Ultracold Bosons in Optical Lattices for Quantum
Measurement and Simulation
by

William Cody Burton
Bachelor of Arts in Physics, Princeton University (2012)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2019
c Massachusetts Institute of Technology 2019. All rights reserved.
○

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
September 12, 2018
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nergis Mavalvala
Associate Department Head, Physics

2

Ultracold Bosons in Optical Lattices for Quantum
Measurement and Simulation
by
William Cody Burton
Submitted to the Department of Physics
on September 12, 2018, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Ultracold atoms provide a platform that allows for pristine control of a physicalsystem, and have found uses in both the fields of quantum measurement and quantum
simulation. Optical lattices, created by the AC Stark shift of a coherent laser beam,
are a versatile tool to control ultracold atoms and implement novel Hamiltonians.
In this thesis, I report on three experiments using the bosonic species Rubidium-87
trapped in optical lattices.
I first discuss our work in simulating the Harper-Hofstadter Hamiltonian, which
describes charged particles in high magnetic fields, and has connections to topologicalphysics. To simulate the charged particles, we use laser-assisted tunneling to add
a complex phase to tunneling in the optical lattice. For the first time, we havecondensed bosons into the ground state of the Harper-Hofstadter Hamiltonian. In
addition, we have demonstrated that we can add strong on-site interactions to theeffective Hamiltonian, opening the door to studies of interesting states near the Mott
insulator transition.
Next, I present a novel technique to preserve phase coherence between separated
quantum systems, called superfluid shielding. Phase coherence is important for bothquantum measurement and simulation, and is fundamentally limited by projection
noise. When an interacting quantum system is split, frozen-in number fluctuations
lead to fluctuations of the relative phase between separated subsystems. We cancel
the effect of these fluctuations by immersing the separated subsystems in a common
superfluid bath, and demonstrate that we can increase coherence lifetime beyond the
projection noise limit.
Finally, I discuss our efforts in simulating magnetic ordering in the spin-1 Heisenberg Hamiltonian. It is hard to adiabatically ramp into magnetically ordered groundstates, because they often have gapless excitations. Instead, we use a spin-dependentlattice to modify interspin interactions, allowing us to ramp into the spin Mott insulator, which has a gap and can therefore act as a cold starting point for explorationof the rest of the phase diagram. We have achieved a cold spin temperature in thespin Mott insulator, and I discuss plans to also achieve a cold charge temperature3and then ramp to the the xy-ferromagnet, which has spin-charge separation.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

4

Acknowledgments
In this thesis, I use the first-person singular “I” when discussing the work, but nothesis is created by a single person. Through my time at MIT, I have been supported
and helped by more people than I can list here.
I first wish to acknowledge my advisor Wolfgang Ketterle. He has undoubtedly had
the greatest influence on how I think about and approach physics. It always amazes
me when Wolfgang can take something that I thought was irreducibly quantum andgive a classical analogy that neatly explains it. Of course, I would not be where
I am today without a host of teachers and mentors who nurtured and supported
my love of physics, including Robert Donaldson, Ralph Bunday, James Schafer, Jim
Restorff, Marilyn Wun-Fogle, Bobby Berg, David Huse, Mike Romalis, Trey Porto,and Andrew Houck.
When I first began working in BEC4, Hiro Miyake and Georgios Siviloglou were
the senior graduate student and postdoc, respectively. Both were kind and patientin helping me learn not only atomic physics, but the special quirks of our particularexperimental apparatus. I think that Georgios’s most important advice was always
to remember “do not panic.” Colin Kennedy and Woo Chang Chung were my fellow
grad students and companions for most of my time here. Colin’s energy and Woo
Chang’s focus made for a stimulating and efficient research environment. The second
postdoc that I worked with, Wenlan Chen, was a great addition to the lab. She had
a broad knowledge base and a view for a good picture. I especially want to thank her
for keeping us from going down rabbit holes of research and keeping our eyes on the
prize! Finally, I must acknowledge the assistance of our undergraduate and visiting
students: Yuri Lenskiy, Benedikt Groever, and Samarth Vadia.
Of course, BEC4 is not an island, and the other labs, both in Wolfgang’s groupand in the CUA, have been immensely helpful. The ability to just walk down the
hallway if you have technical questions, need supplies, or even just want to chatabout physics is truly unique and has led to many friendships over the years. I wantto especially point out the close connection between BEC4 and BEC5, who both want5

to use bosons in optical lattices to study magnetic ordering. This has led to many
conversations and deep friendships with Niki Jepsen, Ivana Dimitrova, Jesse AmatoGrill, and Will Lunden. The postdocs in Wolfgang’s other labs, Alan Jamison andChris Lee, provided mentorship, a sounding-board, and friendship to all the labs.
Outside of work, the people I probably spent the most time with were my roommates Tom Cooper and Matt Nichols, and the quasi-roommate Ian Counts. I’m glad
I got to live with good friends and colleagues for five years, and I’ll miss coming home
to a house full of physicists. I would like to thank all of my friends from the Physics
Department for helping to keep me sane, whether through pub socials, hiking, board
games, or just hanging out! A special thanks goes out to Zak Vendeiro, Alban Urvoy, Boris Braverman, Lawrence Cheuk, Sergio Cantu, John Hardin, Gabriel Collin,
Mukund Madhav, Nancy Aggarwal, Jenny Schloss, Tom Mahony, Alex Leder, AlexJi, and Keaton Burns!
Working as a volunteer Emergency Medical Technician with MIT Emergency Medical Services provided a welcome complement to my work in grad school. Whilephysics research can take months or years and alternates between tedious and sublime, working on an ambulance provides a much faster feedback loop — within anhour of getting a call, you’ve done your best to make someone’s life better. I appreciate all members of MIT EMS for giving me a second home on campus, and especiallywant to thank my qCC partner Andrew Tang, the Training Officer before me Shivangi
Goel, and my successors Vivian Hu and Ann Liu.
Finally, none of this would be possible without the support and love of my family.
My parents, Bill and Suzanne Burton, raised me to be intellectually curious and havesupported my interest in physics since childhood. They have been examples of how
to work hard and be organized without losing sight of the importance of family. I
would also like to thank my grandparents Bill and Jean Eckert, who were always there
for me, and were always excited to hear about what I was working on. My siblings,
Tuck, M-E, and Mindy, have always been my best friends — it makes me happy tohave such a close family, and I am proud of what they have accomplished so far andexcited to see what their future holds!
6

Contents
1 Introduction

15

2 Optical Lattices

19

2.1

Making an Optical Lattice . . . . . . . . . . . . . . . . . . . . . . . .

19

2.2

Theory of Particles in Lattice Potentials . . . . . . . . . . . . . . . .

21

2.2.1

Bloch’s Theorem and Bandstructure . . . . . . . . . . . . . .

21

2.2.2

Wannier Functions . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3

Bose-Hubbard Hamiltonian. . . . . . . . . . . . . . . . . . . . . . .

25

2.3.1

Superfluid to Mott Insulator Transition . . . . . . . . . . . . .

26

2.3.2

Wannier-Stark Ladder . . . . . . . . . . . . . . . . . . . . . .

27

3 Simulating Strong Magnetic Fields
3.1

31

Harper-Hofstadter Hamiltonian . . . . . . . . . . . . . . . . . . . . .

32

3.1.1

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

34

3.2

Raman tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.3

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.3.1

Experimental Upgrades . . . . . . . . . . . . . . . . . . . . . .

42

3.3.2

Final Experimental Sequence . . . . . . . . . . . . . . . . . .

43

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.4.1

Apparent Gauge Dependent Observables . . . . . . . . . . . .

48

3.4.2

Population Imbalance . . . . . . . . . . . . . . . . . . . . . . .

49

3.4.3

Superfluid Peak Fitting . . . . . . . . . . . . . . . . . . . . . .

52

3.4.4

Adding Strong Interactions . . . . . . . . . . . . . . . . . . . .

53

3.4

Hofstadter’s Butterfly

7

3.5

3.6

Lifetime of the Harper Superfluid . . . . . . . . . . . . . . . . . . . .

55

3.5.1

Floquet Fermi’s Golden Rule . . . . . . . . . . . . . . . . . . .

57

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4 Superfluid Shielding

63

4.1

Atom Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.2

Bloch Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.3

Superfluid Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.3.1

Gross-Pitaevskii Description . . . . . . . . . . . . . . . . . . .

70

4.3.2

Quantum Mechanical Description . . . . . . . . . . . . . . . .

71

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.4.1

Contrast Fitting . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Projection Noise Limit Analysis . . . . . . . . . . . . . . . . . . . . .

74

4.5.1

Two-Dimensional Discrete Thomas-Fermi Profile . . . . . . . .

75

4.5.2

Two-Mode Number Squeezing Model . . . . . . . . . . . . . .

76

4.6

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

4.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.4

4.5

5 Spin-1 Heisenberg Hamiltonian
5.1

83

Heisenberg from Bose-Hubbard . . . . . . . . . . . . . . . . . . . . .

84

5.1.1

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

86Spin-1 Heisenberg Hamiltonian Phase Diagram . . . . . . . . . . . . .

89

5.2.1

𝑥𝑦 -Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5.2.2

Spin Mott . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.3.1

On-Site Correlations . . . . . . . . . . . . . . . . . . . . . . .

93

5.3.2

Off-Site Correlations . . . . . . . . . . . . . . . . . . . . . . .

95

5.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5.5

Spontaneous Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2

5.3

Spin-Dependent Lattice8

6 Conclusion

109

A Lattice Noise Measurements

113

A.1 Empirical Measure of Condensate Fraction Decay . . . . . . . . . . . 114
A.1.1 Intensity Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.1.2 High Frequency Phase Noise . . . . . . . . . . . . . . . . . . . 115
A.1.3 Low Frequency Phase Noise . . . . . . . . . . . . . . . . . . . 116
A.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

B Realizing the Harper-Hamiltonian with Laser-Assisted Tunnnelingin Optical Lattices
119
C Spin-Orbit Coupling and Quantum Spin Hall Effect for Neutral
Atoms without Spin Flips
125

9

10

List of Figures
2-1 Bandstructure of an atom in an optical lattice . . . . . . . . . . . . .

23

2-2 Maximally-localized Wannier functions in the lowest band . . . . . .

24

2-3 Phase diagram of the SF to MI transition and the LDA . . . . . . . .

28

2-4 Wannier-Stark wavefunctions at varying tilts . . . . . . . . . . . . . .

29

3-1 Hofstadter’s Butterfly . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3-2 Schematic of our laser assisted tunneling setup. . . . . . . . . . . . .

37

3-3 Demonstration of the amplitude of laser-assisted tunneling . . . . . .

41

3-4 Experimental sequences for two different state preparation protocols
for the Harper-Hofstadter Hamiltonian . . . . . . . . . . . . . . . . .

44

3-5 Observation of Bose-Einstein condensation in the Harper-Hofstadtermodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3-6 Population imbalance of the two ground states of the Harper-Hofstadter
Hamiltonian with 1/2 flux . . . . . . . . . . . . . . . . . . . . . . . .

51

3-7 An example fit for a 1/2 flux superfluid image . . . . . . . . . . . . .

52

3-8 Harper-Hofstadter Hamiltonian with strong interactions . . . . . . . .

54

3-9 Strongly interacting Harper superfluid lifetime . . . . . . . . . . . . .

55

3-10 Decay of Bose-Einstein condensates in modulated lattices . . . . . . .

56

3-11 Lifetime of a Bose-Einstein condensate in a 1D lattice with resonant
amplitude modulation . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4-1 Cartoon of an optical interferometer . . . . . . . . . . . . . . . . . . .

65

4-2 Cartoon of Bloch oscillations as atom interferometry effect . . . . . .

67

4-3 Schematic of superfluid shielding69

11. . . . . . . . . . . . . . . . . . . .

4-4 Superfluid shielding for different atom numbers and densities . . . . .

78

4-5 Contrast lifetime versus shielding fraction . . . . . . . . . . . . . . . .

79

4-6 Superfluid shielding of external fields. . . . . . . . . . . . . . . . . . .

80

5-1 Cartoon of the superexchange process . . . . . . . . . . . . . . . . . .

85

5-2 Phase diagram of the spin-1 Heisenberg Hamiltonian. . . . . . . . . .

90

5-3 Description of our measurement for on-site correlations . . . . . . . .

94

5-4 Spin paired doublon fraction after ramping directly into the Mott insulator at varying 𝑈↑↓ /𝑈 . . . . . . . . . . . . . . . . . . . . . . . . .

98

5-5 Spin paired doublon fraction after ramping from the spin Mott insulator to varying 𝑈↑↓ /𝑈 . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5-6 Cartoon of the effect of hole tunneling on the measured SPDF . . . . 100
5-7 Initial condensate fraction measurements after ramping back to BEC

101

5-8 Calculated excitation fraction at the center of the Mott insulator . . . 101
5-9 Condensate fraction measurements after experimental upgrades . . . 103
5-10 Calculated photon scattering rate of an atom sitting in a purely vector
AC Stark shift lattice of 1 ER, 780 nm . . . . . . . . . . . . . . . . . . . 105
5-11 Scattering fraction of Raman and Rayleigh scatteing off of a 𝜎 − lattice 106
A-1 Intensity spectrum after injecting noise at various frequencies . . . . . 114
A-2 Condensate fraction lifetime with intensity noise applied . . . . . . . 115
A-3 Spectrum after injecting phase noise at -20 dBm at various frequencies 115
A-4 Condensate fraction lifetime with high-frequency phase noise applied

116

A-5 Spectrum after injecting -12 dBm phase noise at low frequencies . . . 117
A-6 Condensate fraction lifetime with low-frequency phase noise applied . 117

12

List of Tables
A.1 1/𝑒 lifetimes of a 24𝐸𝑅 Mott Insulator with 16-20 kHz noise added. . 116
A.2 1/𝑒 lifetimes of a 20 𝐸𝑅 Mott Insulator with 0.6-1.3 kHz noise added.

13

117

14

Chapter 1
Introduction
Ultracold atoms provide a clean and precisely controllable system for studying a wide
range of physical effects. Through the use of lasers, magnetic fields, and microwave
and radio frequency (RF) waves, one can implement nearly any microscopic Hamiltonian to a high accuracy. With the advent of laser cooling and evaporative cooling, it
became possible to create Bose-Einstein condensates (BECs) [1, 2], which were firstpredicted by Bose and Einstein in 1924. In a BEC, the low temperatures lead to a
macroscopic occupation of a single quantum state. Initial work with BECs involved
exploring exotic physics, including matter wave interference [3], superfluidity [4], vortex lattices [5], and superradiance [6]. Later work achieved similarly low temperaturesin fermions, leading to the production of degenerate Fermi gases [7].
Once it had been shown that optical dipole forces could be used to create lattice potentials [8, 9], a new realm of quantum simulation was opened. Fermions inan optical lattice are described by the Fermi-Hubbard model, which is also the simplest model of electrons in a crystal lattice, and is believed to be connected to thephenomenon of high-temperature superconductivity [10, 11]. Bosonic atoms, on theother hand, are most naturally described by the Bose-Hubbard Hamiltonian [12], andinital work studied the superfluid-Mott insulator transition [13, 14]. Since then, muchwork has looked at lattices with lower dimensionality [15], novel geometries [16, 17],or modified tunneling [18, 19, 20, 21, 22, 23, 24].
In addition, because of their cold temperature and precise control, cold atoms
15

have become one of the primary systems for precision measurement [25]. One general method used is that of atom interferometery, which relies on the wave nature ofquantum mechanics and can be explained in analogy with classical optical interferometry. In a traditional interferometer, a beam of light is split and sent along twopaths. When the light is re-combined and measured, the light from the two paths
interfere, and the interference pattern contains information about the difference inpath lengths. For an atom interferometer, one takes advantage of the wave natureof cold atoms, running separate parts of the wavefunction along different paths, andthereby measuring the energy difference between the paths. This method has proved
useful not only in studying fundamental physics [26, 27, 28], but also as a precise
force sensor [29, 30, 31].
The experiments described in this thesis were done in Wolfgang Ketterle’s Rubidium lab at MIT. This lab studies ultracold bosonic 87 Rb atoms in optical lattices, andin the past has been used to explore many topics, including superradiance [6], thequantum Zeno effect [32], and demagnetization cooling [33], and in recent years has
focused on lattice physics. I begin in Chapter 2 by describing the theory of opticallattices and some background theory necessary to understand our system.
In Chapter 3, I discuss our work in simulating strong magnetic fields, where acharged particle in a uniform magnetic field undergoes cyclotron motion with a cyclotron orbit size of the same order as the lattice spacing. Then, the motion is
described by the Harper-Hofstadter Hamiltonian, whose spectrum is given by thefractal Hofstadter’s butterfly. The spectrum is interesting not only for its fractal
nature, but also for its connection to toplogical physics. Where the vacuum has a
Chern number, of exactly zero, many of the subbands of Hofstadter’s butterfly havea non-zero Chern number. Since Chern numbers are topological invariants and must
be integers, it cannot go smoothly from its non-zero value in the bulk of a system
to zero outside it — the bandgap must close at the edge, allowing for gapless edge
modes. Topological invariants are robust to local variations in system parameters, sothese edge modes are “topologically protected” from lattice impurities or distortions.
This inherent robustness makes topological materials interesting in themselves and16

as a potential candidate for use in quantum computing.
In order to see these effects in a traditional material, one would need magnetic
fields ∼10,000 T, which is clearly not feasible. Instead, we simulate the effect of

this field by directly controlling the phase of tunneling elements in our lattice. By

doing so, we can emulate the Aharanov-Bohm phases associated with such a highfield. I present the first realization of a BEC in the Harper-Hofstadter Hamiltonianand demonstrate some of the properties of this Harper superfluid, including gaugedependent observables.
In Chapter 4, I discuss superfluid shielding, which is a novel method to increase
the coherence lifetime in separated quantum systems, like those in atom interferometers. One fundamental mechanism for the decay of phase coherence is quantum
projection noise. When a coherent quantum system is split, there are unavoidable
density fluctuations. If the systems are self-interacting, these density fluctuations
get written into the relative phase between the subsystems, and therefore reduce thephase coherence and signal-to-noise in measurements. Because of this, most groups
using atom interferometry work with atoms above the BEC transition point, where
densities are low enough that interactions are negligible. However, this is the equivalent of using a white light interferometer, rather than a laser interferometer, which
will in general have a much higher brightness.
Superfluid shielding gets around this issue, and allows for long coherence lifetimes,
by using good interactions to zero out the effect of bad interactions. We immerse theseparated quantum systems in a superfluid bath of a different spin state, which because it is superfluid, moves its density to exactly cancel the density fluctuations inthe separated subsystems. Then, since for 87 Rb, all inter- and intra-spin interaction
strengths are equal to the percent level, this cancels the effect of the density fluctuations on the relative phase between the separated subsystems. I present a theoreticalanalysis of this effect and demonstrate its ability to lengthen coherence times beyond
the projection noise limit by using Bloch oscillations in a two-component 87 Rb BEC.
In Chapter 5, I present our work on the Heisenberg Hamiltonian, a paradigmatic
description of systems with magnetic ordering. A system with two spin components
17deep in the 𝑛 = 2 Mott insulator can be mapped onto a spin-1 Heisenberg model,
which has an additional phase that the spin-1/2 model does not have — the spin
Mott insulator. This state is an insulator in both spin and charge space, and has
gapped excitations, which should allow us to keep a lower entropy when we ramp tothis phase from a BEC. Once there, we will be able to use a spin-dpenedent lattice
to vary the inter-spin interaction strength and adiabatically enter other states thathave magnetic ordering and gapless excitations. The first phase that we wish to ramp
to is the 𝑥𝑦 -ferromagnet, which is a spin superfluid while still being an insulator in
charge, demonstrating spin-charge separation. So far, we have achieved spin Mott
insulators that are cold in spin space, but were initially unable to ramp into a cold𝑥𝑦 -ferromagnet because of the large number of charge excitations. I discuss upgrades
made to the experiment that have greatly reduced the charge temperature, along with
future plans for our system.

18

Chapter 2
Optical Lattices
Lattice models are used in much of physics, from condensed matter [34] to high-energy
physics [35, 36]. In the field of ultracold atoms, lattice models have been proposed
and used to study topological physics [37, 38], magnetic phases [39, 40], and even
dynamical gauge fields [38]. Optical lattices, which are created using coherent laser
light, are a versatile tool for creating lattice potentials for ultracold atoms.
In this chapter, I first describe how to create an optical lattice using the AC Stark
shift of a coherent laser beam. I then discuss the basic theory of particles in a latticepotential, including Bloch’s theorem and bandstructure calculations. In deep lattices,
the system can be described by the Bose-Hubbard Hamiltonian, which has superfluidand Mott insulating regions, and can easily be extended by adding external (possibly
time-dependent) potentials or novel types of interactions.

2.1 Making an Optical Lattice
Optical lattice potentials can be formed using the AC Stark Shift [8, 9], which is thepotential felt by atoms in an off-resonant oscillating electromagnetic field. Working
in the dipole approximation and ignoring atomic motion, we write the interaction
between the oscillating field and the atom as⃗
𝐻 ′ = −𝑑⃗ · 𝐸(𝑡)
19

(2.1)

⃗ = 𝐸 𝑥ˆ cos(𝜔𝑡) is the electric field. For
where 𝑑⃗ = −𝑒⃗𝑟 is the dipole operator, and 𝐸
the scalar AC Stark shift, which is spin-independent and the primary component atlarge detunings, we can treat the atom as a two level system with ground state 𝑔 andexcited state 𝑖. We define the dipole matrix element between the two states

𝑀𝑔𝑖 = ⟨𝑖| 𝑑⃗ · 𝑥ˆ |𝑔⟩ .

(2.2)

Note that we have left the electric field strength out of this definition, so ⟨𝑖| 𝐻 ′ |𝑔⟩ =𝐸𝑀𝑔𝑖 . Then with second order perturbation theory, we find that the energy shift is∆𝐸 = 2𝜋𝛼𝑎20 𝐼 |𝑀𝑔𝑖 |2

(︂
)︂
1
1
− +
,
𝛿 𝜔𝑔𝑖 + 𝜔

(2.3)

where 𝛼 is the fine structure constant, 𝑎0 is the Bohr radius, 𝜔𝑔𝑖 is the atomic resonance frequency, and 𝛿 = 𝜔 − 𝜔𝑔𝑖 . We have included the electric field strength in theintensity 𝐼 = 𝑐𝜖0 𝐸 2 /2. Now, if the intensity varies with position, we can treat ∆𝐸 as
an external potential. These potentials, which we create with far-detuned lasers, arethe primary tools used in the experiments described in this thesis.
The sign of the potential changes with detuning — a red-detuned laser beam creates an attractive potential, while a blue-detuned beam creates a repulsive potential.
Our main trapping beams, which provide confinement on the scale of 10-100 𝜇m, aremade by focusing the output of a 1064 nm fiber laser. It is possible to make a latticepotential out of an array of independent dipole traps [41, 42], but doing so requires alarge numerical aperture and precise control over many degrees of freedom. Instead,
we make our optical lattices by retro-reflecting a laser beam focused on the atoms,
which creates an intensity lattice with a spacing that is half of the laser wavelength.
Unlike in [41, 42], we cannot move individual lattice sites around, but a retro-reflectedlattice is much more stable: the spacing between lattice sites is set only by the laser
wavelength, and the relative depths of lattice sites is determined by the Gaussian envelope of the beam, which can be much broader than the atomic cloud. This stability
allows us to reach and maintain the colder temperatures required to study latticephysics mediated by tunneling and on-site interactions.
20

2.2 Theory of Particles in Lattice Potentials
Our neutral atoms in a one-dimensional optical lattice are then described by thesingle-particle Hamiltonian𝐻=

𝑝2+ 𝑉𝐿 cos2 (𝑘𝐿 𝑥)
2𝑚

(2.4)

where 𝑉𝐿 is the height of the optical lattice potential and 𝑘𝐿 is the wavevector 2𝜋/𝜆of the lattice laser. This is the same form as a Hamiltonian describing electrons ina crystal lattice, and is the starting point for much of condensed matter physics.
This connection both allows us to use theory tools first created for condensed mattersystems, as well as use our cold atoms systems as analogues of interesting condensed
matter systems.

2.2.1 Bloch’s Theorem and Bandstructure
Our Hamiltonian breaks continuous translation symmetry, so momentum is not conserved, and the eigenstates do not have a definite momentum. However, there is stilla discrete translation symmetry of 𝑥 → 𝑥 + 𝑎, where 𝑎 = 𝑘𝐿 /𝜋. Bloch’s theorem[34] tells us that eigenstates of the Hamiltonian are also eigenstates of the discrete
translation operator

𝑇 =𝑒𝑖𝑝𝑎/~(2.5)

𝑇 𝜓(𝑥) =𝜓(𝑥 + 𝑎) .
We can explicitly split out this eigenvalue and write the wavefunction as

𝜓(𝑥) = 𝑒𝑖𝑞𝑎 𝜑𝑞 (𝑥)(2.6)

where 𝑞 is the eigenvalue of 𝑇 , known as the quasimomentum, and 𝑢𝑞 (𝑥) is periodic
in 𝑥 with period 𝑎. The quasimomentum is often similar to real momentum, but it
is only defined on (−𝜋/𝑎, 𝜋/𝑎] and is only conserved modulo 2𝜋/𝑎. We find that the21quasimomentum part of the ansatz factors out:𝐻𝜓(𝑥) =𝐻𝑒𝑖𝑞𝑎 𝜑𝑞 (𝑥)
𝑖𝑞𝑎

(2.7)

=𝑒 𝐻𝑞 𝜑𝑞 (𝑥) ,where(𝑝 + 𝑞)2+ 𝑉𝐿 𝑐𝑜𝑠2 (𝑘𝐿 𝑥) ,
𝐻𝑞 =
2𝑚

(2.8)

and 𝑞 is now a c-number, rather than an operator. Thus, each 𝜑𝑞 (𝑥) is given by the
eigenfunctions of the appropriate 𝐻𝑞 , which have the form of Mathieu functions. However, Mathieu functions can be hard to work with, so for our numerical calculations,
we use a plane wave expansion

𝜑𝑞 (𝑥) =∑︁ (𝑞)
𝑐𝑗 𝑒2𝜋𝑖𝑗𝑥/𝑎

(2.9)

𝑗

where we have chosen plane waves that are harmonics of the lattice spacing 𝑎, which
enforces the translation symmetry of 𝜑𝑞 (𝑥) . In this basis, 𝐻𝑞 can be written as a
tri-diagonal matrix with the elements

𝐻𝑞;𝑗,𝑗 = (2𝑗 + 𝑞/𝑘𝐿 )2 𝐸 + 𝑉𝐿 /2
R(2.10)

𝐻𝑞;𝑗,𝑗±1 = − 𝑉𝐿 /4
and all other elements 0. We have defined the recoil energy scale 𝐸R = ~2 𝑘𝐿2 /2𝑚.
By numerically diagonalizing 𝐻𝑞 for each 𝑞 , we can determine the bandstructures asshown in Figure 2-1. We find that using a maximum |𝑗| of 15 provides good results
for the lowest three bands.

An intuitive way to understand the off-diagonal terms is most obvious in an opticallattice, as opposed to a crystal lattice. In Section 2.1, we described the AC Stark
shift in second-order pertubation theory where we ignored atomic motion. We canread the term 𝐸 2 |𝑀𝑔𝑖 |2 /𝛿 as the following proces: A ground state atom accepts a

photon from the laser field with matrix element 𝐸𝑀𝑔𝑖 , and is virtually excited into
state 𝑖 with an energy deficit of 𝛿 . Then, the atom is de-excited back to the ground22

V L/E R = 2

V L/E R = 10

10

8

6

6

6

4

4

4

E/E R

8

2

2

2

0

0

0

-2

-2

-2

-4

-4

-4

-6

-6

-6

-1

0

1

Quasimomentum ( : /a)-1

0

V L/E R = 20

10

8

E/E R

E/E R

10

1

Quasimomentum ( : /a)

-1

0

1

Quasimomentum ( : /a)Figure 2-1: Bandstructure of an atom in an optical lattice. Here we plot the lowest
three bands as a function of quasimomentum at different lattice depths. Note thatat low lattice depths, the bands almost follow the free particle parabolic dispersion,
with small avoided crossings. At higher lattice depths, the bands become flatter andthe bandgaps approach the harmonic oscillator frequency for a single lattice site.
*.
state by emitting a photon into the laser field, this time with matrix element 𝐸𝑀𝑔𝑖

Now in the real system, we need to take momentum conservation into account, and in
an optical lattice composed of two retro-reflected laser beams, there are two options.
First, if the emitted photon is in the same beam as the exciting photon, then theatom does not pick up any momentum. This is where the 𝑉𝐿 /2 on the diagonal of 𝐻𝑞
comes from. On the other hand, the atom could accept a photon from the input beamand emit into the retro-reflected beam, or vice versa, which will give it a momentum
kick of 2~𝑘𝐿 . This takes 𝑗 → 𝑗 ± 1, and gives the off-diagonal terms of 𝐻𝑞 . It is theseterms that give rise to the avoided crossings seen in Figure 2-1.

2.2.2 Wannier Functions
Once we have solved for the bandstructure, we can use the Bloch functions to define
another set of orthogonal basis states, which are equivalent to position eigenstates inthe same way that Bloch states are the lattice equivalent of momentum eigenstates.
These are known as the Wannier functions [43] and for a state localized at lattice site23

1
V /E = 0.5
L0.8

V L /E R = 10

0.6

A

R

V L /E R = 5

0.4
0.2
0
-0.2
-4

-2

0

2

4

x/a

Figure 2-2: Maximally-localized Wannier functions in the lowest band. We havechosen a phase convention so that the wavefunction is purely real, and have plotted
the maximally localized Wannier functions at varying lattice depths. At low lattice
depths, the wavefunction has significant weight outside of the lattice site, but at higher
lattice depths, it rapidly approaches the ground state wavefunction of the harmonic
oscillator.

