Spin-orbit Coupled Bose Gases
byBoris Shteynas
B.Sc., Moscow Institute of Physics and Technology (2010)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Physics
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2019
c Massachusetts Institute of Technology 2019. All rights reserved.
○

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
August 30, 2019

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

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

2

Spin-orbit Coupled Bose Gases
byBoris Shteynas
Submitted to the Department of Physics
on August 30, 2019, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy in Physics

Abstract
Quantum simulation is a very active and growing field. Various quantum systems
can be used to emulate existing materials in an accurate and controllable way, as
well as to generate new states of matter that have not been found in the real world
but the existence of which does not contradict the fundamental laws of physics. Ultracold atoms form a perfect system to realize idealized models and study physicalmechanisms that stand out clearly in them.
Recent efforts have been made to simulate artificial gauge fields with ultracold
atoms, including spin-dependent gauge fields, such as spin-orbit coupling. Motivatedby this goal our lab explored several approaches to generate a one-dimensional spinorbit coupling interaction, which has a rich phase diagram and plays an importantrole for topological insulators, the quantum spin Hall effect and spintronics. The first
method we developed allowed us to detect a stripe phase by dressing Bose-Einsteincondensates with an optical superlattice and Raman beams. The observed density
modulation in the ground state meets the definition of the long-awaited supersolidstate of matter.
The second approach we took was to generate spin-orbit coupling without use oflasers. The method is based on the idea of periodic driving of the quantum systemand dressing its evolution with fast micromotion, often refered to as Floquet engineering. Our experiment provided an insightful pedagogical example of what Floquet
engineering is capable of. In the experiment we endowed a low energy radio-frequency
photon with tunable momentum. When dressed with recoil momentum, the interaction of a radio-frequency photon with an atom occurred in a Doppler-sensitive way.
We also demonstrated how to tune the momentum and flip its direction.
In this thesis, I first describe the experiments done in the optical superlattice.
Then I discuss the behavior of periodically driven classical and quantum systems andprovide detailed analysis of how a radio-frequency photon can be magnetically dressedwith tunable momentum.
The experiments we carried out demonstrated novel methods of generation for
spin-dependent gauge fields and showed pedagogical examples and interpretations ofevolution of periodically driven systems. The scheme of periodically driven atoms
3inspired a theoretical study of heating in Floquet systems.
Thesis Supervisor: Wolfgang Ketterle
Title: John D. MacArthur Professor of Physics

4Dedicated to my parents
and to the memory of
my grandparents.

Acknowledgments
It is hard to find the right words to fully express how grateful I am to the people
who were around me during my PhD and how important the experience of graduate
school is for me. It has been a time of enormous challenges and hard work, but at thesame time it has been an incredibly rewarding period of life. MIT not only pushed
me to my intellectual and physical limits, but also formed my personality. After all,graduate school at MIT made me realize who I am and to what circle of people I
belong.
Absolutely the highest honor of my life was to learn from Professor Wolfgang
Ketterle. The years spent in his group have definitely been, and most likely willremain, the highlight of my life. Observing his striking analytical mind, the way heviews physics, his incredible intuition for sorting out experimental questions – all ofthat is the school of physics of which one can only dream. I’m also very grateful to
Wolfgang for being very kind to me (perhaps, more than I deserved). Without his
timely encouragements I would not be able to make it to the end of graduate school.
During all 7 years at MIT I was happy to be surrounded by brilliant wonderfulpeople. I will not be able to mention all of them here, but I would like to say thatthe intellectual environment at MIT itself is a unique medium which stimulates andinspires personal growth. It has been a great honor to be a part of the MIT Physics
Department and the Center for Ultracold Atoms.
I’d like to thank the other professors at the CUA. Professor Vladan Vuletic was
my academic advisor. I always knew I could rely on his advice in physics and on
the choice of academic path. Professor Martin Zwierlein’s genuine excitement about
physics inspires not only his students, but also those who happen to come acrosshim in the corridor or those like me, who have their office just next door. I will
also remember insightful discussions with Professor Ike Chuang, whose expertise invarious fields of physics and technology is absolutely overwhelming.
I’d like to thank former members of BEC2 lab Ed Su and Wujie Huang, whobuilt the machine I eventually used. Wujie, your guidance helped me make the first6

semi-independent steps in the lab. I also enjoy remembering our road trips including
the last one when Junru and I gave you a ride from Boston towards your new placeand new life.
I’d like to thank Junru Li who started MIT the same year with me. Junru has
always been an example to me of what a successful graduate student should be like.
I always admired his dedication to lab work and his talent for physics. We share a
lot of memories inside and outside of the lab.
I was happy to work with two great postdocs Alan Jamison and Jeongwon (Chris)Lee. Chris and I worked particularly closely on the experiment on which this thesis
focuses. Our styles in the lab were quite different, but we both were determined tomake this project work, and, I think, we not only both enjoyed working together but
also benefited from this collaboration. I learnt a lot from Chris, and there were several
moments when I had no strength to push the experiment, but Chris was ready tocontinue the effort. So, I can unhesitatingly say that without Chris this work would
not be completed. Alan has been in the group during most of my PhD, and I asked
him so many physics questions I was confused about. Every time the answer was
prompt, accurate and comprehensive. It is really important that such people as Alan
keep going the “thorny” path of academia, and I was genuinely happy to hear the news
that Alan got appointed to a well-deserved professorship and very soon will start his
own cold atoms lab. Good luck at the new place, Alan, and thank you for your help
during those many years!
I would like to acknowledge all the members of the Ketterle group, past andpresent. Sometimes the boundaries between our labs vanish and the ideas (along withequipment) flow from one team to another. For example, every electronics project inour group starts with a “pilgrimage” to Will Lunden. I am very grateful to him for all
the very valuable suggestions and ideas which helped to implement our experiment.
I have not worked together with Timur Rvachov from the BEC3 lab, but fromthe very first days in the group he helped me a lot. Even now, many years later,
I remember his transparent explanations of fundamental concepts, not to mention
the many technical skills in optics and other areas I picked up directly from him.
7

Timur was always ready to help even at 5 am when he was busy with running his
own experiment. Our conversations on life issues (happening around 5 am, as well)
are also something I will always remember. Thank you, Timur!
I would like to thank Jesse Amato-Grill from the BEC5 lab. Even before webecame groupmates I remember seeing him for the first time in the atomic physics
class. I thought this guy must be from Harvard: well-dressed and with very goodmanners. . . I was happy to get to know Jesse closer after I joined the group. It
turned out that from the inside he was no less refined a person. After almost sevenyears spent side by side I can conclude that Jesse is one of the smartest and mostknowledgeable people I’ve known. I’ve been happy to have him as a colleague and afriend.
Other members of the BEC5 lab, Niklas Jepsen and Ivana Dimitrova, have also
been around throughout my entire PhD. Niki, your presence in the group made a lotof difference to me. So many times you saved my day from frustration. Just chattingwith you about the lab problems or life helped me to pull through the difficulties of
the PhD. I am also thankful to Ivana. I’ve always been impressed by her sharp mind
and critical thinking.
At the end of my PhD I was happy to overlap with Yair Margalit. He made avery good impression on me when he visited MIT for the interview in our group.
It was immediately clear that Yair is very a smart physicist and a very easy-going
friendly person. I was very glad when he joined our lab and, in my opinion, he is an
excellent team member. I really enjoyed working together with him and it is a pity
this collaboration was not very long.
Sean Burchesky spent his undergraduate years in our lab and I’ve been absolutely
amazed by his skills and talent for experimental physics. Sean revealed all the goodstereotypes about MIT students. He enjoys making things work, and things actuallywork. Thank you, Sean, for being part of our team! Furkan Top is next in line forgraduation in BEC2 lab. I have always been impressed by his diligence and accuracy
in solving problems. I have no doubts Furkan will have new results from BEC2 soon
and will bring our machine to the new stage of experiments with fermions.
8

The most closely connected lab to ours in all aspects is BEC3. Not only do theywork with the same atoms but also we are located right next door from each other. We
borrowed your equipment without asking, and you borrowed ours – and nevertheless
we are still friends. Hyungmok, I have always been impressed by how hard-working
you are. Julie, it’s been such a pleasure to have you as a groupmate. We have had
quite a few conversations on different aspects of physics and those discussions mademe a better physicist. I know how smart and strong you are. With your dedication towork, Lithium and Sodium will have no choice but to surrender and be at your service.
I wish you good luck and look forward to hearing from you about the advances incold molecules.
Pierre Barral has been a relatively recent addition to the group. He is definitely
not hesitant to ask questions, and this is great. Thanks to his curiosity in physics, I
have learned so much myself. I would have learned even more if Pierre spoke a little
slower though. . . Keep having a lot of fun in physics, Pierre! Good luck to Li andMichael with Dysprosium and Pierre. Your lab is really cool, even after it has been
on fire. Li, we had a great time at the last DAMOP conference. I was happy to get
to know you better. Now I agree with you: academia is great (or at least nothing isbetter than it). So, “Vivat academia!”
I’d also like to thank BEC4 team: Woo Chang, Julius and Jinggang. You, guys,
put our group on the new level of understanding optics. I still remember Jinggang’s
talk on microscope design at our group seminar: that was so eye-opening. Good luck
to you, guys!
I would like to thank my physicist friends, and first of all Janos Perczel. We met
at the first gathering of the incoming grad students at MIT, and I am still convincedthat it was one of the most valuable connections I made at MIT. We went throughclasses and exams together; we accompanied each other in our struggles with research,though Janos struggled a little less since he is a theorist. Enough for the foundationof a life-long friendship!
Grisha Tarnopolsky has been my friend forever (since undergrad). Our life paths
keep intertwining, and I can’t be happier to have you as a friend and colleague. Max
9Metlitsky is a brilliant person and an incredible friend. It is a joy to know that such
Renaissance people still exist, and I am particularly happy to meet one of the veryfew (if not unique, who knows. . . )I am also grateful to many people here at the CUA working in other groups. Parth,Matt, Ian, Sergio, Biswaroop and many others. I will have many warm recollections
about our hallway and those who inhabit it. Parth Patel has been a great friend anda roommate during the last couple of years at MIT. I always knew I could rely onyou, Parth. I am sure, wherever life disperses us we will always be in touch.
I would also like to thank Cathy Modica, Sydney Miller, Joanna Welch and Paula
Sack for arranging the research and education to be smooth and thus letting me focus
on physics. At the time I started the graduate program at MIT I was practically a
23 year-old child. Cathy took good care of me and kept an eye on my growth fromthe very first days and up to graduation. Thank you, Cathy!
At the end, I would like to thank my parents, for whom this degree probably
means more than for anyone else in the world including myself. I always had yoursupport. Thank you for everything you did for me.

10

Contents
1 Introduction

15

1.1

Quantum simulations . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.2

Ultracold atoms as a quantum simulator . . . . . . . . . . . . . . . .

17

1.3

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

18

1.4

23

Na Bose-Einstein Condensate Production . . . . . . . . . . . . . . .

19

1.5

Overview of the Experimental Machine . . . . . . . . . . . . . . . . .

20

1.6

Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2 Introduction to spin-orbit coupling

23

2.1

Synthetic spin-orbit coupling in cold atoms . . . . . . . . . . . . . . .

24

2.2

1D spin-orbit coupling. Λ-scheme. . . . . . . . . . . . . . . . . . . . .

25

3 Spin-Orbit Coupling in a Superlattice

29

3.1

States in a superlattice . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.2

Spin-orbit-coupling Hamiltonian . . . . . . . . . . . . . . . . . . . . .

39

3.3

Antiferromagnetic spin texture . . . . . . . . . . . . . . . . . . . . . .

40

3.4

Phase of the Superlattice . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.5

Green light generation . . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.5.1

Generation of double-frequency wave . . . . . . . . . . . . . .

43

3.5.2

Plane wave passing through the crystal . . . . . . . . . . . . .

44

3.5.3

Phase-matching . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.5.4

Gaussian beam analysis . . . . . . . . . . . . . . . . . . . . .

48

3.5.5

Operation of SHG . . . . . . . . . . . . . . . . . . . . . . . . .

49

11

3.5.6

Buyer’s catalog . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.6

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3.7

Stripe phase detection . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4 Introduction to periodically driven systems
4.1

61

Classical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.1.1

Kapitsa pendulum . . . . . . . . . . . . . . . . . . . . . . . .

63

4.1.2

Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.2

Quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.3

Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.4

Extended Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5 Floquet Engineering of Recoil-dressed Photons

71

5.1

Magnetic shaking experiment . . . . . . . . . . . . . . . . . . . . . .

71

5.2

Our scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

5.2.1

Bloch sphere representaion . . . . . . . . . . . . . . . . . . . .

75

5.2.2

Floquet treatment . . . . . . . . . . . . . . . . . . . . . . . .

77

5.2.3

Evolution of the system . . . . . . . . . . . . . . . . . . . . .

80

5.2.4

Canonical vs mechanical momenta. . . . . . . . . . . . . . .

82

Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . .

83

5.3.1

High current . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.3.2

Stability of magnetic field and gradient . . . . . . . . . . . . .

89

5.3.3

Symmetry of the modulated magnetic field gradient . . . . . .

92

5.3.4

Minimize modulation of magnetic bias field . . . . . . . . . . .

94

5.3.5

Timing of the RF pulses with respect to the magnetic modulation 95

5.3.6

Induced shaking . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

5.4

95

Heating due to collisions between the spin states . . . . . . . . . . . . 100

6 Conclusions and Outlook

103

12

List of Figures
1-1 A schematic plot of the BEC2 two-species apparatus . . . . . . . . .

21

2-1 1D spin-orbit coupling in Λ-scheme . . . . . . . . . . . . . . . . . . .

26

3-1 Pseudo-spin states in an asymmetric double-well potential. . . . . . .

31

3-2 Superlattice band structure. . . . . . . . . . . . . . . . . . . . . . . .

34

3-3 Superlattice and Raman beams. . . . . . . . . . . . . . . . . . . . . .

35

3-4 Phase of the superlattice. . . . . . . . . . . . . . . . . . . . . . . . . .

42

3-5 Phase-vectors diagram for the second harmonic generation. . . . . . .

47

3-6 Efficiency of the second harmonic generation as a function of a beam’s
geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3-7 Green light power measurement. . . . . . . . . . . . . . . . . . . . . .

50

3-8 TOF images for 𝑞 = 0 and 𝑞 = 𝜋/𝑑 states in the superlattice. . . . . .

53

3-9 TOF image for equal populations of atoms prepared in the ground andfirst excited bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3-10 TOF images of atoms loaded in the ground and first excited bands
with and without Raman potential. . . . . . . . . . . . . . . . . . . .

54

3-11 Mapped out spin-orbit coupling resonances . . . . . . . . . . . . . . .

55

3-12 Calibration of the offset ∆ between the wells. . . . . . . . . . . . . .

56

3-13 A schematic plot of a stationary density modulation. . . . . . . . . .

57

3-14 Angle resolved Bragg signal. . . . . . . . . . . . . . . . . . . . . . . .

58

5-1 Illustration of the scheme for creating a tunable atomic recoil momentum. 73
5-2 Phase accumulation between the spins. . . . . . . . . . . . . . . . . .
1374

5-3 Bloch sphere representation of magnetic shaking and RF pulses. . . .

76

5-4 Energy-momentum dispersion relations for spin-orbit coupled spin 1/2
states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

5-5 Ramsey fringe overlapped with the thermal cloud distribution. . . . .

84

5-6 Experimental data for spin-flipped atoms. . . . . . . . . . . . . . . .

86

5-7 Doppler shift mapped for different velocity groups of atoms. . . . . .

88

5-8 Scheme of the coils circuit. . . . . . . . . . . . . . . . . . . . . . . . .

89

5-9 Hall probe measurement of the current. . . . . . . . . . . . . . . . . .

905-10 Schematic plot of the atomic cloud in the field of a quadrupole. . . .

92

5-11 An H-bridge with built-in asymmetry. . . . . . . . . . . . . . . . . . .

93

5-12 A position of the cloud when it is more sensitive to inhomogeneous
Zeeman shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5-13 Slicing of the momentum distribution at an angle. . . . . . . . . . . .

96

5-14 An effect of eddy currents. . . . . . . . . . . . . . . . . . . . . . . . .

97

5-15 Steering of polarized spin due to induced field.

9814. . . . . . . . . . . .

Chapter 1
Introduction
1.1

Quantum simulations

The goal of science and physics, in particular, is to understand nature, its fundamental laws, and take it under control. Physicists during the last two centuries achieved
tremendous technological progress while looking at complex systems and phenomena
and decomposing them into simple building blocks. It is striking how few exactly
solvable problems there are in physics. The job of a physicist is to take very complicated problems, decompose and simplify them, and eventually reduce them to basic
concepts and models. If we make the complicated problem look trivial, it means we
did our job well and we really understand what is happening. Even though the number of exactly solvable quantum problems is so small, and the number of methods wecan apply to them is very limited, the beauty of physics is that it can always surpriseus. My journey to atomic physics was like a stroll in a forest: a forest which basically consists of three trees: a two-level system, a hydrogen-like atom and a harmonic
oscillator. And still one can easily go astray in a forest of three trees. It is really
beautiful how simple complexity can be, and how complex is simplicity.
The study of the universe starts from an attempt to understand of a two-bodyproblem – how Earth travels around the Sun. The study of matter that surrounds
us led to the discovery of atoms, nuclei and elementary particles. Understanding theproperties of light required us to look at it at a single-photon level. The interest in15

understanding the laws of nature made us look at it on a single-particle level, andwhen we got there we started to move in the opposite direction. We now want toconstruct matter atom-by-atom, or photon-by-photon. From single-particle physics,
we move to two-body systems, three-body systems, and eventually we want to build
controllable many-body systems.
Understanding physics on the single-particle level mainly confirms the existing
theories. Building new states of matter paves the way to the area of science, where
theoretical and computational methods are limited.
The difficulty of quantum problems is that their complexity grows exponentiallywith their size. A proposed solution to such problems was envisioned by Richard
Feynman [1]. We can simulate quantum phenomena with quantum objects: atoms,
ions, artificial atoms and other quantum systems. One has to think up how to reformulate the problem in terms of the chosen quantum system and then ask nature toprovide us with the answer. In other words we want to ask nature to do the hard
part: to simulate, to perform the “calculation”, whereas on our side we still need toinitialize the problem and do the measurements. Both things can be difficult enoughby themselves.
The idea of a quantum simulator can be generally formulated as 1) mapping
the initial state of the system |𝜓(0)⟩ to the initial state of the simulator |𝜑(0)⟩;ˆ and evolving the2) initializing the state |𝜑(0)⟩; 3) engineering the Hamiltonian 𝐻^

state of the simulator with the unitary operator 𝑈 = 𝑒−𝑖𝐻𝑡/ℎ̄ ; 4) measuring the state|𝜑(𝑡)⟩ = 𝑈 |𝜑(0)⟩; 5) mapping the obtained final state of the simulator |𝜑(0)⟩ on
the state of the system. Thus, the major questions we pose before setting up theexperiment are how to map the Hamiltonian and the state of the system onto the
Hamiltonian and state of the simulator.
In the present work I would like to show two examples how this mapping can bedone in the case of the spin-orbit coupling Hamiltonian, a fundamental interactionthat combines both internal and motional degrees of freedom of a particle. Two
different approaches to simulating spin-orbit coupling were developed in our lab, bothhaving their own advantages and shortcomings. The scheme I want to focus on in16

this thesis is based on utilizing a method of engineering Hamiltonians in periodicallydriven quantum systems, known as Floquet engineering. Our experiment provides alucid example of how powerful Floquet engineering is and what it is capable of.

1.2

Ultracold atoms as a quantum simulator

The field of quantum simulations has been growing rapidly over the past decade. The
demand in analysis of complicated many-body problems in condensed matter physics
and chemistry stimulates advances in the development of coherent control of various platforms for quantum simulations. Different systems, such as ultracold atoms,
trapped ions, quantum dots, superconducting circuits, and nuclear spins provide tools
for the simulation of a broad range of problems. Each of these quantum simulators
has its own strengths and can tackle special problems, which are particularly suitable
for them.
Ultracold atoms have already proved to be a great platform for quantum simulations. Neutral atoms in optical lattices mimic the behavior of electrons in crystals.
However, unlike real crystal structures, optical lattices are easily tunable and different
geometries can be explored (such as triangular [2] and honeycomb [3] lattices). By
controlling the depth of the optical lattice potential it has become possible to observe
the Mott insulator to superfluid quantum phase transition [4]. In general, opticallattices are natural systems to explore both Bose- and Fermi-Hubbard models [5].
In our experiments atoms are not only cold, but also very dilute. Interactions between the atoms are determined by a single parameter — the scattering length, which
in turn can be controlled by the magnetic field through Feshbach resonances. By tuning the interaction between two fermionic spin states, one can make a smooth transition from BEC molecules to BCS pairs and investigate the formation of fermionicsuperfluid [6].
Recent addition to the advantagous qualities of ultracold atoms, such controllability and accuracy, was the development of single-site resolution microscopes [7, 8]. It
is now possible to study the dynamics of atoms in an optical lattice and perform new17

measurements, which are not accessible in other fields (for example, measurement ofentanglement entropy [9]).
An important direction of research in the field of ultracold atoms is the simulation of artificial gauge fields. Neutral atoms emulate the motion of an electron in amagnetic field, when its wave function acquires a phase while moving in space. Thisapproach allowed to simulate incredibly high magnetic fields (equivalent to 10000 T
in real life) [10, 11]. Our recent work in the BEC2 lab belongs to the same themeof developing new methods for engineering exotic gauge fields. We have realized aspin-dependent gauge field by two very different methods. One of them led to theobservation of the new state of matter - a supersolid, while the other method createda new physical concept — a recoil-dressed photon. Both projects explored fundamental aspects of quantum mechanics and resulted in the creation of new man-madesystems, which are not forbidden by the laws of nature, but have not been realized
in the real world.

1.3

Thesis Outline

This thesis will present the results of the experimental realization of a spin-orbit
coupling Hamiltonian in an optical superlattice and with periodically driven atoms.
The superlattice model allowed us to explore the spin-orbit coupled Bose-Einstein
Condensate (BEC) in new regimes, and led to the subsequent detection of density
modulation in the ground state, the so-called stripe phase, which serves as an example
of supersolid state matter. The approach of engineering spin-orbit coupling as aneffect of periodic drive realized an alternative laser-free scheme. It allowed us todemonstrate an exotic phenomenon of recoil-dressed radio-frequency (RF) photon.
The thesis is organized as follows:∙ Chapter 1 overviews BEC2 lab experimental apparatus.
∙ Chapter 2 explains what spin-orbit coupling interaction is. This chapter discusses spin-orbit coupling in the context of solid state physics and earlier real18

izations with ultracold atoms.
∙ Chapter 3 discusses physics in an optical superlattice. It starts with the desctiption of the states in the superlattice, followed by the physical implementationof the optical superlattice and experimental data on the realization of the spinorbit coupling Hamiltonian and stripe phase detection.
∙ Chapter 4 reviews the principles of the description of periodically driven classical and quantum systems.
∙ Chapter 5 presents the idea of Floquet-engineered recoil-dressed radio-frequency
photon, its connection to spin-orbit coupling and its experimental realization
with oscillating magnetic force and the synchronized sequence of RF pulses.
∙ Chapter 6 provides the conclusion and an outlook.

1.4

23

Na Bose-Einstein Condensate Production

In 2013 when I walked into the BEC2 lab for the first time, I was absolutely shocked by
the complexity of the apparatus built by the previous generation of graduate students.
Hundreds of cables from the floor to the ceiling, forests of mirrors, lenses and otherelements on the optical tables. It takes a lot of effort and expertise to maintain theapparatus and run experiments with cold atoms. However, it took even more effort tobuild this machine and conduct the first experiments: produce the first BEC, set-up
lattices and carry out the test experiments. These difficult steps were done by thegraduate students before I joined the group. My goal was to keep performance ofthe machine on the same level and possibly modify, improve if needed, and adjust it
for the upcoming scientific agenda of our lab. Due to the increase of the laser power
needs in our lab, our laser system experienced significant changes. We have also had
two vacuum failures in the course of the last 6 years. Brining the machine back towork required us to go through the basic steps of laser cooling in order to produce
BEC again. In this chapter I would like to overview the existing BEC machine. For
19

the more detailed description of the apparatus I recommend early theses from our lab[12, 13, 14].

1.5

Overview of the Experimental Machine

Our BEC2 apparatus is a general-purpose machine for experiments with ultracold
23

Na and 6 Li. Fig.1-1 shows a schematic plot of the whole apparatus. On the rightis the two-species oven which is the source of 23 Na and 7 Li. The oven is connected
to a small intermediate chamber through the differential pumping tube. The VAT
gate valves before and after the intermediate chamber allow the isolation of different
segments of the apparatus in order to protect the vacuum. The intermediate chamberis connected to the Zeeman slower another differential pumping tube. Atoms which
travel through the Zeeman slower arrive to the main chamber, where they are trappedand cooled to quantum degeneracy. Each of the chambers are pumped by their own
ion pump and the pressure is monitored by ion gauges or by the RGA (Residual Gas
Analyzer) in the main chamber.
The magnetic coils are mounted in the top and bottom buckets of the main chamber. A detailed description of the coils and winding procedure can be found in [12].
Here I just want to mention that the coils are water cooled and can carry high current
up to 500A. Being connected in a circuit with mechanical H-bridge, they can work inboth Helmholtz and anti-Helmholtz configurations. When running current throughall the segments of the coils, one can achieve magnetic bias fields up to 1 kGs andmagnetic gradient up to 1 kGs/cm. The ability to handle high currents and produce
high magnetic gradient was a critically important point for the experiment we did on
magnetically generated spin-orbit coupling described in Chapter 6. The strong gradient is also important to do efficient evaporative cooling, since we want to compress
atoms to high densities by loading them into a tight magnetic trap. For the future
experiment with 6 Li, high bias magnetic field is needed to reach s-wave Feshbachresonance around 830 Gs, which is the way to control the interactions between theatoms.
20

Figure 1-1: A schematic plot of the two-species apparatus. The cups on the rightcontain sodium and lithium. They are heated to produce atomic flux. The atoms
undergo the first cooling stage while traveling through the Zeeman slower. Once they
arrive to the main chamber, atoms are trapped and cooled further. The custom-madestainless steel chamber provides optical access through many viewports. High-current
coils are mounted in the top/bottom buckets to provide strong magnetic fields. The
figure is adapted from [12]1.6

Laser systems

Resonant light for sodium 23 Na is provided by a laser from MPB Photonics. 1178 nm
light from Topitca DL Pro diode laser is seeded to a Raman amplifier, which gives
around 6W of power. Then the light is frequency-doubled to produce 1.4W of 589 nm
yellow light. Over the years, Toptica diode degraded and currently our lab is using atemporary replacement laser diode and our home-built delivery system which shapes
the beam and provides the minimal required power (18-20mW) to the next stage Raman Amplifier. Back-up diode from the company Innolume were ordered to beshared between BEC2 and BEC3 labs for the future replacement in order to solve
this problem permanently.
Green 532 nm light is needed for several purposes in the lab. First, it is required
to plug the low field region in the magnetic trap [15]. Second, it was needed for theexperiment with optical superlattice. While the second application required single21

mode light phase-locked with the 1064 light source, the plug beam application israther simple - a multimode laser can be used. However, due to the failure of our oldMillenia from Spectra-Physics lasers, we had to use the same light source (describedin details in the Chapter 4) as we used for the superlattice as a plug laser. Eventually,a new 10W Sprout was purchased and installed specifically to serve only as a pluglaser.
High power 1064 infra-red (IR) light is needed to trap atoms in optical dipole
traps and to generate optical lattices. Usually, several Watts of IR is needed for eachof the beams used for these application. Our home-made 532nm frequency-doubling
setup required around 30W of IR. Our lab is equipped with two NUFERN amplifiers,
each of them can safely produce 40W of power. In addition to them we have an old
IPG fiber laser giving around 15W. As seed source for NUFRN amplifiers, we used a
2W Mephisto laser from Coherent.
Over the course of several years we have tried different parameters of sample
preparation. The typical path to BEC is the following: we load 109 atoms from
MOT to the magnetic quadrupole trap with a gradient of 50 Gs/cm. Then the trap
is ramped up to 500 Gs/cm for further evaporation either by driving microwave(︀)︀(︀|𝐹 = 1, 𝑚𝐹 = −1⟩ → |𝐹 = 2, 𝑚𝐹 = −2⟩ or radio-frequency |𝐹 = 1, 𝑚𝐹 = −1⟩ →)︀|𝐹 = 1, 𝑚𝐹 = 0⟩ transitions. After evaporation, about 1 − 10 · 106 condensed atoms
can be loaded into the optical dipole trap.

22

Chapter 2
Introduction to spin-orbit coupling
One of the major motivations for the research in our lab has been the study ofspin-orbit coupling. Why is it important to simulate spin-orbit coupling for neutral
atoms? Implementation of the interaction which includes both internal degree of
freedom (spin) and motional (orbital) degrees of freedom is interesting by itself, andthe 𝑝ˆ · 𝑠ˆ term is the simplest combination of the two operators. It naturally appears

from the Hamiltonian of a particle in spin-dependent gauge field (ˆ𝑝 − 𝐴ˆ
𝑠)2 . The goalof quantum simulations is to implement various hamiltonians and this is one of themost generic ones.
In addition to being important as a fundamental concept spin-orbit coupling(SOC) also plays an important in many phenomena. It first appears as an interaction between electrons spin and the field of nucleus in an atom, leading to a finestructure of the energy levels. In the field of utracold atoms we find motivationfor our experiment in phenomena inspired by solid state physics. SOC also plays afundamental role there.
A paradigmatic example is the velocity-dependent energy splitting in heterostructures [16]. An electric field intrinsically appears due to the charge redistribution at
the interface of a bilayer system. In the frame moving with electrons at velocity ⃗𝑣
the electric field is transformed n into magnetic field, which depends on ⃗𝑣 throughthe Lorentz transform.
𝐵𝑆𝑂 =

𝐸0 ℎ̄
(𝑘𝑥 𝑒𝑦 − 𝑘𝑦 𝑒𝑥 )𝑚𝑐2
23

This magnetic field in the frame of electrons leads to a momentum dependent Zeeman
effect. The interaction of the form 𝑘𝑥 𝑒𝑦 − 𝑘𝑦 𝑒𝑥 is known as Rashba SOC. One can saythat the asymmetry of the structure in the growth direction leads to Rashba SOC.
Another type of the spin-orbit interaction, which has a form of 𝑘𝑥 𝑒𝑥 − 𝑘𝑦 𝑒𝑦 , is referred
to as Dresselhaus interaction, and it originates from lack of inversion asymmetry inthe bulk of semiconductors.
Solid-state devices which rely on electron charge are ubiquitous parts our everyday
life. Another boost for the modern electronics, memory storage and computations,
could be the use of the spin degree of freedom of electrons. Devices based on spintransfer are studied in the field of spintronics [17]. One of the advantages of using
spin transistors is that in order to change the spin-state one does not need to provide
electrical current as for the commonly used field transistors. The smaller size, reduced
consumption of power and better sensitivity would enable a significant improvementof current electronics technology. The spin degree of freedom can also be used as nonvolatile storage of memory. Spin-orbit interaction is essential for spintronics devices.
It allows electrical control of the spin by perturbing the motional degree of freedom.
SOC also plays an important role in physics of topological insulators [18]. The
spin degree of freedom enriches the band structure of electrons in solids. Due tothe coupling between the spin and momentum a spin texture appears in momentum
space. Winding of the spin in momentum space when the quasimomentum is swept
over the Brillouin zone can give nonzero winding numbers and topological bands.

2.1

Synthetic spin-orbit coupling in cold atoms

For neutral atoms coupling between motional degree of freedom and atom’s spin does
not happen naturally like for charged particles. However, the effective spin-orbit
coupling can be mimicked with the help of external fields. Simulation of 2D spinorbit coupling is particularly interesting [19], since it is equivalent to generation of
non-Abelian gauge fields (proposals [20, 21]) and will give rise to study topologically
nontrivial phenomena. For instance, realization of Rashba-type spin-orbit coupling24

is very exciting because, because it would emulate a solid states system in a controllable way. But cold atoms experiments aim not only to serve solid state physics byuntangling its problems and reproducing analogous systems, but can also create thestates of matter which do not naturally occur in life. In this sense, simulating a onedimensional spin-orbit coupling, which is equivalent to Abelian gauge, is interestingby itself and not only as a stepping stone towards 2D spin-orbit coupling. In somesense engineering the systems which do not have an existing material counterpart
may be even more exciting than trying to mimic nature.
1D spin-orbit coupling was proposed to be seen in Λ-type schemes [22]. It was
shown that the ground states of the Λ-scheme coupled via two-photon transition gives
rise to the interaction with equal combination of Rashba and Dresselhaus terms, whichin the convenient basis can be written as 𝑘𝑧 𝜎ˆ𝑧 .
Gauge potentials can be simulated by affecting atom’s dynamics with external
forces (for instance, by rotating the condensate [23]) or fields through atom-lightinteraction [24]. The idea of generating Abelian and non-Abelian gauge fields goesback to the work of Wilczek and Zee [25].

2.2

1D spin-orbit coupling. Λ-scheme.

Apparently, the simplest for of spin-orbit coupling can be realized and very simple
Λ-type scheme. It was shown that 1D spin-orbit coupling 𝑘𝑧 𝜎
ˆ𝑧 , which is a sum of
Rashba and Dresselhaus terms, arises as a synthetic gauge field in three-level system
dressed with two Raman laser beams [26]. This gauge field is spin-dependent, but
Abelian, unlike 2D spin-orbit coupling.
The Λ-type configuration can be realized with the hyperfine states of alkali atoms.
In the work of Ian Spielman’s group 87 Rb atoms were used [27]. The Hamiltonian
corresponding to the dressed level structure on the Fig.2-1 is
𝐻 = ℎ̄∆ − (ℎ̄Ω1 sin |𝑔1 ⟩ ⟨𝑒| + ℎ̄Ω2 |𝑔↓ ⟩ ⟨𝑒| + ℎ.𝑐.)
25

(2.1)

Figure 2-1: 1D spin-orbit coupling in Λ-scheme. (a) A three-level system dressed with
two Raman laser beams. (b) Level structure for 87 Rb used in the of the Spielmangroup. (c) Raman beams provide moementum transfer which accompanies spinflip
which shifts the parabolic dispersions and opens the gap at the level crossing. The
image is adopted from [22], where the strengths of the coupling between the ground states and the excited state
manifold Ω1 = Ω sin 𝜃𝑒𝑖𝜑1 and Ω = Ω cos 𝜃𝑒𝑖𝜑2 are determined by the two-photon Rabi
frequencies of the Raman beams Ω, their alignment captured by angle 𝜃 and thephases of the laser beams.
In the case of large detuning ∆ ≫ |Ω|, the single-photon transition from theground states to the excited state |𝑒⟩ is suppressed, and the excited state then can be
adiabatically eliminated. So, the Λ scheme is reduced to effective two-level system,
coupled by a two-photon process. After adiabatic elimination the eigen states of the
2-by-2 Hamiltonian are:|𝜒1 (r)⟩ = cos 𝜃𝑒𝑖𝑆2 (r) |𝑔↑ ⟩ − sin 𝜃𝑒𝑖𝑆1 (r) |𝑔↓ ⟩
26

(2.2)

|𝜒2 (r)⟩ = sin 𝜃𝑒𝑖𝑆2 (r) |𝑔↑ ⟩ + sin 𝜃𝑒𝑖𝑆1 (r) |𝑔↓ ⟩(2.3)

In the simplest case when the Raman beans are propagating along 𝑧ˆ, i.e. the Rabi
frequencies are Ω1 = Ω𝑒𝑖𝑘0 𝑧 and Ω2 = Ω𝑒−𝑖𝑘0 𝑧 , where 𝑘 = 2𝜋/𝜆 is the wave-number ofthe Raman beams. After adiabatic elimination of |𝑒⟩ original Hamiltnonian is reduced
to the effective one⎛𝑝^2𝑧− ℎ̄𝛿2𝑅𝐹
2𝑚⎝
𝐻𝑒𝑓 𝑓 =ℎ̄Ω20 −𝑖2𝑘0 ·𝑧𝑒2Δ⎞ℎ̄Ω20 𝑖2𝑘0 ·𝑧
𝑒
2Δ
⎠,
𝑝^2𝑧
ℎ̄𝛿𝑅𝐹
+ 2
2𝑚

(2.4)

where 𝛿 is a two-photon detuning. The strength of the interaction is characterizedℎ̄Ω2

now by 2-photon Rabi frequency Δ0 . Thus, we obtained a Hamiltonian of a two-levelsystem driven by an optical field with the wave number 2𝑘0 .
ˆ 𝑧 = 𝑒−𝑖𝑘0 𝑧^𝜎𝑧 the effective
After applying a position-dependent transformation 𝑅
Hamiltonian 𝐻𝑒𝑓 𝑓 is transformed into 1D spin-orbit coupling Hamiltonian 𝐻𝑆𝑂𝐶 =ˆ 𝑧 𝐻𝑒𝑓 𝑓 𝑅
ˆ † , where
𝑅𝑧

𝐻𝑆𝑂𝐶 =(𝑝𝑧 + 𝑘0 𝜎ˆ𝑧 )2 𝛿
Ω𝑅
+ 𝜎
ˆ𝑧 +
𝜎
ˆ𝑥
2𝑚
2
2

(2.5)

The fact that the Hamiltonians 2.4 and 2.5 are equivalent up to the unitary transformation gives us a hint how to interpret spin-orbit coupling. It is intrinsicallyrelated to the recoil momentum. Transition from one state to another is carried out
by absorbing a photon from one of the Raman beams and reemitting a photon intoanother Raman beam in a stimulated way. The momentum of atom changes by 2𝑘0 .
Depending on the direction of the process the atom gains or looses 2𝑘0 . As a result,the parabolas of the dispersion relation for spin-up and spin-down are shifted by 2𝑘0(Fig.2-1). The two-photon Rabi frequency characterizing the strength of transition
between |𝑔↓ ⟩ and |𝑔↑ ⟩ enters the spin-orbit coupling Hamiltonian in 𝜎ˆ𝑥 term and opens
the gap between the parabolas.
One has to notice that adding another pair of Raman beams in a different direction
does not create the "missing" term to make 2D spin-orbit coupling out of 1D. It merely
27changes the direction of recoil momentum. In order to fundamentally change the typeof the gauge field and make it non-Abelian one needs to consider more complicatedsystems such as, for instance, tripod level configurations [28].

28Chapter 3
Spin-Orbit Coupling in a Superlattice
As was discussed in the previous chapter, spin-orbit coupling is a profound interaction
mechanism relevant for various phenomena such as topological insulators, Majorana
fermions, and spintronics. Realization of SOC with ultracold atoms enables exploration of new phases of matter and allows control of the motional degree of freedom
by the internal degree of freedom of an atom, or the other way around. For example,
in quantum dots, by talking to the motional degree of freedom, one can control its
spin [29].
Spin-orbit coupling makes the atom’s motion spin-dependent.

