Quantum Simulation of Spin-1 Physics with Bosons
in Optical Latticeby

Woo Chang Chung
B.A., Columbia University (2013)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Physics
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2021
c Massachusetts Institute of Technology 2021. All rights reserved.
○

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
May 15, 2021

Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deepto Chakrabarty
Associate Department Head of Physics

2

Quantum Simulation of Spin-1 Physics with Bosons in Optical
Latticeby
Woo Chang Chung
Submitted to the Department of Physics
on May 15, 2021, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy in Physics

Abstract
Mott insulators of ultracold atoms in optical lattices are widely used as an experimental platform for simulating and studying many-body physics. A topic at the frontierof quantum simulation with Mott insulators is the study of quantum spin models,
which are intimately connected to other modern research topics such as the study ofquantum phase transitions and quantum thermalization. While quantum spin models
are also realized and have traditionally been studied with complex magnetic materials in solid-state physics, the advantages of a cold atom quantum simulator are its
wide tunability of model parameters and its capability of preparing a variety of initialstates, some of which may not even be possible in solid-state settings.
So far, most of the spin models realized with two-component atom clouds in optical lattices are based on Mott insulators with singly occupied sites. In this thesis,
I describe how a Mott insulator with doubly occupied sites gives rise to a qualitatively new spin model. In particular, the addition of the on-site interaction in doubly
occupied sites gives rise to a new magnetic anisotropy term known as a single-ion
anisotropy, which is inaccessible to models with singly occupied lattice sites. The
thesis describes in detail the mapping from a doubly-occupied Mott insulator to aneffective spin-1 Heisenberg model with a tunable single-ion anisotropy, along withthe details of the experimental setup and the benchmarks that need to met in order to probe low-energy physics such as those studied in the effective spin model. I
demonstrate that the experimentally realized spin chains in our optical lattice features coherent spin-1 exchange dynamics and also demonstrate a remarkable interplay
between the spin exchange and the single-ion anisotropy term.
The atomic species used is 87 Rb, and its internal structure allows the use of a statedependent lattice with an appreciable detuning from resonances. I explain how thestate-dependent lattice is used to control the spin-dependent interaction and how it
can be used to initialize a two-component atom cloud into a spin Mott insulator, which
is a spinful state with no density and spin fluctuation. The spin Mott insulator stateis highly interesting because it can be used as an initial state for adiabatic passages todifferent spinful states, such as the spin superfluid state or an antiferromagnetically3

textured spin state. I describe the on-going experimental efforts for making adiabatic
passages from the spin Mott to the aforementioned target states and provide an
outlook on how the experimental setup can be upgraded for a cleaner and moreversatile study of spin-1 physics within the optical lattice.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

4

Acknowledgments
During my time at MIT I got to know and work with many smart and intellectually
motivated people; I consider this the biggest perk of studying physics in a graduate school. Without their help and willingness to stop and listen for questions andanswers, I do not think I would have made this far.
I would first like to acknowledge my advisor Wolfgang Ketterle. He was willing totake me into his group when I had no prior experience with atomic physics or laser
optics, and his style of teaching has had a profound impact on my understandingof physics and perhaps on my life as well. I used to be a student who was so easily impressed by complicated/convoluted mathematical equations and knowledge oftechnical details when searching for answers to scientific problems. I also used to lookdown upon “obvious” problems and concepts and not study them carefully. However,
Wolfgang’s insistence on simplicity and intuitive pictures of the given problem thatcan convince any other scientist has helped me analyze problems from a more effective
angle and communicate my understanding better to my colleagues. As a result I feelmuch more confident about communicating scientific (and also non-scientific) ideas
than I used to.
Julius de Hond joined our group as a postdoc in late October 2018 and has overlapped with me just under three years, but he has had a great influence on me andothers within the group with his diverse set of skills and very nice personality. He
helped me move forward when I was not feeling too optimistic about the experimental
progress within the lab, and his proficiency with numerical simulations has helped usgain better insight to the spin model we are studying. I feel like our working stylescomplemented each other and I really enjoyed my last several years of doctoral studies
because of his help and frequent exchange of ideas in and outside the lab. And thisthesis would have not come to fruition without his extensive proofreading.
Jinggang Xiang also joined our group several months earlier than Julius, and he
was eager to get things done on the first day he joined. I think all of us in the groupwere impressed by his construction of a home-made white-light interferometer and5characterization of optical aberrations in an objective, all of which were done in his
first year. I feel sorry that I could not finish building the microscope setup with him,
which he hoped to do early in his graduate school career, but I hope that I will enjoy
his new role and responsibility as the new senior student of the group. Enid CruzCólon is the latest member of the rubidium team and while she is yet inexperienced
in atomic physics research like I was when I was a junior student, I am happy to see
the progress she has been making lately in the lab and also in her participation in
our discussions. I would also like to thank her for keeping the Thanksgiving dinnerparty a tradition of the lab. Thanks to her, Jinggang, and Julius, the atmosphere in
the rubidium lab has been collegial.
Of course, I also spent a long time as a junior student in the lab, and Colin
Kennedy and William Cody Burton were the senior students above me, both fromwhom I have learned a lot. Colin had a great energy and also a lot of courage when
tweaking things and sensitive alignments; this is something I tried to emulate when
I became a senior student. Cody was a calm presence that every lab needs, and henever seemed to be disturbed when something wrong happened in the lab while I
would panic and feel unsure what to do. I think I absorbed some of that quality
when I eventually became the chief debugger of the experiment. It is an interesting
fate that the three of us will convene again in the Boulder area when I start mynew job after graduation. Wenlan was the postdoc before Julius and she brought a
great set of knowledge and skills that she gained from Vladan Vuletic’s group. The
idea of a microscope upgrade became an actual plan when Colin, Cody, and Wenlan
were still in the group and they also contributed to our on-going experimental effort
towards the realization of an adiabatic passage within the spin-1 model. I should
also mention that Samarth Vadia was an exchange student in our lab for a year andgreatly increased the productivity of the lab by providing a new database system.
Outside the rubidium lab, there are many other Ketterle group and CUA membersthat I learned a lot from and enjoyed talking to, and it would be very difficult to list
them all here. I would like to thank in particular the original members of the lithiumlab (BEC5) - Jesse Amato-Grill, Ivana Dimitrova, Niki Jepsen and Will “Goreblaster”
6

Lunden - for sharing materials and ideas with the rubidium lab and also going out
to lunches and dinners together. I would also like to acknowledge the current andpast members of the Dypole lab (Pierre Barral, Michael Cantara, Li Du, Alan Jamison), BEC3 (Hyungmok Son, Julie Park, Timur Rvachov), and BEC2 (Yair Margalit,
Yukun Lu, Furkan Top, Junru Li, Boris Shteynas, Jeongwon Lee) for stimulating
discussions and sharing optics. Outside the Ketterle group I would like to acknowledge Lawrence Cheuk, Matthew Nichols and Carsten Robens in Martin Zwierlein’sgroup for sharing important tips for the microscope upgrade and Yu-Ting Chen andJoonseok Hur from Vladan Vuletic’s group for sometimes stopping by the far-away
rubidium lab office and exchanging ideas or just hanging out. Outside the lab, I
enjoyed the weekend potluck dinners that I had with Jinwook Lee and Changhoon
Oh, both of whom I met through the Samsung Scholarship program.
Finally, I would like to thank my parents and my older brother for supportingme while I was studying far away from them. They always have thought so proudlyof me and unconditionally invested a lot of emotional and financial support for mywell-being and education. I feel a bit sad that my continued life in the United States
will not allow me to see them in Korea more frequently after my graduation, but weknow that our bonds are the strongest. I cannot wait to see them in person and tellthem I made it through the finish line.

7

8

Contents
1 Introduction

19

1.1

Quantum simulation and ultracold atoms . . . . . . . . . . . . . . . .

19

1.2

Why quantum magnetism? . . . . . . . . . . . . . . . . . . . . . . . .

20

1.3

Overview of research efforts in BEC4 . . . . . . . . . . . . . . . . . .

21

1.4

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

23

2 The BEC4 87 Rb machine
2.1

2.2

25

MOT chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.1.1

Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.1.2

MOT loading and molasses. . . . . . . . . . . . . . . . . . .

28

2.1.3

Loading of the atom cloud to the magnetic trap . . . . . . . .

28

2.1.4

Forced RF evaporation in the magnetic trap . . . . . . . . . .

29

2.1.5

Transfer to the transport optical dipole trap . . . . . . . . . .

29

Old science chamber . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.2.1

Optical setup for dipole traps . . . . . . . . . . . . . . . . . .

30

2.2.2

Optical setup for optical lattices . . . . . . . . . . . . . . . . .

31

2.2.3

Mitigation of interference fringes. . . . . . . . . . . . . . . .

32

2.2.4

Magnetic field control . . . . . . . . . . . . . . . . . . . . . . .

34

2.2.5

Atom number calibration . . . . . . . . . . . . . . . . . . . . .

37

3 Non-scalar Light Shift

41

3.1

Classical explanation of light shift . . . . . . . . . . . . . . . . . . . .

41

3.2

Anisotropy in atomic structure . . . . . . . . . . . . . . . . . . . . . .

44

9

3.3

Calculation of atomic polarizability . . . . . . . . . . . . . . . . . . .

46

3.4

Wavelength selection . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4 Optical Lattices

53

4.1

Making an optical lattice . . . . . . . . . . . . . . . . . . . . . . . . .

53

4.2

State-dependent optical lattice . . . . . . . . . . . . . . . . . . . . . .

57

4.2.1

lin-𝜃-lin scheme . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.2.2

𝜎 + − 𝜎 − scheme . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5 The Bose-Hubbard Hamiltonian

71

5.1

Review of the derivation . . . . . . . . . . . . . . . . . . . . . . . . .

71

5.2

Mean-field phase diagram . . . . . . . . . . . . . . . . . . . . . . . .

73

5.2.1

Perturbative approach . . . . . . . . . . . . . . . . . . . . . .

74

5.2.2

Numerical diagonalization . . . . . . . . . . . . . . . . . . . .

77

5.2.3

Comparison to experiments . . . . . . . . . . . . . . . . . . .

79

Corrections due to higher bands . . . . . . . . . . . . . . . . . . . . .

81

5.3

6 Quantum Magnetism
6.1

91

Derivation of an effective Hamiltonian. . . . . . . . . . . . . . . . .

92

6.1.1

Kinetic Exchange . . . . . . . . . . . . . . . . . . . . . . . . .

92

6.1.2

On-site anisotropy . . . . . . . . . . . . . . . . . . . . . . . .

93

6.1.3

Formal mapping . . . . . . . . . . . . . . . . . . . . . . . . . .

93

6.2

Magnetic phases in spin-1 . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3

Spin-1/2 vs spin-1 realizations . . . . . . . . . . . . . . . . . . . . . . 102

7 Spin-1 physics with a two-component Bose gas

107

7.1

Measurement of local spin pairing . . . . . . . . . . . . . . . . . . . . 107

7.2

Observation of exchange in spin-1 model . . . . . . . . . . . . . . . . 111
7.2.1

Idea behind the experiment . . . . . . . . . . . . . . . . . . . 112

7.2.2

Experimental sequence . . . . . . . . . . . . . . . . . . . . . . 114

7.2.3

Result and analysis . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2.4

Possible effects of magnetic gradient . . . . . . . . . . . . . . . 116
10

7.2.5
7.3

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Investigation of the spin Mott phase . . . . . . . . . . . . . . . . . . . 121
7.3.1

Adiabatic passage with a state-dependent lattice . . . . . . . . 121

7.3.2

Adiabatic passage with a magnetic gradient . . . . . . . . . . 129

8 Conclusion

133

A Quantum Gas Microscope

137

A.1 Optical aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.2 Strategy for a high-flatness vacuum window . . . . . . . . . . . . . . 141
B Tunable Single-Ion Anisotropy in Spin-1 Models Realized with Ultracold Atoms

145

C Calculation of the effective spin-1 model parameters

155

D Spherical tensor identities

159

D.1 Spherical basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

D.2 Reduced matrix element . . . . . . . . . . . . . . . . . . . . . . . . . 161
D.3 Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

11

12

List of Figures
2-1 Experimental sequence for the production of a BEC . . . . . . . . . .

26

2-2 Laser beams propagating in the science chamber. . . . . . . . . . . .

32

2-3 Examples of atom clouds affected by accidental interference fringes, asviewed from the vertical direction. . . . . . . . . . . . . . . . . . . . .

33

2-4 Ramsey coherence measurement before and after shimming residual
magnetic gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2-5 Effect of coil current ripple on the magnetic bias field. . . . . . . . . .

36

2-6 Optical pumping during absorption imaging at weak imaging beam
intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3-1 Atomic level structure with and without spin -orbit coupling. . . . . .

45

3-2 Ratio of vector shift to total light scattering rate. . . . . . . . . . . .

50

3-3 Raman scattering rate vs. total scattering rate. . . . . . . . . . . . .

52

4-1 Comparison between Wannier and Bloch functions. . . . . . . . . . .

56

4-2 Plot of the state-dependent lattice potential and physical separation
between different hyperfine states. . . . . . . . . . . . . . . . . . . . .

60

4-3 Polarization rotation control for state-dependent lattices. . . . . . . .

61

4-4 Imbalance in the state-dependent lattice potential depth caused by afolding mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4-5 The 𝜎 + -𝜎 − scheme for state-dependent lattices. . . . . . . . . . . . .

66

5-1 Mean-field phase diagram of the Bose-Hubbard Hamiltonian. . . . . .

76

13

5-2 Calculation of the average filling factor ⟨ˆ𝑛⟩ around the 𝑛 = 1 and 𝑛 = 2
Mott lobes, for 𝑈/𝑧𝑡 = 11. . . . . . . . . . . . . . . . . . . . . . . . .

78

5-3 Simulation of the Mott lobe populations in 3D 20 𝐸𝑅,1064 nm lattice .

87

5-4 Perturbative calculation of correction to Hubbard interaction energy
𝑈 due to inclusion of higher-band Wannier orbitals. . . . . . . . . . .

88

5-5 Perturbative calculation of correction to tunneling amplitude 𝑡 due toinclusion of higher-band Wannier orbitals and the bond-charge effect.

89

6-1 Kinetic exchange mechanism in a two-component Mott insulator withfilling factor ⟨ˆ𝑛⟩ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

6-2 On-site anisotropy in a two-component Mott insulator with filling factor ⟨ˆ𝑛⟩ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

6-3 Table of second-order matrix elements for a ⟨ˆ𝑛⟩ = 2 Mott insulator. .

98

6-4 Comparison of energy spectrum between the Bose-Hubbard model andthe effective spin-1 model. . . . . . . . . . . . . . . . . . . . . . . . . 100
6-5 Magnetic phases of a ferromagnetic spin-1 chain . . . . . . . . . . . . 101
7-1 Measurement of local spin pairing by inducing spin-selective decay.

. 110

7-2 Variation of -𝐽 and 𝐷 with respect to the longitudinal lattice depth ina three-dimensional optical lattice. . . . . . . . . . . . . . . . . . . . 112
7-3 Experimental sequence for the measurement of spin alignment and doublon fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7-4 Coherent dynamics of the spin alignment 𝐴 after a quench in 𝐷/𝐽.
Varying the hold time at characteristic lattice depths for both positive
and negative values of 𝐷/𝐽. . . . . . . . . . . . . . . . . . . . . . . 117

7-5 Coherent spin oscillations in a two-site model. . . . . . . . . . . . . . 118
7-6 Effect of magnetic gradient on a two-site spin-1 model. . . . . . . . . 120
7-7 Phase diagram of the spin-1 model . . . . . . . . . . . . . . . . . . . 122
7-8 Demonstration of an increase in condensate fraction due to the spin
entropy from spin dephasing. . . . . . . . . . . . . . . . . . . . . . . . 123
14

7-9 Experimental sequence for the measurement of spin alignment after aramp into a state-dependent lattice. . . . . . . . . . . . . . . . . . . . 125
7-10 Measurement of spin pairing fraction as a function of 𝑈𝑎𝑏 /𝑈 = 1 − 𝐷/𝑈 .128
7-11 Energy diagram of the spin-1 model augmented by a magnetic gradient∑︀term ∆ 𝑖 𝑖𝑆𝑧,𝑖 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A-1 Objective as a perfect blackbox imaging system . . . . . . . . . . . . 139
A-2 Effect of vacuum window deformation on wavefront error. . . . . . . . 143
A-3 An example vacuum viewport structure. . . . . . . . . . . . . . . . . 144

15

16

List of Tables
4.1

Comparison of LC and EOM performance . . . . . . . . . . . . . . .

17

64

18

Chapter 1
Introduction
The research topic considered in this thesis is at the intersection of three broad fieldsof study in physics: quantum simulation, atomic gases in optical lattices, and quantum magnetism. In this chapter I provide a brief and personal summary of the historyof the experimental research performed within the rubidium lab. While the lab has
worked on a variety of research topics, the study of quantum magnetism using ultracold atoms has been a long standing goal of the lab. The motivation for studying
quantum magnetism is briefly explained. The chapter concludes with a structural
outline of the thesis.

1.1

Quantum simulation and ultracold atoms

The field of ultracold quantum gases has seen a tremendous growth since the firstrealization of Bose-Einstein condensation in 1995 [1, 2] and the first realization of adegenerate Fermi gas in 1999 [3]. Within the next decade following the achievementof quantum degeneracy in atomic gases, researchers intensely studied the properties
and the dynamics of quantum gases in various regimes of temperature and interaction
strengths, developing new techniques, tools and data analysis methods to control andmonitor quantum gases with more ease and precision [4, 5]. At the same time, it wasrecognized early that quantum gases themselves could be ingredients for building new
and possibly more complex quantum systems and studying new quantum phenomena.
19

The origin of this idea dates back to Richard Feynman’s famous keynote speechin 1982, where he proposed using quantum-mechanical objects to build a quantum
computer to simulate quantum mechanical laws of nature [6]. The first observationof the superfluid-Mott insulator transition, a quantum phase transition, in a BoseEinstein condensate in an optical lattice in 2002 confirmed the enormous potential ofcold atomic gases as a new platform for quantum simulations [7]. The interest andthe excitement surrounding quantum simulations with ultracold quantum gases has
only grown stronger in the following two decades.

1.2

Why quantum magnetism?

The history of quantum magnetism dates back to the early days of quantum mechanics, when Heisenberg and Dirac discovered the exchange interaction effects forindistinguishable particles [8, 9]. In that regard it is a rather old subject, but it still
plays a central role in modern research topics such as quantum phase transitions [10]and quantum integrability [11]. It is also challenging to observe and study quantum
magnetic phenomena because the associated energy scales are typically very smalland hence require low temperature and entropy. On the other hand, the strict experimental requirements drive experimenters to improve their methods and come up
with new ideas to construct quantum systems with extremely low entropy.
As a platform for studying quantum magnetism, ultracold atoms offer a distinct
advantages compared to others such as solid crystals with magnetic ions. The internal level structures of atoms are relatively simple compared to those of molecules or
chemical compounds, and thus are well-understood, especially for alkali atoms. Thishas helped experimenters develop a wide variety of techniques for preparing pure initial quantum states with external laser beams, RF/microwave pulses and/or magnetic
fields. The initial states can be ground states that are relevant to the study of phasetransitions between quantum magnetic phases, or they can be excited states1 thatare relevant to the study of non-equilibrium dynamics. The variety of initial states
1

Here I mean the excited states of a quantum spin model, not of an atom.

20

that can be prepared with ultracold atoms enables a wide range of experiments thatprobe different physics within the same model.
In this thesis, we consider both the ground state properties and the nonequilibrium
dynamics of a spin-1 model. We study different ground states of a spin-1 model byusing a state-dependent lattice - an optical lattice that provides different potential
energy for different spin states of an atom - and tune the lattice to induce a phasetransition from a magnetic ground state with an energy gap (“spin Mott”) to a groundstate without an energy gap (“spin superfluid”). We also study how exchanges between
spins occur in the presence of a magnetic anisotropy by initializing the system intoa superposition of different spins and observing how pairing between different spinschanges. It should be pointed out that the study of nonequilibrium physics is not
something that we originally planned to do, but we had the freedom to quickly design
an experiment to probe it and eventually observed some unexpected and interesting
effects. This is an example of the power of quantum simulation with ultracold atoms;
sometimes you make discoveries by simply “trying things out,” and having the freedom
to do so is important.

1.3

Overview of research efforts in BEC4

The history of the rubidium lab in the Ketterle group, also known as BEC4, has followed a shift of interest from studying Bose-Einstein condensates themselves to using
them as a tool. In the beginning, which was only a few years after the achievement of
Bose-Einstein condensation was recognized with a Nobel Prize in physics in 2001, thelab contributed to a deeper understanding of superradiance in matter waves within
the short probe time regime [12]. The research on the reaction of matter waves toexternal probe pulses continued for some time, until the lab took a major interest in
studying spin physics in optical lattices, whose energy scale is very small and hence
requires an extremely low entropy in the system. In 2009 the lab began its progress
toward simulating spin physics in optical lattices by developing a new form of in-situ
thermometry inside an optical lattice (spin gradient thermometry), which was capa21

ble of measuring temperature as low as 1 nK [13]. The lowest temperature measured
was further reduced to 350 pK with the help of an adiabatic demagnetization coolingtechnique in 2011 [14]. While the experiments performed did not simulate a quantum
spin Hamiltonian, they showed that quantum gases can be loaded to optical lattices
and studied at a very low temperature associated with the magnetic exchange energy
scale.
When I joined BEC4 as a young student with no previous experience of atomic
physics in the fall of 2013, the lab was already exploring a new avenue in quantum
simulation, which was the simulation of artificial gauge fields for neutral atoms in
optical lattices [15], with hopes of studying quantum Hall physics with ultracoldbosons. I was fortunate to join this research effort and to contribute to the observationof bosonic condensation in the presence of a large artificial gauge field [16]. During ourefforts toward the artificial gauge field experiment, for which a tilted optical lattice
was a key ingredient, we observed an interesting shielding of matter wave coherence
lifetime (dubbed superfluid shielding) in a tilted optical lattice due to the interaction
between a mobile spin component and an immobile spin component [17]. While I
take pride in having made measurable contributions to both experiments, I will notelaborate on those experiments in this thesis as they have been thoroughly describedin the PhD theses of Colin Kennedy [18] and William Cody Burton [19]. During thistime I also had a chance to work on a laser frequency lock between a 532 nm and a
1596 nm source, which was supposed to generate an optical triple-well superlattice,which would have been useful for extending the experimental effort in the directionof artificial gauge fields. While this research plan was ultimately shelved, I had avaluable time learning about and characterizing laser frequency noise.
Soon after the conclusion of the superfluid shielding experiment, we finally set out
to simulate the quantum Heisenberg spin-1 model following the early theoretical proposals ( [20,21]) from 2003 and guided by a theoretical analysis of the spin-1 magnetic
phase diagram in [22, 23]. We obtained some preliminary results on the realizationof a spin Mott phase after installing the first working state-dependent optical latticein our lab [18, 19], but at the time we did not find conclusive evidence that we were22

probing the physics of magnetic exchange. At the same time, an amorphous idea of a
new science chamber upgrade (on which I took the lead) fully developed into a plan
for a new quantum gas microscope for the future, so that we would be able to probe
off-site spin correlations in our simulated spin chains. As we kept working on thespin chain experiment and the microscope upgrade, we discovered many things in ourold experimental setup that were not explained well previously or not appropriately
adjusted. It was a frustrating experience to work on them one by one, but at thesame time helped me improve and become confident in problem-solving skills.
At the conclusion of my prolonged doctoral studies, we finally found conclusive
evidence of magnetic exchange within our simulated spin-1 chains and observed aninteresting interplay between the magnetic exchange and the magnetic anisotropy
within our spin chain, which we reported in [24]. Also, while we have stocked mostof the ingredients to make the upgrade for a quantum gas microscope, we identified
key issues in our imaging setup. This thesis focuses on how we realize a spin-1 chain
in an optical lattice, and the troubleshooting we had to undertake to improve ourexperiment and our future upgrade for the quantum gas microscope. While it isunfortunate that so many of the insights to the problems arrived rather late in mycareer as a graduate student, I am glad that I discovered some of them and hope thatthe thesis serves as a useful guide for the future generation of students in BEC4.

1.4

Thesis outline∙ Chapter 2 describes the current status of the BEC4 87 Rb machine after many
years of adjustments. It also gives some suggestions for further improvements.
∙ Chapter 3 describes the general theory of non-scalar light shift. Most of thematerial is from [25] but I tried to avoid the heavy machinery of sphericaltensor algebra in the introduction to the theory. The focus is on motivating thenotion of a non-scalar polarizability at a more intuitive level. This chapter is
helpful for understanding how a state-dependent optical lattice works.
23∙ Chapter 4 gives a short introduction to what an optical lattice is and gives aquick demonstration on how to compute the Wannier basis, which is importantto calculating interaction parameters. The rest of the chapter discusses andanalyzes the different realizations of a state-dependent optical lattice.
∙ Chapter 5 reviews the Bose-Hubbard Hamiltonian as a model for neutral atoms
in optical lattices and gives a detailed description of how to simulate the meanfield distribution of the local occupancy number in the optical lattice. The
simulation is important in understanding how we characterize our Mott insulator before using it as a background for simulating spin physics.
∙ Chapter 6 attempts to explain a specific quantum spin model, which is thequantum Heisenberg spin model, and discusses how it is realized with a twocomponent Bose gas in an optical lattice, which can be either state-independent
or state-dependent. The chapter concludes with a discussion of experimental
observables in the spin model.
∙ Chapter 7 describes the two experiments performed on the simulated spin-1
chain. The first experiment is a study of quench dynamics in the spin chain,
where we observe an interesting interplay between spin exchange and spin
anisotropy. The main result is also reported in [24], whose copy is availale
in Appendix B. The second experiment studies the properties of the spin Mott
state and how it can be used as a starting point for adiabatic passages to different spin states with the help of a state-dependent lattice.
∙ Chapter 8 serves as the conclusion to the thesis and provides some scientificoutlook for the study of spin models in doubly occupied Mott insulators.

24Chapter 2
The BEC4 87Rb machine
The rubidium quantum gas setup was conceived and built in the early 2000’s andhas seen many repairs and upgrades. The setup consists of two chambers: the mainchamber, where magneto-optical trapping (MOT) and magnetic trapping are used to
produce a nearly degenerate rubidium cloud, and the science chamber, to which thecloud is transported for evaporation to Bose-Einstein condensate (BEC) and further
manipulation. The design of the main chamber can be found in [26, 27] and that of
the science chamber in [28]. In this chapter I will provide an up-to-date descriptionof the setup and things to be improved for future experiments. The experimental
sequence for the production of a BEC is summarized in Fig. 2-1.

2.1

MOT chamber

2.1.1

Optical setup

The optical setup for the MOT chamber remains similar to what is already describedin [26]. It has been almost two decades since the creation of the first MOT inside
the vacuum chamber, and operations inside the chamber work reasonably well to
produce a nearly degenerate atom cloud within an acceptable experimental cycle
time (approximately 27 s), but it is likely that the parameters have deviated from theoptimal conditions.
25

Figure 2-1: Experimental sequence for the production of a BEC. The steps with thelight background take place inside the MOT chamber, whereas the steps with thedark background take place inside the science chamber.
The main cooling and trapping light power is derived from a frequency-doubled
fiber laser (Quantel EYLSA780), which is set to output 800 mW out of 1000 mW
maximum possible output. This high-power, turn-key system obviated the need for
injection-locked slave diode lasers, which were prone to losing locks due to small
mechanical shocks and temperature drifts. However, after five years of stable operation, the fiber laser began to frequently show problems that were usually notuser-serviceable (internal monitoring photodiode losing efficiency, faulty voltage regulators, enhanced back-reflection of 1560 nm pump power). Based on this experience,a system of tapered amplifier (TA) seeded by an external cavity diode laser (ECDL)
could still be a good option for a stable high-power source at 780 nm in the longrun [29].

87

The fiber laser is locked to −2×93 MHz from the 𝐷2 𝐹 = 2 → 𝐹 ′ = 3 transition of
Rb (also known as the cycling transition) via polarization rotation spectroscopy [30].

The fiber laser output is split into two separate paths, one for the MOT and one for
the Zeeman slower. Each path consists of an acousto-optic modulator (AOM) in the26

double-pass configuration. From the laser table we ultimately deliver 150 mW to the
MOT optics and 30 mW to the Zeeman slower optics on the machine table. Despite
the initial high power and good spatial mode quality of the fiber laser output, theultimate power delivered to the atoms is substantially smaller than 800 mW due tolimited diffraction efficiencies of AOMs in the double-pass configuration, limited fibercoupling efficiencies (60 ∼ 70%), and reflection losses through various beam cubes andlenses. It is suspected that transmission through the AOMs causes distortion to thespatial mode, possibly due to the smallness of the active aperture or thermal effects
from RF carrier power. Note that the actual MOT power delivered to the atoms is
also smaller than 150 mW due to losses during propagation through the optics.
The repumper beam is derived from an old TA (Toptica TA-100) seeded by an
ECDL (Toptica DL-100). The ECDL is locked to +80 MHz from the 𝐷2 𝐹 = 1 →𝐹 ′ = 0/1 crossover transition. The reason for locking to a crossover transition is that it
displays the strongest anisotropy in the polarization rotation signal for lower hyperfine
states (𝐹 = 1 for 87 Rb), where the optical pumping dynamics is more complicatedthan the upper hyperfine states (𝐹 = 2) [30]. In the original setup as described in [26],the ECDL was supposed to be locked to −80 MHz from the 𝐷2 𝐹 = 1 → 𝐹 ′ = 1
transition. We have double-checked that the current locking point is indeed thecrossover transition by comparing to the saturated absorption signal and also bymeasuring the frequency with a high-resolution wavemeter1 . The MOT repumper
beam is then expected to be −33 MHz from the 𝐹 = 1 → 𝐹 ′ = 2 transition, after
passing through an AOM. We have not checked whether the Zeeman slower and
MOT performances are optimal with the current frequency lock point, so this is one
parameter that can be tweaked in a general optimization search. We deliver a totalof 12 mW to the MOT repumper, and 4 mW to the slower repumper.

1

We realized this fact about the repumper locking point around the end of September 2020, which
is only half a year before the time of this writing.

27

2.1.2

MOT loading and molasses

The MOT beams also function as 𝜎 + − 𝜎 − type polarization gradient cooling beams
during the 4 ms optical molasses stage that succeeds the MOT loading. Due tothe design of the Zeeman slower, there is a fraction of the background bias field for
the Zeeman slower that is present during the MOT loading, but absent during themolasses stage. The MOT cloud shifts visibly in position when the current to the
Zeeman slower is switched off, so this makes the beam alignment optimization of the
MOT/molasses beams tricky. Note that the residual field cannot be left on during themolasses as this will degrade the optical molasses performance [31]. Over the course of
the doctoral studies described here, we only had to re-align the MOT/molasses beams
a handful of times, such as when water from the magnetic coil cooling line sprayedonto the optics, or when one of the beams got slightly misaligned after stepping on
the 80/20 structures and replacing the 87 Rb ampoule in the oven. Due to a current
lack of appropriate imaging setup for a time-of-flight imaging in the MOT chamber,
we optimized the loading by monitoring the final number of atoms left in the optical
dipole trap after evaporative cooling in the magnetic trap. In contrast to the sequencedescribed in [26,27], neither the intensity of the molasses beam or the repumper beamis adjusted during the compressed MOT and the molasses stage.

2.1.3

Loading of the atom cloud to the magnetic trap

In the original setup there used to be a 𝐹 = 2 → 𝐹 ′ = 2 depumper beam that
optically pumps atoms into the 𝐹 = 1 manifold after the molasses step and before
loading to the magnetic trap. It turned out that what was labeled as a depumper
was not causing any effect; the depumping process relied on the small off-resonant
scattering from the 𝐹 = 2 → 𝐹 ′ = 2 transition by the molasses beam, which takes
2 ms to complete. After the depumping, the atom cloud is loaded into the magnetic
trap, first by forming a weak spherical trap to mode-match the density distributionof the atom cloud and then transforming into a cigar-shaped trap with a very highgradient in the radial direction. This “catch-and-compress” step is crucial and needs28to be well-optimized. We found that a suboptimal step leads to a noticeably lower
atom number in the optical dipole trap before the transport to the science chamber.
Prior to this step, the capacitors of the high-current power supplies are fully charged,
and provide a fast rush of current to the magnetic trap once the insulated-gate bipolartransistor (IGBT) switches are closed. It was empirically found that slightly delaying
the rapid switch-on of the “axial” power supply (which pushes current to the curvature
and anti-bias coils in our cloverleaf magnetic trap) while the “squeeze” power supply(which pushes current to the curvature coil only) rapidly turns on yields the mostnumber of atoms caught. A plausible explanation is that the initial absence of current
in the anti-bias coil makes sure that the radial gradient is not too strong comparedto the axial gradient.

2.1.4

Forced RF evaporation in the magnetic trap

One of the previous lab members (Colin Kennedy) found out that we could pushcurrent (527 A) into the radial coils beyond the nominal rating of the power supply(500 A), helping us achieve a tighter magnetic trap and a faster evaporation. The
evaporation in the magnetic trap now takes 7 seconds and probably does not needfurther optimization at this point.

2.1.5

Transfer to the transport optical dipole trap

Ideally, the optical dipole trap beam that transports the atom cloud from the MOT
chamber to the science chamber is aligned simultaneously to the center of the magnetic
trap and to the crossing of the laser beams that define the dipole trap in the sciencechamber. However, that cannot be satisfied in general. Because the optical transport
is done by moving a single lens and thereby moving the focus of the dipole trap beam,alignment at one focal position fixes the alignment at other focal positions. If wetreat the moving lens as an ideal lens, the angle at which the laser beam hits the lensdetermines the position on the focal plane, and the angle is fixed. We discovered that
our current center of alignment for our lattice beams and the objective in the science29

chamber constrains the position of the optical transport beam to be ≈ 500𝜇m below
the magnetic trap. The transfer from the magnetic trap to the optical trap is done
by weakening the magnetic gradient and letting the cloud sag down due to gravity.
However, this also reduces the trapping frequencies, such that the confinement along
the axis of the transport beam does not enjoy enhancement by the magnetic trap, andthis prolongs the required transfer time. We currently have to wait 6 s to completely
fill up the dipole trap volume in the sagged trap position. If we shim the magnetic
field to move the magnetic trap center down, we can reduce the transfer time to lessthan 1 s. Given the strength of the peak gradient (250 G/cm), we need 12 G of extra
shim field along the transport axis.
An unsolved problem in the optical transport is the appearance of |𝐹 = 1, 𝑚𝐹 = 0⟩
impurities within the originally pure |𝐹 = 1, 𝑚𝐹 = −1⟩ cloud as the atoms get
transported from the MOT chamber to the science chamber. It is suspected that thisdiabatic transition happens somewhere in the connection (gate valve) between thetwo chambers; we remove the impurities in the science chamber by flipping them to|𝐹 = 2, 𝑚𝐹 = 0⟩ and blowing them out with a resonant light pulse.

2.2

Old science chamber

2.2.1

Optical setup for dipole traps

The optical transport beam (“ODT”) moves the nearly-degenerate atom cloud roughly
38 cm with 80% single-pass transfer efficiency over 3 seconds. The atom cloud istransferred from the ODT beam to a crossed dipole trap [see Fig. 2-2(a)]. Previously,
the crossed dipole trap was formed by a beam anti-parallel to the transport beam(“anti-ODT”) and a beam perpendicular to the transport beam (“XODT”) [28]. Now it
is formed by a highly elliptical XODT beam from the side, and a vertical dipole beam(“VD”) along the gravity direction. Both beams are derived from an externally seeded
1064 nm fiber amplifier. The XODT beam has a 1/𝑒2 beam waist that is roughly(𝑤𝑥 , 𝑤𝑦 ) = (500 𝜇m, 50 𝜇m) at the center of the trap and provides levitation against30

gravitational sag. The VD beam has a 1/𝑒2 beam waist of 95 𝜇m and provides radialconfinement when viewed from the top. A precise control of the tilt of the XODT
beam allows us to null projected optical gradient along the axis of the beam andlevitate the atom cloud optically without horizontal drift2 . The absence of horizontal
gradients while holding the atom cloud with just the XODT beam allows us to alignthe horizontal directions of lattice beams by monitoring the displacement of the intrap position of the atom cloud. To control the atom number, we introduce a tightly
focused beam (“Dimple”) that keeps a desired number of atoms while a high magnetic
gradient in a horizontal direction kicks out the remaining number of atoms. The
Dimple beam is derived from a temporally incoherent light source (superluminescent
diode, Superlum M-S-850) with its center wavelength around 850 nm. The temporal
incoherence minimizes occurrences of interference fringes and helps achieve a tightfocal spot.

2.2.2

Optical setup for optical lattices

The three-dimensional optical lattice is generated by propagating three independent
laser beams in three orthogonal directions and then retroreflecting them. The directions for the horizontal lattice beams are marked “Closet” and “Tool”, which areroughly −45∘ and +45∘ rotated from the direction of the XODT beam [see Fig. 22(b)]. The wavelength of the Vertical and Closet lattice beams is 1064 nm (Nufern
fiber amplifier) while that of the Tool beam can be either 1064 nm (from the same
fiber amplifier that provides power for Vertical and Closet beams) or 810 nm (from a
Ti:Sapphire laser), depending on the experiment being performed. For every latticebeam, we have a set of two photodiodes that receive some light from a beam sampler
and are used for feedback stabilization and monitoring of the optical power output3 .
After the beam sampler, an optical isolator is placed so that the retroreflected light
Since the gravitational force is equivalent to 2130 Hz/𝜇m for 87 Rb, a 1∘ tilt results in a projectedgradient of 37 Hz/𝜇m, which becomes noticeable at small radial trapping frequencies.
3
The second photodiode for monitoring serves to measure out-of-loop intensity noise. The closed
feedback loop suppresses shot noise seen by the feedback photodiode, so an independent measurementof the output power is needed to see the true noise level of the output light.
231

does not reflect off the optical fiber tip and reach the photodiodes, introducing noisein the process4 . It has been found out that such backreflection is deleterious for thestabilization of the optical lattice intensity at acoustic frequency range [18].

Figure 2-2: Laser beams propagating in the science chamber. The arrow labeled “g”
denotes the gravity direction. The arrow labeled “B” denotes the bias magnetic field
direction. The beam labels are attached to the side where the beam is launched.
(a) Optical dipole trap beams. For the two vertical beams, the low-divergence beam is
the VD beam and the high-divergence beam is the Dimple beam. The VD beam formsa crossed optical dipole trap with the XODT beam that traps the BEC. All beams
except the Dimple have a wavelength of 1064 nm. The Dimple beam is derived froma temporally incoherent source with a center wavelength of 850 nm and a coherence
length of 𝒪(10 𝜇m). (b) Optical lattice beams. The wavelength of the Tool beam
can be either 1064 nm or 810 nm.

2.2.3

Mitigation of interference fringes

Over the years, a recurrent problem in the optical setup around the science chamber
was the unintentional formation of interference fringes. Because of the extremely
high coherence length of the lasers being used (e.g. 300 km for our 1064 nm master4

Such a problem was described by Markus Greiner at a CUA winter retreat. I later found out
the problem in our setup by noticing a small but visible difference in the oscilloscope trace of thephotodiode while I blocked and unblocked the retroreflecting mirror.

