Spin-orbit Coupling and Supersolidity in Ultracold
Quantum Gases
byJunru Li
B. S., Physics (2012)
Tsinghua University
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
October 31, 2018
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacArthur Professor of Physics
Thesis Supervisor

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

2

Spin-orbit Coupling and Supersolidity in Ultracold Quantum
Gases
byJunru Li
Submitted to the Department of Physics
on October 31, 2018, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Ultracold quantum gases provide a clean, isolated, and controllable platform for simulating and characterizing complex physical phenomena. In this thesis, I present several
experiments on realizing one-dimensional spin-orbit coupling in ultracold 23 Na gases
and the creation of a new form of matter with supersolid properties using interacting
spin-orbit coupled Bose-Einstein condensates.
The first part describes the realization of spin-orbit coupling in optical superlattices which consist of a stack of pancakes of imbalanced double-wells. The orbital levels, individual pancakes, in a superlattice potential are used as pseudospin
states. Spin-orbit coupling was induced by two-photon Raman transition between
the pseudospin states, and was experimentally characterized by the spin-dependent
momentum structure from this dressing. The realization suppresses heating due tospontaneous emission. The system is highly miscible, allowing the study of novel
phases in interacting spin-orbit coupled systems.
Next, spin-orbit coupling was demonstrated by synchronizing a fast periodicallymodulating magnetic force with the Radio-Frequency (RF) pulses. The modulation
effectively dressed the RF photons with tunable momentum. The consequent Doppler
shifts for RF transitions were observed as velocity-selective spin flips. The scheme isequivalent to Floquet engineered one-dimensional spin-orbit coupling.
Finally, I report experiments on creating a new form of matter, a supersolid, inultracold quantum gases. An interacting spin-orbit coupled Bose-Einstein condensate in the stripe phase spontaneously breaks two continuous symmetries : the U (1)symmetry, observed as sharp interference peaks in momentum space, and the continuous translational symmetry, observed as a spontaneously formed density modulation.
The density modulation is measured and characterized with Bragg scattering. A
system spontaneously breaking these two symmetries is a crystal and a superfluid
simultaneously, and is considered as a supersolid.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics
3

Acknowledgments
I was once told that when acknowledging someone, it’s always better to be specificand detailed. But it is not the case when you own some people so much that you’re
not able to capture all the different aspects. That’s what my feeling was when I was
writing this.
First and foremost, I would like to express my gratitude to my supervisor Professor
Wolfgang Ketterle. I thank him for teaching me how to do physics, for guiding andencouraging me through this journey, and for babysitting the lab etc. Besides all of
the physics knowledge I am privileged to have learned from him, his true enthusiasm
for physics amazes and moves me all the time. I cannot remember how many times
it had happened that wherever and whenever you met him, he was always ready to
discuss physics with you if you wanted, and he was never tired of it. Because of thisunending enthusiasm and optimism, he possessed the power to cheer me up no matter
what happened in the lab and discussions with him during the group meeting has
always been enjoyable and exciting.
I thank the CUA community where I found a lot of true scientists and physicists.
I thank Professor Martin Zwierlein for being my academic adviser. 10-min of conversation with him each semester is enough to get me charged for doing physics. I
thank Professor Vladan Vuletić for taking me as a visiting student in my senoir year.
I thank people in Vladan’s group Marko Cetina, Xudong Chen, Alexei Bylinskii, Dorian Gangloff, Wenlan Chen, Kristi Beck, and Hao Zhang for their patience when
they are continuously bothered by my naive questions. I thank the students in CUA
who had made the lab a place where people what to come to instead of escape from.
Especially I would like to thank people from BEC3, 4, 5, and Dypole:Tout, Timur,
Hyungmok, Julie, Sepehr, Colin, Cody, Georgies, Woochang, Samarth, Ivana, Niki,
Jesse, Will, Michael, Li, Pierrel. It makes a huge difference if every time you havesome weird ideas on no matter what, you can always find people to talk to. And ofcourse, thank you for lending us things.
I thank the people I have worked with: Edward Su who has launched my Ph.
4

D. when I just joined the lab. Wujie Huang, with whom I had spent most of mytime with,who has become more like a brother to me. I thank Boris Shteynas who
fed me with enjoyable memories on science and music. Sean Burchesky for sharing
his love for doing experiments. I thank Alan Jamison for making my life betterin all different aspects one could imagine. I thank Furkan Top for the assistance inexperiments and tea discussions and Jeongwon Lee for collaborations on experiments.
I also thank previous BECII members Christian Sanner and Aviv Keshet for theirconstant supports on the BECII machine even six years after their graduation.
I thank my girlfriend Di Wu who has always been supportive to me and to mycareer outside, even inside, the lab. I thank my friends from college, especially Wenbo
Fu, Yingfei Gu, Fan Feng, Chenghui Yu, Wenjie Ji, Linda Ye etc. for their constantsupports during the past six years. The Ph. D. years were filled with ups and downs.
It was especially comforting that there are someone you can go anytime and have abeer with. I thank my other friends in the physics department especially Alex, Janosfor making my life much easier here in Boston.
I thank my previous physics teachers especially Professor Li You from Tsinghua
University, and Xiaozhen Li from high school who have shown me the beauty ofphysics when I was young.
I thank my parents for their unconditional support.
I thank the Chinese poet Yuanming Tao (365?–427) whose works have been amoral
supports to me during my time here. I would like to end with my naive translationof my favorite quote from him”Life is just a blink of time, why not follow your heart? ”

5

6

This doctoral thesis has been examined by a Committee of the
Department of Physics as follows:

Professor Wolfgang Ketterle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chairman, Thesis Committee
John D. MacAurthur Professor of Physics

Professor Martin W. Zwierlein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Member, Thesis Committee
Thomas A. Frank Professor of Physics

Professor Aram W. Harrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Member, Thesis Committee
Associate Professor of Physics

8Contents
1 Introduction

19

1.1

Spin-orbit Coupling, a historical review . . . . . . . . . . . . . . . . .

21

1.2

Supersolidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

1.3

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

24

2 Bose-Einstein Condensates in Optical Superlattices
2.1

2.2

2.3

2.4

27

Theory of Optical Superlattices . . . . . . . . . . . . . . . . . . . . .

28

2.1.1

Double-well Potentials . . . . . . . . . . . . . . . . . . . . . .

28

2.1.2

Optical Superlattices Potentials . . . . . . . . . . . . . . . . .

30

2.1.3

Bloch Bands and Wannier States in Optical Superlattices . . .

32

2.1.4

Bose-Einstein condensates in an Optical superlattice . . . . .

34

Creating an Optical Superlattice . . . . . . . . . . . . . . . . . . . . .

35

2.2.1

Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.2.2

Manipulating the Superlattice Configuration . . . . . . . . . .

38

2.2.3

Superlattice Alignment . . . . . . . . . . . . . . . . . . . . . .

43

Calibration of an Optical Superlattice . . . . . . . . . . . . . . . . . .

46

2.3.1

Superlattice Kapitza-Dirac Calibration . . . . . . . . . . . . .

46

2.3.2

Inter-well Interferometry . . . . . . . . . . . . . . . . . . . . .

48

Stability of an Optical Superlattice . . . . . . . . . . . . . . . . . . .

49

2.4.1

Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . .

49

2.4.2

Long-term stability - Galvo System Stability . . . . . . . . . .

51

2.4.3

Long-term stability - Ambient Environment Stability . . . . .

54

9

3 Spin-orbit Coupling

57

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

3.2

One-dimensional Spin-orbit Coupling . . . . . . . . . . . . . . . . . .

58

3.2.1

Spin-orbit Coupling as a Doppler Effect . . . . . . . . . . . . .

60

3.2.2

Spin-orbit Coupling as a Spin-flip with momentum transfer . .

60

3.2.3

Realizing spin-orbit coupling in Cold atoms . . . . . . . . . .

61

Spontaneous Heating . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.3

4 Spin-orbit coupling in optical superlattices
4.1

4.2

4.3

4.4

67

Spin-orbit Coupling with Orbital Pseudospins . . . . . . . . . . . . .

67

4.1.1

Perturbative Wavefuntions and Hamiltonian . . . . . . . . . .

69

4.1.2

On-site and the Spin-orbit Coupling Term . . . . . . . . . . .

70

4.1.3

Extended to Optical Superlattices . . . . . . . . . . . . . . . .

71

Experimental Realization . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.2.1

Pseudospin States Preparation . . . . . . . . . . . . . . . . . .

73

4.2.2

Detection of Pseudospin states . . . . . . . . . . . . . . . . . .

75

4.2.3

Lifetime of the Pseudospins . . . . . . . . . . . . . . . . . . .

78

Observation of the Spin-orbit Coupling in Optical Superlattice . . . .

83

4.3.1

Raman Setup . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.3.2

Dispersive and Population Signal . . . . . . . . . . . . . . . .

89

4.3.3

Broadening and Width . . . . . . . . . . . . . . . . . . . . . .

97Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5 Floquet Engineered Spin-orbit Coupling

101

5.1

Periodically Driven Systems - Floquet Engineering . . . . . . . . . . . 102

5.2

Spin-Orbit Coupling with Floquet Engineering . . . . . . . . . . . . . 105
5.2.1

5.3

Sequence and Effective Hamiltonian . . . . . . . . . . . . . . . 105

Experimental Realizations . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.1

Generation of the Modulating Magnetic Field Gradient . . . . 108

5.3.2

Signal and Measurements . . . . . . . . . . . . . . . . . . . . 110

5.3.3

Effects of Eddy Currents . . . . . . . . . . . . . . . . . . . . . 112
10

5.3.4
5.4

Effects of Anisotropic Trap . . . . . . . . . . . . . . . . . . . . 117

Micromotion Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6 Interacting Spin-orbit Coupled Systems and Supersolidity
6.1

6.2

6.3

Supersolidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.1.1

Phase and Symmetry Breaking . . . . . . . . . . . . . . . . . 134

6.1.2

Symmetry Breaking in a Supersolid . . . . . . . . . . . . . . . 135

6.1.3

Elementary Excitations - Goldstone Mode and the Higgs Mode 137

The Stripe Phase and Supersolidity . . . . . . . . . . . . . . . . . . . 138
6.2.1

Interacting Spin-orbit Coupled Bose-Einstein Condensates . . 138

6.2.2

Mean-field Ground State Wave Function . . . . . . . . . . . . 145

6.2.3

Mean-field Phase Diagram . . . . . . . . . . . . . . . . . . . . 149

6.2.4

Symmetry Breaking properties in the Stripe Phase . . . . . . 150

The Stripe Phase: why at large? . . . . . . . . . . . . . . . . . . . . . 152

7 Observation of the Supersolid Stripe Phase
7.1

7.2

7.3

7.4

133

155

Theory of Bragg Diffraction . . . . . . . . . . . . . . . . . . . . . . . 155
7.1.1

Bragg Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.1.2

Quantum Description . . . . . . . . . . . . . . . . . . . . . . . 157

7.1.3

Multi-scattering Effects - Heating . . . . . . . . . . . . . . . . 160

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.2.1

Beam Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.2.2

Optical Setup and Alignment . . . . . . . . . . . . . . . . . . 163

7.2.3

Alignment of the Bragg Beam . . . . . . . . . . . . . . . . . . 165

7.2.4

Fluorescence Signal . . . . . . . . . . . . . . . . . . . . . . . . 166

7.2.5

Few Photon Level Detection . . . . . . . . . . . . . . . . . . . 167

Warm-up : Bragg Scattering along the Superlattice . . . . . . . . . . 171
7.3.1

Detection of the Pseudospin Texture . . . . . . . . . . . . . . 171

7.3.2

Detection of the Band Relaxation Dynamics . . . . . . . . . . 174

Detection of the Supersolid Stripe Phase with Bragg scattering . . . . 175
7.4.1

On-site modulation and the Stripes . . . . . . . . . . . . . . . 180
11

7.5

Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

8 Conclusion and Outlook
8.1

185

Quantum Simulation with Degenerate Fermi Gas . . . . . . . . . . . 186

A Theory of Spin-orbit Coupling in Optical superlattices

189

B Spin-orbit Coupling and Spin Textures in Optical Superlattices

195

C A stripe phase with supersolid properties in spin-orbit-coupled BoseEinstein condensates

201

12

List of Figures
2-1 Eigenstates in a double-well potential.

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

29

2-2 From double-well systems to superlattices. . . . . . . . . . . . . . . .

31

2-3 Band minimum for the first two bands. . . . . . . . . . . . . . . . . .

33

2-4 Superlattice formed by overlapping two optical lattices. . . . . . . . .

36

2-5 Second Harmonic Generation and Periodic Polling. . . . . . . . . . .

37

2-6 Superlattice geometry controlled by a dispersive glass plate. . . . . .

39

2-7 Laser system for the long and short lattice light. . . . . . . . . . . . .

42

2-8 Seed switching with RF switches. . . . . . . . . . . . . . . . . . . . .

43

2-9 Layout of the superlattice. . . . . . . . . . . . . . . . . . . . . . . . .

43

2-10 Momentum kick from a pulse of misaligned ODT beam. . . . . . . . .

44

2-11 Chromatic effects in optical superlattices. . . . . . . . . . . . . . . . .

46

2-12 Intensity of momentum components vs. Kapitza-Dirac pulse duration.

47

2-13 1st order population imbalance of Kapitza-Dirac scattering from a superlattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2-14 Calibration of the superlattice offset ∆. . . . . . . . . . . . . . . . . .

50

2-15 Measurement of long-term drifts of the superlattice phase . . . . . . .

53

2-16 Chromatic effect of the air. . . . . . . . . . . . . . . . . . . . . . . . .

54

2-17 Drift of the air pressure. . . . . . . . . . . . . . . . . . . . . . . . . .

56

3-1 Spin-orbit coupling in one dimension. . . . . . . . . . . . . . . . . . .

59

3-2 Spin-orbit coupling as a Doppler effect. . . . . . . . . . . . . . . . . .

60

3-3 Spontaneous heating in the Raman scheme. . . . . . . . . . . . . . .

63

4-1 Spin-orbit coupling with orbital pseudospin. . . . . . . . . . . . . . .

68

13

4-2 Pseudospin Stern-Gerlach separation. . . . . . . . . . . . . . . . . . .

76

4-3 Stern-Gerlach separation of the spin states. . . . . . . . . . . . . . . .

77

4-4 Population in Stern-Gerlach separation of the spin states. . . . . . . .

77

4-5 Pseudospin interference pattern. . . . . . . . . . . . . . . . . . . . . .

78

4-6 Pseudospin texture. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4-7 Possible schemes for the Raman setup. . . . . . . . . . . . . . . . . .

84

4-8 Optical layout of the Raman beams. . . . . . . . . . . . . . . . . . .

84

4-9 Kapitza-Dirac pattern for Raman lattice with and without superlattice. 87
4-10 Two-photon Raman transfer in the a green optical lattice. . . . . . .

88

4-11 Higher band population from two-photon coupling. . . . . . . . . . .

89

4-12 Momentum components of |↓i state dressed with two-photon process.

90

4-13 Interference between the on-site and the spin-orbit coupling term. . .

92

4-14 Population imbalance between the ±k peaks. . . . . . . . . . . . . . .

93

4-15 Spectrum of the beat notes between the two Raman beams. . . . . .

94

4-16 Characterization of spinor BECs through their momentum distributions. 95
4-17 Spin-orbit coupling resonances. . . . . . . . . . . . . . . . . . . . . .

96

4-18 Interaction induced shift of resonance. . . . . . . . . . . . . . . . . .

96

5-1 Magnetic spin-orbit coupling scheme . . . . . . . . . . . . . . . . . . 106
5-2 Generation of the gradient modulation with H-bridge. . . . . . . . . . 109
5-3 Observation of velocity selective RF transitions. . . . . . . . . . . . . 110
5-4 Doppler shift of the RF photons. . . . . . . . . . . . . . . . . . . . . 111
5-5 Modeling the eddy current. . . . . . . . . . . . . . . . . . . . . . . . . 113
5-6 Impedance of the primary coil near vacuum chamber. . . . . . . . . . 114
5-7 Electric resonance profile of the coil. . . . . . . . . . . . . . . . . . . 114
5-8 Effect of eddy currents on observed velocity-selected atom slices. . . . 117
5-9 Two-dimensional Time-of-flight. . . . . . . . . . . . . . . . . . . . . . 118
5-10 Two-dimensional TOF with perfect velocity selection. . . . . . . . . . 119
5-11 Collisions between two Floquet states. . . . . . . . . . . . . . . . . . 124
5-12 Collisional cross section of two Floquet states. . . . . . . . . . . . . . 12514

5-13 Pauli blocking of the collisional processes. . . . . . . . . . . . . . . . 129
5-14 Floquet heating rate per particle. . . . . . . . . . . . . . . . . . . . . 131
6-1 Phase mixing and Phase separation. . . . . . . . . . . . . . . . . . . . 141
6-2 Interaction in a spin-orbit coupled system. . . . . . . . . . . . . . . . 143
6-3 Energy per particle vs. in an interaction spin-orbit condensate. . . . . 147
6-4 Phases of an interacting spin-orbit coupled condensate system. . . . . 149
6-5 Mean-field phase diagram β vs. U at δ0 = 0. . . . . . . . . . . . . . . 150
6-6 Mean-field phase diagram δ0 vs. beta. . . . . . . . . . . . . . . . . . . 151
6-7 Mean-field phase diagram for the superlattice stripe phase. . . . . . . 153
7-1 Bragg diffraction and Bragg conditions. . . . . . . . . . . . . . . . . . 156
7-2 Numerical calculation of the Bragg pattern. . . . . . . . . . . . . . . 159
7-3 Delivery of the Bragg probe beam. . . . . . . . . . . . . . . . . . . . 163
7-4 Mounting of the final Bragg mirrors. . . . . . . . . . . . . . . . . . . 164
7-5 Alignment of the Bragg beam. . . . . . . . . . . . . . . . . . . . . . . 165
7-6 Fluorescence signal from a condensate. . . . . . . . . . . . . . . . . . 168
7-7 Stray light mode filtering. . . . . . . . . . . . . . . . . . . . . . . . . 169
7-8 Pseudospin and intraband texture. . . . . . . . . . . . . . . . . . . . 172
7-9 Bragg reflection due to pseudospin texture. . . . . . . . . . . . . . . . 173
7-10 Dynamics of the intraband interference texture. . . . . . . . . . . . . 174
7-11 Transient behavior and enhancement. . . . . . . . . . . . . . . . . . . 176
7-12 Effect of Raman detuning on the supersolid stripes. . . . . . . . . . . 177
7-13 Bragg diffraction from the stripe phase. . . . . . . . . . . . . . . . . . 178
7-14 Quantitative study of the Bragg diffracted signal. . . . . . . . . . . . 179
7-15 Moving stripes from the on-site modulation. . . . . . . . . . . . . . . 182

15

16

List of Tables
3.1

Spontaneous Heating Rate. . . . . . . . . . . . . . . . . . . . . . . . .

64

4.1

Amplitudes from first-order perturbation calculation. . . . . . . . . .

72

4.2

Collisional Matrix elements of pseudospins. . . . . . . . . . . . . . . .

80

4.3

Inelastic collision lifetime of pseudospins. . . . . . . . . . . . . . . . .

81

5.1

Effective collisional velocity of Floquet states. . . . . . . . . . . . . . 121

5.2

Floquet heating rate per particle. . . . . . . . . . . . . . . . . . . . . 130

17

18

Chapter 1
Introduction
The concept of Quantum Simulation arose when it was realized that it is fundamentally hard to calculate the behavior of a quantum system with classical calculation
tools [1, 2, 3]. Therefore, instead of solving a Hamiltonian, a more practical way tounderstand the behavior of a system is to use a second system (the simulator) tosimulate the original one. As long as the two systems follow the same Hamiltonian
and the physics laws are universal, the behavior of the two systems should be thesame. This is the core idea of Quantum Simulation.
Three ingredients are needed to implement the idea of quantum simulation
1. The simulator should be a quantum system.
2. The system should be generic and manipulable such that a variety of Hamiltonians can be implemented and studied.
3. The system should be measurable and characterizable by clear measurements.
Recently, several systems been proposed to serve as such a universal simulator. Major
players include ultracold neutral atoms [4], trapped ions [5], photonic systems [6], andsuperconducting circuits [7]. In this thesis, we focus on the ultracold gases, neutral
atomic gases systems.
Ultracold atom systems, atoms cooled down to quantum degeneracy, have provento be one of the promising candidates as a generic platform for simulating complex
19phenomena. The production of Bose-Einstein condensates (BEC) where ultracoldbosons condensate at a single quantum state [8, 9], and the sequencial achievement ofdegenerated fermi gas [10], provide clean macroscopic quantum systems with millionsof atoms which can be controlled and characterized. Since then, the field of using thesecold atoms to simulate other systems has been flourishing. Especially, cold atoms
loaded into periodic potential formed by laser beams have been successfully used todemonstrate condensed matter Hamiltonians where atoms play the role of electrons.
When interaction between atoms are taken into account, complicated and exotic
many-body states can also be studied including the Bose-Hubbard model [11] and the
Fermi-Hubbard model[12] which potentially captures the essence of high-temperature
superconductor. The system can also be easily dynamically modulated, allowing therealization of a large category of dynamic Hamiltonains (Floquet Hamiltonian) [13,14]. This novel direction has shown great potential in creating unconventional mattersand phases such as time-crystals [15, 16].
One of the appealing directions emerging recently is to simulate artificial gauge
field for the ultracold atoms which are usually neutral. In general, simulating gauge
field is challenging for neutral atoms since they do not naturally couple to the electromagnetic field. However, as has been demonstrated by several groups [17, 18, 19], it
is possible to create artificial gauge field for neutral atoms with atomic-light interactions. A special type of gauge field is called spin-orbit coupling where the gauge field
experienced by atoms depends on the internal state of the atom (spin). In this case,the spin of an electron and it’s center-of-mass motion are no longer independent. An
intuitive example could be what happens in the spin hall effect where electrons with
opposite spins move along the opposite directions [20]. Such system displays non-zerospin current while shows no net charge current. Spin-orbit interaction also played an
important role in the field of spintronics and is considered to be an important candidate for making useful electronic devices [21, 20]. Realizing the spin-orbit interaction
with a controllable system from a bottom-up approach would be helpful in understanding how spin-orbit interaction would alter behavior of the system, and further,for example, how many-body physics could be combined with spin-orbit interactions.
20This partially motivates the study carried out in the thesis.
Beside simulating the existing phenomenon, the controllability to an ultracold
atom system could further allow the creation of new forms of matters which do nothave counterpart in nature. Several important examples including the realization of
Haldane phase by modulating the optical lattice in a time-periodic fashion [22] andthe creation of extreme magnetic field [17]. One of the interesting form of matters isthe so called supersolid phase, or more clearly, the “superfluid solid” phase which has
never been found in nature so far. It is a superfluid, but at the same time a crystal.
It is even controversial if there is such possibility for the existence of such a solid at
all [23]. With the tool of ultracold gases, people were able to create such a matterand were able to study it for the first time ( [24] and this thesis work [25]).
In this thesis, I will present works using ultracold quantum gases to simulate spinorbit interactions, and to a create new form of matter, a supersolid, in Bose-Einsteincondensates of 23 Na. As an introduction and overview, I first present some histories
and physics backgrounds for spin-orbit coupling and supersolidity.

1.1

Spin-orbit Coupling, a historical review

Spin-orbit interaction, where a particle’s internal degree of freedom spin is coupled
to its center-of-mass motion, was first revealed and studied in atomic system withthe light emission spectrum of Hydrogen in 1920s. Besides the observed discretelines which triggered the development of the Bohr model and further the Schrödinger
equation, it was observed that each line was further split into finer spectral structures,
known as the fine structure. To explain the splitting, two conclusions were reached:1) electrons have an intrinsic internal degree of freedom which can pick two discretevalues. This is known as the electron spin later. 2) the spin is coupled to the circularorbits of the electron and results in the further splitting of the spectral line depends
on the electrons orbital angular momentum. The coupling is due to the relativistic
effect.
Spin-orbit coupling has also been found in condensed matter fieldIn a condensed
21

matter system, the center-of-mass motion of the electrons are described with Blochwaves and the energy spectrum forms Bloch bands [26]. The interaction between theelectron spin and its center-of-mass motion modifies the band structure [27], consequently changes the transportation properties of the electrons and, therefore, further
modifies a material’s thermal and electric conductance. Such interplay between theelectron spin and charge motion has triggered numerous studies along different directions , for example spin hall effects [28], spintronics [21], exotic spin diffusions etc.
and has been implemented for designing spin-based devices [21]. Recently, Bernevig& Zhang [29], Kane & Mele [30] pointed out that when the intrinsic spin-orbit interaction is strong enough, it can lead to a class of topological materials which do notbreak the time-reversal symmetry. Such systems support robust dissipativeless spincurrents, which makes then intriguing candidates for future electronic devices.
Realizing spin-orbit coupling in cold atoms can help gain insights for how spinorbit interaction affects the system. However, as mentioned before, one of the fundamental issues is that neutral atoms don’t feel either Lorentz force or electric field. It
was not until 2011, the interplay between the spin and the center-of-mass motion of
the atoms was first engineered in 87 Rb system by Lin et. al. [31] through two-photon
Raman process (See chapter II). From then on, studies of spin-orbit coupling havebeen a hot topic in the ultracold gases research.

1.2

Supersolidity

The second theme of the thesis is on realizing a new form of matter, a supersolid,
which is a superfluid crystal.
Superfluidity, when fluids flow with zero viscosity, is one of the most remarkable
phases of matter discoveries in the 20th century. It was first discovered in the Helium4 by Kapitza [32], Allen and Misner [33], and was later related to the formation of
Bose-Einstein condensates citelondon1938. By far, superfluidity has been found in
only liquid and gaseous systems.
The concept of Supersolidity is first brought up in the 60s, when people conceived22

a superfluid which host a crystalline structure. It was conceptually contradictory
since that such a matter would preserve two seemingly mutually exclusive properties: it is a rigid crystal with long-range order, and a superfluid which can flow without
viscosity.
A possible mechanism was proposed for the existance of such a strange matterin 1969 by Andreev and Lifshitz [34]. They key idea is that with on average lessthan one particle per lattice site, the vacancies in a lattice could form a Bose-Einsteincondensate at low enough temperature., giving superfluidity behavior. Therefore, themost probable candidate could be the solid He-4, where the vacancies have the highest
condensate transition temperature. Following that, multi theoretical proposals have
explored the possibility of the existence of the supersolid phase [35, 34, 36]. However,
the experimental observation has been missing thereby.
In 2004, a research group reported the possible observation of a the supersolid
phase in solid He-4 [37]. The signature was that the momentum of inertia reducedwith decreasing temperatures, which was interpreted as the sign of the emergence of
non-dissipative portion in the solid. Though plausible at that time, the results turned
out to be a false result from the non-ideality of the experiment. Nevertheless, this result on He-4 triggered extensive efforts on the system and led to numerous discoveries
on the quantum properties of the solid He-4 at low temperature [38]. Example includes quantum plasticity [39], where He-4 crystals are elestic at low temperature, andunconventional mass transport [40], where superfluid-like mass tranport was foundwhen He-4 atoms were injected from superfluid into soilid He-4.
Ultracold gases systems which provides clean many-body systems and allows thecontrol over the interactions. BECs provide simple superfluid platforms. The question is, is it possible to make a BECs “crystallize” by artificially engineering the
Hamiltonian? If it is possible, maybe we don’t have a condensed matter type piece ofsupersolid, but at least we have things which have the same properties of a supersolid
and therefore studies could be carried out along the direction.
In this thesis, I present the result that by adding interaction to the spin-orbit
coupled system, the ground phase stripe phase is such a phase that share the same23

properties of a supersolid.

1.3

OutlineIn this thesis, I present three separated experiments related to the realization of the
one-dimensional spin-orbit interaction with cold gases of 23 Na and the consequent
realization of a supersolid phase. This thesis is organized as the following:• Chapter 2 is centered around the optical superlattices, which is the primary
tool we used for the work presented in this thesis. The theory of BEC in an
optical superlattice will be discussed first. The experiment part describes how
we construct, calibrate, and characterize ultracold atoms in such potentials.
• Chapter 3 presents theory regarding spin-orbit coupling physics. The chapter
starts from a more technical overview and concludes with a discussion on theprevious works and limitations when spin-orbit coupling in ultracold gases.
• Chapter 4 presents the experimental results on the realization of one-dimensional
spin-orbit coupling in optical superlattices. The chapter includes the specificrelated theory to spin-orbit coupling in optical superlattices and experimental
realizations.
• Chapter 5 is focused on realizing spin-orbit coupling via periodical driving.
Work described in this chapter involves no optical photons and is an independent approach compared with chapter 4. A general framework of the Floquet
engineering is first presented. The framework is then adopt to experimentallyrealize spin-orbit coupling with alternating magnetic field gradient.
• Chapter 6 is a theoretical chapter. Interaction is added to a spin-orbit coupled system. I analysis different resulting phases and point out the connection
between the stripe phase and supersolidity.
• Chapter 7 shows how we create the supersolid stripe phase and observed itwith Bragg diffraction experiments. I talk about the experimental and techni24

cal details on the experimental procedure. The data and results will also be
analyzed.

25

26

Chapter 2
Bose-Einstein Condensates in
Optical Superlattices
A lattice is a spatially periodic structure. It can be found in almost all the materials in
daily life. A lattice is characterized by it periodicity, this special discrete translational
symmetry results in band structures which determines the materials electric andthermal properties.
A superlattice is a periodic structure with multiple layers of materials. Comparedwith regular lattice, the relative arrangement of the different layers allows more exotic
properties of materials and leads to the realization of Hofstadter butterfly [41] andsuperconductivity [42].
In ultracold gases, simulating real materials with lattice formed by standing waveof lights, an optical lattice, has been very successful in the past 20 years [43]. By
directly controlling the lattice parameters, interesting phase transitions and manybody physics has been observed [11]. Optical superlattices have also been implemented and discussed under different context including quantum optics and quantum
simulation [44, 45].
This chapter is focused on theory and realization of a specific type of optical superlattice - double-well optical superlattice with laser lights for ultracold gases experiments. In the theory part, double-well optical superlattice potentials are introduced.
Compared with a regular lattice which can be describe by a single parameter lattice27

depth V0 , an optical superlattice has more degree of freedom therefore presents morefeatures and higher controllability reflected in the eigenstate wavefunctions and theband structure. In the experimental part, we present how we construct, control, andcalibrate an one dimensional optical superlattice. The chapter is concluded with a
discussion on the stability.
Discussions prensented in this work are limited to double-well superlattice potentials. Superlattices with more complicated unit-cell structures have been proposed
to realize intersting physics such as Quantum Spin Hall effects without spin-flip [46].
They will not be explored in this thesis.

2.1

Theory of Optical Superlattices

2.1.1

Double-well Potentials

We start with a single, isolated double well potential which is the fundamental building block for an double-well optical superlattice as shown in Fig.2-1. Double-wellpotential has been used to study various important phenomenon such as the interference of BECs [?],and interferometry experiments. Bosonic atoms in a double-wellpotential has been extensively studied theoretically in the context of “Josephson junctions” [47]. Atomic dynamics in a double-well potential is exactly solvable, and istherefore often used as a toy model for the understanding of more complicated latticephysics such as Bose-Hubbard physics [48], and disorder [49].
A double-well potential can be characterized with two parameters : the interwell
tunneling J and the energy offset between the two wells ∆. The two parameters canbe independently controlled. In the limit of J → 0, a double-well potential will host
two obvious eigenstates, |Li (|Ri) which denotes the state that the atom is maximally
localized in the left(right) well of the potential. Though each well can have multiple
levels as the case in a harmonic oscillator, we consider only the lower energy level ineach of the well. The eigenenergies for the two eigenstates are 0, ∆, the double-welloffset.
28

a

b

J

R

Δ

c

|2>

4
3

J/Δ

J/Δ

2
1

|1>

L

L

R

Figure 2-1: Eigenstates in a double-well potential. a,b. The lowest two eigenstates ina double-well potential are localized in either the left and the right well. The interwell
tunnling introduce a leakage to the other with an amplitude J/∆. the phase for thetwo satellites are opposite. c. A double-well potential separated the first two statesfrom the other, forming an effective two-state manifold.

The interwell coupling J introduces mixing between the two eigenstates. The
single particle eigenstates and the corresponding eigenenergies can be calculated bydiagonalizing the full 2 × 2 Hamiltonian




0 J

H=
J ∆

(2.1)


√
1
J
2
2
∆ − 4J + ∆|1i = |Li + |Ri , E1 =
∆
2


√
1
J
2
2
∆ + 4J + ∆|2i = |Ri − |Li , E2 =
∆
2

(2.2)

and readIn first order, the mixing amplitudes are ±J/∆ and have opposite signs for the twostates.
The two original states |Li and |Ri are pushed away in energy from each other√by 4J 2 + ∆2 . If the two states were originally degenerated, namely ∆ = 0, thedegeneracy is lifted and a gap of 2J opens up.
The mixing is fundamental to future operations since it provides the possibilityof coupling the two eigenstates |1i and |2i with coupling matrix elements. Morespecifically, consider a coupling with the form hx|Ω̂|x0 i = Ω(x)δ(x − x0 ). With J =0, the resulting coupling matrix element between hL| Ω |Ri would be hL| Ω̂ |Ri =29

R

dx ψL† (x)Ω(x)ψR (x) = 0 since there is no spatial overlap between the |Li and |Ristates. When a mixing is introducedZ
h1| Ω̂ |2i =dx ψ1† (x)Ω(x)ψ2 (x)(2.3)

depending on the form of Ω(x) there is possibility that a tunable coupling can be
introduced between |1i and |2i. We will come back to this point in section 4.1.
Another important feature on the signal particle spectrum of a double-well potential is the well isolation of the two lowest levels to the higher energy manifold (1,2from 3,4 shown in Fig.??c). This can be understood by considering the adiabatic
switching on of the middle barrier in a harmonic trap. In the harmonic trap, all thefour states are equally spaced. The barrier lifts the energy of the even-parity statesR
more due to larger wavefunction overlaps V = |φi (x)|2 V (x)dx, causing the spectrum separation between 1,2 manifold from the others. This point will be discussed
carefully in the next section.
Inter-atomic interactions could introduce non-trivial dynamics including atom
number squeezing [50], chaotic behavior, and other non-linear phenomenon [47]. Thisis beyond the scope of this thesis.
A single Double-well potential has been implemented experimentally with, forexample, atomic chips [51] and by combining optical and magnetic potential [?]. Fullcontrol over the double-well configuration can be achieved easily.

2.1.2

Optical Superlattices Potentials

An optical superlattice potential used in the thesis has the form
V (z) = −VI
VG
cos 2kIR z −cos (2kG z + φ)
2
2

(2.4)

It can be understood as a chain of double-wells with non-zero tunneling between twonearest double-wells. In our scenario where there is no transverse confinement, aquasi-1D superlattice can also be understood as a stack of pancakes. The phase φ30

controls the ‘symmetry’ of each double-well, the parameter VI controls the offset ∆,while VG controls the interwell tunneling J. These three parameters fully determine
the configuration and properties of the studied superlattice.
When having periodically arranged isolated identical system, the total eigenstateof the entire system will be a direct product of the eigenstate in each subsystem. Whencoupling between subsystems is introduced, the originally degenerated eigenstates will
be resonantly coupled and the energy spectrum is broadened due to avoided crossing,forming bands. The width of the spectrum is on the order of the coupling. Similarly

a

t2

d

t1

b

Vl = 7EI Vs = 10EI

E/EIR

20

= 0.625

Vl = 7EI Vs = 10EI

20

= 0.5

Vl = 7EI Vs = 10EI

20

15

15

15

10

10

10

5

5

5

0

0

0

0

0.2

q / ( /d)

0.4

0

0.2

q / ( /d)

0.4

0

0.2

q / ( /d)= 0.375

0.4

Figure 2-2: a. Coupling between the two nearest subsystems broadens the eigenstates in a single system into bands. The tunneling matrix elements between |1i
and |2i have opposite signs, which is crucial for understanding the band structure of
the superlattice(lattice). b. First three bands in optical superlattices with different
parameters.
the extra tunneling J2 and J1 introduces band structures, as in the case for regularlattices where the harmonic levels in each sites are broadened into bands due totunneling.
31

2.1.3Bloch Bands and Wannier States in Optical Superlattices

In a lattice, translational symmetry is discrete as Ĥ(x + d) = Ĥ(x) where d is thelattice constant. With this special symmetry, the eigenstates and the corresponding
energies are described by the Bloch theorem [52] which states:
1. Eigenenergies of the particle in a lattice forms bands.
2. Momentum is not a good quantum number. State of a particle is labeled withthe band index m and quasi-momentum ~q can pick − NN−1 πd , − NN−2 πd ... NN−1 πd , − πd .
Note that the total number of possible values is equal to the number of sites
such that the number of states is conserved. Atoms with quasi-momentum q
contains multiple real momentum components q + 2lπ/d where l is an integer.
3. The eigenstates(Bloch states) wavefunctions, the Bloch waves, can be written~ = u(~r).
as ψ(~r) = ei~q·~r u(~r) where u(~r + d)
4. Bloch states form a complete basis.
The Bloch states extend across the entire lattice, it is often useful to construct
the Wannier states which is the maximized localized states on a single lattice site. A
Bloch state is therefore written asm|ψBlochi=nXei~q~ri |ψ m ii .

(2.5)

i

where |ψ m ii is the m-band Wannier state on site i.
Importantly, to the lowest order, the Wannier states for band m is very close to
the mth trapped state of the single subsystem when inter-site tunneling is absent.
This can be understood by considering an adiabatic switching on of the tunneling.
The ground band of the lattice will simply be a superposition of all the ground statesin each subsystems, while the first excited states will correspondingly form the first32

excited band in the new lattice system. Therefore in a superlattice system we have2i=|ψBloch1|ψBloch
i=

n
X
i
n
X

eiqxi |2ii
(2.6)
e

iqxi

|1ii

i

where |2i , |1i are the eigenstates discussed in the previous section. Similar to a

A

E

A

B

B
-4

0

0.2

q / ( 2 /d)

0.4

Wavefunction

-2

0

2

4 �/d

Momentum Components

Figure 2-3: Band minimum states for the first two bands and the corresponding
wavefunction and the experimentally observed momentum components.
regular lattice, the lowest energy state in the first band has quasi-momentum q = 0,
while the lowest energy state on the second band has q = π/d therefore the Blochwave gains a minus sign when moves by one lattice site as shown in fig.2-3. This key
difference is crucial for the experiments described in this thesis and emphasizes theimportance of a superlattice over a single double-well.
The energy gaps between bands are roughly equal to the gaps between the corresponding single site states. Therefore the band gap between the first two band are√approximated to be 4J 2 + ∆2 , while the gap between the second band and higher
bands are much larger than the splitting between the ground and the first band forthe reason discussed in the previous section.
∆ ≈ E2 − E1  E3 − E2

(2.7)

In other words, unlike in a regular lattice where the band gap separates the first band33

from the other, a superlattice provides a low energy Hilbert space where both the first
and the second bands are well separated from other higher bands while these twobands are relatively close to each other. Atoms sitting in the left well in a single unit
cell is mapped to the first band while the atoms in the right well is mapped to thesecond band. This provides an complementary scenario to the regular lattice and willallow the study of physics where two bands are involved for example the extended
Bose-Hubbard model [53].
Parameters of the bands, e.g. gaps, width etc. can be controlled by tuning thesuperlattice configuration, namely VI , VG , φ. We leave the technical implementation
discussion to the next section.