𝑥𝑖 are given by
𝑤𝑛 (𝑥 − 𝑥𝑖 ) =√∫︁ 𝜋/𝑎𝑁(2.11)−𝑖𝑞𝑥𝑖 /~
𝜑(𝑛)𝑑𝑞,
𝑞 (𝑥) 𝑒−𝜋/𝑎(𝑛)where 𝑛 is the band index and 𝑁 is a normalization constant. Note that the 𝜑𝑞 (𝑥) areonly defined up to a phase, and only one phase convention will provide the maximally
localized functions that we want. This can be done by direct minimization [44] orby using the phase convention in [43]. For the ground band and even excited bands,(𝑞)maximally localized Wannier functions are produced by choosing all 𝑐𝑗 to be purely(𝑞)real. For odd excited bands, we choose all 𝑐𝑗

to be purely imaginary. In Figure

2-2, I have plotted the normalized ground band Wannier functions at various lattice
depths. Note that at low lattice depths, the Wannier function has some weight outside
the center well, but these parts quickly decay with increasing lattice depth, and theWannier functions rapidly approach the eigenstates of the Harmonic oscillator.
24

2.3 Bose-Hubbard Hamiltonian
In a deep lattice, i.e. the single band approximation, we can write our Hamiltonian
in terms of the Wannier functions

𝐻 = −𝑡∑︁ †𝑎𝑖 𝑎𝑗(2.12)⟨𝑖,𝑗⟩

where 𝑎𝑗 is the annihilation operator for the lowest band Wannier function on site 𝑗 ,and ⟨𝑖, 𝑗⟩ denotes pairs of nearest neighbors. We find the tunneling parameter, 𝑡, bylooking at the matrix element between neighboring Wannier functions

∫︁
𝑡=

*

(︂

𝑑⃗𝑟 𝑤(⃗𝑟 − ⃗𝑟0 ))︂
𝑝2
2+ 𝑉𝐿 𝑐𝑜𝑠 (𝑘𝐿 𝑥) 𝑤(⃗𝑟 − ⃗𝑟1 ) .
2𝑚(2.13)The system becomes more interesting when we add interactions between the atoms
on a given site. Doing so gives us the Bose-Hubbard Hamiltonian [12]𝐻 = −𝑡∑︁ †𝑈 ∑︁𝑎𝑖 𝑎𝑗 +𝑛𝑖 (𝑛𝑖 − 1)
2 𝑖

(2.14)⟨𝑖,𝑗⟩with the number operator 𝑛𝑖 = 𝑎†𝑖 𝑎𝑖 as usual. The interaction parameter 𝑈 is givenby the squared on-site density

4𝜋~2 𝑎𝑠𝑈=𝑚∫︁𝑑⃗𝑟 𝑤(⃗𝑟 − ⃗𝑟0 )*2 𝑤(⃗𝑟 − ⃗𝑟0 )2 ,(2.15)

where 𝑎𝑠 is the s-wave scattering length.
The Bose-Hubbard Hamiltonian is the starting point for all of the research done inthis thesis, and one of its main advantages is how easy it is to modify to take into account other potentials, interactions, or tunneling types. The most common extension
is that of an external confining potential and the local density approximation. For an∑︀external potential 𝑉𝑒𝑥𝑡 (⃗𝑟), we add a term to the Hamilonian of the form 𝑖 𝑉𝑒𝑥𝑡 (⃗𝑟𝑖 ) 𝑛𝑖 ,∑︀and to take into account the chemical potential, we add a term −𝜇 𝑖 𝑛𝑖 . We can25

combine these two terms in the “local density approximation”

𝐻 = −𝑡

∑︁ †
∑︁
𝑈 ∑︁
𝑛𝑖 (𝑛𝑖 − 1) −𝑎𝑖 𝑎𝑗 +
𝜇𝑖 𝑛𝑖
2 𝑖
𝑖

(2.16)⟨𝑖,𝑗⟩with 𝜇𝑖 = 𝜇 − 𝑉𝑒𝑥𝑡 (⃗𝑟𝑖 ).

2.3.1 Superfluid to Mott Insulator Transition
Now let us look at the phases in a 𝑑-dimensional isotropic lattice with one spincomponent. There are two clear extremes: the first is known as the superfluid phase,and the second as the Mott insulator, both of which are accessible in optical lattices[13, 14].
The superfluid phase occurs when the Hamiltonian is dominated by −𝑡†⟨𝑖,𝑗⟩ 𝑎𝑖 𝑎𝑗∑︀and has all atoms in the same Bloch state, sitting at the bottom of the lowest band.
In the mean-field, the superfluid has an order parameter⟨ ⟩ √𝜓 = ⟨𝑎𝑖 ⟩ = 𝑎†𝑖 = 𝑛𝑖 .

(2.17)At the other extreme, the Hamiltonian is dominated by the interaction term

𝐻=

∑︁
𝑈 ∑︁
𝑛𝑖 (𝑛𝑖 − 1) −
𝜇𝑖 𝑛𝑖
2 𝑖
𝑖

(2.18)

and becomes a sum of purely local energy terms. This leads to a ground state known
as the Mott insulator with exactly 𝑁𝑖 = ⌊(𝜇/𝑈 )⌋ + 1 atoms on each site 𝑖.

To find the phase transition from a superfluid to a Mott insulator, we first write

the Hamiltonian for a single lattice site in the mean field approximation [13](︂
𝐻 = 𝑧𝑡

(︁
)︁
𝜇𝑖
𝑈
†
𝑛𝑖 (𝑛𝑖 − 1) − 𝑛𝑖 − 𝑎𝑖 + 𝑎𝑖 𝜓 + 𝜓 2
2𝑧𝑡
𝑧𝑡

)︂

(2.19)

where 𝑧 = 2𝑑 is the number of nearest neighbors of a given site. According toLandau’s theory of phase transitions, a second order phase transition from 𝜓 = 0 (aMott insulator) to non-zero 𝜓 (a superfluid) will occur when 𝑑2 𝐸/𝑑𝜓 2 = 0. Therefore,26

we will work in the low 𝑡/𝑈 limit, splitting the Hamiltonian into the zeroth order andperturbative parts as

𝑈
𝐻0 = 𝑛𝑖 (𝑛𝑖 − 1) − 𝜇𝑖 𝑛𝑖
2 (︁(︁
)︁)︁
†2
𝑉 =𝑧𝑡 𝑎𝑖 + 𝑎𝑖 𝜓 + 𝜓 .

(2.20)The zeroth-order energy is then just𝑈
𝑁𝑖 (𝑁𝑖 − 1) − 𝜇𝑁𝑖2

𝐸 (0) =

(2.21)

with 𝑁𝑖 defined as above. The second order energy term is found to be

𝐸

(2)

= 𝑧𝑡𝜓

2

(︂

𝑁𝑖 𝑧𝑡
𝑧𝑡 (𝑁𝑖 + 1)+
𝑈 (𝑁𝑖 − 1) − 𝜇𝑖
𝜇 − 𝑈 𝑁𝑖)︂

(2.22)

which leads to a phase boundary at

1
𝜇𝑖
=
𝑈
2

(︂

𝑧𝑡
2𝑁𝑖 − 1 −
𝑈

)︂

1
±
2

√︃

2𝑧𝑡
1−
(2𝑁𝑖 + 1) +
𝑈

(︂

𝑧𝑡
𝑈

)︂2
.

(2.23)

Figure 2-3 shows the phase diagram for a three-dimensional isotropic system. The
Mott insulating lobes are clearly visible and in order of increasing 𝜇𝑖 are associatedwith 𝑁𝑖 = 1, 2, 3, . . . , and as expected, decrease in size as we increase 𝑡/𝑈. Since ouratoms are held in an external harmonic trap, 𝜇𝑖 varies from 0 to 𝜇 across the cloud,and so any given cloud will sample a vertical line in Figure 2-3a. Since we know theform of 𝜇𝑖 , though, we can calculate which Mott shells are present in our system andwhat their relative weight is. For some work done in this thesis, it is important that
we fill the 𝑁𝑖 = 1 or 𝑁𝑖 = 2 shell and not have any atoms in 𝑁𝑖 = 3 or higher. We
use the above work to calculated our expected maximum filling, and then confirm it
via clock shift spectroscopy [45].

2.3.2 Wannier-Stark Ladder
Another useful extension of the Bose-Hubbard Hamiltonian describes particles in atilted lattice, which is used as an integral part of two of the projects in this thesis.
27

(a)

(b)

3

μ and Vext

Ni = 3

2.5

μi/U

2

μi

Ni = 2

1.5
1

Ni = 1

0.5
0

0

Ni
0.05

0.1

0.15

0.2

zt/U
Figure 2-3: Phase diagram of the SF to MI transition and the LDA. (a) Calculatedmean-field phase diagram of the superfluid to Mott insulator transtion, up to a filling
factor of three. Each lobe is labeled with the corresponding filling factor, and thearea outside the lobes is in the superfluid state. (b) Cartoon of the local density
approximation. The cloud has a constant chemical potential 𝜇 and is held in an
external trapping potential 𝑉𝑒𝑥𝑡 . Then for each lattice site, we find 𝜇𝑖 = 𝜇 − 𝑉𝑒𝑥𝑡 .
Once 𝜇𝑖 is known, we can use Figure 2-3a to determine the filling at each lattice site
𝑁𝑖 .

28

1
"/t = 2"/t = 5"/t = 20

A

0.5

0

-0.5
-4

-2

0

2

4

x/a

Figure 2-4: Wannier-Stark wavefunctions at varying tilts. Here, we plot the wavefunction of Wannier-Stark states in a lattice of 10 ER with varying tilt-to-tunnelingratios ∆/𝑡. (As in Figure 2-2, the phase convention is chosen so that the wavefunctions are purely real.) The wavefunctions all have some component on neighboring
lattice sites, but this is strongly suppressed when ∆ ≫ 𝑡.
If we define the energy tilt per lattice site ∆, the single-particle Hamiltonian is given
by [46]𝐻=−)︁∑︁ (︁ †
∑︁ †𝑡 𝑎𝑙 𝑎𝑙+1 + 𝐻.𝐶. + ∆𝑙𝑎𝑙 𝑎𝑙 .
𝑙(2.24)𝑙

For simplicity, we define 𝛼 = 𝑡/∆ and 𝜖 = 𝐸/𝑡. If we write the wavefunction in the∑︀basis of Wannier states |𝜓⟩ = 𝑙 𝑐𝑙 𝑎†𝑙 |0⟩ ,𝐸 |𝜓⟩ =𝐻 |𝜓⟩]︃[︃)︁ 1 ∑︁∑︁ (︁ †††
𝜖 |𝜓⟩ = −𝑎𝑙 𝑎𝑙+1 + 𝑎𝑙+1 𝑎𝑙 +𝑙𝑎𝑙 𝑎𝑙
𝛼 𝑙
𝑙
)︁
∑︁ †
∑︁ (︁
𝜖𝛼
𝑐𝑗 𝑎𝑗 |0⟩ =𝑐𝑗 −𝛼𝑎†𝑗−1 − 𝛼𝑎†𝑗+1 + 𝑗𝑎†𝑗 |0⟩ .
𝑗(2.25)

𝑗

Then by detailed balance,𝜖𝛼𝑐𝑗 = − 𝛼 (𝑐𝑗+1 + 𝑐𝑗−1 ) + 𝑗𝑐𝑗
𝑐𝑗+1 + 𝑐𝑗−1 =𝑗 − 𝜖𝛼𝑐𝑗 .
𝛼29(2.26)

This matches the Bessel function recurrence relation:

𝑍𝑛+1 + 𝑍𝑛−1 =

2𝑛𝑍𝑛 (𝑥)
𝑥(2.27)and so the eigenstate coefficients are

𝑐𝑙 = 𝐴𝐽𝜖𝛼−𝑙 (2𝛼) + 𝐵𝑌𝜖𝛼−𝑙 (2𝛼)(2.28)

where 𝐽𝑙 and 𝑌𝑙 are Bessel functions of the first and second kind, respectively. In
general, 𝐴 and 𝐵 are given by the boundary conditions. For an infinite lattice, we
can set 𝐵 = 0 to get localized states, which we call Wannier-Stark states. For 𝜖 = 0,
the eigenstate takes the form

|𝜓𝜖=0 ⟩ =

∑︁

(︂
𝐽−𝑙

𝑙

2𝑡
∆

)︂

𝑎†𝑙 |0⟩ .

(2.29)⃒ (︀ )︀⃒2⃒.
I.E. It is centered on site 0, and sites 𝑙 away from the center have weights ⃒𝐽𝑙 2𝑡
Δ
If we change the energy by ∆ then the Wannier-Stark state is simply translated onesite left or right. In the case of large ∆ > 𝑡, tunneling is suppressed by energy
conservation and the states become more localized about a single site. Figure 2-4
shows the probability distribution associated with the Wannier-Stark state centeredabout site 0 at several different tilt values.

30

Chapter 3
Simulating Strong Magnetic Fields
Topological states of matter are an active area of research in physics, and are strongly
associated with high magnetic fields. While topological single-particle physics is wellunderstood, there are many open questions about the interaction of topological materials with strong interactions [47, 48], with connections to the fractional quantum
Hall effect [49] and Majorana fermions [50, 51, 52].
In neutral atoms, synthetic magnetic fields have previously been created throughuse of the Coriolis force in rotating systems [53, 5, 54], Berry’s phases in multiple
hyperfine states [18, 19, 20, 21, 22], or lattice shaking [23, 24]. Much recent work has
focused on the Harper-Hofstadter Hamiltonian, which describes particles in a latticesubject to a magnetic field strong enough that the cyclotron orbit length becomescomparable to the lattice spacing [55, 56, 57]. The comparable length scales leadto the fractal structure of Hofstadter’s butterfly, which has sub-bands with non-zeroChern numbers [58]. Previous work on uniform magnetic fields has either been limitedin the magnitude of the synthetic field possible or plagued by heating. Bose-Einsteincondensation has previously been achieved only in staggered flux configurations [59,23] or small ladder systems [60, 61].
In this chapter, I present the first observation of Bose-Einstein condensation in
the bulk Harper-Hofstadter Hamiltonian. I first explain some of the interesting properties of the Harper-Hofstadter Hamiltonian and describe how we use laser-assisted
tunneling to imprint the proper phase on the wavefunction. I then describe the ex31

perimental sequence and upgrades that were required to achieve a cold superfluidin the Harper-Hofstadter Hamiltonian. Our observations directly show the broken
translational symmetry of the vector potential, despite the uniform magnetic field,
and demonstrate the non-gauge invariant momentum distribution and degeneracy of
the ground band [62, 63, 64]. We observe the result of the addition of interactions
and make a Harper superfluid near the Mott Insulating transition. Finally, I end witha discussion of the observed lifetime and some heating processe that may be relevant.
This chapter covers material from [65].

3.1 Harper-Hofstadter Hamiltonian
A two-dimensional system of charged particles in both a lattice potential and a hightransverse magnetic field are described by the Harper-Hofstadter Hamiltonian [55,56, 57]

𝐻=

∑︁ (︁
𝑚,𝑛

)︁
−𝐾 𝑒−𝑖𝜑𝑚,𝑛 𝑎ˆ†𝑚+1,𝑛 𝑎
ˆ𝑚,𝑛 − 𝐽 𝑎ˆ†𝑚,𝑛+1 𝑎
ˆ𝑚,𝑛 + H.c.

(3.1)

where 𝐾 and 𝐽 are real tunneling coefficients and 𝑎𝑚,𝑛 and 𝑎†𝑚,𝑛 are the annihiliationand creation operators for site (𝑚, 𝑛). 𝜑𝑚,𝑛 is a spatially dependent phase1
𝜑𝑚,𝑛 =~∫︁ 𝑥𝑚+1𝑥𝑚⃗ · 𝑑⃗𝑟𝐴(3.2)⃗ . For convenience, we choose awith the charge included in the vector potential 𝐴
gauge which has no component in the 𝑦 direction. We want the vector potential tocorrespond to a constant magnetic field in the 𝑧ˆ direction, so we say⃗ =∇×𝐴⃗𝐵
𝐵0 𝑧ˆ = ∇ × 𝐴ˆ
𝑥
=

(3.3)

𝜕
𝜕
𝐴ˆ
𝑦−
𝐴ˆ
𝑧.
𝜕𝑧
𝜕𝑦

⃗ must be (−𝐵𝑦 + 𝑓 (𝑥)) 𝑥ˆ, where 𝑓 (𝑥) is an arbitrary
This tells us that the form of 𝐴
function of 𝑥 only. This is an example of gauge freedom, and the choice of 𝑓 (𝑥) does32

not change the physics of the system. Note that we later discuss some experimental
results that appear to be gauge dependent, but they rely on the specifics of thesimulation that we use and are not present in the real system discussed here. Most
theoretical work on the Harper-Hofstadter Hamiltonian has used the Landau gauge⃗ = −𝐵𝑦ˆ
𝐴𝑥, which we will use in this section.

For intuition about the form of the Hamiltonian, first recall the Aharanov-Bohm
effect, which is most easily explained by looking at a closed adiabatic path 𝑃 . The
definition of adiabaticity tells us that applying a closed-loop adiabatic transformation
leaves the probability distribution the same |𝜓𝑖 (𝑥)|2 −→ |𝜓𝑓 (𝑥)|2 = |𝜓𝑖 (𝑥)|2 , which𝑃

implies that the only possible change to 𝜓𝑖 (𝑥) is a phase: 𝜓𝑓 (𝑥) = exp[𝑖𝜑AB ] 𝜓𝑖 (𝑥) .
This phase is the Aharanov-Bohm phase, given by

𝜑

AB

1
=
~

∮︁

⃗ · 𝑑⃗𝑟 = Φ ,
𝐴
~
𝑃

(3.4)

where Φ is the enclosed magnetic flux. Now we look at a particle described by theHarper-Hofstadter Hamiltonian in Equation 3.1 that is initially localized at (0, 0),
and then takes a counterclockwise path (0, 0) → (1, 0) → (1, 1) → (0, 1) → (0, 0).

The 𝑦 legs of this path are mediated by real tunneling, and so add no phase to the

wavefunction. On the other hand, the forward move from (0, 0) → (1, 0) and the
backward move from (1, 1) → (0, 1) add a phase of 𝜑0,0 − 𝜑0,1 . In the Landau gauge,∫︁
1 𝑥𝑚+1𝜑𝑚,𝑛 =−𝐵𝑦ˆ
𝑥 · 𝑑⃗𝑟
~ 𝑥𝑚
−𝐵𝑛𝑎2
=
~
−Φ𝑛
=
,
~

(3.5)

and thus, 𝜑0,0 − 𝜑0,1 = Φ~ = 𝜑AB . From here, it is simple to see that this is true forevery closed path in the Harper-Hofstadter Hamiltonian.
33

3.1.1 Hofstadter’s Butterfly
The most striking property of the Harper-Hofstadter Hamiltonian is that it leads toa fractal bandstructure known as Hofstadter’s Butterfly, shown in Figure 3-1 for asymmetric lattice 𝐾 = 𝐽 . We define the normalized flux 𝛼 = 𝐵𝑎2 /Φ0 , or the ratio of
the flux enclosed in a single lattice unit cell to the magnetic flux quantum Φ0 = ℎ/𝑒,
which gives a phase of 2𝜋 when a particle circles it. In a normal material, with𝑎 ∼ 1 Å, we would need a magnetic field of ∼ 10, 000 T to have 𝛼 = 1.
Let us first build some intuition about the form of Hofstadter’s butterfly. At
zero flux, the unit cell is simply given by the lattice, and so the lowest band covers
the normal Brillouin zone, with an energy ranging from −4𝐽 to 4𝐽 . At 𝛼 = 1, anyclosed path that a particle takes encloses an integer number of flux quanta, and so
the particle picks up a phase that is a multiple of 2𝜋 , which is indistinguishable fromzero. We can therefore say that Hofstadter’s butterfly should be the same at 𝛼 and𝛼 + 1. A system at −𝛼 is the same as a system with 𝛼 and 𝑧ˆ → −ˆ𝑧 , so the associatedenergies should be the same. Combining the two previous symmetries tells us thatthe butterfly is symmetric about 𝛼 = 1/2. Finally, for rational fluxes of the form

𝛼 = 𝑝/𝑞 for 𝑝, 𝑞 ∈ Z, we can understand the structure of the subbands. If 𝑝 and𝑞 are relatively prime, then a particle must enclose 𝑞 lattice unit cells to enclose an
integer number of flux quanta. We call this larger cell the magnetic unit cell, andit provides the translation symmetry for the system. Since it is 𝑞 times bigger thanthe lattice unit cell, the new Brillouin zone is then 𝑞 times smaller, and the groundband is split into 𝑞 subbands. In Figure 3-1, this is obvious for 𝛼 = 1/3 and 1/4.
For 𝛼 = 1/2, which is the magnetic field that we have simulated in this chapter, afull bandstructure calculation shows that there are in fact two subbands, which areconnected at Dirac points.
For a more quantitative description of Hofstadter’s butterfly, we follow his original
paper [55, 56, 57], which begins by treating the wavefunction as a sum over Wannier34

Flux, α10-4Eigenenergy, E/J

4

Figure 3-1: The fractal spectrum of the Harper-Hofstadter Hamiltonian, known as
Hofstadter’s butterfly. The vertical axis is the flux per unit cell, or 𝛼, and the horizontal axis is energy. Each filled point represents an eigenstate of the system undera flux 𝛼 with a corresponding eigenenergy.
functions localized at each lattice site 𝑚, 𝑛,

|𝜓⟩ =

∑︁
𝑚,𝑛

𝑐𝑚,𝑛 |𝑚, 𝑛⟩ .

(3.6)

⃗ = −𝐵𝑦ˆ
Using the Landau gauge, 𝐴
𝑥 = −𝐵𝑛𝑎ˆ
𝑥 → 𝜑𝑚,𝑛 = −2𝜋𝛼𝑛, we find thefollowing relation between coefficients:

𝐸
𝑒𝑖2𝜋𝛼𝑛 𝑐𝑚+1,𝑛 + 𝑒−𝑖2𝜋𝛼𝑛 𝑐𝑚−1,𝑛 + 𝑐𝑚,𝑛+1 + 𝑐𝑚,𝑛−1 = − 𝑐𝑚,𝑛 .
𝐽

(3.7)

We make the ansatz that 𝑐𝑚,𝑛 is separable into exp [𝑖𝑘𝑥 𝑎𝑚] 𝑔(𝑛), and then find arecursion relation𝐸
2 cos(2𝜋𝛼𝑛 + 𝑘𝑥 𝑎) 𝑔(𝑛) + 𝑔(𝑛 + 1) + 𝑔(𝑛 − 1) = − 𝑔(𝑛) .
𝐽

(3.8)

Then to numerically calculate the Harper Hamltonian as in Figure 3-1, we assume
a lattice of sufficiently large size 𝑁 with periodic boundary conditions. For rational35

𝛼 ∈ [0, 1] and 𝑘𝑥 𝑎 ∈ [−𝜋, 𝜋], we plot the eigenenergies, revealing the fractal structureof Hofstadter’s butterly.

3.2 Raman tunneling
The Harper-Hoftsadter Hamiltonian normally describes charged particles in a highmagnetic field, but we are working with neutral bosons of 87 Rb. Equation 3.1 tells usthat the dynamics that we are interested in fundamentally come not from the charge,
but from the phase pattern imprinted during tunneling. Enforcing real tunneling isrelatively easy — we just put our atoms in a two-dimensional optical lattice. However,adding a complex phase to the tunneling coefficient is more involved.
In order to remove the real tunneling coefficient along the 𝑥 direction, we apply
a strong energy gradient, as shown in the schematic Figure 3-2. The tilt per latticesite ∆ must be greater than the bandwidth of the untilted system, but smaller thanthe bandgap. Then energy conservation freezes out normal tunneling. We can restore
tunneling by use of a pair of Raman lasers with a frequency difference 𝛿𝜔 = ∆. Then,by accepting a photon from one laser and emitting into the other, an atom can hop
down one site and still conserve energy. When it does so, if the Raman lasers are notcollinear, the atom must also pick up a momentum kick 𝛿⃗𝑘 = ⃗𝑘1 − ⃗𝑘2 , which is what

determines the phase of our laser-induced tunneling. To provide intuititon for how

a momentum kick turns into a complex tunneling element, recall that a momentum[︁]︁⃗
eigenstate has the form 𝜓𝑘 = exp 𝑖𝑘 · ⃗𝑟 , i.e., a spatially dependent phase. Insteadof explicitly thinking of momentum conservation for each tunneling event, we canfactor it into the tunneling term 𝐾 exp [𝑖𝜑𝑚,𝑛 ], where

𝜑𝑚,𝑛 = 𝛿⃗𝑘 · ⃗𝑟𝑚,𝑛(3.9)= 𝑚 𝛿𝑘𝑥 𝑎 + 𝑛 𝛿𝑘𝑦 𝑎.
This gives us an effective flux of 𝛼 = 𝛿𝑘𝑦 𝑎/2𝜋, which is tunable simply by changing
the angle between the two Raman lasers.
For our specific implementation, our Raman lasers are far-detuned from the atomic
36

Figure 3-2: Schematic of our laser assisted tunneling setup. (a) We suppress normal
tunneling in the 𝑥 direction by applying a strong tilt ∆. It is then re-established by a
two-photon process with two Raman beams whose frequencies are offset by 𝛿𝜔 = ∆.
(b) The Raman lasers are at an angle to each other, and so impart momentum witheach 𝑥-tunneling event, which is equivalent to a complex tunneling parameter. The
lattice is untilted along 𝑦 , and so has purely real tunneling. (c) Phase pattern of thetunneling. In our gauge, 𝜑𝑥 = 𝜑𝑦 = 𝜋.

37resonance. This allows us to ignore light scattering, and should lead to long lifetimes[21, 22]. In addition, it allows us to treat the effect of our Raman beams as a classicalexternal potential. The interference of the two Raman lasers gives a “running stand(︁
)︁
𝑘𝑥 𝑎
⃗
ing wave” with an intensity proportional to sin 𝛿 𝑘 · ⃗𝑟 − 2 − ∆𝑡 , oscillating at thedifference of the two frequencies and moving in the direction of 𝛿⃗𝑘 . (Here, the term𝑘𝑥 𝑎is purely for convenience and does not change the physics.) Next, we can quanti2

tatively analyze the physical system and show that it maps to the Harper-Hofstadter
Hamiltonian. The physical singe-particle Hamiltonian is as follows:

𝑝⃗2
𝐻=
+𝑉
2𝑚

latt

(︂
)︂
∆
𝑘
𝑎
𝑥
(⃗𝑟) − 𝑥 + Ω sin 𝛿⃗𝑘 · ⃗𝑟 −− ∆𝑡 ,𝑎2(3.10)

where 𝑉latt is the normal two-dimensional lattice potential, and Ω is the AC Stark
shift associated with the Raman lasers. If we restrict ourselves to the first band, we
can write the Hamiltonian in a localized basis of Wannier functions in the 𝑦 -direction
and Wannier-Stark functions in the 𝑥 direction:

𝐻=

∑︁[︂
𝑚,𝑛

− 𝑚∆ |𝑚, 𝑛⟩ ⟨𝑚, 𝑛| − 𝐽 |𝑚, 𝑛 + 1⟩ ⟨𝑚, 𝑛| + ℎ.𝑐.

(︂
)︂
]︂
𝑘𝑥 𝑎
⃗
Ω |𝑚 , 𝑛 ⟩ ⟨𝑚 , 𝑛 | sin 𝛿 𝑘 · ⃗𝑟 −− ∆𝑡 |𝑚, 𝑛⟩ ⟨𝑚, 𝑛| .
2
′
′
𝑚 ,𝑛
∑︁

′

′

′

(3.11)′

To calculate the Raman matrix element, we shift the center and define 𝜃𝑚,𝑛 = ∆𝑡 −

𝛿⃗𝑘 · ⃗𝑟𝑚,𝑛 = ∆𝑡 − 𝜑𝑚,𝑛 :

(︂
)︂
(︂
)︂
𝑘
𝑎
𝑘
𝑎
𝑥
𝑥
⟨𝑚, 𝑛| sin 𝛿⃗𝑘 · ⃗𝑟 −− ∆𝑡 |𝑚 + 𝑙, 𝑛 + 𝑝⟩ = ⟨0, 0| sin 𝛿⃗𝑘 · (⃗𝑟 + ⃗𝑟𝑚,𝑛 ) −− ∆𝑡 |𝑙, 𝑝⟩ .
22
(3.12)
We split the sin function, and in the tight binding limit, we have the following rela-38

tions:⟨0| sin(𝑘𝑦 𝑦) |𝑝 = 0⟩ = 0⟨0| cos(𝑘𝑦 𝑦) |𝑝 = 0⟩ = 1⟨0| sin(𝑘𝑥 𝑥) |𝑙 = 0⟩ = 0(3.13)⟨0| cos(𝑘𝑥 𝑥) |𝑙 = 0⟩ = 1⟨0| sin(𝑘𝑥 (𝑥 − 𝑎/2)) |𝑙 = 1⟩ = Φ𝑥 ≈ −2𝐽

sin(𝑘𝑥 𝑎/2)∆

⟨0| cos(𝑘𝑥 (𝑥 − 𝑎/2)) |𝑙 = 1⟩ = 0.
Plugging these into the Hamiltonian gives us:

𝐻=

∑︁ [︂
𝑚,𝑛

(−𝑚∆ − Ω sin(𝜃𝑚,𝑛 + 𝑘𝑥 𝑎/2)) |𝑚, 𝑛⟩ ⟨𝑚, 𝑛|]︂+ ΩΦ𝑥 cos𝜃𝑚,𝑛 |𝑚 + 1, 𝑛⟩ ⟨𝑚, 𝑛| − 𝐽 |𝑚, 𝑛 + 1⟩ ⟨𝑚, 𝑛| + ℎ.𝑐. .