Rather trivial

regimes without spin-flips can be achieved with far detuned light [30, 31]. In principle,
this is, for example, enough to emulate Spin Quantum Hall effect.
However, in order to simulate Rashba or Dresselhaus spin-orbit coupling, we needto bring in 𝜎𝑥 and 𝜎𝑦 terms, and therefore, there must be an implementation ofa spin-flip in some form. In the pioneering work of Ian Spielman’s group, spinorbit coupling was engineered using hyperfine states of 87 Rb [27]. Coupling of thetwo states was achieved by a 2-photon Raman process and the atomic spin-orbit
interaction in the excited state. Without fine-structure splitting, caused by the L · S
interaction, changing the hyperfine state in the ground 𝑙 = 0 manifold would not
be possible. Therefore, the experiments using Raman transitions between different
hyperfine states faces two contradicting requirements: for the strong coupling throughthe L · S interaction, one needs to minimize the detuning from the excited states, but
29to reducing the heating, the detuning must be as large as possible. The strength ofthe coupling and heating through the spontaneous emission from the excited state
scales the same way with the two-photon detuning, i.e. 1/∆2 [32].
Our goal is to create a transition between spin-states which is accompanied bymomentum transfer. Two-photon Raman transitions, when the Raman beams havedifferent orientations, seem to be a perfect choice for the source of the momentum
transfer. A photon k1 is absorbed from one beam and reemitted into another beamwith a wave-vector k2 . Then the momentum transfer is k2 − k1 . The order canbe different: a photon is absorbed with wave-vector k2 and reemitted with k1 ina stimulated way. The momentum transfer then has the opposite sign. Since thetwo-photon process and different hyperfine states as a choice of pseudo-spin do notreally work well together, there are two passes towards improving already existing
spin-orbit coupling schemes.
First, one could keep two-photon Raman process but choose different pseudo-spinstates which would not require resonant light to be coupled — for instance, left andright states in a double-well, or different bands in an optical lattice with atoms staying
in the same hyperfine state.
Second, one could continue using hyperfine states as pseudo-spins, but not rely on
Raman process for momentum transfer. The hyperfine states can be easily coupledwith an RF signal, but then one needs to come up with another mechanism to impartrecoil, since an RF photon does not have one (for instance, Floquet engineering).
The problem of coupling different hyperfine states along with providing momentum
transfer can also be eliminated by using species that have nonzero angular momentum
in the ground state, like Dy [33], and do not require near-resonant light.
Our lab has explored both strategies for alkali atoms: a different choice of thepseudo-spin, and a different source of recoil. In the present chapter, I would like tooverview our experiments based on the first strategy [34, 35]. In order to avoid the
problem of spontaneous emission, we decided to consider the two lowest states in theasymmetric double-well potential as pseudo-spin states Fig.5-1. In the tight-binding
regime, when the barrier between the wells can be treated perturbatively, the lowest
30

Figure 3-1: A schematic plot of the pseudo-spin states in an asymmetric double-wellpotential. The overlap between the pseudo-spin-down (the state localized in the leftwell) and the pseudo-spin-up (the state localized in the right well) is controlled by
the tunneling through the barrier 𝐽 and the offset between the wells ∆.

eigenstates can be written as|↓⟩ = |𝑙⟩ + (𝐽/∆) |𝑟⟩ ,|↑⟩ = |𝑟⟩ − (𝐽/∆) |𝑟⟩ ,(3.1)

where |𝑙⟩ and |𝑟⟩ are the state, localized in the left and right wells respectively. Sinceboth states belong to the same hyperfine states, there would be no need for resonant
light to couple them with the Raman process. Another important advantage of suchchoice of pseudo-spins is that we can control interactions between them. The collision
rate between atoms in left and right wells of double-well potential is determined by
the overlap of their wave-functions and can be adjusted by the height of the barrier.
In such systems, intra-spin collisions naturally prevail over inter-spin collisions. The
collisions between the atoms in left and right wells are suppressed by a factor ∼ (𝐽/∆)231

with respect to on-site collisions of atoms. Such a relation between the collision rates
makes the system miscible. This property is important, since we want to realize andprobe many-body Hamiltonians and explore new, unconventional phases of matter.
One of the most intriguing predictions for the spin-orbit coupled Bose gas is theexistence of ground state with a density modulation, called stripe phase [36, 37, 38].
2
< 𝑔↑ 𝑔↓ , i.e. when the
In the regime where the intra-spin interactions dominate, 𝑔↑↓system remains miscible, a spontaneous density modulation forms. This phase ofmatter meets the definition of a supersolid [39], a long-awaited phenomenon which
was sought in superfluid helium [40, 41] and eventually was realized in our lab with
Bose-Einstein condensates [35] and in the Esslinger group [42].
Our scheme addresses the major challenges of spin-orbit coupling: it does not
require resonant light which solves the problem with the heating due to spontaneousemission and the interactions can be controlled so that the system favors miscibility.
In the actual experiment, we created a lattice of double-wells - an optical superlattice. So, the correct model for our system is a sequence of replicated double-wells
along the 𝑧ˆ axis, and free space in the transverse direction. This approach creates anumber of technical advantages. First, it is more convenient to work with a stack ofcoherently coupled double wells because of the improved signal-to-noise ratio. Second,
the interference between the unit cells of superlattice helps to distinguish pseudo-spinstates - since we work with the same hyperfine state of sodium, a traditional methodof distinguishing spins, such as Stern-Gerlach experiment, does not work.

3.1

States in a superlattice

The Hamiltonian for a one-dimensional superlattice, created by overlapping standing
waves of infrared and green light with relative phase 𝜑𝑆𝐿 , is𝐻𝑙𝑎𝑡𝑡𝑖𝑐𝑒 =

𝑝ˆ2𝑧
𝑝ˆ2
+ ⊥ + 𝑉𝐺𝑟 sin2 (𝑘𝐺𝑟 𝑧) + 𝑉𝐼𝑅 sin2 (𝑘𝐼𝑅 𝑧 + 𝜑𝑆𝐿 )
2𝑚 2𝑚

(3.2)

Here the first term describes motion along the lattice direction, the second term
describes the motion in orthogonal plane in free space, and the third and fourth terms
32are the potentials created by the green and IR lattice respectively.
We can use a basis of |𝑙𝑛 (𝑟𝑛 )⟩ states, localized in the left(right) double well of
the unit cell with the number 𝑛, so that in the tight-binding approximation the
Hamiltonian can be rewritten as

𝐻𝑙𝑎𝑡𝑡𝑖𝑐𝑒 =

∑︁
1 ∑︁
𝑝ˆ2⊥
+ ∆0
(|𝑟𝑛 ⟩ ⟨𝑟𝑛 | − |𝑙𝑛 ⟩ ⟨𝑙𝑛 |) − 𝐽(|𝑙𝑛 ⟩ ⟨𝑟𝑛 | + ℎ.𝑐.)
2𝑚 2
𝑛
𝑛
∑︁ ∑︁
−
(𝐽𝑡𝑡′ ′ |𝑡𝑛 ⟩ ⟨𝑡′𝑛+1 | + ℎ.𝑐.), (3.3)
𝑛𝑡=𝑙,𝑟𝑡′ =𝑙,𝑟where |𝑙𝑛 (𝑟𝑛 )⟩ is a wavefunction localized in the left(right) well of the 𝑛𝑡ℎ unit cell,∆0 is the energy separation between the right and the left wells. ℎ̄ is taken to be 1.
Tunneling between neighboring unit cells is important for maintaining coherencein the superlattice: eventually what we observe experimentally is the interference
patterns of matter waves coming from different nodes of the lattice, but not relevant
for describing spin-orbit coupling with double-well states. Thus, all the tunnelingterms beyond one unit cell with 𝐽𝑡𝑡′ ′ in the Hamiltonian can be neglected for the
purpose of describing the pseudo-spin flip and momentum transfer.
When instead of considering an isolated double well we look at the stack of coherently coupled double wells in a superlattice. We have to take lattice band structure
into account (Fig.3-2). The lowest state of the double potential will turn into thelowest band of the superlattice. The minimum energy state of the lowest band occurs
at zero quasimomentum 𝑞 = 0. The first excited state state of the doublewell will now
become the first excited band. But in contrast to the ground band, the first excited
band has an inverted structure in qasimomentum space, i.e. its energy minimum islocated at the edge of the Brillouin zone at 𝑞 = 𝜋/𝑑, where 𝑑 is the period of thelattice.
It is going to be more convenient to use eigenstates of a double-well instead of
localized states for further description. To first order in the small parameter Δ𝐽0 ≪ 1,33

Figure 3-2: Two lowest bands of the superlattice. A Raman process couples the 𝑞 = 0
state of the ground band with edge of the Brillouin zone of the first excited band.
they can be written as|↓𝑛 ⟩ = |𝑙𝑛 ⟩ +𝐽𝐽|𝑟𝑛 ⟩ , |↑𝑛 ⟩ = |𝑟𝑛 ⟩ −|𝑙𝑛 ⟩ .
∆0
∆0

(3.4)

If the system in our experiment is initially prepared in the lower wells of doublewells, which corresponds to the 𝑞 = 0 of the lowest band of the superlattice,(↓)|𝜓𝑞=0 ⟩ =𝑁∑︁1
√ |↓𝑛 ⟩ ,
𝑁
𝑛=1

(3.5)

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

𝑁
∑︁
𝜋
𝑑
1
√ 𝑒𝑖 𝑑 (𝑧𝑛 + 2 ) |↑𝑛 ⟩
𝑁
𝑛=1

34

(3.6)

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

(3.7)

(↓)

(↑)

Now, we want to establish coupling between these to states |𝜓𝑞=0 ⟩ and |𝜓𝑞=𝜋/𝑑 ⟩.
For that we turn on the Raman potential 𝑉𝑅𝑎𝑚𝑎𝑛 and treat it as perturbation. The
complete Hamiltonian of the system is
𝐻 = 𝐻𝑙𝑎𝑡𝑡𝑖𝑐𝑒 + 𝑉𝑅𝑎𝑚𝑎𝑛 ,(3.8)

where 𝑉𝑅𝑎𝑚𝑎𝑛 = Ω cos(𝑘𝑧 𝑧 + 𝑘𝑥 𝑥 − 𝛿𝑡) is a moving lattice potential. One of theRaman beams is aligned along the lattice direction 𝑧ˆ, and is responsible for thequasimomentum transfer from 𝑞 = 0 to the edge of the Brillouin zone 𝑞 = 𝜋/𝑑. The
quasimomentum transfer from this IR beam 𝑘𝐼𝑅 matches half of the reciprocal vectorof the periodic structure, which is 𝜋/𝑑 = 𝜋/(𝜆𝐼𝑅 /2). The second Raman beam isaligned in the orthogonal direction to the lattice 𝑥ˆ (Fig.3-3). After absorbing its
photon, the 𝑥-component of the wave function of the atom becomes a plane wave
with wave number 𝑘 = 𝑘𝐼𝑅 . For the purpose of our experiment, the initial phase of
the Raman potential is not relevant, and is taken to be zero.

Figure 3-3: A schematic plot of the superlattice and Raman beams. The periodicity ofthe superlattice is 𝑑 = 523 nm. One of the Raman beams (k1 , 𝜔1 ) is along the superlatice, while another Raman beam (k2 , 𝜔2 ) is in the transverse direction. Atoms thathave experienced a 2-photon transition acquire a momentum in transverse direction
equal to k2 and quasimomentum transfer along 𝑥ˆ equal to 𝑘1 .
2𝐽We expand the Hamiltonian 3.8 in the new basis 3.4 with ∆ = ∆0 + 2 Δ≈ ∆0 :
0

35

1 ∑︁
𝑝ˆ2⊥
+ ∆
(|↑𝑛 ⟩ ⟨↑𝑛 | − |↓𝑛 ⟩ ⟨↓𝑛 |)+2𝑚 2𝑛(︁ ∑︁ ∑︁
)︁
∑︁
′
′
′
|𝑝⊥ ⟩
|𝑖𝑛 ⟩ ⟨𝑖𝑛 | ⟨𝑝⊥ | Ω cos(𝑘𝑧 𝑧 + 𝑘𝑥 𝑥 − 𝛿 · 𝑡) |𝑝⊥ ⟩ |𝑖𝑛 ⟩ ⟨𝑖𝑛 | ⟨𝑝′⊥ | (3.9)

𝐻=′𝑝⊥ ,𝑝⊥𝑛𝑖=↓,↑𝑖′ =↓,↑

In our experiment 𝑘𝑧 ≈ 𝑘𝑥 ≈ 𝑘𝐼𝑅 = 𝜋𝑑 , where 𝑑 is a period of the superlattice. In
order to consider the effect of the Raman potential with arbitrary phase using firstorder perturbation theory, one needs to calculate overlap integrals for cos(𝑘𝐼𝑅 ·(𝑧−𝑧𝑛 ))𝐽and sin(𝑘𝐼𝑅 · (𝑧 − 𝑧𝑛 )) to first order in Δ:⟨↓𝑛 | cos(𝑘𝐼𝑅 · (𝑧 − 𝑧𝑛 )) |↑𝑛 ⟩ ≈ ⟨𝑙𝑛 | −𝐽𝐽|𝑙𝑛 ⟩ = − ,∆∆⟨↓𝑛 | cos(𝑘𝐼𝑅 · (𝑧 − 𝑧𝑛 )) |↓𝑛 ⟩ ≈ 1, ⟨↑𝑛 | cos(𝑘𝐼𝑅 · (𝑧 − 𝑧𝑛 )) |↑𝑛 ⟩ ≈ 0,⟨↓𝑛 | sin(𝑘𝐼𝑅 · (𝑧 − 𝑧𝑛 )) |↑𝑛 ⟩ ≈ ⟨𝑟𝑛 |𝐽𝐽|𝑟𝑛 ⟩ = ,∆∆⟨↓𝑛 | sin(𝑘𝐼𝑅 · (𝑧 − 𝑧𝑛 )) |↓𝑛 ⟩ ≈ 0, ⟨↑𝑛 | sin(𝑘𝐼𝑅 · (𝑧 − 𝑧𝑛 )) |↑𝑛 ⟩ ≈ 1,

(3.10)(3.11)(3.12)(3.13)

where 𝑧𝑛 = 𝑛𝑑 is a coordinate of the left well in the 𝑛𝑡ℎ unit cell.
Thus, the Raman potential can be expanded in the basis of double-well eigenstates:

∑︁

𝑖=𝑎,𝑏
𝑖′ =𝑎,𝑏|𝑖𝑛 ⟩ ⟨𝑖𝑛 | Ω cos(𝑘𝑧 (𝑧 − 𝑧𝑛 ) + 𝑘𝑧 𝑧𝑛 + 𝑘𝑥 𝑥 − 𝛿 · 𝑡) |𝑖′𝑛 ⟩ ⟨𝑖′𝑛 | =)︀𝐽 (︀|↓𝑛 ⟩ ⟨↑𝑛 | + |↑𝑛 ⟩ ⟨↓𝑛 | + |↓𝑛 ⟩ ⟨↓𝑛 |}+∆𝐽− Ω sin 𝜑𝑛 {|↑𝑛 ⟩ ⟨↑𝑛 | + (|↓𝑛 ⟩ ⟨↑𝑛 | + |↑𝑛 ⟩ ⟨↓𝑛 |)}, (3.14)∆= Ω cos 𝜑𝑛 {−36where 𝜑𝑛 = 𝜋𝑛 + 𝑘𝑥 𝑥 − 𝛿𝑡. Later, we will calculate how 𝑥 as an operator acts on the
momentum states |𝑝⊥ ⟩.
𝑉ˆ𝑅𝑎𝑚𝑎𝑛 =∑︁𝑛Ω(−1)𝑛 cos(𝑘𝑥 𝑥 − 𝛿𝑡) |↓𝑛 ⟩ ⟨↓𝑛 | − Ω(−1)𝑛 sin(𝑘𝑥 𝑥 − 𝛿𝑡) |↑𝑛 ⟩ ⟨↑𝑛 | + (3.15)√ 𝐽𝜋− 2Ω (−1)𝑛 cos(𝑘𝑥 𝑥 − 𝛿𝑡 − ){|↓𝑛 ⟩ ⟨↑𝑛 | + |↑𝑛 ⟩ ⟨↓𝑛 |}∆4(3.16)

The factor (−1)𝑛 represents the phase of the Raman beams. This factor is present
because the wavelength of the Raman beams 𝜆 = 1064 nm, is two times the length of
the unit cell. In our experiment, the atomic sample is prepared in the zero-momentum
state. When the Raman perturbation is applied, the atoms experience a kick in the𝑥-direction. In the 𝑦-direction atoms remain unperturbed, i.e. 𝑝ˆ𝑦 = 0. Since theconfinement along 𝑥 is weak, we can use the basis |↑ (↓), 𝑘⟩ = |↑ (↓)⟩ ⊗ 𝑒𝑖𝑘𝑥 . Herethe states |↑ (↓), 𝑘⟩ correspond not to the individual double wells, but to the lowest
and first exctied band of the lattice. They include all the factors needed for a Blochwave to reflect the symmetry of the Hamiltonian. The Raman interaction gives riseto intra-band coupling terms (3.14), and to the spin-orbit coupling term (3.16).
In addition to coupling the ground band to the first excited band with Raman
potential, we also inevitably create some intra-band excitations, which we refer to as"onsite coupling". Atom only keep the same Bloch function, but acquire quasimomentum along the lattice direction and momentum in the orthogonal plane in orderto fulfil the conservation laws.
The intra-band coupling terms are

⟨

⃒

⃒

⃒
⃒ (↓)
(↓)
𝜓𝑞=𝜋/𝑑 ⃒ 𝑉ˆ𝑅𝑎𝑚𝑎𝑛 ⃒𝜓𝑞=0

⟩

=

∑︁ 1
𝑛,𝑛′

𝑁

(−1)𝑛 ⟨↓𝑛′ | Ω cos(𝑘𝑧 𝑧 + 𝑘𝑥 𝑥 − 𝛿𝑡) |↓𝑛 ⟩

=Ω

∑︁ 1
𝑛

𝑁

37

cos(𝜑𝑛 )(−1)𝑛 = Ω cos(𝑘𝑥 𝑥 − 𝛿𝑡) (3.17)

⟨⃒⃒⟩ ∑︁ 1⃒ (↑)(↑) ⃒𝜓𝑞=0 ⃒ 𝑉ˆ𝑅𝑎𝑚𝑎𝑛 ⃒𝜓𝑞=𝜋/𝑑 =⟨↑𝑛′ | Ω cos(𝑘𝑧 𝑧 + 𝑘𝑥 𝑥 − 𝛿𝑡)𝑖(−1)𝑛 |↑𝑛 ⟩
𝑁
𝑛,𝑛′
∑︁ 1
𝑖(−1)𝑛 (− sin(𝜑𝑛 )) = −𝑖Ω sin(𝑘𝑥 𝑥 − 𝛿𝑡) (3.18)=Ω
𝑁𝑛

While the spin-orbit coupling matrix elements are⟨⃒⃒⟩ ∑︁ 1
′
⃒
⃒ (↓)
(↑)
𝜓𝑞=𝜋/𝑑 ⃒ 𝑉ˆ𝑅𝑎𝑚𝑎𝑛 ⃒𝜓𝑞=0 =(−𝑖)(−1)𝑛 ⟨↑𝑛′ | Ω cos(𝑘𝑧 𝑧 + 𝑘𝑥 𝑥 − 𝛿𝑡)| |↓𝑛 ⟩
𝑁
𝑛,𝑛′
∑︁ 1
(−1)𝑛 ⟨↑𝑛 | cos(𝑘𝑧 (𝑧 − 𝑧𝑛 )) cos(𝜑𝑛 ) − sin(𝑘𝑧 (𝑧 − 𝑧𝑛 )) sin(𝜑𝑛 ) |↓𝑛 ⟩ ,
= −𝑖Ω
𝑁
𝑛

(3.19)

⟨

⃒
⃒
⟩
)︀
𝐽 ∑︁
1 (︀
⃒
⃒ (↓)
(↑)
(−1)𝑛
𝜓𝑞=𝜋/𝑑 ⃒ 𝑉ˆ𝑅𝑎𝑚𝑎𝑛 ⃒𝜓𝑞=0 = 𝑖 Ωcos(𝜑𝑛 ) + sin(𝜑𝑛 )
∆ 𝑛
𝑁
=𝑖

𝐽
Ω(cos(𝑘𝑥 𝑥 − 𝛿𝑡) + sin(𝑘𝑥 𝑥 − 𝛿𝑡)) (3.20)∆

The intra-band coupling matrix elements (3.17) and (3.18) provide recoil kick in2𝑘𝑥
𝑥−direction with recoil energy 𝐸𝑟 = 2𝑚
and along the superlattice, changing the(↓)

quasimomentum by half a reciprocal vector. If the system is initially at |𝜓𝑞=0 , 0⟩, the
new adiabatically connected eigenstate in first order perturbation theory is:
1 Ω
1 Ω
(↓)
(↓)
𝑒−𝑖𝛿𝑡 |𝜓𝑞=𝜋/𝑑 , 𝑘𝑥 ⟩ −𝑒𝑖𝛿𝑡 |𝜓𝑞=𝜋/𝑑 , −𝑘𝑥 ⟩2 𝐸𝑟 − 𝛿
2 𝐸𝑟 + 𝛿
𝑖 𝜋4
−𝑖 𝜋4
𝑒
𝐽Ω/∆
𝑒
𝐽Ω/∆
(↑)
(↑)
𝑒−𝑖𝛿𝑡 |𝜓𝑞=𝜋/𝑑 , 𝑘𝑥 ⟩ − 𝑖 √𝑒𝑖𝛿𝑡 |𝜓𝑞=𝜋/𝑑 , −𝑘𝑥 ⟩ (3.21)−𝑖 √2 𝐸𝑟 + ∆ − 𝛿
2 𝐸𝑟 + ∆ + 𝛿(↓)|Ψ1 ⟩ = |𝜓𝑞=0 , 0⟩ −

38

(↑)

If the system is prepared in |𝜓𝑞=𝜋/𝑑 , 0⟩, instead we find
1 Ω
1 Ω
(↑)
(↑)
𝑒−𝑖(𝛿+Δ)𝑡 |𝜓𝑞=0 , 𝑘𝑥 ⟩ +𝑒𝑖(𝛿−Δ)𝑡 |𝜓𝑞=0 , −𝑘𝑥 ⟩2 𝐸𝑟 − 𝛿
2 𝐸𝑟 + 𝛿
−𝑖 𝜋4
𝑖 𝜋4
𝑒
Ω𝐽/∆
Ω𝐽/∆
𝑒
(↓)
(↓)
+𝑖 √
𝑒−𝑖(𝛿+Δ)𝑡 |𝜓𝑞=0 , 𝑘𝑥 ⟩ + 𝑖 √𝑒𝑖(𝛿−Δ)𝑡 |𝜓𝑞=0 , −𝑘𝑥 ⟩ (3.22)
2 𝐸𝑟 − ∆ − 𝛿
2 𝐸𝑟 − ∆ + 𝛿(↑)|Ψ2 ⟩ = 𝑒−𝑖Δ𝑡 |𝜓𝑞=𝜋/𝑑 , 0⟩ −Our goal in the experiment was to map the resonances by sweeping the detuningof the Raman beams. In the vicinity of the spin-orbit coupling resonances 𝛿 = ∆ +
(↓)

(↑)

(↑)

(↓)

𝐸𝑟 (|𝜓𝑞=0 , 0⟩ → |𝜓𝑞=𝜋/𝑑 , 𝑘𝑥 ⟩) and 𝛿 = ∆ − 𝐸𝑟 (|𝜓𝑞=𝜋/𝑑 , 0⟩ → |𝜓𝑞=0 , 𝑘𝑥 ⟩)the onsite(intra-band) coupling is suppressed and spin-orbit coupling terms dominate. For𝛿 = ∆, which is slightly away from the actual resonance by recoil energy, the spin(↓)

orbit coupling terms form stationary density modulations: the term |𝜓𝑞=0 , 0⟩ from𝑖𝜋(↓)

equation 3.21 with the term 𝑖 𝑒√42 𝐸Ω𝐽/Δ𝑒𝑖(𝛿−Δ)𝑡 |𝜓𝑞=0 , −𝑘𝑥 ⟩ from 3.22, and the term𝑟 −Δ+𝛿
−𝑖 𝜋

(↑)

(↑)

−𝑖 𝑒√24 𝐸𝐽Ω/Δ
𝑒−𝑖𝛿𝑡 |𝜓𝑞=𝜋/𝑑 , 𝑘𝑥 ⟩ from equation 3.21 with the term 𝑒−𝑖Δ𝑡 |𝜓𝑞=𝜋/𝑑 , 0⟩ from𝑟 +Δ−𝛿
equation 3.22. The density modulations arising from the interference of these states
manifest the stripe phase in the language of pseudo-spins.

3.2

Spin-orbit-coupling Hamiltonian

Omitting all the off-resonant and counter-rotating terms from (3.21) (or similar with
(3.22)) we can rewrite the Hamiltonian describing the system as
⎛

𝑝^2𝑥
− Δ2
2𝑚

𝐻𝑆𝑂𝐶 = ⎝ −𝑖𝜋/4
𝑖𝑒 √

2

𝐽
Ω𝑒𝑖(𝑘𝑥 𝑥−𝛿𝑡)Δ⎞𝑖𝜋/4 𝐽−𝑖𝑒√Ω𝑒−𝑖(𝑘𝑥 𝑥−𝛿𝑡)
2 Δ
⎠
𝑝^2𝑥
Δ
+ 2
2𝑚

(3.23)𝜋
After a position-dependent unitary transformation, 𝑈ˆ = 𝑒(−𝑖𝑘𝑥 𝑥+𝑖𝛿𝑡−𝑖 4 )𝜎𝑧 /2 , the Hamil′

tonian turns into 𝐻𝑆𝑂𝐶 = 𝑈 † 𝐻𝑆𝑂𝐶 𝑈 − 𝑖𝑈 † 𝜕𝑈.
𝜕𝑡′

𝐻𝑆𝑂𝐶 =(ˆ𝑝𝑥 + 𝛼𝜎𝑧 )2+ 𝛽𝜎𝑥 + 𝛿0 𝜎𝑧 ,2𝑚(3.24)

where 𝛼 = − 21 𝑘𝑥 , 𝛽 = √𝐽Ωand 𝛿0 = 12 (𝛿 − ∆). This Hamiltonian corresponds to
2Δ
equal contributions of Rashba and Dresselhaus interactions. The parameter 𝛼 defines
39the momentum transfer. It can be tuned by the angle between the Raman beams.
The parameter 𝛽 is the strength of spin-orbit coupling, which can be tuned by thetwo-photon Rabi frequency of the Raman beams.

3.3

Antiferromagnetic spin texture

A spin-1/2 aligned along the cos 𝜑ˆ
𝑥 +sin 𝜑ˆ𝑦 direction has the wavefunction |↓⟩+𝑒𝑖𝜑 |↑⟩.
In our experiment, atoms can be prepared in the left and right sites of the double-wellswith equal population. The wave function is then proportional to
|𝜓⟩ =

∑︁
𝑛

|↓𝑛 ⟩ + (−1)𝑛 𝑒𝑖𝜃 𝑒−𝑖Δ𝑡 |↑𝑛 ⟩ ,which corresponds to spin-states aligned in the x-y plane with opposite directions onneighboring sites, showing x-y antiferromagnetic ordering. The expectation values ofspin evolve as ⟨𝜎𝑥 ⟩ = (−1)𝑛 cos(𝜃 − ∆𝑡), ⟨𝜎𝑦 ⟩ = (−1)𝑛 sin(𝜃 − ∆𝑡) and ⟨𝜎𝑧 ⟩ = 0.
Tunneling between wells within a unit cell causes a density modulation. Local
populations in the 𝑛th cell acquire a density imbalance:(︀)︀𝐽| ⟨𝑙𝑛 |𝜓⟩ |2 ∼ 1 − (−1)𝑛 cos(𝜃 − ∆𝑡) ,
∆
(︀
)︀
𝐽
| ⟨𝑟𝑛 |𝜓⟩ |2 ∼ 1 + (−1)𝑛 cos(𝜃 − ∆𝑡)∆Translating by one superlattice period flips the sign of the imbalance. This showsthat 𝑟𝑛 and 𝑙𝑛+1 wells, and 𝑟𝑛−1 and 𝑙𝑛 wells have the same change in density, but
oscillate out of phase. Therefore, this density wave has twice the wavelength of thesuperlattice and is shifted spatially by half a unit cell.
One can say that the state with both bands in 𝑞 = 0,∑︀𝑛 |↑𝑛 ⟩ + |↓𝑛 ⟩ demonstrates

a ferromagnetic spin-texture. After the excited bands relaxes to 𝑞 = 𝜋/𝑑, the resulted∑︀state |𝜓⟩ = |↓𝑛 ⟩+(−1)𝑛 𝑒𝑖𝜃 𝑒−𝑖Δ𝑡 |↑𝑛 ⟩ shows an antiferromagnetic texture. This state𝑛

has a reduced symmetry with respect to the superlattice.
40

3.4

Phase of the Superlattice

In the experiment, we observed how the optical superlattice — and spin-orbit coupling
on top of it —affected the momentum structure of a BEC. The set-up we created
had two major ingredients: the optical superlattice and the Raman beams. The
superlattice is produced by an overlap of two standing waves, the infrared and green
ones. Combined 1064 nm and 532 nm light propagate together and reflect from a
retro-mirror, forming standing waves with 532 nm and 266 nm respectively. The
superlattice has to be deep enough, with a barrier sufficient to suppress tunnelingwithin the unit cell. It has to have fast dynamic control: the asymmetry of the doublewell potential has to be manipulated on a time scale faster than the atomic motion.
Basically, we had to be capable to go from symmetric double well to asymmetricdouble well before atoms could see the difference.
The shape of the double well was determined by the relative phase 𝜑𝑆𝐿 between
1064 nm and 532nm light. In our experiment 𝜑𝑆𝐿 , was controlled by a dispersive
glass plate and by a small shift of the frequency of the IR light. The motion of
the glass plate was relatively slow, and was used to preset the initial phase of thesuperlattice. We made it possible to rotate the plate by a galvo on a time scale of2-3 ms, but eventually it turned out to be too slow for our needs. Fast control of thephase was achieved by a frequency shift of the IR light, using a double-pass AOM.
The retro-reflecting mirror sets the boundary condition for the superlattice (Fig.3-4).
Both green and IR light must have a node on it. If the frequency of green is notexactly 2 times the frequency of IR, but shifted a small amount (in our case 70 MHz)
then a phase shift 𝜑𝑆𝐿 accumulates between the lattices on the way from the common
node at the mirror to the atoms. Using this method, the phase 𝜑𝑆𝐿 can be changed
very quickly at sub-microsecond times.
If we want to go from the symmetric to the antisymmetric double-well potential,
we have to change the superlattice phase by 𝜋. We have to generate a relative shiftof 133 nm between the green lattice lattice and the IR lattice, which corresponds tohalf the period of green lattice and qartter period of the IR lattice.
41

L=500 mm

Figure 3-4: The relative phase between green and IR light at the location of the atoms
determines the shape of the double-well cell.
The number of nodes in a standing wave between the retro-mirror and the atoms
is
𝑁=

𝐿
,
𝜆/2

(3.25)

where 𝐿 ≈ 0.5 m is the distance from retro-mirror to the center of the vacuum
chamber. The change of the number of nodes due to the frequency detuning of thebeam is
∆𝑓
∆𝑁
=
𝑁
𝑓

(3.26)The required ∆𝑁 for green light is 21 , and for IR is ∆𝑁 = 14 . Thus, we either haveto shift the frequency of the green light by 140 MHz, or shift the IR by 70 MHz. It
was more convenient for us to shift the IR light frequency, which was easily achievedby a double-pass AOM.
Another important aspect of the superlattice is to make 1064 nm and 532 nm lightphase-locked. Without this feature the green lattice would jitter with respect to the
IR and the consecutive runs of the experiment would not be reproducible.

3.5

Green light generation

Our experiment with an optical superlattice requires high-power single-mode lightsources for fundamental and doubled frequencies. The BEC2 lab is equipped witha stable 2 W seed laser of 1064 nm light (Mephisto from Coherenent) and two 50
42

W Nufern fiber amplifiers. Thus, there’s no lack of IR power in the lab, so insteadof buying another single-mode green laser, we decided to use some of the IR lightand generate 532 nm light using a frequency-doubling crystal. One Nufern is used
to provide light for IR lattice, while the output of another one is used for generating
a green light by passing it through a nonlinear crystal. Both of the Nuferns use
common seed beam. Thus, the current setup has an advantage that the frequency
doubled green and fundamental IR are phase-locked when obtained from common
seed, whereas two independent lasers would require additional electronic phase-locking
for the purpose of our experiment.
Second-harmonic generation (SHG) is a well known and commonly used methodof generating light in the blue-green range of the EM spectrum. The general ideaof SHG is to utilize the nonlinear polarizability of crystals. A beam of frequency
𝜔 passes through the crystal and generates a beam of frequency 2𝜔 via a nonlinear
interaction. Since the nonlinear response of the system is a process of "second-order",based on a two-photon interaction, it is supposed to be weak in comparison to linear
effects. This intrinsic weakness has lead to the development of several methods toboost the efficiency of SHG, such as guided-wave interactions, intra-cavity doublingand resonant enhancement.

3.5.1

Generation of double-frequency wave

In Gaussian units, Maxwell’s equations in non-magnetic medium with no free charge
and currents give⃗−∇2 𝐸

1 𝜕2 ⃗
𝐷=0
𝑐2 𝜕𝑡2(3.27)The electric displacement vector is⃗ =𝐸⃗ + 4𝜋 𝑃⃗𝐷
43(3.28)

The wave equation transforms into⃗−
∇2 𝐸

1 𝜕2 ⃗
4𝜋 𝜕 2 ⃗𝐸=𝑃
𝑐2 𝜕𝑡2𝑐2 𝜕𝑡2(3.29)⃗ can be expanded
Any component 𝑃𝑘 of the polarization induced by the electric field 𝐸
in a series
(2)

(3)

𝑃𝑘 = 𝜒(1) 𝐸𝑖 + 𝜒𝑖𝑗𝑘 𝐸𝑖 𝐸𝑗 + 𝜒𝑖𝑗𝑚𝑘 𝐸𝑖 𝐸𝑗 𝐸𝑚 + ...

(3.30)The system is driven by monochromatic field at fundamental frequency 𝜔
E = 𝐸𝑖 𝑒ˆ𝑖 = (E𝑖 𝑒𝑖𝜔𝑡 + E𝑖* 𝑒−𝑖𝜔𝑡 )ˆ
𝑒𝑖(3.31)We are most interested in the second-order contribution to the polarization
(2)

𝑃𝑖

)︀
(2) (︀
= 𝜒𝑗𝑖𝑖 E𝑖2 𝑒𝑖2𝜔𝑡 + E𝑖*2 𝑒−𝑖2𝜔𝑡 + 2E𝑖 E𝑖*(3.32)

One can see that the second order polarization also gives rise to a constant term
2E𝑖 E𝑖* , which is the reason for the phenomenon of optical rectification.

3.5.2

Plane wave passing through the crystal

The problem of the beam passing through a nonlinear medium is very complicatedand requires tedious derivations. However, some intuitive features and scaling laws
can be obtained from a simple model of a plane wave (find references).
(2)

We will search for the solution of the wave equation in the form 𝐸𝑘 (𝑧, 𝑡) =E2𝜔 𝑒−𝑖2𝜔𝑡 + 𝑐.𝑐.. Since we decided to neglect the dynamics of the green beam in thetransverse direction, we can employ slowly-varying envelope approximation.
(2)

E2𝜔 = 𝐴(𝑧)𝑒𝑖𝑘𝑧 𝑧(3.33)

This approximation is valid as long as ∇𝐴 ≪ 𝑘𝑧 𝐴. Substituting this representationof green light into the wave equation 3.27 and omitting the quadratic terms, we are44

left with the equation for the envelope−𝑘 2 𝐴(𝑧)𝑒𝑖𝑘𝑧 + 2𝑖𝑘𝑒𝑖𝑘𝑧

4𝜔 2𝜕16𝜋𝜔 2 (2) *2 𝑖2𝑘0 𝑧
𝐴(𝑧) + 2 𝐴(𝑧)𝑒𝑖𝑘𝑧 =𝜒 E𝜔 𝑒,𝜕𝑧𝑐𝑐2(3.34)

where 𝑘0 corresponds to the principle wave vector of the IR light. The first and thethird terms cancel out and the resulting equation describes the growth of doubled
frequency amplitude as a function of fundamental field and distance.
𝑖𝜕𝐴(𝑧) = 2𝜋𝑘𝜒(2) E𝑖*2 𝑒𝑖(2𝑘0 −𝑘)𝑧𝜕𝑧

(3.35)

Integrating from 𝑧 = 0 to 𝑧 = 𝑙 we come to the result𝑖𝐴(𝑙) = 2𝜋𝑘𝜒(2)2 E𝑖*2 𝑙𝑒𝑖(2𝑘0 −𝑘)𝑙2

sinc((2k0 − k)l)2(3.36)

2k0 − kl)2

(3.37)

Thus, the intensity of the output green beam is𝐼(𝑙) = |𝐴(𝑧)2 | = 4𝜋 2 𝑘 2 𝜒(2)2 𝑙2 𝐼02 sinc2 (In the experiment, the quantity we deal with is power, rather than intensity. So,
putting all the constant factors together and introducing the beam waist 𝑤0 , whichstays constant for a plane wave, we can rewrite the previous equation in terms of
power:
2
2k0 − k
2 𝑃0
l)
𝑃 (𝑙) = 𝐶𝑙 2 sinc2 (𝑤0
2(3.38)

An important condition to maximize doubled frequency power is phase matching
2𝑘0 = 𝑘. This equation basically expresses the conservation of momentum. However,in practice, it is not so easy to meet this condition. In addition to careful alignment,
one needs to keep in mind that the wave vectors for green and IR beams depend on
refractive indexes for frequencies 𝜔 and 2𝜔
2𝑘0 − 𝑘 = 2𝑛(𝜔)𝜔2𝜔− 𝑛(2𝜔)=0𝑐𝑐

45(3.39)

One needs to carefully adjust temperatures and employ some tricks based on crystal
birefringence or periodically polling to meet the condition 𝑛(𝜔) = 𝑛(2𝜔). For optimalphase-matching,
𝑃 (𝑙) = 𝐶

𝑃02 2
𝑙
𝑤02

(3.40)

This result does not take into account many additional effects. First of all, it contradicts conservation of energy, because the depletion of the fundamental beam is notincluded in the model. More accurate analysis including the decrease of the IR poweron the way through the crystal would give [43]2𝑃
𝑃 (𝑙) = 𝐶˜ 02 tanh2 (𝛼𝑙)
𝑤0

3.5.3

(3.41)

Phase-matching

In the previous chapter, we saw how important it is to meet the phase-matchingcondition to reach a high efficiency of SHG. From formula 3.38, one can see thatmaximum power is achieved when 𝑘 = 2𝑘0 , i.e. when 𝑛(𝜔) = 𝑛(2𝜔). Generated green
light must travel at the same phase velocity as the fundamental IR. This condition
is impossible to meet in materials with normal dispersion, where the refractive index
usually increases with the frequency. One of the possible ways to get around thisproblem is to use the birefringence of the crystal. If the polarization of the incident
fundamental beam is along one axis of the crystal, and the polarization of the secondharmonic beam is aligned in orthogonal direction, then the refractive indices for thenormal and anomalous beam could be matched at some temperature or some angle.
However, sometimes it is hard to find materials with sufficient adjustability forsome wavelengths. Another more universal technique is quasi-phase-matching or periodic poling. Quasi-phase-matching relies on artificial structure in the crystal [44]and can be implemented in materials even without birefringence.
The physical idea behind phase-matching can be explained in the following way.
While traveling through an elongated crystal, the fundamental wave excites secondary
radiation sources to emit at double frequency. The amplitudes of the secondary
46

a

b

Figure 3-5: a. Phase-vectors add up constructively. b. Periodically-polled crystal;
when phase-vectors add up to the maximum amplitude the structural insertion in the
crystal adds a 𝜋 phase shift and the spiral continues to grow, instead of rolling down.

radiation add up with the phase factors corresponding to the relative wave pass.
In order to have a strong SHG output of the crystal these amplitudes should add inconstructively. When we calculate the total amplitude from all the spatially separated∫︀𝑙
SHG sources, we need to calculate an integral 𝑒𝑖Δ𝑘𝑥 𝑑𝑥. This type of integral arises in0optics very often. We can utilize the diagram technique of Fresnel optics to visualize
the summation over the secondary emitters with different phases.
In the case of ∆𝑘 = 0 all the sources interfere constructively and all the "phasevectors" add in a straight line giving the maximum total amplitude Fig.3-5.a. If we
don’t meet phase-matching condition, then there will be an angle ∆𝑘∆𝑧 between"phase-vectors" of two neighboring emitters and the resultant interference amplitude
will not be maximal.
For nonzero ∆𝑘, one can find a length realizing the 𝜋 phase shift and equal to𝜋
for which the resultant amplitude vector is the longest. If we can break into𝑙𝜋 = Δ𝑘the crystal structure and bring extra 𝜋 shift at the length 𝑙𝜋 , then the spiral will keep
building up (Fig. 3-5.b) instead of falling off to the 2𝜋 phase and zero total amplitude.
Thus, if we can can insert this extra 𝜋 phase every 𝑙𝜋 , the resultant amplitude willbe less than in perfect constructive interference case, but still pretty big and will be
proportional to the number of 𝑙𝜋 lengths that we can fit into crystal.
47

3.5.4Gaussian beam analysis

The plane wave model is very attractive in its simplicity. However, in practice weusually deal not with plane waves, but Gaussian beams. In their 1968 paper, Boydand Kleinman [45] studied Gaussian beams passing through nonlinear crystal media.
Their research took into account birefringent "walk-off" of the beam, which occurs
when the direction of propagation of the beam does not coincide with the axis of
the crystal. Our case is much simpler, because in periodically-poled crystals (PPLN,PPSLT) the phase-matching does not rely on the birefringence phenomenon. It canbe considered as a special case of the Boyd-Kleinman problem when the birefringent"walk-off" is set to zero.
The plane wave model showed that the second-harmonic intensity scales quadratically with the intensity of the fundamental beam. Thus, if a beam of fixed power isfocused tighter, the efficiency of SHG is expected to be higher. However, we knowthat the tighter we confine the beam, the faster it will diverge. So, even though we are
winning in the minimum waist region, we may be loosing more around the Rayleighrange of the beam.
Boyd and Kleinman examined loose vs tight focusing and realized that there isan optimal geometry of them beam which gives maximum efficiency of the SHG.
For the simplest case of a phase-matched Gaussian beam focused at the center ofthe crystal and propagating without losses and birefringent "walk-off", the secondharmonic power is given by a formula:𝑃 =16𝜋 2 𝑑2𝑒𝑓 𝑓 2
𝑃 𝑙ℎ(𝜉, 𝐵)
𝜖𝑐𝜆3 𝑛1 𝑛3 1

(3.42)

Variable 𝜉, defined as 2𝑧𝑙𝑅 , characterizes the extent of the focusing. Fig.3-6 adopted
from the paper [45] shows the family of curves for different values of the "walk-off"parameter 𝐵. The upper curve corresponds to the case of no "walk-off" (𝐵 = 0), in
which we are particularly interested in. One can see that the curve ℎ(𝜉, 0) has a zeromaximum at the value 𝜉𝑚 = 2.84. Thus the optimal efficiency of SHG is achieved
when the Rayleigh range is 5.68 times shorter than the length of the crystal. For a48

given power of the IR beam 𝑃1 , the maximum of green light power is
𝑃𝑜𝑝𝑡 = 1.068

16𝜋 2 𝑑2𝑒𝑓 𝑓 2
𝑃 𝑙
𝜖𝑐𝜆3 𝑛3 𝑛1 1

(3.43)

Figure 3-6: A family of SHG power dependences of parameter 𝜉 for different values of
the walk-off parameter 𝐵. The case we are interested is depicted by the upper curve(𝐵 = 0). The figure is adopted from [45].
One needs to point out that the maximum green power for a given input IR power
scales linearly with the length of the crystal, unlike the result obtained for the planewave. To confirm the correctness of these results we can consider the limit 𝜉 ≪ 1of the formula (15). This limit will gives us output green power for the case of veryloose focusing, i.e. a plane wave. From reference [45] ℎ(𝜉, 0) → 𝜉 which leads to the
output power
𝑃 =8𝜋𝑑2𝑒𝑓 𝑓
𝑃 2 𝑙2 ,𝜖𝑐𝜆2 𝑛3 𝑛21 𝑤02 1(3.44)

which agrees with the plane wave model.