2.1.4

Bose-Einstein condensates in an Optical superlattice

At T = 0, a cold Bose gas fully condensate to a single state with lowest energy. For
an ensemble of N particles, the many-body wavefunction is the product of identical
single particle wavefunction corresponding to the condensate stateΨN (~r1 , ~r2 , ..., ~rN ) =NYφ(~ri ).

(2.8)

i=1

therefore a condensate can be described by a macroscopic wavefunction√
ψ(~r) =N φ(~r)(2.9)

notice that the many particle coordinate (~r1 , ~r2 , ..., ~rN ) is replaced with a single coordinate ~r.
For the ground band of an optical superlattice, condensate forms at q = 0. The
momentum side band 0, 2π/d, 4π/d... can be resolved by sudden switching off of thetrapping potential and image the spatial density distribution after long enough ballistic expansion of the cloud (Time-of-flight Imaging), as shown in 2-3.
Condensation on the second band happens at the band minimum q = π/d. The
momentum components π/d, −π/d, 3π/d... are resolved in the similar way shown in34

2-3.Though it is a metastable state, e.g. atoms eventually decay to the first band, thelifetime (∼ 500 ms) is long enough for a stable condensate to form at q = π/d. On
the other hand, condensate formed at q = π/d in the first band decay to the band
minimum within a time scale of 1 ms.
Above I have presented the basic theoretical aspects related to a Bose-Einstein in
an optical superlattice. In the following section, I present technical details on how
superlattices were constructed, controlled, and calibrated in the laboratory.

2.2

Creating an Optical Superlattice

The superlattice is constructed by overlapping two regular optical lattice with one
lattice wavelength being exactly twice as the other. Each optical lattice is a standing
laser field formed by retro-reflecting a Gaussian laser beam. An atom sitting in an
oscillating laser field experiences a potential due to the AC stark shift. When theoscillating frequency is far detuned form the atomic transitions, the AC stark shift issimplified to beV =h̄Γ2 I(~r)
8∆ Isat(2.10)

where Γ is the spontaneous emission rate of the transition, ∆ is the detuning of
the laser, and I is the laser intensity. The potential is proportional to the laser
intensity while the property of the potential, repulsive or attractive, can be changed
by changing the detuning ∆ from blue(∆ > 0) to red(∆ < 0). A standing laser field,with periodic I(~r) forms a periodic potential for the atoms, simulating an lattice.

2.2.1

Laser System

Our superlattice was implemented with λl = 1064 nm and λs = 532 nm high power
lasers. For 23 Na which has a primary transition of ∼ 589 nm, we have both the redand blue-detuned lattice.
The 1064 nm light is provided by a 50W Fiber amplifier seeded by a single mode
Mephisto seed laser with a seed power ≈ 150 mW.
35

Figure 2-4: Superlattice formed by overlapping two optical lattices. Each opticallattice is generated by retro-reflecting a Gaussian laser beam, forming a standingwave.

The 532 nm lattice light is obtained by frequency doubling the 1064 nm lightwith a frequency doubling crystal (second Harmonic Generation). Crystals breaking
inversion symmetry have non-zero χ(2) coefficient. When a plane wave of amplitude
E(ω) traveling in such non-linear media. Polarization
P (2ω) = 0 χ(2) E 2 (ω)(2.11)

at doubled frequency is generated.
To obtain efficiency conversion, proper phase matching is required. Frequencydoubled light generated at different positions of the crystal has the same phase as
the seed light at the same position. However, because the green and IR light havedifferent refractive indexes in the crystal, the frequency-doubled light generated at
two different positions will not simply added up in a maximally constructive way.
Instead, they add up with a small relative phase between them - reducing the total
intensity of the final green light. In the worst case, green lights generated at different
positions could completely cancel out due to the destructive interference.
Formally, in the simplest model which ignore depletion of the seed light and assumes plane wave, the field strength of the second harmonic along the direction of
propagation z reads∂E(2ω)∝ E 2 (ω)ei∆kz∂z36(2.12)

where ∆k = k(2ω) − 2k(ω). The intensity at the end of the crystal z = L is therefore
2I(2ω, L) ∝ L

sin (∆kL/2)
∆kL/2

2

I 2 (ω)(2.13)The phase matching condition is therefore clear.
A technique called Periodic Pooling is implemented during manufacturing to compensate the phase mismatch, as shown in 2-5b. In short, the crystal was build withalternating layers between type-A and type-B materials. The differential refractive
index for 1064 nm and 532 nm lights ∆n = n1064 − n532 has opposite sign. In this
scenario, while type-A is providing the frequency doubling, layer made from typeB material will compensate the undesired phase difference accumulation during the
doubling stages such that the total phaser will by stretched instead of forming loops.
We used the MgO:PPLN frequency doubling crystal from Covision.
The refractive indexes of materials depend on the temperature. To minimize
drifting of the doubling efficiency and optimize the phase matching condition, it isimportant to make sure the crystal is stabilized at the correct temperature. Practically, crystals are heated up locally by the high power input seed light. It is therefore
an empirical process to obtain the optimum condition for best efficiency and the
optimum condition could change with the power of the input.

a

c

b

A B A B A B

Figure 2-5: a. Periodic polling phase matching in a frequency doubling crystal, bycompensating the differential phase, generated green light added up constructively.
b. Efficiency of the second harmonic generation vs. input power. Depletion showed
up at high power. c. Optimum crystal temperature vs. input power..
The beam profile of the IR light sent into the crystal was also important. Fre37

quency doubling is a non-linear process so in principle the higher the intensity is,
the higher the efficiency will be. However, too much intensity causes optical damage
to the crystal and depletion. Moreover, if the high intensity is achieved by tightly
focusing, the Rayleigh range will also be shortened and the ‘effective volume’ of thecrystal will drop. Therefore there is always an optimum beam profile. For us, it isaround 42µ m.
With this, we obtained around ∼ 10 W of 532 nm light out of ∼ 30 W input IR
power, giving an efficiency of 33%. The 1064 nm and frequency-doubled 532 nm light
were then sent to the experimental table via high-power photonic bandgap fibers.

2.2.2

Manipulating the Superlattice Configuration

We control the configuration of the superlattice in the following way. The lattice
depths VI and VG are controlled with the standard Proportional-Integral-Differential
Feedback control. The relative phase of the two laser lights needs to be stable andcontrollable.

Phase Control
We did the following to stabilize the phase between the 1064 nm lattice and the532 532nm lattice.

Stability Instead of active feedback as done in the phase lock schemes, we locked
the phase passively. Our scheme consisted of two parts
1. The two fiber amplifiers used to generate the IR light for the long lattice andthe IR light for frequency doubling are seeded with the same Mephisto seed. If
seeded separately, normal relative frequency drifts of the seed lasers at the levelof ∼ 10 MHz/min would change the superlattice phase φ from 0 to 2π with inseveral minutes. By sharing the seed this zero-th order drift becomes common
mode.
38

2. The two lattice are sharing the same retro-mirror. Both the lattices are guaranteed to have a node at the mirror and are therefore automatically phaselocked.
The second part sounds magical. Understanding how actually it works just require
some basic E&M knowledge. The nodes appear at the reflective surface is due to theπ phase difference at the reflection. The field is reflected because the input electric
filed drives some oscillations of the dipoles in the di-electric materials. The time scale
for reaching the steady state of this process is only several optical periods which is∼ THz. So in other words, the retro-mirror is acting as a THz feedforward follower
which faithfully follow the phase of the input light.

Control The relative phase is controlled in two ways : 1) a slow but large dynamic
range way with a dispersive glass plate on the beam path shared by the two latticelight. 2) A fast but small range way by quenching the frequency of the laser light forthe 1064 nm lattice.
The refractive indices of the glass plate for the 1064 nm and 532 nm light are
different, therefore creates different phase shifts for the two lattice lights, as illustrated
in Figure 2-6. The differential phase delay ∆φ depends on the differential refractive
Dispersive Glass PlateᶿCollimating Lens

Shared Retro-mirror

Figure 2-6: The figure shows the setup for controlling the relative phase of the superlattice by a single dispersive glass plate. The thickness of the glass plate is exaggerated
for clearness.
index ∆n = n1064 − n532 , the thickness d of the plate, and the tilting angle θ as
illustrated in Figure 2-6. The general expression for the phase delay caused by the39glass plate for an arbitrary angle θ can be found in [54]. When θ = π/2,expression
simplified to (here I further assume the air to be achromatic to emphasize the effectsof the glass plate)∆φ = d · (n1064 k1064 − n532 k532 )(2.14)

This differential phase delay allow us to control the phase by simply rotating the glass
plate, which can be done by using a galvo or simply a manual mirror mount. It is aneasily-setup scheme.
Material

n1064

n532

Fused Silica

1.44961.4607BK71.50661.5195

The glass plate provides wide range tunability, but was not suitable for dynamic
manipulation for the following two reasons:
1. The glass plate will not only change the relative phase of the two lattice but also
the absolute phase. In other words, the entire lattice will be shifted when theglass plate is rotated. The moving lattice therefore drag the BEC and creates
heating and excitations.
2. The galvo has a bandwidth of < 1kHz. To be able to manipulating the superlattice geometry while maintaining the population in each well, the bandwidth
required should be larger than the inter-well tunneling J ≈ 2π ×2kHz in a single
double-well. It was not possible to use it for fast switching of the superlatticeconfiguration which is crucial for our experiments (see Sec.4.2.1). Since The
atoms would adiabatically follow the changing of the superlattice and maintainthemselves at the ground state.
When fast dynamic control(> 1 kHz) was implemented by shifting the frequencyof the 1064 nm lattice light.
When the frequency of long lattice light is shifted by ∆ν, the extra propagatingphase accumulated across the distance L between the atoms and the retro-reflecting
40

mirror for the 1064 nm lattice is∆φ = ∆k · L =

2π
∆νL.
c

(2.15)

For L = 0.3 m and ∆φ = π/4(which is the range required for our purpose), a change∆ν ≈ 125MHz is needed.
Usually, fast shift of the laser frequency is achieved by changing the injection
current of the laser diode. This allows a bandwidth of ∼ MHz. However, in our case
we intended to change the frequency of only one branch of the seed. This suggested
using an Acousto-opto modulator in a double-pass configuration for only the seedlaser for the fiber amplifier for the long lattice. By changing the frequency of the RF
driving, the frequency of the deflected beam is modified, as illustrated in Figure 2-7.
An Acousto-opto Deflector guarantees decent diffraction efficiency across a tuning
range of 125/2 = 62.5MHz. A deflector has several additional pieces of actuators
along the propagating direction (instead of a single one for AOMs) of the acoustic
wave inside the crystal. These actuators work collectively such that the wave frontof the RF driving is effectively rotated to compensate the changing of the Bragg
condition due to the changing of the RF frequency. The RF source for the deflector
would need to be able to ramp by ∼ 100MHz within several 10µs and the faster the
better. This is hard for a regular function generator. Voltage Controlled Oscillators
are noisy. Since the noise of the RF will be directly printed on the intensity of theseed therefore the lattice light, we tried to avoid using VCOs. Eventually, we preconfigured two function generators to the target RF frequencies corresponding to
the initial and final configurations of the superlattice designed. We then switchedbetween them with RF switches during the experiment. This was a dangerous idea
in two senses:
1. We put several controlling unit on the seed for the Nufern amplifier. It increased
the chance that the amplifier accidentally loses its seed due to either human orsequence error.
2. The dead time during the switching process, as shown in Figure 2-8b, could
41

Mephisto
Seed

Double-pass
ADM

Fiber Amplifier for
532nm lattice light

Frequency
Doubling
CrystalTo Experiment

To Experiment

Fiber Amplifier for
1064nm lattice light

λ/4

Seeding

Amplifing

Doubling

Delivering

Figure 2-7: Laser system for the long and short lattice light. A double-pass ADM was
set up on the seed path for the IR laser, allowing fast switching of the laser frequency.

be dangerous. The dead time ranged from 500ns to several microseconds, depending on the specific choice of hardwares. An RF switch and the amplifier
usually have a rise time on the order of several ns A deflector has an access timeof ∼ 300 ns/mm depending on the beam size due to the limited propagationof the acoustic wave across the beam. We found that the amplifier would still
operate normally only when the dead time is shorter than 1µs,.

To minimize the chance for human or sequence errors, we made sure the switchingsystem was set up in a way that there was guaranteed to be a seed when all thecontrol signals were off.
To minimize the dead time, we first tried different type of Rf switches. Then wecame up with the idea of using two switches, one for each RF source as illustrated
in Fig. 2-8. When appropriately coordinated with the sequence, we guarantee there
was a short overlapping period ∼ 1µs when both the seeds are on. We were notable to completely eliminate the ‘dents’ but the power was enough during the entire
switching process for the amplifier to operate.
42

a

b
FG

c

RF1
RF
swtich141MHz
TTL I

RFswtich174MHzFG

to RF
amplifeir

RF2

TTL II

Seed switching circuit layout

Single Switch dead time

Double switch dead time

Figure 2-8: a. Circuit layout. b.Seed dead time during the switching with a single
switch. c. Seed light dead time during the switching with both the switches.

2.2.3

Superlattice Alignment

After the light preparation, I now discuss the alignment procedure. Alignment ofRetro-mirror

Dispersive
Glass Plate

Dichroic Mirror

Figure 2-9: Optical layout of the superlattice.

the superlattice is in principle similar to the alignment of a single lattice. However,the chromatic effects from the refractive optics makes the alignment harder for an
superlattice. Besides, the 1064 nm and 532 nm lattice shared the retro-reflecting
mirror, which makes the alignment tedious.
43

First We blocked the retro-reflected beam and align the input branch of the latticelight as a new ODT beam. What we want to achieve was that this new ODT beam
was well aligned to the original condensate. A rough alignment was done by firstimaging the BEC in trap and then aligning the ODT to the same spot on the camera.
This has always been good enough to give initial signal for further fine alignment.
Further fine alignment was done by the ‘kicking’ method. The idea is to pulseon the full power lattice beam with the retro-reflected beam blocked and see themomentum kick from the light pulse to the atoms. If the beam is misaligned, thepulse of extra potential will exert an imbalanced impulse to the condensate and acorresponding net momentum kick along a certain direction depending the misalignment. This momentum has a dispersive behavior (Fig. 2-10. When the beam is wellaligned, the net momentum kick is absent, accompanied by a visible dipole oscillationof the condensate due to the balanced compression from the pulse. This signature isoften used to distinguish between the situation of perfect alignment and totally off(the beam is around several beam waists off) since neither of them will give a net
momentum to the atomic cloud. For a blue-detuned light with a small beam waist,
when the beam is getting close to the cloud, the force is so strong that the cloud isblasted and can form a crescent or a ring depending on the alignment.

Momentum Kick (arb. unit)

Momentum Kickfrom a Red- detuned light
0.4
0.2
0.0
- 0.2
- 0.4
-4

-2
0
2
Misalignment (arb. unit)

4

Figure 2-10: Momentum kick from a pulse of misaligned ODT beam.

Second The alignment of the retro-reflected beam was done by back-coupling.
When the mode of the retro-reflected beam is well overlapped with the incoming
beam, it can be back-coupled into the light delivery fiber from its output port and44

be monitored at the input side of the fiber. The retro-mirror and the collimatinglens before it was adjusted to maximize the coupling efficiency. We then repeated the
ODT aligning procedure (adiabatic loading and kicking) without blocking the beam
for further fine alignment.
Chromatic effects from the refractive optics complicates the alignment for superlattices. As presented previously, the focal length for a fused silica lens is different
for the two lattice lights. For 532 nm and 1064 nm we haven1064 − n532
f1064 − f532≈
f1064
n1064 − 1

(2.16)

which is a difference of ∼ 2%. Three direct consequences are:Undesired Focus Displacement Two light beams can not be simultaneously focused on the atomic clouds. This effect is usually negligible. The focal length shift∆f = 6 mm for a 300 mm length. This shift is usually much smaller than the Rayleighω2

range Rayleigh = π λ0 ≈ 60 mm with a beam waist of ω0 = 100 µm, causing only 1%
intensity drop compared with the waist intensity.
Alignment Complication Illustrated in Fig.2-11 b., c. When the lattice optics
are perfectly adjusted for the 1064 lattice light, a displaced d from the optical axis
by a distance d for the green light caused a displacement of δ = d · ∆f /f ≈ 0.02d
at the atoms. A deliberate tilt of the input light is necessary to compensate if themisalignment is larger than the condensate size, ω0 + RBEC ≈ 70 µm. Because of thistilting, the retro-reflected mirror has to be put at a certain distance from the second
collimation lens to ensure that the retro-reflected beam and the input beam overlap
at the atoms. This put an upper limit of d < 3.5 mm.
We minimized the second effect by centering the lattice beams on the optics asmuch as we can. Iris was put on all the lenses shared by the two lattice lights as amarker(including the imaging system along the lattice direction to improve the qualityof the rough alignment procedure). The superlattice performance was immediately
improved. In the future, chromatic effects can be eliminated by using s curved mirrors
45

a

b

d

f1064

f1064

c
f1064

f1064

Figure 2-11: Assuming that the lattice system has been perfectly setup for λ =1064 nm lattice, chromatic effects make it hard to simultaneously align the otherlattice. a. Misfocus. b. Misalignment. c. Failure of back coupling.
for collimation.
This concludes the construction of an optical superlattices.

2.3

Calibration of an Optical Superlattice

We subsequently calibrated superlattice. Understanding the band structure of thesuperlattice is essential for lattice diagnosis, and the study of orbits in the superlattice.
In this section, I describe techniques we used to probe the parameters {VI , VG , φ} andthe band structure of an optical superlattice.

2.3.1

Superlattice Kapitza-Dirac Calibration

Lattice depths VI and VG were calibrated with the standard Kapitza-Dirac scatteringexperiment. When the lattice was pulsed on a BEC for a time much shorter than
the recoil time, the condensate was projected onto different bands. Atoms populating
different bands beat with the beating frequency equal to the band gap. We assume
a single boson with momentum k = 0 at t = 0 and ψ(0, x) = const.. The lattice forcalibration is pulsed on for some short time δt. The state after this evolution |ψ(t)iwill bei|ψ(t)i = e− h̄ H/δt |ψ(0)i
46(2.17)

with H = p2 /(2m) + VI Cos(2kI x) where kI is the wave vector of the laser light used.
When the evolving time δt is much faster than recoil energy ER = h̄2 kI2 /(2m), we canignore the kinetic energy term in ??, the evolution then gives a final wave function
at t = 0 + δti

ψ(δt, x) = e− h̄ VI Cos(2kI x)δt ψ(0, x)i= e− h̄ VI Cin(2kI x)δt
X
VI=
in Jn ( δt)ei2nkI xh̄
n=±1,±2...

(2.18)

Only even multiples of kI are coupled to the BEC and the amplitude of the 2nt hcomponent follows Jn ( Vh̄I δt) depending on the lattice depth and pulse duration δt,where Jn is the nt h order Bessel function. The lattice depth VI is extracted from thepulse duration when the first order diffraction peak zeroed with the relationVδt = 3.831h̄
The situation is the same for all monochromatic lattices after normalize the lattice
depth and evolution time with the corresponding recoil energy. Direct calibration

Figure 2-12: Intensity of momentum components vs. Kapitza-Dirac pulse duration.

of φ can be done by implementing the Kapitza-Dirac diffraction for a superlattice
H = VI cos(2kI x) + VG cos(2kS x + φ). Following the similar procedure, the state at47

time t reads
ψ(δt, x) =X

im+n Jn (VI δt)Jm (VS δt)ei(2nkI +m)kI x+mφ

(2.19)

n,m

φ creates an imbalance between 2n + m = 1 and 2n + m = −1 states. In the lowest
order approximation, we haveA2n+m=1 =X

i1−n Jn (VI δt)J1−2n (VS δt)ei(1−2n)φn(2.20)≈ J0 (VI δt)J1 (VS δt)eiφX
A2n+m=−1 =i1−n Jn (VI δt)J−1−2n (VS δt)ei(−1−2n)φ ≈ J0 (VI δt)J1 (VS δt)e−iφnThe population imbalanceγ=|A2n+m=1 |2 − |A2n+m=−1 |2|A2n+m=1 |2 + |A2n+m=−1 |2(2.21)

can be readily used as a measure for the phase φ. The advantage of this method isthat φ can be calibrated with a single shot. However, the drawback is obviously that
this is not a direct measurement of ∆ which is more experimentally relevant. With
all these considerations, we used the first method for our experiments.

2.3.2

Inter-well Interferometry

The offset ∆ was directly calibrated by observing a beat note within a single doublewell. This is conceptually the same as a double-slit interference experiments. Condensates with equal population of atoms in the left and right well were prepared by
pre-setting the glass plate to the desired value and then rapidly ramping up the IR
lattice to the final offset. Before atoms relaxed to their corresponding band minimum, interference patterns for atom sitting in the lower and upper well overlap andtherefore interfere with each other. The interference pattern evolves periodically withfrequency ω = ∆(Fig.2-14).
48

a

20

c

VI = 10EI , VG = 10EI ,
=0

20

0
0

1

2

3

0
0

Position (λ L/2)

Normalized Population Imbalance

VI = 10EI , VG = 10EI ,= 0.3πPotential (EL)

Potential (EL)

Potential (EL)20b

VI = 10EI , VG = 10EI ,= - 0.3π

1

2

3

0
0

Position (λ L/2)

1

2

3

Position (λ L/2 )

0.4
0
- 0.4
- 0.5

0
(π )

0.5

Figure 2-13: 1st order population imbalance of Kapitza-Dirac scattering from a superlattice vs. superlattice phase φ.

2.4

Stability of an Optical Superlattice

After developing a procedure for superlattice calibration, we examine the stability ofthe superlattice. As already discussed in sec.??, lowest order stability are achieved bysharing the seed and the retro-mirror. However, as I will present and discuss in thischapter, changes in the ambient environment such as air pressure P was identified as
the major contributor to the residual lattice drift.

2.4.1

Sensitivity Analysis

As will be discussed in sec.4.1, the most relevant parameter is the offset ∆. To thelowest order, offset ∆ is determined by the long lattice depths and the superlattice49

OD (a.u.)
Fraction of Population

0
Hold Time ( s)Figure 2-14: Calibration of the superlattice offset ∆. The relative phase between
atoms in the lower and upper wells accumulated linearly with time and results in periodic changes in the time-of-flight interference pattern. The period of the oscillation
is equal to ∆ which is ∼ 23 kHz for the data shown.
phase
VI
1 π
1 3π
(cos [ ( − φ)] − cos [ (− φ)])
2
2 2
2 2
φ
VI
= − √ cos ( )22∆(VI , φ) ≈ −(2.22)

we therefore obtain the relative sensitivity
1
φ
δVI
δ∆
= − tan ( )δφ +
∆
2
2
VI

(2.23)

Therefore we conclude
1. The intensity fluctuation of the IR lattice causes negligible fluctuation and drifting. The offset ∆ is proportional to the IR lattice depth VI . A 0.1% fluctuationsin the IR lattice depth causes 0.1% fluctuation in ∆, corresponding to a change
about 40lHz. With appropriate PID control, the lattice intensity drift is wellcontrolled to be within 0.1% as tested.
2. ∆ has unfavorable dependence on φ. Depending on the superlattice configuration, a relative change in the phase δφ/φ can be either magnified or reduced by
the factor tan φ/2 which could be large.
50Besides, the relative phase is much harder to control as discussed below. From thesereasons, we focused the discussion on the relative phase stability.
The relative phase φ depends on the optical path difference from the retro-mirror
to the atoms for the two lattice lights and can be simply expressed as
φ = φair + φglass(2.24)

φair ∝ (n532 − n1064 )L
where φair and φglass are the path difference introduced by the free propagation in
air and the glass plate respectively. φair is linear to the dif f erential refractive index
of the air and therefore is subjected to the change with ambient environment. φglass
depends on the thickness, temperature and angle of the glass plate and therefore more
related to the mechanical properties of the plate. We now estimate the contribution
from these factors.

2.4.2

Long-term stability - Galvo System Stability

In this section I present the test result for the stability of the Galvo mirror mounts.
In the early stage of the experiments, we designed to use the glass plate as atool for the dynamic control of the dispersive glass plate during the experimental
sequence. Therefore the mount of the glass plate needs to be electrically controllable.
We considered both the galvo mounts and the piezo-controlled rotational mounts.
However, for the reasons presented in 2.2.2, we eventually abandoned the idea ofdynamic control with the glass plate and simply mount the glass plate on a manual
mirror mount to avoid any possible drifts caused by the electronics in the a galvo
mount. The glass plate was used only as a tool to pre-set the superlattice to a certain
configuration.
In spite of this, I still present our test results briefly for future references. Detailed
discussions can be found in thesis [54, 55].
51

Motorized galvos vs. manual mirror mounts
We seek a galvo with the following three properties
1. Long term stability : the galvo can hold at a certain angle assuming the a stable
controlling voltage. It should be characterized with low temperature drift andgain drift.
2. Reproducibility : the galvo angle should be reproducible from run to run.
3. Short term stability : once set, the galvo should not perform any rapid vibrations.
Since we have already decided not to use the galvo dynamically during the sequence,
we did not seek for a galvo with fast ramping speed.
We started with galvo GVS011 from Thorlabs because of its availability and cheap
cost. This model performed well when standing along. However, we noticed that thegalvo showed very strong microwave pick up. In fact, during our BEC preparation
sequence where strong microwave around 1.7 Ghz is used for evaporation, the galvo
shaked rapidly and lost its initial position completely by a, angle of several degrees.
The main reason is the capacitive position sensing system implemented with thismodel. A galvo system usually include three different parts: 1) an actuator controllingthe mechanical motion of the galvo. 2) a position sensing part which detects the
position of the mirror. 3) a PID feedback loop which connects the previous two partsand stabilizes the mirror position. Two major designs are used as the position sensing
component. In capacitive position sensing, the rotation of the galvo changes therelative capacitance of several coupled capacitors and therefore changes the chargebalance which indicates the rotation angle. This is how our GVS001 galvo works
and therefore we can directly understand that any charge accumulation due to themicrowave will strongly disturb the sensor and therefore create a false signal. On theother hand, optical position sensing uses the reflection of a internal reference light todetermine the angular position of the galvo, which is very insensitive to the external
microwave disturbance. We therefore moved to the other models.
52

Several vendor we have looked into included Thorlabs, Nutfield and Cambridge
Technology. The latter two are more specialized in galvanometer productions. Especially, Nutfield offers models with 15 µrad/K and 10 µrad repeatability and are“greatly immune to radio frequency interference”. With all the considerations wechose GVS001 from Thorlabs to fix the microwave pick up problem. We observed
Superlattice phase drifts

0.5
0.4

Symmetric point / V

0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5

0

1

2

3

4

5

6

Time / hourthe phase period is 2.6±0.2 V

Figure 2-15: The measurement of the long-term drifts in the superlattice phase. The
symmetric point of the superlattice phase was monitored over six consecutive hours.
The long-term drifts in the superlattice phase are estimated to be within 10%, mainly
limited by our phase measurement error.
no visible rotation of the galvo during the sequence. However, two observations
motivated us to adopt a manual mirror mounts which did not have any electronic
components in it.
1. The superlattice phase φ still showed measurable drifts from one shot to anotherwhich gave an offset drift of kHz.
2. The galvo gave an audible sound around 10 kHz occasionally. This audible noise
suggested some high frequency noise in the mirror positioning.
We first checked this high frequency noise by reflecting a reference laser off from theglass plate mounted on the galvo and measuring the pointing noise with a quadrant
53photodiode. We did not observe significant glass plate vibration. However, we still
decided to move to a manual rotational mirror mount just for peace of mind.

2.4.3

Long-term stability - Ambient Environment Stability

We identified the major contributor to the superlattice drifts is the ambient environment, especially air pressure.
When electromagnetic radiation propagates through a medium, additional phaseshift and absorption (random scattering) from the medium is expected. These two
effects are captured by the real and the imaginary part of the refractive index n(λ) =n0 (λ)+iκ(λ) which usually depends on the wavelength of the radiation.The absorption
κ is usually very small compared with n0 when far away from absorption resonance andtherefore can be ignored. We then simply let n ≈ n0 . When the 1064 nm and 532 nm
light propagates through the air, chromatic effect of air causes extra differential phaseshift between the long and short lattice. This phase shift is determined by the ambientenvironment and is not well controlled.
The refractive index of the air depends on the constituents of the air, the pressure,and the temperature.

Figure 2-16: Superlattice Configuration with distance. Chromatic effect of hte air
resulting in a changing of superlattice configuration with propagating.

Constituents of the air
In a gas mixture, as in the case for atmosphere, the effective refractive index of the
mixture can be expressed from a weighted summation of the refractive indexes of its
54

components as
n2 − 1 X n2i − 1=ρin2 + 2n2i + 2i

(2.25)

where ni is the refractive index of each constituents and ρi is the partial density of
the constituents. The refractive indices of different major constituents of room air
and their corresponding partial volume is summarized below

FormulaPartial Volume (%)

n532 − 1 (285K,1atm)

n1064 − 1

N2

78.08

0.000284

0.000295

O2

20.94

0.000253

0.000248

Ar

0.934

0.000282

0.000282

CO2

0.04

0.000450

0.00044

Another important constituent of air is the water vapor which is commonly scaledwith relative humidity S defined as S = pH2 O /pH2 O(Sat) . The water vapor sometimes
takes extra attention due to the dense absorption lines in the IR regime where therefractive index could change rapidly.In our case, 1064µm and 532µm are far enoughfrom the absorption lines.

Air pressure P and Temperature T
The density of the air changes with ambient pressure P and temperature T which canbe simply modeled by the idea gas status function P M = ρkT . Therefore we expect
δ(n − 1)
δρ
δP
δT
=
=
+
n−1
ρ
P
T

(2.26)So a 1% fluctuation in the pressure and temperature, the relative change of thedifferential refractive index for the two lattice lights will be 1% which converted to∆(n1064 − n532 ) ≈ 10−7 . This relative drift will introduce a superlattice phase drift
and eventually change the lattice configuration at the position of the atomic cloud55

by.
∆φair /2π = k532 L × ∆(n1064 − n532 )≈ 10%(2.27)(2.28)

which is not negligible. Shown in Fig.2-17 is the monitor ambient pressure drift.

Daily Pressure Drift

Weekly Pressure Drift

Figure 2-17: Daily and Weekly air pressure drifts causing the drifting of the superlattice configuration.
The ambient air pressure strongly depends on the weather and random factors anddrifted constantly. Unfortunately, this drift is very hard to control and compensate,
we eventually took all our data during the plateau of the air pressure.
In future, a possible way to stabilize it is to cover the entire lattice path andpurge nitrogen into the enclosure. A release valve could be installed on the enclosure
to maintain a constant pressure inside.
These discussions conclude this chapter regarding both the theoretical and technical aspects of optical superlattices. In later chapter, we shall see how we use thesuperlattices we construct to realize spin-orbit coupling and to create new forms ofmatters with Bose-Einstein condensates.

56

Chapter 3
Spin-orbit Coupling
In this chapter, I present theory regarding spin-orbit coupling physics. The chapter
starts with a brief technical overview on the spin-orbit coupling physics. Then discussions are focused on one-dimensional spin-orbit coupling which is realized in the work
presented in this thesis. The chapter concludes with a discussion on the previous
experimental works and their limitations on realizing spin-orbit coupling in ultracoldgases, which motivates the work described in the next chapter.

3.1

Overview

In quantum mechanics, the evolution of a system is governed by the system’s Hamiltonian Ĥ through Schrodinger equation. The spin and the center-of-mass motion
are separated. Therefore, the internal and external degree of freedom are naturally
decoupled unless additional interaction terms are introduced.
On the contrary, Dirac equation, which combines the quantum mechanics of aspin-1/2 system and special relativity, provides a unification of the spin and thecenter-of-magic motion [56]. It was realized that the coupling between the spin andthe orbits is a pure relativistic effect. Indeed, in an atom, an electron orbiting in the
Coulomb field V (~r) with a velocity ~v sees an effective magnetic field in its own rest
frame due to the relativistic effect. This magnetic field interacts with the intrinsic
magnetic moment of the electron which is proportional to its spin and therefore
57

couples the spin and the electron motion. This coupling is encapsulated in the term
~ = C(~v × ~r ) · ~σ .
B
r3

(3.1)

The energy of the electron consequently splits into two branches corresponding to
the two spin states. This effect is one of the major contributors to the fine structurein the atomic spectrum. Depending on the atomic species, the splitting could beranging from 100MHz to THz typically.
In a solid, materials with structures breaking inversion symmetry could host intrinsic electric field. Electrons moving in this field experiences relativistic magnetic~ · ~σ isfield. The consequent momentum-dependent Zeeman energy Hsoc ∼ (p × E)
the spin-orbit interaction. Depending on the structure of the materials, two forms ofspin-orbit interaction have been observed
Hsoc = Cd (kx σy + ky σx ) Dresselhaus Type

(3.2)

Hsoc = Cr (kx σy − ky σx ) Reshba Type
Where Cd , Cr characterizes the interacting strength. In materials, Cr is on the order of
100 kHz m which is weak compared with other energy scales in materials. These twobasic types forms the basis for a description of an arbitrary form of the lowest-orderspin-orbit interaction in 2D materials.

3.2

One-dimensional Spin-orbit Coupling

A special case for the spin-orbit interaction is the linear combination of the two withequal weight, giving Hsoc ∝ σx py . The full Hamiltonian is equivalent toH=p2+ ασz pz
2m

(3.3)

Physically, the Hamiltonian describes a system where particles have a preference toeither moving to the left or right depending on their spins. For atoms in|↓i, the lowest
58

energy state corresponds to a constant momentum +α/2 while the |↓i atoms are theopposite. It is also shown as two displaced parabola displaced by α in the dispersion.
Without introducing any spin-flip processes, for example spin flipping scattering byimpurities in materials, the two tracks run independently, as shown in Fig.??. The
splitting of the spin sub-bands has been directly observed in certain materials [57].

Figure 3-1: Spin-orbit Coupling in one dimension. Without the interspin coupling,
the dispersion shows two displaced energy minima. A coupling opens the gap at thecrossing point. At a critical coupling, the double minimum merged into a single one.
When coupling between the two spin states is introduced, the two spin sub-bands
are coupled. The coupling results in a gap opened at the degenerate point. The full
Hamiltonian with the coupling reads
H=p2+ ασz pz + βσx + δ0 σz
2m

(3.4)

where β characterizing the strength of the Rabi coupling. Depending on the couplingstrength β, the dispersion of the system evolves from a double-minimum structure
to a single minimum. A direct diagonalization of the Hamiltonian eq.3.4 gives a
critical coupling strength βc = 2h̄2 α2 /m. When β < βc , energy are minimized atph̄k = α 1 − (β/βc )2 . When β > βc , the system minimize it’s energy at k = 0.
Notice that spin-orbit coupling strength refers to α. while β refers to Rabi couplingstrength.
The coupling is on resonance (set δ0 = 0) at p = 0 and is detuned by 2αp at
a non-zero momentum for transition |↓, pi → |↑, pi. The linear dependence of the59dutuning on momentum is indeed the signature of the Doppler effect.

3.2.1

Spin-orbit Coupling as a Doppler Effect

Doppler effect describes the change of the observed light frequency when there isrelative motion between the observing and the source. Doppler effects for mechanicalwave can be understood by counting the number of wave the observer received ina certain period of time. For lights, Doppler effect is a relativistic effect. Whenthe observer is the moving along the propagating direction of the light, the observed
frequency of the light ν is shifted by δν = ~v · k. Here ~v is the velocity of the observerand ~k is the wave vector for the light.
When using the light to flip the spin of a particle, the frequency of the light seenby the particle in its rest frame depends on the motion of the particle. Therefore thespin and the orbit are coupled.

Figure 3-2: Spin-orbit coupling with half Dresselhaus and half Rashba mixture in
atomic physics can be understood as a Doppler effect.

3.2.2

Spin-orbit Coupling as a Spin-flip with momentum transfer

In a more quantum point of view, Doppler effect can also be understood with momentum and energy conservation. The spin flipping process is accompanied by photon
absorption and emission. When a photon is absorbed (or emitted) by the particle,
60the mechanical momentum of the photon p~ = h̄~k0 is delivered, causing a momentum
recoil of the particle to fulfill momentum conservation. The recoiling particle obtains
an extra kinetic energy
1∆E = h̄~k · ~v +2mh̄|~k|m!2(3.5)

where ~v is the initial velocity of the atom. Therefore to resonantly drive a transition,
the energy of the photon needs to be ∆E +h̄ω0 to compensate the extra kinetic energy
due to recoil. The extra energy depends on the atom’s initial velocity linearly, whichis the essence of the Doppler effect.
It is therefore clear that the momentum of transfer ~k form the photon during
the spin-flipping process is essential for one-dimensional spin-orbit coupling. Indeed,
under unitary transformation Û = e2iαx σz , the eq.?? turns into2

Hsoc =

3.2.3

i2αx



0
e
p

⊗1+β
2m
e−i2αx δ0(3.6)

Realizing spin-orbit coupling in Cold atoms

The connection between one-dimensional spin-orbit interaction and the atom-lightinteraction makes cold atoms a natural platform for simulating such effects.
Pseudospin
In ultracold atoms, the concept of “spin” has been generalized to “pseudospin”. A
pseudospin is a label of the internal states. It can be any two quantum states which
can be coherently manipulated.
The chosen two quantum states can be arbitrary. However, practically they should
satisfy the following two criteria:
1. They should have long enough bare lifetime compared with the experimental
time scale. In atomic physics, this usually means small enough spontaneous
decay rate and spin-relaxation rate ( change of the spin state due to collisions)2. the two states can be detuned far from the other states such that dynamics
61

between the pseudospins are unitary.
In ultracold gases experiments, the typical choices are the Zeeman levels of atoms.
For example, the |F = 1, mF = 1, 0, −1i or |F = 2, mF = 2, 1, 0, −1, −2i states of the
3S1/2 manifold. Zeeman level are usually very long lived. energy separation between
Zeeman levels are at a radio-frequency range. Therefore the spontaneous emission
rate which has the scaling Γ ∝ ω 3 to the frequency of the emission transition ω, isroughly on the order of ∼ (100 MHz/100 THz)3 × 10MHz = 10−11 Hz. In alkaliearth atoms or some lanthanide atoms, there exists some states which are opticallyseparated from each other but at the same time show long enough lifetime. They surealso can be chosen as the pseudospin states. Quadratic Zeeman effects makes it easy
to isolate two Zeeman levels from others. For |F = 1, mF = 1, 0, −1i in 23 N a, only
∼ 10 Gauss is enough to make a separation of 13 kHz between 1 → 0 and 0 → −1
transitions.
Raman Scheme
Subsequently, one-dimensional spin-orbit interaction was successfully demonstrated
by using atom-light interaction in Bose-Einstein condensates [31] and cold fermions [?].
The actual experimental setup was a little bit more complicated since a two-photon
process is used for the spin-flipping instead of a single photon transition. One of thereasons is that the chosen pseudospin states cannot be directly coupled by electricdipole transition due to the selection rule. The other reason is that the transitionis at RF frequencies therefore the k0 for the spin-orbit coupling will be too small toobserve.

