Spin-Orbit Coupling in Optical Superlattices
byWujie Huang
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2016
c Massachusetts Institute of Technology 2016. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
October 19, 2015

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

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nergis Mavalvala
Professor of Physics, Associate Department Head for Education

2

Spin-Orbit Coupling in Optical Superlattices
byWujie Huang
Submitted to the Department of Physics
on October 19, 2015, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Quantum simulation is emerging as an exciting and active frontier in atomic physics.
It allows us not only to verify existing models with high precision, but also to engineer
novel systems with strong correlations and exotic topologies. Recent efforts have been
made to include synthetic gauge fields and spin-orbit couplings into ultracold quantum
gas experiments, which would enable us to study the quantum Hall effect, topologicalinsulators as well as topological superfluids. This thesis will describe the experimental
implementation of a new spin-orbit coupled system using pseudospin-1/2 in an opticalsuperlattices, as well as progress towards detecting the stripe phase in this system.
The first part of this thesis describes the development of a new apparatus for performing quantum simulations with sodium and lithium in optical lattices. A quantum
simulation program is challenging itself, therefore having a stable platform for preparing quantum gases is essential for this task. We’ll describe our development in reliable
and efficient production of sodium Bose-Einstein condensates and lithium degenerate
Fermi gases, as well as the characterization of our optical lattice system in a superfluid
to a Mott-insulator quantum phase transition. The dynamics of a Bloch oscillation ina tilted lattice has also been studied as an important step towards the implementationof synthetic magnetic fields in our system.
The second part of this thesis describes the experimental realization of spin-orbit
coupling in a pseudospin-1/2 system using an optical superlattice. This new scheme
uses orbital states in a tilted double-well as the pseudospins, therefore does not requirenear-resonant Raman light to flip the spins and promise longer lifetimes compared toearlier spin-orbit coupling experiments in atomic gases. It also features a robust miscible ground state with stationary density stripes, which is closely related to the conceptof supersolidity in condensed matter systems. We’ll present our experimental implementation of this new system, signatures of the resonant spin-orbit coupling, as wellas progress toward experimental detection of the stripe phase via Bragg scattering.
This pseudospin-1/2 system could also be used for simulating quantum magnetism,
and potentially novel models with topological properties and Majorana excitations.

3

Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

4

Acknowledgments
First and foremost, I’d like to thank my research advisor Prof. Wolfgang Ketterle forshowing me the wonderful physics in cold atom systems, and providing me this great
opportunity to carry out the experimental research. I want to sincerely thank you forgiving me encouragement, freedom and guidance throughout the years.
During my Ph.D. I’ve worked with a number of very talented and generous people
on the BEC2 experiments. Christian is an expert in optics and electronics as well asa very patient and supportive teacher and friend. He taught me useful experimental
techniques when I joined this lab with little experience, and also guided me to explore
other aspects of life. Aviv has a great amount of positive energy, and commitment tomaking everything great. I’m impressed by his engineering and programming talent
in building this formidable machine and the word generator. Our postdoc Jon’s early
work in developing the frequency doubling for green light turned out to be crucial forour superlattice experiments. In addition, I’d like to thank Jon for making our lab
trip to Foxwoods possible: it was a memorable experience. Thanks to Ed, for staying
patient and positive throughout the difficult times in developing our new apparatus.
There’s no doubt that you are very talented and have good character, and I wish you
success in your new journey.
In the latest spin-orbit coupling experiment carried out with the new apparatus,
I’ve worked with a number of very promising young talents. Junru has always been agreat labmate that one can rely on. He is not afraid to make changes to the existing
setup, and always does his best to bring the machine back to the working condition
when something stops functioning. Fixing the machine can sometimes be frustrating,however, you have all the good qualities one needs to carry out excellent research. I
wish you best of luck! Boris is a great addition to the lab to bring BEC2 social life
alive. Thank you for your efforts to make our lab roadtrip possible. With your hard
work and commitment, I’m sure you will soon become the new master of the BEC2
machine. Furkan’s attention to the details and perfection in his lab projects and ourlatest data analysis surprised me. I see great potential from you. Our UROP student
5

Sean really impressed me with what a MIT undergrad can do in his/her dorm room.
I cannot wait to see you play a more important role in the future experiments.
I’d like to thank Prof. Martin Zwierlein for being my academic advisor throughout
my years at MIT, and also serving my thesis committee. Thank you for the inspiringand encouraging discussions at the start of every semester. I look forward to hearing
your continuing great success and exciting new results from your group.
I’d also like to thank everyone in the Ketterle group for lending me equipments,
sharing technical insights and generating fruitful discussions in our plenary meeting.
I’d like to especially thank colleagues who have considerable overlap with me in myyears at MIT: Thanks to Timur, Tout, Myoung-Sun and Alan in BEC3 for being agreat neighbor, and lending us a hand whenever needed; Thanks to Colin, Cody, Woo
Chang, Hiro and David in BEC4 for stimulating insightful discussions and sharing
experimental techniques they’ve developed in the rubidium lab; Thanks to Niki, Jesseand Ivana in BEC5 for spreading smiles and laughs across the hallway, and sharing
your expertise in designing electronic circuits and reducing laser noise. It has been aprivilege for me to work in this group and I’ll miss all of you.
It is also a great pleasure to work in the CUA community. I met Mark on the firstday I came to MIT, and thank you for being a great friend and offering me invaluable
help along the years. I’d also like to thank Peter, Sebastian, Cheng, Ibon, Payman in
Fermi1 lab, and Ariel, Tarik, Waseem, Lawrence, Melih, Matt in BEC1/Fermi2 lab,
as well as Hao, Wenlan, Molu, Alexei in the Vuletic group. I want to thank each oneof you for your generous help and the great memories together.
I want to thank the Chinese community at MIT to make me feel like home and beable to focus on my study and research. I want to thank my friends in the Departmentof Physics, including Wenjun Qiu, Bo Zhen, Yannan Zheng, Yabi Wu, Wenlan Chen,
Jiazhong Hu, Jiexi Zhang, Hao Zhang, Mingming Yang, Lu Feng, Lei Dai and others.
I’d like to thank everyone in the CSSA basketball team. It was my great pleasure toserve as your captain for two years. It was probably the most memorable highlightof my life at MIT.
I want to thank the ones I had once fallen in love with. You gave me passion, and6made me take a pause to appreciate the beauty of life besides science.
I’d like to thank my advisors and mentors in Tsinghua, Hui Zhai and Prof. Lee
Chang, for guiding me into this exciting field and setting the high standards of beinga physicist.
Last and certainly not least, I would like to thank my parents and brother for theunconditional support. As I said during my thesis defense, “there is only one home,
and that’s the place I miss so much at this moment”.

7

8

Contents
1 Introduction
1.1

1.2

19

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.1.1

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

19

1.1.2

Cold atoms as quantum simulators . . . . . . . . . . . . . . .

20

1.1.3

A brief history. . . . . . . . . . . . . . . . . . . . . . . . . .

20

Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2 Production of Quantum Degenerate Gases

23

2.1

Overview of the Experimental Apparatus for 23 Na-6 Li in Optical Lattices 25

2.2

Condensation of 23 Na in the F = 1 Hyperfine State . . . . . . . . . .

31

2.2.1

Single-species 23 Na MOT . . . . . . . . . . . . . . . . . . . . .

32

2.2.2

Evaporative cooling in a plugged magnetic-quadrupole trap . .

33

2.2.3

BEC in an optical dipole trap . . . . . . . . . . . . . . . . . .

35

Sympathetic Cooling of 6 Li with 23 Na in the F = 2 State . . . . . . .

36

2.3.1

The hyperfine states of 6 Li and 23 Na . . . . . . . . . . . . . .

37

2.3.2

Condensation of 23 Na in the F = 2 hyperfine state . . . . . . .

38

2.3.3

Sympathetic cooling of 6 Li with 23 Na . . . . . . . . . . . . . .

39

Hyperfine State Preparation of 23 Na . . . . . . . . . . . . . . . . . . .

41

2.4.1

Microwave transitions between hyperfine states . . . . . . . .

42

2.4.2

Radio-frequency transitions between hyperfine states . . . . .

43

2.4.3

Two-photon transitions between hyperfine states . . . . . . . .

44

2.3

2.4

9

3 Bose-Einstein Condensates in Optical Lattices

45

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.2

Basic Lattice Physics . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2.1

Band structure and Bloch states . . . . . . . . . . . . . . . . .

47

3.2.2

Tight binding and Wannier states . . . . . . . . . . . . . . . .

50

3.2.3

Lattice calibration via Kapitza-Dirac diffraction . . . . . . . .

53

Superfluid to Mott-Insulator Quantum Phase Transition . . . . . . .

54

3.3.1

Mean-field theory for a Bose-Hubbard model . . . . . . . . . .

54

3.3.2

Observation of the superfluid to Mott insulator transition . . .

62

Tilted Lattices and Bloch Oscillations . . . . . . . . . . . . . . . . . .

64

3.4.1

Bloch oscillation. . . . . . . . . . . . . . . . . . . . . . . . .

65

3.4.2

Dynamical instability . . . . . . . . . . . . . . . . . . . . . . .

66

3.3

3.4

4 Quantum Simulation in Superlattices

71

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.2

Experimental Setup and Techniques . . . . . . . . . . . . . . . . . . .

74

4.2.1

Second-harmonic generation for 532 nm green lattice . . . . .

74

4.2.2

Controlling the phase of the superlattice . . . . . . . . . . . .

77

4.2.3

Considerations for the Raman beams . . . . . . . . . . . . . .

85

4.2.4

Experimental layout . . . . . . . . . . . . . . . . . . . . . . .

86

Experimental Characterization and Calibration of a Superlattice . . .

86

4.3.1

Superlattice band structure . . . . . . . . . . . . . . . . . . .

87

4.3.2

Calibration and dynamical control of the superlattice phase

.

89

4.3.3

Interband population transfer and band mapping . . . . . . .

95

Theory of Spin-Orbit Coupling in a Superlattice . . . . . . . . . . . .

99

4.3

4.4

4.4.1

Single-particle spin-orbit coupling Hamiltonian . . . . . . . . . 100

4.4.2

Spin-orbit coupling and onsite couplings . . . . . . . . . . . . 104

4.4.3

Quantum phase transition of an interacting spin-orbit coupledcondensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.5

Experimental Realization of Spin-Orbit Coupling in a Superlattice . . 108
10

4.5.1

State preparation in a superlattice. . . . . . . . . . . . . . . 108

4.5.2

Dynamics in the double-wells . . . . . . . . . . . . . . . . . . 110

4.5.3

Detection of the spin-orbit coupling in time-of-flight . . . . . . 112

4.5.4

Experimental observation of resonant spin-orbit coupling . . . 120

4.5.5

Detection of the spin-orbit coupling in Bragg scattering . . . . 122

5 Density Fluctuations in Quantum Degenerate Fermi Gases

127

6 Outlook and Conclusion

129

6.1

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A Suppression of Density Fluctuations in a Quantum Degenerate Fermi
Gas

133

B Speckle Imaging of Spin Fluctuations in a Strongly Interacting Fermi
Gas

143

C Correlations and Pair Formation in a Repulsively Interacting Fermi
Gas

149

D Kapitza-Dirac Diffraction Simulation

159

E Superlattice Band Structure Calculation

163

F Simulation of Double-Slit Interference Pattern in a Superlattice

167

11

12

List of Figures
2-1 Overview of BEC2 sodium-lithium apparatus. . . . . . . . . . . . . .

26

2-2 Layout of the optics on the experimental table.

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

31

2-3 Evaporative cooling of F=1 Na with radio-frequency . . . . . . . . . .

34

2-4 Evaporative cooling of F=1 Na with 1.7 GHz microwave . . . . . . .

35

2-5 Hyperfine structures of 23 Na and 6 Li . . . . . . . . . . . . . . . . . .

37

2-6 Gravity-assisted spin purification and optical pumping of F=2 Na . .

38

2-7 Sympathetic cooling of Li with F=2 Na . . . . . . . . . . . . . . . . .

40

2-8 Sympathetic cooling efficiency . . . . . . . . . . . . . . . . . . . . . .

41

2-9 Spin flip via Rapid Adiabatic Passage: the Landau-Zener sweep . . .

42

2-10 Rabi oscillations in spin flip . . . . . . . . . . . . . . . . . . . . . . .

43

2-11 Spin flip with radio-frequency and the Stern-Gerlach experiment . . .

43

2-12 Spin flip via two-photon transition. . . . . . . . . . . . . . . . . . .

44

3-1 Band structure of a one-dimensional optical lattice . . . . . . . . . .

49

3-2 Bloch states in optical lattices . . . . . . . . . . . . . . . . . . . . . .

50

3-3 Wannier states in the lowest band . . . . . . . . . . . . . . . . . . . .

51

3-4 Kapitza-Dirac diffraction patterns . . . . . . . . . . . . . . . . . . . .

53

3-5 Experimental calibration of the lattice depth via Kapitza-Dirac diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3-6 Numerical calculation of tunneling t and on-site interaction U . . . .

57

3-7 Bose-Hubbard model phase diagram . . . . . . . . . . . . . . . . . . .

59

3-8 Bose-Hubbard model phase diagram in an inhomogeneous system . .

60

3-9 Wedding-cake density profile in a harmonic trap . . . . . . . . . . . .

61

13

3-10 Experimental observation of superfluid to Mott insulator transition .

63

3-11 Mott insulator lifetime measurement . . . . . . . . . . . . . . . . . .

64

3-12 Experimental demonstration of Bloch oscillations . . . . . . . . . . .

66

3-13 Experimental observation of dynamical instability . . . . . . . . . . .

67

3-14 Relaxation of quasi-momentum states near the edge of the brillouin zone 69
4-1 The spin-orbit coupling band structure . . . . . . . . . . . . . . . . .

72

4-2 The pseudo-spin scheme to simulate spin-orbit coupling physics . . .

73

4-3 Optimal crystal temperature for 532 nm second-harmonic generation

76

4-4 532 nm second-harmonic-generation conversion efficiency . . . . . . .

77

4-5 Superlattice phase control via AOM frequency shifter . . . . . . . . .

81

4-6 Experimental schematics of the superlattice experiment . . . . . . . .

87

4-7 Diffraction patterns of various beams . . . . . . . . . . . . . . . . . .

88

4-8 Band structure of the superlattice . . . . . . . . . . . . . . . . . . . .

90

4-9 Predicted time-of-flight patterns of the Raman-coupled p = +kL states

91

4-10 Superlattice phase calibration via diffraction . . . . . . . . . . . . . .

91

4-11 Experimental sequence for the dynamical control on superlattice phase92

4-12 Experimental demonstration of dynamical control on the superlatticephase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

4-13 Measurement of long-term drifts of the superlattice phase . . . . . . .

95

4-14 Interband population transfer in a 1064 nm lattice via amplitude modulation and Raman coupling . . . . . . . . . . . . . . . . . . . . . . .

96

4-15 Interband coupling resonances and the band structure in a 1064 nm
lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

4-16 Raman coupling into higher bands in a green lattice in TOF . . . . .

99

4-17 Quantum phases of an interacting spin-orbit coupled condensate . . . 107
4-18 Experimental calibration of the energy offset ∆. . . . . . . . . . . . . 111
4-19 The ground states of the lowest two bands in the superlattice . . . . . 112
4-20 Relaxation into the ground state in the second band . . . . . . . . . . 113
14

4-21 The relative phases and the corresponding interference of the k = 0
component for 50/50 population . . . . . . . . . . . . . . . . . . . . . 116
4-22 The relative phases and the corresponding interference of the k = ±kL
component for 100/0 population with only the OC or SOC . . . . . . 117
4-23 The relative phases and the corresponding interference of the k = ±kL
component for 100/0 population with both the OC and SOC . . . . . 118
4-24 No interference between quasi-momentum q = 0 and q = kL states . . 119
4-25 Typical time-of-flight image in the spin-orbit coupling experiment . . 120
4-26 Resonant spin-orbit coupling measurement . . . . . . . . . . . . . . . 121
4-27 The stationary stripes and moving density modulations . . . . . . . . 123
4-28 Bragg scattering experiment for detecting the stripe phase . . . . . . 124

15

16

List of Tables
4.1

The index of refraction for 532/1064 nm in glass . . . . . . . . . . . .

78

4.2

The natural units in 532/1064 nm superlattices . . . . . . . . . . . .

89

4.3

The Bragg angles for detecting the stripe phase . . . . . . . . . . . . 125

17

18

Chapter 1
Introduction
1.1

Background1.1.1

Quantum simulation

The idea of quantum simulation originates from remarks made by Richard Feynman in
1982, that physical systems can be hard to simulate with a computer. This statement
is particularly true for quantum systems, as the memory required to represent a state(space complexity) and the time required to do computations (time complexity) withthem increase exponentially with the number of degrees of freedom being simulated.
One way to overcome the difficulties is to design and build novel quantum computers. They perform unitary operations instead of logic gates, and operate on quantum
bits instead of classical bits. Because of their vastly larger state space and greater
computational power, quantum computers could allow us to tackle difficult problems
that are not tractable with classical machines. To make scalable and robust quantum
computers is currently a field of enormous activity and interest.
Another promising solution is to simulate the same problem in a different quantum
system. Assuming physics is universe, studying problems in a clean and well controlled
setting allows us to gain an edge in solving them, as well as to engineer novel quantum
models to be explored.
19

1.1.2

Cold atoms as quantum simulators

Since the first realization of Bose-Einstein condensation and later Fermi degeneracy ofneutral atomic gases, ultracold atoms have shown themselves to be an ideal candidatefor quantum simulations. Bose condensates and degenerate Fermi gases are pure formsof quantum matter. Furthermore, the diluteness of the atomic clouds and their weakcouplings to the environment allow them long coherence times for quantum simulation
experiments.
The powerful toolbox for preparing, manipulating and probing atoms for quantum
simulations has been developed over the years. Here I’d like to mention a few examples
to demonstrate the powerfulness and convenience of these tools. First, optical lattices
are created by interfering laser beams. By changing the geometry and polarization of
the beams, we are able to make cubic, triangular, checkerboard or other very different
lattices in a single experimental setup. Second, Feshbach resonances would allow us totune the two-body interactions, ranging from strong attractions to strong repulsions.
Finally, atoms can be imaged in absorption, phase contrast, or single atom detection
via fluorescence [5]. All of these tools make the ultracold atom system particularly
attractive, and greatly facilitate the quantum simulation program.

1.1.3

A brief history

The field of quantum simulation using ultracold atoms has become diverse in recent
years. To me still, there are focused efforts in the following directions. The first one isto simulate the Bose-Hubbard model with a single-component Bose gas in 3D opticallattices. The quantum phase transition from a superfluid to a Mott insulator has been
observed in this sytem [15]. With newly developed microscopy techniques and singlesite resolution imaging, one can characterize the system with high precision [16, 17].
The second direction is to simulate the Fermi-Hubbard model with an interacting
two-component Fermi gas in optical lattices. This is partly motivated by the high-Tc
superconductivity and quantum magnetism. Experimental progress in this direction
includes observation of fermionic superfluid [3], band insulator and Mott insulator [19,20

20], and most recently results on probing the anti-ferromagnetic correlations near theNéel transition [21, 22].
In the past five years or so, a third direction has emerged to simulate gauge fields,
spin-orbit couplings and topological states. New lattice modulation and laser-assisted
techniques have been developed to generate complex tunneling amplitudes. Highlights
in this direction so far include creating strong synthetic magnetic fields [23, 24, 25, 26],
and realizing novel Hamiltonians with topological properties [27, 28]. These exciting
developments have opened up a whole new phase of quantum simulations in ultracold
atoms, and motivate the work that will be presented in this thesis.

1.2

Thesis Outline

This thesis will present progress towards experimental realization of exotic Hamiltonians in optical lattices, including spin-orbit couplings in an optical superlattice andsubsequent experimental detection of the stripe phase, as well as synthetic magnetic
fields. It is organized as follows.
• Chapter 2 presents an overview of the new apparatus, and discusses technicaldetails of the development of the machine after its initial construction, including
the creation of its first Bose-Einstein condensates and degenerate Fermi gases,
as well as the techniques for spin state preparation.
• Chapter 3 introduces the physics in optical lattices, and describes the development of building and characterizing our optical lattice system. It also discusses
our studies of dynamical instability in tilted lattices and progress towards implementing synthetic magnetic fields.
• Chapter 4 discusses the experimental realization of spin-orbit coupling in thepseudospin-1/2 system in an optical superlattice. It will detail how to create,control and characterize the superlattice, as well as the experimental detectionof the spin-orbit couplings and the stripe phase.
21• Chapter 5 briefly describes three density fluctuation measurements in a Fermi
gas. These experiments were performed in an apparatus that was later decommissioned.
• Chapter 6 presents an outlook for the future directions of the experiment andthe conclusion.

22

Chapter 2
Production of Quantum
Degenerate Gases
When I first joined this lab in 2009, I was totally shocked to see the seemingly fragile
glass cell surrounded by complicated magnetic coils, and cables and fibers were overthe place tangled up on the ceiling and floor. It is still a mystery to me how I managed
to learn and operate such a formidable apparatus in the following two years without
any prior experience, and together with my labmates to carry out the atomic density
fluctuation experiments with it.
This old apparatus, originally built in 1998, was the second of its kind on thishallway to perform BEC experiments [1]. It had been upgraded to a dual-species
sodium-lithium machine [2], and become one of the first machines to achieve quantum degeneracy in fermions. In the following years, old generations of BEC2 people
had produced a series of pioneering works, including the first observation of fermionicsuperfluidity in optical lattices [3] and the study of critical velocity of fermionic superfluidity in the BCS-BEC crossover [4].
However, by the time I joined this lab, the old BEC2 apparatus already started toshow limitations for our long-term goal, which is to carry out quantum simulation in
optical lattices. It often required a few hours of tweaking and optimizing for a proper
performance, and was only stable enough to operate late during the night. In addition,it had many layers of unnecessary complexity, partially because of its long history23

of upgrades, making it difficult to adapt changes for new experiments. Furthermore,vacuum in the main chamber started to degrade, as a few key components such as gate
valves and titanium filaments begun to fail. These would ultimately limit our cloud
lifetime for studying superexchange physics in optical lattices. In the end, the decision
was made to develop a next-generation apparatus for optical lattice experiments.
The primary focuses of designing the new apparatus are the reliability, versatility,
and low complexity. Consistent performance and low maintainance of the new apparatus could allow us to work more efficiently and concentrate on the experiment itself.
For example, we have built enclosures with individual temperature stabilization tohave much better control of ambient temperature fluctuations. We have also switched
to a solid state laser system from the notorious dye laser for producing 589 nm light,
and carefully considered the mechanical stiffness and stability of mounting techniques
against beam drifts.
In addition, the design of our new apparatus aimed for versatility rather than for
one particular strength or area of research such as single-site resolution microscopeexperiments [5] or multi-species cold molecule experiments (e.g., Na-Li-K [6]). We
decided to continue the Ketterle group tradition of making general purpose cold atom
machines, which may not be the ultimate tool for any specific study, but are suitable
to a wide variety of experiments.
Finally, in the new apparatus we have also tried to design built-in features that
reduce unnecessary complexity and facilitate future upgrades. For instance, we have
MOT light prepared on the back side of the main breadboards, and delivered withtranslation stages so that it does not interfere with future use of high value surfacesof these breadboards. We selected a much simpler plugged quadrupole magnetic trap
design, compared with the more complicated Ioffe-Pritchard design used previously.
Almost all laser beams are fiber coupled and delivered so that the experimental table
is decoupled from light preparation.
While the noise measurements were being carried out in 2011, the next-generation
apparatus was already in the process of being designed and prepared by the membersof the lab. Its initial design and construction are described in detail in Aviv Keshet’s
24thesis [7]. Edward Su and I took over the machine from the point of its first magnetooptical trap, and eventually made our first sodium BEC in the new apparatus a year
after. Some technical details of the design and implementation for the developmentin this phase can be also found in Edward Su’s thesis [8]. After another year of workon cooling lithium and testing the new optical lattice system, now we are on the vergeof pushing the frontier of quantum simulation in optical lattices again.
This chapter will describe our efforts to reliably and efficiently produce quantum
gases for optical lattice experiments. First we’ll provide a brief overview of our new
apparatus in its current state. We’ll then discuss a few key developments in the process of achieving quantum degenerate gases of sodium and lithium in this apparatus.
Finally, we’ll describe the techniques to manipulate the hyerfine states of sodium as away to prepare the states. This chapter is by no means a complete guidebook for designing and constructing a new apparatus for quantum gases experiments. For that,the reader is advised to refer to early theses in Ketterle lab [1, 2, 7, 9] and references
therein.

2.1

Overview of the Experimental Apparatus for23

Na-6Li in Optical Lattices

The new BEC2 apparatus is a general-purpose machine for 23 Na and 6 Li dual-species
experiments. It shares some features with machines that were built around the sametime in Fermi1 (for 23 Na-6 Li-40,41 K) and BEC5 (for 7 Li) on the hallway of the CUA.
The following contains a brief description of this apparatus.
Fig. 2-1 presents an overview of the whole apparatus. A two-species oven on the
far right is the source for 23 Na and 6 Li atoms. The oven chamber is connected througha differential pumping tube to a small intermediate chamber, which is then connectedto the main chamber at the beginning of the Zeeman slower through a second stage ofdifferential pumping. The oven-to-intermediate and intermediate-to-main interfaces
can be sealed off with VAT gate valves, allowing the main chamber to be kept under25

ultra-high vacuum during oven refills. The intermediate chamber provides additional
differential pumping and acts as a buffer protection, so that the intermediate-to-maingate valve sits in a cleaner environment and is less likely to fail. Each of these vacuum
chambers is monitored by its own vacuum ion gauge, and constantly pumped by its
own ion pump.

Figure 2-1: Overview of our sodium-lithium apparatus. The oven cups (right) containing sodium and lithium are heated up to produce atomic beams, which are then cooled
through the Zeeman slower (middle) and trapped in the main chamber (left). The
main chamber on the left is a custom-made steel chamber with numerous viewports
for optical access and high-current magnetic coils mounted in the vertical buckets.
This figure is adapted from [7].

Main vacuum chamber
The main chamber is designed to prioritize optical access, and to accomodate thebucket windows, with two on the top and bottom, and one on the side. The bucket
window design allows us to bring coils and optics much closer to the atomic cloud, andtherefore achieve strong magnetic fields and high resolution imaging. All viewports
are given a multi-wavelength coatings optimized for 532 nm, 589 nm, 671 nm, and1064 nm, except the side bucket window and its facing viewport were given a single-V
26

anti-reflection coating for 671 nm, intended to be the best-performing imaging axis
for lithium.
Magnetic coils
The high-current magnetic coils are mounted inside the top and bottom buckets toprovide strong magnetic fields. Each of them was wound with five individual layers,which were then wired in series and combined into an outer and inner segment. Bothsegments in the top-bucket coil can be configured, via a mechanical relay H-bridge,
into a Helmholtz or anti-Helmholtz configuration with its counterpart in the bottom
bucket, making the new coil system really easy to switch and reconfigure. These
high-current magnetic coils are constantly cooled with pressurized water, designed tohandle the power comsumption at 500 A. When running this current in both segments,
the Helmholtz configuration yields a bais field over 1000 G, and the anti-Helmholtz
configuration yields a magnetic field gradient of 1000 Gauss/cm, readily covering thebroad s-wave Feshbach resonance of 6 Li near 830G, and the strong magnetic gradientneeded for fast evaporative cooling.
Temperature stabilization
Ambient temperature and humidity stabilization is crucial for the success of atomic
physics experiments. The old BEC2 experiments suffered terribly from this problem,as the room temperature could drift a few degrees over the day, and the only practicalhours of data taking were in the middle of the night.
To address this problem in the new apparatus, we have upgraded the room temperature stabilization system, and more importantly, have built enclosures and individual temperature control for the experiment and laser tables. The details of our
new temperature stabilization system can be found in [7]. In practice, the daily room
temperature fluctuates within a couple of degrees, and the ambient temperature on
the tables fluctuates around tenth of a degrees, when the enclosure doors are fully
closed. To keep the machine away from extreme high humidity, we have also installeda stand-alone dehumidifier to supplement the existing dehumidification performed by27

the air conditioners, and maintain the room humidity always below 35%.
Solid-state 589 nm laser system
For our sodium BEC experiments, a laser source at 589 nm of at least 1.4 Watts andwith a linewidth of about 1 MHz or better is required. Historically, this was achievedby using a tunable dye laser with Rhodamine 6G dye in BEC2. However, the dye laser
was notoriously difficult to operate, and was a major source of experiment downtime.
Finally, it was replaced by a turnkey all-solid-state system from MPB Photonics.
This scheme was made possible by recent advances in the Raman amplifiers. In thissystem, the 1178 nm light from a Toptica DL pro diode laser is amplified through theRaman amplifier to about 5.8 Watts, and then frequency-doubled to create the 589
nm yellow light.
This compact package offers an almost maintainance-free yellow laser solution, andwe are in general very happy about it. There is one caveat that I want to mention:even though MPB claims the minimal threshold of seed power is 13 mW, this value
is way below the saturation limit. This means, a stronger seed always generates moreyellow light. On the other hand, a weaker seed requires the Raman amplifer to run ata higher current to maintain the same final yellow output, which shortens the lifetimeof the pump diodes.
In practice, we run this sytem at a seed power greater than 25 mW, and will geta total output of 1.35 W yellow light. The degradation from 30 mW to 20 mW of the
maximum seed from the Toptica laser is observed over the past two years. Therefore
to overcome this degradation, one has to either replace the Toptica laser diode in thenear future, or to look for alternatives of the seed, such as the promising distributed
feedback lasers (DFB) that provide high power at low cost.
532 nm plug laser
In the new apparatus, we abandoned the Ioffe-Pritchard trap in favor of the quadrupole
trap, because of its simplicity in design and convenience in use. However, different
from the Ioffe-Pritchard trap in which a harmonic trap is created on top of a bias field,28the simplest quadrupole trap with a pair of coils run current in opposing directions
always creates a zero magnetic field at the trap center. As the magnetically trapped
atoms pass through the trap center of zero field, their magnetic dipoles cannot adiabatically follow the field direction, causing them to undergo Majorana spin flips tothe untrapped states and get ejected from the trap.
The problem was solved by either moving the zero field point around to create atime-averaged trapping potential, the so-called TOP trap in JILA experiments [10],or by combining the magnetic potential with a repulsive [11] or attractive [12] opticalpotential so that the atoms are trapped away from the region of zero field.
In our experiment, we have a 10-Watt 532 nm laser Millennia from Spectra Physics
used as a repulsive optical potential to plug the zero magnetic field. It’s an old laserand has already shown its aging. With a degraded output of 6 Watts, we are still able
to keep the machine running. However, if the old pump diodes fail, one could either
get diode replacement for the old laser directly from Spectra Physics if the cost is low,or consider buying newer cost-effective lasers discussed in section 4.2.1, or multitask
the frequency-doubled green light for both the plug and the superlattices.
High power 1064 nm system
Far-off-resonant light is used to generate optical dipole traps and optical lattices forquantum gases experiments. To study exotic states in quantum simulation with optical lattices, especially physics deep in the Mott-insulator such as quantum magnetism,
it is often desirable to work at low densities near one particle per site, and we wantlarge lattice beams with low curvature and simultaneously enough intensity to reachdeep into the Mott insulator regime. This in practice means a few Watts per beam
for sodium, and up to 10 Watts per beam for lithium. Safely, reliably and efficiently
making and controlling high-power light is essential for our experiments..
With high-power laser beams, it’s important to carefully choose optics with highoptical damage threshold and low thermal drifts, since these damages are irreversible
and some of these drifts are hard to trouble-shoot. It is highly recommended to takethese factors into considerations at the designing phase. Based on our experience, it
29

is preferred to use high-power optics for a millimeter beam with more than a coupleof Watts. These include high-power AOMs, high-power fibers and connectors, fusedsilica lenses and air-spaced polarizing beamsplitting cubes.
We have several water-cooled fused-silica AOMs from AA Opto-Electronic to intensity stablize 20-Watt 1064 nm beams. These AOMs have large aperture (2-3 mm),high diffraction efficiency (∼ 90%) and relatively low thermal drifts. However, fusedsilica AOMs have a low figure-of-merit in the acousto-optic effect, and therefore needs
a strong RF signal to drive (in our case about 30 Watts). The best alternative seems to
be AOM made in Tellurium dioxide (TeO2), which is rated at a lower maximum CW
laser power of 5 W/mm2 but requires less RF. We get these AOMs from IntraAction.
For high power fibers, we use photonic crystal fibers from NKT Photonics, withthe maximum available mode area, which are the LMA-25 series without polarization
maintaining, and the LMA-PM-15 series with polarization maintaining. These fibers
are all end-sealed and custom connectorized to SMA-905 high-power connectors by aFrench technology center called Alphanov. These high-power connectors are rated tohandle beams up to 50 Watts. In practice, we found no major thermal issues for a
10 Watts CW input without any active cooling. By installing thermoelectric coolers(TEC) on these connectors to transfer away the excessive heat, other labs reported
that no severe thermal drifts were found with more than 20 Watts CW light. Finally,
we use fiber collimators with fused silica components from Micro Laser Systems (FC5Y-FS) or large area collimators from Schäfter+Kirchhoff GmbH (60FC-SMA series).