(3.14)

Finally, we move into the rotating frame given by(︂)︂)︂]︃∑︁ (︂Ω𝑘𝑥 𝑎cos 𝜃𝑚,𝑛 +|𝑚, 𝑛⟩ ⟨𝑚, 𝑛| ,𝑈 = exp 𝑖
𝑚∆𝑡 −
~∆
2
𝑚,𝑛
[︃

(3.15)

which cancels the diagonal terms, and gives off-diagonal terms:⟨𝑚 + 1, 𝑛| 𝐻 |𝑚, 𝑛⟩ =ΩΦ𝑥 cos(∆𝑡 − 𝜑𝑚,𝑛 ) 𝑒⟨𝑚, 𝑛 + 1| 𝐻 |𝑚, 𝑛⟩ = − 𝐽

∑︁
𝑟

(︂
𝐽𝑟

∑︁

−𝑖𝜔𝑡

𝑟

(︂
𝐽𝑟

(︂
)︂)︂
𝑘𝑥 𝑎
2Ω
sin
𝑒𝑖𝑟(Δ𝑡−𝜑𝑚,𝑛 )
~∆
2

(︂
)︂)︂
2Ω
𝑘𝑦 𝑎
sin
𝑒𝑖𝑟(Δ𝑡+(𝑘𝑥 −𝑘𝑦 )𝑎/2−𝜑𝑚,𝑛 ) .
~∆2(3.16)

Here, we have used the identity 𝑒𝑖𝑥 sin 𝜃 =𝑖𝑟𝜃𝑟 𝐽𝑟 (𝑥) 𝑒 ,∑︀where 𝐽𝑟 are the Bessel

functions of the first kind. Averaging over a several ∆ periods, and using Bessel

39

function identities, we find(︂)︂)︂ (︂(︂)︂)︂−1
𝑘𝑥 𝑎
2Ω
𝑘𝑥 𝑎
2Ω
sin
sin
=Ω𝑒
Φ𝑥 𝐽1
~∆
2
~∆
2
(︂
(︂
)︂)︂
2Ω
𝑘𝑥 𝑎
= − 𝐽𝐽1
sin
∆
2
(︂
(︂
)︂)︂
2Ω
𝑘𝑦 𝑎
𝐽 = − 𝐽𝐽0
sin
.
~∆
2

−𝑖𝜑𝑚,𝑛𝐾𝑒−𝑖𝜑𝑚,𝑛(︂(3.17)

It is important to note here that, although we have previously stated that changing 𝑘𝑥
merely leads to a gauge transformation, and does not change 𝛼, its value is important
here. In the Landau gauge, with 𝑘𝑥 = 0, the magnitude of the tunneling matrix
element in the 𝑥-direction would be exactly zero. A non-zero 𝑘𝑥 is necessary to theoverlap integral that allows 𝑥-tunneling to happen!

3.3 Experiment
For our specific experiment, we use a 1064 nm laser to make the optical lattices,leading to either a two- or three-dimensional lattice potential with a spacing of 532
nm. For the Raman lasers, we use the same 1064 nm source, and we send one beamalong each lattice direction, which provides a momentum kick with 𝛿𝑘𝑦 𝑎 = 𝛿𝑘𝑥 𝑎 = 𝜋,leading to an effective flux of 𝛼 = 1/2. We attempted several experimental sequences,which I will discuss below, all of which follow the same basic formula. We start witha BEC in an optical lattice before applying a tilt and turning on resonant Raman
beams. After some hold time, we snap off all potentials and use time-of-flight imaging
to view the momentum distribution. The form of the superfluid peaks seen in the
momentum distribution should demonstrate the symmetries of the Harper-Hofstadter
Hamiltonian, and their visibility will tell us the ground state fraction.
For our initial attempts, we tried two different methods of creating a gradient.
First, we attempted to ramp on a magnetic field gradient of about 1 kHz per latticesite. However, because of the configuration of our magnetic gradient coils, this was aslow process, taking a few hundred milliseconds. During the gradient ramp time, the40

Figure 3-3: Demonstration of the amplitude of laser-assisted tunneling. Here we plot
the increase in the squared width of a cloud 1500 ms after turning off dipole confinement and turning on laser-assisted tunneling through the use of Raman beams. The
blue circles are the width in the tilt direction, and the red squares are in the transverse direction. The widths qualitatively agree with the Bessel function predictions
shown in the inset.
system had time to undergo Bloch oscillations and totally dephase.
A faster method was to quickly turn off the confining dipole traps. After thisramp-off, the atoms are then exposed to the full force of gravity, which is also around
1 kHz per lattice site. However, this method had several limitations. First, withthis method we were restricted to gradients less than or equal to the force of gravity,
leaving us much closer to the tunnelling rate of around 50 Hz than the bandgap near
10 kHz. Second, there is not an easy way to shim the gradient, meaning that if the 𝑥
lattice is not exactly aligned to true vertical, this will also induce Bloch oscillationsin the 𝑦 -direction. Finally, turning off the dipole traps imprints on the cloud not only
the gradient that we want, but also curvature that is the inverse of the initial dipole
confinement. This leads to the Raman beams moving in and out of resonance acrossthe cloud, creating unwanted Bloch oscillations in the system.
Because of these non-idealities, we were initially unable to load the BEC into
the ground state of the Harper-Hofstadter Hamiltonian. We were, however, able
to demonstrate laser-assisted tunneling in Figure 3-3. We did this by turning off
the dipole confinement at the same time as we turned on the tilt and laser-assisted
41

tunneling. We held for 1500 ms and measured the intrap width. We report 𝜎𝑓2 −

𝜎𝑖2 , where 𝜎𝑓,𝑖 are the initial and final intrap widths, because we expect diffusive

rather than ballistic expansion. As the Raman intensity is varied, we see qualitative
agreement in both the laser-assisted tunneling amplitude and the transverse tunneling
amplitude with the Bessel functions predicted in Equation 3.17.

3.3.1 Experimental Upgrades
In order to reach the ground state, we needed to make several upgrades to the experiment. The first two deal with problems alluded to above, while the third is of amore technical nature.

Fast Gradient Turn-on
In order to keep from having Bloch oscillations dephase during the tilt turn-on, wewant to to be able to turn the gradient on faster than a Bloch oscillation. This isdone using a hybrid gradient. First, we use a magnetic field gradient to levitate a
BEC in the |𝐹 = 1, 𝑚𝑓 = −1⟩ spin state against gravity. At this point, the BEC feels

a force of −𝑚𝑔 + 𝑚𝑔 = 0, from gravity and the magnetic field gradient respectively.

After ramping up the optical lattice, we perform a Landau-Zener RF sweep to flip all
atoms into the |2, −2⟩ state in 0.29 ms. The magnetic moment of |2, −2⟩ is −2 times

that of |1, −1⟩, so the atoms now feel a force of −𝑚𝑔 + −2 × 𝑚𝑔 = −3𝑚𝑔, which forour atoms and lattice spacing is 3.420 kHz per lattice site. This method of creatinga gradient gives us much higher gradients with much faster turn-on times.

Gradient Shimming
Now that we have a strong, fast gradient, it is important to ensure that it is exactlyaligned with the 𝑥-lattice to prevent slow Bloch oscillations and dephasing in thetransverse directions. To do this, we use amplitude modulation of the 𝑥-lattice as acalibration tool [66, 67]. First, we apply a tilt as described above. Instead of restoringtunneling using Raman lasers, we restore it by modulating the amplitude of the 𝑥42

lattice at the tilt frequency. It is possible to fully analyze the system as in Section 3.2,
but here I will give only an intuitive explanation. We can view a lattice modulatedat 𝛿𝜔 as two sidebands at ±𝛿𝜔 superposed on then normal lattice beam. If an atom

accepts a photon from one of the sidebands and emits one into the main lattice beam,
it will either move up or down one lattice site while conserving energy if 𝛿𝜔 = ∆.
This will restore real tunneling in the 𝑥-direction, and the lifetime of the superfluid inthis system is then limited by any projection of the gradient along transverse latticedirections. We optimize the superfluid lifetime and therefore minimize the transverse
gradient by using small shim coils along the two transverse lattice axes.

Phase Locking
The final major improvement of the experiment was to ensure the phase stability of
the Raman lasers, since any relative phase slips are equivalent to randomly shifting
the origin of the system. To provide passive stability, we sourced both Raman beams
from the same fiber, and split the two beams on the machine table. Both beams are
frequency shifted by identical 80 MHz acousto-optic modulators, set at 80 MHz and
80 MHz + ∆. After the beams propogate along the 𝑥- and 𝑦 -directions, they are recombined on another beam cube, and their interference is measured on a photodiode.
A lock-in amplifier set to ∆ measures the phase fluctuations caused by non-commonmode vibrations and gives an error signal that is fed back to the phase of the RF
drive of one of the acousto-optic modulators.

3.3.2 Final Experimental Sequence
The experiment begins with a nearly pure |1, −1⟩ BEC of ∼ 1 × 105 87 Rb atoms
confined in a crossed dipole trap. We turn on a magnetic field gradient to levitate theatoms against gravity. From here, there are two possible sequences, which we term
the “non-adiabatic” and “adiabatic” sequences, shown in Figure 3-4.
In the nonadiabatic sequence, the condensate is first adiabatically loaded into a
two-dimensional optical lattice in the 𝑥 (vertical) and 𝑦 (horizontal) directions, with43

50
20 Adiabatic
89 ± 16 ms

0Time-of-Flightz-Lattice (Er)x/y-Lattice (Er)Non-adiabatic
10 71 ± 8 ms

120

Time (ms)

160

2

0
2

Raman (Er)Time-of-Flight

Tilt

0

Figure 3-4: Experimental sequences for two different state preparation protocols. The
non-adiabatic scheme switches suddenly from a standard 2D lattice to the HarperHofstadter lattice. The adiabatic protocol uses a quantum phase transition to the
3D Mott insulator as an intermediate step. For this, lattice beams are adiabaticallyramped up in all three directions to 20 𝐸𝑟 to enter the Mott insulating phase, after
which the hyperfine state is flipped with a 0.29 ms RF sweep as done in the nonadiabatic scheme. The lattices are then ramped down to their final values while theRaman lattice is ramped up in 35 ms. Lifetimes of both methods are given in the
top corner of each figure; the lifetime of the adiabatic approach is measured from theend of the 35 ms ramp. Errors are statistical and given by the exponential fit to the
decay of the peak visibility.

44

a depth of 11 recoil energies (ER ) in each direction. The third direction is confined
only by the overall harmonic confinement, and so we have a two-dimensional array oftubes. At the same time, we raise a very weak Raman lattice (< 0.1ER ) and hold fora short time to allow phase lock to be achieved as discussed above.
Next, we apply a gradient in the method of Section 3.3.1 and linearly ramp theintensity of the Raman beams in 0.58 ms to 2Ω/∆ = 1.6, where Ω is the two-photon
Rabi frequency. We hold for a variable time before snapping off all laser beams to allow
a 20 ms time-of-flight and absorption imaging to reveal the momentum distribution.
The non-adiabatic sequence is rather violent and quenches the system from a
Hamiltonian where the ground state has a constant phase to one with a complex
phase pattern. It is surprising that we can even form a Harper superfluid at all withthis method, but we have found it to give consistently crisp superfluid peaks. Thiscan be explained by looking at the third dimension. When we take absorption images,
we automatically integrate over this dimension and therefore cannot see excitationsin the 𝑧 -direction. Since the system is only weakly trapped in 𝑧 , we should treat eachlattice site as a tube of about 500 atoms with gapless excitations along 𝑧 . When weapply the tilt and laser-assisted tunneling, we perform a quench and create excitationsin the phase between the tubes. As the system thermalizes, the 𝑧 -direction acts as abath and absorbs entropy created during the non-adiabatic sequence.
While this sequence works well for studying non-interacting physics, it does notallow a lattice in the third dimension, which is used to modify interactions and studying the strongly-interacting regime. In order to do this, we developed what we callthe “many-body adiabatic” sequence. In this sequence, before applying the tilt, we
ramp up a three-dimensional lattice to 20 ER in all directions. The system is thenin a Mott Insulator, where there is no phase coherence between the sites. In fact,
in the deep lattice limit of 𝑈 ≫ 𝐽, 𝐾 the ground state wavefunction for the HarperHofstadter Hamltonian is the same as in the standard Bose-Hubbard Hamiltonian.

Therefore, once we are in the deep Mott Insulator, we can apply the tilt and turnup the laser-assisted tunneling with the proper phase as quickly as we like without
adding entropy. Then we ramp the lattice depths down to the final values for the45experiment before holding and imaging as usual. We call this sequence “many-bodyadiabatic” because it is adiabatic at all points except one, where it is protected by
the many-body gap of the Mott Insulator.
We used both sequences in this work and found that the adiabatic sequence wasless robust against technical noise, leading to a higher shot-to-shot variability. Thisis likely because of the higher specific heat of a system with gapless excitations in one
direction. We do, however, find that the coherence lifetimes of the system after eachturn-on procedure are comparable.

3.4 Results
For our first experiment, we use the non-adiabatic sequence to prepare the Harper
superfluid and observe its momentum distribution in Figure 3-5h-j. The sharp peaks
are the hallmark of a superfluid BEC in a lattice potential and demonstrate that wehave prepared a low entropy state in the bulk Harper-Hofstadter Hamiltonian.
We first look at the pattern of peaks in Figure 3-5, which directly indicate thereduced symmetry of the Hamiltonian, despite the translational symmetry of theuniform magnetic field. The way to understand the pattern of peaks in the momentum
distribution of a superfluid begins with the real-space symmetries of the Hamiltonian.
In a square lattice with real tunneling in both directions, the Hamiltonian has thediscrete symmetry vectors 𝑎ˆ
𝑥 and 𝑎ˆ𝑦 and has a unit cell of size 𝑎 × 𝑎. In momentum

space, the symmetry vectors map to 2𝜋/𝑎 𝑘ˆ𝑥 and 2𝜋/𝑎 𝑘ˆ𝑦 , and the unit cell to a

Brillouin zone of size 2𝜋/𝑎 × 2𝜋/𝑎, as shown in Figure 3-5c and g. In our analysisof the Harper Hofstadter Hamltonian, Figure 3-5a, we must also take the tunnelingphase into account, which leads to two new unit cells — gauge and magnetic. For
our gauge, the new symmetry vectors are 2𝜋/𝑎 (ˆ𝑥 ± 𝑦ˆ) , and both the magnetic andgauge unit cells are twice the size of the bare lattice unit cell. The magnetic unit cellis gauge independent and is the smallest unit cell that contains an integer number offlux quanta [68]. For a magnetic flux 𝛼 = 𝑝/𝑞 , the magnetic unit cell is 𝑞 times larger
than the original unit cell and contains 𝑞 indistinguishable sites [64]. The gauge unit46

a)Real Space
2π

3π

π

2π

0

π

c) Trivial Superfluid d) Single Minimumb)π/a Momentum Space E/J g)

i)

h)

-0.2

e) Both Minima (0) f) Both Minima (π/2)

j)

-0.6

ky 0

y

-1.0

-π/a
-π/a

kx 0

π/a

-1.4

kL=2π/ax
0 Column Density (arb.) 1

Figure 3-5: Observation of Bose-Einstein condensation in the Harper-Hofstadtermodel. (a) Spatial structure of the cubic lattice with the synthetic vector potential– (dashed) x-bonds feature a spatially dependent tunneling phase, whereas tunnelingalong (solid) y-links is the normal tunneling. The synthetic magnetic field generates
a lattice unit cell that is twice as large as the bare cubic lattice (green diamond).
(b) The band structure of the lowest band shows a twofold degeneracy of the groundstate. The magnetic Brillouin zone (green diamond) has half the area of the original Brillouin zone. Due to the twofold degeneracy, the primitive cell of the band
structure is even smaller (doubly reduced Brillouin zone, brown square). These lattice symmetries are both revealed in time-of-flight pictures showing the momentum
distribution of the wavefunction. (c-f) Schematics of the momentum peaks of a superfluid. The dominant momentum peak (filled circle) is equal to the quasimomentum of
the ground state. Due to the spatial periodicity of the wavefunction, additional momentum peaks (empty circles) appear, separated by reciprocal lattice vectors (green
arrows) or vectors connecting degenerate states in the band structure (brown arrows).
(g-j) Time-of-flight images. The superfluid ground state of the normal cubic lattice
is shown in (g) in comparison to different repetitions of the same sequence for thesuperfluid ground state of the HH lattice (h-j). In (h), only one minimum of the band
structure is filled, directly demonstrating the symmetry in our chosen gauge. The
number of momentum components in (i-j) is doubled again due to population of bothdegenerate ground states. The micromotion of the Floquet Hamiltonian is illustratedin (e-f, i-j) as a periodically shifted pattern in the x-direction, analogous to a Blochoscillation. All diffraction images have a field of view of 631 𝜇m × 631 𝜇m and weretaken at a lattice depth of 11 Er and 2.7 kHz Raman coupling with at least 30 ms
hold in the HH lattice.

47

cell takes into account the specific gauge implemented in an experiment, and in highly
symmetric gauges like ours can entirely overlap with the magnetic unit cell. Thereis a further symmetry caused by the indistinguishability of the 𝑞 sites contained ineach magnetic unit cell, which leads to a 𝑞 -fold degeneracy of each state within themagnetic Brillouin zone, seen clearly in Figure 3-5b.
If the condensate occupies only one of the two degenerate momenta, we see amomentum distribution (Figure 3-5d and h), which clearly shows that the gauge
symmetries are in the 2𝜋/𝑎 (ˆ𝑥 ± 𝑦ˆ) directions. Figure 3-5e and i show the full sym-

metry of the bandstructure. Here, both energy minima are equally populated, andwe can understand the image as the sum of Figure 3-5h with a second copy shiftedby the gauge-invariant scale 2𝜋/ (𝑞𝑎) in either 𝑥
ˆ or 𝑦ˆ. The fourfold increase in visible
peaks [69] should be thought of as a factor of two from the reduced symmetry of
the lattice vectors and an additional factor of two from the degeneracy of the groundband. This is the first direct observation of the several symmetries of a square latticein a uniform magnetic field.
Finally, in Figure 3-5e, f, i, and j, we comment on a subtlety of our method of
simulation. In Section 3.2, we moved into a rotating frame for our analysis. However,we actually do our measurements in the stationary lab frame! Shifting from therotating frame to the lab frame manifests itself as a kick in the 𝑥-direction that isdependent on the hold time, the initial phase of the stationary and Raman lattices [22],and the relative position of the harmonic confinement beams [70]. In our experiment,these wavelength scale drifts are not controlled from shot-to-shot, and we present two
example shots with a phase of 0 and 𝜋/2.

3.4.1 Apparent Gauge Dependent Observables
In our above dicussion, we stated the time-of-flight images show the symmetries of thegauge, which seems to contradict the common wisdom that all observables are gaugeindependent. There are two ways to square this with what we see in our systems withsynthetic magnetic fields. First, we can view time-of-flight imaging, where we snap
off all potentials, as preserving the form of the wavefunction at snap-off, but then48

allowing it to evolve under the untrapped Hamiltonian 𝐻 = 𝑝⃗2 /2𝑚. This maps thegauge-dependent canonical momentum to mechanical momentum, which we observe[71, 62, 63].
A second, equivalent, description looks at this from the perspective of Maxwell’sequations. When we perform time-of-flight measurements, we are turning off the⃗ at the same time as all other fields. By Maxwell’s equations, thevector potential 𝐴⃗ = −𝜕 𝐴/𝜕𝑡,⃗changing vector potential creates an electric field 𝐸
which leads to an∫︀⃗ · 𝑑𝑡. Thus, the mechanical momentum immediately after the field
impulse ∆⃗𝑝= 𝐸
is turned off is given by a combination of the initial mechanical momentum and thestarting vector potential in the natural gauge. This natural gauge is the one that has
the same symmetry as whatever is physically creating the vector potential. In ourcase, this is the symmetry of the Raman lasers, while for a real magnetic field, it is
the symmetry of the current used to create it. It is possible to analyze the system ina gauge with a different symmetry, but the gauge transformation will also add a term
to the electric field pulse that cancels any physical effect of the gauge transformation.
Thus, when we say that we have implemented a system with a specific gauge, wemean that our implementation has symmetries that make a given gauge the mostnatural for our analysis.

3.4.2 Population Imbalance
Nearly all iterations of the experiment result in roughly equal populations in bothminima, and less than 1% of the shots show a single minima filled as in Figure 3-5h.
To quantify this, we use a bandmapping technique before time-of-flight imaging. In
bandmapping, we ramp down the lattice potentials slower than the timescale givenby the bandgap, but faster than that given by the external harmonic confinemnt.
This maps a quasimomentum 𝑞 in lattice band 𝑛 to a momentum state 2𝜋𝑛/𝑎 + 𝑞 ina harmonic trap. We then rapidly snap off the harmonic confinement and perform
time-of-flight imaging. This technique would be useful for us because it allows us to“collapse” all the peaks seen in Figure 3-5 into only one or two, depending on how
many minima are filled.
49

However, there are two subtleties of our bandmapping procedure for an 𝛼 = 1/2
Harper superfluid that differ from the standard bandmapping. First, the two subbands are connected at Dirac points, and there is no bandgap! Bandmapping would
fail at quasimomentum at or near the Dirac points, but since our atoms are condensed
into the minima, they are sufficiently far from the Dirac points, and the gap wecare about is the local gap. Note that this means that this bandmapping techniquewould not work for fermions or thermal atoms that fill a significant part of the lowest
band. The second subtlety is that this rampdown is occuring in a real magnetic field
gradient, which will cause potentially destructive Bloch oscillations. Thus, the wholeprocedure needs to be done on a time of the order of a Bloch oscillation time. We
empirically found that first ramping down the Raman beams in 0.88 ms, then thelattice beams in 0.43 ms, and finally snapping off the harmonic confinement provides
a good signal-to-noise.
Figure 3-6 shows the results of this bandmapping. As a control, we first applied the bandmapping sequence to a superfluid with only real tunneling created byamplitude modulation (labeled “AM” in the inset). As expected, we see only one
quasi-momentum occupied. For a Harper superfluid, we nearly always see two peaks,
one for each degenerate minima. We integrate the density in each peak and plot ahistogram of the absolute population imbalance between the two for 30 shots, eachtaken at a 29.4 ms hold time.
We find that the population imbalance is sharply peaked around zero, demonstrating the degeneracy of the two minima. In some gauges, such as the Landau gauge,
it is predicted that a superposition of the two single particle minima avoids density
modulation and is therefore the ground state for a weakly interacting system [63, 64].
However, in our gauge, the quasimomentum eigenstates already have uniform occupation of the lattice sites, and so the weakly interacting ground state should be asingle minimum. Therefore, we interpret the equal populations to be attributable todomain formation, likely as a result of technical fluctuations or non-adiabatic state
preparation. Domain formation in a different Floquet system has been studied recently, and it was shown that single domains can be achieved with proper adiabatic
50

Lattice (Er)a)

Raman (Er)100Time-of-Flight+
Imaging0Time (ms)Probabilityb) 0.3

0

1

AM

2

HH

0.2
y

0.1
0

x

0.0

0.2

0.4
0.6
Population Imbalance

0.8

1.0

Figure 3-6: Population imbalance of the two ground states of the Harper-Hofstadter
Hamiltonian with 1/2 flux. (a) Band mapping sequence adiabatically connecting
quasimomentum to free space momentum. The Raman beams were ramped downfrom the initial strength of 1.4𝐸𝑟 to zero in 0.88 ms, followed by a linear ramp
down of the lattice beams from 11𝐸𝑟 to zero in 0.43 ms. (b) The histogram shows
the relative population imbalance of the two degenerate minima. Equal populationin the two diffraction peaks is suggestive of domain formation due to spontaneous
symmetry breaking but can also be driven by lattice noise and technical fluctuations.
The data consists of 30 shots taken after a hold time of 29.4 ms in the HH lattice.
The inset displays a raw image for the band mapping of the 1/2 flux superfluid withtwo degenerate ground states compared to a topologically trivial superfluid with one
ground state (see text). The Brillouin zones of the cubic lattice (grey) and the gauge(green) are overlaid for clarity.

51

y

x

Figure 3-7: An example fit for a 1/2 flux superfluid image after a hold time of 34
ms. The thermal fraction fit is shown in yellow with the components of the superfluid
diffraction fit in cyan, purple, and green. Their sum is shown in dark blue laid over
the raw data in orange. The residuals are shown in dark red. A visibility of 𝑉 = 0.40
is derived from the amplitudes of this fit.
state preparation [72].

3.4.3 Superfluid Peak Fitting
In order to move forward with our experiments, we want a measure of the coherenceof the system, which we quantify using peak visibility. All of our absorption images
are the sum of the superfluid peak pattern with 25 peaks and a broad incoherent
background, which we associate with the thermal portion of the system. In analogy
with Michelson’s definition of optical fringe contrast, we define the visibility of agiven superfluid peak as its amplitude divided by twice the thermal background at
its center.
The full two-dimensional image has too many degrees of freedom to fit reliably,
so our fitter works by fitting to two one-dimensional line cuts, one in each latticedirection. We make these line cuts by first finding the center of mass and thenaveraging several rows or columns of pixels around this. We then fit the 𝑥 and 𝑦 line
cuts to the sum of six Gaussian peaks, which correspond to the one thermal and five52

superfluid peaks mentioned above. We report the average visibility of the peaks inboth directions.
Even restricting ourselves to one lineout requires us to fit a high-dimensionalparameter space — we need to find the center of the thermal background, its widthand amplitude, the phase of the micromotion, the distance between superfluid peaks,
the width of the superfluid peaks, and each peak’s amplitude. We can constrain
the fit so that the superfluid peak differences are equal and that they all have thesame width, but this still leaves 11 fitting parameters. Therefore, our nonlinear leastsquares fit requires good initial guesses. Find our initial guesses for the thermal
bakground first by fitting a single Gaussian to the image with frequencies of orderthe normal superfluid peak spacing and higher filtered out. Next, we subtract thisfit from the raw data and pick the highest five remaining peaks, whose position andamplitude become the initial guess for the superfluid peaks in the final fit. Finally,
we fit the full model using the initial conditions collected from the pre-processing.
An example of a typical fit is shown in Figure 3-7.

3.4.4 Adding Strong Interactions
Now that the single-particle physics of the Harper-Hofstadter Hamiltonian has beenverified, we wish to study the strongly interacting limit, near the Mott Insulatortranstion. This transition for 𝛼 = 1/2 and a filling factor of five occurs at 𝐽/𝑈 ∼ 0.016[73]. For an isotropic HH lattice, the transition happens around 15-16 Er , with theprecise value depending on multiple lattice parameters such as anisotropy, Raman
drive strength, and tilt strength. For lower filling factors, the critical 𝐽/𝑈 ∼ 0.059,

and the transition occurs around 12 Er , so our experiments with a three-dimensionallattice can be in the strongly interacting regime.

We use the many-body adiabatic sequence, with a lattice in the third dimension, toreach this regime. We keep the final lattice depths (and therefore tunneling strenghts)
along the 𝑥- and 𝑦 -directions constant, while varying the 𝑧 lattice depth, which cangive us a Hubbard 𝑈 parameter from nearly zero to ∼450 Hz, which with the filling
factor of five mentioned above, brings us near the Mott Insulator transition.
53

11 Er

14 Er

17 Er

20 Er

Figure 3-8: Harper-Hofstadter Hamiltonian with strong interactions. Here we present
a selection of shots with the 𝑥-and 𝑦 -lattices fixed at 11 Er and a variable 𝑧 lattice of11, 14, 17, and 20 Er .

54

Visibility Lifetime (ms)

120

2 210

Hubbard Parameter U (Hz)
290
390
420
350

450

80

40

0

0

5
z-Lattice Depth (Er)10

Figure 3-9: Strongly interacting Harper superfluid lifetime. Shown is the lifetime ofthe visibility of the diffraction pattern versus z-lattice depth. The top axis shows the
Hubbard interaction parameter U. All lifetimes are measured from the end of the 35
ms ramp exiting the Mott insulator. Uncertainty is given by the statistical error of
the mean of five repetitions of the experiment, added in quadrature to uncertainty inthe peak visibility fitting.
In Figure 3-8, we present representative shots of a BEC in the Harper-Hofstadter
Hamiltonian with strong interactions. We see visible, if weak, superfluid peaks for𝑧 -lattice depths up to 20 E , while the other two lattices are held at 11 E . Above arr𝑧 -lattice of 11 E , we found significant shot-to-shot variation, likely due to technicalrfluctuations, so in Figure 3-9, we measured the visibility lifetime of strongly interacting Harper superfluids with interaction parameters up to 𝑈 = 450 Hz. We findthat the lifetime is reduced in the strongly-interacting system in compariston to the
non-interacting system, but only by a factor of ∼ 2 − 4.