3.3

Spontaneous Heating

However, a major draw back from the two-photon scheme is the spontaneous heating
issue. The issue arises form the the fact that a two-photon process will always needan intermediate level as a bridge. Though only virtually populated, this intermediate level gives undesired destructive interference if more of these intermediate levels
62Figure 3-3: Spontaneous heating caused by near resonant light.

present. Let me elaborate with the energy level of a real atom, for example 23 Na, as
shown in Fig.3-3.
In 23 Na we define |↓i = |1, −1i and |↑i = |1, 0i. In a two-photon process, thecoupling matrix element from the starting state to the final state via intermediate
states ish↓| Ĥ 0 |↑i =X h↓| Ĥ1 |M i hM | Ĥ2 |↑i
∆M

M

(3.7)

where ∆M is the detuning from the certain intermediate state |M i. When running thesummation, we roughly ignore the hyperfine splitting within the same fine structure
manifold assuming that ∆HF  ∆F S , then we haveh↓| Ĥ 0 |↑i =X h↓| Ĥ1 |M i hM | Ĥ2 |↑i
M≈

X
M ∈2P1/2∆MXh↓| Ĥ1 |M i hM | Ĥ2 |↑i
+
∆
M ∈2P

h↓| Ĥ1 |M i hM | Ĥ2 |↑i∆ + ∆F S

(3.8)

3/2

therefore to the lowest order, we see thath↓| Ĥ 0 |↑i =∆F S 2Ω∆2(3.9)

The intermediate states |M i have spontaneous decay rate of Γ. When virtually63

Table 3.1: The ratio of the fine structure splitting and the spontaneous emission rate
for different atomic species.
Species132
Cs∆F S /Γ
3 × 106

87

106
5 × 104
103

Rb
23
Na
6
Li

populated, the effective spontaneous decay rate for the ground states will beΓg ∝ (Ω 2)Γ
∆(3.10)In the regular dipole trapping or optical lattice where the trapping potential comesfrom the AC stark effect and is therefore a single photon process, the trapping potential scale with V ∝ Ω2 /∆ while the spontaneous emission scales with Γg ∝ Ω2 /∆2 .
Therefore, by increase the detuning and also the beam intensity at the same time, it
is possible to suppress the heating due to spontaneous emission while maintaining acertain trap depth. However, in this spin-orbit coupling case, the spin-orbit couplingstrength and the spontaneous heating have the same scaling with detuning ∆. In fact
we have
ΩSOC
∆F S
∝
Γg
Γ

(3.11)

This suggests that we can never win by detuning the Raman lasers. From this argument, we would prefer atoms with as large ∆F S /Γ as possible for the scheme, assummarized in Tab:3.1. This explains the short lifetime in the Lithium implementation [58, 59].
The hard limit on the fine structure splitting timescale can be understood pictorially. When using two Zeeman levels as the pseudospin states, a pseudospin flipping
process require the flipping of a real outermost shell electron. However, since electricdipole transition cannot couple to the electron spin, the only way to achieve the~ of the electron
actual electron spin flipping is by adding orbital angular momentum L~ ·S~ do thevia a s − p transition and let the spin-orbit coupling within the atoms L
64

job. Then it makes sense that the “speed” of this flipping is strongly limited by the~ ·S~ coupling which is nothing but the fine structure splitting ∆F S .
strength of the L
Acknowledging this, we also understand that there are no other tricks one could do
to improve the scaling unless one picks different pseudospin states or atoms.
To move forward, there are two approaches can be taken to tackle the spontaneousemission problem without switching to another atomic species. The first is to encode
the pseudospin into another two states which can be dressed without using nearresonant lights. As will be shown in chapter 4, we use the left and the right well in
double well potentials as the pseudospin and drive the pseudospin-flipping with far
off-resonant lights. Another approach is to use the Radio-Frequency photons to couple
the Zeeman states. However, Radio-frequency photon has 10−6 smaller momentum
than an optical photon. Therefore, extra procedure is needed to engineer a large recoil
momentum for RF photons. We show how to achieve this with Floquet engineering
which will be described in Chapter 5.

65

66

Chapter 4
Spin-orbit coupling in opticalsuperlattices
In this chapter, I describe the first experimental results of the thesis - the realization of
one-dimensional spin-orbit coupling in an optical superlattice [60]. I first formulate
our scheme. Then discussions on the experimental result starts with the characterization of the new pseudospin states we implemented. I will discuss about how
atoms with different pseudospins are prepared and distinguished, and the lifetime.
The major results on the observation of the spin-orbit coupling is then presented.
The signature of one-dimensional spin-orbit coupling is a spin-flip with momentum
transfer. The transferred momentum reversed when the direction of the flip changed,
preserving time-reversal symmetry. The chapter ends with a general discussion on
realizing two-dimensional spin-orbit coupling in our system.

4.1

Spin-orbit Coupling with Orbital Pseudospins

Considering an asymmetric double-well potential along z-direction which hosts two
wells, as shown in Fig.4-1. The transverse direction is free space. The two wells
are offset by an energy difference h̄∆ and are connected with interwell tunneling J.
Both the parameters are tunable. Atoms loaded into the double-well potential cansit either in the left or the right well, encoded as the pseudospin down state |↓i (left67

well, |1i state) and up state |↑i (right well, |2i state)1 .

|↓i = |Li +

J
∆




|Ri , |↑i = |Ri −

J
∆


|Li

(4.1)

The two pseudospin states are connected by a two-photon process stimulated by( k1, ω1 )( k2, ω2 )
J

y

x

z

Δ

Figure 4-1: Spin-orbit coupling with orbital pseudospin. The left and the right wellof an double-well potential is used as pseudospin. The spin flip is provided by a two
photon process, providing the required momentum transfer.

two Raman beams. The pseudospin is flipped when an atom is transferred fromone well to the other. When an atom is transferred from one well to the other, it
absorbs a photon from one beam and emits a photon into another, receiving a photon
recoil during the process when the two beams are appropriately arranged. A spin flip
associated with a momentum transfer is spin-orbit coupling (See Chap.3.
Unlike the Zeeman pseudospins, spin flipping porcess with orbital pseudospins
does not involve real electron spin flip. Therefore far-detuned light can be used. This
process is similar to the laser-assisted tunneling where the suppressed tunneling of
atom between two ajacent lattice sites in an tilted optical lattice is restored by thetwo-photon processes, which has been done in [19].
1

For necessary background knowledge for superlattices, please refer to Chap.2

68

4.1.1

Perturbative Wavefuntions and Hamiltonian

We write the two-photon Raman field as Ωeikz z+iky y e−iδt . The eiky y provides a momentum transfer along the transverse y direction. This is the momentum associatedwith the spin-orbit interaction. eikz z couples the atoms from the two wells, exerting
the spin flip.
Written on the base of |↓, ki , |↑, ki, the two-photon Raman field reads
HRaman =M e−iδt |↓, k + ky i h↓, k| + iM e−iδt |↑, k + ky i h↑, k|(4.2)+ βe−iδt |↑, k + ky i h↓, k| + h.c.
Here we introduce the second index k to denote the momentum along the transverse
direction y. β = h↓| eikz z |↑i ∼ ΩJ/∆ describes the strength of spin flip whereas
M = h↓| eikz z |↓i ≈ h↑| eikz z |↑i ∼ Ω describes the on − site coupling.
If we start with a mixture state |↓, 0i + e−i∆t |↑, 0i and adiabatically ramp up theRaman beams, with the first order perturbation theory, the state is connected to|F i = |↓, 0i + e−i∆t |↑, 0iββ∗e−iδt |↑, ky i −eiδt |↑, ky i
δ − (∆ + Er )
δ + (∆ + Er )β∗β∗+ei(δ−∆)t |↓, ky i +e−i(δ+∆)t |↓, ky i−δ + (∆ − Er )
δ + (∆ − Er )M ∗ iδtM+
e−iδt |↓, ky i −e |↓, −ky iδ − Erδ + EriM −i(δ+∆)tiM ∗ i(δ−∆)t+e|↑, ky i +e|↑, −ky i
δ − Erδ + Er
+(4.3)

Here Er = h̄2 ky2 /2m is the recoil energy due to the recoil momentum. Terms inthe second lines describe the spin flipping processes. They include both the rotating
and counter rotating terms. Terms on the last lines capture the on − site coupling.
The spin flipping resonance happens at δ = ∆+Er (δ = ∆−Er ) for |↓i → |↑i(|↑i →|↓i) process. At these frequencies, the final state |F i contains two resonant terms|↑i + e−iEr t |↑, ky i and |↓i + eiEr t |↓, −ky i describing propagating density modulation
moving with the Raman beams.
69

The Spin-orbit resonance is reached when δ = ∆. At the frequency, the propagating Raman beams creates stationary density modulations.
The experiments were done in the regime where δ ≈ ∆  Er . The counterrotating terms are far detuned and are neglected in future discussions.
With δ tuned close to ∆, only the spin-orbit coupling terms are near resonant, wetherefore reach the Hamiltonian
H0 = 

h̄2 k22m∗ −iky y+iδtβ eβeiky y−iδth̄2 k2
+ h̄∆
2m




(4.4)

which exactly describe one-dimensional spin-orbit coupling with δ0 = δ − ∆ andα = ky .

4.1.2

On-site and the Spin-orbit Coupling Term

The on − site coupling terms have resonances at δ = ±Er . when tuned on resonance,on-site coupling describes a propagating wave directly imprinted on the initial spinstate by the Raman beams.
With our experimental condition where ∆  Er , the on − site coupling terms areusually far-detuned. However, the on − site coupling strength M is much larger than
the strength of the spin-orbit coupling β due to the stronger wavefunction overlap, theamplitude of the on − site coupled state is not necessarily weaker than the spin-orbit
coupling.
More specifically, when tuned on the spin-orbit resonance, the on-site amplitudeis Ω/δ while the spin-orbit coupling amplitude is ΩJ/(δEr ). In our experiments wechose Raman detuning δ and offset ∆ roughly equal to 50 kHz. With J ∼ 3 kHz,Ω ∼ 15 kHz, and Er ≈ 7.6 kHz, the amplitude of the on-site term Ω/δ ∼ 0.3 while thespin-orbit coupling amplitude is 0.15. Further increasing J will cause the atoms in
the right well to decay. Therefore, the on-site coupling cannot be totally diminished.
The on-site coupling needs to be avoided because the atoms in the state have theexactly same momentum components as the outcoupled atoms from the spin-orbit70

coupling. Therefore they cannot be distinguished from the time-of-flight imaging. I
will discuss about the strategy we used to filter the spin-orbit coupling signal out
from this background.
For even more detailed calculations, please refer to the thesis [55].

4.1.3

Extended to Optical Superlattices

The above discussions on the spin-orbit coupling can be readily extended to therealistic optical superlattice.
A superlattice host the band structure. As has been discussed in the chap.2, the|↓i and |↑i states in an optical superlattice are the band minima of the first and thesecond band, which have quasi-momentum q = 0 and q = π/d respectively. The
perturbation wavefuntions with both the on-site and the spin-orbit coupling terms is
written as|Ψ1 i = |↓, 0, 0i + K1 e−iδt |↑, π/d, kx i
+ M1 e−iδt |↓, π/d, kx i + M10 eiδt |↓, −π/d, −kx i (4.5)
For the system initially prepared in the state |↑, π/d, 0i, the new state will be:|Ψ2 i = e−i∆t |↑, π/d, 0i + K2 ei(δ−∆)t |↓, 0, −kx i+ M2 ei(δ−∆)t |↑, 0, −kx i + M20 e−i(δ+∆)t |↑, 0, kx i (4.6)
with the amplitudes summarized in Tab.4.1 . The frequency dependence is very
similar to the double-well case. (Detailed calculation can be found in Appendix.A)
With proper unitary transformation, the Hamiltonian achieved is indeed describing one-dimensional spin-orbit coupling.
(p̂z + ασz )2
HSOC =+ βσx + δ0 σz ,
2m
0

(4.7)

where α = − 21 ky , β = √JΩand δ0 = 12 (δ − ∆). At this point, we have theoretically2∆
71

Table 4.1: The amplitudes of the wavefunctions in eqs. 4.6 and eq.4.5 obtained from
1st order perturbation theory. (i = 1,2)
|Ψ1 i state

Amplitude

Population

−i π

(↓)

(↑)

+ky

− 12 ErΩ−δ |ψq=π/d i − i e√24 EJΩ/∆|ψq=π/d ir +∆−δ

−ky− 21 ErΩ+δ |ψq=π/d i1| Ω |24 Er +δAmplitude

Population(↓)|Ψ2 i state

(↑)

+ky

− 12 ErΩ−δ |ψq=0 i

−ky

i
(↑)
(↓)
1 Ω
|ψq=0 i + i e√42 EΩJ/∆|ψq=0 i2 Er +δr −∆+δ

π1J∆| Ω |2 + 21 | ∆|2
4 Er −δEr +∆−δ1| Ω |24 Er −δ1J∆| Ω |2 + 21 | ∆|2
4 Er +δEr −∆+δ

prove that by using an orbital degree as the pseudospin combined with two-photon
process, it is possible to realize spin-orbit coupling in optical superlattices2 . I now
present the experimental results.

4.2

Experimental Realization

To experimentally demonstrate the feasibility of the newly chosen pseudospin states,
it is important to solve the following four issues:
1. Populating atoms in different Zeeman levels is straightforward by using RadioFrequency or microwave transitions. How do we selectively load the atoms ineither potential wells? We need to be able to prepare an arbitrary mixture of
the two pseudospin states.
2. We need to have a strategy to selectively detect atoms in different pseudospin
states. Since all the atoms are in the same internal states, regular Stern-Gerlach
methods with magnetic gradient would not work. It is also impossible to do
Zeeman selective imaging. How do we tell if the atoms were sitting in either
2

For detailed calculations, please refer to Appendix A.

72

the left or the right well considering that the spatial separation is only 266 nm
which is far beyond our imaging resolution.
3. The new pseudospins may not be collisionally stable. As we mentioned before
and in many literature [61], bosons suffer from short lifetime when populating
the excited bands due to collisions. The lifetime could be improved by implementing strong confinement in the transverse direction. When the energy gap
in the transverse direction ωt is much larger than the band gap in the longitudinal direction, decay could be suppressed. However, in a quasi-1D system as
in our case where the transverse direction is nearly free, the lifetime could beshort. Our |↑i state is exactly on the excited band of the superlattice, we needto know how long it will live.
4. A lattice will host usually more than two bands. In the harmonic oscillator the
spacing between two adjacent bands is exactly the same, so we actually are notable to isolate our pseudospin states from other states. In an optical lattice,
the band spacing is slightly different because of the anharmonicity of the lattice
site, but not by a lot. We need to find the correct parameter regime such thatthe first two bands are well isolated from higher band, fulfilling a close spin-1/2
system.
This section is mainly focused on these four questions.

4.2.1

Pseudospin States Preparation

We first address the question on the pseudospin preparation. |↓i state is the groundstate in an optical superlattice. Atoms can be prepared in the |↓i state by adiabatically loading a Bose-Einstein condensate into the superlattice preset to the required
configuration. This was achieved by a slow ramping up of the superlattice power
within a time scale much longer than both the band gaps of the lattice and trapping
frequency of the combined trap of the optical dipole trap and the superlattice, avoiding both the uncontrolled excitation to higher bands and distortion of the atomic73

cloud. When ∆  µ the chemical potential, it is guaranteed that all the atoms will
be loaded into the well with lower energy, creating a 100:0 mixture of |↓i and |↑istate.
To prepare atoms in the |uparrowi state, there are two options
1. Adiabatic way : Prepare the atoms in the |↓i state and then flip the spin witheither an extra pair of Raman beams or by modulating the lattice.
2. non-adiabatic way :Prepare atoms in the |↓i states followed by a non-adiabaticchange of the sign of the offset ∆. After a relaxation, atoms settle at |↑i state.
We took the second way for simplicity and efficiency. We first prepared all theatoms in the |↓i state, and immediately flip the the sign of the superlattice detuning ∆by non-adiabatically change the phase φ of the superlattice. As long as the quenchingis much faster than the inter-well tunneling time scale J, atoms were then lifted up
and ended up in the |↑i state. The process can be understood in multiple ways. One
way is that the atoms were simply lifted up before they could tunnel to the lower state.
Another way to understand is that we project the atoms from one spin state to theother, similar to the scenario where the spin is reversed if the magnetic field flips thesign much faster than the Lamour frequency. The most formal way of understanding
is a fast Landau-Zener sweep where the sweeping is much faster than the minimum
gap between the two state which is the inter-well tunneling J. The transfer efficiency
was as high as 80%.
To gain a rough idea on fast we need to change the phase, we measured the interwell tunneling J. We first prepare the atoms in the |↓i when some non-zero detuning∆ then immediately changed the detuning ∆ to 0. When then hold the sample forvarious time during thold before a time-of-flight measurement. When th = 0, we saw
normal d = 532 nm diffraction patter with Brillouin Zone size kBZ = π/d. After acertain holding time th ≈ 600 µs, the Brillouin Zone doubled with kBZ = 2π/d. Thissuggest that the atoms have already tunneled to both wells and the superlattice wasmore like a d = 266 nm lattice. This timescale suggested a interwell tunneling ofJ ∼ 2π × 2 kHz for the green lattice depth we used. We therefore know that we need74

to change the superlattice phase within 10 µs.

4.2.2

Detection of Pseudospin states

In many previous studies of double-well superlattice systems, selective imaging ofatoms in the two wells were done by band mapping technique [61, 62]. Supposethat the two wells in a double-well system are already populated. When the barrier
between the two wells is adiabatically ramped down,the superlattice evolves into aregular lattice. Atoms initially in the left well evolve to the ground band whereas
the atoms in the right well to the first excited band. Then the lattice is adiabaticallyremoved will map the atoms in the ground band to the first Brillouin zone and firstexcited band atoms to the 2nd Brillouin zone. By this procedure we count the numberof atoms initially in different wells by counting the atom number in the first and second
Brillouin zone respectively.
Collisional loss or decay should be avoided for robust band mapping. That is
also the main reason why so far robust band mapping has only been demonstrated inthree-dimensional optical lattices. For the quasi-1D lattice as in our case, atoms inhigher bands decay during the adiabatic mapping process. We therefore decided not
to go along this direction.

Pseudospin Stern-Gerlach Separation
We developed an equivalent Stern-Gerlach effect for our pseudospin system. In SternGerlach experiments, atoms are separated by a state-dependent potential. In themagnetic case, atoms possessing different magnetic moments as separated with a DC
magnetic field gradient. In our situation, the two spin states experience different
transverse confinement from the 1064 nm lattice. When the lattice is slightly misaligned, the atoms in the two spin states experience different momentum kicks inthe transverse direction when the IR lattice is suddenly increased. This leads tooscillations of the two spin states relative to each other with different frequencies
which can be observed in ballistic expansion(Fig.4-3). The frequency of the oscilla75

Figure 4-2: Pseudospin Stern-Gerlach Separation. Atoms are loaded into the green
lattice , then the IR lattice is suddenly ramped up. After different hold times, thevertical displacements of atoms in different spin states is measured with 10ms ballistic
expansion. y = 0 corresponds to the position of atoms without the sudden ramp.
The spin states are displaced due to of transverse oscillations in the superlattice. The
oscillating frequencies are equal to the transverse trapping frequencies which the spins
experience in the superlattice.

tion is equal to the corresponding transverse trapping frequency of each spin state.
Mathematically, a full optical lattice potential including the transverse confinement22

is V (x, r) = −V0 exp(− σr 2 ) exp(− l2x ) sin (2k1064 x) giving a transverse trapping freRay

quencyrωr =2V (x)
σ2m(4.8)

which brought us torω↓ − ω↑ =

1
∆
ω↓
2
V↓

(4.9)

Assuming ∆ ∼ 40 kHz, V↓ ∼ 10 µK, the difference is around 10%. as calculated anddemonstrated in Fig.4-3. The number of atoms in each state is therefore counted
directly from the time-of-flight images. However, this method depends on the alignment of the superlattice which suffers from daily drift. It is obviously not a robustway.
76

Figure 4-3: Transverse oscillation.In the data shown, atoms in |↑i and |↓i oscillateswith frequencies of 356 Hz and 417 Hz correspondingly.

Figure 4-4: The figure shows atom number counting in a 40:60 sample.

Direct Time-of-Flight
Eventually, the atoms in different spin states are distinguished by simple imaging
after time-of-flight. Time-of-flight imaging resolves only the momentum components.
all the spatial information are lost. For a single asymmetric double-well potential,
momentum components possessed by atoms in different states areZJ|hk |↓i | = | hk |xi hx |↓i dx|2 = |(1 + eikd |2∆ZJ|hk |↑i |2 = hk |xi hx |↑i dx = |(eikd − )|2∆
2

77(4.10)

Therefore the two spins are indistinguishable for small J/∆. However, after considering that we have a chain of double-wells in the superlattice and there is coupling
between the two nearest double-wells, things are different. Before we fully understood the band character of the system, we had the picture that all the left (right)
wells formed their own lattice, and therefore the spin down and up states are in their
quasimomentum q = 0 states in their respective lattice. Therefore they will havethe same inter-site interference pattern. The only difference between the interference
patterns comes from the different single site envelope as in Eq.4.10. However, if theinter-well tunneling J is larger than the inter-site tunneling, physically atoms in theright well will have enough time to realize that they were in the higher bands, therefore the minimum energy state in the first excited band should have quasimomentum
q = π/d. The inter-site interference pattern will have a phase shift π and therefore
be totally orthogonal to the one of the other spin state. It is then possible to use
time-of-flight to distinguish between the two pseudospin states.

Figure 4-5: Interference pattern for atoms in two pseudospins are orthogonal to eachother.

4.2.3

Lifetime of the Pseudospins

We subsequently examined the lifetimes of the atoms in each pseudospin states after
developing a robust preparation and detection procedure.
78

We measured : 1) atom number lifetime Tt describing the atom loss. 2) Superfluid
lifetime Ts describing the phase coherence across the sample. Tt was obtained bymeasuring the atom number after a certain amount of holding time in the superlattice.
The coherence is inferred by the sharpness of the interference peaks after tof. For any
mixture of |↓i and ↑, we didn’t observe significant loss of coherence before the atomloss. Therefore we did not do further measurement on Ts .
When all atoms are prepared in the |↓i state, both Tt and Ts are comparable tothe lifetimes in a simple d = 532 nm lattice which are several seconds. This is not
surprising since it was the ground state. As long as the blue-detuned green lattice
was not seriously misaligned, we should not expect serious atom loss or heating.
When atoms were prepared in the |↑i state which is metastable, we expected thelifetime to be shorter. we observed lifetime of ∼ 500 ms. We suspect that collisional
decay would be the main cause. When two atoms in the |↑i state collide, they decay
to the |↓i state with extra energy transferred into the transverse direction, causing
excitations.
When a spin mixture is prepared, similar mechanism was expected except for thatthe collision could happen between two pseudospin states, which further reduces the
lifetime. We observed ∼ 50 ms.
Quantitatively, the decay rate is calculated with Fermi’s gold ruleΓ=

2π
| hf | V̂ |ii |2 ρ(Ei − Ef )h̄

(4.11)

depending on the coupling matrix element hf | V̂ |ii and the final density of states
ρ(Ei − Ef ).

With s-wave pseudo potential V (r) = g0 δ(r), the matrix element

corresponds to the process |s1 , s2 i → |s01 s02 i can be written as hs01 , s02 | V̂ |s1 , s2 i =g(s01 s02 s1 s2 ) whereg(s01 s02 s1 s2 ) = Cg0

Z

φ∗s01 (x)φ∗s02 (x)φs1 (x)φs2 (x) dx(4.12)

g0 = 4πh̄2 a/m is the contact interaction strength. Tab.4.2 summarizes the matrix
elements: The first three terms guuuu , gdddd , guddu are conservative and therefore re79

Table 4.2: Collisional Matrix elements for different processes.
Initial States s1 , s2
|↑↑i
|↓↓i
|↓↑i
|↑↑i
|↑↑i
|↓↑i

Final States s01 , s02
|↑↑i
|↓↓i
|↑↓i
|↓↓i
|↑↓i
|↓↓i

g(s01 s02 s1 s2 )
n2↑
n2↓
J 2
( ∆ ) n↓ n↑
J 2(∆) n↓ n↑
1/2 3/2J( ∆ )n↓ n↑
3/2 1/2J(∆)n↓ n↑

Ef − Ei
0
0
0
−2∆
−∆
−∆

sponsible for effective energy shifts. In the language of spin dependent interactions,these three terms correspond to the inter-spin and intra-spin interaction. It is worth
noticing here that the inter-spin interaction is supposed to be smaller than the intraspin interaction by a factor of ∼ (J/∆)2 ∼ 1%. This is a complementary scenario
compared with 87 Rb case where interactions have almost SU (3) symmetry. Thisfeature is crucial to our further experiment and will be discussed later.
The last three terms transfer energies from the z direction to the transverse direction r, resulting in pseudospin relaxation.
The general form of the density of state ρ depends on the energy, dimension
and the dispersion relationship. As we mentioned before, the pseudospin relaxation
generates excitation in the transverse x − y direction. The energy released ∼ ∆ ismuch smaller than the trapping frequency along the lattice z direction in a single well,
it is therefore justified to ignore the excitation to the higher bands and regard thetransverse direction as a two-dimensional plane. Another assumption is that since ∆is much larger than the on-site interaction, we assume the excitation to be particle like
instead of phonons. Therefore we have E2d (k) = (h̄2 /2m)(kx2 + ky2 ), giving a energym
independent density of state ρ(∆) = πh̄
2 . We therefore conclude that the decay rate

as summarized below Our observed lifetimes for the equal mixture is roughly 10 times
shorter than the |↑i sample with a J/∆ ≈ 20, suggesting that the collision model isprobably capturing the major physics.
It is clear that he relaxation rate can be varied by changing J/∆. In terms ofsystem stability, the smaller J/∆ the better. However, it will also reduce the spin80

Table 4.3: Inelastic collision lifetime.
Process
Matrix element
ΓJ 2|↑↑i → |↓↓i( ∆ ) n↓ n↑−2∆
1/2 3/2J|↑↑i → |↑↓i( ∆ )n↓ n↑−∆
3/2 1/2J|↓↑i → |↓↓i( ∆ )n↓ n↑−∆orbit coupling strength. In principle, since they have different scaling over J/∆, thespin-orbit coupling can always be compensated by stronger Raman beams. But practically, too strong Raman beams introduces unwanted heating, higher order processes
etc. Therefore it is still meaningful to find the most appropriate window for J/∆.
After checking the lifetime of the pseudospin states, we now could look a little
deeper into the 50:50 mixture of the pseudospin states. It turned out that the system
is already interesting enough even without any spin-orbit coupling.
…

…
-4

-2

0

2

4

n-1

n

n+1

Figure 4-6: Illustration of the pseudospin texture.

For a spin-1/2 system, a state |ψi = |↓i + eiθ |↑i means a spin aligned along thecos θx̂ + sin θŷ direction. In our experiment atoms can be prepared in left and rightsites of the double wells with equal population. The wavefunction is proportional to
|ψi =

X

|↓n i + (−1)n eiθ e−i∆t |↑n i(4.13)nwhere θ is the phase at t = 0. The state ψ corresponds to spin-states aligned in thespin x − y plane with opposite direction on neighboring sites, showing x − y antiferromagnetic ordering due to this quasimomentum difference as shown in Fig.4-6.
81

The expectation values of spin evolve as
hσx i = (−1)n cos(θ − ∆t)hσy i = (−1)n sin(θ − ∆t)(4.14)

hσz i = 0
which are constantly rotating at frequency ∆ with a locked neighbour phase.
The spin texture created a density modulation due to the tunneling between thetwo wells within a unit cell. Local populations in the nth cell acquire a density
imbalance:Jcos(θ − ∆t)∆2nJ| hrn |ψi | ∼ 1 + (−1)
cos(θ − ∆t)∆| hln |ψi |2 ∼ 1 − (−1)n(4.15)

Translating by one superlattice period flips the sign of the imbalance. This shows thatrn and ln+1 wells, and rn−1 and ln wells have the same change in density, but oscillate
out of phase. Therefore, this density wave has twice the wavelength of the superlattice
and is shifted spatially by half a unit cell. This doubling of lattice periodicity canbe detected through the Bragg scattering experiments, where the spatial periodicity
is revealed by the direction of the enhanced light scattering from the sample. In thenext chapter, I will describe the corresponding experiments in Chapter.7.
At this point, we had done the following:
1. We have proposed and tested a new set of pseudospin states in optical superlattices. |↓i and |↑i are the q = 0 and q = π/d state in the ground and the firstexcited band in the superlattice.
2. We were able to prepare and detect the pseudospin states.
3. The lifetime of the newly implemented spins are long enough for further experiments.
We are ready to move forward to implement spin-orbit coupling.
82

4.3

Observation of the Spin-orbit Coupling in Optical Superlattice

In this section, I present the results demonstrating spin-orbit coupling in opticalsuperlattices. I described the setup for the Raman beams, experimental parameters
we used, and the signals we observed with detailed analysis. The section is concluded
with a discussion on the heating and higher-order effect related to the Question 4 at
the beginning of this chapter.

4.3.1

Raman Setup

Two-photon process imprints a momentum of h̄∆~k = h̄~k1 − h̄~k2 and an energy ofh̄∆ω = h̄ω1 − h̄ω2 to the system. The two questions we need to answer are whatgeometry we should use to give the needed momentum transfer and what wavelengthof laser we should be using. The quasi-momentum π/d difference between the two
spin states requires that(∆~k)z = kIR .

(4.16)√ J
With this hard constraint, the coupling matrix element is h↓| ei∆kx x |↑i ∝ eiπ/4 2 ∆.
Though maximum coupling is obtained when (∆~k)z ≈ kGreen , it is only smaller by a√factor of 2 with (∆~k)z = kIR ,.
A non-zero transverse momentum transfer (~k1 − ~k2 )y 6= 0 is essential. Based on
the laser resource in our lab, we have two options:
1. Using two 532 nm beams as shown in Fig.4-7.
2. Using two 1064 nm beams as shown in Fig.4-7.
We decided to go with 1064 nm configuration with the geometry shown in 4-8. The
532 nm is easier to set up since we just need to retro reflect the beam. However,since we would need to detune the two beams, we would need to have independent
control of the two beams. Other considerations is that Green light has always been
harder to manipulate. First of all, the green light power is valuable. Secondly, for the83Superlatticerz

Figure 4-7: Possible schemes for the Raman set up. a) by retro-reflecting a 532nm
green beam. b) by arranging two 1064nm beams.
same intensity of Green and IR light, the green light always has more severe thermal
effects. It is important to keep the relative phase noise of the Raman beams as low

Dispersive
Glass Plate
Retro-mirror
Irisk2
Dichroic
Mirrork1

Figure 4-8: Optical layout of the Raman beams.

as possible. The interference pattern for the two Raman beams, with aligned linear
polarization and same intensity, reads
1
E(y, z) = 2E0 ẑ sin { [kIR (y + z) − (ω1 + ω2 )t + φ1 + φ2 ]}2
1× sin { [kIR (y − z) − (ω1 − ω2 )t + φ1 − φ2 ]} (4.17)
284

giving an intensity of¯ z) = 2E02 sin [kIR (y − z) − (ω1 − ω2 )t + φ1 − φ2 ] + const.
I(y,(4.18)

where φ1 and φ2 are the phase of Raman beams at the atoms. The relative phasenoise causes jitter of the Raman lattice, creating unnecessary heating.
The phase noise has several major sources : the phase noise from the laser, phasenoise from the optical fiber and mechanical vibrations of the optics which changes the
optical path length. We implemented the following to minimize the relative phasenoise:
Fiber phase noise Optical fiber introduces phase noise to the laser light. We useda single fiber to deliver the IR light from our source laser to the experiment table
and then split it into two for the two beams. This procedure made sure that thephase noise induced by the optical fiber is a common mode for φ1 and φ2 and will becanceled at the fiber output (notice that it does not necessarily mean that they willcancel at the atoms unless we make sure that the path length difference for the two
Raman beams is small enough).
Mechanical Noise To minimize the phase noise due to mechanical vibrations, we
did the following besides tightening the optics hard:
1. All the optics for both the Raman beams were mounted on a single piece ofthick breadboard.
2. All the core optics were setup as spatially close to each other as possible tominimize the relative vibrations. More quantitatively, a breadboard, with given
dimensions, mechanical parameters such as stiffness, and mounting, gives acertain mechanical vibration pattern which can in principle be simulated. Optics
should be sitting at the vibration nodes and the spatial scale of the whole opticalsetup should be much smaller than wavelength of the mechanical vibration of
the board.
85

3. Possible vibration sources are either decoupled or removed from the experimental table. For example, the fan of the amplifier close to the table. It isalso worth pointing out that air turbulence could also create quite noticeable
vibrations since the mirrors have large area to pick it up.
4. The experimental table was floated.
After split at the output of the fiber, each Raman beams was sent through an
Acousto-opto Modulator (AOM) for frequency tunability. The frequency difference∆ω can be tuned between 0 to 100 kHz. As mentioned before, the Raman beams
form a extra lattice in the −ẑ + ŷ direction with a wave vector ~k = kIR (−ẑ + ŷ). We
aligned the two Raman beams and calibrated the lattice with the following procedure:
1. To check the alignment of the Raman beam, we set ∆ω = 0 and used KapitzaDirac experiment with the Raman lattice. This is the quickest experiment to
do and is also a coarse alignment check.
2. Set ∆ω = 0 and load the BEC into this Raman lattice. See if we can have adecent lifetime. This is to further check the phase noise and the alignment of
the Raman lattice.
3. Set ∆ω 6= 0 to the Raman beams and see Resonant two-photon process withthe condensate.
4. Set ∆ω 6= 0 and use the Raman beams to do population transfer between bands.
With the above procedure, we moved on to the spin-orbit coupling. We first did
the Kapitza-Dirac scattering. We can see from these Time-of-flight pattern that our
Raman beams are providing the correct momentum transfer. The peak intensity are
symmetric in the +r̂ and −r̂ direction. It will be asymmetric if ∆ω × tKD 6= 0 which
is not important for the momentum transfer calibration. We then quickly loaded theatoms into our superlattice and then do the KD scattering for the Raman lattice.
We saw expected interference peaks. So a check mark for the coarse alignment andmomentum transfer.
86

a

b

z

r

c

With Retro blockeddxzWith Retrorr

superlatticesuperlattice with Raman

Figure 4-9: Kapitza-Dirac pattern for the Raman lattice with and without superlattice. a. Kapitza-Dirac pattern for the Raman lattice. b. Kapitza-Dirac pattern for
the Raman lattice with the Retro unblocked. c. Kapitza-Dirac pattern of the superlattice. d. Kapitza-Dirac pattern with superlattice and Raman lattice combined.
We then load the atoms adiabatically into the stationary Raman lattice (∆ω = 0)
and checked the coherence lifetime. We observed lifetime comparable to the casewhere the BEC is loaded into a regular IR lattice formed by retro-reflected beam forthe same lattice depth. This suggested that we didn’t do a bad job.
After this, we set ∆ω 6= 0 to see if we can resonantly excite the condensate. In
the very early days, two-photon Bragg process has been used to selectively excited a
Bose-Einstein condensate. Here the momentum transfer we have is ~k = kIR (−ẑ + ŷ),corresponding to a resonance at h̄∆ω = 2ER = 15.8 kHz. Now we showed that it
worked for a bare BEC. Step by step, we load BEC in green lattice and use the Raman
beams to do band transfer. For a bare green lattice, the band gap is so much larger
87

V l = 40E I V s = 0E I = 0

50
45
40
35

E/E IR

30
25
20
15
10
5
0
0

0.2

0.4

0.6

0.8

1

q / ( /d)Figure 4-10: Two-photon Raman transfer in the an optical lattice. The grey line
illustrate the lattice potential corresponding to the band structure which is shown incolor.
than the ER therefore the on-site coupling or the intraband coupling is negligible. As
we have shown before, the band-mapping technique didn’t work well in our quasi-1D
system, we did our detection with Time-of-flight.
1. Raman transferred atoms gained a momentum along the transverse direction
and are therefore automatically separated from the untransferred atoms.
2. Though have the same quasimomentum along the lattice direction, the distribution of the real momentum will be different depending on the band index.
But comparing with the calculated Bloch states, we can tell which band it is in.
As shown in Fig.4-11, from the momentum pattern after TOF imaging, we showedthat we successfully transfer the atoms between different bands which the correct
amount of transverse momentum transfer.
This is the last check point before the experiments.
88Figure 4-11: Higher band population from two-photon coupling.

4.3.2

Dispersive and Population Signal

We now move to the spin-orbit coupling in an optical superlattice, our main goal. The
spin-orbit coupling is nothing but a band transfer between the ground and the firstexcited band of the optical superlattice with a transverse momentum kick. However,the complication compared with our test experiment is the on-site coupling. As
explicitly shown in Eq.A.15 and Eq.A.16, the on-site coupling will always have thesame momentum components as the spin-orbit coupling, making it hard for us todistinguish. Especially in our case where the superlattice offset ∆ is not much larger
than the recoil Er . Re-examining the amplitude of different momentum components
from Eq.A.15 and Eq.A.16, we have already ignored the counter-rotating spin-orbit
coupling terms which have Er + ∆ + δ and Er − ∆ − δ in the denominator :A typical image we have obtained as data is shown in Fig.4-12, as we can see all of
the momentum component related to |ψ1 i. The on-site and spin-orbit coupling were
perfectly mixed together. Eventually, we figured out two ways to extract the SOC
89

Figure 4-12: Momentum components of |↓i state dressed with two-photon process in
Eq.4.5. All the diffraction peak from the on-site coupling and spin-orbit coupling are
visible.

signal from the on-site coupling signal.

1. Population Signal. Due to the fact that δ  Er , the on-site coupling is
suppose to give symmetric population in both the +k and −k direction with apopulation ∼ (Ω/δ)2 . The spin-orbit coupling will only enhance the populationin one direction depending on the spin flipping process. This gave an asymmetricfeature in the population of atoms in the +k and −k direction. When the Raman
beam detuning δ is scanned across the spin-orbit coupling resonance, the createdimbalance is quadratic in the spin-orbit coupling strength β and therefore is only
clear when we have some significant spin-orbit coupling strength.