Optics layout
Because of its superior mechanical stability, the horizontal axes tend to be the busiest
in terms of optics density, as sensitive beams are preferred to be installed in this plane.
A schematic diagram of the optics layout in the horizontal plane is shown in Fig. 2-2.
The two 45◦ axes with large viewports are shared by the MOT beams and the lattice
beams, and the two smaller 22.5◦ windows are used by the plug and optical repumping
beam separately. Imaging is available along most major axis for easy beam alignment
and flexible detection. In addition, there are a pair of MOT beams, an optical lattice30

Figure 2-2: Layout of the optics on the experimental table. This diagram illustrates
the layout of the optics in the horizontal plane, including the plug beam and twolattice beams in orthogonal axes shared by MOT beams. Imaging are available along
the plug axis, both horizontal lattice axes, the vertical axis, as well as the side axis.
This figure is adapted from [8].
beam, as well as the primary imaging along the vertical axis, which are not shown inthis diagram.

2.2

Condensation of 23Na in the F = 1 Hyperfine
State

In the following two sections, we will describe the development of making quantum
degenerate gases of sodium and lithium in our new apparatus. These discussions will
be mainly focused on the benchmark results in every stage for this particular machine,31

as well as the experimental techniques that we’ve found useful in troubleshooting andperformance evaluation. For general introduction and broader aspects of cooling 23 Na
and 6 Li into quantum degeneracy, please also refer to the pioneering works in [2, 9].

2.2.1

Single-species 23 Na MOT

The traditional dark-spot MOT of sodium is used in our experiment to trap as manysodium atoms as possible [13]. Typically, our MOT loading takes about 8 seconds,
and then we transfer the sodium atoms from the dark-spot MOT into the magnetic
trap with a vertical axial gradient of 50 G/cm,
Once the first MOT is obtained, critical components such as the slower frequency
and repumping sidebands, the MOT light polarization, and the dark-spot alignment
in the repumping beam are within the working range and unlikely to change. However,there are pieces that might potentially drift over the weeks and deteriorate the MOT.
Based on our experience, the top candidates are the power balance in each pair of thecounter-propagating MOT beams, the current in the last section of the slower which
defines the final velocity of the decelerated atoms in the main chamber, and of course
the laser lock and total yellow light output. In most cases, by checking through thislist, one could recover a decent sodium MOT for the experiment.
When something has drifted away from its proper settings, it might be confusing
to figure out what’s wrong and how to make improvement. In this case, we find setting
up a MOT loading monitor very helpful for debugging the system [13]. A very weakprobe beam is sent through the cloud with a countinuous frequency scan of over 100
MHz (via two double-passed AOMs) near the sodium D2 line. For a decent dark-spot
MOT with very high atomic density, a plateau of over 50 MHz with nearly complete
absorption is expected, and the excited-state hyerfine structure could no longer be
resolved. Similarly, a MOT fluorescence detector can be set up to diagnose the slower
performance in the MOT loading, as well as the transfer efficiency from the dark-spot
MOT to the magnetic trap.
If the MOT loading is very slow, another possibility could be the alkali in the ovenis depleted. Usually, we need to refill the sodium oven almost every year, while the32lithium oven is running at a conservative temperature and is expected to last for theentire lifetime of the apparatus. An independent check of whether the oven needs arefill is to measure the atomic flux. A laser beam near the sodium D2 line is sent into
the intermediate chamber perperdicular to the atomic beam, and a narrow resonant
absorption of 3 - 5% corresponds to a good atomic flux for laser cooling and trapping.
In the end, a high density cloud in the magnetic trap is needed for the subsequentevaporative cooling. For us, several billion of trapped sodium atoms is a good starting
point. To correctly estimate the atom number with such high density, an off-resonanceimaging after a short time-of-flight is used, using the scaled absorption cross sections.

2.2.2

Evaporative cooling in a plugged magnetic-quadrupole
trap

After loading the atoms from the dark-spot MOT, the magnetic trap is ramped up
to 750 G/cm for fast evaporation. While we are able to achieve the BEC in 8 seconds
without trap decompression, we obtain a larger condensate of about 5×106 atoms after
reducing the trap gradient to 75 G/cm with further evaporation in the decompressed
trap for an additional 3 seconds. The first BEC in the new apparatus was obtainedin the magnetic trap and the corresponding radio-frequency evaporation sequence isillustrated in Fig. 2-3.
The plug beam initially came from a Spectra-Physics Millenia Xs (diode-pumped
solid-state laser) with a full output of 9.5 W at 532 nm. This unit unfortunately failed
later and was replaced with a similar unit but at a degraded ouput at 6 Watts. The
plug beam is in a free-space setup, prepared on the bottom layer of the experimental
table and then sent to the horizontal breadboard with a periscope. The plug has awaist around 100 µm, and its position is controlled by motorized actuators (Newport
TRA12CC) on the last mirror before the vacuum chamber for beam alignment. The
motorized actuator is convenient in aligning the plug to the magnetic field zero point,
however, there are always some unexplained fluctuations in the plug beam pointing.
Preliminary investigation suggests that it is from the laser itself, and therefore could33be eliminated by coupling through a fiber, with an improvement in beam shape but
a sacrifice in power.

Figure 2-3: Evaporative cooling of sodium in the F = 1 state with radio-frequency.
From left to right are the images of the cloud at different stages in the evaporation
after 1 ms time-of-flight from a plugged quadrupole trap. A plugged hole can be seenat the center of the cloud. By the end of the evaporative cooling, a sodium condensate
in the F = 1 state emerges in the plugged magnetic trap, shown on the far right witha 10 ms time-of-flight.
Forced evaporative cooling of the sodium typically occurs on either the microwave
transition |1, −1i → |2, −2i, or the radio-frequency (RF) transition |1, −1i → |1, 0i.
The RF evaporation is often easier to implement and has a better performance, but
the strong RF radiation can interfere with the normal operation of electronic devices
ranging from power supplies and motorized actuators to flow and temperature sensors.
Since evaporation on the microwave transition near 1.7 GHz is necessary anyway forthe sympathetic cooling of lithium, in which only sodium atoms are evaporated, it isused as our primary scheme for evaporative cooling.
We have an in-vacuum antenna that is connected via a 2-wire electrical feedthroughand designed mainly for driving RF hyperfine transitions. There is also an outside
antenna mounted inside the top bucket, mainly for driving microwave hyperfine transitions. Both of them have a single loop copper wire design.
The results of the microwave and radio-frequency evaporation can be comparedin Fig. 2-3 and Fig. 2-4. The final atom number at the end of the evaporation seems
sensitive to the Rabi frequency of the transition. For our primary microwave evaporation, a 50W microwave amplifier (EMPOWER 1119) is located as close as possible
and connected to the outside antenna by a short SMA cable to avoid unintended at34

Figure 2-4: Evaporative cooling of sodium in the F = 1 state with 1.7 GHz microwave.
Evaporative cooling using microwave produces similar results as the case using radiofrequency. At the end, a sodium condensate in the F = 1 state is produced in theplugged magnetic trap, and subsequently transferred into an optical dipole trap.
tenuation. Unfortunately, it’s somewhat impractical to boost the microwave transfer
and the Rabi frequency further, by either using a resonant circuit on the antenna
since the evaporation frequencies are ranging from 1.4 GHz to 1.8 GHz, or switching
to a horn antenna design because horn antenna at these frequencies gets very big andwon’t fit in the space.

2.2.3

BEC in an optical dipole trap

Even though we could get a BEC in the plugged magnetic trap, as almost all quantum
simulation experiments are carried out in the optical dipole traps nowadays, efficiently
transferring the atoms from the magnetic trap into a robust dipole trap is an essential
landmark for our quantum gases sample preparation.
We decide to have a crossed dipole trap, because compared to a single-beam dipole
trap, it doesn’t require extremely focused beams and create elongated cloud. To save
space on the optical table and minimize sloshing motion when ramping up the lattices,originally we intended to use LCD waveplates to swich the same beams between dipole
traps and lattices. However, it turns out that, these nematic liquid crystal waveplates
are typically controlled by amplitude-modulating a square wave at 5-20 kHz, which
creates intrinsic intensity noise and heats up the cloud. If optical access permits,
it is recommended to have separate lattice and ODT beams, decoupling the sample
preparation from the subsequent experiments.
35

We’ve set up imaging along the optical dipole beams (as well as all lattice beams)
to conveniently focus the beam at the cloud and achieve coarse alignment. For the finealignment, atoms caught in each of the dipole beams are locating the beam position,
and are used to align the beams to the magnetic trap center using in-trap imaging.
The cloud lifetime in the crossed trap is sensitive to overlap between the dipole beams,
and is maximized to fine tune their relative position. Finally, the plug beam could beslightly tweaked to maximize the transfer from the magnetic trap to the dipole trap.
Since there is always heating associated with the transfer into the dipole trap, we
prefer to be more conservative near the end of evaporation in the magnetic trap, to beable to load more atoms into the ODT, and then perform further evaporation there.
In pracitce, typically with dipole beams of 2 Watts in power each and of around 100µm waist, we are always able to catch a decent atom number in the dipole trap. The
crossed ODT is then slowly ramped down to its half depth for evaporation, in order tocreate a pure BEC. The typical atom number of the condensate in our crossed dipole
trap is about several million, good enough for most quantum simulation experiments.
An example of the F = 1 state BEC in the crossed dipole trap is shown in Fig. 2-4.

2.3

Sympathetic Cooling of 6Li with 23Na in the F= 2 State

Evaporative cooling would not work for a spin polarized Fermi gas, because the Pauli
statistics strongly suppresses s-wave collisions and thermalization process at low temperature. The way to overcome this problems is to employ some form of sympathetic
cooling between two distinguishable particles, which could be two spin states of thesame atom, or two different isotopes of the same chemical element, or two different
chemical elements. We continue the Ketterle lab tradition of using 23 Na to cool 6 Li,
because the large number of sodium from the dark-spot MOT and the favorable collision properties between these two species have proved that the bosonic sodium is agreat coolant for the fermionic lithium.
362.3.1

The hyperfine states of 6 Li and 23 Na

The hyperfine structures of 23 Na and 6 Li are shown in Fig. 2-5. The hyperfine splittingof 23 Na and 6 Li is at 1.7 GHz and 228 MHz, respectively. The lowest two hyperfine
states |1/2, −1/2i and |1/2, 1/2i of 6 Li are usually used as spin up and spin down in a
two-component Fermi gas experiment. In the previous section, we have demonstrated
the Bose-Einstein condensation in |1, −1i state of the 23 Na F = 1 hyperfine manifold.

Figure 2-5: The hyperfine structures of 23 Na and 6 Li. The hyperfine states are labeledin their low-field |F, mF i basis. In the previous section, sodium was cooled to BEC in
the |1, −1i lower hyperfine state. In this section, to sympathetically cool lithium, thecombination of lithium in |3/2, 3/2i and sodium in |2, 2i states is used, because bothstates are magnetically trappable and stable against spin-exchange collisions. This
figure is adapted from [2].
Among these hyperfine states, the combinations of 23 Na and 6 Li that are both magnetic trappable and stable against spin exchange collisions are |1, −1i + |1/2, −1/2i
and |2, 2i + |3/2, 3/2i. Earlier studies [2] have shown that the latter produce the best
results in sympathetic cooling of sodium and lithium, and is used in our experiment37

to achieve a quantum degenerate Fermi gas.

2.3.2

Condensation of 23 Na in the F = 2 hyperfine state

Most sodium atoms in a dark-spot MOT are in |1, −1i state. To sympathetically coollithium, we need to first transfer the sodium atoms from |1, −1i state to |2, 2i state,
and then evaporatively cool |2, 2i state sodium with 1.7 GHz microwave radiation.

Figure 2-6: Gravity-assisted spin purification and optical pumping of sodium into the
F = 2 upper hyperfine state. An additional stage of mF optical pumping at F = 2
hyperfine states boosts the transfer efficiency by about 20 percent. Gravity-assisted
spin purification removes the unwanted |F = 2, mF = 0, 1i states, preventing inelastic
spin-exchange collisions and consequent heating.
The transfer from F = 1 to F = 2 states in sodium is achieved by optical pumping.
The σ + polarized light couples the F = 1 states into the excited states, and then theatoms partially decay into the F = 2 states in spontaneous emission. To optical pump
the atoms onto the preferred Zeeman state |2, 2i and improve the pumping efficiency,

an mF pumping with σ + polarized light at F = 2 → F 0 = 2 transition is applied tomake |2, 2i a dark state and avoid unnecessary heating of the cloud. This additionalmF optical pumping boosted the transfer efficiency into the correct Zeeman state bymore than 20 percent, illustrated in Fig.2-6.
Atoms in the other Zeeman sub-levels |F = 2, mF = 0, 1i of F = 2 hyperfine states
are susceptible to spin exchange collisions and consequent heating, and need to be
removed from the trap before evaporation. Traditionally with an Ioffe-Pritchard trap,38

this is accomplished by applying a bias field to split the F = 2 Zeeman sub-levels, andthen selectively transferring these unwanted states into untrapped |F = 1, mF = 0, 1ilower hyperfine states using microwave transitions. In our new apparatus with thequadrupole magnetic trap, we’ve developed a new gravity-assisted spin purification
scheme, demonstrated in Fig. 2-6. This method uses the fact that the Zeeman states
have different magnetic moments, and therefore experience different trapping forcesin the magnetic trap. As we lower the trap, magnetic force is not able to hold up theatoms against gravity in the unwanted states |F = 2, mF = 0, 1i, and therefore only
atoms in |F = 2, mF = 2i are preserved in the end.
Finally, the microwave evaporation in the F = 2 state is fairly similar to the case of
F = 1, except that the microwave frequency is ramped down, instead of being ramped
up. The three-body collision loss rate is a bit higher for F = 2 states compared with
F = 1 states, and therefore the magnetic trap needs to be decompressed earlier before
further evaporation. The evaporation sequence is empirically chosen to optimize thefinal atom number in the condensate.

2.3.3

Sympathetic cooling of 6 Li with 23 Na

While there were major upgrades on the sodium table and infra-red setup, the lithium
table was almost kept the same. Most efforts were made to improve the temperaturestability, by moving the hot lithium cell onto a separate breadboard outside the maintable, and by building small enclosures to confine the air circulation. In addition,a New Focus Vortex II tunable laser (TLB-6900) was bought and phase locked tothe main laser. This laser has a total output of 15 mW and a large mode-hop freetuning range (>120 GHz), perfect for the lithium high-field imaging near the Feshbachresonance.
In making a lithium MOT in our new apparatus, a tapered amplifier was added toboost the slower light power and the MOT loading. In order to optimize the Zeeman
slower for a sodium-lithium dual-species MOT, first the current in its last section is
tweaked for the sodium MOT, and then the lithium slower frequency is scanned to findits optimal value for the lithium MOT. Currently with things tweaked and optimized,39

we have a good performing dual-species MOT. However, future generation is advised
to consider adding tapered amplifier to the MOT/repump light for compensating thelarger beam sizes in our new MOT system.
During the transfer from dual-species MOT into magnetic trap, we also opticallypump 6 Li into the stretched |3/2, 3/2i state. Again, two σ + polarized laser beams
are used, resonant with the transitions from F = 1/2 and 3/2 ground states. In caseof 6 Li, the excited states are not resolved, so the discussion of the dark state during
the optical pumping in 23 Na does not arise here. The optical pumping performancecan be checked by imaging the F = 1/2 and 3/2 state population independently. The
gravity-assisted spin purification method doesn’t work for lithium when both species
are loaded in the magnetic trap, due to the much lighter weight of the lithium atoms.
One could first load lithium alone and purify its spin states in the magnetic trap, andthen load the sodium subsequently. However, we find this procedure too cumbersome
and in practice unnecessary.

Figure 2-7: The sympathetic cooling of lithium with sodium in the F = 2 state. The
lithium cloud shows a decreased temperature as we evaporate the sodium. However,with a fully loaded MOT, the lithium gas doesn’t reach the deep quantum degeneracy
even when all sodium atoms are evaporated.
This is pretty much all we have to do. At this point, we simply evaporate sodium
the same way as in Section 2.3.2. The sympathetic cooling of lithium is observed inFig. 2-7, as we evaporate sodium, the temperature of the lithium cloud also decreases.
40

Figure 2-8: The efficiency of the sympathetic cooling. The two images on the left show
the lithium cloud at the end of sympathetic cooling with initially (a) more lithium
atoms or (b) fewer lithium atoms loaded into the magnetic trap. Image (c) shows alithium cloud loaded into the optical dipole trap after the sympathetic cooling.
However, it seems that for a fully loaded lithium MOT, we are not able to reach thequantum degeneracy of lithium even when all the sodium atoms are evaporated. To
trade in the 6 Li atom number for deeper quantum degeneracy, we deliberately switch
on the lithium slower light only at the end of the MOT loading. This tradeoff turned
out to be quite favorable, as we were able to get a very cold lithium cloud after the
sympathetic cooling in the magnetic trap, and subsequently loading it into the optical
dipole trap created a quantum degenerate Fermi gas of 6 Li, shown in Fig. 2-8.

2.4

Hyperfine State Preparation of 23Na

In the final section, I’ll describe how we use microwave and radio-frequency radiations
to drive transitions between the hyperfine states in sodium. These experiments are
performed in order to benchmark the performance of our antennas by measuring thecorresponding Rabi frequencies, as well as to develop the required techniques for state
preparation in future experiments, such as a spin mixture of lithium |1/2, −1/2i and|1/2, 1/2i states in strongly interacting Fermi gases, or a pure sodium condensate in
the stretched |2, ±2i state for the synthetic gauge field experiments in the optical lattices, where very homogeneous tilts are created by applying a magnetic field gredientacross the cloud. Finally, two-photon transitions are also demonstrated in the F = 1
to F = 2 transition in sodium, as it has been proved to be a powerful way to measure
the filling number in the Mott insulator state [14].
412.4.1

Microwave transitions between hyperfine states

Landau-Zener transition is a powerful method to prepare spin mixture states in quantum gases. In the first experiment, we have performed a Landau-Zener sweep between
F = 1 and F = 2 hyperfine states in sodium, shown in Fig. 2-9. A weak bias magnetic
field is turned on to split the Zeeman sub-levels, and the microwave radiation centeredat 1762.9 MHz is linearly swept over 0.3 MHz in 10 milliseconds, slow enough to keep
the sodium atoms following the adiabatic passage. In the first one-way sweep, almost
all sodium atoms flip to the other hyperfine states, and the second return-trip sweep
confirms that this process is fully reversible and does not create much heating.

Figure 2-9: Spin flips between sodium F = 1 and F = 2 hyperfine states via LandauZener transition with microwave radiations around 1.763 GHz. The transfer efficiency
is close to unity shown in the single sweep, and this process is reversible without muchheating demonstrated in the double sweep.
In the second experiment, we drive the microwave transition on resonance to makecoherent transfer in the Rabi oscillation between F = 1 and F = 2 hyperfine states.
Again, a weak bias field is applied to split the Zeeman sub-levels, before the microwave
pulse is switched on at the resonance frequency for a variable time. The population in
the initial and final state can be measured independently, and are shown as a functionof the microwave pulse time in Fig. 2-10. A full Rabi oscillation was observed, andthe Rabi frequency of this microwave transition is estimated to be around 4 kHz.
42

Figure 2-10: Rabi oscillations in the microwave transition between |1, −1i and |2, 0i
hyperfine states. A weak bias field is switched on to split the Zeeman sub-levels, anda microwave radiation pulse at the resoance refrequency of 1768.738 MHz is applied
to the sodium condensate in the dipole trap for a variable duration. These images
show an resonant Rabi oscillation at the estimated Rabi frequency of about 4 kHz.

2.4.2

Radio-frequency transitions between hyperfine states

Similarly, radio-frequency can be used to drive transitions between Zeeman sub-levelsin the hyperfine states. For F = 1 hyperfine state in sodium, there are three Zeeman
sub-levels |F = 1, mF = 0, ±1i. However, the selection rules predict that there is no
direct one-photon transition between the |1, −1i and |1, 1i states.

Figure 2-11: Spin flips between F = 1 hyperfine states with radio-frequency radiations
and the Stern-Gerlach experiment. A radio-frequency pulse at 2.933 MHz is switchedon for a variable time to drive the two-photon transition between |1, −1i and |1, 1i. A
magnetic field gradient is rapidly switched on during the time-of-flight to separate the
Zeeman sub-levels and perform a Stern-Gerlach experiment to read out the populationin each hyperfine state. These images show an estimated Rabi frequency of 25 kHz.
To drive the RF transition, again a weak bias field is turned on to split the Zeeman
sub-levels. The hyperfine coupling in 23 Na is so strong that, this bias field is far inside
the low field regime of the Breit-Rabi problem, and therefore these three Zeeman sublevels have linear Zeeman shifts and are equally spaced. We drive this RF transitionat the resonance frequency of 2.933 MHz for a variable time. Due to the linear Zeeman
shifts, effectively all three states are coupled, and a way to selectively read out eachstate is needed. In contrast to the microwave transition, this splitting is much smaller43

than the sodium natural linewidth of 9.8 MHz, and thus cannot be resolved optically.
Instead, we perform a Stern-Gerlach type experiment, where a strong magnetic field
gradient is suddenly switched on during the time-of-flight to spatially separate these
Zeeman sub-levels, and a single absorption image is taken to measure the populationin each state, as demonstrated in Fig. 2-11. The measured Rabi frequency for the RFtransitions is about 25 kHz in our system.

2.4.3

Two-photon transitions between hyperfine states

Finally, we perform a two-photon transition between the F = 1 and F = 2 hyperfine
states in sodium. This is relevant because the two-photon techniques can be applied tothe clock transition, which is between |1, −1i to |2, 1i states in sodium, to measure thefilling number in a Mott insulator [14]. This transition is called the clock transition,
because both the initial and final states have the same first-order Zeeman shift, andtherefore it is widely used in atomic clocks to avoid the Zeeman shifts and broadening
from magnetic field fluctuations.

Figure 2-12: Two-photon Rabi oscillations between F = 1 and F = 2 hyperfine states
in sodium. The two-photon coupling is provided by 1.77 GHz microwave photons and
2.4 MHz radio-frequency photons. The observed resonant Rabi oscillations indicate
a two-photon Rabi frequency of about 200 Hz.
In Fig. 2-12, we have shown the Rabi oscillation of a two-photon transition between
the |1, −1i to |2, 0i states in sodium. The final state is identified by the relative position of the nearby resonances. This two-photon coupling is provided by a microwave
at 1769.237 MHz and a radio-frequency at 2.408 MHz. The resonant Rabi frequencyof the two-photon transition is measured to be around 200 Hz.
44Chapter 3
Bose-Einstein Condensates in
Optical Lattices
3.1

Introduction

Quantum simulation using ultracold gases in the optical lattices has become an very
active field in the past decade. This system has not only been demonstrated as a great
platform for testing fundamental concepts in quantum physics, but also has promised
a set of novel Hamiltonians that are far beyond the reach of other systems. The early
success of quantum simulations has also attracted a lot of interests from other fieldsof physics in condensed matter and high energy communities, as it has proved itself
a powerful tool to tackle some long-standing unsolved problems in physics.
The field of quantum simulation in optical lattices has become very diverse lately.
However to me, the progress of the mainstream and the milestones in this field can bedivided into roughly speaking three phases. The first phase is focused on the behaviorof the condensates in optical lattices, from the first experimental observation of thequantum phase transition from a superfluid to a Mott insulator from the interference
peaks during time-of-flight [15], to the recent microscopic studies of this phase transition using single-site resolution imaging [16, 17], and thorough characterization of thequantum criticality near the transition [18]. The second phase is regarding fermions in
optical lattices, including the experimental observation of fermionic superfluidity [3],45band insulator and Mott insulator [19, 20], and most recent efforts towards probing
the anti-ferromagnetic state in a Fermi-Hubbard model [21, 22]. In the past five years
or so, there has been a wave of research works on bringing gauge fields into the system with complex tunnelings [23, 24, 25, 26], and simulating novel Hamiltonians with
topological properties [27, 28]. These exciting progresses have opened up a whole newphase of quantum simulations in optical lattices.

During the design and construction of our new apparatus, the goal we envisioned
initially was to observe the antiferromagnetic ordering in the Fermi-Hubbard model.
However, another opportunity later arose, as a novel scheme for implementing synthetic gauge fields became available. A different lab in our group working with 87 Rb
was in the process of implementing this scheme, but they observed anomalous heatingthat had prevented them from reaching the ground state of the Harper Hamiltonian.
It appeared that the fast tunneling of sodium and the ease of making large magnetic
gradients in our apparatus might be the solution to solve this heating problem. Thesethoughts have led to an extended study of the relationship between interactions andthe Bloch oscillations described at the end of this chapter. Now the rubidium lab havefound strong evidence of observing the ground state of the Harper Hamiltonian [26],
we have largely turned our focus towards implementing spin-orbit coupling in superlattices and synthetic gauge fields with fermions, presented in the next chapter.

In this chapter, we will first introduce the basic picture and concepts of the physics
in optical lattices. Then we experimentally demonstrate the quantum phase transition
from a superfluid to a Mott insulator, as a way to characterize our new optical latticesystem. Finally, we will describe our experiments of making very strong tilts in an
one-dimensional optical lattice to drive the Bloch oscillations, as well as our studies
on the interaction effects in the Bloch oscillations, such as dynamical instability andrelaxation in the quasi-momentum states. All of these studies pave the road to a new
implementation of spin-orbit coupling in a superlattice described in the next chapter.
46

3.2

Basic Lattice Physics

In general, the Hamiltonian of a quantum gas consists of a single-particle term andan interaction term,
H = H0 + Hint

(3.1)

Let us start with the single-particle Hamiltonian, which includes the kinetic energy−h̄2 ∇2 /2m and the external potential V (r). In solid-state systems, the externalpotential is usually from the static electric field of crystalline ions. In an opticallattice system, the external potential is from the AC-Stark shift via photon-atominteraction, and consists of two parts
H0 = −h̄2 ∇2 /2m + VL (r) + VT (r) = H0L + VT (r)(3.2)

the former VL (r) is a periodic lattice potential created by the standing wave, while thelatter VT (r) is any additional potential, for example, the harmonic confinement fromoptical dipole traps. As the extra potential is often slowly-varying compared to theoptical lattices, one can make the assumption that it is constant locally and first study
H0L in a homogeneous setting, and then consider the inhomogeneous effects of VT (r)with a local chemical potential µ(r) = µ − VT (r) in the local density approximation(LDA).