32

Figure 2-3: Examples of atom clouds accidental interference fringes, as viewed fromthe vertical direction. (a) Fringes formed by Vertical lattice beam input and a backreflection of a backreflection (double backrflection) from a lens. The two images show
in-trap atomic density profiles with and without fringes. (b) Fringes formed by the
XODT beam and a backreflection from a vacuum window. A ghost diffraction peakalong the XODT axis is visible.

laser Coherent Mephisto), it is easy to form accidental fringes if there are many
surfaces or if the laser beams are not angled sufficiently away from normal incidence.
Due to anti-reflection coatings, the reflected power will usually be 1% or less, butconstructive interference in amplitude with the incident beam will result in up to±20% spatial modulation of the original optical intensity. If the length scale of themodulation is large enough to be resolved by the imaging system and the atomic gas
is degenerate and in the Thomas-Fermi regime, the corrugation in the potential canbe seen directly in an in-trap image of the atom cloud [Fig. 2-3(a)]. If the lengthscale is too short to be resolved, but the modulation depth is sufficiently high, it may
leave detectable “ghost” diffraction peaks in a time-of-flight image [Fig. 2-3(b)]. The
corrugation in the potential can be dynamic if the optical elements vibrate at acoustic
frequencies and it can possibly heat the atom cloud while ramping up optical lattices.
In general, it is advised that the laser beams are angled by 1∘ or more with respect
to surfaces/windows/flats, depending on the beam divergence and power.
33

2.2.4Magnetic field control

The magnetic coils around the science chamber are described in [28]. The bias magnetic field is set to be roughly parallel to the Tool beam, which generates the spindependent lattice in our experiments. The residual gradients are shimmed by flowing
extra current in one of two coils in each of the Tool coil pair and the Closet coilpair. The extra bias generated from this shimming contributes 3 G in quadrature to
8 G generated by the Tool coil pair, resulting in a net bias field of ∼ 9 G which isslightly tilted away from the Tool lattice beam direction. We check the quality of
the gradient shimming by performing a Ramsey interferometry measurement on the|𝐹 = 1, 𝑚𝐹 = −1⟩ → |𝐹 = 2, 𝑚𝐹 = −2⟩ transition (see Fig. 2-4).
The biggest source of noise in the magnetic field is from the ripple current of theTool coil power supply (Agilent 6651A) that runs at a superposition of harmonics ofthe mains frequency (Fig. 2-5). The RMS ripple current according to the datasheet
is 25 mA and it results in a 8.8 mG ripple in the bias. In principle, this bias ripple
can be nulled by connecting a feedforward shunt running out of phase with respect tothe mains line [32]. However, we have left the bias ripple uncompensated and simply
switch on the RF or microwave power on a mains line trigger when performing spin
flips. Besides current noise, ferromagnetic components around the chamber can havea hysteretic effect on the bias field. Applying large horizontal gradients requires
running 𝒪(100 A) through our coils, and we measured that the bias changes by𝒪(100 mG) after a large gradient was ramped up and down. This bias change issemi-permanent in the sense that subsequent experiment cycles without a gradient
ramp show a similar bias change that decays slowly, ruling out the possibility of thechange being induced by eddy currents. It is likely that there are some ferromagnetic
domains within the science chamber or some nearby bolts, so it is advised that a newchamber is checked for magnetic hysteresis before the final assembly.
A nuisance of the current magnetic field control is the lack of Earth’s-field-cancelingcoil pairs. At a low bias field (< 2 G), which is required for efficient optical pumping
during imaging, the Earth’s magnetic field (≈ 0.5 G) interferes with the field strength34

Figure 2-4: Ramsey coherence measurement before and after shimming residual magnetic gradient. The transition used is |𝐹 = 1, 𝑚𝐹 = −1⟩ → |𝐹 = 2, 𝑚𝐹 = −2⟩ atthe bias field of 8.916 G (transition frequency 𝜔0 = 2𝜋 × 6815.962 MHz). During themeasurement, the atom cloud is held in a crossed optical dipole trap with no lattices,
so some motion-induced averaging of the gradient by the atoms might have occurred.
The blue solid dots and the red open dots correspond to data obtained before andafter shimming, respectively. The blue solid line and the red dashed line correspondto fit models with estimated gradient values before and after shimming, respectively.
The fit models are based on the averaged values of cos(𝜋𝑡𝛿)2 over a spherical volume
in which a linear gradient runs along the z-axis. We assume the extent of the cloud is
10 𝜇m. The scattering of the raw data away from the fit function envelopes is due tothe uncompensated bias fluctuations at the harmonics of mains frequency (Fig. 2-5),
which can add extra rotation by 𝜋 or more during the hold time. If the AC line
trigger is set such that the average of the bias deviation over a ripple period is zero,and the hold times are exactly integer multiples of the ripple period, we expect the
scattering to disappear.

35

Figure 2-5: Effect of coil current ripple on the magnetic bias field. A 𝜋-pulse addressing the |1, −1⟩ → |2, −2⟩ transition is applied to a |1, −1⟩ BEC after a variable
hold time with respect to the AC line trigger edge. The gray band denotes the +/standard error of the mean of the |2, −2⟩ atom number measurements. The power
supply providing most of the bias field is Agilent 6651A, which has a 25 mA current
ripple. The specified current ripple is estimated to produce an amplitude of 8.8 mG
in bias fluctuation at harmonics of the mains frequency. The estimated maximum
detuning of the transition is 0.65Ω, where Ω = 2𝜋 × 28.63 kHz is the Rabi frequencyof the transition. The estimated maximal detuning corresponds to a maximal bias
deviation of 8.9 mG.

36

and orientation control. Something as modest as wrapping flexible wires multiple
times around the vacuum chamber and flowing a small current through them shouldcompensate the Earth’s magnetic field5 .

2.2.5

Atom number calibration

Absorption imaging
Absorption imaging measures integrated column density 𝑛𝑙 of the atom cloud, and it
requires a precise knowledge of the effective absorption cross section 𝜎abs to convert
the integrated column density into the observed atom number in an image. The
formula for the total atom number 𝑁atoms is
𝑁atoms =
=

1
𝜎abs

∫︁

(︂
)︂
𝐴pixel ∑︁𝐴pixel ∑︁
𝐼𝑃 𝑊 𝐴(𝑛𝜎𝑙) 𝑑𝐴 =(𝑛𝜎abs 𝑙) = −log
𝜎abs pixels
𝜎abs pixels
𝐼𝑃 𝑊 𝑂𝐴

𝐴pixel(Ncount)
𝜎abs

where 𝐼PWA and 𝐼PWOA denote pixel values of the picture-with-atoms image and thepicture-without-atoms images respectively. The effective absorption cross section willreach the ideal value of 𝜎0 = 3𝜆2 /(2𝜋) only if the imaging is performed on the cyclingtransition, the imaging beam polarization is of a pure circular polarization, and theimaging intensity is far below the saturation intensity, which is 1.67 mW/cm2 for the𝐷2 cycling transition [33]. For best results, there should be a small homogeneous bias
to fix the quantization axis. We currently flow a small current in one of the Closet
coils to establish an imaging bias field.
The cleanliness of the imaging beam polarization beam can be checked by imaging
atom clouds prepared in different hyperfine states (e.g. |𝐹 = 2, 𝑚𝐹 = 0, −1, −2⟩)
but under otherwise same conditions, and comparing the ratios of the absorption
signals to the ratios of the transition strengths [see Fig. 2-6(b)], which are given bysquares of Clebsch-Gordan coefficients for respective transitions [34]. Note that such
5

As for why this has not been done, it is difficult to wrap anything around the chamber, especiallythe bottom side, due to the proximity of the optical breadboards and RF/microwave antennae.

37

measurements should be done at a very low imaging beam intensity and with short
exposures to prevent optical pumping. A CCD camera in a low-noise readout mode
and with hardware pixel binning enabled should be able to detect such low signals.
For an accurate count of the atom numbers, it is best to keep the peak opticaldensity (𝑂𝐷) well below 1. We have seen that the same atom cloud will register alower total count if it does not expand enough during time-of-flight and hence has ahigher optical density. A detailed look at the evolution of the absorption signals during
imaging is provided for two different ranges of 𝑂𝐷 in Fig. 2-6. We suspect that at
high optical densities, re-absorption of spontaneously emitted photons becomes nonneglible and provides a small optical pumping out of the stretched state, lowering theeffective absorption cross section.

BEC expansion
An atom number calibration method that does not rely on a precise knowledge ofthe effective absorption cross section is the measurement of cloud widths as a BEC
expands during time of flight. Instead, it requires a precise knowledge of the trapping
parameters and the velocities at which the cloud expands. The method also assumes
that the BEC is trapped in a harmonic trap and is deep in the Thomas-Fermi regime.
In the Thomas-Fermi regime, when the harmonic trap is abruptly switched off, thetotal energy released is dominated by the mean-field interaction energy, which isconverted into the kinetic energy of the expanding atom cloud [35]. Also, in thisregime the density distribution is expected to maintain its initial parabolic shapeduring the expansion [36], so that the time-dependent density distribution can be
written as
(︂
)︂
𝑥2
𝑦2
𝑧2
𝑛(𝑥, 𝑦, 𝑧, 𝑡) = 𝑛0 (𝑡) 1 − 2−−𝑅𝑥 (𝑡) 𝑅𝑦2 (𝑡) 𝑅𝑧2 (𝑡)(𝑖 ≤ 𝑅𝑖 (𝑡), 𝑖 ∈ {𝑥, 𝑦, 𝑧})where 𝑅𝑥 (0), 𝑅𝑦 (0), 𝑅𝑧 (0) correspond to the initial Thomas-Fermi radii of the cloudand 𝑛0 (0) is the initial peak density. Note that the expression holds only where 𝑛 ≥ 0.
38

Figure 2-6: Optical pumping during absorption imaging at weak imaging beam intensities. A pulse time of 20 𝜇s corresponds to roughly one photon absorbed per
atom. The imaging bias is around 2 G. (a) Calibration of the imaging beam polarization. A linear polarizer (Thorlabs NIR100) and a quarter-wave plate are installed
before the vacuum chamber. A maximal absorption by |2, −2⟩ atoms is achieved
when the polarization is 𝜎 − . The non-sinusoidal behavior is possibly due to a birefringent effect from the vacuum window. After modeling the vacuum window as afixed retarder at some angle and fitting the model to the data, we estimate that the
maximum absolute value of the circularity of the input polarization is around 0.9,
which is close to the maximum value of one. We set the quarter-wave plate at theangle specified by the dashed line. (b) Optical pumping dynamics at different opticaldensities. The ratios of the early-time absorption signals agree nicely with the ratiosof the relative line strengths, which are (24,40,60) for the ∆𝑚𝐹 = −1 transitions
from |𝐹 = 2, 𝑚𝐹 = 0, −1, −2⟩ [34]. However, we see a noticeable decay of the signal
for |2, −2⟩ at high optical densities, whereas it stays relatively flat for low opticaldensities. We suspect there is a small optical pumping rate out of |2, −2⟩ at highoptical densities due to reabsorption of spontaneously emitted photons, which is to
be expected at high OD.

39

From this we can derive that the observed kinetic energy of the expansion 𝐸rel is𝐸rel =

8𝜋 𝑚 2𝑣 𝑛0 (𝑡)𝑅𝑥 (𝑡)𝑅𝑦 (𝑡)𝑅𝑧 (𝑡)105 2 𝑏where 𝑚 is the mass of a single atom and 𝑣𝑏 =√︀ 2
𝑅𝑥 + 𝑅𝑦2 + 𝑅𝑧2 /𝑡 is proportional to

the root mean square of the expansion velocity of the cloud boundaries. On the other
hand, the total mean-field energy 𝐸int of a trapped cloud is𝐸int =16𝜋
𝑔𝑛0 (0)2 [𝑅𝑥 (0)𝑅𝑦 (0)𝑅𝑧 (0)]105

where 𝑔 = 4𝜋~2 𝑎𝑠 /𝑚 and 𝑎𝑠 is the 𝑠-wave scattering length. If we set 𝐸rel = 𝐸int , weobtain√2 𝑚2
𝑣𝑏5
𝑁atoms =120 ~2 𝑎𝑠 𝜔𝑥 𝜔𝑦 𝜔𝑧where 𝜔𝑥,𝑦,𝑧 are the angular trapping frequencies in the 𝑥, 𝑦 and 𝑧 directions, which
can be measured accurately with induced dipole oscillations after kicking the atom
cloud in the trap. Because the expression contains a fifth power of 𝑣𝑏 and hence
is sensitive to its precise value, long times of flight should be used to measure 𝑣𝑏 .
For 87 Rb, 𝑚 = 86.9 u and 𝑎𝑠 = 100 𝑎0 , where 𝑎0 = 5.29 × 10−11 m is the Bohrradius, so for (𝜔𝑥 𝜔𝑦 𝜔𝑧 )1/3 = 2𝜋 × 40 Hz, 𝑣𝑏 = 2.5 mm/s yields 𝑁atoms = 25650,

while 𝑣𝑏 = 2.8 mm/s yields 𝑁atoms = 45200. We have checked that this method
agrees nicely with perturbative absorption imaging performed on the stretched state|𝐹 = 2, 𝑚𝐹 = −2⟩ prepared with the cleanest imaging beam polarization possible.

40Chapter 3
Non-scalar Light Shift
Understanding the non-scalar or state-dependent light shift is crucial for many applications. For example, when atoms are used as qubits and stored in tightly-focused
beams, the electric field can have a non-zero longitudinal polarization near the focus
and give rise to an inhomogeneous state-dependent potential that dephases any internal state superposition unless it is suppressed by an orthogonal and real magnetic
field [37, 38].
On the other hand, a state-dependent potential can be used as a beneficial tool
for state preparation and manipulation of cold atoms. There have been numerous
experimental works using state-dependent potentials in different platforms, such as
optical nanofiber traps [39] and optical lattices [40–42]. Our quantum simulation
experiment also makes a prominent use of state-dependent lattice potential. In thischapter we give a brief review of the theory of non-scalar light shift.

3.1

Classical explanation of light shift

In this section, we neglect the details of the internal structure of an atom, and concentrate on how atoms interact with a polarized light in general. Atoms can be thoughtof as miniature dielectric objects. They can be polarized by external electric fields,
and hence they experience an energy shift due to the interaction between the induced
dipole moment and the external electric field. If the external field is sufficiently small,41

⃗ holds:
a linear relationship between the induced dipole moment 𝑑⃗ and external field 𝐸⃗⃗𝑑(𝜔)= 𝛼(𝜔)𝐸(𝜔),where the transfer function 𝛼(𝜔) is called the polarizability of theatom. The argument of 𝜔 in the relationship denotes that the terms are functions of⃗frequency. Note that 𝛼(𝜔) and 𝐸(𝜔)are complex in general. For a fixed frequency−𝑖𝜔𝑡⃗⃗𝜔, complex 𝐸(𝜔)is related to the real electric field E(𝑡) as 𝐸(𝜔)𝑒+ c.c where2⃗c.c stands for complex conjugate and to its time-averaged intensity 𝐼 as 2𝜀0 𝑐|𝐸(𝜔)|where 𝜀0 is the vacuum permittivity.
The energy of a real dipole moment d in a real electric field E is −d · E. Because
the dipole moment of an atom is induced from zero to its final value as the external
field is switched on, the net energy shift that an atom experiences under the field is1
1
2⃗
𝑈 = − ⟨d · E⟩ = −Re[𝛼(𝜔)]|𝐸(𝜔)|=−Re[𝛼(𝜔)]𝐼
2
2𝜀0 𝑐

(3.1)

where the angle brackets denote time-averaging.
We see that the light shift is proportional to the intensity of the laser but is notdependent on its polarization. To allow such dependence, we have to generalize thecomplex polarizability 𝛼(𝜔) from a scalar to a rank-2 tensor 𝛼(𝜔)𝑖𝑗 , (𝑖, 𝑗 = 𝑥, 𝑦, 𝑧)∑︀so that the light shift is expressed as − 𝑖,𝑗 Re[𝛼𝑖𝑗 (𝜔)𝐸𝑖 (𝜔)* 𝐸𝑗 (𝜔)]. We suppress thesummation symbol in the discussions to follow.

We can make use of rotational symmetry to decompose that expression into a
series of terms, one of which couples to intensity only (the scalar term), and others
that have additional couplings to polarization. Any rank-2 tensor can be decomposed
into three irreducible tensors that respect rotational symmetry:
(0)

(1)

(2)

𝛼𝑖𝑗 = 𝛼𝑖𝑗 + 𝛼𝑖𝑗 + 𝛼𝑖𝑗
1
(0)
𝛼𝑖𝑗 =
𝛼𝑘𝑘 𝛿𝑖𝑗
3
1
1
(1)
𝛼𝑖𝑗 =
(𝛼𝑖𝑗 − 𝛼𝑗𝑖 ) = 𝜀𝑘𝑖𝑗 (𝜀𝑘𝑎𝑏 𝛼𝑎𝑏 )
2
2
1
1
(2)
(𝛼𝑖𝑗 + 𝛼𝑗𝑖 ) − 𝛼𝑘𝑘 𝛿𝑖𝑗
𝛼𝑖𝑗 =
2
3
The superscripts (0), (1), and (2) on the three irreducible tensors mean that they42

transform as a scalar, a vector and a (symmetric and trace-less) tensor respectively
under rotation. They have 1, 3, and 5 degrees of freedom respectively1 , which sum up
to the total of 9 as expected for a general rank-2 tensor. The tensor 𝜀𝑖𝑗𝑘 is the LeviCivita symbol2 and 𝛿𝑖𝑗 is the Kronecker delta function. Following the same approach,the tensor 𝐸𝑖* 𝐸𝑗 can be decomposed into scalar, vector, and tensor parts:
𝐸𝑖* 𝐸𝑗 = (𝐸𝑖* 𝐸𝑗 )(0) + (𝐸𝑖* 𝐸𝑗 )(1) + (𝐸𝑖* 𝐸𝑗 )(2)⃗ 2
|𝐸|(𝐸𝑖* 𝐸𝑗 )(0) =𝛿𝑖𝑗
31⃗ * × 𝐸)⃗ 𝑘
(𝐸𝑖* 𝐸𝑗 )(1) =𝜀𝑖𝑗𝑘 (𝐸
2 (︂
)︂
1
2 ⃗ 2
*
(2)
*
*
(𝐸𝑖 𝐸𝑗 )=𝐸𝑖 𝐸𝑗 + 𝐸𝑗 𝐸𝑖 − |𝐸| 𝛿𝑖𝑗
2
3
If we combine the decompositions for 𝛼𝑖𝑗 and 𝐸𝑖* 𝐸𝑗 together by taking their product, it is not difficult to see that the scalar part of 𝛼𝑖𝑗 couples to the scalar part of∑︀(𝑘)
𝐸𝑖* 𝐸𝑗 only and so on: 𝛼𝑖𝑗 𝐸𝑖* 𝐸𝑗 = 2𝑘=0 𝛼𝑖𝑗 (𝐸𝑖* 𝐸𝑗 )(𝑘) . This decomposition principle
looks slightly different in the spherical basis, due to the transformation between the
Cartesian basis and the spherical basis (see Appendix D).
We see that the vectorial and tensorial components of the polarizability couples
to the polarization of the laser as well as its intensity. In particular, the vectorial partis proportional to the circularity of the light 𝒜, where 𝒜 is defined as𝜖ˆ* × 𝜖ˆ = 𝑖𝒜𝑘ˆ
where 𝜖ˆ is the complex polarization of the laser and 𝑘ˆ is the propagation directionof the laser. For linear polarization, 𝒜 is zero, while circular polarizations 𝜎+ =√√(−ˆ
𝑥 − 𝑖ˆ
𝑦 )/ 2 and 𝜎− = (ˆ𝑥 − 𝑖ˆ
𝑦 )/ 2 have values of +1 and −1 respectively for 𝒜.
Just as the vectorial part of 𝐸𝑖* 𝐸𝑗 is proportional to the circularity of the po1

In fact, these degrees of freedom correspond to those of spherical tensors with rank 𝐽 = 0, 1, 2
respectively. We have chosen to express the tensor components in the Cartesian basis, but one may
as well express them in the spherical basis, which will facilitate evaluations of expectation values
with respect to angular momentum eigenstates.
2
A useful identity is 𝜀𝑘𝑖𝑗 𝜀𝑘𝑎𝑏 = 𝛿𝑖𝑎 𝛿𝑗𝑏 − 𝛿𝑗𝑎 𝛿𝑖𝑏 .

43

larization, the vectorial part of the polarizability tensor 𝛼𝑖𝑗 carries some notion ofhandedness as well. We will see that this is related to the projection of the electron
angular momentum to the wavevector of the light. Intuitively, one needs a non-zero
angular momentum to have some anisotropy in the response of the atom to the external electric field.

3.2

Anisotropy in atomic structure

Before we delve into the details of the quantum-mechanical calculation of the atomic
polarizability tensor, it is instructive to consider how changes in the atomic levelstructures will lead to changes in the anisotropy of the polarizability. First consider the atomic structure that arises from the kinetic energy of the electron and
the Coulomb potential, 𝑝2 /2𝑚 + 𝑉 (𝑟). This Hamiltonian has eigenstates that arelabeled by the orbital angular momentum 𝐿 and for alkali atoms the ground state is
𝐿 = 0, for which the atom-light coupling is isotropic [Fig. 3-1(a)]. The coupling isisotropic in the sense that the transition strength is independent of the polarization⃗ · 𝑆)⃗ between the electron spin 𝑆 and theof the light field. If spin-orbit coupling (𝐴 𝐿electron orbital angular momentum 𝐿 is switched on, the new eigenstates are labeled⃗ + 𝑆,⃗ and the excited states before theby the total electron angular momentum 𝐽⃗ = 𝐿
switch-on (𝐿′ = 1) now splits into a fine structure of two 𝐽 ′ manifolds (also known as
the 𝐷 lines in alkali spectroscopy). The non-zero angular momentum of the groundstate gives rise to an anisotropic atom-light coupling [Fig. 3-1(b)].
For atoms with zero orbital angular momentum in the ground state, the strengthof the anisotropic atom-light coupling depends on the detuning of the laser frequency
from the center of the 𝐷 lines. If the laser detuning ∆𝐿 is much larger than thefine-structure splitting ∆𝐹 𝑆 , then in the rotating frame of the laser, the spin-orbit
coupling term that gives rise to ∆𝐹 𝑆 appears as a small off-diagonal perturbationin comparison to the large diagonal energy term of ∆𝐿 . Colloquially we say thatthe highly-detuned laser cannot “resolve” the fine structure of the atom. Hence, thesystem can be approximated as an isotropic system in the large detuning limit [Fig. 344

Figure 3-1: (a) Atomic level structure without spin-orbit coupling. Because theground state has zero angular momentum (𝐿 = 0), its coupling to the excited states(𝐿′ = 1) is isotropic (it has equal Clebsch-Gordan coefficients 𝐶𝜖 for any polarization 𝜖ˆ
of the electric field). (b) Atomic level structure with spin-orbit coupling. Each state islabeled by the total electron angular momentum 𝐽, which is non-zero for both groundand excited states. Coupling from the ground state to the excited states is no longer
isotropic. (c) If the laser detuning ∆𝐿 is much larger than the fine-structure splitting∆𝐹 𝑆 due to the spin-orbit coupling interaction, the anisotropy cannot be resolved andthe system approximates an isotropic system. The light-red and light-blue coloring of
the states denotes the spin component with a higher weight within the total angularmomentum state (light-red and light-blue refer to spin up and down, respectively).

45

1(c)].
In general, light shift scales as 1/∆𝐿 and light scattering rate as 1/∆2𝐿 . However, the vector light shift for states with zero orbital angular momentum scales with∆𝐹 𝑆 /∆2𝐿 [43] so that, asymptotically, the ratio of the vector light shift to the total light scattering rate becomes independent of the detuning. However, such anargument does not apply to states with non-zero orbital angular momentum. An
example is a hypothetical four-level system consisting of |𝐿 = 1, 𝑚𝐿 = −1, 0, 1⟩ as
the ground state manifold and |𝐿 = 0, 𝑚𝐿 = 0⟩ as the excited state. In this example,

the |𝑚𝐿 = −1, 0, 1⟩ states respond only to 𝜎 + , 𝜋, and 𝜎 − polarizations respectively,
so that the light shift is clearly state-dependent but scales with 1/∆𝐿 . Indeed, foratomic species with non-zero orbital angular momentum in the ground state, such astitanium, which is a transitional metal, or erbium, which is a lanthanide, significant
non-scalar light shift per unit light scattering rate can be achieved far away fromatomic resonances [44].

3.3

Calculation of atomic polarizability

The general quantum-mechanical expression for light shift experienced by an atom
in an external electric field is provided by second-order perturbation theory. To derive the expression for atomic polarizability, it is sufficient to work in the semiclassicalpicture, where we work with quantized energy levels of the atom but treat the electromagnetic field classically. Furthermore, in the long-wavelength approximation (alsoknown as dipole approximation) where the wavelength of the light is much larger
than the localization extent of the addressed electron, the atom-light interaction 𝐻int
resembles the expression for the energy of a classical dipole in an electric field: −d·E,
with d = 𝑞r now a quantum-mechanical operator. Assuming a monochromatic fieldwith angular frequency 𝜔, we can split 𝐻int into two parts, one rotating at positive46

frequency (𝑉− ) and the other rotating at negative frequency (𝑉+ )3 :𝐻int = −(d · 𝜖ˆ)𝐸(𝜔)𝑒−𝑖𝜔𝑡 − (d · 𝜖ˆ* )𝐸(𝜔)𝑒+𝑖𝜔𝑡= 𝑉− 𝑒−𝑖𝜔𝑡 + 𝑉+ 𝑒𝑖𝜔𝑡Note that the complex phase is assumed by the polarization vector 𝜖, and the amplitude of the oscillation 𝐸(𝜔) is real. We obtain the second-order energy shift due
to this time-dependent perturbation by calculating the first-order correction to the
wavefunction using standard time-dependent perturbation theory. Since 𝐻int is timeperiodic, we can also apply a more systematic approach such as Floquet perturbation
theory, which is described in Appendix D of Ref. [25]. The second-order energy shift
given by this approach closely resembles the expression for the shift given by timeindependent second-order perturbation theory:𝛿𝐸 =∑︁ ⟨𝑛|𝑉+ |𝑛′ ⟩⟨𝑛′ |𝑉− |𝑛⟩
𝐸𝑛 − 𝐸𝑛′ + ~𝜔𝑛′+∑︁ ⟨𝑛|𝑉− |𝑛′ ⟩⟨𝑛′ |𝑉+ |𝑛⟩𝑛′

𝐸𝑛 − 𝐸𝑛′ − ~𝜔The state |𝑛⟩ is the initial and final state of the atom, and |𝑛′ ⟩(̸= |𝑛⟩) are the intermediate states connected by the atom-light interaction. The energy shift may be
rewritten as
𝛿𝐸 =
𝑅± =

(︀

)︀
⟨𝑛|(d · 𝜖ˆ* )𝑅+ (d · 𝜖ˆ) + (d · 𝜖ˆ)𝑅− (d · 𝜖ˆ* )|𝑛⟩ 𝐸(𝜔)2∑︁|𝑛′ ⟩⟨𝑛′ |
𝑛′

(3.2)

𝐸𝑛 − 𝐸𝑛′ ± ~𝜔We can express (d·ˆ𝜖* )𝑅+ (d·ˆ𝜖) as (𝑑𝑖 𝑅+ 𝑑𝑗 )ˆ𝜖*𝑖 𝜖ˆ𝑗 and apply the decomposition into three
irreducible tensor components for 𝑑𝑖 𝑅+ 𝑑𝑗 and 𝜖ˆ*𝑖 𝜖ˆ𝑗 . Similar decomposition should
be done for 𝑑𝑖 𝑅− 𝑑𝑗 and 𝜖ˆ𝑖 𝜖ˆ*𝑗 . We assume that |𝑛⟩ and |𝑛′ ⟩ are eigenstates of thetotal angular momentum operator J, so it is better to perform the irreducible tensor
decomposition in the spherical basis. From here now on, we refer to the tensor
𝑑𝑖 𝑅± 𝑑𝑗 as d ⊗ 𝑅± d and the spherical components of its irreducible rank-𝑘 tensors as
3Note that Ref. [25] uses the symbol 𝑉− for the positive-frequency term. The symbol may makemore sense if we take the minus sign to suggest photon absorption [45].

47

(𝑘)

(d ⊗ 𝑅± d)𝑞 with −𝑘 ≤ 𝑞 ≤ 𝑘, and similarly for 𝜖ˆ*𝑖 𝜖ˆ𝑗 . In the spherical basis, we canre-organize the terms further in using Eq. D.2 (see Appendix D and Ref. [25]):2

𝛿𝐸 = 𝐸(𝜔)

2
∑︁
𝑘=0

(−1)

𝑘

𝑘
∑︁

𝑞=−𝑘

⟨ ⃒(︀
)︀(𝑘) ⃒⃒ ⟩ *⃒(𝑘)𝑘(−1) 𝑛 ⃒ d ⊗ (𝑅+ + (−1) 𝑅− )d 𝑞 ⃒ 𝑛 (𝜖 ⊗ 𝜖)−𝑞𝑞

Assuming |𝑛⟩ = |𝐽, 𝑚𝐽 ⟩, we can apply the Wigner-Eckart theorem4 to⟨𝑛|d ⊗ 𝑅± d|𝑛⟩:⟨𝐽, 𝑚𝐽 | (d ⊗ 𝑅± d)(𝑘)𝑞 |𝐽, 𝑚𝐽 ⟩

⃦ ⟩
⟨ ⃦
⃦
(𝑘) ⃦
= 𝐽 ⃦(d ⊗ 𝑅± d) ⃦ 𝐽 ⟨𝐽, 𝑚𝐽 |𝐽, 𝑚𝐽 ; 𝑘, 𝑞⟩

The Clebsch-Gordan coefficient forces 𝑞 = 0. We can substitute explicit expressions
for the Clebsch-Gordan coefficient for 𝑘 = 0, 1, 2:⟨𝐽, 𝑚𝐽 |𝐽, 𝑚𝐽 ; 0, 0⟩ = 1⎧⎨ √ 𝑚𝐽𝐽 ≥ 12𝐽(𝐽+1)⟨𝐽, 𝑚𝐽 |𝐽, 𝑚𝐽 ; 1, 0⟩ =
⎩
0
else
⎧
2
⎨ √ 3𝑚𝐽 −𝐽(𝐽+1)𝐽 ≥1
𝐽(1+𝐽)(2𝐽−1)(2𝐽+3)⟨𝐽, 𝑚𝐽 |𝐽, 𝑚𝐽 ; 2, 0⟩ =
⎩
0
else

We immediately see why a non-zero angular momentum is required for non-scalareffects. In particular, if 𝐽 = 1/2, as is the case for alkali atoms, the vector term
is non-zero but the tensor term is zero. A hand-waving explanation is that one
can establish a non-zero orientation with two sublevels, but one needs at least threesublevels to establish a non-zero alignment, which is associated with tensorial effects5 .
The minimum requirement of 𝐽 = 1/2 for a vector effect and 𝐽 = 1 for a tensor
effect still holds when the hyperfine interaction between the electron total angular4

Note that the normalization of reduced matrix element depends on the choice of convention; see
Appendix D. Also, reduced matrix element does not behave like a normal matrix element under
conjugation [46].
5
A visual explanation for the difference between orientation and alignment is an asymmetric 2D
Gaussian distribution centered at zero with a small variance along the x-axis and a large variancealong the y-axis. The zero average represents zero orientation and the asymmetric variances represent
a non-zero alignment towards the y-axis.

48momentum 𝐽 and the nuclear spin 𝐼 is included.
To arrive at a simple expression for the second-order light shift, we also provide(𝑘)explicit expressions for (𝜖* ⊗ 𝜖)0 [25]:(0)(𝜖* ⊗ 𝜖)0(1)

(𝜖* ⊗ 𝜖)0(2)

(𝜖* ⊗ 𝜖)0

1
= −√
3
𝑖
1
= √ (ˆ𝜖* × 𝜖ˆ) · 𝑧ˆ = − √ 𝒜 cos 𝜃𝑘
2
2
(︀
)︀
)︀
1 (︀
1
= √ 3|ˆ𝜖 · 𝑧ˆ|2 − 1 = √ 3 cos2 𝜃𝑝 − 1
66

We have introduced two angles related to polarization: cos 𝜃𝑘 = 𝑘ˆ · 𝑧ˆ and cos 𝜃𝑝 = 𝜖ˆ· 𝑧ˆ.
The first angle is the angle between the laser wavevector and the quantization axis,
and the second angle is the angle between the polarization and the quantization axis.
We see that in order to take advantage of the vectorial effect, the wavevector needs
to be parallel to the quantization axis. The final expression for the second-order lightshift is2

𝛿𝐸 = −𝐸(𝜔)

[︂

𝑚𝐽 𝑉
𝛼𝐽 +
𝛼𝐽𝑆 + 𝒜 cos 𝜃𝑘
2𝐽(︂

3 cos2 𝜃𝑝 − 12)︂3𝑚2𝐽 − 𝐽(𝐽 + 1) 𝑇
𝛼𝐽
𝐽(2𝐽 − 1)

]︂

(3.3)

where the scalar, vector, and tensor polarizabilities 𝛼𝑆 , 𝛼𝑉 , 𝛼𝑇 are defined as
1 (0)𝛼𝐽𝑆 = √ 𝛼𝐽3
√︂2𝐽 (1)
𝛼𝐽𝑉 = −𝛼𝐽 +1 𝐽√︃
2𝐽(2𝐽 − 1)(2)𝛼𝐽𝑇 = −𝛼3(𝐽 + 1)(2𝐽 + 3) 𝐽⟨ ⃦(︀⟩)︀(𝑘) ⃦⃦⃦(𝑘)𝛼𝐽 = 𝐽 ⃦ d ⊗ (𝑅+ + (−1)𝑘 𝑅− )d ⃦ 𝐽(3.4)

Note that the frequency dependence is built in 𝑅± . For the computation of non-scalar
light shift for hyperfine states, where we assume that the detuning is sufficiently large
enough to leave the hyperfine structure unresolved, we replace |𝐽, 𝑚𝐽 ⟩ with |𝐹, 𝑚𝐹 ⟩
(𝑘)

(𝑘)

and make the substitutions 𝐽 → 𝐹, 𝛼𝐽 → 𝛼𝐹 in Eq. 3.3 and Eq. 3.4. The explicit
49

(𝑘)

(𝑘)

expressions for 𝛼𝐽 and 𝛼𝐹 are provided in Appendix D.

3.4

Wavelength selection

For state-dependent control with off-resonant laser beams, we are interested in having
a large vector shift per unit of optical power. However, since that requires a smaller
detuning for atomic species without non-zero orbital angular momentum in the groundstate manifold, one has to make a compromise between maximum achievable vector
shift and limited lifetime due to light scattering. For atomic species such as rubidium
and cesium, the presence of a large splitting between the 𝐷1 and 𝐷2 lines allows one
to achieve a reasonable (vector shift/scattering rate) ratio by setting the wavelength
between the 𝐷 lines. See Fig. 3-2 for the ratio calculated for 87 Rb. However, between

Figure 3-2: Ratio of vector shift to total light scattering rate for |𝐹, 𝑚𝐹 ⟩ = |1, −1⟩
for different polarizations. The vector shift is defined here as the difference between
the light shift of the chosen circular polarization and that of the opposite circularpolarization. The assumed beam parameters are incident power of 0.1 W and 1/𝑒2
beam radius of 125 𝜇m.
the 𝐷 lines the nature of the light scattering is largely spontaneous Raman scatteringfrom one hyperfine state to another, as we see in Fig. 3-3. Since spin impurities may
be deleterious for quantum simulation of spin chains in an optical lattice, it is best50

to select a wavelength where Raman scattering is minimized. Thus we decided to
set the wavelength of our state-dependent optical lattice to 810 nm where Rayleighscattering is dominant.

51

Figure 3-3: Light scattering rate calculation for |𝐹, 𝑚𝐹 ⟩ = |1, 1⟩ with incident powerof 0.1 W and 1/𝑒2 radius of 125 𝜇m. The dotted lines show the locations for 𝐷1 and𝐷2 resonances. (a) 𝜎 + polarization. (b) 𝜎 − polarization.

52

Chapter 4
Optical Lattices
An optical lattice is a standing wave of light that provides a periodic potential foratoms. It is widely used as a primary tool in quantum simulation and atom-based
metrology experiments. A small sample of the spectacular applications of this tool
includes the observation of long-range anti-ferromagnetic ordering in fermions [47] andthe realization of 10−18 frequency stability by an optical lattice clock [48]. A generalsurvey of modern research topics being investigated in optical lattice experiments canbe found in [49–51]. A review of the dynamics of matter waves inside an opticallattice can be found in [52]. Some of the non-cubic lattice geometries are discussed
in [53]. Our lab is interested in using a state-dependent optical lattice for controllingthe interaction between atoms in different hyperfine states1 . Hence this chapter will
review the properties of a state-dependent lattice and its implementations.

4.1

Making an optical lattice

We first review the simplest form of an optical lattice: the state-independent or scalarlattice with cubic symmetry. Assume we have a laser whose electric field can beidealized as a plane wave that propagates in the +𝑥 direction with angular frequency
𝜔, wavevector 𝑘 = 𝜔/𝑐 and complex amplitude 𝐸0 . Without loss of generality, we
1

In this context, the hyperfine states all belong to the ground state manifold. Another example
of a state-dependent control is the provision of a differential light shift between an excited state anda ground state [54].