2. Dispersive Signal. The weights for different real x momentum components
would change with the parameter ER + ∆ − δ due to the interference between
the on-site coupling term and the spin-orbit coupling term. For example, for
(↑)

(↓)

|ψ1 i, it is between |ψq=0 , +ki and |ψq=0 , +ki.
90

Dispersive Signal
The dispersive signal can be understood as follows. According to Eq.A.15, we haveπe−i 4 JΩ/∆
1 Ω
(↓)
(↑)
hkx | ψ1 i = −hkx |ψq=π/d i − i √hkx |ψq=π/d i
2 Er − δ
2 Er + ∆ − δ(4.19)In an optical lattice, the q = π/d states in the ground and 1st excited band corresponds to the symmetric and asymmetric superposition of |k = π/di and |k = −π/dito the lowest order. Therefore, approximately,π11 Ω
e−i 4 JΩ/∆
hkx | ψ1 i = √ [(−−i √|kx = π/di
2 Er − δ22 Er + ∆ − δπ1 Ω
e−i 4 JΩ/∆
+ (−
+i √
) |kx = −π/di]
2 Er − δ
2 Er + ∆ − δ(4.20)

with
π

1
1 Ω
e−i 4 JΩ/∆ 2| hkx = π/d| ψ1 i|2 = | −−i √|22 Er − δ
2 Er + ∆ − δ
π11 Ω
e−i 4 JΩ/∆2
| hkx = −π/d| ψ1 i| = | −
+i √
)|2
2
2 Er − δ
2 Er + ∆ − δ(4.21)

we can immediately see that| hkx = π/d| ψ1 i|2 − | hkx = −π/d| ψ1 i|2 ∝Ω JΩER − δ ∆ ER + ∆ − δ(4.22)

Eq.4.22 is the major result. It shows that the population imbalance between thekx = π/d, ky and kx = −π/d, ky switched sign when the Raman beams are swiped
across the spin-orbit coupling resonance due to the interference between the spin-orbit
coupling term and the on-site coupling term, as shown in Fig.4-13. The dispersive
signal scales linearly with the small parameter J/∆ instead of quadratic. The onsite coupling term now acts as local oscillator for the spin-orbit term. Our effectively
homodyne detection enhanced the signal.
The dispersive signal was indeed the first signal we obtained for this scheme, as
shown in Fig.4-14. We saw the clear feature of the sign change in the population91q= /d, E R +

- >0
1

0.8

0.8

0.6

0.6

Np /N

Np /N

1

0.4
0.2

0.4

0
-5 -3 -1

1

3

5

7

-5 -3 -1

k / ( /d)q=0, E R +

1

3

5

7

k / ( /d)

- >0
1

0.8

0.8

0.6

0.6

Np /N

Np /N

- <0

0.2

0

1

q= /d, E R +

0.4
0.2

q=0, E R +- <0-6 -4 -2

2

0.4
0.2

0

0
-6 -4 -2

0

2

4

6

k / ( /d)0

46k / ( /d)Figure 4-13: Interference between the on-site and the spin-orbit coupling term. Whenthe detuning of the Raman beams is scanned across the resonant point, interference
between the on-site and the spin-orbit coupling term cause the population difference
between the q = ±π/d states to flip sign on the first band.
imbalance for +kz and −kz peak when scanned across the resonant point. The total
atom number also dropped because when the population transfer due to spin-orbit
coupling was too small, the total population in the +k peaks was dominated by theon-site coupling term which was detuned further with increasing δ.

Population Signal
The dispersive signal is more sensitive and is the first signal we observed. However,when started with the |↑i state, the dispersive signal was not reliable anymore fortwo main reasons. One reason was that the |↑i state was harder to prepare and morefragile against misalignment of the superlattice. Therefore usually our sample was92

Figure 4-14: Measure population imbalance between the ±k peaks. the populationdifference changes sign when the Raman detuning is swept across the spin-orbit coupling resonance.

worse than the |↓i state. The second reason was that the heating due to the moving
Raman lattice was more sever for the |↑i state. A possible reason was that whensitting in the 1st excited band, the atoms could be couple to both the ground and the
2nd excited band through two-photon process, while the |↓i could only be coupled to
the 1st excited band. Since the 2nd excited band is very lossy, we expected that the
coupling would make the lifetime of the atoms in |↑i shorter. The dispersive signal
was therefore very hard to see, considering that it was not that clear compared withthe |↓i case even theoretically. Also there was no reason that the population transfer
shouldn’t work. So we decided to fix the problem and search for the population signal.
We first focused on improving the lifetime of our sample. The key factor turned out
to be simply the alignment of the superlattice and picking the correct parameters.
The core parameters in the experiments were J/∆ and Ω. The superlattice phaseturned out to be not crucial for stability as long as J and ∆ were maintained.
Too large J/∆ caused severe collisional decay of the pseudospin states. Also toostrong Raman beams immediately introduced high order process to higher bands andenormous heating. We then improved the spectral purity of the Raman beat notes,
mainly by redesigning the optical layout of the Raman beams 1) to reduce the leakageof the light from the wrong order of the TTL AOM 2) to simplify and minimize theoptical path length. Ideally, the beat notes of the two Raman beams should be purely
93

Figure 4-15: Typical spectrum of the beat notes between the two raman beams at a
detuning of 20 kHz.The top one is accompanied by a strong unwanted 10 kHz peak.
The bottom one is improved in the spectral purity.
∆ω. Any other components, especially around Er /h̄ would cause unnecessary heating.
We then improved the purity of the Raman spectrum as shown in Fig.4-15.
We then observed clear population signal for the |↓i state after these improvements
and better alignment (now we have enough power to use the Raman beam to actuallycatch atoms. These helped the precise alignment). We further improved the qualityof sample in the |↑i states by finding the sweet combination of superlattice phase φ,
J and ∆, and better alignment. Then we successfully observed the |↑i → |↓i spinorbit coupling. As shown in Fig.??. We clearly see the two important features of thespin-orbit coupling:
1. Population imbalance showed resonance feature.
2. The ‘polarization’ (+y or −y) is opposite for different initial spin state, demonstrating time-reversal symmetry.
A more quantitative measurement was done by counting population the imbalance. A
quantitative study is done by measure the population imbalance. Shown in Fig.4-17
is the population imbalance between the “+y” and “−y” momentum peaks versus
Raman detuning for |↓i → |↑i(blue) and |↑i → |↓i(red) processes. The two sets94

OD (a.u.)

Off Res.
On Res.

(a)

(e)

(b)

(f)

(c)

(g)

(d)

(h)

-4

-2

0

2

4

-4

-2

0

2

4

(i)

y

z

-4

-2

0

2

4

Figure 4-16: (a),(e)TOF images of |↓i and |↑i states, respectively. (b),(f)Schematicsof the momentum peaks for |↓i and |↑i with Raman coupling. Both the spin-orbit
coupling (solid arrows) and the density modulation (dashed arrows) are shown. The
main peak (filled circle) is equal to the quasimomentum of the state. Extra peaks(opencircles) appear due to the periodic potential. (c),(d),(g),(h)Same as in (a) and (e), but
now with Raman coupling at different detunings δ. Momentum components createdby Raman process are vertically shifted compared to (a) and (e) due to the transverse
momentum kick. The momentum shift along the superlattice (z direction) reflects
the π/d quasimomentum of the Raman lattice. The off-resonant density modulation
creates momentum peaks which are symmetric along +y and −y (Figs. (c),(g)),
whereas resonant spin-orbit coupling creates unidirectional momentum transfer (Figs.
(d),(h)). (i) Spin-orbit coupled BECs with equal population in spin up and spin downstates.

of data were measured for the same superlattice parameters VIR = 7.5(2)Er , VGr =20(2)Er and φSL ≈ 0.22(1)π which gives ∆ ≈ 37(1) kHz. The spin-orbit couplingstrength β was calculated to be 0.40(5) kHz. The solid lines are Gaussian fits to theresonances centered at 32.2(3)kHz and 43.2(3) kHz. The resonances are broadened
due to inhomogeneous offset ∆ from the Gaussian beam profile. The error bars
represents one σ statistical uncertainty. Inset: Resonance center frequencies versus
IR lattice depth VIR for fixed φSL . The resonances are linear in VIR with a constantsplit equal to twice the recoil energy. The slope of the linear fit reveals the superlatticephase φSL . Error bars are the uncertainties of the fit.
The displacement between the two resonance for the same superlattice detuning
95

Res.
Frequency (kHz)Normalized Population Imbalance

Raman Detuning

VIR
Er

(kHz)

Figure 4-17: Spin-orbit coupling resonances.
∆ was ∼ 13 kHz, close to the theoretical value ∼ 2ER . In the inset, we showed thatno matter what the ∆ was, this displacement was always the same, further proved
our theory.
However, two remaining issues were First, the displacement was not exact 2Er =15.2 kHz. Second, the resonant peaks were broad.
We attribute the discrepancy between measured and the theoretical resonancesplitting to the following two main reasons:Intra-spin interactions As illustrate in Fig.4-18, U↓↓ will shift |↓i up and therefore
reduce the resonant frequency for |↓i → |↑i while U↑↑ increase the energy splitting
between |↓i and |↑i. With intraspin interactions added to the single particle Hamil-

J

J

guunu
gddnd

Δ

Δ

Figure 4-18: Interaction induced shift of resonance.

96tonian on mean level, the modified resonant conditions are|↓i → |↑i : δ = ER + ∆ − U↓↓(4.23)|↑i → |↓i : δ = ∆ − ER + U↑↑
From the measured chemical potential of the condensate, we estimated the shift∼ 2µ ≈ 5 kHz.
Imperfect alignment of the Raman beams The splitting between the two resonances is calculated to be 2h̄2 ky2 /2m − (U↓↓ + U↑↑ ) which is quadratic in ky . The
actual transverse momentum transfer can be experimentally measured by the TOF
images to be 1.07kR which will give shift of ∼ 10% in the recoil energy. This part
covers ∼ 800 Hz of the shift. This deviation comes from our Raman beam geometry.
The broadening of the lineshape came from three major sources:
1. Doppler broadening due to the finite size of the trapped condensate.
2. inhomogeneous mean-field shift due to inhomogeneous condensate density
3. inhomogeneous shift of the superlattice offset ∆ due to the Gaussian beamprofile.

4.3.3

Broadening and Width

Doppler Broadening The two-photon process in our scheme is Doppler sensitive based on our scheme and geometry, the finite size of the condensate implies a
distribution of momenta along the transverse direction which Doppler broadens theresonance. A momentum p along x direction will shift the resonance by kIR · p/m
considering the geometry of our setup. Therefore a condensate size x0 results in a
broadening of ∼ kIR h̄/mx0 . Assuming a Thomas-Fermi distribution in the transverse
direction, we obtain an rms width of [63]rδνD =21 kIR h̄
8 2π mx097(4.24)

where x0 =p2µ/m(2πνx )2 . This gives a broadening of ∼ 400 Hz. This broadening

is very natural and is actually a real effect of spin-orbit coupling. As we discussed
before, the 1D spin-orbit coupling is nothing but a momentum dependent detuning.
Therefore if there is a distribution in the atoms’ momentum, there is expected to bea detuning and therefore a broadening in the lineshape.
The resonance is also broadened by the inhomogeneous density distribution of thecondensate as
r
δνM =

8 µ
147 h

(4.25)

with µ being the chemical potential. This effect broadens the resonance by ∼ 600 Hz.
The Gaussian beam profile of the long lattice implies an inhomogeneous offset ∆within the sample, therefore broadening the transition. The resulted shift is estimatedto beδνI ≈ 2∆(

x0 2
)
σ

(4.26)

where σ is the Gaussian beam waist parameter. This effect broadens the resonanceline by ∼ 1.10 kHz. However, the broadening can be much larger for small displacement between the green and IR lattice.
The three widths add up quadratically to a value of 1.30 kHz. The linewidthof the frequency difference of the Raman beams was negligible — it was monitored
through a beat note and was ∼ 100 Hz without actively phase-locking the two. We
conclude that observed resonance width of 2 kHz is most likely dominated by the
spread in ∆ due to slight misalignment of lattice beams.

4.4

Conclusion

In conclusion, we demonstrate the feasibility of using an double-well optical superlattice to realize one-dimensional spin-orbit coupling. With this choice of pseudospins,no near-resonant light is needed and therefore heating due to spontaneous emission
is suppressed.
Another feature hasn’t been fully covered is the imbalanced inter- and intra- spin98

interaction. Our system creates a situation where the inter-spin interaction is muchsmaller than the intra-spin interaction allows us to study the stripe phase presented
in an interacting spin-orbit coupled system. Related discussions will be present in
Chapter.6 and Chapter.7.

99

100

Chapter 5
Floquet Engineered Spin-orbit
Coupling

Beside implementing different pseudospins, another approach to realize spin-orbit coupling while suppressing the spontaneous heating is to use Radio-Frequency photons
with engineered photon momentum. In the chapter I describe how we realized spinorbit coupled Hamiltonian by using Radio-Frequency (RF) photons. When RF pulses
are synchronized with a fast oscillating force created by a modulating magnetic field
gradient, an effective momentum is dressed to the RF photons. The amplitude andthe direction of the dressed momentum can both be tuned by changing the synchronization sequence. For this system, the effective low-frequency effective Hamiltonian
is equivalent to a one-dimensional spin-orbit Hamiltonian.
I first present a general discussion on engineering the Hamiltonian and the dynamics of a system by periodical driving, namely the Floquet engineering. Then thespecific theory and sequence for generate one-dimensional spin-orbit coupling implemented in the thesis work is carried out. The results are presented in the experimental
section with the focus on the experimental non-idealities and how we deal with them.
Before concluding, a discussion on the heating due to micromotions is presented.
101

5.1

Periodically Driven Systems - Floquet Engineering

Controlling the dynamics of a system by dressing is not a new idea in physics.
In classical physics, a prominent example is the Kaptiza pendulum where thepivot point of a rigid pendulum vibrates vertically along the direction of gravity. If
the vibration is fast enough, the pendulum can balance stably at the inverting point
where the bob is above the pivot. This is an example that a fast modulation, thevibrating of the pivot points, could change the slow motion, the oscillating of thependulum, of the system.
Similar phenomenon can also be found in quantum mechanics. When a two-levelsystem is dressed with off-resonant laser light, the dressing of the excited state into
the ground state caused AC stark shift, therefore changing the low energy dynamics
of the system.
The general way to deal with the problems involving high frequency modulationand low-frequency dynamics is by taking the time average within time scale associated with the fast modulation. The consequence of this fast motion, after the time
averaging, appears as an effective potential steering the low-frequency motions. For
example, when the pendulum pivot is vibrating as y(t) = A sin νt, the effective potential for the oscillation of the pendulum isVeff (φ0 ) = −mgl cos φ0 + m

Aν
sin φ02

(5.1)

where φ0 is the coordinate of the pendulum. It is therefore clear that the fast modulation drastically change the dynamics of the pendulum by allowing an unconventional
stable point at φ0 = π.
In quantum mechanics, dynamics of a system is governed by the Hamiltonian andis described by the evolution operator Û (ti , tf ). When the system is under modulation,
the Hamiltonian is time-dependent. However, if the modulation is fast and only
the slow dynamics is concerned, it is possible to average out the fast modulation.
102

The result is an effective Hamiltonian which governs the slow dynamics and is timeindependent.
This approach of engineering the slow Hamiltonian through fast modulation iscalled Floquet engineering. The fast modulation averaged out is named micromotion.
Mathematically, if a Hamiltonian Ĥ(t) is periodic Ĥ(t + T ) = Ĥ(t), the evolutionof the system Û (ti , tf ) has the simple form [64, 65, 66]Û (ti , tf ) = e−iK̂(tf ) e−Heff (tf −ti )h̄ eiK̂(ti )(5.2)

where K̂(t) = K̂(t + T ) is a periodic operator describing the micromotion, Heff is theeffective Hamiltonian for the slow motion.
It worth pointing out that the standard way to calculate the evolution under atime-dependent Hamiltonian uses the mathematically complex time-ordered integralRt
U (ti , tf ) = T tif exp(−iH0 + V (t)) dt . However, the special temporal symmetry
in a periodically driven system simplifies the results, allowing the separation between
the fast micromotion and the slow effective dynamics

Energy Scales

The key requirement for the validity of the effective Hamiltonian is that the modulation frequency ω = 2π/T satisfies
ω  Edyn(5.3)

where Edyn is the energy scale involved in the system evolution. It is the intrinsic
frequency of the pendulum in the Kapitza pendulum case, and the rate of the spontaneous emission in the case of AC stark effect. Mathematically, this can be seen by103

writing the full form of the effective Hamiltonian [64]:∞

1 X 1 (j) (−j)
Heff = H0 +[V , V]ω j=1 j
∞

1 X 1
(j)
(−j)
+ 2
[[V
,
H
],
V
]
+
H.c.
0
ω j=1 2j 2

(5.4)

∞

1 X 1
[V (j) , [V (l) , V (−j−l) ]] − [V (j) , [V (−l) , V (−j+l) ]] + H.c.
+ 2
ω l,j=1 3jl+ O(1/ω 3 )The Heff is organized according to the order of 1/ω, emphasizing the energy structurein the system.
The energy scales for the dynamics of the system should be well separated fromthe driving energy h̄ω. There are two ways to understand this. First, if we calculate hφ| Heff |φi where |φi is the eigenstates, we immediately see that the expansion parameter should be (Eφ /ω). This means that the expansion converges whenthe modulation is fast. The second way is purely mathematical. In the formalismof Magnus expansion, the sufficient condition for the expansion to converge is that
RT||H(t)||2 dt. Here ||H(t)||2 is the second matrix normal which in our case is noth0
ing but the largest eigenenergy of H(t). Therefore either pictures or treatments givethe same criteria for the Floquet picture to be valid. However, when the modulation
frequency hits a resonance in the system, the expansion presented above would notwork and the effective Hamiltonian will have more complicated form [13]. This isbeyond the scope of the thesis.

Oscillating ForceTo exemplify, we consider a Hamiltonian describing a particle’s motion under a periodically modulating force can be written as H(t) = p2 /(2m) + Ax sin (ωt). Under
the framework of Floquet engineering, the effective Hamiltonian and the micromotion
operator reads
Heff = p2 /(2m)andK(t) =
104

Ax
cos t
ω

(5.5)

Therefore an oscillating force give an effective Hamiltonian equivalent to a free particle
Hamiltonian. True, under the oscillating force, the BEC could drift with a certain
velocity. However, The drifting is a trivial frame transformation. In the frame of thecondensate, no strange physics is expected.

5.2

Spin-Orbit Coupling with Floquet Engineering

Following the last part of the discussion in the previous section. If now we consider
two condensates with opposite spin with the oscillating force being spin dependent.
the oscillating force makes the spin drift to opposite directions. If the two condensates
do not interact with each other, the oscillating force is again trivial. However, when an
coupling, say Radio-Frequency transition) between the two spin states is introduced,
every time the RF is pulsed on, each spin states will check itself with the reference tothe other. Then they are able to tell that they are moving away from each other sincein quantum mechanics, drifting is equivalent to an extra kinetic phase. Then, we areable to tell by the transferred spin population that they are moving. The drifting
has a real measurable effect. The spin transfer depending on the relative drifting of
the cloud is essentially a Doppler effect. In other words, an effective momentum andspin-orbit coupling is engineered for RF photons.
Indeed, it is possible to dress an RF photon with a tunable momentum by Floquet
engineering. The dressed momentum is 106 larger than the intrinsic momentum of the
RF photon. The scheme is equivalent to one-dimensional spin-orbit coupling. The
discussion in this section will be focused along this direction.

5.2.1

Sequence and Effective Hamiltonian

Figure 5-1 shows the time sequence of our scheme, which consists of a sinusoidal
spin-dependent force f (t) = gF µB B00 sin( 2πt + φRF )σz , where gF is the Lande factor,TµB is the Bohr magneton and B00 is the magnitude of the magnetic gradient, and asynchronized sequence of short RF pulses at times t = 0, T, 2T...
The timing of the pulses with respect to the periodic force is described by the105phase φRF which will determine the magnitude of the photon recoil. Each of the
RF pulses couples the spin-up and spin-down states with the same velocity vRF . For
φRF = 0 the velocities averaged over a full cycle of the oscillating force, hv↑ i andhv↓ i are different. By flipping the spin, atoms experience an “extra” half-cycle ofthe magnetic acceleration (hatched area in Fig. 5-1(a)), which transfer them to thestate with a different averaged velocity, and, therefore, provides recoil. For the caseφRF = π/2, the time-averaged velocities for spin-up and spin-down are identical tovRF . Therefore, an RF transition will not change the time-averaged velocities, andthere is no recoil.

B’tf�-1 ti�
Ω

tf�

ti�+1

tf�+1

Ω
t

v
t
Figure 5-1: Illustration of our scheme for creating a tunable atomic recoil momentum with RF transitions using magnetic forces. (a) & (b) shows the experimental
conditions for φRF = 0 and φRF = π/2, respectively. The spin-dependent forcesand velocities are shown (as thick solid lines) for the amplitude of the wavefunction
which is transferred from spin down (red) to up (blue) by the RF pulse marked by
the gray dashed line. For φRF = 0, the average velocities hv↓ i and hv↑ i are different,
which implies a finite recoil associated with the spin-flip. In contrast, hv↓ i = hv↑ i for
φRF = π/2 and there is no recoil.
Using this semi-classical picture, we obtain for the amount of momentum transferh̄k = m(hv↑ i − hv↓ i) = h̄k0 cos φRF , where k0 = gFπh̄µB B00 T .
Effective Recoil Energy
For an optical transition with recoil h̄k and an atom moving at initial velocity vin ,the resonance frequency is shifted by the Doppler shift kvin and recoil shift (h̄k)2 /2m
which ensures energy conservation. In the current situation, energy can also come106

from the time-dependent magnetic force. Indeed, if we would apply a single RF π
pulse at phase φRF = 0, the time-averaged velocity would change by h̄k0 /m, but the
RF resonance frequency would be independent of velocity and k0 . However, if a seriesof RF pulses is used, as in Fig.5-1, the resonance is Doppler shifted and becomesvelocity selective.
Between pulses, the phase evolution for spin up/down is solely determined by theR2/2)dt, leading to a phase difference δα = h̄1 (m(hv↑ i −kinetic energy α↑↓ = h̄1 (mv↑↓hv↓ i)vRF )T = kvRF T after one period of shaking, where vRF = (v↑ + v↓ )/2 is thecommon velocity at the moment of RF pulse. With hv↓ i = vRF − h̄k/2m we find thatfor resonant excitation, the RF frequency has to compensate for this phase shift by
the Doppler detuning khv↓ i and the recoil shift (h̄k)2 /2m.

Effective Hamiltonian
We can show more formally, that our scheme creates an RF photon with recoil. For
our scheme, the time-dependent Hamiltonian of the system in the frame rotating with
the RF drive after the rotating-wave approximation is

Ĥ =

π
2π
p̂2z
+ h̄k0 ẑ sin( t + φRF )σ̂z
2m
T
T
X
1
− h̄δRF σ̂z + h̄Ωσ̂x T
δ(t − nT ), (5.6)
2nwhere δRF is the RF detuning with respect to the atomic resonant frequency and mis the atomic mass. The short RF pulses are represented as a series of delta-functions
with effective Rabi frequency Ω.
Through the derivation shown in the Appendix, we obtainĤef f =2 2p̂2z
1 h̄ k0+ 16− h̄δ2RF
2m
m


h̄Ωe

ik0 cos φRF ·z

K̂(t) = −ik0 zσ̂z cos(107h̄Ωe−ik0 cos φRF ·z
2 2p̂2z
1 h̄ k0++ h̄δ2RF
2m
16 m

2π
t + φRF ).
T




(5.7)

The effective Hamiltonian is identical to the one for a two level atom driven by aphoton field at frequency ωRF and with wavevector k, which confirms our discussionh̄2 k2

1
0
above about recoil momentum and Doppler shift. The term 16
is the kineticmenergy due to micromotion.

5.3

Experimental Realizations

The two pseudospin states considered are |↑i = |F = 1, mF = −1i and |↓i = |F = 1, mF = 0i.
A DC magnetic field B0 ≈ 10 G is applied along x̂, providing the quantization axis,
polarizing the pseudospins. The |F = 1, mF = 1i state is detuned due to quadratic
Zeeman effect and is therefore neglected.
The spin-dependent oscillating force is implemented by combining an oscillating
quadrapole magnetic field together with the bias B0 x̂, effectively creating an oscillating magnetic field gradient along x̂.
~ =
|B|

p
(B0 + B 0 x sin (ωt + φ))2 + (B 0 y sin (ωt + φ))2 + (2B 0 z sin (ωt + φ))2
(5.8)

B0
≈ B0 +sin (ωt + φ)x

The amplitude of the force f = B 0 µ gives an effective recoil momentum k0 = B 0 µ/ω.
The experiments were performed with thermal clouds of ∼ 105 23 Na atoms at
a temperature T ≈ 0.5 µK loaded into an optical dipole trap. We used thermal
clouds for two reasons. First, the atoms thermally populated a velocity range withpa width of σv = kT /m. Therefore, it samples a wide range of velocity groups at
one time. Second, thermal clouds have low density therefore ruling out the effects ofinteractions.

5.3.1

Generation of the Modulating Magnetic Field Gradient

A large dressed momentum requires a high amplitude of magnetic shaking B 0 . Our
main magnetic coil can generate roughly 0.5 Gauss/cm magnetic field gradient at thecenter of the chamber with 1 Amp. To create an effective recoil momentum of k0 =108

0.1kL , where kL = 2π/589nm is the photon recoil for the major transition in 23 Na,
the required peak of magnetic field gradient is ∼ 50 G/cm at a modulation frequencyof 5 kHz. The coil has an inductance of 70 µH and a DC resistance of Rcoil ≈ 70 mΩ.
In principle, the required gradient can be generated with a P = I 2 Rcoil ∼ 700 W
power source. However, as will be discussed in section.5.3.3, the magnetic coupling to
the vacuum chamber gives an effective dissipation of 1 Ω. Therefore a 10 kW powersource is needed.
We generated the sinusoidal gradient modulation B 0 (t) = B 0 sin ωt + φ with an
H-bridge setup shown in Fig.5-2. A DC power supply is connected to an H-bridgewith four IGBT switches A1, A2, B1, B2. By switching on and off group A and group
B alternatively, polarization of the DC power supply is switched therefore acting as
a square wave source. When driving the coil with an extra capacitor C, the resulting
steady state current has a periodic form without DC components. This setup provides
enough power for generating the current for k0 ≈ 0.1kL

a

P
B1

A1

A2

B2
N

C

C

Rcoil

Lcoil

Lcoil

Magnetic Coil

Magnetic Coil

b

Rcoil

c
B’
T

2T

t

Figure 5-2: Generation of the gradient modulation with H-bridge. A DC power
supply connected to an H-bridge serves as a square wave source. When driving an
RLC circuit, the current has a sinusoidal form at the frequency of the square wave.

109

5.3.2Signal and Measurements

The Doppler effect, therefore the spin-orbit coupling, was demonstrated with velocityselective RF spin flip. This was done by first implementing the sequence with theatoms trapped by an optical dipole trap and then measuring the spin populationof atoms in different velocity groups. The velocity information is revealed with thestandard time-of-flight imaging while atoms in different spin states are separated withthe regular Stern-Gerlach separation with a pulse of magnetic field gradient.

aIntegrated column density (arb. units)8δRF = - 300 Hz=
0 Hz= +300 Hz
Without shaking

6

4

2

- 0.4

- 0.2

0

0.2

0.4

Position (mm)bWithout shaking

With shaking (δRF= 0)

Shaking direction

Figure 5-3: Observation of velocity selective RF transitions. (a) Absorption imagesof the spin-flipped atoms after 12 ms of TOF with and without magnetic shaking. The
yellow dashed ellipses have major and minor axes obtained as FWHM of Gaussianfits. After TOF, the thermal could expands by a factor of 2.13, thus a single-velocity
class is narrower than the thermal cloud by 1/2.13 ≈ 0.47. The Fourier limit ofour velocity selection increases this to 0.50, and inclusion of eddy currents further
modifies it to 0.45 (dashed-dotted line). The field of view is 1 mm by 1 mm. (b)Integrated column density distribution obtained from absorption images like those in(a), for different detunings of the RF frequency. The solids lines are Gaussian fits tothe data points. The RF phase was at φRF = 0 to maximize Doppler sensitivity.
With bare RF pulses, atoms initially prepared in |↓i were addressed globally by the110radio-frequency photons. All the velocity groups were resonantly transferred to theother spin state simultaneously, characterized by the isotropic distribution of atomsin |↑i after the long free expansion.
The modulating magnetic field gradient shifts the averaged kinetic energy in aspin-dependent way, causing Doppler shift. When φ = 0, maximum effective recoil
kso is obtained. Only atoms with the initial x−velocity fulfilling the Doppler conditionvx = δ/kso were resonantly transferred to the other spin state. This velocity-selectivityof spin transfer is observed as a narrowing of the transferred velocity slice, shown in
Fig. 5-3(c). The resonant velocity, measured by the center-of-mass position of theatomic profile after expansion, shifted linearly with δ, featuring an effective Doppler
effect. The slope flipped sign when φ changed from 0 to π, (Figure 5-4), consistent
with theory.

abDoppler shift (Hz)400

ɸRF= 0

Doppler shift (Hz)
400
200

200

ɸRF= �
Velocity (cm/s)

-0.2

0.2

-200

-0.2

Velocity (cm/s)

0.2
-200
-400

-400

B’

B’

v

v

Ω

Ω
t

t

Figure 5-4: Observation of RF transitions with Doppler shifts. (a) & (b) Central
velocities of the spin-flipped atomic distribution (as in Fig. ??(b)) are shown as
a function of RF detuning for φRF = 0 and φRF = π, respectively. Shifting the
RF phase changes the sign of the Doppler shift and therefore the direction of therecoil momentum. The solid line represents the predicted Doppler shifts based on
the calibration of recoil momentum. The dashed line takes into account the effects ofeddy currents (see Supplement). The error bars are 1σ.

The resolution of the velocity-selectivity is determined by the recoil momentum
111

kso , the instrumentation resolution limited by the coherence time of our system, andfurther diminished by other technical imperfections, e. g. residual DC gradient along
x̂.

5.3.3

Effects of Eddy Currents

Eddy currents are loops of electrical current inside the conductors induced by thechanging magnetic flex inside the conductor. When the magnetic field is modulatingnear our stainless steel vacuum chamber, the induced eddy currents inside the chamber materials gave unwanted stray fields and extra dissipation. The stray fields are~ 0 , effectively gives a velocity selectivity
along the direction perpendicular to the bias B~ 0 . The dissipation increase the power consumpalong the direction deviating from B
tion when resonantly driving the gradient oscillation. As we detail below, the eddy
currents have profound effects on our measurements.
The discussion is quite general and is applied to any experiments which involve
modulating magnetic field at high frequencies.
Eddy Currents in Vacuum Chambers
The essence of the eddy currents can be captured by the model of a conductive
ring sitting the magnetic field. A time-various magnetic flux going through the ring
induces current inside the ring, with
Ieddy = −

dΦ 1
.
dt Z

(5.9)

Here Φ is the magnetic flux, and Z = R + iωL is the impedance of the ring. The realpart Re(Z) = R is the DC resistance of the ring and is responsible for the dissipationof the system. Im(Z) = ωL is the reactance determined by the frequency of themodulation and the self-inductance. It is responsible for the delay of the response.
The eddy current is delayed from the drive by a phase of arctan (ωL/R) with an
1√
amplitude dφ.
dt R2 +ω 2 L2

In practical scenario, eddy currents are induced by the magnetic field generated
112

by the set of magnetic coils giving the oscillating force, therefore the process can be
described by the mutual inductance between the chamber and the coil, as shown in
5-5. In this model, the stainless steel chamber is simply modeled by an RL circuit
k

Rcoil

τ
I
Magnetic Coil

Lcoil

Lch

Rch

Reff

ch

Leff

II

Stainless Steel Chamber

Magnetic CoilFigure 5-5: The primary circuit describes the coils creating the oscillating gradient.
The secondary circuit captures the environment (chamber, other coils) that was coupled to our primary coil due to mutual inductance. The effects of eddy current isequivalent to extra dissipation and inductance.
with effective inductance Lch and resistance Rch . The coupling between the main√
coil and the chamber is captured by a single parameter k = M/ Lch Lcoil which is apurely geometric factor. M is the mutual inductance.
The current flows through the main coil I˜coil driven by Ṽ is modified from I˜coil =Ṽ /(Rcoil + iωLcoil ) toI˜coil = Ṽ /(Reff + iωLeff )(5.10)

ωτ
Reff (ω) = Rcoil + k 2 · ωLcoil ·1 + (ωτ )2(ωτ )2
2
Leff (ω) = 1 − k ·· Lcoil
1 + (ωτ )2(5.11)

where

where τch = Lch /Rch is the characteristic damping time for the eddy current.
This model is verified by measuring the impedance of the main coil at different
driving frequencies. In a regular RL circuit, the impedance is quadratic at low frequencies where the DC resistance dominates. Then it turns linear with the slope
equal to the inductance. The spectacular feature of this model is that at the frequency higher than ω ≈ 1/τ , the slope is reduced from Lcoil to (1 − k 2 )Lcoil . The
result is shown in Fig. 5-6. The measured impedance showed a change of the system’s
113

a

b

Figure 5-6: a. Impedance of the primary coil near vacuum chamber. when f >8 kHz, the effective DC resistance and the inductance of the coil sharply changes. b.
Impedance of a testing coil in different electric environments.

inductance at a corner frequency of ≈ 8 kHz. The parameters Rcoil , Lcoil , τ and k 2 are
extracted from the fitting
Rcoil = 42 mΩ, Lcoil = 40 µH, τ = 16 µs, k2 = 0.52.

(5.12)

Rcoil was also measured from a DC measurement which gave consistent result of
Rcoil = 37 mΩ.
For further verification, we carried out measurements of the impedance of a test
coil sitting in different environments including another coil, a vacuum flange, etc. The
results, shown in Fig. ??, supported the model.

Figure 5-7:
Resonance profile of the coilpled(blue)/decoupled(orange) from the environment.

114

when

mutuallycou-

Dissipation and Shielding
The effects of eddy currents are clear through Eq. 5.11 and can be summarized as adissipation effect and as a shielding effect.
1. Reff > Rcoil . The dissipation of the coupled system is always larger than that of
an isolated coil. Therefore to drive the main coil with a peak current of ∼ 100A,
the input power from the power supply needs to be I 2 Reff ≈ 10 kW instead of
I 2 Rcoil ≈ 500 W. The dissipated energy is dumped into the chamber, heating
it up. This is exactly the working principle of an inductive oven. It also meansthat the RLC resonant circuit has a small Q-factor.
2. Leff < Lcoil . The magnetic field generated by the coil is shielded by the chamber
at high frequencies. The effect can be verified by comparing the magnetic
field gradient generated by a DC current and AC current. We did not observe
significant different mainly due to the asymmetry of the chamber which will bebriefly discussed in the section 5.3.3. The reduced inductance also shifts theresonant frequency of the RLC circuit for a certain capacitor. However, thiscan be corrected by using the inductance extracted from the model.
At last, it worth noting that the coupled RL circuits model can also be used to describe
the shielding effects of the vacuum chamber to the EM waves in the Radio-frequency
and the microwave regime.
Stray Field and Two-dimensional Velocity Selectivity
The geometry of the chamber determines whether the stray field generated by the eddy
currents is gradient dominating or bias dominating depending on the sensitivity of theexperiment. From the machine geometry and our observations, we inferred that the~ ec (t) = Bec sin (ωt + φ)ey along y
main stray field is an induced oscillating bias field B
with the same modulation frequency ω and a relative phase delay φ. This oscillating
bias field led to a y-component of the oscillating force. As a result, the effective recoiland velocity selection are tilted away from the z direction. The selected velocity slices
are rotated in the y − z plane.
115

In a simplified model, the total magnetic field experienced by the atoms is~ = [B0 + B00 sin (ωt)z] ez + [Bec sin (ωt + φ) + B00 sin (ωt)y] ey − 2B00 sin (ωt)xex ,B(5.13)
with a magnetic field strengthq[B0 + B00 sin (ωt)z]2 + [Bec sin (ωt + φ) + B00 sin (ωt)y]2 + [2B00 sin (ωt)x]2q
B00 z + γ sin (ωt + φ)B00 y2
2≈ B0 1 + γ sin (ωt + φ) + p
sin (ωt)
1 + γ 2 sin2 (ωt + φ)

~ =
|B|

(5.14)
here γ = Bec /B0 . The first term corresponds to a time varying homogeneous bias
field resulting in a velocity-independent effective detuning of the RF driving. The
oscillating magnetic field gradients along the z and y directions are
~
∂|B|
B00
p
sin (ωt),=∂z
1 + γ 2 sin2 (ωt + φ)
~
∂|B|
γB00
=p
sin (ωt) sin (ωt + φ).
∂y
1 + γ 2 sin2 (ωt + φ)

(5.15)It should be noted that the gradient in ey oscillates at 2ω, twice the frequency of thedriving.
The phase delay φ dramatically changes the system’s behavior. For φ = π/2, theinduced field is an oscillating gradient with zero DC bias. For φ = 0, the averaged
DC gradient along ey due to the stray field gives rise to an inhomogeneous Zeeman
broadening which can dominate over the Doppler broadening which is the signal wewant to measure. Fortunately our chamber gives a phase of φ ≈ π/2. Therefore theoscillating bias field along y causes a non-zero oscillating gradient in that direction,
leading to velocity y selectivity. The Doppler shift is modified asδω = ky vy + kz vz ,(5.16)

directly observed as a rotation of the velocity slice with an angle θ = arctan (ky /kz )116

in the time-of-flight images, as shown in Fig. 5-8. We verified two predictions of

Positive kso,x

Negative kso,x+60˚ tilt-60˚ tilt

Total Bias Field
Oscillating Byθ
B0= 6 G
Bx

(Small γ)+40˚ tilt-40˚ tilt

Oscillating By

B0=11.4 G
Bx(Large γ)

Figure 5-8: The induced bias field along ey led to a y-component of the oscillating
force, resulting in velocity selectivity in ey and therefore tilting of the resonant velocity
slice in the y − z plane. The tilt angle depends on the static bias field B0 and the RFphase φRF . The dashed lines are guides to the eye.
~0
this model : the angle θ of the rotation decreased with stronger static bias field B
which lowered γ (Fig.5-8a. and Fig.5-8 c). Due to the 2ω oscillating frequency of they force, ky did not change sign when the RF phase φRF was shifted from 0 to π incontrast to kz , and therefore the rotation angle flipped from θ to −θ, as suggested by
Eq. 5.16 and shown in Fig. 5-8.
In the future, the effects of the induced eddy current can be suppressed by using~ 0 or by conducting the experiment in a glass cell.
an even stronger static bias field B

5.3.4

Effects of Anisotropic Trap

What we have described so far applied in free space or in an isotropic trap. However, the optical trap in the experiment is anisotropic. For zero time-of-flight, in they − z plane, the minor axis of the ellipsoidal cloud is oriented along y, θ = π/2.
For long time-of-flight, the angle is solely determined by the velocity selection θ =117arctan (ky /kz ). For intermediate time-of-flight, as used in the experiment, the observed angle interpolates between these values. The cloud is observed as rotating
during the time-of-flight as shown in Fig.5-9. The bias direction is x. The thermalacy (y’)yx’

x (x’)y’

No Velocity Selectivitybθ

y (y’)
t=0αt asymptotic

t asymptotic

t

t=0

t

x

Velocity Selectivity tilted from Trapping Direction

x (x’)
Velocity Selectivity aligned with Trapping Direction

Figure 5-9: Three scenarios for two-dimensional time-of-flight.
cloud has an initial 2D Gaussian distribution

vy2
x20
y02
vx2
ρ(r0 , v, t = 0) ∝ exp − 2 − 2 − 2 − 2 .
2σx0 2σy0 2σvx 2σvy(5.17)

the density distribution after time-of-flight t isZρ(r, t) ∝dr0 dpρ(r0 , p, 0)δ 3 (r − r0 − pt/m)(5.18)

By calculating the integral we obtain (see Appendix ?? for details) the rotation angle
at time t to be
2
2
(σx0
− σy0
) sin 2θtan 2α = −2B/(A − C) = − 2 s2
22
2t (σvx0 − σvy0 ) + (σx0 − σy0 ) cos 2θ

(5.19)s
We plug in our experimental parameter θ ∼ π/4, σvx
0 ≈ σvy 0 /3, σx0 ≈ 2σy0 , ourfar field condition is σvy0 t  σx0 .
118

The observed tilt angles were used to infer the induced eddy currents. We calculatethat the observed tilt angles of 60◦ and 40◦ (Fig.5-8) correspond to tilt angles of thebias field arctan (ky /kz ) of 53◦ and 32◦ , respectively, with correction to the time-offlight rotation.
The effect can be simulated with Monte-Carlo simulations.
= -40, tof=2ms t tr = 0ms

15

5
0
-5

10

5
0
-5

-10

-10

-15
-15

-15
-15

-10

-5

0

5

10

15

= -40, tof=12ms t tr = 0ms

-10

15

0

5

10

-15
-15

15

5
0
-5
-10

= -40, tof=16ms t tr = 0ms

15

-5

0

5

10

15

5
0
-5

-15
-15

-5

0

5

10

15

= -40, tof=20ms t tr = 0ms
No-Selectivity
Initial
Velocity Selectivity
Direction of the Selectivity

10

-10

x in sqrt(2kt/m/wx 2)-10x in sqrt(2kt/m/wx 2)

y in sqrt(2kt/m/wx 2)

y in sqrt(2kt/m/wx 2)

y in sqrt(2kt/m/wx 2)

-5

10

-10

0
-5

x in sqrt(2kt/m/wx 2)

10

-15
-15

5

-10

x in sqrt(2kt/m/wx 2)

15= -40, tof=8ms t tr = 0ms

1510

y in sqrt(2kt/m/wx 2)

y in sqrt(2kt/m/wx 2)

10= -40, tof=4ms t tr = 0ms

y in sqrt(2kt/m/wx 2)

15

5
0
-5
-10

-10

-5

0

5

x in sqrt(2kt/m/wx 2)

10

15

-15
-15

-10

-5

0

5

10

15

x in sqrt(2kt/m/wx 2)

Figure 5-10: Two-dimensional TOF with perfect velocity selection. We see clear
rotation of the cloud in the near-field regime.