3.5.5

Operation of SHG

In the previous section, we concluded that neither a plane wave nor very tight focusing
give us maximum SHG efficiency, and there is an optimal focusing 𝜉 = 2.84 which49

maximizes the power output. At the same time we should keep in mind that we wantto keep the intensity of light below the damage threshold of the crystal. The typical𝑊
value of the damage threshold for SLT crystals 1 𝑀(ref. HC-photonics). Therefore,
𝑐𝑚2

the minimum spot size one could afford, using 50 W beam, is 𝑤0* = 40𝜇𝑚. This
beam waist corresponds to the Rayleigh range 𝑧𝑅 = 4.7𝑚𝑚. If the crystal length is𝑙 = 30𝑚𝑚 then the focusing parameter 𝜉 ≈ 3.2 > 𝜉𝑜𝑝𝑡 = 2.84. Therefore, it appears
that we can focus the beam even a little tighter than the optimal focusing and we
still would not reach the damage threshold of SLT crystal. This estimation is very
promising and means that we should not we be excessively careful with operating
crystal at high power. The optimal Rayleigh range and beam waist for a 30 mm long
crystal are 𝑧𝑅* ≈ 5.3𝑚𝑚 and 𝑤0* ≈ 42𝜇𝑚
The parameters of the optical setup were chosen to match the optimal minimum
waist and divergence of the beam for SHG. The use of the variable beam expander
allowed us to to carefully tune the beam waist and optimize the power of SHG. The
optimal beam parameter appeared to be in a very good match with the theoreticalprediction.
532 nm Output Power, W

8

6

4

2

5

10

15

20

25

30

35

1064 nm Input Power, W

Figure 3-7: Green power generated as a function of the input IR power. As a twophoton process, second-harmonic generation scales quadratically at low powers. At
high powers the output was not stable and the points characterize momentary maximum values.

50

3.5.6

Buyer’s catalog

During the upgrade of our SHG setup I had to reach many sellers and collect information from companies which manufacture or sell nonlinear optics. Our old green
light setup had been using a 30 mm long PPMgSLT crystal (MgO: Stoichometric
LiTaO3) along with an oven and temperature controller from Taiwanese company
HC-photonics. Unfortunately, this company no longer sells this kind of crystals.
When I contacted them regarding backup crystals, they offered PPLN crystals, whichhave higher efficiency than PPSLT, but lower photorefractive damage threshold. We
would not be able to obtain more than 3 W of green out of 20 mm crystal on a single
pass (according to HC-photonics). Another acknowledged company in our community
is Covesion (from the UK). However, their crystals have a low damage threshold aswell. Covesion offered 40 mm long PPLN crystals which are supposed to provide 2
W of green with 10 W of IR input. Their customers claimed they could obtain up
to 5 W of green light without observable signs of photorefractive damage, but thatpower would still not be sufficient for our applications. Even though Covesion crystals would not work well for our experiment, I decided to buy an oven (PV40) andtemperature controller (COV461) from this company. The temperature resolution of
the oven+controller was 0.01∘ , which was a step forward from the old BEC2 oven
(0.1∘ resolution). One cannot overstate the importance of stabilizing the temperature along the crystal. This is highly important for maintaining the phase-matchingcondition and keeping the efficiency of SHG high. After a search, we realized thecompany Oxide from Japan sells exactly what our lab needed: PPMgSLT crystals(part Q-STD-1064C-S01) with high damage threshold. The unit price for them isaround $2000, which is a lot cheaper than typical purchases from HC-photonics andCovesion. We successfully operated their crystals in BEC2 for several years and werequite satisfied with their performance. Other companies which were explored andmay be found useful in the future include Raicol crystals from Israel, Deltronic
from California, Alphalas from Germany.
At the end of the day, after having a lot of experience with operating nonlinear
51

crystals, I must say that the choice of the oven I made was not optimal. The Covesion’soven would do the job well for the normal operating regime of their crystals (2 W ofgreen out of 10 W of IR). We worked in the different range of parameters ( 10W of
green out of 30 W of IR). At such high input power of IR, a phenomenon known asGreen Induced Infrared Absorption (GRIIRA) [46] becomes important. Basically, a
lot of heat is deposited from the strong IR beam and the crystal’s temperature grows
without assistance from the oven. The temperature of the crystal and the output
green power never stabilizes. It can grow as high as 12 W but then after several
seconds it can drop down to 2-3 W. The temperature of the crystal was determinedby the thermal equilibrium of the deposited heat from the IR beam and through theoven cover and input/output holes. The active stabilization practically did not work,
since the oven could only heat and could not cool the crystal. Thus, it was moreconvenient to operate the crystal in the more stable medium power range of (8-10 W
of green output). Ideally, I should have considered buying an oven which would also
have a cooling option through Peltier thermoelectric effect, such ovens are available
on the market.

3.6

Experimental Results

The spin-orbit coupling experiment was done with a sodium BEC in |𝐹 = 1, 𝑚𝐹 = −1⟩
state. The BEC was adiabatically loaded within 250 ms from an optical dipole
trap to the lowest band of the superlattice, which is pseudo-spin down of our ef𝑁
∑︀
(↓)
(↑)
√1 |↓𝑛 ⟩. The pseudo-spin-up state |𝜓fective two-level system |𝜓𝑞=0 ⟩ =𝑞=𝜋/𝑑 ⟩ =
𝑁
𝑛=1

𝑁
∑︀

𝑛=1

√1 (−1)𝑛 |↑𝑛 ⟩, at the edge of the Brillouin zone, has an alternating sign pattern. In
𝑁time-of-flight (TOF), where we look at the state in momentum space, the interference
(↓)

(↑)

patterns for |𝜓𝑞=0 ⟩ and |𝜓𝑞=𝜋/𝑑 ⟩ will be different Fig.(3-8).
After atoms are loaded in the lowest band, we can quickly change the phase of thesuperlattice, creating an offset ∆ and lifting the atoms from the ground band to thefirst excited band. The quasimomentum of the atoms will remain the same (𝑞 = 0),
so they will be at highest point of the the first excited band and then will relax to the
52

(↓)

(↑)

Figure 3-8: patterns for |𝜓𝑞=0 ⟩ (top) and |𝜓𝑞=𝜋/𝑑 ⟩ (bottom). The state at the edge of
the Brillouin zone 𝑞 = 𝜋/𝑑 is characterized by alternating sign pattern.
edge of the Brillouin zone. The process of relaxation will be reflected in TOF images
3-9.

Figure 3-9: The top pattern shows a TOF image for equal populations of atoms
prepared in the ground and first excited state with the same quasimomentum 𝑞 = 0.
|↓⟩ and |↑⟩ are deliberately vertically displaced by an analog of the Stern-Gerlach
effect. After the |↑⟩ state relaxes to 𝑞 = 𝜋/𝑑, the periodicity of the wavefunction
doubles, and therefore the momentum peaks in the inverse space two times as frequent.
The relaxation time is about 2 ms.
When atoms are prepared in both ground and excited band at the same quasimomentum 𝑞 = 0 the periodicity of the wave function is going to be 532nm, i.e. thelength of the unit cell of the superlattice. After atoms in the excited band relax to𝑞 = 𝜋/𝑑 the periodicity of the wavefunction changes to 1064nm. In the TOF image,
the number of momentum peaks will double.
The adiabatically ramped Raman coupling 𝑉𝑅𝑎𝑚𝑎𝑛 = Ω cos(𝑘𝑥 𝑥 + 𝑘𝑧 𝑧 − 𝛿𝑡) makes
53

e)

b)

f)

c)

g)

d)

h)

On Res.

Off Res.

OD (a.u.)

a)

-4

-2

0

2

4

-4

-2

0

2

4

i)
x
z

-4

-2

0

2

4

Figure 3-10: TOF interference for atoms loaded in (a) lower wells and (b) upper
wells. Schematic plots for the expected SOC signal for atoms prepared in (c) pseudospin-down state and in (g) pseudo-spin-up state. Observed signal for both initiallyprepared states resonant (c, g) and off-resonant (d, h) Raman two-photon detuning.
Interference pattern for atoms equally loaded in both pseudo-spin states with Raman
interaction tuned between two resonances.

the wavefunction acquire satellites according to equations 3.21 and 3.22. Becauseof the momentum kick in the direction orthogonal to the superlattice, in TOF themomentum satellites will be displaced along 𝑥ˆ.
↓(↑)

We can describe the wave function as |↓, 𝑞, 𝑘𝑥 ⟩ = |𝜓𝑞⟩⊗𝑒𝑖𝑘𝑥 𝑥 — a tensor productof a Bloch wave (characterized by the quasimomentum 𝑞 and quasi-spin) and theplane wave in the transverse direction 𝑥ˆ. In the experiment we studied three regimes:(1) atoms loaded in the lower well |↓, 0, 0⟩, where the Raman potential creates spinorbit side peak |↑, 𝜋/𝑑, 𝑘𝑥 ⟩ along with other off-resonant terms; (2) atoms preparedin |↑, 𝜋/𝑑, 0⟩ with the corresponding spin-orbit coupling satellite is |↓, 0, −𝑘𝑥 ⟩; (3)equal population of atoms loaded in the lower and upper wells, wher both types of54

spin-orbit peaks are observed simultaneously. All three scenarios are depicted in theFig.3-10.
The momentum components created by 2-photon Raman processes are displacedalong 𝑥ˆ by an amount of the IR photon recoil ℎ̄𝑘𝐼𝑅 . The off-resonant peaks are
symmetric for +𝑥 and −𝑥. The resonant spin-orbit coupling peaks are one sided: weare either tuned to the resonance in forward direction from the lower to the upper
well, 𝛿 = ∆ + 𝐸𝑟 , or to the backward direction from upper to lower well 𝛿 = ∆ − 𝐸𝑟 .
The two resonances are supposed to separated by 2𝐸𝑟 = 15.3 kHz. One can see thatthe forward and backward resonances are indeed displaced with respect to each other:Fig.3-11 shows a plot of the normalized population imbalance, where the fractionsof atoms with 𝑘𝑥 = +𝑘𝐼𝑅 and 𝑘𝑥 = −𝑘𝐼𝑅 are mapped as a function of the Raman
detuning 𝛿. The discrepancy from the 2𝐸𝑟 values can be epxlained by the mean fieldRes.
Frequency (kHz)Normalized Population Imbalanceshift ∼ 2𝜇 ≈ 5 kHz.

Raman Detuning

VIR
Er

(kHz)

Figure 3-11: Mapped out spin-orbit coupling resonances as a function of Raman
detuning. The population inbalance is plotted for forward (blue) and backward (red)spin flips. The inset is the resonance the center frequency of the resonances as afunction of the IR lattice depth 𝑉𝐼𝑅
The offset ∆ can be measured by observing a beat note between the atoms in|↑⟩ and |↓⟩. The condensate was loaded with equal populations into the symmetricdouble well and then the potential was rapidly changed to the asymmetric one. De55

pending on the hold time in that potential, atoms in the first excited band acquired atime-dependent phase which can be observed through oscillations of the interference
pattern in TOF. This beat note measurement must be completed before atoms in the
excited band relax in quasimomentum to the edge of Brilluan zone. The offset used
for the final data taking was 23 kHz, so the period of oscillation was 44 𝜇𝑠. The
relaxation from 𝑞 = 0 to 𝑞 = 𝜋/𝑑 happens within 2 ms.

Figure 3-12: Calibration of the offset ∆ between lower and upper wells. On a shorttime scale (relative to 2ms of relaxation in the excited band) the atoms in the upper
well have a phase accumulated linearly with time, 𝑒−𝑖Δ𝑡 . From the period of theoscillation of the interference pattern ∆ ≈ 23 kHz was deduced.
The presence of satellites for the wavefunctions localized in the left or right wells
not only provides possibility for interference between them and future Raman coupling, but also gives a source of heating through collisions. The lifetime observed forthe |↑⟩ state and for the state with 50/50 population in |↑⟩ and |↓⟩ was about 200ms
for both cases. However, the collision rate for the 50/50 state should have been suppressed by a factor of 4(𝐽/∆)2 . The similarity in the lifetimes indicates that the main
heating mechanism comes not from the collisions between spin-up and spin-down, butfrom rather some technical noise.
56

3.7

Stripe phase detection

As explained in the previous paragraph, the adiabatically ramped Raman potential
dresses the atoms in the left and right wells with with momentum satellites according
to (3.21) and (3.22).
The states |Ψ1 ⟩ and |Ψ2 ⟩ produce interference anddensity modulations.

The state |↓, 0, 0⟩ in |Ψ1 ⟩ inter-

feres with ∼ 𝑒𝑖(𝛿−Δ)𝑡 |↓, 0, −𝑘𝑥 ⟩, resulting in a stationary density modulation at 𝛿 = ∆ along 𝑥ˆ (orthogonal
to the superlattice direction), and this density modulation is the sought-for stripe phase (Fig.3-13).

|↓, 0, 0⟩also iterferes with the on-site terms 𝑀1 𝑒−𝑖𝛿𝑡 |↓, 𝜋/𝑑, 𝑘𝑥 ⟩and 𝑀1′ 𝑒𝑖𝛿𝑡 |↓, −𝜋/𝑑, −𝑘𝑥 ⟩ giving rise to moving density modulations.

In the geometry of our experiment𝑘𝑥 = 𝜋/𝑑, so these stripes are formed at 45 degrees
to the lattice and move with a phase velocity of 𝑣 =
√ 𝛿
.
2𝜋/𝑑

Similarly, the term 𝑒−𝑖Δ𝑡 |↑, 𝜋/𝑑, 0⟩ from |Ψ2 ⟩ in-terferes with 𝐾1 𝑒−𝑖𝛿𝑡 |↑, 𝜋/𝑑.𝑘𝑥 ⟩, 𝑀2 𝑒𝑖(𝛿−Δ)𝑡 ↑, 0, −𝑘𝑥 and𝑀2′ 𝑒−𝑖(𝛿+Δ)𝑡 |↑, 0, 𝑘𝑥 ⟩. As a result of this interference, there
is one stationary density modulation (stripe phase) at Figure 3-13: A stationary𝛿 = ∆ and two moving modulations.
density modulation andThus, the stationary stripe phase occurs at the same the probe meeting thedetuning 𝛿 = ∆ for both spin-up and spin-down in direc- Bragg condition.
tion perpendicular to the superlattice with the periodicity
inverse to the momentum transfer 2𝜋/𝑘𝑥 , and can be detected by Bragg spectroscopy.
The experimental conditions for stripe phase detection followed the parameters
mentioned in the previous chapter. A Bragg probe beam to detect the stripes wasadded to the existing setup. The Bragg detection had to be done with sodium resonant
light 𝜆𝐵𝑟𝑎𝑔𝑔 = 589𝑛𝑚 at an incident angle of 𝜃 = 16∘ in order to meet the Bragg
condition 𝜆𝐵𝑟𝑎𝑔𝑔 = 4𝑑 sin 𝜃.
57

Figure 3-14: Anlge-resolved Bragg signal. The density modulation gives a sharp
feature in the center on the top of diffuse Rayleigh scattering. Without spin-orbit
coupling there is no density modulation and only the Rayleigh scattering is observed.
The Fig.3-14 shows the angular distribution of Rayleigh-scattered resonant light.
Due to the stripe phase formation, the scattering is enhanced in the direction of
the Bragg condition. A sharp feature in the angular distribution of the scattered
light with zero detuning is a direct proof of the existence of the stationary density
modulation.
The FWHM of the observed peak is 9 ± 1 mrad, which is consistent with thediffraction limit 𝜆𝐵𝑟𝑎𝑔𝑔 /𝐷, where 𝐷 is the size of the cloud. For the same experimental
parameters we observed superfluid peaks of the BEC in TOF images, which suggeststhat the state of matter we have been working with is superfluid. Thus, we canconclude the that the observed phase is a superfluid with a long-range order. This
state of matter is referred to as a supersolid.
Since the stripe phase is not the only density modulation formed in our system,
we have to make sure that other periodic structures do not contribute to the detected
Bragg signal. The other density modulations emerging from the onsite coupling areoriented at 45∘ to the superlattice, parallel to 𝑥ˆ + 𝑧ˆ direction, whereas the stationary
stripes form with the wave vector along 𝑥. The Bragg condition cannot be fulfilled
for them simultaneously, and therefore the detected signal is background-free.
The detected Bragg signal can be quantified. The contrast of the stripes is estimated to be 𝜂 = 2𝛽/𝐸𝑟 , which is about 8% for 𝛽 = 300 Hz. Denoting the Raleighscattering isotropic rate per atom per solid angle as 𝛾, the enhanced Bragg signal is58

2going to be 𝛾(𝜂𝑁𝐵𝐸𝐶
/4), where 1/4 comes from Debye-Waller factor for a sinusoidal

modulation. The detected signal confirmed the anticipated scaling with the BEC
2number ∼ 𝑁𝐵𝐸𝐶.

One can also look at the Bragg signal as "gain" of the Raleigh intensity. We varied
the BEC fraction 𝑓 = 𝑁𝐵𝐸𝐶 /𝑁𝑡𝑜𝑡𝑎𝑙 and confirmed that the Bragg intensity scales as
𝑁𝑡𝑜𝑡𝑎𝑙 (𝑓 𝛽/𝐸𝑟 )2 , which also proves that the Bragg enhancement comes only from thesuperfluid component.
We also mapped the detuning dependence of Bragg and Raleigh signals. By
sweeping the detuning of the Raman beams we saw the effect of resonant coupling
between spin-up and spin down — the broad resonance separated by 2𝐸𝑟 and the
sharp Bragg resonance from the stationary stripes at 𝛿0 = 0.
The periodicity of the stripes depends not only on external parameters such as
the momentum transfer from Raman beams but also on the interactions between the√︀atoms. The length of the period is modified to be 2𝑑/ 1 − (𝛽/𝐹 )2 , where 𝛽 is thespin gap and 𝐹 = (2𝐸𝑟 + 𝑛(𝑔 + 𝑔↑↓ ))/4 reflects the effect of interactions. However, in
our schemes this correction is only 0.4% and can be neglected.

59

60

Chapter 4
Introduction to periodically driven
systems
In this chapter I’ll review the basic theory describing the behavior of periodicallydriven systems, both classical and quantum. Dynamics of classical systems, moving
in rapidly oscillating fields, splits into slow and fast motion. Fast motion (or micromotion) averages into effective potential which in turn affects the slow variables of thesystem. These effects do not have a profound connection with discrete time translational symmetry, so Floquet treatment, applicable to quantum systems, is not necessary for them. However, they provide a lucid example of separation of time scales, ofwhat micromotion is, and how important its impact can be. In quantum mechanics,
the time-periodicity of Hamiltonian imposes constraints on the wave function. The
knolwedge about the symmetry of Hamiltonian allows to reduce the time-dependentproblem of evolution of quantum system to a time independent one. The quantum
picture is more complicated than the classical, however, the main idea is very similar.
The system’s evolution splits into two types of dynamics: fast micromotion and slow
time-averaged evolution. The method of controlling the slow dynamics of the system
by time-dependent fields is called Floquet engineering. The variety of the phenomena
which can be realized by this technique is really impressive ref.. The goal of thischapter is to show a powerful application of the Floquet engineering and endow a
low energy radio-frequency photon with peculiar paradoxical properties: enhance its
61momentum by orders of magnitude.

4.1

Classical systems

There are two paradigmatic examples of classical systems where micromotion contributes to the slow dynamics of the systems in a nontrivial way: Kapitsa pendulum
and Paul trap. The former is well described in [47] along with the classical theory of
motion in rapidly oscillating fields.
Let’s consider a particle moving in an external potential 𝑈 and driven by a rapidly
oscillating force

𝑓 (𝑡) = 𝑓1 cos 𝜔𝑡 + 𝑓2 sin 𝜔𝑡(4.1)

The force has to be much faster than particle’s motion in the potential 𝑈 , i.e.
√︀∫︀ 𝑑𝑥
2𝜋/𝜔 ≪ 𝑚/2 𝐸−𝑈. We will imply that the displacement 𝜒 caused by the force(𝑥)

𝑓 is only a small perturbation to the particle’s trajectory 𝑋 in the time-independent
potential 𝑈 . The equation of motion of the particle is𝑑
𝑈 + 𝑓 (𝑡)
𝑑𝑥

(4.2)

𝑥(𝑡) = 𝑋(𝑡) + 𝜒(𝑡)

(4.3)

𝑚¨
𝑥=−
We will seek the solution in the form

assuming that the fast periodic force causes some jittering 𝜒(𝑡) on the top of slow motion 𝑋(𝑡) given by the potential U. The average of 𝜒(𝑡) over drive period 2𝜋/𝜔 is zeroand 𝑋(𝑡) during the same time barely changes, so 𝑥¯(𝑡) = 𝑋(𝑡), and 𝑋(𝑡) describes
slow smooth motion averaged over fast oscillations. After substituting equation 4.3
into 4.2 and grouping separately fast oscillating terms and smooth terms we arrive to𝑚𝜒¨ = 𝑓 (𝑋, 𝑡)
62

(4.4)

Since 𝑓 is a harmonic function the equation is easily integrated, and the instantaneous value of the fast oscillatory variable is determined by the value of the force1𝜒(𝑡) = − 𝑚𝜔
2 𝑓 (𝑡).

The equation of motion for the smooth variable 𝑋 averaged over drive period is
¨ =−
𝑚𝑋

𝑑
𝑈𝑒𝑓 𝑓 ,
𝑑𝑋

(4.5)

1
¯2 = 𝑈 + 𝑚 𝜒˙ 2 . One can see that when averaged over thewhere 𝑈𝑒𝑓 𝑓 = 𝑈 + 2𝑚𝜔
2𝑓
2

fast oscillations the particle moves in a potential which is a sum of the initial timeindependent potential and the kinetic energy associated with rapid oscillations - themicromotion energy.

4.1.1

Kapitsa pendulum

One of the mechanical systems every physicist has known since the school is a simple
pendulum: a body suspended from a support, moving under the force of gravity. If
the pendulum is reclined from its equilibrium position at 𝜑 = 0 to some small angle 𝜑and then released, it starts doing small harmonic oscillations around the equilibrium
position. If some friction is added to the system, then eventually the pendulum willstop at its equilibrium point 𝜑 = 0 - this is why we say that this equilibrium isstable. Another equilibrium position 𝜙 = 𝜋 is unstable. When deflected from it, thependulum first shows large-amplitude unharmonic oscillations, which then die out,
and the pendulum eventually finds its stable equilibrium at 𝜙 = 0. All of that is
trivial and very intuitive. What is not trivial is that the unstable equilibrium 𝜙 = 𝜋can change its character and become stable — after we make the support jitter! By
driving the system fast enough we can actually make it more stable. Isn’t it amazing?
Following one of the examples in [47] the Lagrangian of the pendulum with verticallyoscillating support point with position 𝑎 cos 𝛾𝑡 can be written as

𝐿=

𝑚𝑙2 2
𝜙˙ + 𝑚𝑔𝑙 cos 𝜙 + 𝑚𝑙𝑎𝛾 2 cos 𝛾𝑡 cos 𝜙
2

(4.6)

The generalized oscillating force is 𝑓 = −𝑚𝑙𝑎𝛾 2 cos 𝛾𝑡 sin 𝜙. Then the the effective
63

potential that includes micromotion energy is going to be)︀(︀
𝑎2 𝛾 2cos2 𝜙𝑈𝑒𝑓 𝑓 = 𝑚𝑔𝑙 − cos 𝜙 +
4𝑔𝑙

(4.7)

Under the condition that 𝑎2 𝛾 2 > 2𝑔𝑙 the upper unstable equilibrium point 𝜙 = 𝜋changes its character and becomes stable. Physically, when the body is deflected fromthe 𝜙 = 𝜋 the reduction in potential energy in the gravity field is smaller than theacquired kinetic energy due to micromotion. It is interesting to point out that if the
support point jitters horizontally the character of the equilibrium does not change —it remains unstable. This happens because the upper point 𝜙 = 𝜋 is also the point of
maximum micromotion and deviation from it reduces both potential energy of gravity
and the kinetic energy of micromotion.
Kapitsa pendulum is a good transparent example of how counterintuitive thebehavior of periodically driven system could be. By employing the micromotion andtaking it under control we can engineer the systems which do not naturally happen
in the real life.

4.1.2

Paul trap

Another paradigmatic example of the importance of micromotion is Paul trap [48],a powerful technique of traping charged particles. From Maxwell’s equations in the⃗ = 0,
absence of charge we know that the divergence of the field must be zero 𝑑𝑖𝑣 𝐸
which implies that the static electric field cannot confine ions in all the directions.
If the created potential is trapping along 𝑥 and 𝑦, then it must be untrapping in
the remaining direction 𝑧. It seems that trapping of charged particles is facing avery fundamental obstacle. However, if the electric field oscillates fast enough the
emerging micromotion will help to trap ions and produce confining potential in all
the directions.
It is enough to see the role of micromotion in one dimension. The Hamiltonian ofa particle in a quadratic potential which oscillates at the frequency 𝛾 is
64

𝐻=

1
𝑝2
+ 𝑚𝜔02 𝑥2 sin 𝛾𝑡
2𝑚 2

(4.8)

Using the same approach as for the Kapitsa pendulum we decompose particle’s
motion into a sum of smooth motion 𝑋(𝑡) and fast oscillations around it 𝜒(𝑡). The
average of 𝜒 over the drive period is zero: ⟨𝜒(𝑡)⟩2𝜋/𝛾 = 0𝑥(𝑡) = 𝑋(𝑡) + 𝜒(𝑡)(4.9)

The equation of motion splits into two for the fast and the slow variables:𝑑2 𝜒= −𝜔02 𝑋 sin 𝛾𝑡𝑑𝑡2(4.10)

𝑑2 𝑋= −𝜔02 ⟨𝜒 sin 𝛾𝑡⟩2𝜋/𝛾
2𝑑𝑡(4.11)

After integrating of the first equation and plugging the result for 𝜒 into the second
one we arrive at
𝜔02
𝑑2 𝑋
2
=
−𝜔
·
𝑋
0
𝑑𝑡2
2𝛾 2

(4.12)

From this equation one can conclude that the slow dynamics of the particle follows(︀ 𝜔2 )︀2 2
0
the motion in a harmonic potential 𝑈𝑒𝑓 𝑓 = 21 𝑚 √2𝛾
𝑋

The idea of oscillating ion trap also found an application in transverse strong

focusing (also called alternating-gradient focusing) of the beams of charged particles
in accelerator physics. To prevent transverse divergence of the beam in an accelerator
engineers use, so called, FODO systems. They alternate the focusing and defocusing
lenses (quadrupoles) on the beam path. From the particles’ point of view, theysee how the attractive potential flips its sign and becomes repulsive: changes fromfocusing to defocusing, then to focusing again and so on. Eventually, it results ineffective overall confining transverse potential for the moving particles [49]
65

4.2

Quantum systems

In this chapter we investigate quantum systems that have time-periodic Hamiltonians

𝐻(𝑡) = 𝐻(𝑡 + 𝑇 )(4.13)The discrete time translational symmetry allows us to apply Floquet formalism
to the problem of finding the evolution of the system. The Floquet theory was firstdeveloped in as a mathematical theory. In the second half of the 20th century it wasrevisited in the context of solving time-dependent Schrodinger equation [50].
In the literature there are two ways how the main statement of the Floquet theory is formulated. First, since the Hamiltonian posseses discrete time translational
symmetry its eigen states can be written as|𝜓𝑛 (𝑡)⟩ = 𝑒−𝑖𝜖𝑛 𝑡 |𝑢𝑛 (𝑡)⟩ ,(4.14)

where 𝑢𝑛 (𝑡) is called Floquet mode, and it’s periodic in time 𝑢𝑛 (𝑡) = 𝑢𝑛 (𝑡 + 𝑇 ), and𝜖𝑛 is called quasienergy. After the period time T passes the wave-function acquires aphase factor 𝑒−𝑖𝜖𝑛 𝑇 , just like for a Bloch state in spatially periodic potential: the wave
function acquires a phase factor, when the particle travels a period of distance. This
formulation of the Floquet theory is almost trivial, it simply reflects the symmetry of
the Hamiltonian and it is a direct analog of the Bloch theorem in solids.
Another formulation of the Floquet theorem is that the evolution of the system
under time-periodic Hamiltonian can be decomposed into initial kick, evolution undersome effective time-independent Hamiltonian and final kick^^^𝑈 (𝑡𝑓 , 𝑡𝑖 ) = 𝑒−𝑖𝐾(𝑡𝑓 ) 𝑒−𝑖𝐻𝑒𝑓 𝑓 (𝑡𝑓 −𝑡𝑖 ) 𝑒𝑖𝐾(𝑡𝑖 )(4.15)

ˆ is periodic in time 𝐾(𝑡)ˆ
ˆ + 𝑇 ) andThe kick (micromotion) operator 𝐾= 𝐾(𝑡describes the initial kick and the subsequent micromotion.
This formulation of the Floquet theorem is less trivial than the first one. Hereˆ 𝑒𝑓 𝑓 and theit is guaranteed the existence of such time-independent Hamiltonian 𝐻
66

kick operator 𝐾(𝑡). The proof of this theory along with the perturbative methods ofˆ 𝑒𝑓 𝑓 and 𝐾
ˆ and comprehensive review of the Floquet formalism can be
determining 𝐻
found in the references [51, 52].
In this chapter I’m going to show how these two formulations are interconnected.
While the Floquet state (the quasienergy and the Floquet mode) is defined as asolution of time-dependent Schrodinger equation, it may be more convenient to refer
to an eigen value problem in order to have another view on the definition of Floquet
states. One can easily see that the Floquet state is the eigen state of the evolution
operator over a period of the drive 𝑇 :𝑈ˆ (𝑡0 + 𝑇, 𝑡0 ) |𝜓𝑛 (𝑡0 )⟩ = 𝑒−𝑖𝜖𝑛 𝑇 |𝜓𝑛 (𝑡0 )⟩(4.16)The starting time 𝑡0 is a parameter which defines the gauge: the evolution operator
for one period and the Floquet state are interconnected. However, the eigenvalue
𝑒−𝑖𝜖𝑛 𝑇 does not depend on choice of the starting time 𝑡0 . So, the quasienergy spectrum
can be determined from diagonalizing the evolution operator 𝑈ˆ (𝑡0 +𝑇, 𝑡0 ) for arbitrary
𝑡0 . Also one can see that even though the eigenvalue 𝑒−𝑖𝜖𝑛 𝑇 itself is well-defined, thequasienergy 𝜖𝑛 is only known by modulo 𝜔 = 2𝜋/𝑇 . Quasienergies separated by adriving frequency correspond to the same eigenvalue of the evolution operator.
In the basis of Floquet states 4.15 the evolution operator can be written as𝑈ˆ (𝑡2 , 𝑡1 ) =∑︁𝑛

𝑒−𝑖𝜖𝑛 (𝑡2 −𝑡1 )/ℎ̄ |𝑢𝑛 (𝑡2 )⟩ ⟨𝑢𝑛 (𝑡1 )|(4.17)The Floquet states form a complete basis states, and generally the time evolutionof a state is given by|𝜓(𝑡)⟩ =∑︁𝑛𝑐𝑛 𝑒−𝑖𝜖𝑛 (𝑡−𝑡0 ) |𝑢𝑛 (𝑡)⟩ .

(4.18)The time-independent coefficients 𝑐𝑛 can be deduced from 𝑐𝑛 = ⟨𝑢𝑛 (𝑡0 )| 𝜓(𝑡0 )⟩.
To add a little more intuition into this description, one can say that if the system
is prepared in a single Floquet mode, i.e. |𝜓(𝑡)⟩ = 𝑒−𝑖𝜖𝑛 𝑡 |𝑢𝑛 (𝑡)⟩, it’s time evolution

is going to be periodic disregarding the obvious phase factor 𝑒−𝑖𝜖𝑛 𝑡 Evolution of the67

system prepared in a superposition of several Floquet states is no longer periodic,
but it can be formulated in terms of two constituents. First is the periodic timedependence of the Floquet mode 𝑢𝑛 (𝑡) is associated with micromotion. On top ofthat the dynamics of the system is governed by quasienergies, as though we would
have an evolution of an eigenstate of a time-independent Hamiltonian. So, the second
constituent is the deviation from periodic evolution described by dephasing factors
𝑒−𝑖𝜖𝑛 𝑡 .

4.3

Effective Hamiltonian

Now let’s approach this problem from a little different direction. In the reference^ˆ is self-adjoint operator and P̂ is[53] it is proved that 𝑈ˆ (𝑡) = P̂(𝑡)𝑒−𝑖𝑡𝐺/ℎ̄ , where 𝐺

unitary operator periodic in time with the same periodicity as the Hamiltonian.
^Let’s find a unitary gauge transformation 𝑒𝑖𝐹 (𝑡) where 𝐹ˆ (𝑡) is a Hermitian oper-

ator periodic in time, such that in the new gauge the Hamiltonian becomes timeindependant. Applying the gauge transformation to the time-dependent Schrodinger
equation:
𝑖ℎ̄

𝜕 ^
𝜕 𝑖𝐹^
^ ˆ
(𝑒 𝜓) = 𝑒𝑖𝐹 𝐻(𝑡)+ 𝑖ℎ̄( 𝑒𝑖𝐹 )𝜓
𝜕𝑡
𝜕𝑡(4.19)^In the new gauge the wave function is 𝜑 = 𝑒𝑖𝐹 𝜓. With the new Hamiltonianˆ 𝑒𝑓 𝑓 = 𝑒𝑖𝐹^ 𝐻𝑒
ˆ −𝑖𝐹^ + 𝑖ℎ̄( 𝜕 𝑒𝑖𝐹^ )𝑒−𝑖𝐹^𝐻𝜕𝑡(4.20)

ˆ 𝑒𝑓 𝑓 time-independent,
Let’s assume that such transformation 𝐹ˆ , that makes 𝐻
ˆ 𝑒𝑓 𝑓 are also time-independent. We will call themexists, then the eigenstates of 𝐻
|𝑣𝑛 (𝑥)⟩ and the corresponding eigenvalues 𝜖𝑛 . They evolve according to |𝜑(𝑡, 𝑥)⟩ =𝑒−𝑖𝜖𝑛 𝑡 𝑣(𝑥). In the original gauge the corresponding states are going to be^^|𝜓𝑛 (𝑥, 𝑡)⟩ = 𝑒−𝑖𝐹 |𝜑𝑛 (𝑥, 𝑡)⟩ = 𝑒−𝑖𝜖𝑛 𝑡/ℎ̄ 𝑒−𝑖𝐹 |𝑣𝑛 (𝑥)⟩⏟⏞(4.21)|𝑢(𝑥,𝑡)⟩^Here’s the state 𝑒−𝑖𝐹 |𝑣𝑛 (𝑥)⟩ is the Floquet mode |𝑢𝑛 (𝑥, 𝑡)⟩ introduced earlier in68the text. The evolution operator after transformed from the new gauge to the originalis^^^𝑈ˆ (𝑡2 , 𝑡1 ) = 𝑒−𝑖𝐹 (𝑡2 ) 𝑒−𝑖𝐻𝑒𝑓 𝑓 𝑡/ℎ̄ 𝑒𝑖𝐹 (𝑡1 ) .

(4.22)The review paper [51] along with the references [54, 53, 55] discuss how to find^the gauge transformation 𝑒𝑖𝐹 and corresponding effective Hamiltonian and its uniqueness. The reference [52] derives it from the stroboscopic evolution and strobosocopic
Hamiltonian. It is important to emphasize that approach of effective Hamiltonian ismore generic since it does not depend on the phase of the drive or initial time. Thereexist a class of stroboscopic Hamiltonians connected through the transformation ??,but there is only one effective Hamiltonian related to this class.

4.4

Extended Hilbert space

Another prospective on Floquet formalism which may provide more intuition on periodically driven system is based on idea that we can treat time just like other independent variables of the Hamiltonian. The time-dependent Schrodinger equation
ˆ − 𝑖ℎ𝜕𝑡 ) |𝑢𝑛 (𝑡)⟩ = 𝜖 |𝑢𝑛 (𝑡)⟩(𝐻(𝑡)(4.23)

can be viewd as an eigenvalue problem in extended Hilbert space ℱ = ℋ ⊗ ℒ𝑇

ˆ = 𝐻(𝑡)ˆThe hermitian operator 𝑄− 𝑖ℎ𝜕𝑡 is called quasienergy operator. Itseigenstate are the Floquet modes and eigenvalues are the quasienergies
ˆ |𝑢𝑛𝑚 ⟩ = 𝜖𝑛𝑚 |𝑢𝑛𝑚 ⟩𝑄(4.24)

Diagonaliazation of the quasienergy operator is another way of approaching the
problem of finding the evolution of periodically driven system.
The description in terms of quasienergies and Floquet modes is crucial for studing
the excitation of the systems or collisional properties. However, the direct methods of deducing the effective Hamiltonian and the micromotion operators can be
more helpful for the purpose of Floquet engineering: the simulation of some effective
69

Hamiltonian with system’s slow evolution. In the case of our scheme the effective
Hamiltonian and the micromotion operators can be obtained by employing several
methods. These methods, both perturbative and exact are discussed in the next
chapter.

70

Chapter 5
Floquet Engineering of Recoil-dressed
Photons
5.1

Magnetic shaking experiment

Our initial motivation was to demonstrate a laser-free spin-orbit coupling scheme.
The scheme which would inherently avoid the problem of heating due to spontaneous emission. However, while working on this project we realized that the ideas
we were developing are more general than just the context of spin-orbit coupling.
The scheme we experimentally realized provides a transparent and insightful view onrapidly developing field of Floquet engineering and its application of creating artificial
gauge-fields [24] for quantum gases.
A photon is an object which carries energy (frequency) and momentum. How do
we know that? An atom, playing a role of a frequency filter, can absorb a photon atcertain frequency and experience a recoil kick to conserve momentum brought withthe photon. The frequency of the absorbed photon must be detuned from the levelspacing frequency of the atom by the amount of Doppler shift and recoil shift.