3.5 Lifetime of the Harper Superfluid
There are many open questions about topological physics in the presence of strong
interactions, which often involve small energy scales and therefore require longer
coherence lifetimes than we have achieved. Since our system is relatively complex,
there are several categories of heating that can limit our lifetimes. First, there is what
we call “generic technical heating,” which encompasses the sorts of heating which are
55

Diffraction Visibility

0.8

Amplitude Modulation: 142 ± 15 ms
Raman Modulation: 71 ± 8 ms

0.6
0.4
0.2
0.0
10

50
100
Hold Time (ms)500

Figure 3-10: Decay of Bose-Einstein condensates in modulated lattices. The figurecompares the decay of the 1/2 flux Harper superfluid (red circles) to the decay of
the amplitude-modulated superfluid (blue squares). Note that the lower visibility of
the Harper superfluid is due to the peak doubling, which at the same condensate
fraction, leads to lower visibility. Exponential fits to the decay of the visibility ofthe diffraction patterns give lifetimes of 142 ± 15 ms and 71 ± 8 ms, respectively.
Data were taken with 11 𝐸𝑟 cubic lattice with either 2.7 kHz Raman coupling or
20% amplitude modulation and start after a 10 ms hold time after switching on
the final Hamiltonian using the non-adiabatic procedure. Uncertainty is given by thestatistical error of the mean of five repetitions of the experiment, added in quadrature
to uncertainty in the peak visibility fitting.
common to all lattice systems, e.g., lattice and amplitude modulation, beam pointing
drifts, and magnetic field fluctuations. The second category is “Floquet heating,”
which include all heating processes that occur in periodically driven Floquet systems
as ours is. The final category is heating specific to the Harper-Hofstadter Hamiltonian,which is still an open field.
In order to disentagle these different sorts of heating, we compared our noninteracting Harper superfluid to a normal superfluid created using amplitude modulation [66, 67]. This system is also a Floquet system, and should have many heating
sources in common with the Harper superfluid. In Figure 3-10, we show the lifetimeof the superfluid peak visibility for both the Harper superfluid and the amplitudemodulated superfluid. At first, the Harper superfluid’s lifetime was notably shorter
than that of the amplitude-modulated superfluid, but the experimental upgrades discussed in Section 3.3.1 improved the Harper superfluid lifetime until it was comparable56

to the amplitude-modulated lifetime.
By looking at atom number loss, we can rule out several possible sources of noise.
We observed no discernable number loss out to 500 ms, which immediately tells usthat neither two-body dipolar collisions nor three-body recombination are the limiting
source of heating. Because the tilt ∆ is so strong, higher Bloch bands are coupled
to the continuum and would rapidly be lost from our system, so the lack of atom
loss during experimental timescales makes it clear that the majority of the heating is
intra-band.

3.5.1 Floquet Fermi’s Golden Rule
We next look to a Floquet heating explanation for intra-band heating. The lowest
order decay path in a Floquet system is via overlap of neighboring Wannier-Starkstates with offset energy ∆, which is transferred to excitations of the lowest bands,
or to the free particle motion along the tubes orthogonal to the two-dimensionallattice. We can analyze this by using Fermi’s golden rule for Floquet systems [74, 75].
We follow and extend [75], which calculates the heating rate for a one-dimensional
amplitude-modulated lattice. Here, the ground state of the corresponding Floquet
Hamiltonian is a superfluid in a lattice in one dimension and free motion in the other
two.
There are two main processes that contribute to the decay of the condensate.
Wannier-Stark states – which are mostly localized in a given well 𝑛 – have small
components in lower wells 𝑛 − 1, with amplitude ∝ 2𝐽/∆. This small component,

which has energy ∆ higher than the Wannier-Stark state localized in the well 𝑛 − 1,

can collide with atoms in the lower lattice site, thus creating two excitations withenergy ∆/2 in the transverse direction (the creation of two exitations is necessary forconservation of transverse momentum) . The second mechanism involves the counterrotating term of the phonon or photon assisted tunneling process. The co-rotatingterm enables resonant tunneling to a lower lattice site by stimulated emission of aphonon. In the counter-rotating process, the atoms tunnels to a lower lattice site,
but absorbs a lattice phonon. Therefore, it has now an excess energy of 2∆ which57

1Visibility (arb.)

Atom Number (Arb.)

800

400

0
0.3

3

30
Hold Time (ms)

300

3000

0

Figure 3-11: Lifetime of a Bose-Einstein condensate in a 1D lattice with resonant
amplitude modulation. Shown are the decay of the number of trapped atoms, andthe decay of the visibility of the diffraction pattern observed after ballistic expansion.
can create two transverse excitations where each of them has energy ∆.
These two-body elastic collisions convert modulation or tilt energy into motion
along the transverse directions. The characteristic relaxation times of the two processes are given by 𝜏1 and 𝜏2 :1
4𝐽 2 ∑︁(0)2𝜋𝛿(𝐸𝑘 − ∆/2)
∝ 2
𝜏1
∆ 𝑘
1
𝐾 2 ∑︁(0)∝ 22𝜋𝛿(𝐸𝑘 − ∆).
𝜏2∆ 𝑘
We can associate 𝜏1 with a process that occurs even in the absence of modulation– the rate is proportional to the tunnel coupling squared between neighboring sites,
whereas the rate of the 𝜏2 process is proportional to the square of the phonon induced
tunneling rate. Both decay processes are rather general and should apply to bothtilted lattices and lattices modulated by superlattices.
As a test of these predictions, we implemented this Hamiltonian experimentallyusing the same experimental procedure as used for a two-dimensional amplitudemodulated superfluid. We measured the coherence and lower band population lifetimes as shown in Figure 3-11. Approximate values in our experiment were 𝑔𝑛 =ℎ × 550 − 830 Hz, 𝑎𝑠 = 5.03 nm, 𝑑 = 254 nm, 𝐾 = ℎ × 10 Hz, 𝐽 = ℎ × 30 Hz, and ∆ =58

ℎ × 3420 Hz, with 𝑔𝑛 as the chemical potential, 𝑎𝑠 the s-wave scattering length, 𝐾

laser-assisted tunneling rate, 𝐽 the bare tunneling rate, and ∆ the energy tilt per lat∫︀
tice site. 𝑑−1 = d𝑥 𝜑𝑥 (𝑥)4 is the size of the Wannier state in the x-direction 𝜑𝑥 (𝑥).
These give us the values of 1/𝜏2 = 0.003 − 0.005 s−1 and 1/𝜏1 = 0.13 − 0.20 s−1 , which

are longer than our observed lifetime. Uncertainty is driven by uncertainty in ourdensity measurements due to redistribution during lattice ramp-up. However, during
the long coherence lifetime, significant population loss was observed (due to inelatic
two- or three-body collisions, or transfer to higher bands), and detailed further study
is needed to identify the contributions of different processes to the decay rate.
The mechanism proposed in [75] appears to give a reasonable upper limit forcollisional decay in 1D lattices. We expect the same mechanism to explain at least
part of the decoherence observed for both amplitude modulation in a two-dimensionallattice and the HH Hamiltonian. For a quantitative comparison, we modify thecalculation in [75] to reflect that a 2D lattice has only one orthogonal direction offree motion, which changes the density of states for decay processes. For resonant
modulation, in the Floquet basis, the Hamiltonian is𝐻=∑︁𝜖k (𝑡)𝑏†k 𝑏k +𝑘𝑔 ∑︁ † †𝑏 𝑏 𝑏k 𝑏k
2𝑉 k ,k ,k k1 k2 3 4123with k4 = k1 + k2 − k3 , V as the system volume, and 𝑏k and 𝑏†k as the annihiliationand creation operators for states with quasimomentum k. The interaction parameteris given by

4𝜋~2 𝑎𝑠 2
𝜆
𝑔=
𝑚

∫︁

d𝑥 d𝑦𝜑𝑥 (𝑥)4 𝜑𝑦 (𝑦)4 =

4𝜋~2 𝑎𝑠 𝜆2
𝑚 𝑑2

where 𝜑𝑥 (𝑥) and 𝜑𝑦 (𝑦) are the Wannier functions in the tilted (x) and untilted (y)lattice directions, 𝜆 is the lattice spacing, and 𝑑 is the size of the Wannier state. The
instantaneous single particle dispersion 𝜖k (𝑡) is given by~2 𝑘𝑧2.
𝜖𝑘 (𝑡) = −2𝐾 cos(𝑘𝑥 ) − 2𝐽𝑦 cos(𝑘𝑦 ) − 2𝐾 cos(𝑘𝑥 − 2∆𝑡) − 2𝐽𝑥 cos(𝑘𝑥 − ∆𝑡) +2𝑚
where 𝐽𝑥 and 𝐽𝑦 are the bare tunneling rates in the 𝑥 and 𝑦 directions, and 𝐾
59

is the effective tunneling in the tilt direction (𝑥). We ignore the overall harmonic
confinement and treat motion along 𝑧 as free. Following a similar argument, we findthat the scattering rate is given by

1
1
1
=
+
𝜏
𝜏2 𝜏1
1
2(𝑔𝑛/2)2 𝐾 2 ∑︁
(0)=2𝜋𝛿(𝐸𝑘 − ∆)2𝜏2𝑁~ ∆ 𝑘2(𝑔𝑛/2)2 4𝐽 2 ∑︁
1(0)=2𝜋𝛿(𝐸𝑘 − ∆/2)2
𝜏1
𝑁~ ∆ 𝑘with the effective dispersion:√︃(0)
𝐸𝑘 = 2𝐾[1 − cos(𝑘𝑥 )] + 2𝐽𝑦 [1 − cos(𝑘𝑦 )] +𝑔𝑛(~𝑘𝑧 )2 +𝑚(︂(~𝑘𝑧 )22𝑚)︂2.

Note that ref. [75] uses a Hartree-Fock excitation energy with free particle dispersion.
The high mean field energy in the one dimensional tubes created by a two dimensionallattice, 𝑔𝑛, is similar in energy scale to the tilt frequency, and we thus use the full
Bogoliubov spectrum for the energy of quasi-particles.

(0)

Since 𝐾 and 𝐽𝑦 are small, we neglect the dependence of 𝐸𝑘 on 𝑘𝑥 and 𝑘𝑦 , giving

𝜌(𝜈) =

∑︁
𝑘

=

𝑉
𝜆2

(0)

(3.18)

2𝜋𝛿(𝐸𝑘 − 𝜈)

∫︁

⎛√︃

d𝑘𝑧
2𝜋𝛿 ⎝
2𝜋

𝑔𝑛
(~𝑘𝑧 )2 +𝑚√︁(︂(~𝑘𝑧 )2
2𝑚

⎞

)︂2

− 𝜈⎠

2~2

𝑉𝑚
𝜈
𝑚
√︁
= 2 2 √︀√︀~𝜆(𝑔𝑛)2 + 𝜈 2 −(𝑔𝑛) + (𝑔𝑛)2 + 𝜈 2
and thus√︁~21
𝑔𝑛 4𝜋𝑎𝑠 𝐾 2
∆
2𝑚𝑑2
√︀
√︁
=
√︀
𝜏2
~ 𝑑 ∆2 (𝑔𝑛)2 + ∆2−(𝑔𝑛) + (𝑔𝑛)2 + ∆2
60

(3.19)(3.20)

√︁
21
𝑔𝑛 4𝜋𝑎𝑠 4𝐽
∆/2
√︀
√︁
=
2
𝜏1
~ 𝑑 ∆
(𝑔𝑛)2 + (∆/2)2−(𝑔𝑛) +

~2
2𝑚𝑑2

√︀
(𝑔𝑛)2 + (∆/2)2

For our system, representative numbers are 𝑔𝑛 = ℎ × 740 − 1740 Hz, and ∆ =ℎ × 3420 Hz, 𝐾 = ℎ × 10 Hz, 𝐽 = ℎ × 30 Hz, 𝑎𝑠 = 5.03 nm, 𝑑 = 254 nm, and𝑚 = 1.4 × 10−25 kg, resulting in 1/𝜏2 = 0.006 − 0.014 s−1 and 1/𝜏1 = 0.29 − 0.67 s−1 .

This is less than the decay rates measured in the experiment by a factor of ∼ 10 − 20.

A three-dimensional lattice should be helpful for reducing heating described by the

Floquet Fermi’s golden rule. If a three-dimensional lattice has a bandbap at energies∆ and ∆/2, we expect to see a “gapping out” of this heating and a strong increasein the lifetime [76]. However, in our experiment, we do not see this strong change
when changing the bandgap. We therefore believe that the heating we see is either
technical heating common to both amplitude-modulated and Harper superfluids, orsome as-yet-unexplained heating process for Floquet systems.

3.6 Conclusion
In conclusion, for the first time, we have demonstrated Bose-Einstein condensation
in a strong synthetic magnetic field. The momentum distribution of the ground stateclearly shows how the vector potential and gauge break the translation symmetryof the bare lattice and uniform magnetic field. We have added strong interactions
to our system and have demonstrated reasonable coherence times near the Mott
Insulating transition. With longer lifetimes, modifications of our system could lead tothe realization of Weyl points [77], the quantum spin Hall Hamiltonian and ultracold
atom topological insulators [78], and bosonic Laughlin states [73, 79, 38, 80]. A fullunderstanding of Floquet heating is still an open area of research in the field [81, 82],
while it is possible that a tripled-superlattice system could avoid some of the heatingprocesses in this work [78].

61

62

Chapter 4
Superfluid Shielding
A central feature of quantum mechanics is the wave-particle duality, where matterthat we normally think of as composed of particles can be described by a wavefunction

𝜓(𝑥). Since the probability of finding a particle is given by |𝜓(𝑥)|2 , it is tempting
to think of the wavefunction as merely a probability distribution. However, we mustnote that the probability distribution is the squared modulus of the wavefunction,and that 𝜓(𝑥) has an additional degree of freedom in its complex phase 𝜓(𝑥) =exp[𝑖𝜑(𝑥)] |𝜓(𝑥)| . This quantum phase is directly relevant for many applications inin quantum information, quantum simulation [83, 84, 65], atom interferometry [25],and force sensing [29, 30, 31].

Important applications rely on maintaining phase coherence between spatially separated quantum subsytems. However, our ability to split a quantum system, perform
an operation on the separated subsystems, and use interference to read out phase information is limited. For now, I will ignore technical or environmental limitations andfocus on the fundamental limits. In a non-interacting system of size 𝑁 , classical shot√noise leads to fluctuations of ∼ 1/ 𝑁 in the number of particles in each subsystem,

which determine the final signal-to-noise ratio and precision of the measurement of
the phase, but do not affect the coherence lifetime.
In an interacting system, on the other hand, the coherence time is also set by shotnoise. The phase of each subsystem evolves at a rate given by the total energy 𝐸total =𝐸external + 𝜇, where 𝜇 is the chemical potential. The shot noise fluctuations mentioned63

above give rise to fluctuations of the chemical potential 𝛿𝜇 = 𝛿𝑁 × |𝜕𝜇/𝜕𝑁 | , leading
to decoherence. Previous work has focused on either one of the two terms in theformula for 𝛿𝜇. In [85], the researchers use Feshbach resonances to modify interatomic
interactions and therefore |𝜕𝜇/𝜕𝑁 |, which lead to very long coherence times. Other
work has modified 𝛿𝑁 by using non-classical squeezed states, which can be createdusing nonlinear interactions [86, 87] or quantum measurements [88]. While mostwork on squeezing has been done with relatively small numbers of atoms, highly
spin-squeezed states have been observed for pairs [89] and arrays [90] of independent
Bose-Einstein condensates.
In this chapter, I present a new method for increasing the phase coherenc timeof separated BECs beyond the projection noise limit. If we immerse the separated
BECs in a common superfluid bath, we find that the interactions between the bathand the system can compensate for technical and number fluctuations that normallyshorten the phase coherence time. I begin with a review of atom interferometry, which
provides an intuitive way to think about the effects of phase coherence and separated
quantum objects. I then discuss the phenomenon of Bloch oscillations, where a BEC
in a tilted lattice precesses through the Brillouin zone, and demonstrate how this effectcan be thought of as a form of atom interferometry. I provide a more quantitative
explanation of superfluid shielding, and report on several experiments that explore
the effect in various parameter regimes that demonstrate that superfluid shielding isa robust effect that can in principle reduce chemical potential fluctuations in atominterferometers by two orders of magnitude.
This chapter covers work covered in [91].

4.1 Atom Interferometry
The importance of phase coherence is most obvious in the technique of atom interferometry. This technique exemplifies the wave nature of ultracold atoms by making ananalogy with optical interferometry. I will first review classical optical interferometry
and then point to the analogs of each piece in atom interferometry.
64

Figure 4-1: A basic optical interferometer. A coherent light source (red arrows) is
split by a 50:50 beamsplitter and travels along two different paths. The two paths
are recombined on a second beamsplitter and interfere. The path length difference
between the two arms becomes a phase difference and determines the photodiodereading.
The simplest form of an optical interferometer (cartoon in Figure 4-1) starts witha single coherent light source (i.e. a laser), which is split on a 50:50 beamsplitter. The
light travels in two arms, of length 𝑙1 and 𝑙2 , and then is re-combined at the second
beamsplitter. The two arms will interfere with a phase given by 2𝜋 (𝑙1 − 𝑙2 ) /𝜆, where

𝜆 is the wavelength of the light, and we can thus use a photodiode or camera to readout the difference in path-length. The conceptually simplest atom interferometers
map exactly to this optical interferometer. Instead of a laser, start with a BEC,
which is a coherent state and therefore has a well-defined phase. The beam splittercan be implemented with Raman pulses, which provide a momentum kick that separates the clouds, or optical lattices, which can remove tunneling between different
sites, effectively separating the condensates. The phases of each now-separated condensate evolve at a rate given by −𝐸(1,2) /~, where 𝐸(1,2) are the energies of the twocondensates. The condensates are re-combined and interfered by physically overlapping the two condensates. If they were split using a Raman pulse, subsequent Raman
pulses can reverse the momentum and push the clouds back together. On the other
hand, if a lattice was used to separate the condensates, we use time-of-flight imaging,where we snap off all trapping potentials. Then the condensate on each lattice site
acts as an emitter of a spherical matter wave with a phase given by the phase of
the wavefunction on that site. Once they have expanded to several times the initialsystem size, the interference pattern can be used to read out the phase information
65

from the interferometer.
An important difference between an atom interferometer and its optical counterpart is the role of interactions. As discussed above, inter-particle interactions (which
are not present for an optical interferometer) can limit the coherence lifetime anddecrease the contrast of the interferometer. Thus, most atom interferometers use
atoms just above the BEC transition point, where interactions are negligible, and are
therefore more similar to a white-light interferometer. Superfluid shielding on theother hand allows us to use a high-brightness coherent source without being limited
by interaction effects, and should lead to better overall contrast and coherence times
in atom interferometers.

4.2 Bloch Oscillations
In our experiments, we used the coherence of Bloch oscillations as our measure ofcoherence of the separated condensates. Since Bloch oscillations are not normallydescribed as an atom interferometry effect, it makes sense to take some time to
discuss the effect now. We should emphasize that the following discussion and ourexperiments are in the high-tilt regime, in contrast to most previous work on Blochoscillations, which have focused on the low-tilt regime [92, 93, 94] where adjacent
Wannier-Stark states overlap and tunneling is still allowed.
As discussed in Section 2.1, the single-particle eigenstates in a one-dimensionallattice potential are described by Bloch wavefunctions |𝜓𝑘 ⟩ with quasimomentum 𝑘

and energy 𝐸𝑘 = −4𝐽 cos (𝑘𝑎). The traditional explanation of Bloch oscillations isthat if we apply a constant energy tilt per lattice site of ∆, we can write this as a
66

(a)

(b)

Figure 4-2: (a) Cartoon of Bloch oscillations as an atom interferometry effect. Blochoscillations occur when a BEC is placed into a tilted optical lattice. After the tilt, eachlattice site has a separated condensate whose phase is represented by the clocks. Each
phase evolves at a different rate, given by the on-site energy, which leads to different
interference patterns after different evolution times. (b) Interference patterns of 87 Rb
atoms in a lattice the course of a Bloch cycle at times 0, 𝜋/2, 𝜋, 3𝜋/2, 2𝜋 .

force 𝐹 = −∆/𝑎, and quasi-classical dynamics then tells us

~

𝑑𝑘
=𝐹
𝑑𝑡
1 𝑑𝐸
𝑣=
~ 𝑑𝑘
4𝐽𝑎
sin (𝑘𝑎)
=
~
(︂
)︂
4𝐽𝑎
𝐹 𝑎𝑡
=
sin
~
~

(4.1)

which shows that a constant force begets oscillations whose frequency provide a measure of the applied force. However, this traditional explanation does not make obvious
the connection to atom interferometry and the importance of phase coherence.
To do this, first note that in the tight-binding limit, we can write |𝜓𝑘 ⟩ as the∑︀
sum of Wannier states centered on site 𝑗 , with |𝜓𝑘 ⟩ = 𝑗 exp [𝑖𝑘𝑎𝑗] 𝑎†𝑗 |0⟩ where 𝑎†𝑗 is

the Wannier state creation operator. To see Bloch oscillations, we start in with anuntilted lattice, so the atoms are in some |𝜓𝑘 ⟩. This is the equivalent to the initial
single BEC in traditional atom interferometry. We then rapidly turn on a constant67

tilt ∆. The Hamiltonian is simply given by

𝐻 = −𝐽∑︁ †∑︁ †𝑎𝑗+1 𝑎𝑗 + ∆𝑗𝑎𝑗 𝑎𝑗 .
𝑗

(4.2)

𝑗

Since we assume the high-tilt regime of ∆ ≫ 4𝐽 , the lattice sites are decoupled, whichis equivalent to physically separating the two sub-condensates. The state evolves to|𝜓 (𝑡)⟩ =

∑︁
𝑗

⃒
⟩︀
exp [𝑖𝑗 (𝑘𝑎 − ∆/~)] 𝑎†𝑗 |0⟩ = ⃒𝜓𝑘+𝐹/~ .

(4.3)

We can picture this, as in Figure 4-2 as each lattice site having a clock attached to it,that runs at the phase evolution rate. When all the clocks have evolved by a multipleof 2𝜋 , we are back at the starting point, leading to oscilliatory motion. Finally, to
see this effect, we can interfere parts of the wavefunction from different lattice sites
by performing time-of-flight imaging, as discussed above.

4.3 Superfluid Shielding
If we are using an interacting system for the atom interferometer, number fluctuationsin the separated condensates lead to chemical potential fluctuations. These chemical
potential fluctuations change the phase evolution of the systems, leading to dephasing and a shortened coherence time. Rather than attempting to reduce the number
fluctuations using sqeezed states, or reduce the interactions via a Feshbach resonance,
we present a new technique. In superfluid shielding, the separated condensates are
immersed in a superfluid bath. The interactions between the bath and the interferometer cancel out the effect of projection noise on the phase evolution and increase
the coherence time.
Fig. 4-3a-c provides a graphical view of superfluid shielding in our experiment.
After the sudden tilting of the |↓⟩ atoms, each lattice site is a separate condensate,with no resonant tunneling from site to site. Then, if there is no shielding component,
the evolution of the phase on each site is given only by the local chemical potential
68Unshielded
(d)

(b)

(a)

m-1

Contrast+ Apply Force

0

0

40

|

(f)

+

80 120Time (cycles)

m+1

Shielded
(e)

(c)
(g)

0.4

m

|

m

m-1

m+1

|

Figure 4-3: Schematic of superfluid shielding. (a) Before applying a tilt, the atoms
are in a superfluid, which is approximately described by a coherent state on each site.
The chemical potential is constant across the cloud. (b) In the limit of a strong tilt(∆ ≫ 𝐽 ), the wavefunction at each lattice site is projected onto the number basis,
leading to fluctuations in the number of atoms and chemical potential from site tosite. (c) If the gas has two components, one which is localized by the tilt, and one
which remains superfluid, the itinerant component can compensate for fluctuationsin the localized component. (d-f) Momentum distribution over the course of a single
Bloch oscillation after ten cycles. (d) Without superfluid shielding, the diffuse cloudindicates decoherence of the condensate. (e) The itinerant component feels no force
and does not Bloch oscillate. (f) For the shielded component, the Bloch oscillation
contrast is high. (g) Exponential decay of the Bloch oscillation contrast for a onecomponent (blue dots) and two component (red squares) gas, for a transverse lattice
depth of 11 𝐸𝑟 and ∼8 × 103 atoms.

69and its fluctuations, 𝜇↓𝑗 , and has a relatively short coherence time. On the other hand,when the system is immersed in a superfluid of |↑⟩ atoms, which are not subject tothe tilt, we see superfluid shielding. The |↓⟩ atoms and the |↑⟩ atoms on a givensite interact repulsively. However, because |↑⟩ atoms remain superfluid, these atoms

can freely adjust their local density in order to counteract the fluctuations in |↓⟩atoms and thus maintain a uniform global chemical potential. This leads to an anticorrelation in the densities of the two components, which mitigates the fluctuationsin the chemical potential of the |↓⟩ component. Previous work in the low-tilt regime
has shown that the presence of disorder applied before the tilt leads to a shortened

coherence time of Bloch oscillations, and that this can be partially compensated forby weak interactions [95, 96]. This work demonstrates a different regime, where aseparate superfluid bath can compensate for disorder created during and after thetilt, including projection noise disorder.
We use two more quantitative descriptions of the effect of superfluid shielding:one is based on the Gross-Pitaevskii equation and provides a useful intuitive picture,
while the other treats the effect fully quantum mechanically.

4.3.1 Gross-Pitaevskii Description
In the Thomas-Fermi approximation, the Gross-Pitaevskii equation for a two component gas in a lattice is given by the following:𝜇↓𝑗 = 𝑔 ↓↓ 𝑛↓𝑗 (𝑥, 𝑦) + 𝑉 ↓ ,𝑗 (𝑥, 𝑦) + 𝑔 ↑↓ 𝑛↑𝑗 (𝑥, 𝑦)
ext

(4.4)

𝜇↑𝑗 = 𝑔 ↑↑ 𝑛↑𝑗 (𝑥, 𝑦) + 𝑉 ↑ ,𝑗 (𝑥, 𝑦) + 𝑔 ↑↓ 𝑛↓𝑗 (𝑥, 𝑦),

(4.5)

ext

(↑,↓)

where 𝜇𝑗(↑,↓)

and 𝑛𝑗are the chemical potential and number density, for a given

component and lattice site index 𝑗 . The interaction terms 𝑔 ↑↑,↓↓,↑↓ are the standard
contact interactions 4𝜋~2 𝑎(↑↑,↓↓,↑↓) /𝑚 with 𝑎(↑↑,↓↓,↑↓) as the 𝑠-wave scattering lengths
for intra- and inter-component collisions. In our experiment, before the tilt is applied,both spin components are superfluid, and are subject to a common trapping potential

𝑉

(↑,↓)
(↑,↓)
,𝑗 = 𝑉trap,𝑗 . Then the initial densities are such that the chemical potentials 𝜇𝑗

ext

70

are constant, and the trapping potential is fully compensated by an inhomogeneous
Thomas-Fermi density profile [97].
After our experiment is initialized, we apply an additional potential with a spinindependent part 𝛿𝑉𝑗 , and a spin-dependent force −𝑗∆, which is only felt by |↓⟩atoms. The spin-dependent force separates the condensates of |↓⟩ atoms, which feelan energy shift of ∆ per lattice site, while leaving |↑⟩ atoms, which feel no shift, ina superfluid state with a constant chemical potential. We label the frozen-in number
fluctuations 𝛿𝑛↓𝑗 and the |↑⟩ chemical potential 𝜇↑𝑗 = 𝜇↑ ∀ 𝑗 . The constraint on 𝜇↑𝑗allows us to determine the induced density fluctuations in |↑⟩:

𝛿𝑉𝑗
𝑔 ↑↓
𝛿𝑛↑𝑗 = − ↑↑ − ↑↑ 𝛿𝑛↓𝑗 .
𝑔
𝑔

(4.6)

If we plug this into Equation 4.4, we find that the |↓⟩ chemical potential after theapplied tilt is

(︁
)︁
(︁
)︁
𝜇↓𝑗 = 𝑔 ↓↓ 𝑛↓𝑗 + 𝛿𝑛↓𝑗 + 𝑉 ↓ ,𝑗 + 𝑔 ↑↓ 𝑛↑𝑗 + 𝛿𝑛↑𝑗
ext

=𝜇

↓,0

− 𝑗∆ + 𝛿𝜇↓𝑗

(4.7)

where 𝜇↓,0 is defined as the constant chemical potential of |↓⟩ before the tilt is applied.
Here 𝛿𝜇↓𝑗 are the final fluctuations of the |↓⟩ chemical potential:𝛿𝜇↓𝑗 = 𝜂1 𝑔 ↓↓ 𝛿𝑛↓𝑗 + 𝜂2 𝛿𝑉𝑗 ,(︁which are reduced by the factors 𝜂1 = 𝑔 𝑔

↑↑ ↓↓

(4.8)

(︀ ↑↓ )︀2 )︁ ↑↑ ↓↓
(︀
)︀
− 𝑔
/𝑔 𝑔 and 𝜂2 = 𝑔 ↑↑ − 𝑔 ↑↓ /𝑔 ↑↑ .

For the case of 87 Rb, 𝑔 ↑↑ ≈ 𝑔 ↓↓ ≈ 𝑔 ↑↓ , to the percent level, so the shielding factors

can be around 100. Since the chemical potential is then dependent only on the linear
tilt −𝑗∆, we should see long-lived Bloch oscillations.