5.4

Micromotion Heating

As has discussed before, the magnetic spin-orbit coupling scheme does not involve
single particle heating from the spontaneous emissions. However, intuitively, when
the atoms are shaken and elastically colliding with each other, the micromotion along
the shaking direction can be randomized and the extra energy due to the micromotion
is transferred to the secular motion (thermalization). The missed energy along the
micromotion direction is immediately added back by the driving. The net effectis that the system absorbs energy from the driving field and the temperature of the119system increases. The heating rate associated can be estimated with the micromotionenergy E0 = h̄2 k02 /(4m) and the two-body elastic collision rate ρσv simply asΓ ∼ ρσvE0(5.20)

Here ρ is the atomic density, σ the elastic cross section and v the velocity of theatoms. The energy transferred ranges continuously between 0 and E0 , there is noquantization involved. This is the semi-classical description.

However, from the Floquet perspective, the energy transferred can only be amultiple of the modulation energy h̄ω [67]. The reason is the following: assuming aparticle with an initial energy E, the carrier energy, without driving. The driving
field now dresses each atom with energy sidebands of energies E + lω. Here l is an
integer. Because of the sidebands, the elastic collisions now can couple two stateswith different carrier energies different by a multiple of h̄ω. The system’s total energy
therefore changes. These is sometimes called the Floquet inelastic collisions, as shownin Fig.5-11.

Should the energy transferred be discrete or continuous? At which regime thesemi-classical picture works? The following calculations are intended to show that thetwo pictures correspond to two different limits and are consolidated with each other.
We started from a full Floquet description and show that under certain limits, theFloquet picture resembles the semi-classical picture. More specifically, we conclude
that the micromotion heating of the system can be described with a general expression
by introducing the effective collisional velocity vcol parametrizing the density of states:Γ ∼ ρσvcol E0 .

(5.21)The velocity vcol is determined by the dominating energy in the system considered
and is summarized in table.5.1.
120

Table 5.1: Effective collisional velocity vcol in different systems.
p vcolp h̄ω/m Fast modulation regime when h̄ω dominates
kb T /m Thermal motion dominating as in a thermal
ensemblep
E0 /m Micromotion dominating as in condensatesand cold atomic samples.
p
Ef /m Fermi energy dominating as in a degenerate
Fermi gas. The heating rate is further modified by Pauli blocking.

Exact Wavefunction of the Floquet mode
We consider the collision between two atoms with opposite spin in an equal spin
mixture with densities n↑ = n↓ = ρ/2. To formulate the idea of scattering between
the sidebands and estimate quantitatively the scattering strength, we first calculated analytically the expression of the sidebands by exactly solve the time-dependent~ ~r, t) = H(R,~ ~r, t)Ψ(R,~ ~r, t) where
Schrödinger equation ih̄∂t Ψ(R,~ 2 h̄2~k 2h̄2 K~
H(R, ~r, t) =++ h̄ωk0 z sin ωt.
2M
2µ(5.22)~ ~r are the coordinates for the center-of-mass motion and the relative motion
Here R,~ is the center-of-mass momentum and ~k is the relative momentum.
respectively. K
M = 2m is the total mass and µ = m/2 is the reduced mass.
The exact solution is~ 2~ ~r, t) =V −1/2 exp iK~ ·R~ − i h̄|K| t
Ψ(R,
2M

!

× V −1/2 exp (i(kz − k0 cos ωt)z + ikx x + iky y)


Z t
h̄
0 2
2
2
0
× exp −i[(kz − k0 cos ωt ) + kx + ky ]dt
2µ 0(5.23)~ t)φ~ (~r, t).
=ΦK~ (R,kHere kz is the time-averaged relative momentum in the z direction. The center121

~ t) is unaffected by the two-body collisions, and is therefore
of-mass motion ΦK~ (R,omitted in the rest of the notes. The wave function describing the relative motion
φ~k (~r, t) is rewritten as
φ~k (~r, t) =V −1/2 exp[i(kx x + ky y) + i(kz − k0 cos ωt)z]
× exp (−iE~k t/h̄) exp (−ih̄k02h̄kz k0h̄k 2t) exp (isin ωt) exp (−i 0 sin 2ωt).
4µ
µω
8µω
(5.24)

h̄2
kx2 + ky2 + kz2 .
with the secular kinetic energy E~k = 2µ
The modulation of the kinetic energy at the frequency ω implies that the wavefunction φ~k (~r, t) is a superposition of components with frequencies differing by ω.
Using the definition of the modulation index
αkz =and the expansion eiz sin ωt =φ~k (~r, t) = V−1/2h̄kz k0
,
µω

P∞

inωt
, we obtain−∞ Jn (z)e

∞
X
h̄k02
Jq (αkz ) exp(−iEk t/h̄ − i
exp{i[kx x + ky y + (kz − k0 cos ωt)z]} exp (−isin 2ωt)8µωl=−∞(5.25)
Jn is the nth Bessel function. The term with the sin 2ωtmodulation is the common
mode and does not lead to any transition between Floquet states. In analogy with
optical phase modulators, it is useful to distinguish the carrier at l = 0 and sidebands
for l 6= 0, especially in the limit of small modulation indices, where only the plus andminus first sidebands are relevant. Our experiments were done with small α ≈ 0.1,
therefore the Floquet wavefunction has the simple limit
φ~k (~r, t) ≈V −1/2 exp{i[kx x + ky y + (kz − k0 cos ωt)z]}αk
αk
× e−iEk t (1 + z e−iωt − z e−iωt )22(5.26)In the following discussion, we use this limit. Large modulation indices lead to many122

sidebands. The results under this limit will be presented numerically at the end of
the discussions.

Collision between Two Floquet States
We now consider the scattering between two Floquet states with momentum ~k and
k~0 defined in Eq.5.25 due to elastic collisions. Collisions are described by the s-wave
pseudo-potential gδ(~r) where g = 4πh̄2 a/m. Without the periodic driving, elastic
collisions couple only states with the same energy E~k = Ek~0 . However, the energy
sidebands introduced by the periodic driving, formulated in Eq. 5.25, allow the
scattering between states for which the quasi energy difference is equal to a multipleof the modulation energy h̄ω (shown in Fig.5-11). The associated energy change,quantized in h̄ω, is transferred to the secular motion, leading to heating (or cooling).
The heating rate is obtained from a generalization of Fermi’s Golden rule, called
Floquet Fermi Golden rule [67] which we quickly rederive now. Using time-dependent
perturbation theory, the transition rate between two states coupled by collisions isdetermined by the coupling matrix element and the density of states of the final state
which depends on the energy of the final state. Formally, we consider the transition
from an initial Floquet state φ~k to a final Floquet state φk~0 . The transition amplitudeis written as
Z tigM (~k → k~0 , t) = −
h̄
=g

0

dt0 ghφk~0 (0, t0 )|φ~k (0, t0 )i

X−i(Ek +nh̄ω−E ~0 )t/h̄eMn (~k → k~0 )n=0,−1,1k−1(5.27)

Ek + nh̄ω − Ek~0

The periodic driving allows the transition between the two states satisfying Ek~0 =E~k + nh̄ω, in other words, allows energy exchanges between the system and thedriving field. The transition amplitudes for n photon exchange process is
Mn (~k → k~0 ) =X

Jl+n (αkz )Jl∗ (αkz0 ) = Jn (αkz − αkz0 )

l

123

a

Ei-ћω Ei Ei+ћω

Ei

Ef-ћω Ef Ef+ћω
Ef

b

Ef+ћω

Ei+ћω

Ef+ћω

Ei+ћω

Ef

Ei

Ef

Ei

Ef-ћω

Ei-ћω

Ef-ћω

Ei-ћω

Floquet elastic process

Floquet inelastic process

Figure 5-11: a) Without modulation, each atom preserves a certain energy Ei .Elastic
collisions change the momentum of the atom, but do not change the energy Ef = Ei .
With periodic driving, each particle gains energy sidebands E = Ei + lh̄ω besides theoriginal carrier energy. b) In the Floquet elastic collision process, the carrier energiesof the initial and the final states are the same. However, elastic collisions can also
cause Floquet inelastic process, where the carrier energy changes by a multiple of h̄ω,
results in heating.

and are listed below for n = 0, ±1.
M0 (~k → k~0 ) = 1 − (αkz − αkz0 )2 /2
M±1 (~k → k~0 ) = ±(−αkz + αkz0 )/2(5.28)

Note that although the total scattering amplitude is isotropic, as expected for an
isotropic potential, the scattering is non − isotropic for each scattering channel Mn ,as shown in Fig.5-12. Scattering with photon exchange n 6= 0 occurs when the momentum projection along the micromotion direction z changes, describing the transfer
of energy from the micromotion to the secular motion.
124

n=0

n=1

Figure 5-12: Collisional cross section for processes involving different photon exchange.

We obtain the transition rate of n photon process as
2π
Γn (~k) = g 2h̄X|Mn (~k → k~0 )|2 δ(E~k + nh̄ω − Ek~0 )(5.29)

k~0

with the matrix elements given by Eq.5.28. The transition rate from the initial state~k to all the possible final states Γ(~k) = P Γn (~k) isni
2π g 2 h
2 ~
2 ~
2 ~
ρ3D (E~k )γ (k, 0) + ρ3D (E~k + h̄ω)γ (k, 1) + ρ3D (E~k − h̄ω)γ (k, −1)
Γ~k =
h̄ V
(5.30)q
where γ (k, n) =|~k|2 + 2µnω/h̄ cos θ)|2 is a dimensionless√
2µ 3/2
1
factor characterizing the coupling matrix element and ρ3D (E) = (2π)
E is
2 ( h̄2 )
2 ~

1
2

Rπ

dθ sin θ |Mn (~k →0the three-dimensional density of state of a free particle with energy E.
The three terms correspond to the three following processes 1) elastic scattering
between states with the same quasi energy 2) an energy quantum h̄ω is absorbed fromthe driving field, causing heating. 3) an energy quanta h̄ω is emitted into the driving
field.
125

Heating
We now calculate heating rates defined as energy transfer rates Γheating = Ė.
Γheating (~k) =XΓn (~k) nh̄ωni
2π g 2 h
ρ3D (E~k + h̄ω)γ 2 (~k, 1) − ρ3D (E~k − h̄ω)γ 2 (~k, −1) h̄ω=h̄ V(5.31)We explicitly calculate γ 2 (~k, ±1) and obtainσ X
Γheating (~k) = 4
V n=±1r

E~k + nh̄ωmh̄2 kz2 h̄2 |~k|2 2n
+
+
µh̄ω
3µh̄ω
3

!
nE0

(5.32)

with σ = 4πa2 being the s-wave elastic scattering cross section.
For the following discussion, we distinguish three different regimes depending on
the hierarchy of energies:
1. Fast modulation regime where h̄ω is larger than any other energy scale in thesystem.
2. Single-sideband regime where E~k > h̄ω but only the ± first sidebands are populated.
3. Semi-classical regime where h̄ω is the smallest energy scale in the system. In
this case, the system approaches the semi-classical limit.
The heating rate of the system behaves qualitatively different at the limit ofE~k  h̄ω and Ek  h̄ω where the energy of the final state is dominated by h̄ωand E~k respectively. In the limit of Ek < h̄ω, only the photon absorption process isallowed. The energy of the final state is dominated by h̄ω with a density of states√of ρ3D (E~k + nh̄ω) ≈ h̄ω. The heating rate is therefore independent of Ek in lowest
order:
4σΓheating (~k) ≈
3V

s

2h̄ω
E0
µ

(5.33)In the limit of Ek > h̄ω, as realized in our experiments, the final density of statesp
is approximated to be ρ3D (E~k + nh̄ω) ≈ E~k (1 + nh̄ω/(2E~k )). The heating of the126system now comes from the imbalance between absorption and emission of photons,
due to the higher density of states and the larger value of the scattering matrix
element for the photon absorption process. We obtain from Eq.5.32
σΓheating (~k) ≈ 8
V

s

2Ek
E0
µ

(5.34)

showing the same dependence on parameters as the semi-classical result. For evensmaller values of ω, many sidebands will be populated. Although we haven’t shown
this mathematically, one would naturally expect the heating to converge to the semiclassical limit.
We now extend the discussion from a single initial momentum state ~k to an ensemble by taking the ensemble average over the initial states ~k and the center-of-mass~
momentum K.
Γensemble
heating =Z~ d3~kd3 K~ ~k)Γheating (~k)
f (K,(2π)3 (2π)3(5.35)For a thermal ensemble, the relative momentum h̄~k and the center-of-mass momentum~ fulfill the Boltzman distributionh̄K~ ~k) =f (K,

8π 3/28π 3/2(4mkb T /h̄2 )3/2 (mkb T /h̄2 )3/2



2~

~

2~ ~

e−h̄ K·K/(4mkb T ) e−h̄ k·k/(mkb T ). The heating rate of the ensemble is therefore
2ΓThermalheating = N ρσrh̄ωE0 , for fast modulation regime
3
m
r
16
kT
ΓThermalN ρσE0 , for single-sideband regime
heating =3πm(5.36)

agrees with the numerical calculations.

For Bose-Einstein condensates, Ek = 0, and therefore Mn = Jn (αkz0 ). In this case,the micromotion energy E0 is the relevant energy scale. For fast modulation regime,127E0  h̄ω,2
ΓBEC
heating = N ρσ

r

3

h̄ω
E0
m

(5.37)For E0  h̄ω, |n| > 1 processes have to be involved.
!
r
Z
∞
X
2π
sin
θ
nE
0
ΓBEC
ρ3D (nh̄ω)nh̄ω dθ
g2|Jn 4
cos θ |2
heating =h̄ n=0
2
h̄ω
!
r
∞ Z
X
nE0
=
dθ sin θ |Jn 4
cos θ |2 (nh̄ω)3/2h̄ω
n=0(5.38)

We numerically confirmed that now the heating rate ΓBEC
heating = 3.36N ρσp
E0 /mE0

which corresponds to the semi-classical scaling.

For degenerate Fermi gases, Pauli blocking prevents the atoms to be scattered toalready occupied states which can suppress heating. Formally, the heating rate is
calculated with
Γheating
Z 3 ~ 3 ~ 3 ~0 3 ~0
d k1 d k2 d k1 d k2=f↓ (~k1 )f↑ (~k2 )(1 − f↓ (~k10 ))(1 − f↑ (~k20 ))(2π)3 (2π)3 (2π)3 (2π)3× Γheating (k~1 , k~2 → ~k10 , ~k20 )Z 3~
d K d3~k d3 k~0~ + ~k)f↑ (K/2
~ − ~k)(1 − f↓ (K/2
~ + ~k 0 ))(1 − f↑ (K/2
~ − ~k 0 ))=f↓ (K/2(2π)3 (2π)3 (2π)3× Γheating (~k → ~k 0 )(5.39)
where f (~k) is the Fermi-Dirac distribution, and we have used the substitution ~k1,2 =~ ± ~k and ~k 0 = K~ 0 /2 ± ~k 0 . For h̄ω  EF Pauli blocking is absent, and Eq.5.39
K/2
1,2
becomes similar to Eq. 5.36 with h̄ω  kb T .
rΓFermiheating ∝ N ρσ↑↓
128

h̄ω
E0
m

(5.40)

When h̄ω < Ef , heating is suppressed. In the single sideband regime with E0 
~ < (h̄ω/Ef ) kf .
h̄ω  Ef , scattering is only allowed between two atoms with |K|√
Mn ≈ ( nk0 /(2kf )). At T = 0, Eq. 5.39 suggests a heating rate ofΓFermiheating ∝



h̄ω
Ef

5

r
N ρσ↑↓

Ef
E0m(5.41)

The factor (h̄ω/Ef )5 captures the effect of Pauli blocking. Three contributions add up~ are allowed, contributes (h̄ω/Ef )3 .
to the order of 5: 1) The fact that only certain |K|~ only the scattering from the surface is allowed, giving an additional
2) With each |K|,
factor of (h̄ω/Ef ). 3) The scattering matrix element gives (h̄ω/Ef ). The process is
illustrated in Fig.5-13.

a)

b)
k’

k

k

k’

K/2

K/2
k’

k

k’

k

Figure 5-13: Illustration of the effects of Pauli blocking. Pauli blocking constrains
the available final and initial states to be near the Fermi surface. In addition, theallowed center-of-mass momentum is limited to be within (h̄ω/Ef ) kf .

Numerical Calculations
The calculations and discussions above for relevant limiting cases can be readilyextended numerically to include higher orders in Eq.5.25 and therefore processes with|n| > 1, corresponding to the scenarios with large modulation indices.
In this case, we numerically calculate the heating rates for various modulating
frequency ω and k0 which are presented in the tables below. The numbers presented
are calculated with ρ = 1012 cm−3 and a scattering length a = 53.8a0 . The heating129

rate Ė is further converted to the increase of temperature Ṫ assuming a single particle
heat capacity CV = 3kb .
The heating rate for a Bose-Einstein condensate can be readily obtained by using
the corresponding condensate density and assuming T = 0 nK. As shown, for BoseEinstein condensate, the heating rate scales with k03 when h̄ω  E0 . Therefore,
the magnetic scheme is preferable for small spin-orbit coupling strength k0 and low
density.
Table 5.2: Heating rates per particle for various spin-orbit coupling strength h̄k0 /(2m)and modulation frequency ω. The numbers presented are calculated with ρ =1012 cm−3 and scattering length a = 53.8a0 . The heating rate for a Bose-Einsteincondensate can be readily obtained by using the condensate density and assuming
T = 0 nK.
Ṫ = Ė/(3kb ) (nK · s−1 )ω (2π × kHz) k0 (kL ) E0 /h (kHz) T = 380 nK T = 0 nK
0.1
0.12
8.95
1.75
5
1
12.4
1541
1255
0.1
0.12
8.34
5.03
50
1
12.4
1445
1124
0.1
0.12
16.69
15.78
500
1
12.4
1848
1753

For the experimental parameters adopted in our experiments with k0 = 0.05 kL
and ω = 2π × 5 kHz, we numerically calculated the heating rate for a condensate
at ρ = 1014 cm−3 to be Ė ≈ 2.5 kHz/s. We have not characterized all the technicalheating and evaporative cooling in our experiment, but the observed 8 s condensate
lifetime is consistent with only weak Floquet heating.
As shown in the Fig.5-14 a), the heating rate scales with√h̄ω in the fast mod-ulation regime and plateaus at slow modulation frequencies at the value calculatedwith the single sideband approximation. The calculated heating rate holds even forlower frequencies where multiple sidebands are populated due to the contribution
from larger |n| processes.
The heating rate due to micromotion is general for any similar schemes for generating spin-orbit coupling or synthetic gauge field such as ones described in Ref.[68,13050
T = 380 nK
T = 800 nK20(2/3)ρσ ћω / m E 0

single-sideband

10
5
3

0.1

1

10
ћω/(kb T)

100

1000

Heating Rate per Particle (nK/s)(b)

Heating Rate per Particle (E0 /s)

(a)

104
Numerics

10(2/3)ρσ ћω / m E 0
3.36ρσ E0 / m E 0

2

100
10- 2

0.01

0.1
k0 /kL

1

Figure 5-14: Numerical calculation of the heating rate involving many sidebands|n| > 1 for A thermal cloud at temperatures T = 380 nK and T = 800 nK (shownin a.) and for a Bose-Einstein condensate at ω = 2π × 5 kHz (shown in b). For
the thermal
cloud, the heating rate is indepedent of ω at low frequencies while scale√with ω at high frequencies. The orange and gray dashed line represent the result
obtained with the single sideband approximation Eq.5.36. The gray guide line is theresult with singleband approximation but extended to the full range of frequency
concerned. For the condensate, the heating rate scales quadratically with the spinorbit coupling strength h̄k0 /(2m) when h̄ω  E0 and resembles the semi-classicalscaling k03 when h̄ω  E0 . The black line is the numerical result . The gray dashedline is the heating rate calculated with the single sideband approximation (Eq.5.37)
which matches the numerical result in the regime h̄ω  E0 .
69, 70].
It worth pointing out that the cross-over from the discrete to continuous energy
transfer is conceptually similar to the Doppler shift of an atom confined in a tighttrap [71]. In free space, the Doppler shift of an atom is continuous, and the resonant
frequency satisfies δ = ω0 +~k·~v /h̄. However, when the atom is held in a tight trap, theresonant frequencies are discretized to be δ = ω0 + nωtrap where ωtrap is the frequencyof the trapping. The two pictures converge when ωtrap approaches 0.

131

132

Chapter 6

Interacting Spin-orbit Coupled
Systems and Supersolidity

Experiments presented above regarding spin-orbit coupling physics are limited tosingle particle physics where interactions between particles are omitted.

When interactions are introduced, the interplay between the interaction and the
kinetic energy leads to different phases and phase transitions. One of the phases fora two-component BEC, which is specifically interesting, is the stripe phase which isa crystal and, at the same time, a superfluid, in other words, a supersolid.

In this and the following chapter.7, I show how we create a new form of matter, asupersolid, with interaction spin-orbit coupled condensates. The experiment is basedon the implementation of spin-orbit coupling in optical superlattices as described inchapter.4. This chapter will focus on the theory of interacting spin-orbit coupledsystem, with an introduction to the the physics of supersolid and supersolidity. Different phases, their properties, and the requirement for observations will be discussed.
Experimental details and results will be presented in chapter.7.
133

6.1

Supersolidity

6.1.1

Phase and Symmetry Breaking

Matters come with different phases. Liquid flows around while solid is hard. Whatmakes them so different? Through long experience, people discovered that mostphases are distinguished by their symmetry.
In a narrow sense, symmetry describes how the constituents of a matter are organized. It is usually referred to the spatial configuration as in the case for crystals.
Recently, the concept of symmetry has been extend to a much broader definition.
A system is said to preserves a certain symmetry when it is invariant under a certain operation. The operation could be a practical operation like spatial rotation,or could be an abstract operation as in the case for time-reversal operation. By invariant, we mean that the Hamiltonian describing the system is unchanged under~ ·L~ preserves time-reversal
certain operations. For example, spin-orbit interaction S
symmetry. Atomic Hamiltonians preserve inversion symmetry, until broken by theexternal electric field.
Physical properties of materials, such as mechanical, electrical, thermal properties,
in different phases are strongly associated with its symmetry breaking properties.
Materials lacking of inversion symmetry can host intrinsic electric fields, leading tospin-orbit interaction. Crystals have broken translational symmetry and are therefore
resistant to shear, while a liquid is not. Superfluid breaks U (1) symmetry, giving it
the unconventional and phenomenal property of frictionless flow.
A system hosting more than one symmetry breaking properties could demonstrate
many exotic macroscopic properties. A prominent example is the liquid crystal which
consists of thin molecules. Liquid crystals can flow like regular liquid, but with themolecules aligned to a certain direction, breaking the rotational symmetry. This
special property grants liquid crystals special properties.
Because of the strong connection between the system’s symmetry and its physicalproperties, a large variety of phases are in fact defined by the symmetry it exhibits. On
the other hand, two system sharing the same symmetry breaking properties could also
134

share many common properties despite different microscopic constituents. Therefore,
studying and creating system with novel symmetry breaking properties has been oneof the interests in fundamental research.

6.1.2

Symmetry Breaking in a Supersolid

A supersolid is a “superfluid solid” combining the symmetry breaking properties forthe crystal solid and superfluid.

Symmetry Breaking in Solid
A crystalline solid spontaneously breaks the continuous translational symmetry. Insuch solids, the system is invariant only when translated by certain spatial vectors,
known as the lattice vectors. More formally, consider the density of a crystalline
solid ρ(~r) and δρ(~r) = ρ(~r) − ρ is the deviation from the spatial average. We havefor a crystal δρ(~r) = δρ(~r + T~ ) where T~ belongs to the lattice vectors. The Fouriertransformδρ(~k) ∝Z~

d3 rρ(~r)e−ik·~r(6.1), and at its square S(~k), known as the Static Structure Factor, then displayed peaks~ satisfying
at the Reciprocal wavevectors G~ · T~ = 2nπ.
G

(6.2)

In a perfect lattice with infinite size, the peaks at the reciprocal vectors are infinitely
narrow. With fluctuations of the positions of the constituent, either thermal or quantum mechanical, the peaks of S(~k) in the reciprocal vector space are broadened and~ Depending on the specific man|S(~k)| at the peaks decay with the increase of |G|.
ner it decays, the correlation is categorized as either short, featured by faster-thanexponential decay, or long range correlation, usually polynomial. Crystals further
require a long range correlation. A system displays long-range translational symmetry is also said to posses Diagonal Long Range Order. The term “diagonal” comes135

from the fact density correlations are usually described by the diagonal terms of thedensity matrix.
Breaking continuous translational symmetry, the crystalline, structure immediately leads to the crystal’s resistance to shear and deformation.

Symmetry Breaking in a Superfluid
The fact that crystals break the translational symmetry is intuitive and pictorial.
Superfluidity is a pure quantum effect. It breaks U (1) symmetry. [72]In quantum mechanics, U (1) symmetry refers to the arbitrariness in choosing a
global phase for the wavefunction, that is to say, a quantum state |ψi is the same aseiα |ψi. For a superfluid, the system acquired a homogeneous global phase, showing
coherence across a long range.
In a three dimensional system, superfluidity is acompanied by Bose-einstein condensation [?]. A Bose-einstein condensate acquires a global phase across the phasetransition because of interaction. A common way to characterize the BEC superfluid
is done by Penrose and Onsager [73] using the first-order density matrix defined asρ(~r0 , ~r)† (~r0 )ψ̂(~r)]i. Using the eigenstates of the Hamiltonian k, the density matrix
readsρ(~r0 , ~r) =Xnk φk (~r)φ∗k (r~0 )

(6.3)

k

when BEC occurs, the ground state is macroscopically occupied even in the thermodynamic limit N0 /Ntot 6= 0. The system obtains an off-diagonal long-range orderρ0 = limr→∞ ρ(r, 0) = N0 /V . The order shows up as a sharp peaks in its momentum
components.

Symmetry Breaking in a Supersolid
A supersolid then combines the symmetry breaking properties mentioned above. It
has discrete translational symmetry as a crystal, and a broken U (1) symmetry as asuperfluid. Systems spontaneously acquire these two symmetry breaking properties
had not been experimentally found before.
1366.1.3Elementary Excitations - Goldstone Mode and the Higgs
Mode

The symmetry breaking properties of a system is also reflected in the spectrum of its
elementary excitation.
As described in the Goldstone theorem, any spontaneous continuous symmetry
breaking corresponds to a massless excitations. A pictorial understanding of theGoldstone theorem is the following. Assuming that the continuous symmetry is labeled by a parameter θ. It can be a global phase obtained by a condensate, or the
pointing of the spins in a 1D spin chain with ferromagnetic interaction. The system
is symmetric because the ground energy is independent of this parameter, showing
infinite degeneracy. |φiθ
When the system spontaneously picks a certain value of θ in the ground state
during a phase transition, this continuous symmetry is spontaneously broken. The
picked θ is a global phase assuming no domains are formed. We can easily constitutea form of excitation by considering a field of θ(x). Therefore a system displaying
spontaneous continuous symmetry breaking hosts at least one excitation mode whichis called the Goldstone Mode. The excitation costs energy, but at a long wave limit∂x θ(x) → 0, the energy cost goes to zero. Therefore, the excitation is gapless (sometimes also called soft) at ∂x θ(x) → 0.
Another possible excitation is the Higgs mode which corresponds to an excitationin the amplitude of the order parameter. In the case of a Bose-Einstein condensate,
the Goldstone mode is the excitation of the condensate phase, while the Higgs excitation is the excitation in the condensate number. It has been shown that the Goldstone
modes are usually massless, featured by a linear dispersion at k → 0 while the Higgs
mode display massive feature with a quadratic dispersion at the long wave limit.
A supersolid shows two continuous symmetry breaking therefore hosts two Goldstone modes : one related to the superfluid U (1) symmetry which is nothing but
the Bogoliubov excitation in the condensate. The other is the related to the lattice
structure which is the longitudinal phonon mode in the crystalline lattice.
137

6.2

The Stripe Phase and Supersolidity

We now discuss the connection between a spin-orbit coupled system and supersolidity.
The message is : an interacting one-dimensional spin-orbit coupled Bose-Einsteincondensate hosts a stripe phase which shares the same symmetry breaking properties
as a supersolid. It also hosts the expected excitation modes.
The competition between the interactions and the kinetic energy drives phasetransitions. Water solidifies to ice when the Van der Waals interaction beats thethermal motion of the water molecules when the temperature is lowered. At T =0, the quantum superfluid to Mott insulator transition happens when the on-site
interaction U suppresses the tunneling J.
In an interacting spin-orbit coupled system, besides the kinetic energy, the competition between the interaction between atoms with the same spin (interspin interaction) and with different spin (intraspin interaction) could cause the two spin statesto either be mixed or separated. This transition happens at T = 0 and is a purequantum effect. When the two spin states are miscible, the system develops a density
modulation from the spin-orbit coupling. This is the essence of the stripe phase.
We now formulate the process by explicitly calculating the different phases in aninteracting spin-orbit coupled system at the mean field level. Then the symmetry
breaking properties and the excitations of the stripe phase will be discussed.

6.2.1

Interacting Spin-orbit Coupled Bose-Einstein Condensates

An interacting spin-orbit coupled Bose-Einstein condensate involves the following
three energy scales• Intra- and inter-spin interactions.
• Psudospin Zeeman energy δ0 : the detuning between the two pseudospin states.
• Coupling energy : Energy shift due to the coupling between the two pseudospin
states.
138

The competition between these energies rises different phases in this system. Morespecifically• the stripe phase : two spin states are miscible and developed density modulations.
• magnetized phase : two spin states prefer minimum mixing. No density modulations.
• single minimum phase.
We now formulate this.
Weakly interacting Bose-Einstein Condensate
The Hamiltonian describing N interacting bosons in an external potential Vext iswritten as
Z
Ĥ =

†



d~rψ̂ (~r, t)Zp2+ Vext (~r) ψ̂(~r, t) + g d~rψ̂ † (~r, t)ψ̂ † (~r, t)ψ̂(~r, t)ψ̂(~r, t) (6.4)
2mwhen the interacting is weak na3  1, quantum depletion is weak. The field operatorψ̂(~r) = hψ̂(~r)i + δ ψ̂(~r) is approximated with the average value hψ̂(~r, t)i = ψ(~r, t).
Assuming spontaneous broken U (1) symmetry, this value is non-zero. This is thecalled the mean-field approximation.
The key of the mean-field approximation is the omitting of the effect of quantum
fluctuations which is valid when the quantum fluctuations are much smaller than themean-field. This usually related to the specific systems or dimensions.
With this approximation, Bose-Einstein condensates with s-wave interaction are
described by the Gross-Pitaevskii equation∂ih̄ ψ(~r, t) =
∂t



h̄∇2
2
−
+ Vext (r) + g|ψ(~r, t)| ψ(~r, t),
2m

(6.5)

which is the non-linear Schrodinger equation. |ψ(~r, t)|2 = n(~r) is the density of thecondensate at position ~r. g > 0(< 0) describes the case with repulsive (attractive)
139interaction. We limited the discussion to the repulsive case.
In the limit where the kinetic term p2 is ignored, a direct prediction from theGP equation is that the density profile will follow the external trapping potential
n0 (~r) = (µ − Vext )/g where µ is the chemical potential. Due to the interaction, thecondensate will maintain a smooth and flat density distribution. A density modulationδn(~r) = n0 (~r) + A sin kr causes the interaction to increase by A2 /2 and is therefore
unfavorable.

Inter and Intra-spin Interaction
In an interacting spin mixture, two types of interactions involved are the intra-spin
interaction, describing the two-body interaction between atoms in the same spin state,
and the inter-spin interaction, describing the interaction between atoms with opposite
spin.
The inter- and intra-spin interacting strength for the same atomic species areusually different. With the s-wave pseudo-potential in mind, both interactions can be
described with the same form of s-wave pseudopotential but with different associatedscattering length a.
The difference between these interactions could be either small or large, dependingon the specific spin states and atomic species. For 87 Rb, the interaction is very
balanced in the |F = 1i manifold with a difference between the inter and intra-spin
interaction less than 1%.

Miscibility/immiscibility Phase Transition
The imbalance of the inter- and intra-spin interaction determines whether atoms
with different spins in a spin mixture would like to stay spatially mixed or spatiallyseparated. Assuming that we have conserved N↑ and N↓ atoms mixed together in abox with a volume V . The total energy of the system is
Etot = g↑↑ n↑ N↑ + g↓↓ n↓ N↓ + g↑↓ (n↑ N↓ + n↓ N↑ )
140

(6.6)

with the full spatial overlap between the |↑i and |↓i states. When intraspin inter-

Figure 6-1: Phase mixing and Phase separation driven by the competition between
the inter- and intra-spin interaction.

action is dominating, or equivalently g↑↓ is small compared with g↑↑ , g↓↓ , there is nointeraction between the two spin states. The system minimizes its energy by fillingthe entire box with the each spin states and to reduce the density n↑ , n↓ . We therefore
obtain a full mixture of the two spin states. On the other hand when the interspin
interaction dominates, the system reduces its energy by de-mixing the two spin states
spatially such that they won’t spatially overlap and therefore avoid inter-spin interaction. The result is domain formation as illustrated in Fig.6-1. The spatial separation
effective reduces the volume each spin state could occupy.
Mathematically, when the spins are mixed and both occupy the entire space, thetotal energy is
E mix = g↑↑

N↑2
N↓2
N↑ N↓
+ g↓↓
+ 2g↑↓
V
V
V

(6.7)

we assume the number of atoms for each spin states to be N↑ ,N↓ and N↑ +N↓ = N .
We also assume the equilibrium volume in the phase separated case to be V↑ and V↓
respectively which add up to V . The total energy in the demixed phase is
E

sep

N↑2
N↓2
+ g↓↓
(V↑ , V↓ ) = g↑↑
V↑
V↓

(6.8)

by minimizing the total energy E sep (V↑ , V↓ ) with respect to V↑ , we obtained the equi141

librium volume of each component
VV↑ = pg↓ /g↑ (N↓ /N↑ ) + 1
VpV↓ = V − V↑ =1 + g↑ /g↓ (N↑ /N↓ )(6.9)

We therefore obtain the total energy at the phase separation to be
E

sep

√
2N↑ N↓ g↑↑ g↓↓ + g↑↑ N↑2 + g↓↓ N↓2(V↑ , V↓ ) =V(6.10)In the spin Therefore the final configuration is determined by the competition between
these two effects. We have√
2N↑ N↓ g↑↓ + g↑↑ N↑2 + g↓↓ N↓2
2N↑ N↓ g↑↑ g↓↓ + g↑↑ N↑2 + g↓↓ N↓2
compared withVV
(6.11)√g↑↓ compared with g↑↑ g↓↓
When g↑ >√g↑↑ g↓↓ , system prefer phase separation. When g↑ <√g↑↑ g↓↓ , systemstays mixed.