𝜔 = 𝜔0 + 𝑘𝑣 +

ℎ̄𝑘 2
2𝑚

(5.1)

This condition is a direct consequence of the energy conservation during atom71

photon interaction. For a given photon the Doppler shift and the recoil energy arestrictly determined by the conservation laws. For example, if we take RF (MHz)photon, it’s intrinsic momentum is tiny and the associated Doppler and recoil shifts
are negligible. Is there a loophole which we can use to tune the recoil and enforce
Doppler shift even on RF photon? It turns out, yes. We should sacrifice with theconservation of energy. But how can we get rid of one of the most universal ubiquitousconventions which follows us through out our entire journey in physics? In fact, in a
fairly easy way. One can say that conservation of energy comes from the invarianceof time translation [47], from uniformity of time. If the invariance of translation oftime is broken, the energy is no longer conserved, it is not an integral of the systemanymore. It may sound trivial: if the Hamiltonian explicitly depends on time, then theenergy is changing. But even for a Hamiltonian which periodically depends on time
(when the energy averaged over period does not change, but the time translational
symmetry is still reduced) the energy is not an integral of motion.

5.2

Our scheme

Our scheme has two ingredients: an oscillating magnetic force and the sequence of the
RF pulses. The time sequence is shown on Fig.5-1. Atoms are driven by a sinusoidal
spin-dependent force 𝑓 (𝑡) = 𝑔𝐹 𝜇𝐵 𝐵0′ sin(2𝜋/𝑇 · 𝑡 + 𝜑𝑅𝐹 )ˆ
𝜎𝑧 . The RF pulses are fired
at times 0, 𝑇, 2𝑇.... The relative phase between the oscillating force and the RF is
defined by 𝜑𝑅𝐹 . Each of the RF pulses couples spin-up and spin-down with the same
instantaneous velocity. Both of the ingredients do not provide a momentum kick:the integral of the magnetic force of the period is zero, so there is no net momentum
transfer, and momentum associated with the RF photon is negligible in comparison
to any relevant atomic scale. However, the beauty of the Floquet engineering is thatwhen these two ingredients are put together and properly synchronized, the result
is going to be a an RF photon endowed with tunable momentum. Our experimentdemonstrates not only the power of Floquet engineering, but also a lucid semiclassicalinterpretation of the physics of a periodically driven atom.
72

(a)Magnetic
gradienttime(b)Magnetic
gradient

RF pulses(ɸRF=0)RF pulses(ɸRF=!/2)

Spindependent
velocity

Spindependent
velocity‹ ›‹ ›≠‹ ›‹ ›

Force fora selectedtrajectory

Force fora selectedtrajectory=

Figure 5-1: Illustration of our scheme for creating a tunable atomic recoil momentum with RF transitions using magnetic forces. (a) & (b) shows the experimental
conditions for 𝜑𝑅𝐹 = 0 and 𝜑𝑅𝐹 = 𝜋/2, respectively. The spin-dependent forcesand velocities are shown (as thick solid lines) for the amplitude of the wavefunction
which is transferred from spin down (red) to up (blue) by the RF pulse marked by
the gray dashed line. For 𝜑𝑅𝐹 = 0, the average velocities ⟨𝑣↓ ⟩ and ⟨𝑣↑ ⟩ are different,
which implies a finite recoil associated with the spin-flip. In contrast, ⟨𝑣↓ ⟩ = ⟨𝑣↑ ⟩ for
𝜑𝑅𝐹 = 𝜋/2 and there is no recoil.

Each of the RF pulses couples spin-up and spin-down with the same instantaneous
velocity, so atoms while being spin-flipped do not change their velocity. However, they
make a "jump" from one velocity trajectory to another, and the average velocity forthese trajectory can be different. It is like passengers in metro: on the station, when
they change the trains their velocities along the track are not changeing. The spin flip
happens when a passenger walks out of the doors of one train, goes across the platform
and sits into the train which will move in the opposite direction. The instantaneous
velocities of the trains were the same, otherwise the passenger would not be able
get from one train to another. But after this connection the second train will havedifferent average velocity than the first one.
The velocity trajectories for two cases (𝜑𝑅𝐹 = 0 and 𝜑𝑅𝐹 = 𝜋/2) are shown on theFig. 5-1. The moment 𝑡 = 0 happens at the first RF pulse, and the synchronization
phase 𝜑𝑅𝐹 determines how we are going to drive atoms with the oscillating force.
73

One can easily see that 𝜑𝑅𝐹 determines the difference between the average velocities
for spin-up and spin-down. For 𝜑𝑅𝐹 the difference ⟨𝑣↑ ⟩ − ⟨𝑣↓ ⟩ is maximal, whereas
for 𝜑𝑅𝐹 = 𝜋/2 it is zero. So, if we imagine a spin-flip, for the case on the Fig.51.(a) from the atoms point of view they have experienced two positive half-periodsof a spin-dependent force back to back. For the case depcited on the Fig.5-1.(b) the
spin-dependent force is discontinuous, but still does not provide extra kick. So, we
can conclude that depending on the phase of the drive the spin-dependent force may
provide a momentum transfer in terms of the average values.
So, even if we have a single 𝜋-pulse, the atoms will experience a recoil and willbe transferred in a state with different average velocity. However, our goal is not
only to simulate recoil, but also a Doppler shift to exactly replicate the propertiesof atom-photon interaction in time-independent systems. With a single RF pulse
the frequency of RF is not any different from the case when the shaking is off. To
observe the effects we intend to simulate having only one RF pulse is not enough. The"magic" of Floquet engineering will work only if the Hamiltonian is truly periodic intime, and a single-pulse example would break the discrete time translation symmetry.
Only when the RF-pulses happen every period one can apply the Floquet formalismand describe the dynamics of the system with the effective Hamiltonian.

Figure 5-2: Phase accumulation between the spins due to magnetic shaking. On the
figure the average velocity for spin up is larger than the average velocity for spindown. Difference in average kinetic energies during a period of shaking results inphase accumulation between the spins.
A period of magnetic shaking results in relative phase accumulation between spinup and spin-down. The spin-up and spin down atoms which start from the same
velocity after a period of shaking return back to the same velocity since the integral of the force is zero over the period. However, the spin-up and spin-down
74

had different histories of their motion during that period of time. If the spin-up
was first accelerated and then decelerated back to its original velocity, the spindown was first decelerated and then accelerated. Therefore, the average kinetic energy for spin-up was higher than for spin-down. Quantum mechanically it meansthat a relative phase being accumulated during a period of motion. This phasecan be calculated semiclassically as the integral of the difference of kinetic energies)︀∫︀ 𝑇 (︀𝛼 = 1/ℎ̄ 0 𝐸↑𝑘𝑖𝑛 (𝑡) − 𝐸↓𝑘𝑖𝑛 (𝑡) 𝑑𝑡. Here we choose integration limits as 0 and 𝑇 , as-suming that the timing of the RF pulses sets the zero of time, and the synchronizationof the shaking force with respect to the RF is defined through the phase 𝜑𝑅𝐹 . The accumulated relative phase between spin-up and down depends on the synchronization
phase as 𝛿𝛼 = ℎ̄1 (𝑚(⟨𝑣↑ ⟩ − ⟨𝑣↓ ⟩)𝑣𝑅𝐹 )𝑇 = 𝑘0 cos 𝜑𝑅𝐹 𝑣𝑅𝐹 𝑇 , where 𝑘0 = (𝑔𝐹 𝜇𝐵 /𝜋ℎ̄)𝐵0′ 𝑇
and 𝑣𝑅𝐹 is the common velocity for spin-up and spin-down at 𝑡 = 0, 𝑇, 2𝑇.... It’s
already here in the semiclassical treatment one can see that if we enforce RF detuning to be equal to the rate of phase accumulation 𝛿 = 𝛿𝛼/𝑇 then the expression for
it will look like Doppler shift 𝛿 = 𝑘𝑣𝑅𝐹 . If the common velocity 𝑣𝑅𝐹 is rewritten as𝑣𝑅𝐹 = ⟨𝑣↓ ⟩+ℎ̄𝑘/2𝑚 then we recover the Doppler relation we have seen in atom-photon
interaction
𝛿 = 𝑘⟨𝑣↓ ⟩ + ℎ̄𝑘 2 /2𝑚(5.2)

The RF detuning compensates the Doppler shift and recoil shifts in the time-averaged
picture. It is important to emphasize the role of the synchronization between the shaking and the RF pulses parametrized as 𝜑𝑅𝐹 . The momentum transfer 𝑘 = 𝑘0 cos 𝜑𝑅𝐹
is determined by the strength of the drive and by timing of the RF pulses. For a given
magnitude of the drive, the momentum transfer is maximized when the RF pulses
are fired at the zeros of the force (𝜑𝑅𝐹 = 0) and it is equal to zero when RF is fired
at the points of maximum force (𝜑𝑅𝐹 = 𝜋/2).

5.2.1

Bloch sphere representaion

The semiclassical approach can be visualized on the Bloch sphere. A period of magnetic shaking does not change the spin populations and results only in the relative75

phase accumulation √12 (|↓⟩ + |↑⟩) → √12 (|↓⟩ + 𝑒𝑖𝛿𝛼 |↑⟩), i.e. 𝜎𝑧 −rotation in the frame
rotating at the frequency of the spin-splitting 𝜔0 (RF-resonant frequency). The RF
pulses coupling the spins rotate the Bloch vector with 𝜎
ˆ𝑦 around the instantaneous𝑦ˆ(𝑡) axis. In the current frame the 𝑦ˆ-axis rotates with the rate of RF detuning 𝛿around 𝑧ˆ−axis. If the rate of precession of 𝑦ˆ−axis delta compensate the phase accu(a)(b)ɸRF= 0››ɸRF= !/2›

Off-resonant›

On-resonant

Figure 5-3: Bloch sphere representation of magnetic shaking and RF pulses. (a) &(b) Trajectories on the Bloch sphere for several periods of magnetic shaking (green
solid lines representing 𝛼) and RF pulses for 𝜑𝑅𝐹 = 0 and 𝜑𝑅𝐹 = 𝜋/2, respectively.
Fig. (a) shows the trajectories for atoms with a finite initial velocity when the RF
frequency is at 𝜔0 , the atomic resonance, and when it is detuned by the Doppler andrecoil shift. In (b), the RF frequency is at 𝜔0 , the trajectory is independent of theatomic velocity, and there is no net rotation around the z-axis during a magnetic
shaking cycle. The red (blue) dot represents the initial (final) spin state.
mulation during the magnetic shaking, then the subsequent RF pulses will move theBloch vector along the meridians, transferring population to the south pole in themost "efficient’ way. Since the phase accumulation due during shaking depends onthe velocity of the atoms, the spin flip is going to be efficient for some velocity groupof atoms (when RF pulses add up constructively), and for some velocities the Blochvector will come back to the north pole. This is the way we impose Doppler selectivity
on RF photons. For a given RF detuning, the RF signal will match Doppler condition
for some velocity group and these atoms will experience resonant Rabi oscillations.
When we fire RF pulses at the maxima of the periodic force, then the momentum
transfer is 0. It means that in the frame rotating with 𝜔0 the shaking does not move76

the Bloch vector. Then the RF pulses will add up constructively when the RF isresonant with the spin-splitting for all the velocity groups. The Doppler selectivity
is completely lost in this case.

5.2.2

Floquet treatment

According to Floquet theory, the evolution of a system with time-periodic Hamiltonian can be factorized as^^^𝑈ˆ (𝑡𝑓 , 𝑡𝑖 ) = 𝑒−𝑖𝐾(𝑡𝑓 ) 𝑒−𝑖𝐻𝑒𝑓 𝑓 (𝑡𝑓 −𝑡𝑖 ) 𝑒𝑖𝐾(𝑡𝑖 ) ,(5.3)

where 𝐾(𝑡) is a micromotion operator, which describes the initial and final kick dueˆ 𝑒𝑓 𝑓 is time-independent effective Hamiltonian, which
to turn on and turn off, and 𝐻
governs slow time evolution of the system averaged over fast micromotion with period
𝑇 . A powerful formalism of Floquet theory provides us with methods to determine
the 𝐻𝑒𝑓 𝑓 and the micromotion operator 𝐾, and solve the evolution of the system.
The time-dependent Hamiltonian for our scheme in the frame rotating with the
RF drive after the rotating-wave approximation is2∑︁1
ˆ = 𝑝ˆ𝑧 + ℎ̄𝑘0 𝑧ˆ 𝜋 sin(2𝜋𝑡/𝑇 + 𝜑𝑅𝐹 )ˆ
𝐻
𝛿(𝑡 − 𝑛𝑇 ), (5.4)
𝜎𝑧 − ℎ̄𝛿𝑅𝐹 𝜎
ˆ𝑧 + ℎ̄Ωˆ
𝜎𝑥 𝑇
2𝑚
𝑇
2
𝑛

where 𝛿𝑅𝐹 is the RF detuning with respect to the atomic resonant frequency. The
short RF pulses are represented as a series of delta-functions with effective Rabi
frequency Ω.
The choice of RF pulses to be short and approximated with the delta-functions
is consistent with our experimental settings (a typical RF pulse duration is 4 𝜇𝑠,
whereas the period 𝑇 is usually set to be 100 or 200 𝜇𝑠) and the reason of this choice
will be discussed later.
We now rederive the same effective Hamiltonian and micromotion operator using
a rigorous high-frequency 1/𝜔 expansion. Appendix K of the reference [51] discusses
77

Hamiltonians of a general formˆ
ˆ 0 + 𝐴𝑓ˆ (𝑡) + 𝜔 𝐵𝑔(𝑡)
ˆ
𝐻(𝑡)
=𝐻

(5.5)

ˆ 𝑒𝑓 𝑓 and the kick operator 𝐾.
ˆand derives expansions for an effective Hamiltonian 𝐻ˆ 𝑒𝑓 𝑓 =
𝐻

∞
∑︁
1

𝜔
𝑛=0

ˆ (𝑛) , 𝐾(𝑡)ˆ𝐻=𝑛 𝑒𝑓 𝑓

∞
∑︁
1

𝜔𝑛
𝑛=0

ˆ (𝑛) (𝑡)𝐾

(5.6)

ˆ in equation 5.5 is of this form withThe Hamiltonian 𝐻2
ˆ 0 = 𝑝 − 1 𝛿𝑅𝐹 + Ωˆ
𝜎𝑥 ,
𝐻
2𝑚 2

𝑔(𝑡) = sin(𝜔𝑡 + 𝜑𝑅𝐹 ),ˆ = 1 𝑘0 𝑧ˆ𝜎𝐵ˆ𝑧 ,𝐴ˆ = Ω𝜎𝑥 ,2∑︁𝛿(𝑡 − 𝑛𝑇 ) − 1
𝑓 (𝑡) = 𝑇

(5.7)
(5.8)

𝑛

Functions 𝑓 (𝑡) and 𝑔(𝑡) meet the requirement of having zero mean value over a
period 𝑇 .
The kick operator is in 0-th order:

ˆ (0)

𝐾

ˆ
= 𝐵𝐺(𝑡),𝐺(𝑡) = 𝜔∫︁ 𝑡𝑔(𝜏 )𝑑𝜏 = − cos(𝜔𝑡 + 𝜑𝑅𝐹 )(5.9)

The effective Hamiltonian to the lowest order in 1/𝜔 can be expanded asˆ 𝑒𝑓 𝑓 = 𝐻
ˆ0 +
𝐻

∑︁ 𝑖𝑛
𝑛=1

𝐺𝑛 𝑓 [𝐵...[𝐵, 𝐴]] +
⏟ ⏞
𝑛!
n

∑︁ 𝑖𝑛

𝑛!
𝑛=1

𝐺𝑛 [𝐵...[𝐵, 𝐻0 ]] + 𝑂(1/𝜔)⏟ ⏞(5.10)nAfter calculating all commutators and time-averaged coefficients before them, andgrouping the terms proportional to 𝜎
ˆ𝑥 and 𝜎ˆ𝑦 , the expansion reduces to

𝑝ˆ2𝑧
1
1 𝑘02
ˆ
𝐻𝑒𝑓 𝑓 =− 𝛿𝑅𝐹 𝜎ˆ𝑧 + Ω cos(𝑘0 𝑧 cos 𝜑𝑅𝐹 )ˆ
𝜎𝑥 + Ω sin(𝑘0 𝑧 cos 𝜑𝑅𝐹 )ˆ
𝜎𝑦 ++ 𝑂(1/𝜔)2𝑚 2
16 𝑚(5.11)
It can be rewritten in the matrix form as78

ˆ 𝑒𝑓 𝑓 =
𝐻

⎛

2 2
𝑝^2𝑧
1 ℎ̄ 𝑘0+− ℎ̄𝛿2𝑅𝐹⎝ 2𝑚 16 𝑚ℎ̄Ω𝑒−𝑖𝑘0 cos 𝜑𝑅𝐹 ·𝑧

ℎ̄Ω𝑒
𝑝^2𝑧

2 2

1 ℎ̄ 𝑘0+ 16+ ℎ̄𝛿2𝑅𝐹2𝑚𝑚

𝑖𝑘0 cos 𝜑𝑅𝐹 ·𝑧

The corresponding micromotion operator is

⎞
⎠

2𝜋
ˆ
𝐾(𝑡)
= −𝑖𝑘0 𝑧ˆ
𝜎𝑧 cos( 𝑡 + 𝜑𝑅𝐹 ).
𝑇(5.12)(5.13)

This effective Hamiltonian describes a two-level atom with energy spacing 𝜔0driven by a photon field at frequency 𝜔𝑅𝐹 with a wave-vector 𝑘 cos 𝜑𝑅𝐹 . The common
term ℎ̄2 𝑘02 /2𝑚 is associated with the kinetic energy of micromotion. The form of theeffective Hamiltonian is a solid proof that our scheme simulates an RF photon withtunable recoil.
The effective Hamiltonian is one step away from the Spin-orbit Coupling Hamiltonian. If we apply an spatially-dependent unitary transformation

ˆ 𝑧1 = exp (−𝑖𝑧𝑘0 cos 𝜑𝑅𝐹 𝜎𝑅ˆ𝑧 /2)(5.14)The transformed Hamiltonian becomes translationally invariant and acquires thestandard form of 𝐻𝑆𝑂𝐶 for one-dimensional spin-orbit coupling:1
𝛿𝑅𝐹ˆ 𝑆𝑂𝐶 = 1 (ˆ𝐻𝑝𝑧 − 𝑘0 cos 𝜑𝑅𝐹 𝜎ˆ𝑧 )2 + Ωˆ
𝜎𝑥 −
𝜎
ˆ𝑧 ,
2𝑚
2
2

(5.15)

where the spin-orbit coupling strength is described by the momentum shift 𝑘0 cos 𝜑𝑅𝐹 /2.
With the new kick operator
[︂ (︂
)︂
]︂
2𝜋
ˆ 𝑆𝑂𝐶 (𝑡) = −𝑘0 𝑧ˆ
𝑡 + 𝜑𝑅𝐹 − cos 𝜑𝑅𝐹𝐾
𝜎𝑧 cos𝑇(5.16)

the time evolution can be written as^^^

𝑈 (𝑡2 , 𝑡1 ) = 𝑒−𝑖𝐾𝑆𝑂𝐶 (𝑡2 ) 𝑒−𝑖𝐻𝑆𝑂𝐶 (𝑡2 −𝑡1 ) 𝑒𝑖𝐾𝑆𝑂𝐶 (𝑡1 ) .

(5.17)

ˆ 𝑆𝑂𝐶 (𝑡) has a non-zero temporal average, so it cannot
In that case the operator 𝐾
79

ˆ 𝑆𝑂𝐶 can not be considered
be treated as a pure micromotion operator. Similarly 𝐻
as an effective Hamiltonian for the time-periodic Hamiltonian (5.4). It is rather a
Hamiltonian related to the true effective Hamiltonian by the unitary transformation:† ˆˆ 𝑆𝑂𝐶 = 𝑅
ˆ 𝑧1
ˆ 𝑧1 . Note that reference [56] has obtained 𝐻
ˆ 𝑆𝑂𝐶 as a stroboscopic𝐻
𝐻𝑒𝑓 𝑓 𝑅

Floquet Hamiltonian.

5.2.3

Evolution of the system

In the previous section we solved the evolution of the system by summing infinite
number of terms in the expansion given us by Floquet theory. The reference [51]proves that the result of this summation is the effective Hamiltonian which is uniquewith respect to time transformations. The perturbative methods are typical for thetreatment of periodically driven systems. However, our example is very special because it is exactly solvable. We can find the evolution of the system and deduce
the effective Hamiltonian from it by regrouping time-depend terms in the evolution
operator.
We start with the same Hamiltonian describing our system 5.5. The reason why
we can’t integrate it to find the evolution operator is that it has both momentum
and position-dependent terms, so the Hamiltonian does not commute with itself atˆˆ ′ )] ̸= 0. However, it is not hard to find a position-dependent
different times [𝐻(𝑡),𝐻(𝑡˜ =𝑅ˆ 𝑧 (𝑡) such that in the new frame |𝜓(𝑡)⟩ˆ † (𝑡) |𝜓(𝑡)⟩ the transformed
spin rotation 𝑅𝑧ˆ˜Hamiltonian 𝐻(𝑡)will still be time-dependent, but without non-commuting terms:ˆ˜ˆ˜ ′ )] = 0.
[𝐻(𝑡),
𝐻(𝑡Let’s applying a position-dependent spin rotationˆ 𝑧 (𝑡) = exp [−𝑖𝑘0 𝑧𝛾 (𝑡) 𝜎𝑅ˆ𝑧 /2](5.18)

where the rotation angle 𝛾 is

𝛾 (𝑡) = 𝜔

∫︁ 𝑡
0

sin

(︂

2𝜋 ′
𝑡 + 𝜑𝑅𝐹𝑇)︂

′𝑑𝑡 − 𝐶 = − cos

80

(︂

2𝜋
𝑡 + 𝜑𝑅𝐹𝑇)︂

(5.19)

The integration constant 𝐶 entering 𝛾 (𝑡) has been taken to be 𝐶 = cos 𝜑𝑅𝐹 , sothat 𝛾 (𝑡) averages to zero over a period. The reason for this choice will be discussed
later on.
ˆ 𝑧 (𝑛𝑇 ) = exp [𝑖𝑘0 𝑧 cos 𝜑𝑅𝐹 𝜎
At the RF pulses where 𝑡 = 𝑛𝑇 the transformation 𝑅
ˆ𝑧 /2]
describes a spin rotation by an angle 𝑘0 𝑧 cos 𝜑𝑅𝐹 around the 𝑧 axis. As a result, theˆ˜ (𝑡) = 𝑅
ˆ (𝑡) takes the form
ˆ𝑅
ˆ (𝑡) − 𝑖𝑅ˆ † (𝑡) 𝜕 𝑅
ˆ † (𝑡) 𝐻
transformed Hamiltonian 𝐻𝑧𝑧ˆ˜ (𝑡) = 1
𝐻
2𝑚

𝑧

𝑡

𝑧

(︂
)︂2
1
1
𝑝ˆ𝑧 − 𝑘0 𝛾 (𝑡) 𝜎ˆ𝑧 − 𝛿𝑅𝐹 𝜎ˆ𝑧 +22Ω [cos (𝑘0 𝑧 cos 𝜑𝑅𝐹 ) 𝜎
ˆ𝑥 + sin (𝑘0 𝑧 cos 𝜑𝑅𝐹 ) 𝜎
ˆ𝑦 ] 𝑇∑︁𝑛𝛿 (𝑡 − 𝑛𝑇 ) (5.20)

Note that unlike the spin-dependent potential gradient featured in the original
Hamiltonian (5.5), the oscillating momentum shift term 𝑘0 𝛾 (𝑡) 𝜎ˆ𝑧 /2 is no longer proportional to the driving frequency and hence can be considered as a small perturbationin the limit of high frequency driving where 𝑘0 𝛾 (𝑡) ≪ 𝜔 and also Ω ≪ 𝜔. In that case
it is appropriate to describe the evolution of the system in terms of the zero-orderˆ˜ (𝑡) over a single driving period,
effective Hamiltonian obtained by time averaging of 𝐻ˆ˜ (𝑡), giving
i.e. by the zero frequency component of the Hamiltonian 𝐻2
1 𝑘02
ˆ 𝑒𝑓 𝑓 = 𝑝ˆ𝑧 − 1 𝛿𝑅𝐹 𝜎𝐻
ˆ𝑧 + Ω cos (𝑘0 𝑧 cos 𝜑𝑅𝐹 ) 𝜎
ˆ𝑥 + Ω sin (𝑘0 𝑧 cos 𝜑𝑅𝐹 ) 𝜎ˆ𝑦 +, (5.21)2𝑚 2
16 𝑚

where the momentum shift has averaged to zero. The obtained effective Hamiltonianˆ 𝑒𝑓 𝑓 is in exact agreement with 5.12.
𝐻
The full dynamics includes also the micromotion. In the present situation there
are two origins of the micromotion. The first kind comes from the time-dependenceˆ˜ (𝑡). However, in the limit of the large drivingof the transformed Hamiltonian 𝐻
frequency this kind of micromotion is negligibly small compared to the second typeof micromotion emerging due to the time-dependence of the unitary transformation˜
ˆ 𝑧 (𝑡). In fact, returning to the original representation |𝜓(𝑡)⟩ = 𝑅
ˆ 𝑧 (𝑡) |𝜓(𝑡)⟩,𝑅onearrives at the following time-evolution of the state-vector from the initial to the final
81

time
|𝜓(𝑡2 )⟩ = 𝑈ˆ (𝑡2 , 𝑡1 ) |𝜓(𝑡1 )⟩ ,

ˆ 𝑧 (𝑡2 ) 𝑒−𝑖𝐻^ 𝑒𝑓 𝑓 (𝑡2 −𝑡1 ) 𝑅ˆ 𝑧† (𝑡1 ) , (5.22)
where 𝑈ˆ (𝑡2 , 𝑡1 ) = 𝑅

ˆ 𝑧 (𝑡) represents a micro-motion operator.
where the unitary transformation 𝑅
The time evolution operator can be rewritten as^^^𝑈ˆ (𝑡2 , 𝑡1 ) = 𝑒−𝑖𝐾(𝑡2 ) 𝑒−𝑖𝐻𝑒𝑓 𝑓 (𝑡2 −𝑡1 ) 𝑒𝑖𝐾(𝑡1 ) ,whereˆ (𝑡) = −𝑘0 𝑧ˆ
𝐾𝜎𝑧 cos(︂2𝜋
𝑡 + 𝜑𝑅𝐹𝑇)︂(5.23)(5.24)

is a Hermitian micromotion (kick) operator. The choice of the integration constant𝐶 = cos 𝜑𝑅𝐹 in the unitary transformation (5.19) ensures that the micromotion operˆ (𝑡) averages to zero over the driving period. Thus, the effective Hamiltonian
ator 𝐾and the micromotion operators are defined in a unique way through the condition
𝐶 = cos 𝜑𝑅𝐹 .

5.2.4

Canonical vs mechanical momenta

In the presence of the vector potential two important variables a distinguished: canonical momentum and mechanical momentum. Canonical momentum is conjugate to
the coordinate, mechanical momentum is quantity needed to evaluate kinetic energy
in the system. Our scheme provides a lucid interpretation of what the difference
between canonical and mechanical momenta is. The spin-orbit coupling Hamiltonian
ˆ 𝑆𝑂𝐶 = 1 (ˆ𝐻
𝑝 − 12 𝐴ˆ
𝜎𝑧 )2 +ℎ̄Ωˆ
𝜎𝑥 − ℎ̄𝛿2𝑅𝐹 𝜎
ˆ𝑧 . is an example of spin-dependent gauge field.
2𝑚 𝑧
Spin-dependent vector potential 12 𝐴ˆ
𝜎𝑧 shifts the mechanical momentum for spin-up

and spin-down from the canonical momentum.
In our scheme the canonical momentum is the instantaneous momentum at the
moment of RF pulse 𝑝𝑧 = 𝑚𝑣𝑅𝐹 , which is the same for spin-up and spin-down.
The mechanical momenta for spin-up and spin-down is the average momenta 𝑝𝑘𝑖𝑛↑(↓) =𝑚⟨𝑣↑(↓) ⟩. The mechanical momenta for spin-up and spin-down 𝑝𝑧 ± 21 𝐴 are shifted by82

the vector potential 𝐴 = 𝑘0 cos 𝜑𝑅𝐹 , i.e. momentum transfer which depends on therelative phase between the RF pulses and the magnitude of shaking force.
The difference between canonical and mechanical momenta can be seen on thedispersion relation (Fig.5-4). Using canonical momentum as the coordinate of horizontal axis, all couplings and transitions between the two spin states are vertical.
The dashed lines illustrate the transitions observed in our experiment. Away fromthe spin gap the energy separation is dominated by Doppler and recoil shifts.
Energy››z

Canonical
momentumz

Figure 5-4: Energy-momentum dispersion relations for spin-orbit coupled spin 1/2
states. The two minima are separated by the recoil momentum ℎ̄𝑘. The verticaldashed arrows show spinflip transitions. Their lengths are given by the Doppler andrecoil shifts.

5.3

Experimental realization

Experimentally what we wanted to do is to demonstrate that the RF transition can bemade velocity-selective. In a thermal cloud different velocity groups of atoms acquire
velocity-dependent phase due to magnetic shaking. This accumulated phase can beaddressed by the RF detuning 𝛿 = 𝑘𝑣. One can also view this effect in the following
way. The sequence of the RF pulses results in a Ramsey fringe in Fourier space. The
thermal cloud velocity distribution can be mapped out through the 𝛿 = 𝑘𝑣 relation.
We convolute the Ramsey fringe with the velocity distribution, the major peaks of83

the fringe cut a slice out of the thermal cloud (Fig.5-5). By changing the detuning of
the RF signal we move the Ramsey fringe across the velocity distribution, resonantly
exciting different velocity groups of atoms. In TOF, after the cloud is released fromthe trap, we should see how the slice of the atoms in the excited states moves asa function of the RF detuning. The stronger we can shake the atoms, the higher
momentum transfer 𝑘 is, which means the wider thermal cloud distribution looksin the frequency space. Thus, we can cut more slices with a given Ramsey fringe
from the thermal cloud velocity distribution, if we shake stronger. Alternative way of
improving resolution is to shake slower (since 𝑘0 ∼ 𝐵 ′ /𝜔), but then we soon hit theconstraint of limited coherence time. We do not want to have the sequence of the RF
pulses to be much longer than 1 ms, and if the shaking frequency 𝜔 is too small wewould not be able to fit many periods of shaking into this time duration.

1/T

Figure 5-5: Ramsey fringe of the sequence of the RF pulses - blue, thermal cloud
distribution mapped through the enforced Doppler relation 𝛿 = 𝑘𝑣 - yellow. Bychanging the detuning of the RF we make the Ramsey fringe move through thevelocity distribution.
We prepared a thermal cloud of approximately 105 23 Na atoms at the temperature
380 nK - close to the onset of condensation. If the atoms were prepared at higher
temperature it would, of course, allow us to cut more of the velocity slices out of
the cloud, but at the same time it would reduce the signal-to-noise ratio. So, for84the purpose of cleaner detection we preferred to prepare atoms right above the Bosecondensation.
Atoms were loaded into a shallow optical dipole trap with trapping frequencies(𝜔𝑥 , 𝜔𝑦 , 𝜔𝑧 ) = 2𝜋(98, 94, 25) Hz, corresponding to Gaussian radii of 19.5, 20 and 68
𝜇m, respectively. The trap was made shallow in the shaking direction 𝑧ˆ because it was
important to separate the time scales of the dynamics in the trap and the experimental
sequence. In the trap the momentum distribution is converted to spatial distribution
every quarter period. For us it was important to address only momentum informationand prevent any coordinate-dependent phase accumulation. So, the duration of thesequence of the Floquet engineering (1-2 ms) was deliberately made much shorter
than the trap period 40 ms.
As pseudospin-1/2 system we used |𝑚𝐹 = −1⟩ and |𝑚𝐹 = 0⟩ of the 𝐹 = 1 hyperfine manifold of the atoms. The state |↓⟩ = |𝑚𝐹 = 0⟩ is magnetically insensitive, itdoes not see any oscillating force. So, when atoms were prepared in |↓⟩ they were
not affected by any transient magnetic effect such as "pre-shaking", when it takes a
few periods for the ac-current generating circuit to stabilize. In the experiment we
did not have two states moving out of phase with respect to each other as it was
described in the theory part. Instead we had a |↓⟩ state standing still, and |↑⟩ state
shaken around it. So, the maximum momentum transfer ℎ̄𝑘0 was reduced by a factorof 2 compared to example discussed above. The third hyperfine state of the 𝐹 = 1
manifold |𝑚𝐹 = +1⟩ was decoupled from the two-level system via quadratic Zeeman
effect.
The oscillating magnetic force was created by a time-dependent 3D quadrupole(︀)︀
field 𝐵 ′ (𝑡) 𝑧 𝑧ˆ + 𝑦 𝑦ˆ − 2𝑥ˆ
𝑥 . The direction of the effective 1D force was determinedby the bias field 𝐵0 along 𝑧ˆ. Orthogonal to the bias field, the periodic potential isquadratic, which only creates a small modulation to the confinement:

𝑈 = 𝜇𝐵√︁(︀𝐵0 + 𝐵 ′ (𝑡)𝑧)︀2(︀)︀2 (︀
)︀2+ 𝐵 ′ (𝑡)𝑦 + 4𝐵 ′ (𝑡)𝑥 ≈(︀)︀𝐵 ′2 (𝑡) 2(𝑥 + 4𝑦 2 ) (5.25)
𝜇𝐵 𝐵0 + 𝐵 ′ (𝑡)𝑧 +2𝐵085

In the presented results the amplitude of the magnetic field was 48 G/cm at afrequency of 5 kHz, which led to the momentum transfer 𝑘0 = 0.07𝑘𝐿 , where ℎ̄𝑘𝐿 is
the recoil of the 𝜆 = 589 nm Na transition with the recoil velocity ℎ̄𝑘𝐿 /𝑚 = 2.9 cm/s.
We pulsed a sequence of 9 RF pulses at 8 MHz. The duration of the single pulse
was 4 𝜇s. The Rabi frequency of the RF signal is CW regime was 10 kHz. The
effective Rabi frequency with the duty cycle (4 𝜇 out of 200 𝜇s) was Ω = 200 Hz.
Each single 4𝜇s pulse resulted in approximately 𝜋/12 rotation on a Bloch sphere.
(a)Without shaking

With shaking (δRF= 0)

Shaking direction(b)

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

6

4

2

- 0.4

- 0.2

0

0.2

0.4

Position (mm)Figure 5-6: (a) images of the atoms in the excited state after being spin-flipped by
the sequence of the RF pulses. Without shaking the cloud is isotropic (a feature ofthermal cloud expansion). With shaking the cloud is narrowed, because the atoms
were chosen to be exited based on their velocities. The yellow line dashed ellipses
have their major and minor axes according to FWHM of a Gaussian fit. The dasheddotted line corresponds to a Gaussian fit with the effect of the induced currents taken
into account. (b). Integrated column densities for different detunings of RF pulses
fired at the positive slope of the magnetic force (𝜑𝑅𝐹 = 0). The assymetry between
the ±300 Hz profiles most likely comes from the field drifts estimated to be ∼ 70 Hz
or small residual magnetic field gradients.
The Doppler width of the thermal cloud due to the momentum transfer 𝑘 = 0.07𝑘𝐿
was 3 kHz (FWHM), whereas the Fourier width of the central feature in the Ramsey
86fringe was 625 Hz (given the total duration of the pulse sequence 1.6 ms). Thus,
according to our estimation we could resolve 5 velocity slices out of the thermal cloud
velocity distribution.
In the cold atoms experiments the standard way to probe momentum distributionof the atomic sample is long TOF expansion. In the limit of long expansion time
the spatial information is lost and only momentum distribution is revealed. However,since we worked with the thermal cloud we could not really afford using long TOF,
because of the quality of the images being deteriorated. The presented data weretaken at 𝜏 = 12 ms TOF (only twice the inverse of 𝜔𝑧 ).
For such relatively short TOF the size of the cloud is mainly determined by theoriginal spatial size. So, instead of seeing a sharp slice we could only see the partial
narrowing of the width of the spin-flipped atoms Fig.5-6.(a).
However, even for short TOF the displacement of the center of the spin-flipped
atoms is still exactly 𝑣𝜏 . So as long as we could do a reliable fit into the image of spin
flipped atoms and resolve 𝑣𝜏 displacement for different velocity groups Fig.5-6.(b) we
could still claim that we enforced Doppler sensitivity of RF.
The dependence of the RF detuning on the center-velocity of the slice is shown on
the Fig.5-7. The slope of the line gave the momentum transfer which was in a good
agreement with the predicted 𝑘 = 0.07𝑘𝐿 . The dependence of the momentum on thesynchronization of the RF pulses with the oscillating force was checked for 𝜑𝑅𝐹 , i.e.
when the RF pulses were fired at the negative slope of the magnetic force. Effectively
it flipped the sign of the momentum transfer 𝑘 = −0.07𝑘𝐿 and the corresponding
dependance of the RF detuning on the velocity of the atoms is on the Fig.5-7(b).
It was impossible to explore phases 𝜑𝑅𝐹 besides 0 and 𝜋 due to technical reasons
discussed further in the section.

5.3.1

High current

In order to dress RF photon with momentum comparable to the recoil from opticalphoton we had to generate relatively high currents at audio frequencies (5-10 kHz).
The benchmark was 100A at the frequency of 10 kHz. It was clear that if we can reach
87

(a)

ɸRF= 0

Doppler shift (Hz)
400

200

-0.2

0.2

Velocity (cm/s)
-200

-400

(b)

ɸRF= !
400
200

-0.2

0.2
-200
-400

Figure 5-7: Doppler shift mapped for different velocity groups. (a) RF detuning as afunction of the target resonant center-velocity of the atoms for the case of RF firedat positive slope 𝜑𝑅𝐹 = 0. (b) The same for negative slope 𝜑𝑅𝐹 = 𝜋. The solid lines
represent the predicted Doppler shifts based on the calibration of recoil momentum.
The dashed line takes into account the model of induced currents. The error bars apurely statistical and correspond to 1𝜎. The inferred field fluctuation from 1𝜎 is 70
Hzit, then we can simulate recoil on the order of 0.1𝑘𝐿 . The main challenge we faced
was that at audio frequencies the impedance of the coils turned out to be much higher
than for dc current, 1 Ohm in contrast to 0.06 Ohm. Such dramatic increase of the
impedance came from eddy currents induced in the stainless still vacuum chamber.
It turned out that we had to work in the regime which was quite different from thetypical settings of atomic physics experiment. We usually use high current low voltage
power supplies in the lab to create strong gradients and bias fields. To generate highcurrent at audio frequency we had use high-current high voltage source. We had toefficiently use output power, i.e. ideally to make power supply to be both current andvoltage limited. Also in order to produce ac current from dc power supply we had to88

build our own H-bridge, which can tolerate high power and work as a fast switch.

1

4

IGBT

IGBT
+100 V

0V

IGBT

IGBT
3

2

LcoilsC

Figure 5-8: Scheme of the coils circuit. Two DC power supplies connected in series
and placed in the "bar" of the H-bridge. IGBTs are triggered in pairs: 1 and 3, 2
and 4. Triggering of the pairs is out of phase. When pair 1-3 is on, pair 2 an 4 is off.
The resonant frequency of the cicruit is controlled by the capacitor 𝐶. The IGBTs
are triggered at twice the frequency of the cicruit thus realizing parametric pumping.
As power supplies we used model RST-5000-48 from Mean Well. Two power
supplies were connected in series to produce the voltage of 100A. For the H-bridge
we used four IGBT switches, which met both requirements: they switched fast andthey stand high currents. IGBTs and power supplies were protected by varistors. To
tune the resonance of the circuit we had to add the 𝜇𝐹 capacitors compatible withac current (model ..). It is a pleasure to acknowledge that during extensive use of
our homemade ac-current generator at the limit of its capacity it worked reliably andnever failed.