3.2.1

Band structure and Bloch states

In this section, we include a solid-state textbook calculation of the band structure in
an one-dimensional optical lattice. The more general question can be raised as, how
do we solve the eigen-problem for a periodic potential?
The eigen-states of an optical lattice, also known as the Bloch states φnk (r) (withband index n and quasi-momentum k), need to satisfy
H0L |φnk (r)i = εnk |φnk (r)i
47

(3.3)

where φnk (r) = eik·r unk (r) and unk (r) has the same periodicity as the lattice by Bloch’s
theorem.
One needs to diagonalize the Hamiltonian to find the eigen-states. In the basis ofplane waves, the kinetic energy term is diagonal. A key observation is that, for the caseof a sinusoidal potential, only plane waves with momentum shift ±2kL are coupled.
This dramatically reduces the size of the Hilbert space we need to consider. Finally,
the eigenvalues of the Hamiltonian tell us the band structure, and superposition of
the plane waves weighted by the eigenstates give us the Bloch states.
For the rest of this section, we’ll walk through some details of this calculation.
First of all, since unk (r) has the same periodicity of the lattice, one can expand unk (r)in its Fourier components,
1 X n
unk (r) = √ak (G)eiG·r
Ω G

(3.4)

substitute φnk (r) with the Fourier expansion into the eigen-problem,
H0L

X
ei(G+k)·r
ei(G+k)·r= εnk
ank (G) √ank (G) √
Ω
Ω
G
G

X

(3.5)

√
0
We convert the equations into a matrix form by multiplying e−i(G +k)·r / Ω on bothsides and applying integration over space,X

ank (G)HG0 ,G =GX

ank (G)εnk δG0 ,G

(3.6)

Z

(3.7)

G

with matrix element of the Hamiltonian1
HG0 ,G =ΩZe−i(G0 +k)·rh̄2 2 i(G+k)·r
1
(−
∇ )e
dr +
2m
Ω

0

e−i(G +k)·r VL ei(G+k)·r dr

The first term is non-vanishing only when G0 = G, meaning that the kinetic energy isdiagonal in momentum. The second term corresponds to an optical-lattice potential
V = V0 sin2 (kL r), which only couples plane waves with G0 = G ± 2kL , in addition to48

an overall energy offset.
HG0 ,G =V0V0h̄2|G + k|2 δG,G0 + δG,G0 − (δG+2kL ,G0 + δG−2kL ,G0 )
2m
2
4

(3.8)

In summary, the single-particle Hamiltonian in an optical lattice in the basis ofplane waves is written as


···


 ···


H(k) = 





2

···

h̄
|k − 2kL |2 + V0 /22m−V0 /4
−V0 /4h̄22|k| + V0 /2−V0 /42mh̄2
2−V0 /4|k + 2kL | + V0 /2 · · · 
2m

···
···
(3.9)

For actual numerical calculation, one needs to cut off this infinite matrix at a
certain size, because couplings into those high momentum states are far detuned
and thus negligible. With efficient commerical matrix eigen-solver from Matlab orMathematica, one finds the band structure and the corresponding Bloch states.

Figure 3-1: The band structures of a one-dimensional lattice are plotted for lattice
depths between 0ER and 12ER . Even bands are shown in solid lines, odd bands areshown in dotted lines, and the lowest band is shown in red. As the lattice gets deeper,
the lowest band becomes flat and the band gap becomes large.
In Fig. 3-1 and 3-2, we’ve plotted the Bloch bands for various lattice depths, and49

the Bloch states at the center and the edge of the Brillouin zone.

Figure 3-2: The real part and probability (absolute-value squared) of Bloch statesφq (r) with quasi-momentum q = 0 (in purple) and q = h̄kL (in green) in the lowest
band, for a one-dimensional lattice with depths of V0 /ER =3, 5, 10 and 30, respectively.

3.2.2

Tight binding and Wannier states

To study the physics of localized particles, such as superfluid to Mott insulator phasetransition and quantum magnetism in optical lattices, it is often more convenient toconsider the tight binding limit and use a set of localized states, the so-called Wannier
states.
It’s worth mentioning that Bloch states and Wannier states are both complete
orthonormal basis for the single-particle Hilbert space, as the field operator can be
decomposed asψσ† (r) =Xi,nwn∗ (r − Ri )c†inσ =X

φnk (r)c†knσ(3.10)

k,n

The difference is that Bloch states describe the itinerant behavior in the metallic orsuperfluid phase, while Wannier states features localization in the insulator phase.
This is in analogy to the wave-particle duality in quantum physics, in which the wave
representation and the particle repensentation complement each other.
50

n
One can construct a set of Wannier states wR(r) = wn (r−Ri ), which are localizediat every lattice site Ri , using linear combinations of the Bloch states in the same band,
1 X −ik·Ri n
1 X ik·(r−Ri ) n
n
√
√
wR
(r)
=
w
(r
−
R
)
=
e
φ
(r)
=
e
uk (r)
n
i
k
i
N k
N k

(3.11)

One caveat here is that, there are actually many different ways of choosing the linear
combinations, namely the relative phases of the Bloch states in the superposition.
The consensus regarding this procedure is to choose the superpostition that gives the
maximally-localized Wannier states.

Figure 3-3: The Wannier states in the lowest band. The left figure plots the Wannier
states in a one-dimensional lattice for various lattice depths, whereas the right figure
plots the Wannier state in a two-dimensional square lattice with equal strength V0 =10ER . Both figures show that Wannier states have secondary maxima at the nearestneighbors, which provide tunneling between them.

In Fig. 3-3, we plot the Wannier states in one-dimensional and two-dimensionallattices. Each of them is mainly localized in a single lattice site, and has a small
overlap with its nearest-neighbors. This overlap is responsible for the amplitude ofnearest-neighbor tunneling.
51

Second quantization of the single-particle Hamiltonian in the Wannier basis gives
H0L =

XZ
σ

=

X

drψσ† (r)[−h̄2 ∇2 /2m + VL (r)]ψσ (r)k,n c†knσ cknσ =

X

i,n,σ

k,n,σ

n0 c†inσ cinσ −(3.12)

XXnσtnij c†inσ cjnσ(3.13)

ijwith the tunneling amplitude and onsite single-particle energies given by
tnij = −Z

drwn∗ (r − Ri )[−h̄2 ∇2 /2m + VL (r)]wn (r − Rj ) = −
n0 =

X

X

eik·(Ri −Rj ) k,n (3.14)

k

k,n

(3.15)k

There are a few comments to be made about this derivation. First of all, because
the Wannier states are superposition of Bloch states in the same band, the Hamiltonian does not couple Wannier states in different bands and thus there is no interband
tunneling. Furthermore, for separable optical lattices in higher dimensions, for example a cubic lattice, tunneling is non-vanishing only along the principle axes. In otherwords, there is no tunneling along diagonal directions.
For deep lattices, Wannier states are well localized and their amplitude goes verysmall after a couple of lattice constants. Deep lattices also exhibit large band gaps,
and low-temperature quantum gases usually only populate the lowest band. Therefore, we will only keep the nearest-neighbor tunneling in the lowest-band. The singleparticle Hamiltonian in the single-band Wannier states simply reads
H0L =

X
i,σ

0 c†iσ ciσ −XXσ

tc†iσ cjσ

(3.16)<ij>

where < ij > includes all nearest-neighbors i and j, and the nearest-neighbor tunneling amplitude in the lowest band ist=−Z

drw1∗ (r − Ri )[−h̄2 ∇2 /2m + VL (r)]w1 (r − Rj ) = −52Xkeik·(Ri −Rj ) k,1 (3.17)

3.2.3

Lattice calibration via Kapitza-Dirac diffraction

The depth of an optical lattice in principle can be calculated based on the power
and waist of the beam. Nonetheless, there are often practical imperfections from theoverlap and polarization, and the atoms themselves are always the best probe of thelattice depth.
The depths of optical lattices are routinely calibrated in our lab via Kapitza-Diracdiffraction. During the experiment, a very short pulse of optical lattice is applied to acondensate, and then after a few milliseconds of time-of-flight, the diffraction patternis recorded to analyze the strength of the optical lattice potential.
When the short pulse is diabatically switched on, the initial zero-momentum stateis projected onto the zero-quasimomentum states in each band of the lattice, which
evolve at different energies during the pulse. The zero-quasimomentum state at eachband has different contribtions from ±2h̄kL , ±4h̄kL , and higher momentum components. As time evolves, these states in different bands become in and out of phase,and this results in oscillations in the diffraction pattern.

Figure 3-4: Kapitza-Dirac diffraction for a shallow lattice at 10ER and a deep lattice
at 30ER . The time evolution of the relative strength of the diffraction peaks areshown for 532 nm lattices.
Fig. 3-4 shows different time-evolution of the diffraction patterns for a shallow
lattice and a deep lattice. For shallow lattices, only the ground and first excited
band are populated, and the diffraction pattern shows oscillations at the band gap
frequency. For deep lattices, more momentum states are excited, the time-dependence
becomes more complicated, and experimental data are usually more difficult to fit.
53

In practice, the number of visible peaks in the Kapitza-Dirac diffraction gives areliable rough estimate of the lattice depth. Precision calibration is usually performed
first with a shallow lattice by extracting the oscillation period, and then the depthof a deep lattice is extrapolated by scaling the optical power. For example, Fig. 3-5
shows an experimental calibration of our 532 nm lattice. An oscillation at about 4µs in the diffraction pattern gives an estimate of 10ER in green, corresponding to theleft plot in Fig. 3-4. Careful calibration is done by fitting the diffraction pattern as afunction of time and matching the time evolution at a certain lattice depth.

Figure 3-5: Kapitza-Dirac diffraction for various pulse durations in a 532 nm opticallattice. An oscillation period around 4 µs indicates a depth of 10 green recoils.

3.3

Superfluid to Mott-Insulator Quantum Phase
Transition

3.3.1

Mean-field theory for a Bose-Hubbard model

Short-range interactions in quantum gases
Now let’s turn to the many-particle Hamiltonian. In quantum gas experiments withlow density and low temperature, the inter-particle distance and lattice spacing are
54

usually much larger than the hardcore radius of the inter-atomic potential. In thislimit, the short-range details of this potential could be ignored, and inter-particle
interaction is characterized by a single parameter, the s-wave scattering length as .
In most cases, an effective pseudo-potential captures most part of the effects of theshort-range interaction between atoms,
Vint (r − r0 ) = gδ(r − r0 )(3.18)

with parameter g defined as g ≡ 4πh̄2 as /m. A more stringent form of this contact∂
pseudo-potential contains a regularization operator ∂rr, which can be left out as longas the wavefunction is slowly varying at short range.
The s-wave scattering length depends only on the details of inter-atomic potential,
and is fixed once the internal atomic structure is given. However, if couplings exist
between the collision channels and bound state channels of this potential, one could
apply a magnetic field to shift the potentials of two channels with different magnetic
moment with respect to each other. Resonant scattering behavior would occur when
the two channels are near degeneracy. This phenomenon is called magnetic Feshbachresonance, and was first observed in [29]. Unfortunately, most BEC systems sufferfrom strong inelastic collision losses near a Feshbach resonance, which limits theatomic density and lifetime. In contrast, strongly interacting fermions are generally
longer-lived due to their different quantum statistics, and examples include broad
Feshbach resonances found in 6 Li and 40 K. Feshbach resonance is a useful techniqueto change atomic interaction in quantum gas experiments, whenever it is available.
The Bose-Hubbard model
Second quantization of the atomic interaction in the Wannier basis goes as follows
Hint = gXZdrψσ† (r)ψσ† 0 (r)ψσ0 (r)ψσ (r)

σ,σ 0

=

X

ν1 ν2 ν3 ν4
X UR
1 R2 R3 R4 †cR1 ν1 σ c†R2 ν2 σ0 cR3 ν3 σ0 cR4 ν4 σ
2σ,σ 0 R R R R ν ν ν ν
1

X
2

3

4

1 2 3 455

(3.19)

where the interaction strength is a product of the Wannier state overlap,Z
4πh̄2 asν1 ν2 ν3 ν4
UR1 R2 R3 R4 =mdrwν∗1 (r − R1 )wν∗2 (r − R2 )wν3 (r − R3 )wν4 (r − R4 ) (3.20)

Again, a few simplifications can be applied here. First of all, since Wannier states
are localized, the dominant contribution corresponds to the on-site interaction whenmore than one particle occupy the same lattice site. Furthermore, for deep lattices,higher bands are separated from the lowest one and can be neglected. This leaves us
only the on-site interaction in the lowest band,
Hint =X X U0 † †cRi σ cRi σ0 cRi σ0 cRi σ
2
0
R σ,σ

(3.21)iwhere on-site interaction is
4πh̄2 asU0 =mZ

dr|w1 (r − Ri )|4(3.22)

Together with the single-pariticle Hamiltonian, we have derived the general formof a single-band Hubbard model,
H=

X
i,σ

0 c†iσ ciσ −XXσ

tc†iσ cjσ +<ij>

X X U0 † †ciσ ciσ0 ciσ0 ciσ
2
0
i
σ,σ

(3.23)For a single-component bosonic system, this is known as the Bose-Hubbard model(setting the energy offset to zero)
H=−

X

<ij>

tc†i cj +X †X U0 † †U0 X
ci ci ci ci = −t
ci cj +ni (ni − 1)
2
2 i
<ij>
i

(3.24)The two parameters in this model, the nearest-neighbor tunneling t and on-site interaction U , are numerically calculated for an equal-strength three-dimensional cubic
lattice as a function of lattice depths V0 /ER in Fig. 3-6.
The kinetic and interaction terms favor quantum states with very different characters. When the kinetic energy dominates, that is t  U , the system minimizes its
56

Figure 3-6: Tunneling amplitude t and on-site interaction U , as a function of lattice
depth V0 , for a three-dimensional cubic lattice. Tunneling amplitude t is plotted inunits of ER , and on-site interaction U is in units of 100ER as /d, with as the scattering
length and d the lattice spacing. As we increase the lattice depth, tunneling t decreases
because of the reduced overlap between neighboring sites, and interaction U slightly
increases because of the squeezing of localized Wannier states.
energy by having particles delocalize and be at zero momentum. The ground state isa superfluid that can be written as
1 X † Na ) |0i|ΨSF iU/t→0 = (M i=1 iM

(3.25)

where M is the total number of lattice sites, N is the total number of particles, andlocal creation operator a†i |ni = |n + 1i at ith site. In a grand canonical ensemble, it
is a coherent state at each site with an average density n = N/M , and occupationshows Poissonian statistics.
When the on-site interaction dominates, that is U  t, the system minimizes its
energy by having particles uniformally distribute and suppress particle-hole excitation. The ground state is a Mott insulator, which consists of a product of Fock stateswith the same number of particles on each site,|ΨM I it/U →0 =MY(a†i )n |0i(3.26)

i=1

A quantum phase transition from a superfluid to a Mott insulator occurs when
the ratio t/U is tuned across the critical value in the Bose-Hubbard model. This is57

usually done experimentally by varying the lattice depth becuase of the convienence.

Mean-field theory for Bose-Hubbard model
Let’s start with the single-component Bose-Hubbard model in a grand canonical ensemble,
H=−

X

<ij>

tij c†i cj +X
U0 X
ni (ni − 1) − µ
ni
2 i
i

(3.27)

A mean-field theory is formulated by replacing operators by their expected values.
In this case, they are the expected values of creation and annihilation operator of theneighboring sites acting as an effective field.
In this mean-field approach, the tunneling term can be decomposed asc†i cj → const + hc†i icj + c†i hcj i + quantum fluctuations(3.28)

then the effective Hamiltonian can be written as
†
i
∗
Hef
f = −λi ci − λi ci +U0n(n − 1) − µn2(3.29)

This mean field, or sometimes referred as the “Weiss field”, is characterized byparameter λi . It is no longer an quantum operator but a constant, and it is evaluatedby taking the average of expected values of nearest-neighbor tunneling
λi =

X
j

tij hcj i(3.30)For a homogeneous and isotropic Bosonic system, all nearest-neighbor couplings
have equal strength. This means that the “Weiss field” is proportional to the total
number of nearest-neighbors, or the connectivity z,
λ = zthci(3.31)In the mean-field theory, we assume different lattice sites are coupled only via
58

Figure 3-7: Phase diagram of the Bose-Hubbard model at zero temperature is plotted
as a function of chemical potential µ/U and normalized tunneling zt/U . The predicted
phase transition is shown in solid purple line for a first-order perturbation theory,and in dashed blue line for numerical simulation. The dotted-dashed gray lines show
the trajectory for a gas with various integer fillings in the superfluid phase. The
critical ratio of (U/zt)c '5.83, 9.90 and 13.93, for the filling number n =1, 2 and 3,
respectively, and it asymptotically reaches 4n when n is large [30].

this effective field, and quantum fluctuations and short-range correlations can beneglected. This effectively reduces the Bose-Hubbard model into a local problem at
each site with self-consistent requirement. One could use the Fock states |0i, |1i, |2i,
etc., as the basis, the effective Hamiltonian in the mean-field theory is written as0−zthci√ −zthci−µ− 2zthci


√
√


Heff = H − µN = − 2zthci U − 2µ − 3zthci


√



− 3zthci 3U − 3µ · · · 


···
···
59

(3.32)

To solve this problem, one first diagonalizes the Hamiltonian and finds the groundstate, which is a superposition of Fock states. This calculated ground state is thenused to evaluate the expected value hci, and subsequently the Weiss field z 0 . Iterate
this procedure until z converges, that’s when the self-consistent condition is satisfied.
The Weiss field λ is also an order-parameter to characterize the quantum phasetransition from a superfluid to a Mott insulator. It is non-vanishing in the superfluid
phase, and zero in the Mott insulator phase. The mean-field predictions of the BoseHubbard model phase diagram are summarized in Fig. 3-7.

Density distribution in an inhomogeneous system
Optical lattice experiments are usually carried out in a finite-size and inhomogeneous
setting. For example, there are often residual harmonic confinements associated withthe lattice beams, or from additional dipole traps.

Figure 3-8: Left: Phase diagram of the Bose-Hubbard model in a three-dimensionallattice. An incompressible Mott insulator is found inside each lobe with integer filling.
Right: Atomic density distribution plotted as a function of t/U in a harmonic trap.
The density distribution is obtained by taking the cross section at fixed t/U from theleft phase diagram.
Local density approximation (LDA) is valid when these additional potentials are
slowly-varying in space compared with the lattice potential. Therefore, locally the60Figure 3-9: The density distribution of the Bose-Hubbard model in a harmonic trap.
The “wedding cake” structure results from the incompressibility in the Mott insulator.
Four density profiles are simulated with 6.0×104 atoms in a 200×200 two-dimensionallattice, with parameter t/U = 0.001, 0.01, 0.02 and 0.03, respectively, covering thephase transition from a Mott insulator to a superfluid.

atomic cloud can be regarded as a homogeneous system, with a local chemcial potentialµ(r) = µ0 − VT = µ0 − mω02 r2 /2(3.33)

With the local density approximation, one can extract the density profile in a trapfrom a cross section in the phase diagram shown in Fig. 3-8, as the local chemical
potential is higher in the trap center and lower near the edge. Depending on t/U ,the density profile shows dramatic difference across the phase transition. When t/Uis large and the superfluid phase prevails, the density profile is smoothly-varying,similar to a weakly interacting Bose gas in the trap with a modified effective mass.
When t/U is very small and the Mott insulator phase dominates, the density profile
61

shows flat Mott plateaus with integer filling, and sharp edges between these plateaus.
The density distribution of a Mott insulator with discretized plateaus is sometimes
referred as the “wedding-cake” structure, illustrated in Fig. 3-9. It is a clear signature of the Mott insulator, and has been experimentally detected via high-resolution
in-situ imaging [16, 17]. The Mott insulator also features a finite energy gap andexponentially-vanishing compressibility, and was previously observed in [15, 31].

3.3.2

Observation of the superfluid to Mott insulator transition

Quantum phase transition from a superfluid to a Mott insulator is one of the recent
highlights in quantum simulation with ultra-cold atoms. It was first reported from theMunich group in 2002 [15], and still regarded as a solid starting point for optical latticeexperiments. For this reason, we have characterized our new apparatus by observing
this quantum phase transition and measuring the lifetime of the Mott insulator, anddemonstrated the capability of simulating new interesting Hamiltonians and exotic
quantum states in our optical lattice system.

Detecting the phase transition in the interference pattern
A loss of coherence in the time-of-flight interference pattern for a Mott insulator, as
demonstrated in the very first experiment [15], is a clear and robust signal of thephase transition.
To drive the system from a superfluid to a Mott insultor, one typically could either
reduce the tunneling by ramping up the lattice, or increase the on-site interaction via
Feshbach resonance. The latter option is not available for us, due to the lack of broad
Feshbach resonance in 23 Na, and the limited condensate lifetime caused by strong
inelastic collisions near a Feshbach resonance [29].
In this experiment, after a pure condensate is prepared in the optical dipole trap, athree-dimensional lattice is adiabatically ramped up to its final depth in 100 ms. Afterwards, the lattice and dipole trap are simultaneously turned off, and an absorption
62

image of the cloud is taken after 10 ms free expansion in the time of flight.

Figure 3-10: Experimental observation of a superfluid to a Mott insulator quantum
phase transition. An optical lattice is adiabatically ramped up to its final depth in
100 ms, then simultaneously turned off with the dipole trap to release the gas forfree expansion. The sharp peaks in the expanded cloud indicate superfluid coherence
across the sample in shallow lattices, and they disappear as the sample goes acrossthe phase transition into a Mott insulator.
In Fig. 3-10, we’ve shown the time-of-flight image of the condensate after being released from the optical lattice with variable depths. The sharp peaks in the expanded
cloud correspond to the ±2kL components of the groud state with quasi-momentum
q = 0, indicating a global coherence and a uniform phase across the sample. These
interference peaks disappear in deep lattices, as the sample goes across the phasetransition into a Mott insulator and loses the global coherence.
Measuring the Mott insulator lifetime
To demonstrate that the loss of coherence is in fact due to the Mott insulator phasetransition, rather than heating of the sample caused by the deeper lattice, and to characterize the technical heating in our apparatus, we perform a lifetime measurement of
the sample in the Mott insulator phase. The experimental sequence is that, we startwith a superfluid, ramp it into a Mott insulator and hold it for a variable amount oftime, and then ramp it back to a superfluid to measure the residual coherence.
The results are shown in Fig. 3-11, from where we see the coherence of the sample
almost disappears after 150 ms hold in the Mott insulator at 22ER . This measured
63

lifetime is mostly limited by three-body collision loss, and reducing the atomic density
should significantly improve the lifetime of the Mott insulator to be around a second,
suitable for studying super-exchange physics in optical lattices in the future.

Figure 3-11: The lifetime of a Mott insulator at 22ER . A superfluid is first preparedin a lattice at 8ER , which is then ramped up to 22ER in 100 ms and held for a variable
time, and finally ramped back down to 8ER in 20 ms. The loss of the coherence as
we increase the hold duration in the Mott insulator occurs as a result of heating,typically from three-body collisions or mechanical vibrations of the optics.

3.4

Tilted Lattices and Bloch Oscillations

When a uniform gradient potential is applied to the condensate in an optical lattice,
the atoms experience a constant force and undergo Bloch oscillations. This tilt creates an energy offset ∆ between every neighboring sites, providing another energy
scale besides tunneling t and on-site interaction U . Tilted optical lattices is an ideal
platform to study the dynamics of interacting particles in Bloch oscillations, or to
implement Raman-assisted tunneling and realize synthetic gauge fields for quantum
gases in optical lattices [25].
643.4.1Bloch oscillation

Atoms in a tilted lattice have their quasi-momentum increase linearly until they areresonantly reflected at the edge of the Brillouin zone. Bloch oscillations can be directly
observed from the time evolution of the momentum distribution in time-of-flight, orfrom the periodic center-of-mass motion in position space.
For small tilts, the semiclassical equations for the Bloch oscillation arek̇ = h̄F(3.34)∂Ek∂k(3.35)ṙ =where k is the quasi-momentum in the lattice. In the tight-binding limit and with aconstant external force, the center-of-mass motion isx − x0 =2Jdcos[(k0 + F t)d]F

(3.36)For strong tilts, the Hamiltonian with nearest-neighbor tunneling is diagonalized
by the following Wannier-Stark states,|φ(m)i =X

Kn−m (n2J)w(x − xn )∆(3.37)

where the Ki are the Bessel functions. The Wannier-Stark states are localized at eachsite, when the tilt ∆ is much larger compared to the bandwidth J. Bloch oscillationsin this regime can be understood from the ladder of accumulated phases along thelattice, which are linearly increasing in time and equally spaced in space.
Bloch oscillations are experimentally observed in time-of-flight images, shown in
Fig. 3-12. In this measurement, a strong magnetic field gradient is suddenly switchedon by rapidly turning off one of the magnetic coils running in Helmholtz configuration.
Bloch oscillations up to four cycles are recorded, before the sample heats up and loses
the coherence. The measured oscillation period at T ' 0.5 ms gives an estimate of
the tilt at ∆ ' h × 2 kHz. However, the tilt strength seems to vary in time, possibly
65Figure 3-12: Bloch oscillations in a three-dimensional optical lattice, with 20ER along
the tilt direction and 5ER in the transverse directions. A tilt potential of ∆ ' h × 2
kHz is applied for a variable duration from 0 to 2 ms.
due to drifts from the sudden switch-on of the magnetic field gradient.
The technical reason for switching on the tilt by turning off the coils is to reach amaximum switch-on speed. Given the harmonic confinement in a dipole trap, if thetilt is switched on gradually, the atomic cloud simply follows an adiabatic shift to
new equilibrium in the trap. However, the large inductance of the coils means thatthe rate of change of the magnetic field gradient is proportional to the voltage acrossthe coils. By turning the coils off, rather than on, it is possible to generate a hugetransient voltage across the coils and switch on the tilt faster, without the need ofbuilding a capacitor circuit to boost the switch-on speed.

3.4.2

Dynamical instability

In the presence of interactions, Bloch oscillations can be damped due to a dynamical
instability that occurs for states above a certain critical value of the quasi-momentum.
The damping process goes as follows: for two particles at quasi-momentum q, a twobody collision results in one particle at q+∆k and the other at q−∆k, which conserves
the quasi-momentum but may release excessive energy. The excessive energy can be66

transferred into the transverse directions in the case of a one-dimensional lattice where
the transverse directions have a free-particle dispersion relation. However, in the caseof a three-dimensional lattice where the transverse directions are strongly confinedand have flat bands, the excessive energy has no place to go and therefore the losses
are strongly suppressed.

Figure 3-13: Experimental observation of dynamical instability. Top: condensates at
a variable quasi-momentum are prepared by accelarating the lattice, held for 2 ms,
and then ramped back to zero quasi-momentum before being released from the trap.
Bottom: condensates at a variable quasi-momentum are prepared by accelarating thelattice, and immediately released from the trap to measure the momentum distribution. Image captions in this figure are in arbitrary units proportional to the value ofthe quasi-momentum during the hold on the top. Quasi-momentum states near theedge of the brillouin zone (from 0.35 to 0.5 in the figure) show rapid heating and lossof coherence. This could be explained by dynamical instability.

When the dispersion relation is concave upward, the energy that is lost fromthe particle at q − ∆k is not enough to excite the other particle at q + ∆k. Thesequasi-momentum states, such as free particles or quasi-momentum states near q = 0
in an optical lattice, are therefore stable against two-body collisions. However, the67dispersion relation becomes convex upward above the inflection point. In this regime,
two-body collisions are allowed as the final states under energy conservation become
available, and the sample undergoes rapid dephasing and heating.
We study dynamical instability in an alternate experimental configuration, where
we create a moving lattice with a constant acceleration and the fictitious force in themoving frame drives the atoms undergo Bloch oscillations. A pair of non-retroreflected
Raman beams are used to create this moving lattice, and the velocity of the moving
lattice is a product of the wavelength and the frequency difference between the two
beams. A constant acceleration of the moving lattice is generated by applying a linear
frequency ramp in one of the Raman lattice arms.
This method has the disadvantage that the cloud can move out of the trap or thefield of view in a matter of milliseconds, because the imaging system and transverse
confinement exist in the stationary lab frame. However, it has a distinct advantagethat the tilt can be easily stopped, restarted, or reversed, and can be programmed inarbitrary waveforms. As a result, compared with the magnetic field gradient, the tilt
in an accelerated lattice is more controllable and uniform in time, although the time
window for experimental observation is limited to a few cycles of Bloch oscillations.
In Fig. 3-13, we hold the condensate at a variable quasi-momentum to observe
the onset of the dynamical instability. The sample is first prepared at a target quasimomentum by accelarating the optical lattice, held for 2 ms, and then ramped back tozero quasi-momentum before release from the trap. Quasi-momentum states near theedge of the brillouin zone have shown rapid heating and loss of coherence, consistent
with the theory of dynamical instability.
To our surprise, further experiments have shown a decay of the cloud down to theground state at zero quasi-momentum, even though dynamical instability could notoccur in these states. These measurements, shown in Fig. 3-14, were carried out in astationary lattice with magnetic field gradient tilts, where we first turn off one of thebias coils to initiate the Bloch oscillation, and subsequently turn off the second coil inthe pair to ‘freeze’ the quasi-momentum. The decay of these finite-momentum statesoccurs at a timescale much faster than the nearest-neighbor tunneling, and the cloud68

Figure 3-14: Rapid decay of quasi-momentum states near the edge of the brillouinzone. A tilt pulse is applied to prepare the condensate at the edge of the Brillouinzone. This tilt is then switched off, and time-of-flight images are taken after a variable
hold up to 2.5 ms. Compared with the ground state at q = 0, quasi-momentum state
at q = kL shows rapid decay into the ground state, which could not be explained bysimple two-body collisions alone.
always relaxes into the zero-quasi-momentum state. A complete understanding of theunderlying mechanism for this decay is still unknown: our experimental observations
could not be fully explained by two-body collisions alone, and suggest many-bodyeffects in this process.