Ground States
With the discussion presented above, we now include the spin-orbit coupling into thesystem. The difference is that since the spin-orbit coupling will dress each spin statewith the other with a non-zero relative momentum, say we dress the |↑i with |↓i, themoving component, here |↓i, will interfere with the initial stationary |↓i atoms in thesystem. The intraspin interaction within |↓i now is different because of the density
modulation created, effectively change the g↓↓ . The effective g↓↓ depends on how
much propogating |↓i is mixed and therefore depends on β. The transition between
the mixed to the separated state now could also be driven by tuning β besides g↑↓ /g↓↓ .
We now formulate the idea,.
Starting from a 50/50 mixture of the |↑, k = 0i and |↓, k = 0i states with atom
number N↑ = N↓ = N/2 in a total volume V . Spin-orbit coupling with detuning δ142

gddβ2

guuβ2
- β2

- β2

gud

guu

gdd

Figure 6-2: Interaction processes involved in a spin-orbit coupled system. Solid lines
represent coupling. Dashed line represent interactions.

and Rabi frequency β couples |↓, k = k0 i and |↑, k = −k0 i into the system.
N↑
N↓
N↑ N↓
N↑ + g↓↓ N↓ + 2g↑↓
. . . (Bare interaction)
V
V
V
β2
−
(N↑ + N↓ ) · · · (Coupling Energy)
ER,k0
β 2 N↑
β 2 N↓+ g↑↑ 2N↑ + g↓↓ 2
N↓ER,k0 VER,k0 V

mix
Etot
= g↑↑

(6.12)· · · (Extra Mean field energy due to density modulation)
Equ.6.12 can be regouped into a mathematically neater form with n = N/V =(N↑ + N↓ )/V
mix
Etot
/N =

n
n
n
β2
n
n
[g↑↑ + g↓↓ + g↑↓ ] + 2 (g↑↑ + g↓↓ − ER,k0 )
4
4
2
ER,k0
4
4

(6.13)

Here ER,k0 = h̄2 k02 /2/m is the recoil energy associated with the 1D spin-orbit coupling.
It is therefore clear that the interacting strength g↑↑ , g↓↓ , g↑↓ are modified by the spin143

orbit coupling0g↑↑= g↑↑ 1 +



β
ER,k0

2 !

0
, g↓↓
= g↓↓ 1 +



2 !

β
ER,k0

0
, g↑↓
= g↑↓

(6.14)

If the two spin states are spatially separated, the total energy
N↓N↑N↑ + g↓↓N↓ . . . (Bare interaction)
V /2
V /2
β2
−
(N↑ + N↓ ) · · · (Coupling Energy)
ER,k0

sep
Etot
= g↑↑

(6.15)

If we compare Eq.6.13 and Eq.6.15, whether the system will be mixed or phase separated is determined by
sep
mix
Etot
compared with Etot

(6.16)The result is the similar to Eq.6.11, we obtain the conditiong↑↓ compared with

1−

β
ER,k0

2 !
g

(6.17)

We further simplified that g↑↑ = g↓↓ = g. Though Eq.6.17 is not a rigorous result, it
is already showing several important features
1. If the system stays mixed at β = 0, in other words g↑↓ < g, it is possible toq
g−g↑↓
ER,k0 .
drive phase separation by increasing β. The critical strength βc =g
2. The coupling energy terms in Eq.6.13 and Eq.6.15 is a common mode between
the mixed state and the separation state. Therefore, the coupling energy is notplaying an important role in driving the mix-to-demix transition.
3. The maximum amplitude of the stripe is determined by the interaction imbalance g − g↑↓With this we conclude this section which captures the fundamental physics of different
physics involved in an interaction spin-orbit coupled syste. The next section provides
a rigorous and detailed calculation on the different phases and the phase transitions
144

by adopting the mean-field Hamiltomian and minimize the energy of the system with
ansatz wave functions. In the end, the phase diagrams are obtained and presented.

6.2.2

Mean-field Ground State Wave Function

At T = 0, the ground state is obtained by minimizing the energy functional of thesystem. We follow the calculations presented in [74]. We take the full Hamiltonian
with interactions and minimize the energy functional with the assumed Ansatz of amean-field wavefunction. We consider the following Hamiltonian
Z
Ĥ =

†

d ~r Ψ̂ H0 Ψ̂ +Z

d ~r (g↑↑ n̂2↑ + g↓↓ n̂2↓ + 2g↑↓ n̂↑ n̂↓ )(6.18)

where H0 is the single particle spin-orbit coupling Hamiltonian, and Ψ̂ = (Ψ̂↑ , Ψ̂↓ ) is
the spin-1/2 spinor. The first simplification would be implementing the mean-fieldapproximation where we take Ψ̂ = hΨ̂i + δ Ψ̂ withhΨ̂i = (hΨ̂↑ i, hΨ̂↓ i)T r = (ψ↑ , ψ↓ )T r

(6.19)and keep the 0th order. We eventually reach the mean-field energy functionalZ
E(ψ↓ , ψ↑ ) =d~r [(ψ↑ , ψ↓ )H0 (ψ↑ , ψ↓ ) + (g↑↑ |ψ↑ |4 + g↓↓ |ψ↓ |4 + 2g↑↓ |ψ↑ |2 |ψ↓ |2 )] (6.20)

with(p̂x + ασz )2
HSOC =+ βσx + δ0 σz ,2m(6.21)

where α = − 21 kx , β = √JΩand δ0 = 12 (δ − ∆). Now we pick the Ansatz wavefunction.
2∆
In principle, there are in total four states involved with the spin-orbit dressing|↓, Ii = |↓, k = 0i + (β/δ) |↑, k = k0 i|↓, IIi = −(β/δ) |↓, k = 0i + |↑, k = k0 i|↑, Ii = |↑, k = 0i + (β/δ) |↓, k = −k0 i|↑, IIi = −(β/δ) |↑, k = 0i + |↓, k = −k0 i
145(6.22)

States with label |IIi have higher energypδ 2 + β 2 , therefore we adiabatically elim-

inate the states in the Hilbert space |IIi and assume that the ground state of oursystem is a superposition of only |↓, Ii and |↑, Ii. We therefore have a trial wavefunctionψ ↓ =
ψ↑

r







−ik1 ycos θ1sin θ2 eN] +C2 [C1 V
− cos θ2− sin θ1 eik1 y
|
|
{z
}
{z
}
|↓,Ii

(6.23)|↑,Ii

where θ1 and θ2 are just the mixing angles. C1 and C2 are the complex amplitudes of
the two states respectively. Now we transform to the Canonical momentum Ki whichis equivalent to a Gauge transformationk0

ei 2 y00−i 20 yek

(6.24)

we have a more symmetric form




  r
sin θ2cos θ1ψ e−iK2 y ] eiK1 y + C2  ↓  = N [C1 V− cos θ2− sin θ1ψ↑

(6.25)

We now have six parameters C1 , C2 , θ1 , θ2 , K1 , K2 with a constraint |C1 |2 + |C2 |2 = 1
due to normalization. We plug it into Eq.6.23 and minimize E(C1 , C2 , θ1 , θ2 , K1 , K2 ).

For a special spin-symmetric case with δ = 0 and g↑↑ = g↓↓ , an analytic solution
is possible. In this case, we have K1 = K2 and θ1 = θ2 = arccos (K1 /k0 )/2. We havethe energy per particle
 = E/N = −β2+ G1 + G2 (1 − 4γ)2[(α2 − 2G2 ) + 4(G1 + 2G2 )γ]withs
K1 = K2 = α

1−

β2
.
α2 − 2G2 + 4γ(G1 + 2G2 )146(6.26)(6.27)

Here we define G1 = n(g + g↑↓ )/4 and G2 = n(g − g↑↓ )/4 and an important parameterγ = G2 /G1
which characterizes the imbalance of of the inter- and intra-spin interaction.
We see immediately that this rigorous results is almost exactly the same as Eq.6.12
obtained by our physical pictures in the limit of Gi  α2 . We then analyze possible
Energy per particle vs. q

0.2

(ER)

0.1
0

-0.1
0

q

0.25

Figure 6-3: Energy per particle vs.q. system minimizes it’s energy either at q = 0 or
q = 1/4 depending on the interaction and spin-orbit coupling strength.
phases related to this Hamiltonian. First, we have the single minimum phase
where the spin-orbit coupling is so strong such that the two minima merge into asingle one, as discussed in Chap.3 and illustrated in Fig.6-4. The interaction changes
the threshold spin-orbit coupling strength β from α2 to
βt2 = α2 − 2G2 + 4γ(G1 + 2G2 )(6.28)

Now assuming β < βt . As we have discussed, in a spin mixture interaction could cause
the two spin states to spatially separate. Therefore, when we initially have doubleminimum with a population in both minima, the interaction could drive a transition
from the Stripe Phase where all the states are spatially mixed with each other tothe Magnetized Phase where spin states are spatially separated to reducing the147

mixing energy. Mathematically, the single particle energy  depends on q = |C1 C2 |.
Within certain regime,  is minimal for q = 1/4, indicating a ground state of an equal
mixture of the two spin states. Otherwise  minimizes itself with q = 0, indicating aphase separation or domain formation, as shown in Fig.6-3.
We now closely look into the two phases, as illustrated in Fig.6-4. We use threeparameters to further demonstrate the difference between the two phases : 1) the
local polarization hσz iLocal 2) atomic density ny,l = |ψl |2 and ny = |ψ↓ |2 + |ψ↑ |2 . Pluginto the most general wavefunction Eq.6.25, we havehσz i =

K1(|C1 |2 − |C2 |2 )αp
α2 − K12
cos (2K1 y + φ)hny,↓ i = |C1 |2 cos2 θ + |C2 | sin2 θ + |C1 C2 |p α
α2 − K12
cos (2K1 y + φ)hny,↑ i = |C2 |2 cos2 θ + |C1 | sin2 θ + |C1 C2 |αpα2 − K12
ny = n[1 + 2|C1 C2 |cos (2K1 y + φ)]α(6.29)In the Magnetized phase, we have either |C1 | = 0 or |C2 | = 0. Therefore themagnetized phase is featured with non-zero local magnetization and zero density
modulations. Pictorially, the condensate tends to condensate into one of the minima
to reduce the inter-spin interaction. However, with δ = 0, it is totally spontaneous
which minimum the condensate will go to. Therefore for the whole system, we have in
average zero magnetization with non-zero local magnetization in the form of domains
as illustrated in Fig.6-1.

In the stripe phase, we have |C1 | = |C2 | = 1/2 with an arbitrary relative phase
φ. The system’s ground state wave function then reads




ψ
 ↓
ψ↑

=
stripe

1
2

r









N iξ  cos θ  iK1 y  sin θ  −iK2 y+iφ
e [
e
+
e
]
V
− sin θ− cos θ(6.30)

we therefore conclude that in the stripe phase : 1) there will be no local magnetization(therefore no global magnetization). 2) there exist a density modulation n(y) along148

the direction of the one-dimensional spin-orbit coupling.
The density modulation is the most interesting feature and deserves a deeper
discussion which will be presented in the next section. We summarize the phases

Figure 6-4: Phases of an interacting Spin-orbit coupled condensate system.
below
1. When β > βc , there will be only a single minimum.
2. When β < βc , there will be two minimum. If the inter-spin interaction is toolarge, two spins separate.
3. When β < βc , if the inter-spin interaction is small, two spins are spatially mixed
and there is density modulation.
We point out again that by increasing β, one could drive the phase transition between
the magnetized phase and the stripe phase. However, this phase transition is not
driven by the competition between the coupling energy and the interaction. The
phase transition is fully driven by the competition between the intra-spin and theinter-spin interactions and β induced the transition by changing the extra mean-fieldinteraction due to the density modulation.

6.2.3

Mean-field Phase Diagram

The above results can be summarized as a phase diagram for interaction spin-orbit
systems. One axis to be the coupling strength β, the other axis to be the normalized
g−g↑↓interaction imbalance gn g+g↑↓/Eα , as shown in Fig.6-5a, b. The phase diagrams

149

assumes δ = 0. The system goes from the stripe phase to the magnetized phase
before eventually evolved into the single minimum phase with increase spin-orbit
coupling Rabi frequency. The critical Rabi frequency βc for the transition between
the stripe phase and the magnetized phase, and therefore the maximum contrast of
the stripes, depends on the imbalance. A larger interaction imbalance (larger gamma)
is preferred of high contrast stripes. [75, 76]abPhase Diagram (β vs. U)
with γ = 0.009

Phase Diagram (β vs. U)
with γ = 0.991
0.5

0.5

Magnetizedsingle minimum

Magnetized

0.25

β/ER

β/ER

single minimum

0.25

Stripe0Stripe

0
0

10

20

30

40

0

0.1

0.2

0.3

0.4

0.5

U/ER

U/ER

Figure 6-5: Mean-field phase diagram of an interacting spin-orbit coupled systemwith δ0 = 0 for balanced interaction γ = 0.009 and imbalanced interaction γ = 0.991.

From another aspect, we draw the phase diagram with β vs. δ, shown in Fig.66 . This phase diagram describes stable the stripes are against fluctuations in the
detuning. For a magnetic system, e.g. Zeeman level pseudospins, the phase diagram
describes how sensitive the system is to the ambient magnetic field fluctuation. As
can be seen, for balanced inter- and intra-spin interaction, stable stripes requires verysmall ambient magnetic field fluctuations.

6.2.4

Symmetry Breaking properties in the Stripe Phase

According to 6.29, system in the stripe phase develops a density modulation1
ny = n[1 +2pα2 − K12
cos (2K1 y + φ)]α150(6.31)

Figure 6-6: Mean-field phase diagram of an interacting spin-orbit couple system δ0
vs. β. The robustness of the stripe phase against ambient magnetic field fluctuation
depends on γ[76].

with a periodicityd= p2d
1 − (β/F )2

with F = (2Er + n(g + g↑↓ ))/4

(6.32)

the scale of the periodicity is set by the momentum transfer ky associated with thespin-orbit coupling, but further modified by the interaction of the system. To obtaina sense on the elementary excitations associated with the stripe phase but with the
spontaneously broken parameter ξ and φ.




ψ
 ↓
ψ↑

=
stripe

1
2

r









N iξ  cos θ  iK1 y  sin θ  −iK2 y+iφ
e
e
e [
+
]
V
− sin θ− cos θ(6.33)The spontaneously broken phase ξ is associated with the continuous U (1) symmetry breaking. the corresponding excitation is the regular phonon modes similar tothe case in a bare condensate. The phase φ is related to the position of the stripes.
The corresponding excitation can be obtained by consider a spatially dependent φ(x)which gives a longitudinal wave in the stripes. This is similar to the phonon excitation in a crystal lattice. both excitations are gapless. The Higgs mode in this case isrelated to the excitation imposed upon the stripes describing a spatially variation of151

the stripe contrast.
The excitations of a spin-orbit coupled bose-einstein condensate in the magnitizedphase and the single minimum phase has been observed and thoroughly measured in
[77].

6.3

The Stripe Phase: why at large?

As mentioned in the very beginning of the chapter, a material or a system’s physical properties are strongly associated with it symmetry breaking properties. Since a
supersolid hasn’t been discovered in nature, the stripe phase with the same symmetry breaking properties could reveal important information about the properties of asupersolid. However, though spin-orbit coupling in one-dimension has been demonstrated for years, the stripe phase hasn’t been directly observed.
One of the reason is the spontaneous heating issue discussed in chapter.3. To
prepare a sample in the stripe phase, the lifetime of the system should be at least
longer than the energy gap between the stripe phase and the plane wave phase.
This gap is extremely small when the interactions are balanced, putting a stringent
constrain on the lifetime of the system.
Another reason is that the stripe phase is fragile to technical magnetic field noise
when using Zeeman levels as pseudospins. Illustrated in fig. 6-6, for a system withbalance inter- and intra-spin interaction, a small amount of detuning δ0 results inspin polarization. In the rubidium system demonstrated in [31], the allowed windowfor δ ≈ 2π × 10 Hz. Therefore the ambient magnetic field fluctuation needs to becontrolled down to δ/µB ≈ 10−5 Gauss which is technically difficult.
Finally, when the interaction is balanced, the allowed mixing between the spin
down and spin up atoms is small. Since the stripes require the mixing between thetwo spin states, balanced interaction means a small contrast before phase separation,
making it difficult to experimentally detect the stripes. Formally, the critical spinorbit Rabi frequency βc between the stripe phase and the magnetized phase is small
when the interaction is balanced. since the contrast of the stripe is proportional to152the Rabi frequency β, smaller β means weaker modulation.
In conclusion, to observe stable and high contrast stripes, the key is to increase themiscibility of the two spin state. As discussed in Chapter.4 and shown in Table.4.2,
our pseudospin exhibit very imbalanced inter- and intra-spin interaction and signature
by a phase diagram with the stripe phase dominating (Fig.6-7. With this, our system
is suitable for the study of the stripe phase. In the next chapter, I present ourexperimental result on realizing and detecting the stripe phase, a new form of matter.

0.0
1.0

0.5

Spin-Orbit Coupling Strength, β (kHz)1
1.5 7.5
10Magnetized
Phase

Detuning (δ0 /Er)

0.5

0.0

12.5

15

Phase
Separation

Stripe Phase
this work

SingleMinimum
Phase-0.5
Magnetized
Phase
-1.0
0.0

0.1

0.2 1
1.5
Spin-Orbit Coupling Strength (β/Er)

2

Figure 6-7: Mean-field Phase Diagram for superlattice stripe phase.

153

154

Chapter 7
Observation of the Supersolid
Stripe Phase
In this chapter I present results on the detection of the supersolid stripe phase in aspin-orbit coupled system. The spin-orbit coupling is realized in an optical superlattice described in chapter.4. The stripe phase is featured by a spontaneously formed
density modulation along the direction of the momentum transfer associated withthe spin-orbit coupling. This density modulation has a periodicity of about 1 um
and is therefore beyond the system imaging resolution. We detected the stripes byusing Bragg diffraction. The chapter described the theory of Bragg diffraction, ourdetection optical setup, and presents the results.

7.1

Theory of Bragg Diffraction

Bragg scattering, or Bragg diffraction, has been widely used in crystallography to obtain information on the crystal structure. From the very early implementation which
led to the discovery of DNA double-helix structure [78], to modern studies on thephase transtions in the layered materials for batteries during operations [79], Bragg
diffraction served as a fundamental tool to reveal underlying periodic structures incrystals. Compared with in-situ microscopy which is focused on obtaining local information with high spatial resolution, Bragg diffraction reveals the global information155

of the material, Bragg diffraction analyzes the sample in momentum space, suitable
for the detection of possible long-range orders.

7.1.1

Bragg Diffraction

Bragg scattering uses the interference of light waves, as shown in Fig.7-15a). The
re-emitted light from each emitter, the atom, under a coherent illumination interferewith each other. When the relative phase of the light emitted from the two emittersequal to 2nπ, depending on the emitted angle α and the incident angle θ, constructive
interference is achieved. When the atoms are spatially arranged in an orderly fashion,
there exists a global Bragg condition for the sample, giving an enhanced scattering
along the direction.

a)

b)
Ѳ

α
Ѳ

α

Figure 7-1: The Bragg scattering from a periodic structure. a) from a 3D lattice. b)from a stack of layers.
For a simple cubic lattice with lattice constant d, the Bragg condition can be
expressed asd(sin α − sin θ) = nλ,(7.1)

where λ is the wavelength of the detection beam, and n is the order of the reflection.
More formally, the Bragg scattering can be calculated with156

In atomic physics, Bragg diffraction has been used in interfereometry experiments [80, 81, 82] and to probe the atomic wave function and quantum phase transitions in optical lattices described by the Bose-Hubbard model [83]. In the superfluid
phase, atoms move freely across the lattice and therefore no enhanced scattering isobserved. In the Mott insulating phase, showing diagonal long-range-order, atoms
are localized in each lattice site with a homogeneous filling across a decent volume.
The resulted periodic pattern in the wave function enhanced the scattering along thedirections satisfying the Bragg condition Eqn.7.1.

7.1.2

Quantum Description

A full picture of the diffraction process can be found by considering the light-atominteraction [84, 85, 86]. Consider a trap with the trap states |ji. The differential
light scattering cross section for incident lights with wave vector ~ki is
X
X
X
dσ
~
~
~
Ni Nf | hi| ei∆k·~r |f i |2
Ni | hi| ei∆k·~r |f i |2 + C=C|Ni hi| ei∆k·~r |ii |2 + C
dΩ
i
i6=f
i6=f
(7.2)
The second term is the incoherent term featured by the summation of intensity insteadof amplitude. The second term, the Rayleigh background is a purely single particle
effect. With all the final states summed up, this term can be simply understood as thelight scattered by a semi classical particle. Therefore the scattering pattern shows no
structures besides the dipole antenna pattern. This part of light has a broad angular
distribution, filling the detecting aperture, and is used as convenient reference forother scattering processes.
The third term describes the inelastic process where the atom’s external stateis changed due to the light scattering. The scattering is enhanced due to bosonicstatistics. When a boson is scattered to an already occupied state, the scattering
process is enhanced.
The first term corresponds to the elastic scattering related to the Bragg diffraction.
The light scattered by different atoms added up coherently. The intensity of the157scattered lights along a certain direction strongly depends on the spatial arrangementof the atoms as| hi| ei∆~k·~rZ

2|ii | = |~d~rn(~r)ei∆k·~r |2(7.3)

which is the Static structure factor discussed in Sec.6.1.2.
For a condensate in the ground state of a 3D harmonic trap, the wavefunction has
the form h~r |ii ∝ exp (−x2 /σx2 − y 2 /σy2 − z 2 /σz2 ). The scattered light therefore has anangular distribution
σdiff
2 2
2 2
2 2
2
e−∆kx σx /2 e−∆ky σy /2 e−∆kz σz /2= CNBECdΩ(7.4)

The coherent scattering is in the forward direction with ∆kx = ∆ky = ∆kz = 0 withan angular spread δθ ≈ 2π(λ/σx , λ/σy , λ/sigmaz ).
The stripe phase displays a density modulation hny i = n[1 + η cos (2K1 y + φ)]along y, the scattering matrix element has a formZd~rn(~r)ei∆~k·~rZ=i∆~k·~
r

d~rne

Z
+η

~

d~rn cos (2K1 y + φ)ei∆k·~r(7.5)

The angular distribution has the form
σdiff
2 2
2 2
2 2
2
= CNBEC[e−∆kx σx /2 e−∆ky σy /2 e−∆kz σz /2
dΩ
η2
2 2
2 2
2 2
+ e−∆ky σy /2 e−∆kz σz /2 e−(∆kx −2K1 ) σx /2 ]
4

(7.6)

Here we leave only the ∆k − 2K1 term since |∆k + 2K1 | > ki and therefore is not
physical. A bragg diffraction peak appears at ∆kx = ∆kz = 0j, ∆ky = 2K1 with asimilar angular spread as in the case for the bare BEC.

Debye-Waller Factor
The factor 41 in front of the η 2 is also called the Debye-Waller factor, which is formallydefined as
D = hexp (i∆~k · ~u)i2 .
158

(7.7)

where the average could be thermal or time average. The Debye-Waller factor characterizes the fluctuations in a periodic structure, either by thermal or other effects. A
pure sinusoidal modulation gives a Debye-Waller factor of 1/4. Thermal fluctuations
could reduce this factor, giving smaller Bragg diffraction signal (see section.7.1.3).
The homogeneous background from the second term gives a fluorescence of

a

b
0.1

Incident
5.0
0.0

I(Ѳ)

5.0

�Ѳ

0.1

Bragg
Diffraction

0.1

5.0

0.0

5.0

0.1

y

g

z

x

Figure 7-2: The Bragg scattering from a periodic structure. a) Illustration of the
Bragg scattering geometry b) Numerical calculation of the intensity distribution of
the scattered light with an homogeneous Rayleigh background. The parameter η ispicked to be 0.5 for clearer display. The strong peak is the forward scattering of theprobe light while the small side peak is the Bragg scattered light.

dσhom= CNtotdΩ(7.8)

therefore we define “gain” of due to the Bragg diffraction to beG=2η 2 NBEC
4Ntot

(7.9)

We conclude
1. The stripes in the stripe phase gives Bragg diffraction along the direction of∆k = 2K1 .
159

2. The angular width of the diffraction peak is determined by the coherent length
which is the size of the condensate if no more than one domain formed.
3. The intensity of the Bragg scattered light is determined by the number of atoms
involved in the collective scattering process.
In fact, the Bragg diffraction peak is a direct detection of the static structure
factor in solid state physics. In some literatures, the Bragg diffraction strength iseven used as the order parameter for a crystal. Therefore, the detection of the Bragg
diffraction peak is a direct signature of the existence of the Diagonal Long Range
Order in a supersolid.

7.1.3

Multi-scattering Effects - Heating

It is important to make sure that each atom would not scatter too many photons
during the Bragg pulse to minimize the effect of atom recoil.
In crystallography, since the energy of the incident photons are much smaller thanthe binding energy of the crystal atoms, the recoil heating due to the scattering is
negligible, in other words, the atoms stand still and the crystal structure is rigid
during the scattering process. Therefore, more scattered photons is usually preferred
for better signal-to-noise ratio.
On the contrary, the density modulation in the stripe phase is fragile. Thereis no external potential which pins the density modulation to its initial position.
The recoil exerted by the scattered photon causes diffusion of the atoms. When an
atom diffuses further than the period of the density modulation before it scatters thenext photon, the initial position information is lost. In other words, each scatteredphoton changes the shape of the density modulation. Photons scattered afterwards
will simply show as an overall background, diminishing the signal-to-noise ratio of
the Bragg diffraction.
Quantitatively, the diffusion of the atoms due to the recoil diminishes the Debyep
Waller factor.
∆x2 (N ) denotes the diffused distance of an atom after it scattered160

N photons. Assuming a constant photon scattering rate Γsc , we have
2
k−1
1−
N
.
(N + 1)(2N + 1)
N 2=
× (vrec )
6N
Γsc

N 2
)
∆x2 (N ) = ΣNi (vrecΓsc

(7.10)

Therefore, the best signal-to-noise ratio is obtained when each atom scatters only
one photons. We took this conclusion as the constraint for picking the probe beam
parameters.
Rτ ≈ 1(7.11)

where τ is the Bragg pulse duration and
R=

I/Is
Γ
2 1 + I/Is + 4(∆/Γ)2(7.12)

is the spontaneous scattering rate. For 23 N a, we have Γ = 2π × 9.7 MHz andIs = 9.3 mW/cm2 .

7.2

Experimental Setup

In this section I describe the experimental parameters and physical setup of the Bragg
probe and detection section.

7.2.1

Beam Parameters

We determined ∆, τ, I under the constraint Rτ ≈ 1.

Detuning of the Bragg Beam
When the Bragg beam propagates through a atomic cloud, the interaction of atoms
with light involves : the absorption of photons, re-emission of photons, and shifting161

the phase of the transmitted light. The processes are described by the refractive index


i
σ0 λ
δ
nref − 1 = n+4π 1 − δ 2 1 + δ 2(7.13)

The absorption of the Bragg beam causes an inhomogeneous illumination of thecloud. It scales with ∼ 1/δ 2 . Re-emission of the photons gives fluorescence and in
our case also the Bragg scattered light. The absorption and re-emission are capturedby the imaginary part of the refractive index.
The shift of phase distorts the wave front of the Bragg probe beam. The
extra phase shift of light caused by the atoms on the probe path previous to thescatterer changes the Bragg condition and stirs the Bragg scattered light. This effectis captured by the real part of the refractive index and therefore scales with 1/δ.
Based on the above analysis, we not only need the atomic cloud to be opticallytransparent, but also phase transparent as well. For a cloud with an optical densityof ∼ 100, this requires a detuning ∆ = δ × (Γ/2) = 100(Γ/2). We eventually chosea detuning of ∆ = 1030 MHz. Further detuning is always beneficial. However, thatwould require higher intensity of the Bragg beam which we did not have and give
more stray light.

Intensity and Beam Size
The larger the better. There are several benefits for using a large beam:
1. Larger beam makes it easier for the alignment. At the same time, the alignment
is more robust against daily drifting.
2. Due to the Gaussian beam profile, the Bragg beam acts as an near-resonant
dipole trap for the cloud. The Bragg pulse therefore changes the trapping
for the condensate cloud causing dipolar excitation of the condensate. The“impulse” from the Bragg beam is expressed as F (r) = −2V0 r/σ 2 . For a fixed
number of photon to be scattered, a large beam is preferred to minimize thecurvature at the center of the beam.
162

7.2.2

Optical Setup and Alignment

Determined by the spin-orbit coupling strength k0 = 2π/1064 nm, equation 7.1 suggests the first order Bragg angle θ1 ≈ 16◦ . To deliver the probe light (also to detect
the scattered light), a numerical aperture N A = 0.27 is required. Limited by our
superlattice and vacuum chamber geometry, only (and luckily) the top and bottom
bucket can provide barely enough optical access. It is not the easiest job to mount all

a

M3
M2

M1

Bragg Path

M1

T2

b

M3

Atmoic Sample

M2

M2

Aux Path

GimbleMount

M3
Atmoic Sample

T1

AOM

AOM
λ/2

λ/2

Figure 7-3: Delivery of the Bragg probe beam. a, The Bragg beam and the auxiliary
beam are prepared by splitting the output from an optical fiber. The output beam
is PID stabilized. A telescope (T1) is used to modify the beam size. T2 imaged thecenter spot of the gimbal mount to the atomic cloud, allowing convenient stirring of
the incident angle without changing the spatial alignment. b, mounting of the final
mirrors. The last mirror M3 is mounted by a specially machined mount on one of thepost inside the bucket in order to cover the incident angle at 16◦ .

the optics needed to deliver the probe light. All the optics for delivering the Bragg
beam are mounted on a separate breadboard.
To cover the incident angle of 16◦ , the final mirror (M3) has to catch the light inside
the bucket, otherwise the light will just clip on our vacuum chamber. We machineda special mirror mount from the standard Thorlabs cage part as shown in Fig. a.
Limited by the geometry, a slope of 7◦ is machined from a 60mm cage plate LCP01B
163

and is clamped on the cage rod already existing on the machine. We specifically use
brass set screws since the original screws are magnetic and causes misalignment of
the mirror when high magnetic fields are present during the magnetic evaporation
process.
For the mirror itself, a thin and large mirror is preferred for more tolerance inpositioning. Thorlabs offer those thin silver coated mirror. The coating doesn’t
matter that much, the shape matters. These mirrors have 36 mm × 25 mm × 2 mm
in dimension, which is the best fit we can find on the market. The mirror is epoxyedon the mount.
To acquire some tunability, we adopt a technique from previous related work [83].
The idea is to use a gimbal mirror mount combined with a telescope. The centerof the mount is imaged on the condensate through the telescope. When the probe
beam is directed at the gimbal center, it is automatically spatially aligned with thecondensate. The incident angle can be continuously changed by tweaking the gimbal
with no need to realign the beam. The angular tuning range δθ = (f1 /f2 )∆θ where
f1 , f2 are the focal lengths of the telescope length, and ∆θ is the tuning range of thegimbal mount. Our scheme provided us a range of ±1◦ . The layout of the setup isab

Figure 7-4: Mounting of the Final Bragg Mirrors M3, M2.

shown in Fig. ??.
164

7.2.3

Alignment of the Bragg BeamWe first aligned the center line of this telescope to the atomic cloud. There are threedifferent requirements for the alignment
1. The probe beam should first go through the condensate cloud.
2. The incident angle should be around the Bragg angle θ ≈ 16◦ .
3. The gimbal point should be focused on the cloud.
Step I Since the Bragg setup is completely fiberized, we are able to switch the Bragg
beam source, and therefore the frequency, freely without affecting the final alignment.
The typical diameter of the Bragg beam is around one millimeter, it was therefore
not challenging to make it overlap with the cloud. To achieve this, we used resonant|F = 1i → |F 0 = 2i light as the Bragg beam for alignment purpose only. Then we let
our dark spot MOT expand freely for 1 ms and pulsed on the Bragg beam during the
expansion. The |F = 1i → |F 0 = 2i acted as a repumper. It would pump the |F = 1i
atoms to |F = 2i state, leaving a hole on the imaged cloud in the |F = 1i state. Byaligning the burnt hole to the marked position of the crossed ODT, the Bragg beamis spatially aligned (Figure.7-5). The burnt hole has a elliptical shape although theab

Figure 7-5: Alignment of the Bragg beam. a) Rough alignment for overlapping the
Bragg beam and the condensate. b) Kapitza-Dirac pattern for aligning to the Bragg
angle with an auxiliary beam.
Bragg beam profile is perfectly Gaussian. This is simply because the beam is tilted
from the imaging direction.
165

Step II The focusing is done by locally moving the position of one of the telescope
lenses until the spatial overlap between the probe beam and the cloud doesn’t change
drastically over a wide range of gimbal angle.
Step III We aligned the probe beam to the expected incident angle with an auxiliary beam Aux as shown in Fig.??.The auxiliary beam and the probe beam are set
up to be symmetric to each other with respect to the Bragg planes. The two beams
have the same frequency and form a stationary lattice. When the angle θ = θ0 areproperly aligned to 16◦ , the recoil momentum kick from this stationary lattice shouldbe 2π/1064 nm, the reciprocal vector of the periodic stripe modulation. The recoil
is directly observable in time-of-flight images similar to the Kapitza-Dirac method
discussed in Chapter. III.
In principle, the detection of the Bragg scattered light is straightforward. In crystallography, the diffracted light is collected with a film which covers a wide range ofsolid angle. Therefore no precise alignment is necessary and the film can simultaneouscollect multiple orders, showing the diffraction pattern.
In our setup, limited by the geometry of the machine and our vacuum chamber,the largest detecting solid angle is only about 1 inch/30 cm ≈ 0.021 Sr corresponding
to ±4◦ . Therefore the imaging system should be pre-aligned to the expected directionof the scattered light. At this point, the auxiliary beam we used to align the probe
beam provides a useful guide. We simply centered the auxiliary beam at all the optics
on the detection side. This accomplished the alignment procedure.

7.2.4

Fluorescence Signal

We tested our system with the homogeneous Rayleigh scattered light from a barecondensate. When the detecting camera was in the focus of the detecting system, theposition of the Rayleigh scattered light imagined on the camera overlapped with the
expected position of the Bragg scattered light. At that point, we did three things toincrease the signal to noise ratio
1. We started with long detection pulses and resonant light to make sure the atoms
166

scattered as many photons as they can before heated up and lost from the trap.
2. We used as large optics on the path as possible to make sure that we covered
the largest solid angle and therefore collected as many photons as we can.
3. We focused the collected scattered photons down to a single pixel on the camera.
As a conclusion from Gaussian optics, the diffraction limited size of the focus
from a lens with a focal length f and diameter D is
ω0 =

2f
λ
πD

where λ is the wavelength of the light being focused. We would like the sizeof the focal spot to be smaller than the physical size of a single pixel of thecamera to maximize the signal-to-noise ratio. The size of a single pixel. for thecameras we used are 7 µm for Pixelfly and 5 µm for Andor EMCCD. Both areachievable by using a final lens with f /D < 10. At this level, achromatic lenses
were used to minimize the spherical distortion.
The Rayleigh scattered light from atoms held in the magnetic trap was easy toobserve since each atom could scatter hundreds of photons before escaping the trap,
as shown in Fig. 7-6. For a condensate held in a typical optical dipole trap witha trapping depth of several µK, each atom scatters ∼ 10 589 nm photons before
escaping. Considering our limited collecting solid angle, we expected approximately
several hundreds of photons in total to be collected for a condensate with 105 atoms.

7.2.5

Few Photon Level Detection

From the fluorescent measurement, we learned two messages:
1. The stray light should be controlled down to single photon level.
2. A camera with single photon detection ability is needed for the experiment.
I describe how we process in parallel along the two directions.
167

Figure 7-6: Collected fluorescence signal from a Condensate.

Stray Light Suppression to Single Photon Level

Several general methods are used to block the stray light down to single photon level.
First, the entire imaging setup is carefully covered by black anodized aluminum foil.
The joints are sealed with black tape. The breadboards with through holes arecovered by the aluminum foil to prevent leakage into the imaging system from theholes. Second, we use optical tubes as much as we can to connect optics. The tubes
reduce the collecting solid angle for the stray lights.
However, the input aperture can not be sealed and the light coming through the
input aperture will not be distinguished from the signal if not mode-filtering is done.
To filter out the stray light from the input, we used a relay imaging setup, insteadof direct imaging, with an iris sitting at the relay image point at shown in Fig. 77. The filtering achieved the best performance when placed at the image spot at theintermedium stage and closed to the minimum until it blocks the signal mode (shaded
grey in Fig.7-7).
For typical lens parameters D = 2” and f = 200mm, the signal mode at the
focal spot FI has a beam diameter of 3 µm. It is therefore necessary to use either azero-close iris or a fixed aperture pinhole.
168

Iris

SourceMode Filter

Downstream Optics

Figure 7-7: Mode filter for the stray light.
Detectors : PMT and EMCCD
Weak light detection has been thoroughly studied in the community of astrophysics [87].
A Photon Counting Head or a Photomultiplier Tube (PMT) is a device capable of single photon detection. Each photon hitting the photocathode in the PMT
generate an electron. The electron is accelerated by a high voltage and amplified
when hitting the dynode inside the electron multiplier. after several similar cascade
processes, the generated pulse of electrons reaches the anode and the PMT output asingle TTL pulse as a single detection event. For the PMT we used (HAMAMATSUH7421), the resolution of the PMT is 70 ns with a single pulse width of 30 ns.
The pulses are then sent into a counter which counts the number of TTL pulses
received during the gate time. A typical counter can handle the resolution of 70 ns,
however, it is difficult of obtain a short gate time (the model we use has a gate timeof 10 ms. Our Bragg pulse is ∼ µs wide therefore we would like a similar gate time.
We solved the problem by using an RF switch between the counter and the PMT.
The switch acts as a external gate for the counter and can provide fast enough gatetime.
The PMT is internally sealed and cooled to reduce possible thermal electrons
generated at the cathode.
In our experiments, we set the PMT to collect the homogeneous Rayleigh background scattered photons at an angle avoiding the Bragg scattered light. The countfrom the PMT served as a reference for the condensate number and the scatteredintensity. We collected on average ∼ 20 photons during each Bragg pulse.
169

An ’Electron-multiplying’ CCD (EMCCD) camera is a CCD camera withan attached solid state electron amplifier.
Each pixel is a separate device that converts photons into electrons with the ratio
called quantum efficiency. Depending on the wavelength of the photons, the quantum
efficiency ranges from roughly 40% to 90%. The electrons are first held within the well.
During the readout phase of the camera, electrons within each well are sent throughthe same counter and then the Analog-to-Digital Converter in an order defined by thecamera itself. An internal clock synchronizes this process to make sure the camera
correctly link the counts to the position of the pixel. For an EMCCD camera, theelectrons are amplified right before sent into the counter.
The major contributors of the noise in an EMCCD camera are the thermal noise(dark noise), clock induced charges, and the readout noise. Among them, readout
noise can be conveniently overcome by using higher EM gain or binning. Thermal
noise and Clock-induced charges, on the other hand, are amplified together with thesignal electrons, therefore setting the limiting noise floor.