4.3.2 Quantum Mechanical Description
The previous section is useful for intuition and calculations in the limit of large
atom numbers, but it raises a question. Atom number and condensate phase areconjugate variables, and so cannot both be definite, but the above derivation appears71

to treat them as such. In this section, I present a more complete quantum mechanicaldescription that resolves this tension. For this analysis, I will look at the wavefunction
on a single lattice site, written in the basis |𝑛↓ , 𝑛↓ + 𝑛↑ ⟩ , and assume that all scattering
lengths are equal.

First, look at the situation without superfluid shielding, where the atoms begin
in a |↓⟩ coherent state∑︁ 𝛽 𝑁⃒ (︀ − )︀⟩︀2
⃒𝜓 0
√ |𝑁, 𝑁 ⟩ .
= 𝑒−|𝛽| /2
𝑁!
𝑁

(4.9)

Then the only dynamics are phase evolution given by the on-site interaction energy

𝑈 𝑁 (𝑁 − 1) and at later times, the wavefunction is2|𝜓(𝑡)⟩ = 𝑒−|𝛽| /2∑︁ 𝛽 𝑁√ 𝑒−𝑖𝑈 𝑁 (𝑁 −1)𝑡 |𝑁, 𝑁 ⟩ ,𝑁!
𝑛(4.10)

which diverges from a coherent state and therefore loses its phase information at latertimes.
On the other hand, in the situation of superfluid shielding, we begin with a coherent state of |↓⟩ atoms within each component of the coherent state of overall atom
number∑︁ 𝛽 𝑁 𝛼𝑛⃒ (︀ + )︀⟩︀22
⃒𝜓 0
√ √ |𝑛, 𝑁 ⟩ .
= 𝑒−|𝛽| /2 𝑒−|𝛼| /2
𝑁 ! 𝑛!
𝑛<𝑁(4.11)We allow the wavefunction to evolve under the same operator as above, and at latertimes the density matrix is2

𝜌(𝑡) =𝑒−|𝛽| 𝑒−|𝛼|2∑︁ 𝛽 𝑁 𝛼𝑛𝛼*𝑚 𝛽 *𝑀√ √ 𝑒−𝑖𝑈 𝑁 (𝑁 −1)𝑡 |𝑛, 𝑁 ⟩ ⟨𝑚, 𝑀 | 𝑒𝑖𝑈 𝑀 (𝑀 −1)𝑡 √ √ .
𝑁 ! 𝑛!
𝑚! 𝑀 !
𝑛<𝑁

𝑚<𝑀(4.12)

Our measurement looks only at |↓⟩ atoms, so we trace over the total atom number

to get2

Tr𝑁 𝜌 = 𝑒−|𝛽| 𝑒−|𝛼|2∑︁ ∑︁ 𝛽 𝑃 𝛽 *𝑃 𝛼𝑛 𝛼*𝑚√|𝑛⟩ ⟨𝑚| ,
𝑃
!
𝑛!𝑚!
𝑃 𝑛<𝑃
𝑚<𝑃

72

(4.13)

which if the fraction in |↓⟩ is small enough (I.E. |𝛼|2 is sufficiently smaller than|𝛽|2 ), is close to the density matrix for a pure coherent state in 𝑛↓ for all time. In
our experiments, where |𝛽|2 ∼ 1000, we need only a small proportion of superfluid
component (|𝛽|2 − |𝛼|2 ) to see good superfluid shielding.

4.4 Experiment
In order to demonstrate the effect of superfluid shielding, we use two spin components
in a spin-independent lattice with a spin-dependent force. The experiments start witha BEC in the |𝐹, 𝑚𝑓 ⟩ = |1, −1⟩ state, levitated against gravity by a magnetic field

gradient and confined in a crossed dipole trap made with 1064 nm light. We raise athree-dimensional lattice potential with a lattice spacing of 532 nm in 100 ms. The
vertical lattice is always raised to 12 𝐸𝑟 , and the transverse lattice height is varied,
which allows us to control the on-site density and therefore interaction parameters.
To create a spin-dependent force, we perform an RF sweep to transfer a fractionof the atoms from the |↑⟩ ≡ |1, −1⟩ state to the |↓⟩ ≡ |2, −2⟩ state. Using an RF

sweep makes us robust against fluctuations in the bias field, and allows us to control
the fraction of atoms flipped into |↓⟩ by varying the intensity of the RF drive.
The magnetic moment of |↓⟩ atoms is equal to −2 times the magentic moment of

the levitated |↑⟩ atoms, and so they feel a force equal to three times gravity, which

gives a tilt per lattice site of ∆ = ℎ × 3410 Hz along the vertical direction. tunnelingis ∼ℎ × 24 Hz, so for |↓⟩, we are deep in the separated condensate regime. Based on
our atom number and harmonic trap, we have about ten pancake shaped condensates,

each with up to 3500 atoms. The |↑⟩ atoms remain untilted and act as the superfluidbath. After we prepare the two component system, we allow it to evolve for a variable
time before switching off all confining potentials for time-of-flight imaging. During the
expansion, we perform a Stern-Gerlach pulse to separate the two spin components,
allowing us to image the interference pattern of the |↓⟩ atoms only.
73

4.4.1 Contrast Fitting
The figure of merit for our experiment is the contrast of the interference pattern.
We obtain the contrast from two-dimensional absorption images, which have alreadyintegrated the atomic density along the imaging axis (one of the transverse directions).
We then integrate along the other transverse direction to get a one-dimensional plotof two-dimensional density as a function of momentum along the tilt direction. For
a perfectly coherent cloud, we expect to see a series of peaks whose spacing is givenby the lattice spacing, width by the initial spatial extent of the cloud, and overallenvelope by the shape of the on-site Wannier function. Our lattice depth is such that
only the zeroth and first order peaks are non-negligible. On the other hand, for a
perfectly incoherent cloud, we expect to see a single broad peak given just by theenvelope function from the Fourier tranform of the Wannier function.
We fit the integrated atomic density to a sum of four Gaussians with appropriate
constraints on the fitting parameters to ensure that the fitting routine picks out
which three are associated with the coherent part and which one is associated with
the incoherent part. We define contrast as the fitted number of atoms under thethree coherent peaks divided by the total fitted number of atoms. For most of thefollowing experiments, we report the 1/𝑒 lifetime of the contrast, which has a one-toone correspondence to the coherence lifetime.

4.5 Projection Noise Limit Analysis
In order to demonstrate that superfluid shielding can move the coherence lifetime
beyond the projection noise limit, we must first calculate what the projection noise
limit actually is. We do this by numerically simulating the discrete Gross-Pitaevskii
equation in the direction of the tilt. Along the other two directions, we use a discreteThomas-Fermi profile to find the average occupation number in each two-dimensionalpancake. We then project onto states with a definite number, drawn from one oftwo distributions. If we were to snap the lattice up from a low value, the atom
number distribution would be Poissonian. On the other hand, if the lattice ramp
74

were perfectly adiabatic, the distribution should exhibit some number squeezing (seeSection 4.5.2). Since we know our lattice ramps are not perfect, we run the simulations
for each distribution, and plot both results, which should place bounds on the true
projection noise limit.
Once we have added the number fluctuations to the model, we time evolve thesystem. Since tunneling is suppressed, we can assume that the only relevant dynamics are given by the two-dimensional Gross-Pitaevskii equation as |𝜓𝑗 (𝑡)⟩ =exp[−𝑖𝜇𝑗 𝑡/~] |𝜓𝑗 (0)⟩ for each pancake at a lattice site 𝑗 . The chemical potential
𝜇𝑗 is given by the two-dimensional Thomas-Fermi profile to take into account the

pancake shape of the condensate on each site, as discussed in Section 4.5.1. Aftera variable evolution time, we take a Fourier tranformation of the wavefunction tosimulate time-of-flight imaging. We average twenty simulated images, and then fit
them using the same fitting method as our experimental data. In this chapter, thesimulated projection noise limits are marked by dashed black lines for both the upperand lower bounds.

4.5.1 Two-Dimensional Discrete Thomas-Fermi Profile
Since we have lattices in the two transverse dimensions, we slightly modify the normal
Gross-Pitaevskii equation in the Thomas-Fermi approximation to(︀)︀1
𝜇𝑗 = 𝑈 𝑛𝑗𝑘𝑙 + 𝑚𝑎2 𝜔𝑧2 𝑗 2 + 𝜔𝑥2 𝑘 2 + 𝜔𝑦2 𝑙2 ,2

(4.14)

where 𝑈 is the Hubbard interaction parameter, 𝑎 is the lattice spacing, 𝜔𝑥,𝑦,𝑧 are
the harmonic trapping frequencies, 𝑗 is the index along the tilt direction, and 𝑘
and 𝑙 are the indices along the two transverse directions. To solve for 𝜇𝑗 , we write75

𝑛𝑗𝑘𝑙 = 𝑛𝑗 (𝑥, 𝑦) and find the total atom number in a pancake 𝑁𝑗 :(︃𝑛𝑗 (𝑥, 𝑦) = Max

(︀
)︀ )︃
𝜇 − 12 𝑚𝑎2 𝜔𝑧2 𝑗 2 + 𝜔𝑥2 𝑘 2 + 𝜔𝑦2 𝑙2
,0
𝑈

∫︁ ∫︁
𝑁𝑗 =

𝑛𝑗 (𝑥, 𝑦) 𝑑𝑥𝑑𝑦
)︂2(︂𝜋
1 2 2 2=.
𝜇𝑗 − 𝑚𝑎 𝜔𝑧 𝑗𝑚𝜔𝑥 𝜔𝑦 𝑈 𝑎2
2

(4.15)

Inverting this expression shows us that 𝜇𝑗 ∝ (𝑁𝑗 𝑈 )1/2 .

4.5.2 Two-Mode Number Squeezing Model
In order to determine the maximum number squeezing expected, we use a two-mode
model [98], but modify it to take into account the two-dimensional nature of thelattice sites.
This model looks at a double-well system, with a two-dimensional BEC in eachwell and tunnel-coupling 𝐽 between them. For large total atom number 2𝑁 , theeffective Hamiltonian is

(︁
)︁
)︁
2𝜇0 (︁ 3/2
𝐻 = −𝐽 𝑎†𝑙 𝑎𝑟 + 𝑎†𝑟 𝑎𝑙 + √𝑛𝑙 + 𝑛3/2
𝑟
3 𝑁0(4.16)

with 𝑎𝑙 and 𝑎𝑟 as the annihilation operators for the two wells, and 𝑛𝑙 and 𝑛𝑟 as
the respective number operators. At first, the interaction term looks unusual. The√
form of the interaction term is chosen so that 𝜇 = 𝜕𝐸/𝜕𝑁 ∝ 𝑁 for a single well

as required by the two-dimensional Gross-Pitaevskii equation. We have defined the
prefactor to make the chemical potential 𝜇0 when 𝑁0 atoms are in the well.
We can write the Hamiltonian in the Fock state basis of |𝑛⟩ ≡ |𝑁 + 𝑛⟩𝑙 |𝑁 − 𝑛⟩𝑟 :

𝐻

∑︁
𝑛

𝑐𝑛 |𝑛⟩ =∑︁
𝑛√√[−𝐽( 𝑁 + 𝑛 𝑁 − 𝑛 + 1𝑐𝑛−1 +√√𝑁 − 𝑛 𝑁 + 𝑛 + 1𝑐𝑛+1 )+
(︂
)︂
2𝜇0
3 (︁ 𝑛 )︁2
3/2
√ 𝑁
2+
𝑐𝑛 ] |𝑛⟩ ,
4 𝑁
3 𝑁0
76

(4.17)

dropping terms of order 𝑂(︁(︀ )︀ )︁
𝑛 4
𝑁and higher. In the spirit of [98], we write 𝑐𝑛±1 as

𝑐𝑛±1 ≡ 𝐶(𝑛 ± 1) = 𝐶(𝑛) ± 𝐶 ′ (𝑛) + 21 𝐶 ′′ (𝑛) + . . . and then expand the coefficients

𝐶(𝑛), 𝐶 ′ (𝑛), 𝐶 ′′ (𝑛) in powers of 1/𝑁 . Dropping constant terms and all orders higher
than 1/𝑁 gives us(︂𝜖𝐶(𝑛) =

−𝐽𝑁 𝜕𝑛2 +

(︂

𝐽
3 2𝜇0
1
√
+
𝑁
4 3 𝑁0 𝑁 1/2

)︂

2

𝑛

)︂
𝐶(𝑛).

(4.18)

This has the same form as the Schroedinger equation for the simple harmonic oscillator, and so the ground state is a Gaussian with the width⎞1/2⎛
𝛿𝑛2 = 𝑁 ⎝
𝐽+

𝐽
√

𝜇0

𝑁/𝑁0
2

⎠

.

(4.19)

4.6 Results
In order to demonstrate the efficacy of superfluid shielding, we performed several
different experiments. In Figure 4-3d-f, we present the time-of-flight patterns of thetenth Bloch oscillation after applying a tilt in two scenarios. Figure 4-3d is ourcontrol, with all atoms in the tilted state, and no superfluid shielding. It is clearthat there is no appreciable phase coherence left in the system. The time-of-flightpatterns for a sequence with superfluid shielding are shown in 4-3e-f. Subfigure e isthe shielding component, which we can see is not oscillating, demonstrating that |↑⟩atoms do not feel the gradient. Subfigure f is the oscillating component, whose crisp
peaks make clear that the phase coherence is protected by the superfluid shielding.
Finally, Figure 4-3g shows the contrast decay for an unshielded (red, for all figures)and shielded (blue, for all figures) system, which we have fit to an exponential toobtain the coherence lifetime. Even by eye, it is clear that the shielded system has alonger coherence lifetime.
Next, we quantitatively compare coherence lifetimes, with and without superfluid
shielding, over a range of chemical potentials. In the absence of technical fluctuations, the fundamental limit on coherence lifetime is given by the projection noise
77

Lifetime (cycles)(a) 7550

Projection Noise

25
0
500

600

700

800

900

1000

Chemical Potential (Hz)Lifetime (cycles)(b) 7550

Projectio
n

Noise

25
0
500

550

600

650

Chemical Potential (Hz)Figure 4-4: Superfluid shielding for different atom numbers and densities. Shown are
the exponential decay lifetimes of spin |↓⟩ Bloch oscillating component for unshielded(blue dots) and shielded (red squares) cases versus chemical potential. In (a), thechemical potential is changed by varying atom number from ∼6 × 103 to ∼2 × 104
while keeping the lattice depths at 10 𝐸𝑟 in both transverse directions. In (b), thechemical potential is varied by changing the transverse lattice depth from 4 𝐸𝑟 to
11 𝐸𝑟 . In both plots, the dashed lines represent the projection noise limit given by
two theoretical models.

78

Lifetime (cycles)

100
75
50

Projection Noise

25
0
0

0.25
|

0.5

0.75

Fraction

Figure 4-5: Contrast lifetime versus shielding fraction. Exponential decay lifetime ofspin |↓⟩ Bloch oscillating component upon varying the number of atoms in spin |↑⟩state (i.e. fraction of atoms in |↑⟩ state over total number of atoms) indicates thatshielding is effective beyond projection noise limits once more than 25% atoms arein the |↑⟩ state.
discussed above. An upper and lower bound for the projection noise limits are plotted
as dashed lines in the following figures. In Figure 4-4a, we vary the chemical potential by changing the total atom number. Recall that for a two-dimensional cloud in−1/2

a harmonic oscillator, |𝜕𝜇𝑗 /𝜕𝑁𝑗 | ∝ 𝑁𝑗

𝑈 1/2 . Since the lower bound is given by
1/2

Poissonian number fluctuations, which are ∝ 𝑁𝑗 , the lower limit is actually inde-

pendent of atom number, while the upper limit slightly increases with increasing atom

number. We observe that the shielded lifetimes in 4-4a are all above the projection
noise limit in either model, and are constant to within experimental uncertainty. In
Figure 4-4b, we vary the transverse (non-tilted direction) lattice depths while keeping
the atom number constant. This varies the chemical potential via 𝑈 , and we see thatthe projection noise limit decreases with increasing 𝑈 . For low 𝑈 , the shielded andunshielded lifetimes are consistent with the projection noise limits, but at higher 𝑈 ,the unshielded lifetimes are clearly below the projection noise limit, and the shielded
lifetimes clearly above it.
Having demonstrated that superfluid shielding with equal weight in both theshielding and oscillating components works over a range of parameters, we next ask
79

Contrast

0.4

0.2

0

120

140

160

180

Vertical Trap Frequency (Hz)Figure 4-6: Superfluid shielding of external fields, applied by increasing the trap
frequency. The vertical dashed line is the initial trap frequency that corresponds to alinear chemical potential for the |↓⟩ atoms. We compare the contrast after 25 Blochoscillation cycles for unshielded (blue dots) and shielded (red squares) components.
In this figure only, for technical reasons, one of the transverse lattices had a spacingof 392.5 nm.
how much of the shielding component is actually needed. If we are limited to a constant total atom number, then there must be a tradeoff: as we increase the fractionin the shielding component 𝑓 , the signal-to-noise of the measurement decreases by a√factor of 1 − 𝑓 . In Figure 4-5, we hold the total atom number constant and vary

the fraction that is in the shielding component to see that as long as the shielding
fraction is at least 20-30%, we get the full effect of superfluid shielding., and thatat 𝑓 ≈ 1/2, we find an increase in lifetime of a factor of 3.2. Thus, the increasein coherence time because of superfluid shielding is much more important than thedecrease in signal-to-noise because of the reduced atom number in the measurement
component. In principle, the chemical potential of the shielding component needs
only be of order the residual fluctuations in the oscillating component to provide fullsuperfluid shielding.
Since 𝛿𝑛↓𝑗 /𝑛↓𝑗 ≈ 0.02 is much less than the observed fraction required to achieve fullshielding, we most likely have technical sources of dephasing in our system. For unshielded samples in Figure 4-4, we may have introduced chemical potential curvature
if the lattice ramp was not fully adiabatic with respect to global density redistribu80

tion. However, this curvature would be common mode to both the |↑⟩ and |↓⟩ atoms,

and in principle, superfluid shielding should be able to compensate for this. To test
this, we intentionally added common-mode curvature to the system after applying thetilt, when the |↓⟩ density is already frozen in. In Figure 4-6, we increase the verticaltrapping frequency from 𝜔𝑖 = 2𝜋 × 128 Hz to a variable 𝜔𝑓 , adding a quadratic term)︀(︀𝛿𝑉𝑗 = 12 𝑚 𝜔𝑓2 − 𝜔𝑖2 𝑧𝑗2 to the chemical potential. It is clear that added curvature

drastically reduces the lifetime of an unshielded system, and that adding superfluid
shielding cancels much of this effect.
However, even taking this into account, the increase in shielded lifetimes seen in
Figure 4-4 are still lower than the factor of ∼ 100, as Equation 4.8 suggests, tells

us that there is some non-common-mode disorder in our system. We estimate thecurvature of the applied magnetic field to be ∼ 100 Hz across the sample, which isconsistent with the shielded lifetimes observed.

4.7 Conclusion
In this chapter, I have presented a new method to increase the phase coherence lifetimeof separated quantum systems — superfluid shielding. The shielding component,
which stays superfluid acts to “fill in the gaps” caused by technical and inherent
fluctuations in the chemical potential of the measuring component. We have used
Bloch oscillations as a measure of phase coherence to demonstrate that superfluid
shielding can increase the coherence lifetime beyond the projection noise limit. Not
only is this work directly applicable to force sensors based on Bloch oscillations [30],but it is also part of a growing body of work in atomic clocks [99, 100, 101] and atominterferometers [87, 89] demonstrating that interactions can be useful in precision
measurement.
While this work has focused on the phase coherence of the oscillating component,
there are also interesting dynamics in the superfluid component. For example, since
the mean field of the |↓⟩ atoms acts as disorder for the |↑⟩ superfluid, in certain

regimes, we should see localization of the |↑⟩ atoms as well. In addition, adding laser81

assisted tunneling to the tilted component [83, 84, 65] (see Chapter 3) would lead toan interacting system with a spin-dependent synthetic magnetic field.

82

Chapter 5
Spin-1 Heisenberg Hamiltonian
A major theme of research in ultracold atoms in recent years has been to attempt toachieve low-entropy spin-ordered ground states of magnetic Hamiltonians [102, 103,
104, 105, 106]. The paradigmatic magnetic Hamiltonian is the Heisenberg Hamiltonian, which describes the interactions of quantum spins pinned to a lattice and has
many interesting phases. Recently, there has been much success using fermions inquantum gas microscopes to study anti-ferromagnetic order [104, 105, 106], but despite the fact that BECs can achieve much lower temperatures than degenerate fermi
clouds, there has been less progress on the bosonic side.
In this chapter, I present a method for using a spinful bosonic Mott insulator
to study the spin-1 Heisenberg Hamiltonian [39, 40] and discuss preliminar results.
This Hamiltonian has a gapped magnetically ordered state known as the spin Mott
insulator, which can be used as a cold starting point for explorations of the phasespace. The next state that is most natural for us to look at is the 𝑥𝑦 -ferromagnet,
which is a spin superfluid and exhibits spin-charge separation. I begin by discussing
how we map our Bose-Hubbard Hamiltonian onto the Heisenberg Hamiltonian. I
then discuss the phase diagram and some special properties of the spin-1 Heisenberg
Hamiltonian. I conclude with dicussion of the current state of the experiment andproposals for moving forward.
835.1 Heisenberg from Bose-Hubbard
In order to study the spin-1 Heisenberg Hamiltonian, we use a two-component Mott
Insulator with two atoms per site. At first, this seems counter-intuitive — the groundstate of 87 Rb is in 𝐹 = 1 and so is already a spin-1 system. However, as we describe
here, using a pair of spin-1/2 bosons instead of a single spin-1 particles gives us more
flexibility and control over the system parameters.
First, we restrict our analysis to sites with exactly two atoms, both in the lowest
band. Then because of the symmetry requirements for bosons, the pair of atoms
must be in the triplet spin state. For this chapter, we use |↑⟩ and |↓⟩ to represent thephysical atomic spins and |+1⟩, |−1⟩, and |0⟩ for the effective spins in the Heisenberg
Hamiltonian. We identify the effective spins with the physical states as follows:|+1⟩ = |↑, ↑⟩1|0⟩ = √ (|↑, ↓⟩ + |↓, ↑⟩)
2

(5.1)

|−1⟩ = |↓, ↓⟩ .
In a one-dimensional lattice, the physical Hamiltonian is given by the multi-spin
Bose-Hubbard Hamiltonian [39, 40])︁)︁∑︁ (︁ †
∑︁ (︁ †𝑎↑,𝑗 𝑎↑,𝑗+1 + ℎ.𝑐. − 𝑡↓𝑎↓,𝑗 𝑎↓,𝑗+1 + ℎ.𝑐.
𝐻 = − 𝑡↑
𝑗

𝑗

)︂ (︂
)︂
∑︁ (︂
1
1
+ 𝑈↑↓
𝑛↑,𝑗 −𝑛↓,𝑗 −
2
2𝑗
𝑈 ∑︁+[𝑛↑,𝑗 (𝑛↑,𝑗 − 1) + 𝑛↓,𝑗 (𝑛↓,𝑗 − 1)]
2 𝑗
∑︁
∑︁
− 𝜇↑
𝑛↑,𝑗 − 𝜇↓
𝑛↓,𝑗
𝑗

(5.2)

𝑗

where 𝑎𝜎,𝑗 and 𝑛𝜎,𝑗 are the annihilation and number operators for a given spin andlattice site. 𝑈 is the intra-spin interaction, which is assumed to be symmetric, while
the inter-spin interaction 𝑈↑,↓ can vary. The chemical potentials 𝜇↑ and 𝜇↓ are set
by the total atom number. To find the effective Hamiltonian, we restrict ourselves to
84

(a)

(b)

t↑

t↓

t↓

t↑

t↑

t↑

t↓

t↓

Figure 5-1: Cartoon of the superexchange process. Superexchange is a second orderprocess, with two tunneling events an intermediate virtual particle-hole excitation,
which is mapped to terms in the spin Hamiltonian. (a) Superexchange that leads tospin changes give 𝐽⊥ . (b) Superexchange terms that leave spin unchanged give 𝐽𝑧 .
sites with exactly two atoms and define the spin operators

𝑆𝑗+ =𝑎†↑,𝑗 𝑎↓,𝑗𝑆𝑗− =𝑎†↓,𝑗 𝑎↑,𝑗

(5.3)

1
𝑆𝑗𝑧 = (𝑛↑,𝑗 − 𝑛↓,𝑗 ) .
2
We can use second-order perturbation theory to find the forms of the coefficients inthe spin-1 Heisenberg model:]︂∑︁ [︂ 𝐽⊥ (︀∑︁ (︀ )︀2
∑︁
)︀
+ −
− +
𝑧 𝑧
𝐻𝑒𝑓 𝑓 = −𝑆𝑗 𝑆𝑗+1 + 𝑆𝑗 𝑆𝑗+1 + 𝐽𝑧 𝑆𝑗 𝑆𝐽+1 + 𝑢
𝑆𝑗𝑧 − ℎ𝑆𝑗𝑧 . (5.4)
2𝑗𝑗𝑗Figure 5-1 shows examples of the sort of processes that go into 𝐽⊥ and 𝐽𝑧 , known assuperexchange. Figure 5-1a shows one process that lowers the spin in the right well by
one while raising the spin in the left well. In the intermediate state, there is an energy
deficit of 𝑈 , so the matrix element 𝐽⊥ ∝ 𝑡↑ 𝑡↓ /𝑈. In Figure 5-1b, the state goes back(︀)︀
to the initial state through two different paths, so we can see that 𝐽𝑧 ∝ 𝑡2↑ + 𝑡2↓ /𝑈.
85

(︀ )︀2
The 𝑆𝑗𝑧 term 𝑢 is the energy difference between having two atoms of the same spin
on one site and one of each, and ℎ is the difference in chemical potentials. The fullanalysis [39, 40] gives the following coefficients:

4𝑡↑ 𝑡↓
𝑈𝑡2↑ + 𝑡2↓
𝐽𝑧 =2
𝑈

𝐽⊥ =

(5.5)

𝑢 =𝑈 − 𝑈↑↓ℎ =𝜇↑ − 𝜇↓ ,and so when 𝑡↑ = 𝑡↓ = 𝑡, the spin term is symmetric:)︀𝐽⊥ (︀ + −4𝑡2 ⃗ ⃗+𝑧
𝑆𝑗 · 𝑆𝑗+1 .
𝑆𝑗 𝑆𝑗+1 + 𝑆𝑗− 𝑆𝑗+1
+ 𝐽𝑧 𝑆𝑗𝑧 𝑆𝐽+1
=
2
𝑈

(5.6)

5.1.1 Spin-Dependent Lattice
Our experiments are all done in an optical lattice, which allows us to easily vary 𝑡
and 𝑈 , but the novel piece of the experiment comes from our control over 𝑈↑↓ . If the
Wannier functions for the two atomic species are 𝑤↑,↓ (⃗𝑟), we can write

4𝜋~2 𝑎↑↓
𝑈↑↓ =
𝑚

∫︁

𝑑⃗𝑟 |𝑤↑ (⃗𝑟)|2 |𝑤↓ (⃗𝑟)|2 ,(5.7)

where for 87 Rb, the S-wave scattering length 𝑎↑↓ is approximately spin-independent.
In other atoms, including 7 Li, Feshbach resonances are available to vary the ratio of𝑎↑↓ /𝑎↑↑ , but for 87 Rb, the Feshbach resonances are either too lossy or at too high of afield. Instead, we focus on the second part of Equation 5.7, the overlap between the
Wannier functions of the two spins.
To do this, we use an effect known as the vector AC Stark shift. In Section 2.1,
we introduced the scalar AC Stark shift by treating an atom as a two-level system
exposed to an oscillating electromagnetic field. The AC Stark shift associated with a86specific intermediate state is given by

𝑉 = −2𝜋𝛼𝑎20 𝐼|𝑀𝑔𝑖 |2,𝛿(5.8)

where 𝑀𝑔𝑖 is the dipole matrix element between the ground state and the intermediate
state being off-resonantly addressed and 𝛿 is the detuning from that resonance. (Forsimplicity, we have dropped the counter-rotating term.) If 𝛿 is large, we can generally
ignore the fine and hyperfine structure, and treat the laser field as addressing one
combined resonance. However, closer to the atomic resonance, this assumption breaks
down and we have to take different 𝑀𝑔𝑖 and 𝛿𝑖 into account. In general, the matrix
elements will depend on the starting state, the intermediate state, and the polarizationof the light. We will restrict ourselves to looking at circularly polarized light 𝜎 + or 𝜎 −and will look at the combined effect for the D1 line and the D2 line. We will denote+,−the possible matrix elements as 𝑀(↑,↓)(1,2). We also assume that |↑⟩ and |↓⟩ form apair in the ground state manifold |𝐹, ±𝑚𝑓 ⟩ . Finally, we define the energy difference

between the D1 and D2 lines ∆𝐹 𝑆 and let the detuning 𝛿 be the detuning from the
midpoint between the lines.
We write the potentials as

⎛⃒
⃒
⃒
⃒ ⎞
⃒ +,− ⃒2
⃒ +,− ⃒2
⃒𝑀(↑,↓),2) ⃒ ⎟⎜ ⃒𝑀(↑,↓),1) ⃒+,−𝑉(↑,↓)= −2𝜋𝛼𝑎20 𝐼 ⎝+⎠𝛿 + ∆𝐹 𝑆 /2 𝛿 − ∆𝐹 𝑆 /2

(5.9)

and by symmetry, we can say

𝑉↓+ =𝑉↑−𝑉↓− =𝑉↑+ .