5.3.2

Stability of magnetic field and gradient

Coherent manipulation of hyperfine-states requires substantial stability of magnetic
field. Our first attempts to do the experiment convinced us that the control of magnetic fields in our apparatus was not sufficient. Initially we used audio amplifiers89

Figure 5-9: An example of a typical trace of Hall probe measuring the current generated by the H-bridge parametric triggering. Our benchamrk is 120 A at 5 kHz (onthe plot), 90 A at 10 kHz
model Behringer EP4000, which seemed to work in desirable for us regime. However,they appeared to be the major sources of the 60 Hz noise, which for the time scale ofour experiment cannot be viewed only as a dc shift of magnetic field (the duration of
the quantum simulation was 1-2 ms in comparison to 13 ms of the 60Hz noise period).
The instabilities in operation of the audio amplifiers were hard to fix due to the absence of documentation and control of the non-scientific equipment. Even though theaudio amplifiers proved to be not suitable for for generation of the accurate symmetricoscillating gradient, they still potentially can be used for implementing our scheme
in the TOP-trap configuration. Instead of sending oscillating current through theanti-Helmholtz coil and creating dc bias field which polarizes the spin and defines the
direction of the force one can, in principle, run dc current through the anti-Helmholtz
pair and rotate the bias field by driving oscillating currents through two other perpendicular Helmholtz pairs with a 𝜋/2 phase shift. In this scheme there are lessrequirements on the symmetry of the ac-current since it only sets the instantaneous
direction of the polarizing bias. The symmetry of the magnetic force will depend on
the stability of the dc current through the anti-Helmholtz pair, which is much easierto accomplish than for the ac-current. For this application the audio-amplifiers may
90be a good choice.
The observation of Doppler shifts at the 200 Hz level required careful control of
Zeeman shifts. Three critical adjustments were done.
We had to protect our system from the potential causes of the false signal. Inhomogeneous Zeeman shift was one of our most dangerous enemies. We had to makesure the signal we observe comes from Doppler shift. In principle the phase acquiredby atoms may be not only velocity dependent, but also position dependent. Unless wewould go to infinite TOF these two mechanisms cannot be fully separated. Thus, we
put a lot of effort in minimizing all the position-dependent effects in order to observea pure Doppler signal.
Here’re the precautions we took to eliminate the false Zeeman effect. The RF
pulses can be safely fired only in the vicinity of the zero crossing of the time-dependent
gradient, i.e. for the phases 𝜑𝑅𝐹 = 0 and 𝜑𝑅𝐹 = 𝜋. Since the clouds’ size is finite
a non-zero gradient across the cloud leads to inhomogeneous Zeeman effect, which
is very undesirable for us. It may sound like this circumstance limits the tunabilityof scheme, that the momentum transfer in SOC cannot be changed continuously bychanging the timing of the Rf pulses. However, this constraint is only technical. The
shorter RF pulses, the broader their Fourier width, and if the Fourier broadening
prevails over the inhomogeneous Zeeman shift the RF pulses can be fired not only at
the zero crossings, but also at finite gradients. The duration of the RF pulse itself
is limited only by the RF power (in the case of our experiment) and the switching
electronics. We found that the optimal pulse duration in the experiment was 4𝜇𝑠. It
was short enough for the Fourier width to be broader than than the inhomogeneous
Zeeman effect, and long enough to transfer substantial population of atoms in 9-10
pulses.
An important requirement on the field control is the symmetry of the gradient.
The absence of it would hurt the velocity-dependent Doppler effect. If the magnetic
gradient is asymmetric in time it averages over the period 𝑇 as the dc-gradient ⟨𝐵 ′ ⟩𝑇which again leads to inhomogenous Zeeman effect.
Another side effect comes from the imperfection of the anti-Helmholtz coils. In
91addition to the oscillating gradient, we at the same time excite small oscillating bias
magnetic field. The bias field quickly sweeps through the resonance during the RF
pulse and effectively shortens its duration. Also when the induced bias is not perfectly
symmetric in time, it again results in unwanted phases accumulation due to Zeeman
effect.
The most peculiar and unexpected side effect came from the current induced in
the surrounding stainless steal vacuum chamber. The eddy currents caused shaking ofatoms at doubled frequency in the perpendicular direction with respect to the chosen
axis, which tilted the velocity slice in TOF. This effect is going to be discussed further.

Bz

Figure 5-10: Atomic cloud is center with respect to the zero of the quadrupole. Dueto the finite size of the cloud atoms displaced from the origin of the quadrupole
experience oscillating bias magnetic field.

5.3.3

Symmetry of the modulated magnetic field gradient

If inhomogeneous Zeeman shifts across the cloud (Fig.5-10) are comparable or larger
than Doppler shifts, the spinflips are no longer velocity selective since there is always alocal Zeeman shift to compensate for the Doppler shift. Therefore, the magnetic field
gradient averaged over one modulation cycle ⟨𝐵 ′ ⟩, had to be zeroed: 𝑔𝐹 𝜇𝐵 ⟨𝐵 ′ ⟩𝐷 ≪92

𝑘𝑣, where 𝐷 is the length of the cloud. To avoid transient asymmetries from theturn-on process of the periodic magnetic gradient, we added a pre-shaking period of
3 ms before the spectroscopic sequence. This didn’t affect the trapped atom cloud,since the atoms were initially in the non-magnetic |𝑚𝐹 = 0⟩ state. After the preshaking, we achieve ⟨𝐵 ′ ⟩ ≈ 20 mG/cm, implying a time-averaged differential Zeeman

shift across the cloud of less than 100 Hz. ⟨𝐵 ′ ⟩ was determined from converting
the measurement of time-averaged current asymmetry to the time-averaged magnetic
gradient asymmetry using the Stern-Gerlach calibration. As a final check, we added
asymmetries (Fig.5-11) on either the positive or negative side of the sinusoidal current
to create ⟨𝐵 ′ ⟩ ≈ ± 100 mG/cm, and for both cases observed a slight increase in thewidth of the velocity-selected atom slice confirming that the residual asymmetry of
the magnetic gradient modulation was negligibly small.

1

R

4

IGBT

IGBT
+100 V

0V

IGBT

IGBT
3

2

LcoilsC

Figure 5-11: An H-bridge with built-in asymmetry. A resistor in one of the legs
controlled the asymmetry of current at the per cent level.
The following two adjustments addressed the issue that the RF pulses were notdelta functions, but had a duration of 4 𝜇s. The presence of Zeeman shifts comparable
93

B

z

Figure 5-12: Atomic cloud displaced from the origin of the quadrupole. Magnetic
fields experienced by the atoms sweeps through larger range of values and effectively
shortens the duration of the RF pulses.
or larger than the Fourier width of a single pulse would reduce the RF pulse area.
For our parameters, a 45 kHz detuning will reduce the pulse area by 5 percent (andtherefore the single pulse excitation probability by 10 percent).

5.3.4

Minimize modulation of magnetic bias field

The time-dependent gradient creates also a time-dependent bias field given by the gradient times the displacement of the atoms from the origin of the magnetic quadrupole
field. 60 𝜇m away from the origin, the bias field changes by 30 mG during the 4 𝜇s
RF pulse. To minimize the reduction of the RF pulse area, the optical trap wasaligned with the center of the quadrupole field to within 1 𝜇m like on Fig.5-10 (asituation which we wanted to avoid is depicted on the Fig. 5-12). This was done byminimizing the shift in the RF resonant frequency when a stationary gradient field
was added to the constant magnetic bias field. In addition, the eddy currents createda time-dependent bias field, which was compensated by RF detuning. The detuning
and the timing of the RF pulses (described below) were adjusted together in order to94

maximize the fraction of spin-flipped atoms.

5.3.5

Timing of the RF pulses with respect to the magnetic
modulation

The goal was to pulse on the RF while the magnetic field gradient crosses zero. A 5 𝜇s
offset would imply a gradient of 7.5 G/cm and a differential magnetic field along thecloud of 50 mG. In the presence of strong gradients, the short RF pulse is resonant
only for a small part of the cloud. Therefore, we could find the optimum conditionby scanning both the timing and the detuning of the RF pulses until the measured
total fraction of the spin-flipped atoms is maximized. The optimum time was offset
by 2 𝜇s from the zero-crossing of the current through the gradient coils, possibly due
to eddy currents.
To summarize, we optimized three parameters, which are trap position, timing of
the RF pulse, and RF detuning. The optimal position minimizes temporal variationof the bias field, optimal timing of RF minimizes 𝐵 ′ during the pulse, and optimal
detuning compensates for any bias field at the time of the pulse.

5.3.6

Induced shaking

One of the striking features of our experiment was that the narrowing of the atoms
in the excited state of the TOF happened not in the direction we expected to see it.
The slicing of the cloud happened not in the shaking direction 𝑧ˆ which was definedby the bias field, but at the some angle to it (Fig.5-13).
It looked like the Doppler effect we simulated was𝛿 = 𝑘𝑦 𝑣𝑦 + 𝑘𝑧 𝑣𝑧
The tilt angle we first observed was 𝜃 = arctan(𝑘𝑦 /𝑘𝑧 ) ≈ 60∘ (Fig. 5-14). So, we had
to find out what may cause Doppler selectivity in the 𝑦ˆ direction with the magnitudeof the effective momentum transfer comparable to one along 𝑧ˆ.
95

ky

y
z

kZ

kZ

Figure 5-13: Slicing of the momentum distribution at an angle.

The first check we did was changing the RF phase to 𝜑𝑅𝐹 = 𝜋, i.e. fire the RF
pulses at zero crossing on the negative slope of the magnetic drive. The change ofthe phase to 𝜋 flips the sign of the momentum transfer. So, we had an expectation
to simulate Doppler shift 𝛿𝜔 = −𝑘𝑦 𝑣𝑦 − 𝑘𝑧 𝑣𝑧 . It would keep the orientation of theslice the same. However, to our surprise, in the experiment the orientation of the slicechanged. After fitting the data we realized that 𝑘𝑧 has changed to −𝑘𝑧 as we expected,

but 𝑘𝑦 stayed the same, so the new tilt was 𝜃 = −60∘ . This peculiar observation led
us to a possible model which provided qualitatively accurate explanation to the effect.
We inferred that in addition to the bias field 𝐵0 we created along 𝑧ˆ, the eddy
currents in the stainless steel vacuum chamber induce oscillating bias field along 𝑦ˆ.
This magnetic field from the eddy currents should oscillate at the same frequency 𝜔as magnetic gradient we create (and as the current we drive through the coils), and⃗ 𝑒𝑐 (𝑡) = 𝐵𝑒𝑐 sin(𝜔𝑡 + 𝜑)ˆ
may have some phase shift 𝜑 with respect to it 𝐵𝑦.
This assumption has the following intuition behind it. In order to produce shaking
in the 𝑦ˆ-direction there must a component of oscillating force in the same direction.
While the magnitude of the force is determined by the amplitude of the oscillating
gradient, the direction of the force comes from the polarization of the spin, i.e. fromthe instantaneous direction of bias magnetic field (Fig.5-15). If the bias field has only
dc component along 𝑦ˆ it causes some polarization of the spin along 𝑦ˆ and therefore
a component of the force oscillating at the same frequency 𝜔 as the gradient. That96ɸRF= #, kz < 0ɸRF= 0, kz > 0(a)+60˚ tilt(b)-60˚ tilt

Total bias field
Oscillating Byθ
B0= 6 G
(Small γ)

Bz
y
z

(c)

+40˚ tilt(d)-40˚ tilt

Oscillating By

B0=11.4 G
Bz(Large γ)

Figure 5-14: Effect of eddy currents on observed velocity-selected atom slices. The
induced bias field along e𝑦 led to a 𝑦-component of the oscillating force, resulting invelocity selectivity in e𝑦 and therefore tilting of the resonant velocity slice in the 𝑦 − 𝑧plane. The tilt angle depends on the static bias field 𝐵0 and the RF phase 𝜑𝑅𝐹 . The
dashed lines are guides to the eye.

would explain a tilt of the slice we cut in TOF, but would not agree with the observation that for the pulses at the negative slope the tilt changes from 𝜃 to −𝜃. If we
imagine that there is an oscillating bias field at frequency 𝜔 along 𝑦ˆ, the polarizationof the spin, its projection on 𝑦ˆ, will also oscillate at 𝜔. The force along 𝑦ˆ, which is
proportional to the product of the spin-projection (oscillating at 𝜔) and the gradient(oscillating at 𝜔) will rectify and have a component oscillating at 2𝜔. Well, so now wehave atoms being shaken at 𝜔 along 𝑧ˆ and 2𝜔 along 𝑦ˆ. The RF pulses are fired every
𝑇 = 2𝜋/𝜔, i.e. every period for the 𝑧ˆ force and every other period for 𝑦-force. When
we change the phase of the the RF to 𝜋 and fire it on the negative slope for 𝑧-force,with respect to the 𝑦-force the phase does not change. In turns out that 𝑘𝑦 does notchange whereas 𝑘𝑧 changed to −𝑘𝑧 . Thus, shaking at double frequency caused by thebias oscillating at the single frequency may explain the observed reflection of the slicewith respect to 𝑦ˆ.
Here’s a more quantitative analysis of the model. The total magnetic field expe97

y
z

By

S
Bz

Figure 5-15: Due to the induced bias field along 𝑦ˆ oscillating at the drive frequency
𝜔 the polarized spin is steered along the instantenous direction of the total bias field,
i.e. 𝐵0 𝑧ˆ + 𝐵𝑒𝑓 (𝑡)ˆ𝑦rienced by the atoms comes from the two bias components and the gradient:⃗ = [𝐵0 + 𝐵0′ sin (𝜔𝑡)𝑧] 𝑧ˆ+[𝐵ec sin (𝜔𝑡 + 𝜑) + 𝐵0′ sin (𝜔𝑡)𝑦] 𝑦ˆ −2𝐵0′ sin (𝜔𝑡)𝑥ˆ
𝐵
𝑥, (5.26)
with a magnetic field strength√︁[𝐵0 + 𝐵0′ sin (𝜔𝑡)𝑧]2 + [𝐵ec sin (𝜔𝑡 + 𝜑) + 𝐵0′ sin (𝜔𝑡)𝑦]2 + [2𝐵0′ sin (𝜔𝑡)𝑥]2√︁𝐵0′ 𝑧 + 𝛾 sin (𝜔𝑡 + 𝜑)𝐵0′ 𝑦sin (𝜔𝑡)≈ 𝐵0 1 + 𝛾 2 sin2 (𝜔𝑡 + 𝜑) + √︀
1 + 𝛾 2 sin2 (𝜔𝑡 + 𝜑)

⃗ =
|𝐵|

(5.27)

here 𝛾 = 𝐵ec /𝐵0 . The first term corresponds to a time varying homogeneous bias
field resulting in a velocity-independent effective detuning of the RF transition. The
oscillating magnetic field gradients along the 𝑧 and 𝑦 directions are
⃗
𝜕|𝐵|
𝐵0′
= √︀
sin (𝜔𝑡),𝜕𝑧
1 + 𝛾 2 sin2 (𝜔𝑡 + 𝜑)
⃗
𝜕|𝐵|
𝛾𝐵0′
= √︀
sin (𝜔𝑡) sin (𝜔𝑡 + 𝜑).
𝜕𝑦
1 + 𝛾 2 sin2 (𝜔𝑡 + 𝜑)

(5.28)The gradient in 𝑦ˆ oscillates at 2𝜔, twice the frequency of the driving. The phase delay
𝜑 is determined by the magnetic properties of the vacuum chamber. We modeled thechamber as an LC circuit with a self inductance 𝐿Ch and a resistance 𝑅Ch , and obtain98

𝜑 = arctan (𝜔𝐿Ch /𝑅Ch )+𝜋/2. Our observations imply 𝑅Ch ≫ 𝜔𝐿, 𝜑 ≈ 𝜋/2, resulting
in an effective recoil component in the 𝑦 direction with1𝑘so,𝑦 =
𝑇

∫︁ 𝑇 (︃∫︁ 𝑡

)︃

𝛾𝐵0′

√︀
sin (𝜔𝑡′ ) cos (𝜔𝑡′ ) d𝑡′ d𝑡2
2′
1 + 𝛾 cos (𝜔𝑡 )00)︃∫︁ 𝑇 (︃∫︁ 𝑡
1
𝛾𝐵0′
√︀
=
sin (2𝜔𝑡′ ) d𝑡′ d𝑡.
2
2
′
𝑇 0
0 2 1 + 𝛾 cos (𝜔𝑡 )(5.29)

Consequentially, the Doppler shift is modified as𝛿 = 𝑘𝑦 𝑣𝑦 + 𝑘𝑧 𝑣𝑧 ,(5.30)

directly observed as a rotation of the velocity slice with an angle 𝜃 = arctan (𝑘𝑦 /𝑘𝑧 )in the time-of-flight images, as shown in Fig. .
A natural way to verify these assumption is to reduce the effect of the transverse
shaking by stronger polarizing the spin along 𝑧ˆ. We increased the bias filed 𝐵0 from 6
Gs to 11.4 Gs and saw how the angle 𝜃 of the rotation decreased with stronger static
bias tp 𝜃 ≈ 40∘ . Due to the 2𝜔 oscillating frequency of the 𝑦 force, 𝑘𝑦 did not change
sign when the RF phase 𝜑𝑅𝐹 was shifted from 0 to 𝜋 in contrast to 𝑘𝑧 , and therefore
the rotation angle flipped from 𝜃 to −𝜃, as suggested by and shown in Fig.5-14 .
The effects of the induced currents can be completely eliminated if the experimentis conducted in a glass cell. In our apparatus, which was design for optical lattices
experiments, and has a stainless steel chamber we could only suppress the effects
by ramping the bias field 𝐵0 further up. However, using higher bias fields has its
own incidental problems. First of all, using higher bias fields means that we needto change the frequency of the radio signal resonant with the level splitting of thehyperfine states. Higher frequency would lead to the loss of the efficiency of theantenna and therefore a decrease in Rabi frequency of the RF transition. Also, ourbias coils were not water-cooled, so eventually we would also be limit by the power
we dissipate in them at high currents.
What we have described so far applies to free space or to an isotropic trap.
99

However, the optical trap in the experiment is anisotropic. For zero time-of-flight,
in the 𝑦 − 𝑧 plane, the minor axis of the ellipsoidal cloud is oriented along 𝑦,𝜃 = 𝜋/2. For long time-of-flight, the angle is solely determined by the velocity selection 𝜃 = arctan (𝑘𝑦 /𝑘𝑧 ). For intermediate time-of-flight, as used in the experiment,
the observed angle interpolates between these values. We calculate that the observed
tilt angles of 60∘ and 40∘ correspond to tilt angles of the bias field arctan (𝑘𝑦 /𝑘𝑧 ) of
53∘ and 32∘ , respectively.

5.4

Heating due to collisions between the spin states

One of the advantages of magnetically generated spin-orbit coupling is that it inherently does not suffer from spontaneous emission, since no lasers are used to produce
it. Since the scheme we realized is laser-free, it is promising as an experimental tooland may potentially make the experiments realizing spin-dependent gauges easier tobuild and maintain.
However, instead of heating via spontaneous emission like in two-photon schemes,
the use of periodic drive brings another heating mechanism: heating due to collisions.
Micromotion of the different spin states can be transferred into secular motion andheat the sample. Our experiment was done with rather humble momentum transfer
𝑘 ∼ 0.7𝑘𝐿 ., characterizing the strength of the drive. The lifetime of the two colliding
condensates in |𝐹 = 1, 𝑚 = 0⟩ and |𝐹 = 1, 𝑚 = −1⟩ states was very long — about 8s.
The follow-up theoretical study [57] describes heating due to collisions in the presence of periodic drive for various system. In this work it was shown that energy pumping into the system of two colliding species can be expressed as 𝐸˙ ∼ 𝜌𝑣𝑐𝑜𝑙 𝐸0 , where 𝑣𝑐𝑜𝑙
reflects an effective density of states, 𝜌 is the density and 𝜎 is two-body 𝑠-scattering√︀cross section. For instance, in the case of colliding condensates 𝑣𝑐𝑜𝑙 = ℎ̄𝜔/𝑚 forlarge modulation frequencies 𝜔. It was estimated that if the momentum transfer isscaled to the optical recoil 𝑘0 , the collisional heating will increase significantly unlessthe density of atoms is reduced to ∼ 1012 cm−3 . Otherwise it will not be possible to100reach long lifetime of the order of 100 ms. For degenerate Fermi gas the expectations
are more optimistic. Due to Pauli blocking in 𝑠-wave collisions for driving frequency
below the Fermi energy ℎ̄𝜔 < 𝐸𝐹 the heating is suppressed by a factor of (ℎ̄𝜔/𝐸𝐹 )2 .

101

102

Chapter 6
Conclusions and Outlook
Two methods of generating one-dimensional spin-orbit coupling were presented inthis thesis. In the first approach, we used a two-photon Raman process as a source of
momentum transfer and applied it to couple pseudo-spin states, defined as the groundand first excited state of a double-well potential. In the second approach we pickedconventional hyperfine states of sodium as pseudo-spin states, but developed a completely new mechanism of momentum transfer: we took a low-energy RF transition,
which inherently does not carry any recoil momentum, and endowed it with tunable
momentum in the time-averaged picture.
Both schemes shared as a motivation the goal of simulating spin-orbit couplingwith minimal heating. In addition, the first approach was targeted to explore a new
point on the phase diagram – the stripe phase, a long-awaited supersolid state ofmatter.
The method based on dressing RF photons with recoil was conceived in the context of spin-orbit coupling, but in fact, goes beyond it. It also provided a lucid
pedagogical interpretation of what Floquet engineering is. This technique can also
be used for velocimetry and accelerometry. The new fundamental concept of recoildressed photons can become a new building block for future quantum simulations.
Our experiment was carried out in an apparatus which was not perfectly suitable forthe use of high oscillating fields. The humble value of momentum transfer that we
generated ∼ 0.1𝑘𝐿 was only technically limited by the induced eddy currents in the103stainless steel chamber. If the same scheme would be realized in a glass cell, thislimitation would be lifted. Also, much higher magnetic gradients than ours could be
realized in atom-chip experiments, which would also allow one to boost the momentum transfer. Thus, seeing our work as the first step and proof-of-principle result,
I hope it has provided a new look at the quantum mechanics of periodically driven
systems. This work can inspire future development in the next-generation of quantum
simulation experiments. Application of the scheme to fermionic gases was found tobe particularly promising in the follow-up theoretical study [57].

104PRL 117, 185301 (2016)

week ending
28 OCTOBER 2016

PHYSICAL REVIEW LETTERS

Spin-Orbit Coupling and Spin Textures in Optical Superlattices
Junru Li, Wujie Huang, Boris Shteynas, Sean Burchesky, Furkan Çağrı Top, Edward Su,
Jeongwon Lee, Alan O. Jamison, and Wolfgang Ketterle
Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 10 June 2016; revised manuscript received 17 August 2016; published 27 October 2016)
We propose and demonstrate a new approach for realizing spin-orbit coupling with ultracold atoms.
We use orbital levels in a double-well potential as pseudospin states. Two-photon Raman transitions
between left and right wells induce spin-orbit coupling. This scheme does not require near resonant light,features adjustable interactions by shaping the double-well potential, and does not depend on special
properties of the atoms. A pseudospinor Bose-Einstein condensate spontaneously acquires an antiferromagnetic pseudospin texture, which breaks the lattice symmetry similar to a supersolid.
DOI: 10.1103/PhysRevLett.117.185301

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

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

185301-1© 2016 American Physical Society

PRL 117, 185301 (2016)

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

week ending
28 OCTOBER 2016

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

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

185301-2

TABLE I.

week ending
28 OCTOBER 2016

PHYSICAL REVIEW LETTERS

PRL 117, 185301 (2016)

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

States

Mi

M 0i

jΨ1 i
jΨ2 i

− 12 ðΩ=Er − δÞþ 12 ðΩ=Er þ δÞ

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

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

ð1Þ

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

ð2Þ

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

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

185301-3

PRL 117, 185301 (2016)

PHYSICAL REVIEW LETTERS

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

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

week ending
28 OCTOBER 2016

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

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

185301-4

PRL 117, 185301 (2016)

PHYSICAL REVIEW LETTERS

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

[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045
(2010).
[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
[3] R. Wei and E. J. Mueller, Phys. Rev. A 86, 063604 (2012).
[4] I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76,
323 (2004).
[5] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A.
Fisher, Nat. Phys. 7, 412 (2011).
[6] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton,and W. Ketterle, Phys. Rev. Lett. 111, 185302 (2013).
[7] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B.
Paredes, and I. Bloch, Phys. Rev. Lett. 111, 185301 (2013).
[8] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).
[9] G. Dresselhaus, Phys. Rev. 100, 580 (1955).
[10] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and
R. A. Duine, Nat. Mater. 14, 871 (2015).
[11] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature(London) 471, 83 (2011).
[12] N. Q. Burdick, Y. Tang, and B. L. Lev, Phys. Rev. X 6,031022 (2016).
[13] I. B. Spielman, P. R. Johnson, J. H. Huckans, C. D. Fertig,
S. L. Rolston, W. D. Phillips, and J. V. Porto, Phys. Rev. A
73, 020702 (2006).
[14] M. A. Khamehchi, C. Qu, M. E. Mossman, C. Zhang, and P.
Engels, Nat. Commun. 7, 10867 (2016), article.
[15] D. Hügel and B. Paredes, Phys. Rev. A 89, 023619 (2014).
[16] M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B.
Paredes, and I. Bloch, Nat. Phys. 10, 588 (2014).
[17] T. L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011).

week ending
28 OCTOBER 2016[18] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett.
105, 160403 (2010).
[19] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 108,
225301 (2012).
[20] M. Boninsegni and N. V. Prokof’ev, Rev. Mod. Phys. 84,
759 (2012).
[21] G. I. Martone, Y. Li, and S. Stringari, Phys. Rev. A 90,
041604 (2014).
[22] More specifically, our scheme realizes 1D SOC in the 2D
free space orthogonal to the superlattice.
[23] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.117.185301 for derivations and experimental details, which includes Ref. [24].
[24] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn,
D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 82, 4569
(1999).
[25] An alternative view is that this system is a superposition of
two BECs with two spontaneously broken U(1) symmetries(the overall phase, and the relative phase θ). The phase θdetermines the orientation of the antiferromagnetic pseudospin texture in the spin x-y plane. The spin texture breaks
the lattice symmetry by doubling the lattice constant. Thissystem therefore does not fulfill a stronger definition ofsupersolidity that demands breaking a continuous translational symmetry.
[26] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger,
Nature (London) 464, 1301 (2010).
[27] Y. Chen, J. Ye, and G. Tian, J. Low Temp. Phys. 169, 149(2012).
[28] X. Luo, L. Wu, J. Chen, Q. Guan, K. Gao, Z. Xu, L. You,and R. Wang, Sci. Rep. 6, 18983 (2016).
[29] Collisional losses could be further diminished by reducing
the overlap between the pseudospin states. The consequently reduced spin-orbit coupling strength can be compensated by higher intensity Raman beams up to the point
where higher order effects and additional heating occur due
to nonperturbative driving. The parameters of Fig. 4 wereclose to this limit.
[30] H. Miyake, G. A. Siviloglou, G. Puentes, D. E. Pritchard, W.
Ketterle, and D. M. Weld, Phys. Rev. Lett. 107, 175302(2011).
[31] S.-W. Su, S.-C. Gou, Q. Sun, L. Wen, W.-M. Liu, A.-C. Ji, J.
Ruseckas, and G. Juzeliūnas, Phys. Rev. A 93, 053630 (2016).
[32] J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, Rev.
Mod. Phys. 83, 1523 (2011).
[33] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M.
Lewenstein, Phys. Rev. Lett. 95, 010403 (2005).
[34] A. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. Julienne, J.
Ye, P. Zoller, E. Demler, M. Lukin, and A. Rey, Nat. Phys. 6,289 (2010).
[35] M. Foss-Feig, M. Hermele, and A. M. Rey, Phys. Rev. A 81,
051603 (2010).
[36] M. Nakagawa and N. Kawakami, Phys. Rev. Lett. 115,
165303 (2015).

185301-5

Supplemental Material for“Spin-Orbit Couping and Spin Textures in Optical Superlattices”
Junru Li,∗ Wujie Huang,∗ Boris Shteynas, Sean Burchesky,
Furkan Çağrı Top, Edward Su, Jeongwon Lee, Alan O. Jamison, and Wolfgang Ketterle
Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Dated: August 17, 2016)

SUPERLATTICE HAMILTONIAN WITH RAMAN COUPLING

The Hamiltonian for a one-dimensional superlattice, created by standing waves of infrared and green light withrelative phase φSL is

Hlattice =

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

In the tight-binding limit, it can be rewritten as
Hlattice =

X
X
X X
p̂2⊥
1
0
0
(Jtt
+ ∆0
(|rn i hrn | − |ln i hln |) − J(|ln i hrn | + h.c.) −0 |tn i htn+1 | + h.c.),
2m 2
n
n
n
t=l,r
t0 =l,rwhere |ln (rn )i is a wavefunction localized in the left(right) well of the nth unit cell, ∆0 is the energy separation
between the right and the left wells. ~ is taken to be 1. Tunneling between neighboring unit cells is important for
maintaining of coherence in the superlattice, but not relevant for the physics of spin-orbit coupling. Thus, tunneling0terms with Jtt
0 in the Hamiltonian can be neglected.
The complete Hamiltonian of the system is
H = Hlattice + VRaman ,where VRaman = Ω cos(kz z + kx x − δt) is a moving lattice potential. For the purpose of our experiment, the initial
phase of the Raman potential is not relevant and taken to be zero.
We prefer to use eigenstates of a double well for the description. To first order in small parameter ∆J0  1 they
can be written as|↓n i = |ln i +JJ|rn i , |↑n i = |rn i −
|ln i
∆0
∆0
2

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

H=

X X

X
1 X
p̂2⊥
+ ∆
(|↑n i h↑n | − |↓n i h↓n |) +|p⊥ i
|in i hin | hp⊥ | Ω cos(kz z + kx x − δ · t) |p0⊥ i |i0n i hi0n | hp0⊥ |
2m 2
0
n
n
p⊥ ,p⊥i=↓,↑i0 =↓,↑

In our experiment kz ≈ kx ≈ kIR = πd , where d is a period of the superlattice. In order to estimate the effect of theRaman potential with arbitrary phase we need to know the overlap integrals for cos(kIR ·(z−zn )) and sin(kIR ·(z−zn )).
J
To first order in ∆:h↓n | cos(kIR · (z − zn )) |↑n i ≈ hln | −JJ|ln i = − ,∆∆h↓n | cos(kIR · (z − zn )) |↓n i ≈ 1, h↑n | cos(kIR · (z − zn )) |↑n i ≈ 0,

2h↓n | sin(kIR · (z − zn )) |↑n i ≈ hrn |JJ|rn i = ,∆∆h↓n | sin(kIR · (z − zn )) |↓n i ≈ 0, h↑n | sin(kIR · (z − zn )) |↑n i ≈ 1,
where zn = nd is a coordinate of the left well in the nth unit cell.
Thus, the Raman potential can be expanded in the basis of double-well eigenstates:
Xi=a,bi0 =a,b|in i hin | Ω cos(kz (z − zn ) + kz zn + kx x − δ · t) |i0n i hi0n | == Ω cos φn {−J|↓n i h↑n | + |↑n i h↓n | + |↓n i h↓n |}+∆− Ω sin φn {|↑n i h↑n | +J(|↓n i h↑n | + |↑n i h↓n |)},∆

where φn = πn + kx x − δt. Later, we will calculate how x as an operator acts on the momentum states |p⊥ i.
V̂Raman =XnΩ(−1)n cos(kx x − δt) |↓n i h↓n | − Ω(−1)n sin(kx x − δt) |↑n i h↑n | +√ Jπ− 2Ω (−1)n cos(kx x − δt − ){|↓n i h↑n | + |↑n i h↓n |}
∆
4

(1)

(2)

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

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

N
X
π
d
1
√ ei d (zn + 2 ) |↑n iN
n=1

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

For the calculation of matrix elements we assume that overlap is nonzero only for n = n0 .
Intra-band coupling terms:
D
E X 1
(↓)
(↓)
ψq=π/d V̂Raman ψq=0 =(−1)n h↓n0 | Ω cos(kz z + kx x − δt) |↓n i
N
0
n,n

=Ω

X 1
n

N

cos(φn )(−1)n = Ω cos(kx x − δt)

(3)

3

E X 1
D
(↑)
(↑)
h↑n0 | Ω cos(kz z + kx x − δt)i(−1)n |↑n iψq=0 V̂Raman ψq=π/d =
N
0
n,n

=Ω

X 1
n

N

i(−1)n (− sin(φn )) = −iΩ sin(kx x − δt)(4)Spin-orbit coupling matrix element:
E X 1
D
0
(↓)
(↑)
ψq=π/d V̂Raman ψq=0 =(−i)(−1)n h↑n0 | Ω cos(kz z + kx x − δt)| |↓n i
N
0
n,n

= −iΩ

X 1
n

N

(−1)n h↑n | cos(kz (z − zn )) cos(φn ) − sin(kz (z − zn )) sin(φn ) |↓n i ,

E
D

J X
1
J
(↓)
(↑)
ψq=π/d V̂Raman ψq=0 = i Ω(−1)ncos(φn ) + sin(φn ) = i Ω(cos(kx x − δt) + sin(kx x − δt))
∆ n
N
∆
k2

x
Intra-band coupling matrix elements (3) and (4) provide recoil kick in x−direction with recoil energy Er = 2m
and(↓)
along the superlattice, changing the quasimomentum by half a reciprocal vector. If the system is initially at |ψq=0 , 0i,
the new adiabatically connected eigenstate in first order perturbation theory is:(↓)|Ψ1 i = |ψq=0 , 0i −

1 Ω
1 Ω
(↓)
(↓)
e−iδt |ψq=π/d , kx i −eiδt |ψq=π/d , −kx i +2 Er − δ
2 Er + δ
π
π
e−i 4
JΩ/∆
ei 4
JΩ/∆
(↑)
(↑)
−iδt
−i √
e
|ψq=π/d , kx i − i √eiδt |ψq=π/d , −kx i
E
+
∆
−
δ
E
+
∆
+
δ
2 r
2 r
(↑)

If the system is prepared in |ψq=π/d , 0i:(↑)|Ψ2 i = e−i∆t |ψq=π/d , 0i −

1 Ω
1 Ω
(↑)
(↑)
e−i(δ+∆)t |ψq=0 , kx i +ei(δ−∆)t |ψq=0 , −kx i +2 Er − δ
2 Er + δ
π
π
e−i 4
ei 4
ΩJ/∆
ΩJ/∆
(↓)
(↓)
+i √
e−i(δ+∆)t |ψq=0 , kx i + i √ei(δ−∆)t |ψq=0 , −kx i
2 Er − ∆ − δ
2 Er − ∆ + δFor δ close to ∆ intra-band coupling is off-resonant and both co- and counter-rotating terms contribute at comparable
strengths, whereas for spin-orbit coupling the co-rotating term is resonant and, therefore, much stronger than the
counter-rotating term.
SPIN-ORBIT-COUPLING HAMILTONIAN

Keeping only the near-resonant spin-orbit coupling term, the Hamiltonian describing the system is


iπ/4
p̂2x
∆
−ie
J
−i(kx x−δt)
√
−
Ωe
2
∆
2

HSOC =  ie−iπ/42mp̂2xJ∆i(kx x−δt)√Ωe2m + 22 ∆πAfter a unitary transformation with a position-dependent rotation, Û = e(−ikx x+iδt−i 4 )σz /2 , the Hamiltonian turns
0
into HSOC = U † HSOC U − iU † ∂U
∂t .
0

HSOC =(p̂x + ασz )2+ βσx + δ0 σz ,2mwhere α = − 21 kx , β = √JΩand δ0 = 12 (δ − ∆). For the total Hamiltonian we have to add the intra-band coupling,2∆
which causes a density modulation.

4
PSEUDOSPIN STERN-GERLACH EFFECT

FIG. 1: (color)Stern-Gerlach separation of the spin states. Atoms are loaded into the green lattice , then the IR lattice issuddenly ramped up. After different hold times, the vertical displacements of atoms in different spin states is measured with
10ms ballistic expansion. y = 0 corresponds to the position of atoms without the sudden ramp. The spin states are displaced
due to of transverse oscillations in the superlattice. The oscillating frequencies are equal to the transverse trapping frequencies
which the spins experience in the superlattice. In the data shown, atoms in |↑i and |↓i oscillates with frequencies of 356 Hz
and 417 Hz correspondingly.

Atoms in different pseudospin states can be spatially separated by a pseudospin Stern-Gerlach effect. The two spinstates experience different transverse confinement from the 1064 nm lattice. When this lattice beam is displaced fromthe green lattice, the atoms in the two spin states experience different momentum kicks in the x − y plane when the IR
lattice is suddenly increased. This leads to transverse oscillations of the two spin states relative to each other and canbe used to separate them in ballistic expansion(Fig.1). The frequency of the oscillation is equal to the correspondingtransverse trapping frequency of each spin state.
SUPERLATTICE CALIBRATION

Fraction of Population

OD (a.u.)

The superlattice potential V (x) = VIR sin2 (kIR z + φSL ) + VGr sin2 (kGr z) was produced by overlapping two onedimensional lattices with spacing λIR /2 = 532 nm(long) and λGr /2 = 266 nm(short). The 532 nm light was generated
by frequency doubling a high power λ = 1064 nm laser seeded by the same seed laser as the laser for the long lattice.
This eliminated the need for any phase or frequency locking of two independent lasers. The long and short latticealso shared the same retroreflective mirror to minimize noise and drifts in the relative phase φSL .

0
Hold Time ( s)FIG. 2: (color)Calibration of the superlattice offset ∆. The relative phase between atoms in the lower and upper wells
accumulated linearly with time and results in periodic changes in the time-of-flight interference pattern. The period of theoscillation is equal to ∆ which is ∼ 23 kHz for the data shown.

The relative phase φSL was controlled by introducing different phase shifts for the λ = 1064 nm and 532 nm latticelights from a rotatable dispersive glass plate and by switching the frequency of the 1064 nm light with an acoustooptical modulator. The glass plate allowed a wide tuning range for φSL . This design minimizes the optical path5
length, reducing the sensitivity to atmospheric pressure changes. The frequency switching allowed rapid but small
phase shifts. For our geometry, a 70 MHz shift for the 1064 nm light corresponded to a π/4 change in φSL
The offset ∆ was directly calibrated by observing a beat note within a single double-well. Condensates with equal
population of atoms in the left and right well were prepared by pre-setting the glass plate to the desired value andthen rapidly ramping up the IR lattice to the final offset. Interference patterns for atom sitting in the lower andupper well overlap and therefore interfere with each other before they relax to orthogonal quasi-momentum states.
The interference pattern evolves periodically with frequency ω = ∆(Fig.2).
The relaxation to the band minimum occurs in less than 2 ms as shown in Figure 3. The observation of therelaxation also indicates the population to be transferred to the first excited band.
z

OD (a.u.)

150 µs250 µs1600 µs

-4

-2

2

0

4

FIG. 3: Relaxation process for atoms in the first excited band of the superlattice. After preparation of the |↑i state withquasimomentum q = 0, it relaxes to the bottom of the band at q = π/d within a time scale of 2 ms.

ANTIFERROMAGNETIC SPIN TEXTURE

A spin-1/2 aligned along the cos φx̂ + sin φŷ direction has the wavefunction |↓i + eiφ |↑i. In our experiment atoms
can be prepared in left and right sites of the double wells with equal population. The wave function is proportional
to
X
|ψi =
|↓n i + (−1)n eiθ e−i∆t |↑n i ,nwhich corresponds to spin-states aligned in the x-y plane with opposite direction on neighboring sites, showing x-y
antiferromagnetic ordering. The expectation values of spin evolve as hσx i = (−1)n cos(θ−∆t), hσy i = (−1)n sin(θ−∆t)and hσz i = 0.
Tunneling between wells within a unit cell causes a density modulation. Local populations in the nth cell acquirea density imbalance:| hln |ψi |2 ∼ 1 − (−1)nJcos(θ − ∆t) ,∆| hrn |ψi |2 ∼ 1 + (−1)nJcos(θ − ∆t)∆Translating by one superlattice period flips the sign of the imbalance. This shows that rn and ln+1 wells, and rn−1
and ln wells have the same change in density, but oscillate out of phase. Therefore, this density wave has twice thewavelength of the superlattice and is shifted spatially by half a unit cell.
RESONANCE LINE BROADENING

The spin-orbit coupling shows resonant behavior as a function of the frequency difference between the Raman
beams. We attribute the broadening of the resonant line shape to 1) Doppler broadening due to the finite size of thetrapped condensate 2) inhomogeneous mean-field shift due to inhomogeneous condensate density 3) inhomogeneous
shift of the superlattice offset ∆ due to the Gaussian beam profile.