53assume the field is linearly polarized in the 𝑧-direction. Then the total electric field
after a retroreflection is⃗𝐸(𝑥,𝑡) = (𝐸0 𝑒𝑖𝑘𝑥−𝑖𝜔𝑡 + 𝐸0 𝑒−𝑖𝑘𝑥−𝑖𝜔𝑡 )ˆ
𝑧
The time-averaged intensity of this electric field in vacuum with permittivity 𝜀0 written as1⃗ 2 = 𝜀0 𝑐|𝐸0 |2 cos(𝑘𝑥)2
𝐼(𝑥) = 𝜀0 𝑐|𝐸|2
The potential seen by atoms with scalar dynamic polarizability 𝛼(𝜔) (Eq. 3.1) atangular frequency 𝜔 is
𝑈 (𝑥) = −1Re[𝛼(𝜔)]Re[𝛼(𝜔)]𝐼(𝑥) = −|𝐸0 |2 cos(𝑘𝑥)22𝜀0 𝑐
2One can generalize the one-dimensional lattice to three dimensions by adding
retro-reflected laser beams along other directions. By using orthogonal polarizations
and introducing reasonable frequency offsets, the potential is separable and we canwrite the total potential as 𝑈3𝐷 (𝑥, 𝑦, 𝑧) = 𝑈 (𝑥) + 𝑈 (𝑦) + 𝑈 (𝑧). Intricate interference
between electric fields of lasers with different orientations of wavevectors can give rise
to non-separable potentials, such as hexagonal [55], honeycomb [56], or Kagome [57]lattices in two dimensions.
The behavior of a single atom trapped in periodic potential can be found by diagonalizing the Hamiltonian 𝐻 = 𝑝2 /2𝑚 + 𝑈 (𝑥) with the help of Bloch’s theorem [58].
The eigenstates |𝜓𝑛,𝑘 ⟩ are labeled by band index 𝑛 where 𝑛 = 0, 1, 2, . . . and crystal
momentum (also known as quasimomentum) ~𝑘, where −𝜋/𝑎𝐿 ≤ 𝑘 ≤ 𝜋/𝑎𝐿 and 𝑎𝐿
is the spatial periodicity or the lattice constant of the potential. By Bloch’s theorem,
the eigen-wavefunctions 𝜓𝑛,𝑘 (𝑥) take the form 𝑒𝑖𝑘𝑥 𝑢𝑛,𝑘 (𝑥) where 𝑢𝑛,𝑘 (𝑥) is periodic
with periodicity 𝑎𝐿 . These so-called Bloch functions are delocalized in space.
However, for the study of interacting atoms, the Bloch state basis is rather inconvenient to work with, because for many atomic species the dominant form of54

interaction is a short-range interaction by 𝑠-wave collisions and hence interaction
takes place locally at individual lattice sites. Thus, it is preferable to transform to abasis where the wavefunctions are maximally localized, and such a basis is known as
the Wannier basis. Each Wannier function is labeled by band index 𝑛 and coordinate
number 𝑅, which is the spatial coordinate of the center of the function. Without lossof generality, the transformation from the Bloch basis to the Wannier basis in one
dimension is defined as

𝐿|𝑤𝑛,𝑅 ⟩ =
2𝜋

∫︁

BZ

𝑑𝑘 𝑒−𝑖𝑘𝑅 𝑒𝑖𝜃(𝑘) |𝜓𝑛,𝑘 ⟩where 𝐿 is the length of the lattice and BZ stands for integration over the first Brillouin zone. A good choice of “gauge” function2 𝑒𝑖𝜃(𝑘) yields a wavefunction that is
maximally localized, i.e., yields minimal ⟨𝑥2 ⟩ [59]. See Figure 4-1 for a comparison
between a Bloch function and a Wannier function. Although it is customary for solidstate physics textbooks to introduce the Wannier basis states as simple superpositions of Bloch states [58], with 𝑒𝑖𝜃(𝑘) appearing to be unity3 , a direct application ofsuch a definition to the numerically obtained Bloch wavefunctions leads to Wannier
wavefunctions that are not well-localized over single or several lattice sites. For onedimensional periodic potentials which are even under inversion, it has been provedthat for every band there exists a unique Wannier function satisfying the following
three properties [60]:1. It is real.
2. It is either even or odd under inversion about either 𝑥 = 0 or 𝑥 = 𝑎𝐿 /2.
3. It falls off exponentially.
One can generalize the transformation to involve multiple bands by promoting 𝑒𝑖𝜃(𝑘) to a unitarymatrix [59], although this is not necessary for lattices with simple band structures such as cubic
lattice.
3
Technically there is nothing wrong with this, as the Bloch basis states can be defined in such a
way that they absorb the phase 𝑒𝑖𝜃(𝑘) .
255

Figure 4-1: Comparison between Wannier and Bloch functions. The black solid line
represents a one-dimensional lattice with lattice depth 12 𝐸𝑅 where 𝐸𝑅 = ℎ2 /(2𝑚𝜆2 )is the recoil energy of the lattice laser. The solid red line denotes the modulus of theprobability amplitude of a ground band Wannier function centered at 𝑅 = 0. The
dotted red line denotes the modulus of the probability amplitude of a ground band
Bloch function with wavevector 𝑘 = 𝜋/𝑎𝐿 . The gray bands represent the three lowest
energy bands of the lattice potential, where the bandwidth is inversely related to theeffective mass inside the lattice.

Furthermore, the exponential decay rate of such a Wannier function is maximal, wherein the tight-binding limit the decay rate is proportional to the inverse of the length√︀scale given by the band energy gap ~/ 2𝑚𝐸gap . Properties 1 and 2 suggest that thedesired Wannier function for the 𝑛-th band can be simply obtained by choosing 𝜃(𝑘)to be conjugate to the phase angle of either 𝜓𝑘,𝑛 (𝑥 = 0) or 𝜕𝑥 𝜓𝑘,𝑛 (𝑥 = 0), dependingon the symmetry of the 𝑛-th band orbitals (see Section 6 in Ref. [60]).
56

4.2

State-dependent optical lattice

If the atoms have a non-zero vector polarizability, we can generate a state-dependentlattice: atoms in different internal states experience different light shifts under thesame laser, as we saw in Section 3.3. By introducing a polarization gradient along
the lattice beam, we can adjust the differential light shift between two internal states.
In the present context, a polarization gradient is a periodic spatial modulation of
the circularity of the local polarization. There are two schemes to implement such
modulation: lin-𝜃-lin and 𝜎 + − 𝜎 − .
It is worth mentioning that there are important similarities and differences between polarization gradients used for state-dependent optical lattices and those used
for polarization gradient cooling, which is a sub-Doppler laser cooling technique [61].
The type of polarization gradient created in the lin-𝜃-lin scheme is related to theone used in the lin-⊥-lin polarization gradient cooling method, where interference between a linear polarized beam and a counter-propagating beam with a rotated linear
polarization creates a periodic modulation in the circularity of the local polarization.
On the other hand, for the 𝜎 + -𝜎 − cooling method, a 𝜎 + beam propagates against a
counter-propagating 𝜎 − beam, and the local polarization always stays linear but its
angle rotates as it travels along the beam axis. However, in the 𝜎 + -𝜎 − scheme forstate-dependent lattices, the 𝜎 + and 𝜎 − beams copropagate in the same direction andcreates a spatial modulation in the circularity of the local polarization.
Each scheme comes with two variants. The lin-𝜃-lin scheme requires the use of avariable retarder, and one has the option of placing the variable retarder before orafter the atom cloud. The 𝜎 + − 𝜎 − scheme requires two orthogonally polarized andindependent beams, and one has the option of introducing either a phase offset or afrequency offset between the input beams. In our experiment, we explored the uses ofthe secondary option for each scheme. Below we analyze in detail the implementationof the secondary option for each scheme but also make connections to the implementations of the primary option for each scheme realized by other groups [41, 62].
57

4.2.1lin-𝜃-lin scheme

In this scheme, the input lattice beam is linearly polarized. The input beam goes
through a variable waveplate and other fixed waveplates so that the retroreflectedbeam has a linear polarization that is rotated with respect to the input polarization. We will discuss the physical implementations of the polarization rotation by the
waveplates later in this subsection.
Assume we have a total electric field described by
⎤
⎛
⎞
⎡⎛ ⎞
cos 𝜃
1
⃗ = 𝐸0 𝑒−𝑖𝜔𝑡 ⎣⎝ ⎠ 𝑒𝑖𝑘𝑧 + ⎝⎠ 𝑒𝑖𝜑 𝑒−𝑖𝑘𝑧 ⎦ + c.c
𝐸
sin 𝜃
0
The phase offset 𝑒𝑖𝜑 of the retroreflected light results in a spatial offset of the intensityand polarization gradients [𝑥 → 𝑥 − 𝜑/(2𝑘)], so we set 𝜑 = 0 without loss of generality4 . If we denote the ratio of the vector polarizability to the scalar polarizability
𝛼𝐹𝑉 /𝛼𝐹𝑆 to be 2𝑅, then using Eq. 3.3 we can write the total light shift as(︁𝑚𝐹 )︁
𝛿𝐸(𝜃, 𝑧) = −|𝐸0 |2 𝛼𝐹𝑆 2 + 2 cos(2𝑘𝑧) cos 𝜃 − 2𝑅 cos 𝜃𝑘 sin(2𝑘𝑧) sin 𝜃𝐹where 𝜃𝑘 is the angle between the quantization axis and the lattice wavevector. If wechoose two hyperfine states with 𝑚𝐹 = ±𝐹 , we can simplify the expression using thetrigonometric identity 𝐴 cos(2𝑘𝑧 + 𝐵) = 𝐴 cos(𝐵) cos(2𝑘𝑧) − 𝐴 sin(𝐵) sin(2𝑘𝑧):
(︁
(︁
)︁)︁
𝑚𝐹
𝛿𝐸(𝜃, 𝑧) = −|𝐸0 |2 𝛼𝐹𝑆 2 + 2𝐴(𝜃) cos 2𝑘𝑧 +
𝐵(𝜃)
𝐹
√︁
𝐴(𝜃) =
cos2 𝜃 + (𝑅 cos 𝜃𝑘 )2 sin2 𝜃𝐵(𝜃) = arctan(𝑅 cos 𝜃𝑘 tan(𝜃))See Fig. 4-2 for an example of how different hyperfine states are spatially separated
by the state-dependent lattice. Note that a smaller vector-to-scalar ratio results in anarrow tuning range for the spin separation. The product of the local intensity and4However, this shows that 𝜑 needs to be stable, or the optical lattice has a phase noise and cangive rise to a nonzero transition rate from the ground band to the first excited band [63].

58the local circularity of the polarization 𝒜(𝑥) acts a source for a fictitious magnetic
field that separates atoms with opposite magnetic moments. The gradient of thefictitious magnetic field is strongest at the nodes and the antinodes of the intensity
modulation. If −2|𝐸0 |2 𝛼𝐹𝑆 is equivalent to a potential depth of 10 kHz, then thefictitious magnetic gradient at the intensity extrema seen by |𝐹 = 1, 𝑚𝐹 = −1⟩ isequivalent to 277 G/cm for (𝑅 cos 𝜃𝑘 , 𝜃) = (1/8, 𝜋/2).
The simplest method to achieve polarization rotation of a retroreflected laser beam
is to insert a quarter-waveplate in front of the retroreflecting mirror. However, one
would have to mechanically rotate the quarter-waveplate to dynamically change thespin separation, and that is not ideal in terms of speed and stability. A common way toachieve electric control of polarization rotation is to use a voltage variable waveplate,
in which the retardance along the slow axis of the waveplate can be adjusted by an
external voltage. However, with a variable waveplate alone, the polarization of theretroreflected light is not linear in general, except for special values of retardances.
Hence the variable waveplate needs to be combined with other fixed waveplates.

Variable waveplate after the atom cloudA general setup for polarization rotation control with the variable waveplate placed
after the cloud is described in Fig. 4-3(a). The setup has two fixed waveplates and asingle variable waveplate. This setup features an extra half-waveplate (HWP) added
to the setup described in [40]. The HWP controls the rotation angle of the polarization
at zero voltage applied to the variable waveplate. The quarter-waveplate (QWP),
whose slow axis is oriented 45∘ with respect to that of the variable waveplate, ensuresthat the output is linearly polarized. With the input polarization set to a verticallyoriented linear polarization [(𝐸𝑥 , 𝐸𝑦 ) = (0, 1)], and the angles of the slow axes asshown in Fig. 4-3, the polarization of the retroreflected beam (𝐸𝑥 , 𝐸𝑦 ) can be shownusing Jones matrices [64] to be
⎛

⎞

⎛

⎞
𝐸
− cos (4𝜃𝐻 + 𝜂)
⎝ 𝑥 ⎠ = 𝑒𝑖𝜂 ⎝⎠,𝐸𝑦sin (4𝜃𝐻 + 𝜂)
59

(︀
)︀
Figure 4-2: (a) Plot of 2𝐴(𝜃) cos 2𝑘𝑧 + 𝑚𝐹𝐹 𝐵(𝜃) for (𝑅 cos 𝜃𝑘 , 𝜃) = (1/4, 𝜋/4). The
red and blue lines are the modulation of the total potential seen by |𝐹 = 1, 𝑚𝐹 = 1⟩
and |𝐹 = 1, 𝑚𝐹 = −1⟩ hyperfine states. The gray dotted line is the local circularityof the polarization 𝒜(𝑥). The green dashed line is the local intensity. The productof the local intensity and the local circularity acts as a fictitious magnetic field that
separates atoms with opposite magnetic moments. Note that the extrema of thecircularity do not exactly match the extrema of the potentials seen by the different
hyperfine states. (b) Separation between 𝑚𝐹 = 𝐹 and 𝑚𝐹 = −𝐹 states given by
2𝐵(𝜃) vs. rotation angle of the polarization, for different values of 𝑅 cos 𝜃𝑘 .

60

Figure 4-3: Polarization rotation control for state-dependent lattices. (a) Variable
retarder after the atom cloud. The setup consists of a half-waveplate (HWP), quarterwaveplate (QWP), and a variable retarder such as a electro-optic modulator (EOM)or a liquid crystal (LC) device. The dotted lines denote the vertical axis. The thicksolid lines denote the slow axes of the waveplates. With the input polarization (bluearrow) vertically oriented and the slow axes oriented as in the diagram, the output
polarization (red arrow) is linear and rotated by 4𝜃𝐻 + 𝜂 from the input polarization,
where 𝜃𝐻 is the HWP angle and 𝜂 is the retardance by the LC slow axis. The output
electric field also has a phase delay (not shown) of 𝑒𝑖𝜂 with respect to the input field.
(b) Variable retarder before the atom cloud. Figure adapted from Fig. 1(a) in [41].

61where 𝜃𝐻 is the rotation angle of the HWP and 𝜂 is the retardance of the slow
axis of the variable waveplate. If the QWP does not provide exactly 𝜆/4 retardanceor its slow axis is not oriented 45∘ with respect to that of the variable waveplate,then the polarization of the retroreflected beam is elliptical in general. If 𝛼 is thedeviation of the QWP retardance from 𝜆/4 and 𝛽 is the deviation of the relative
angle between the QWP and the variable waveplate from 45∘ , then one can showthat 𝐸𝑦 has an amplitude cos(𝜂) sin(𝛼) + cos(𝛼) sin(2𝛽) sin(𝜂) in quadrature to therest of the amplitudes. This residual quadrature amplitude forms a secondary stateindependent optical lattice with the input polarization that is added in quadrature to
the intensity modulation of the main optical lattice. Such a secondary optical latticeleads to a noticeable imbalance between the potential depths seen by hyperfine stateswith opposite 𝑚𝐹 .

Variable waveplate before the atom cloud
One can also place the variable waveplate before the atom cloud, as it was demonstrated in the state-dependent optical superlattice experiment in Ref. [41]. The main
advantage of this setup is that the variable plate does not introduce any intensity imbalance between the input beam and the retroreflected beam and the variable plate
selectively rotates the polarization of one beam while not affecting that of the otherbeam, making it suitable for an optical superlattice experiment. Another advantage
is that if the variable waveplate were to introduce fluctuations around the retardance
setpoint 𝜂, it would not affect the optical path length between the atom cloud andthe retroreflecting mirror5 . The setup is described in Fig. 4-3(b). We assume theslow axes of the two QWP’s are oriented vertically and the EOM slow axis is oriented+45∘ with retardance of 𝜂. After a horizontally polarized beam passes through the
EOM and the two QWP’s, the input polarization and the output polarization at the5

An LC device is susceptible to introducing fast fluctuations around the retardance setpoint
because of its operation mechanism.

62

location of the atom cloud is
⎛
⎝

𝐸𝐻
𝐸𝑉

⎞

cos 𝜂2𝑖𝜂/2 ⎝⎠ =𝑒in⎛− sin 𝜂2

⎞ ⎛
⎠,⎝

𝐸𝐻
𝐸𝑉

⎞
⎠

out

= −𝑒

⎛

⎞

cos 𝜂2𝑖𝜂/2 ⎝⎠sin 𝜂2Thus, the output polarization is rotated from the input polarization by 𝜋+𝜂. Similarly
to the setup with the variable waveplate after the cloud, the relative angle between
the EOM and the QWP must be ±𝜋/4 to ensure the linearity of the polarization.
Residual elliptical polarization due to a folding mirror
Besides the imperfection in the QWP retardance and in the relative angle alignment
between the QWP and the LC/EOM device, optical elements between the waveplates
can impact the fidelity of the polarization control. In particular, folding mirrors canimpart a nominally small phase shift between the 𝑠 and 𝑝 polarizations that nevertheless results in a substantial imbalance in the state-dependent potential. Figure 4-4
shows a comparison between a simulation of an additional phase shift of 𝜆/35 between𝑠 and 𝑝 polarizations due to a folding mirror (a protected silver mirror from Thorlabs)
before the QWP in Fig. 4-3(a) and measured imbalance in the 810 nm lattice potential
seen by |𝐹, 𝑚𝐹 = 1, −1⟩ and |1, 1⟩. We see that the added phase shift nicely explainsthe 20% potential imbalance we observed, which is significant in that a phase shiftof 𝜆/35 is probably negligible for most other purposes. A lattice depth imbalance of
20% results in a large imbalance in the second-order tunneling amplitudes for the two
hyperfine states, due to the exponential dependence of tunneling amplitude on lattice
depth. Thus it is strongly advised that the state-dependent lattice optics be kept ona single optical axis as was done in our setup for the experiments described below.
LC vs EOM
The device choice for the variable waveplate is also important for high fidelity polarization control. One can use a liquid crystal (LC) variable retarder or an electro-optic
modulator (EOM). A qualitative comparison between the two devices is summarizedin Table 4.1. It is immediately clear that the EOM is a superior device in terms of63

Figure 4-4: Imbalance in the state-dependent lattice potential depth caused by afolding mirror (Thorlabs ultrafast-enhanced silver mirror UM10-AG) in front of the
QWP in Fig. 4-3(a). The blue dots are the estimated imbalance ratio caused by anadditional phase lag of 𝜆/35 by the 𝑠 polarization. The red dots are imbalance ratiosof the lattice depths estimated from Kapitza-Dirac scattering data.
Table 4.1: Comparison of LC and EOM performance

Response time
Voltage control
Intensity noise
Refractive index uniformity

LC
𝒪(100 ms)
nonlinear
large
good

EOM
< 1 mslinearsmall
poor

ease of control, but its major disadvantage is the refractive index inhomogeneity. As
the voltage applied to the EOM changes, the curvature of the refractive index alsochanges and results in a deflection of the beam6 [18]. The inhomogeneity also leads
to a worse polarization extinction ratio. It is possible to minimize the effect of therefractive index inhomogeneity by making the beam spot passing through the EOM
as small as possible. However, nonlinear crystals (e.g. lithium niobate) used for the
EOM may suffer from photorefractive damage at high intensity [65]. We were able tominimize the beam deflection by putting the EOM after a focusing lens and before the6

The device tested was Newport electro-optic amplitude modulator (Model 4102) [discontinued].

64

retroreflecting mirror, but the quoted threshold intensity for photorefractive damage
was not high enough for our application. In principle, the beam displacement canalso be minimized if the EOM is constructed with multiple units of carefully matchedcrystals [66]. Typical amplitude modulators are equipped with two matched crystals
oriented at ±45∘ , which cancels thermal drift due to the natural birefingence while
doubling the polarization rotation [65]. If another pair of matched crystals is added7 ,the beam displacement due to a gradient in the natural birefringence may be canceled. Such a four-matched-crystal modulator is sold by QiOptiq/LINOS [67], but
its main downside is that the device is rather heavy and quite bulky in size. Due to
its long lead time and our limited optical breadboard space, we decided to continue
using the device of our default choice, which is the LC waveplate.
Since polarization purity is of utmost importance for dynamic tuning of a statedependent lattice, we decided to use a LC device. The main problem with the LC
device is its control mechanism, which relies on reducing the inherent birefringenceof the LC molecules by rapidly modulating the direction of the electric field. This
unavoidably leads to intensity noise added at the modulation frequency and causes
hysteresis in the switching speed (increasing the voltage leads to a faster retardancechange than decreasing the voltage). It can also produce lattice phase noise by modulating the global phase 𝑒𝑖𝜂 in the polarization of the laser but this can be prevented
by placing the LC in front of the atom cloud. It is possible to change the retardanceof the LC faster than its natural switching speed by forcing the control voltage higher
than the desired value and then reducing it - this is known as the transient nematiceffect [68]. To make use of this, we manually tune the control voltage pulse shape forevery combination of initial and final retardances and ramping times.

4.2.2

𝜎 + − 𝜎 − scheme

In this scheme, the state-dependent lattice is formed by a combination of two independent lattice beams with 𝜎 + and 𝜎 − polarizations respectively. One has the option
to add either a frequency offset or a phase offset between the two beams (Fig. 4-5).
7

Possibly oriented at 180∘ ± 45∘ .

65

We first analyze the scheme with a frequency offset.

Figure 4-5: The 𝜎 + -𝜎 − scheme for state-dependent lattices. A 𝜎 + beam (red arrow)and a 𝜎 − beam (blue arrow) copropagate in the same direction. (a) Scheme withfrequency offset. There is no lattice phase offset at the surface of the retroreflecting
mirror, but a frequency offset results in an accumulated lattice phase offset at thelocation of the atom cloud. (b) Scheme with phase offset. Instead of a retroreflecting
mirror, a linearly polarized beam (black arrow) counterpropagates against the input
lattice beams. A phase offset of 𝜑 between the 𝜎 + beam and the 𝜎 − beam results ina lattice phase offset of 𝜑/(2𝑘), where 𝑘 is the wavevector of the input beams. The
value of 𝜑 can be continuously ramped by an AOM [62].

Frequency offset between 𝜎 + and 𝜎 − beams
A large tunable frequency offset is introduced between the two beams so that the
separation between the antinodes of the two lattices can be adjusted while making
sure that there is no interference between the two beams (Fig. 4-5(a)). Assume that
one of the lattice beams (𝑉𝜎− ) has frequency 𝑓 while the other lattice beams (𝑉𝜎+ )has frequency 𝑓 + ∆𝑓 . We also assume that ∆𝑓 is much smaller than 𝑓 while muchlarger than the largest energy scale associated with the external states of the atom inthe optical lattice, so that we can time-average away any interference formed between
66𝑉𝜎+ and 𝑉𝜎− . Then the total potential can be expressed as𝑉total = 𝑉𝜎+ cos2 ((𝑘 + ∆𝑘)𝑧) + 𝑉𝜎− cos2 (𝑘𝑧)
= 𝑉𝜎+ cos2 (𝑘(𝑧𝑎 + 𝐿) + ∆𝑘(𝑧𝑎 + 𝐿)) + 𝑉𝜎− cos2 (𝑘(𝑧𝑎 + 𝐿))𝑉𝜎−𝑉𝜎 +(1 − cos(2𝑘𝑧𝑎 + 2𝑘𝐿 + 2∆𝑘(𝑧𝑎 + 𝐿)) +(1 − cos(2𝑘𝑧𝑎 + 2𝑘𝐿))=22
where 𝑧 = 𝑧𝑎 + 𝐿 is the distance between the observation point and the retroreflecting
mirror, 𝑧𝑎 is the distance between the observation point and the center of the atom
cloud, and 𝐿 is the distance between the center of the atom cloud and the retroreflecting mirror. Since atom cloud size is very small, we can ignore 2∆𝑘𝑧𝑎 . Then we
see that 𝑉𝜎+ has a lattice phase offset of 2∆𝑘𝐿 = 4𝜋(∆𝑓 /𝑐)𝐿. In order to scan thefull range of spin separation, the phase offset needs to have a tuning range of at least
𝜋. For 𝐿 = 1 m, this requires that ∆𝑓 have a tuning range of 75 MHz. Similar towhat we have shown for the lin-𝜃-lin scheme, we can rewrite the total potential as
single cosinusoidal term, so that the expression for the physical separation between
different hyperfine states becomes clear. If we denote the lattice phase offset as 𝜃,
then
𝑉𝜎+
𝑉𝜎−
(1 − cos(2𝑘𝑧 + 𝜃/2)) +(1 − cos(2𝑘𝑧 − 𝜃/2))22√︀ 2
𝑉𝜎+ + 𝑉𝜎2− + 2 cos(𝜃)𝑉𝜎+ 𝑉𝜎−
𝑉𝜎+ + 𝑉𝜎−
=
−
2
(︂2
(︂
)︂)︂
𝑉𝜎 + − 𝑉𝜎 −𝜃× cos 2𝑘𝑧 + arctantan.
𝑉𝜎 + + 𝑉𝜎 −2

𝑉total =Recalling that we defined 2𝑅 to be the ratio 𝛼𝐹𝑉 /𝛼𝐹𝑆 , we can write 𝑉𝜎+/− in termsof input electric field and 𝑅. Assume that the electric field amplitude for 𝑉𝜎+ is
𝐸𝜎+ 𝑒−𝑖𝜔𝑡 (𝑒𝑖𝑘𝑧 + 𝑒−𝑖𝑘𝑧 ) + c.c. and similarly for 𝑉𝜎− . Then the maximum intensity foreach lattice is 8𝜀0 𝑐|𝐸𝜎+/− |2 . Also,
(︁
𝑚𝐹 )︁
.
𝑉𝜎+/− = 4|𝐸𝜎+/− |2 𝛼𝐹𝑆 1 ± 𝑅 cos 𝜃𝑘𝐹67If we assume that 𝑉𝜎+ and 𝑉𝜎− have equal input power, then the above expression
helps us to rewrite the state-dependent position shift as
arctan(︂𝜃𝑉𝜎+ − 𝑉𝜎−tan
𝑉𝜎+ + 𝑉𝜎−
2

)︂

(︂

𝑚𝐹
𝜃
= arctan 𝑅 cos 𝜃𝑘
tan
𝐹
2

)︂

.

If 𝑚𝐹 = ±𝐹 , then the separation between the two hyperfine states is
𝜆arctan(𝑅 cos 𝜃𝑘 tan(𝜃/2)).
2𝜋
At 𝜃 = 𝜋, we achieve the maximal separation of 𝜆/4. Just as in the lin-𝜃-lin scheme,
we see that the physical separation of the hyperfine states is a nonlinear function of
the control parameter 𝜃, even though the separation between the antinodes of 𝑉𝜎+ and𝑉𝜎− is linear in 𝜃. The separation becomes a linear function of 𝜃 if |𝑅 cos 𝜃𝑘 | = 1. Insuch a situation, hyperfine states respond to only one of the two circular polarizations
and not to the other.
The 𝜎 + − 𝜎 − scheme is implemented by combining two orthogonally polarized
input beams on a polarizing beam splitter and then passing the output through aquarter waveplate. One of the beams (with frequency 𝑓 + ∆𝑓 ) is controlled by adouble-pass AOM. Because of the limited double-diffraction efficiency at high RF
frequency and lack of constructive interference between the two beams, the 𝜎 + − 𝜎 −scheme requires more optical power than the lin-𝜃-lin scheme. For the configuration
where the two hyperfine states are merged, the 𝜎 + − 𝜎 − requires twice as much input
power as the lin-𝜃-lin scheme because the circular polarization components interferewith their retro-reflections individually.

Phase offset between 𝜎 + and 𝜎 − beams
So far, all of the schemes discussed are limited in the range of polarization rotation.
When using an LC or EOM the retardance change is bounded, and the window overwhich the frequency of an AOM can be tuned is relatively narrow. While one only
needs a rotation of 𝜋 to go from a minimal separation to a maximal separation between
68different hyperfine states, additional rotation beyond 𝜋 can separate the atoms evenmore, resulting in a differential transport of atoms by an arbitrary distance. Such ascheme may be useful for measuring off-site correlations between different hyperfine
states. In principle, a single retarder can achieve arbitrary displacement of a latticesite by doing a stick-slip motion of slow rotation followed by a rapid initialization tozero rotation. However, such control is highly diabatic and will increase the motionaltemperature of the atoms. It turns out that a smooth, endless polarization rotation
control can be achieved with a series of multiple retarders controlled by a complex
algorithm [69], but it may be difficult to accommodate all the devices for a quantum
gas experiment. Robens et al. demonstrated a relatively simpler approach in [42, 62]
where a phase ramp between a 𝜎 + beam and a 𝜎 − beam counterpropagating against
a linearly polarized beam (with all frequencies of the beam being equivalent) achieves
state-dependent transport of atoms over an arbitrary number of lattice sites. A linear
polarization can be seen as a superposition of 𝜎 + and 𝜎 − polarizations, and hence
the interference of the linearly polarized beam with the two independent 𝜎 + and𝜎 − beams forms a superposition of a 𝜎 + lattice and a 𝜎 − lattice (Fig. 4-5(b)). By
working at a wavelength where |𝑅 cos 𝜃𝑘 | = 1, lattice sites can be directly controlled

by changing the phase offset between the input 𝜎 + and 𝜎 − beams. A linear change
in the phase offset as a function of time (𝜑 = 2𝜋∆𝑓 𝑡) can be simply implemented
by introducing a tiny frequency offset ∆𝑓 between the 𝜎 + and the 𝜎 − beams (e.g.
∆𝑓 = 1 kHz translates one lattice site per millisecond). However, this scheme is notwithout technical challenges as it involves not only intensity stabilization but also
phase stabilization of individual lattice beams [62].

69

70

Chapter 5
The Bose-Hubbard Hamiltonian
The Bose-Hubbard Hamiltonian describes the dynamics of cold bosonic atoms in
optical lattices. It was proposed by Jaksch et al. in Ref. [70] that a BEC loaded
to an optical lattice realizes a Bose-Hubbard model studied in condensed matterphysics [71], where at zero temperature and without disorder, the bosonic system canundergo a quantum phase transition from a superfluid state to a Mott insulating stateas the strength of the repulsive interaction reaches a threshold value. The predictionof the superfluid-to-insulator quantum phase transition was validated experimentallyin Ref. [7] in 2002, and since then there have been numerous experiments studying
the properties of the Bose-Hubbard Hamiltonian and its extensions, along with theiranalogues in fermions [51]. Most importantly at present, however, the Bose-Hubbard
Hamiltonian is the starting ground for the effective quantum spin Hamiltonian that
we explore experimentally. In this chapter, we explain how to obtain the mean-field
phase diagram of the Bose-Hubbard Hamiltonian realized in an optical lattice anduse it to estimate the volume of the Mott phase within the trapped atom cloud.

5.1

Review of the derivation

Because it is crucial to understanding corrections to the Hamiltonian that will be discussed later, we briefly review the derivation of the (spinless) Bose-Hubbard Hamilonian with single-band approximation as shown in Ref. [70]. The starting point is the71field operator description of the Hamiltonian of bosonic atoms in a periodic potential
𝑉 (𝑥), interacting with a two-body interaction term 𝑔𝑈 (𝑥 − 𝑥′ ):
)︂
(︂
~2 2
†
ˆ
ˆ
∇ + 𝑉 (𝑥) 𝜓(𝑥)𝐻 =𝑑 𝑥 𝜓 (𝑥) −2𝑚∫︁𝑔ˆ ′ )𝜓(𝑥)ˆ+𝑑3 𝑥 𝑑3 𝑥′ 𝜓ˆ† (𝑥)𝜓ˆ† (𝑥′ )𝑈 (𝑥 − 𝑥′ )𝜓(𝑥
2
∫︁

3

(5.1)

ˆ
Here 𝜓(𝑥)(𝜓ˆ† (𝑥)) is the bosonic field annihilation (creation) operator at position 𝑥.
For many atomic species the interaction is short-ranged, so we use the pseudopotential1 𝑔𝑈 (𝑥 − 𝑥′ ) = (4𝜋~2 𝑎𝑠 /𝑚)𝛿(𝑥 − 𝑥′ ). Then we make the ansatz that the field
operator can be expanded in the basis of ground band Wannier orbitals 𝑤𝑖 (𝑥), whose
centers are indexed by lattice sites 𝑖:
ˆ
𝜓(𝑥)
=

∑︁

𝑎
ˆ𝑖 𝑤𝑖 (𝑥)𝑖This ansatz is valid if the atomic cloud is sufficiently cold such that all the atoms
occupy the ground band. Note that the Wannier orbital 𝑤(𝑥) is actually a product of three one-dimensional Wannier orbitals for a three-dimensional cubic lattice2 .
However, interaction may virtually admix some higher band Wannier orbitals [72–74],
which will be discussed later. With this single-band approximation, and after using
the bosonic commutation relations for 𝑎
ˆ𝑖 and 𝑎ˆ†𝑖 , we arrive at
𝐻 = −𝑡)︁ 𝑈 ∑︁∑︁ (︁ †𝑎
ˆ𝑖 𝑎
ˆ𝑗 + c.c +𝑛ˆ 𝑖 (ˆ
𝑛𝑖 − 1),
2 𝑖
⟨𝑖,𝑗⟩

1

This form does not work if the wavefunction is singular at the origin, in which case one needs
to regularize the delta function, but we do not worry about it here.
2
Three-dimensional Wannier orbitals are not expected to be separable for a general lattice geometry.

72where ⟨𝑖, 𝑗⟩ denotes neighboring pair of indices and the tunneling amplitude 𝑡 andthe interaction strength 𝑈 are defined as
(︂
)︂
~2 2
∇ + 𝑉 (𝑥) 𝑤(𝑥)𝑡 = − 𝑑 𝑥 𝑤(𝑥) −
2𝑚
∫︁
4𝜋~2
𝑈 =
𝑎𝑠 𝑑3 𝑥 |𝑤(𝑥)|4 .
𝑚
∫︁

3

*

Note that we have neglected next-nearest neighbor tunneling and interaction between
neighboring atoms because such effects are very small due to the localization of the
Wannier orbitals.
When 𝑡 ≫ 𝑈 , the ground state of the system is a superfluid state (|𝜓𝑈 =0 ⟩ ∝∑︀
exp(𝛼 𝑖 𝑎ˆ𝑖 † )|0⟩), where there is phase coherence between lattice sites. The local

occupation number has a well-defined average with some variance around the averagevalue. On the other hand, in the opposite limit 𝑡 ≪ 𝑈 , the ground state is a Mott∏︀ † 𝑛𝑎𝑖 ) |0⟩) which has no phase coherence between lattice
insulating state (|𝜓𝑡=0 ⟩ ∝ 𝑖 (ˆsites and has integer occupation number with zero number fluctuation. The superfluid
phase has a gapless excitation spectrum where the elementary excitation is a phonon.
On the other hand, the Mott insulator phase has a gap in the excitation spectrum,
where the elementary excitation is a quasiparticle-quasihole pair. Its excitation energy
at zero momentum is zero at the phase transition but quickly grows to ≈ 𝑈 as 𝑈/𝑡increases beyond the critical value [75].

5.2

Mean-field phase diagram

It is instructive to use a mean-field approximation to obtain the phase diagram of the
Bose-Hubbard Hamiltonian and see where the superfluid-insulator boundary lies in
the parameter space. We posit that the order parameter for the phase transition is⟨ˆ𝑎𝑖 ⟩, which is zero in the Mott insulating phase and nonzero in the superfluid phase.
The operator 𝑎
ˆ𝑖 can be expanded around this expectation value: 𝑎ˆ𝑖 = ⟨ˆ𝑎𝑖 ⟩+𝛿ˆ
𝑎𝑖 . Then73the tunneling term can be approximated as𝑎ˆ†𝑖 𝑎ˆ𝑗 = (⟨ˆ𝑎†𝑖 ⟩ + 𝛿ˆ𝑎†𝑖 )(⟨ˆ𝑎𝑗 ⟩ + 𝛿ˆ𝑎𝑗 )= ⟨ˆ𝑎†𝑖 ⟩⟨ˆ𝑎𝑗 ⟩ + 𝛿ˆ𝑎†𝑖 ⟨ˆ
𝑎𝑗 ⟩ + ⟨ˆ𝑎†𝑖 ⟩𝛿ˆ
𝑎𝑗 + 𝛿ˆ
𝑎†𝑖 𝛿ˆ𝑎𝑗≈ ⟨ˆ
𝑎†𝑖 ⟩⟨ˆ𝑎𝑗 ⟩ + (ˆ𝑎†𝑖 − ⟨ˆ𝑎†𝑖 ⟩)⟨ˆ𝑎𝑗 ⟩ + ⟨ˆ𝑎†𝑖 ⟩(ˆ𝑎𝑗 − ⟨ˆ𝑎𝑗 ⟩)= 𝑎ˆ†𝑖 ⟨ˆ
𝑎𝑗 ⟩ + 𝑎ˆ𝑗 ⟨ˆ
𝑎†𝑖 ⟩ − ⟨ˆ𝑎†𝑖 ⟩⟨ˆ𝑎𝑗 ⟩The effect of this approximation is that we neglect correlation between neighboringsites, which comes from 𝛿ˆ𝑎†𝑖 𝛿ˆ
𝑎𝑗 . The on-site correlation between occupation number
basis states will be still present in general. The approximation allows us to rewrite theBose-Hubbard Hamiltonian as a sum of local Hamiltonians. We assume the system
is homogeneous so that the mean field is uniform across the sites: ⟨ˆ𝑎𝑖 ⟩ = 𝛼. Then
𝐻≈

∑︁

ℎ𝑖 =

𝑖

∑︁
𝑖

(︀
)︀ 𝑈
ˆ 𝑖 (ˆ
𝑛𝑖 − 1)−𝑧𝑡 𝛼𝑎ˆ𝑖 † + 𝛼* 𝑎
ˆ𝑖 − |𝛼|2 + 𝑛2where 𝑧 is the coordination number of the lattice. We insert the chemical potential
𝜇 as a Lagrange multiplier that controls the average occupation number, so that we
do not have to diagonalize the Hamiltonian under the constraint of a fixed numberof particles. Then to find the ground state of the system, it remains to minimize theenergy of the on-site Hamiltonian ℎ − 𝜇ˆ
𝑛 with 𝛼 as our variational parameter, under
the constraint ⟨ˆ𝑎⟩ = 𝛼:ℎ − 𝜇ˆ
𝑛 = −𝑧𝑡(𝛼ˆ𝑎† + 𝛼 * 𝑎
ˆ − |𝛼|2 ) +

5.2.1

𝑈
𝑛
ˆ (ˆ
𝑛 − 1) − 𝜇ˆ𝑛2Perturbative approach

We start the analysis from a Mott insulating state with ⟨ˆ𝑛⟩ = 𝑛, 𝛼 = 0 and apply

perturbation theory to find the energy shift due to the (𝛼ˆ𝑎† + 𝛼 * 𝑎ˆ) term [75]. It isclear that only the even-order perturbation terms survive. We apply the perturbation74

theory up to the fourth order.
𝐸(𝛼) = 𝑎0 + 𝑎2 |𝛼|2 + 𝑎4 |𝛼|4 + 𝒪(|𝛼|6 )𝑈
𝑎0 = −𝜇𝑛 + 𝑛(𝑛 − 1)
2
(︂
)︂
𝑛+1
𝑛
𝑎2 =
+
(𝑧𝑡)2 + 𝑧𝑡
𝜇 − 𝑛𝑈−𝜇 + (𝑛 − 1)𝑈(︂(𝑛 − 1)𝑛4
𝑎4 = (𝑧𝑡)(2𝑛𝑈 − 3𝑈 − 2𝜇)(𝜇 − 𝑛𝑈 + 𝑈 )2(𝑛 + 1)(𝑛 + 2)−
(2𝑛𝑈 + 𝑈 − 2𝜇)(𝜇 − 𝑛𝑈 )2
(︂
)︂ )︂
𝑛
(𝜇 + 𝑈 )𝑛+1−+(𝜇 − 𝑛𝑈 )(𝜇 − 𝑛𝑈 + 𝑈 ) (𝜇 − 𝑛𝑈 + 𝑈 )2 (𝜇 − 𝑛𝑈 )2The expansion of the variational energy in terms of powers of |𝛼|2 is reminiscent of thephenomenological Landau theory of second order phase transitions, which predicts aphase transition when 𝑎2 changes sign if 𝑎4 stays positive [76]. More specifically,|𝛼|2 = 0 for 𝑎2 > 0 and |𝛼|2 = −𝑎2 /(2𝑎4 ) for 𝑎2 < 0. This is the spontaneous
symmetry breaking of a continuous symmetry, which is the global U(1) symmetry(ˆ𝑎 → 𝑒𝑖𝜃 𝑎ˆ) in our problem. Before we demonstrate the positivity of 𝑎4 , first consider
the equation 𝑎2 = 0 to find the phase boundary line. The equation yields the boundarylines defined by1
𝜇± /(𝑧𝑡) =2(︃𝑈(2𝑛 − 1) ±
𝑧𝑡

√︃

𝑈
𝑧𝑡

(︂

)︃
)︂
𝑈
−4𝑛 + − 2 + 1 − 1𝑧𝑡These boundary lines define lobe-like regions in which the system enters a Mott
insulator phase with integral filling factor, as shown in Fig. 5-1. The positivity of thequartic term 𝑎4 can be checked by looking at Fig. 5-1(a). For any choice of integer𝑛 ≥ 1, the corresponding 𝜇 is bounded as (𝑛 − 1)𝑈 < 𝜇 < 𝑛𝑈 , where (𝑛 − 1)𝑈 and𝑛𝑈 define the zeroth-order phase boundary lines. Given a fixed 𝑛 one can apply thisbound for 𝜇 to the expression of 𝑎4 (𝑛, 𝑈, 𝜇) and confirm that 𝑎4 > 0, although thecheck is tedious and can be done more conveniently with a computer algebra system(e.g. using Mathematica’s Reduce function for checking compatibility of inequalities).