(5.10)For a given polarization, the two spins will feel different potentials. We define thecommon-mode part of the potential to be the scalar AC Stark shift and the differential
potential to be the vector AC stark shift. In general, when we interfere two linearlypolarized laser beams of the same frequency, we will get both an intensity lattice, as
usual, and a polarization lattice, where the polarization changes from 𝜎 + to linear87and then 𝜎 − and back. In this case, we can associate the intensity lattice with thescalar shift and the polarization lattice with the vector shift. With a few symmetry
relations, we can say

11−𝛿 + ∆𝐹 𝑆 /2 𝛿 − ∆𝐹 𝑆 /2
∆𝐹 𝑆
∝ 2
𝛿

𝑉𝑣𝑒𝑐 ∝(5.11)

for large 𝛿 .
Now if we use the states |1, ±1⟩ and a polarization lattice, we can separate thetwo spins and therefore reduce 𝑈↑↓ . There are two ways to do this. The conceptuallysimplest way has us creating two independent lattices at the same wavelength, onewith a polarization 𝜎 + , and one with 𝜎 − . We retro-reflect both of them off thesame mirror, which forces the standing waves to both have a zero at that point. We
can give them a slight frequency offset 𝛿𝑓 , and if the distance between the atoms
and the retro-reflecting mirror is 𝑙, then at the atoms, the two lattices will have aphase difference of 2𝜋 2𝑙𝛿𝑓
. This is most easily implemented by using a double-pass𝑐
AOM on one of the arms to give a variable frequency offset. Unfortunately, this isrelatively power inefficient, so for most of this work, we use a second method. In
this method, the input beam is linearly polarized, and in the retro-reflecting path,
we have a quarter-waveplate and liquid crystal phase retarder, which when combinedand double-passed rotate the linear polarization by a variable amount. Two linearlypolarized beams with their polarizations at an angle to each other create lattices of

𝜎 + and 𝜎 − light with a phase offset between the sublattices that varies from 0 when
the polarizations are parallel to 𝜋 when they are perpindicular.
Finally, we note that while the vector Stark shift is non-zero at large detunings,
the scalar-to-vector ratio scales as 𝛿 . At high scalar-to-vector ratios, two effects areimportant. First, the effect of the curvature of the gaussian envelope of the latticebeam, which is usually negligible, is given by the scalar Stark shift, and so becomesdominant at large 𝛿 . In addition, if we have any polarization impurity when we areattempting to separate the sublattices by 𝜋 , this will lead to a small intensity lattice
whose strength scales with the scalar Stark shift. Therefore, at large detunings, we88are extremely sensitive to polarization fluctuations or impurities, leading to a practial
upper limit on the detuning at which we can work.

5.2 Spin-1 Heisenberg Hamiltonian Phase Diagram
Before going into the details of the experiment, we wish to discuss the phase diagramof our system, as well as the states that are available to us, the 𝑥𝑦 -ferromagnet andthe spin Mott insulator. Following [40], we can make the ansatz that the ground state
wavefunction of the spin-1 Heisenberg Hamiltonian will have the form

|Ψ⟩ =

∏︁ (︂
𝑗

(︂ )︂
(︂ )︂ (︁(︁ 𝜒 )︁
(︁ 𝜒 )︁
)︁)︂
𝜃
𝜃
𝑖𝜑
−𝑖𝜑
𝑖𝜂
cos
𝑒 cos
|1⟩𝑗 + 𝑒 sin|−1⟩𝑗|0⟩𝑗 + 𝑒 sin
.
2
2
2
2
(5.12)

We wish to look for the phase transition between the 𝑥𝑦 -ferromagnet, which is aspin superfluid, and the spin Mott insulator, so in analogy with charge superfluids,
which use ⟨𝑎⟩ as their order prarameter, we look at(︁ 𝜒 )︁)︁
(︁(︁ 𝜒 )︁
⟨︀ − ⟩︀1
𝑆𝑗 = √ 𝑒𝑖𝜂 sin 𝜃 𝑒−𝑖𝜑 sin+ 𝑒𝑖𝜑 cos22
2

(5.13)and define the order parameter 𝜓 = sin 𝜃. We then plug our ansatz into the Hamiltonian to get the energy functional

𝜃
𝜃
𝜃
𝐽⊥
sin2 𝜃 (1 + sin 𝜒 cos 2𝜂) − 𝐽𝑧 sin4 cos2 𝜒 + 𝑢 sin2 − ℎ sin2 cos 𝜒
2
2
2
2
)︁2
(︁
)︁
(︁
√︀
√︀
𝐽⊥ 2
=−
𝜓 (1 + sin 𝜒 cos 2𝜂) − 𝐽𝑧 1 − 1 − 𝜓 2 cos2 𝜒 + (𝑢 − ℎ cos 𝜒) 1 − 1 − 𝜓 2
(︂ 2
)︂
𝐽⊥
𝑢 ℎ
= −
(1 + sin 𝜒 cos 2𝜂) + − cos 𝜒 𝜓 2 ,2
4 4(5.14)

𝐸 =−where in the last line, we have dropped all terms higher order than 𝜓 2 . Our experiments are done with equal |↑⟩ and |↓⟩ atom number, so we can set ℎ = 0. In addition,
we see that sin 𝜒 cos 2𝜂 is bounded in [−1, 1] , and so the energy is minimized whensin 𝜒 cos 2𝜂 = 1 and(︁𝑢 )︁ 2𝐸 = −𝐽⊥ +𝜓 .
489(5.15)

(a)

(b)

U↑↓/U

+
+

+
+

(c)

U/t
Figure 5-2: (a) Phase diagram of the spin-1 Heisenberg Hamiltonian with DMRG
calculations from [39]. The color scale gives the calculated gap in units of 𝑡 for asystem of 12 particles on 6 sites. (b) At high 𝑈↑↓ , 𝑢 is small and the ground state isan 𝑥𝑦 -ferromagnet, with a superposition of |0⟩ and |±1⟩ on each lattice site. (c) When𝑈↑↓ is small, 𝑢 dominates the system and the ground state is a spin Mott insulator,
with |0⟩ on every site.
The phase transition occurs where the energy switches from being minimized by 𝜓 = 0
to being minimized by a non-zero 𝜓, or in other words, when

1
𝐽⊥
= ,
𝑢
4

(5.16)

which is the black line plotted in Figure 5-2a, along with DMRG calculations from[39]. We should note that the DMRG calculations give a phase transition at a higher𝐽⊥ /𝑢 than the mean field calculations done here, but this is not surprising becausemean field calculations are known to be innacurate for one-dimensional systems.

5.2.1

𝑥𝑦 -Ferromagnet

The variational ansatz, despite its slight disagreement with some DMRG calculations,provides intuition for the form of the wavefunction and can be used to calculate
observables useful for our experiment. We first want to look at the properties of the“deep” 𝑥𝑦 -ferromagnet.
90

First, we set 𝜒 = 𝜋/2 and 𝜂 = 0, which minimize Equation 5.14 as discussed
above. For the deep 𝑥𝑦 -ferromagnet, the order parameter 𝜓 is maximized, and since𝜓 = sin 𝜃, we set 𝜃 = 𝜋, giving us a wavefunction of
(︂ 𝑖𝜑
)︂)︂
∏︁ (︂ 1𝑒1𝑒−𝑖𝜑
√ |0⟩𝑗 + √√ |1⟩𝑗 + √ |−1⟩𝑗.
|𝜓𝑥𝑦 ⟩ =22
2
2
𝑗(5.17)

From this, we immediately see that the population on any site is in a superpositionof |0⟩ and |±1⟩, with weights 1/2, 1/4, and 1/4, respectively. We also see that there
is a free phase 𝜑, which does not appear in the energy function Equation 5.14. This

leads to Goldstone modes and gapless excitations from long-wavelength variations in

𝜑 across the system, and thus Equation 5.17 gives the ground state when all 𝜑𝑗 areequal. In order to get a better intuition of the form of these excitations, we can write
the wavefunction in the atomic basis|𝜓𝑥𝑦 ⟩ =

∏︁
𝑗

[︃ (︀

)︀ (︀ 𝑖𝜑 /2)︀ ]︃
𝑒𝑖𝜑𝑗 /2 |↑⟩ + 𝑒−𝑖𝜑𝑗 /2 |↓⟩𝑒 𝑗 |↑⟩ + 𝑒−𝑖𝜑𝑗 /2 |↓⟩
√
√
⊗
.
2
2

(5.18)

From this, it is clear that the 𝑥𝑦 -ferromagnetic ground state has exactly two atoms
on a site, each with a spin pointing along the equator of the Bloch sphere with adirection given by 𝜑. Thus, variations in 𝜑 should be interpreted as gapless spinwaves, and we can think of the 𝑥𝑦 -ferromagnet as a superfluid in the spin sector.
Interestingly, the state always has exactly two atoms per site, and normal charge
tunneling is forbidden, so the state is still a charge insulator. Therefore, the 𝑥𝑦 ferromagnet shows spin-charge separation, a very unusual and interesting property.

5.2.2 Spin Mott
The 𝑥𝑦 -ferromagnet occurs in both the spin-1/2 and the spin-1 Heisenberg Hamiltonian, but we chose to implement the spin-1 version for the following state, which(︀ )︀2
occurs when the 𝑆𝑗𝑧 term is dominant. We call this state the spin Mott, and in
the same way that the 𝑥𝑦 -ferromagnet is a superfluid in spin space, this is a Mott
91

insulator. Deep in the spin Mott, the wavefunction is given by

|𝜓⟩ =

∏︁
𝑗

|0⟩𝑗 =)︁∏︁ 1 (︁√ |↑, ↓⟩𝑗 + |↓, ↑⟩𝑗2
𝑗(5.19)and the state has a gap of 𝑢.
The form of the ground state is clearest in the limit of 𝑢 = 𝑈 , or when 𝑈↑↓ = 0,
which occurs when the 𝜎 + and 𝜎 − lattices are offset by 𝜋. Then, we can think of thesystem as two interleaved, but non-interacting Mott insulators, each with exactly one
atom per site. It is obvious then that we must have exactly one atom of each type
per site, and that we are protected by the normal Mott insulator gap 𝑈 , which shouldmake adiabatic ramps into the spin Mott easier than into the 𝑥𝑦 -ferromagnet.
In our experimental system, we have an additional harmonic confining potential,
which forces us to use the local density approximation. Thus, when the center of thecloud is in the spin Mott state, it will be surrounded by a superfluid, which has nogap. For a system in thermal equilbrium, entropy will be pushed from the gapped
to the gapless part of the cloud, leaving the center at a locally lower entropy than
average. This acts as additional cooling if we are later able to selectively measure theMott insulating shell only.

5.3 Experiment
Our goal is to ramp into the ground state of the 𝑥𝑦 -ferromagnet by using the spin Mott
insulator as a cold starting point. We start with a BEC of |𝐹 = 1, 𝑚𝑓 = −1⟩ atoms ina bias field of ∼ 9 G, after using a dimple trap and side evaporation to precisely control

the atom number. We then convert the atoms into an equal mixture of |1, 1⟩ = |↑⟩and |1, −1⟩ = |↓⟩ atoms via two Landau-Zener transitions through |2, 0⟩ . A three-dimensional lattice is ramped up to take the system into a Mott insulating state. The
two transverse arms are made by spin-independent 1064 nm beams, while the latticearm along the direction of the magnetic field bias is our spin-dependent lattice. We use
a Ti-sapphire laser with a wide tuning range to create the spin-dependent lattice and92

determine the separation between the 𝜎 + and 𝜎 − sublattices by one of the methods
discussed in Section 5.1.1. We can then move the sublattices to ramp 𝑈↑↓ /𝑈 throughthe phase diagram.

5.3.1 On-Site Correlations
Once we have prepared a state, we first wish to measure the on-site correlations. For
this, we will ignore the phases 𝜑𝑗 and look only at the spin population on a site.
Since we know that we have an equal number of |↑⟩ and |↓⟩ atoms, measuring thefraction of sites that are in |0⟩ will also tell us how many are in |±1⟩ . We also need tonote that because the harmonic traps cause density variation across the cloud, not all
atoms are in an 𝑛 = 2 Mott insulating state. Thus, we need to measure the fractionof atoms in |0⟩ relative to those in sites which have exactly two atoms, which we callthe “spin-paired doublon fraction,” or SPDF.

Figure 5-3 demonstrates how we measure the SPDF in three different experimental
runs. At the end of each run, we ramp up the lattice height and merge the spindependent sublattices together, projecting the system to a set of independent sites,
each with a fixed atom number and spin configuration. In the first run, we measure
the total atom number 𝑁 using an intrap absorption image. For the second run, wewant to measure the number of atoms in |0⟩. This is done through the use of a lossy

Feshbach resonance [107, 108]. We use an RF sweep to convert the |1, −1⟩ atoms to|2, −2⟩, which have a Feshbach resonance with |1, 1⟩ at 9.045 G. We then sweep thebias field to this value and modulate it for 25 ms. During this time, the sites thatstarted in |0⟩ (i.e. the sites with one atom of each spin) will decay through Feshbachloss, while the |±1⟩ sites stay stable. Measuring the remaining atom number andsubtracting from 𝑁 gives us the number of atoms in |0⟩ . Finally, we need to measure

the fraction of atoms that are on an 𝑛 = 2 site. We first use an RF sweep to flip
all of the |1, −1⟩ atoms to |2, 0⟩, which has a lossy Feshbach resonance with |1, 1⟩ at

9.091 G. As before, we shift and modulate the bias field, but that is not enough tokill all of the 𝑛 = 2 sites, so we additionally perform about ten 50:50 Landau-Zenersweeps between |1, 1⟩ and |2, 0⟩ . If a site started in |0⟩ , then it will decay after the93Measure |0⟩|1, +1⟩|1, -1⟩|1, +1⟩|2, -2⟩|1, -1⟩|1, -1⟩|2, -2⟩|2, -2⟩

Measure n = 2|1, +1⟩|2, 0⟩|2, 0⟩|1, +1⟩(b)
Atom Number (arb.)(a)Measure spin pairs
Measure all doublons

600
500
400
300
0

5

10

15

Hold Time (ms)Figure 5-3: Description of our measurement for on-site correlations. (a) To measure
the spin-paired doublon fraction (SPDF), we used three experimental runs. First, wemeasure the total atom number. In the second run, we flip the |1, −1⟩ atoms into
|2, −2⟩, which has a lossy Feshbach resonance with |1, 1⟩, so all of the sites with one
spin of each type will decay. For the third run, we flip the |1, −1⟩ atoms into |2, 0⟩
and then perform several incomplete Landau-Zener sweeps between it an |1, 1⟩. This
causes all of the sites with two atoms to decay. (b) Here we plot the atom number
remaining as a function of hold time during the Feshbach loss process for a spin Mott
initial state. We see that for both of the processes, the loss saturates by 10-15 ms,
which is much shorter than any other relevant time scales in the system.

first sweep. On the other hand, a site that started in |±1⟩ has a 50% chance to turnhave one of each spin and therefore decay after a 50:50 Landau-Zener sweep. Afterthe ten sweeps, all but a negligible fraction ((1/2)10 ∼ 0.1%) of the 𝑛 = 2 sites havedecayed, while 𝑛 = 1 sites are unaffected. Again, by measuring the atom number

remaining and subtracting from 𝑁 , we can get the total number of atoms in 𝑛 = 2
sites. The ratio of these two results gives us the SPDF.
There are three important possible results that we could get when we measure the
SPDF. First, look at a system in a thermal state with a temperature 𝑇, larger than the
Heisenberg energy scales, but lower than the Mott insulator gap, 𝑢, 𝐽⊥ , 𝐽𝑧 ≪ 𝑇 ≪ 𝑈.

Then the three possible spin states should be equally populated, and the SPDF will be

1/3. For a spin Mott insulator or an 𝑥𝑦 -ferromagnet, the wavefunctions from Section
5.2 give an SPDF of 1 and 1/2 respectively.
94

5.3.2 Off-Site Correlations
While the spin Mott is a relatively simple system, with no off-site correlations, muchof the 𝑥𝑦 -ferromagnet’s interesting properties come from offsite correlations. Recall
that the ground state wavefunction is(︂ 𝑖𝜑𝑗)︂]︂
∏︁ [︂ 1𝑒
1𝑒−𝑖𝜑𝑗
√ |0⟩ + √√ |1⟩ + √ |−1⟩ .
|𝜓𝑥𝑦 ⟩ =22
2
2
𝑗(5.20)

If all of the 𝜑𝑗 are equal, then we are in the ground state. On the other hand, nonconstant 𝜑𝑗 are caused by some excitations or spin waves. We can quantify the level⟨︀⟩︀of excitation with the spin correlation 𝑆0+ 𝑆𝑗− + 𝑆0− 𝑆𝑗+ . There are two ways in which
we can measure the correlation length.

Fluctuation Analysis
To understand the first way to measure correlation length, write the wavefunction in
the atomic basis|𝜓𝑥𝑦 ⟩ =

∏︁
𝑗

[︃ (︀

)︀ (︀ 𝑖𝜑 /2)︀ ]︃
𝑒𝑖𝜑𝑗 /2 |↑⟩ + 𝑒−𝑖𝜑𝑗 /2 |↓⟩𝑒 𝑗 |↑⟩ + 𝑒−𝑖𝜑𝑗 /2 |↓⟩
√
√
⊗
2
2

(5.21)and note that we can interpret this wavefunction as having two atoms on each site,each of which is on the equator of the Bloch sphere, with a direction given by 𝜑𝑗 .
To measure the correlation length, we apply a 𝜋/2-pulse and measure the resulting
polarization, defined as (𝑁↑ − 𝑁↓ ) /𝑁. Even in the perfect case of constant 𝜑𝑗 acrossthe cloud, the result will be random, as it depends on 𝜑𝑚𝑤 − 𝜑𝑗 , where 𝜑𝑚𝑤 is theuncontrolled phase given by our microwave source and 𝜑𝑗 is spontaneously chosen
by the system. However, looking at the fluctuations in the polarization over many
iterations can tell us the number of domains and therefore the correlation length.
Each domain, after the 𝜋/2-pulse, becomes a patch of constant polarization. Since
the domains are uncorrelated, the fluctuation of the average polarization will scale as√√
1/ 𝑁domains ∼ 𝑙𝑐𝑜𝑟𝑟 , where 𝑙𝑐𝑜𝑟𝑟 is the correlation length. Note that other fluctua-

tions (e.g. fluctuations in initial polarization) will add to this measure, and so this95

can only give us an upper bound on the correlation length.

Arbitrary Order Spin Correlations
While the above method can give us a measure of the correlation length, it makes
no distinction between true random domains and something like a phase gradientcaused by a slight magnetic field gradient. In order to distinguish between the two,
we present a method that uses the spin-dependent lattice to measure arbitrary orderspin correlation functions in an 𝑛 = 1 𝑥𝑦 -ferromagnet.
Look at the wavefunction at sites 0 and 𝑗 of the 𝑥𝑦 -ferromagnet

𝑒𝑖𝜑0 |↑⟩0 + |↓⟩0 𝑒𝑖(𝜑0 +𝛿) |↑⟩𝑗 + |↓⟩𝑗|𝜓𝑥𝑦 ⟩ =⊗⊗ |𝜓𝑒𝑙𝑠𝑒 ⟩22(5.22)

where 𝛿 = 𝜑𝑗 − 𝜑0 and |𝜓𝑒𝑙𝑠𝑒 ⟩ is the wavefunction on all sites except for 0 and 𝑗 . Thenthe spin correlation can be related to 𝛿 as⟨︀⟩︀
𝐶𝑆 = 𝑆0+ 𝑆𝑗− + 𝑆0− 𝑆𝑗+𝑒𝑖𝛿 𝑒−𝑖𝛿+44cos(𝛿).
=
2=(5.23)We define 𝑅𝜋/2 to be a 𝜋/2-rotation operator, and the shift operator 𝑆𝑗 to be theoperator that shifts |↓⟩ atoms over by 𝑗 sites, so that |↓⟩ atoms that start on site 𝑘 end

up at the same position as |↑⟩ atoms starting at site 𝑘 + 𝑗. This can be implemented

by adding 2𝜋𝑗 to the phase difference between 𝜎 + and 𝜎 − lattices [109]. Finally,

we denote the effect of interaction term of the Bose Hubbard Hamiltonian 𝑊 (𝑡) =∑︀
exp[−𝑖𝑈 𝑘 (𝑛↑,𝑘 + 𝑛↓,𝑘 ) ((𝑛↑,𝑘 + 𝑛↓,𝑘 ) − 1)] , where we have assumed that 𝑈↑↓ = 𝑈,
as is valid when the phase difference between the sublattices is a multiple of 2𝜋.
We can measure 𝛿 by use of a Ramsey-style sequence:|𝜓𝑓 ⟩ = 𝑅𝜋/2 𝑆𝑗† 𝑊 (𝑡) 𝑆𝑗 𝑅𝜋/2 |𝜓𝑥𝑦 ⟩ .
96(5.24)

Let us go step-by-step to understand this sequence. First, the 𝜋/2-rotation transforms the state on each site into a mixture of |↑⟩ and |↓⟩ with weights sin2 (𝜑𝑗 /2)and cos2 (𝜑𝑗 /2) , respectively. We have converted phase information into population

information. 𝑊 (𝑡) 𝑆𝑗 allows the |↓⟩ component on site 0 to interact with the |↑⟩component on site 𝑗 . If there is a non-zero 𝛿 , then the shift operation will move more
atoms away from site 𝑗 and the site receives from site 0 or vice versa. Then, since
the action of 𝑊 (𝑡) depends only on the total atom number per site, a phase will be
imparted that directly depends on the value of 𝛿 . We then shift the lattice back,and perform a second 𝜋/2-rotation, which converts the phase information created by

𝑊 (𝑡) into population on site 0.
Measuring the polarization on site 0 gives us⟨𝑛↑,0 − 𝑛↓,0
𝑛↑,0 + 𝑛↓,0

⟩

2

[︂

= −cossin[𝛿]=
4]︂
𝜑0 + 𝛿
sin[𝑈 𝑡] sin[𝜑0 ]2(5.25)

where for the second line we have assumed that 𝑡 is chosen such that sin[𝑈 𝑡] = 1,
and we have averaged over 𝜑0 . Thus, we can measure spin correlations to arbitrary
distances, limited only by how quickly we can apply the shift operator without losing
phase coherence.

5.4 Results
Our first experiments looked at the on-site spin correlations of a Mott insulator withvarying distances between the spin-dependent sublattices. The spin-dependent lattice
was given a polarization structure using the liquid crystal phase retarder and quarterwaveplate setup discussed in Section 5.1.1, and was held at a wavelength of 810 nm,
slightly red-detuned of both the D1 and D2 line. With the liquid crystal phaseretardance held constant, we raised up a three-dimensional lattice to 25 ER in thetransverse directions, and 10 ER in the spin-dependent lattice direction. In Figure 54, we plot the spin-paired doublon fraction versus the ratio of inter-spin to intra-spin
97

1

SPDF

0.8
0.6
0.4
0.2
0

0

0.2

0.4

0.6

0.8

1

U↑↓/U

Figure 5-4: Spin paired doublon fraction after ramping directly into the Mott insulator at varying 𝑈↑↓ /𝑈 . For this experimental run, we used two 25 ER lattices in thetransverse directions and an 10 ER spin-dependent lattice. The dashed line marks the
expected phase transition point calculated in [39]. We see that deep in the spin Mott,
we have a SPDF near one, and that at 𝑈↑↓ = 𝑈 , we have a thermal Mott insulator.
interactions 𝑈↑↓ /𝑈 at the final lattice depth.
Near 𝑈↑↓ /𝑈 = 0, we are in the deep spin Mott, and we see that the SPDF isapproximately 1. The phase transition from a spin Mott to an 𝑥𝑦 -ferromagnet occurs
at 𝑈↑↓ /𝑈 = 0.94 [39] and is marked by the dashed line. While we would expect the
SPDF to stay near one up until the phase transition, we instead see that it starts
to drop around 𝑈↑↓ /𝑈 = 0.4, indicating that our ramp is not perfectly adiabatic forsmall 𝑢. On the other end, at 𝑈↑↓ /𝑈 = 1, if we were perfectly cold, we would be in
the 𝑥𝑦 -ferromagnet, but we instead achieve a SPDF that is consistent with 1/3, athermal Mott insulator, as discussed above.
Instead, in order to prepare the 𝑥𝑦 -ferromagnet, we want to use the spin Mott
as a cold starting state. To do this, we first ramp the lattices up in the deep spin
Mott 𝑈↑↓ /𝑈 = 0 configuration before ramping the liquid crystal retardance and spindependent lattice intensity to keep the lattice depth constant but change the sublattice
separation and therefore 𝑈↑↓ /𝑈. In Figure 5-5, we plot the final SPDF at different final𝑈↑↓ /𝑈. The SPDF begins to drop before the phase transition, but at 𝑈↑↓ /𝑈 = 0.6,
rather than 0.4, and the SPDF at 𝑈↑↓ /𝑈 = 1 is between 1/3 and the 1/2 that we
98

1

SPDF

0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6

0.8

1

U↑↓/U

Figure 5-5: Spin paired doublon fraction after ramping from the spin Mott insulator
to varying 𝑈↑↓ /𝑈 . As before, we used two 25 ER lattices in the transverse directions
and an 10 ER spin-dependent lattice, and the dashed line marks the expected phasetransition point calculated in [39]. At 𝑈↑↓ = 𝑈 , we see that the SPDF is between 1/3
and 1/2, which tells us that we are doing some cooling with this sequence, but notenough to be truly in the 𝑥𝑦 -ferromagnet.
expect for an 𝑥𝑦 -ferromagnet. Our sequence is doing some cooling, but not enough.
At first, this is confusing, since we can consistently make what appear to behigh-purity spin Mott insulators. However, our measurements until now have looked
exclusively at spin temperature. The paradigmatic charge excitation in a Mott insulator is a “hole,” or a site with one fewer atom than expected. Our spin measurement
procedure cannot distinguish a hole from a site that is in the surrounding superfluid
or 𝑛 = 1 shell, and therefore normalizes them out. First, look at the effect of a holein the deep spin Mott, as in Figure 5-6a. In this regime, we can think of the system
as two interleaved 𝑛 = 1 Mott insulators — one for each spin. Therefore, a hole issimply an empty site in one of these two Mott insulators, and that site is unmeasured.
A hole on the |↑⟩ sublattice can only tunnel to another site on that lattice, and so
also with a |↓⟩ hole. Therefore, a hole tunneling event will simply change the positionof the excitation, but will not change the total number of sites with |0⟩ on them.

Figure 5-6b shows the effect of holes for a shallow spin Mott or an 𝑥𝑦 -ferromagnet.

If 𝑢 is comparable to the hole bandwidth ∼ 4𝑡, then from a site next to a hole, eithera |↑⟩ atom or a |↓⟩ atom can tunnel to fill the hole, regardless of the second atom
99

(a)
t↑

(b)

t↑

t↓
Figure 5-6: Cartoon of the effect of hole tunneling on the measured SPDF. (a) In
a deep spin Mott, a hole tunneling event will not change the SPDF, and therefore
charge and spin excitations are decoupled. (b) When 𝑢 / 4𝑡, some tunneling events(top) preserve SPDF, but others (bottom) can cause a spin excitation.

sitting on the hole site. If the tunneling atom happens to be the one “missing” on theother site, then the SPDF is unchanged. However, if it is the wrong spin state, thiswill create a spin excitation.
To measure charge excitations, after creating the state we are interested in, we
adiabatically ramp the lattices down to 0 ER and perform time-of-flight imaging. We
fit to a bimodal distribution and extract the condensate fraction, which is a measureof the entropy in the system. In Figure 5-7 we see the measured condensate fraction
for one- and two-component clouds at various spin-dependent sublattice differences.
We note that both adding a second spin component and moving towards 𝑈↑↓ /𝑈 = 0
significantly hurt the condensate fraction.
In order to get a quantitative understanding of the numbers involved in Figure
5-7, we calculate the relationship between the measured condensate fraction 𝐶𝐹 , andthe hole fraction at the center of the Mott insulator. For an ideal BEC of 𝑁 atoms,
100Condensate Fraction

1.

|1, -1⟩0.8|1, -1⟩+ |1, +1⟩

0.6
0.4
0.2
0.

0

π

π

π

6

3

2

2π
3

5π
6

π

Sublattice Phase Difference

Figure 5-7: Initial condensate fraction measurements after ramping back to BEC. For
these measurements, we ramped up a three-dimensional 20 ER lattice, held for 10 ms,
and ramped back to BEC before measuring the condensate fraction in time-of-flight.
The spin-dependent lattice was made with 810 nm wavelength light, and we varied
the phase difference between the 𝜎 + and 𝜎 − sublattices. At a phase difference of 0,𝑈↑↓ = 𝑈 , while a phase difference of 𝜋 corresponds to 𝑈↑↓ = 0. We report condensate
fractions measured for one component (blue) and two component (red) systems.

MI Excitation Fraction

0.6
0.5
0.4
0.3
0.2
0.1
0

0

0.2

0.4

0.6

0.8

1

Measured BEC Condensate Fraction

Figure 5-8: Calculated excitation fraction at the center of the Mott insulator as afunction of measured condensate fraction when ramped down to BEC.

101

the total entropy is given by

𝑆 = 3.602 𝑘𝐵 𝑁 (1 − 𝐶𝐹 ) .