The thermal noise also known as dark current noise, refers to the thermal electrons generated inside the well independent of lights. The thermal electrons createdduring the exposure time are mixed with the signal electrons and are totally indistinguishable. They can onlybe suppressed either by cooling the CCD chip down to
low temperatures or by reducing the exposure time. An Andor iXon EMCCD camera
with the CCD chip cooled to −80◦ has a dark current of 0.00015; e− /pixel/sec, giving
0.001 electrons over the typical exposure time 40 µs. Due to short exposure time, thethermal noise of EMCCD is usually negligible for our purpose.

Clock-induced charges , similar to the thermal electrons, occur when electrons
are inadvertently generated without being induced by incident photons. They are
generated by the clock voltages exerted to transfer the collected electrons from eachpixels to the amplifier and readout. The Clock-induced charge features itself withspurious hot pixels on the image. It is usually optimized by choosing the correct170

clock cycle or shift speed settings of the camera.

Readout noise is added at the final stage of the electron counting. It is a random number of electrons added independent of the number of the signal electrons.
Therefore, the readout noise can be suppressed by either amplifying the signal before
the counting process what the Eletronmultiplier does, or by using binning where theelectrons captured by several pixels are combined and counted as a whole.
Practically, the noise feature of a camera depends strongly on the settings of thecamera. It is necessary to experimentally carry out statistical measurements to findthe optimal settings for the purpose.

7.3

Warm-up : Bragg Scattering along the Superlattice

When the Bragg probe and detection setup aligned along the superlattice direction
at an incident angle of 16◦ , Bragg scattering reveal structures with periodicity of1064 nm which doubles the superlattice periodicity.
We therefore detected the pseudospin texture discussed in section 4.2.1 and eq.??.
In addition, we also detected what we called transient texture originating from theinterference between |↑, q = 0i and |↑, q = π/di states.

7.3.1

Detection of the Pseudospin Texture

BECs were adiabatically loaded into a symmetric superlattice with ∆ = 0, populating
both lattice wells equally. The superlattice phase φ is suddenly changed to lift one of
the wells up. After relaxation, system reached the metastable plateau as a equal mixture of |↑i and |↓i. The Bragg probe beam was sequentially pulsed and the diffracted
photons are detected at the Bragg diffracting angle. The interference between thepseudospin down and up states alternates between constructive and destructive foradjacent unit cells in the lattice. This generates an oscillating density modulation,
171

a

b

E

Transient Texture

E

Pseudospin Textures

0

0.2

0.4

0

q / ( /d)

0.2

0.4

q / ( /d)Figure 7-8: Pseudospin texture a and intraband texture b. The grey arrow indicatesthe direction of the incident and the Bragg diffracted beam. The pseudospin texture
is formed by the interference between |q = 0, di and |q = π/d, ui state. While thetransient texture is caused by the interference between |q = 0, ui and |q = π/d, ui
state. Both textures double the periodicity of the superlattice. The intraband texture
shows deeper density modulation than the pseudospin texture. However, the transient
texture was not metastable and vanished within ≈ 1 ms after all atoms decayed to|q = π/d, ui.

which has a spatial periodicity of 2d along the superlattice direction and depends on
the relative phases between the orbital pseudospins as φ = φ0 + ∆t, with φ0 as the
spontaneous phase. The density modulations at different phase conditions are shown
below (in the dashed box). This breaks the discrete symmetry of the lattice potential.
Figure 7-9 shows the Bragg signal for independent measurements with atomic density n ≈ 1.5 × 1014 cm3 . The Bragg signal, which is set up for detection of a spatial
periodicity of 2d along the superlattice, depends on the relative phase φ when the
Bragg pulse width is shorter than 1/(2∆). Black crosses show the Rayleigh-scatteredsignal integrated over 40 mrad before the anti-ferromagnetic spin texture develops.
Grey filled circles show Bragg-enhanced scattering. The Bragg enhancement fluctu172

Figure 7-9: Bragg Reflection from the density modulation caused by Pseudospin
Texture.

ated between zero and a factor of two, which indicates variations of the spontaneous
phase φ0 between independent measurements.
The maximum contrast of the modulation is J/∆ ≈ 1/20 for our experimental
parameters. With a total number of N ∼ 105 and a condensate fraction of ∼ 50%,
eq.7.12 gives a gain of 15. However, since the Bragg diffracted beam is directional,
the collecting solid angle for the homogeneous Rayleigh background is larger, andtherefore is smaller.
We observed a gain of ∼ 2 in the experiments, as shown in Fig.7-9.
173

7.3.2Detection of the Band Relaxation Dynamics

We also detected the density modulation caused by the intraband interference, illustrated in Fig.7-10b. As described in 4.2.1, the preparation of the pseudospin |↑i state
required a relaxation from |↑, q = 0i to |↑, q = π/di. During the relaxation, a mixtureof the two states creates the transient texture which has the same periodicity as thepseudospin texture but with much larger depth of density modulation 7-10. This
modulation was detected by the Bragg setup with the same configuration for detecting the pseudospin texture. Therefore, the dynamics of the intraband relaxation can

Figure 7-10: Dynamics of the Intraband Interference Texture. The intraband texture
peaked when the system has equally population in |↑, q = 0i and |↑, q = π/di.

be partially extracted from the intensity of the Bragg scattered light. The relaxation
time could be depending on the transverse confinement and the tunneling J of theband [88, ?].
The details of this relaxation process is unclear. In general, collision processes aredescribed by the interaction termV̂ = gX †â−q â†q â0 â0 .
q

174(7.14)

From the Fermi’s golden rule, the transition rate from state |φq=0 i to |φq i is
Γ(i → f ) =2π| hφq=0 | V̂ |φq i |2 ρ(E)h̄

Though the coupling matrix element | hφq=0 | V̂ |φq i | is independent of the quasimomentum q of the final state, the density of the accessible final states, for a given
total energy, maximizes at the energy extremes of the dispersion which are at theband minima with q = π/d. Therefore, the atoms will be more likely to accumulate
at q = π/d, giving the enhanced Bragg signal.

7.4

Detection of the Supersolid Stripe Phase with
Bragg scattering

In this section I present the main results of using Bragg scattering to detect the stripe
phase.
As previous discussed at the end of Chapter.??, our spin-orbit coupling schemefeatures imbalanced inter- and intra spin interaction. With the parameter (J/∆) ∼1/10, we expect g↑↓ ≈ 0.01g↑↑ . The imbalanced interaction is complementary to thecases realized in 87 Rb. The calculated phase diagram [?], shown in Fig.7-12a exhibitsa large region for the stripe phase.
In addition, our pseudospin is encoded into orbital degree of freedom instead of
Zeeman levels which are magnetically sensitive. This makes our system immune tothe ambient magnetic field fluctuations.
The above two reasons makes our system preferable for studying the stripe phase.
In order to detect the stripes, the Bragg probe beam and the detection setup
were simply rotated by 90 degree to be aligned to the transverse direction y of thesuperlattice.
Following the detection of the Bragg diffraction along the superlattice direction,
we first looked for the enhancement of the intensity of scattered light along the expected Bragg diffraction direction. The EMCCD camera was setup in the focus of the175

imaging system to obtain a minimized imaging spot and therefore improved signalto-noise ratio. The detection parameters we used for the probe beam were 1030 MHz
detuning, τ = 5 µs, and I ≈ 500 µW with a beam width of ≈ 300 µm.
After preparing an equal mixture of the |↑i and ↓ state, we adiabatically ramp
up the intensity of the Raman coupling β within 20 ms. The final coupling strength
β was chosen empirically to balance between acceptable heating and large enoughdensity modulation amplitude q. Since our system is highly miscible, driving a phasetransition to the magnetized phase due to too strong coupling β was not a concern.
We eventually picked the β between 0 and 300 Hz as stronger driving caused excessive
heating within the experimental time scale and destroyed the signal, as shown in Fig.714. When the detuning of the Raman beams is scanned, the spin-orbit coupling is

Figure 7-11: Transient behavior of the enhanced scattering when the Raman coupling
were suddenly switched on. The oscillating frequency is close to the the couplingstrength around 300 Hz.

tuned on and off-resonance. When the detuning is tuned on resonant with the spin-flip
process δ = ∆±Er , one spin state is resonantly flipped to the other with a momentum
transfer with a Rabi frequency of β. This caused a transient Bragg diffraction signal
oscillating at an oscillating frequency roughly equal to β if the intensity of the Raman
beam were not ramped up adiabatically. When the duration of the Bragg pulse isshorter than 1/β ∼ 3ms, this oscillation is resolvable. The oscillation decayed after
several ms.
With an adiabatic ramp, We obtained Bragg signal as the function of detuning.
176

0.5

Spin-Orbit Coupling Strength, β (kHz)1
1.5 7.5
10Magnetized
Phase

Detuning (δ0 /Er)

0.5

0.0

12.5

Phase
Separation

Stripe Phase
this work

SingleMinimum
Phase-0.5
Magnetized
Phase
-1.0
0.0

0.1

0.2 1
1.5
Spin-Orbit Coupling Strength (β/Er)

Detuning (δ0/Er)

15-1
Bragg + Rayleigh Signal (arb. units)

0.0
1.0

a Phase diagram of spin-orbit coupled BEC

1

20

10

0

2

0

-10

-5

0
Detuning, δ0 (kHz)5

10b Detuning dependence

Figure 7-12: a. Mean-field phase diagram of spin-orbit coupled BECs as a functionof detuning and spin-orbit-coupling strength with a density n ≈ 1.5 × 1014 cm−3 .
b.Detuning dependence. Black filled circles show the total light scattering signal(Bragg and Rayleigh) as a function of the frequency detuning. The light was detected
within a solid angle of 10 mrad. The grey dashed line and shaded area show the meanand standard deviation of the Rayleigh scattered light for the same conditions. . Error
bars represent 1σ standard error of the mean with a sample size of 3 to 6.

The result is shown in Fig.7-12b. The light-grey solid line is a triple Gaussian fit to
the total light scattering where the widths and center positions of the two outer peaks
are constrained to be identical to the spin-flip resonances studied in chapter.4. Whenthe detuning is on resonant with the spin-orbit coupling δ = ∆, the formed stripes
are stationary though the lattice from the Raman beams is moving. The intensity ofthe scattered light plateaued after the ramp-up of the spin-orbit coupling and slowly
decay after ≈ 30 ms accompanied by the visible heating up of the spin-mixture.
The parameter space explored is crosshatched in grey. We explore a regime ofδ0 varying within ±10 kHz. However, detection of the stripes is possible even forlarge detunings. This is because population relaxation between the two spin states
is very slow. For our parameters, the equal population of the two pseudospin stats isconstant during the lifetime of the system for all detunings δ0 studied.
A special feature about Fig. 7-12b is the triple-peak feature. When the detuningof the Raman beams δ was tuned to be on resonant with the spin-flipping processesδ = ∆ ± ER , the amplitude of the density modulation was achieved and therefore the177enhancement of the Bragg diffraction, giving the two peaks at side. We haven’t fully
understood why the Bragg diffracted light was stronger at the frequency δ = ∆. A
possible explanation could be the following : when the Raman detuning is on resonantwith either spin-flip process, the stripes presented mainly in one of the wells in the
double-well potential. When δ = ∆, stripes presented in both the wells and werealigned relatively. Therefore the Bragg diffracted lighted were more directed along
the superlattice direction, giving stronger enhancement. Nevertheless, we did nothave further information on explaining the central peak.

a)M3
SourcePinholeMode Filter

EMCCD

b)
Light intensity

1

0

10mradBragg + RayleighRayleigh

Figure 7-13: The Bragg scattering from a periodic structure. a) from a 3D lattice.
b) from a stack of layers.
A more quantitative characterization is done by first separating the homogeneousRayleigh background from the Bragg diffracted light. This was done by moving thecamera out of the focus of the imaging system by a little bit. The Rayleigh background178filled the entire detecting solid angle while the Bragg diffracted light filled only partof it. Therefore when the camera is moved out of focus, the Rayleigh background andthe Bragg enhanced diffraction signal are spatially separated.
We fixed the detuning to be δ0 = 0 and observed several quantitative features of
the Bragg scattered light. Ideally, the angular separation would be perfectly implemented in the Fourier plane. However, due to the limited signal intensity, we were
not able to obtain high quality Fourier plane images. For a pure condensate, the

b
0
600

1

NBEC2
2

3

0

4×109

Bragg Signal (arb. units)

Bragg Signal (CCD counts / pixel2)

a

400

200

0

100

β (Hz)
200

300

400

40

20

0
0

0.025

0.05

β/Er

c Spin-orbit coupling dependenceb BEC number dependence

Gain / Ntotal

2×10-4

10-4

0

0.2

0.3

0.4

(NBEC/Ntotal)2

Figure 7-14: The Bragg diffracted signal versus a. Square of the condensate number.
b. spin-orbit coupling strength. c. Gain versus the condensate fraction.
contrast of the density modulation is predicted to be η = 2β/Er which is about 8%
for β ≈ 300 Hz. Following the discussion in sec.7.1.2, a sinusoidal density modulation of ηNBEC atoms gives rise to a Bragg signal equivalent to γ(ηNBEC )2 /4 whereγ is the independently measured Rayleigh scattering signal per atom per solid an179

gle. As shown in 7-14, we observed the expected behavior of the Bragg signal to be
2
proportional to NBEC
with the appropriate pre-factors.

Two observations confirmed the long-range coherence of the system. First, theprediction for the signal assumes a long-range order throughout the whole cloud, inorder words, single domain. If the system developed M domains either perpendicularor parallel to the strips, the enhanced signal will be simply M times smaller. Anothersign to tell the coherence length is the sharpness of the Bragg peak.
We also quantify the Bragg signal by examining the ‘gain’ of the Bragg diffraction
light compared with the Rayleigh background within the same solid angle. The ‘gain’is calculated to be Ntotal (f β/Er )2 where f = (NBEC /Ntotal ) is the condensate fraction.
Shown in Fig.7-14c, the normalized gain is proportional to the condensate fraction
squared and intercepted at zero. This shows that the observed gain comes only fromthe superfluid component of the atomic sample.
We were not able to detect the correction to the stripe periodicity due to interactions. With our parameters, the correction is only about 0.4% [?], which means thatat least 250 stripes are required to resolve this shifts from Bragg diffraction angle.
This implies that a condensate with coherence length of hundreds microns is needed,which is beyond our machine’s ability. Another way to improve angular resolution isby spreading out the Bragg scattering signal to more pixels, say Npix on the camera.
Then it is possible to determined the center of the Bragg peak better than its angularpwidth by a factor of 1/ Npix . However, this inevitably reduces the number of photon
collected by each pixel and therefore the signal-to-noise ratio.
With all the results presented above, we conclude the observation of the stripe
phase.

7.4.1

On-site modulation and the Stripes

It is fair to raise the question that what the difference is between the stripes associatedwith the stripe phase and the density modulation that are externally imprinted bypotentials? More specifically, in our case, when Raman beams present, they directly
imprinted a moving density modulation on to our system, along the direction of the180

moving lattice created by the interference of the two Raman beams. The moving
stripes mathematically corresponds to the on-site coupling terms discussed in the
Sec.4. Therefore, to claim the detection of the stripe phase, we need to be able todistinguish between the density modulations associated with the spin-orbit coupling
and the on-site coupling which presented simultaneously.
In fact, it turned out that the moving stripes and the stationary stripes satisfy
different Bragg conditions.This moving stripes has the form of
e−i[(k1 −k2 )·~r+(ω2 −ω1 )t] ,(7.15)

corresponding to moving stripes along an angle of 45◦ relative to the lattice. Therefore, the first order Bragg diffraction peak will be in the plane formed by the Bragg
incident light and a vector normal to the on-site stripes, which will not lie in thesolid angle of our detection setup. The enhancement from the on-site modulation istherefore angularly filtered out by our detection setup.

7.5

Superfluidity

Above I have presented the observation of the crystalline structure, the diagonal longrange order, of the stripe phase. The other spontaneously broken U (1) symmetry
associated with a superfluid is observed simply as sharp momentum peaks in thetime-of-flight images with a sample prepared in the stripe phase. It is also revealedin the measurement of the Bragg diffraction intensity vs. the condensate fraction.
We showed that the enhanced scattering is indeed due to the superfluid part in thesample.
The two results with the observation of the associated density modulation andthe superfluid fraction conclude the observation of the supersolid stripe phase.
181

J

J

Superlattice

Superlattice

d

2d

v

kRamanStationary stripes
from Raman induced
spin-orbit coupling

Moving density modulationfrom Raman potential

Figure 7-15: The moving stripes corresponding to the on-site modulation forms inthe direction of the Raman moving lattice. It, therefore, does not fulfill the same
Bragg condition as the stripes related to the stripe phase.

7.6

Conclusions

In conclusion, we have observed the long-predicted supersolid stripe phase of spin–orbitcoupled BECs. This is a system which is a superfluid and a crystal simultaneously.
Two broken symmetries breaking properties, continuous translational symmetry andcontinuous U (1) symmetry, are combined together. We studied and characterized oursystem with Bragg diffraction which proves to be a versatile tool to detect long-range
properties of a system.
In the future, it will be interesting to characterize this system’s collective excitations. For example, the two Goldstone modes and the Higgs mode related to the182

two spontaneous continuous symmetry breaking [?]. Especially, by using stimulated
Bragg process, excitations with tunable momentum and energy can be imposed on
the system. By resolving the phase and the energy of the Bragg diffracted beam,for example through homodyne detection, it is also possible to reveal the dynamic
properties of the stripe phase and characterize the dynamic response of the system.

183

184

Chapter 8
Conclusion and Outlook

The theme of this thesis has been to use ultracold gas systems to simulate known
phenomena, and create new form of matter.
The major tool we have used is the optical superlattice. In an optical superlattice
we have implemented, each lattice site is consist of two mutually coupled wells. The
two wells in each unit cell provide new pseudospin states and are used to realize
one-dimensional spin-orbit coupling.
We then realized one-dimensional spin-orbit coupling in Bose-Einstein condensateswith two different approaches : 1) With the newly defined pseudospins in the opticalsuperlattices, we used two-photon process to flip the pseudospin and at the same time
provided the momentum transfer required by spin-orbit coupling. 2) We synchronizeda periodically modulated magnetic field gradient with radio-frequency photons. The
field gradient provides the momentum transfer while the RF photons flip the spins.
Finally, we created a new form a matter, a supersolid, in spin-orbit coupled BoseEinstein condensates. The two broken continuous symmetry were measured by directly observing the momentum components in the condensate and with Bragg scattering methods.
185

8.1

Quantum Simulation with Degenerate Fermi
Gas

The three experiments presented in this thesis were carried out with 23 Na BoseEinstein Condensates. Below the critical temperature, bosons condensate on a single
quantum state, allowing the study certain quantum states of a Hamiltonian macroscopically.
However, many interesting phenomena in condensed matter physics requires Fermi
statistics. Examples include the formation of cooper pairs in free space with weaklyinteracting Fermi gases [89, 90] and the Fermi-Hubbard model which is believe tocapture the essence of high temperature superconductivity [12].
From a fundamental perspective, Fermi-Dirac distribution and the resulting Fermi
sea structure in momentum space have been directly demonstrated in ultracold gases
in both harmonic potential [91] and the box potential [92]. Changing of the collisional
properties of a degenerated Fermi gas due to Pauli blocking has also been explored [93,94]. In fact, Pauli blocking can also modify the atom-light interaction. When an atom
is first excited to the excited state and sequentially spontaneously emits a photon,
the atom’s internal states remains the same, but could obtain a momentum recoil
from the emitted photon due to momentum conservation. When the recoil energy
from this recoil is much smaller than the Fermi energy, the highest kinetic energyof a Fermion in a filled Fermi sea, the final state of the atom after the scattering
event has already been populated by another identical Fermion. For this reason,
Pauli blocking will hinder the scattering event. Therefore spontaneous scattering
process is suppressed. It would be interesting to observe this suppression and further
see how interactions between atoms could change the behavior. To see measurable
suppression, the requirement is that the Fermi energy is much larger than the recoil
energy, which would mean a cold and dense sample. The could be achieved by using
strong optical dipole trap for trapping, possibly with near-resonant dipole trap.
The BECII machine is a dual species machine which has the capability of preparing
both degenerated Fermi gas of 6 Li and bosonic 23 Na. The machine is featured with a186

dual species oven [95], shared Zeeman slower for both species, and a single chamber
for both Magneto-optical trapping and science (for details about the configuratiionof the machine, please see Ref.[96]). The fermionic 6 Li is sympathetically cooled by23

Na [97]. By the time of this thesis, we achieved T /Tf ∼ 0.5 with ∼ 104 Lithium

atoms. The magnetic coils are capable of providing ∼ 1000 Gauss bias magnetic field,
allowing us to explore the broad Feshbach resonance of lithium at ∼ 834 Gauss. In
future, by adding optical lattice to the system, the BECII machine can be used to
study degenerated fermi gas in both free space and in optical lattice.

187

188

Appendix A
Theory of Spin-orbit Coupling in
Optical superlattices
Now we show a detailed calculation of the theory of spin-orbit coupling in optical superlattices. The Hamiltonian for a one-dimensional superlattice, created by standing
waves of infrared and green light with relative phase φSL is
Hlattice =

p̂2z
p̂2
+ ⊥ + VGr sin2 (kGr z) + VIR sin2 (kIR z + φSL )
2m 2m

(A.1)

It can be rewritten as
p̂2⊥
+
2m
1 X
∆0
(|rn i hrn | − |ln i hln |)−
2
n
X
XX
J
(|ln i hrn | + h.c.) −(Jtt0 0 |tn i ht0n+1 | + h.c.),

Hlattice =

n

n

(A.2)

t=l,r
t0 =l,rwhere |ln (rn )i is a wavefunction localized in the left(right) well of the nth unit cell,∆0 is the energy separation between the right and the left wells. h̄ is taken to be 1.
Tunneling between neighboring unit cells is important for maintaining of coherencein the superlattice, but not relevant for the physics of spin-orbit coupling. Thus,
tunneling terms with Jtt0 0 in the Hamiltonian can be neglected.
189

The complete Hamiltonian of the system is
H = Hlattice + VRaman ,(A.3)

VRaman = Ω cos(kz z + kx x − δt)

(A.4)

where

is a moving lattice potential. For the purpose of our experiment, the initial phase of
the Raman potential is not relevant and taken to be zero. We prefer to use eigenstatesof a double well for the description. To first order, the small parameter ∆J0  1 they
can be written as
J
|rn i
∆0
J
|ln i
|↑n i = |rn i −∆0|↓n i = |ln i +

(A.5)

2

J
We expand the Hamiltonian in the new basis with ∆ = ∆0 + 2 ∆≈ ∆0 :0

1 X
H= ∆(|↑n i h↑n | − |↓n i h↓n |)
2
n
X X

X
0
0
0
|in i hin | hp⊥ | Ω cos(kz z + kx x − δ · t) |p⊥ i |in i hin | hp0⊥ |
|p⊥ i
+
0

p⊥ ,p⊥ni=↓,↑i0 =↓,↑

p̂2⊥
+
2m
(A.6)
In our experiment kz ≈ kx ≈ kIR = πd , where d is a period of the superlattice. In
order to estimate the effect of the Raman potential with arbitrary phase we need toknow the overlap integrals for cos(kIR · (z − zn )) and sin(kIR · (z − zn )). To first order

190

J
in ∆
:

h↓n | cos(kIR · (z − zn )) |↑n i ≈ hln | −JJ|ln i = − ,∆∆h↓n | cos(kIR · (z − zn )) |↓n i ≈ 1, h↑n | cos(kIR · (z − zn )) |↑n i ≈ 0,JJh↓n | sin(kIR · (z − zn )) |↑n i ≈ hrn | |rn i = ,
∆
∆

(A.7)

h↓n | sin(kIR · (z − zn )) |↓n i ≈ 0, h↑n | sin(kIR · (z − zn )) |↑n i ≈ 1,
where zn = nd is a coordinate of the left well in the nth unit cell. Thus, the Raman
potential can be expanded in the basis of double-well eigenstates:X|in i hin | Ω cos(kz (z − zn ) + kz zn + kx x − δ · t) |i0n i hi0n |i=a,bi0 =a,bJ|↓n i h↑n | + |↑n i h↓n | + |↓n i h↓n |}∆J− Ω sin φn {|↑n i h↑n | + (|↓n i h↑n | + |↑n i h↓n |)},∆=Ω cos φn {−(A.8)

where φn = πn + kx x − δt. Later, we will calculate how x as an operator acts on the
momentum states |p⊥ i.
V̂Raman =XΩ(−1)n cos(kx x − δt) |↓n i h↓n | − Ω(−1)n sin(kx x − δt) |↑n i h↑n | + (A.9)

n

√ J
π
− 2Ω (−1)n cos(kx x − δt − ){|↓n i h↑n | + |↑n i h↓n |}∆4(A.10)

The factor (−1)n represents the phase of the Raman beams, which have a wavelength
two times the length of the unit cell. In our experiment the atomic sample is preparedin the zero-momentum state. When the Raman perturbation is applied the atoms
experience a kick in the x-direction. In the y-direction atoms remain unperturbed,
i.e. p̂y = 0. Since the confinement along x is weak, we can use the basis |↑ (↓), ki =|↑ (↓)i ⊗ eikx . The Raman interaction gives rise to intra-band coupling terms (1), andto the spin-orbit coupling term (2). The system in our experiment initially prepared in191the lower wells, which corresponds to the q = 0 of the lowest band of the superlattice:(↓)|ψq=0 i =NX
1√ |↓n iN
n=1

N is the number of unit cells in the lattice. When all the atoms are confined in theupper wells, the lowest state is the q = πd state of the first excited band, due to theinverted dispersion relation:(↑)|ψq=π/d i =

N
X
π
d
1
√ ei d (zn + 2 ) |↑n iN
n=1

With zn = nd, the state becomes:(↑)|ψq=π/d i =NX1√ i(−1)n |↑n iN
n=1

For the calculation of matrix elements we assume that overlap is nonzero only forn = n0 . Intra-band coupling terms:
D

E X 1
(↓)
(↓)
ψq=π/d V̂Raman ψq=0 =(−1)n h↓n0 | Ω cos(kz z + kx x − δt) |↓n i
N
n,n0
X 1
=Ω
cos(φn )(−1)n = Ω cos(kx x − δt) (A.11)
N
n

D

E X 1
(↑)
(↑)
ψq=0 V̂Raman ψq=π/d =h↑n0 | Ω cos(kz z + kx x − δt)i(−1)n |↑n i
N
n,n0
X 1
=Ω
i(−1)n (− sin(φn )) = −iΩ sin(kx x − δt) (A.12)
N
n

192

Spin-orbit coupling matrix element:D=(↑)ψq=π/dX 1(↓)
V̂Raman ψq=0E0(−i)(−1)n h↑n0 | Ω cos(kz z + kx x − δt)| |↓n i

NX 1(−1)n h↑n | cos(kz (z − zn )) cos(φn ) − sin(kz (z − zn )) sin(φn ) |↓n i ,= − iΩ
Nnn,n0(A.13)
DE(↑)(↓)
ψq=π/d V̂Raman ψq=0
=i


J X
1
Ω
(−1)n
cos(φn ) + sin(φn )
∆ n
N

=i

J
Ω(cos(kx x − δt) + sin(kx x − δt))∆

(A.14)

Intra-band coupling matrix elements (3) and (4) provide recoil kick in x−direction
2kx
with recoil energy Er = 2m
and along the superlattice, changing the quasimomentum(↓)

by half a reciprocal vector. If the system is initially at |ψq=0 , 0i, the new adiabaticallyconnected eigenstate in first order perturbation theory is:
1 Ω
1 Ω
(↓)
(↓)
e−iδt |ψq=π/d , kx i −eiδt |ψq=π/d , −kx i +2 Er − δ
2 Er + δππ
e−i 4 JΩ/∆ei 4 JΩ/∆
(↑)
(↑)
−iδt
−i √
e
|ψq=π/d , kx i − i √eiδt |ψq=π/d , −kx i (A.15)
E
+
∆
−
δ
E
+
∆
+
δ
2 r
2 r
(↓)

|Ψ1 i = |ψq=0 , 0i −(↑)

If the system is prepared in |ψq=π/d , 0i:
1 Ω
1 Ω
(↑)
(↑)
e−i(δ+∆)t |ψq=0 , kx i+ei(δ−∆)t |ψq=0 , −kx i +2 Er − δ
2 Er + δ
−i π4
i π4
e
ΩJ/∆
e
ΩJ/∆
(↓)
(↓)
+i √
e−i(δ+∆)t |ψq=0 , kx i + i √ei(δ−∆)t |ψq=0 , −kx i (A.16)
2 Er − ∆ − δ
2 Er − ∆ + δ(↑)|Ψ2 i = e−i∆t |ψq=π/d , 0i−

For δ close to ∆ intra-band coupling is off-resonant and both co- and counter-rotating
terms contribute at comparable strengths, whereas for spin-orbit coupling the corotating term is resonant and, therefore, much stronger than the counter-rotating
term.
193

Keeping only the near-resonant spin-orbit coupling term, the Hamiltonian describing the system is




p̂2x
− ∆2
2m

HSOC =  −iπ/4
ie √

2

J
Ωei(kx x−δt)∆iπ/4 J−ie√Ωe−i(kx x−δt)
2 ∆

p̂2x
∆
+
2m
2

π

After a unitary transformation with a position-dependent rotation, Û = e(−ikx x+iδt−i 4 )σz /2 ,0.
the Hamiltonian turns into HSOC = U † HSOC U − iU † ∂U∂t
0

HSOC =(p̂x + ασz )2+ βσx + δ0 σz ,2m

(A.17)

where α = − 21 kx , β = √JΩand δ0 = 12 (δ − ∆). For the total Hamiltonian we have to2∆
add the intra-band coupling, which causes a density modulation.

194

Appendix B
Spin-orbit Coupling and Spin
Textures in Optical Superlattices
This appendix contains a reprint of Ref. [60]: Junru Li, Wujie Huang, Boris Shteynas,
Sean Burchesky, Furkan Çağrı Top, Jeongwon Lee, Alan O. Jamison, and Wolfgang
Ketterle, Spin-orbit Coupling and spin Textures in Optical Superlattices, Phys. Rev.
Lett. 117, 185301 (2016).

195

PRL 117, 185301 (2016)

week ending
28 OCTOBER 2016

PHYSICAL REVIEW LETTERS

Spin-Orbit Coupling and Spin Textures in Optical Superlattices
Junru Li, Wujie Huang, Boris Shteynas, Sean Burchesky, Furkan Çağrı Top, Edward Su,
Jeongwon Lee, Alan O. Jamison, 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 10 June 2016; revised manuscript received 17 August 2016; published 27 October 2016)
We propose and demonstrate a new approach for realizing spin-orbit coupling with ultracold atoms.
We use orbital levels in a double-well potential as pseudospin states. Two-photon Raman transitions
between left and right wells induce spin-orbit coupling. This scheme does not require near resonant light,features adjustable interactions by shaping the double-well potential, and does not depend on special
properties of the atoms. A pseudospinor Bose-Einstein condensate spontaneously acquires an antiferromagnetic pseudospin texture, which breaks the lattice symmetry similar to a supersolid.
DOI: 10.1103/PhysRevLett.117.185301

Spin-orbit coupling is the mechanism for many intriguing phenomena, including Z2 topological insulators,
the spin quantum Hall effect [1,2], Majorana fermions [3],and spintronics devices [4]. Realizing controllable spinorbit coupling with ultracold atoms should make it feasible
to explore fundamental aspects of topology in physics andapplications in quantum computing [5].
Spin-orbit coupling requires the atom’s motion to be
dependent on its spin state. Spin-orbit coupling without
spin flips is possible for schemes that are diagonal in thespin component σ z . Such spin-dependent vector potentials,
which are sufficient for realizing quantum spin Hall physics
and topological insulators, can be engineered using fardetuned laser beams to completely suppress spontaneousemission [6,7].
However, spin flips (i.e., spin-orbit coupling terms
involving σ x or σ y operators) are necessary for Rashba[8] and Dresselhaus [9] spin-orbit coupling [10].
Experiments with ultracold atoms couple pseudospin statesusing optical dipole transitions, which couple only to theorbital angular momentum of the atom. Most realizations,
including the first demonstration [11], use hyperfine statesof an alkali atom as pseudospins. In this case, the couplingof the two states occurs due to internal spin-orbit couplingin the excited state of the atom, which causes the finestructure splitting between the D1 and D2 lines. The
optimum detuning of the lasers is comparable to thissplitting, leading to heating. Special atomic species withorbital angular momentum in the ground state can avoid
this problem, as recently realized with dysprosium [12].
Here, we present a new method that can be applied to any
atomic species, using an external orbital degree of freedom
as the pseudospin to avoid the need for near-resonant light.
An external degree of freedom as the pseudospin could
be realized for a two-dimensional system by using the
ground and first excited states of the confinement along
the third dimension as pseudospin states. However, the0031-9007=16=117(18)=185301(5)excited state would rapidly relax due to elastic collisions,typically on a millisecond time scale [13]. This is also thecase for the recent implementation of spin-orbit coupling(SOC) with hybrid s-p Floquet bands in a one-dimensional
optical lattice [14]. To solve this issue, we choose an
asymmetric double-well potential (Fig. 1). Pseudospins up
and down are realized as the two lowest eigenstates of the
double-well potential. For J=Δ ≪ 1, they can be expressedby the tight-binding states jli and jri localized in the left
and right wells, respectively: j↓i ¼ jli þ ðJ=ΔÞjri and

FIG. 1. Realization of orbital pseudospins in a superlattice.
(a) The unit cell of the superlattice is a double well with offset Δ
and tunneling J. The two lowest eigenstates (pseudospin up anddown) are coupled via a two-photon Raman process. (b) Ramanprocess in the band structure of the superlattice. The ground statewith quasimomentum q ¼ 0 is coupled to the edge of theBrillouin zone q ¼ ðπ=dÞ of the first excited band. (c) Top viewof the superlattice with period d ¼ λIR =2 ¼ 532 nm. Raman
coupling is implemented by two λIR beams: one along thesuperlattice (z direction), the other along the x direction. SOC(curved arrows) transfers transverse recoil in the x direction to theatoms (dashed arrows).

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PRL 117, 185301 (2016)

PHYSICAL REVIEW LETTERSj↑i ¼ jri − ðJ=ΔÞjli. The tunneling J and offset Δ
between the two wells are used to adjust the overlap—
and therefore the interactions and the collisional relaxation
rate—between the two pseudospin states. We couple thetwo states via a two-photon Raman transition with large
detunings to achieve SOC with spin flips. (For convenience, we will refer to pseudospin as spin in this Letter.)
Recent work on two-leg ladders can be mapped to SOC
between the two legs of the ladder [15,16]. Our scheme isqualitatively different from other realizations of orbital
pseudospin since it realizes spin-orbit coupling in freespace as compared to lattice models.
An intriguing prediction for spin-orbit coupled BoseEinstein condensates (BECs) is the existence of a stripe
phase [17–19], a spontaneous density modulation thatrealizes a supersolid [20]. However, when the interspin(g↑↓ ) and intraspin (g↑↑ , g↓↓ ) interaction strengths are thesame, the increased interaction energy of the density
modulation drives spatial phase separation, eliminating
the stripes. The system can be kept in the miscible phasewhen interspin interactions are weaker than intraspin
interactions, g2↑↓ < g↑↑ g↓↓ [19]. In our realization, g↑↓ isproportional to the overlap squared of the wave functions
on the two sides of the double well. An analogous scheme
can be realized with hyperfine pseudospins and spindependent lattices [21], but requires near-resonant light.
Our scheme does not depend on specific atomic properties
and addresses three challenges to realizing the stripe phase:(1) spin-orbit coupling without near resonant light, (2) amiscible system with adjustable interspin interactions, (3) along lifetime against collisional relaxation.
Instead of one double-well system, we create a lattice of
double wells using an optical superlattice [Fig. 1(c)]. The
advantages of working with a stack of coherently coupled
double wells are twofold: the increased signal to noise ratio
and the use of interference between the double wells toseparately observe the two spin states. In the present work,
the degree of freedom along the superlattice direction ispurely an aid to observation [22].
Our main result is the observation of the momentum
structure of a BEC modified by a superlattice and spin-orbit
coupling. We first describe the effects of the superlatticewithout adding SOC. A one-dimensional superlattice of
double wells was realized by combining lattices of λIR ¼
1064 nm light and λGr ¼ 532 nm light obtained by frequency doubling the λIR ¼ 1064 nm light. The shape of the
double-well unit cell is determined by the relative strengthand spatial phase ϕSL between the two lattices. The phase iscontrolled by a rotatable dispersive glass plate and an
acousto-optical modulator for rapidly switching the IR
lattice frequency.
The experiment starts with a BEC of ∼3 × 105 23 Na
atoms in the jF ¼ 1; mF ¼ −1i state in a crossed optical
dipole trap. The superlattice is adiabatically ramped up
within 250 ms. For an offset Δ ≫ J, all the atoms

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FIG. 2. Spontaneous formation of an antiferromagnetic spin
texture. (a),(b) Time-of-flight (TOF) pattern of atoms in the
ground (first excited) band of the superlattice. After preparationof the j↑i state with quasimomentum q ¼ 0, it relaxes to thebottom of the band at q ¼ π=d. (c) An equal mixture of spinstates is prepared by rapidly switching the superlattice parameters. The two spin states can be separated in the TOF by apseudospin Stern-Gerlach effect [23]. The figure shows that bothspin states are in q ¼ 0 before the relaxation. (d) After relaxation,spinor BECs with states j↓i, q ¼ 0 and j↑i, q ¼ π=d areobserved. The momentum pattern implies a periodic structure
at 2d, twice the lattice constant, indicating that an antiferromagnetic spin structure with a doubled unit cell has formed. The plus
and minus signs indicate (one possible choice for) the phase of
the BEC wave function. n is the site index.

equilibrate at the band minimum q ¼ 0 of the lowest
superlattice band, putting 100% of the population in thej↓i state. The relative population of the two spin states canbe controlled by first adjusting Δ for the loading stage toachieve a desired state population and then rapidly lifting
one well up to the final offset [23]. The upper wellcorresponds to the first excited band, which has its
minimum energy at quasimomentum q ¼ π=d with d ¼
λIR =2 [Figs. 2(a) and 2(b)]. Since the lowest energy j↑i andj↓i states have different quasimomenta and experience
different transverse confinement, they can be separately
observed in ballistic expansion images without the bandmapping techniques [23].
The π=d quasimomentum difference also leads to aninteresting spin texture for an equal population of the j↑i
and j↓i states. For this, atoms are prepared in both bands
with q ¼ 0 [Fig. 2(c)], corresponding to a wave function
periodicity of 532 nm, i.e., the lattice constant. However,after relaxation, the periodicity has doubled to 1064 nm, as
indicated by the doubled number of momentum components in ballistic expansion images [Fig. 2(d)]. Specifically,the
P system was prepared in the symmetric staten ðj↓n i þ j↑n iÞ, where n denotes the lattice site, which
is a ferromagnetic spinPstate in the x-y plane. Afterrelaxation into the state n ½j↓n i þ ð−1Þn eiθ e−iΔt j↑n i anantiferromagnetic spin texture has developed, which
reduces the translational symmetry of the lattice. This

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TABLE I.