75

Figure 5-1: Mean-field phase diagram of the Bose-Hubbard Hamiltonian. (a) Phase
diagram on the (𝑈/𝑧𝑡, 𝜇/𝑧𝑡) parameter space. The dotted red lines show the phaseboundary lines for the zeroth-order calculation, where the lines are determined by𝑎0 (𝜇, 𝑈, 𝑛) = 𝑎0 (𝜇, 𝑈, 𝑛 + 1). In the zeroth-order level, the filling factor 𝑛 is always
an integer. The solid blue lines show the phase boundary lines for the second-order
calculation, where the lines are determined by 𝑎2 (𝜇, 𝑈, 𝑛) = 0. Outside the lobe
regions the superfluid order parameter takes a non-zero value and the filling factor
is no longer an integer in general. (b) Phase diagram on the (𝑧𝑡/𝑈, 𝜇/𝑈 ) parameter
space. Outside the lobe regions the superfluid order parameter is nonzero. The
black dotted line shows an example of the local chemical potential variation in an
inhomogeneous system. Under the local density approximation, the local filling factor
is determined by local (𝑧𝑡/𝑈, 𝜇/𝑈 ) sampling the phase diagram.

76

The deviation of the filling factor away from an integral value in the superfluid
region can be also calculated in this framework. We make use of the definition of thechemical potential as a Lagrange multiplier for the average occupation number: ⟨ˆ𝑛⟩ ≡−𝜕⟨𝐻⟩/𝜕𝜇 = −𝜕𝐸(𝛼)/𝜕𝜇. When the order parameter is zero, 𝐸(𝛼) = 𝐸(0) = 𝑎0 and⟨ˆ𝑛⟩ = 𝑛 = 0, 1, 2, . . .. When the order parameter is nonzero, 𝐸(𝛼) = 𝑎0 − 𝑎22 /(4𝑎4 )and thus
(︂

)︂
(︂
)︂2
𝑎2 𝜕𝑎2𝑎2
𝜕𝑎4⟨ˆ𝑛⟩ = 𝑛 +−2𝑎4 𝜕𝜇
2𝑎4
𝜕𝜇
𝜕𝑎
𝜕𝑎
2
4
= 𝑛 − |𝛼|2
− |𝛼|4𝜕𝜇𝜕𝜇
The slope 𝜕𝑎2 /𝜕𝜇 near the phase transition can be visually inferred from the phaseboundary lines in Fig. 5-1. The value of 𝑎2 inside the lobe regions is negative, while
it is positive outside those regions. By fixing (𝑡, 𝑈 ) and varying 𝜇, we scan thephase diagram vertically. As we increase 𝜇, if we enter a lobe region from outside,𝜕𝑎2 /𝜕𝜇 < 0 and the filling factor is slightly less than the integral value associated withthe lobe. On the other hand, if we exit a lobe region as we increase 𝜇, 𝜕𝑎2 /𝜕𝜇 < 0and the filling factor is slightly larger than the integral value associated with the lobe.
These two regions of ⟨ˆ𝑛⟩ < 𝑛 and ⟨ˆ𝑛⟩ > 𝑛 eventually merge near the tip of the ⟨ˆ𝑛⟩ = 𝑛lobe, whose location is defined by 𝜕𝑎2 /𝜕𝜇 = 0. Figure 5-2 shows a perturbative
calculation of ⟨ˆ𝑛⟩ around the 𝑛 = 1 and 𝑛 = 2 Mott lobes for 𝑈/𝑧𝑡 = 11. Note that
only the 𝜕𝑎2 /𝜕𝜇 contribution is kept, as the 𝜕𝑎4 /𝜕𝜇 term is highly divergent near
the point 𝜇 = 𝑛𝑈 . The high-order non-degenerate perturbation theory we employed
does not work near the zeroth-order phase boundaries 𝜇 = 𝑛𝑈 because of energy
degeneracy between 𝑛 and 𝑛 + 1 phases at those boundaries [75].

5.2.2

Numerical diagonalization

The perturbative approach to the mean field phase diagram yields closed-form expressions that give many insights into the Bose-Hubbard Hamiltonian, but in orderto obtain more accurate results, especially near the degeneracy points 𝜇 = 𝑛𝑈 , we
77

Figure 5-2: Calculation of the average filling factor ⟨ˆ𝑛⟩ around the 𝑛 = 1 and 𝑛 = 2
Mott lobes, for 𝑈/𝑧𝑡 = 11. The blue solid line represents the perturbative calculationand the red dots are from the numerical diagonalization as described in Section 5.2.2,
which agrees well with the perturbative calculation. However, for the (nondegenerate)
perturbative calculation, the 𝜕𝑎4 /𝜕𝜇 contribution is ignored as it has a highly singularbehavior near the degeneracy point 𝜇 = 𝑛𝑈 . The dotted lines mark the Mott lobe
regions where the average filling factor is an integer. In an inhomogeneous system,
the local chemical potential varies from a maximal value at the center to zero at
the edge, and thus the radial profile of the system has a “wedding cake” structure ofalternating superfluid and Mott-insulating regions [77–80].

have to employ numerical diagonalization of the Hamiltonian matrix. We assume∏︀ ∑︀
a variational wavefunction of the form |Ψ𝑀 𝐹 ⟩ = 𝑖 ∞𝑛=0 𝑓𝑛 |𝑛⟩. The Hamiltonian
matrix takes the form

⎞
⎛
2
|𝛼|
−𝛼
0
···
0
⎟
⎜
√
..
⎟
⎜ *
⎟
⎜𝛼
···
.
−˜
𝜇 + |𝛼|2
− 2𝛼
⎟
⎜
√
..
⎟
⎜
*
2
˜
𝑧𝑡 ⎜ 0⎟−2˜
𝜇 + 𝑈 + |𝛼|
···
− 2𝛼
.
⎟
⎜
√
⎟
⎜ ..
..
..
...
⎟
⎜ .
.
.
− 𝑛𝛼
⎠
⎝
√ *
˜
𝑈
2
0
···
···
− 𝑛𝛼 −𝑛˜
𝜇 + 2 𝑛(𝑛 − 1) + |𝛼|where (˜𝜇, 𝑈˜ ) = (𝜇/𝑧𝑡, 𝑈/𝑧𝑡). The idea is to truncate the matrix at 𝑛 = 𝑛max andvary 𝛼 until the lowest eigenvalue of the matrix reaches a minimal value for a given(𝜇, 𝑈 ) input [81]. The maximum occupation number 𝑛max is also increased until the78

optimal value of 𝛼 found converges to a steady value. For low filling numbers 𝑛max
does not need to be higher than 5 unless 𝑈/𝑧𝑡 is set to be very low (deep into thesuperfluid phase). An example of the calculation for 𝑈/𝑧𝑡 = 11 is shown in Fig. 5-2,
which actually agrees pretty well with the perturbative calculation.

5.2.3

Comparison to experiments

Because of the Gaussian envelopes of the lattice beams and the harmonic traps for confining atoms, the system is inhomogeneous, and the parameters (𝑡, 𝑈, 𝜇) vary acrossthe atom cloud. As long as the variation is not too strong (relative changes being
small over a lattice constant length scale), we can invoke a local density approximation and use the mean-field phase diagram to estimate quantities such as ⟨ˆ𝑛⟩. With
the numerical diagonalization technique, we can generate a detailed map of ⟨ˆ𝑛⟩ as afunction of 𝜇/𝑈 and 𝑧𝑡/𝑈 . In many BEC experiments, the BEC is initially trappedin a harmonic trap and is eventually loaded to an optical lattice. The variation of
the local chemical potential, which follows the shape of the harmonic trap in theThomas-Fermi regime, will be present in the atomic cloud at the onset of the phasetransition. We can estimate how many distinct Mott lobes are present and how many
atoms are within each lobe by integrating the average filling factor across the phase
diagram. To do the integration, we traverse the (𝜇/𝑈, 𝑧𝑡/𝑈 ) trajectory made when
one moves from the center of the trap towards the edge, such as the dotted line in
Fig. 5-1(b) or the 𝑥-axis in Fig. 5-2. The computation of the estimated atom numbersin the different Mott lobes is summarized as follows:∫︀
1. Construct a lookup table of lattice parameters (𝑉𝐿 , 𝑡, |𝑤(𝑥)|4 𝑑𝑥). 𝑉𝐿 denotes

optical lattice depth in units of recoil energy 𝐸𝑅 . The lookup table can be gen-eralized by interpolation to a function with 𝑉𝐿 as the argument. The quantity∫︀|𝑤(𝑥)|4 𝑑𝑥 is used to compute the local value of 𝑈 once the three-dimensional
coordinate in the optical lattice is provided.

2. Construct a lookup table of Bose-Hubbard parameters (𝑧𝑡/𝑈, 𝜇/𝑈, ⟨ˆ𝑛⟩) with thenumerical diagonalization technique.
793. Provide the harmonic confinements due to the Gaussian laser beams and thebackground magnetic field. To find the Gaussian beam contributions to theoverall confinement, one needs an accurate knowledge of one of either combination: input power and Gaussian beam waist, or input power and dipole
oscillation frequencies. The dipole oscillation frequencies can be easily measured by kicking the atom cloud with a displaced version of each beam. Since
frequency can be more easily and precisely measured than beam waist, we usually use the latter combination. We can also infer the corresponding Gaussian
beam waist from the said combination, and we make sure the inferred value isconsistent with the optical layout.
4. Construct a function that calculates the local variation of (𝑧𝑡/𝑈, 𝜇/𝑈 ) from thevalue at the center of trap. The 𝜇/𝑈 variation is directly given by the change inthe total potential depth, and 𝑧𝑡/𝑈 is computed from the change in the lattice
depth.
5. Integrate ⟨ˆ𝑛⟩ as a function of local (𝑧𝑡/𝑈, 𝜇/𝑈 ) throughout the volume inside
the trap.
Many quantum simulation experiments that use a Mott insulating phase as the restricted Hilbert space require a strict bound on the maximal filling factor. Thereare experimental techniques such as interaction-induced clock shift spectroscopy [77],spin-changing collisions [78], interaction blockade [79], and quantum gas microscopy[80], that can directly show the in-situ distribution of the filling factor. However, suchtechniques require special setups or time-consuming data acquisitions. The value ofthe numerical simulation of the filling factor distribution is that it gives a quick estimate of the maximal filling factor based on the total atom number measured byimaging.
We have checked the computation procedure by using the provided lattice depths
and beam profile data in Ref. [79] and reproducing the same simulation result as shownin Fig. 4(a) of the reference. For comparison against our own experiment, see Fig. 5-3.
In the Figure, we estimate the Mott lobe populations in a three-dimensional lattice80

formed by 1064 nm beams with lattice depths of 20 𝐸𝑅 . Although the experimentmakes use of a deeper lattice depth (35 𝐸𝑅 ), a lower lattice depth was chosen as thedensity is expected to be frozen before the ramp-up of the lattice depths completes. By
using the measurement protocol described in Section 7.1, the result of the simulation
can be compared to the experiment by measuring the atom number at which ⟨ˆ𝑛⟩ = 2lobe disappears, and also by measuring the rate at which the ⟨ˆ𝑛⟩ = 2 lobe fractionor the total doublon fraction grows as a function of total atom number. However,as Fig. 5-3(b) shows, we could not bring the measured non-singlon fraction of theMott insulator into an agreement with the estimated value as a function of measured
total atom number - we had to introduce a “fudge factor” of two into the atom
number calibration to produce a better agreement. One could argue that an incorrecteffective absorption cross section was used for the number calibration, but we madeindependent estimates of the atom number by measuring the released energy from
BEC expansion, and the independent estimates were in good agreement with theabsorption imaging esimates. A likely explanation for the discrepancy is that thelattice loading procedure is not adiabatic and hence the density distribution frozen
in the Mott insulating phase is not in equilibrium, resulting in higher non-singlon
fractions.
It must be emphasized that the mean-field approximation becomes unreliable at
low dimensions where correlations become more important [76]. Indeed, the onedimensional superfluid to Mott insulator transition happens much earlier at (𝑡/𝑈 )𝑐 ≈0.3 instead of ≈ 0.086 as predicted by the mean-field calculation [82]. Also, the Mott
lobe regions appear sharply pointed.

5.3

Corrections due to higher bands

We are interested in the corrections to Hubbard parameters because the parametersof the effective spin models we would like to study depend sensitively on 𝑡 and 𝑈 .
In particular, the spin exchange parameter is 𝐽 = 4𝑡2 /𝑈 . Typically the calculationof 𝑡 and 𝑈 uses Wannier orbitals 𝑤(𝑥) calculated from the lowest Bloch band of the81

lattice in the absence of interaction. However, on-site interactions modify the Wannierorbitals if a site is occupied by more than one atom [73,74]. An intuitive picture is thatthe repulsive interaction energy encourages atoms to separate, flattening the density
distribution near the center of the lattice site. This modified density distribution
can also be interpreted as the on-site wavefunction being a superposition of Wannierorbitals from different Bloch bands. In the case of a cubic lattice, the expansion of
the field operator in Section 5.1 is now generalized toˆ 𝑟) =
𝜓(⃗

∑︁
𝑖

𝑚
max
∑︁

𝑎
ˆ(𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧 ),𝑖 𝑤(𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧 ),𝑖 (⃗𝑟)

(5.2)

𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧

where 𝑖 is the lattice site index, and 𝑤(𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧 ),𝑖 (⃗𝑟) = 𝑤𝑚𝑥 ,𝑖 (𝑥)𝑤𝑚𝑦 ,𝑖 (𝑦)𝑤𝑚𝑧 ,𝑖 (𝑧) is thethree-dimensional Wannier orbital at lattice site 𝑖 constructed from the 𝑚th
𝛼 Blochband for 𝛼 = 𝑥, 𝑦, 𝑧. For practical purposes we introduce a cutoff for the maximum
band number in each direction, denoted by 𝑚max . Because of the three-dimensional
nature of the interaction, inclusion of Bloch bands beyond the ground band leads toan exponentially growing on-site Hilbert space. For 𝑛 atoms in a single lattice site,each atom can occupy any of the 𝑚3max available states. The total number of distinct
many-body states of 𝑛 atoms in a single lattice site is(𝑛 + 𝑚3max − 1)!
𝑛!(𝑚3𝑚𝑎𝑥 − 1)!
If 𝑚max = 4, then the dimension of the on-site Hilbert space is 2080 for 𝑛 = 2 and45760 for 𝑛 = 3. A physical argument for setting 𝑚max low is that the higher Blochbands are not classically bound (e.g. the third band in Fig. 4-1) and are prone to loss
due to lack of confinement,and hence they contribute minimally to the renormalizationof the on-site density distribution [83].
The full exact diagonalization of the on-site problem is computationally challenging3 . However, standard second-order perturbation theory can be used, and its use isjustified because the final answer yielded by the calculation does not show a strong
3

For more information on how to implement the numerical diagonalization, see Ref. [84]82

modification of the Hubbard parameters for typical experimental conditions [85]. The
on-site interaction energy modified by the inclusion of higher-band Wannier orbitals
is
𝑈

(2)

= 𝑈

(0)

−

∑︁

𝑔2

(0)
𝑚
⃗ 1 ,𝑚⃗ 2 4(𝐸𝑏𝑔 − ∆𝐸𝑈 )

(︂∫︁

*
*
2 3
𝑤𝑚
⃗ 1 (𝑟)𝑤𝑚⃗ 2 (𝑟)[𝑤(0,0,0) (𝑟)] 𝑑 𝑟

)︂2

†
†
× |⟨1𝑚𝑎𝑚ˆ𝑚
ˆ(0,0,0) 𝑎ˆ(0,0,0) |𝑛(0,0,0) ⟩|2⃗ 1 , 1𝑚
⃗ 2 , (𝑛 − 2)(0,0,0) |ˆ
⃗ 1𝑎
⃗ 2𝑎

where 𝑛 is the number of atoms at the site, 𝑚⃗ 𝑖 = (𝑚𝑥,𝑖 , 𝑚𝑦,𝑖 , 𝑚𝑧,𝑖 ) are the new band
indices for two atoms after the interaction, 𝐸𝑏𝑔 is the sum of the energy band gaps
(0)

from all three directions, and ∆𝐸𝑈 is the magnitude of the change in the zerothorder interaction energy of 𝑚 atoms after promoting 2 atoms to (possibly) different
Wannier orbitals. Note that 𝑚⃗ 1 and 𝑚⃗ 2 cannot be simultaneously be equal to (0, 0, 0).
Also note that the provided expressions are only valid for spin-polarized systems. For
spinful systems, 𝑔 is generalized to 𝑔𝜎𝜎′ and appropriate changes also need to bemade to the bosonic stimulation factor ⟨. . . |ˆ
𝑎† 𝑎
ˆ† 𝑎
ˆ𝑎
ˆ| . . .⟩. Note that 𝐸𝑏𝑔 is typically∫︀much larger than 𝑈 (0) = 𝑔𝑛(𝑛 − 1)/2 |𝑤(𝑟)(0,0,0) |4 𝑑3 𝑟 and the resulting smallness∫︀of 𝑔 2 ( . . .)2 /(𝐸𝑏𝑔 ) ensures that we stay in the regime of validity for second-order
perturbation theory.

The calculation can be simplified by making use of the inversion symmetry of the
Wannier orbitals in a cubic lattice. The Wannier orbitals are even under inversion
if the band index is even, and odd if the band index is odd. Therefore, only theband indices 𝑚⃗ 1, 𝑚⃗ 2 that satisfy 𝑚𝛼,1 + 𝑚𝛼,2 ≡ 0 mod 2 (𝛼 = 𝑥, 𝑦, 𝑧) enter into thecalculation. Also, the calculation of the bosonic stimulation factor ⟨. . . |ˆ
𝑎† 𝑎
ˆ† 𝑎
ˆ𝑎
ˆ| . . .⟩can be reduced to four special cases:
1. 𝑚
⃗1=𝑚
⃗ 2 = (0, 0, 0): the factor is 𝑛(𝑛 − 1).
2. 𝑚
⃗1=𝑚
⃗ 2 and 𝑚⃗ 1 ̸= (0, 0, 0): the factor is√︀
2𝑛(𝑛 − 1).

3. 𝑚⃗ 1 ̸= 𝑚⃗ 2 and [𝑚⃗ 1 = (0, 0, 0) or 𝑚
⃗ 2 = (0, 0, 0)]: the factor is
4. 𝑚⃗ 1 ̸= 𝑚⃗ 2 and 𝑚⃗ 1,2 ̸= (0, 0, 0): the factor is83√︀𝑛(𝑛 − 1).

√𝑛(𝑛 − 1).

(0)The computation of ∆𝐸𝑈 is straightforward in principle but can be lengthy in formulation into a code. However, calculations suggest that its contribution to the final(0)

answer is negligible (𝐸𝑏𝑔 ≫ ∆𝐸𝑈 ) and can be ignored for simplicity.
We show the result of the perturbative calculation for the correction to 𝑈 in
Figure 5-4. We see that for two (𝑛 = 2) or three atoms (𝑛 = 3) per site, the change
in 𝑈 is within 10%. The dominant contributions to the energy correction come fromprocesses where one of the atoms jumps to a band with index number 2 or two of
the atoms jump to a band with index number 1. As for a comparison to an exact
diagonalization solution, we note that if we use 𝑉transverse = 𝑉longitudinal = 35 𝐸𝑅,754nm ,𝑎𝑠 /𝜆 = 0.007, and 𝑚max = 4, we obtain the correction to 𝑈 be 0.93, 0.91, 0.89 for
𝑛 = 2, 3, 4 whereas Ref. [72] obtains 0.9, 0.875, 0.85 for 𝑛 = 2, 3, 4. So second order
perturbation theory misses 3 to 4% extra correction but nonetheless yields a good
estimate of the magnitude of the corrections.
We now consider corrections to the tunneling parameter 𝑡. There are two maincorrections: the first correction is the so-called bond-charge tunneling [73] and thesecond correction is the multi-orbital tunneling due to the admixture of higher-bandorbitals. The bond-charge effect arises from the terms of the form (𝑔/2)ˆ
𝑎†𝑖 (ˆ
𝑎†𝑗 𝑎
ˆ𝑗 )ˆ
𝑎𝑗
when we insert Eq. 5.2 into Eq. 5.1. We note that the bond-charge tunneling is stillpresent without the inclusion of higher bands and was simply ignored in the derivationof the Bose-Hubbard Hamiltonian in Sec. 5.1 in order to keep the parameters densityindependent. The contribution of the bond-charge effect to the tunneling 𝑡 without
inclusion of the higher-band orbitals is
−𝑔

(︂∫︁

*

*

3

)︂

𝑤(𝑟 − 𝛿𝑟) 𝑤(𝑟) 𝑤(𝑟)𝑤(𝑟) 𝑑 𝑟 (ˆ𝑛𝑖 + 𝑛ˆ 𝑗 − 1)

where 𝑖, 𝑗 are lattice site indices for nearest neighbors and 𝛿𝑟 is a translation by one
lattice constant along one of the three directions. For the lattice parameters specified
in Fig. 5-4, the bond-charge effect enhances 𝑡 by 17 to 18% for 𝑛𝑖 = 𝑛𝑗 = 2 as thelongitudinal lattice depth is varied. The inclusion of higher-band orbitals into the onsite wavefunctions allows multi-orbital tunneling and also corrects the bond-charge
84effect. The admixture of higher-band orbitals can determined again by second order
perturbation theory, and the modified onsite wavefunction |˜𝑛(0,0,0) ⟩ is expressed as:|˜𝑛(0,0,0) ⟩ = |𝑛(0,0,0) ⟩
)︂
(︂∫︁
∑︀
𝑔
†
†
2 3
*
*
ˆ𝑚
ˆ𝑚
𝑎(0,0,0) )2 |𝑛(0,0,0) ⟩− 𝑚𝑤𝑚⃗ 2 (𝑟)[𝑤(0,0,0) (𝑟)] 𝑑 𝑟 𝑎⃗ 1 (𝑟)𝑤𝑚⃗ 1 ,𝑚⃗2⃗ 1𝑎⃗ 2 (ˆ
(0)2(𝐸𝑏𝑔 − ∆𝐸𝑈 )Note that 𝑚⃗ 1 and 𝑚⃗ 2 cannot be simultaneously be equal to (0, 0, 0). For our typicallattice conditions (see Fig. 5-4) with two atoms per lattice site, the total populationin the excited bands is between 0.1 and 0.2%. For the multi-orbital tunneling, we only
consider tunneling processes between equal orbitals (equal band indices)4 . With thisconstraint, the matrix element of the multi-orbital tunneling process for |˜𝑛𝐿 , 𝑛˜𝑅⟩ →|˜
𝑛𝐿 − 1, 𝑛
˜ 𝑅 + 1⟩ can be expressed as follows:
∑︁
1
𝑂
𝑡𝑀
𝑡𝑚
𝑛𝑅 + 1|ˆ𝑎†𝑅,𝑚𝑛𝑅 ⟩⟨˜𝑛𝐿 − 1|ˆ
𝑎𝐿,𝑚
𝑛𝐿 ⟩
⃗ ⟨˜
⃗ |˜
𝑛
˜ 𝐿 ,˜𝑛𝑅 = √︀⃗ |˜𝑛𝐿 (𝑛𝑅 + 1) 𝑚⃗

where 𝑎ˆ𝑚
⃗ and 𝑡𝑚
⃗ is the annihilation operator for a Wannier orbital with band index 𝑚⃗is the zeroth-order tunneling matrix element for Wannier orbitals with band index 𝑚,⃗which becomes exponentially larger as the band index increases along the considered𝑂tunneling direction. For 𝑛
˜𝐿 = 𝑛˜ 𝑅 = 2, 𝑡𝑀
2,2 is evaluated to be

𝑡⃗0 +

∑︁

(2)

(3)*

𝑐 ⃗
8𝑡𝑚
⃗ 𝑐⃗0,𝑚⃗ ⃗0,𝑚

𝑚̸
⃗ =0
(𝑛)

where ⃗0 = (0, 0, 0) and 𝑐⃗0,𝑚is the short-hand symbol for the admixture coefficient⃗
calculated for 𝑛 atoms per site:
−

𝑔
(0)

2(𝐸𝑏𝑔 − ∆𝐸𝑈 )

(︂∫︁

*
2 3
𝑤0* (𝑟)𝑤𝑚⃗ (𝑟)[𝑤(0,0,0) (𝑟)] 𝑑 𝑟

4)︂

Ref. [73] claims that tunneling between unequal orbitals is zero because of orthogonality relations
between the Wannier orbitals. However, this is not true; orthogonality relations only hold for overlaps
between Wannier orbitals. The insertion of the tunneling Hamiltonian −~2 /2𝑚∇2 + 𝑉 (𝑟) inside theoverlap integral no longer yields zero for unequal orbitals. However, we expect tunneling between
unequal orbitals to be highly off-resonant because of the large energy band gaps, and hence it isappropriate to ignore them.

85

(𝑛)

The dependence of 𝑐⃗0,𝑚on the atom number 𝑛 is hidden in the denominator 𝐸𝑏𝑔 −⃗(0)∆𝐸𝑈 . The result is plotted in Fig. 5-5 as a function of the longitudinal lattice depth.
As a result of the correction, the spin exchange 4𝑡2 /𝑈 is likely to be enhanced
by 20 to 30% compared to the bare value in typical lattice depths. However, sucha deviation can be also explained by less than 10% uncertainty in the lattice depth,
since 𝑡 depends exponentially on the lattice depth. Therefore, we do not consider thecorrections in the following discussions given the noisiness of the data, but they need
to be taken into account once the data is precise enough and if one wants to checkthe scaling of the exchange parameter with respect to the lattice depth.

86Figure 5-3: Simulation of the Mott lobe populations in three-dimensional 20 𝐸𝑅,1064 nm
lattice. The 1/𝑒2 beam waists for the lattice beams are: 𝑤Tool = 145 𝜇m, 𝑤Closet =147 𝜇m, 𝑤Vertical = 268 𝜇m. The 1/𝑒2 beam waists for the crossed dipole trap are
𝑤XODT = (528 𝜇m, 50 𝜇m), 𝑤VD = 94 𝜇m (see Fig. 2-2 for the beam geometries). (a)Total atom number vs. Mott lobe populations. (b) Measured non-singlon (⟨ˆ𝑛⟩ > 1)
fraction as a function of atom number vs. simulated non-singlon fraction based onbeam parameters. A fudge factor has to be applied to the atom number conversion
to bring the experiment and the simulation to a better agreement, although there isno evidence to suggest that the absorption imaging calibration is off.
87

Figure 5-4: Perturbative calculation of correction to Hubbard interaction energy
𝑈 due to inclusion of higher-band Wannier orbitals. The lattice depth is fixed to
35 𝐸𝑅,1064 nm along the transverse directions and is varied along the longitudinal direction, where the 810 nm lattice is installed. The highest band index 𝑛max we allow
is 4, and the number of atoms occupying a single lattice is either 2 (red) or 3 (blue).

88

Figure 5-5: Perturbative calculation of correction to tunneling amplitude 𝑡 due toinclusion of higher-band Wannier orbitals and the bond-charge effect. We do notconsider multi-orbital correction to the bond-charge effect. The assumed atom number per site is two.

89

90

Chapter 6

Quantum Magnetism

Quantum magnetism is a vast field of study that investigates magnetic phenomena where quantum effects play an import role. It is highly relevant to the studyof strongly correlated systems with strong interactions and has inspired many new
theoretical methods as well as experiments [86, 87]. Often the physics of quantum
magnetism is distilled into spin models with several parameters, such as exchangeinteraction strength and anisotropy strength. A spin model is classical if the spin istreated as a classical vector, whereas it is quantum mechanical if the spin is treatedas a quantum spin operator (e.g. Pauli matrices for spin-1/2). Because of the noncommutativity of the spin operators, quantum fluctuations play an import role inspin ordering within quantum spin models, whereas classical spin models have nofluctuation at zero temperature, by definition.

While classical spin models constitute a rich subject on their own [76], quantum
spin models intersect with many modern research areas of physics such as quantum
phase transitions [10], quantum integrability [11], and quantum simulation [88]. This
chapter will focus on introducing the quantum Heisenberg spin model and its realization with neutral atoms in optical lattices [20, 21].
91

6.1

Derivation of an effective Hamiltonian

We are interested in deriving a Heisenberg spin model. For spin-1/2 a canonical modelis
𝐻 = 𝐽𝑥𝑦∑︁(𝑆𝑖,𝑥 𝑆𝑗,𝑥 + 𝑆𝑖,𝑦 𝑆𝑗,𝑦 ) + 𝐽𝑧

⟨𝑖,𝑗⟩

∑︁
⟨𝑖,𝑗⟩

𝑆𝑖,𝑧 𝑆𝑗,𝑧 − ℎ𝑧

∑︁

𝑆𝑖,𝑧

𝑖

which is known as the XXZ spin-1/2 model, and for spin-1 a canonical model is
𝐻=𝐽

∑︁
⟨𝑖,𝑗⟩

⃗𝑖 · 𝑆⃗𝑗 + 𝐷𝑆∑︁𝑖(𝑆𝑖,𝑧 )2 − ℎ𝑧

∑︁

𝑆𝑖,𝑧

𝑖

where 𝑆𝑖,𝛼=𝑥,𝑦,𝑧 are spin operators in the three orthogonal directions at the site 𝑖.
Note that the spin-1/2 model can have an anisotropy in the exchange strength (i.e.
𝐽𝑥𝑦 ̸= 𝐽𝑧 ), whereas the spin-1 model has a type of anisotropy known as the single iontype anisotropy (SIA) in material science literature [89]. The anisotropy term (𝑆𝑖,𝑧 )2

cannot exist in spin-1/2 because it evaluates to a constant term. For completeness, the
2
2
),
−𝑆𝑖,𝑦
(𝑆𝑧 )2 term is known as uniaxial single ion anisotropy; another SIA term is (𝑆𝑖,𝑥which evaluates to a null term in spin-1/2. While previous studies on the effective
spin-1 model for a ⟨ˆ𝑛⟩ = 2 bosonic Mott insulator considered an isotropic exchangeinteraction in combination with the SIA term [21–23], we show in Section 6.1.3 thatan exchange anisotropy is also possible. We mention in passing that there are otherbut less common interactions such as the Kitaev exchange interaction on a honeycomb
lattice [90] or the Dzyaloshinskii-Moriya interaction [91] but we do not consider thoseterms here.

6.1.1

Kinetic Exchange

When atoms in two distinct internal states (|𝑎⟩ and |𝑏⟩) are loaded into a deep opticallattice such that they enter a Mott insulating phase, first-order tunneling is prohibitedby the high on-site interaction energy cost. The local density at each site is fixed to
be an integer, meaning that if there are 𝑛 atoms, it would require an energy of 𝑛𝑈
92to add an atom, where 𝑈 is the on-site interaction energy. However, the spatial
distribution of the |𝑎⟩ and |𝑏⟩ states can freely change by second-order tunneling, asshown in Fig. 6-1. This gives rise to a mapping from the low energy Hilbert space ofthe two-component Bose-Hubbard Hamiltonian to a quantum spin Hamiltonian. It
is not difficult to see that if there are 𝑛 = 𝑛𝑎 + 𝑛𝑏 atoms per site, the on-site Fockstate can be mapped to a spin with 𝑆𝑧 = (𝑛𝑎 − 𝑛𝑏 )/2. Second-order tunneling allows
swapping of one |𝑎⟩ and one |𝑏⟩ between neighboring sites, and this process can bemapped to a spin flip-flop process described by the operator 𝑆𝑖+ 𝑆𝑗− + c.c. In the fieldof quantum simulation with ultracold atoms, this exchange process by second-ordertunneling has become to be known as superexchange [92], but in the field of solidstate physics the term superexchange technically refers to a fourth-order tunnelingprocess, where electrons from separated transition metal ions hop to an intermediateoxygen atom bridging the two ions to exchange spins [93]. In accordance with theconvention in solid state physics, we use the term kinetic exchange to describe thesecond-order tunneling process.

6.1.2

On-site anisotropy

In the limit of a very deep lattice where even the second-order tunneling becomes negligible, the many-body ground state is a product of 𝑆ˆ𝑧 eigenstates. Each 𝑆ˆ𝑧 eigenstateis associated with a spin-dependent on-site interaction energy 𝑈𝜎𝜎′ , (𝜎, 𝜎 ′ ∈ 𝑎, 𝑏). In
the spin-model, the difference in the on-site interaction energies can be expressed by
the sum of a single-ion anisotropy term and a fictitious magnetic field, 𝐷(𝑆𝑧,𝑖 )2 −ℎ𝑧 𝑆𝑧,𝑖(Fig. 6-2). Hence the realization of the (𝑆𝑧 )2 term with ultracold atoms is rather simple, whereas in material science, the origin of the single-ion anisotropy term is more
complicated as it involves second-order perturbation by spin-orbit coupling [89].

6.1.3

Formal mappingWith an intuitive idea of how a spin exchange operation is realized by second-ordertunneling, we are ready to understand the formal procedure for deriving an effective
93

Figure 6-1: Kinetic exchange mechanism in a two-component Mott insulator withfilling factor ⟨ˆ𝑛⟩ = 2. Each atom can be in two internal states, |𝑎⟩ (red) or |𝑏⟩ (blue).
The longitudinal spin 𝑆𝑧 is given by (𝑛𝑎 − 𝑛𝑏 )/2. The left well and the center well can
exchange spins by second-order tunneling, for which there are two processes: either
the |𝑏⟩ atom from the center tunnels to the left and then an |𝑎⟩ atom from the left
tunnels to the center, or an |𝑎⟩ atom from the left tunnels to the center and then the|𝑏⟩ atom from the center tunnels to the left. For both processes, the second-ordertunneling amplitude is −2𝑡𝑎 𝑡𝑏 /𝑈𝑎𝑏 where 𝑈𝑎𝑏 is the interaction energy between |𝑎⟩and |𝑏⟩, and the net interaction can be summarized as a ferromagnetic spin exchange
coupling with with strength 𝐽 = −4𝑡𝑎 𝑡𝑏 /𝑈𝑎𝑏 .

Figure 6-2: On-site anisotropy in a two-component Mott insulator with filling factor⟨ˆ𝑛⟩ = 2. In general, 𝑈𝑎𝑎 ̸= 𝑈𝑏𝑏 . By applying a fictitious magnetic field ℎ𝑧 = (𝑈𝑎𝑎 −𝑈𝑏𝑏 )/2 to the model, the energies of the different 𝑆𝑧 states with respect to the referenceenergy 𝑈𝑎𝑏 can be expressed as 𝐷(𝑆𝑧 )2 , where 𝐷 = (𝑈𝑎𝑎 + 𝑈𝑏𝑏 )/2 − 𝑈𝑎𝑏 .
spin Hamiltonian from the underlying Bose-Hubbard Hamiltonian1 .
1Derivation of an effective Hamiltonian can be understood as a form of degenerate perturbation
theory [94], and there are many different approaches. Here we just use one of them.

94First, we can perform a unitary transformation from the original Hamiltonian
to an effective Hamiltonian 𝐻eff that does not have matrix elements between statesthat belong to different energy manifolds [95]. In the current context, the lowest
energy manifold 𝛼 consists of many-body states that do not have any quasiparticlequasihole excitations. The higher energy manifolds 𝛽 consist of states with different
numbers of elementary excitations in the Mott phase, separated by an energy scale 𝑈 .
Assuming that the appropriate unitary transformation operator exists, we can express𝛼
the matrix elements of the effective Hamiltonian for the lowest energy manifold 𝐻eff

up to second order in the off-diagonal term 𝑉 ,𝛼⟨𝑖, 𝛼|𝐻eff|𝑗, 𝛼⟩ = 𝐸𝑖 𝛿𝑖𝑗 + ⟨𝑖, 𝛼|𝑉 |𝑗, 𝛼⟩
(6.1)
(︂
)︂
∑︁
1
1
1
+
⟨𝑖, 𝛼|𝑉 |𝑘, 𝛽⟩⟨𝑘, 𝛽|𝑉 |𝑗, 𝛼⟩++ 𝒪(𝑉 3 )2 𝛽̸=𝛼𝐸𝑖 − 𝐸𝑘 𝐸𝑗 − 𝐸𝑘

where the notation |𝑖, 𝛼⟩ denotes the 𝑖-th state in the energy manifold 𝛼. In thecurrent context, the zeroth-order energy 𝐸𝑖 refers to the sum of chemical potential
and interaction energy from all sites, and the off-diagonal operator 𝑉 refers to thenearest-neighbor tunneling operator. This expansion has the same condition of validity as standard second-order perturbation theory, which is that the matrix elementsof 𝑉 be far smaller than the energy gap between the different energy manifolds. It
𝛼must be noted that although 𝐻eff
has the same eigenenergy spectrum for the low

energy subspace as the original Hamiltonian, its eigenstates will not be the same,as the excited states have been adiabatically eliminated in some sense. In the current context, even though the admixture of virtual particle-hole pairs is essential forsecond-order tunneling process to happen, it does not appear in the eigenstates of theeffective Hamiltonian. Although this seems rather trivial, it can affect the validityof the effective Hamiltonian if the excited states are known to be lossy, e.g. due tothree-body collisions.
We can view the basis states |𝑗, 𝛼⟩ in two ways: as a basis state for the BoseHubbard Hilbert space with zero particle-hole pairs, or as a basis state for a spin
Hamiltonian. For example, in a one-dimensional double-well system, |𝑗, 𝛼⟩ could be95

a Fock state with one |𝑎⟩ atom and one |𝑏⟩ atom per each site. If we use the mapping
𝑆𝑧 = (𝑛𝑎 −𝑛𝑏 )/2, then this Fock state is equivalent to a |𝑆 = 1, 𝑆𝑧 = 0⟩ on each site of𝛼|𝑗, 𝛼⟩ with respect to the Fock states using thethe double-well. If we evaluate ⟨𝑖, 𝛼|𝐻eff

second-order expansion above, the same number can be equated to a matrix elementof a spin Hamiltonian between 𝑆ˆ𝑧 eigenstates [96]. Thus for each matrix element, weestablish an equation between the Bose-Hubbard parameters and the Heisenberg spinmodel parameters:
𝑓 (𝑡𝑎 , 𝑡𝑏 , 𝑈𝑎𝑎 , 𝑈𝑎𝑏 , 𝑈𝑏𝑏 ) + 𝑁 𝑐 = 𝑔(𝐽𝑥𝑦 , 𝐽𝑧 , ℎ𝑧 , 𝐷, . . .)(6.2)

where 𝑐 is an energy offset we can add and 𝑁 is the number of sites2 . Since interaction
only happens between nearest neighbors, the set of equations for matrix elements
between many-body states can be reduced to a set of equations for matrix elements
between states spanning over two sites. For example, consider a diagonal matrix
element where we reduce the triple well in Fig. 6-1 to the left-center double well andthe final state is the same as the initial state shown. The equation we obtain is
𝑈𝑎𝑎 + 𝑈𝑎𝑏 − 𝑡2𝑏 /𝑈𝑎𝑏 − 4𝑡2𝑎 /𝑈𝑎𝑏 − 3𝑡2𝑎 /(2𝑈𝑎𝑎 − 𝑈𝑎𝑏 ) + 2𝑐 = −ℎ𝑧 + 𝐷
Using such an approach, one can arrive at an exact mapping between the secondorder expansion for the Bose-Hubbard Hamiltonian and the XXZ spin-1/2 Heisenberg
Hamiltonian. The spin parameters for the spin-1/2 Hamiltonian are expressed as [97]
𝑡𝑎 𝑡𝑏
𝑈𝑎𝑏
(︂ 2
)︂
2
2(𝑡𝑎 + 𝑡2𝑏 )
𝑡𝑎
𝑡2𝑏
−4
+
𝐽𝑧 =
𝑈𝑎𝑏
𝑈𝑎𝑎 𝑈𝑏𝑏
(︂ 2
)︂
𝑡𝑎
𝑡2𝑏
ℎ𝑧 = 2𝑧−
𝑈𝑎𝑎 𝑈𝑏𝑏
)︂
(︂ 2
𝑡2𝑎
𝑡2𝑏
𝑡𝑎 + 𝑡2𝑏
𝑐 = 𝑧
+
+
4𝑈𝑎𝑏
2𝑈𝑎𝑎 2𝑈𝑏𝑏

𝐽𝑥𝑦 = −42This offset only appears on the diagonal matrix elements.