(5.26)

To find the entropy associated with the Mott insulator, we ignore tunneling and lookat each individual site, which has an energy 𝐸𝑛 = 𝑈2 𝑛 (𝑛 − 1) − 𝑛𝜇. We can find thepartition function and average filling for a given temperature and chemical potential:

𝑍=

∞
∑︁
− 𝐸𝑛
𝑒 𝑘𝐵 𝑇(5.27)

𝑛=0− 𝑘𝐸𝑛𝑇∑︀∞⟨𝑛⟩ =𝑛=0 𝑛𝑒𝐵𝑍.

To find the entropy on a given site, we recall

𝐹 = − 𝑘𝐵 𝑇 ln[𝑍]𝑆 = − 𝑘𝐵𝜕𝐹.
𝜕𝑇

(5.28)

Finally, given a known harmonic confinement, we can make the local density approximation 𝜇 → 𝜇 − 𝑉 (⃗𝑟) and sum over all lattice sites to get a total entropy and atom
number. We numerically find the chemical potential and temperature required tohave the entropy and atom number in the Mott insulator match that which is measured in the BEC. From there, it is easy to find the excitation fraction at the center ofthe Mott insulator. Figure 5-8 plots the Mott insulator excitation fraction versus themeasured condensate fraction for representative parameters for a single-component𝑛 = 1 Mott insulator, which should be equivalent to our system at 𝑈↑↓ /𝑈 = 0. We
see that a measured condensate fraction of ∼ 0.65 is equivalent to about 10% charge
excitation fraction in the Mott insulator, and that lower condensate fractions arecorrespondingly worse.
Over the past year, we have performed several upgrades to the experimental sequence and apparatus, which have led to increased condensate fraction in the single
component system. In Figure 5-9, we plot the condensate lifetime of the system with102

Condensate Fraction

1

Phase shift 00.8

Phase shift π

0.6
0.4
0.2
0

0

200

400

600

800 1000 1200 1400

Hold Time (ms)Figure 5-9: Condensate fraction measurements after experimental upgrades. Here,we plot the condensate fraction lifetime measurements for a one-component systemwith 20 ER in all three directions and a spin-dependent lattice wavelength of 810.
These lifetimes were measured after the upgrades discussed in the text.
all three lattices at 20 ER , and a spin-dependent lattice wavelength of 810 nm. The
starting condensate fraction and 1/𝑒 lifetime at a phase shift of 0 are 0.91 and 1,400
ms, respectively, and are 0.81 and 280 ms for a phase shift of 𝜋. Both of these arenotably better than the previous condensate fractions, leading us to believe that wemay be able to ramp into the 𝑥𝑦 -ferromagnet as described above.
The most important upgrades are as follows:∙ We empirically found the optimal shape for a piecewise linear lattice ramp with[︀]︀
ten pieces. The optimal ramp closely matches a form of exp (𝑡/𝜏 )7 , which weuse in order to reduce the search space to one parameter whenever we change
the lattice depth or sublattice phase.

∙ We set all lenses at a small angle to the optical axis, to ensure that strayreflections from the lens do not hit the atoms.

∙ We ensure that all dipole beams, lattice beams, and retro beams are aligned tothe center of the cloud to within 2 𝜇m.

∙ We added passive low frequency vibration isolation with a corner frequency ∼10
Hz to the breadboard that supports the two non-vertical retroreflection mirrors.
103

The last two items seem initially surprising, since the conventional wisdom is that a
Mott insulator is robust against any noise with a frequency less than its gap at 𝑈.
While this is true, the initial purity of a Mott insulator state is extremely sensitive
to excitations during the lattice ramp up, where we cross a phase transition and theexcitation gaps close.

5.5 Spontaneous Scattering
It appears that in the spin Mott configuration, when the lattice intensity is highest,
the condensate fraction lifetime is limited in part by spontaneous scattering, which
is a process where an atom will accept a photon from the spin-dependent lattice to
go into a virtual excited state, which then decays to some spin state (maybe not
the same one) and emits a real photon. The emitted photon can destroy any phasecoherence the atom initially had and can scatter atoms into higher bands. For a single
intermediate state, the scattering rate takes the form

8𝜋𝛼2 𝑎40 𝐼 3
𝑅=
𝜔 |𝑀𝑖→𝑓 |22
3~𝑐(5.29)

where 𝛼 and 𝑎0 are the fine structure constant and the Bohr radius. The local
intensity of the lattice beam and its frequency are denoted 𝐼 and 𝜔 , and the second
order perturbation theory matrix element for the transition from intial spin state 𝑖to the final spin state 𝑓 is 𝑀𝑖→𝑓 . It is tempting to treat scattering from the D1 andD2 lines separately, and simply add the rates given by analyzing each one by itself.
This will work for large detunings, but is not valid close to the atomic resonance.
For a set 𝑖 and 𝑓 , we must linearly add the matrix elements associated with different
intermediate states, and then square the magnitude. To find the total scattering rateof an atom in state 𝑖, we can then add in quadrature the matrix elements associatedwith different final states 𝑓 , as these processes cannot interfere.
For far detunings 𝛿 , the scattering rate drops as ∼ 1/𝛿 2 , which is the same scaling

as the vector AC Stark shift discussed in Section 5.1.1. This seems to imply that104

Scattering Rate(s- 1 /ER, 780 nm )

0.2
0.15
0.1
0.05
0

750

800

850

900

λ (nm)

Figure 5-10: Calculated photon scattering rate of an atom sitting in a purely vector
AC Stark shift lattice of 1 ER, 780 nm . We see that at large detunings, the scattering
rate is flat with wavelength, but near resonance there is non-trivial structure andoptimization to be had.
the spin-dependent lattice wavelength does not matter, and that the heating rate will
always be the same. The analysis is more subtle for several reasons. First, look at
Figure 5-10, which shows the calculated scattering rate for a vector Stark shift of 1
ER, 780 nm . At wavelengths greater than ∼ 820 nm, the scattering rate dependence on
wavelength is flat, but closer to resonance, we can find scattering rates that are muchgreater.
In addition, not all spontaneous scattering is the same. As shown in Figure 5-11, at
far detunings, most of the scattering is Rayleigh scattering, which preserves the atomic
spin and will only add band excitations. On the other hand, at closer detunings andbetween the D1 and D2 lines, scattering is dominated by Raman scattering, where
the atom is flipped to a different spin state. If the new spin state has a different
magnetic moment than the initial spin state, after scattering, the atom will be sittingat the maximum rather than the minimum of the potential. This highly excited state
has much more energy than a band excitation.
Finally, although much of our work so far has been in the fully separated regime of𝑈↑↓ /𝑈 = 0, we have seen evidence that we can make a relatively cold spin Mott with𝑈↑↓ /𝑈 up to at least 0.4. For this configuration, the lattice potential has both vector
105

Scattering Ratio

1

Rayleigh Scattering

0.8

Raman Scattering

0.6
0.4
0.2
0
750

800

850

900

λ (nm)

Figure 5-11: Scattering fraction of Raman and Rayleigh scatteing off of a 𝜎 − lattice. We see that when the lattice wavelength is between the D1 and D2 lines,
spin-changing (Raman) scattering events are dominant, while at far detunings, spinpreserving (Rayleigh) scattering dominates.
and scalar Stark shift components, and so the required intensity scales somewhere
between 𝛿 and 𝛿 2 , leading to a scattering rate that decreases sub-linearly with 𝛿 at far
detunings. Therefore, we believe that the optimal wavelength for the spin-dependentlattice is as far detuned as possible given power constraints and the curvature andstability considerations in Section 5.1.1.

5.6 Conclusion
In this chapter, I have presented a method for using a two-component Mott insulator
and a spin-dependent lattice to create a highly tunable spin-1 Heisenberg Hamiltonian. We have already shown that we can create a spin Mott insulator that is coldin spin space and discussed possible upgrades to reduce the charge excitations. Once
this is done, we will be able to use the measures discussed here to read out bothon-site and off-site spin correlations through the clever use of RF fields, Feshbachresonances, and the spin-dependent lattice.
For future work, there are two main directions that could increase the flexibility
and stability of the system. First, we are currently working on designing and installing106

a quantum gas microscope that will have single site resolution. In addition to singlesite readout, this allows us to project arbitrary potentials, which can be used as a“cookie cutter” to remove entropy from the system once the spin Mott has pushed it
into the superfluid wings.
The second possible upgrade is the addition of a large gradient along the spindependent lattice direction. If we are in the spin Mott state, and then quickly apply
a gradient ∆ ≫ 𝑡, normal tunneling is frozen out. This will localize all the charge
excitations, which travel with 𝑡 and will not allow entropy flow from the superfluidedges to the center. In addition, for a strong enough tilt, the first excited band will
be strongly coupled to the continuum, and any band excitations will escape. This
will mean that instead of depositing a bandwidth and possibly a bandgap of energy
and the corresponding entropy into the system, each band excitation will result in
only one lost particle, which is much lower entropy. Finally, we note that this tilt,
while it blocks normal tunneling, still allows for superexchange. Instead of the normal

𝐽 = 𝑡2 /𝑈, the superexchange tunneling will be renormalized to 𝑡2 / (𝑈 − ∆), which

can be much higher than the normal 𝐽 and even change sign. This will allow us to
explore new regimes of the spin-1 Heisenberg Hamiltonian phase space, including thepossibility of observing the Haldane phase [110].

107

108

Chapter 6
Conclusion
The work in this thesis has covered applications of ultracold bosons in optical lattices
to both precision measurement and quantum simulation. It is unified in its focus on
the phase coherence between different lattice sites.
Chapter 4 presented the work most closely related to precision measurement andhad the most obvious connection to phase coherence. When condensates sitting on
different sites in a lattice are separated, either by applying a strong energy gradientor by rapidly increasing the lattice depth, each condensate’s phase evolves independently of the others, with a phase evolution given by the local energy. Since ina condensate, interactions are generally non-negligible, density fluctuations lead tonoise in the phase evolution and thereby dephasing between the condensates on separate sites. We presented and explored the technique of superfluid shielding, whichuses the interactions with a superfluid bath to cancel out the effects of density fluctuations, which allows the separated condensates to maintain phase coherence. We
have demonstrated that superfluid shielding can protect against fundamental projection noise as well as common-mode external curvature, and this method should
allow for greater coherence in BEC atom interferometers. In addition, the systemused here, where one spin feels a strong magnetic tilt and the other does not, shouldhave dynamics that are interesting in themselves, like localization of the superfluid
component or potentially a spin-dependent synthetic gauge field.
In Chapters 2 and 5, we present work on quantum simulation. Chapter 2 dis109

cusses our simulation of the Harper-Hofstadter Hamiltonian, which describes charged
particles in a high magnetic field. The ground state of a simple lattice superfluid,
without a magnetic field, has a simple phase pattern — every site has a coherent stateof atoms on it, all of which have the same phase. On the other hand, the groundstate of a Harper superfluid has the same constant density, but a complex phase pattern, like that of a vortex lattice. In our system, this phase pattern is given by thecomplex tunneling that we have added to the system using a tilt and laser-assisted
tunneling. However, in general, if we start with a trivial superfluid and immediately
switch to a novel Hamiltonian, we will end in some highly excited mixture of states
without phase coherence between lattice sites. We have explored two ways to getaround this and transform a trivial superfluid into a ground-state Harper superfluid.
In the first, we use a two-dimensional lattice, so that the condensate on each site isa long one-dimensional tube. Immediately after switching from the Bose-Hubbard
to the Harper-Hofstadter Hamiltonian, there is no phase coherence between the lattice sites, as we feared. However, the excitations in the lattice direction quickly
thermalize with the tube direction, which has many low-lying excited states. Eachtube needs only accept ∼1 𝑘𝐵 of entropy to fully restore phase coherence, and so

we still achieve a cold sample. This method allowed us to study the single particle
physics of the Hamiltonian, including demonstrating the gauge-dependent nature ofsome observables. To observe the result of strong interactions, we needed a threedimensional lattice, so this loading technique was unavailable. Instead, we createda many-body adiabatic sequence. In this loading sequence, we started with a trivial
superfluid and adiabatically ramped into the deep Mott insulator, where the groundstate has a definite number on each site, and therefore no well-defined phase. Then,we switch the Hamiltonian to the Harper-Hofstadter Hamiltonian with interactions,which also has a Mott insulating regime. From there, we “melt” the Mott insulator
into a relatively cold Harper superfluid. This allowed us to study the lifetime of theinteracting Harper superfluid near the Mott insulating transtion and will be a useful
tool in future simulations of interacting systems.
The work on simulating the spin-1 Heisenberg model is discussed in Chapter 5.
110

Since phase is a continuous variable, the presence of phase correlations is generally
associated with gapless excitations. Because we work exclusively in a Mott insulatingshell, one may think that there is no phase coherence, and so we have some measureof protection. While this is true in the charge sector, the 𝑥𝑦 -ferromagnet has a phasein the spin sector. One can picture the ground state 𝑥𝑦 -ferromagnet as having a spinon every lattice site pointing in the same direction along the equator of the Bloch
sphere. The direction within the plane is given by the phase, so any phase gradients orfluctuations are interpreted as spin waves. These gapless excitations make it difficult
to adiabatically ramp directly from a BEC into the 𝑥𝑦 -ferromagnet, so we propose
and study a ramping sequence similar to the many-body adiabatic sequence used
above. We first wish to ramp from a BEC into the spin Mott insulator, which has nophase coherence in either spin or charge space and has a gapped excitation spectrum.
This protects our system from non-adiabaticities and noise during the initial ramp-up.
Then, once we are in the spin Mott phase, we wish to ramp into the 𝑥𝑦 -ferromagnet,
where phase correlations will be created by superexchange processes. In this work, wehave focused on the first part, and have begun by creating a spin Mott insulator thatis cold in spin space. Technical upgrades, as discussed in Chapter 5, have reduced
our charge temperature to the point that adiabatic ramps to the 𝑥𝑦 -ferromagnet are
in reach.
In the near future, our lab is planning a major upgrade — adding a quantum gas
microscope to the setup, which will greatly expand our capabilities. We are using acustom-made objective with an NA of 0.8 and a long working distance of 11 mm. The
objective will sit directly on the top window, which also provides a reference for thevertical lattice. This removes three degrees of freedom from the objective alignment(focus and two angles from the optical axis) and so should be more stable against
long-term alignment drifts.
The most obvious application of the quantum gas microscope is for single-site
imaging. By allowing us to directly image particle-hole excitations, it will speed the
optimization of lattice ramps and Mott insulator preparation. In addition, singlesite readout will make measuring nearest neighbor correlations in the 𝑥𝑦 -ferromagnet
111

much easier. Single-site imaging is not the only benefit of a quantum gas microcsope— perhaps the more important one is that it allows for single-site addressing of theapplied potential. Ramping to a Mott insulator in a harmonic trap pushes entropy
out into the superfluid wings of the system. With a microscope, we can add a sharp
wall at the edge of the Mott insulating region of the cloud to act as a “cookie-cutter”
and selectively remove the high entropy superfluid wings, leaving us with an evencolder system to start our simulations in. Another use of projected potential walls
is to create sharp edges, which will allow us to image the topological edge currents
associated with a Harper superfluid. By combining our lab’s expertise in laser-assisted
tunneling and spin-dependent lattices with the new tools provided by a microscope,
we will open new doors in the realm of quantum simulation.

112

Appendix A
Lattice Noise Measurements
In order to achieve a low entropy in our spin Mott insulating state, we must minimize
excitations driven by external noise. In this appendix, I report on emprical studies
of noise coming from our lattices, as measured in Spring 2018.
We assume that all of our heating in the deep lattice is parametric heating ofseparate harmonic oscillators, with frequency 𝜔0 , equal to the bandgap. Then we are
only sensitive to intensity noise at 2𝜔0 and phase noise at 𝜔0 .
The calculated excitation rates 𝑅0→2 and 𝑅0→1 are given by

𝑅0→2 =𝜋𝜔02
𝑆 (2𝜔0 )8(A.1)

𝑅0→1 =𝜋𝑚𝜔03
𝑆𝑥 (𝜔0 )2~(A.2)

where 𝑆 (𝜔) is the power spectral density of the relative intensity noise of the laser
measured in 1/s−1 , and 𝑆𝑥 (𝜔) is the power spectral density of the phase noise of thelaser and retro mirror, measured in nm2 /s−1 .
113Figure A-1: Intensity spectrum after injecting noise at various frequencies. This plotshows that our injection is linear up to at least tens of kiloHertz.

A.1 Empirical Measure of Condensate Fraction Decay
Regardless of the calculations that we do, an empirical way to tell whether we arelimited by noise is to first measure the noise present in the system. Then add noise
until the injected noise is a few dB above background, and see if this affects thecoherence lifetime. Note that for the phase noise measurements so far, we havemeasured with the interferometer and the mirror on the same breadboard, so there
may be common-mode vibration that we do not measure.

A.1.1 Intensity Noise
We inject intensity noise into the closet arm (1064 nm light) simply by directly adding
intensity noise to the output of the PID controller of our intensity stabilization. Figure
A-1 shows that this process is linear up to tens of kiloHertz. We see that injecting 0
dBm noise with a 1kHz bandwidth gives a response of about −50 dBc/Hz, while thebackground noise is around −110 dBc/Hz at twice the bandgap, which is about 26

kHz for a 19 𝐸𝑅 lattice. If we plug these numbers into Equation A.1, we calculate a
rate of 4 × 10−3 s−1 for the background noise.

In Figure A-2, we inject intensity noise at twice the bandgap in a 19 𝐸𝑅 Mott

Insulator for 100 ms. We see that we have to inject noise above -20 dBm to significantly heat our atoms. This corresponds to −70 dBc/Hz, which is 40 dB above
114

Figure A-2: Condensate fraction lifetime with intensity noise applied.

Figure A-3: Spectrum after injecting phase noise at -20 dBm at various frequencies.
Note that at 16-20 kHz, we are about 10 dB above background.
background in heating, where, theory predicts an excitation rate of 40 s−1 . Since weneeded to inject noise that is so much higher than the background to see an effect,
we can conclude that the intensity noise at the bangap is not the primary heating
source.

A.1.2 High Frequency Phase NoiseIn Figure A-3, we see that injecting noise at -20 dBm in boxed white noise from 16-20
kHz gives us noise that is about 10 dB above background in amplitude. The bandgap
for a 1064 nm lattice at 24𝐸𝑅 is around 18 kHz, and the background phase noise is√about 3 × 10−5 nm/ Hz. If we plug these numbers into Equation A.2, we calculate

a rate of 5 × 10−4 s−1 .

We measured the lifetime of the condensated fraction with different noise intensi-

ties, as seen in Figure A-4, and fit to exponential decays with lifetimes given in Table
115

Figure A-4: Condensate fraction lifetime with high-frequency phase noise applied.
Table A.1: 1/𝑒 lifetimes of a 24𝐸𝑅 Mott Insulator with 16-20 kHz noise added.
Noise Intensity (dBm) dB above background Lifetime (ms)None02480 ± 170-25101970 ± 190-2020990 ± 90-1030400 ± 40
A.1. We see that if we go about a factor of ten in amplitude (which should be a factor
of a hundred in heating) above the background, we only get a factor of about twochange in lifetime. If we add a further factor of ten in noise intensity, to -10 dBm, we
get only an additional factor of two in lifetime.

A.1.3 Low Frequency Phase NoiseIn Figure A-5, we see that mirrors mounted like the vertical retro have about 10 dB
extra amplitude noise at low frequencies, but are similar to the horizontal-type retro
mirrors at bandgap frequencies. We want to see if this higher low-frequency phasenoise possibly heats the atoms. We can add -12 dBm of phase noise from 0.6-1.3
kHz, which raises the low frequency noise by more than 10 dB in amplitude above
background. As seen in Figure A-6 and Table A.2, we see little increase in the heating
rate when we add -12 dBm noise. When we increase the noise injection by 20 dB in
intensity (which should be a factor of a hundred in heating) we only see a decreasein lifetime of a factor of four or five.
116

Figure A-5: Spectrum after injecting -12 dBm phase noise at low frequencies. The
legend gives the lattice arm-length.

Figure A-6: Condensate fraction lifetime with low-frequency phase noise applied.

Table A.2: 1/𝑒 lifetimes of a 20 𝐸𝑅 Mott Insulator with 0.6-1.3 kHz noise added.
Noise Intensity (dBm) dB above background Lifetime (ms)None01890 ± 170-12201720 ± 270+840
390 ± 100117

A.1.4 Conclusion
Given the above data, we can conclude that intensity noise on the lattices is definitely
not limiting our Mott insulator lifetime. For phase noise both at the bandgap andat low frequencies, the situation is complicated by the sub-linear scaling that we seein heating rate. We expect the inverse lifetime to be insensitive to added noise while
the heating rate associated with it is lower than the background heating, and for itto then scale linearly with the intensity of added noise once we pass the backgroundthreshold. Instead, we see non-zero but strongly sublinear behavior. This may point
to a saturation effect, but further study is needed before definitive statements can be
made.

118

Appendix B
Realizing the Harper-Hamiltonian
with Laser-Assisted Tunnneling in
Optical Lattices
This appendix contains a reprint of [22], which was touched on, but not discussed indetail in the main text.

119PRL 111, 185302 (2013)

Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS

week ending
1 NOVEMBER 2013

Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices
Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William Cody Burton, 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 6 August 2013; published 28 October 2013; publisher error corrected 28 October 2013)
We experimentally implement the Harper Hamiltonian for neutral particles in optical lattices using
laser-assisted tunneling and a potential energy gradient provided by gravity or magnetic field gradients.
This Hamiltonian describes the motion of charged particles in strong magnetic fields. Laser-assisted
tunneling processes are characterized by studying the expansion of the atoms in the lattice. The band
structure of this Hamiltonian should display Hofstadter’s butterfly. For fermions, this scheme should
realize the quantum Hall effect and chiral edge states.
DOI: 10.1103/PhysRevLett.111.185302

PACS numbers: 67.85.d, 03.65.Vf, 03.75.Lm

Systems of charged particles in magnetic fields have led
to many discoveries in science—including both the integer[1] and the fractional [2,3] quantum Hall effects—and havebecome important paradigms of quantum many-body physics [4]. Generalizations have led to important developments
in condensed matter physics, including topological insulators [5,6], fractional Chern insulators [7,8], and Majorana
fermions [9,10]. At high magnetic fields, exotic new phenomena like the fractal energy spectrum of Hofstadter’s
butterfly [11] are predicted to emerge. Its direct observation
would require an inaccessibly high magnetic field of one
flux quantum per unit cell—corresponding to 10000 T ina traditional condensed matter system. Recently, some
aspects of Hofstadter’s butterfly were addressed using
superlattices in high magnetic fields [12–15].
Neutral atoms provide an excellent platform to simulate
the physics of charged particles in magnetic fields freefrom disorder. Rotating quantum gases realize the limit
of weak magnetic fields, exploiting the equivalence
between the Lorentz force and the Coriolis force. The
observed vortex lattices [16,17] are analogous to magnetic
flux lattices. A more general method to create synthetic
magnetic fields for neutral atoms is based on the insightthat vector potentials introduce spatially varying phases 
into the wave function when the particle propagates  ¼
H
A  ds=@, where the charge is included in the vector
potential. For neutral atoms, such a phase structure canbe realized through Berry phases, when two hyperfine
states of the atom are coupled by Raman lasers withinhomogeneous intensity or detuning [18,19]. This concept
of coupling of two or several internal states to realize
synthetic magnetic fields was also suggested in opticallattice geometries [20–22]. Here, the crucial element islaser-assisted hopping between neighboring sites which
imprints the phase of the laser into the atomic wave function. Alternatively, instead of using Raman laser beams,
lattice modulation techniques can generate complex tunneling matrix elements in optical lattices [23,24].
Experimentally, these techniques have been used so far
0031-9007=13=111(18)=185302(5)only to realize staggered magnetic fields [24,25]. In the
Munich experiment, the two internal states in the proposed
schemes [20,22] were replaced by doubling the unit cell of
the optical lattice using superlattices [25].
So far, all proposals for generating high magnetic fields
are based on the coupling of different internal states. For
alkali atoms, this involves different hyperfine states [20].
Spin flips between such states require near-resonant lightwhich heats up the sample by spontaneous emission. At
least for staggered fluxes, the realizations with lattice
shaking and superlattices demonstrate that internal structure of the atom is not essential. Here, we suggest andimplement a scheme which realizes the Harper
Hamiltonian [26], a lattice model for charged particles inmagnetic fields, the spectrum of which is the famous
Hofstadter’s butterfly [11]. Our scheme requires only faroff-resonant lasers and a single internal state. It is anextension of a scheme suggested by Kolovsky [27], which
was shown to be limited to inhomogeneous fields [28], but
as we show here, an additional momentum transfer in thelaser-assisted hopping process provides a simple solution.
While this work was in progress [29], an identical scheme
was proposed by the Munich group [30]. In this Letter, we
describe the features and implementation of this scheme,
and characterize the laser-assisted hopping process.
We start with the simple Hamiltonian for noninteracting
particles in a 2D cubic latticeX
H ¼  ðJx a^ ymþ1;n a^ m;n þ Jy a^ ym;nþ1 a^ m;n þ H:c:Þ; (1)m;nwhere JxðyÞ describes tunneling in the x (y) direction anda^ ym;n (a^ m;n ) is the creation (annihilation) operator of aparticle at lattice site (m, n). Tunneling in the x directionis then suppressed by a linear tilt of energy  per lattice
site, where =h is the Bloch oscillation frequency. This tilt
can be created with magnetic field gradients, gravity, or an
ac Stark shift gradient. Resonant tunneling is restored with
two far-detuned Raman beams of two-photon Rabi frequency , frequency detuning ! ¼ !1  !2 , and momentum transfer k ¼ k1  k2 , as shown in Fig. 1(a).

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The spatially dependent phase imprinted by the Raman
lasers, given by m;n , can be intuitively understood in a
perturbative regime, where Jpert ¼ Jy andZ 2 
^d rw ðr  Rm;n Þeikr wðr  Rm;n  axÞ
2¼ KeikRm;n :(3)Kpert ¼(c)(b)

FIG. 1 (color online). (a) Raman-assisted tunneling in thelowest band of a tilted lattice with an energy offset  between
neighboring sites and two-photon Rabi frequency .
(b) Experimental geometry to generate uniform magnetic fields
using a pair of far-detuned laser beams and a uniform potential
energy gradient. Tunneling along the x direction with amplitude
K imprints a complex, spatially varying phase m;n —with site
indices (m, n)—into the system due to the momentum transfer inthe y direction. (c) A schematic depicting the position-dependent
phases of the tunneling process. The equivalent number of flux
quanta per unit cell is  ¼ y =2.

Note that the two Raman beams couple different sites but
do not change the internal state of the atoms. For resonant
tunneling ! ¼ =@, time averaging over rapidly oscillating terms [20] yields an effective Hamiltonian which istime independent. As a result, the tilt has disappeared
because, in the dressed atom picture, site (m, n) with jand k photons in the two Raman beams is degenerate withsite (m þ 1, n) and j þ 1 and k  1 photons in the two
beams. This effective Hamiltonian describes the systemwell, assuming that  is larger than the bandwidth J andsmaller than the band gap Egap . In addition, the effects ofpower broadening can be avoided if we choose  less than. The resulting Hamiltonian is equivalent to one that
describes charged particles on a lattice in a magnetic fieldunder the tight-binding approximation [11,26]—thesingle-band Harper HamiltonianX
H ¼  ðKeim;n a^ ymþ1;n a^ m;n þ J a^ ym;nþ1 a^ m;n þ H:c:Þ (2)m;nwith a spatially varying phase m;n ¼ k  Rm;n ¼mx þ ny where Rm;n denotes the position of eachlattice site. Solutions in this model are periodic withrespect to the number of flux quanta per unit cell . If
the frequencies of the Raman beams are similar to thoseused for the optical lattice, one can tune  over the fullrange between zero and one by adjusting the angle between
the Raman beams, and consequently ky . A similar
Hamiltonian can be realized for the tunneling of phonons
between ion microtraps [31].