6
First, the finite size of the condensate implies a distribution of momenta along the transverse direction which
Doppler broadens the resonance. A momentum p along x direction will shift the resonance by kIR · p/m considering
the geometry of our setup. Therefore a condensate size x0 results in a broadening of ∼ kIR ~/mx0 . Assuming a
Thomas-Fermi distribution in the transverse direction, we obtain an rms width of [1]r
21 kIR ~δνD =8 2π mx0pwhere x0 = 2µ/m(2πνx )2 . This gives a broadening of ∼ 400 Hz.
The resonance is also broadened by the inhomogeneous density distribution of the condensate as
r
8 µ
δνM =
147 h
with µ being the chemical potential. This effect broadens the resonance by ∼ 600 Hz.
The Gaussian beam profile of the long lattice implies an inhomogeneous offset ∆ within the sample, therefore
broadening the transition. The resulted shift is estimated to beδνI ≈ 2∆(

x0 2
)
σ

where σ is the Gaussian beam waist parameter. This effect broadens the resonance line by ∼ 1.10 kHz. However, thebroadening can be much larger for small displacement between the green and IR lattice.
The three widths add up quadratically to a value of 1.30 kHz.
The linewidth of the frequency difference of the Raman beams was negligible — it was monitored through a beatnote and was ∼ 50 Hz without actively phase-locking the two. We conclude that observed resonance width of 2 kHz
is most likely dominated by the spread in ∆ due to slight misalignment of lattice beams.

∗
These two authors contributed equally.
[1] J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 82, 4569
(1999).

Letterdoi:10.1038/nature21431

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

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

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

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

(1)

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

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

1

Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, MIT, Cambridge, Massachusetts 02139, USA.
*These authors contributed equally to this work.
2 m a rc h 2 0 1 7 | VO L 5 4 3 | NAT U R E | 9 1© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

RESEARCH Letterak = kIR⎥ ↓〉Spin–orbit
coupling

k=0

k=0

⎥ ↑〉

k = –kIR

2π/kIR

Bragg

b

Light intensity

1

0

10 mradBragg + RayleighRayleigh

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

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

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

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

Letter RESEARCH
b

Δ

2d

v

1

0
600

a

NBEC2 (×109)
2
3

0.0
1.0

4

200
2×10–4

kRaman

10–4
0

0.2

0.3

Phase
separation

E (Hz)
0

100

200

300

400

40

20

0
0

0.025
E/Er

Singleminimum
phase

Stripe phase
This work⎥ ↑ 〉 Magnetized
phase

–1.0
0.0

0.05

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

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

0.1

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

2

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

c

0.0

0.4

(NBEC/Ntotal)2

Bragg signal (arbitrary units)

15

–0.5

Moving density modulationfrom Raman potential

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

0.5

400

0

2d

0.5

⎥ ↓ 〉 Magnetizedphase
Detuning, G0 /Er

J

Gain / Ntotal

SuperlatticedBragg signal (CCD counts
per square pixel)

a

0

1

20

10

0

–10

–5

0
Detuning, G0 (kHz)5

10

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

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

RESEARCH Letterd

JSuperlattice

Δ
⎥ ↓〉
⎥ ↑〉
z

b
Bragg + Rayleigh signal(arbitrary units)

a

6

1.

4

2

0
Constructive

Destructive

received 26 October 2016; accepted 14 January 2017.

0

5
10
Independent measurements

x
z
2d
M=0

M = π/2
Density modulation from superlattice

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

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

Boninsegni, M. & Prokof’ev, N. V. Supersolids: what and where are they?
Rev. Mod. Phys. 84, 759–776 (2012).
2. Chester, G. V. Speculations on Bose–Einstein condensation and quantum
crystals. Phys. Rev. A 2, 256–258 (1970).
3. Kim, E. & Chan, M. H. W. Probable observation of a supersolid helium phase.
Nature 427, 225–227 (2004).
4. Kuklov, A. B., Pollet, L., Prokof’ev, N. V. & Svistunov, B. V. Quantum plasticity andsupersolid response in helium-4. Phys. Rev. B 90, 184508 (2014).
5. Li, Y., Pitaevskii, L. P. & Stringari, S. Quantum tricriticality and phase transitions in
spin-orbit-coupled Bose–Einstein condensates. Phys. Rev. Lett. 108, 225301 (2012).
6. Ho, T.-L. & Zhang, S. Bose–Einstein condensates with spin-orbit interaction.
Phys. Rev. Lett. 107, 150403 (2011).
7. Wang, C., Gao, C., Jian, C.-M. & Zhai, H. Spin-orbit coupled spinor Bose–Einsteincondensates. Phys. Rev. Lett. 105, 160403 (2010).
8. Han, W., Juzeliūnas, G., Zhang, W. & Liu, W.-M. Supersolid with nontrivial
topological spin textures in spin-orbit-coupled Bose gases. Phys. Rev. A 91,
013607 (2015).
9. Li, Y., Martone, G. I., Pitaevskii, L. P. & Stringari, S. Superstripes and theexcitation spectrum of a spin-orbit-coupled Bose–Einstein condensate. Phys.
Rev. Lett. 110, 235302 (2013).
10. Lin, Y.-J., Jiménez-García, K. & Spielman, I. B. Spin–orbit-coupled Bose–Einsteincondensates. Nature 471, 83–86 (2011).
11. Ji, S.-C. et al. Softening of roton and phonon modes in a Bose–Einsteincondensate with spin-orbit coupling. Phys. Rev. Lett. 114, 105301 (2015).
12. Ji, S.-C. et al. Experimental determination of the finite-temperature phase
diagram of a spin–orbit coupled Bose gas. Nat. Phys. 10, 314–320 (2014).
13. Giovanazzi, S., O’Dell, D. & Kurizki, G. Density modulations of Bose–Einsteincondensates via laser-induced interactions. Phys. Rev. Lett. 88, 130402 (2002).
14. Henkel, N., Nath, R. & Pohl, T. Three-dimensional roton excitations andsupersolid formation in Rydberg-excited Bose–Einstein condensates. Phys. Rev.
Lett. 104, 195302 (2010).
15. Ostermann, S., Piazza, F. & Ritsch, H. Spontaneous crystallization of light andultracold atoms. Phys. Rev. X 6, 021026 (2016).
16. Wessel, S. & Troyer, M. Supersolid hard-core bosons on the triangular lattice.
Phys. Rev. Lett. 95, 127205 (2005).
17. Léonard, J. et al. Supersolid formation in a quantum gas breaking a continuoustranslational symmetry. Nature http://dx.doi.org/10.1038/nature21067 (2017).
18. Bulgac, A. & Forbes, M. M. Unitary Fermi supersolid: the Larkin–Ovchinnikov
phase. Phys. Rev. Lett. 101, 215301 (2008).
19. Chen, Y., Ye, J. & Tian, G. Classification of a supersolid: trial wavefunctions, symmetry
breaking and excitation spectra. J. Low Temp. Phys. 169, 149–168 (2012).
20. Stanescu, T. D., Anderson, B. & Galitski, V. Spin-orbit coupled Bose–Einsteincondensates. Phys. Rev. A 78, 023616 (2008).
21. Miyake, H. et al. Bragg scattering as a probe of atomic wave functions and quantum
phase transitions in optical lattices. Phys. Rev. Lett. 107, 175302 (2011).
22. Martone, G. I. Visibility and stability of superstripes in a spin-orbit-coupled
Bose–Einstein condensate. Eur. Phys. J. Spec. Top. 224, 553–563 (2015).
23. Burdick, N. Q., Tang, Y. & Lev, B. Long-lived spin-orbit-coupled degenerate
dipolar Fermi gas. Phys. Rev. X 6, 031022 (2016).
24. Cheuk, L. W. et al. Spin-injection spectroscopy of a spin-orbit coupled Fermi
gas. Phys. Rev. Lett. 109, 095302 (2012).
25. Wang, P. et al. Spin-orbit coupled degenerate Fermi gases. Phys. Rev. Lett. 109,095301 (2012).
26. Song, B. et al. Spin-orbit coupled two-electron Fermi gases of ytterbium atoms.
Phys. Rev. A 94, 061604(R) (2016).
27. Li, J. et al. Spin-orbit coupling and spin textures in optical superlattices. Phys.
Rev. Lett. 117, 185301 (2016).
28. Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phasetransition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 (2010).
29. Sun, Q., Wen, L., Liu, W.-M., Juzeliūnas, G. & Ji, A.-C. Tunneling-assisted
spin-orbit coupling in bilayer Bose–Einstein condensates. Phys. Rev. A 91,
033619 (2015).
30. Zhai, H. Degenerate quantum gases with spin–orbit coupling. Rep. Prog. Phys.
78, 026001 (2015).
Acknowledgements We thank S. Stringari for discussions and W. C. Burton
for reading the manuscript. We acknowledge support from the NSF throughthe Center for Ultracold Atoms and from award 1506369, from ARO-MURI
Non-equilibrium Many-body Dynamics (grant W911NF-14-1-0003) and from
AFOSR-MURI Quantum Phases of Matter (grant FA9550-14-1- 0035).
Author Contributions J.-R.L., W.H., J.L., B.S., S.B., F.C.T. and A.O.J. contributed tothe building of the experiment. J.-R.L. led the experimental efforts. J.L. led thedata analysis and simulations. W.H., J.-R.L. and W.K. conceived the experiment.
All authors contributed to the writing of the manuscript.
Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial
interests. Readers are welcome to comment on the online version of the paper.
Correspondence and requests for materials should be addressed to
J.-R.L. (junruli@mit.edu).
Reviewer Information Nature thanks K. Hazzard and the other anonymous
reviewer(s) for their contribution to the peer review of this work.

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

Letter RESEARCH
Methods

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

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

PHYSICAL REVIEW LETTERS 123, 033203 (2019)

How to Dress Radio-Frequency Photons with Tunable Momentum

1

Boris Shteynas,1,‡ Jeongwon Lee,1,*,‡ Furkan Çağrı Top,1 Jun-Ru Li,1,† Alan O. Jamison,1Gediminas Juzeliūnas,2 and Wolfgang Ketterle1

Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, Vilnius 10257, Lithuania(Received 18 July 2018; published 16 July 2019)
We demonstrate how the combination of oscillating magnetic forces and radio-frequency (rf) pulses
endows rf photons with tunable momentum. We observe velocity-selective spin-flip transitions and theassociated Doppler shift. Recoil-dressed photons are a promising tool for measurements and quantum
simulations, including the realization of gauge potentials and spin-orbit coupling schemes which do not
involve optical transitions.
DOI: 10.1103/PhysRevLett.123.033203

The field of cooling and trapping atoms depends onmechanical forces exerted by light through photon recoil[1]. Since photons can be scattered only by admixing
electronically excited states, the mechanical forces due tolight always involve dissipation by spontaneous emission.
This is desirable in laser cooling but causes heating andatom loss in other situations where it is often suppressed byusing far off resonant light (e.g., in optical lattices).
In this work we show how to dress radio-frequency (rf)photons with tunable recoil momentum by combining rf
pulses with an oscillating magnetic force. This is a new
application of Floquet engineering: periodically driven
systems can have time-averaged properties which cannot
be achieved with constant fields. Well-known examples are
the Kapitza pendulum, Paul traps for ions, and the realizationof Hamiltonians with complex tunneling matrix elements forultracold atoms in optical lattices [2–4].
The question of how to replace photon recoil by otherforces was raised in the context of spin-orbit coupling forultracold atoms [5]. The well-established two-photon
scheme is limited by heating due to spontaneous emissionof photons. This limitation has motivated the developmentof alternative schemes which use time-dependent magnetic
fields [6–9] to realize spin-dependent synthetic gauge
fields. Some of those schemes are fairly complex, andhave motivated the following question: Is it possible to
Floquet engineer a rf or microwave transition between two
spin states in such a way that it shows all aspects of recoil
momentum?
With this motivation, we propose and demonstrate the newconcept of recoil-dressed rf photons. This scheme allows us
to conduct Doppler-sensitive spectroscopy and velocimetryof molecules when suitable optical transitions are notavailable. It is a building block for quantum simulations
and offers a new approach for spin-orbit coupling using
time-dependent magnetic forces. In our scheme, we drive rf
0031-9007=19=123(3)=033203(5)transitions between two different hyperfine states in thepresence of an alternating magnetic field gradient. The timeaveraged evolution is a rf transition where recoil momentumis transferred. The sign and magnitude of the momentum kickis adjustable via the magnetic fields, and we observe a recoil
momentum for the dressed photon which is 6 × 106 higher
than the (usually negligible) momentum of a bare rf photon
around 8 MHz frequency.
Our scheme shows the power of Floquet engineering: we
combine a rf transition, which has negligible momentum
transfer, with a sinusoidally oscillating magnetic field
gradient, which has no time-averaged momentum transfer,and the result is a rf photon with recoil, depending on how rf
pulses are synchronized with the time-dependent magnetic
field gradient.
Figure 1 shows the time sequence of our scheme, which
consists of a sinusoidal spin-dependent force fðtÞ ¼
gF μB B00 sin½ð2π=TÞt þ ϕrf σ z , where gF is the Lande factor,
μB is the Bohr magneton, and B00 is the magnitude of themagnetic gradient, and a synchronized sequence of short rf
pulses at times t ¼ 0; T; 2T; …. The timing of the pulses
with respect to the periodic force is described by the phaseϕrf which determines the magnitude of the photon recoil.
Each of the rf pulses couples spin-up and spin-down stateswith the same momentary (i.e., at the time of the rf pulse)
velocity vrf. For ϕrf ¼ 0, the velocities averaged over a fullcycle of the oscillating force, hv↑ i and hv↓ i, are different.
By flipping the spin, atoms experience an “extra” half-cycleof the magnetic acceleration [hatched area in Fig. 1(a)],which transfers them to the state with a different averaged
velocity, and, therefore, provides recoil. For the caseϕrf ¼ π=2, the time-averaged velocities for spin-up andspin-down are identical to vrf . Therefore, a rf transition will
not change the time-averaged velocities, and there is no
recoil.

033203-1© 2019 American Physical Society

PHYSICAL REVIEW LETTERS 123, 033203 (2019)(a)Magnetic
gradient

time

(b)

Hamiltonian is not unique and may depend on the initial time
when the periodic drive is switched on. We adopt theapproach of Ref. [2], where the evolution of the quantum
system with periodic drive is expressed by an effective
Hamiltonian independent of initial and final times ti , tf and akick (micromotion) operator K̂, which describes the initialkick due to a sudden switch on and the subsequent micromotion, shown as Ûðtf ; ti Þ ¼ e−iK̂ðtf Þ e−iĤeff ðtf −ti Þ eiK̂ðti Þ .
For our scheme, the time-dependent Hamiltonian of thesystem in the frame rotating with the rf drive after therotating-wave approximation is

Magnetic
gradient

rf pulses( RF=0)

rf pulses( RF

Spindependent
velocity

Spindependent
velocity=

Force fora selectedtrajectory

Force fora selectedtrajectory

FIG. 1. Illustration of our scheme for creating a tunable atomic
recoil momentum with rf transitions using magnetic forces.
Panels (a) and (b) show the experimental conditions for ϕrf ¼ 0
and ϕrf ¼ π=2, respectively. The spin-dependent forces andvelocities are shown (as thick solid lines) for the amplitude of
the wave function which is transferred from spin-down (red) to -up(blue) by the rf pulse marked by the gray dashed line. For ϕrf ¼ 0,
the average velocities hv↓ i and hv↑ i are different, which implies afinite recoil associated with the spin flip. In contrast, hv↓ i ¼ hv↑ ifor ϕrf ¼ π=2, and there is no recoil.

Using this semiclassical picture, we obtain for the amountof momentum transfer ℏk ¼ mðhv↑ i − hv↓ iÞ ¼ ℏk0 cos ϕrf ,where k0 ¼ ðgF μB =πℏÞB00 T. Next we discuss where thechange in kinetic energy comes from. For an opticaltransition with recoil ℏk and an atom moving at initial
velocity vin, the resonance frequency is shifted by the
Doppler shift kvin and recoil shift ðℏkÞ2 =2m, which ensuresenergy conservation. However, in the current situation,energy can also come from the time-dependent magnetic
force. Indeed, if we would apply a single rf π pulse at phase
ϕrf ¼ 0, the time-averaged velocity would change by ℏk0 =m,
but the rf resonance frequency would be independent ofvelocity and k0 . However, if a series of rf pulses is used, as in(Fig. 1), the resonance is Doppler shifted and becomesvelocity selective. This can be seen by regarding the pulses as
Ramsey pulses and considering the phase evolution of the
wave function between two pulses. The rf pulses create asuperposition of spin-up and spin-down. Between pulses, thephase evolution for spin-up or -down
R is2solely determined by
the kinetic energy α↑↓ ¼ ð1=ℏÞ ðmv↑↓ =2Þdt, leading to aphase difference δα ¼ ð1=ℏÞðmðhv↑ i − hv↓ iÞvrf ÞT ¼ kvrf T
after one period of shaking, where vrf ¼ ðv↑ þ v↓ Þ=2 is thecommon velocity at the moment of the rf pulse. Withhv↓ i ¼ vrf − ℏk=2m, we find that for resonant excitation,
the rf has to compensate for this phase shift by the Doppler
detuning khv↓ i and the recoil shift ðℏkÞ2 =2m (seeSupplemental Material for more details [13]).
Periodic Hamiltonians are formally treated by Floquet
theory [2,3,15–17], which provides an expression for aneffective Hamiltonian Ĥeff describing the slow time evolution of the system averaged over the fast micromotion withperiod T. However, in the standard treatment the effective

Ĥ ¼

π
1
p̂2z
þ ℏk0 ẑ sinð2πt=T þ ϕrf Þσ̂ z − ℏδrf σ̂ zT22mXþ ℏΩσ̂ x T δðt − nTÞ;ð1Þnwhere δrf is the rf detuning with respect to the atomic
resonant frequency and m is the atomic mass. The short rf
pulses are represented as a series of delta functions witheffective Rabi frequency Ω.
Through the derivation shown in the Supplemental
Material [13], we obtain an explicit expression for theeffective Hamiltonian and the kick operator defined above: p̂2zĤeff ¼

2 2

ℏδrf
1 ℏ k02m þ 16 m − 2ℏΩe−ik0 cos ϕrf z
2

ℏ2 k2

p̂z
ℏδrf
1
0
ℏΩeik0 cos ϕrf z
2m þ 16 m þ 22πt þ ϕrf :K̂ðtÞ ¼ −ik0 zσ̂ z cos
T


;
ð2Þ

The effective Hamiltonian is identical to the one for a
two-level atom driven by a photon field at frequency ωrf
and with wave vector k, which confirms our discussion
above about recoil momentum and Doppler shift. The term
ℏ2 k20 =16m is the kinetic energy due to micromotion.
We implemented this scheme using a thermal cloud of
approximately 1 × 105 23Na atoms at 380 nK in a crossed
optical dipole trap with trapping frequencies ðωx ; ωy ; ωz Þ ¼2πð98; 94; 25Þ Hz corresponding to Gaussian radii of 19.5,
20, and 68 μm, respectively. The jmF ¼ −1i and jmF ¼ 0i
states of the F ¼ 1 hyperfine manifold of the atoms wereused to form a pseudospin-1=2 system, which will bereferred to as j↑i and j↓i states, respectively. The jmF ¼ 1i
state was decoupled from this two-level system through thequadratic Zeeman effect at a bias field of 11.4 G. Since
there is no micromotion in the “nonmagnetic” jmF ¼ 0i
state, the maximum momentum transfer ℏk0 is reduced by afactor of 2 compared to the discussion above.
The oscillating magnetic force was created by a timedependent 3D quadrupole field. Along the bias field directionz, this provides a 1D periodic force. Orthogonal to the bias
field, the periodic potential is quadratic—there is no net
force, only a (negligible) modulation of the confinement.

033203-2

PHYSICAL REVIEW LETTERS 123, 033203 (2019)
The amplitude of the magnetic field gradient was 48 G=cm ata frequency of 5 kHz, implying a recoil k0 ¼ 0.07kL , whereℏkL is the recoil of the resonant transition at 589 nm, with arecoil velocity ðℏkL =mÞ ¼ 2.9 cm=s.
To resolve Doppler shifts of 200 Hz, sub-mG stability
was needed. Any asymmetry of the periodic magnetic field
gradient leads to a time-averaged dc field gradient resulting
in an inhomogeneous Zeeman shift which had to be
suppressed at the 100 Hz level. Finally, the appliedmagnetic fields were modified by eddy currents in thestainless steel chamber, which had to be accounted for (seeSupplemental Material [13]).
The goal of the experimental demonstration was to showthat the rf transition is now Doppler sensitive due to therecoil transfer. The spin-flip transitions were driven by 4 μslong rf pulses at 8 MHz with a Rabi frequency of 10 kHz
resulting in approximately π=12 pulses and an average Rabi
frequency of Ω ¼ 200 Hz, which is defined as the Rabi
frequency of the pulse times the duty cycle. Since it was notpossible to switch off the shaking coils on microsecond
timescales, the rf pulses had to be applied with the magnetic
shaking present, which required several steps of spatial andtemporal alignments (see Supplemental Material [13]).
The rf pulses and the shaking were applied while theatoms were trapped to ensure that the velocity distribution
is independent of position. In time of flight (TOF), this is nolonger the case, and any residual Zeeman shift gradients
could lead to velocity selection. To avoid broadening of the
Doppler selected velocity groups by the trapping potential,
the total interrogation was chosen to be 1.6 ms, much
shorter than the trap period along the z direction. This timeis also comparable to the coherence time due to the ambient
magnetic field stability. Based on these considerations, weapplied a pulse sequence of 2 ms consisting of 10 magnetic
shaking cycles with 9 rf pulses across them.
The temperature of the cloud was chosen to be highenough that the Doppler width of 3 kHz (FWHM) was larger
than our spectral resolution, mainly Fourier limited to 625 Hz
by the 1.6 ms pulse sequence. Because of the Doppler
shift, different detunings of the rf selected different velocity
groups which were observed in ballistic expansion (Fig. 2).
The width of the observed spin-flipped slices is almost
completely determined by the original spatial size of thecloud since the expansion time of τ ¼ 12 ms was only twice
the inverse of ωz . The TOF was limited by the signal-to-noise
ratio, given the constraints discussed above. Fortunately,even for small TOF, the displacement of the center of thespin-flipped atoms is exactly vτ, which could be accurately
measured as a function of rf detuning, as shown in Fig. 3.
The observed Doppler shift is in agreement with thetheoretical treatment above and confirms that rf photons
have been Floquet engineered to have recoil of k ¼ 0.07kL .
The dependence of the recoil on the rf phase wasdemonstrated by shifting the rf phase from 0 to π
[Fig. 3(b)]. The Doppler shift and therefore the direction(a)Without shaking

With shaking ( rf = 0)

Shaking direction(b)

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

6

4

2

- 0.4

- 0.2

0

0.2

0.4

Position (mm)FIG. 2. Observation of velocity-selective rf transitions. (a)Absorption images of the spin-flipped atoms (i.e., in the jmF ¼−1i state) after 12 ms of TOF with and without magnetic shaking.
The yellow dashed ellipses have major and minor axes obtainedas FWHM of Gaussian fits. After TOF, the thermal could expand
by a factor of 2.13; thus, a single-velocity class is narrower than
the thermal cloud by 1=2.13 ≈ 0.47. The Fourier limit of ourvelocity selection increases this to 0.50, and inclusion of eddy
currents further modifies it to 0.45 (dash-dotted line). The field ofview is 1 × 1 mm2 . (b) Integrated column density distribution
obtained from absorption images like those in (a), for different rf
detunings. The solid lines are Gaussian fits to the data points. The
rf phase was at ϕrf ¼ 0 to maximize Doppler sensitivity. The
asymmetry between the 300 Hz is most likely caused by bias
field drifts (estimated in Fig. 3 to be 70 Hz) or small residual
magnetic field gradients.

of the recoil changed sign. This observation confirmed thatthe selection of slices in Fig. 2 is not due to time-averaged
magnetic field gradients, which do not depend on the rf
phase. We could not experimentally explore ϕrf ¼ π=2,
since this would have required to pulse on the rf at the
maximum field gradient, which would have caused large
spatially dependent detunings.
Our scheme can be used to implement one-dimensional
spin-orbit coupling of ultracold atoms with magnetic forces
and without lasers. The Hamiltonian [Eq. (3)] which we have
implemented is, by a unitary transformation, equivalent to a
Hamiltonian with spin-dependent gauge fields [4]:

2
1
1
ℏδ
Ĥ SOC ¼p̂z − Aσ̂ z þ ℏΩσ̂ x − rf σ̂ z : ð3Þ
2m
2
2
We note that Ref. [14] obtains the same Hamiltonian as
the stroboscopic Floquet Hamiltonian. The gauge field

033203-3

PHYSICAL REVIEW LETTERS 123, 033203 (2019)
(a)
rf

=0

Doppler shift (Hz)
400

200

-0.2

0.2

Velocity (cm/s)
-200

-400

(b)
rf

400
200

-0.2

0.2
-200
-400

FIG. 3. Observation of rf transitions with Doppler shifts. (a),(b) Central velocities of the spin-flipped atomic distribution [as inFig. 2(b)] are shown as a function of rf detuning for ϕrf ¼ 0 and
ϕrf ¼ π, respectively. Shifting the rf phase changes the sign of the
Doppler shift and therefore the direction of the recoil momentum.
The solid line represents the predicted Doppler shifts based on thecalibration of recoil momentum. The dashed line takes into
account the effects of eddy currents (see Supplemental Material[13]). The error bars are purely statistical based on five data
points and correspond to 1 standard deviation. The inferred 1σ
fluctuations for the frequency are 70 Hz.

A ¼ ℏk0 cos ϕrf is equal to the recoil momentum transfer ℏk,which depends on the rf phase ϕrf . Previous experimental
studies claimed the realization of spin-orbit coupling andgauge fields purely by magnetic shaking, without rf transitions [8,18]. These claims are ambiguous based on our
discussion here: without rf coupling, the momentum transfer
and the gauge field are not defined and can be transformedaway with a gauge transformation. According to Eqs. (1)and (3), pure magnetic shaking leads only to a kick operator
for the micromotion, and the effective Hamiltonian is thefree-particle Hamiltonian. Therefore, all observations in
Refs. [8,18] are related to an initial kick and micromotion
and not to a modified effective Hamiltonian.
In the presence of gauge fields, there are two momenta:
the mechanical or kinetic momentum ðpz  12 AÞ and thecanonical momentum pz ¼ mvrf . In our scheme, they canboth be directly observed and have a very transparent
meaning: the kinetic momenta are the time-averaged
momenta mhv↑ i, mhv↓ i. The canonical momentum isthe instantaneous momentum during the rf pulse (seeSupplemental Material [13]).
Our demonstration of rf dressed photons was done witha modest recoil k0 of 0.07kL due to technical limitations(see Supplemental Material [13]). The recoil could havebeen increased by using a glass cell, miniature coils, or
atomic chips for which 12 times larger magnetic field
gradients have been assumed [6]. Given the small valueof k0 , we did not look for recoil effects in Bose-Einsteincondensates, as in Ref. [19], and rather focused on Doppler
shifts in thermal clouds.
Dressed-photon recoil has several features different fromoptical photon recoil: The maximum recoil of a dressed
photon is only technically limited, and can be tuned via thestrength of the magnetic gradient and in principle also via
the rf phase, whereas the recoil in a two-photon transition
can be tuned via the angle between the two laser beams.
Heating for optical recoil transfer is independent of
momentum transfer and depends on the Rabi frequency,
whereas the reverse applies to the magnetic scheme.
As in any Floquet schemes, micromotion can lead toheating when the associated kinetic energy is transferred tothe secular motion by elastic collisions between the two
spin states (which is equivalent to transitions between
Floquet states of different quasienergies). In Ref. [20],we describe various cases (high and low temperature limit,
bosons, and fermions) and conclude that the energy transfer
can be expressed by E_ ∝ ρσvcol E0, where vcol reflects aneffectivepdensity
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ of states. For Bose-Einstein condensates,
vcol ¼ ℏω=m for large modulation frequencies ω, implemented in our experiments, and vcol ¼ ℏk0 =m in the semiclassical regime. Here, ρ is the density and σ is the two-bodys-wave scattering cross section. For a sodium condensate
with two spin components and density ρ∼1014 cm−3, we
observed a condensate lifetime of ∼8 s at k0 ¼ 0.05kL ,consistent with weak Floquet heating [20]. If the momentum
transfer k0 is scaled up to kL , losses increase proportional
to the second or third power of k0 , depending on the regime[20]. For momentum transfers of kL , lifetimes larger than
100 ms will require low density clouds on the order of∼1012 cm−3 or small scattering lengths. For degenerate
Fermi gases with EF ≫ ℏω, heating is Pauli suppressedby a factor of ðℏω=EF Þ2 .
There are possible extensions of generating magnetic
recoil. One is to use the time-averaged, orbiting potential(TOP) trap configuration [21], where a constant gradient iscombined with a rotating bias field in the x-y plane, which
creates a rotating force. A sequence of rf pulses generates“dressed photons” with recoil k along the cos ϕrf ex þsin ϕrf ey direction. The rf phase now controls the directionof the recoil. The concept of dressed rf photons should be
useful for a more general class of quantum simulations. For
instance, it applies to spin-dependent forces created by thevector ac Stark shift. Using focused laser beams or lattices tocreate spin-dependent potentials, the effective recoil is
spatially localized, and can easily be time modulated. In
comparison to magnetic field gradients, the forces due to thevector ac Stark shift can be larger, and much faster modulation frequencies are possible. Using optical spin-dependent

033203-4

PHYSICAL REVIEW LETTERS 123, 033203 (2019)
forces instead of magnetic forces may eliminate several
limitations of our scheme, especially for atoms like cesium
and rubidium, where large spin-dependent optical forces canbe realized without major heating by spontaneous scattering.
In conclusion, we demonstrated how magnetic shaking
can be used to endow a rf photon with large and tunable
recoil. This scheme illustrates many aspects of Floquet
engineering, including heating in both the quantum andclassical limit. This technique is a building block forquantum simulations including spin-dependent gauge
fields and measurements such as Doppler velocimetry.
It can be applied to any atom or molecule with nonzero
spin in the ground state, and is independent of the structureof electronically excited states.
We would like to acknowledge Will Lunden for experimental assistance, Ivana Dimitrova for critical readingof the manuscript, and Viktor Novičenko for discussions.
We acknowledge support from the NSF through the Centerfor Ultracold Atoms and Grant No. 1506369, from AROMURI Non-equilibrium Many-body Dynamics (Grant
No. W911NF-14-1-0003), from AFOSR-MURI Quantum
Phases of Matter (Grant No. FA9550-14-1-0035), fromONR (Grant No. N00014-17-1-2253), and a VannevarBush Faculty Fellowship. Part of this work was performedat the Aspen Center for Physics, which is supported by NSF
Grant No. PHY-1607611.

*

jwlee0705@gmail.com
Present address: The Institute for Advanced Study, The
Hong Kong University of Science and Technology, Clear
Water Bay, Kowloon, Hong Kong.
†
Present address: JILA, NIST and University of Colorado,
440 UCB, Boulder, Colorado 80309, USA.
‡
B. S. and J. L. contributed equally to this work.
[1] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,
Atom-Photon Interactions: Basic Process and Applications(Wiley, New York, 2008).

[2] N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027(2014).
[3] M. Bukov, L. D’Alessio, and A. Polkovnikov, Adv. Phys.
64, 139 (2015).
[4] J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, Rev.
Mod. Phys. 83, 1523 (2011).
[5] V. Galitski and I. B. Spielman, Nature (London) 494, 49(2013).
[6] B. M. Anderson, I. B. Spielman, and G. Juzeliūnas, Phys.
Rev. Lett. 111, 125301 (2013).
[7] Z.-F. Xu, L. You, and M. Ueda, Phys. Rev. A 87, 063634(2013).
[8] X. Luo, L. Wu, J. Chen, Q. Guan, K. Gao, Z.-F. Xu, L. You,and R. Wang, Sci. Rep. 6, 18983 (2016).
[9] For relevant work with modulated magnetic field in opticallattices, see Refs. [10–12].
[10] J. Struck, J. Simonet, and K. Sengstock, Phys. Rev. A 90,
031601(R) (2014).
[11] G. Jotzu, M. Messer, F. Görg, D. Greif, R. Desbuquois, andT. Esslinger, Phys. Rev. Lett. 115, 073002 (2015).
[12] J. Yu, Z.-F. Xu, R. Lü, and L. You, Phys. Rev. Lett. 116,
143003 (2016).
[13] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.123.033203 for thederivation of the effective Hamiltonian, which includes
Refs. [2,6,14].
[14] See Supplemental Material in X. Luo, L. Wu, J. Chen, Q.
Guan, K. Gao, Z.-F. Xu, L. You, and R. Wang, Sci. Rep. 6,
18983 (2016).
[15] A. Eckardt and E. Anisimovas, New J. Phys. 17, 093039(2015).
[16] V. Novičenko, E. Anisimovas, and G. Juzeliūnas, Phys. Rev.
A 95, 023615 (2017).
[17] A. Eckardt, Rev. Mod. Phys. 89, 011004 (2017).
[18] L.-N. Wu, X.-Y. Luo, Z.-F. Xu, M. Ueda, R. Wang, and L.
You, Sci. Rep. 7, 46756 (2017).
[19] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature(London) 471, 83 (2011).
[20] J.-R. Li, B. Shteynas, and W. Ketterle, arXiv:1906.08747.
[21] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell,
Phys. Rev. Lett. 74, 3352 (1995).

033203-5

Supplemental Material for“How to dress radio-frequency photons with tunable momentum”(Dated: June 21, 2019)

EXPERIMENTAL IMPLEMENTATION OF MAGNETIC SHAKING

To realize magnetic shaking, we drove a sinusoidal current through a pair of anti-Helmholtz coils along the x-axis
while there was a fixed bias field of 11.4 G aligned to the z-axis. The sinusoidal current was provided by a DC power
supply and four insulated gate bipolar transistors connected in an H-bridge configuration. The transistors created asquare wave voltage modulation, which resulted in a sinusoidal current due to the frequency response of the coils. A
capacitor was connected in series to eliminate the imaginary component of the impedance coming from the inductanceof the coils. The amplitudes of the current and the voltage were 70 A and 70 V. The real impedance of 1 Ω is mainly
due to eddy currents in the stainless steel vacuum chamber and is much larger than the resistance of the coils (0.1 Ω).
The combined fields result in a periodic 1D magnetic force along the z-axis Bz0 (t) = B00 sin(ωt + φRF ), where B00 = 48
G/cm, ω = 2π× 5 kHz, and φRF is determined from the relative phase between the magnetic gradient modulation andthe radio frequency (RF) pulses. Larger recoil momenta can be realized by either increasing B00 or decreasing ω. The
field gradient was calibrated using Stern-Gerlach deflection during ballistic expansion of a Bose-Einstein condensate.
We also calibrated the recoil k0 directly by measuring the momentum transfer to a cloud in the |mF = −1i state
during a half-cycle of the magnetic shaking. The two calibrations agreed to within the accuracy of measurement.
ADJUSTMENTS TO THE MAGNETIC FIELD PROFILE

The observation of Doppler shifts at the 200 Hz level required careful control of Zeeman shifts. Three criticaladjustments were done.
(1)Symmetry of the modulated magnetic field gradient: If inhomogeneous Zeeman shifts across the cloud are comparable or larger than Doppler shifts, the spinflips are no longer velocity selective since there is always a local Zeeman
shift to compensate for the Doppler shift. Therefore, the magnetic field gradient averaged over one modulation cycle
hB 0 i, had to be zeroed: gF µB hB 0 iD  kv, where D is the length of the cloud. To avoid transient asymmetries fromthe turn-on process of the periodic magnetic gradient, we added a pre-shaking period of 3 ms before the spectroscopic
sequence. This didn’t affect the trapped atom cloud, since the atoms were initially in the non-magnetic |mF = 0i
state. After the pre-shaking, we achieve hB 0 i ≈ 20 mG/cm, implying a time-averaged differential Zeeman shift acrossthe cloud of less than 100 Hz. hB 0 i was determined from converting the measurement of time-averaged current asymmetry to the time-averaged magnetic gradient asymmetry using the Stern-Gerlach calibration. As a final check, we
added asymmetries on either the positive or negative side of the sinusoidal current to create hB 0 i ≈ ± 100 mG/cm,
and for both cases observed a slight increase in the width of the velocity-selected atom slice confirming that theresidual asymmetry of the magnetic gradient modulation was negligibly small.
The following two adjustments addressed the issue that the RF pulses were not delta functions, but had a durationof 4 µs. The presence of Zeeman shifts comparable or larger than the Fourier width of a single pulse would reduce
the RF pulse area. For our parameters, a 45 kHz detuning will reduce the pulse area by 5 percent (and therefore thesingle pulse excitation probability by 10 percent).
(2)Minimize modulation of magnetic bias field : The time-dependent gradient creates also a time-dependent bias
field given by the gradient times the displacement of the atoms from the origin of the magnetic quadrupole field. 60µm away from the origin, the bias field changes by 30 mG during the 4 µs RF pulse. To minimize the reduction of
the RF pulse area, the optical trap was aligned with the center of the quadrupole field to within 1 µm. This wasdone by minimizing the shift in the RF resonant frequency when a stationary gradient field was added to the constantmagnetic bias field. In addition, the eddy currents created a time-dependent bias field, which was compensated by
RF detuning. The detuning and the timing of the RF pulses (described below) were adjusted together in order tomaximize the fraction of spin-flipped atoms.
(3)Timing of the RF pulses with respect to the magnetic modulation: The goal was to pulse on the RF while themagnetic field gradient crosses zero. A 5 µs offset would imply a gradient of 7.5 G/cm and a differential magnetic
field along the cloud of 50 mG. In the presence of strong gradients, the short RF pulse is resonant only for a smallpart of the cloud. Therefore, we could find the optimum condition by scanning both the timing and the detuning of

2
the RF pulses until the measured total fraction of the spin-flipped atoms is maximized. The optimum time was offset
by 2 µs from the zero-crossing of the current through the gradient coils, possibly due to eddy currents.
To summarize, we optimized three parameters, which are trap position, timing of the RF pulse, and RF detuning.
The optimal position minimizes temporal variation of the bias field, optimal timing of RF minimizes B’ during thepulse, and optimal detuning compensates for any bias field at the time of the pulse.
EFFECTS OF INDUCED EDDY CURRENTS

The modulated magnetic field gradient B 0 (t) = B00 sin ωt induced eddy currents in the stainless steel vacuum
chamber. From our observations, we inferred that the main effect was caused by an induced oscillating bias field~ ec (t) = Bec sin (ωt + φ)ey along y with the same modulation frequency ω and a relative phase delay φ. ThisB
oscillating bias field led to a y-component of the oscillating force. As a result, the effective recoil and velocity selection
are tilted away from the z direction, and the selected velocity slices are rotated in the y − z plane.
ɸRF= #, kz < 0ɸRF= 0, kz > 0(a)+60˚ tilt(b)-60˚ tilt

Total bias field
Oscillating Byθ
B0= 6 G
(Small γ)

Bz
y
z

(c)

+40˚ tilt(d)-40˚ tilt

Oscillating By

B0=11.4 G(Large γ)

Bz

FIG. S1: Effect of eddy currents on observed velocity-selected atom slices. The induced bias field along ey led to a y-componentof the oscillating force, resulting in velocity selectivity in ey and therefore tilting of the resonant velocity slice in the y − zplane. The tilt angle depends on the static bias field B0 and the RF phase φRF . The dashed lines are guides to the eye.