96Note 𝑧 is the number of nearest neighbors. However, that is not the case for spin1, where the failure of an exact mapping is caused by having more constraints ofthe form 𝑓 (. . .) = 𝑔(. . .) than there are independent variables in the Heisenberg
Hamiltonian. It may seem daunting to derive an effective spin-1 Hamiltonian that stillholds approximately true. However, Jepsen et al. showed that the mapping to an XXX
spin-1 Hamiltonian is approximately true at the degeneracy point defined by 𝑡𝑎 = 𝑡𝑏 =𝑡, 𝑈𝑎𝑏 = (𝑈𝑎𝑎 + 𝑈𝑏𝑏 )/2 if the relative interaction imbalance 𝛿 = (𝑈𝑎𝑎 − 𝑈𝑏𝑏 )/(𝑈𝑎𝑎 + 𝑈𝑏𝑏 )remains small [97]. Furthermore, the error term to the spin Hamiltonian scales with
𝛿 2 . If we define 1/𝑈 = 1/(2𝑈𝑎𝑎 ) + 1/(2𝑈𝑏𝑏 ), the spin parameters for the spin-1 models
at the degeneracy point 𝑈𝑎𝑏 = (𝑈𝑎𝑎 + 𝑈𝑏𝑏 )/2 are expressed as
4𝑡2
𝑈
𝑈𝑎𝑎 + 𝑈𝑏𝑏
𝐷 =− 𝑈𝑎𝑏 = 0
2(︂
)︂
6𝑡2
ℎ𝑧 = −𝑈 𝛿 1 + 𝑧 2
𝑈24𝑡𝑐 = −𝑈 + 𝑧
𝑈
𝐽 = −

(6.3)

The full expressions for the second-order matrix elements calculated in Ref. [85] areshown in Fig. 6-3.

For our experiments, 𝛿 = 0 but 𝐷 can vary over a large range. As 𝐷 deviates from∑︀zero, the SIA term ∝ 𝑖 (𝑆𝑧,𝑖 )2 becomes active. We define 𝑑 = (𝑈𝑎𝑎 + 𝑈𝑏𝑏 )/2 − 𝑈𝑎𝑏 as
the deviation from the degeneracy point (𝑈𝑎𝑎 +𝑈𝑏𝑏 )/2 = 𝑈𝑎𝑏 , and obtain an expression

for the strength of the SIA term 𝐷 in terms of 𝑑 and 𝑡2 /𝑈 . The full set of parameters,
97

consistent with Eq. 6.2 up to first order in 𝑑 and zeroth order in 𝛿 are as follows:
(︂
)︂
(︂
)︂
𝑑
4𝑡2
𝑈𝑎𝑏
4𝑡2
1+
=−
2−
−
𝑈
𝑈
𝑈
𝑈
(︂
)︂
(︂
)︂
4𝑡2
𝑑
4𝑡2 𝑈𝑎𝑏
−
1−
=−
𝑈
𝑈
𝑈
𝑈
(︂
)︂ (︂
)︂ (︂
)︂
2
2𝑡
𝑈𝑎𝑎 + 𝑈𝑏𝑏
2𝑡2
𝑑 1−𝑧 2 =
− 𝑈𝑎𝑏
1−𝑧 2
𝑈
2
𝑈
0
4𝑡2
4𝑡2
= −𝑈𝑎𝑏 + 𝑧
𝑑−𝑈 +𝑧
𝑈
𝑈

𝐽𝑥𝑦 =
𝐽𝑧 =
𝐷 =
ℎ𝑧 =
𝑐 =

What is noticeable is the appearance of an exchange anisotropy, where 𝐽𝑥𝑦 ̸= 𝐽𝑧 , anda relative correction of order (𝑡/𝑈 )2 to the bare SIA strength (𝑈𝑎𝑎 + 𝑈𝑏𝑏 )/2 − 𝑈𝑎𝑏 used
in Refs. [21–23, 97].
|n

initial|m

m|Heﬀ |n

final

−
−

−

−





m|Heﬀ |n

12t2
U↑↑

−

2

express as function of delta

12t2
U↑↑

−

adding overall energy offset E0

m|Heﬀ |nm|Heﬀ |n
substitute J

add fictitious magnetic field

4t2(1 − 3δ)U

J (1 − 3δ)

J

0

0

12t2
(1 − δ)
U

−

4t
4t2
+
U↑↑
U↓↓

8t2
−
U

8t2
−
U

0

12t2
−
U↓↓

12t2
−
U↓↓

12t2
−
(1 + δ)U

4t2−(1 + 3δ)
U

−



3t2
5t2
+
2U↑↑ − U↑↓
U↑↓



−



−



3t2
5t2
+
2U↓↓ − U↑↓
U↑↓



−





m|Heﬀ |n

at degeneracy point

2t2
2t2
+
2U↑↓ − U↑↑
2U↑↓ − U↓↓



3t2
5t2
+
2U↑↑ − U↑↓
U↑↓
3t2
5t2
+
2U↓↓ − U↑↓
U↑↓
−





t2 
8 − 6δ + 4δ 2 + ...
−
U
−


t2 
8 + 6δ + 4δ 2 + ...
U

4t2
U

−

4t2
U
−



4t2
U

4t2
U

J (1 + 3δ)JJ −3δ/2 + δ 2 + ...

J δ 2 + ...

J 3δ/2 + δ 2 + ...

J δ 2 + ...

4t2
U

−J

−J

3δ
− δ 2 + ...
2







3δ
+ δ 2 + ...
2+

m|Heﬀ |n

−

4t2
U↑↓

−

4t2
U↑↓

−


4t2 
1 − δ2
U

−


4t2 
1 − δ2U
J 1 − δ2
J 1 − δ2

−

4t2
U↑↓

−

4t2
U↑↓

−


4t2 
1 − δ2
U

−


4t2 
1 − δ2U
J 1 − δ2
J 1 − δ2

4t2
U

J

J

2t2
2t2
+
U↑↓ + (U↑↑ − U↑↑ )/2 U↑↓ + (U↓↓ − U↑↑ )/2



−

4t2
U

−

4t2
U

−

Figure 6-3: Table of second-order matrix elements for a ⟨ˆ𝑛⟩ = 2 Mott insulator.
2
Figure5adapted
from
Ref.
[97].
The
arrow
pointingupcorresponds
tothethestate−4t /U . We deﬁned an effective interaction 1/U := (1/U↑↑ + 1/U↓↓ )/2. On top of that there are a few corrections to
Heisenberg
model which arise from the fact that U↑↑ = U↓↓ . They are on the order δ 2 where δ := (U↑↑ − U↓↓ )/(U↑↑ + U↓↓ ).
|𝑆𝑧 = 1⟩ = |𝑎⟩|𝑎⟩ and the arrow pointing down corresponds to the state |𝑆𝑧 √= −1⟩ =|𝑏⟩|𝑏⟩. The filled circle corresponds to the state |𝑆𝑧 = 0⟩ = (|𝑎⟩|𝑏⟩ + |𝑏⟩|𝑎⟩)/ 2.
.
For more general points in the parameter space where 𝛿 ̸= 0 and 𝑑 ̸= 0, one cannot arrive at a mathematically consistent set of parameters for the diagonal elements(processes described in the first six rows of Fig. 6-3) because Eq. 6.2 becomes overdetermined. A heuristic approach is to obtain an approximate solution to the overdetermined system of linear equations 𝐴⃗𝑥 = ⃗𝑏 by minimizing the difference |𝐴⃗𝑥 − ⃗𝑏|.
98

The least-square solution obtained in such a way can be regarded as minimizing theoverall energy difference between the spectrum of the proposed effective Hamiltonian
and that of the true Hamiltonian. Using this approach, we obtain a set of parametersthat are approximately true up to first orders of 𝛿 and 𝑑:
𝐽𝑥𝑦 =
𝐽𝑧 =
𝐷 =
ℎ𝑧 =
𝑐 =

(︂
)︂
4𝑡2
𝑑
−
1+
𝑈
𝑈
(︂
)︂
2
4𝑡
𝑑
−
1−
𝑈
𝑈
)︂
(︂
2
2𝑡
𝑑 1−𝑧 2𝑈(︂)︂
6𝑡2
12 𝑡2 𝑑−𝑈 𝛿 1 + 𝑧 2 − 𝑧
𝑈
5 𝑈2 𝑈24𝑡
𝑑−𝑈 +𝑧𝑈(6.4)

See Appendix C for the derivation. The only major difference from the (𝑑 ̸= 0, 𝛿 = 0)
result is the appearance of a nonzero ℎ𝑧 , which scales linearly with 𝛿. Indeed, if we
set 𝑑 = 0, we obtain the same result as Eq. 6.3. With this approximate solution, theerror terms in the Hamiltonian scale with 𝑑2 , 𝛿 2 or 𝑑𝛿. In Fig. 6-4 we demonstrate
the effectiveness of the derived spin-1 model by comparing the numerically obtained
low-energy eigenvalues to those of the corresponding Bose-Hubbard model. We seethat after introducing the correct energy offset to the Bose-Hubbard eigenvalues, thelow energy spectra from the two models agree well.

∑︀In the actual experiments, the number of |𝑎⟩ and |𝑏⟩ atoms are conserved, so

𝑖 𝑆𝑖,𝑧 is preserved.

If the system is homogeneous, we are allowed to drop the bias

term ℎ𝑧 as it acts a global energy offset. However, if the system is inhomogeneous,the inhomogeneous bias term has a true physical effect and cannot be neglected
anymore [85]. The effect of the inhomogeneity is more pronounced in the spin-1/2
models as all parameters scale with 𝑡2 /𝑈 , which can vary substantially in a tightly
focused lattice beam.
99

Figure 6-4: Comparison of energy spectrum between the Bose-Hubbard model (leftcolumn) and the effective spin-1 model (right column). The respective plots show
the ten lowest eigenenergies for the given parameters and the boundary condition.
The physical system considered is a chain of four∑︀sites, filled with four |𝑎⟩ and four|𝑏⟩ atoms, which is equivalent to the constraint 𝑖 𝑆𝑧,𝑖 = 0 for the spin model. The
dimension of the Hilbert space for the Bose-Hubbard model is 1225 whereas it is 19
for the effective spin-1 model. We assume the Bose-Hubbard parameter of 𝑡/𝑈 = 0.1.
The spin-1 model uses parameters from Eq. 6.4. Here we assume 𝛿 = 0 but allow
non-zero 𝑑 = 𝑈 (1 − 𝑈𝑎𝑏 /𝑈 ). We also consider two different boundary conditions(open for the top row and periodic for the bottom row). An additional energy offset
expressed in Eq. 6.4 has been added to the Bose-Hubbard specta. The spectra agree
well, although there is more discrepancy between the higher energy spectra for theperiodic boundary condition.
.

6.2

Magnetic phases in spin-1

The magnetic phase diagram of spin-1 differs from that of spin-1/2 in that it featuresa phase transition from a gapped to a gapless phase [22]. In fact, the phase transition
is analogous to the superfluid-to-Mott insulator transition in the charge sector. Whenthe anisotropy energy 𝐷 is much higher than the exchange strength 𝐽, the groundstate is a product state of 𝑆𝑧 = 0. This state is called a spin Mott state. When the100exchange strength dominates over 𝐷, the energy can be lowered by increasing thefluctuations away from 𝑆𝑧 = 0. This state, where the average value of 𝑆𝑧 is still zerobut its variance is nonzero, is called a spin superfluid state or a spin-1 xy ferromagnetic
state, where we expect to see a quasi-long range order (characterized by an algebraic
decay of the order parameter) in the transverse spin [23]. In both magnetic phases,
there is no density transport but superfluid spin transport becomes possible within thespin superfluid phase. Such a motion is also known as counterflow superfluidity [21].
The spin Mott state is a useful initial state from which we can perform an adiabatic

Figure 6-5: Magnetic phases of a ferromagnetic spin-1 chain. (a) Top: spin Mott state(𝐷/𝐽 ≪ 1)). All sites are occupied by |𝑆𝑧 = 0⟩, which is a perfect pairing between|𝑎⟩ and |𝑏⟩ atoms. Bottom: spin superfluid or xy-ferromagnetic state (𝐷/𝐽 < 1).
Quantum fluctuation develops around the 𝑆ˆ𝑧 eigenstate with 𝑆𝑧 = 0. There is adefinite phase coherence between the neighboring spins. (b) One can attempt anadiabatic passage from one state to the other by ramping the control parameter 𝐷/𝐽.
A state-dependent lattice can control the value of 𝐷 by changing the wavefunction
overlap between |𝑎⟩ and |𝑏⟩ and thereby changing 𝑈𝑎𝑏 . Note that such control isbounded by |𝐷| ≤ |𝑈𝑎𝑎 + 𝑈𝑏𝑏 |/2.
passage toward the spin superfluid state because the spin Mott phase features anenergy gap between the ground state and a band of excited states (Fig. 6-4) that
reduces to zero as 𝑈𝑎𝑏 /𝑈 ramps from zero to one. Density matrix renormalizationgroup (DMRG) calculation for an one-dimensional Bose-Hubbard chain shows thatfor 𝑡/𝑈 = 0.1, a phase transition is expected to occur around 𝑈𝑎𝑏 /𝑈 = 0.956 [23]. If
our constituent atoms for the pseudo-spin-1 have opposite magnetic moments, we can101use a state-dependent optical lattice to control the separation between the two atoms
and thereby control the spin-dependent interacton 𝑈𝑎𝑏 . To initialize into a spin Mott
state, the two-component atom cloud is loaded into a state-dependent lattice with alarge separation between the different hyperfine states so that a very large 𝐷/𝐽 isachieved while the lattices ramp up to their final value and the atom cloud enters into
the Mott insulator phase. The spin Mott state is a state that features perfect pairing
between two different hyperfine states, and in the extreme anisotropy limit (𝑈𝑎𝑏 ≈ 0)
it can be regarded as a sum of two independent ⟨ˆ𝑛⟩ = 1 Mott insulators. Anothermethod to obtain the spin Mott state in a deep optical lattice is to start with a spinpolarized Mott insulator with |𝐹 = 1, 𝑚𝐹 = 0⟩⊗2 occupying the ⟨ˆ𝑛⟩ = 2 Mott phase,and then use resonant spin-changing collisions that transform a |𝐹 = 1, 𝑚𝐹 = 0⟩⊗2pair into a pair of |𝐹 = 1, 𝑚𝐹 = −1⟩ and |𝐹 = 1, 𝑚𝐹 = 1⟩ [98]. However, thistechnique requires a low bias field and a careful tuning of microwave Stark shift,which is necessary for reducing the detuning of the oscillation3 .

6.3

Spin-1/2 vs spin-1 realizations

Interaction and anisotropy
We note that the most important difference between the spin-1/2 model and thespin-1 model is the presence of a diagonal on-site interaction. In the spin-1/2 model,
which is realized within a ⟨ˆ𝑛⟩ = 1 Mott insulator phase, all on-site interactions are
virtual. On the other hand, the anisotropy in the on-site interaction directly shows up
as a magnetic anisotropy in the spin-1 model. Since on-site interactions are usuallymuch greater than second-order tunneling amplitudes in the Mott-insulating phase,even a small anisotropy in the on-site interaction can easily become comparable tothe exchange strength. Indeed, for 87 Rb, 𝑈𝑎𝑎 ≈ 𝑈𝑏𝑏 ≈ 𝑈𝑎𝑏 within 1% [99], and thespin-1/2 model is mapped to the isotropic XXX model [100, 101]. However, in our3

The very same spin-dependent interaction responsible for the spin-changing collision is also
⃗1 · 𝑆⃗2 term, theresponsible for this detuning. If we regard the spin-dependent interaction as a 𝑆transverse coupling 𝑆𝑥,1 𝑆𝑥,2 + 𝑆𝑦,1 𝑆𝑦,2 gives rise to the Rabi frequency of the oscillation whereas
𝑆𝑧,1 𝑆𝑧,2 gives rise to the detuning.

102

experiment we make use of the 1% difference between the interactions to access awide range of values for |𝐷/𝐽| in the spin-1 model.

Observables
In the exploration of ordered magnetic phases, we are interested in measuring spincorrelation functions such as the longitudinal correlation ⟨𝑆𝑧,𝑖 𝑆𝑧,𝑗 ⟩ or the transverse

correlation ⟨𝑆𝑖+ 𝑆𝑗− ⟩, which can be measured under a quantum gas microscope. Totransform a longitudinal correlation measurement 𝑆𝑧,𝑖 𝑆𝑧,𝑗 into a transverse correlation
measurement, one needs the transformation𝜋𝜋

𝑆𝑧 → 𝑒−𝑖 2 𝑆𝑦 𝑆𝑧 𝑒𝑖 2 𝑆𝑦 = 𝑆𝑥
For both spin-1/2 and spin-1 models, the unitary transformation is realized by a 𝜋/2pulse that puts the atoms into a superposition of |𝑎⟩ and |𝑏⟩ states. In terms of theactual atom density being detected by imaging, the longitudinal correlation can be
expressed as1⟨(ˆ𝑛𝑎 − 𝑛ˆ 𝑏 )𝑖 (ˆ𝑛𝑎 − 𝑛ˆ 𝑏 )𝑗 ⟩41=(⟨ˆ
𝑛𝑎,𝑖 𝑛ˆ 𝑎,𝑗 ⟩ + ⟨ˆ
𝑛𝑏,𝑖 𝑛ˆ 𝑏,𝑗 ⟩ − ⟨ˆ𝑛𝑎,𝑖 𝑛ˆ 𝑏,𝑗 ⟩ − ⟨ˆ
𝑛𝑏,𝑖 𝑛ˆ 𝑎,𝑗 ⟩)
4

⟨𝑆𝑧,𝑖 𝑆𝑧,𝑗 ⟩ =We can make use of the identity
𝑛2 = ⟨(ˆ𝑛𝑎 + 𝑛ˆ 𝑏 )𝑖 (ˆ𝑛𝑎 + 𝑛ˆ 𝑏 )𝑗 ⟩= ⟨ˆ
𝑛𝑎,𝑖 𝑛ˆ 𝑎,𝑗 ⟩ + ⟨ˆ
𝑛𝑏,𝑖 𝑛ˆ 𝑏,𝑗 ⟩ + ⟨ˆ𝑛𝑎,𝑖 𝑛ˆ 𝑏,𝑗 ⟩ + ⟨ˆ
𝑛𝑏,𝑖 𝑛ˆ 𝑎,𝑗 ⟩
where 𝑛 is the local occupancy of the Mott insulator. To summarize, the longitudinal
spin correlation can be connected to intra-spin density correlations ⟨ˆ𝑛𝑎,𝑖 𝑛ˆ 𝑎,𝑗 ⟩ and⟨ˆ
𝑛𝑏,𝑖 𝑛ˆ 𝑏,𝑗 ⟩ as follows:⟨ˆ𝑛𝑎,𝑖 𝑛ˆ 𝑎,𝑗 ⟩ + ⟨ˆ
𝑛𝑏,𝑖 𝑛ˆ 𝑏,𝑗 ⟩ = 2⟨𝑆𝑧,𝑖 𝑆𝑧,𝑗 ⟩ +
103

𝑛2
2

(6.5)

On the other hand, we can also express the longitudinal spin correlation in terms ofinter-spin density correlations ⟨ˆ𝑛𝑎 𝑛ˆ 𝑏 ⟩:⟨ˆ
𝑛𝑎,𝑖 𝑛ˆ 𝑏,𝑗 ⟩ + ⟨ˆ
𝑛𝑏,𝑖 𝑛ˆ 𝑎,𝑗 ⟩ =𝑛2
− 2⟨𝑆𝑧,𝑖 𝑆𝑧,𝑗 ⟩2

We provide an example calculation for the spin-1/2 model. If we assume a product√⨂︀ †state of the form 𝑖 (ˆ𝑎 + 𝑒𝑖𝜑ˆ𝑏† )/ 2|0⟩𝑖 , then Eq. 6.5 yields
1 1
1+ = 2⟨𝑆𝑧,𝑖 𝑆𝑧,𝑗 ⟩ +4 42which shows that the longitudinal spin correlation is zero, as expected. On the other
hand, if we apply a 𝜋/2 rotation4 before measuring the density-density correlations,
we obtain
(︂ )︂
(︂ )︂
1
𝜑
𝜑
4
sin
+ cos
= 2⟨𝑆𝑧,𝑖 𝑆𝑧,𝑗 ⟩ +22
2
4

If we average over the spin superfluid phase5 𝜑, then we obtain ⟨𝑆𝑥,𝑖 𝑆𝑥,𝑗 ⟩ = ⟨𝑆𝑦,𝑖 𝑆𝑦,𝑗 ⟩ =1/8 and thus ⟨𝑆𝑖+ 𝑆𝑗− ⟩ = 1/4, which is what we expect for a chain of spin-1/2 lying on
the equator of the Bloch sphere and pointing in the same direction. The same result
can be obtained by calculating the inter-spin density correlations.
The intra-spin density correlations can be measured directly under a microscope.
In principle, one can obtain some information about the intra-spin density correlations
without a microscope, by performing spatial quantum noise interferometry6 on the |𝑎⟩atom cloud and the |𝑏⟩ atom cloud separately after time of flight [103, 104]. Anothermethod to detect transverse spin correlation without high-resolution imaging is themeasurement of an enhanced fluctuation in 𝑛𝑎 − 𝑛𝑏 after applying a 𝜋/2 pulse. If
4Here we assume it is realized as a transformation exp (−𝑖(𝜋/2)𝑆𝑦 ) in order to measure ⟨𝑆𝑥,𝑖 𝑆𝑥,𝑗 ⟩.
A rotation with 𝑆𝑥 operator yields a different expression for the density correlations but the final
result for the spin correlation does not change, which is consistent with the axial symmetry of the
input state.
5
One could also average over the rotation direction and obtain the same answer, which is independent of 𝜑.
6
To be specific, one obtains a Fourier transform of the density correlation, assuming translational
invariance of the correlation ⟨^𝑛𝑖 𝑛^ 𝑗 ⟩ = 𝑓 (|𝑖 − 𝑗|). See Section 5.1.1 in Ref. [102].

104

there were no spin correlation, the fluctuation is described by shot noise, where thevariance scales with the atom number, but if all parts of the cloud are correlated,
one could see the variance scale with the square of the atom number. However, themeasured signal can also be affected by parts of the atom cloud that are not mapped
to the spin model under study, such as superfluid shells or the outer ⟨ˆ𝑛⟩ = 1 Mott
plateau in the case of the spin-1 model realization. The advantage of a quantum gas
microscope is that one can take a direct look at the core of the cloud and obtain ameasurement of the correlation operator without worrying about the surroundings ofthe core.
Finally, spin-1 has an observable that is not available to spin-1/2, and that is ⟨𝑆𝑧2 ⟩,which amounts to a constant in spin-1/2. We make use of this observable heavily toinfer the nature of the quantum state inside our spin-dependent lattice.

105

106

Chapter 7
Spin-1 physics with a two-component
Bose gas
In the preceding chapters we have extensively explained how we realize a spin-1
model with two-component bosons in a deep three-dimensional optical lattice, where
the bosons reside in a Mott insulating phase at the center of the trap. By making
one axis of the optical lattice to be state-dependent, we can control the interaction
between atoms in different hyperfine states. It is also possible to study spin-1 physics
in a state-independent lattice by choosing an appropriate set of hyperfine states andmaking use of the small anisotropy in the spin-dependent interactions between atoms.
In this chapter we summarize our experimental observations of spin-1 physics in astate-independent lattice and also in a state-dependent lattice.

7.1

Measurement of local spin pairing

A spin observable that we can measure reliably is the average spin pairing fraction
𝑓 , or the fraction of |𝑆𝑧 = 0⟩ = |𝑎⟩|𝑏⟩ pairs among all pairs in the ⟨ˆ𝑛⟩ = 2 Mott
insulator. Formally 𝑓 is equivalent to 1 − ⟨𝑆𝑧2 ⟩, where ⟨. . .⟩ represents averaging overall sites in a spin chain. Ideally, we would like to image and then count just the |𝑎⟩|𝑏⟩pairs in the system. However, that requires many experimental prerequisites. First,
the |𝑎⟩|𝑏⟩ pairs need to be transferred to a different state before imaging while the107|𝑎⟩|𝑎⟩ and |𝑏⟩|𝑏⟩ pairs remain in their original states and stay invisible to the imaging
beam. Furthermore, the new state to which the |𝑎⟩|𝑏⟩ pairs are transferred needs tobe collisionally stable so that the pairs do not decay after the transfer and before theimaging step. Finally, because not all the atoms in the optical lattice form pairs (seeFig. 5-2), we need to normalize the number of |𝑎⟩|𝑏⟩ pairs by the number of all pairs,
and we would like to have a single experimental technique that can detect pairs inboth state-sensitive and state-insensitive ways.
A relatively simple technique that can measure pairs is detection by an induced
loss, where the induced loss can be either state-sensitive or state-insensitive. Thenthe number of pairs is estimated by taking the difference between the atom numbers
before and after the induced loss. We induce losses in pairs by using narrow Feshbachresonances for |𝐹 = 1, 𝑚𝐹 = 1⟩ and |𝐹 = 2, 𝑚𝐹 = −2, −1, 0⟩ around a bias fieldof 9 G [105]. Around the resonances, the inelastic two-body collision rate is greatly
enhanced. After such a collision, the initial pair of a 𝐹 = 1 atom and a 𝐹 = 2 atomexits as a pair of two 𝐹 = 1 atoms that maintain the total magnetic quantum number∑︀𝑚𝐹 , and the large hyperfine energy splitting (ℎ × 6.8 GHz) between the 𝐹 = 2 and𝐹 = 1 manifolds is released as a kinetic energy, promptly kicking the pair out of thesystem1 .

In our experiment we make use of two Feshbach resonances: the |1, 1⟩|2, −2⟩
resonance at 9.0918 G and the |1, 1⟩|2, 0⟩ resonance at 9.0448 G. Both resonances
have narrow widths (<2 mG) [105]. Our experiments are performed at a bias field of
8.916 G, and we provide an additional current to the bias coil from a high precision
power supply (Keysight E3616A) to reach the Feshbach resonances. Before inducing
the loss, we make sure that all the atoms are either in state |1, −1⟩ or |1, 1⟩, andthen transfer |1, −1⟩ to either |2, −2⟩ or |2, 0⟩ (Fig. 7-1). After the transfer, we turnon the additional current to reach the desired resonance and modulate the bias fieldaround the resonances with a peak-to-valley amplitude of 10 mG, which is larger than
the resonance widths. The modulation is done at the frequency of 300 Hz while the1Such a mechanism is called spin-exchange relaxation. To understand how a 𝐹 = 2 state isflipped to a 𝐹 = 1 state during the collision, consult Chapter 5 in Ref. [106].

108

atoms are held inside the optical lattices. After some hold time (<100 ms), all thepairs sensitive to the selected Feshbach resonance have been removed from the systemand the bias field is reduced back to 8.916 G, at which any remaining 𝐹 = 2 atom istransferred back to a 𝐹 = 1 state. Then the remaining number of atoms is measured.
For the |1, 1⟩|2, −2⟩ resonance, which is used to selectively remove |𝑎⟩|𝑏⟩ pairs, all|𝑎⟩|𝑏⟩ are removed within 10 ms of hold time inside a deep three-dimensional opticallattice with lattice depth of 35 recoil energy in each direction. The |𝑎⟩|𝑎⟩ = |1, 1⟩|1, 1⟩
and |𝑏⟩|𝑏⟩ = |2, −2⟩|2, −2⟩ pairs remain intact as they are collisionally stable. On theother hand, for measuring all pairs, we have to apply a special trick as a Feshbachresonance itself is state-sensitive. We employ a continuous and diabatic Landau-Zenersweep between |1, 1⟩ and |2, 0⟩ so that all pairs (|1, 1⟩|1, 1⟩, |2, 0⟩|2, 0⟩, and |1, 1⟩|2, 0⟩)
are admixed with a |1, 1⟩|2, 0⟩ pair wavefunction and gain a non-zero decay rate. The
center of the diabatic sweep is set to the |1, 1⟩ → |2, 0⟩ resonance at 9.0448 G andthe sweep scans a range of ±140 kHz around the center with 1 kHz sweep rate in atriangular waveform2 . All pairs are seen to decay within 30 ms of hold time at the
Feshbach resonance with this technique.
The average local spin pairing fraction is estimated from measurement of threenumbers. One first measures the number of all atoms 𝑁𝑎 before the induced losses.
Then one measures 𝑁𝑝 , the number of atoms left after removing |𝑎⟩|𝑏⟩ pairs, and 𝑁𝑑 ,the number of atoms left after removing all pairs. The pairing fraction can then be
expressed as𝑓=𝑁𝑎 − 𝑁𝑝
𝑁𝑎 − 𝑁𝑑

For absorption imaging, we have to measure 𝑁𝑎 , 𝑁𝑝 , and 𝑁𝑑 separately as the imaging
process is destructive. Hence, a single datum of the pairing fraction measured thisway requires three experimental cycles. Because 𝑓 is a ratio of differences, number
fluctuations measured in each of 𝑁𝑎 , 𝑁𝑝 , 𝑁𝑑 can lead to a large error once propagated.
So it is crucial that the atom number preparation is as stable as possible. A possible
2

The optimal sweep range and sweep rate will in general depend on the Rabi frequency for the|1, 1⟩ → |2, 0⟩ transition, which is estimated to be 𝒪(10 kHz) in our experiment.

109

Figure 7-1: Measurement of local spin pairing by inducing spin-selective decay. We
make use of narrow Feshbach resonances around 𝐵 = 9 G [105]. Each resonanceoccurs for a pair of |𝐹 = 1, 𝑚𝐹 = −1⟩ state and one of |𝐹 = 2, 𝑚𝐹 = 0, −1, −2⟩ states(relevant states highlighted in yellow). (a) Measurement of |𝑎⟩|𝑏⟩ pairs by removal.
First, |𝑏⟩ = |1, −1⟩ is mapped to |2, −2⟩. Then the bias is tuned to the Feshbachresonance for the |1, 1⟩, |2, −2⟩ pair. (b) Measurement of all pairs by removal. First,|𝑏⟩ = |1, −1⟩ is mapped to |2, 0⟩. Then the bias is tuned to the Feshbach resonance forthe |1, 1⟩, |2, 0⟩ pair. With a continuous diabatic Landau-Zener sweep between |1, 1⟩
and |2, 0⟩, we ensure that every pair (|𝑎⟩|𝑎⟩, |𝑏⟩|𝑏⟩, |𝑎⟩|𝑏⟩) is admixed with a decaying
component.

110improvement would be to use a nondestructive in-trap imaging, such as polarization
rotation imaging or phase shift imaging, that we can use to image the same sample
consecutively and obtain (𝑁𝑎 , 𝑁𝑝 , 𝑁𝑑 ) whose fluctuations are correlated. We actuallytried both techniques, but a persistent problem was a small loss of atoms from thelattice after each exposure, as nondestructive imaging does impart recoil momenta to
the atoms, and we could not ascertain any correlation between the measured 𝑁𝑎 − 𝑁𝑝
and 𝑁𝑝 − 𝑁𝑑 . After several months of investment into the nondestructive imaging
setup, we were forced to fall back on absorptive imaging. To obtain a decent estimateof 𝑓 , we take three to four measurements each for 𝑁𝑎 , 𝑁𝑝 , and 𝑁𝑑 . On a passing note,the most consistent data for pairing fraction was obtained when the lattice alignments
were done with utmost precision, which suggests that the fidelity of the spin physics
inside the lattice may be sensitive to the initial lattice loading conditions. If there
were misalignments, it is possible that the density distribution prepared is not in
equilibrium and does not match the expect distribution for a Mott insulator.
For convenience, in the following discussion we use a variable named spin alignment 𝐴 = 𝑆(𝑆 + 1) − 3⟨𝑆𝑧2 ⟩ in lieu of 𝑓 . The variable is equivalent to 3𝑓 − 1. By
definition, spin alignment is zero when the spin system is at infinite temperature(𝑓 = 1/3).

7.2

Observation of exchange in spin-1 model

In this section we summarize our study of non-equilibrium dynamics in the spin-1
model, where we observe exchange-induced relaxations in the spin alignment after aa quench in the anisotropy-to-exchange ratio 𝐷/𝐽 (with 𝐽 changing from zero to a
non-zero value). An unexpected result is that the relaxation is most pronounced when
the anisotropy and the exchange are on an almost equal footing (|𝐷/𝐽| ≈ 2). We
note that this experiment is performed in a state-indepedent lattice (all lattice beams
have wavelength of 1064 nm), and hence 𝐷/𝐽 is changed not by changing 𝐷 but
primarily by changing 𝐽 through a variation of the lattice depth. Although Eq. 6.4
predicts a nonzero exchange anisotropy and a correction to the single-ion anisotropy111

in general, because both 𝑑/𝑈 and (𝑡/𝑈 )2 are less than 1% in this experiment, suchcorrections can be ignored. The main result has been published in Ref. [24], a copyof which is provided in Appendix B. In the following we also provide more details forthe experimental conditions not mentioned in the publication.

7.2.1

Idea behind the experiment

The original idea behind the experiment was to find a signature of spin exchange inour ⟨ˆ
𝑛⟩ = 2 Mott insulator by preparing a non-equilibrium distribution of |𝑎⟩|𝑏⟩ =|𝑆𝑧 = 0⟩ and monitoring its relaxation toward the equilibrium. If the dynamics inside
the Mott insulator is governed by kinetic exchange, then we expect the relaxation
rate to scale with |𝐽| = 4𝑡2 /𝑈 as a function of lattice depth along the spin chain
direction. Because 𝐽 scales exponentially with lattice depth (Fig. 7-2), we expecteda monotonic change in the relaxation rate as a function of lattice depth and hoped
to see a constant coefficient after normalizing the different relaxation rates by 𝐽.
However, as it is explained later, such a conclusion was erroneous because we did notconsider the effect of the single-ion anisotropy term in our model at all3 .

J/h+]

102|a = |1, 1 |b = |1, 1|a = |1, 1 |b = |1, 0

100

D/h+]

10 2











/RQJLWXGLQDOODWWLFHGHSWK(ER)

Figure 7-2: Variation of -𝐽 and 𝐷 with respect to the longitudinal lattice depth. The
transverse lattices are set to 35 ER . The lattice laser wavelength is 1064 nm.
3Julius de Hond, our postdoctoral associate, noted first that the SIA term might be important
after performing numerical simulations.

112

Figure 7-3: Experimental sequence for the measurement of spin alignment and doublon fractions. (i) The lattices are ramped up to initialize a single-component Mott
insulator with a maximal site occupancy of two.√(ii) Microwave pulses prepare a superposition of two hyperfine states (|𝑎⟩ − 𝑖|𝑏⟩) / 2. Ramping down the longitudinallattice initiates spin exchange dynamics. (iii) Ramping up the lattices stops the exchange dynamics. Microwave pulses transfer the two components to a pair of stateswith a Feshbach resonance. (iv) Either |𝑎𝑏⟩ doublons or all doublons are removed
with the help of Feshbach-enhanced inelastic losses. Remaining atoms are transferredback to the 𝐹 = 1 hyperfine states and are counted via absorption imaging to measure
𝑁𝑝 or 𝑁𝑑 .

113

7.2.2

Experimental sequence

The experimental sequence is described in Fig. 7-3. A spin-polarized Mott insulator
with all the atoms in the |𝑎⟩ state receives a 𝜋/2 pulse that puts |𝑎⟩ into a superposition with |𝑏⟩. Then the initial average fraction of |𝑎⟩|𝑏⟩ pairs in the ⟨ˆ𝑛⟩ = 2 Mott
insulator is 0.5, and hence the initial spin alignment is 3(0.5) − 1 = 0.5. During
this state preparation, all lattices are deep and hence 𝐽 is negligible. We have two
possible choices4 for |𝑎⟩ and |𝑏⟩. The first choice realizes a negative value of 𝐷, which
is |𝑎⟩ = |1, −1⟩ and |𝑏⟩ = |1, 1⟩. The second choice realizes a positive value of 𝐷,
which is |𝑎⟩ = |1, −1⟩ and |𝑏⟩ = |1, 0⟩. The scattering lengths for different combinations of the hyperfine states can be found in Appendix B. For both choices, thestate preparation begins with a diabatic Landau-Zener sweep of |1, −1⟩ to |2, 0⟩ thattransfers roughly 50% of the population into the latter state. Then the |2, 0⟩ state canbe adiabatically transferred to either |1, 1⟩ or |1, 0⟩. Because |2, 0⟩ can undergo spinexchange relaxation when partnered with a 𝐹 = 1 atom, the second and adiabatic
sweep is performed quickly after the first and diabatic sweep. The typical two-bodydecay rate 𝜅 for |2, 1⟩ is 𝒪(10−13 cm3 /s) [107], and we assume that |2, 0⟩ decays ata similar rate. For a peak on-site density of 1.4 × 1014 /cm3 in a three-dimensionallattice with depth 30 ER,1064nm , we expect roughly a 5% loss during a 5 ms sweep5 .
Once the initial state preparation is finished, the lattice depth is quickly reduced
to a lower value along the longitudinal direction, so that the Mott insulator can be
regarded as a two-dimensional array of isolated one-dimensional spin chains. Thisquench step happens over several milliseconds, so that it is highly diabatic withrespect to the exchange 𝐽 but adiabatic with respect to the on-site interaction energy
𝑈 so that we do not create quasiparticle-quasihole excitations from the Mott insulator
ground state. A reverse quench to a deep three-dimensional lattice is performed aftera variable hold time in the lowered longitudinal lattice, in order to carry out the4Note that we previously referred to |1, −1⟩ as |𝑏⟩, e.g. in Fig. 7-1. For convenience, we now refer
to it as |𝑎⟩.
5
The main limitation was the relatively small Rabi frequency for the |2, 0⟩ → |1, 1⟩ transition.
The Landau-Zener sweep for |2, 0⟩ → |1, 0⟩ transition can be replaced by a short Rabi 𝜋 pulse due
to its relative insensitivity to bias fluctuation.

114

Feshbach-enhanced loss measurement of pairs.