69

70

Chapter 4
Quantum Simulation inSuperlattices
4.1

Introduction

In this chapter, we will present the experimental implementation of a new scheme using pseudo-spins in a superlattice to simulate spin-orbit coupling physics with ultracold atoms. This system is ideal to prepare and study the stripe phase which simultaneously breaks the gauge symmetry and translational symmetry, and will potentiallyallow us to explore exotic topological states of matter.
In contrast to the spin-orbit coupling in atomic physics which describes the interactions between the orbital and spin angular momentum of an electron L · S, and inpart gives rise to the fine structure splitting in the atomic structure, the spin-orbit
coupling in condensed matter systems couples particle momentum with its spin k · σ
and modifies the band structure. Two-dimensional spin-orbit couplings play a crucialrole in spin-Hall materials, topological insulators and topological superfluids [32, 33].
To simulate spin-orbit coupling with ultracold atoms, we need to couple two spinstates with a momentum transfer. For simplicity, let’s consider the one-dimensional
problem and the spin states have the free-particle dispersion relation. After a spatial
and a temporal unitary transformation, the effective spin-orbit coupling Hamiltonian
to be simulated is written as
71

H0 =

h̄2 (k + kL /2 · σz )2+ Kσx + δ0 σz
2m

(4.1)

where Kσx is the spin-orbit coupling term, and δ0 σz is the Zeeman term.
The single particle band structure of this Hamiltonian is shown in Fig. 4-1. As weturn on the spin-orbit coupling, the eigenstate will be a superposition of opposite spinswith shifted momentum. When spin-orbit coupling is weak, there are two degenerate
ground states. Depending on the miscibility of the system, particles can stay miscible
and form density stripes or phase separate [39]. In contrast, when spin-orbit couplingis very strong, only a unique ground state is available.

Figure 4-1: The single particle band structure of the spin-orbit coupling Hamiltonian.
The spin-orbit coupling strength increases from left to right, and the colors represent
the two spin components.
Previously in the pioneering experiment in Ian Spielman’s group, hyperfine statesof 87 Rb were used to simulate a one-dimensional spin-orbit coupling Hamiltonian [34].
In this experiment, the single particle spin-orbit coupling band structure was studied.
Later, similar techniques were used in fermionic systems to create spin-orbit coupledFermi gases [35, 36].
However, there are two major limitations with the hyperfine state scheme. Firstof all, to drive a two-photon transition between hyperfine states, near-resonant lightneeds to be used. This severely limits the lifetime of the system. Second, for the bareinteractions between hyperfine states, g↑↓ ' g↑↑ , g↓↓ , and the system prefers to stayimmiscible. This results in a small window for the stripe phase in the phase diagram72

and weak density stripes, making the experimental detection very difficult.

Figure 4-2: A new scheme to simulate spin-orbit coupling physics. The pseudo-spinsin a double well are coupled via a two-photon process to compensate the energy offset
and provide momentum transfer. Because no real spin flip is needed, far-detuned laser
beams can be used.
To overcome these limitations, we have implemented a new scheme using pseudospins in a double well potential to simulate spin-orbit couplings, illustrated in Fig. 4-2.
In this scheme, only one hyperfine state is involved and real spin flips are not required,
far-detuned laser beams can be used. Furthermore, because of the spatial separation
between opposite spin states, g↑↓  g↑↑ , g↓↓ , the system prefers to stay miscible andtherefore is ideal to prepare and study the stripe phase.
In this chapter, we will first describe the experimental techniques to create andcontrol an optical superlattice with 532 nm and 1064 nm laser beams. Then we’ll experimentally characterize and calibrate our superlattice system, in particular its phaseand band structure. Finally, we’ll describe the spin-orbit coupling experiment in detail. First, the theory of spin-orbit coupling in this system is developed, including theexact derivation of the single-particle Hamiltonian, the comparison of the spin-orbit
coupling and onsite coupling, and the phase transitions in an interacting spin-orbit
coupled system. Then our experimental realization of the spin-orbit coupling Hamiltonian is presented, where we discuss the state preparation techniques, experimental
results on detecting the resonant spin-orbit coupling signal and the dynamics in theupper state, as well as proposal to use Bragg scattering to directly probe the stripes.
73

4.2

Experimental Setup and Techniques

In this section, we will describe the experimental techniques for the second-harmonic
generation of 532 nm green light, the phase control of the superlattice, as well as the
preparation of our Raman beams. Besides the implementation and performance of ourchosen scheme, other available solutions and their pros and cons are also discussed.

4.2.1

Second-harmonic generation for 532 nm green lattice

Laser source selection
Single-mode high-power 1064 nm fiber lasers have been available for quite a while, andhave been used for optical lattice experiments at the CUA in the past decade. Basedon our experience, the best configuration for building the high-power infra-red system
is to dedicate a high-quality single-mode laser as the seed, combined with powerfulfiber amplifiers downstream. In this case, one could use a seed laser powerful enoughto drive multiple fiber amplifiers when high power is in demand, or have the option toswap seed lasers for diagnosis purposes and special needs, or implement light control
such as frequency modulation involving only the low power seed light, instead of theamplified light with full power.
For 532 nm green light, there are mainly two options for us. The first one is to finda turnkey high-power single-mode laser that is commerically available. We have founda few candidates: IPG Photonics offers the GLR-20 which is a 20-Watt single-mode
fiber laser, Coherent Inc offers the Verdi V series that can go up to 18 Watts and was
claimed to be a single-frequency laser, and Spectra-Physics offers the single-frequency
Millennia Edge that has a maximum output of 6 Watts. Some other options include
the 18-Watt Sprout from Lighthouse Photonics, and the 25-Watt Millennia eV fromSpectra-Physics, but both are multi-mode lasers and thus not suitable for opticallattice experiments.
The other option is to frequency double the 1064 nm light in house, using relatively
cost-effective and high-power sources at this wavelength. The advantage of the former
solution is the convenience and low maintainance of a turnkey equipment, and possibly74

a higher optical power they could provide. On the other hand, one needs to phaselock the IR and green lasers to stablize the relative frequency against long-term drifts
in a superlattice experiment. Furthermore, there are still some concerns about thesingle-mode quality and linewidth of aforementioned commencial products. In theend, we decided to make our own frequency-doubled green light.
In our system, a 2-Watt Coherent Mephisto laser is used as the seed laser, which is
powerful enough to simultaneously drive multiple fiber amplifiers. For fiber amplifiers,
we have two 50-Watt units from Nufern, with one dedicated for 1064 nm IR lattices,and the other one for frequency doubling and generating 532 nm green lattices. Thereis another older 20-Watt 1064 nm laser from IPG Photonics, which has an integrated
seed. This laser has been used mostly for optical dipole traps, due to concerns on its
single-modeness.
Second-harmonic generation for green lightSingle-frequency continuous-wave 532 nm light can be generated via second-harmonicgeneration (SHG) of a high-power 1064 nm light. With the full output from a 50-Watt
Nufern fiber amplifier, one could generate about 15 Watts of 532 nm in a single passfrequency doubling through a nonlinear crystal [37].
HC Photonics in Taiwan provided us the nonlinear cystals. There are two types ofthese cystals: PPMSLT is periodically-poled MgO (1 mol.%) dope SLT, and PPMCLT
is periodically-poled MgO (1 mol.%) dope CLT. From the specifications, PPMSLT has
a higher damage threshold and a slightly higher second-harmonic conversion efficiency,
but unfortunately HC Photonics could no longer find vendors to consistently supply
them quality wafers, and may have to discontinue this product line soon. We havereceived one PPMSLT crystal from their last batch, and have permanently set it up
in our second-harmonic generation for the superlattice experiment. There are several
PPMCLT crystals for backup, in case the current PPMSLT crystal is damaged or has
degraded.
We have selected the single-pass frequency doubling scheme, because of the convenience in setting it up, as well as the sufficient output that it could generate for our
75

Figure 4-3: The optimal crystal temperature set point for second-harmonic-generationof 1064 nm light. When we increase the fundamental power, weak absorption from thebeam creates local heating inside the crystal. This thermal effect can be compensatedby reducing the set point of oven temperature, indicating there is a small temperature
gradient between the sensor and the core of the crystal.
application. The SHG efficiency is highly sensitive to the phase-matching conditions,
therefore the focus and beam waist of the fundamental beam are carefully chosen andwell tweaked. The crystal needs to be maintained in a temperature-stabilized environment for the optimal performance. Therefore, we’ve mounted it in an commercial
oven from HC Photonics as well. Although somewhat expensive, it is well-built anddoes stablize the ambient temperature for the crystal.
Even with the over temperature stabilized, we still have seen several-degree dropin the optimal oven temperature for second-harmonic conversion, as we increase thefundamental power from a few Watts to 30 Watts, shown in Fig. 4-3. This observation
could be explained by heating from weak absorption of the fundamental light inside
the crystal, and also indicates that there is a small temperature gradient between thesensor and the core of the crystal.
With the oven temperature retuned and optimized at each fundamental power, the76conversion efficiency for our 532 nm second-harmonic generation is measured in Fig. 44. With the fundamental beam at 25 Watts, we could get more than 6 Watts of 532 nm
light, sufficient for our superlattice experiment. The characteristics and performancesof the second-harmonic-generation of 1064 nm light in other configurations can befound in [37] for more details.

Figure 4-4: The conversion efficiency of second-harmonic-generation of 1064 nm light.
The SHG efficiency is measured as a function of the fundamental 1064 nm power. The
measured conversion efficiency starts to deviate from theoretic prediction above 15Watt fundamental power. This reported conversion efficiency has taken into accountthe thermal effect in Fig. 4-3, as it was measured after tuning the optimal temperature
at each fundamental power.

4.2.2

Controlling the phase of the superlattice

The phase of the superlattice, namely the relative position of the green and IR lattice,needs to be experimentally controlled to change the superlattice between all possible
configurations. The major considerations for the phase control are the full range, themaximum speed, the long-term drifts, and the possibility of changing it dynamicallyduring the experimental sequence.
77

n532
BK7
1.5195
Fused Silica 1.4607
Sapphire
1.7718ZnSe
2.6754
CaF2
1.4354MgF2
1.3789n1064
1.50661.44961.75452.4823
1.42851.3733

∆n
0.013
0.011
0.017
0.19
0.007
0.006

Table 4.1: The indices of refraction for 532 nm and 1064 nm light in various types ofglass. The dispersion between green and IR light is around one percent for these glass
except ZnSe. Overall, BK7 and fused silica are most suitable for our phase plate.
We have mainly considered three options for the phase control. They are rotatinga dispersive glass plate on a galvo mount, sweeping the frequency of one lattice beamusing an acousto-optic modulator (AOM), and changing the refractive index for one
lattice beam using an electro-optic modulator (EOM). We’ll discuss the pros and consof each method in the following, meanwhile we’ve implemented the first two schemesin our experiment.

Method 1: Glass Plate Phase Shifter on a Galvo
One way to control the phase of the superlattice is to use a piece of dispersive glass.
By rotating the glass plate and adding different path lengths to the two wavelengths,
we create a relative position shift of the two lattices at the cloud.
The index of refraction in a medium depends on the wavelength of the light, and it
usually decreases for longer wavelength. This dispersive effect is about a few percent
over the visible spectrum for most transparent materials. For example, for dry air at
15 ◦ C, one atmosphere and with 450 ppm CO2 , the index of refraction is 1.0002782 for
532 nm and is 1.0002740 for 1064 nm. For glass such as BK7, the index of refractionis 1.5195 for 532 nm and is 1.5066 for 1064 nm. In table 4.1, we have listed the indicesof refraction for 532 nm and 1064 nm in common optical glasses.
Among these glasses, ZnSe has the strongest dispersion, but the transmission is
low for visible and near-infrared light. Uncoated CaF2 and MgF2 glass have fairly low
reflection of a few percent, but their dispersions are relatively weak. Overall, standard78

BK7 and fused silica (high optical damage threshold) are the favorable options, and alayer of anti-reflection coating for 532nm and 1064nm can be applied on both surfacesof the phase plate.
Edmund Optics offers fused silica windows of thickness between 3 mm and 5 mmwith various diameters. We’ve being using the 3 mm glass plate in our experiments,
because of its light weight for fast switching. An estimated rotation about 2 degrees at
30-degree incident angle gives 266 nm shift of IR and green lattices, which corresponds
to a full period of the superlattice potential.
The rotation of the glass plate needs to be computer-controlled by the Cicero Word
Generator, so that the phase ramp can be synchronized in the experimental sequence.
The maximum speed and repeatability of this mechanical phase control, as well as
the sensitivity to electronic noise, such as radio-frequency or microwave radiations,require careful considerations. All of these aspects can be tested independently using
a position sensing detector (e.g., Thorlabs PDQ80A), before integrating it into theexperiment.
There are two options for mechanically controlling the glass plate. One way is
to choose a piezo-based device. We find the Agilis piezo-driven rotation stage AGPR100 from Newport that features 360-degree continuous rotation and ultra-highadjustment sensitivity. It was tested to be reasonably fast, but it has shown clear
hysteresis, and the bigger headache is that they do not offer a convenient interface to
Cicero. Therefore, it is not suitable for our experiment.
The other approach is to use a galvometer scanning system. The galvometer has
two core components: an actuator that drives the load, and a position detector thatmonitors the load position for fast tuning and stabilization in a closed loop system.
There are two types of position detectors in a galvometer. In the traditional dielectric
capacitive design, a radio frequency source is driving two variable capacitors, and theresulting rectified differential currents measure the position of the galvo actuator andload. In recent years, there has been a new design based on optical position detection.
A light source illuminates parts of four photocells, and a moving butterfly-like shape
between the light source and the receivers casts more or less shadow onto the pairs of79

the receiver cells. As a result, the currents read out the relative position of the galvo
actuator and load.
In our first attempt, we used the galvo GVS011 from Thorlabs, which seemed to beof the dielectric capacitive design. It shakes rapidly and loses its initial position during
the microwave evaporation phase, and needs to be reset after every shot. On the other
hand, a galvo based on optical position detection is far less sensitive to microwave andshould solve this problem for our experiments. The vendors for the galvos that we havelooked at include Nutfield Technology, Cambridge Technology and Thorlabs. Nutfield
and Cambridge Technology have more in-depth knowledge and experience about galvo
design and applications. Specifically, Nutfield offers galvos that “significantly reduce
temperature based gain and offset drift and is greatly immune from radio frequency
interference”.
Up to now, we have been using the galvo system Thorlabs GVS001 for the superlattice experiments, because of its short lead time and lower cost. After careful tuningof the closed loop, it can shift the superlattice phase over 2π within one millisecond.
It still picks up some microwave, and may undergo a random shift equivalent to about
10 percent of the superlattice period during the microwave evaporation, but it is able
to settle back to its initial position within a deviation equivalent to one percent of thesuperlattice period by the end of evaporation when the microwave is turned off. This
one-percent level phase uncertainty is comparable to phase drifts from other sources,
such as the ambient temperature and pressure fluctuations, therefore is acceptable interms of control precision. Overall, the one-millisecond maximum speed, one-percent
precision in repeatability and the convenient interface of this galvo have met our needs
for the static and slow dynamical control of the superlattice phase.

Method 2: AOM Frequency Shifter
The phase of the superlattice can also be controlled by shifting the light frequency.
The reason is, a one part per million shift in the frequency changes the lattice spacing
by one part per million as well. This small difference might be hard to notice locally,
but a finite position shift can be built up for the lattice after a distance, since a node80

of the standing wave is always locked at the retro-reflection mirror. We estimate therequired frequency shift to control our superlattice in the following.
The superlattice potential in the general case is
U (x) = −VR
VG
cos(2kR x + 2πφR ) −cos(2kG x + 2πφG )
2
2

(4.2)

Figure 4-5: The phase control of the superlattice via AOM frequency shifter. Whenthe frequency of one lattice beam is swept, its lattice spacing is sligthly modified, butwith a node fixed at the retro-mirror. This results in an effective shift in the relative
position between the two lattices. The case of shifting green lattice is plotted on theleft, while the case of shifting IR lattice is plotted on the right. Although the absolute
shift in the relative position is the same for the two cases, shifting IR lattice requiresa smaller frequency change, and also keeps the atomic wavefunction more stationary,
therefore it is the favorable option.
Let’s aim to shift the superlattice phase by a half of the period, which corresponds
to changing from symmetric double-wells to asymmetric double-wells. As illustrated
in Fig. 4-5, this is translated into a relative motion of 133 nm between the two lattices,which is half of the green lattice spacing, ∆φG = 0.5 or ∆NG = 1/2, and a quarter of
the IR lattice spacing, ∆φR = 0.25, or ∆NR = 1/4. It is intuitive for a sanity check,since the superlattice potential is identical if either green lattice or IR lattice moves
by 266 nm, the green lattice spacing.
81The number of nodes in the standing wave between the retro-reflection mirror andthe cloud is
N=

l
λ/2

(4.3)

in our experiment, l ' 500 mm is the distance from the retro-mirror to the cloud.
It is easy to find out all relative changes are equal
∆N
∆λ
∆f
=−
=
N
λ
f

(4.4)

If we are shifting the green lattice,NG =500 mm∼ 2 × 106266 nm(4.5)

the relative frequency shift for the green lattice needs to be
∆fG
∆NG
1/2
=
=
= 1/4 × 10−6
fGNG
2 × 106(4.6)

we know the frequency of 532 nm light isfG =3 × 108 m/s= 5.6 × 1014 Hz
532 nm

(4.7)

therefore, the absolute frequency shift for the green lattice needs to be∆fG = 1/4 × 10−6 × 5.6 × 1014 Hz = 140 MHz(4.8)

On the other hand, if we are shifting the IR lattice,
∆NR
1/4
∆NG
=
=
NR
NG /2
NG

(4.9)

the absolute frequency shift for the IR lattice needs to be∆fR ∝ fR = ∆fG /2 = 70 MHz
82(4.10)

which is half of the frequency needed for shifting green lattice, and feasible by using
an AOM in the double-pass configuration. Shifting the IR lattice has the additional
advantage that the atomic wavefunction is more stationary during the sweep, which
reduces heating and other side effects.
In short, controlling the superlattice phase by changing the frequency of IR lattice
is a favorable choice, when speed is the priority and the phase only needs to be tuned
within half of a period.
Method 3: EOM Phase Shifter
The third way to control the superlattice phase is to use an electro-optic modulator.
Depending on the type and orientation of the nonlinear crystal, as well as the directionof the applied electric field, the refractive index of an electro-optic crystal will dependon the polarization of the light. Based on this result, one could make the green andIR lattices in orthogonal polarizations, and use the electro-optic modulator to shift
the phase of only one beam.
This scheme can be in practice very fast, and may allow us to shift the superlatticephase over several periods. However, standard electro-optic modulator devices usually
have a small aperature of 2 mm in diameter, which require us to shrink the beam size.
Furthermore, for high power beams at small beam waist, optical damage becomes anissue. For the EOMs that we’ve looked at, at their maximum aperature and for abeam at the laser power required for our experiments, the intensity of the beam will
be fairly close to the specified optical damage threshold for these devices. There areother EOM devices which offer larger aperatures. However, they are rather expensive,
have longer lead time, require very high voltage drivers, and we are still unsure about
the overall phase shift from the crystal expansion and the residual phase shift for theother polarization beams. In the end, we decide not to pursuit this scheme.
Estimated phase drifts from the changes in ambient condition
The phase of the superlattice is determined by the path length difference of the green
and IR lattices from the retro-reflection mirror to the atomic cloud. This path length83

is subject to changes in the ambient conditions such as temperature and air pressure.
In this section, we will make estimates for these effects.
First, let’s consider the path length in the air. The refractive index of air is linear
in its density,n(P, T, λ) ' 1 + cλ ×

P
T0
×
P0
T

(4.11)

where cλ is the linear coefficient that depends on the wavelength of the light. The
difference in the refractive index for 532 nm and 1064 nm light is∆nair ' 4 × 10−6 ×

P
T0
×
P0
T

(4.12)

Let’s assume the maximum relative changes for room temperature and air pressue
are both about 1%. If the atomic cloud is half a meter away from the retro-reflection
mirror as in our experiment, the effective phase drift of the superlattice is
L × ∆nair× relative change
λ532 /2
0.5 × 4 × 10−6
× 1%=
0.266 × 10−6

∆φair /2π =' 10%(4.13)(4.14)(4.15)

which is not completely negligible.
For a temperature stabilized environment, such as in the enclosure system that wehave, and in a day with steady weather, the superlattice phase should be stable for a
full day of experiment running, as demonstrated in the superlattice phase calibration
section later. However, because these drifts accumulate over time, we cannot expect
the superlattice phase to be stable more than a week. In practice, we have not seen
severe drifts in the superlattice phase in our experiments. In case the drifts from airtemperature and pressure become a problem, it can be reduced by shortening the
distance between the retro-reflection mirror and the chamber.
Finally, let’s consider drifts from thermal effects on the glass plate and viewports.
The thermal expansion coefficient is about 10 × 10−6 K −1 in standard glasses. The
84

phase drift of the superlattice due to thermal expansions in glass is
αD × ∆T × ∆nglass
λ532 /2
10 × 10−6 × 10−2 × 1 × 0.013=
0.266 × 10−6∆φglass /2π =' 0.5%(4.16)(4.17)(4.18)

which is small and negligible.

4.2.3

Considerations for the Raman beams

Raman beams are used to drive the spin-orbit coupling between the double wells. In
our experiment, one beam of the Raman pair is along the lattice direction to guarantee
a non-vanishing coupling matrix element, due to the fact that eigenstates are always
orthogonal! The other beam of the Raman pair is perpendicular to the lattice, which
provides the transverse momentum kick.
There are a few special considerations for the Raman beam preparation. First ofall, the Raman beams are split after the same optical fiber on the experimental table.
The motivation is to minimize relative frequency noise of the two beams, because
any fluctuation in the path length inside the fiber is now common mode. Second,each of the Raman beams is going through a double-pass AOM for switching on/off
and intensity stabilization. This is intended to minimize the pointing shifts while
scanning the Raman frequency. Lastly, to be able to look at the frequency spectrum
and make sure our Raman coupling is narrow and at signal frequency, we have built
up a beat note monitor between the Raman beams for diagnosis.
Initially, our Raman beams were set up for the synthetic gauge fields in opticallattices experiment, as one beam was along the vertical direction, in which a strong
and uniform tilt would be created, and the other was horizontal to provide transverse
momentum kick and consequent vector potential. Later after we turned to the spinorbit coupling experiment in superlattices, this vertical Raman beam has been moved
to the horizontal plane for mechanical stability and detection convenience.
85For the experiments that have been carried out in this thesis, Raman beams of 1064
nm light and in orthogonal directions are used. The choice of Raman beams is not
restricted to this particular configuration at all, as long as they are far detuned fromthe atomic transition, have the correct frequency offset, and provide an non-vanishingcoupling matrix element. As we will see later, in certain situations, a different Raman
configuration (both in wavelength and geometry) could come in handy for the state
preparation and experimental detection.

4.2.4

Experimental layout

To give the reader a graphic view about the geometry and layout of the optical setup
in our superlattice experiment, a schematic drawing looking from the top is shown in
Fig. 4-6,
The green and IR beams are combined along one of the 45-degree axes, and thensimultaneously retro-reflected to make the superlattice. A glass plate is inserted into
the retro-path to control the phase of the superlattice. A pair of IR beams with a small
frequency offset are used for the Raman coupling, with one along the superlattice andthe other perpendicular to it.
Another way to see the geometry of beams in action is to look at the corresponding
diffraction patterns. This is illustrated in Fig. 4-7, as a fun aspect of the experiment.

4.3

Experimental Characterization and Calibrationof a Superlattice

In this section, we will experimentally characterize the superlattice, including testing
our experimental control of the superlattice phase, and probing the band structure
via amplitude modulation (AM) and two-photon Raman process (Raman).
86

Figure 4-6: Experimental schematics of the superlattice experiment viewing from thetop. The green beam is sent collinearly with the IR beam through the atomic cloud,and then both are retro-reflected to create a superlattice. A glass plate is mounted ona galvo in the retro-path to control the superlattice phase. One of the Raman beams
is sent along with the superlattice, and the other is perpendicular to the superlattice,both without retro-reflection.

4.3.1

Superlattice band structure

The potential of a one-dimentional superlattice can be written as
U (x) = −VG
VRcos(2kR x) −cos(2kG x + φ0 )22

(4.19)

where kR , kG are the wavenumbers of 1064 nm and 532 nm light, and VR , VG are thecorresponding lattice depths.
The natural units for the physical quantities of interest in a superlattice are listed
in table 4.2. Please note that kL = π/d is also the size of the Brillouin zone in a 1064
nm lattice.
With all physical quantities in their natural units, the single-particle Hamiltonian87

Figure 4-7: The diffraction patterns for relevant beams in the superlattice experiment.
These images are taken by looking at the cloud from the bottom. To stay consistent
with the schematics in Fig. 4-6, these images need to be flipped vertically due to themirrors in the imaging path.
in the superlattice is simply
H0 = −∇2 −VG
VRcos(2x) −cos(4x + φ)22

(4.20)The band structure calculation for a superlattice is a straightforward extension of
the solid-state textbook problem that we’ve derived earlier in section 3.2.1. The only
difference is that, in addition to the IR component in cos(2kL x) which couples stateswith momentum transfer ±2kL , there is also a green component cos(4kL x + φ) which
couples states with momentum transfer ±4kL .
88Physical Quantity
Unit
MomemtumkL = kR = 2π/λR
EnergyDistanceh̄2 k2EL = ER = 2mL
d/π = 1/kL

Meaning1064 nm wavenumber
1064 nm recoil energy
1064 nm lattice spacing

Table 4.2: The natural units of the physical quantities in a 532/1064 nm superlattice.
The superlattice band structure as a function of the phases is shown in Fig. 4-8.
In the case of very deep symmetric double wells, the lowest two bands correspond to
the symmetric and anti-symmetric states, and they are splitted by the tunneling J
between the double wells. As the phase is tuned away from the symmetric point, anenergy offset is created between the left and right wells, and the lowest two bands are
separated, roughly speaking, by this offset ∆.
Quasi-momentum states of each band in a superlattice can be calculated as well.
For example, quasi-momentum states at q = kL in the lowest three bands are shownin Fig. 4-9. Please notice that for a superlattice without IR component, there is no
gap opened at q = kL , therefore the lowest three bands in a green lattice shown on
the top right correspond to the 1st, 3rd and 5th lowest band in the other superlatticeconfigurations.
They are the final states when we couple the ground state into higher bands via
a Raman process, and the Raman beams provide a momentum transfer kL along thesuperlattice as in our experiment. The momentum distribution of the final state canbe measured in the time-of-flight images and is used to identify the band index of the
final state.

4.3.2

Calibration and dynamical control of the superlatticephase

To have the full control of the superlattice in our experiments, we need to calibrate anddial in the superlattice phase anywhere between 0 and 2π. In addition, it would very
helpful to be able to dynamically change the phase during the experimental sequence.
Finally, the superlattice phase needs to be stable against long-term ambient drifts.
89Figure 4-8: The band structure of the superlattice under various conditions, with thelowest two bands shown in red and blue. The relative phase φ between the IR andgreen lattices are 0 for the top row, π/2 for the middle row and π for the bottom row.
The green lattice becomes deeper from left to right, while the IR lattice depth is kept
constant. The symmetric point of the superlattice corresponds to the bottom row.
We will discuss all of these aspects in the following.

Superlattice phase calibration
The phase of the superlattice can be read out in the time-of-flight image. When thephase is at a symmetric point, the superlattice is an array of symmetric double wells,
and the first-order peaks will disappear due to destructive interference in the doublewell structure factor. The other phases of the superlattice can be linearly interpolated
from the nearest symmetric points, using either the glass plate angle set point or the
AOM frequency.
90

Figure 4-9: The momentum components of quasi-momentum states at p = +kL forthe lowest three bands in an superlattice. The units on the horizontal axis are the IR
wavenumber kL . This figure also predicts the time-of-flight images of the final statesin Fig. 4-16, when we couple the ground state into higher bands via a Raman process.
The time-of-flight images can be taken by adiabatically ramping up a superlattice,or by applying a superlattice pulse and then looking at the diffraction in time-of-flight.
We find that in practice the second method usually has better contrast and robustness,possibly because of a much shorter interaction time (a few µs).
A typical superlattice phase calibration series is shown in Fig. 4-10. The superlattice is symmetric when first-order peaks are minimal in the time-of-flight diffraction.

Figure 4-10: Calibration of the superlattice phase via diffraction in time-of-flight. The
image captions are the set points for the glass plate, which is linear in the superlatticephase. The symmetric point in the superlattice phase shows minimal first-order peaks,
which is at 3.3V in this series.

91

Load BEC into a superlattice
A pure condensate can be loaded into a superlattice by ramping up the green and IR
lattices simultaneously to reach the desired values. When both lattice beams are wellaligned, we are able to load the condensate into a superlattice with various presetted
phases in 100 ms without noticeable heating, shown Fig. 4-11. When the lattice
beams are slightly misaligned from the dipole trap (especially for the repulsive green
lattice beam), the sample may experience sloshing motion in the trap. In this case, aslower ramp-up in a few hundred milliseconds can prevent the sample from heating
up, and sometimes additional confinement may be necessary to keep the cloud in the
trap.

Figure 4-11: Experimental sequence for the dynamical control on superlattice phase.
Left: a condensate is loaded into a symmetric superlattice with a static phase. Right:
The experimental sequence for a single ramp and a double ramp in the slow dynamicalphase control experiment.

