmonic trapping potential. The maximum scattering length which can be used is the one for which thethree body loss rate becomes comparable to the inverse
lattice loading time.
PREPARATION OF THE SPIN SPIRAL

The spin spiral is created by turning on a magnetic
field gradient of 50.8 Gauss/cm and then quickly applying a π/2 pulse to rotate the spins from the | ↑i to the(| ↑i + | ↓i) /2 state on each site. During the next 550 µs
of free evolution, the spins precess by a different amounton each site because of the magnetic field gradient andthe differential magnetic moment of 31 kHz/Gauss between the | ↑i and the | ↓i states at 882 Gauss. Another

0

-20

-10

0

10

Distance along the cloud ( m)20

FIG. S4. Fitted density distribution from a single image ofspin | ↑i atoms after averaging along the horizontal direction.

We analyze the spin spiral by imaging one of the two
spin states. Fig. S4 shows the spin density distributionof the | ↑i atoms. We take 5 images and after averaging
along the direction perpendicular to the stripe pattern,
we fit them simultaneously with a 1D function of the
form:
2
2
a
sj (x) = [1 + C sin(Qx + φj )] e−(x−x0 ) /2w0
(S4)
2
where only the phase φj is allowed to vary from image
to image. Here Q is the spiral wavevector, C is the contrast, a is an overall scaling factor, and x0 and w0 are the
midpoint and the width of the Gaussian envelope respectively. The phase variation comes from magnetic field
fluctuations which are currently 1 × 10−5 at 880 Gauss.
Once we have calibrated the wavelength of the spiral asa function of evolution time, we also fix the wavevector
Q in the fit. We extract the contrast C and rescale it to
the initial contrast in Fig. 3. The value of the starting
contrast is limited by our imaging system.
HEISENBERG MODEL

The Heisenberg Hamiltonian is derived from the twocomponent Bose-Hubbard model with one particle persite as in [2].
XXH = JzSiz Sjz + Jxy
Six Sjx + Siy Sjy
(S5)
hi,ji

hi,ji

9
matrices Siα are defined as
andSix = (a†i↑ ai↓ + a†i↓ ai↑ )/2,

where the spin
Siz = (ni↑ − ni↓) /2,

Siy = − i(a†i↑ ai↓ − a†i↓ ai↑ )/2. The coefficients are:
4t2 1
Jz =
U↑↓ 2
4t2 1
−
U↑↑ 2

4t2 1
−
U↓↓ 2
Jxy = −

4t2 1
U↑↓ 2








1
1
+
1 − ∆/U↑↓
1 + ∆/U↑↓

1
1+
1 − ∆/U↑↑
1 + ∆/U↑↑




(S6)



(S7)

1
1
+
1 − ∆/U↓↓
1 + ∆/U↓↓

1
1+
1 − ∆/U↑↓
1 + ∆/U↑↓

NUMERICAL SIMULATIONS

For simulating the dynamics of spin |↑i in a chain of
spin |↓i, we use the two-component Bose-Hubbard model:
H =−t
+