7.2.3

Result and analysis

To our surprise, we discovered that the spin alignment changed noticeably faster not
at the lowest lattice depth at which we can hold onto the Mott insulator, but at someintermediate lattice depth (Fig. 7-4). Furthermore, the sign of the change in the spinalignment follows the sign6 of 𝐷/|𝐽|. The latter point can be intuitively understoodby considering that spins prefer to be in |𝑆𝑧 = 0⟩ (higher spin alignment) when𝐷 > 0 and |𝑆𝑧 = ±1⟩ (lower spin alignment) when 𝐷 < 0. If there were any sourceof external heating rate that can disorder the spin chain, the spin alignment would
reduce from the initial value of 0.5 to zero and produce a signal that looks similar
to the 𝐷 < 0 result. Thus, the observation of an increase in the spin alignment for
𝐷 > 0 is strong evidence of exchange dynamics within the spin chain.
The intermediate lattice depths at which the spin alignment is seen to have extremal changes correspond to |𝐷/𝐽| ≈ 2. This can be qualitatively understood byconsidering a two-site model. In a two-site model, the only states of the Hilbert spacethat can contribute to a change in the spin alignment are |𝑆𝑧,1 , 𝑆𝑧,2 ⟩ = |0, 0⟩ and√(|1, −1⟩ + |−1, 1⟩)/ 2. Thus, the dynamics is reduced to that of a two-level system.
If we picture the dynamics of the two-level system on a Bloch sphere (Fig. 7-5), with
|0, 0⟩ acting as the north pole, we see that the Hamiltonian is represented by a ficti√
tious field of strength −(𝐽 +2𝐷) on the z-axis and −2 2𝐽 on the x-axis. Thus, the to√︀√
tal fictitious field has a strength of 8𝐽 2 + (𝐽 + 2𝐷)2 = 9𝐽 2 + 4𝐷𝐽 + 4𝐷2 and its√
axis is oriented away from the north pole by an angle 𝜃 where tan 𝜃 = 2 2𝐽/(𝐽 +2𝐷).
Our initial state, when projected to the reduced Hilbert space, corresponds to)︃√︂√ (︃√︂
6
2
1 |1, −1⟩ + |−1, 1⟩√|0, 0⟩ +4
3
3
2
On the Bloch sphere, this projected state precesses around the axis of the total
fictitious field, and thus the total fraction in |0, 0⟩ oscillates with a peak-to-valley
6⃗𝑖 · 𝑆⃗𝑗 there.
Note 𝐽 = 4𝑡2 /𝑈 in Ref. [24] as the exchange interaction is defined as −𝐽 𝑆

115

amplitude of
2

(︂

(𝐷/𝐽)
9 + 4(𝐷/𝐽) + 4(𝐷/𝐽)2)︂

This amplitude is zero when 𝐷/𝐽 = 0 and has maximal values at 𝐷/𝐽 = ±3/2. Atzero anisotropy, the initial state is simply a rotated version of the ferromagnetic state
where all spins point up (𝑆𝑧 = 1) and hence it is an eigenstate of the isotropic spin-1
model, meaning that there is no temporal change in the spin alignment. Indeed,this situation corresponds to the projected initial state being parallel to the total
fictitious field on the Bloch sphere. On the other hand, if the anisotropy is extreme(|𝐷/𝐽| ≫ 1), the total fictitious field on the Bloch sphere points along the z-axis,
and hence the projection onto the north pole (population in |0, 0⟩) does not change.
Physically, a large anisotropy makes spin exchange off-resonant and prevents |1, −1⟩from converting into |0, 0⟩ and vice versa. So it is natural to conclude that an extremalchange in the spin alignment occurs when |𝐷/𝐽| is somewhere between zero and alarge value. Without anisotropy, no dynamics can be induced by a simple rotation of
the initially spin-polarized state, but the anisotropy has to be small enough to allow
spin exchange to occur. As you increase the number of sites in the spin chain, we
expect a variety of precession frequencies arising from Hilbert subspaces with different
total 𝑆𝑧 values, and hence a single-frequency oscillation in the two-site model changes
into a relaxation.

7.2.4

Possible effects of magnetic gradient

A source of perturbations that can couple to the spin dynamics is the residual magnetic gradient, whose strength we denote as ∆. It creates a magnetic energy offset
between neighboring sites. If the gradient is strong enough, it can reverse the sign ofthe change in the spin alignment, so that the spin alignment is reduced for 𝐷 > 0 andenhanced for 𝐷 < 0. This can be also understood by considering the two-site model(Fig. 7-6). In the presence of a gradient, the Hilbert subspace that is responsible fora change in the spin alignment consists of three states: |0, 0⟩, |1, −1⟩, and |−1, 1⟩.
116

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Figure 7-4: Coherent dynamics of the spin alignment 𝐴 after a quench in 𝐷/𝐽. Varying the hold time at characteristic lattice depths for both positive and negative valuesof 𝐷/𝐽 (top and bottom pairs of panels, respectively) reveals that strong transientsin 𝐴 only occur at intermediate lattice depth for which 𝐷 and 𝐽 are comparable.
Dashed lines are the results of the MPS-TEBD simulation. The shaded regions denote the MPS-TEBD results with ±0.5 𝐸𝑅 uncertainty in the lattice depths, andinclude exponential decay towards a thermal spin state with 𝐴 = 0 with empirical
1/𝑒 times of 350 ms (𝐷 > 0) and 175 ms (𝐷 < 0), the ratio of which reflects therelative sensitivity to ambient magnetic field fluctuations of the pairs.

117

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Figure 7-5: Coherent spin oscillations in a two-site model. While the full basis contains nine states, the oscillations in the spin alignment 𝐴 involve only a 2 × 2 blockof the Hamiltonian. This allows us to illustrate the dynamics on Bloch spheres (left),
where the initial state is represented by the open circle. If 𝐽 = 0 the effective magnetic field points along 𝑧ˆ, and the purely azimuthal precession will not change 𝐴. If
𝐽 > 0 the effective magnetic field is tilted, resulting in a precession along the dashedcircle which is observed as an oscillation√︁ in 𝐴 (right). The frequency of the oscilla-tion, in units of 𝐽/~, is given by Ω = 9 + 4𝐷/𝐽 + 4 (𝐷/𝐽)2 , and its peak-to-peak[︀]︀amplitude is 6 (𝐷/𝐽) / 9 + 4𝐷/𝐽 + 4 (𝐷/𝐽)2 (see inset). This shows that the direction of oscillation depends on the sign of 𝐷/𝐽 (compare top and bottom rows, where
𝐷/𝐽 = 0.5 and −0.5, respectively). Note that while the initial value of 𝐴 for thissubspace is 1, the contribution of other states sets the initial 𝐴 of the whole system
to 1/2.

118

At zero gradient, |1, −1⟩ and |−1, 1⟩ are degenerate and the transitions from |0, 0⟩ tothose states are equally detuned. However, if ∆ = ±(𝐽 + 2𝐷), we obtain a resonanttransition between |0, 0⟩ and either |1, −1⟩ or |−1, 1⟩. Similarly to the zero gradientcase, we can restrict ourselves to a two-level system to obtain a better analysis. In
this case the Bloch sphere corresponding to the two-level system consists of |0, 0⟩
on the north pole and either |1, −1⟩ or |−1, 1⟩ on the south pole. Without loss ofgenerality, we assume |1, −1⟩ is on the south pole. On the Bloch sphere, the fictitious
field is (𝐵𝑥 , 𝐵𝑧 ) = (−2𝐽, ±∆ − 𝐽 − 2𝐷). The orientation (𝑣𝑥 , 𝑣𝑧 ) of the projectionof the initial state onto the Bloch sphere is (4/5, 3/5). If (±∆ − 𝐽 − 2𝐷) = −3/2𝐽,
then the fictitious field becomes anti-parallel to the projection of the initial state. So
if we set the gradient to ±∆ = 2𝐷 − 𝐽/2, we expect no dynamics other than a highly
detuned oscillation between |0, 0⟩ and the leftover state |−1, 1⟩. By perturbing thegradient value around this point, we can tilt the axis of the fictitious field away fromthe initial state orientation and guide its axis either closer to or farther away from thenorth pole, so that one may expect, for example, an increase in the spin alignment for
𝐷 < 0 if the gradient is strong enough. Indeed, numerical simulations show that for
longer chains, ∆/𝐽 ≈ 2|𝐷/𝐽| + 2 provides a noticeable reversal of the spin alignmentchange. In our experiment, the value 2|𝐷| + 2𝐽 for |𝐷/𝐽| = 2 is 2𝜋 × 6 Hz for 𝐷 > 0
and 2𝜋 × 36 Hz for 𝐷 < 0. Such gradient strengths correspond to 0.16 G/cm for
𝐷 > 0 and 0.5 G/cm for 𝐷 < 0, given the experimental conditions. We make surethat our residual gradient level is far below such numbers after shimming the fields
and checking the Ramsey coherence lifetime of a field-sensitive superposition state(Fig. 2-4) and thus we conclude that our main observation is not affected by theresidual magnetic gradient.

7.2.5

Conclusion

In conclusion, our observation of the correspondence of the sign of the spin alignment change to the sign of the anisotropy 𝐷 and of an enhanced change in the spinalignment near 2𝐽 ≈ |𝐷| strongly suggests that we have observed a coherent spin-1
dynamics within our ⟨ˆ𝑛⟩ = 2 Mott insulator. Because of the smallness of the exchange
119Figure 7-6: Effect of magnetic gradient on a two-site spin-1 model. (a) No gradient.
(b) Finite gradient. If ∆ ≈ 𝐽 + 2𝐷, one can see a large change in the |0, 0⟩ state
population even in the presence of a large anisotropy. (c) Simulation of spin alignment∑︀evolution in a two-site spin-1 model augmented by a magnetic gradient −∆ 𝑖 𝑖𝑆𝑧,𝑖 .
∏︀2 (︁ |𝑎⟩𝑖 +|𝑏⟩𝑖 )︁⊗2√
Here ∆/𝐽 = 2|𝐷/𝐽| + 2. The initial state is 𝑖=1. The dynamics with2
negative (positive) anisotropy now produces an upward (downward) trend in the spinalignment change, which is opposite to the trend shown in the zero gradient case.

120

strength (< 2𝜋 × 10 Hz in our experiment), one expects a very low spin entropy inthe system as a prerequisite for studying the physics of coherent exchange, and our
results show that the spin entropy stays low for at least an exchange period. Ourdata analysis also demonstrated the usefulness of simple models such as effective twolevel systems for studying coherent spin dynamics. On a broader outlook, simulationof non-equilibrium spin-1 physics with arbitrarily prepared initial states using ultracold atoms may be quite useful for providing a clean study of complicated effects
observed in magnetic materials, such as the renormalization of interactions between
spin excitations (magnons) at high densities in a spin-1 ferromagnet [108].

7.3

Investigation of the spin Mott phase

In this section we summarize our study of the ground state properties of the spin-1
model. Primarily, we focus on the spin Mott state (𝐷/𝐽 ≫ 1) and we rely on astate-dependent lattice along the bias field direction to tune 𝐷/𝐽 over a wide range.
In the limit of extremely high 𝐷/𝐽, the spin Mott can be thought of as an overlayof two independent spin-polarized ⟨ˆ𝑛⟩ = 1 Mott insulators. While this is no longer
true for a finite interaction between different spins (𝑈𝑎𝑏 > 0), the analogy to a regularcharge Mott insulator is useful, as we show in the following discussions. However, thespin Mott state is more than just an analogue to the density Mott state, as it allows
an adiabatic passage toward the fragile spin superfluid state (Fig. 7-7). Since thethe spin superfluid phase is characterized by a very low energy scale, the adiabatic
ramp is also a form of adiabatic cooling within the Mott insulating state. We havecharacterized the properties of the spin Mott phase in various ways.

7.3.1

Adiabatic passage with a state-dependent lattice

The primary role of the spin Mott state is to use its high energy gap to prevent the
spin entropy from increasing while performing an adiabatic ramp to another spinfulstate. Within the deep spin Mott phase, spin entropy can be measured by counting
how many sites do not have a pairing between two opposite hyperfine states. However,121

Figure 7-7: Phase diagram of the spin model. Figure adapted from Fig. 1 in Ref. [23].
The white dashed line is the phase boundary between the spin Mott phase and thespin superfluid phase calculated by 1D DMRG.
.
one can also demonstrate the effect of spin entropy in a more macroscopic sense, by
measuring the reduction of the condensate fraction of a BEC in a spin superposition
state which then undergoes decoherence. We present both approaches and we first
expound on the condensate fraction measurement experiment.
Spin entropy and BEC condensate fraction
A BEC in a spin superposition state without any dephasing can be considered aspin-polarized cloud. However, after dephasing, the BEC turns into a mixture of two
smaller BECs, and the entropy of mixing leads to a decrease in the overall condensate
fraction. To probe this effect, a spin polarized cloud is initialized into a ⟨ˆ𝑛⟩ = 1
Mott insulator. To create a superposition state with a long coherence time, we usedthe clock state pair |1, −1⟩ + |2, 1⟩ around the bias field of 3.23 G, where the state
becomes first-order insensitive to ambient field fluctuations. The superposition state
was created by a 𝜋/2 two-photon pulse. On the other hand, other superposition statessuch as |1, 1⟩ + |2, −1⟩ or |1, −1⟩ + |1, 1⟩ are sensitive to ambient field fluctuations
and dephase rather quickly. By ramping down the Mott insulator back to BEC and122

Figure 7-8: Demonstration of an increase in condensate fraction due to the spin
entropy from spin dephasing. A spin polarized cloud of |1, −1⟩ (or |1, 1⟩) is preparedinto a ⟨ˆ𝑛⟩ = 1 Mott insulator held in a 3D 25 𝐸𝑅,1064 nm lattice (first column). The
condensate fraction is measured after ramping down the lattice and transforming
the Mott insulator back to BEC. After a 𝜋/2 pulse, each atom is put into a spin
superposition state (second column). The pair |1, −1⟩ + |2, 1⟩ is insensitive to field
fluctuations in first order around bias field of 3.23G, while other combinations dephasequickly under field fluctuations. The field sensitive superposition states result in avisible drop in the retrieved condensate fraction. On the other hand, if we complete
the spin flip with a 𝜋 pulse (third column) before the dephasing can take place, theoriginal condensate fraction is largely recovered.

123

measuring the retrieved condensate fraction, we noticed a visible reduction in thecondensate fraction in the case of field-sensitive superposition states, whereas thefield-insensitive superposition state suffers less reduction (Fig. 7-8). The reason for
the rather low initial values of the condensate fraction is due to the very low atom
number required to reduce ⟨ˆ𝑛⟩ and the limited time over which the lattices can beramped down smoothly. Because of spin exchange relaxation between 𝐹 = 1 and𝐹 = 2 states, as soon as the Mott insulator melts and the atoms can interact witheach other, the atom cloud undergoes a fast decay. Hence, the Mott insulator had to
be ramped down as quickly as possible while allowing a few tunneling times for phasecoherence to build up between the atoms [109].
The condensate fraction measurement demonstrates that without a state-dependent
potential, atoms in different hyperfine states are free to mix around, resulting in alarge spin entropy (lack of spin ordering). In the next subsection, we demonstrate
how the spin entropy of the two-component atom cloud can stay minimized by loading
the cloud into a state-dependent potential.

Spin entropy and state-dependent lattice control
We use a state-dependent lattice at the wavelength of 810 nm, which provides a vectorto-scalar ratio 2𝑅 = 𝛼𝑉 /𝛼𝑆 = 1/4 (see Eq. 3.3 and Sec. 4.2.1). The 810 nm latticebeam is generated by a continuous-wave, narrow linewidth (<50 kHz) Ti:sapphirelaser (M Squared SolsTiS 5W). The hyperfine states used are |𝐹 = 1, 𝑚𝐹 = −1⟩
and |𝐹 = 1, 𝑚𝐹 = 1⟩. The experimental sequence is described in Fig. 7-9. If wewere to ramp an atom cloud of spin mixture directly into a deep optical latticewithout any state-dependent potential, the atoms will in general require some density
redistribution (for the formation of a Mott insulator) and also spin redistribution (forthe formation of a spin superfluid if the inherent 𝐷 is smaller than 𝐽) in order tostay in the ground state of the system. The presence of a large fictitious magnetic
gradient from the state-dependent lattice ensures that the atoms find it energeticallyfavorable to organize themselves in a simple spin density distribution, where different
hyperfine states are interleaved with each other [step (ii) in Fig. 7-9]. Within the spin
124

Figure 7-9: Experimental sequence for the measurement of spin alignment after aramp into a state-dependent lattice. The longitudinal lattice is the state-dependentlattice. (i) Before the lattice ramp, an initially spin-polarized atom cloud is transformed into a superposition of two different hyperfine states with opposite magnetic
moments. (ii) The atom cloud is ramped into a deep three-dimensional optical lattice, with a non-zero polarization gradient along the state-dependent lattice direction.
The wavefunction overlap between the different hyperfine states is reduced. After the
ramp the variable retarder can further change the polarization gradient or stay at
the present setting. (iii) After a variable hold time, the variable retarder ramps thepolarization gradient to zero and the initially separated states are merged. (iv) Spinpairing fraction measurement is performed as described in Sec. 7.1.

125

Mott phase, the spin entropy of the system can be inferred from the deviation of thespin pairing fraction from the ideal value of one7 .
The experimental data is shown in Fig. 7-10(a), where we initialize a spin Mott ina very deep optical lattice with depths 35 𝐸𝑅,1064nm in the transverse directions and
15 𝐸𝑅,810nm in the longitudinal direction, along which we have 𝑈/𝑡 = 58. After the
initialization we adiabatically ramp down the anisotropy to a final value (parametrizedby 𝑈𝑎𝑏 /𝑈 ). The ramp happens over many tens of milliseconds, and we compensate thelattice intensity so that the state-dependent lattice stays constant while the contrast of
the polarization gradient is reduced by the variable retarder. We see that we maintaina very high pairing fraction characteristic of the spin Mott phase until 𝑈𝑎𝑏 /𝑈 = 0.85.
According to a full numerical simulation of the one-dimensional Bose-Hubbard model[Fig. 7-10(b)], we expect to maintain a high pairing fraction until we get very close
to the isotropy point 𝑈𝑎𝑏 /𝑈 = 1 in this deep lattice, but we start to lose the pairing
fraction early on. If the ramp were an adiabatic process, we expect to recover the loss
in the pairing fraction when we ramp back to the high anisotropy region (low 𝑈𝑎𝑏 /𝑈 ).
However, we have seen no sign of such recovery, suggesting that the loss in the pairing
fraction is a real increase in spin entropy within the state-dependent lattice. Someexperimental concerns so far are
1. A large change in the overall confinement during the lattice-depth-compensatedpolarization gradient ramp due to the tight Gaussian envelope of the 810 nm.

2. Highly non-equilibrium density loading of the Mott insulator. The mismatch
between the estimated ⟨ˆ𝑛⟩ = 2 Mott insulator size and the actual atom number
is even bigger for the 810 nm lattice than it was for the 1064 nm lattice (Fig. 5-3).

3. Light-scattering limited lifetime. We expect that every atom will have scattereda photon on average after a 1 s hold time in a lin-90∘ -lin lattice of 810 nm [see7

For a finite exchange strength, the ground state spin alignment will be slightly less than one.

126

Fig. 3-2].

4. Noise from the liquid crystal (LC) variable retarder. The LC device can impart
both lattice intensity noise and lattice phase noise. The lattice phase noise issue
can be solved by placing the retarder before the atom cloud (see Sec. 4.2.1).

Although the current state-dependent lattice in use is not fully adequate, it does
provide some degree of protection from an increase in spin entropy. If we load thespin mixture atom cloud directly into the state-dependent lattice with a final 𝑈𝑎𝑏 /𝑈value, without any spin Mott initialization at a lower 𝑈𝑎𝑏 /𝑈 , then we see that it
always returns a reduced spin pairing fraction [red data points in Fig. 7-10(a)]. When
we load the cloud at the isotropy point 𝑈𝑎𝑏 /𝑈 = 1, we expect to see a pairing fractionof 1/3, which is the value consistent with a uniformly random spin ordering insidethe ⟨ˆ
𝑛⟩ = 2 Mott insulator. However, we see deviations from that value, and thedeviation depends on the time between the end time of the lattice ramp and the end
time of the mixture creation, suggesting that some complicated spin density dynamics
is triggered within the BEC after the creation of the mixture8 .
In conclusion, we have demonstrated a basic feature of the spin Mott state, whichis the protection from spin entropy (maintenance of a high spin pairing fraction) at
high anisotropy values (low 𝑈𝑎𝑏 /𝑈 values). While the goal is to use the spin Mott
state as a stepping stone to reach the spin superfluid state, more work needs to bedone to improve the robustness of the spin Mott state where the anisotropy strength
is relatively weak (𝑈𝑎𝑏 /𝑈 ≈ 0.9). Based on the listed experimental concerns, it ispossible that the background Mott insulating state needs to be carefully examined tosee if the ⟨ˆ𝑛⟩ = 2 phase within the cloud is in equilibrium.
8

In particular, because |𝐹 = 1, 𝑚𝐹 = −1⟩ and |𝐹 = 1, 𝑚𝐹 = 1⟩ are very close to the miscibility/immiscibility boundary, and the effective potential seen by each hyperfine state is very small.
Thus the Thomas-Fermi approximation no longer holds for individual spin densities; it only holds
for the total density [110].

127

Figure 7-10: Measurement of spin alignment as a function of 𝑈𝑎𝑏 /𝑈 = 1 − 𝐷/𝑈 . (a)Blue: the atom cloud is loaded into a lin-𝜃-lin polarization gradient with 𝜃 = 76∘ andthen 𝜃 is reduced to a final value. Reduction in 𝜃 is equivalent to an increase in 𝑈𝑎𝑏 /𝑈 .
[see the curve for 𝑅 cos 𝜃𝑘 = 1/8 in Fig. 4-2(b)]. The spin pairing fraction stays highuntil 𝑈𝑎𝑏 /𝑈 = 0.85. Red: the atom cloud is loaded directly into a state-dependentlattice with the final value of 𝜃. The spin pairing fraction monotonically decreases as
𝑈𝑎𝑏 /𝑈 increases. At 𝑈𝑎𝑏 /𝑈 = 1, we expect a pairing fraction of 1/3 for a uniformly
random spin ordering. (b) Numerical simulation of the full Bose-Hubbard model on a
six-site chain. At higher 𝑈/𝑡, the transition from the spin Mott (pairing fraction ≈ 1)
to the spin superfluid (pairing fraction ≈ 0.5) happens more abruptly near 𝑈𝑎𝑏 /𝑈 = 1.
Data kindly provided by Julius de Hond.

128

7.3.2

Adiabatic passage with a magnetic gradientIn the previous subsection, we have described our efforts on making an adiabatic passage toward the spin superfluid state by tuning the state-dependent lattice. However,
one can also perform an adiabatic passage to a different but still interesting state, and
it is performed by tuning an external magnetic gradient. We start with an analogy
between the spin Mott state and the density Mott state to motivate the experiment
to be described. Unlike the superfluid state, the density Mott state is robust against
a potential gradient. However, as the strength of the gradient increases to the point
where the energy offset between neighboring sites equals the Hubbard interaction 𝑈 ,then an initial ⟨ˆ𝑛⟩ = 1 Mott state becomes energetically degenerate with a statewith an alternating pattern of doublons and holons: |2, 0, 2, 0, . . .⟩. If the gradient is
swept adiabatically, it is possible to induce a transition between the two states. Whatis interesting is that this physics can be mapped to an antiferromagnetic transverse
Ising model [111], where the effective Ising interaction is provided by the constraintthat atoms are free to hop onto a neighboring site if the neighbor atom has not yet
tunnelled (the constraint energy scale is equivalent to the Hubbard interaction 𝑈 )and the transverse field is provided by the tunneling amplitude. Another way to describe such a motion is that the restored first-order tunnelings are correlated by the
aforementioned constraint.
Analogously, we can expose the spin Mott state, which is visualized as a chain
of |𝑆𝑧 = 0⟩, to a steep magnetic gradient. When the magnetic gradient strengthis such that the energy offset becomes comparable to 2(𝐽 + 𝐷), two neighboring|𝑆𝑧 = 0⟩’s will flip to pairs of |𝑆𝑧 = −1⟩ and |𝑆𝑧 = 1⟩, which can be detected asa reduction in the measured pairing fraction. The spin flip is performed by secondorder tunneling, which is robust against the presence of a gradient. Similar to thecase of the density Mott, the second-order tunneling or magnetic exchange has to becorrelated in the sense that a spin can freely exchange with its neighbor as long as theneighbor has not already exchanged with some other spin. Thus, the demonstrationof an adiabatic sweep from a spin Mott state to an antiferromagnetic texture will129

be an interesting example of correlated magnetic exchanges and also accentuate theclose analogy between the density Mott and the spin Mott.
The energy diagram for a six-site spin chain described by the anisotropic spin∑︀
1 model augmented by a magnetic gradient ∆ 𝑖 𝑖𝑆𝑧,𝑖 is shown in Fig. 7-11. The
initial spin Mott state gets lost in a forest of intermediate states as we increase thegradient from zero. However, if we directly jump to intermediate values of the gradient(∆/𝑈 ≈ 0.2 or ∆/𝑈 ≈ 0.5), the spin Mott state can be mapped to an excited statethat has a relatively high pairing fraction. We can then make a transition toward
the state with an antiferromagnetic spin texture (represented by the red coloring)either by starting from ∆/𝑈 ≈ 0.3 and increasing the gradient, or by starting from∆/𝑈 ≈ 0.5 and decreasing the gradient. We note that the center of the largest
avoided crossing is roughly around ∆/𝑈 = 0.3, which is close to 2(𝐷 + 𝐽)/𝑈 .
Since the Hubbard interaction energy scale 𝑈 is typically 𝒪(1 kHz), in order toaccess a gradient strength of ∆/𝑈 = 0.5, we need to provide an energy offset of
approximately 500 Hz between neighboring sites. For |𝑎⟩ = |𝐹 = 1, 𝑚𝐹 = −1⟩ and|𝑏⟩ = |𝐹 = 1, 𝑚𝐹 = 1⟩ trapped in an 810 nm optical lattice, we need a magnetic
gradient strength of 8.8 G/cm. While this is a modest gradient by any standard,
because our side coils are located relatively far away from the atom cloud [28], we
require a huge current [𝒪(100 A)] to achieve the required gradient. Nonetheless,
we installed an extra power supply with a maximum rating of 250 A to the bias
coil connections. The bias coil connections are set up so that before the gradientis enabled, a switch stops the current running in half of the Helmholtz pair, andthen another switch enables the addition of the huge extra current running in ananti-Helmholtz fashion. Note that this leads to a reduction of the total bias anda possible rotation of the bias field. Because our total field consists of 8 G in the
Tool direction and 3 G adding in quadrature, we expect the bias field to rotate
by arctan(3/4) − arctan(3/8) = 16∘ . We expect the effective vector-to-scalar ratio
𝑅 cos 𝜃𝑘 (see Sec. 4.2.1) to drop by 16%. At the time of this writing, we are stillsearching for an optimal gradient ramp.

130

Figure 7-11:∑︀ Energy diagram of the spin-1 model augmented by∑︀a magnetic gradientterm ∆ 𝑖 𝑖𝑆𝑧,𝑖 . The number of sites is 6, and 𝑈𝑎𝑏 /𝑈 = 0.85.
𝑖 𝑆𝑧,𝑖 is constrained
to zero. The Bose-Hubbard parameters are converted to the spin model parameters
using Eq. 6.4. The coloring represents the average spin pairing fraction of the state.
Top: 𝐽𝑥𝑦 = 0.007 𝑈 . The avoided crossings are very narrow. Bottom: 𝐽𝑥𝑦 = 0.046 𝑈 .
The avoided crossings are visibly wider for certain transitions. If the transverse
lattice depths are 35 𝐸𝑅,1064nm , 𝑈/𝑡 = 25 (𝑈/𝑡 = 10) corresponds to 11.5 𝐸𝑅,810nm(8 𝐸𝑅,810nm ). The interaction energy 𝑈 at such lattice depths is typically 2𝜋 ×1.2 kHz.

131

132

Chapter 8
Conclusion
From an experimenter’s point of view, quantum simulation with ultracold atoms
requires many prerequisites to yield successful1 results. It requires smooth and consistent operations of many different hardware devices (optics, electronics, and evensomething as mundane as air conditioners) and an acceptable understanding of a wide
range of theoretical subjects in physics so that the signal (observable) of interest is
defined and carefully measured. This thesis tries to convey some sense of the effort
required to observe and report original studies of quantum many-body physics withultracold atoms.
Chapter 2 describes in detail how the current rubidium quantum gas machine isrun and has been incrementally improved over the years. It also points out many issues
that need to be addressed in a future upgrade of the vacuum science chamber, such as
the need for a precise feedback/feedback control of currents flowing through magnetic
coils and prevention of hysterestic materials around the chamber. The chapter also
has some useful details for beginning graduate students, such as independent methods
to carefully calibrate the number of atoms in the prepared atom cloud.
Chapter 3 and 4 describes the theory of non-scalar light shift and how the resultant
non-scalar light shift is used to control the interaction between different spin states
within an optical lattice. The derivation of the non-scalar polarizability presented in
1

We generally define success here as an acceptable agreement between theory and experiment.
Of course, not every disagreement is a failure! It could be a sign of a new discovery.

133

this thesis is unique in the sense that it tries to semi-rigorously explain the couplingof the circular polarizations to a vectorial light shift by analyzing the minimal mathematical structure for a light shift in the Cartesian basis, without delving directly
into the formalism of spherical tensors. Chapter 4 provides a short but more detailed
explanation of how to compute the Wannier basis for optical lattices and a widesurvey of currently known experimental techniques for generating a state-dependentlattice potential. It is hoped that the details presented in Chapter 4 will be useful for
future researchers trying to do high-fidelity state-dependent control with polarization
gradients within optical lattices.
Chapter 5 describes the basic physics of the Bose-Hubbard model and presents an
useful guide on how to compute the estimated volumes of the Mott phases within theatom cloud given the experimental settings. It is important that the atoms are initialized into the ground state of the Mott insulating phase before effective spin models
can be studied within the Mott phase, and one way to verify that is to compare theequilibrium density distribution inside a Mott insulator to the measured density distribution inside the optical lattice. The discrepancy between the calculation and themeasurement mentioned in the chapter shows that more work needs to be done on
optimal lattice ramps. The chapter also provides some calculation of the corrections
to the Hubbard parameters due to the interaction-induced admixture of higher-bandorbitals into the on-site wavefunction. While the corrections are not taken into account for our spin model experiment, it will need to be taken into account when aquantitatively accurate analysis is due for experimental data with low noise.
Chapter 6 describes how the low energy physics of a Mott insulator is capturedby an effective spin model. The chapter explains how to obtain the spin model
parameters from the Bose-Hubbard parameters in a systematic manner and derives
corrections to the spin-1 model parameters that have not been discussed elsewhere.
The chapter concludes with a discussion of how one may obtain spin-1 correlations
from density correlation measurements without high-resolution imaging, althoughexperimental hurdles are likely to be just as high while only returning a reduced
signal.
134

The derivations and detailed explanations of the preceding five chapters are put
into good use in Chapter 7, which describes two different studies of the spin-1 model:
non-equilibrium dynamics in a state-independent lattice and ground state control
in a state-dependent lattice. Study of coherent spin exchange was a long standing
goal of the rubidium lab, and the accumulated efforts in hardware improvement andtheoretical understanding lead to the first observation of magnetic exchange in aspin-1 model realized with ultracold atoms. The chapter also discusses the on-goinginvestigation of the spin Mott phase and provides useful calculations and suggestions
for the remaining work to be done.
There are still many discrepancies between measurements and calculations to beresolved. It is hoped that the imminent upgrade of the vacuum science chamber andthe surrounding optical layouts will transform the experiment into a more consistentand stable platform. The jewel of the planned upgrade is the new high-resolution
objective that will allow us to see individual atoms in lattice sites and give us abetter idea of what is actually going on inside the optical lattice. Also, single-site
addressing capability enabled by such an objective will be a useful tool for studying
spin physics. A unique capability of quantum simulation experiments with ultracold
atoms is the ability to prepare a variety of initial states not accessible in other settings
such as traditional condensed matter experiments. The objective can be used to inject
carefully calibrated spin excitations into the spin system and perform a precise studyof non-equilibrium spin dynamics. Although I have been involved in the planning forthe proposed upgrade to a quantum gas microscope for a long time, it will be in thehands of the junior and future graduate students of the rubidium lab to implement
the upgrade - and the lab is in good hands for the upgrade.
In terms of the scientific outlook, the main contribution of the rubidium lab willcontinue to be the study of two-component physics in the ⟨ˆ𝑛⟩ = 2 Mott insulator
phase, which has not received as much as attention as the ⟨ˆ𝑛⟩ = 1 Mott phase as
an experimental platform. The presence of the on-site interaction is essential forthe single-ion anisotropy (SIA), and it turns out that the SIA leads to qualitatively
something new in the spin model. Specifically, for a negative value of the SIA strength,135

magnons (visualized as a spin flip to |𝑆𝑧 = 0⟩ in a ferromagnetic background of|𝑆𝑧 = 1⟩) can form a bound state (a pair of neighboring |𝑆𝑧 = 0⟩) where an effectively
attractive interaction is provided by the SIA, via tunneling of the paired |𝑆𝑧 = 0⟩⊗2

into |𝑆𝑧 = 1⟩|𝑆𝑧 = −1⟩ and |𝑆𝑧 = −1⟩|𝑆𝑧 = 1⟩ and hence lowering of the totalenergy [108]. This is a new type of magnon interaction that is not possible in the often
studied spin-1/2 model. There is experimental evidence from solid state physics thatthe strength of this effective magnon-magnon interaction is renormalized at higher
magnon densities [108], and it would be interesting to study the dynamics of the spin-1
model with a precisely calibrated density of magnons injected into the chain. For this,local addressing by a quantum gas microscope will be essential. The local addressing
capability may be also crucial for improving the understanding of our adiabatic ramp
from the spin Mott state to the spin superfluid state. We may be actually able tocarve out a single chain of doubly-occupied sites, so that we do not have to worry
about the effects of surrounding ⟨ˆ𝑛⟩ = 1 Mott phase or superfluid shells during theadiabatic ramp. Another useful tool to add in the future is resonant spin-changing
collision [98], which can initialize a spin Mott state in a deep lattice with no physicalseparation between the two-component atoms. Such an initialization avoids all thecomplications of ramping a two-component atomic gas into a Mott insulator, andone can still use the state-dependent lattice to control the spin-dependent interaction
after the initialization. However, one will need a flexible magnetic field control thatcan change the bias field for more than 8 G without any hysteresis or substantial eddy
currents. Some care must be put into the microwave control as well in order to makethe spin changing process fully resonant. The sum of all the suggested additions to theexperiment makes up a challenging upgrade, but for a “conventional” atomic species
like rubidium, it would be good to have all the “unconventional” tools at disposal tohave a chance of discovering new physical phenomena with the quantum simulator.

136

Appendix A
Quantum Gas Microscope
In this appendix we describe some of the technical challenges for the installment of aquantum gas microscope system and provide some suggestions for solutions.

A.1

Optical aberration

High numerical aperture imaging requires stringent minimization of wavefront aberration as the collected fluorescent light goes through various optical surfaces. Not
only must the microscope objective be of high quality and must it be tested to bediffraction-limited in its performance, but surfaces such as vacuum windows, mirrors
and filters must impart as little wavefront deformation as possible. It is possible thatwhile the optical component itself has good flatness, its quality may be degraded bymechanical stress from poor mounting or bonding. While high-quality mirrors, filtersand optomechanical components can be bought from vendor catalogs, high-quality
vacuum windows in vacuum flanges require custom orders and may not preserve theirflatness despite the vendors’ best efforts. Therefore, it is important to understandhow much degradation can be caused by vacuum window deformation.
Before we explain the subtleties of vacuum window deformation, it is important tounderstand the different types of aberration that may be present. In geometric optics,
the wavefront error of a rotationally symmetric imaging system can be shown to only
depend on object height and exit pupil polar coordinates [112], yielding a power137

series expansion of the wavefront error function in those three variables. The first
two terms, which are quadratic in the variables, are focus and tilt, which can be always
adjusted away. The next five terms, which are quartic in the variables, are known asprimary or Seidel aberrations and consist of spherical aberrations, coma, astigmatism,field curvature and distortion. Their effects on geometrical beam spot are describedin [112]. Spherical aberration is special in that it only depends on the exit pupil
radius. Because of its independence from objective height and exit pupil angle, asingle corrector plate at the exit pupil can compensate the aberration for all locations
on the object plane. On the other hand, the other primary aberrations are dependent
on both objective height and exit pupil coordinates, and thus a corrector plate at theexit pupil can only correct the image of an object at a specific location. The conclusion
is that axially symmetric deformation is acceptable to some degree, but asymmetricdeformation must be avoided if possible as it cannot be easily compensated.
Unfortunately, vacuum windows are likely to have axially asymmetric deformations. Vacuum windows are typically bonded to a metal alloy sleeve, which is thenwelded to the stainless steel flange. If there is asymmetric radial stress during either
glass-to-metal bonding or sleeve-to-metal welding, the window will bend preferentially along some direction, yielding a saddle-like deformation. The window will still
maintains its thickness and hence preserves its transmission wavefront error, so thatthe deformation poses no problem for collimated light. However, since the vacuum
window is the first optical surface through which the diverging cone of fluorescentlight from atoms passes, the deformation, as measured by reflection wavefront error
in interferometers, can impart a substantial effect.
Without understanding the internals of a high numerical aperture (NA) imaging
objective, we can derive the acceptable level of window surface deformation given
the working distance and the specified NA of the objective, and the thickness of
the window. Consider the geometry in Fig. A-1, where fluorescing atoms are placed
working distance away from the vacuum side of the window. Our assumption isthat for any given ray emanating from the atoms, the objective as a blackbox adjust
the total optical path lengths (OPL) so that the ray arrives at the focal point by138

a fixed but practically unknown amount of time. In other words, we assume thatthe objective itself is a perfect imaging system and we apply Fermat’s principle forpropagation of light.