Slow dynamical phase control using glass plate
To test our dynamical control of the superlattice phase using a glass plate, we first loada condensate into a symmetric superlattice, and then change its phase in either a single
ramp or a double ramp with the galvo rotation, illustrated in Fig. 4-11. These phaseramps are done in tens of milliseconds, which is slow compared to the estimated interwell tunneling about kilohertz, and enough to keep the condensate in the ground state.
Time-of-flight images after the one-way ramp show the corresponding ground state92

in a tilted double-wells, and round-trip images confirm that the process is adiabatic
and without severe heating.
It’s important to point out that, as we rotate the glass plate to change the phase,the superlattice itself has an overall motion of hundreds of lattice constant, dragging
the condensate along. This effect may require even slower speed in the phase ramp tomaintain adiabaticity. Therefore, even though the galvo has the capability of rotating
our plate within a millisecond, the glass plate phase shifter scheme is limited for only
static and slow dynamical control of the superlattice phase.

Figure 4-12: The dynamical control of the superlattice phase using a glass plate. The
time-of-flight images are shown on the left after a one-way phase ramp, and on theright after a round-trip phase ramp in the superlattice. These phase ramps are slow
enough so that the condensate remains in the ground state of the superlattice.

93

Fast dynamical phase control using AOM
The superlattice phase can also be tuned by changing one of the lattice frequencies,
which can go as fast as a microsecond, and doesn’t come with the unwanted motion of
the superlattice. As discussed in earlier sections, changing IR frequency is favorable
compared to green, because it requires a smaller frequency shift, and keeps the atomic
wavefunction more stationary. By suddenly switching the IR frequency, we freeze theatomic population in each well between two superlattice phases, critical for the 50/50
sample preparation in the spin-orbit coupling experiment.
In our experiment, a double-passed AOM is used to shift the frequency of the seed
light for the Nufern fiber amplifier. In this way, we only need to shift the frequency
with a low power beam, and the double-pass scheme maintains the fiber coupling
efficiency. The RF frequency and power for this AOM need to be carefully chosen
so that the power of the seed light into the amplifier stays constant before and after
the frequency switch, otherwise the fiber amplifer may complain and shut down itself
from a built-in safety feature. Experimentally, we have observed a shift of the phasesymmetric point by 0.7V in the galvo set point when we shift the IR frequency by 40
MHz, consistent with our estimates in section 4.2.2.

Long-term superlattice phase drifts
One could characterize the long-term phase drifts by measuring the superlattice symmetric point over time. In Fig. 4-13, we’ve located a symmetric point by the time-offlight diffraction, and monitored its position over half a day with a half-hour interval.
In the earlier calibration, the period of the superlattice phase between neighboring
symmetric configurations is measured to be at 2.6±0.2 V in the unit of glass plate set
point. Long-term drifts of the superlattice phase is measured to be within 10 percentof the full period over six consecutive hours. This upper bound is mainly limited by
the measurement error bar of locating the symmetric phase. In practice, we have notexperienced severe phase drifts in the superlattice, although as a routine we’ll checkand calibrate the superlattice phase everytime we start the experiment.
94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 4-13: 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 bar.

4.3.3

Interband population transfer and band mapping

The band structure of the superlattice can be probed by coupling the condensate fromthe ground state into higher bands. Understanding the band structure is essential for
lattice diagnosis, the study of orbit physics and excited state dynamics, and sample
preparation for quantum simulation with pseudospin states in superlattices as well.
In this section, we’ll develop the techniques for transferring atoms into higher bands,
as well as for measuring the population in each band afterwards.
Population transfer into higher bands is achieved by amplitude modulating thelattice, or by switching on a pair of Raman beams through the two-photon process.
Amplitude modulation preserves the quasi-momentum of the initial state, while Raman coupling may shift the quasi-momentum depending on the wavelength and beamgeometry. In our experiment with orthogonal 1064 nm beams, the Raman coupling
shifts quasi-momentum by kL , and also provides a momentum transfer of kL perpendicular to the lattice, illustrated in Fig. 4-15.
95

Figure 4-14: Interband coupling in a 1064 nm lattice via amplitude modulation and
Raman process. Population in each band is individually read out with band mapping,as the bands are resolved with spatially separated Brillouin zones in the time-of-flightimages. The image captions are the corresponding resonance frequencies.

Population in different bands could be directly read out by ramping down the lattice and subsequently taking the time-of-flight image, a technique commonly known asband mapping. During this slow ramp, quasi-momentum states in different bands are
mapped into the corresponding real-momentum states in the free-particle dispersion
relation in an adiabatic connection process, and population in each band shows up in
individual Brillouin zone spatially separated in the time-of-flight image. Populationin higher bands transferred via Raman coupling can also be identified in time-of-flightwithout band mapping, as they are spatially shifted from the ground state due to theperpendicular momentum transfer, and show different momentum distribution. We
96will demonstrate both methods in the following.
For amplitude modulation, we are modulating the RF power of the AOM. The
modulation depth is usually around 30% to balance the strong coupling and excessive
heating, and the duration is about 100 milliseconds. For Raman coupling, its couplingstrength is proportional to the depth of the moving Raman lattice, which is usuallychosen to be a few lattice recoils, and the pulse duration is less than a millisecond.

Figure 4-15: Interband coupling resonances and the corresponding band structure in a1064 nm lattice. The solid lines show the AM resonances, while the dotted lines show
the Raman resonances. The Raman resonance frequencies have been subtracted by arecoil energy EL = 7.6 kHz for the transverse momentum transfer. Within experimental precision, these measured resonances are in agreement with the band structure of
an IR lattice of 27EL , calibrated independently via Kapitza-Dirac diffraction, except
for the AM resonance of the 5th band.
In band mapping, the lattices are simultaneously slowly ramped down. In practice,
we find out that the maximum speed of this ramp so that the condensate could still97

follow the adiabatic passage is around 0.5 ms for the green lattice, and around 2 ms
for the IR lattice. A ramp-down faster than that might result in partial projection
onto the higher bands and create a false signal in the band mapping detection. The
time-of-flight images of a condensate in the ground state are used to calibrate the sizeof the Brillouin zone in band mapping.
First, we’ve tested the population transfer and band mapping in a 1064 nm lattice,
shown in Fig. 4-14. Population transfers up to the 5th band have been identified, andthe resonance frequencies are consistent with the case of a lattice depth of 27 EL , after
taking into account the additional recoil energy due to the perpendicular momentum
transfer, illustrated in Fig. 4-15. This inferred lattice depth is in agreement with our
independent calibration using Kapitza-Dirac diffraction.
There are a few findings and comments regarding this experiment. First of all, intheory, amplitude modulation only couples the ground state into higher bands withthe same parity, which in our notation are the odd bands in Fig. 4-15. The fact that
we seem to see coupling into even bands might due to deviations from the ideal case,or simply heating from the modulation. Second, we find that AM coupling is usuallybroader than 5 kHz, while Raman coupling is about a couple of kilohertz wide. Third,
Raman couplings usually transfer population with much less heating compared to AM
couplings. Finally, the lifetime of higher band population is about 10 ms in the 1064
nm lattice, as the atoms collide and rapidly decay into the ground state.
The band of the final state in the Raman coupling scheme can also be identified in
the time-of-flight images. As shown in Fig. 4-16, the Raman transferred population in
the higher bands is shifted in the transverse direction, and has shown distinguishable
momentum distributions. This separation in time-of-flight on lattice bands is possible
only with the assistance of perpendicular Raman momentum transfer. Time-of-flightdetection relies on analyzing the relative strength of the momentum peaks, therefore
it is less direct and robust compared to the band mapping scheme. However, it doesn’t
require additional time for the slow ramp as in band mapping, and is useful when the
lifetime of the final state is limited.
98Figure 4-16: Raman coupling into higher bands with time-of-flight images in a green
lattice. The final states of the Raman coupling are offset to the upper-right diagonally because of the perpendicular momentum transfer. In this experiment, wedemonstrated the distinguishable momentum distributions of quasi-momentum states
at q = kL for the lowest three bands in a green lattice. The measured resonance frequences indicate the lattice depth around 10EG .

4.4

Theory of Spin-Orbit Coupling in a Superlattice

In this chapter, we will develop the theory for a spin-orbit coupled condensate in the
superlattice. First, we will discuss the Raman couplings in superlattices, and derive
the single-particle Hamiltonian with spin-orbit coupling in this system. Then we willcompare the relative strength of the spin-orbit coupling which couples states between
the wells, and onsite coupling with couples states within the same well. The desirable
experimental parameters are also estimated. Finally, we discuss the phase transition
from the miscible stripe phase to the immiscible separated phase for the interacting
spin-orbit coupled system.
99

4.4.1

Single-particle spin-orbit coupling HamiltonianIn this section, we derive the Raman coupling contributions for a double-well system.
We will focus on only the single-particle physics in this section.
Raman couplings in a double-well system
First of all, let’s consider the general case where we apply a pair of Raman beams inthe superlattice, and discuss the possible coupling processes.
In the tight-binding approximation, the basis functions are the localized Wannier
states labelled as |im , ki, where i can be left a or right b in the double well, representing
our pseudospin-1/2 states, m is the superlattice double-well unit-cell index, and k is
the transverse momentum. If the coupling between double-wells can be neglected for
now and only a single double-well is considered, index m is omitted.
The two-photon Raman field can be written as Ωeikx x+iky y e−iδt . The spatial phasecomponent eiky y provides the transverse momentum kick ky , while component eikx x is
critical to maintain a non-vanishing matrix element for nearest-neighbor couplings.
As the Wannier states are localized and vanishing after a few lattice sites, all othercouplings are negligible except the onsite and nearest-neighbor couplings,
HRaman =Γe−iδt |am , k + kL i ham , k| + iΓe−iδt |bm , k + kL i hbm , k|
+ Ke−iδt |bm , k + kL i ham , k| + h.c.

(4.21)(4.22)

where Γ is the strength of coupling between states in the same well, as we will call it
the “onsite coupling”; while K is the strength of coupling between states in different
wells, as we will call it the “spin-orbit coupling”. K is complex in this notation, thephase is 3π/4, from the propagating phase of the Raman coupling,
In the case where an energy offset ∆ is applied between the double wells to makewell-defined pseudospins, first-order perturbation theory gives the onsite couplingstrength Γ ' Ω, and the spin-orbit coupling strength K ' JΩ/∆.

If we start with a coherent equal superposition in the double-well |a, 0i+e−i∆t |b, 0i,

and then slowly ramp up the Raman beams, this state will be adiabatically connected100

to the following state (in the perturbative limit),|Si = |a, 0i + e−i∆t |b, 0i(4.23)

Γ∗Γ
e−iδt |a, kL i −e+iδt |a, −kL i
δ − EL
δ + ELiΓ
iΓ∗ +i(δ−∆)t+e−i(∆+δ)t |b, kL i +e|b, −kL i
δ − EL
δ + ELK∗
K∗+
e−i(∆+δ)t |a, kL i +e+i(δ−∆)t |a, −kL i∆ + δ − EL
∆ − δ − ELKK+
e−iδt |b, kL i −e+iδt |b, −kL i−∆ + δ − EL
∆ + δ + EL+

(4.24)(4.25)(4.26)(4.27)

where Eq. 4.24 and 4.25 are the onsite coupling contributions, and Eq. 4.26 and 4.27
are the spin-orbit couplings and their counter-rotating terms.
When the Raman detuning δ is set to be equal to the double well offset ∆,K∗
K −i∆t|Si = |a, 0i −|a, −kL i + e−i∆t |b, 0i −e|b, kL iELELΓ∗Γ
e−i∆t |a, kL i −e+i∆t |a, −kL i+∆ − EL∆ + ELiΓiΓ∗+e−2i∆t |b, kL i +|b, −kL i∆ − EL∆ + ELK∗
K+
e−2i∆t |a, kL i −e+i∆t |b, −kL i
2∆ − EL2∆ + EL(4.28)(4.29)(4.30)(4.31)In this equation, Eq. 4.28 is the spin-orbit coupling state, which shows the stationary
stripe phase. Eq. 4.29 and 4.30 are the onsite coupling states in each well, which givea time-dependent density modulation moving with the Raman beams. The last terms
in Eq. 4.31 are the counter-rotating contributions from the spin-orbit coupling.
As we will see, our experiment is usually done in the regime where ∆ is about anorder of magnitude bigger than the recoil energy EL , therefore the effects from thecounter-rotating terms are negligible. The strength and effects of the onsite couplings
will be discussed in section 4.4.2.
101

Single-particle spin-orbit coupling Hamiltonian
When the Raman detuning δ is very close to double-well offset ∆, one can make therotating-wave approximation, and neglect the onsite couplings. The Hamiltonian of
the spin-orbit coupled spin-1/2 system in the lab frame is

H0 = 

h̄2 k22m∗ −ikL y+iδt

K e

ikL y−iδtKeh̄2 k2
+ h̄∆
2m




(4.32)

after applying a temporal and spatial unitary transformation, the effective Hamiltonian in the transverse direction is
Heff =h̄2 (k + kL /2 · σz )2+ Kσx + δ0 σz
2m(4.33)

where δ0 = δ − ∆ is the offset of the pseudospin-1/2 in the new frame.

Effects of the couplings between double-wells
In the last two sections, we’ve only considered the Raman couplings within a doublewell, as the normal tunnelings are strongly suppressed by the energy offset. In thissection, we’ll discuss the effects of the Raman couplings between neighboring doublewells, as well as the resonant normal tunnelings between wells of the same kind.
Firstly, the resonant tunneling between wells of the same kind in the neighboring
double-wells, which is excepted to be small, is maintaining the relative phases between
double-wells and therefore the global coherence. This is critical for both the time-offlight interference detection and the Bragg scattering detection. As we’ll see in theexperimental section, the upper-well states, which correspond to the second band,
have the ground state at quasi-momentum q = kL , rather than q = 0. This insight,
which in retrospect totally makes sense, came initially as a little surprise, and would
completely change the expected signals in our detection for the spin-orbit coupling.
Therefore, when we prepare a coherent superposition with equal population in the102

double-wells, the initial ground state can be written as|Sm i = |am , 0i + (−1)m e−i∆t |bm , 0i(4.34)

Secondly, let’s also consider the propagating phase of Raman couplings, as well as
the effects of the Raman couplings between double-wells. The latter is written as
HRaman =Xm(−1)m [K2 e−iδt |bm−1 , k + kL i ham , k| + h.c.](4.35)

where K2 is the coupling strength between states in neighboring wells but from different unit cells. Again, this coupling strength is approximately K2 ' J 0 Ω/∆, and J 0 is
the normal tunneling corresponding to the barrier between neighboring double-wells,
which could be much higher than the barrier within a double-well. Therefore, in mostexperimental configurations where the IR lattice is not very weak, K2 is much weaker
than K.
If one carefully counts the phases of the Raman couplings, and keep the spin-orbit
coupling between double-wells, the full ground state that is adiabatically connected toour initial state, in the presence of Raman couplings at balanced spin-orbit coupling
condition δ = ∆, is0|Sm
i = |am , 0i + (−1)m e−i∆t |bm , 0i(4.36)KK∗|am , −kL i − (−1)m e−i∆t |bm , kL i
EL
EL
∗
K
K2
+ 2 |am , −kL i + (−1)m e−i∆t |bm , kL i
EL
EL
Γ
Γ∗
+
(−1)m e−i∆t |am , kL i −(−1)m e+i∆t |am , −kL i∆ − EL∆ + ELiΓiΓ∗+e−2i∆t |bm , kL i +|bm , −kL i∆ − EL∆ + EL
−(4.37)(4.38)(4.39)(4.40)

where K and K2 are complex if we assume Γ is real in this notation, and have phase of
π/4 and −π/4 in approximation. This complete wavefunction contains the stationary
stripes from the spin-orbit coupling as well as the moving density modulation from the103onsite couplings, and is also used to make predictions in the time-of-flight interference
measurement.

4.4.2

Spin-orbit coupling and onsite couplings

In this section, we’ll discuss the relative strength of spin-orbit coupling and the onsite
couplings. The optimal parameters for creating the spin-orbit coupling for experimental detection are also proposed.
First of all, the onsite coupling couples the |a, 0i and |a, ±kL i states,


0

Ω




HOC = Ω ±δ + ER
In this case, we have approximated the Rabi frequency as Γ ' Ω. If Raman detuningδ is much greater than Ω, the onsite coupling in first-order perturbation theory is|a, 0i ±Ω|a, ±kL iδ(4.41)

In our experiment, we choose the Raman detuning δ ≈ ∆ ∼ 50 kHz, and the Rabi frequency in this case is roughly the Raman lattice depth, which we assume to be around15 kHz, the amplitude of the onsite component is about 30%, and the corresponding
population is around 10 percent.
In contrast, the spin-orbit coupling couples the |a, 0i and |b, ±kL i states,HSOC = 

0

J
Ω
∆

J
Ω
∆

ER + ∆ − δ

the Rabi frequency in this case is JΩ/∆ due to wavefunction overlap between left andright wells. J is the normal tunneling rate mainly determined by the barrier between
left and right wells, and is about a few kilohertz. Let’s say we use a maximum valueof J to boost the spin-orbit coupling signal, which corresponds to a green barrier of104

3EG ∼ 100 kHz, still large enough to separate the left and right wells,
3 kHz × 15 kHzJΩ∼∼ 1 kHz∆
50 kHz(4.42)

When we set the Raman detuning at the double-well offset δ = ∆ for the balanced
spin-orbit coupling and the stationary stripes, the spin-orbit coupling in the first-order
perturbation theory is|a, 0i −JΩ|b, kL i∆ER(4.43)

therefore, the spin-orbit component is about 12% in amplitude and less than a coupleof percent in population.
In short, although the Raman-assisted tunneling does provide a finite spin-orbit
coupling in the double-wells, it also comes with an unwanted onsite coupling component that is a few times stronger than the spin-orbit coupling in the balanced case.
To demonstrate the Raman-assisted spin-orbit coupling, one could tune the coupling close to the resonance. The lower-to-upper transition has a resonance frequency
at δ = ∆ + ER , while the upper-to-lower transition has a resonance frequency atδ = ∆ − ER , therefore we expect to see a significant increase on one side of theRaman-coupled final-state population when the Raman detuning is near one of theresonances. This is what we have seen in experiment that, as we scan the Raman detuning, one side of the Raman-coupled components shows a resonance feature about
1 kHz wide, while the other side does not. However, the stationary stripe phase requires the balanced spin-orbit coupling at δ = ∆. In this regard, a sensitive detection
is needed to measure the spin-orbit coupling. One way is to implement the Bragg
scattering to detect the stationary density modulation, which is the linear cross termin the amplitude. Another possible way is to look at the time-of-flight interference,
not only the total population but also the interference between the spin-orbit coupling
component and the onsite coupling component. Both methods will be discussed in
the experimental detection section.
105

4.4.3

Quantum phase transition of an interacting spin-orbit
coupled condensate

In this section, we will consider a spin-orbit coupled condensate with repulsive interactions. The mean-field Hamiltonian in this system can be written as,H=ψa∗ ψb∗





H0 

ψa



 + gaa |ψa |4 + gbb |ψb |4 + gab |ψa |2 |ψb |222
ψb(4.44)

where H0 is the single-particle spin-orbit coupling Hamiltonian as before,h̄2 (k + kL /2 · σz )2
H0 =+ Kσx + δ0 σz
2m(4.45)The spin-orbit coupling mixes spin states with different momentum components,
as a result, it creates a spatial density modulatio and costs kinetic energy. Therefore,
for a balanced spin-1/2 system, spin-orbit coupling favors an immiscible state where
opposite spins are spatially separated. On the other hand, if the repulsive interaction
between different spins is much weaker than the intra-spin counterpart, gab  gaa , gbb ,the excessive interaction energy forces the spins to say miscible. Therefore, dependingon the relative strength of the kinentic energy in the stripes, and the asymmetry in theintra-spin and inter-spin interactions, a quantum phase transition from a stripe phase
to a separated phase is predicted for an interacting spin-orbit coupled condensate [38,39], and the relevant quantum phases are illustrated in Fig. 4-17.
Up to today, the stripe phase, which is related to supersolidity in condensed matter, has not been observed in experiment. In the first spin-orbit coupling experimentwith 87 Rb [34], the interaction strength gaa ' gbb = g, and the difference (gab − g)/g
is less than 1%, therefore the window for detecting the stripe phase is very small, as
the system prefers to stay in the separated phase.
In our spin-orbit coupled pseudospin system, the interspin interaction is controlled
by the overlap between the states from the left and right wells. In the lowest-order approximation, gab ' (J/∆)2 g ≤ 0.1g. There is a huge mismatch between the intraspin
106

Figure 4-17: Illustration of the quantum phases of an interacting spin-orbit coupledcondensate. For the weak couplings, two degenerate minima are created in the single
particle dispersion relation. Depending on the interaction, both minima are occupied
in the miscible phase with density stripes, or in an immiscible phase, the condensate
falls into one of the minima via spontaneous symmetry breaking. For strong couplings,
only a single minimum is present in the dispersion relation.
and interspin interactions, and naturally the system prefer to stay in the stripe phase.
Quantitative predictions are made on the phase diagram for an interacting spinorbit coupled condensate in [39]. For a low density atomic cloud, in which the meanfield energy is less than the recoil of the Raman beams, ng < EL /4, as we increase thespin-orbit coupling strength, a first transition from the stripe phase to the separated
phase happens at the critical Raman couplingEL
Kc1−2 =2r2γ
1 + 2γ(4.46)

where γ = (g − gab )/(g + gab ) is a dimensionless interaction parameter that measures
the asymmetry between interspin and intraspin interactions. In our experiment, γ ≈ 1gives the critical Raman coupling of Kc1−2 ' 0.4EL .
When we increase the spin-orbit coupling further, a second phase transition fromthe separated phase to the single minimum phase happens at
Kc2−3 = 0.5EL(4.47)

Finally, the phase transition from the stripe phase to the separated phase can be
driven by changing the Raman detuning away from the symmetric spin-orbit couplingat δ = ∆ as well, although the stripes will be moving in the detuned scenario.
107

In summary, compared with other spin-orbit coupled systems, the pseudospin-1/2
system in the superlattice has the desirable feature that, the interspin interaction istunable and could be dramatically different from the intraspin interactions. Therefore,
it is an ideal system to prepare and detect the stripe phase.

4.5

Experimental Realization of Spin-Orbit Coupling in a Superlattice

4.5.1

State preparation in a superlattice

To implement the spin-orbit coupling Hamiltonian and study its phase diagram suchas the stripe phase, one needs to prepare the initial quantum state in the double wells.
Conceptually, if every step in the preparation is adiabatic, the condensate will remainin the ground state, which is the lower well in an asymmetric double-well. Therefore,
some form of the diabatic process, or additional external couplings, are needed in thestate preparation for spin-orbit coupling experiment.
Fast switching of the superlattice phase
By fast switching the superlattice phase and changing its configuration in a timescale
much faster than the double-well tunneling, the atomic wavefunction is frozen in eachwell. However, special care needs to be taken to make sure that there’ll be no residual
overall motion of the superlattice so that the atomic wavefunction is kept stationary.
Even with a fairly high barrier, the tunneling rate is still on the order of hundred
Hertz between the double-well, which means we need to switch the superlattice phasein less than a millisecond. It is very challenging for mechanical switchs like the glass
plate. Furthermore, in the glass plate scheme, the relative shift of the two lattices isabout just one percent of the overall shift of the superlattice. Assuming the atomic
cloud will not be able to follow the overall motion in such a short timescale, it creates
dragging and undefined matching of the atomic wavefunction.
In contrast, fast switching of the lattice frequency won’t have the same problem.
108

As discussed earlier, switching the IR frequency keeps the atomic wavefunction morestationary, and it could be done far less than a milliesecond. We have been using thisscheme to prepare initial states with balanced population, or 100-percent populationin the upper-well, in an asymmetric double-well.
Sudden switch-on of the offset
Another way of preparing a balanced population in the asymmetric double-well is
to first load the condensate into a green lattice, and subsequently switch on the IR
lattice with a proper relative shift that is presetted by the glass plate. This is analternative option when the fast dynamical control of the superlattice phase is notavailable.
However, as we found out experimentally, this method is quite sensitive to latticealignment. If the IR lattice is not well-centered, this additional dipole potential also
creates sloshing motion and breathing mode excitations, demonstrated later in theexperimental section. Better beam alignment would reduce these effects.
On the other hand, these side effects from sudden switch-on of the IR lattice mightturn into our favor in the experimental detection phase. Because of the different trap
frequencies for the two states in the double well, the sloshing motion of them results ina spatial separation in the time-of-flight images. Therefore, this scheme could be used
as a Stern-Gerlach type experiment for an independent readout of the pseudospin-1/2
states.
Raman process
Both of the aforementioned methods prepare the upper-state with a constant phaseacross the atomic cloud. As we found out later, the upper state in fact has the groundstate at q = kL , or its wavefunction has alternating phases in the lattices. The above
two schemes rely on the assumption that there will not be too much heating created
while the atoms relax into the ground state in the upper well, which seems to be thecase in our experiment.
Motivated by this observation, another way of preparing the initial state is to use a109

direct two-photon process to couple the ground state in the bottom well to the groundstate in the upper well. To compare with the Raman-assisted spin-orbit coupling, thisRaman-assisted state preparation does not provide the extra momentum transfer inthe perpendicular direction. For state preparation, the required momentum transfer
of kL along the superlattice could be provided by a pair of 1064 nm Raman beams at
30◦ angle of incidence, or a pair of 532 nm Raman beams at 14.5◦ angle of incidence.
This Raman process without perpendicular momentum transfer could potentially be
useful in an interferometric readout of the atomic wavefunction in the double-well as
well.
Due to space constraints near the apparatus, we’ve implemented the first two state
preparation schemes, but not yet the last one.

4.5.2

Dynamics in the double-wells

The energy offset of ∆ can be calibrated by measuring the accumulated phase between
the A-state (the lower well) and B-state (the upper well) as a function of time in thetime-of-flight interference images. In this measurement, a BEC is prepared in a green
lattice, and subsequently a weak IR lattice is suddenly switched on with a properly
pre-selected superlattice phase to provide the energy offset. After a variable hold, thetime-of-flight image is taken to record the interference pattern, shown in Fig. 4-18.
As we see in Fig. 4-18, the population in the time-of-flight images oscillate between
the zeroth-order and first-order peaks, corresponding to the case with relative phaseof 0 and π. The oscillation frequency in the time-of-flight interference pattern, or theslope of the accumulated phase, is a measure of the energy offset ∆. This measurement
was performed before the upper-well state relaxed into its ground state at q = π.
If we continue to hold the condensate in this superlattice configuration, interesting
dynamics occur. This is better demonstrated from the horizontal imaging, where the
A-state and B-state are separated vertically due to a differential force from the slightmisalignment of the IR beam. The A-state on the top stayed in the same pattern ofq = 0, whereas the B-state on the bottom developed a different pattern corresponding
to a quasi-momentum of q = π, shown in Fig. 4-19.
110

Figure 4-18: The experimental calibration of the offset ∆. The relative phase between
the left and right wells is accumulated linearly in time, and shows up as the oscillationin the time-of-flight interference. This oscillation calibrates the offset frequency ∆ inour experiment. This measurement was performed before the upper-well state relaxed
into its ground state at q = kL .

There are two strong pieces of evidence to support our findings. First of all, theA-state and B-state correspond to the lowest two bands in a superlattice. Therefore,
A-state naturally has the ground state at q = 0, whereas B-state has the ground state
at q = π, based on the shape of the band structure. Second, a sloshing motion of theatomic cloud is induced by the sudden switch-on of the IR beam, and A-state would
have a faster oscillation in the sloshing motion because of the higher IR intensity andtherefore a higher trap frequency in the A-state. This is confirmed in Fig. 4-20, where
A state, or the lower state, does show a faster sloshing oscillation. Furthermore, therelaxation process in B-state is an indirect measure of the resonant tunneling between
the neighboring upper wells, whereas the tunneling between the A-state and B-state
within the same double-well is strongly suppressed by the energy offset, as the relative
population stay constant in Fig. 4-20.
111

Figure 4-19: The ground states of the lower- and upper-well in the superlattice. The
separation in the time-of-flight images is induced by the sloshing motion and different
trap frequency for the two states. The initial state is prepared by suddenly switchingon an energy offset ∆. After 750 µs, upper-well relaxes to its ground states at q = kL .

4.5.3

Detection of the spin-orbit coupling in time-of-flight

Structure factor in optical lattices
The interference pattern in the time-of-flight image shows momentum distribution of
the atomic cloud. One could also say, it is the measurement of static structure factorof the atomic wavefunction in the trap, and thus can be used to measure the relative
strength and phase of the superposed components in the superlattice. In this section,
we’ll derive the static structure factor for a periodic potential in the general case, anddiscuss the corresponding interference patterns in a few relevant phase configurations.
This general discussion about the static structure factor will show its important in
the Bragg scattering section later on as well.
Let’s assume that the distribution of interest is ψ(r). The momentum distribution
measured in a time-of-flight image, or the scatter signal measured in a scattering exR
periment, is proportional to I(q) ∼ S(q) ≡ |ψ(q)|2 , where ψ(q) = V ψ(r) exp(−iqr) dr
is the Fourier transform. This defines the structure factor in the most general context.
112

Figure 4-20: The separation of the lowest two bands in the time-of-flight images andthe relaxation of the second band into its ground state. The upper-well and lower-wellstates show oscillations with different trap frequencies. During the relaxation process,the relative population stays constant, which confirms that the tunneling between the
double-well is strongly suppressed by the energy offset ∆.