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PHYSICAL REVIEW LETTERS

PRL 117, 185301 (2016)

The amplitudes of the wave functions in Eqs. (1) and (2) obtained from first order perturbation theory (i ¼ 1, 2).

States

Mi

M 0i

jΨ1 i
jΨ2 i

− 12 ðΩ=Er − δÞþ 12 ðΩ=Er þ δÞ

− 12 ðΩ=Er þ δÞ
− 12 ðΩ=Er − δÞsystem breaks both U(1) symmetry (the phase of the BEC)
and the translational symmetry of the superlattice. In
addition to the spin-density wave, it also has a density
wave with the same period due to the interference of the j↑i
and j↓i satellites. The position of the spin and density
modulations is determined by the spontaneous phase θ andoscillates at frequency Δ [23]. It is a simple systemfulfilling one definition of supersolidity [25–27].
The small satellites allow spin-orbit coupling, but also
lead to the collisional decay of the j↑ > state. We observed
lifetimes on the order of 200 ms for both the j↑i and equally
mixed states at a density of n ≈ 2.5 × 1014 cm−3 . The
similar lifetimes for both states and the sensitivity to dailyalignment indicate the lifetime being limited by technicalnoise and the misalignment of the lattice rather than bycollisions. Collisions would lead to a shorter lifetime for themixed state by a factor of 4ðJ=ΔÞ2 . Adding Raman beams(with the parameters presented in Fig. 4) increases the loss
rate by ∼10=s, probably caused by technical issues. Whileprevious work with 87 Rb reports a lifetime of seconds [11],the Raman hyperfine spin flip scheme is not promising forlighter atoms because of the substantially higher heating
rates compared with 87 Rb, which are 103 (105 ) times higher
for 23 Na (6 Li) [28]. Even without major improvements, thelifetimes achieved in our work are longer than any relevant
dynamic time scale and should be sufficient for further
studies, including the observation of the stripe phase [29].
Coupling between the two spin states is provided by two
λIR beams: one along the superlattice direction z, the otherorthogonal to it (along x). The frequency difference of thetwo beams is close to the offset in the double well, allowing
near-resonant population transfer. The recoil kz along thelattice is necessary to couple the two orthogonal spin statesin the double well, and was chosen to be kz ¼ π=d. The
recoil kick kx in the transverse plane provides the coupling
between the free-space motion in the transverse plane andthe spin. It has opposite signs for the transition j↓i to j↑iand the reverse transition.
The Raman coupling can be described as a moving
potential V Raman ¼ Ω cosðkx x þ kz z − δtÞ, characterized bya two-photon Rabi frequency Ω, a detuning of Raman
beams δ, and a wave vector ðkx ; 0; kz Þ. We characterize thestates by their spin, quasimomentum q, and x momentum
kx (the y momentum is always zero).
If the system is initially prepared in the statej↓; q ¼ 0; kx ¼ 0i, the adiabatically ramped Raman beams
will transfer it to a new eigenstate:

K
pﬃﬃﬃ
−i½e
=pﬃﬃ2ﬃ ðJ=ΔÞðΩ=Er þ Δ − δÞþi½eiðπ=4Þ = 2ðJ=ΔÞðΩ=Er − Δ þ δÞ−iðπ=4ÞjΨ1 i ¼ j↓; 0; 0i þ K 1 e−iδt j↑; π=d; kx i þ M1 e−iδt j↓; π=d; kx i
þ M1 0 eiδt j↓; −π=d; −kx i:

ð1Þ

If prepared in j↑; π=d; 0i, the new state will bejΨ2 i ¼ e−iΔt j↑; π=d; 0i þ K 2 eiðδ−ΔÞt j↓; 0; −kx iþ M2 eiðδ−ΔÞt j↑; 0; −kx iþ M2 0 e−iðδþΔÞt j↑; 0; kx i:

ð2Þ

The amplitudes obtained from first order perturbation
theory appear in Table I. The spin-orbit coupling is
described by the second term in Eqs. (1) and (2). In
addition, the Raman beams act as a comoving lattice and (inthe limit δ ≫ Er ) create a moving density modulation in thetwo spin states, described by the third and fourth terms.
The spin-orbit coupling shows a resonant behavior forδ ≈ Δ—the range of interest for SOC—where the moving
density modulation is nonresonant. Both contributions areproportional to Ω=Δ. The off-resonant counterrotating spin
flip term is proportional to Δ−2 and has been neglected. For
δ ≫ Er , all off-resonant amplitudes M i , M 0i become ≈Ω=δ.
For δ ¼ Δ and both spin states populated, the spin-orbit
admixture of jΨ1 i is expected to form a stationary interference pattern with jΨ2 i along x with wave vector kx, andvice versa, which constitutes the stripe phase of spin-orbit
coupled BECs in the perturbative limit. (In general, theperiodicity of the stripes depends on β and the atoms’interactions [19].)
The resonant Raman coupling leads to the standard spinorbit Hamiltonian [23]:ĤSOC ¼ðp̂ þ ασ̂ z Þ2þ βσ̂ x þ δ0 σ̂ z ;2m

ð3Þwhich can be considered as equal contributions of Rashba
and Dresselhausinteractions. The parameters α ¼ −kx =2,
pﬃﬃﬃ
β ¼ ð1= 2ÞΩJ=Δ, and δ0 ¼ ðδ − ΔÞ=2 are independently
tunable in our experiment.
To characterize all the components of the wave functions
above, the Raman coupling was adiabatically switched on
by ramping up the intensity of the two Raman beams. The
momentum space wave function was observed by suddenly
switching off the lattice and trapping beams and measuring
the resulting density distribution with absorption imaging
after 10 ms of ballistic expansion (Fig. 3).

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PHYSICAL REVIEW LETTERS

FIG. 3. Characterization of spinor BECs through their momentum distributions. (a),(e) TOF images of the j↓i and j↑i states,
respectively. (b),(f) Schematics of the momentum peaks for j↓i
and j↑i with Raman coupling. Both the SOC (solid arrows) andthe density modulation (dashed arrows) are shown. The main
peak (filled circle) is equal to the quasimomentum of the state.
Extra peaks (open circles) appear due to the periodic potential.
(c),(d),(g),(h) Same as in (a) and (e), but now with Raman
coupling at different detunings δ. The momentum components
created by the Raman process are vertically shifted compared to(a) and (e) due to the transverse momentum kick. The momentum
shift along the superlattice (z direction) reflects the π=d quasimomentum of the Raman lattice. The off-resonant density
modulation creates momentum peaks that are symmetric alongþx and −x [(c) and (g)], whereas resonant spin-orbit couplingcreates unidirectional momentum transfer resulting in asymmetry[(d) and (h)]. (i) Spin-orbit coupled BEC with equal population in
the spin up and spin down states.

The momentum components created by the Raman
beams are displaced in the x direction by the recoil shiftℏkIR . For off-resonant Raman beams, the pattern is symmetric for the þx and −x directions—signifying themoving density modulation [see Eqs. (1) and (2)]. The
resonant spin-orbit coupling is one sided, with opposite
transfer of x momentum for the two spin states—asobserved in Fig. 3. We separate the momentum peaks
due to the moving density modulation from SOC byevaluating the difference between the momentum peaks
along the þx and −x directions. Figure 4 shows theresonance feature of the SOC when the Raman detuning
was varied. The resonances for the two processes j↓i → j↑i
and j↑i → j↓i should be separated by 2Er ≈ 15.3 kHz. The
observed discrepancy is consistent with mean field interactions, which reduce the separation by ∼2μ ≈ 5 kHz,
where μ is the single site chemical potential. The observed
widths of the resonances are probably dominated by theinhomogeneity of Δ due to the Gaussian beam profile of the
IR lattice laser [23].

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FIG. 4. Spin-orbit coupling resonances. Shown is the population imbalance between the “þx” and “−x” momentum peaks
versus Raman detuning for the j↓i → j↑i (blue) and j↑i → j↓i(red) processes. The two sets of data were measured for the same
superlattice parameters V IR ¼ 7.5ð2ÞEr , V Gr ¼ 20ð2ÞEr , andϕSL ≈ 0.22ð1Þπ, which gives Δ ≈ 37ð1Þ kHz. The spin-orbit
coupling strength β was calculated to be 0.40(5) kHz. The solidlines are Gaussian fits to the resonances centered at 32.2(3) and43.2(3) kHz. The Gaussian profile of the IR lattice inhomogeneously broadens the resonances. The error bars represent 1σstatistical uncertainty. Inset: resonance center frequencies versus
the IR lattice depth V IR for fixed ϕSL . The resonances are linear in
V IR with a constant split equal to twice the recoil energy.
The slope of the linear fit reveals ϕSL . The error bars are theuncertainties of the fit.

Having established spin-orbit coupling at the singleparticle level, the next step is to explore the phase diagramof spin-orbit coupled Bose-Einstein condensates withinteractions [17,19,21], particularly the stripe phase. The
clear signature of the stripe phase is the stationary, periodic
density modulation on the BEC mentioned above. The
periodicity is tunable through the spin-orbit couplingstrength and can be directly observed via Bragg scattering[30]. In contrast to experiments carried out with 87 Rb,
which has similar inter- and intraspin scattering lengths,
our system has an adjustable interspin interactiong↓↑ ≈ ðJ=ΔÞ2 g↓↓ ¼ ðJ=ΔÞ2 g↑↑ . Small values of g↓↑ =g↑↑lead to a large window of β for observing the stripe phaseand enable higher contrast stripes [19]. Figure 3(i) shows
the momentum distribution of an equal spin mixture with
SOC. We observed an ∼40 ms lifetime for the parameterspresented in Fig. 4. After adding Bragg detection, theobservation of the stripe phase is in reach.
In conclusion, we proposed and demonstrated a newscheme for realizing spin-orbit coupling using superlattices. An asymmetric double-well potential provides attractive features for pseudospins, including long lifetimes,
adjustable interactions, and easy detection. This scheme
can be applied to a wide range of atoms including lithium
and potassium, which suffer from strong heating whenhyperfine pseudospins are coupled. On the other hand, by

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PHYSICAL REVIEW LETTERS

combining multiple hyperfine states with the orbital
degree of the double well, our scheme can realize twodimensional Rashba spin-orbit coupling [31] and suggestions made for alkaline-earth atoms, for example, synthetic
non-Abelian gauge potentials [32,33], and Kondo latticemodels [34–36].
We acknowledge helpful discussions with S. Stringari,
E. Mueller, and N. Cooper. We thank A. Keshet forcontributions to building the experiment and J. AmatoGrill, C. J. Kennedy, and P. N. Jepsen for suggestions anda critical reading of the manuscript. We acknowledge
support from the NSF through the Center for Ultracold
Atoms and by Grant No. 1506369, from Army Research
Office –Multidisciplinary University Research Initiative Non-equilibrium Many-body Dynamics (Grant
No. W911NF-14-1-0003), and from Air Force Officeof Scientific Research – Multidisciplinary University
Research Initiative Quantum Phases of Matter (Grant
No. FA9550-14-1-0035).
J. L. and W. H. contributed equally to this work.

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Appendix C
A stripe phase with supersolid
properties in spin-orbit-coupled
Bose-Einstein condensates
This appendix contains a reprint of Ref. [25]: Jun-Ru Li, Jeongwon Lee, Wujie Huang,
Sean Burchesky, Boris Shteynas, Furkan Çağrı Top, Alan O. Jamison, and Wolfgang
Ketterle, A stripe phase with supersolid properties in spin-orbit-coupled Bose-Einsteincondensates, Nature. 543, 91-94 (2016).

201

Letterdoi:10.1038/nature21431

A stripe phase with supersolid properties in
spin–orbit-coupled Bose–Einstein condensates

Jun-Ru Li1*, Jeongwon Lee1*, Wujie Huang1, Sean Burchesky1, Boris Shteynas1, Furkan Çağrı Top1, Alan O. Jamison1 &Wolfgang Ketterle1

Supersolidity combines superfluid flow with long-range spatial
periodicity of solids1, two properties that are often mutually
exclusive. The original discussion of quantum crystals2 andsupersolidity focused on solid 4He and triggered extensive
experimental efforts3,4 that, instead of supersolidity, revealed exotic
phenomena including quantum plasticity and mass supertransport4.
The concept of supersolidity was then generalized from quantum
crystals to other superfluid systems that break continuoustranslational symmetry. Bose–Einstein condensates with spin–orbit
coupling are predicted to possess a stripe phase5–7 with supersolid
properties8,9. Despite several recent studies of the miscibilityof the spin components of such a condensate10–12, the presenceof stripes has not been detected. Here we observe the predicted
density modulation of this stripe phase using Bragg reflection(which provides evidence for spontaneous long-range order in one
direction) while maintaining a sharp momentum distribution (thehallmark of superfluid Bose–Einstein condensates). Our work thus
establishes a system with continuous symmetry-breaking properties,
associated collective excitations and superfluid behaviour.
Supersolids are defined as systems that spontaneously break two­continuous U(1) symmetries: the global phase of the superfluid
breaks the internal gauge symmetry, and a density modulation
breaks the translational symmetry of space. Starting from ­superfluid
Bose–Einstein condensates (BECs), several forms of s­ upersolidity
have been predicted to occur when the condensates feature d­ ipolar­i nteractions 13, Rydberg interactions 14, superradiant Rayleigh­scattering15, ­nearest-neighbour interaction in lattices16 or spin–orbit
interactions5–7. Work simultaneous with ours used light scattering into
two cavities to realize a BEC with supersolid properties17. For fermions, the predicted Fulde–Ferrell–Larkin–Ovchinnikov states havesupersolid properties18,19. Several of these proposals lead to solidity
along a single spatial direction maintaining gaseous or liquid-like
properties along the other directions. These systems are different
from quantum crystals, but share the symmetry-breaking properties.
Spin–orbit coupling occurs in solid-state materials when an electron
moving at velocity v through an electric field E experiences a Zeeman
energy term −​μBσ·(v ×​  E) owing to the relativistic transformation ofelectromagnetic fields. Here σ is the spin vector and μB is the Bohr magneton. The Zeeman term can be written as αijvjσi/4, where the strengthof the coupling α has the units of momentum. The vxσz term, together
with the transverse magnetic Zeeman term βσx, leads to the Hamiltonian
H =​  ((Px +​  ασz)2 +​  Py2 +​  Pz2)/2m +​  βσx, where m is the atomic mass.
A unitary transformation can shift the momenta by ασz, resulting inH=

β e 2iαx 
P 2  0+  −2iαx
0 
2m  β e

(1)

The second term represents a spin-flip process with a momentum
transfer of 2α, which is therefore equivalent to a form of spin–orbit

coupling. Such a spin-flip process can be directly implemented for
ultracold atoms using a two-photon Raman transition between thetwo spin states10,20.
Without spin–orbit coupling, a BEC populating two spin states showsno spatial interference, owing to the orthogonality of the states. With
spin–orbit coupling, each spin component has now two momentum
components (0 and either +​2α or −​2α, where the sign depends on theinitial spin state), which form a stationary spatial interference patternwith a wavevector of 2α (Fig. 1a). Such spatial periodicity of the atomic
density can be directly probed with Bragg scattering21, as shown in
Fig. 1b. The position of the stripes is determined by the relative phaseof the two condensates. This spontaneous phase breaks continuoustranslational symmetry. The two broken U(1) symmetries are reflected
in two long-wavelength collective excitations (the Goldstone modes),one for density (or charge), the other one for spin transport9. Adding
a longitudinal Zeeman term δ0σz to equation (1) leads to a rich phase
diagram6,22 as a function of δ0 and β. For ­sufficiently large δ 0 , theground state is in a plane-wave phase. This phase has a roton gap9,11,
which decreases when δ 0 is reduced, causing a roton instability andleading to a phase transition into the stripe phase.
Most experimental studies of spin–orbit coupling with ultracold
atoms used two hyperfine ground states coupled by a two-photon
Raman spin-flip process10–12,23–26. So far direct evidence of the ­spatial
modulation pattern has been missing, possibly suppressed by straymagnetic fields detuning the Raman transitions and low miscibility
between the hyperfine states used (see Methods). Both limitations
were recently addressed by a new spin–orbit coupling scheme in whichorbital states (the lowest two eigenstates in an asymmetric double-wellpotential) are used as the pseudospins27. Since the eigenstates mainly
populate different wells, their interaction strength g↑↓ is small and canbe adjusted by adjusting their spatial overlap, improving the miscibility(see Methods). Furthermore, since both pseudospin states have thesame hyperfine state, there is no sensitivity to magnetic fields. The
scheme is realized with a coherently coupled array of double wells using
an optical superlattice, a periodic structure with two lattice sites perunit cell with intersite tunnelling J (Fig. 2a). The superlattice has two
low-lying bands, split by the energy difference Δ between the double
wells, each hosting a BEC in the respective band minima. The BECs
in the lower and upper band minima are the pseudospin states in oursystem. Spin–orbit coupling and the supersolid stripes are created forthe free-space motion in the two-dimensional plane orthogonal to
the superlattice. The physics in a single two-dimensional plane is not
modified in a stack of coherently coupled double wells. However, thisincreases the signal-to-noise ratio and suppresses the background to
the Bragg signal (see below).
Experiments started with approximately 1 ×​  105 23Na atoms forming
a BEC loaded into the optical superlattice along the z direction,equally split between the two pseudospin states with a density
n ≈​  1.5 ×​  1014 cm−3. The superlattice consists of laser beams at

1

Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA.
*These authors contributed equally to this work.
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RESEARCH Letterak = kIR⎥ ↓〉Spin–orbit
coupling

k=0

k=0

⎥ ↑〉

k = –kIR

2π/kIR

Bragg

b

Light intensity

1

0

10 mradBragg + RayleighRayleigh

Figure 1 | Origin of supersolid stripes and detection via Bragg
scattering. a, Supersolid stripes from spin–orbit coupling. Spin–orbit
coupling adds momentum components +ħkIR or −ħkIR of the opposite
spin state to the spin-up and spin-down BECs (at the top are spin statesin momentum space). Matter wave interference leads to a spatial density
modulation of period 2π​/kIR (at the bottom are spin states in real space).
The spatial periodicity can be directly probed by Bragg scattering.
b, Angle-resolved Bragg signal. The supersolid stripe phase is detected byangle-resolved light scattering. A sharp specular feature in the left panel isthe Bragg signal due to the periodic density modulation. The diffuse signal
is Rayleigh scattering filling the round aperture of the imaging system.
Without spin–orbit coupling, only Rayleigh scattering is observed (rightpanel). The figure is the average over seven shots.

­ avelengths of 1,064 nm and 532 nm, resulting in a lattice constant ofw
d =​ 532 nm. Spin–orbit coupling was induced by two infrared (IR)Raman laser beams λIR =​ 1,064 nm along the x and z axes, providing amomentum transfer ħkRaman =​  ħ(kIR, 0, kIR) and spin flip from one wellto the other with two-photon Rabi frequency Ω. Here ħkIR =​  2π​ħ/λIR is
the recoil momentum from a single infrared photon (see ref. 27 andMethods). The scheme realizes the spin–orbit Hamiltonian in equation (1)with α =​  k IR /2, β =​  (1/ 2 )JΩ /∆ , and an extra Zeeman term
δ0σz =​  (δ −​  Δ)/2σz, depending on the Raman-beam detuning δ and thesuperlattice offset Δ. The parameters J, Ω and Δ are determined fromcalibration experiments27. A separate laser beam was added in the x–yplane to enable detection of the stripes, which form perpendicularly tothe superlattice with a periodicity of approximately 2d =​ 1,064 nm.
Their detection requires near-resonant yellow light (Bragg probe lightwavelength λBragg =​ 589 nm) at an incident angle θ =​ 16°, fulfilling the
Bragg condition λBragg =​  4dsinθ.

Figure 1b shows the angular distribution of the Rayleigh-scattered
light induced by the 589-nm laser at δ0 =​ 0 in the Bragg direction(see Methods). The spin–orbit coupling leads to supersolid stripes andcauses a specular reflection of the Bragg beam, observed as a sharp
feature in the angular distribution of the Rayleigh-scattered light(Fig. 1b). The angular width (full-width at half-maximum, FWHM)
of the observed peak of 9 ±​ 1 mrad is consistent with the diffraction
limit of λBragg/D, where D is the FWHM size of the cloud, demonstrating phase coherence of the stripes throughout the whole cloud.
This observation of the Bragg-reflected beam is our main result, andconstitutes a direct observation of the stripe phase with long-rangeorder. For the same parameters, we observe sharp momentum peaks intime of flight27—the signature of BECs—which implies superfluidity.
Our detection of the stripe phase is almost background-free, since allother density modulations have different directions, as depicted inFig. 2a. The superlattice is orthogonal to the stripes, along the ẑ axis. The
Raman beams form a moving lattice and create a propagating ­density
modulation at an angle of 45° to the superlattice, parallel to xˆ + zˆ. The
pseudospin state in the upper band of the superlattice forms at the minimum of the band at a quasimomentum of q =​  π​/d. The wavevector of
the stripes is the sum of this quasimomentum and the momentum transfer that accompanies the spin-flip of the spin–orbit coupling interaction27,
resulting in a stripe wavevector in the x direction. Since the differencein the wavevectors between the off-­resonant density modulation andthe stripes is not a reciprocal lattice vector, the Bragg condition cannot
be simultaneously fulfilled for both density modulations. Thisbackground-free Bragg detection of the stripes uniquely depends on therealization of a coherent array of planar spin–orbit-coupled systems.
For a pure condensate, the contrast of the density modulation ispredicted5,6 to be η =​  2β/Er, which is about 8% for β ≈​ 300 Hz. Here
Er =​ 7.6 kHz is the 23Na recoil energy for a single 1,064-nm photon.
A sinusoidal density modulation of ηNBEC (where NBEC is the numberof atoms in the BEC) atoms gives rise to a Bragg signal equivalentto γ(ηNBEC)2/4, where γ is the independently measured Rayleigh­scattering signal per atom per solid angle, and the factor ¼ is the
Debye–Waller factor for a sinusoidal modulation. In Fig. 2b, weobserved the expected behaviour of the Bragg signal to be ­proportional
to NBEC2 with the appropriate pre-factors. The prediction for the signal
assumes that the stripes are long-range-ordered throughout the wholecloud. If there were m domains, the signal would be m times smaller.
Therefore, the observed strength of the Bragg signals ­confirms thelong-range coherence already implied by the sharpness of the a­ ngular
Bragg peak. Another way to quantify the Bragg signal is to define theratio of the peak Bragg intensity to the Rayleigh intensity as ‘gain’,
which is calculated to be Ntotal(fβ/Er)2, where f =​  NBEC/Ntotal is the condensate fraction. The inset of Fig. 2b shows the normalized gain asa ­function of condensate fraction squared. The linear fit to the data
points is c­ onsistent with a y-axis intercept of zero. This shows that theobserved gain comes only from the superfluid component of the atomic­sample. Figure 2c shows that the Bragg signal increases with larger
spin–orbit-coupling strength up to β ≈​ 300 Hz, and starts to decrease
owing to heating from the Raman driving (see Methods).
Figure 3a shows the phase diagram for spin–orbit-coupled BECs for theparameters implemented in this work. The stripe phase is wide, owing to
the high miscibility of the two orbital pseudospin states. Our spin–orbit
coupling scheme and the one previously used10,11 with 87Rb are complementary. In 87Rb, the phase-separated and the single-­minimum states
were easily observed10,11, whereas our scheme favours the stripe phase.
Exploring the phase diagram in the vertical direction requires­varying δ0 with the two Raman beams detuned. For δ0 =​  0, spin–orbit
coupling leads to two degenerate spin states. For sufficiently large valuesof δ 0 , the ground state is the lower spin state. The vertical width of the
stripe phase in Fig. 3a depends on the miscibility of the two spin­components6,22. However, population relaxation between the two spinstates is very slow10. For our parameters, the equal population of thetwo pseudospin states is constant during the lifetime of the system for

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Letter RESEARCH
b

Δ

2d

v

1

0
600

a

NBEC2 (×109)
2
3

0.0
1.0

4

200
2×10–4

kRaman

10–4
0

0.2

0.3

Phase
separation

E (Hz)
0

100

200

300

400

40

20

0
0

0.025
E/Er

Singleminimum
phase

Stripe phase
This work⎥ ↑ 〉 Magnetized
phase

–1.0
0.0

0.05

Figure 2 | Density modulations from Raman beams, and quantitative
studies of the supersolid stripes. a, Effect of Raman beams. The twolowest bands of the superlattice are mapped into orbital pseudospins,
where the pseudospin down state (localized in the left wells of thesuperlattice) is shown in blue and the pseudospin up state (localized in theright wells of the superlattice) is shown in red. Coupling the pseudospinswith Raman laser beams causes two different types of density modulations;
one is a moving density modulation caused by the moving lattice potential,
and the other is the stationary stripes from Raman-induced spin–orbit
coupling between the pseudospins. The stationary stripes along the freespace x direction break the continuous translational symmetry. b, BEC
number dependence. Bragg signal is plotted versus the BEC number,showing the count rate integrated over the Bragg peak. The grey wedge
is the theoretical prediction without any adjustable parameters, using
independently measured values of β and γ (and the corresponding 1σerrors) along with the theoretical Debye–Waller factor, assuming fullphase coherence of the stripes. The simple theory (see main text) predicts
the peak angular amplitude of the Bragg signal. To compare it to the total
count rate, we assumed a Gaussian lineshape with a constant linewidth.
The linewidth was obtained by averaging the widths obtained from twodimensional Gaussian fits to the data for each condensate number. The
inset shows the normalized gain as the BEC fraction is varied. The greysolid line is a linear fit, where the y-axis intercept is consistent with zero,
within 2σ fitting error. c, Spin–orbit-coupling dependence. Bragg signal
versus Raman coupling strength β at zero Raman detuning is shown. Error
bars represent 1σ standard error of the mean with a sample size of 3 to 4.

all detunings δ 0 studied (up to ±10 kHz). Therefore, detection of the
stripes is possible even for large detuning.
We observed peaked Bragg reflection at δ0 ≈​  ±​0.7Er, which was­characterized previously as spin-flip resonances coupling ↑, P = 0〉 to↓, P = − ħkIR and ↓, P = 0〉 to ↑, P = − ħkIR〉 (Fig. 3b). These peaks
show that density modulations are resonantly created in either the ↑〉or ↓〉 states. In addition, we observed a third peak around δ0 =​  0, where
the stripe pattern is stationary. For finite δ0, it moves at a velocity ofδ0/kIR. Our observation shows that the stationary stripe ­pattern is either
more stable or has higher contrast compared to a moving stripe. Since
the tunnel coupling along the superlattice ­direction is weak (about
1 kHz) it seems possible that the alignment of moving stripe patterns

0.1

0.2 1
1.5
Spin–orbit coupling strength, E/Erb

2

Detuning, δ0/Er–1
Bragg + Rayleigh signal (arbitrary units)

c

0.0

0.4

(NBEC/Ntotal)2

Bragg signal (arbitrary units)

15

–0.5

Moving density modulationfrom Raman potential

Stationary stripes
from Raman-induced
spin–orbit couplingSpin–orbit coupling strength, E (kHz)1
1.5 7.5
10
12.5

0.5

400

0

2d

0.5

⎥ ↓ 〉 Magnetizedphase
Detuning, G0 /Er

J

Gain / Ntotal

SuperlatticedBragg signal (CCD counts
per square pixel)

a

0

1

20

10

0

–10

–5

0
Detuning, G0 (kHz)5

10

Figure 3 | Phase diagram for spin–orbit-coupled BECs, and effect ofRaman detuning on the supersolid stripes. a, Mean-field phase diagram ofspin–orbit-coupled BECs as a function of detuning and spin–orbit-couplingstrength with n ≈​  1.5 ×​  1014 cm−3. The parameter space explored in thiswork is crosshatched in grey. However, owing to metastability, our effective
detuning is always zero (see text). b, Detuning dependence. Black filled circles
show the total light scattering signal (Bragg and Rayleigh) as a function of thefrequency detuning. The light was detected within a solid angle of 10 mrad.
The grey dashed line and shaded area show the mean and standard deviationof the Rayleigh scattered light for the same conditions. The light-grey solidline is a triple Gaussian fit to the total light scattering where the widths andcentre positions of the two outer peaks are constrained to be identical to thespin-flip resonances studied in our previous work27. Error bars represent 1σ
standard error of the mean with a sample size of 3 to 6.

is more sensitive to perturbations than for s­ tationary stripes and leads
to a reduced Debye–Waller factor for ­moving stripes.
The periodicity of the supersolid density modulation can depend on
external, single-atom, and two-atom parameters. In the present case,the periodicity is given by the wavelength and geometry of the Raman
beams. It is then further modified by the spin gap parameter β and theinteratomic interactions 5,6 to become 2d/ 1 − (β /F )2 , where
F =​  (2Er +​  n(g +​  g↑↓))/4. For β ≈​ 300 Hz, the correction due to the interactions is only 0.4% and was not detected in our work. In contrast, forthe dipolar supersolid13 and a quantum crystal with vacancies1,2, theperiodicity dominantly depends on atomic interactions.
So far, we have presented a supersolid that breaks the continuoustranslational symmetry in the free-space x direction (Fig. 2a). Unrelated
to the presence of spin–orbit coupling, our superlattice system also
breaks a discrete translational symmetry along the lattice direction ẑ
by forming a spatial period that is twice that of the external latticeowing to the interference between atoms in the two pseudospin stateswith quasimomentum difference Δq =​  π​/d (Fig. 4a) (see ref. 27). Thisfulfils the definition of a lattice supersolid19,28. This 1,064-nm-period
2 m a rc h 2 0 1 7 | VO L 5 4 3 | NAT U R E | 9 3© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

RESEARCH Letterd

JSuperlattice

Δ
⎥ ↓〉
⎥ ↑〉
z

b
Bragg + Rayleigh signal(arbitrary units)

a

6

1.

4

2

0
Constructive

Destructive

received 26 October 2016; accepted 14 January 2017.

0

5
10
Independent measurements

x
z
2d
M=0

M = π/2
Density modulation from superlattice

M=πFigure 4 | Bragg detection of a lattice supersolid caused by an
antiferromagnetic spin texture. a, Phase-sensitive density modulation of
the lattice supersolid. The interference between the pseudospin down and up
states alternates between constructive and destructive for adjacent unit cells
in the lattice. This generates an oscillating density modulation, which has aspatial periodicity of 2d along the superlattice direction ẑ and depends on
the relative phases between the orbital pseudospins as ϕ =​  ϕ0 +​  Δt, with ϕ0
as the spontaneous phase. The density modulations at different phaseconditions are shown below (in the dashed box). This breaks the discrete
symmetry of the lattice potential. b, The Bragg signal for independentmeasurements with n ≈​  1.5 ×​  1014 cm−3. The Bragg signal, which is set up for
detection of a spatial periodicity of 2d along the superlattice, depends on therelative phase ϕ when the Bragg pulse width is shorter than 1/(2Δ). Blackcrosses show the Rayleigh-scattered background integrated over 40 mrad
before the antiferromagnetic spin texture develops. Grey filled circles show
Bragg-enhanced scattering. The Bragg enhancement fluctuated between
zero and a factor of two, which indicates variations of the spontaneous phase
ϕ0 between independent measurements.

density modulation has a maximum amplitude of (J/Δ) and oscillates
at frequency 2Δ temporally with spontaneous initial phase and can bedetected with the same geometry of the Bragg beam and camera, butrotated to the y–z plane. Figure 4b shows the observed enhanced lightscattering due to Bragg reflection. The enhancement was absent immediately after preparing an equal mixture of the two pseudospin states,
both in q =​ 0, and appeared spontaneously after the upper pseudospin
state relaxed to the band minimum at q =​  π​/d. With the Bragg pulse
duration shorter than 1/(2Δ), the Bragg signal varied between 0% and100%, depending on the phase of the oscillation of the density­modulation when probed. The increased fluctuation in Fig. 4b shows
the random nature of the initial phase, which is consistent with­spontaneous symmetry breaking.
In conclusion, we have observed the long-predicted supersolid stripe
phase of spin–orbit-coupled BECs. This realizes a system that simultaneously has off-diagonal and diagonal long-range order. In the future,
it will be interesting to characterize this system’s collective excitations9and to find ways to extend it to two-dimensional spin–orbit coupling,which leads to a different and rich phase diagram29. Another direction
for future research is the study of vortices and the effects of impurities
and disorder in different phases of spin–orbit-coupled condensates30.
Online Content Methods, along with any additional Extended Data display items andSource Data, are available in the online version of the paper; references unique tothese sections appear only in the online paper.

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Acknowledgements We thank S. Stringari for discussions and W. C. Burton
for reading the manuscript. We acknowledge support from the NSF throughthe Center for Ultracold Atoms and from award 1506369, from ARO-MURI
Non-equilibrium Many-body Dynamics (grant W911NF-14-1-0003) and from
AFOSR-MURI Quantum Phases of Matter (grant FA9550-14-1- 0035).
Author Contributions J.-R.L., W.H., J.L., B.S., S.B., F.C.T. and A.O.J. contributed tothe building of the experiment. J.-R.L. led the experimental efforts. J.L. led thedata analysis and simulations. W.H., J.-R.L. and W.K. conceived the experiment.
All authors contributed to the writing of the manuscript.
Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial
interests. Readers are welcome to comment on the online version of the paper.
Correspondence and requests for materials should be addressed to
J.-R.L. (junruli@mit.edu).
Reviewer Information Nature thanks K. Hazzard and the other anonymous
reviewer(s) for their contribution to the peer review of this work.

9 4 | NAT U R E | VO L 5 4 3 | 2 m a rc h 2 0 1 7© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

Letter RESEARCH
Methods

Experimental setup and sample preparation. The superlattice (SL) potential
V(z) =​  VIRsin2(kIRz +​  φSL) +​  VGreensin2(kGreenz) was generated by overlapping two
one-dimensional lattices with spacing λIR/2 =​ 532 nm (long) and λGreen/2 =​ 266 nm(short). The relative phase φSL is controlled by the frequency of the 1,064-nm latticelight for rapid switching and a rotatable dispersive glass plate in the lattice beampath. The offset Δ is determined by VIR and φSL and is calibrated by observing
the oscillation of the interference pattern between atoms released from the two
wells27. We estimate the fluctuations of the offset parameter Δ to be within 1 kHz,
caused by drifts in the air pressure that affect the relative phase φSL between the
two lattices.
The two Raman beams were generated from the same laser and then split into
two parts. The relative frequency was tuned by two independent acousto-optic
modulators in each path. We prepared N ≈​  4 ×​  105 23Na atoms in a crossed optical
dipole trap. An equal mixture of spin-up and spin-down states was created by first
adiabatically ramping up the superlattice with an offset Δ =​ 0 and then rapidly
setting Δ to the final value by a fast frequency change of the infrared lattice27.
Subsequently, the Raman lasers inducing the spin–orbit coupling are adiabaticallyramped up within about 10 ms followed by a variable hold time, after which the
Bragg probe beam is applied.
Miscibility and the stripe phase. Achieving stable and high-contrast stripes
requires miscibility of the two spin components. The difference in energy density
between a BEC in the stripe phase and a phase-separated phase is gδn2 −​  (g −​  g↑↓)n2
where g =​  4π​ħ2a/m and g↑↓ =​  4π​ħ2a↑↓/m parameterize the interaction energy
strengths between atoms in the same and in different spin states, respectively. Herea (a↑↓) is the interspin (intraspin) s-wave scattering length. The extra mean-fieldenergy due to a modulation of the density n with amplitude δn leads to phaseseparation when the contrast of the stripes η =​  δn/n exceeds (g − g↑↓)/g . All
previous studies with bosons used 87Rb atoms10–12. For 87Rb atoms in the
F = 1, mF = 0 and F = 1, mF = − 1 states, (g −​  g↑↓)/g =​  10−3 is extremely small.
In addition, the full width in δ0 for stable stripes is W =​  2n(g −​  g↑↓), which is about
10 Hz for 87Rb and requires extreme control of ambient magnetic field fluctuations.
For these reasons the stripe phase has not yet been observed in previous studies ofspin–orbit-coupled rubidium atoms10–12.
Using the orbital degree of freedom as pseudospin, Raman lights can be fardetuned from atomic transitions, suppressing spontaneous emission heating. In addition, the orbital overlap controls the intra-spin interaction g↑↓ ≈​  (J/Δ)g, where J is theinterwell tunnelling and Δ is the well offset27. For typical experimental ­parameterswith (J/Δ) ≈​ 1/20, (g −​  g↑↓)/g ≈​ 1. The highly imbalanced interaction enhances the
miscibility and therefore allows higher contrast and more stable stripes.
Bragg beam parameters and detection. The Bragg beam was chosen to beblue-detuned about 1,030 MHz from the sodium 3S1 / 2, F = 1 to 3P 3 / 2, F = 2
transition with a linear polarization along the superlattice direction. The detuning
was chosen such that the Bragg beam can propagate through the entire condensate
without much absorption or wave-front distortion.
The alignment of the Bragg beam required accurate prealignment by­triangulation. A major challenge was the alignment of this beam to an accuracy of
better than about 0.5°, the angular width of the Bragg signal, without any auxiliary
density modulation at the same periodicity, given that creating such a density
modulation would have required a standing wave of laser light at 2,128 nm.
Experimentally, the lattice supersolid is more robust than the stripe phase while
having the same periodicity of the density modulation. Therefore, the alignment
procedure was first developed for the lattice supersolid. The same setting was
rotated around the axis ŷ by 90° to probe for the stripe phase.
The Bragg reflected beam and the Rayleigh fluorescence were recorded with
an electron multiplying charge-coupled device (CCD) camera. The angular­distribution was recorded by first focusing an imaging system onto the camera andthen moving the camera out of focus. The signal was normalized for f­ luctuating
atom numbers using the fluorescence intensity monitored by a photomultiplierusing a separate viewport. The Bragg signal was obtained by integrating the countsof the CCD pixels around the Bragg-matched angle. The Rayleigh signal was
obtained from fitting the diffuse background with a two-dimensional Gaussian
fit. The detected Bragg signal was of the order of only ten photons.
We observed a lifetime of about 20 ms for the Bragg signal after ramping up the
spin–orbit coupling, accompanied by a clearly visible reduction in the n­ umber ofatoms in the BEC. We believe that it is limited by the heating due to the Raman
driving27. At values of β ≈​ 300 Hz, the moving Raman lattice has a depth of about
3Er, which is comparable to the stationary lattice at around 10Er. When the spin–
orbit coupling was increased further, the Bragg signal decreased, as shown in
Fig. 2c, with noticeable atom loss. In addition, the observed heating may still havea contribution from technical sources, since the observed lifetime is sensitive toalignment.
Data availability. The data that support the findings of this study are available
from the corresponding author upon reasonable request.

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

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