Figure A-1: Objective as a perfect blackbox imaging system. While the finite thickness of the front vacuum window can impart aberrations such as spherical aberration,
the blackbox compensates for such effects so that all the rays from the atoms thatcan fit inside the exit pupil of the blackbox are focused onto the focal point. In otherwords, the blackbox does whatever it needs to do so that Fermat’s principle holds for
all the rays with the start point at the atom and the end point at the focal point.
Once the vacuum window starts to deform as in Fig. A-2(b), the balance of OPL
performed by the blackbox is broken, and some rays do not focus onto the previous
focal point anymore. We assume that the deformation is small enough so that therays exiting the window do not significantly change in their angles and positions. In
experience, high-quality vacuum windows in metal flanges come with peak-to-valley
deviation of a few wavelengths over the clear aperture, so this is a reasonable assumption. If the ray angle and position were to change greatly after the deformed window,
then the blackbox output cannot be easily predicted as it was built to compensate for
specific aberrations caused by the originally flat window. In other words, we assume
the change in the OPL inside the window glass is the only substantial perturbationby the deformation.
If a ray strikes a flat window at an angle 𝜃 as in Fig. A-2(a), the OPL inside theglass is𝑂𝑃 𝐿𝑔 (𝜃) = √︁𝑛𝑔 𝑑
1 − sin2 𝜃/𝑛2𝑔

139

where 𝑛𝑔 is the refractive index of the glass and 𝑑 is the thickness of the window.
If the window starts to bend, the incidence angle of the ray at the window deviatesfrom 𝜃. Now we assume the deformation of the window is described by a parabola(𝑃 𝑉 )𝑟2 /𝑅2 , where 𝑃 𝑉 is the peak-to-valley deformation in units of wavelengths, 𝑅 =𝑊 𝐷 tan (arcsin NA) is the radius of the clear aperture over which the deformation ismeasured, and 𝑟 = 𝑊 𝐷 tan 𝜃 is the radial position where the ray strikes the glass.
The slope of the parabolic deviation is𝛿𝜃 = 2𝑃𝑉
𝑊 𝐷 tan 𝜃𝑅2. Thus, the change in the OPL inside the glass is∆𝑂𝑃 𝐿𝑔 (𝜃) = 𝑂𝑃 𝐿𝑔 (𝜃) − 𝑂𝑃 𝐿𝑔 (𝜃 − 𝛿𝜃)
If the surface deformation is parabolic and axially symmetric, the resulting aberration
is a defocus term that can be fully compensated. However, because of the aforementioned saddle-shaped deformation, we assume the sign of the parabola is different
between the two orthogonal axes on the surface. This is the essence of astigmatism:
different focal lengths between x-axis and y-axis. Figure A-2(c) plots ∆𝑂𝑃 𝐿𝑔 (𝜃) in
units of wavelength for 𝑃 𝑉 = 1𝜆 and for two different values of WD: 6.5 mm (that of
the new BEC4 objective) and 2.5 mm. The Rayleigh criterion1 for diffraction-limited
imaging states that a the wavefront deformation must be less than 𝜆/4 [112]. Notethat this condition is necessary but not sufficient for diffraction-limited imaging [113].
We see that for WD = 6.5 mm we lose diffraction-limited imaging performance around
NA = 0.57, which is close to what Zemax simulation of the actual objective and theaberrated window predicts (around NA ≈ 0.6). On the other hand, WD = 2.5 mmshows a higher likelihood of maintaining diffraction-limited imaging up to the specified NA of 0.8. The conclusion is that for high-NA objectives with long working
distance, peak-to-valley deformation must be less than 𝜆/4, and if this specification
1

A heuristic argument is that wavefront deviation beyond 𝜆/4 leads to destructive information
between Huygens wavelets at the focus.

140cannot be met, the objective needs to be designed for a shorter working distance, atthe cost of a reduced optical access to the atoms from the side.

A.2

Strategy for a high-flatness vacuum window

Usually the vacuum window glass is bonded to a metal alloy that forms a ferrule, andthe ferrule is later joined to a metal alloy sleeve that is connected to the rest of themetal flange (Fig. A-3). The selection of the ferrule material is important, as the useof more flexible materials reduces the possibility of deformations. For example, using
a tantalum sleeve resulted in a much better deformation [(reflection wavefront errorof 𝒪(𝜆)] than an Inconel sleeve [𝒪(100𝜆)], although the improvement could have alsocome from a smaller welding area between the sleeve and the ferrule for the tantalum
sample. The choice of materials is severely constrained by the requirements that thematerial be non-magnetic and easily weldable.
If the windows end up deformed, the next option is to polish the glass. In principle,
it is straightforward and fairly standard to polish a glass down to peak-to-valley
deviation of 𝜆/20 or better. However, the requirement is that the polishing tool canaccess the entire surface of the glass and have the room to move from one edge tothe other [114]. Magnetorheological fluid (MRF) polishing can correct the surface
deformation over an arbitrary aperture in principle, but the final quality and ease ofpolishing still depends on the initial flatness of the window. In our experience, all
MRF polishers have refused to work on our deformed windows because it is hard toaccess the full area of the glass and would take too much time (and hence opportunitycost) to polish the glass down to the specified level. There is a contact-free polishingtechnique called ion beam figuring (IBF) but convincing a vendor to work on thesample despite the opportunity cost was also an issue.
If the high-end polishing techniques are not available, then making a home-madevacuum window is an option, as some other cold atom groups have done by gluinga glass to a metal flange [115]. We are currently pursuing this option, but the main
challenge is to prevent a slow rise in the vacuum pressure, possibly due to some141

components in the epoxy.

142

Figure A-2: Effect of vacuum window deformation on wavefront error. (a) Geometryof the setup. “WD” stands for working distance, “d” thickness of the vacuum window,
and “R” the radius of the area illuminated by the light cone from the atom. (b)When the vacuum window curves under stress, the optical path length (OPL) inside
the glass deviates from the original value (black dotted line). Note the terminal
separation between the new trajectory and the old trajectory is greatly exaggerated
for clear view. (c) Calculation of the change in the OPL inside the glass induced by aparabolic deviation from a flat surface (peak-to-valley error of 𝜆 over 2𝑅 = 17.3 mm).
For WD = 6.5mm, the objective is no longer diffraction limited around 𝑁 𝐴 = 0.55.
For a smaller WD of 2.5 mm, diffraction-limited performance may be maintained up
to high NA.
143

Figure A-3: An example vacuum viewport structure. Picture courtesy of UKAEA.

144

Appendix B
Tunable Single-Ion Anisotropy in
Spin-1 Models Realized with
Ultracold Atoms
This appendix contains a reprint of Ref. [24]: Woo Chang Chung, Julius de Hond,Jinggang Xiang, Enid Cruz-Colón, and Wolfgang Ketterle, Tunable Single-Ion Anisotropy
in Spin-1 Models Realized with Ultracold Atoms, Physical Review Letters 126, 163203(2021).

145

PHYSICAL REVIEW LETTERS 126, 163203 (2021)

Tunable Single-Ion Anisotropy in Spin-1 Models Realized with Ultracold Atoms
Woo Chang Chung , Julius de Hond , Jinggang Xiang (项晶罡) , Enid Cruz-Colón, and Wolfgang Ketterle
Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 4 January 2021; accepted 18 March 2021; published 23 April 2021)
Mott insulator plateaus in optical lattices are a versatile platform to study spin physics. Using sites
occupied by two bosons with an internal degree of freedom, we realize a uniaxial single-ion anisotropy
term proportional to ðSz Þ2 that plays an important role in stabilizing magnetism for low-dimensional
magnetic materials. Here we explore nonequilibrium spin dynamics and observe a resonant effect in thespin alignment as a function of lattice depth when exchange coupling and on-site anisotropy are similar.
Our results are supported by many-body numerical simulations and are captured by the analytical solutionof a two-site model.
DOI: 10.1103/PhysRevLett.126.163203

Mott insulators of ultracold atoms in optical lattices
comprise a widely used platform for quantum simulationsof many-body physics [1]. Since the motion of atoms isfrozen out, the focus is on magnetic ordering and spin
dynamics in a system with different (pseudo)spin states. As
suggested in 2003, Mott insulators with two-state atoms
realize quantum spin models with tunable exchange interactions and magnetic anisotropies [2,3]. Experimental
achievements for spin-1=2 systems include the observationof antiferromagnetic ordering of fermions [4] and the studyof spin transport in a Heisenberg spin model with tunableanisotropy of the spin-exchange couplings [5]. Spin
dynamics for S > 1 has also been investigated [6].
However, all studies thus far have exclusively addressedspin systems with occupations of one atom per site. This
limits spin Hamiltonians to spin-exchange
betweenP termsk kdifferent sites i, j proportional toS
Shiji i j (wherek ∈ fx; y; zg) and to Zeeman
couplings
toeffective magP
netic fields, proportional to i Szi . For Mott insulators with
two or more atoms per site, the Hubbard model has direct
on-siteP interactions that can give rise to a nonlinear term
D i ðSzi Þ2 , where D is the so-called single-ion anisotropy
constant. ðSz Þ2 terms, which are present for S ≥ 1 only, areimportant for establishing nontrivial correlations such as inspin squeezing [7]. In spin-1 models, such terms can lead toa qualitatively new magnetic phase diagram [8,9]. For
example, for ferromagnetic spin-1 Heisenberg models, thesingle-ion anisotropy gives rise to a gapped spin state(the “spin Mott insulator”) that can be used as an initial
low-entropy state for an adiabatic ramp toward a highly
correlated gapless spin state (the XY ferromagnet) [10,11].
For antiferromagnetic systems in one dimension, thesingle-ion anisotropy leads to a quantum phase transition
between a topologically trivial phase and a nontrivial phaseas predicted by Haldane [12–14]. The magnetic propertiesof many materials crucially depend on crystal field
0031-9007=21=126(16)=163203(5)anisotropies that break rotational symmetry and can stabilize ferromagnetism in two-dimensional materials by
avoiding the Mermin-Wagner theorem, which forbids
long-range order for continuous symmetries [15,16]. Theinterest in spin-1 systems is demonstrated by various
studies on different platforms [17–19].
In this Letter, we use cold atoms in optical lattices toimplement a spin-1 Heisenberg Hamiltonian using a
Mott insulator of doubly occupied sites and demonstrate dynamical features that arise from the single-ion
anisotropy. For spin-exchange interactions studied thus
far in optical lattices, the only timescale for dynamics issecond-order tunneling (i.e., superexchange), whichmonotonically slows down for deeper lattices. In contrast,
as we show here, the single-ion anisotropy introduces a
new timescale, and we find a dynamical behavior that isfaster in deeper lattices due to a resonance effect when theenergies of superexchange and single-ion anisotropy arecomparable.
We present a protocol to directly measure the anisotropy
in the spin distribution and find a pronounced transient
behavior of this quantity when the resonance condition ismet. Transients change sign along with the single-ion
anisotropy. We find good agreement with theoreticalsimulations and explain the most salient features using a
two-site model with an exact solution.
In the Mott insulator regime, the optical lattices are
sufficiently deep that the on-site interaction suppresses
first-order tunneling, and exchange processes are only
possible via second-order tunneling. For two atoms per sitewith two internal states, the Bose-Hubbard Hamiltonian isapproximated by an effective spin Hamiltonian

163203-1

H ¼ −J

X
X
X
Si · Sj þ D ðSzi Þ2 − B Szi ;
hiji

i

ð1Þ

i

© 2021 American Physical Society

PHYSICAL REVIEW LETTERS 126, 163203 (2021)
where Si are spin-1 operators, hiji are pairs of nearestneighboring sites, J is the exchange constant, D is theuniaxial single-ion anisotropy constant, and B is a fictitious
magnetic bias field. The spin-1 operators are related tothe boson creation and annihilation operators via Szi ¼††−
ða†i ai − b†i bi Þ=2, Sþi ¼ ai bi , Si ¼ bi ai under the constraint††
ai ai þ bi bi ¼ 2, where ai and bi are boson annihilation
operators at site i for state a and state b, respectively. In
terms of the tunneling amplitude t and interaction energies
Uσσ 0 : J ¼ 4t2 =Uab and D ¼ ðUaa þ Ubb Þ=2 − Uab , whereUσσ 0 represents the on-site interaction energy between atoms
in two states σ; σ 0 ∈ fa; bg. The term proportional P
to B canbe dropped if the total longitudinal magnetization i Szi isconstant, as it is in the experiment.
For the species studied here, 87Rb, all U σσ0 differ by lessthan 1%, and therefore all spin-exchange couplings arealmost equal, resulting in isotropic spin Hamiltonians forsite occupancy ν ¼ 1. However, for ν ¼ 2, we can tune therelevant anisotropy parameter D=J over a large range ofvalues because J decreases exponentially with lattice depth
while D—a differential on-site energy—slowly increases.
The experimental sequence begins by preparing a
Bose-Einstein condensate (BEC) of 87Rb atoms in thejF ¼ 1; mF ¼ −1i hyperfine state inside a crossed optical
dipole trap. The sequence proceeds by loading the BEC
into a deep three-dimensional optical lattice formed byretroreflected lasers with wavelengths of λ ¼ 1064 nm.
The lattices are ramped to final depths of 30ER in 250 ms,
where ER ¼ h2 =ð2mλ2 Þ is the recoil energy for atomic
mass m. Experimental parameters are chosen to maximize
the size of the ν ¼ 2 Mott-insulator plateau without
significant population of sites with ν ¼ 3 [see Fig. 1(a)and Ref. [20]].
To allow for spin dynamics, all atoms are rotatedinto an equalpﬃﬃﬃ superposition of two hyperfine statesðjai − ijbiÞ= 2 using a combination of microwave pulses[20]. This initial state is a simple product state. Negativeand positive values of D are realized with the pairs
jai ¼ j1; −1i, jbi ¼ j1; 1i and jai ¼ j1; −1i, jbi ¼ j1;0i,
respectively [20]. The spin-exchange dynamics in onedimensional chains is initiated by a 3-ms quench, during
which we ramp down the longitudinal lattice to a variable
depth while the transverse lattices are ramped up to 35ER
[Fig. 1(b)]. After a variable evolution time, the final spin
configuration is “frozen in” by ramping the longitudinallattice to 35ER as well [Fig. 1(c)].
Our observable for the anisotropy in the spin distribution
is the longitudinal spin alignment A ¼ SðSPþ 1Þ − 3hðSz Þ2 i,
measured in the ν ¼ 2 plateau. hðSz Þ2 i ¼ Ni¼1 hðSzi Þ2 i=N is
the average on-site longitudinal spin correlation. A is defined
to be zero for a random distribution of spins. Since Sz ¼
1; 0; −1 for the jaai, jabi, and jbbi doublons, respectively,
A can be obtained by measuring the relative abundance ofthe different doublons. Specifically, we refer to the fractionof jabi doublons as the “spin-paired doublon fraction” f.

(a)

(b)

(a)

(c)

(b)

(d)

(c)

(d)

FIG. 1. Experimental sequence for the measurement of spinalignment and doublon fractions. (a) The lattices are ramped up to
initialize a single-component Mott insulator with a maximal site
occupancy of two. (b) Microwave pulses
pﬃﬃﬃ prepare a superposition of
two hyperfine states ðjai − ijbiÞ= 2. Ramping down the longitudinal lattice initiates spin-exchange dynamics. (c) Ramping up
the lattices stops the exchange dynamics. Microwave pulses
transfer the two components to a pair of states with a Feshbachresonance. (d) Either jabi doublons or all doublons are removed
with the help of Feshbach-enhanced inelastic losses. The remaining atoms are transferred back to the F ¼ 1 hyperfine states and arecounted via absorption imaging to measure N p or N d.

Since hðSz Þ2 i ¼ 1 − f, we obtain A ¼ 3f − 1. The doublon
statistics can be measured by selectively introducing a fastloss process that targets a specific type of doublon and bycomparing the remaining total numbers of atoms, which aremeasured via absorption imaging. Specifically, if N a is theaverage total atom number in the whole cloud, N p the
average number of remaining atoms after removing jabi
doublons, and N d the average number of remaining atoms
after removing all doublons, then f ¼ ðN a − N p Þ=ðN a − N d Þ[Fig. 1(d)]. Fast losses of doublons are induced by transferring the atoms to hyperfine states for which inelastic twobody loss is enhanced near two narrow Feshbach resonances
around a magnetic field of 9 G [20,28]. Since f and A are
obtained from the ratio of differences in atom numbers, good
atom number stability in the experiment (the deviation frommean being typically < 4%) was crucial to measure A with
sufficiently small uncertainties.
For the initial state, f ¼ 1=2 and A ¼ 1=2. Over times
that are long compared to the spin-exchange timescaleℏ=J, heating processes drive the system toward thermal
equilibrium with A ¼ 0. At short times, coherent spin
dynamics is observed: If D is negative, the jaai and jbbi
doublons are energetically favorable, and we expect f and
A to decrease. If D is positive, the jabi doublons are

163203-2

PHYSICAL REVIEW LETTERS 126, 163203 (2021)
favorable, and we expect f and A to increase. If D is zero,
the system is described by an isotropic spin-1 Heisenberg
Hamiltonian of which the initial state is an eigenstate. By
fixing the hold time and scanning the value of the lattice
depth for the spin chains, we can monitor the impact of
D=J on the dynamical change in A. For positive (negative)
D, we chose a hold time of 70 ms (25 ms). Thesehold times are chosen to be comparable to ℏ=J when
jD=Jj ∼ 1 [20].
Figure 2 shows that for jD=Jj ≪ 1 or jD=Jj ≫ 1, A stays
near its initial value of 1=2. However, when D=J ∼ 2,
which corresponds to a longitudinal lattice depth of 14ER
(11ER ) for positive D (negative D), we see that A reaches amaximum (minimum). This nonmonotonic change of A
with lattice depth is indicative of the interplay between
spin-exchange and single-ion anisotropy. In addition, we
observe that the change in A is smaller for positive D thanfor negative D.

FIG. 2. Transient enhancement and reduction of the spinalignment A by coherent spin dynamics. The change in A isstrongest when jD=Jj ∼ 2. Measurements were done for both
positive (top) and negative (bottom) values of D=J. The atoms
were held for 70 ms and 25 ms, respectively (also see Fig. 4). The
top axis in both figures indicates the D=J ratio. Solid lines are theresults of matrix-product state–time-evolving block decimation(MPS-TEBD) calculations. The error bars represent the standard
error of the mean for A, obtained by error propagation after
averaging three measurements for each of N a , N p , and N d . We
found the error bars to be dominated by fluctuations in preparedatom number over systematic errors. For the lowest lattice depths,
the spin model may not fully represent the Bose-Hubbard model.

Several aspects of the observed spin dynamics can be
captured by a two-site model. Although states on two spin1 sites span a 9-dimensional Hilbert space, we can reduce
the spin dynamics to a beat note between two states. Since
exchange
do not change the total magnetizaPN interactionsztion i¼1 Si , the Hilbert space factorizes to subspaces withthe same total magnetization (although Szi can differ withina subspace). Furthermore, the initial superposition state issymmetric between the left and right wells, and any change
in A comes from the two coupledpﬃﬃﬃ states: jabiL jabiR andðjaaiL jbbiR þ jbbiL jaaiR Þ= 2, for which A equals 2 and−1, respectively (Fig. 3). By describing these two states as
two poles on a Bloch sphere, we see that the initial state isrepresented by a vector pointing somewhere between thenorth pole and the equator with a vertical fictitious external
field. The quench in J and D suddenly changes the strengthand the orientation of this external field and induces aprecession of the state vector around the new external field[20]. This results in an oscillation of A with amplitude

FIG. 3. Coherent spin oscillations in a two-site model. While
the full basis contains nine states, the oscillations in the spinalignment A involve only a 2 × 2 block of the Hamiltonian. This
allows us to illustrate the dynamics on Bloch spheres (left), where
the initial state is represented by the open circle. If J ¼ 0, theeffective magnetic field points along ẑ, and the purely azimuthal
precession will not change A. If J > 0, the effective magnetic
field is tilted, resulting in a precession along the dashedcircle, which is observed as an oscillation in A (right). The
frequencyof the oscillation, in units of J=ℏ, is given by
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ω ¼ 9 þ 4D=J þ 4ðD=JÞ2 , and its peak-to-peak amplitudeis 6ðD=JÞ=½9 þ 4D=J þ 4ðD=JÞ2  (see inset). This shows thatthe direction of oscillation depends on the sign of D=J (compare
top and bottom rows, where D=J ¼ 0.5 and −0.5, respectively).
Note that while the initial value of A for this subspace is 1, thecontribution of other states sets the initial A of the whole system
to 1=2.

163203-3

PHYSICAL REVIEW LETTERS 126, 163203 (2021)
6ðD=JÞ=½9 þ 4D=J þ 4ðD=JÞ2 . This function has local
extrema for D=J ¼ 3=2 but is not symmetric around
D=J ¼ 0. This explains the nonmonotonic behavior as afunction of lattice depth and shows why the contrast issmaller for positive D=J than for negative D=J.
One would expect that, for a larger number of sites,
additional precession frequencies appear, turning the periodic oscillation for two sites into a relaxation toward an
asymptotic value, which, according to the eigenstate
thermalization hypothesis, represents a low-temperature

FIG. 4. Coherent dynamics of the spin alignment A after aquench in D=J. Varying the hold time at characteristic lattice
depths for both positive and negative values of D=J (top andbottom pairs of panels, respectively) reveals that strong transientsin A only occur at intermediate lattice depth for which D and J arecomparable. The vertical dash-dotted lines indicate the hold times
used for these pairs in Fig. 2. Dashed lines are the results of the
MPS-TEBD simulation. The shaded regions denote the
MPS-TEBD results with 0.5ER uncertainty in the lattice depths
and include exponential decay toward a thermal spin state with
A ¼ 0 with empirical 1=e times of 350 ms (D > 0) and 175 ms(D < 0), the ratio of which reflects the relative sensitivity toambient magnetic field fluctuations of the pairs. The error bars
are computed in the same manner as those in Fig. 2.

equilibrium [29]. Comparison between the two-site modeland a many-site model numerically simulated using thetime-evolving block decimation for matrix-product states(MPS-TEBD) shows that the initial change in A is indeed
well captured by the two-site model [20]. Because of thespin dynamics, the system evolves from a product state intoa highly correlated state with entanglement between sites;
this has been the focus of recent theoretical works [30,31].
In the two-site model, the von Neumann entanglement
entropy can reach up to ∼0.9 × lnð3Þ due to the interplay
between single-ion anisotropy and exchange terms. This
corresponds to an almost maximally entangled state sincelnð3Þ is the maximum entropy for a spin-1 site.
To show that changes in the spin alignment A result fromcompetition between the exchange interaction and thesingle-ion anisotropy, we study the time evolution of A
at two different lattice depths (Fig. 4). For positive D,
MPS-TEBD simulations predict very little change in A at alower lattice depth, where the exchange constant is relatively large, but the anisotropy is small, while it predicts a
noticeable change in A at a higher lattice depth, where theexchange constant and the anisotropy term become comparable. While the simulation predicts equilibration of A toan asymptotic value (thin lines), measurements show that it
decays toward a lower value for positive D and does notdecrease as much as the simulation predicts for negative D.
The measurements are consistent with the fact that at highspin temperatures, the spin distribution becomes isotropic
and A vanishes. Indeed, when we ramp down the lattices
and retrieve a BEC, we observe a significant reduction of
the condensate fraction after 300 ms. This represents thetimescale over which entropy is either transported throughthe cloud or created by heating. It could possibly be
extended by better stabilizing ambient field fluctuations
or by adding a tilt to suppress entropy transport byholes [32].
In conclusion, we have implemented a spin-1 Heisenberg
model with a single-ion anisotropy using the ν ¼ 2 plateau
of a Mott insulator and have observed the subtle interplay
between spin-exchange and on-site anisotropy in coherent
spin dynamics. Much larger values of D can be implemented with spin-dependent lattices, which will allow us toobserve much faster anisotropy-driven dynamics and will
also enable mapping out the phase diagram of the anisotropic spin Hamiltonian [11]. It should also be noted that it
is possible to change the sign of J with the gradient of an
optical dipole potential [32,33], which will permit exploration of the antiferromagnetic sector with bosons.
Interesting dynamical features of anisotropic spin models
have been predicted [34], including transient spin currents,
implying counterflow superfluidity.
Regarding quantum simulations, single-ion anisotropies
play a crucial role in magnetic materials (e.g., monolayers
containing chromium [35,36]). In such materials, crystal
field effects lift the degeneracy of d orbitals, and spin-orbit

163203-4

PHYSICAL REVIEW LETTERS 126, 163203 (2021)
interaction transfers this anisotropy to the electronic spins
responsible for the magnetism [37]. Here we have simulated
this anisotropy by selecting a pair of atomic hyperfine states
where the interspecies scattering length is different from theaverage of the intraspecies values. This illustrates thepotential for ultracold atoms in optical lattices to implement
idealized Hamiltonians describing important materials.
We thank Colin Kennedy, William Cody Burton, andWenlan Chen for contributions to the development ofexperimental techniques and Ivana Dimitrova andAndrew Daley for their critical reading of the manuscript.
We acknowledge support from the NSF through the Centerfor Ultracold Atoms and Grant No. 1506369, ARO-MURI
Non-equilibrium
Many-Body
Dynamics(GrantNo. W911NF14-1-0003), AFOSR-MURI Quantum
Phases of Matter (Grant No. FA9550-14-1-0035), ONR(Grant No. N00014-17-1-2253), and a Vannevar-Bush
Faculty Fellowship. W. C. C. acknowledges additional
support from the Samsung Scholarship.
W. C. C. and J. d.-H. contributed equally to this work.

[1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80,
885 (2008).
[2] L.-M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett.
91, 090402 (2003).
[3] A. B. Kuklov and B. V. Svistunov, Phys. Rev. Lett. 90,
100401 (2003).
[4] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M.
Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greiff,and M. Greiner, Nature (London) 545, 462 (2017).
[5] P. N. Jepsen, J. Amato-Grill, I. Dimitrova, W. W. Ho, E.
Demler, and W. Ketterle, Nature (London) 588, 403 (2020).
[6] A. de Paz, A. Sharma, A. Chotia, E. Maréchal, J. H.
Huckans, P. Pedri, L. Santos, O. Gorceix, L. Vernac, andB. Laburthe-Tolra, Phys. Rev. Lett. 111, 185305 (2013).
[7] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).
[8] Y. Li, M. R. Bakhtiari, L. He, and W. Hofstetter, Phys. Rev.
B 84, 144411 (2011).
[9] Y. Li, L. He, and W. Hofstetter, Phys. Rev. A 93, 033622 (2016).
[10] E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin,
New J. Phys. 5, 113 (2003).
[11] J. Schachenmayer, D. M. Weld, H. Miyake, G. A. Siviloglou,
W. Ketterle, and A. J. Daley, Phys. Rev. A 92, 041602(R)(2015).
[12] F. D. M. Haldane, Phys. Lett. 93A, 464 (1983).
[13] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).
[14] F. D. M. Haldane, Rev. Mod. Phys. 89, 040502 (2017).
[15] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133
(1966).

[16] J. Strečka, D. Ján, and L. Čanová, Chin. J. Phys. 46,
329 (2008), https://www.ps-taiwan.org/cjp/issues.php?vol=46#=3.
[17] J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C.
Erkelens, J. Rossat-Mignod, and W. G. Stirling, Europhys.
Lett. 3, 945 (1987).
[18] P. Chauhan, F. Mahmood, H. J. Changlani, S. M. Koohpayeh,and N. P. Armitage, Phys. Rev. Lett. 124, 037203 (2020).
[19] C. Senko, P. Richerme, J. Smith, A. Lee, I. Cohen, A.
Retzker, and C. Monroe, Phys. Rev. X 5, 021026 (2015).
[20] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.126.163203 for the calculation of D and J values, the confinement parameters, thedetails of state preparation and doublon measurement, thederivation of the two-site model, and a description of ourmatrix-product state simulations, with Refs. [21–27] therein.
[21] N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847(1997).
[22] W. Kohn, Phys. Rev. 115, 809 (1959).
[23] P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger, M.
Moreno-Cardoner, S. Fölling, and I. Bloch, Phys. Rev. Lett.
101, 090404 (2008).
[24] G. K. Campbell, J. Mun, M. Boyd, P. Medley, A. E.
Leanhardt, L. G. Marcassa, D. E. Pritchard, and W. Ketterle,
Science 313, 649 (2006).
[25] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85,
1191 (2013).
[26] J. Hauschild and F. Pollmann, SciPost Phys. Lect. Notes 5(2018).
[27] G. Vidal, Phys. Rev. Lett. 93, 040502 (2004).
[28] A. M. Kaufman, R. P. Anderson, T. M. Hanna, E. Tiesinga,
P. S. Julienne, and D. S. Hall, Phys. Rev. A 80, 050701(R)(2009).
[29] J. M. Deutsch, Rep. Prog. Phys. 81, 082001 (2018).
[30] I. Morera, A. Polls, and B. Juliá-Díaz, Sci. Rep. 9, 9424
(2019).
[31] A. Venegas-Gomez, J. Schachenmayer, A. S. Buyskikh, W.
Ketterle, M. L. Chiofalo, and A. J. Daley, Quantum Sci.
Technol. 5, 045013 (2020).
[32] I. Dimitrova, N. Jepsen, A. Buyskikh, A. Venegas-Gomez,
J. Amato-Grill, A. Daley, and W. Ketterle, Phys. Rev. Lett.
124, 043204 (2020).
[33] H. Sun, B. Yang, H.-Y. Wang, Z.-Y. Zhou, G.-X. Su, H.-N.
Dai, Z.-S. Yuan, and J.-W. Pan, arXiv:2009.01426.
[34] A. Venegas-Gomez, A. S. Buyskikh, J. Schachenmayer,
W. Ketterle, and A. J. Daley, Phys. Rev. A 102, 023321(2020).
[35] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,
W. Bao, C. Wang, Y. Wang et al., Nature (London) 546, 265(2017).
[36] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, npj Comput.
Mater. 4, 57 (2018).
[37] D. Dai, H. Xiang, and M.-H. Whangbo, J. Comput. Chem.
29, 2187 (2008).

163203-5

Supplemental Material for
Tunable Single-Ion Anisotropy in Spin-1 Models Realized with Ultracold Atoms
Woo Chang Chung,∗ Julius de Hond,∗ Jinggang Xiang (项晶罡), Enid Cruz-Colón, and Wolfgang Ketterle
Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Dated: March 19, 2021)

Calculation of D and J

Confinement parameters

The three-dimensional lattice is created by retroreflecting three 1064-nm wavelength laser beams. The
two horizontal beams have Gaussian beam waists of
150 µm, while the vertical lattice beam has a waist of
270 µm. During the entire experiment the atoms arebeing held in a crossed-beam optical dipole trap. Thisconsists of a vertical beam (which has isotropic trap frequencies of 2π × 24 Hz) intersecting a highly elongatedhorizontal beam that is at a 45◦ angle with respect tothe horizontal lattices. The latter primarily serves tohold the atoms against gravity, and it has trap frequencies of 2π × 13 Hz and 2π × 130 Hz along its horizontal
and vertical axes, respectively.
Using these parameters, we were able to calculate
the occupation statistics of the Mott insulator, and obtained plateau fractions analogous to those presented inRef. [S3]. We desire a large ν = 2 Mott insulator plateau,
while avoiding any population in the ν = 3 shell as thatwould interfere with the doublon measurements. Occasionally, we have monitored the population in the different shells using clock-shift spectroscopy [S4]. On a
day-to-day basis, however, we use the total atom number or the doublon fraction as indicators (note that ourdoublon detection scheme detects all the atoms on sites

J/h+]

|a = |1, 1 |b = |1, 1|a = |1, 1 |b = |1, 0

100
10 2

D/h+]

The superexchange parameter J and single-ion
anisotropy D were calculated using maximally localized
Wannier functions for a simply cubic lattice [S1, S2] andthe scattering lengths in Table I.
The sign of D is important for the qualitative behavior. Of the F = 1 states, the only combination with
D < 0 is that of the |1, −1i and |1, 1i states. Any pair
involving the |1, 0i state has a positive value of D; wechose the |1, −1i and |1, 0i combination because it was
the easiest to prepare from the initial |1, −1i state. As
mentioned in the main text, the value of D is proportional to the various onsite interactions, which have alinear dependence on the scattering lengths. This meansthat D ∝ (aaa + abb ) /2 − aab which equals −0.93a0 and
0.23a0 for the two chosen pairs. Through the Wannierfunctions, D and J depend on the lattice depth, which
dependence is shown in Fig. S1.

102












/RQJLWXGLQDOODWWLFHGHSWK(ER)

FIG. S1. Values of D and J as a function of longitudinallattice depth. The results are based on the scattering lengths
given in Table I, and assume transverse lattice depths of 35 Er .

|1, −1i|1, 0i|1, 1i|1, −1i
100.4|1, 0i
100.4
100.867|1, 1i101.333
100.4
100.4

TABLE I. Scattering lengths in units of a0 calculated using
the values tabulated in Ref. [S5].

with ν ≥ 2.). To be safe, the doublon fraction is kept below 0.5, and the atom number below 40 × 103 ; for theseparameters the population in ν = 3 should be negligible.

State preparation & doublon measurement

The initial state is prepared by a diabatic Landau–
Zener sweep from the initial |1, −1i state to the |2, 0i
state. The sweep parameters are set in such a way that
we robustly create an equal superposition of the twostates. Depending on whether we want to probe positive or negative D/J we either transfer the population
fraction in |2, 0i to |1, 0i using a π pulse (which has small

2
sensitivity to magnetic-field fluctuations), or to |1, 1i using an adiabatic Landau–Zener sweep.
As described in the main text, the doublon statistics are derived from three separate measurements of theatom number, two of them after inducing selective losses
that depend on the doublon type. To measure the total
doublon fraction, all doublons are removed, regardless of
their internal states. Dipolar relaxation is too slow, so
a Feshbach resonance between the |1, 1i and |2, 0i states
can be used. For this, the |1, −1i component of the pairis transferred to the |2, 0i state using a Landau–Zenersweep, while the other pair component is left in or put
into the |1, 1i state. The pairs are removed by modulating the magnetic bias field by ∼ 10 mG (peak-to-peak) at
300 Hz around the narrow Feshbach resonance at 9.045 G[S6]. Since the composition of the pairs we want to remove is arbitrary (they can be either |aai, |abi, or |bbi),
we continuously employ a diabatic Landau–Zener sweepthat transfers the atoms back and forth between the |2, 0i
and |1, 1i states during the bias modulation, to make sureany doublon spends some time in the Feshbach pair statein order to be removed. In practice, a removal time of
80 ms is sufficient.
In order to specifically remove paired doublons (i.e.
those of the |abi type), we transfer the |1, −1i componentof the pair to the |2, −2i state, and ensure that the othercomponent is in the |1, 1i state. To remove these pairs,
the bias field is modulated by ∼ 10 mG (peak-to-peak) at
300 Hz around the 9.092 G Feshbach resonance between
the |1, 1i and |2, −2i states [S6].
For both removal steps, the sweep range is a few times
the resonance width [S6]. This makes the process insensitive to small field fluctuations.

Two-site model

In the limit of two sites, the spin Hamiltonian (1) reduces tohi22
H = −JS1 · S2 + D (S1z ) + (S2z ) .
(1)

The initial state is a product state between site 1 andsite 2: |Ψi = |ψi1 ⊗ |ψi2 , where the single-site state isgiven by:
|ai − i|bi|ai − i|bi
√
√
⊗
|ψi =
2
2
atom 1
atom 2√1
=|1i − i 2 |0i − |−1i(2)2

The full Hilbert space describing the two spin-1 sites isnine-dimensional. However, the Hamiltonian is block diagonal in the total spin projection, S1z +S2z , and also withregard to odd and even symmetry between the two sites.
For the state prepared initially, all the dynamics takes

place in the symmetric S1z+ S2z = 0 subspace, √which contains only two states: (|1, −1i + |−1, 1i) / 2, |0, 0i .
The Hamiltonian is given by√ J+
2D − 2J
√
H=
.
(3)
− 2J
0
The projection of the initial state into this subspace isrr
1
2(|1, −1i + |−1, 1i) +|0, 0i ,
(4)
|ψi =
6
3
2

also see Fig. 3. Since the √|0, 0i state has (Siz ) = 0,2
and the (|1, −1i + |−1, 1i) / 2 has (Siz ) = 1, a Rabi
oscillation between them leads to an oscillation of thespin alignment A. Note that the components of the initialstate in other subspaces contribute a constant value toA.
Inspection of the Hamiltonian (3) identifies √J + 2D asa z field, which is added to an x field equal to 2J. In a
deep lattice with J ∼ 0, the field is parallel to the z axis,
but lowering the lattice adds an x field, which tilts thefield vector and initiates a precession of the state vector
around it (see Fig. 3).
The Rabi frequency of this oscillation is given bypΩ = 9J 2 + 4JD + 4D2 /h̄,(5)while the peak-to-peak amplitude of the oscillation in A
is 6JD/Ω2 , which is maximized for |D/J| = 3/2.
Matrix-product state simulations

We implemented the time-evolution block decimation algorithm for matrix-product states (MPS-TEBD)[S7, S8] on 100 sites, using a maximum bond dimension
χ of 20. This was found to give results consistent withpublished data [S9]. The modest bond dimension is sufficient because the transient behavior in A occurs withina few exchange times (h̄/J), during which correlations
only build up between clusters of sites. This has the
additional benefit that the calculation can be run on adesktop computer.
The simulated evolution of the spin alignment A as
a function of lattice depth is shown in Fig. S2. These
results form the basis of the simulations presented in themain text. Comparing to the two-site model, we observethat the early time behavior is dominated by nearestneighbor physics. Specifically, the first minima seen in
Fig. S2 occur at the periodof the Rabi oscillation given√
in Eq. (5) divided by 2 to account for the fact that asite in the chain has not one but two neighbors.
This also allows us √to understand the choice of holdtimes in Fig. 2 as π/ 2Ω at the lattice depth where|D/J| ∼ 3/2. This number equals 67 and 17 ms for the
positive and negative D/J pair, respectively, while the

100

200

Time (ms)

300

0.3

-50
-20
-5
-2
0

100

200

Time (ms)

-0.5
300

0.0
-0.3
-0.6

von Neumann entropy (kB)

0.6

0.2
0

0.75

Spin alignment A

Lattice depth (Er)

2
1

0.9

D/J

20
18
16
14
12
10
8

MPS

20
10

Spin alignment A

20
18
16
14
12
10
8

D/J

Lattice depth (Er)3

Exact Diagonalization
L=2
L=4
L=6
L=8

3
2
= 20
= 30
= 40
= 50
= 60

1
0
0

5

Time ( /J)

10

0

5

Time ( /J)10

FIG. S3. Comparison of the von Neumann entanglement entropy between matrix product state calculations on L = 100
sites for varying bond dimension χ (left), and exact diagonalization for varying chain length (right); both calculatedfor D/J = 1. Note that the bond dimensions in the MPS
calculations are not sufficiently large to capture the full entanglement at long evolution times, indicated by a lack ofconvergence there.

FIG. S2. Time evolution of the spin alignment A for various lattice depths, calculated using the TEBD algorithm for
matrix-product states. The top and bottom figures are calculated for pairs with positive (|1, −1i and |1, 0i) and negative(|1, −1i and |1, 1i) values of D/J, respectively. The solid lines
indicate the inverse√(lattice depth dependent) Rabi frequency
of Eq. (5) times π 2, showing that the initial behavior is
dictated by nearest neighbors.

bounded, because an MPS with a maximum bond dimension of χ can only represent states with a maximal
entanglement entropy of ln (χ). Nevertheless, the simulation is expected to provide an accurate result for the
first few exchange times h̄/J.

actual values we use are 70 and 25 ms. It can be thoughtof as half a Rabi oscillation to ensure the largest possible
contrast in the signal.
Finally, we investigate the entanglement properties ofthe spin chain in greater detail. See Fig. S3 for the von
Neumann entropy calculated for both long (L = 100)spin chains using MPS-TEBD, and shorter ones that canbe solved exactly. The von Neumann entropy is evaluated between the equally long left and right halves of thechain. Both calculations show an initial linear growthin entanglement entropy, which implies that the bonddimension required to accurately capture the evolution
grows exponentially. Naturally, this leads to convergenceissues after a few exchange times. Note, though, that theinitial growth rate is independent of system size.
Physically, if the chain contains an even number ofsites, the maximum entanglement entropy between
 thetwo halves, in units of kB , is given by ln 3L/2 . In
the case of the MPS simulations this will be artificially∗
These authors contributed equally to this work.
[S1] N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847(1997).
[S2] W. Kohn, Physical Review 115, 809 (1959).
[S3] P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger,
M. Moreno-Cardoner, S. Fölling, and I. Bloch, Phys.
Rev. Lett. 101, 090404 (2008).
[S4] G. K. Campbell, J. Mun, M. Boyd, P. Medley, A. E.
Leanhardt, L. G. Marcassa, D. E. Pritchard, and
W. Ketterle, Science 313, 649 (2006).
[S5] D. M. Stamper-Kurn and M. Ueda, Reviews of Modern
Physics 85, 1191 (2013).
[S6] A. M. Kaufman, R. P. Anderson, T. M. Hanna,
E. Tiesinga, P. S. Julienne, and D. S. Hall, Phys. Rev.
A 80, 050701(R) (2009).
[S7] J. Hauschild and F. Pollmann, SciPost Physics Lecture
Notes , 5 (2018).
[S8] G. Vidal, Phys. Rev. Lett. 93, 040502 (2004).
[S9] A. Venegas-Gomez, A. S. Buyskikh, J. Schachenmayer,
W. Ketterle, and A. J. Daley, Phys. Rev. A 102, 023321(2020).