To make it clear, the distribution ψ(r) corresponds to the atomic wavefunction in thetime-of-flight interference measurement, while it is actually the atomic density in the
Bragg scattering experiment.
If the system of interest is composed of identical constituents, such as unit cells
in a lattice, it is convenient to explicitly capture the variation in ψ by the sum of thecontribution from each constituent,ψ(r) =

N
X
j=1

N
X
f (r − Rj ) =(f ∗ δ)(r − Rj )
j=1

113

(4.48)

with Rj , j = 1, . . . , N the position of each constituent. In the second equality, thefield is decomposed as a sum of the convolution product ∗ of the function f and Diracdelta functions depending only on the positions. Using the property that the Fouriertransform of a convolution product is simply the product of the Fourier transformsof the two factors, we have22

S(q) = |ψ(q)| = |f (q)| ×

N
X

2

e

−iqRj

(4.49)

j=1= S(q)unit cell × N S(q)grating(4.50)In short, time-of-flight interference images measure the structure factor of the wholesystem, which is a product of the structure factor of a grating and the structure factorof each unit cell.

For a one-dimensional superlattice, the structure factor of the grating with grating
constant a (also the size of the unit cell) isN
1 X −iqRjeS(q)grating =
N j=1

2

1 1 − e−iN qa=
N 1 − e−iqa22
1 sin(N qa/2)=
N sin(qa/2)(4.51)The reciprocal lattice has a spacing 2π/a, the intensity of the maxima increases with
the number of particles as S(q = 2nπ/a) = N , and the peak width decreases as 1/N .
In the large N limit, the peaks become infinitely sharp Dirac delta functions,lim S(q)grating =

N →∞

X
n

δ(q − 2nπ/a)(4.52)

which is a Dirac comb as a infinite series of Dirac delta functions spaced at intervalsof 2π/a.

Now let’s consider the structure factor in the unit cell S(q)unit cell . As before, one
could decompose the unit cell structure factor into a product of the structure factor ofeach constituent, or the single-slit structure factor, and the multi-slit structure factor
114

based on the arrangement of the constituents in each unit cell,
S(q)unit cell = S(q)single slit × S(q)multi slit

(4.53)

the single slit in the case of our superllattice corresponds to the Wannier state in eachwell. For simplicity, the single-slit structure factor is chosen to be a Gausssian profile
centered at q = 0 with a proper width for the following discussion in this section.
In the following, we’ll apply the general results to a few special cases for the spinorbit coupling experiment in a superlattice. The natural length unit in this system isthe superlattice period d = λ1064 /2, and the corresponding momentum unit is 2π/d.
We label k as the transverse momentum perpendicular to the superlattice, and q as
the quasi-momentum along the superlattice.

Case 1. k = 0 for 50/50 population
In the first case, we consider the interference between a 50/50 population with transverse momentum k = 0 in the double wells. This is relevant for the double-well offset
calibration, when the relative phase between left and right population is accumulated
linearly as a function of time, before the upper-state population relax into q = kL .
In this case, the structure factor in a unit cell is simply in a double-slit form,
S(q)multi slit = S(q)double slit = 1 + ei(φ−qb)2= 2[1 + cos(φ − qb)]

(4.54)

where φ is the relative phase between the right and left wells. The double-slit structure
factor is a sinusoidal function with the period 2π/b. In this case, the grating constantor the size of the unit cell is a = d, and the double-slit spacing is b = d/2.
The predicted time-of-flight images as a function of the relative phase φ are shownin Fig. 4-21. As time evolves, we expect to see oscillations between the centered φ = 0
pattern and the symmetric φ = π pattern at the frequency of offset ∆.
115

Figure 4-21: Case 1: The relative phases and the corresponding interference pattern of
the k = 0 component for 50/50 population. The relative phase is linearly accumulated
in time, therefore the time-of-flight interference will show oscillations at the frequency∆.

Case 2. k = ±kL for 100/0 population with only OC or SOC

The second case applies when we have only on-site coupling or spin-orbit coupling in
a 100/0 population in the superlattice. The propagating phase of the Raman coupling
is imprinted onto the atomic wavefunction to create an alternating phase.
In this case, the size of the unit cell is a = 2d, the double-slit spacing is b = d,and the relative phase in the double-slit structure factor is fixed at φ = π. Therefore,
we expect to see a stationary pattern of φ = π in time.
116

Figure 4-22: Case 2: The relative phases and the corresponding interference patternof the k = ±kL component for 100/0 population with only the OC or SOC component.
The relative phase is locked by the propagation of the Raman coupling, and therefore
the time-of-flight interference will be stationary in time.

Case 3. k = ±kL for 100/0 population with both OC and SOC
The third case applies when we have both the on-site coupling and spin-orbit coupling
in a 100/0 population in the superlattice. In this case, besides the structure factors
considered above, there is an additional double-slit structure factor for the doublewell with spacing b = d/2, due to the interference between the onsite component andthe spin-orbit component.
The relative phase between these two comes from the propagating phase of theRaman coupling, and also depends on the sign of the two-photon detuning δ −∆±ER .
When the Raman frequency of ∆ is increased across the resonance, the relative phasebetween them (the right respect to the left) goes from −π/4 to 3π/4, while the spinorbit coupled component has a resonant strength. Therefore, we expect to see adispersive signature in the population difference between the left and right peaks.
The predicted time-of-flight images as a function of the relative phase φ are shownin Fig. 4-23, assuming on-site coupling and spin-orbit coupling have equal strengths.
117

Figure 4-23: Case 3: The relative phases and the corresponding interference pattern of
the k = ±kL component for 100/0 population with both the OC and SOC component.
The relative phase is finite from the propagation of the Raman coupling, and flips its
sign when across the SOC resonance, and therefore the time-of-flight interference willshow asymmetry and a dispersive signature in the difference between the plus/minus
first-order peaks.

Case 4. No interference between quasi-momentum q = 0 and q = kL states

Let’s consider the last situation, where we have a condensate in the left wells at quasimomentum q = 0, which has a constant phase in each well, and a condensate in theright wells at quasi-momenum q = kL which has a relative phase of φ with alternating
sign. This is motivated by our experimental observation that the upper well, or thesecond band in the superlattice, has the ground state at q = kL , rather than q = 0.
118

Figure 4-24: Case 4: There is no interference between quasi-momentum q = 0 andq = kL states. The total interference pattern is just the sum of the individual patterns.

The multi-slit structure factor in the unit cell in this case isS(q)multi slit = 1 + ei(φ+qd/2) + eiqd − ei(φ+3qd/2)

2(4.55)= 2 + 4 sin(qd/2)2 + 2 cos(qd) + 4[sin(φ) + sin(φ + qd)] sin(qd/2) (4.56)= 4 + 4 sin(φ + qd/2) sin(qd)(4.57)

Although the multi-slit structure factor shows dependence on φ, the grating structure
factor is non-vanishing only when at q = πn/d, where the multi-slit structure factor
is a constant of S(q = πn/d)multi slit = 4. Therefore, there is no interference in φ, andwe are not table to extract the value of it from the time-of-flight interference images,
as the final interference pattern is just the sum of the individual patterns.
Another easier way to see the result without explicit calculation is that, particles
with different momentum simply do not interfere. This is simple and straightforward,
since we are measuring the momentum-distribution in the time-of-flight images.
To take the point one step further, we claim that all the onsite coupling and spinorbit coupling from the left wells do not interfere with couplings from the right wells.
This conclusion could be made by noticing that the same Raman coupling shifts the119left-well state and right-well state by the same amount in momentum, and therefore
they still do not interference, or by carefully checking the relevant phases. As a result,the cases that we’ve discussed in this section have covered all the scenarios that we’ll
see in our experiment.

4.5.4

Experimental observation of resonant spin-orbit coupling

In this section, we present the experimental data on detecting the spin-orbit couplingsignal across the resonance. The experimental results presented here will be focusedon a sample with 100 percent population loaded into the A-state.

Figure 4-25: A typical time-of-flight image in the spin-orbit coupling experiment, andthe notation used in our data analysis. The relevant peaks are explained in the text.
First of all, a typical time-of-flight image in this configuration is shown in Fig. 4-25.
The time-of-flight pattern for the A-state with q = 0 shows up in the middle. After the
Raman coupling, there is a +k component separated from the main component, and it
has contributions from both on-site coupling and spin-orbit coupling. Similarly, there
is a −k component which corresponds to the counter-rotating terms in the Ramanprocess, and it is usually weaker than the +k component. In the following discussion,
we are mainly looking at the +k component.
120

Figure 4-26: The experimental detection of spin-orbit coupling near resonance. The
top plot shows the difference between the left and right peaks, where a clear dispersive
signal was observed. The dashed line is the zero crossing to guide the reader’s eyes.
The bottom plot shows the total of the left and right peaks where a bump showed up
around 43 kHz, which could be explained by the enhanced spin-orbit coupling near
the resonance. The dashed curve is a 1/∆2 fit from the tail for the on-site coupling.
Both +k and −k components show q = π time-of-flight pattern in Fig. 4-25. Thisis the result of interference from the alternating phase in the final state imprinted by
the propagating Raman coupling. From the momentum picture, the initial state is
the A-state with q = 0. Since the Raman coupling in our experiment also provides amomentum transfer along the superlattice, this results in the final state at q = π.
Since both on-site coupling and spin-orbit coupling have the final state at q = π,they will interfere in the time-of-flight depending on their relative phase. This relative
phase comes from the propagation of the Raman coupling, and will change only when
the spin-orbit coupling resonance is crossed. With careful count on the relative phase,
we expect to see a stronger left peak below the resonance, and a stronger right peakabove the resonance.
The experimental data are shown in Fig. 4-26, as the Raman detuning was scanned121across the spin-orbit coupling resonance. As we expected, the difference between theleft and right population showed a clear dispersive signature, and it went back to zeroaway from the resonance. In addition, there was a bump in the totoal population due
to the enhanced spin-orbit coupling near the resonance. The broadened width can be
explained by the inhomogeneity of the offset ∆ across the atomic cloud.

4.5.5

Detection of the spin-orbit coupling in Bragg scattering

For a spin-orbit coupled condensate in the miscible state, different momentum components of the same spin in the superposition interfere and result in density modulation,or the stripe. In our experiment, even though the pseudospins are in fact in the sameatomic state, they are spatially separated in the double-wells, and form stripes along
the wells perpendicular to the lattice.
The density modulation comes from the interference between different momentum
components. Let’s consider the general case with a superposition of two momentum
states shifted by momentum kL and energy ω,
f (x) = (a + beikL x−iωt+iφ0 )e−iµt(4.58)

the atomic density is just the norm of the wavefunction,n(x) = (a + beikL x−iωt+iφ0 )(a + beikL x−iωt+iφ0 )∗(4.59)= a2 + b2 + 2ab cos(kL x − ωt + φ0 )(4.60)

In our experiment, the interference between the main component Eq. 4.36 and thespin-orbit coupling component Eq. 4.37 gives the stationary stripes with periodicity of1064 nm, when δ = ∆ and therefore ω = 0. It is stationary because the relative phase
does not change in time. The stripes are in phase within the double-well, as well as
between the neighboring double-wells, as the initial phase φ0 is constant throughout
the cloud. Therefore, the stripe pattern is orthogonal to the superlattice as shown in
Fig. 4-27.
122

Figure 4-27: The stationary stripes from the spin-orbit coupling, and the moving density modulations from the onsite coupling, and their corresponding structure factors.
Notice they have different Bragg planes, and therefore could be detected individuallyby sending the probe beam from the corresponding Bragg angles. Higher-order Bragg
scatterings will vanish, because the sinusoidal density modulation has a single Fouriercomponent.
In contrast, the density modulation created by the onsite coupling, comes from theinterference between the main component Eq. 4.36 and the onsite coupling component
Eq. 4.39-4.40. These two momentum components have a frequency difference of δ,and therefore cause the stripes to move with a speed of δ/kL ' 0.1 m/s. The initial
phase of φ0 is alternating due to the propagating phase of the Raman coupling. As aresult, the density modulation is at the 45◦ angle and moving perpendicularly to thelattice, shown in Fig. 4-27 as well.
In fact, an easier way to figure out the stripe pattern is by noticing (again) that it
all comes from the momentum difference of the two components that are interfering.
123

For the onsite coupling, these two are the Raman-coupled and the main component
from the same well, which differ by the momentum transfer of (kL , kL ) in the Ramanprocess. Whereas for the spin-orbit coupling, these two are the Raman-coupled component from left and the main component from right, or vice versa, and because thetwo wells have ground states that are shifted in momentum by kL , the two componentsthat are interfering differ by momentum of (0, kL ). We reach the same conclusion on
the stripe patterns in a direct way.
Finally, let’s discuss how to detect the density stripes. The time-of-flight detection
presented earlier measures the momentum distribution of a cloud, but it does not tell
the miscibility of the system. Directly sensitive to the density stripes, Bragg scattering
becomes the ideal way of probing the miscible state.

Figure 4-28: The Bragg scattering experiment for detecting the stripe phase. A spinorbit coupled condensate in the miscible state creates density stripes with periodicityof 1064 nm. A near-resonant light of 589 nm is sent to the atomic cloud at the Bragg
angle, illustrated in the figure, for detecting the density stripes.
The constructive interference of the scattered photon follows the Bragg’s law,nλ = 2d sin θ(4.61)

where θ is the Bragg angle illustrated in Fig. 4-28. The Bragg scattering experimentcan also be considered as a way to measure the structure factor of the atomic density,124Bragg angle θ

1st order
16.07◦

2nd order
33.61◦

3rd order
56.13◦

Table 4.3: The Bragg angles for detecting the stripe phase using 589 nm probe beam.
Higher-order Bragg scatterings will vanish, because the sinusoidal density modulation
has a single Fourier component.
in which the Bragg photon only gets diffracted with a momentum kick ∆k when the
structure factor of the atomic density S(∆k) is finite.
To generate a sufficient amount of coherently scattered photons, the near-resonant
589 nm light will be used. The Bragg angles of the lowest three order Bragg scattering
have been calculated in Tab. 4.3. However, the higher-order Bragg scatterings vanishin our experiment, because the sinusoidal density modulation has a only single Fouriercomponent.
The moving stripes induced by the onsite coupling can also be detected via Bragg
scattering, with a different set of Bragg angles. Due to the motion of these stripes, thescattered Bragg photons will be Doppler-shifted by frequency δ. This frequency shift
can be detected by beating the scattered beam with the probe beam via a heterodyne
detection.

125

126

Chapter 5
Density Fluctuations in Quantum
Degenerate Fermi Gases
Atomic density fluctuation measurements have been carried out as a proof-of-principle
experiment in an ideal Fermi gas [40], and these developed techniques are then used
to probe many-body physics in strongly interacting Fermi gases [41, 42]. These works
have been summarized with details in earlier theses in this lab [7, 8, 43], and publishedpapers are attached in the appendix.

127

128

Chapter 6
Outlook and Conclusion
This chapter will briefly discuss the extensions of the spin-orbit coupling experimentand future experiment towards direct observation of the stripe phase, as well as otherpromising directions with the new apparatus. It will also conclude the research workthat has been presented in this thesis.

6.1

OutlookIn this section, we’ll discuss future directions of this experiment, including extensionsof the work carried out in this thesis, as well as other interesting and promising topics
that can be studied with this system.
Spin-orbit coupling from the B-state
As we have experimentally observed the resonant spin-orbit coupling signal from the
A-state, the next step is to detect the reverse process of resonant spin-orbit couplingfrom the B-state. Due to the recoils along the transverse direction, the two resonances
will be offset by 2EL .
The fact that A-state and B-state have the ground state with different momentum
leads to a favorable situation for our experimental detection: they are orthogonal anddo not interfere in the time-of-flight measurement. This means, even though spin up
and spin down are in fact in the same atomic hyperfine state in our scheme, we are able129

to independently read them out in a single shot. Having detected the resonant spinorbit coupling from the B-state would complete the study of single particle physics
in the spin-orbit coupling Hamiltonian in this system.
Direct detection of the stripe phase via Bragg scattering
From single particle physics to an interacting system, it is very interesting to probe
the stripe phase and study the phase transition from the stripe phase into a separated
phase. The stripe phase, equivalent of the supersolid phase in condensed matter, has
not been directly observed yet in experiment and is of great interest.
The spacing of the stripes in our experiment is on the order of optical wavelength,
therefore it is very challenging to resolve them in-situ. Alternatively, Bragg scattering
is an ideal tool to directly probe the spatial ordering in the atomic density. Detailed
discussions about using Bragg scattering to detect the stripe phase can be found insection 4.5.5.
Spin-dependent models in quantum magnetism
This pseudo-spin system in optical superlattices is also suitable to simulate quantum
magnetism and study the magnetic phase transition. Because of different barriers, theresonant tunneling is spin-dependent, t↑ 6= t↓ , and due to the spatial separation of thepseudo-spins, U↑↓ ≤ U↑↑ , U↓↓ . This would allow us to simulate spin-dependent models
without using spin-dependent lattices of near-resonant light, and does not require a
Feshbach resonance since the model parameters can be controlled experimentally bytuning the superlattice.
Spin-orbit couplings and gauge fields with fermions
As we have seen, laser-assisted coupling in a one-dimensional superlattice would allow
us to simulate a spin-orbit coupled condensate, and laser-assisted tunneling in a tilted
lattice allows us to simulate a condensate under a strong synthetic magnetic field [26].
These techniques can be readily transferred into a fermionic system, such as 6 Li in our
experiment. Although specific topics still need to be identified, the emergent Fermi
130surface and additional spin degree of freedom in the hyperfine states with Feshbachresonances almost certainly guarantee us new physics waiting to be explored.

6.2

Conclusion

Quantum simulation of condensed matter systems using ultracold atomic gases remains one of the most exciting avenues of atomic physics. Towards this goal, a nextgeneration general-purpose cold atom machine was built and tested to meet the newlevel of complexity and reliability in optical lattice experiments. A new scheme withlong sample lifetime and robust stripe phase to simulate spin-orbit coupling Hamiltonian has been implemented using pseudo-spins in an optical superlattice. Thesedeveloped platforms and techniques could become relevant in simulating exotic quantum states in topological insulators and topological superfluids in the near future.
I hope this work would allow younger generations in this lab to explore interesting
quantum physics towards topological states of matter.

131

132

Appendix A
Suppression of Density
Fluctuations in a Quantum
Degenerate Fermi Gas
This appendix contains a reprint of Ref. [40]: Christian Sanner, Edward J. Su, Aviv
Keshet, Ralf Gommers, Yong-il Shin, Wujie Huang, and Wolfgang Ketterle, Suppression of Density Fluctuations in a Quantum Degenerate Fermi Gas, Phys. Rev. Lett.
105, 040402 (2010).

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Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS

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Suppression of Density Fluctuations in a Quantum Degenerate Fermi Gas
Christian Sanner, Edward J. Su, Aviv Keshet, Ralf Gommers, Yong-il Shin, Wujie Huang, and Wolfgang Ketterle
MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, and Department of Physics,
Massachusetts Institute of Technology, Cambridge Massachusetts 02139, USA(Received 7 May 2010; published 19 July 2010)
We study density profiles of an ideal Fermi gas and observe Pauli suppression of density fluctuations(atom shot noise) for cold clouds deep in the quantum degenerate regime. Strong suppression is observedfor probe volumes containing more than 10 000 atoms. Measuring the level of suppression provides
sensitive thermometry at low temperatures. After this method of sensitive noise measurements has beenvalidated with an ideal Fermi gas, it can now be applied to characterize phase transitions in strongly
correlated many-body systems.
DOI: 10.1103/PhysRevLett.105.040402

PACS numbers: 03.75.Ss, 05.30.Fk, 67.85.Lm

Systems of fermions obey the Pauli exclusion principle.
Processes that would require two fermions to occupy thesame quantum state are suppressed. In recent years, several classic experiments have directly observed manifestations of Pauli suppression in Fermi gases. Antibunching
and the suppression of noise correlations are a direct consequence of the forbidden double occupancy of a quantum state. Such experiments were carried out for electrons [1–3], neutral atoms [4,5], and neutrons [6]. In
principle, such experiments can be done with fermionsat any temperature, but in practice low temperatures increase the signal. A second class of (two-body) Pauli
suppression effects, the suppression of collisions, requires
a temperature low enough such that the de Broglie wavelength of the fermions becomes larger than the range of theinteratomic potential and p-wave collisions freeze-out.
Experiments observed the suppression of elastic collisions[7] and of clock shifts in radio frequency spectroscopy[8,9].
Here we report on the observation of Pauli suppressionof density fluctuations. This is, like the suppression ofcollisions between different kinds of fermions [10], amany-body phenomenon which occurs only at even lower
temperatures in the quantum degenerate regime, where theFermi gas is cooled below the Fermi temperature and thelow lying quantum states are occupied with probabilities
close to 1. In contrast, an ideal Bose gas close to quantum
degeneracy shows enhanced fluctuations [11].
The development of a technique to sensitively measuredensity fluctuations was motivated by the connection between density fluctuations and compressibility through thefluctuation-dissipation theorem. In this Letter, we validate
our technique for determining the compressibility by applying it to the ideal Fermi gas. In future work, it could be
extended to interesting many-body phases in optical lattices which are distinguished by their incompressibility [12].
These include the band insulator, Mott insulator, and also
the antiferromagnet for which spin fluctuations, i.e., fluctuations of the difference in density between the two spinstates are suppressed.
0031-9007=10=105(4)=040402(4)Until now, sub-Poissonian number fluctuations of ultracold atoms have been observed only for small clouds ofbosons with typically a few hundred atoms [13–16] anddirectly [17,18] or indirectly [19] for the bosonic Mott
insulator in optical lattices. For fermions in optical lattices,
the crossover to an incompressible Mott insulator phase
was inferred from the fraction of double occupations [20]or the cloud size [21]. Here we report the observation ofdensity fluctuations in a large cloud of fermions, showing
sub-Poissonian statistics for atom numbers in excess of
10 000 per probe volume.
The basic concept of the experiment is to repeatedly
produce cold gas clouds and then count the number ofatoms in a small probe volume within the extended cloud.
Many iterations allow us to determine the average atom
number N in the probe volume and its variance ðNÞ2 . For
independent particles, one expects Poisson statistics, i.e.,ðNÞ2 =hNi ¼ 1. This is directly obtained from thefluctuation-dissipation theorem ðNÞ2 =hNi ¼ nkB TT ,where n is the density of the gas, and T the isothermal
compressibility. For an ideal classical gas T ¼ 1=ðnkB TÞ,p

Ballistic expansion
p
2pF
x

x
2pF

FIG. 1. Phase space diagram of ballistic expansion of a harmonically trapped Fermi gas. Ballistic expansion conserves
phase space density and shears the initially occupied sphericalarea into an ellipse. In the center of the cloud, the local Fermi
momentum and the sharpness of the Fermi distribution are scaledby the same factor, keeping the ratio of local temperature to
Fermi energy constant. The same is true for all points in the
expanded cloud relative to their corresponding unscaled in-trap
points.

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PHYSICAL REVIEW LETTERS

and one retrieves Poissonian statistics. For an ideal Fermi
gas close to zero temperature with Fermi energy EF , T ¼
3=ð2nEF Þ, and the variance ðNÞ2 is suppressed below
Poissonian fluctuations by the Pauli suppression factor
3kB T=ð2EF Þ. All number fluctuations are thermal, as indicated by the proportionality of ðNÞ2 to the temperaturein the fluctuation-dissipation theorem. Only for the idealclassical gas, where the compressibility diverges as 1=T,
one obtains Poissonian fluctuations even at zerotemperature.
The counting of atoms in a probe volume can be donewith trapped atoms, or after ballistic expansion. Ballistic
expansion maintains the phase space density and therefore
the occupation statistics. Consequently, density fluctuations are exactly rescaled in space by the ballistic expansion factors as shown in Fig. 1 [22,23]. Note that this
rescaling is a unique property of the harmonic oscillator
potential, so future work on density fluctuations in opticallattices must employ in-trap imaging. For the present work,
we chose ballistic expansion. This choice increases thenumber of fully resolved bins due to optical resolution
and depth of field, it allows adjusting the optimum opticaldensity by choosing an appropriate expansion time, and it
avoids image artifacts at high magnification.
We first present our main results, and then discuss
important aspects of sample preparation, calibration ofabsorption cross section, data analysis and corrections forphoton shot noise. Figure 2(a) shows an absorption imageof an expanding cloud of fermionic atoms. The probe
volume, in which the number of atoms is counted, ischosen to be 26 m in the transverse directions, and extends through the entire cloud in the direction of the line ofsight. The large transverse size avoids averaging of fluctuations due to finite optical resolution. From 85 such images, after careful normalization [24], the variance in themeasured atom number is determined as a function of
position. After subtracting the photon shot noise contribution, a 2D image of the atom number variance ðNÞ2 isobtained. For a Poissonian sample (with no suppression offluctuations), this image would be identical to an absorption image showing the number of atoms per probe volume. This is close to the situation for the hottest cloud (thetemperature was limited by the trap depth), whereas thecolder clouds show a distinct suppression of the atom
number variance, especially in the center of the cloud
where the local T=TF is smallest.
In Fig. 3, profiles of the variance are compared totheoretical predictions [25,26]. Density fluctuations atwave vector q are proportional to the structure factor
Sðq; TÞ. Since our probe volume (transverse size 26 m)
is much larger than the inverse Fermi wave vector of the
expanded cloud (1=qF ¼ 1:1 m), Sðq ¼ 0; TÞ has been
integrated along the line of sight for comparison with theexperimental profiles. Within the local density approximation, Sðq ¼ 0; TÞ at a given position in the trap is the
binomial variance nk ð1  nk Þ integrated over all momenta,
where the occupation probability nk ðk; ; TÞ is obtained
from the Fermi-Dirac distribution with a local chemical
potential  determined by the shape of the trap. Figure 4
shows the dependence of the atom number variance on
atom number for the hot and cold clouds. A statisticalanalysis of the data used in the figure is in [24].
The experiments were carried out with typically 2:5 
106 6 Li atoms per spin state confined in a round crossed
dipole trap with radial and axial trap frequencies !r ¼
2  160 s1 and !z ¼ 2  230 s1 corresponding toan in-trap Fermi energy of EF ¼ kB  2:15 K. The sam2

/1000

FIG. 2 (color online). Comparison of density images to variance images. For Poissonian fluctuations, the two images at agiven temperature should be identical. The variance images wereobtained by determining the local density fluctuations from a setof 85 images taken under identical conditions. (a) Two dimensional image of the optical density of an ideal Fermi gas after
7 ms of ballistic expansion. The noise data were taken bylimiting the field of view to the dashed region of interest,
allowing for faster image acquisition. (b) For the heated sample,
variance and density pictures are almost identical, implying only
modest deviation from Poissonian statistics. (c) Fermi suppression of density fluctuations deep in the quantum degenerate
regime manifests itself through the difference between densityand variance picture. Especially in the center of the cloud, there
is a large suppression of density fluctuations. The varianceimages were smoothed over 6  6 bins. The width of images(b) and (c) is 2 mm.

5 (a)

0

0

(b)

(c)

100
Position (bins)

FIG. 3. Comparison of observed variances (black dots) with atheoretical model (black line) and the observed atom number(gray), at three different temperatures (a, b, and c), showing 50,
40, and 15% suppression. Noise thermometry is implemented byfitting the observed fluctuations, resulting in temperatures T=TF
of 0:23  :01, 0:33  :02, and 0:60  :02. This is in good
agreement with temperatures 0:21  :01, 0:31  :01, and 0:6 :1 obtained by fitting the shape of the expanded cloud [32]. The
quoted uncertainties correspond to 1 standard deviation and are
purely statistical.