X †
X
a↑, i a↑, j + a†↓, i a↓, j + Un↑, i n↓, i
i

hi,ji

UX
2

−∆

i

X
i

(n↑, i (n↑, i − 1) + n↓, i (n↓, i − 1))i (n↑, i + n↓, i )(S8)where a†σ, i and aσ, i are the creation and annihilation
operators of spin σ on site i, t is the tunneling matrix
element, U is the on-site interaction, and ∆ is the tilt.
Here U↑↑ = U↑↓ = U↓↓ = U . We perform quench simulations from initial product states with and without a tilt
to investigate the effects of holes and doublons on the
superexchange dynamics.
In particular, we use a one-dimensional lattice of 11
sites with three initial states: (i) unit filled MI state of| ↓i particles with a single | ↑i particle on site 6; (ii) we
replace the | ↓i particles on sites 2 and 10 with holes; (iii)
we replace holes with | ↓↓i doublons. We time-evolve theinitial state either with no tilt ∆ = 0 or with a tilt tuned
close to the resonance ∆ = 1.25. Calculations were performed using the open-source Python package for exact
diagonalization and quantum dynamics QuSpin v0.3.2[45, 46].
ADIABATIC STATE PREPARATION WITH A
TILTOne of the important potential applications of preventing transport of holes and doublons is in isolating entropy,
especially when the entropy is initially dominant at theoutside of a harmonic trap. For example, if we begin witha Mott Insulator plateau in the center of the system, we
can adiabatically prepare interesting spin-ordered states
within that plateau [8, 47]. However, this situation iscomplicated if propagation of entropy into the Mott Insulator plateau (as holes and doublons) is allowed during
the adiabatic ramp. In the presence of a tilt, these become pinned defects, as described in the main text. This
will generally isolate entropy at the edge of the system,
and in the worst case it will generally lead to a break forholes, or weak coupling for doublons in the spin chain.
A full analysis of this situation for a general thermal
state would be an interesting next direction. As a simple
demonstration, here we consider adiabatic state preparation in the presence of a single hole, with and without
a tilt. We aim to begin with all spins aligned with thex-axis, and then prepare an XY Ferromagnet within the
Hamiltonian of Eq. (S5) [48]. This can be accomplishedby switching on an RF field that couples the spins, initially far detuned from resonance, and then tuning thefield into resonance. The difficult part of the adiabatic
process is the removal of this field, which is what wemodel here. The modified Bose-Hubbard Hamiltonian isHa (τ ) = H − Ω(τ )

X

Six ,

(S9)

i

where H is the Heisenberg Hamiltonian (S5) and Ω(τ )is the time-dependent external field, which we decreaseduring the adiabatic ramp in the following waydΩ(τ )∼ ∆E(Ω(τ )),
dτ

(S10)

where ∆E is the energy gap of the Hamiltonian (S9)computed numerically.
In Fig. S5, we present example results, beginning with
spins aligned along the x-axis, and either no holes ora single hole, which can either begin in the center ofthe system (localized), or begin in a superposition of L
states where the hole occupies different sites with equal
amplitude (delocalized). In order to account for holes wecompute the adiabatic ramp with the tunneling terms aswell as the linear tilt terms
X †H̃a (τ ) =Ha (τ ) − t(a↑,i a↑,j + a†↓,i a↓,j )
−∆

X
i

hi,ji

i(n↑,i + n↓,i ),(S11)where the notation is the same as in Eq. (S8).
When the hole is pinned by the tilt, it creates spin
chains of shorter length. As an indicator of the effects
just on the adiabatic state preparation, we consider correlation functions at each possible separation distance
where we remove any contributing state where a hole has

10which neglects the contributions from holes. HereX
Ni,j =|ck |2 F (i, j, k)(S13)Spin correlation function Kd

0.3

k

P
is the renormalization matrix for |ψi =k ck |ki expanded in the Fock basis {|ki} and the numerator
X(S14)
c∗k0 ck F (i, j, k)hk 0 |Si+ Sj− |kiCi,j =

0.1

0.05

k,k0

1

2

3

4

5

6

7

is the conditional correlation matrix. For both functions
the condition is defined asDistance d = |i - j|

F (i, j, k) = 1, iff
FIG. S5.
Averaged over distance correlation function
Kd = h|Ki,i+d |ii of the adiabatically prepared state of 10
sites with the following parameters: Jz /Jxy = 1/5, t/Jxy = 5,Ω(0)/Jxy = 50, τmax Jxy = 500. The results are presented for
3 configurations of the initial state with and without tilt ∆.

broken the chain. This gives a renormalized conditionalcorrelation matrix

Ki,j =

Ci,j
,
Ni,j

(S12)

j
X
l=i

hk|nl |ki = |i − j| + 1,

(S15)

which takes into account only states |ki that do not haveholes between sites i and j.
As shown in Fig. S5, in the absence of holes, the tilt
does not have an effect on the correlations and they decay following the expected algebraic law. When holes
are added, the correlations are suppressed without a tilt,
more prominently so when the hole is localized. This effect of mobile holes becomes significantly weaker in thepresence of the tilt which pins the holes, restoring thestrength of the correlations. For a localized hole (in the
middle of the chain), the tilt effectively splits up the chain
in two parts and prevents the build-up of correlations between them, which leads to suppression of correlations at
distances longer than half the chain.

278

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