Adding up the accumulated phases around a closed path,
one sees that this method leads to an enclosed phase ofy ¼ ky a per lattice unit cell of area a2 , thus realizing theHarper Hamiltonian with  ¼ y =2.
In a cubic lattice, the Wannier function wðrÞ factorizes
into wðxÞwðyÞ which are the localized Wannier-Stark
and Wannier wave functions, Rrespectively. The resulting
expression for K ¼ ð=2Þ dxw ðxÞeikx x wðx  aÞ Rdyw ðyÞeiky y wðyÞ shows that the momentum transfer
in the x direction is necessary to have a nonvanishingtunneling matrix element K without changing the internal
state. The x momentum transfer distinguishes our schemefrom Refs. [20,22,27] and contributes to the vector potential A ¼ ½@ðky y þ kx xÞ=ax^ but does not contribute to theenclosed flux or the value of the synthetic magnetic field B.
Note that this scheme does not realize the simple Landau
gauge for the magnetic field.
For a more comprehensive description, we add themoving lattice—VRM ¼  sinðk  r  !tÞ—of the two
Raman lasers along with a linear tilt to the Hamiltonianin Eq. (1). In addition to the off-diagonal laser-assisted
tunneling term, this moving lattice causes a diagonal term,
which is a temporal modulation of the on-site energies. A
unitary transformation, as in Refs. [29,32], leads to a frame
rotating nonuniformly in time and position that eliminates
the diagonal time dependence. For resonant drive  ¼@!, the on-site energies are all equal and vanish while
the remaining off-diagonal coupling has a timeindependent part, leading to the Harper Hamiltonian, as
in Eq. (2). The resulting expressions for K and J are (seethe Supplemental Material [33])J1 ðx Þ
dJ1 ðx Þ
0
K ¼ y0 x1
þ ix1
;
x
dx
(4)
 
2y0 x0
ki a
;
sin
J ¼ Jy J0 ðy Þ;
i ¼
2where i0 ¼ h0j cosðki xi Þj0i is the on-site matrix
element, and x1 ¼ h0j sin½kx ðx  a=2Þj1i and 0x1 ¼
h0j cos½kx ðx  a=2Þj1i are the off-diagonal matrix elements. This result is more general than the case of phasemodulation [32] and the tight-binding limit inRefs. [30,34], where K is proportional to J1 ðxÞ.
We implement the Harper Hamiltonian with each Raman
laser aligned along one of the two lattice directions x and ycorresponding to momentum transfer in both directions of@kL —the single-photon recoil of the lattice laser. The magnetic flux per unit cell resulting from ky ¼ kL is  ¼ 1=2. In
the tight-binding limit for this momentum transfer, i0  1185302-2

Width (µm)

Width (µm)

20
10
6

10
14
18
Lattice Depth (ER)

12

(a)
1.0

Jn(2 /∆)

Corrected Width Squared (µm2)

Experimentally, the system is prepared by starting with a
Bose-Einstein condensate of 5  105 87 Rb atoms in thej2; 2i state in a crossed dipole trap. The Raman lasers areramped up to their final intensities in 30 ms at a large
detuning of 200 kHz and are switched to their final detuning after the tilt is applied to the system (see below).
Unwanted interferences between lattice and Raman lasers
are avoided with relative frequency offsets of >30 MHz
using acousto-optic modulators. Next, we adiabaticallyload the condensate in 100 ms into a two-dimensional
cubic optical lattice of spacing latt =2 ¼ 532 nm. For longer hold times, a weak 2Er lattice beam along the third
direction is simultaneously ramped up to provide additional confinement. Here, Er ¼ @2 k2L =2m  h  2 kHz is
the single-photon recoil energy. The two-photon Rabi frequency of the moving Raman lattice is determined using
free-space Rabi oscillations and chosen to be considerably
smaller than the static lattice.
After loading the condensate into the lattice, a uniform
potential energy gradient of mga=h  1:1 kHz between
adjacent lattice sites is applied by turning off the confining
crossed dipole traps in 20 ms. Alternatively, we havesuccessfully used a magnetic field gradient to access abroader range of tilts. The gravitational gradient has theadvantage of a much faster switching time compared to themagnetic gradient. The cloud widths x and y are
obtained by standard absorption imaging along the direction perpendicular to the 2D lattice.
The essential feature of our implementation of theHarper Hamiltonian is that tunneling in the x direction issuppressed by a potential tilt and reestablished by laserassisted tunneling. This is demonstrated in Fig. 2, whichshows the resonance for the laser-assisted process. For this,tunneling is characterized by looking at the expansion of
the cloud within the lattice. Expansion occurs since theconfinement by the optical dipole trap has been switched
off, and due to some heating during the 500 ms hold time.
Note that for fully coherent time evolution, charged particles in a magnetic field will undergo cyclotron motion
which would suppress the expansion. The resonance widthof 60 Hz may have contributions from laser frequency
jitter, inhomogeneous lattice potential, and atomic interactions. The Lorentzian fit suggests a homogenous broadening mechanism.
The dependence of K and J on the intensity of theRaman lasers (described by Bessel functions) allows tuning of the ratio of the two. For low intensities, K increases
linearly with the intensity, and J decreases quadratically.
The latter reflects the depletion of the unperturbed Wannierfunction by the modulation due to the moving Raman
lattice. Figure 3(a) shows experimental results in qualitative agreement with these predictions.
For a quantitative interpretation of the expansion of thecloud, we assume an incoherent diffusion process, where
the square of the width  of the expanded cloud is proportional to the tunneling rate times expansion time. For shorttimes, the expansion of the cloud should be fully coherent,
and the width should increase linearly with time. However,heating by light scattering and intensity fluctuations of theCorrected Width Squared (µm2)

and x1  2Jx =  0x1 , so the resonant tunneling
amplitudes resulting from kx ¼ kL simplify to
 
 
2
2
K ¼ Jx J1
and J ¼ Jy J0
:
(5)



16

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PRL 111, 185302 (2013)

1200

0.0
0

600

2

4

Raman Intensity2 /∆

0
2

0

(b)

4

Raman Intensity (2 / )2 / = 1.9300

2 / = 3.3

200

2 / = 1.2100

2 / =0

0
0

500

1000
Hold Time (ms)

1500

8
1000
1100
1200
Raman Frequency,
π (Hz)

1300

FIG. 2 (color online). In situ cloud width as a function ofRaman detuning ! after an expansion of 500 ms, with aRaman lattice depth of  ¼ =4. The solid line is a
Lorentzian fit to the experimental data (dots) centered at1133 Hz—consistent with the gravitational offset between sites.
Pictures (of size 135  116 m) show typical column densities
on or off resonance. Inset: Dependence of the laser-assisted
tunneling on optical lattice depth.

FIG. 3 (color online). (a) Expansion as a function of resonant
Raman laser intensity shows the laser-assisted tunneling along
the tilt direction (blue circles) and the tunneling rate J along thetransverse direction (red squares). Data taken at lattice depths of
9Er and hold time of 1500 ms. Inset: Theoretical prediction for
the tunneling rates K and J in terms of Bessel functions [Eq. (5)].
(b) Time evolution of the squared width for different Raman
laser intensities. From the slope of the lines, we obtain the laserassisted tunneling rates and their statistical errors: 0:2 0:08
(red squares), 4 0:5 (blue circles), 12 1 (black diamonds),
and 8 0:5 Hz (blue triangles).

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laser beams lead to an incoherent, diffusive expansion. For
finite time, we correct for the initial size 0 by assumingthat the expansion and initial size add in quadrature, andplot the corrected squared width 2corr ¼ 2  20 versus
time. The slope is proportional to the laser-assisted tunneling rate. Absolute tunneling rates are obtained by comparing this result to the expansion of the cloud in the y
direction with the Raman beams far off resonance, whennormal tunneling occurs. The ratio of the slopes is then
K=Jy , with Jy calculated from the calibrated lattice depth
to be h  48 Hz. Figure 3(b) shows the time evolution of
the square of the corrected size for various Raman intensities. The linear fits support the assumption of incoherent
diffusion and allow a determination of tunneling rates, assummarized in the figure caption.
Laser-assisted tunneling is a powerful tool to manipulate
the motion of atoms in optical lattices and to create novel
Hamiltonians. We now describe different tunneling processes observed by a wide scan of the Raman detuning,
shown in Fig. 4. A strong peak near 568 Hz fulfills theresonance condition 2! ¼ =@ for a four-photon
nearest-neighbor tunneling process. This resonance is similar to the one observed in Ref. [35] by shaking the lattice.
Note that the four-photon resonance is narrower (20 Hz
versus 60 Hz) than the two-photon resonance, indicativeof a higher-order process. Broad features at even lower
frequency are most likely due to higher-order tunnelingresonances and low-lying excitations within the first band.
(a)
4γ

16
Width (µm)

2γ
12

8

(b)

Width (µm)

10

9

500
1000
Raman Frequency, δω/2π (Hz)
Corr. Width Sq. (µm2)

0

1500

60
30
0
0
0.4
0.8
Raman Int. (2 /∆)

8
1800

2000
2200
2400
2600
Raman Frequency, δω/2π (Hz)2800

FIG. 4 (color online). Spectrum of excitations and tunnelingresonances. (a) A strong, four-photon, nearest-neighbor tunneling resonance appears at =2h along with the K resonance at
=h. These data were taken at a lattice depth of 9Er , for a twophoton Raman Rabi frequency of 1092 Hz and 500 ms expansion
time. (b) Observation of next-nearest-neighbor laser-assisted
tunneling at 2 and the expansion of the cloud as a function of
. Inset: No saturation is reached. Expansion time of 1500 ms.

week ending
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Next-nearest-neighbor tunneling occurs at ! ¼ 2=@,
twice the frequency of the fundamental resonance. For ashaken lattice (no Raman beams), this was studied inRef. [36]. Analyzing the expansion of the cloud gives atunneling rate of 0:4 0:1 Hz, comparable to the nextnearest-neighbor tunneling rate in an untilted lattice,0:8 Hz in our system. However, in an untilted lattice,next-nearest-neighbor tunneling is typically a hundred
times slower than nearest-neighbor tunneling, whereas inlaser-assisted tunneling, the two processes can be independently controlled by the laser power at the two resonant
frequencies. Tunneling rates below 1 Hz are too slow for
pursuing many-body physics, but the same scheme can be
implemented for lighter atoms such as lithium in a shorter
wavelength lattice, where the relevant scale factor, therecoil energy, is increased by a factor of 50.
After realizing and characterizing all parts of the Harper
Hamiltonian, the next goal is to map out its band structure
as a function of quasimomentum and magnetic field —
the Hofstadter butterfly. The ground state for a given should be accessible by adiabatically transferring a condensate into this Hamiltonian. The ground state of theHarper Hamiltonian for  ¼ 1=2 has a clear signature inthat its magnetic unit cell is twice as large as the lattice unit
cell and its wave function has a unit cell that is 4 times aslarge, so time of flight imaging will reveal the resulting
reduction of the Brillouin zone in momentum space by afactor of 4 [25,29,37–39]. So far, we have not been able topreserve the low entropy of the initial condensate andobserve the ground state.
Preliminary experiments have shown less heating by the
Raman beams at larger frequency detuning, requiring
larger magnetic field gradients. An optimum detuning
should be near half the band gap, avoiding inter- andintraband transitions. Possibly, an extension of the treatment of light scattering in optical lattices [40] could predict
if there is a fundamental lower limit to the ratio of heating
rate over K. Another potential source of heating is atomic
interactions. Instabilities of certain quasimomentum statesin optical lattices have been studied in Refs. [41,42].
Interaction-induced heating effects can in principle beavoided by using Feshbach resonances to tune the scattering length to zero or by using a single spin component of afermionic gas. Once the ground state of the Harper
Hamiltonian is established, different quasimomentum
states can be populated through Bloch oscillations which
occur at frequency  ¼ !  =@, when the Raman
lasers are slightly detuned from the resonance studied here.
The Harper Hamiltonian established in this work will be
the starting point for many exciting explorations, including
the quantum Hall effect, Dirac points, and novel topological phenomena [8,43]. Interactions between atoms may
also lead to bosonic Laughlin states [44] and nonlinear
Hofstadter eigenmodes [45]. The lowest band is topologically nontrivial with a Chern number of 1 [46] and should
show chiral edge states. Most importantly, our scheme issimpler and potentially more robust than other suggestions,

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since it does not require near-resonant light for connecting
hyperfine states. It can be implemented for any atom—
including the workhorse fermionic atoms lithium andpotassium—which has small fine structure splittings, making it impossible to couple different spin states with negligible heating by spontaneous emission.
We acknowledge Yuri Lensky for experimental assistance. We thank Wujie Huang for helpful discussions andcritical reading of the manuscript. This work was supported by the NSF through the Center of Ultracold
Atoms, by NSF Grant No. PHY-0969731; under ARO
Grant No. W911NF-13-1-0031 with funds from the
DARPA OLE Program; and by ONR.
Note added.—Recently, we became aware of similar
work carried out by Bloch and co-workers [34].

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185302-5

Appendix C
Spin-Orbit Coupling and Quantum
Spin Hall Effect for Neutral Atoms
without Spin Flips
This appendix contains a reprint of [78], which was touched on, but not discussed indetail in the main text.

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Spin-Orbit Coupling and Quantum Spin Hall Effect for Neutral Atoms without Spin Flips
Colin J. Kennedy, Georgios A. Siviloglou, Hirokazu Miyake, William Cody Burton, and Wolfgang Ketterle
MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 28 August 2013; revised manuscript received 25 October 2013; published 25 November 2013)
We propose a scheme which realizes spin-orbit coupling and the quantum spin Hall effect for neutral
atoms in optical lattices without relying on near resonant laser light to couple different spin states. The spinorbit coupling is created by modifying the motion of atoms in a spin-dependent way by laser recoil. The spin
selectivity is provided by Zeeman shifts created with a magnetic field gradient. Alternatively, a quantum spin
Hall Hamiltonian can be created by all-optical means using a period-tripling, spin-dependent superlattice.
DOI: 10.1103/PhysRevLett.111.225301

PACS numbers: 67.85.d, 03.65.Vf, 03.75.Lm

Many recent advances in condensed matter physics are
related to the spin degree of freedom. The field of spintronics[1], the spin Hall effect [2], and topological insulators [3] all
rely on the interplay between spin and motional degrees of
freedom provided by spin-orbit coupling. Quantum simulations with neutral atoms have started to implement spin-orbit
coupling using Raman transitions between different hyperfine states [4–8]. Since the Raman process transfers momentum to the atom, the resonance frequency is Doppler
sensitive, and thus couples motion and spin.
The possibility of using spin-flip Raman processes tocreate interesting gauge fields was first pointed out in[9–11], and extended to non-Abelian gauge fields, which
imply spin-orbit coupling, in [12,13]. With the exception of
an atom chip proposal where the spin flips are induced withlocalized microwave fields [14], all recently proposed
schemes are based on spin-flip Raman processes [8,15–20].
The major limitation of these Raman schemes is thatspin-flip processes are inevitably connected with heatingby spontaneous emission if they rely on spin-orbit couplingin the excited state, as in alkali atoms or other atoms withan S orbital ground state. Since laser beams interact withatoms via the electric dipole interaction, they do not flip the
spin. Spin flips occur only due to intrinsic spin-orbit interactions within the atoms; therefore, spin-orbit coupling by
spin-flip Raman processes relies on the spin-orbit coupling
within the atom. Since the spontaneous emission rate andthe two-photon Rabi frequency for Raman spin-flip processes scale in the same way with respect to the ratio oflaser power to detuning, for a given atom the couplingstrength relative to the spontaneous emission rate is fixedby the fine structure splitting compared to the naturallinewidth. This has not been a limitation for the demonstration of single-particle or mean-field physics [4–8], butwill become a severe restriction for many-body physics
where interactions introduce a smaller energy scale andtherefore require longer lifetimes of the atomic sample.
Some authors have considered transitions involving metastable states of alkaline earth atoms to reduce the effects of
spontaneous emission [21,22].
0031-9007=13=111(22)=225301(5)Here we present a spin-orbit coupling scheme that does
not involve spin flips, is diagonal in the spin component z ,and corresponds to an Abelian gauge field. This scheme canbe implemented with far-off-resonant laser beams, thus
overcoming the limitation of short sample lifetimes due tospontaneous emission. In the field of cold atoms, many
discussions of spin-orbit coupling emphasize its close relationship to non-Abelian gauge fields [17,23] which are nondiagonal for any spin component and therefore mix spin andmotion in a more complicated way. However, a scheme
diagonal in the spin component is sufficient for quantum
spin Hall physics and topological insulators [24,25], and its
implementation has major experimental advantages. In thetheoretical proposals and demonstration of the spin Hall
[23,26] and quantum spin hall [27] effects for quantum
gases, Raman spin flips are used to create an Abelian gauge
field diagonal with respect to one spin component.
The physical principle of the spin-orbit coupling scheme
presented here is very different from spin-flip schemes. It
does not require any kind of spin-orbit coupling within
the atom. Rather, spin-dependent vector potentials areengineered utilizing the Zeeman effect in a magnetic
field—atoms in the spin-up and -down states interact
with different pairs of laser beams, or differently with thesame pair, and the photon recoil changes the atom’s motion
in a spin-dependent way. This results in spin-orbit coupling
which is diagonal in the spin basis.
To begin, we summarize the relationship between spinorbit coupling and spin-dependent vector potentials. For
charged particles, the origin of spin-orbit coupling is therelativistic transformation of electromagnetic fields. Whenan electron moves through an electric field E it experiencesa magnetic field B in its moving frame which interacts withthe spin  (described by the Pauli spin matrices). Spinorbit coupling contributes a term proportional toðp  EÞ   in the Hamiltonian. As such, an electric fieldin the z direction gives rise to the Rashba spin-orbit coupling ð  pÞz ¼ x py  y px .
Assuming a 2D system confined to the x, y plane, and an
in-plane electric field, the spin-orbit interaction conserves

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z . Following [25], a radial electric field E  Eðx; y; 0Þleads to a spin-orbit coupling term in the Hamiltonian of
the form Ez ðxpy  ypx Þ, or equivalently to E  L,
where L is the angular momentum of the atom. Such aradial field could be created by a uniformly charged cylinder, or can be induced by applying stress to a semiconductor sample [25]. This spin-coupling term is identical to
the A  p term for the Hamiltonian describing a spin in amagnetic field, z B. Using the symmetric gauge for thevector potential A ¼ ðz B=2Þðy; x; 0Þ, one obtains a termin the Hamiltonian proportional to z Bðxpy  ypx Þ.
Therefore, this form of spin-orbit coupling is equivalent
to a spin-dependent magnetic field which exerts opposite
Lorentz forces on spin-up and -down atoms. This leads tothe spin Hall effect which creates a transverse spin current
and no charge or mass currents [24,25]. The A2 term
constitutes a parabolic spin-independent potential whichis irrelevant for the spin physics discussed here.
We now present a scheme which realizes such an
Abelian gauge field [28] and manifests itself as a spindependent magnetic field. Recently, the MIT group [30,31]
and the Munich group [32,33] have suggested and implemented a scheme to generate synthetic magnetic fields forneutral atoms in an optical lattice. The scheme is based on
the simple Hamiltonian for noninteracting particles in a 2D
cubic lattice,X
H ¼  ðJx a^ ymþ1;n a^ m;n þ Jy a^ ym;nþ1 a^ m;n þ H:c:Þ; (1)We now extend this scheme to the spin degree of freedom,
and assume a mixture of atoms in two hyperfine states,labeled spin up and down. If the potential energy gradientis the same for the two states, then the two states experience
the same magnetic field. This is the situation when the tilt isprovided by gravity, a scalar ac Stark shift gradient, or amagnetic field gradient if both states have the same magnetic moment—the phase m;n is independent of z .
If the two states have the same value of the magnetic
moment, but opposite sign, then the potential gradient is
opposite for the two states. This can be realized by usingstates of the same hyperfine level F, but with opposite
magnetic quantum number MF (e.g., in 23 Na or 87 Rb, thejF; MF i ¼ j2; 2i and j2; 2i states), or by picking anothersuitable pair of hyperfine states. In this case, for laserassisted tunneling between two sites m and m þ 1, theroles of the two laser beams—absorption of a photon
versus stimulated emission of a photon—for the Ramanprocess are reversed, as depicted in Fig. 1. Therefore, thetwo states receive opposite momentum transfer, and thissign change leads to a sign change for the phase,where JxðyÞ describes tunneling in the x (y) direction anda^ ym;n (a^ m;n ) is the creation (annihilation) operator of aparticle at lattice site (m, n). The setup is detailed in [30]and summarized as follows: a linear tilt of energy  perlattice site is applied using a magnetic field gradient in thex direction, thus suppressing normal tunneling in thisdirection. Resonant tunneling is restored with two fardetuned Raman beams of two-photon Rabi frequency ,frequency detuning ! ¼ !1  !2 , and momentum transfer k ¼ k1  k2 . Considering only the case of resonant
tunneling, ! ¼ =@, rapidly oscillating terms time average out [34], yielding an effective Hamiltonian which istime independent [30]:X
H ¼  ðKeim;n a^ ymþ1;n a^ m;n þ J a^ ym;nþ1 a^ m;n þ H:c:Þ: (2)creates the spin-orbit coupling discussed in the introduction, although in a different gauge. The x dependence in the
x component of A is necessary for a non-negligible tunneling matrix element for the laser-assisted process [30].

m;nm;n ¼ ðmkx a þ nky aÞz ;(3)and also for the vector potential and the magnetic field. The
vector potential realized by this scheme,@ðk x þ ky yÞx^ z ;
(4)
A ¼
a x

m;n

This effective Hamiltonian describes charged particles
on a lattice in a magnetic field under the tight-binding
approximation [35,36]. The gauge field arises from thespatially varying phase m;n ¼ k  Rm;n ¼ mkx a þ nky a,where a is the lattice constant, and has the form A ¼^ One can tune the flux per unit cell ½@ðkx x þ ky yÞ=ax.
for a given spin state over the full range between 0 and 1 byadjusting the angle between the Raman beams, and consequently ky .

FIG. 1 (color online). Spin-dependent tunneling in an opticallattice tilted by a magnetic field gradient. When the two spinstates have opposite magnetic moments, the role of absorbtion
and emission of the two photons is exchanged. The result is thatthe two states have tunneling matrix elements with opposite
phases, leading to opposite synthetic magnetic fields and realizing spin-orbit coupling and the quantum spin Hall effect.

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This system now has time-reversal symmetry, in contrast
to the system with the same synthetic magnetic field forboth states (since a magnetic field breaks time-reversal
symmetry). This scheme implements spin-orbit coupling
both in two and three dimensions and with bosons andfermions. For fermions, in two dimensions it realizes thequantized spin Hall effect consisting of two opposite quantum Hall phases. It is protected by a Z topological index
due to the fact that z is conserved [24,25].
When the values of the two magnetic moments are different, and the potential energy gradient is provided by amagnetic field gradient, then the two states have different
Bloch oscillation frequencies, =h. Each state now needs
two separate Raman beams for laser-assisted tunneling (orthey can share one beam). This implies that the synthetic
magnetic field can now be chosen to be the same, to be
opposite, or to be different for the two spin states. One option
is to have zero synthetic magnetic field for one of the states.
Atoms in this state can still tunnel along the tilt direction byusing a Raman process without y-momentum transfer, or,equivalently, by inducing tunneling through lattice modulation [37–39]. In the case of two different magnetic moments,
one could also perform dynamic experiments, where laserparameters are modified in such a way that one switches
either suddenly or adiabatically from the quantum Hall effectto the spin quantum Hall effect.
An intriguing possibility is to couple the two states. Sincez is no longer conserved, the system should become atopological insulator with the Z2 classification [3,40], provided that the coupling is done in a time-reversal invariant
way. This can be done with a term which is not diagonal inz —i.e., a x py term—by adding spin-flip Raman lasers to
induce spin-orbit coupling, or by driving the spin-flip transition with rf or microwave fields. A coherent rf drive fieldwould not be time-reversal invariant, but it would be interesting to study the effect of symmetry breaking in such astate [41]. A drive field where the phase is randomized
should lead to a time-reversal invariant Hamiltonian.
Our scheme implements the idealized scheme for a quantum spin Hall system consisting of two opposite quantum
Hall phases. This is a starting point for breaking symmetries and exploring additional terms in the Hamiltonian.
References [41,42] discuss a weak quantum spin Hall phase,
induced by breaking the time-reversal symmetry by a magnetic field—this can be achieved by population imbalance
between the two spin states. A spin-imbalanced quantum
Hall phase can turn into a spin-filtered quantum Hall phase[41,42] where only one component has chiral edge states.
This can be achieved by realizing a finite synthetic magnetic
field for one component and zero for the other. Changing thespin-orbit coupling can induce topological quantum phasetransitions between a helical quantum spin Hall phase and achiral spin-imbalanced quantum Hall state. This can probably be achieved in a population-imbalanced system by adding additional Raman spin-flip beams [41,42].

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Thus far, we have discussed single-particle physics.
Adding interactions, by increasing the density with deeper
lattices or through Feshbach resonances, will induce interesting correlations and may lead to fractional topologicalinsulators [43]. Another option is spin-drag experiments[44,45], transport experiments where one spin component
transfers momentum to the other component. For the situation mentioned above, where the synthetic magnetic fieldis zero for one component (e.g., spin up), a transport
experiment revealing the Hall effect [46] for spin down
would show a nonvanishing Hall conductivity for spin up
due to spin drag. In addition, one would expect that spinexchange interactions destroy the two opposite quantum
Hall phases, and should lead to the quantum spin Hall
phase with Z2 topological index.
We now present another way of realizing the physics
discussed above, using optical superlattices instead of apotential energy gradient. This has the advantage of purely
optical control and avoids possible heating due to LandauZener tunneling [47] between Wannier Stark states. Thusfar, optical superlattices have allowed the observation of
the ground state with staggered magnetic flux [48], in
contrast to experiments with magnetic tilts [30,33].
Figure 2 summarizes the new scheme. The superlatticehas 3 times the period of the basic lattice, thus distinguishing sites A, B, C in energy. Resonant tunneling is reestablished using three pairs of Raman beams with frequencies!1 þ AB =@, !2 þ BC =@, and !3  ðAB þ BC Þ=@ collinear in one arm and !1 , !2 , and !3 collinear in anotherarm at an angle to the first. Consequently, there is always
the same momentum transfer for tunneling in the y direction, leading to the same flux as the scheme with themagnetic tilt, and Eqs. (3) and (4) apply. This is in contrast

FIG. 2 (color online). Superlattice scheme for realizing thequantum Hall and quantum spin Hall effect. A superlattice with
3 times the spatial period as the fundamental lattice leads to three
distinguishable sites A, B, C. For the quantum spin Hall effect, thesuperlattice operates at a magic wavelength where the ac Stark
effect is opposite for the two spin states. For rubidium, this isachieved at a wavelength of 790 nm. When filled with fermionsthe quantum Hall and spin quantum Hall effects are realized.

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to schemes with two distinguishable sites A and B (byusing internal states [21,34] or a superlattice [48]), whichlead to a staggered magnetic field. Rectification of themagnetic flux in a staggered configuration by adding atilt [34,48] or a superlattice [21] has also been proposed.
In the latter scheme, this would result in four distinguishable sites (two internal states A, B, doubled up by thesuperlattice). Another rectification scheme uses three internal states [49]. Our scheme avoids spin-flip transitions
between internal states, and has the minimum number ofingredients of three different sites to provide directionality.
Furthermore, by adjusting the spatial phase shift between
the fundamental and the superlattice, one can choose theenergy offsets AB ¼ BC ¼ CA =2 (see Fig. 2). The
scheme can then be implemented by shining Raman beams
from two directions, each beam having two frequencies.
This scheme would realize Hofstadter’s butterfly and thequantum Hall effect. For the quantum spin Hall effect, one
has to choose the superlattice laser to be at the magic
wavelength where the scalar ac Stark shift vanishes, andonly a vector ac Stark shift remains, corresponding to a socalled fictitious magnetic field [50,51]. By detuning the laser
between the D1 and D2 lines, one can achieve a pure vector
ac Stark shift, which is equal in magnitude but opposite insign when the atoms in the two hypefine states have opposite
magnetic moments. In this case, the superlattice will provide
opposite potentials for the two states, resulting in opposite
momentum transfers due to the Raman beams and opposite
vector potentials. The superlattice period is =½2 sinð=2Þ,where  is the angle between the two superlattice beams,
which is adjusted to make the superlattice period 3 times theperiod of the basic lattice. This scheme realizes the quantum
spin Hall effect and a topological insulator with two opposite quantum Hall phases with a purely optical scheme andno Raman spin-flip transitions.
To replace the magnetic field gradient by a superlatticethat generates a fictitious magnetic field, the laser detuning
has to be on the order of the fine structure splitting, resulting in heating due to spontaneous emission. For atoms likerubidium, the lifetime is many seconds [50]. To be specific,
we consider a low-density gas 87 Rb atom in the F ¼ 2,
MF ¼ 2 states in a lattice with a depth of 10 photon
recoils at the wavelength of 1064 nm. A superlattice with alattice depth of 10 kHz is created by interfering two laser
beams at 790.0 nm of 1.0 mW of laser power and a beam
waist of 125 m. The resulting offset AB and AC will beapproximately 4 and 8 kHz, respectively, well placed in theband gap of the basic lattice. The spontaneous scattering
rate induced by the superlattice beams is less than 0.1 persecond. Alternatively, the superlattice producing the fictitious magnetic field can be replaced by a sinusoidal (real)magnetic field generated by an atom chip [52].
There have been several suggestions of how to detect
properties of the quantum Hall and quantum spin Hall
phases. Time-of-flight pictures will reveal the enlarged

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magnetic unit cell due to the synthetic magnetic field[48,53–55]. Hall plateaus can be discerned in the density
distribution [56]. The Chern number of a filled band can bemeasured interferometrically [57] or using ballistic expansion [58]. Topological edge states can be directly imaged[59,60] or detected by Bragg spectroscopy [61–63].
Topological insulators can be realized by engineering
appropriate lattice Hamiltonians with ultracold atoms [64].
This may allow studying classes of topological insulators[65,66] which have not been realized in nature. Topologicalinsulators with strong interactions [64] can be explored incold atom systems using Feshbach resonances.
Our work maps out a route towards spin-orbit coupling,
the spin Hall effect, and topological insulators which does not
require coupling of different internal states with spin-flipping
Raman lasers. The Hamiltonian describing the system isdiagonal in the z spin component. This follows closely
the two original Letters on the spin Hall effect [24,25]. In
addition, we have presented two configurations for realizing aquantum spin Hall Hamiltonian. The scheme with the magnetic tilt completely avoids near resonant light, and thesuperlattice scheme provides a purely optical approach.
This work was supported by the NSF through the Centerof Ultracold Atoms, by NSF Award No. PHY-0969731,
under ARO Grant No. W911NF-13-1-0031 with funds
from the DARPA OLE program, and by ONR. This work
was completed at the Aspen Center for Physics (supported
in part by the National Science Foundation under Grant
No. PHYS-1066293), and insightful discussions with Hui
Zhai, Jason Ho, and Nigel Cooper are acknowledged. We
thank Wujie Huang for a critical reading of the manuscript.
Note added.—After most of this work was completed[67], we became aware of similar work carried out by
Bloch and colleagues in Munich [33,68].

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