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bin, the atom number variance ðNÞ2 is obtained by subtracting the known photon shot noise from the variance inthe optical density ðODÞ2 [24]:

4000

1
1
2

ðNÞ2 ¼ ðODÞ2 
2
hN
hN
i
A
1
2i

2000

0

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0

2000

4000

Atom number

FIG. 4. Atom number variance vs average atom number. For
each spatial position, the average atom number per bin and its
variance were determined using 85 images. The filled and open
circles in the figure are averages of different spatial bin positions
with similar average atom number. For a hot cloud at T=TF ¼
0:6 (filled circles), the atom number variance is equal to theaverage atom number (dotted line, full Poissonian noise) in thespatial wings where the atom number is low. The deviation fromthe linear slope for a cold cloud at T=TF ¼ 0:21 (open circles) isdue to Pauli suppression of density fluctuations. There is also
some suppression at the center of the hot cloud, where the atom
number is high. The solid and dashed lines are quadratic fits forthe hot and cold clouds to guide the eye.

ple was prepared by laser cooling followed by sympathetic
cooling with 23 Na in a magnetic trap. 6 Li atoms in thehighest hyperfine state were transferred into the opticaltrap, and an equal mixture of atoms in the lowest two hyperfine states was produced. The sample was then evaporatively cooled by lowering the optical trapping potential ata magnetic bias field B ¼ 320  5 G where a scattering
length of 300 Bohr radii ensured efficient evaporation.
Finally, the magnetic field was increased to B ¼ 520  5 G,
near the zero crossing of the scattering length. Absorption
images were taken after 7 ms of ballistic expansion.
We were careful to prepare all samples with similar
cloud sizes and central optical densities to ensure thatthey were imaged with the same effective cross section
and resolution. Hotter clouds were prepared by heating thecolder cloud using parametric modulation of the trapping
potential. For the hottest cloud this was done near 520 G toavoid excessive evaporation losses.
Atomic shot noise dominates over photon shot noise
only if each atom absorbs several photons. As a result,the absorption images were taken using the cycling transition to the lowest lying branch of the 2 P3=2 manifold.
However, the number of absorbed photons that could be
tolerated was severely limited by the acceleration of theatoms by the photon recoil, which Doppler shifts the atoms
out of resonance. Consequently, the effective absorption
cross section depends on the probe laser intensity andduration. To remove the need for nonlinear normalization
procedures, we chose a probe laser intensity corresponding
to an average of only 6 absorbed photons per atom during a
4 s exposure. At this intensity, about 12% of the 6 Li
saturation intensity, the measured optical density was
20% lower than its low-intensity value [24]. For each

(1)Here, hN1 iðhN2 iÞ are the average photon numbers per bin ofarea A in the image with (without) atoms and  is theabsorption cross section.
The absorption cross section is a crucial quantity in theconversion factor between the optical density and thenumber of detected atoms. For the cycling transition, theresonant absorption cross section is 2:14  1013 m2 .
Applying the measured 20% reduction mentioned above
leads to a value of 1:71  1013 m2 . This is an upper limit
to the cross section due to imperfections in polarization andresidual line broadening. An independent estimate of theeffective cross section of 1:48  1013 m2 was obtained bycomparing the integrated optical density to the number of
fermions necessary to fill up the trap to the chemical
potential. The value of the chemical potential was obtained
from fits to the ballistic expansion pictures that allowed
independent determination of the absolute temperature andthe fugacity of the gas. We could not precisely assess theaccuracy of this value of the cross section, since we did not
fully characterize the effect of a weak residual magnetic
field curvature on trapping and on the ballistic expansion.
The most accurate value for the effective cross section wasdetermined from the observed atom shot noise itself. The
atom shot noise in the wings of the hottest cloud is
Poissonian, and this condition determines the absorption
cross section. Requiring that the slope of variance of theatom number ðNÞ2 vs atom number N is unity (see Fig. 4)
results in a value of ð1:50  0:12Þ  1013 m2 for theeffective cross section in good agreement with the two
above estimates.
The spatial volume for the atom counting needs to belarger than the optical resolution. For smaller bin sizes (i.e.,
small counting volumes), the noise is reduced since thefinite spatial resolution and depth of field blur the absorption signal. In our setup, the smallest bin size without
blurring was determined by the depth of field, since thesize of the expanded cloud was larger than the depth offield associated with the diffraction limit of our opticalsystem. We determined the effective optical resolution bybinning the absorption data over more and more pixels of
the CCD camera, and determining the normalized central
variance ðNÞ2 =N vs bin size [24]. The normalized variance increased and saturated for bin sizes larger than
26 m (in the object plane), and this bin size was usedin the data analysis. We observe the same suppression
ratios for bin sizes as large as 40 m, corresponding tomore than 10 000 atoms per bin.
For a cold fermion cloud, the zero temperature structure
factor SðqÞ becomes unity for q > 2qF . This reflects the040402-3
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PHYSICAL REVIEW LETTERS

fact that momentum transfer above 2qF to any particle will
not be Pauli suppressed by occupation of the final state. In
principle, this can be observed by using bin sizes smaller
than the Fermi wavelength, or by Fourier transforming the
spatial noise images. For large values of q, Pauli suppression of density fluctuations should disappear, and the noise
should be Poissonian. However, our imaging system loses
its contrast before q  2qF [24].
Observation of density fluctuations, through thefluctuation-dissipation theorem, determines the productof temperature and compressibility. It provides an absolute
thermometer, as demonstrated in Fig. 3 if the compressibility is known or is experimentally determined from theshape of the density profile of the trapped cloud [17,27].
Because variance is proportional to temperature for T 
TF , noise thermometry maintains its sensitivity at very low
temperature, in contrast to the standard technique of fitting
spatial profiles.
Density fluctuations lead to Rayleigh scattering of light.
The differential cross section for scattering light of wave
vector k by an angle  is proportional to the structure factor
SðqÞ, where q ¼ 2k sinð=2Þ [26]. In this work, we havedirectly observed the Pauli suppression of density fluctuations and therefore SðqÞ < 1, implying suppression of lightscattering at small angles (corresponding to values of q
inversely proportional to our bin size). How are the absorption images affected by this suppression? Since thephoton recoil was larger than the Fermi momentum of the
expanded cloud, large-angle light scattering is not suppressed. For the parameters of our experiment, we estimatethat the absorption cross section at the center of a T ¼ 0
Fermi cloud is reduced by only 0.3% due to Pauli blocking[28]. Although we have not directly observed Pauli suppression of light scattering, which has been discussed forover 20 years [28–30], by observing reduced density fluctuations we have seen the underlying mechanism for suppression of light scattering.
In conclusion, we have established a sensitive techniquefor determining atomic shot noise and observed the suppression of density fluctuations in a quantum degenerate
ideal Fermi gas. This technique is promising for thermometry of strongly correlated many-body systems and forobserving phase-transitions or cross-overs to incompressible quantum phases.
We acknowledge Joseph Thywissen and Markus Greiner
for useful discussions. This work was supported by NSF
and the Office of Naval Research, AFOSR (through the
MURI program), and under Army Research Office grant
no. W911NF-07-1-0493 with funds from the DARPA
Optical Lattice Emulator program.
Note added in proof.—Results similar to ours are reported in Ref. [31].

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Suppression of Density Fluctuations in a Quantum Degenerate Fermi Gas:
Supplementary Information

EXPERIMENTAL DETAILSTo accurately measure the atom number variance it isnecessary to eliminate patterns in the absorption images
whose ﬂuctuations increase the observed noise. Weakreﬂections of the probe beam from the walls of the glass
cell and from optical elements in the imaging system caninterfere with the probe beam itself, leading to spatial
ﬂuctuations in its intensity proﬁle. To reduce interference
fringes and ensure uniform illumination, the central areaof the probe beam is imaged onto the sample througha 2 mm aperture. Even though the residual fringes are
small, there are two signiﬁcant eﬀects of inhomogeneous
illumination which must be addressed.
First, if the time elapsed between the image with atoms
and the reference image without atoms is too large, mechanical vibrations of the optics will cause the intensity
proﬁle of the probe to change between the two images,
creating artifacts in the absorption image. To reduce
this eﬀect, we operate our CCD in fast kinetics mode,with a time interval ≈ 500µs between exposures. Since
there is no longer enough time for the atoms to exit theframe between images, before taking the reference image
we optically pump the atoms from the |1⟩ state to the |6⟩state, and from the |2⟩ state to the |5⟩ state (|1⟩ refers
to the lowest hyperﬁne state, etc.), by exciting them tothe mJ = 1/2 excited state manifold. At the magnetic
ﬁelds used in the experiment, these levels are separated
in frequency from the |2⟩ state used for imaging by ≈ 2
GHz and contribute negligibly to resonant imaging.
Second, if the average probe intensity is too high, theatoms subjected to higher intensities will have a lower
eﬀective cross-section, and so any spatial ﬂuctuations in
the beam intensity will be ‘imprinted’ onto the absorption images. As a result, in our experiment we use aprobe beam with maximum intensity of 0.12 of the saturation intensity Isat =2.54 mW/cm2 , where these eﬀects
are relatively small. The variation of optical density withintensity is shown in Fig. 1.
Additionally, the exposure time must be kept very
short to prevent the atoms from moving between pixels during the exposure. The expected motion of atoms
during the 4 µs exposure is on the order of 1 µm, muchsmaller than the eﬀective pixel size.

Optical density

1.4

1.2

1
0.05

0

Probe laser intensity (I / ISAT)
FIG. 1: Determination of the absorption cross section. The
observed optical density decreases with increasing probe lightintensity. The line is a quadratic fit to the data. The reduction of the cross section is mainly due to the Doppler
effect caused by acceleration of the atoms by radiation pressure; a smaller reduction results from the partial saturation of
the optical transition. At the probe light intensity chosen inthis study (shaded bar), the number of photons absorbed per
atoms is about 6. The decrease of the cross section is slightly
larger than that predicted by simple models.

must ﬁrst eliminate the eﬀect of ﬂuctuations in the total
atom number between experimental cycles. Initially, we
select the 85 images used in the analysis from a larger
group of ≈ 150 images, using an automated procedure tochoose the images whose total atom numbers are closest
to the center of the distribution. A very small number ofimages (< 1%) are manually excluded because of obvious
artifacts in the frame due to dust particles or other large
perturbations. Then, we subtract a ﬁtted proﬁle fromeach OD image before computing the variance. Initially
we subtracted a ﬁtted 2D Thomas-Fermi proﬁle, but wereplaced this with a Gaussian ﬁt which had an insignificant eﬀect on the variances, while taking considerably
less computation time.
We then compute the variance in optical density ateach position. That variance has contributions from photon and atom shot noise, given by the following formula:(∆(OD))2 =NOISE DETERMINATIONIn this experiment the local atom number variance isdetermined by comparing the measured number of atoms
in the same bin across a series of images. To do this, we

138

0.1

1
1
σ2
+
+ 2 (∆Natom )2⟨N1 ⟩ ⟨N2 ⟩ A(1)This equation holds bin by bin: N1 is the average numberof photons measured in a given bin position for the image with atoms, and N2 is the average number of photons
measured in that bin for the reference image. (∆Natom )2is the variance in atom number for that bin, σ is the2

Atom number variance

(a)

(b)

6000

6000

4000

4000

2000

2000

0

0

0

2000

4000

0

Atom number

2000

4000

Atom number

FIG. 2: Atom number variance vs. atom number. (a) Data for all of the resolution elements is plotted. Red points are from the
hot cloud at T /TF = 0.6, blue points from the cold cloud at T /TF = 0.21. There is significant scatter in the variance data, andthere are many “cold” pixels which actually have higher variance than their corresponding “hot” pixel. (b) The red and blue
shaded regions indicate the expected 2σ scatter in sample variance that is expected due to atom and photon counting statistics.
The large circles are variance data averaged over pixels with similar atom number for hot (red) and cold (blue) cloud. The bars
show the measured 2σ scatter of the data points. The measured scatter agrees very well with the expected scatter, indicatingthat the scatter of the data is fully accounted for by counting statistics. Negative values of the observed atom number varianceresult from the subtraction of photon shot noise.

Atom number variance

5000

0

0

Position (bins)

100

FIG. 3: Determination of profiles of the atom number variancefor a cold cloud. For each bin, the total photon count is determined, and its contribution (red) to the total variance of the
optical density (blue) is subtracted. The obtained atom number variance (green) is compared to the average atom number(black). The displayed trace reveals 50% noise suppression in
the center of the cloud. The apparently high suppression of
atom variation in the wings is a statistical fluctuation. Fig. 2
shows that the suppression is monotonic in atomic density.

139

absorption cross section, and A is the eﬀective bin area.
The atom number variance is isolated by calculating theﬁrst two photon shot noise terms and subtracting them.
The analysis used in the paper also subtracts contributions from detector read noise and photon shot noise in
the dark ﬁeld, but these are fairly small contributions.
The determination of N1 and N2 depends on the CCD
gain, which is measured to be 1.18 (counts/electron) from≈ 240 pairs of images without atoms, employing the assumption that the detector statistics are Poissonian. After the subtraction of photon shot noise (and technicalnoise), the remaining variance in optical density is dueto the atom number variance. Fig. 3 shows the contributions of photon and atom number variance to the overallnoise in optical density.
The large scatter of the measured atom number variance, as depicted in Fig. 2, is not primarily due to technical noise, but instead a statistical property of the sampling distribution of the variance. The shaded areas arederived from theoretical values for the variance of thesample variance. This is given by
Var(Var(N )) =(m − 1)(m − 3)(m − 1)2
µ4 −
µ2
m3
m3

(2)

where m is the number of observations in each sample.
The moments µ2 and µ4 are the central moments of the
population distribution. For a Poisson distribution, µ2 =⟨N ⟩3 and µ4 = ⟨N ⟩(1 + 3⟨N ⟩), and for m, ⟨N ⟩ ≫ 1,
this expression reduces to 2⟨N ⟩2 /m. Fig. 2b shows the3comparison between the expected and measured variancein the sample variance.
IMAGING SYSTEM CHARACTERIZATIONThe blurring of adjacent pixels due to ﬁnite opticalresolution eﬀectively decreases the measured atom number variance. This eﬀect is avoided by binning the datausing a suﬃciently large bin size (Fig. 5). In our experiment, this bin size is determined by the extension of
the cloud along the optical axis, which is much larger
than the depth of focus of the diﬀraction limit of the lenssystem.
Atom noise allows us to characterize the transfer function of our imaging system. Fig. 4 shows the average
power spectrum (modulus squared of the spatial Fourier

140

transform) of the optical density images. Because theFourier transform of uncorrelated ﬂuctuations is ﬂat, thedeviation from ﬂatness of the density noise corresponds
to blurring induced by the lens, barring the central peakcorresponding to the shape of the cloud. For wavevectors
q much larger than the resolution limit of the detection
scheme, the atom number ﬂuctuations are no longer imaged, and the power spectrum is the photon shot noise.
For our experiment this happens for q < qF . Comparison of the power spectra for the cold and the hot cloudshows, at small values of q, a 50% suppression, consistent with the results obtained using spatial bins. If theimaging system still had contrast at q > 2qF , we would
expect the ratio of the power spectra to approach unity,
since momentum transfer q > 2qF to a Fermi cloud has
negligible Pauli suppression.

4Power (arb units)

(a)
200
100
0

0

0.2

0.4

0.6

0.8

1

q / qF
(b)

(c)

-1
0

qy / qF
1-1

0

1

qx / qF
FIG. 4: (a) Radially averaged power spectra of optical density
images for hot (solid line) and cold (dashed line) samples (b)Power spectrum of cold sample (arbitrary units) (c) Power
spectrum of hot sample (arbitrary units). A constant offset isadded to the power spectrum for the hot sample to equalize
the levels of photon shot noise.

1

Var(N)/N

0.8

0.6

0.4

0.2

0
0

10

20

30

40

Bin size (μm)
FIG. 5: Observed atom number variance versus bin size forheated (dashed line) and cold (solid line) samples, normalized
to 1 for Poissonian statistics. A plateau is reached when theblurring of the bins due to finite optical resolution is negligible.

141

142

Appendix B
Speckle Imaging of Spin
Fluctuations in a Strongly
Interacting Fermi Gas
This appendix contains a reprint of Ref. [41]: Christian Sanner, Edward J. Su,
Aviv Keshet, Wujie Huang, Jonathon Gillen, Ralf Gommers, and Wolfgang Ketterle,
Speckle Imaging of Spin Fluctuations in a Strongly Interacting Fermi Gas, Phys. Rev.
Lett. 106, 01040202 (2011).

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Speckle Imaging of Spin Fluctuations in a Strongly Interacting Fermi Gas
Christian Sanner, Edward J. Su, Aviv Keshet, Wujie Huang, Jonathon Gillen, Ralf Gommers, and Wolfgang Ketterle
MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, and Department of Physics,
Massachusetts Institute of Technology, Cambridge Massachusetts 02139, USA(Received 8 October 2010; revised manuscript received 10 December 2010; published 6 January 2011)
Spin fluctuations and density fluctuations are studied for a two-component gas of strongly interacting
fermions along the Bose-Einstein condensate-BCS crossover. This is done by in situ imaging of dispersive
speckle patterns. Compressibility and magnetic susceptibility are determined from the measured fluctuations. This new sensitive method easily resolves a tenfold suppression of spin fluctuations below shotnoise due to pairing, and can be applied to novel magnetic phases in optical lattices.
DOI: 10.1103/PhysRevLett.106.010402

PACS numbers: 05.30.Fk, 03.75.Ss, 67.85.Lm

One frontier in the field of ultracold atoms is the realization of quantum systems with strong interactions andstrong correlations. Many properties of strongly correlated
systems cannot be deduced from mean density distributions. This has drawn interest toward novel ways of probing cold atoms, e.g., via rf spectroscopy [1,2], Bragg andRaman scattering [3], interferometric methods [4,5], andby recording density correlations [6–8]. Further insight
into quantum systems is obtained by looking not only at
expectation values, but also at fluctuations. Several recent
studies looked at density fluctuations, either of bosons
around the Mott insulator transition [9–11], or of a gas ofnoninteracting fermions [12,13].
In this Letter, we extend the study of fluctuations ofultracold gases in several ways. First, we introduce thetechnique of speckle imaging as a simple and highly
sensitive method to characterize fluctuations. Second, weapply it to a two-component Fermi gas across the BoseEinstein condensate (BEC)-BCS crossover. Third, we directly measure fluctuations in the magnetization, i.e., thedifference of the densities in the two different spin components, bypassing the need to measure the individual
densities separately.
Our work is motivated by the prospect of realizing wideclasses of spin Hamiltonians using a two-component gas ofultracold atoms in an optical lattice [14,15]. An important
thermodynamic quantity to characterize two-component
systems is the spin susceptibility, which provides a clear
signature of phase transitions or crossovers involving theonset of pairing or magnetic order [16–19]. At a ferromagnetic phase transition the susceptibility diverges, whereas
in a transition to a paired or antiferromagnetic phase thesusceptibility becomes exponentially small in the ratio of
the pair binding energy (or antiferromagnetic gap) to thetemperature. The fluctuation-dissipation theorem relates
response functions to fluctuations, consequently the spin
susceptibility can be determined by measuring the fluctuations in the relative density of the two spin components.
In our experiment, we image the atom clouds using lightdetuned from resonance so that each atom’s real
0031-9007=11=106(1)=010402(4)polarizability, which contributes to the refractive index, ismuch larger than its imaginary polarizability, which contributes to absorption. Since the detunings for the two spinstates are different, spin fluctuations lead to fluctuations in
the local refractive index, resulting in phase shifts of theimaging light that vary randomly in space. We measurethese phase shifts by imaging the resulting speckle
patterns.
These speckle patterns are created by propagation,
which converts the spatially varying phase shifts of theimaging light into an intensity pattern on our detector
without the use of a phase plate. Spin and density fluctuations occur on all spatial scales down to the interatomic
separation; the smallest observable fluctuations have awavelength equal to the imaging system’s maximum resolution. In our system that length has a Rayleigh range, andhence a depth of field, smaller than the cloud size, so therecorded image is necessarily modified by propagation
effects. Propagation mixes up amplitude and phase signals[Fig. 1]. This can be easily seen in the case of a phasegrating, which creates an interference pattern further
downstream; after propagating for a distance equal to theRayleigh range of the grating spacing, the imprinted phaseis converted into an amplitude pattern. This feature of
speckle makes our imaging technique both simple androbust. It is insensitive against defocusing, and allows us
to image fluctuations of the real part of the refractive index(i.e., a phase signal) without a phase plate or other Fourier
optics.
Similar physics is responsible for laser speckle when arough surface scatters light with random phases [20], andoccurs when a Bose-Einstein condensate with phase fluctuations develops density fluctuations during expansion[21], or when a phase-contrast signal is turned into an
amplitude signal by deliberate defocusing [22].
The experiments were performed with typically 106 6 Li
atoms in each of the two lowest hyperfine states j1i and j2i
confined in an optical dipole trap oriented at 45 to theimaging axis with radial and axial trap frequencies !r ¼
2  108:9ð6Þ s1 and !z ¼ 2  7:75ð3Þ s1 . For the010402-1
144

Ó 2011 The American Physical SocietyTransmission image variance (arb. units)

PRL 106, 010402 (2011)d

PHYSICAL REVIEW LETTERS

2f

2f

0
0

50

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100

150

Defocus distance d (µm)

FIG. 1. Simulation of propagation effects after light has passed
through a Poissonian phase noise object. Shown are the variancemeasured in the amplitude or in-phase quadrature (black line)
and the out-of-phase quadrature (gray line) as a function of
defocus distance, for an imaging system with a numericalaperture of 0.14. Within a distance less than 5% of our cloudsize, noise becomes equally distributed between the two quadratures and the variances in transmission and phase-contrast
images become the same. (Top inset) For small phase fluctuations, an in-focus phase noise object gives no amplitude contrast,
but when it is out of focus it does. (Bottom inset) Sample
intensity patterns for a defocused phase object.

samples imaged at 527 G, the sample preparation was
similar to that described in [13], with a temperature of
0:14ð1ÞTF . The samples imaged at other magnetic fields
were prepared in a similar fashion, except that evaporation
was performed at 1000 G to a final temperature of T ¼
0:13ð1ÞTF before ramping the magnetic field over 1.5 s toits final value. The temperature at 1000 G was determinedby fitting a noninteracting Thomas-Fermi distribution intime of flight. The temperatures at other points in the
crossover were related to that value assuming an isentropic
ramp, using calculations presented in [23]. Using thismethod we obtain temperatures of 0:13ð1ÞTF at 915 G,
0:19ð1ÞTF at 830 G, and 0:19ð3ÞTF at 790 G where additional evaporation was performed to achieve a central
optical density similar to that at the other magnetic fields.
The extent of the cloud along the imaging direction was
135 m, much larger than the Rayleigh range of 8 m forour imaging system with a NA of 0.14.
The superfluid to normal phase boundary was determined by measuring condensate fraction [Fig. 2] using
the standard magnetic field sweep technique [24,25]. For
this, the magnetic field was rapidly switched to 570 G totransfer atom pairs to more deeply bound pairs (molecules)
which survive ballistic expansion. For resonant imaging of
the molecules, the field was ramped back to 790 G over
10 ms. The condensate fraction was determined by fitting
the one-dimensional density profiles with a bimodal
distribution.

FIG. 2. Measured condensate fraction as a function of dimensionless interaction strength 1=ðkF aÞ. Insets show typical images
from which the condensate fraction was extracted by fitting abimodal distribution. The dashed line is a sigmoidal fit to guidethe eye.

As previously described, propagation converts spatial
fluctuations in the refractive index into amplitude fluctuations on the detector. For different choices of the probe
light frequency, the two atomic spin states will have different real polarizabilities and the local refractive index will
be a different linear combination of the (line-of-sight integrated) column densities n1 and n2 . To measure the susceptibility we choose a probe light frequency exactly
between the resonances for states j1i and j2i, so that thereal polarizabilities are opposite and the refractive index is
proportional to the magnetization (n1  n2 ). The intensity
fluctuations on the detector after propagation are consequently proportional to the fluctuations in magnetization.
Since a refractive index proportional to (n1 þ n2 ) occurs
only in the limit of infinite detuning, we measure thefluctuations in the total density by exploiting the fact thatthe fluctuations in total density can be inferred from thefluctuations in two different linear combinations of n1 andn2 . For convenience, we obtain the second linear combination using a detuning that has the same value, but opposite sign for state j2i, and therefore three times the value forstate j1i. With this detuning, we record images of thefluctuations in (n1 =3 þ n2 ).
In principle, this information can be obtained by taking
separate absorption images on resonance for states j1i andj2i. However, the images would have to be taken on a timescale much faster than that of atomic motion and there
would be increased technical noise from the subtraction oflarge numbers. The use of dispersive imaging has the
additional advantage over absorption in that the numberof scattered photons in the forward direction is enhanced
by superradiance. As a result, for the same amount ofheating, a larger number of signal photons can be collected[26]. This is crucial for measuring atomic noise, which
requires the collection of several signal photons per atom.
The choice of detuning between the transitions of the twostates has the important feature that the index of refraction
for an equal mixture fluctuates around zero, avoiding any
lensing and other distortions of the probe beam. This is not

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PRL 106, 010402 (2011)

The observed ratio of 2 =2þ ¼ 1:56ð14Þ reflects excess
noise contributing to 2þ due to residual systematic dispersive effects and is accounted for by setting c0 =c ¼
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1:8=1:56. For high temperatures, the atomic noise of thenoninteracting gas approaches shot noise; for lower temperatures we observe a reduction in noise due to Pauliblocking as in our previous work [13]. With our new
method, we easily discern spin fluctuations with a varianceof less than 10% of atom shot noise.
The fluctuation-dissipation theorem connects the variances ½ðN1  N2 Þ2 and ½ðN1 þ N2 Þ2 to the susceptibility~ and the compressibility ~ via ½ðN1  N2 Þ2 ¼~ and ½ðN1 þ N2 Þ2 ¼ 3N=2ðT=TF Þ~ with3N=2ðT=TF Þ
N ¼ N1 þ N2 and T=TF being the temperature measured
in units of the Fermi temperature TF . Recomposing thevariances from the two experimentally accessible
linear combinations these relations become 2 =Nc2 ¼~
3=2ðT=TF Þand
9=42þ =Nc02  1=42 =Nc2 ¼
. The constants c and c0 are determined using
3=2ðT=TF Þ~
the noise measurements at 527 G for a noninteracting~¼
Fermi gas for which ~ ¼ 1 þ OððT=TF Þ2 Þ. This
analysis ignores line-of-sight integration corrections.
Figure 4 shows the spin susceptibility, the compressibility, and the ratio between the two quantities for the interacting mixtures as the interaction strength is varied throughthe BEC-BCS crossover. The susceptibility and compressibility reproduce the expected qualitative behavior: for the
sample at unitarity, where the transition temperature issufficiently high that a sizable portion of the sample issuperfluid, and for the sample on the BEC side, the spin
susceptibility is strongly suppressed relative to the compressibility. This reflects the fact that the atoms form bound
1/kF a1.2

100 µm8

1/(kF a)= ∞

6
4
2
0

5

10
15
20
25
Pixels (1 pixel = 2.6 µm)

30

1/kF a
-1.0

−4

x 10

∞

-1.5

1.0 0

1/kF a
-1.0

∞

-1.5

1.0 0

-1.0

-1.5

4

1.0
0.8
3

0.6

0.0

2

1

0.2

1/(kF a)= 0

3

0.6
0.4

0.4

4

κ / κ0

χ/κ

0.8

0.2

5

OD Variance

OD Variance

−4

1.0 0

1.0

Optical Density

0.6
0.4
0.2
0

∞

χ / χ0the case for other choices of detuning, and indeed, we
observe some excess noise in those images (see below).
At the detunings chosen, 10% residual attenuation is observed, some due to off-resonant absorption, some due todispersive scattering of light out of the imaging system bysmall scale density fluctuations. The contribution to thevariance of the absorption signal relative to the dispersive
signal scales as ð2Þ2 =2  0:006 and can be neglected in
the interpretation of the data.
The noise analysis procedure was nearly identical to that
performed in [13]. A high-pass filter with a cutoff wavelength of 13 m was applied to each image of the cloud tominimize the effect of fluctuations in total atom number.
Then, for each pixel position, the variance of the opticaldensities at that position in the different images was computed. After the subtraction of the contribution of photon
shot noise, the resulting variance image reflects the noise
contribution from the atoms.
The goal of our noise measurements is to determine atvarious interaction strengths the normalized susceptibility~ ¼ =0 and compressibility ~ ¼ =0 , where 0 ¼
3n=2EF and 0 ¼ 3=2nEF are the susceptibility and compressibility of a zero-temperature noninteracting Fermi gas
with the same total density n and Fermi energy EF . Before
studying spin fluctuations through the BEC-BCS crossover,
we therefore calibrate our measurement by measuring thespin fluctuations in a noninteracting mixture, realized at
527 G where the scattering length between the two statesvanishes. Figure 3 shows raw profiles of the variances 2
and 2þ measured at the two detunings. These fluctuationsin the speckle pattern are proportional to number fluctuations in the specified probe volume V: 2 ¼ ½cðN1 
N2 Þ2 and 2þ ¼ ½c0 ðN1 =3 þ N2 Þ2 . In these relations c
and c0 are factors which have to be calibrated. Without
interactions, N1 and N2 are uncorrelated, and one predicts½ðN1  N2 Þ2 =½ðN1 =3 þ N2 Þ2 ¼ 2=½1 þ ð1=3Þ2  ¼ 1:8.

x 10

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PHYSICAL REVIEW LETTERS

0.0
527

800

900

1000

0
527

800

900

1000

527

800

900

1000

Magnetic Field (G)Magnetic Field (G)Magnetic Field (G)

(a)

(b)

(c)

2
1
0

5

10
15
20 25
30
Pixels (1 pixel = 2.6 µm)

FIG. 3 (color online). (Top panel) Example speckle noise
image, with white box indicating analysis region. (Bottom
panels) Noise data for noninteracting (left panel) and resonantlyinteracting (right panel) cold clouds, showing 2 (black dots)and 2þ (gray dots). Solid lines are Gaussian fits to the data, anddotted lines illustrate the expected full Poissonian noise for thecorresponding quantities based on density determined from offresonant absorption.

FIG. 4. (a) The ratio =, (b) the normalized susceptibility
=0 , and (c) the normalized compressibility =0 in the BECBCS crossover. The variances derived from sequences of images
are converted into thermodynamic variables using the measured
temperatures and a calibration factor determined from the noninteracting gas. The vertical line indicates the onset region ofsuperfluidity, as determined via condensate fraction measurements. The curves show theoretical zero temperature estimates
based on 1st (dotted) and 2nd order (solid) perturbative formulas
obtained from Landau’s Fermi-liquid theory integrated along theline of sight, and results from a Monte Carlo calculation (dashed)
for the compressibility in a homogeneous system [32].

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PHYSICAL REVIEW LETTERS

molecules or generalized Cooper pairs; the spin susceptibility should be exponentially small in the binding energy,
while the enhanced compressibility reflects the bosoniccharacter of the molecular condensate. At 915 G and1000 G, where the sample is above the superfluid criticaltemperature, the susceptibility is larger but still below its
value for the noninteracting gas, reflecting the persistenceof pair correlations even in the normal phase of the gas.
Above the Feshbach resonance, for attractive interactions, we compare our results to first and second order
perturbation theory in the small parameter kF a. This
ignores the instability to the superfluid BCS state at exponentially small temperatures. The perturbation theory is
formulated for the Landau parameters for a Fermi liquid[16,27]. The susceptibility and compressibility are givenby 0 = ¼ ð1 þ F0a Þm=m , 0 = ¼ ð1 þ F0s Þm=m ,where m ¼ mð1 þ F1s =3Þ is the effective mass, and Fls ,Fla are the lth angular momentum symmetric and antisymmetric Landau parameters, respectively. Although the experimental data are taken for relatively strong interactions
outside the range of validity for a perturbative description,
the predictions still capture the trends observed in thenormal phase above the Feshbach resonance. This showsthat more accurate measurements of the susceptibility, anda careful study of its temperature dependence, are required
to reveal the presence of a possible pseudogap phase.
In our analysis we have neglected quantum fluctuations
which are present even at zero temperature [16,28]. They
are related to the large-q static structure factor SðqÞ measured in [29] and proportional to the surface of the probe
volume, scaling with N 2=3 logðNÞ. For fluctuations of thetotal density, their relative contribution is roughly
N 1=3 =ðT=TF Þ, and at most 40% for our experimental
parameters. Attractive interactions and pairing suppress
both the thermal and quantum spin fluctuations, but it is
not known at what temperature quantum fluctuations become essential.
Spin susceptibilities can also be obtained from the equation of state which can be determined by analyzing the
average density profiles of imbalanced mixtures [30]. Our
method has the advantage of being applicable without
imbalance, and requires only local thermal equilibrium.
Moreover fluctuations can be compared with susceptibilities determined from the equation of state to perform
absolute, model-independent thermometry for strongly interacting systems [31].
In conclusion, we have demonstrated a new technique todetermine spin susceptibilities of ultracold atomic gases
using speckle imaging. We have validated and calibrated
this technique using an ideal Fermi gas and applied it to astrongly interacting Fermi gas in the BEC-BCS crossover.
This technique is directly applicable to studying pairingand magnetic ordering of two-component gases in opticallattices.