In a simplified model, the total magnetic field experienced by the atoms is~ = [B0 + B 0 sin (ωt)z] ez + [Bec sin (ωt + φ) + B 0 sin (ωt)y] ey − 2B 0 sin (ωt)xex ,
B
0
0
0

(S1)

with a magnetic field strength
q
2
2
2
~
|B| = [B0 + B00 sin (ωt)z] + [Bec sin (ωt + φ) + B00 sin (ωt)y] + [2B00 sin (ωt)x]q
B00 z + γ sin (ωt + φ)B00 y≈ B0 1 + γ 2 sin2 (ωt + φ) + q
sin (ωt)
1 + γ 2 sin2 (ωt + φ)(S2)

here γ = Bec /B0 . The first term corresponds to a time varying homogeneous bias field resulting in a velocityindependent effective detuning of the RF transition. The oscillating magnetic field gradients along the z and y3directions are
~
B00
∂|B|
=q
sin (ωt),∂z
1 + γ 2 sin2 (ωt + φ)

~
∂|B|
γB00
=q
sin (ωt) sin (ωt + φ).
∂y
1 + γ 2 sin2 (ωt + φ)(S3)It should be noted that the gradient in ey oscillates at 2ω, twice the frequency of the driving.
The phase delay φ is determined by the magnetic properties of the vacuum chamber. We modeled the chamber
as an LC circuit with a self inductance LCh and a resistance RCh , and obtain φ = arctan (ωLCh /RCh ) + π/2. Our
observations imply RCh  ωL, φ ≈ π/2, resulting in an effective recoil component in the y direction with
!
Z t
Z
γB00
1 T
p
sin (ωt0 ) cos (ωt0 ) dt0 dt
kso,y =
T 0
1 + γ 2 cos2 (ωt0 )
0
!
(S4)
Z t
Z
γB00
1 T
0
0
p
sin (2ωt ) dt dt.
=
2
2
0
T 0
0 2 1 + γ cos (ωt )Consequentially, the Doppler shift is modified asδω = ky vy + kz vz ,(S5)

directly observed as a rotation of the velocity slice with an angle θ = arctan (ky /kz ) in the time-of-flight images, as
shown in Fig. S1.
~0
We verified two predictions of this model: the angle θ of the rotation decreased with stronger static bias field B
which lowered γ (Fig.S1a. and Fig.S1 c). Due to the 2ω oscillating frequency of the y force, ky did not change signwhen the RF phase φRF was shifted from 0 to π in contrast to kz , and therefore the rotation angle flipped from θ to−θ, as suggested by Eq. S5 and shown in Fig. S1.
In the future, the effects of the induced eddy current can be suppressed by using an even stronger static bias field~ 0 or by conducting the experiment in a glass cell.
B
What we have described so far applies to free space or to an isotropic trap. However, the optical trap in theexperiment is anisotropic. For zero time-of-flight, in the y − z plane, the minor axis of the ellipsoidal cloud is oriented
along y, θ = π/2. For long time-of-flight, the angle is solely determined by the velocity selection θ = arctan (ky /kz ).
For intermediate time-of-flight, as used in the experiment, the observed angle interpolates between these values. We
calculate that the observed tilt angles of 60◦ and 40◦ (Fig.S1) correspond to tilt angles of the bias field arctan (ky /kz )of 53◦ and 32◦ , respectively.
The observed tilt angles were used to infer the induced eddy currents. Equation S4 provided the dashed line forthe predicted recoil k in Figure 3 of the main text.
BLOCH SPHERE REPRESENTATION OF MAGNETIC SHAKING AND RF PULSESThe evolution of the quantum system under magnetic shaking and RF pulses can be visualized using the Bloch
sphere (Fig. S2). In the frame rotating at the atomic RF resonance frequency ω0 , each RF pulse of area β rotates the
Bloch vector around the y-axis by an angle β. In the absence of magnetic shaking, subsequent pulses would continue
the rotation all the way down to the south pole of the Bloch sphere and up again, resulting in Rabi oscillations at a
rate β/(2πT ). However, due to the phase evolution discussed in the main text, the Bloch vector rotates around thez axis by an angle δα, and therefore, the following RF pulse increases the polar angle by less than β. After several
cycles, the Bloch vector returns to the north pole without having ever reached the south pole, realizing off-resonant
Rabi oscillations (Fig.S2(a)). However, if the RF frequency is shifted by the Doppler and recoil shift, the Bloch vector
reaches the south pole again. In contrast, for the phase φRF = π/2, kinetic energies of the coupled spin up and downstates are the same, irrespective of velocity, and therefore all atoms perform resonant Rabi oscillations (Fig. S2(b)).
It should be noted that the evolution of the atomic wavefunction is the same if the RF frequency is detuned by an
integer multiple of 2π
T , similar to the situation in Ramsey spectroscopy.

4

(a)

›

›

ɸRF= !/2(b)ɸRF= 0›Off-resonant›

On-resonant

FIG. S2: Bloch sphere representation of magnetic shaking and RF pulses. (a) & (b) Trajectories on the Bloch sphere for several
periods of magnetic shaking (green solid lines representing α from the main text) and RF pulses (yellow solid lines representingβ) for φRF = 0 and φRF = π/2, respectively. Fig. (a) shows the trajectories for atoms with a finite initial velocity when the
RF frequency is at ω0 , the atomic resonance, and when it is detuned by the Doppler and recoil shift. In (b), the RF frequency
is at ω0 , the trajectory is independent of the atomic velocity, and there is no net rotation around the z-axis during a magnetic
shaking cycle. The red (blue) dot represents the initial (final) spin state.

DERIVATION OF EFFECTIVE HAMILTONIAN

In our scheme, the Hamiltonian in the frame rotating with the RF drive isp̂2
1
1
Ĥ(t) = z − δRF σ̂z + ωk0 ẑ sin
2m 2
2



2π
t + φRFTσ̂z + Ωσ̂x TXnδ (t − nT ) ,(S6)

where we have applied the rotating-wave approximation and set ~ = 1. To deal with the dynamics of such a periodicallydriven system we shall apply two alternative approaches described below.
To eliminate the spin-dependent potential slope featured in the Hamiltonian (S6), we go to the spin-dependent
˜ = R̂† (t) |ψ(t)i, similar
co-moving frame via a time-dependent unitary transformation to the new state-vector |ψ(t)iz
to the one used in refs. [1, 2]:R̂z (t) = exp [−ik0 zγ (t) σ̂z /2] ,γ (t) = ω

Zt
0

sin



2π 0
t + φRFT

dt0 − C = − cos

2πt + φRF
T



,

(S7)

where the integration constant C entering γ (t) has been taken to be C = cos φRF , so that γ (t) averages to zero overa period. The reason for this choice will be discussed later on.
At the RF pulses where t = nT the transformation R̂z (nT ) = exp [ik0 z cos φRF σ̂z /2] describes a spin rotation by anˆ (t) = R̂† (t) Ĥ R̂ (t)−iR̂† (t) ∂ R̂ (t)angle k z cos φaround the z axis. As a result, the transformed Hamiltonian H̃
0

RF

z

z

z

t

z

takes the form
ˆ (t) = 1
H̃
2m



1
p̂z − k0 γ (t) σ̂z
2

2

X
1
− δRF σ̂z + Ω [cos (k0 z cos φRF ) σ̂x + sin (k0 z cos φRF ) σ̂y ] Tδ (t − nT ) . (S8)2nNote that unlike the spin-dependent potential gradient featured in the original Hamiltonian (S6), the oscillating
momentum shift term k0 γ (t) σ̂z /2 is no longer proportional to the driving frequency and hence can be considered as5a small perturbation in the limit of high frequency driving where k0 γ (t)  ω and also Ω  ω. In that case it isappropriate to describe the evolution of the system in terms of the zero-order effective Hamiltonian obtained by timeˆ (t) over a single driving period, i.e. by the zero frequency component of the Hamiltonian H̃ˆ (t), giving
averaging of H̃
Ĥef f =

1
1 k02
p̂2z
− δRF σ̂z + Ω cos (k0 z cos φRF ) σ̂x + Ω sin (k0 z cos φRF ) σ̂y +
,
2m 2
16 m

(S9)

where the momentum shift has averaged to zero. The effective Hamiltonian can be represented in a matrix form as:
!
2
p̂2z
1 k0
1
−ik0 z cos φRF
+
−
δ
Ωe
RF
2
Ĥef f = 2m 16 m
.
(S10)
2
p̂2z
1 k0
1
Ωeik0 z cos φRF
2m + 16 m + 2 δRF
The full dynamics includes also the micromotion. In the present situation there are two origins of the micromotion.
ˆ (t). However, in the limit of the large
The first kind comes from the time-dependence of the transformed Hamiltonian H̃
driving frequency this kind of micromotion is negligibly small compared to the second type of micromotion emerging
due to the time-dependence of the unitary transformation R̂z (t). In fact, returning to the original representation˜|ψ(t)i = R̂z (t) |ψ(t)i,one arrives at the following time-evolution of the state-vector from the initial to the final time|ψ(t2 )i = Û (t2 , t1 ) |ψ(t1 )i ,whereÛ (t2 , t1 ) = R̂z (t2 ) e−iĤef f (t2 −t1 ) R̂z† (t1 ) ,(S11)where the unitary transformation R̂z (t) represents a micro-motion operator.
The time evolution operator can be rewritten asÛ (t2 , t1 ) = e−iK̂(t2 ) e−iĤef f (t2 −t1 ) eiK̂(t1 ) ,whereK̂ (t) = −k0 zσ̂z cos

2π
t + φRF
T



(S12)

(S13)

is a Hermitian micromotion (kick) operator. The choice of the integration constant C = cos φRF in the unitarytransformation (S7) ensures that the micromotion operator K̂ (t) averages to zero over the driving period. Thus, theeffective Hamiltonian and the micromotion operators are defined in a unique way through the condition C = cos φRF .
We now rederive the same effective Hamiltonian and micromotion operator using a rigorous high-frequency 1/ω
expansion. Appendix K of ref. [3] discusses Hamiltonians of a general formĤ(t) = Ĥ0 + Âf (t) + ω B̂g(t)(S14)

and derives expansions for an effective Hamiltonian Ĥef f and the kick operator K̂.

Ĥef f =

∞
X
1

ω
n=0

(n)

Ĥ , K̂(t) =n ef f

∞
X
1

ωn
n=0

K̂ (n) (t)(S15)

The Hamiltonian Ĥ in equation (S6) is of this form withp2
1
− δRF + Ωσ̂x ,2m 2
1
B̂ = k0 ẑσ̂z ,g(t) = sin(ωt + φRF ),2
XÂ = Ωσx ,f (t) = Tδ(t − nT ) − 1

Ĥ0 =

(S16)
(S17)
(S18)

n

Functions f (t) and g(t) meet the requirement of having zero mean value over a period T .
The kick operator is in 0-th order:K̂(0)= B̂G(t),G(t) = ωZtg(τ )dτ = − cos(ωt + φRF )

(S19)

6
The effective Hamiltonian to the lowest order in 1/ω can be expanded asĤef f = Ĥ0 +

X in

X in
Gn f [B...[B, A]] +Gn [B...[B, H0 ]] + O(1/ω)| {z }| {z }
n!
n!
n=1
n=1
n

(S20)

n

After calculating all commutators and time-averaged coefficients before them, and grouping the terms proportionalto σ̂x and σ̂y , the expansion reduces toĤef f =

p̂2z
1
1 k02
− δRF σ̂z + Ω cos(k0 z cos φRF )σ̂x + Ω sin(k0 z cos φRF )σ̂y ++ O(1/ω)
2m 2
16 m

(S21)

The resulting effective Hamiltonian and micromotion operator are in exact agreement with the above equations(S9), (S10) and (S13).
If we apply an additional spatially-dependent unitary transformation R̂z1 = exp [−izk0 cos φRF σ̂z /2] corresponding
to the choice C = 0 of the constant in Eq.(S7), the transformed Hamiltonian becomes translationally invariant andacquires the standard form of HSOC for one-dimensional spin-orbit coupling:
ĤSOC =1δRF
1(p̂z − k0 cos φRF σ̂z )2 + Ωσ̂x −
σ̂z ,
2m
2
2

(S22)

where the spin-orbit coupling strength is described by the momentum shift k0 cos φRF /2. With the new kick operator




2π
K̂SOC (t) = −k0 zσ̂z cos
t + φRF − cos φRF(S23)
T
the time evolution can be written as
U (t2 , t1 ) = e−iK̂SOC (t2 ) e−iĤSOC (t2 −t1 ) eiK̂SOC (t1 ) .

(S24)

In that case the operator K̂SOC (t) has a non-zero temporal average, so it cannot be treated as a pure micromotion
operator. Similarly ĤSOC can not be considered as an effective Hamiltonian for the time-periodic Hamiltonian (S6). It†
is rather a Hamiltonian related to the true effective Hamiltonian by the unitary transformation: ĤSOC = R̂z1
Ĥef f R̂z1 .
Note that ref. [2] has obtained ĤSOC as a stroboscopic Floquet Hamiltonian after applying a unitary transformationof the form (S7) with C = 0.
MECHANICAL AND CANONICAL MOMENTA

The difference between canonical and mechanical momenta can be seen on the dispersion relation (Fig.S3). Using
canonical momentum all couplings and transitions between the two spin states are vertical. The dashed lines illustrate
Energy››z

Canonical
momentumz

FIG. S3: Energy-momentum dispersion relations for spin-orbit coupled spin 1/2 states. The two minima are separated by therecoil momentum ~k. The vertical dashed arrows show spinflip transitions. Their lengths are given by the Doppler and recoil
shifts.

7
the transitions observed in our experiment. Away from the spin gap the energy separation is dominated by Doppler
and recoil shifts. Energy-momentum dispersion relations for spin-orbit coupled spin 1/2 states. The two minima areseparated by the recoil momentum ~k. The vertical dashed arrows show spinflip transitions. Their lengths are givenby the Doppler and recoil shifts.

[1] B. M. Anderson, I. B. Spielman, and G. Juzeliūnas, Phys. Rev. Lett. 111, 125301 (2013), URL https://link.aps.org/
doi/10.1103/PhysRevLett.111.125301.
[2] X. Luo, L. Wu, J. Chen, Q. Guan, K. Gao, Z.-F. Xu, L. You, and R. Wang, Scientific Reports 6, 18983 EP (2016),
supplementary material, URL http://dx.doi.org/10.1038/srep18983.
[3] N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027 (2014), URL https://link.aps.org/doi/10.1103/PhysRevX.4.
031027.

Floquet Heating in Interacting Atomic Gases with an Oscillating Force
Jun-Ru Li,1, ∗ Boris Shteynas,1 and Wolfgang Ketterle1
1

arXiv:1906.08747v1 [physics.atom-ph] 20 Jun 2019

Research Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms,
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Dated: June 21, 2019)
We theoretically investigate the collisional heating of a cold atom system subjected to timeperiodic forces. We show within the Floquet framework that this heating rate due to two-bodycollisions has a general semiclassical expression P ∝ ρσvcol E0 , depending on the kinetic energy
E0 associated with the shaking, particle number density ρ, elastic collision cross section σ, and an
effective collisional velocity vcol determined by the dominant energy scale in the system. We further
show that the collisional heating is suppressed by Pauli blocking in cold fermionic systems, andby the modified density of states in systems in lower dimensions. Our results provide an exactly
solvable example and reveal some general features of Floquet heating in interacting systems.
I.

INTRODUCTION

Engineering novel Hamiltonians is central to quantum simulations. In general, Hamiltonians can be implemented directly and statically, or in a time-averaged
way. The latter implies periodic driving of the system. If
the fast modulation can be neglected, an effective timeaveraged static Hamiltonian is realized as formally captured by Floquet theory [1, 2]. With proper driving,
dynamically generated Floquet Hamiltonians can be designed. Such Floquet systems potentially exhibit novel
properties which are difficult or impossible to be realized in static settings. Examples include synthetic gauge
fields [3–6], spin-orbit coupling [7–9], and topologicalbands and materials [10–13]. Experimental progress includes creation of the Hofstadter-Hamiltonian in opticallattices for neutral atoms [14–17], realization of the topological Haldane model with shaken optical lattice [18],and the demonstration of dressed recoil momentum forradio-frequency photons in ultracold gases with modulated magnetic fields [19].
However, higher order terms beyond the time average, related to fast micromotion, can cause heating via
interactions, limiting experimental studies of many-body
physics. In general, a driven system constantly exchanges
energy with the driving field. Interactions redistribute
this energy into other degrees of freedom, leading to anincrease of the total entropy and energy. Although thisheating can be suppressed in specific scenarios, e.g. via
many-body localization [20–22], a generic closed quantum system will eventually thermalize at infinite temperature when driven [23], limiting the experimental studies
of many-body Floquet systems. Therefore, understanding and potentially controlling the heating in Floquet
systems has triggered both theoretical [22–28] and experimental efforts [29, 30].
The dynamics of a Floquet system are studied with
Floquet theory. Systems heat by absorbing energy from∗ Present

address: JILA, Department of Physics, Universityof Colorado, 440 UCB, Boulder, Colorado 80309, USA.;junru.li@colorado.edu

the driving field in multiples of the energy quanta related
to the modulation frequency ω, caused by the scatteringof the driven particles [26]. The heating rate reads
XP=Γn n~ω,(1)nwhich is determined by the transition rates Γn for theprocesses of absorption/emission of n energy quanta. In
this description, the energy exchange is quantized.
On the other hand, in the limit of low modulation frequency where the system’s intrinsic energy scales dominate, the quantization of the driving field should not havea prominent effect. The system’s behavior can be described semiclassically. As a result, it is anticipated thatthe heating dynamics of a Floquet system have a corresponding semiclassical counterpart in this low-frequency
regime. Moreover, the quantized and the semiclassicaldescription should exhibit a continuous crossover as afunction of the modulation frequency ω and amplitude.
In this work, we investigate the Floquet heating andthe crossover between the quantum and semiclassicalregimes for systems subjected to periodic forcing in freespace, motivated by the recent experimental demonstration of Floquet-dressed recoil momentum for photons ina two-spin mixture of cold gases [19] where the two spinsare shaken relative to each other. Such a setting is the key
ingredient of many Floquet schemes proposed for generating synthetic gauge fields and topological matter [2, 7–10]. The corresponding semiclassical description of theheating in such a system is the following: the force modulates the particles’ velocities and consequently generates
extra kinetic energy E0 . This micromotion energy E0 canbe transferred into the secular motion of the particles
via inter-particle collisions when the micromotion is outof-phase for the colliding particles, causing an increaseof the system’s total energy, and consequently heating.
The resulting heating rate can be estimated with the twobody elastic collision rate ρσv and the associated energy
E0 as
P ∝ ρσvE0(2)with the atomic density ρ, elastic collisional cross section
σ, and the relative speed of the two particles v. This

2
heating rate is continuously variable depending on thestrength of the driving, characterized by E0 , and is independent of the driving frequency ω, which seems tocontradict the Floquet description of quantized energy
transfer.
In this paper, we calculate the collisional heating rates
in periodically shaken atomic gases with a full Floquet
treatment. We identify several distinct regimes determined by the energy hierarchies in the system and showthat the semiclassical and the Floquet picture are twolimiting cases of a unified general description of the heating rate as P ∼ ρσvcol E0 . The key parameter vcol is aneffective collisional velocity parametrizing the final density of states. Thisp can be, for example, the averaged
thermal velocity
kb Tp/m with kb being the Boltzmannpconstant, ~ω/m, or E0 /m, depending on the dominant energy scales. In addition, we show that collisional
heating is suppressed in a cold fermionic system by Pauliblocking, and due to the modified density of states insystems in lower dimensions.
The paper is organized as follows: Sec. II is a concise
review on the Floquet theory and scattering of FloquetBloch states, which serves as the theoretical basis for themain results presented in Sec. III. We first analyze theFloquet heating for two atoms in Sec. III A. We thenanalyze different regimes of the collisional heating in
Sec. III B and subsequently extend the analysis to atomic
ensembles in Sec. III C, including a specific discussion
on fermionic systems in Sec. III D. A discussion of heating rates in lower-dimensional systems is presented inSec. III E, followed by a summary and outlook in Sec. IV.

II.

FLOQUET THEORY AND FLOQUETHEATING

Our work is based on Floquet theory, which describes
the evolution of a periodically driven system. Evolutionof a Floquet system has been studied in different scenarios with different approaches, for example through highfrequency expansion [1, 2, 31], Floquet-Magnus expansion [32], and extended Hilbert space [33]. We summarize
here the basic concepts and formalism in Floquet theory
and the scattering of the Floquet-Bloch states. This section mainly follows the description in Ref. [26]; Comprehensive discussions can be found in Refs. [2, 26, 32, 33].

A.

General Aspects of Floquet Theory

Floquet theory describes the behavior of a system governed by a time-periodic Hamiltonian Ĥ(t + T0 ) = Ĥ(t).
This temporal translational symmetry allows simple descriptions of the time evolution. Solutions of the timedependent Schrödinger equationĤ(t) |Φ(t)i = i~∂t |Φ(t)i ,(3)known as Floquet-Bloch states, can be decomposed into
Fourier modes asX|Φ(t)i =e−iEt/~+ilωt Cl |φl i .
(4)lHere, ω = 2π/T0 is the modulation frequency and E isthe eigenenergy of the corresponding non-driven system.
The amplitude of each of the Fourier modes Cl generally depends on parameters such as the strength and thefrequency of the driving.
The system does not conserve energy, due to the external drive. In the literature, two different conventions are adopted to describe the energy structure ofsuch a system [26]. Some authors define quasienergies
E q = E mod ~ω lying between (−~ω/2, ~ω/2). Others distinguish between the carrier energy E, describing
the secular motion, and the energy sidebands E ± l~ω,
describing the micromotion. This distinction can be understood by considering an adiabatic ramp of the amplitude of the driving. In this work, we adopt the second
convention.
B.

Scattering of Floquet-Bloch States

The dressed energy sidebands of the Floquet-Blochstates modify the scattering between two states caused
by interactions. Scattering can occur not only between
the carriers but also from the carrier of the initial state
to the sidebands of the final state. In the latter case, thefinal and the initial carrier energy are different by multiples of the energy quanta ~ω, representing the energy
exchange between the driving field and the system via
scattering. This process is formulated with the so-called
Floquet Fermi’s golden rule [26]. The transition amplitude between two Floquet-Bloch states is calculated using time-dependent purterbation theory [26]:Z
i t 0
A(i → f, t) = −dt hΨf (t0 )| V̂ |Ψi (t0 )i~ 0Z
i X t 0 i[Ef −Ei +~(p−q)ω]t/~ p,q
=−
dt e
V ,
~ p,q 0(5)where V p,q = hφqf | V̂ |φpi i is the coupling between two
Fourier modes p, q belonging to the final and the initialstate respectively via, for example, collisions. The corresponding transition rate is readily obtained as|A(i → f, t)|2
Γ(i →f ) = lim
t→∞
t


X
X
2π

=
V l,l+n V ∗m,m+n  δ(Ef − Ei − n~ω).
~ n
l,m

(6)

The sum over the index n explicitly reveals an important feature of the scattering between two Floquet-Bloch3states. In the n = 0 scattering channel, the initial and final states have the same carrier energy, and therefore no
net energy is exchanged between the colliding particles
and the driving field (Floquet elastic processes), which
resembles the conventional elastic scattering. Scatteringchannels with n 6= 0 characterize the processes where theenergy of the atomic system is changed by exchanging
n quanta with the driving field (Floquet inelastic processes), leading to Floquet heating.

C.

We define the heating rate P of the system as the rateof the average increase in the system’s total energy Ėtot ,which can be expressed with the scattering rate Γn andthe related energy transfer n~ω asP=

1.

Single-particle Floquet-Bloch states

The Hamiltonian we consider isĤ(t) =

~2 k̂2+ ~k0 zσ̂z sin (ωt + φ),2m1Ψk (r, t) = √ exp [ik(t) · r − iΦ(t)],VΓn n~ω.

(8)

(7)

n

COLLISIONAL HEATING FOR PERIODICFORCES

We apply the method described in Sec. II to the system of interest: a spin mixture of atoms with different
magnetic moments for a periodically modulated magnetic
field gradient, as implemented in Ref. [19] (Fig. 1). For
simplicity, we assume the two spins to have equal but
opposite magnetic moments, such that they experience
opposite forces. We first derive the exact analytic formof the corresponding Floquet-Bloch state wave function,then calculate the collisional heating for a single pair ofatoms with opposite spins, before generalizing the results
to atomic ensembles. In this section, we focus on threedimensional systems. The results are extended to lower
dimensions in Sec. III E.

(9)

where
~k(t) = ~k + ~k0 êz

As shown above, heating of a periodically driven system originates from the absorption of energy from thedriving field through inter-particle interactions. The
heating rate of a Floquet system can be calculated byfirst finding the exact Floquet-Bloch state wave function. Then the energy exchange rate can be obtainedby calculating the transition rates for all quantized absorption/emission processes and their associated energychange. Generally, the explicit form of the wave functionof the Floquet-Bloch state |Φ(t)i is obtained by inserting
Eq. (4) into Eq. (3) and solving the infinite number ofcoupled equations for amplitudes Cl . However, for somespecial cases, the exact solutions have a simple form, suchas the system presented in Sec. III.

III.

Collisional Heating for Two Atoms

where the time dependent term arises from a spindependent periodic force F = ~k0 σz sin (ωt + φ)êz . The
corresponding Floquet-Bloch states defined in Eq. (3)have the compact and intuitive form

Heating Rates

X

A.

Z t

sin ωt0 dt0 ,

(10)

0

and
Z t

~2 k(t0 ) · k(t0 ) 0dt
2m0Φ(t) =(11)

are the instantaneous momentum and the cumulative dynamic phase at time t. The physical interpretation of
the wave function Eq. (9) is made transparent by consideringa stationary Gaussian wave packet |φit=0 =R
dk exp (−k · k/σk2 ) |φik at the origin. The expectation
values of the position r and the momentum ~k of the
wave packet at time t under the periodic driving
hrit =~m

Z t0

k(t0 )dt0 , h~kit = ~k(t)(12)

are identical to those of a driven classical particle. We
further identify the secular motion of the particle withthe time average of hrit , h~kit over a period T0 .
The periodic modulation at the driving frequency
ω appears in both the dynamic phase Φ(t) and thewave vector k(t). The amplitude Cl of each Fouriermode defined in Eq. (4) can
obtained via
P∞ be readily inωt
ia sin ωttheexpansion
e
=
J
(a)e
as Cl =
n=−∞ n
P
J
(k
z)J
(α)J
(β)
with
the
First-order
Besseljk
i+j+2k=l i 0functions Jν and two parameters defined as
αk =~kz k0
~k02
, β=
.
mω
8mω

(13)

These motional sidebands dressed by the periodic driving have been directly observed via resonant fluorescence
spectroscopy in trapped-ion systems [34]. The result isalso conceptually similar to an optical modulator where
the carrier frequency is dressed with frequency sidebands
due to the periodic modulation of the medium’s opticalproperties.

4

(a)

(b)
θ
Ei- ћω

B(t)EiEi+ ћω
Ef- ћω

Ef+ћωEf

Ei- ћω Ei Ei+ ћω

(c)

z

(d)

Ef = Ei

Ef = Ei + ћω
Ef+ ћω

Ei+ ћω

Ef+ ћω

Ei+ ћω

Ef

Ei

Ef

Ei

Ef- ћω

Ei- ћω

Ef- ћω

Ei- ћω

Floquet elastic process

Floquet inelastic process

FIG. 1. Scattering between two particles subjected to a periodic force. Atoms with different spins are represented by
red and blue. a) The oscillating force dresses each particle
with energy sidebands spaced by ~ω. The scheme has been
implemented in Ref. [19] where atoms with different magnetic
moments are driven by a periodic magnetic field gradient. b)
Elastic collisions couple two Floquet-Bloch states. Gray andblack arrows represent the incoming and the outgoing states
respectively and θ is the scattering angle. c - d) Illustrationof the Floquet elastic (c) and Floquet inelastic (d) process.
Besides the regular elastic collisions where the final and theinitial state have the same carrier energy (Floquet elastic process), the existence of the energy sidebands allows transitions
between states whose carrier energies are different by a multiple of ~ω (Floquet inelastic process), leading to the exchangeof energy between the system and the driving field.

V̂ couples two Floquet-Bloch states in Eq. (9). Without
the periodic driving, elastic collisions couple only stateswith the same kinetic energy Ek = Ek0 = ~2 |k|2 /2µ.
However, the energy sidebands introduced by the periodic driving, formulated in Eq. (9), allow the scattering
between states whose carrier energies differ by a multiple of ~ω, giving Ek0 − Ek = n~ω (Fig. 1(b)-(d)). The
associated quantized energy change n~ω is transferred tothe secular motion, leading to heating (or cooling). The
transition rate from the ingoing state |Ψik i to the outgoing state |Ψfk0 i can be readily calculated from Eq. (6).
By combining Eqs. (6) and (9), we derive the coupling
matrix element
XZ
Mn (k → k0 ) =gdr δ(r)Cl (k)∗ Cl+n (k0 )(15)l=gJn (αk − αk0 ),which gives the total transition rate from |Ψik i to |Ψfk0 i
Γ(k → k0 )X=Γn (k → k0 )

(16)

n

=

X 2π
n

~

0

2

|Mn (k → k )| δ(Ek0 − Ek − n~ω),

explicitly showing the scattering rate of channels with different numbers of energy quanta ~ω exchanged. The scattering matrix element Mn reveals the microscopic processof the energy exchange with the driving field (n 6= 0 processes): it occurs only when αk 6= αk0 , i.e. when the
projection of the relative momentum k to the shaking
axis changes.
(a)

120 °

(b)

60 °

120 °

60 °

n=0

2.

Two-particle collisions and the heating rates

The two-body problem is reduced to a single particle
problem by decomposing the dynamics into relative andcenter-of-mass parts. The center-of-mass motion is unaffected by collisions and is therefore omitted in further
calculations. The two-body Hamiltonian reads
Ĥ(R, r, t) =~2 K̂2~2 k̂2++ ~k0 σ̂z z sin (ωt + φ)
(14)
2M
2µ
+ gδ(r),with µ = m/2 being the reduced mass. r = (r1 − r2 )/2,k = (k1 − k2 )/2 are the relative coordinate and momentum.
The wave functions for the relative motion have thesame structure as Eq. (9), except that the mass is replaced by the reduced mass µ and the momentum by the
relative momentum ~k.
Collisions are captured by the s-wave pseudopotential
V̂ = gδ(r) described by Eq. (14). Here g = 4π~2 a/m isthe strength of the interaction and a is the s-wave scattering length between the two spin states. The interaction

n=1
180 °

n=-1

0.5

n=0

1.
0 180 °

0.5

1.
0

n=1

FIG. 2.
Differential scattering cross sections for Floquet
scattering processes with n = 0(black), n = 1(gray), andn = −1(light gray). The angular coordinate corresponds
to θ as depicted in Fig. 1. (a) For Ek > ~ω, both absorption (n > 0) and emission (n < 0) processes are allowed.
Scattering between the Floquet-Bloch states is anisotropic
in angle due to the sidebands. The figure is plotted for
α = 0.4, β = 0.003. (b) For strong driving the micromotion
dominates, and the forward and backward scattering are symmetric as expected. The maximum scattering cross sections
are normalized to unity, except for the process with n = −1
in (a) which is normalized with the maximal cross section of
the n = 1 process in (a).

A feature of the scattering between two dressed particles is the anisotropy in the scattering cross section, as5shown in the differential scattering cross section (Fig. 2)d2 σn2σ |k0 |=√|Mn |2 δ(Ek0 − Ek − n~ω).
dΩ dE
m |k|

(17)

Though the potential V̂ is isotropic, the scattering crosssections are anisotropic for each channel. This anisotropy
in the scattering could be potentially observed in thetime-of-flight pattern of a spin-mixed driven
P condensate.
We calculated the heating rate P = n Γn n~ω for asingle pair of colliding particles with a relative momentum ~k by summing over the allowed final states k0 andscattering channels n, leading toXXΓn (k → k0 )n~ω
Pk =
n

=

k0

X 2π g 2
n

~ V

(18)

D3D (Ek + n~ω)γ 2 (k, n)n~ω,where
γ 2 (k, n) =
Z π
p
1
dθ
sin
θ|M
(|k|
→
|k|2 + 2µnω/~ cos θ)|2n
2g 2 0(19)
characterizes
amplitude, and D3D (E) =√ the transition
22 3/2
E/(2π) is the three-dimensional density of(2µ/~ )
states of a free particle with energy E. Here we haveassumed for simplicity that the initial relative momentum
k is along the direction of modulation.
B.

Regimes and Crossovers

One of the major results of this paper is to show theconnection between the Floquet picture, where energy
transfer is quantized, and the semiclassical picture, where
the energy transfer is continuous. The consolidation of
the two pictures can be demonstrated already by examining the two-particle calculation presented above.
We recognize three fundamental energy scales in thesystem: 1) E0 = ~2 k02 /(4m) characterizing the micromotion, and the strength of the modulation. 2) ~ω characterizing the modulation quanta, and 3) Ek characterizing
the relative motion between the two colliding particles,
e.g. kb T for a thermal system or EF for a cold Fermi gas.
The heating behavior of the system is qualitatively different depending on the relationship between these quantities.
We identify the following regimes from Eq. (18) :a. Rapid-modulation Regime ~ω  Ek , E0 : A system with this condition has three features. First, only
energy absorption is allowed. Second, the energy of the
final states, Ek + ~ω, is now dominatedby the modu√
lation energy and D3D (Ek + ~ω) ∼ ~ω. Finally, since
Mn ∼ 1/ω n/2 , the transition rate for the multi-quanta
processes scales with 1/ω n and is therefore negligible.

The heating rate can be obtained analytically by considering only the n = +1 process and readss
4 σ 2~ω
Pk ≈
E0 .
(20)
3V
µ
This is the regime where the quantum and the semiclassical picture diverge. Though the amplitudes of the sidebands drop with increasing modulation frequency, thesystem’s heating rate increases due to the larger final
density of states and the energy transfer ~ω.
b. Semiclassical Regime Ek  ~ω, E0 : In thisregime, as realized in Ref. [19], the final densityof√states is approximated to be D3D (Ek + n~ω) ≈ Ek [1 +n~ω/(2Ek )]. Both the energy absorption (heating) andemission (cooling) processes are allowed. The heating of
the system comes from the imbalance between absorption and emission of energy quanta ~ω, due to the higher
density of states and the larger value of the scatteringmatrix element for the energy absorption
process. If a√stronger criterion Ek  ~ω  E0 Ek is fulfilled suchthat αk  1, only sidebands with l = ±1 are relevant.
In this case, we obtain from Eq. (18)
s
σ 2Ek
E0 ,
(21)
Pk ≈ 8Vµshowing the same dependence on parameters as the semiclassical picture. The heating of the system can be understood in the semiclassical picture where the collision
rate is proportional to the initial relative velocity and themodulation energy gets transferred as heat to the secularmotion.
Contributions from multi-quanta transfer processes|n| > 1 can be important. Indeed, as shown in Fig. 3, results obtained with the single-quantum transfer assumption deviate at small ω from the results where higher
energy transfer processes are considered. However, theheating rate at lower ω with all the higher energy transfer processes included still converges to the semiclassicallimit Eq. (21) obtained from the single-sideband approximation.
c. Strong-drive Regime E0  ~ω, Ek : In this limit,
the strongly driven oscillation dominates over the particle’s initial motion. The scattered particles, therefore,
p behave as if each particle were moving at velocity
E0 /m. Multi-quanta processes contribute significantly
to thep heating rate due to the large modulation index
β ∼ E0 /(~ω) of the final state. The heating rate reads∞ 
2π 2 X
Pk=0 = g
D3D (n~ω)n~ω
~
n=0
! 
r
Z
sin θ
nE0
(22)
× dθ
|Jn 4
cos θ |2
2
~ω
r
E0
E0 ,
=3.36 σmwhere the coefficient 3.36 is found numerically.

6

We now apply the two-particle results above to thermal
ensembles with total atom number N by averaging overthe ensemble as
Z 3
d k1 d3 k2 d3 k01 d3 k0 2
δk +k ,k0 +k0
Pens =(2π)3 (2π)3 (2π)3 (2π)3 1 2 1 2(23)00
× f (k1 )f (k2 )P(k1 , k2 → k1 , k2 ),where f (k) is the particles’ velocity distribution andP(k1 , k2 → k01 , k02 ) = P(k → k0 ) with k = (k1 −k2 ), k0 =(k01 − k02 ). We calculate the heating rates at different
regimes for 1) thermal clouds at temperature T , 2) a
Bose-Einstein Condensate, and 3) a degenerate Fermi
gas.
The analytic results presented in this section are obtained by assuming the single-sideband approximation
unless otherwise stated, as justified in the previous Section. Multi-quanta results are presented as numericalresults in Fig. 3.
a. Thermal Ensemble For a classical ensemble at atemperature T , the distribution f (k) is the Boltzmann
distribution. The resultant ensemble heating rate isr2~ω
ThermalPens= N n3D σ
E0
(24)
3
m
for the rapid-modulation regime, andr
16
kT
ThermalPens=
N n3D σ
E0
3
πm

(25)

for the semiclassical regime. Along with thepgeneral expression P ∼ ρσvcol E0 , we identify vcol ∝ kb T /m tobe the ensemble averaged thermal velocity, which reproduces the semiclassical picture where the micromotionenergy is transferred to the secular motion via elastic collisions at a rate proportional to the thermally averaged
relative speed between the two colliding particles.
b. Bose-Einstein Condensates A Bose-Einstein condensate at T = 0 is treated as an ensemble with f (k) =δ(0). Following Eqs. (20) and (22), the heating rate reads
r
E0
BEC
Pens = 3.36N n3D σ
E0
(26)
m
at E0  ~ω and
2
BEC
Pens
= N n3D σ
3

r

~ω
E0
m

(27)

at E0  ~ω, as shown in Fig.
suggests
p 3(b). The result p
a collisional velocity vcol ∝ E0 /m and vcol ∝ ~ω/m
respectively.
To compare with the experimental results in Ref. [19],we numerically calculate the heating rates for various
cases by directly calculating the sum Eq. (18) for the experimental parameters (Fig. 3). Our result is consistent
with the weak Floquet heating observed.

ThermalPens/N (E0 /s)

Ensemble Heating Rates
20(a)kb T / (πm) E0(16/3)n3 D σ(2/3)n3 D σ ℏω / m E0

10

T = 800nK5

T = 380nK
Single-Sideband

3

BEC
Pens
/N (nK/s)

C.

100

0.1

1

10
ℏω/(kb ×380nK)

100

(b)

10
1
(2/3)n3 D σ ℏω / m E0

0.1
0.01

3.36n3 D σ E0 / m E00.1

1
E0 /(ℏω)

10

100

FIG. 3. Numerical calculations of the heating rate involving
multi-quanta process |n| ≥ 1 for a thermal cloud at temperature T = 380 nK (experimental condition in [19]), 800 nK
and for T = 0 (i.e. an ideal Bose-Einstein condensate) atω = 2π × 5√kHz. a) For the thermal cloud, the heating rates
scale with ω at high frequencies and become independent ofω around ~ω ≈ 4kb T which is the ensemble averaged Ek . The
heating rates, then plateau at the semiclassical value given by
Eq. (25), illustrated by the orange lines. The solid gray andthe dashed gray lines represent the results obtained from thesingle-sideband approximation Eq. (24) for T = 380 nK. The
blue dot indicates the parameter implemented in [19]. b) For
a condensate, the heating rate scales quadratically with themodulation strength characterized by E0 when ~ω  E0 andshows semiclassical behavior at ~ω  E0 . The black lineis the numerical result, the dashed gray line is the heating
rate calculated with the single-sideband approximation whichmatches the numerical result in the regime ~ω  E0 . In thiscalculation we use n3D = 1012 cm−3 and scattering lengtha = 53.8a0 .

D.

Fermionic Systems

The heating effect studied in this work relies on atomic
collisions which can be affected by particle statistics.
For deeply degenerated Fermi gases, Pauli blocking effectively reduces the elastic collisional cross section σ, as
experimentally demonstrated in [35–37]. As a result, collisional heating from periodic driving is suppressed in afermionic system.
Fermi statistics dominates when EF is the largest energy scale. At this condition, collisions occur on the
Fermi surface. More formally, the heating rate for a7103Here, f = 1/(gi + 1) is the occupation number withg = exp [(~2 k · k/(2m) − µ)/kb T ], and µ is the chemical
potential. Pauli blocking is captured by the extra factor(1 − f↓ (k01 ))(1 − f↑ (k02 )), accounting for the occupationof the final states. Here we consider an equal mixture ofspin-up and spin-down atoms.
When ~ω is the largest energy scale in the system,
the entire Fermi sea is involved in the collisional heatingprocess since all possible final states are unoccupied. The
heating rate scales with ω in a similar way as in the casewhere Fermi statistics is absent as in Eq. (24).
When ~ω < EF , Pauli blocking occurs. At T = 0, f
approaches a step function with µ = EF . Collisions occur
in a shell with a thickness of ∼ (~ω/EF )kF around the
sharp Fermi surface. We show in Appendix B that theheating rate of the 3D system becomes

2 r
π
~ω
EF
F
Pens ≈ √ N n3D σ
E0
(29)
EF
m
2
when ~ω < EF , where the factor (~ω/EF ) characterizes the effect of Pauli blocking. The power law of ~ω
originates from three effects: 1) scatterings occurs in ashell at the surface of the Fermi sphere, accounting fora factor of (~ω/EF )3 , 2) the scattering matrix elements
contribute 1/(~ω)2 , 3) the energy transfer per Floquet
inelastic scattering process gives ~ω.
For T 6= 0, thermal excitations smear the Fermi surface, affecting the number of states involved in the collisional processes. When kb T < ~ω, the modulation energy still dominate the scattering. The result is similar
to the T = 0 cases, as shown in Fig. 4. When kb T > ~ω,the thickness of the collisional shell in momentum spaceis on the order of (kb T /EF )kF . We show in Appendix B
and also numerically that the heating rate scales as 2 rT
EF
F
Pens ∝ N n3D σ
E0
(30)
TF
m

101

0.10TF

PFens /N (arb. uni.)(28)

PFens /N (arb. uni.)

fermionic system is expressed as
Z 3
d k1 d3 k2 d3 k01 d3 k02
Fδk +k ,k0 +k0
Pens =(2π)3 (2π)3 (2π)3 (2π)3 1 2 1 2
× f↓ (k1 )f↑ (k2 )[1 − f↓ (k01 )][1 − f↑ (k02 )]× P(k1 , k2 → k01 , k02 ).