154

Appendix C
Calculation of the effective spin-1
model parameters
In this appendix we show that the effective spin-1 model for a ⟨ˆ𝑛⟩ = 2 bosonic Mott
insulator contains an exchange anisotropy as well as an on-site anisotropy (single-ion
anisotropy). We make use of a computer algebra system (Mathematica) to derive themain results.
The spin model does not contain any anisotropy when 𝑈𝑎𝑎 = 𝑈𝑏𝑏 = 𝑈𝑎𝑏 and 𝑡𝑎 = 𝑡𝑏 .
We track the appearance of anisotropies by defining the following variables [97]:
(︂
)︂
1
1
1
1
=
+
𝑈
2 𝑈𝑎𝑎 𝑈𝑏𝑏
𝑑 = (𝑈𝑎𝑎 + 𝑈𝑏𝑏 )/2 − 𝑈𝑎𝑏𝑈𝑎𝑎 − 𝑈𝑏𝑏
𝛿 =
𝑈𝑎𝑎 + 𝑈𝑏𝑏
With these variables, the spin-dependent interaction energies can be re-expressed asfollows:
𝑈
1−𝛿
𝑈
=
1+𝛿
𝑑𝛿 2 − 𝑑 + 𝑈=
1 − 𝛿2

𝑈𝑎𝑎 =
𝑈𝑏𝑏
𝑈𝑎𝑏

155

We also denote by 𝑧 the number of nearest neighbors. While Eq. 6.2 expresses
the matrix element equivalence between a 𝑁 -site Bose-Hubbard model and a 𝑁 -sitespin-1 model, we can reduce the equivalence to that between a two-site Bose-Hubbard
model and a two-site spin-1 model, because all the interactions only involve nearestneighbors. The total interaction energy is just the sum of interaction energies foundin all possible two-site pairs within the given many-body state, so it suffices to simply
track distinct two-site pairs. However, during this reduction we have to be careful toavoid double-counting of single-body terms such as the energy offset or the fictitious
bias field, so such terms need to be divided by 𝑧. For example, consider the equation
for two neighboring |𝑆𝑧 = 1⟩ (first row in Fig. 6-3). The equation is written as
𝑈𝑎𝑎 𝑈𝑎𝑎 12𝑡2
𝑐
𝑐
ℎ𝑧 ℎ𝑧
+
−
+ +
= 𝐽𝑧 −
−
𝑧1
𝑧2
𝑈𝑎𝑎
𝑧1 𝑧2
𝑧1
𝑧2

(C.1)

To check the validity of this example, consider a three-site chain filled with |𝑆𝑧 = 1⟩.
The corresponding equation is
3𝑈𝑎𝑎 −

24𝑡2+ 3𝑐 = 2𝐽𝑧 − 3ℎ𝑧
𝑈𝑎𝑎

Since the value of 𝑧 is 1,2,1 for the three sites, we see that summing up Eq. C.1 twice(the number of links in the chain) with (𝑧1 , 𝑧2 ) = (1, 2) and (𝑧2 , 𝑧3 ) = (2, 1) returns
the correct total energy for the three-site chain. For the remainder of the calculation
we assume 𝑧1 = 𝑧2 . The solution (or the least-square solution) to the system of linear
equations is searched with the attached Mathematica code.

156

System of Linear Equations
In[383]:=

With

order

1 ,

Clear c, hz, Jz, Jp, SIA ;eq1
Series 2

U
1

1

U

d

Series 2

eq2

12 t2

z

U

2 c,

2

d

z 4

2

1

, 0, order , d, 0, 1

eq3

Series 2

eq4

Series

U

z

1
U

d

1

12 t2

t2

1

4

Normal

0;

U

1

U
U

Series

U

d

1

U

2

Series

U

U

1

1

2

, 0, order , d, 0, ordereq7and8eq9

4 t2

Seriesd U d 2
1 2

,

U
1

2

3 t2
1

Normalhz

d U d 2
1 2

1

SIA;

d U d 2
1 2

2

Normal

z Jz

1

d U d
1 2

2

2 c,

U
1

2 SIA;

1

U
1

U
1

2

2 hz

2 c,

d U d 2
1 2

2 t2
U

z Jz

2 c,

5 t2

d U d 2
1 2

2 SIA;

SIA;, 0, order , d, 0, order

1

Series t2

Normald U d 2
1 2

hz

U

2 hz

2 c,

2 t2

z

z Jz

5 t2

Normal2, 0, order , d, 0, order

eq6

1

z

2

1

1

U

d U d 2
1 2

U

2

d

t2

3 t2

z

2

Normal, 0, order

2

d

1

2 c,

, 0, order , d, 0, ordereq5, 0, order

d U d
1 2

Printed by Wolfram Mathematica Student Edition

Normal

1
2

U
1

,

Jp;

2 SIA;

2, 0, order , d, 0, order

NormalJp;

Simplify , Assumptions
U
Print Reduce eq7and8, eq9 , Jp , Reals
Evaluate, SIA, Jz, hz, c , Reals
Reduce eq1, eq2, eq3, eq4, eq5, eq6
Simplify , Assumptions
t 0, U 0,0, z 0&
ReduceFullSimplify
4 t2 dJp
Out[383]=, Assumptions

SIA

0

0

& ;

&

U

U2

False

Approximate Solution to an overdetermined system
The equations for the diagonal elements constitute an overdermined system.
b and A is not invertible , then we can obtain an

If we express our system of linear equations as A x
2approximate solution by minimizing A x b (least-square solution). Let’s assume the solution
vector x represents c, hz , Jz , SIA .
If AT A 1 is nonsingular, then the pseudoinverse of A is AT A 1 AT .
]:=

]:=

Clear c, hz, Jz, SIA
sys
1, Coefficient 2, c, hz, Jz, SIA
eq1, eq2, eq3, eq4, eq5, eq6 ;sys ;; , 2 ;A
sys ;; , 1 ;bPseudoInverse A&

AddSides,2c&

MatrixForm

Out[ ]//MatrixForm=

1
27
1
5
7
27 z
4
27
]:=

Out[ ]=

19
54

1
27
1
5
7
27 z
4
27

0
1
27 z
11
27

4
27
1
10
1
27 z
5
54

c, hz, Jz, SIA
dU

4 t2 z
U

,

4
27
1
10
1
27 z
5
54

2
27

0
13
27 z
8
27

PseudoInverse A .b
U

12 d t2 z

6 t2 z

5 U2

U,FullSimplify

Expand4 d t24 t2

2 d t2 z

U2

U

,d

U2

We can check the deviation of the true Hamiltonian from the spin model by feeding the solution above
back into the linear set of equations and inspecting the leftover on the LHS of the equations.
]:=

Map

1

24 d t2 z
Out[ ]=

5 U2

2
, 0,

&, eq1, eq2, eq3, eq4, eq5, eq6
24 d t2 z
5 U2,

48 d t2 z
5 U2,

48 d t2 z
5 U2

FullSimplify,0

Printed by Wolfram Mathematica Student Edition

Appendix D
Spherical tensor identities
In this appendix, we review important definitions and identities related to the algebra(𝑘)of spherical tensors. Explicit expressions for 𝛼𝐽(𝑘)and 𝛼𝐹 defined in Eq. 3.4 areprovided. For a general introduction to spherical tensors, we refer to Ref. [46].

D.1

Spherical basis

With respect to the Cartesian basis defined by unit vectors e𝑥 , e𝑦 , e𝑧 , the sphericalbasis vectors can be written as
e±1 = ∓

e𝑥 ± 𝑖e𝑦√2e0 = e𝑧In the spherical basis, the sclar product A · B between two vectors A and B is defined
as
A·B=

∑︁
𝑞

𝐴𝑞 𝐵𝑞* =∑︁(−1)𝑞 𝐴𝑞 𝐵−𝑞𝑞where the vector components 𝐴𝑞 are defined as A · e𝑞 and similarly for 𝐵𝑞 . This∑︀means the vector A is expressed as 𝑞 𝐴𝑞 e*𝑞 .
159

For rank-𝑘 spherical tensors 𝐴(𝑘) and 𝐵 (𝑘) , the scalar product is expressed as
(𝑘)

·𝐵

𝐴

(𝑘)

=

𝑘
∑︁

𝑞=−𝑘

(𝑘)

(−1)𝑞 𝐴(𝑘)
𝑞 𝐵−𝑞

(D.1)

If two spherical tensor operators 𝑃 (𝑘1 ) and 𝑄(𝑘2 ) commute, we can derive a useful identity. For the purpose of the derivation, denote the Clebsch-Gordan (CG) coefficient⟨𝑘1 , 𝑞1 ; 𝑘2 , 𝑞2 |𝑘𝑞⟩ by 𝐶𝑘𝑘𝑞1 𝑞1 ,𝑘2 𝑞2 . The CG coefficients satisfy an orthogonality relation:
𝑘∑︁
1 +𝑘2

𝑘
∑︁

𝐶𝑘𝑘𝑞1 𝑞1 ,𝑘2 𝑞2 𝐶𝑘𝑘𝑞1 𝑞′ ,𝑘2 𝑞′ = 𝛿𝑞1 ,𝑞1′ 𝛿𝑞2 ,𝑞2′
12

𝑘=|𝑘1 −𝑘2 | 𝑞=−𝑘and they also satisfy the following permutation relation:𝑘(−𝑞)𝐶𝑘𝑘𝑞1 𝑞1 ,𝑘2 ,𝑞2 = (−1)𝑘1 +𝑘2 −𝑘 𝐶𝑘1 (−𝑞1 ),𝑘2 ,(−𝑞2 )
(︀
)︀ (︀
)︀
Now consider the expression 𝑃 (𝑘1 ) · 𝑄(𝑘1 ) 𝑅(𝑘2 ) · 𝑆 (𝑘2 ) , where 𝑄 and 𝑅 commute as
mentioned. We use Eq. D.1 and the two aforementioned properties of CG coefficients:=(︀𝑃 (𝑘1 ) · 𝑄(𝑘1 )∑︁)︀ (︀ (𝑘2 ) (𝑘2 ) )︀ ∑︁(𝑘 ) (𝑘 )𝑅·𝑆=(−1)𝑞1 +𝑞2 𝑃𝑞(𝑘1 1 ) 𝑅𝑞(𝑘2 2 ) 𝑄−𝑞11 𝑆−𝑞22
𝑞1 ,𝑞2

𝑞1 +𝑞2(−1)

𝑞1 ,𝑞2 ,𝑞3 ,𝑞4

=

𝑘∑︁
1 +𝑘2

𝑘
∑︁

(𝑘 ) (𝑘 )𝑃𝑞(𝑘1 1 ) 𝑅𝑞(𝑘2 2 ) 𝑄−𝑞13 𝑆−𝑞24 𝛿𝑞1 ,𝑞3 𝛿𝑞1 ,𝑞4∑︁

𝑘=|𝑘1 −𝑘2 | 𝑞=−𝑘 𝑞1 ,𝑞2 ,𝑞3 ,𝑞4

=

𝑘∑︁
1 +𝑘2

𝑘1 +𝑘2 −𝑘

(−1)

(︁
)︁ (︁
)︁
(𝑘 ) (𝑘 )(−1)𝑞1 +𝑞2 𝑃𝑞(𝑘1 1 ) 𝑅𝑞(𝑘2 2 ) 𝐶𝑘𝑘𝑞1 𝑞1 ,𝑘2 𝑞2 𝑄−𝑞13 𝑆−𝑞24 𝐶𝑘𝑘𝑞1 𝑞3 ,𝑘2 𝑞4

𝑘 ∑︁
∑︁

𝑞=−𝑘 𝑞1 ,𝑞2

𝑘=|𝑘1 −𝑘2 |(︁)︁(−1)𝑞1 +𝑞2 𝑃𝑞(𝑘1 1 ) 𝑅𝑞(𝑘2 2 ) 𝐶𝑘𝑘𝑞1 𝑞1 ,𝑘2 𝑞2)︁∑︁ (︁ (𝑘 ) (𝑘 ) 𝑘(−𝑞)×𝑄−𝑞13 𝑆−𝑞24 𝐶𝑘1 (−𝑞3 ),𝑘2 (−𝑞4 )𝑞3 ,𝑞4

=

𝑘∑︁
1 +𝑘2

𝑘=|𝑘1 −𝑘2 |

=

𝑘∑︁
1 +𝑘2

𝑘=|𝑘1 −𝑘2 |(−1)𝑘1 +𝑘2 −𝑘

𝑘
∑︁

𝑞=−𝑘

(︀
)︀(𝑘) (︀)︀(𝑘)(−1)𝑞 𝑃 (𝑘1 ) ⊗ 𝑅(𝑘2 ) 𝑞 𝑄(𝑘1 ) ⊗ 𝑆 (𝑘2 ) −𝑞

)︀(𝑘)
(︀
)︀(𝑘) (︀ (𝑘1 )(−1)𝑘1 +𝑘2 −𝑘 𝑃 (𝑘1 ) ⊗ 𝑅(𝑘2 )· 𝑄⊗ 𝑆 (𝑘2 )
160

(D.2)

{︀
}︀(𝑘)
(𝑘)
This result is used in the derivation of 𝛼𝐽 = ⟨𝐽‖ d ⊗ (𝑅+ + (−1)𝑘 𝑅− )d‖𝐽⟩ inSec. 3.3.

D.2

Reduced matrix element

A common source of confusion is the different conventions being used for the definitionof the reduced matrix. In this thesis, the reduced matrix element is defined such that
the Wigner-Eckart theorem is expressed as⟨𝐽, 𝑚𝐽 |𝑇𝑞(𝑘) |𝐽 ′ , 𝑚′𝐽 ⟩ = (−1)2𝑘 ⟨𝐽‖𝑇 (𝑘) ‖𝐽 ′ ⟩⟨𝐽, 𝑚𝐽 |𝐽 ′ , 𝑚′𝐽 ; 𝑘, 𝑞⟩This is also the convention used in Refs. [33, 46]. On the other hand, a common
alternative convention, used by Refs. [25, 116], is(−1)2𝑘⟨𝐽, 𝑚𝐽 |𝑇𝑞(𝑘) |𝐽 ′ , 𝑚′𝐽 ⟩ = √⟨𝐽‖𝑇 (𝑘) ‖𝐽 ′ ⟩⟨𝐽, 𝑚𝐽 |𝐽 ′ , 𝑚′𝐽 ; 𝑘, 𝑞⟩
2𝐽 + 1⎛⎞′𝐽
𝑘 𝐽⎠ ⟨𝐽‖𝑇 (𝑘) ‖𝐽 ′ ⟩= (−1)𝐽−𝑚𝐽 ⎝−𝑚𝐽 𝑞 𝑚𝐽 ′where (. . .) is the Wigner 3-𝑗 symbol. The advantage of this convention is that the
Wigner 3-𝑗 symbol has convenient permutation relations. However, depending on thefamiliarity with such relations, the presentation of the Wigner-Eckart theorem by thesecond convention may seem less clear. Hence we use the first convention.

D.3

Polarizability(𝑘)Here we provide the full expression for 𝛼𝐽 . In the summation over the angularmomentum of the excited states 𝐽 ′ , we explicitly specify the remaining set of quantum
numbers 𝑛′ that fully describes the excited states. The formula can be obtained by161

simplifying Eq. (5.5) in Ref. [46]:(𝑘)𝛼𝑛,𝐽 =√√
2𝑘 + 1 2𝐽 + 1∑︁(−1)𝑘+𝐽+𝐽

′

⎧
⎨ 1

𝑘

⎫
1 ⎬

⎩ 𝐽 𝐽′ 𝐽 ⎭𝑛′ ,𝐽 ′
(︂
)︂
1
(−1)𝑘
×
+
𝐸𝑛,𝐽 − 𝐸𝑛′ ,𝐽 ′ + ~𝜔 𝐸𝑛,𝐽 − 𝐸𝑛′ ,𝐽 ′ − ~𝜔|⟨𝑛, 𝐽‖d‖𝑛′ , 𝐽 ′ ⟩|2

The symbol {. . .} is the Wigner 6-𝑗 symbol. Tabulated values of ⟨𝑛, 𝐽‖d‖𝑛′ , 𝐽 ′ ⟩ foralkali atoms can be found in e.g. Ref. [33].
For the computation of polarizabilities for hyperfine states |𝐹 (𝐽, 𝐼), 𝑚𝐹 ⟩, we must
(𝑘)

(𝑘)

(𝑘)

use 𝛼𝐹 , which depends on 𝛼𝐽 . The formula for 𝛼𝐹 can be obtained by simplifying
Eq. (5.9) in Ref. [46]:⎧⎫⎨ 𝐹 𝑘 𝐹 ⎬
√
√
(𝑘)
(𝑘)
𝛼𝐽
𝛼𝐹 (𝐽,𝐼) = (−1)𝐹 +𝐽+𝐼+𝑘 2𝐽 + 1 2𝐹 + 1⎩ 𝐽 𝐼 𝐽 ⎭162

Bibliography[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.
Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,”
Science, vol. 269, no. 5221, pp. 198–201, 1995.
[2] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle, “Bose-Einstein condensation in a gas of sodium
atoms,” Phys. Rev. Lett., vol. 75, pp. 3969–3973, Nov 1995.
[3] B. DeMarco and D. S. Jin, “Onset of Fermi degeneracy in a trapped atomicgas,” Science, vol. 285, no. 5434, pp. 1703–1706, 1999.
[4] W. Ketterle, D. S. Durfee, and D. Stamper-Kurn, “Making, probing and understanding Bose-Einstein condensates,” arXiv preprint cond-mat/9904034, 1999.
[5] W. Ketterle and M. W. Zwierlein, “Making, probing and understanding ultracold Fermi gases,” arXiv preprint arXiv:0801.2500, 2008.
[6] R. P. Feynman, “Simulating physics with computers,” International Journal of
Theoretical Physics, vol. 21, pp. 467–488, Jun 1982.
[7] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum
phase transition from a superfluid to a Mott insulator in a gas of ultracold
atoms,” Nature, vol. 415, pp. 39–44, Jan 2002.
[8] W. Heisenberg, “Mehrkörperproblem und resonanz in der quantenmechanik,”
Zeitschrift für Physik, vol. 38, pp. 411–426, Jun 1926.
[9] P. A. M. Dirac and R. H. Fowler, “On the theory of quantum mechanics,”
Proceedings of the Royal Society of London. Series A, Containing Papers of a
Mathematical and Physical Character, vol. 112, no. 762, pp. 661–677, 1926.
[10] S. Sachdev, Quantum Phase Transitions. Cambridge University Press, 2 ed.,2011.
[11] F. Franchini, An introduction to integrable techniques for one-dimensional quantum systems. Springer, 2017.
[12] D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, “The onset of matter-wave amplification in a superradiant Bose-Einsteincondensate,” Science, vol. 300, no. 5618, pp. 475–478, 2003.
163

[13] D. M. Weld, P. Medley, H. Miyake, D. Hucul, D. E. Pritchard, and W. Ketterle,“Spin gradient thermometry for ultracold atoms in optical lattices,” Phys. Rev.
Lett., vol. 103, p. 245301, Dec 2009.
[14] P. Medley, D. M. Weld, H. Miyake, D. E. Pritchard, and W. Ketterle, “Spingradient demagnetization cooling of ultracold atoms,” Phys. Rev. Lett., vol. 106,
p. 195301, May 2011.
[15] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle,“Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices,” Phys. Rev. Lett., vol. 111, p. 185302, Oct 2013.
[16] C. J. Kennedy, W. C. Burton, W. C. Chung, and W. Ketterle, “Observationof bose–einstein condensation in a strong synthetic magnetic field,” Nature
Physics, vol. 11, pp. 859–864, Oct 2015.
[17] W. C. Burton, C. J. Kennedy, W. C. Chung, S. Vadia, W. Chen, and W. Ketterle, “Coherence times of Bose-Einstein condensates beyond the shot-noise limit
via superfluid shielding,” Phys. Rev. Lett., vol. 117, p. 275301, Dec 2016.
[18] C. Kennedy, Creating Novel Quantum States of Ultracold Bosons in Optical
Lattices. PhD thesis, Massachusetts Institute of Technology, 2017.
[19] W. C. Burton, Ultracold Bosons in Optical Lattices for Quantum Measurement
and Simulation. PhD thesis, Massachusetts Institute of Technology, 2019.
[20] L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling spin exchange interactions of ultracold atoms in optical lattices,” Phys. Rev. Lett., vol. 91, p. 090402,
Aug 2003.
[21] A. B. Kuklov and B. V. Svistunov, “Counterflow superfluidity of two-species
ultracold atoms in a commensurate optical lattice,” Phys. Rev. Lett., vol. 90,
p. 100401, Mar 2003.
[22] E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of
two-component bosons on an optical lattice,” New Journal of Physics, vol. 5,
pp. 113–113, sep 2003.
[23] J. Schachenmayer, D. M. Weld, H. Miyake, G. A. Siviloglou, W. Ketterle, andA. J. Daley, “Adiabatic cooling of bosons in lattices to magnetically ordered
quantum states,” Phys. Rev. A, vol. 92, p. 041602, Oct 2015.
[24] W. C. Chung, J. de Hond, J. Xiang, E. Cruz-Colón, and W. Ketterle, “Tunable
single-ion anisotropy in spin-1 models realized with ultracold atoms,” Phys. Rev.
Lett., vol. 126, p. 163203, Apr 2021.
[25] K. Beloy, Theory of the ac Stark effect on the atomic hyperfine structure andapplications to microwave atomic clocks. PhD thesis, University of Nevada,Reno, 2009.
164

[26] E. W. Streed, Rubidium-87 Bose-Einstein condensates: Machine construction
and quantum Zeno experiments. PhD thesis, Massachusetts Institute of Technology, 2006.
[27] E. W. Streed, A. P. Chikkatur, T. L. Gustavson, M. Boyd, Y. Torii, D. Schneble,
G. K. Campbell, D. E. Pritchard, and W. Ketterle, “Large atom number BoseEinstein condensate machines,” Review of Scientific Instruments, vol. 77, no. 2,
p. 023106, 2006.
[28] J. Mun, Bose-Einstein condensates in optical lattices: The superfluid to Mott
insulator phase transition. PhD thesis, Massachusetts Institute of Technology,
2008.
[29] M. Fuchs, “Development of a high power stabilized diode laser system,” Master’s
thesis, University of Oregon, 6 2006.
[30] M. L. Harris, C. S. Adams, S. L. Cornish, I. C. McLeod, E. Tarleton, and I. G.
Hughes, “Polarization spectroscopy in rubidium and cesium,” Phys. Rev. A,
vol. 73, p. 062509, Jun 2006.
[31] M. Walhout, J. Dalibard, S. L. Rolston, and W. D. Phillips, “𝜎 + –𝜎 − opticalmolasses in a longitudinal magnetic field,” J. Opt. Soc. Am. B, vol. 9, pp. 1997–
2007, Nov 1992.
[32] B. Merkel, K. Thirumalai, J. E. Tarlton, V. M. Schäfer, C. J. Ballance, T. P.
Harty, and D. M. Lucas, “Magnetic field stabilization system for atomic physics
experiments,” Review of Scientific Instruments, vol. 90, no. 4, p. 044702, 2019.
[33] D. A. Steck, “Rubidium 87 D line data,” 2001.
[34] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping. New York:
Springer-Verlag, 1999.
[35] M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee,
and W. Ketterle, “Bose-Einstein condensation in a tightly confining dc magnetic
trap,” Phys. Rev. Lett., vol. 77, pp. 416–419, Jul 1996.
[36] Y. Castin and R. Dum, “Bose-Einstein condensates in time dependent traps,”
Phys. Rev. Lett., vol. 77, pp. 5315–5319, Dec 1996.
[37] A. M. Kaufman, B. J. Lester, and C. A. Regal, “Cooling a single atom in an
optical tweezer to its quantum ground state,” Phys. Rev. X, vol. 2, p. 041014,
Nov 2012.
[38] J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin,“Coherence and raman sideband cooling of a single atom in an optical tweezer,”
Phys. Rev. Lett., vol. 110, p. 133001, Mar 2013.
165

[39] R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel,“Exploiting the local polarization of strongly confined light for sub-micrometerresolution internal state preparation and manipulation of cold atoms,” Phys.
Rev. A, vol. 89, p. 063829, Jun 2014.
[40] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch, “Coherent transport of neutral atoms in spin-dependent optical lattice potentials,”
Phys. Rev. Lett., vol. 91, p. 010407, Jul 2003.
[41] B. Yang, H.-N. Dai, H. Sun, A. Reingruber, Z.-S. Yuan, and J.-W. Pan, “Spindependent optical superlattice,” Phys. Rev. A, vol. 96, p. 011602, Jul 2017.
[42] C. Robens, J. Zopes, W. Alt, S. Brakhane, D. Meschede, and A. Alberti, “Lowentropy states of neutral atoms in polarization-synthesized optical lattices,”
Phys. Rev. Lett., vol. 118, p. 065302, Feb 2017.
[43] R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, “Optical dipole traps forneutral atoms,” Advances in atomic, molecular, and optical physics, vol. 42,
pp. 95–170, 2000.
[44] S. Eustice, K. Cassella, and D. Stamper-Kurn, “Laser cooling of transition-metal
atoms,” Phys. Rev. A, vol. 102, p. 053327, Nov 2020.
[45] R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev., vol. 130,
pp. 2529–2539, Jun 1963.
[46] D. M. Brink and G. R. Satchler, Angular Momentum. London: Clarendon
Press, 1994.
[47] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt,
F. Grusdt, E. Demler, D. Greif, and M. Greiner, “A cold-atom fermi–hubbard
antiferromagnet,” Nature, vol. 545, pp. 462–466, May 2017.
[48] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof,
X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock withaccuracy and stability at the 10−18 level,” Nature, vol. 506, pp. 71–75, Feb 2014.
[49] I. Bloch, “Ultracold quantum gases in optical lattices,” Nature Physics, vol. 1,
pp. 23–30, Oct 2005.
[50] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), and U. Sen,
“Ultracold atomic gases in optical lattices: mimicking condensed matter physics
and beyond,” Advances in Physics, vol. 56, no. 2, pp. 243–379, 2007.
[51] I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracoldgases,” Rev. Mod. Phys., vol. 80, pp. 885–964, Jul 2008.
[52] O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys., vol. 78, pp. 179–215, Feb 2006.
166

[53] P. Jessen and I. Deutsch, “Optical lattices,” vol. 37 of Advances In Atomic,
Molecular, and Optical Physics, pp. 95–138, Academic Press, 1996.
[54] A. Heinz, A. J. Park, N. Šantić, J. Trautmann, S. G. Porsev, M. S. Safronova,
I. Bloch, and S. Blatt, “State-dependent optical lattices for the strontium opticalqubit,” Phys. Rev. Lett., vol. 124, p. 203201, May 2020.
[55] C. Becker, P. Soltan-Panahi, J. Kronjäger, S. Dörscher, K. Bongs, and K. Sengstock, “Ultracold quantum gases in triangular optical lattices,” New Journalof Physics, vol. 12, p. 065025, jun 2010.
[56] K. L. Lee, B. Grémaud, R. Han, B.-G. Englert, and C. Miniatura, “Ultracold
fermions in a graphene-type optical lattice,” Phys. Rev. A, vol. 80, p. 043411,
Oct 2009.
[57] G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M.
Stamper-Kurn, “Ultracold atoms in a tunable optical kagome lattice,” Phys.
Rev. Lett., vol. 108, p. 045305, Jan 2012.
[58] N. W. Ashcroft, N. D. Mermin, et al., “Solid state physics,” 1976.
[59] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, “Maximally localized Wannier functions: Theory and applications,” Rev. Mod. Phys.,vol. 84, pp. 1419–1475, Oct 2012.
[60] W. Kohn, “Analytic properties of Bloch waves and Wannier functions,” Phys.
Rev., vol. 115, pp. 809–821, Aug 1959.
[61] J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit bypolarization gradients: simple theoretical models,” J. Opt. Soc. Am. B, vol. 6,
pp. 2023–2045, Nov 1989.
[62] C. Robens, S. Brakhane, W. Alt, D. Meschede, J. Zopes, and A. Alberti, “Fast,
high-precision optical polarization synthesizer for ultracold-atom experiments,”
Phys. Rev. Applied, vol. 9, p. 034016, Mar 2018.
[63] T. A. Savard, K. M. O’Hara, and J. E. Thomas, “Laser-noise-induced heating in
far-off resonance optical traps,” Phys. Rev. A, vol. 56, pp. R1095–R1098, Aug
1997.
[64] E. Hecht, Optics. Addison-Wesley, 4th ed., 1998.
[65] “Technical note: Electro-optic modulator faqs.” https://www.newport.com/n/electro-optic-modulator-faqs. Accessed: 2021-04-26.
[66] R. Paschotta, “article on ’electro-optic modulators’ in the rp photonics encyclopedia.” https://www.rp-photonics.com/electro_optic_modulators.html.
Accessed: 2021-04-26.
167

[67] “Linos
laser
modulators
laser
modulators
introduction.”
https://www.qioptiq-shop.com/en/Precision-Optics/
LINOS-Laser-Modulators-Pockels-Cells/LINOS-Laser-Modulators/
Laser-Modulators-Introduction/. Accessed: 2021-04-26.
[68] Meadowlark Optics, Inc., “Response time in liquid-crystal variable retarders.”
Application Notes, 2005.[Online].
[69] N. Walker and G. Walker, “Polarization control for coherent communications,”
Journal of Lightwave Technology, vol. 8, no. 3, pp. 438–458, 1990.
[70] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold bosonicatoms in optical lattices,” Phys. Rev. Lett., vol. 81, pp. 3108–3111, Oct 1998.
[71] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B, vol. 40, pp. 546–
570, Jul 1989.
[72] M. Łącki, D. Delande, and J. Zakrzewski, “Dynamics of cold bosons in optical lattices: effects of higher bloch bands,” New Journal of Physics, vol. 15,
p. 013062, jan 2013.
[73] D.-S. Lühmann, O. Jürgensen, and K. Sengstock, “Multi-orbital and densityinduced tunneling of bosons in optical lattices,” New Journal of Physics, vol. 14,
p. 033021, mar 2012.
[74] S. Will, T. Best, U. Schneider, L. Hackermüller, D.-S. Lühmann, and I. Bloch,“Time-resolved observation of coherent multi-body interactions in quantum
phase revivals,” Nature, vol. 465, pp. 197–201, May 2010.
[75] H. T. Stoof, K. B. Gubbels, and D. B. Dickerscheid, Optical Lattices, pp. 391–
430. Dordrecht: Springer Netherlands, 2009.
[76] M. Kardar, Statistical Physics of Fields. Cambridge University Press, 2007.
[77] G. K. Campbell, J. Mun, M. Boyd, P. Medley, A. E. Leanhardt, L. G. Marcassa,
D. E. Pritchard, and W. Ketterle, “Imaging the Mott insulator shells by using
atomic clock shifts,” Science, vol. 313, no. 5787, pp. 649–652, 2006.
[78] S. Fölling, A. Widera, T. Müller, F. Gerbier, and I. Bloch, “Formation of spatial
shell structure in the superfluid to Mott insulator transition,” Phys. Rev. Lett.,
vol. 97, p. 060403, Aug 2006.
[79] P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger, M. Moreno-Cardoner,
S. Fölling, and I. Bloch, “Counting atoms using interaction blockade in an optical superlattice,” Phys. Rev. Lett., vol. 101, p. 090404, Aug 2008.
[80] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr,“Single-atom-resolved fluorescence imaging of an atomic Mott insulator,” Nature, vol. 467, pp. 68–72, Sep 2010.
168

[81] K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, “Superfluid and insulating phases in an interacting-boson model: Mean-field theory
and the RPA,” Europhysics Letters (EPL), vol. 22, pp. 257–263, may 1993.
[82] T. D. Kühner, S. R. White, and H. Monien, “One-dimensional Bose-Hubbard
model with nearest-neighbor interaction,” Phys. Rev. B, vol. 61, pp. 12474–
12489, May 2000.
[83] S. Will, From Atom Optics to Quantum Simulation: Interacting Bosons andFermions in Three-Dimensional Optical Lattice Potentials. Springer Science &Business Media, 2012.
[84] D.-S. Lühmann, Multiorbital Physics in Optical Lattices. PhD thesis, Universityof Hamburg, 2009.
[85] P. N. Jepsen, W. W. Ho, J. Amato-Grill, I. Dimitrova, E. Demler, and W. Ketterle, “Transverse spin dynamics in the anisotropic Heisenberg model realizedwith ultracold atoms,” arXiv preprint arXiv:2103.07866, 2021.
[86] J. Keeling, “Quantum magnetism,” Lecture notes. University of St. Andrews,
2008.
[87] A. Auerbach, Interacting electrons and quantum magnetism. Springer Science& Business Media, 2012.
[88] I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod.
Phys., vol. 86, pp. 153–185, Mar 2014.
[89] D. Dai, H. Xiang, and M.-H. Whangbo, “Effects of spin-orbit coupling on magnetic properties of discrete and extended magnetic systems,” Journal of Computational Chemistry, vol. 29, no. 13, pp. 2187–2209, 2008.
[90] A. Kitaev, “Anyons in an exactly solved model and beyond,” Annals of Physics,
vol. 321, no. 1, pp. 2–111, 2006. January Special Issue.
[91] T. Moriya, “Anisotropic superexchange interaction and weak ferromagnetism,”
Phys. Rev., vol. 120, pp. 91–98, Oct 1960.
[92] S. Trotzky, P. Cheinet, S. Fölling, M. Feld, U. Schnorrberger, A. M. Rey,
A. Polkovnikov, E. A. Demler, M. D. Lukin, and I. Bloch, “Time-resolved observation and control of superexchange interactions with ultracold atoms in opticallattices,” Science, vol. 319, no. 5861, pp. 295–299, 2008.
[93] E. Koch, “Exchange mechanisms,” Correlated electrons: from models to materials, vol. 2, pp. 1–31, 2012.
[94] D. J. Klein, “Degenerate perturbation theory,” The Journal of Chemical Physics,
vol. 61, no. 3, pp. 786–798, 1974.
169

[95] C. Cohen-Tannoudji, G. Grynberg, and J. Dupont-Roc, Atom-Photon Interactions, ch. 1: Transition Amplitudes in Electrodynamics, pp. 5–66. John Wiley& Sons, Ltd, 1998.
[96] E. Demler, “Strongly correlated systems in atomic and condensed matterphysics, lecture notes for physics 284, chapter 9.” http://cmt.harvard.edu/
demler/TEACHING/Physics284/chapter9.pdf, February 2018. [Online; accessed 27-April-2021].
[97] N. Jepsen, J. Amato-Grill, I. Dimitrova, W. Lunden, and W. Ketterle, “Spin-1
Heisenberg model with lithium-7.” Internal report.
[98] F. Gerbier, A. Widera, S. Fölling, O. Mandel, and I. Bloch, “Resonant control ofspin dynamics in ultracold quantum gases by microwave dressing,” Phys. Rev.
A, vol. 73, p. 041602, Apr 2006.
[99] D. M. Stamper-Kurn and M. Ueda, “Spinor bose gases: Symmetries, magnetism,
and quantum dynamics,” Rev. Mod. Phys., vol. 85, pp. 1191–1244, Jul 2013.
[100] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P. Schauß, S. Hild,
D. Bellem, U. Schollwöck, T. Giamarchi, C. Gross, I. Bloch, and S. Kuhr,“Quantum dynamics of a mobile spin impurity,” Nature Physics, vol. 9, pp. 235–
241, Apr 2013.
[101] S. Hild, T. Fukuhara, P. Schauß, J. Zeiher, M. Knap, E. Demler, I. Bloch, andC. Gross, “Far-from-equilibrium spin transport in Heisenberg quantum magnets,” Phys. Rev. Lett., vol. 113, p. 147205, Oct 2014.
[102] S. Fölling, Probing strongly correlated states of ultracold atoms in optical lattices. PhD thesis, Johannes Gutenberg University of Mainz, 2008.
[103] E. Altman, E. Demler, and M. D. Lukin, “Probing many-body states of ultracold
atoms via noise correlations,” Phys. Rev. A, vol. 70, p. 013603, Jul 2004.
[104] S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial
quantum noise interferometry in expanding ultracold atom clouds,” Nature,
vol. 434, pp. 481–484, Mar 2005.
[105] A. M. Kaufman, R. P. Anderson, T. M. Hanna, E. Tiesinga, P. S. Julienne, and
D. S. Hall, “Radio-frequency dressing of multiple Feshbach resonances,” Phys.
Rev. A, vol. 80, p. 050701, Nov 2009.
[106] J. T. M. Walraven, “Quantum gases.” https://staff.fnwi.uva.nl/j.t.m.
walraven/walraven/Publications_files/2019-Quantum-Gases.pdf, October 2019. [Online; accessed 09-May-2021].
[107] K. M. Mertes, J. W. Merrill, R. Carretero-González, D. J. Frantzeskakis, P. G.
Kevrekidis, and D. S. Hall, “Nonequilibrium dynamics and superfluid ring excitations in binary bose-einstein condensates,” Phys. Rev. Lett., vol. 99, p. 190402,
Nov 2007.
170

[108] P. Chauhan, F. Mahmood, H. J. Changlani, S. M. Koohpayeh, and N. P. Armitage, “Tunable magnon interactions in a ferromagnetic spin-1 chain,” Phys.
Rev. Lett., vol. 124, p. 037203, Jan 2020.
[109] S. Braun, M. Friesdorf, S. S. Hodgman, M. Schreiber, J. P. Ronzheimer, A. Riera, M. del Rey, I. Bloch, J. Eisert, and U. Schneider, “Emergence of coherence
and the dynamics of quantum phase transitions,” Proceedings of the National
Academy of Sciences, vol. 112, no. 12, pp. 3641–3646, 2015.
[110] L. Pitaevskii and S. Stringari, Bose-Einstein condensation and superfluidity,
vol. 164. Oxford University Press, 2016.
[111] J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, “Quantum simulation of antiferromagnetic spin chains in an optical lattice,” Nature,
vol. 472, pp. 307–312, Apr 2011.
[112] H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Aberrations, ch. 29, pp. 1–
70. John Wiley & Sons, Ltd, 2006.
[113] R. Barakat, “Rayleigh wavefront criterion,” J. Opt. Soc. Am., vol. 55, pp. 572–
573, May 1965.
[114] QED Technologies, Inc. personal communication.
[115] S. Brakhane, W. Alt, D. Meschede, C. Robens, G. Moon, and A. Alberti, “Note:
Ultra-low birefringence dodecagonal vacuum glass cell,” Review of Scientific
Instruments, vol. 86, no. 12, p. 126108, 2015.
[116] F. Le Kien, P. Schneeweiss, and A. Rauschenbeutel, “Dynamical polarizabilityof atoms in arbitrary light fields: general theory and application to cesium,”
The European Physical Journal D, vol. 67, p. 92, May 2013.

171