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7 JANUARY 2011

We acknowledge Qijin Chen and Kathy Levin for providing calculations of condensate fraction, Gregory
Astrakharchik and Stefano Giorgini for providing
Monte Carlo results for the compressibility, Sandro
Stringari and Alessio Recati for discussions, and Yong-il
Shin for experimental assistance. This work was supported
by NSF and the Office of Naval Research, AFOSR(through the MURI program), and under Army Research
Office Grant No. W911NF-07-1-0493 with funds from the
DARPA Optical Lattice Emulator program.

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[22] L. D. Turner et al., Opt. Lett. 29, 232 (2004).
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Fermi, Varenna, 1998 (IOS, Amsterdam, 1999).
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Press Inc., Oxford, 1980);[28] G. E. Astrakharchik, R. Combescot, and L. P. Pitaevskii,
Phys. Rev. A 76, 063616 (2007).
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148

Appendix C
Correlations and Pair Formation ina Repulsively Interacting Fermi
Gas
This appendix contains a reprint of Ref. [42]: Christian Sanner, Edward J. Su, Wujie
Huang, Aviv Keshet, Jonathon Gillen, and Wolfgang Ketterle, Correlations and Pair
Formation in a Repulsively Interacting Fermi Gas, Phys. Rev. Lett. 108, 240404(2012).

149

PRL 108, 240404 (2012)

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Correlations and Pair Formation in a Repulsively Interacting Fermi Gas
Christian Sanner, Edward J. Su, Wujie Huang, Aviv Keshet, Jonathon Gillen, and Wolfgang Ketterle
MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, and Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 9 August 2011; published 13 June 2012)
A degenerate Fermi gas is rapidly quenched into the regime of strong effective repulsion near a
Feshbach resonance. The spin fluctuations are monitored using speckle imaging and, contrary to several
theoretical predictions, the samples remain in the paramagnetic phase for an arbitrarily large scattering
length. Over a wide range of interaction strengths a rapid decay into bound pairs is observed over times on
the order of 10@=EF , preventing the study of equilibrium phases of strongly repulsive fermions. Our worksuggests that a Fermi gas with strong short-range repulsive interactions does not undergo a ferromagnetic
phase transition.
DOI: 10.1103/PhysRevLett.108.240404

PACS numbers: 03.75.Ss, 67.85.Lm, 75.10.Lp

0031-9007=12=108(24)=240404(5)time evolution of the system [19,20]. Theoretical studies
concluded that the pairing instability is somewhat fasterthan the ferromagnetic instability [21]. The second major
result of this paper is to show that pair formation occurs
indeed on a very short time scale. The measured time
constant of 10@=EF (where EF is the Fermi energy) indicates that the metastable repulsive state will never reachequilibrium and that, even in a metastable sense, a Fermi
gas with strong short-range repulsive interactions does notexist. The fast pair formation could not be observed previously due to limited time resolution [3]. Instead, a muchslower second phase in the conversion of atoms to pairs
was observed leading to the wrong conclusion that theunpaired atoms have a much longer lifetime.

2

Energy / atom (E F)

Many-body systems can often be modeled using contact
interactions, greatly simplifying the analysis while maintaining the essence of the phenomenon to be studied. Suchmodels are almost exactly realized with ultracold gases dueto the large ratio of the de Broglie wavelength to the rangeof the interatomic forces [1]. For itinerant fermions withstrong short-range repulsion, textbook calculations predicta ferromagnetic phase transition—the so-called Stoner instability [2].
Here we investigate this system using an ultracold gas offermionic lithium atoms, and observe that the ferromagnetic phase transition does not occur. A previous experimental study [3] employing a different apparatus foundindirect evidence for a ferromagnetic phase, but did notobserve the expected domain structure, possibly due to thelack of imaging resolution. Here we address this shortcoming by analyzing density and spin density fluctuations
via speckle imaging [4]. When spin domains of m atoms
form, the spin density variance will increase by a factor ofm [5], even if individual domains are not resolved. One
main result of this paper is the absence of such a significant
increase which seems to exclude the possibility of a ferromagnetic state in the studied system.
The Stoner model assumes a two-component Fermi gas
with a repulsive short-range interaction described by asingle parameter, the scattering length. The predicted
phase transition to a ferromagnetic state requires large
repulsive scattering lengths on the order of the interatomic
spacing. They can be realized only by short-range attractive potentials with a loosely bound state with binding
energy @2 =ðma2 Þ, with m being the atomic mass and abeing the scattering length [6]. However, as shown schematically in Fig. 1, the repulsive gas is then by necessity
only metastable with respect to decay into the bound state.
Many theoretical studies of a Fermi gas with strong shortrange repulsive interactions assume that the metastablestate is sufficiently long-lived [7–18]. In recent Monte
Carlo simulations, the paired state is projected out in the

B
1

A

0
magnetic
field-1

speckle trap
image
off
B
A

350 µs

-2

t

10

5

0

-5

-10

Interaction parameter (1/k Fa)

FIG. 1. Diagram showing energy levels and timing of theexperiment. The upper (repulsive) and lower (attractive) branchenergies, near a Feshbach resonance, are connected by threebody collisions. In our experiment, we quickly jump from aweakly interacting Fermi gas (A) to a strongly interacting one(B) with a rapid magnetic field change. The evolution of correlations and domains and the molecule formation (population of
the lower branch) are studied as a function of hold time t.
Adapted from [42].

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Ó 2012 American Physical Society

week ending
15 JUNE 2012PHYSICAL REVIEW LETTERS

The experiments were carried out with typically
4:2  105 6 Li atoms in each of the two lower spin states
j1i and j2i confined in an optical dipole trap with radial andaxial trap frequencies !r ¼ 2  100ð1Þ s1 and !z ¼
2  9:06ð25Þ s1 . The sample was evaporatively cooled
at a magnetic bias field B ¼ 320 G, identical to the procedure described in [22]. Then the magnetic field was slowly
ramped to 730 G (kF a ¼ 0:35) in 500 ms. The fraction of
atoms being converted to molecules during the ramp was
measured (see below for method) to be below 5%. The
temperature of the cloud was typically 0:23ð3ÞTF at 527 Gwith a Fermi energy of EF ¼ kB TF ¼ h  6:1 kHz. After
rapidly switching the magnetic field from 730 G to the final
value in less than 350 s, spin fluctuations were measured
by speckle imaging. Optionally an appropriate rf pulse wasapplied directly before imaging to rotate the spin orientation along the measurement axis. Due to the use of 20 cm
diameter coils outside the vacuum chamber, the inductanceof the magnet coils was 330 H and the fast switching
was accomplished by rapidly discharging capacitors
charged to 500 V.
Experimentally, spin fluctuations are measured using thetechnique of speckle imaging described in Ref. [4]. For an
appropriate choice of detuning, an incident laser beam
experiences a shift of the refractive index proportional tothe difference between the local populations of the two
spin states N1 and N2 . Spin fluctuations create spatial
fluctuations in the local refractive index and imprint aphase pattern into the incoming light, which is then converted into an amplitude pattern during propagation. The
resulting spatial fluctuations in the probe laser intensity areused to determine the spin fluctuations in the sample.
In Ref. [4] we prepared samples on the lower branch of
the Feshbach resonance, where positive values of kF a
correspond to a gas of weakly bound molecules. At
kF a ¼ 1:2, we observed a sixfold suppression of spin
fluctuations and a fourfold enhancement of density fluctuations. Typical fluctuations in the speckle images of a noninteracting Fermi gas at T ¼ 0:23TF amount to 5% of theaverage optical signal per pixel, corresponding to about
50% of Poissonian fluctuations. Those fluctuations are
modified by factors between 0.2 and 1.6 due to pairingand interactions.
In this study, on the upper branch of the Feshbach resonance, the situation is reversed. For unbound atoms, as theinteraction strength increases, the two spin components
should develop stronger and stronger anticorrelations andenhanced spin fluctuations. Previous experimental work [3]and several theoretical studies [10,11,13–15,18,23] predicteda phase transition to a ferromagnetic state where the magnetic
susceptibility and therefore the spin fluctuations diverge.
Recent Monte Carlo simulations [19] predict such a divergence around kF a ¼ 0:83. We therefore expected an increase of spin fluctuations by one or several orders ofmagnitude, related to the size of magnetic domains.

Figure 2 shows the observed spin fluctuations enhancement compared to the non-interacting cloud at 527 G. The
variance enhancement factor reaches its maximum value of
1.6 immediately after the quench, decreasing during the
2 ms afterward. The absence of a dramatic increase showsthat no domains form and that the sample remains in theparamagnetic phase throughout. Similar observations weremade for a wide range of interaction strengths and wait
times. Note that first-order perturbation theory [24] predicts an increase of the susceptibility by a factor of 1.5 at
kF a ¼ 0:5 and by a factor of 2 at kFa ¼ 0:8 (i.e., no
dramatic increase for kF a < 1). Therefore, our data show
no evidence for the Fermi gas approaching the Stonerinstability.
Before we can fully interpret these findings, we have totake into account the decay of the atomic sample on theupper branch of the Feshbach resonance into bound pairs.
We characterize the pair formation by comparing the total
number of atoms and molecules Na þ 2Nmol (determinedby taking an absorption image after ballistic expansion athigh magnetic field where molecules and atoms have thesame absorption resonance) to the number of free atoms(determined by rapidly sweeping the magnetic field to 5 G
before releasing the atoms and imaging the cloud, converting pairs into deeply bound molecules that are completely
shifted out of resonance) [25].
The time evolution of the molecule production (Fig. 3)
shows two regimes of distinct behavior. For times less than
1 ms, we observe a considerable number of atoms converted into molecules, while the total number Na þ 2Nmol
remains constant. The initial drop in atom number becomeslarger as we increase the final magnetic field, and saturates
at around 50% near the Feshbach resonance.
We attribute this fast initial decay in atom number torecombination [26,27] into the weakly bound moleculark Fa 0Relative spin fluctuations

PRL 108, 240404 (2012)

1.6

0.65 1.14 2.27 ∞

(a)

1.6

1.4

1.4

1.2

1.2

1

1
527 750800

Magnetic field (G)

850

(b)

0

1

2

3

Hold time (ms)FIG. 2. Spin fluctuations (a) after 350 s as a function ofmagnetic field and (b) on resonance as a function of hold timescaled to the value measured at 527 G. Even at strong repulsive
interactions, the measured spin fluctuations are barely enhanced,
indicating only short-range correlations and no domain formation. The spin fluctuations were determined for square bins of
2:6 m, each containing on average 1000 atoms per spin state.

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PHYSICAL REVIEW LETTERS

PRL 108, 240404 (2012)
Hold time [ms]
20

40

60

80

100

120

140

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

Molecule Fraction

Surviving fraction

1

0Surviving fraction

1
0.8
0.6
0.4
0

1

2

3

4

5

6

Hold time [ms]FIG. 3 (color online). Characterization of molecule formation
at short and long hold times, and at different values of theinteraction strength. The closed symbols, circles (black) at
790 G with kF a ¼ 1:14, squares (blue) at 810 G with kF a ¼
2:27 and diamonds (red) at 818 G with kF a ¼ 3:5 represent thenormalized number of free atoms, the open symbols the total
number of atoms including those bound in Feshbach molecules(open circles at 790 G with kF a ¼ 1:14). The crosses (green)show the molecule fraction. The characteristic time scale is set
by the Fermi time @=EF ¼ 43 s, calculated with a cloud
averaged Fermi energy.

state. We obtain an atom loss rate N_ a =Na ¼ 250 s1
at 790 G in the first 1 ms after the magnetic field switch.
Assuming a three-body process we estimate the rate
coefficient L3 at this field to be 3:9  1022 cm6 s1 ,though the interaction is already sufficiently strong for
many-body effects to be significant. For stronger interactions, about 30% of atom loss occurs already during therelevant 100 s of ramping through the strongly interacting region, indicating a lower bound of around 3  103 s1
for the loss rate which is 13% of the inverse Fermi time
EF =@, calculated with a cloud averaged Fermi energy.
After the first millisecond, the molecule formation rate
slows down, by an order of magnitude at a magnetic
field of 790 G (and even more dramatically at higher
fields) when it reaches about 50%. It seems likely thatthe molecule fraction has reached a quasi-equilibrium
value at the local temperature, which is larger than theinitial temperature due to local heating accompanying
the molecule formation. Reference [28] presents a simple
model for the equilibrium between atoms and molecules(ignoring strong interactions). For phase space densities
around unity and close to resonance, the predicted
molecule fraction is 0.5, in good agreement with ourobservations [29].

week ending
15 JUNE 2012For longer time scales (hundred milliseconds) we observe a steady increase of the molecule fraction to 90% for
the longest hold time. This occurs due to continuous
evaporation which cools down the system and shifts theatom-molecule equilibrium towards high molecule fractions. During the same time scale, a slow loss in both
atom number and total number is observed caused byinelastic collisions (vibrational relaxation of molecules)
and evaporation loss.
Is the rapid conversion into molecules necessarily fasterthan the evolution of ferromagnetic domains? Our answer
is tentatively yes. First, for strong interactions with kF a
around 1, one expects both instabilities (pair formation andStoner instability) to have rates which scale with the Fermi
energy EF and therefore with n2=3 . Therefore, one cannotchange the competition between the instabilities by working at higher or lower densities. According to Ref. [21] thefastest unstable modes for domain formation have a wave
vector q  kF =2 and grow at a rate of up to EF =4@ when
the cloud is quenched sufficiently far beyond the criticalinteraction strength. Unstable modes with such wave vectors will develop ‘‘domains’’ of half a wavelength or size
 ¼ =q ¼ 2=kF containing 5 atoms per spin state in a
volume 3 . This rate is comparable to the observed conversion rates into pairs of 0:13EF . Therefore, at best,‘‘domains’’ of a few particles could form, but before they
can grow further and prevent the formation of pairs (in afully polarized state), rapid pair formation takes over andpopulates the lower branch of the Feshbach resonance.
Based on our observations and these arguments, it seems
that it is not possible to realize ferromagnetism with strong
short range interaction, and therefore the basic Stonermodel cannot be realized in nature.
One possibility to suppress pair formation is provided bynarrow Feshbach resonances. Here the pairs have dominantly closed channel character and therefore a muchsmaller overlap matrix element with the free atoms.
However, narrow Feshbach resonances are characterized
by a long effective range and do not realize the Stonermodel which assumes short-range interactions. Other interesting topics for future research on ferromagnetism andpair formation include the effects of dimensionality[30,31], spin imbalance [32,33], mass imbalance [34],lattice and band structure [35,36].
We now discuss whether ferromagnetism is possible
after atoms and molecules have rapidly established local
equilibrium. In other words, starting at T ¼ 0, one could
heat up the fully paired and superfluid system and create agas of atomic quasiparticles which are similar to free atoms
with repulsive interactions. Density and temperature of theatoms are now coupled. It is likely that such a state isrealized in our experiments after a few ms following thequench, until evaporative cooling converts the system intoa molecular condensate over  100 ms. The possibility
that such a quasiparticle gas could become ferromagnetic

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has not been discussed in the literature. Our experiments do
not reveal any major increase in spin fluctuations which
seems to exclude a ferromagnetic state. In the simplest
picture, we could regard the atomic quasiparticles as free
atoms, and then apply the Stoner model to them.
Ferromagnetic domain formation is analogous to phaseseparation between the two spin components [3]. Since
dimers interact equally with the two spin components, one
might expect that even a noticeable dimer fraction should
not suppress the tendency of the atomic gas to form domains. Therefore, in a simple model, one may neglect
dimer-atom interactions.
If the Stoner model applies to this quasiparticle gas, thenext question is whether the temperature is low enoughfor the ferromagnetic phase transition. Available theoretical treatments do not predict an exact maximum transition
temperature to the ferromagnetic state and obtain an
unphysical divergence for large scattering lengths. Since
the only energy scale is the Fermi temperature, one would
expect a transition temperature which is a fraction of
the Fermi temperature [37], higher or around thetemperature scale probed in our experiments. However,even above the transition temperature, the susceptibility
is enhanced. A simple Weiss mean field or Stonermodel leads to the generic form of the susceptibilityðTÞ ¼ 0 ðTÞ=ð1  w0 ðTÞÞ, where 0 ðTÞ is the Paulisusceptibility of the non-interacting gas and w the interaction parameter. This formula predicts a twofold increasein the susceptibility even 50% above the transition temperature, which is well within the sensitivity of ourmeasurements.
Therefore, our experiment can rule out ferromagnetism
for temperatures even slightly lower than the experimental
temperatures. Temperatures are very difficult to measure
in a transient way for a dynamic system which may not be
in full equilibrium. For example, cloud thermometry
requires full equilibration and lifetimes much longer thanthe longest trap period. We attempted to measure thetemperature after the hold time near the Feshbach resonance by quickly switching the magnetic field to weakinteractions at 527 G and then performing noise thermometry using speckle imaging [4]. We measure columnintegrated fluctuations that are 0.61(8) of the Poisson value
which implies an effective temperature well below TF ,around 0.33(7) TF , not much higher than our initial
temperature of 0.23 TF . Although the cloud is not in fullequilibrium, an effective local temperature can still beobtained from noise thermometry.
Alternatively, we can estimate the temperature increase
from the heat released by pair formation. A simple model[38] accounting for the relevant energy contributions
predicts for kF a ¼ 1 that molecule fractions of higher
than 20% result in a final temperature above 0:4TF , anestimate which is higher than the measurement reported
above. One may hope that closer to resonance many-bodyweek ending
15 JUNE 2012

effects lower the released energy; however, as we show
in the Supplemental Material (Fig. 1 of [38]) this is
not necessarily the case due to the repulsive interactionenergy.
Our experiment has not shown any evidence for a possible ferromagnetic phase in an atomic gas in ‘‘chemical’’equilibrium with dimers. This implies one of the following
possibilities. (i) This gas can be described by a simple
Hamiltoninan with strong short range repulsion.
However, this Hamiltonian does not lead to ferromagnetism. This would be in conflict with the results of recent
quantum Monte Carlo simulations [19,20] and second
order perturbation theory [11], and in agreement withconclusions based on Tan relations [39]. (ii) The temperature of the gas was too high to observe ferromagnetism.
This would then imply a critical temperature around or
below 0:2T=TF , lower than generally assumed. (iii) The
quasiparticles cannot be described by the simple model ofan atomic gas with short-range repulsive interactions dueto their interactions with the paired fraction.
A previous experiment [3] reported evidence for ferromagnetism by presenting non-monotonic behavior of atom
loss rate, kinetic energy and cloud size when approaching
the Feshbach resonance, in agreement with predictions
based on the Stoner model. Our measurements confirm
that the properties of the gas strongly change near
kF a ¼ 1. Similar to [3], we observe features in kinetic
and release energy measurements near the resonance (seeSupplemental Material [38]). However, the behavior ismore complex than that captured by simple models. The
atomic fraction decays non-exponentially (see Fig. 3), andtherefore an extracted decay time will depend on the detailsof the measurement such as time resolution. Reference [3]found a maximum of the loss rate of 200 s1 for a Fermi
energy of 28 kHz. Our lower bound of the decay rate of3  103 s1 is 15 times faster at a five times smaller Fermi
energy. Our more detailed study rules out that Ref. [3] has
observed ferromagnetic behavior.
Our conclusion is that an ultracold gas with strong shortrange repulsive interactions near a Feshbach resonanceremains in the paramagnetic phase. The fast formation ofmolecules and the accompanying heating makes it impossible to study such a gas in equilibrium, confirming predictions of a rapid conversion of the atomic gas to pairs[21,40]. The Stoner criterion for ferromagnetism obtains
when the effective interaction strength times the density ofstates is larger than one. This is a at least an approximately
valid criterion for multi-band lattice models [41]. We haveshown here that this criterion cannot be applied to Fermi
gases with short-range repulsive interactions (the basic
Stoner model) since the neglected competition with pairing
is crucial.
This work was supported by NSF and ONR, AFOSR
MURI, and under ARO Grant No. W911NF-07-1-0493
with funds from the DARPA Optical Lattice Emulator

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PRL 108, 240404 (2012)PHYSICAL REVIEW LETTERS

program. We are thankful to Eugene Demler, David
Pekker, Boris Svistunov, Nikolay Prokof’ev, andWilhelm Zwerger for valuable discussions and to David
Weld for critical reading of the manuscript.

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standard
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in atom number of ðm N=mÞ2 ¼ mN.
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1=ð1=a þ re k2 =2  ikÞ has a pole on the imaginary axis
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[29] Note that the drop in molecule formation after 1 ms cannot
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Correlations and Pair Formation in a Repulsively Interacting Fermi Gas:
Supplementary Information

KINETIC ENERGY AND RELEASE ENERGY

The energy of a trapped interacting gas is the sum of
three contributions: the kinetic energy, the interactionenergy, and the potential energy in the trapping potential.
E = T + Uint + Utrap(1)

By suddenly releasing the atoms from the trap andmeasuring the radius of the cloud, it is possible to measure either the release energy (T + Uint ) or the kinetic
energy T, depending on whether the interactions are left
on or are switched off, respectively, at the time of release, by leaving the external magnetic field constant or
rapidly switching it to a value away from the Feshbachresonance.
The system is prepared with variable interaction
strength by rapidly switching the magnetic field to avalue near the Feshbach resonance. For the kinetic energy measurement, the field is again rapidly switched to
5G, after which the atoms are released from the trap.
After 8 ms of free expansion, the size of the cloud reflects
the width of the in-trap momentum distribution and theaverage in-trap kinetic energy. The observed increasein kinetic energy with increasing interaction strength reflects the onset of pair (anti-)correlations of opposite-spin
atoms — those anti-correlations reduce the repulsive potential energy at the price of increased kinetic energy.
The observed increase in kinetic energy is consistent withthe observations in Ref. [3]. We do not observe a minimum in kinetic energy as in [3], since the interactions are
suddenly increased and the cloud cannot adiabatically
expand during the ramp as was the case in the earlier
work. For fully spin-polarized domains, the kinetic energy would increase by a factor of 22/3 which provides anupper bound (since the true ground state must have anenergy lower than or equal to the fully phase-separated
state). The smaller observed energy increase (factor of
1.3) implies that fully spin polarized domains have notformed. It should be noted that the kinetic energy increase is insensitive to the correlation length or size of

155

0.36

0.50

0.72

kFa

1.14

2.27

∞

Relative cloud size squared

In this supplement we present measurements of therelease energy of fermions with strong repulsive interactions, derive estimates for sample heating due to molecule
formation, adapt predictions for ferromagnetic domain
formation after a quench to our experimental parameters, and discuss our temporal and spatial resolution forspin fluctuations.

Magnetic field [G]FIG. 1: Measured transverse mean square cloud size after time-of-flight expansion as a function of the interaction
strength before the release in units of the value of the noninteracting cloud. The interactions are switched off (squares)or left on (circles) at the time of release. For sufficiently longtime-of-flight the transverse 2D release energy is directly proportional to the measured mean square width of the cloud.
The expansion is either isotropic (in the case of switched off
interactions) or mostly transverse (in case of strong interactions leading to hydrodynamic expansion) [1, 2]. Uncertainties as indicated by the error bars are purely statistical.

domains and cannot clearly distinguish between ferromagnetic domains and strong anti-correlations [4, 5].
In-trap kinetic and interaction energies are measured
when the magnetic field is left at its value near the Feshbach resonance while the trap is switched off. After 4
ms of the 8 ms free expansion the field is switched to 5G
for imaging. The resulting cloud size is directly related
to the release energy of the cloud, the sum of the in-trap
kinetic and interaction energies. Around kF a = 0.5, we
observe a strong increase of the transverse release energy by a factor of about 2. This implies that the totalenergy has increased by a factor of 4/3 compared to the
non-interacting case since the interactions modify the expansion from ballistic to hydrodynamic. The comparisonof kinetic energy and release energy at 790G and 810G
shows that the extra energy due to repulsive interactions
is clearly dominated by repulsive potential energy andnot by kinetic energy. The latter would occur for ferromagnetic domains. In case of fully spin-polarized domains, the repulsive interaction energy would vanish, and2kinetic energy and release energy should be the same.
Therefore, the energy measurements also rule out a ferromagnetic phase.
The interaction-strength dependence of the release energy shows a weak maximum at 790G. The ground stateenergy has to vary monotonically with the strength of
the repulsive interactions. This follows from a simple
variational argument where the exact ground state wavefunction at strong interactions is used as a trial wavefunction at weaker interactions. Therefore, the observed
maximum is most likely related to non-equilibrium excitations caused by the sudden jump in the scattering
length.

HEATING DUE TO MOLECULE FORMATION

Molecule formation heats the sample by transferring
the binding energy and excess kinetic energy of themolecules to the remaining atoms, and also by creating
holes in the Fermi sea. For small values of kF a, the totalenergy release per molecule is h̄2 /ma2 + 2EF . From theenergy per particle for an ideal homogeneous degenerate
Fermi gas E = 0.6EF [1 + (5π 2 /12)(T /TF )2 ] we obtain

T
=
TF

s

4η
π2



1+

1
(kF a)2(2)

for the final temperature with a small molecule fraction η (assuming initially η=0 and T=0). Evaluating thisresult at kF a = 1, where the two-body binding energy is
2EF , one finds that molecule fractions of higher than
20 % result in a final temperature above 0.4TF , an estimate which is somewhat higher than the measurement
reported in the main article.

DOMAIN GROWTH PREDICTIONS AND
EXPERIMENTAL RESOLUTIONAs described in the main paper, we have observed two
dynamic time scales - a sub-ms timescale for rapid pairformation, followed by a slower time scale of tens of milliseconds where further cooling led to a full conversion of
the atomic into a molecular gas. We now want to discussover what time scale we could have observed ferromagnetic domain formation if it had happened, and someexperimental aspects regarding temporal and spatial resolution.
Pekker et al. [6] predicted in uniform systems fast
growth rates for small domains. Unstable modes withwavevector q ≈ kF /2 grow at a rate of up to EF /4h̄ when
the cloud is quenched sufficiently far beyond the critical interaction strength. This corresponds to a growthtime of around 100µs. For a wide range of interactions

156

and wavevectors, the predicted growth time is faster than
10h̄/EF or 250µs. During this time one would expect thethermal fluctuations to increase by a factor of e.
Wavevectors q ≈ kF /2 will develop “domains” of halfa wavelength or size ξ = π/q = 2π/kF , which is 2.3 µm
at the experimental density of 3 × 1011 cm−3 corresponding to kF = 2.7 × 104 cm−1 . For speckle imaging thesmallest effective probe volume is given by the nominal optical resolution or the optical wavelength times theaperture of the lens system which is equal to the bin sizeof 2.6 µm in our experiment. For a bin size d, the measured fluctuations are an integral over fluctuations at all
wavevectors q with an effective cutoff around 1/d. Due toa mode density factor q 2 the largest contribution comesfrom wavevectors around π/d which is equal to 0.4 kF
for our experimental parameters, fortuitously close to the
wavevector of the fastest growing unstable modes. Therefore, for a quench across a ferromagnetic phase transition,
we would have expected a sub-millisecond growth time
for the spin fluctuations, which was not observed.
Another analysis focuses on domains. As discussed in
the main paper, random domains containing m atoms
will increase the variance of spin fluctuations by a factor of m. The fastest growing unstable modes containaround 5 atoms per spin state in a volume ξ 3 . At our temperatures, the thermal fluctuations correspond to Poisson fluctuations reduced by the Pauli suppression factor (3/2) T /TF , i.e. in a small volume with about ten
particles, the spin fluctuations correspond to plus/minus
one particle, and will approach saturation (i.e. fully spinpolarized domains) after a few growth times (when, ofcourse, the unstable mode analysis is no longer applicable). If these domains randomize through spin diffusion,
it will increase the variance of the spin fluctuations at alllength scales larger than ξ by a factor of m, without requiring optical resolution of the domain size ξ. Spin diffusion effectively “converts” initial fluctuations at wavevector q to fluctuations at all wavevectors smaller than q.
Since we start with a balanced two-state mixture withzero spin density, formation of random (i.e. uncorrelated)domains requires spin diffusion. Full sensitivity to domains requires spin transport over a distance equal tothe bin size. Assuming diffusive motion at resonant interactions with a spin diffusivity Ds ≈ h/m as observedin [7] results in a corresponding minimum wait time ofabout 300µs. This means that after crossing a ferromagnetic phase transition, formation of domains of a few totens of particles should have occurred on ms scales andshould have resulted in an observable increase of spin
fluctuations. Therefore, we conclude that we have notentered any ferromagnetic phase within our experimental parameters.

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Appendix D
Kapitza-Dirac Diffraction
Simulation

159

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161

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Appendix E
Superlattice Band Structure
Calculation

163

164

165

166

Appendix F
Simulation of Double-Slit
Interference Pattern in a
Superlattice

167

168

169

170

171

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