0.05TF
0.02TF

50●

10∝(T/TF )

10

10-1

●

●

●

●

●

●

●

1

●

●

●

●
●

0.1

●

T/TF =0

-1

●

●
●

●

●

2

0.03

0.1

0.3

T/TF

100

101

102

ℏω/EF
FIG. 4.
Collisional heating rates in a periodicallydriven spin mixture of degenerated Fermi gases at T =0.1TF , 0.05TF , 0.02TF . The heating effects are Pauli suppressed for a cold Fermi gas when the energy quantum ~ω issmaller than the Fermi energy EF . The discrepancies at highmodulation frequencies are numerical artifact. Inset: heating
rates at ~ω = 0.016 EF and fixed EF for various temperatures
where ~ω < kb T < EF . The heating rate is proportional to(T /TF )2 .

E.

Lower-Dimension Systems

2

at low temperature and is independent of the modulation
frequency.
When the micromotion energy E0 is large comparedwith both the thermal energy kb T and the modulationenergy ~ω, the heating rate of the system becomes

r
E0
EF
F
Pens ∝ N n3D σ
E0 ,
(31)
EF
m
which can be explained by considering the collision between two Fermi spheres displaced by k0 from each other.
Collisions are only allowed within a Fermi shell with athickness of k0 . The number of available states is there2
fore ∝ (k0 /kF ) N 2 which implies the Pauli blocking fac2
tor (E0 /EF ) .

Though our calculations are done for a specific system, several√ conclusions are generally valid. The resultthat P ∼ ~ω at high frequencies is valid for any threedimensional systems in free space with quadratic particle
dispersion due to the density of states. Since the Floquet
elastic collisional rate is bounded, such Floquet systems
cannot be studied in thermal equilibrium in the limit offast modulation frequencies. However, it is often desirable to have the modulation energy scale ~ω greater than
all the other dynamic energy scales.
One possible solution is to go to lower dimensions, suggested by the observation that heating at high frequency
originates from√the increased density of states with higher
energy D3D ∼ ~ω. In a 2D system, the density of statesis independent of the energy, so that excessive collisional
heating can be suppressed.
We consider here quasi-2D scenarios where the atomic
motion is constrained in a two-dimensional pancake but
scattering is still described by the 3D s-wave pseudopotential. This can be achieved in the case where the scattering length a is much larger than the interactionrangepbut smaller than the oscillator length l0 = ~/(mν⊥ ) in
the strongly confined direction, with trapping frequency
ν⊥ . The modulation is in-plane.
In these cases, the Floquet-Bloch wave function is written as1
Ψ(x, ρ , t) = φ⊥ (x) √ exp [ik(t) · ρ − iΦ(t)],A(32)

where ρ = {y, z} and k = {ky , kz } are the 2D radial vector and wave vectors. A is the system area.

8
The component along the strongly confined direction x
has been explicitly separated as φ⊥ . We assume that1/2φ⊥ (x) = π 1/4 l0 exp (−x2 /2l02 ) and particles stay in the
ground state wave function.
With these parameters, results obtained for the 3D
case can be readily extended to quasi-2D by replacing
the scattering strength g with the effective 2D scatteringstrength [38]√2 2π~2 a
.
(33)
g2D =
m
l0
Together with the 2D density of states D2D = µ/(2π~2 )and σ = 4πa2 , the heating rate can be expressed as (seeAppendix A 2 for details)~ σ 8kb T
P2D = 16N n2D
+
1
E0
(34)
m l02
~ω
in the rapid-modulation regime andP2D = 32N n2D

~ σ
E0
m l02

(35)

1

(a)

P1 D/N (arb. uni.)

P2 D/N (arb. uni.)in the semiclassical regime. This can be interpreted asa 3D density n3D = n2D /l0 = N/(Al0 ), and a velocity
vcol = ~/(ml0 ). The heating rates are bounded in bothregimes.

800nK
380nK
0.5
100
102
ℏω/(kb ⨯380nK)

104

1

(b)
380nK

0.1

3800nK100
101102ℏω/(kb ⨯380nK)

FIG. 5.
Numerical calculations for the collisional heating
rates in a quasi-2D (a) and quasi-1D system (b). The heating
rate is independent of ω at high modulation frequencies inquasi-2D regime, and is suppressed with the increasing modulation frequency ~ω in quasi-1D. The oscillations in (b) are
numerical artifacts.

Similarly, for a quasi-1D system with length L, g1D =~2 a/(ml02 ) is the interaction strength. We obtained
P1D ∝ N n1D
P1D ∝ N n1D





~
m
~
m

2 r
2 r

m σ
E0 (~ω  kb T ),~ω l04m σ
E0 (~ω  kb T ),kb T l04(36)

with n1D = N/L. The heating is now suppressed at highfrequencies.
However, we note that the modulation frequency ω isassumed to be smaller than the trapping frequency along
the direction of the strong confinement ω  ν⊥ to avoid
excitations to higher oscillation states. In the opposite

regime, systems are expected to reduce to the 3D case.
These effects are addressed in the studies of collisional
heating in modulated optical lattices [25, 27, 30], which
is beyond the scope of this work.

IV.

DISCUSSIONS AND SUMMARY

In this work we have shown how a Floquet system acquires energy from the external drive and heats up via
inter-particle interactions. Using the scattering theory
of Floquet-Bloch states, we have calculated the collisional heating rates for a cold atomic gas driven by timeperiodic oscillating forces. We have shown that the heating of such systems can be described by a general expression by introducing the effective collisional velocity vcol
parametrizing the density of states :P ∝ ρσvcol E0 .

(37)

The velocity vcol is determined by the dominant energy
in the system and is summarized in Table I. For fermionicsystems, the collisional heating is further suppressed by
Pauli blocking. In systems with lower dimensions, collisional heating is also reduced due to the modified densityof states.
Our calculation can also help to understand the collisional heating in other similar Floquet systems by using
appropriate interparticle potentials and Floquet-Blochstates wave functions. One such system is a combined
trap for ions and neutral atoms, where ions are sympathetically cooled by atoms, limited by heating effects dueto the micromotion of the ions [39–42]
In this work, we considered collisions between particles
which are periodically driven by opposite forces. Thisis different from radio-frequency ion traps or the TimeOrbital Potential (TOP) trap [43], where all particles experience the same periodic force. In those cases, heating
occurs due to non-adiabatic motion and the inhomogeneous strength of the drive or long-range Coulomb interactions [44].
A major motivation for studying Floquet heating is
to assess the feasibility of preparing interesting Floquet
many-body states. An essential question is whether aquantum state can be prepared before excessive heating
occurs. This is often captured by a dimensionless parameter η defined as:
η∼

τ
,
τev

(38)

which characterizes the number of cycles of evolution thesystem can experience before the system’s total energy
increases by its characteristic energy E due to the heating. Here τev is the timescale for the system’s evolutionand τ ∼ (E/P) is the system’s lifetime. Floquet engineering of quantum states requires η  1. We canuse the results of this paper to estimate the parameter η for various systems. For a thermal ensemble with

9
TABLE I. Effective collisional velocities vcol for various systems. The heating rates follow an unified description P ∝ ρσvcol E0
where E0 is the strength of the drive and ρ is the corresponding particle density.
Dominating Energy Scalevcol

Conditionp~ω/mRapid-modulation regime, where ~ω dominatespkb T /m~ω  E0 , kb T
kb T  ~ω, E0

Semiclassical regime, where the thermal motion dominates

p
E0 /m

E0  ~ω, kb T

Strong-drive regime where micromotion dominates as in, for example, condensates andcold atomic samples.

p
EF /m

EF  ~ω, kb T, E0

Fermi energy dominant as in a degenerate Fermi gas. The heating rate is further suppressed by Pauli blocking with a suppression factor, (~ω/EF )2 , (E0 /EF ), or (kb T /EF )2 ,depending on the relation between ~ω, kb T, E0 .

~/ml0~ν⊥  ~ω, kb T, E0

Confinement energy dominant as in systems of lower dimension.

τev ∼ n3D σvth and E = kb T , we obtain η ∼pkb T /E0
in the semiclassical regime and η ∼ kb T /E0 kb T /~ω
for rapid modulation. For condensates, evolution of
the system is characterized by the mean-field interaction strength ~/τev ∼ U = p
gn3D in 3D,√ which leads toη ∼ U 2 /P ∼ n3D /k03 or ∼ ~2 /ma2 / ~ω(U/E0 ). For
1/2
Fermi gases, ~/τev ∼ EF , giving η ∼ EF /(~2 ω 2 E0 ).
This illustrates the benefit of using systems with large
Fermi energy.
The purpose of this paper was a transparent treatment of heating in different regimes for a particularly
simple Floquet system. Our discussion provides a starting point for more complex systems where we expect similar regimes depending on the hierarchy of the relevantenergy scales.

Appendix A: Ensemble Averaged Heating Rates
1.

3D Systems

We present the detailed calculation for Eq. (25).
Rewriting Eq. (23) with the center-of-mass and relative coordinates K = (k1 + k2 )/2 = (k01 + k02 )/2, andΓ(k1 , k2 → k01 , k02 ) = Γ(k → k0 ) with k = (k1 − k2 ), k0 =(k01 − k02 ) one obtains:Z 3
d K d3 k d3 k0 k
kP=f ( + K)f (− + K)P(k → k0 )(2π)3 (2π)3 (2π)3 2
2(A1)with the Boltzmann distribution function.
3/2



~2 k · k~2
exp −f (k) = N
2πmkb T
2mkb T
As discussed in Sec. III B, we consider only the processes where n = ±1. The coupling matrix elements Mn ,to the lowest orders in αk , are shown in Table II.

ACKNOWLEDGMENTS

We would like to thank useful discussions with Nigel
Cooper, Erich Mueller, Alan O. Jamison, Jeongwon
Lee, Furkan Çağrı Top, Yair Margalit, and Gediminas Juzeliūnas. We would like to thank Jesse AmatoGrill for critical reading of the manuscript. We acknowledge support from the NSF through the Centerfor Ultracold Atoms and award 1506369, from AROMURI Non-equilibrium Many-body Dynamics (Grant
No. W911NF-14-1-0003), from AFOSR-MURI Quantum
Phases of Matter (Grant No. FA9550-14-1-0035), fromONR (Grant No. N00014-17-1-2253) and a VannevarBush Faculty Fellowship.

TABLE II. Approximate coupling matrix elements Mn forvarious Floquet collision processes.
M0 /g M1 /g
10~k0 (kz −kz)2mω

M−1 /g−0~k0 (kz −kz
) 1
2mω
2

M2 /g
h

0
~k0 (kz −kz)2mω

M−2 /g
i2

1
2

h

0
~k0 (kz −kz
)
2mω

i2

With the single-sideband assumption, we obtain theanalytic expression of Eq. (18) by explicitly calculating
γ 2 (k, ±1) :
r


σ X
Ek + n~ω ~2 kz2~2 |k|22n
Pk = 4++nE0 .
V n=±1
m
µ~ω
3µ~ω
3
(A2)

10
with n = ±1 for the semiclassical regime, and n = 1 forthe rapid-modulation regime. The total heating rate P
can be readily obtained as
P =N 23/2 Z~2 k·k4π~2d3 k − 4mkbT P ,
e
k
mkb T
(2π)3

(A3)

which gives Eq. (20) and Eq. (25) at the corresponding
limits.

2.

Quasi-2D Systems

We start with the Floquet-Bloch states wave function
for quasi-2D system written as Eq. (32). The scatteringstrength, defined asZ
g = dr φi (r)V (r)φf (r),has the form [38]:√2 2π~2 a
g2D =
m
l0

Mn (k → k0 ) = g2D Jn (αk − αk0 ),(A4)

and obtain0dθ |Mn (k →p|k|2 + 2µnω/~ cos θ)|2 .

(A5)Following similar procedures as in the 3D calculation,and using the 2D density of states lead to2
πµg2D
Pk2D =
4~3 A

For a quasi-1D system, we use the 1D wave function,a scattering strength g1D , and Mn
1
Ψ(x, y, z, t) = φ⊥ (x, y) √ exp [ik(t)z − iΦ(t)],
L
~2 a
,
g1D =
ml02
Mn (k → k 0 ) = g1D Jn (αk − αk0 ),r1µD1D (E) =,2π~ 2E

X 

n=±1


~ |k|
+
+ 2n nE0 . (A6)µ~ωµ~ω2~2 kz222

TogetherBoltzmann
distribution f (k) =with the 2D
2π~2~2 k·kN mkb T exp − 2mkb T , we obtain the heating rate of
the ensemble
Z 2
d K d2 k d2 k0 kkP2D =f ( + K)f (− + K)P(k → k0 )(2π)2 (2π)2 (2π)2 2
2
Z

2
2
2
~ k·k
2π~
d k − 4mk
2D
bT P
=2N 2
e
k ,
mkb T
(2π)2
(A7)
which gives Eqs. (34) and (35).

(A8)

to obtain2
γ1D(k, n) =p1|Mn (k → k 2 + 2µnω/~)|2
2
2g1D

p
+ |Mn (k → − k 2 + 2µnω/~)|2 ,Pk1D =(A9)2
X k 2 + nµ~ω/~2
8g1D
p
nE0 .
~2 ωL nk 2 + 2µn~ω/~2(A10)

With the 1D Boltzmann distribution f (k)
=


1/2

2
2 2
2π~
~ k
N mk
exp − mk
, we calculate the heatbTbT
ing rate of the ensembleZkdK dk dk 0 kf ( + K)f (− + K)P(k → k 0 )2π 2π 2π 22
1/2 Z22 k2~
π~dk − 4mk T 1D
b P
=N 2
e
k ,
mkb T
2π
(A11)

P1D =

2
γ2D
(k, n) =Z π

Quasi-1D Systems

yielding

in quasi-2D. The results obtained for 3D cases can bereadily extended to quasi-2D by replacing g with g2D
and using the corresponding 2D density. We therefore
obtain the coupling matrix element between two quasi2D Floquet-Bloch states

1
2
2g2D

3.

which results in Eq. (36).
√
We add the note that the scaling ∼ 1/ kb T in theregime ~ω  kb T cannot be directly obtained in thesame way as for other dimensions. In 1D, terms in zeroth, first, and second order in ~ω/Ek cancel between the
n = ±1 processes if both processes are allowed for twoparticles with k 2 > 2µn~ω/h2X

k 2 + nµ~ω/~2pnk 2 + 2µn~ω/~2
n=±1"  #
 3  2 2
4
~ω
~
~ω
=−
+O
.
Ek
mEk
Ek

(A12)

Therefore, even for kb T  ~ω, the leading contribution
comes from the regime k 2 < 2µn~ω/h2 , where only the

11k2 − k01 |2 . The heating rate Eq. (B1) now has the form

n = +1 process is allowed. We obtain1/2 Z~2 k2
dk − 4mk
π~2
1D
bT P
e
k
mkb T
2π
 3
r
~
σ E0
mπ
2
≈N
4
2
m
l0 π ωL kb TZ √2µ~ω/~22 2
k 2 + µ~ω/~2
− ~ k
× √
dke 4mkb T p
k 2 + 2µ~ω/~2− 2µ~ω/~2

P1D =N 2√  2 r
2
mπ σ
~
∼N 2
E0
π L m
kb T l042(~ω  kb T ).
(A13)

The obtained scaling is consistent with the numerical results.

Appendix B: Derivation of the Pauli Blocking Factor
1.

Z

d3 k1 d3 k2 d3 k01 d3 k02
δk +k ,k0 +k0(2π)3 (2π)3 (2π)3 (2π)3 1 2 1 2
× f↓ (k1 )f↑ (k2 )(1 − f↓ (k01 ))(1 − f↑ (k02 ))× P(k1 , k2 → k01 , k02 ).

(B1)

We denote i = ~2 |ki |2 /2m, 0i = ~2 |k0i |2 /2m,and rewrite the integral in spherical coordinateswith ki = ki (sin θi cos φi , sin θi sin φi , cos θi ), k0i =
ki0 (sin θi0 cos φ0i , sin θi0 sin φ0i , cos θi0 ).
Momentum conservation k1 + k2 = k01 + k02 gives
P(k1 , k2 → k01 , k1 + k2 − k01 )2(k1 − k2 )z − (k01 − k02 )z= CΓδ(1 + 2 + ~ω − 01 − 02 )4
2
k1 cos θ1 − k10 cos θ10= CΓ
δ(1 + 2 + ~ω − 01 − 02 ),2(B2)

where CΓ =where q = k1 − k01 and q = |q|. Integrating over the solid
angle dΩ2 = sin θ2 dθ2 dφ2 gives

Zero-Temperature Fermi Gases

In this part we present the calculation for Eq. (29).
The interspin collision rate of a two-component Fermi
mixture is written as

F
Pens
=

Z

d3 k1 d3 k2 d3 k01
δ(1 + 2 + ~ω − 01 − 02 )(2π)3 (2π)3 (2π)3×f↓ (k1 )f↑ (k2 )(1 − f↓ (k01 ))(1 − f↑ (k1 + k2 − k01 ))2
k1 cos θ1 − k10 cos θ10
×CΓ
2


3Z p
1 2 01(2m)3/2=CΓd1 d2 d01 dΩ1 dΩ2 dΩ01
2~3
(2π)9
×f↓ (1 )f↑ (2 )(1 − f↓ (01 ))(1 − f↑ (1 + 2 + ~ω − 01 ))2
k1 cos θ1 − k10 cos θ10×2~2~2 q 2+ q · k2 ),×δ(1 + ~ω − 01 −
2m
m
(B3)

F
Pens
=

2 3 2
2π g ~ k0~ µ2 ω

~is a constant, and 02 = 2m
|k1 +2

3 Z p1 2 01(2m)3/2d1 d2 d01 dΩ1 dΩ01
3
2~
(2π)9
× f↓ (1 )f↑ (2 )(1 − f↓ (01 ))(1 − f↑ (1 + 2 + ~ω − 01 ))2
k1 cos θ1 − k10 cos θ10
×
2
2 2
Z 1
q
−1 − ~ω + 01 + ~2mm
× 2 √dx δ(x −),√2~~ q 2 −1
m q 2(B4)F
Pens= (2π)CΓ

with x = cos θ2 . We subsequently integrate over Ω1 andΩ01 . When k1 , k01 vary over their respective
p solid angles√Ω1 , Ω01 with fixed lengths k1 = 1 , k10 = 01 , the differential vector q also varies overpthe entire solidp angle√Ωq√with the length varying from 01 − 1 to 01 + 1 .
Therefore, we have the equivalencedΩ1 dΩ2 = Cq q 2 dq sin θq dθq dφq dφl ,p√where Cq = 6/( 01 + 1 )3 is the normalization factor, and φl is the angle between the planes expanded by12{k1 , k01 } and {q, ez }. The heating rate follows asp
3 ZCq 1 2 01(2m)3/2F
3d1 d2 d01
Pens = (2π) CΓ
2~3
(2π)9

× q 2 dq sin θq dθq× f↓ (1 )f↑ (2 )(1 − f↓ (01 ))(1 − f↑ (1 + 2 + ~ω − 01 ))2 2
Z1q−1 − ~ω + 01 + ~2mm(q cos θq )2
dx
δ(x
−
)
×
√
√
4~2 q 2
~2 q 2 /m−1

3/2 3 Zp
Cq 1 2 01(2m)d1 d2 d01
= CΓ
2~3
(2π)6
× f↓ (1 )f↑ (2 )(1 − f↓ (01 ))(1 − f↑ (1 + 2 + ~ω − 01 ))√0 √1 + 12 2
ZZ1q−1 − ~ω + 01 + ~2m
mq 3 dq
×
dx
δ(x
−
)
√
√
2
~2 q 2 /m√ 0 √ 6~ 2 −1

1 − 1p
3 Z
Cq 1 2 01(2m)3/2d1 d2 d01
2~3
(2π)6
× f↓ (1 )f↑ (2 )(1 − f↓ (01 ))(1 − f↑ (1 + 2 + ~ω − 01 )) Z √0 +√1
1
mq 3 dq
× √ √
√
2 
2
01 − 1 6~

p
p
√
√
0
0
θ( 22 − 2 + 2 − q)θ(q − ( 22 − 2 − 2 )) .
= CΓ



(B5)
Note that so far the only approximation adopted is the
single-sideband approximation, and that the first sideband is weak, implying J1 (x)p
≈ x/2.
p√
When ~ω < EF , we have 202 − 2 + 2 ≈ 01 +
p
p
p
√
√
 ≈ 2 ~2 /(2m)kF , and 202 − 2 − 2 ≈ 01 −√√ 1
1 ≈ ~ω/(2 EF ). The integral over q in Eq. B5 now
gives
p
3 Z

Cq 1 2 01(2m)3/2F
Pens≈ CΓd1 d2 d01
2~3
(2π)6
× f↓ (1 )f↑ (2 )(1 − f↓ (01 ))(1 − f↑ (1 + 2 + ~ω − 01 )) 2 28mkF4~
× 2√
3~ 2 2m
33(2m)3/2
3/2 (~ω) 8m
≈ CΓ
E
.
F
2~3
8(2π)6 3~2(B6)Here we adopt the approximationZp1
Cq 1 2 01 d1 d2 d01 √2

× f↓ (1 )f↑ (2 )(1 − f↓ (01 ))(1 − f↑ (1 + 2 + ~ω − 01 ))−1/2 (~ω)

≈EF

8

3

.
(B7)

With the definition of the Fermi energy (2mEF )3/2 =3n3D ~3 π 2 , we eventually obtain

2 r
π
EF
~ω
F
Pens ≈ √ N n3D σ
E0 ,
(B8)
EF
m
2
which shows explicitly the Fermi suppression factor2(~ω/EF ) .
The calculations above can be extended to the regimeof small ~ω where multi-quanta transfer processes are
relevant.
2.

Finite Temperature Fermi Gases

In this section we present the calculation for the Pauliblocking factor (T /TF )2 when T /TF  1. We further
assume that ~ω  kb T . The non-zero temperature caseis different from the zero temperature Fermi gas mainly
in two aspects: first, as discussed briefly in the main text,
the active Fermi shell formed by the accessible states hasa thickness of kb T instead of ~ω. Second, energy quantaemission processes are now allowed. In the lowest orderapproximation, the heating rate can be calculated with
only n ± 1 processes and leads toZp1
Cq 1 2 01 d1 d2 d01 √2

× f↓ (1 )f↑ (2 )(1 − f↓ (01 ))(1 − f↑ (1 + 2 + ~ω − 01 ))
00− f↓ (1 )f↑ (2 )(1 − f↓ (1 ))(1 − f↑ (1 + 2 − ~ω − 1 ))Z
p
1= Cq 1 2 01 d1 d2 d01 √2∂f ()
0× f↓ (1 )f↑ (2 )(1 − f↓ (1 )) −~ω∂=1 +2 −0
1

−1/2

∼EF

2

(kb T ) ~ω.
(B9)Together with Eq. (B6), the Fermi suppression factor isreadily recognized to be (kb T /EF )2 .

13

[1] S. Rahav, I. Gilary, and S. Fishman, Phys. Rev. A 68,
013820 (2003).
[2] N. Goldman and J. Dalibard, Phys. Rev. X 4, 031027(2014).
[3] P. Hauke, O. Tieleman, A. Celi, C. Ölschläger, J. Simonet, J. Struck, M. Weinberg, P. Windpassinger,
K. Sengstock, M. Lewenstein, and A. Eckardt, Phys.
Rev. Lett. 109, 145301 (2012).
[4] J. Struck, J. Simonet, and K. Sengstock, Phys. Rev. A
90, 031601 (2014).
[5] C. E. Creffield, G. Pieplow, F. Sols, and N. Goldman,
New Journal of Physics 18, 093013 (2016).
[6] M. Bukov, M. Kolodrubetz, and A. Polkovnikov, Phys.
Rev. Lett. 116, 125301 (2016).
[7] B. M. Anderson, I. B. Spielman, and G. Juzeliūnas,
Phys. Rev. Lett. 111, 125301 (2013).
[8] Z.-F. Xu, L. You, and M. Ueda, Phys. Rev. A 87, 063634(2013).
[9] K. Jiménez-Garcı́a, L. J. LeBlanc, R. A. Williams, M. C.
Beeler, C. Qu, M. Gong, C. Zhang, and I. B. Spielman,
Phys. Rev. Lett. 114, 125301 (2015).
[10] J. Yu, Z.-F. Xu, R. Lü, and L. You, Phys. Rev. Lett.
116, 143003 (2016).
[11] S. K. Baur, M. H. Schleier-Smith, and N. R. Cooper,
Phys. Rev. A 89, 051605 (2014).
[12] J. Cayssol, B. Dra, F. Simon, and R. Moessner, physica status solidi (RRL) Rapid Research Letters 7, 101(2013).
[13] N. H. Lindner, D. L. Bergman, G. Refael, and V. Galitski, Phys. Rev. B 87, 235131 (2013).
[14] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro,
B. Paredes, and I. Bloch, Phys. Rev. Lett. 111, 185301(2013).
[15] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, and W. Ketterle, Phys. Rev. Lett. 111, 185302(2013).
[16] C. J. Kennedy, W. C. Burton, W. C. Chung, and W. Ketterle, Nature Physics 11, 859 EP (2015).
[17] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T.
Barreiro, S. Nascimbène, N. R. Cooper, I. Bloch, and
N. Goldman, Nature Physics 11, 162 EP (2014).
[18] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat,
T. Uehlinger, D. Greif, and T. Esslinger, Nature 515,
237 EP (2014).
[19] B. Shteynas, J. Lee, F. C. Top, J.-R. Li, A. O. Jamison,
G. Juzelinas, and W. Ketterle, “How to dress radiofrequency photons with tunable momentum,” (2018).
[20] J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee,
J. Smith, G. Pagano, I. D. Potirniche, A. C. Potter,
A. Vishwanath, N. Y. Yao, and C. Monroe, Nature 543,
217 EP (2017).

[21] S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya,
F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. vonKeyserlingk, N. Y. Yao, E. Demler, and M. D. Lukin,
Nature 543, 221 EP (2017).
[22] S. A. Weidinger and M. Knap, Scientific Reports 7, 45382
EP (2017).
[23] A. Lazarides, A. Das, and R. Moessner, Phys. Rev. E
90, 012110 (2014).
[24] L. D’Alessio and M. Rigol, Phys. Rev. X 4, 041048(2014).
[25] S. Choudhury and E. J. Mueller, Phys. Rev. A 90, 013621(2014).
[26] T. Bilitewski and N. R. Cooper, Phys. Rev. A 91, 033601(2015).
[27] S. Choudhury and E. J. Mueller, Phys. Rev. A 92, 063639(2015).
[28] M. Genske and A. Rosch, Phys. Rev. A 92, 062108(2015).
[29] M. Weinberg, C. Ölschläger, C. Sträter, S. Prelle,
A. Eckardt, K. Sengstock, and J. Simonet, Phys. Rev.
A 92, 043621 (2015).
[30] M. Reitter, J. Näger, K. Wintersperger, C. Sträter,
I. Bloch, A. Eckardt, and U. Schneider, Phys. Rev. Lett.
119, 200402 (2017).
[31] A. Eckardt and E. Anisimovas, New Journal of Physics
17, 093039 (2015).
[32] F. Casas, J. A. Oteo, and J. Ros, Journal of Physics A:
Mathematical and General 34, 3379 (2001).
[33] V. Novičenko, E. Anisimovas, and G. Juzeliūnas, Phys.
Rev. A 95, 023615 (2017).
[34] C. Raab, J. Eschner, J. Bolle, H. Oberst, F. SchmidtKaler, and R. Blatt, Phys. Rev. Lett. 85, 538 (2000).
[35] B. DeMarco, S. B. Papp, and D. S. Jin, Phys. Rev. Lett.
86, 5409 (2001).
[36] G. Ferrari, Phys. Rev. A 59, R4125 (1999).
[37] M. J. Holland, B. DeMarco, and D. S. Jin, Phys. Rev.
A 61, 053610 (2000).
[38] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov,
Phys. Rev. Lett. 84, 2551 (2000).
[39] R. G. DeVoe, Phys. Rev. Lett. 102, 063001 (2009).
[40] K. Chen, S. T. Sullivan, and E. R. Hudson, Phys. Rev.
Lett. 112, 143009 (2014).
[41] C. Zipkes, S. Palzer, L. Ratschbacher, C. Sias, andM. Köhl, Phys. Rev. Lett. 105, 133201 (2010).
[42] S. Haze, M. Sasakawa, R. Saito, R. Nakai,
andT. Mukaiyama, Phys. Rev. Lett. 120, 043401 (2018).
[43] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A.
Cornell, Phys. Rev. Lett. 74, 3352 (1995).
[44] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried,
B. E. King, and D. M. Meekhof, Journal of research of
the National Institute of Standards and Technology 103,259 (1998).

146

Bibliography[1] R.P. Feynman. Simulating physics with computers. Int J Theor Phys, volume= 21, issue = 6-7, pages = 467–488, year = 1982, month = June, publisher =Springer, doi = https://doi.org/10.1007/BF02650179,.
[2] J. Struck, C. Ölschläger, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger, and K. Sengstock. Quantum simulation of frustrated
classical magnetism in triangular optical lattices. Science, 333(6045):996–999,
2011.
[3] Gregor Jotzu, Michael Messer, Rémi Desbuquois, Martin Lebrat, Thomas
Uehlinger, Daniel Greif, and Tilman Esslinger. Experimental realization of the
topological haldane model with ultracold fermions. Nature, 515:237 EP –, 11
2014.
[4] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hänsch, and Immanuel Bloch. Quantum phase transition from a superfluid to a mott insulator
in a gas of ultracold atoms. Nature, 415(6867):39–44, 2002.
[5] Maciej Lewenstein, Anna Sanpera, Veronica Ahufinger, Bogdan Damski, Aditi
Sen(De), and Ujjwal Sen. Ultracold atomic gases in optical lattices: mimicking
condensed matter physics and beyond. Advances in Physics, 56(2):243–379, 2007.
[6] W. Ketterle and M. W. Zwierlein. Making, probing and understanding ultracold
Fermi gases. Nuovo Cimento Rivista Serie, 31:247–422, May 2008.
[7] Maxwell F. Parsons, Florian Huber, Anton Mazurenko, Christie S. Chiu,
Widagdo Setiawan, Katherine Wooley-Brown, Sebastian Blatt, and Markus
Greiner. Site-resolved imaging of fermionic 6 Li in an optical lattice. Phys. Rev.
Lett., 114:213002, May 2015.
[8] Lawrence W. Cheuk, Matthew A. Nichols, Katherine R. Lawrence, Melih Okan,
Hao Zhang, and Martin W. Zwierlein. Observation of 2d fermionic mott insulators of 40 K with single-site resolution. Phys. Rev. Lett., 116:235301, Jun 2016.
[9] Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin,
Matthew Rispoli, and Markus Greiner. Measuring entanglement entropy in aquantum many-body system. Nature, 528:77 EP –, 12 2015.
147

[10] Colin J. Kennedy, William Cody Burton, Woo Chang Chung, and Wolfgang Ketterle. Observation of bose–einstein condensation in a strong synthetic magnetic
field. Nature Physics, 11:859 EP –, 08 2015.
[11] M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch.
Experimental realization of strong effective magnetic fields in an optical lattice.
Phys. Rev. Lett., 107:255301, Dec 2011.
[12] Aviv Keshet. A Next-Generation Apparatus for Lithium Optical Lattice Experiments. PhD dissertation, MIT, 2012. This is a full PHDTHESIS entry.
[13] Edward Su. Fluctuations and State Preparation in Quantum Degenerate Gasesof Soidum and Lithium. PhD dissertation, MIT, 2014. This is a full PHDTHESIS
entry.
[14] Wujie Huang. Spin-Orbit Coupling in Optical Superlattice. PhD dissertation,
MIT, 2015. This is a full PHDTHESIS entry.
[15] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle. Bose-einstein condensation in a gas of sodium
atoms. Phys. Rev. Lett., 75:3969–3973, Nov 1995.
[16] Victor Galitski and Ian B. Spielman. Spin–orbit coupling in quantum gases.
Nature, 494:49 EP –, 02 2013.
[17] Vinod Kumar Joshi. Spintronics: A contemporary review of emerging electronics devices. Engineering Science and Technology, an International Journal,
19(3):1503 – 1513, 2016.
[18] M. Z. Hasan and C. L. Kane. Colloquium: Topological insulators. Rev. Mod.
Phys., 82:3045–3067, Nov 2010.
[19] Zhan Wu, Long Zhang, Wei Sun, Xiao-Tian Xu, Bao-Zong Wang, Si-Cong
Ji, Youjin Deng, Shuai Chen, Xiong-Jun Liu, and Jian-Wei Pan. Realizationof two-dimensional spin-orbit coupling for bose-einstein condensates. Science,354(6308):83–88, 2016.
[20] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein. Cold atoms
in non-abelian gauge potentials: From the hofstadter "moth" to lattice gauge
theory. Phys. Rev. Lett., 95:010403, Jun 2005.
[21] J. Ruseckas, G. Juzeliūnas, P. Öhberg, and M. Fleischhauer. Non-abelian gauge
potentials for ultracold atoms with degenerate dark states. Phys. Rev. Lett.,
95:010404, Jun 2005.
[22] Xiong-Jun Liu, Mario F. Borunda, Xin Liu, and Jairo Sinova. Effect of induced
spin-orbit coupling for atoms via laser fields. Phys. Rev. Lett., 102:046402, Jan
2009.
148

[23] N.R. Cooper. Rapidly rotating atomic gases. Advances in Physics, 57(6):539–
616, 2008.
[24] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeliūnas, and Patrik Öhberg. Colloquium: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys., 83:1523–
1543, Nov 2011.
[25] Frank Wilczek and A. Zee. Appearance of gauge structure in simple dynamicalsystems. Phys. Rev. Lett., 52:2111–2114, Jun 1984.
[26] Xiong-Jun Liu, Mario F. Borunda, Xin Liu, and Jairo Sinova. Effect of induced
spin-orbit coupling for atoms via laser fields. Phys. Rev. Lett., 102:046402, Jan
2009.
[27] Y. J. Lin, K. Jiménez-García, and I. B. Spielman. Spin–orbit-coupled bose–
einstein condensates. Nature, 471:83 EP –, 03 2011.
[28] J. Ruseckas, G. Juzeliūnas, P. Öhberg, and M. Fleischhauer. Non-abelian gauge
potentials for ultracold atoms with degenerate dark states. Phys. Rev. Lett.,
95:010404, Jun 2005.
[29] Christian Flindt, Anders S. Sørensen, and Karsten Flensberg. Spin-orbit mediated control of spin qubits. Phys. Rev. Lett., 97:240501, Dec 2006.
[30] Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William Cody Burton, and Wolfgang Ketterle. Realizing the harper hamiltonian with laser-assisted
tunneling in optical lattices. Phys. Rev. Lett., 111:185302, Oct 2013.
[31] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch.
Realization of the hofstadter hamiltonian with ultracold atoms in optical lattices.
Phys. Rev. Lett., 111:185301, Oct 2013.
[32] Long Zhang and Xiong-Jun Liu. Spin-orbit coupling and topological phases forultracold atoms. 2018.
[33] Nathaniel Q. Burdick, Yijun Tang, and Benjamin L. Lev. Long-lived spin-orbitcoupled degenerate dipolar fermi gas. Phys. Rev. X, 6:031022, Aug 2016.
[34] Junru Li, Wujie Huang, Boris Shteynas, Sean Burchesky, Furkan Çağrı Top,
Edward Su, Jeongwon Lee, Alan O. Jamison, and Wolfgang Ketterle. Spin-orbit
coupling and spin textures in optical superlattices. Phys. Rev. Lett., 117:185301,
Oct 2016.
[35] Jun-Ru Li, Jeongwon Lee, Wujie Huang, Sean Burchesky, Boris Shteynas,
Furkan Çağrı Top, Alan O. Jamison, and Wolfgang Ketterle. A stripe phasewith supersolid properties in spin–orbit-coupled bose–einstein condensates. Nature, 543:91 EP –, 03 2017.
149

[36] Tin-Lun Ho and Shizhong Zhang. Bose-einstein condensates with spin-orbit
interaction. Phys. Rev. Lett., 107:150403, Oct 2011.
[37] Chunji Wang, Chao Gao, Chao-Ming Jian, and Hui Zhai. Spin-orbit coupledspinor bose-einstein condensates. Phys. Rev. Lett., 105:160403, Oct 2010.
[38] Yun Li, Lev P. Pitaevskii, and Sandro Stringari. Quantum tricriticality andphase transitions in spin-orbit coupled bose-einstein condensates. Phys. Rev.
Lett., 108:225301, May 2012.
[39] Massimo Boninsegni and Nikolay V. Prokof’ev. Colloquium: Supersolids: What
and where are they? Rev. Mod. Phys., 84:759–776, May 2012.
[40] John Beamish. Supersolid helium. Nature, 427(6971):204–205, 2004.
[41] A. B. Kuklov, L. Pollet, N. V. Prokof’ev, and B. V. Svistunov. Quantum plasticity and supersolid response in helium-4. Phys. Rev. B, 90:184508, Nov 2014.
[42] Julian Léonard, Andrea Morales, Philip Zupancic, Tilman Esslinger, and Tobias
Donner. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature, 543:87 EP –, 03 2017.
[43][44] Sergey V. Tovstonog, Sunao Kurimura, and Kenji Kitamura. High power
continuous-wave green light generation by quasiphase matching in mg stoichiometric lithium tantalate. Applied Physics Letters, 90(5):051115, 2007.
[45] G. D. Boyd and D. A. Kleinman. Journal of Applied Physics, 39.
[46] Y. Furukawa, K. Kitamura, A. Alexandrovski, R. K. Route, M. M. Fejer, andG. Foulon. Green-induced infrared absorption in mgo doped linbo3. Applied
Physics Letters, 78(14):1970–1972, 2001.
[47] E.M. Lifshits L.D. Landau. Theoretical Phytsics. Mechanics, volume 1. PhysMathLit.
[48] Wolfgang Paul. Electromagnetic traps for charged and neutral particles. Rev.
Mod. Phys., 62:531–540, Jul 1990.
[49] Ernest D. Courant, M. Stanley Livingston, and Hartland S. Snyder. The strongfocusing synchroton—a new high energy accelerator. Phys. Rev., 88:1190–1196,
Dec 1952.
[50] Jon H. Shirley. Solution of the schrödinger equation with a hamiltonian periodic
in time. Phys. Rev., 138:B979–B987, May 1965.
[51] N. Goldman and J. Dalibard. Periodically driven quantum systems: Effective
hamiltonians and engineered gauge fields. Phys. Rev. X, 4:031027, Aug 2014.
150

[52] Marin Bukov, Luca D’Alessio, and Anatoli Polkovnikov. Universal highfrequency behavior of periodically driven systems: from dynamical stabilization
to floquet engineering. Advances in Physics, 64(2):139–226, 2015.
[53] W. R. Salzman. Quantum mechanics of systems periodic in time. Phys. Rev. A,
10:461–465, Aug 1974.
[54] Saar Rahav, Ido Gilary, and Shmuel Fishman. Effective hamiltonians for periodically driven systems. Phys. Rev. A, 68:013820, Jul 2003.
[55] T. P. Grozdanov and M. J. Raković. Quantum system driven by rapidly varying
periodic perturbation. Phys. Rev. A, 38:1739–1746, Aug 1988.
[56] Xinyu Luo, Lingna Wu, Jiyao Chen, Qing Guan, Kuiyi Gao, Zhi-Fang Xu,
L. You, and Ruquan Wang. Tunable atomic spin-orbit coupling synthesizedwith a modulating gradient magnetic field. Scientific Reports, 6:18983 EP –, 01
2016.
[57] Jun-Ru Li, Boris Shteynas, and Wolfgang Ketterle. Floquet heating in interacting
atomic gases with an oscillating force. arXiv:1906.08747.

151